A Celebration of the Mathematical Legacy of Raoul Bott 0821847775, 9780821847770

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Volume 50

C R M

CRM PROCEEDINGS & LECTURE NOTES Centre de Recherches Mathématiques Montréal

A Celebration of the Mathematical Legacy of Raoul Bott P. Robert Kotiuga Editor

American Mathematical Society

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A Celebration of the Mathematical Legacy of Raoul Bott

Licensed to AMS. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms

Licensed to AMS. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms

https://doi.org/10.1090/crmp/050

Volume 50

C R M

CRM PROCEEDINGS & LECTURE NOTES Centre de Recherches Mathématiques Montréal

A Celebration of the Mathematical Legacy of Raoul Bott P. Robert Kotiuga Editor

The Centre de Recherches Mathématiques (CRM) of the Université de Montréal was created in 1968 to promote research in pure and applied mathematics and related disciplines. Among its activities are special theme years, summer schools, workshops, postdoctoral programs, and publishing. The CRM is supported by the Université de Montréal, the Province of Québec (FQRNT), and the Natural Sciences and Engineering Research Council of Canada. It is affiliated with the Institut des Sciences Mathématiques (ISM) of Montréal. The CRM may be reached on the Web at www.crm.math.ca.

American Mathematical Society Providence, Rhode Island USA

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The production of this volume was supported in part by the Fonds Qu´eb´ecois de la Recherche sur la Nature et les Technologies (FQRNT) and the Natural Sciences and Engineering Research Council of Canada (NSERC). 2000 Mathematics Subject Classification. Primary 01–XX, 18–XX, 19–XX, 35–XX, 55–XX, 57–XX, 58–XX, 81–XX. The compositions For Liv and Invention appear courtesy of the Bott family.

Library of Congress Cataloging-in-Publication Data A celebration of the mathematical legacy of Raoul Bott / P. Robert Kotiuga, editor. p. cm. — (CRM proceedings & lecture notes ; v. 50) Includes bibliographical references. ISBN 978-0-8218-4777-0 (alk. paper) 1. Geometry. 2. Mathematical physics. I. Bott, Raoul, 1923–2005. QA446.C45 516—dc22

2010 2010001317

Copying and reprinting. Material in this book may be reproduced by any means for educational and scientific purposes without fee or permission with the exception of reproduction by services that collect fees for delivery of documents and provided that the customary acknowledgment of the source is given. This consent does not extend to other kinds of copying for general distribution, for advertising or promotional purposes, or for resale. Requests for permission for commercial use of material should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294, USA. Requests can also be made by e-mail to [email protected]. Excluded from these provisions is material in articles for which the author holds copyright. In such cases, requests for permission to use or reprint should be addressed directly to the author(s). (Copyright ownership is indicated in the notice in the lower right-hand corner of the first page of each article.) c 2010 by the American Mathematical Society. All rights reserved.
The American Mathematical Society retains all rights except those granted to the United States Government. Copyright of individual articles may revert to the public domain 28 years after publication. Contact the AMS for copyright status of individual articles. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines


established to ensure permanence and durability. This volume was submitted to the American Mathematical Society in camera ready form by the Centre de Recherches Math´ematiques. Visit the AMS home page at http://www.ams.org/ 10 9 8 7 6 5 4 3 2 1

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Contents Preface

ix

For Liv Raoul Bott

x

Invention Raoul Bott

xi

Introduction P. Robert Kotiuga

1

Montr´ eal, the 1940s, and Mathematical Prehistory My Parents’ Montr´eal Years and Growing Up with Raoul as My Father Candace Bott Raoul Bott, McGill, the 1940s Joachim Lambek

9 13

Iron Rings, Doctor Honoris Causa Raoul Bott, Carl Herz, and a Hidden Hand P. Robert Kotiuga 17 The Bott – Duffin Synthesis of Electrical Circuits John H. Hubbard

33

Early Students and Colleagues Raoul Bott as We Knew Him Peter D. Lax, Friedrich Hirzebruch, Barry Mazur, Lawrence Conlon, Edward B. Curtis, Harold M. Edwards, Johannes Huebschmann, and Herbert Shulman

43

Working with Raoul Bott: From Geometry to Physics Michael Atiyah

51

The Algorithmic Side of Riemann’s Mathematics Harold M. Edwards

63

Actions of Lie Groups and Lie Algebras on Manifolds Morris W. Hirsch

69

PDE from the Point of View of Multiplier Ideals Joseph J. Kohn

79

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vi

CONTENTS

Dirac Operator and K-Theory for Discrete Groups Paul Baum

97

The Lefschetz Principle, Fixed Point Theory, and Index Theory James L. Heitsch

109

A New Look at the Theory of Levels John Cantwell and Lawrence Conlon

125

´ On the Space of Morphisms Between Etale Groupoids Andr´e Haefliger

139

Localization, Equivariance and Outgrowths of Morse Theory and Periodicity Raoul Bott as We Knew Him James A. Bernhard, Nancy Hingston, Jim Stasheff, and Victor Guillemin

153

Loop Products on Connected Sums of Projective Spaces Nancy Hingston

161

Equivariant Cohomology and Reflections James A. Bernhard

177

Connectedness of Level Sets of the Moment Map for Torus Actions on the Based Loop Group Lisa Jeffrey

181

Computing Characteristic Numbers Using Fixed Points Loring W. Tu

185

From Minimal Geodesics to Supersymmetric Field Theories Henning Hohnhold, Stephan Stolz, and Peter Teichner

207

Dualities and Interactions with Quantum Field Theory Raoul Bott as My Math Teacher Cumrun Vafa

277

A Physics Colloquium at McGill that Changed My Life Steven Lu

279

Geometric Langlands from Six Dimensions Edward Witten

281

Duality and Equivalence of Module Categories in Noncommutative Geometry Jonathan Block 311 Generalized Complex Geometry and T-Duality Gil R. Cavalcanti and Marco Gualtieri

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341

CONTENTS

Topological Quantum Field Theories from Compact Lie Groups Daniel S. Freed, Michael J. Hopkins, Jacob Lurie, and Constantin Teleman

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vii

367

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Preface Raoul Bott was a “citizen of the world” in that he was born in Budapest Hungary in 1923, moved to Slovakia soon after, and when he was 16, spent a year being schooled in England. He then lived in Canada for six years, where he obtained two degrees in Electrical Engineering from McGill University, during which time he resolved to become a mathematician. Bott finished a ScD in applied mathematics under Richard Duffin at CMU (which at that time was CarnegieTech), and solved the most challenging synthesis problem in electrical network theory at that time. An encounter with Herman Weyl landed him an invitation to the IAS at Princeton where, after a few relatively intense incubation years, the application of Morse theory to the problem of computing geodesics on Lie groups, and the Bott periodicity theorem for the stable homotopy groups of the classical Lie groups, helped metamorphose him into one of the most influential topologists of the 20th century. In a period of about a decade, he was awarded tenure at the University of Michigan and then at Harvard, and among his graduate students, two (Daniel Quillen and Stephen Smale) would eventually be awarded Fields medals for their work in topology. After four decades at Harvard and many more students, he retired to San Diego where he succumbed to lung cancer late in 2005. Raoul Bott’s collected papers testify to his contributions to mathematics. This book, like the conference that preceeded it, strives to feature prominent researchers who see further by standing on Bott’s shoulders. By looking forward, neither project was a systematic attempt to summarize the mathematics found in the four volumes of his collected works, published over a decade ago. Rather, we are trying to develop a view of how the mathematics Bott mastered is manifested in current mathematical research and in emerging applications to mathematical physics. P. Robert Kotiuga

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For Liv Raoul Bott

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https://doi.org/10.1090/crmp/050/01 Centre de Recherches Math´ ematiques CRM Proceedings and Lecture Notes Volume 50, 2010

Introduction P. Robert Kotiuga

How this book came into being How did an electrical engineer end up editing a book dedicated to an engineer turned mathematician like Raoul Bott? Imagine a graph whose nodes correspond to mathematicians and edges correspond either to coauthorship of a paper or to a thesis advisor/student relationship. In analogy to an “Erd˝ os number” any mathematician can define a “Hilbert number,” “Bott number,” etc., by counting the number of edges on a path joining themselves to a given mathematician in the graph, and then minimizing over paths. If one wants to celebrate the contributions of a particular mathematician called Bott, one can simply try to chat up every living mathematician with a Bott number of one or two, and hope that one ends up with a “coalition of the enthusiastic.” I went through this exercise in the case of Raoul Bott and I can assure you that this was an incredibly rewarding experience. By rounding up a distinguished international advisory committee, tapping into the organizational support of the Clay Mathematics Institute and the incredible logistical support from the Centre des recherches math´ematiques (CRM) in Montr´eal, obtaining financial support from the National Science Foundation of the USA, and tapping into the overwhelming overall enthusiasm that I sensed, I was able to organize a week-long conference, June 9 – 13, 2008, at the CRM entitled “A Celebration of Raoul Bott’s Legacy in Mathematics.” On the heels of the conference I was approached to follow-up with a book having the same title. In this way, the current book is not a conference proceedings but a result of the same “Bott number algorithm” that resulted in the conference. Depending on the travel and writing commitments of all involved, the book and the conference contributions often involve different but overlapping sets of people. The conference focused on the mathematical legacy of Raoul Bott and the extraordinary impact he had on both topology and interactions between mathematics, physics and technology. The forward-looking event brought together some prominent mathematicians with close connections to him, in order to discuss the emerging fields of mathematics and applications. The following focus topics emerged: • Operator algebraic methods, • K-theory, This is the final form of the paper. c
2010 American Mathematical Society

1

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2

P. R. KOTIUGA

• • • • • • • •

Supersymmetry and Quantum Field Theory, Morse theory, String topology, Pseudoholomorphic curves, Floer homology, Equivariant cohomology and localization, Generalized cohomology theories and TQFTs, Networks and modular forms.

The present book follows the lead of the conference.

A word about timing Given that Raoul Bott was born in 1923 [4], many of his contemporaries are still active but finding it more difficult to travel. After Bott was awarded an honorary doctorate from McGill in 1987, and anecdotes from his years in Montr´eal were documented [3, 5, 7], many in the mathematical community became intrigued by his time in Montr´eal. At the conference we were fortunate to have had Jim Lambek reminisce about Raoul Bott as an Engineering student at McGill University in the 1940s. He brought some poorly documented but noteworthy history into the conference without distracting from the forward-looking mathematical agenda. This book, like the conference, is neither billed as a memorial conference nor a historical conference, but is forward looking. However, historical aspects and anecdotes, (see, for example, [2, 5]), impacted both the timing of the conference and the overall coherency of the event. The fact that Bott’s presence is still fresh in the minds of all those involved made for a tremendous amount of enthusiasm. There were several arguments for undertaking this work before memories faded. First, Bott has many prominent colleagues from the 1940s and 50s who are still very active and keen to contribute. They generate a considerable amount of interest amongst two younger generations of mathematicians and it is unreasonable to expect the octogenarians to be as keen and active in future years! Second, from five decades of Bott’s Ph.D. students we needed to recruit those who contribute to the forefront of research. It would be much harder to get excellent forward-looking contributions from his students and coauthors had we waited. Finally, human nature was a key factor: in the case in hand, it was the realization that mathematicians have difficulty talking about another mathematician’s work; they find it easiest to talk about another mathematician’s work in the context of their own! Given this reality, it was really wonderful to have so many of Bott’s students, coauthors, and fellow kindred spirits willing to give talks and to contribute to an inspired “symphony.”

How to read this book All of the contributions to this book can be read independently. However, in order to help a whole to emerge from the parts, the book is broken into four sections. To make the book accessible to a wider audience, each section starts with the easier to read reminiscences and works its way into more involved papers. Of course, those with extraordinary focus and stamina are invited to read the longer papers first! Either way, this book is meant to be enjoyed.

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INTRODUCTION

3

The conference activities in the context of this book This five-day conference was held and the CRM on June 9 – 13, 2008. Montr´eal was a natural venue for such an event since Raoul Bott obtained two degrees in Electrical Engineering at McGill University in the 1940s and an Honorary Doctorate from McGill in 1987. The conference was cosponsored by the Clay Institute, and partially funded by the NSF. The scientific committee consisted of: • • • • • • • • • •

Sir Michael Atiyah (U. of Edinburgh), Octavian Cornea (U. de Montr´eal) David Ellwood (Clay Math Institute), Jacques Hurtubise (McGill U.), Fran¸cois Lalonde (U. de Montr´eal), David Mumford (Brown U.), Graeme Segal (Oxford U.), Stephen Smale (TTI Chicago, UCB Emeritus) Jim Stasheff (U. of Pennsylvania, UNC Emeritus) Edward Witten (IAS, Princeton)

The conference was organized by Robert Kotiuga. We were fortunate to have recruited influential speakers and panelists from three generations in order to cover six decades of Raoul Bott’s research and collectively identify his enduring mathematical legacy. The scientific activities of the conference consisted of two panel sessions, and of twenty-four hours of lectures given by: • • • • • • • • • • • • • • • • • • • • • • • • •

Sir Michael Atiyah (U. of Edinburgh) Paul Baum (Pennsylvania State) James Bernhard (U. of Puget Sound) Octavian Cornea (U. de Montr´eal) Ralph Cohen (Stanford U.) Marco Gualtieri (MIT), James Heitsch (UIC Emitus; Northwestern U.) Nancy Hingston (The College of New Jersey) Morris W. Hirsch (U. C. Berkeley) John Hubbard (Cornell U.) Lisa Jeffrey (U. of Toronto) Nitya Kitchloo (U. C. San Diego) Joseph Kohn (Princeton U.) Robert Kotiuga (Boston U.) Jim Lambek (McGill U.) Peter D. Lax (New York U.) John Morgan (Columbia U.) Stephen Smale (Toyota Technological Institute at Chicago) Andras Szenes (U. of Geneva) Constantin Teleman (U. C. Berkeley) Susan Tolman (U. I. Urbana-Champaign) Loring Tu (Tufts U.) Cumrun Vafa (Harvard U.) Jonathan Weitsman (U. C. Santa Cruz) Edward Witten (Institute for Advanced Study).

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P. R. KOTIUGA

Some of these presentations made it into this volume. The actual titles and abstracts are still archived in the “schedule” on the conference website. A one paragraph summary is as follows. The masterful presentations of Michael Atiyah and other senior colleagues lived up to the audience’s highest expectations. The speakers who talked about localization and singularity theory clearly built on the last two decades of Bott’s research. The talks by C. Vafa and E. Witten dwelled on a stream of dualities which quantum field theory has been offering mathematics in recent decades, and the tantalizing new connections to number theory. Many other talks rounded out the conference in other ways. However, there were several unexpected developments which seemed to have appeared magically, and which were not obvious before the conference. Loosely speaking, one pertains to ChasSullivan “string topology” and its relation to Floer homology via Morse theory. Here, on one hand, the work of N. Hingston and M. Goresky recast string topology in terms of Morse theory as applied to loop spaces by Bott in the 1950s. On the other hand, the works of N. Kitchloo and R. Cohen built on Morse theory in the context of “quantum topology,” and refine the use of Morse theory in low dimensional topology. In the talks given by these speakers, as well as those of O. Cornea and C. Teleman, one could sense where the field of manifold topology was headed in the next few years. While many of the new results would be new to Bott, the connection to his mathematical perspective and legacy is inescapable! Panel sessions were held on the first and last days. The Monday panel session, “Raoul Bott as teacher, mentor, and colleague,” was chaired by Nancy Hingston, and the panelists were: Paul Baum, Jim Stasheff and Loring Tu. Many wonderful anecdotes were shared. Some were reminiscent of those found in the volume on the founders of index theory [8]. The Monday panel session featured the film, “A Peek Into the Book,” by Vanessa Scott. The movie is about her grandfather and Vanessa’s accompanying remarks were read by Candace Bott. The movie was a hit, it brought back many memories, and it was rerun at the end of the conference. Other memorable anecdotes about Bott were also related at the banquet in old Montr´eal, in speeches given by Michael Atiyah, Stephen Smale, and Candace Bott. The Friday panel session “Examining Raoul Bott’s Legacy in Mathematics” was chaired by Michael Atiyah [1], and the panelists were Nancy Hingston, Jacques Hurtubise, Nitya Kitchloo, and Susan Tolman. Unfortunately, it would not do justice to the many threads of the discussion if they were summarized in the space available. However, one aspect pertaining to Bott as an inspiring lecturer must be related. Although this has been documented in many places, the conference has produced some unexpected posthumous testimony! At the end of the Friday panel session, the conference organizer emphasized that the conference was not organized around a mathematical topic but a mathematician, and asked the younger attendees what they thought of the concept. A student who identified himself as a graduate student working in an unrelated field made a remarkable comment: He said he learned more at this conference than at mathematics conferences in his area of expertise since speakers at this conference seemed to make an extraordinary effort to communicate their ideas in the simplest and most visual terms possible. What was more remarkable was that the “instant consensus” in the room was that this was a manifestation of all the speakers being influenced by Bott’s lecturing style and his insistence on understanding

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INTRODUCTION

5

deep mathematical concepts in the simplest terms possible. Hopefully this ease of communication is manifested in this book. Acknowledgements Clearly, my biggest debt is to Raoul Bott. Who else could have cultivated so many extraordinary colleagues and students. When I bought Bott’s collected works [6] over a decade ago, I never thought I’d run into such a group of kindred spirits with whom I could discuss so many of the wide variety of things that I learned from those four wonderful volumes; let alone follow the ideas into the future. The organizational support of the event by the Clay Mathematics Institute and funding from the National Science Foundation (Award ID0805925) were essential ingredients to making the conference a success. Support from the staff at the CMI and the CRM made the whole experience a joy. At every stage the encouragement of David Ellwood lit my path like a beacon. The support of the Montrealers was truly inspiring. Jacques Hurtubise and Peter Russell were useful resources all along, Fran¸cois Lalonde was incredibly supportive from the outset, Suzette Paradis did a wonderful job with the web page and conference poster, Louis Pelletier was a most inspiring conference coordinator, Andr´e Montpetit put the book together, Chantal David took every opportunity to be sure that my efforts were documented in print, and Norbert Schlomiuk did a remarkable job as head cheerleader! All of my interactions with the speakers and members of the Scientific Advisory Committee were enriching experiences which I will cherish. Again, all the personal connections to Raoul Bott had something to do with making this a wonderful event! In particular, my interactions with Michael Atiyah and Jim Stasheff were one of the most positive learning experiences of my life and I am forever grateful for having found such effective mentors. A unique aspect of the conference was the visual memory of Bott; from the “picture gallery” on the website, http://www.crm.math.ca/Bott08, to pictures of him from six distinct decades on the conference poster, to the screening of Vanessa Scott’s film: “A Peek into the Book.” The unique combination of mathematics and an intimate connection to the Bott family would not have been possible without the effort of Raoul Bott’s daughter, Candace Bott, who spoke at the banquet, introduced her niece’s film, and was indispensable in helping with all visual aspects of the conference. Finally, I am most grateful for the tireless work of all the reviewers. Several papers in this volume required multiple reviews and corrections and I am very grateful that this anonymous critical process went so smoothly. Furthermore, Arthur Greenspoon from the Mathematical Reviews went through all the texts of the book and I can only imagine how this helped the look and feel of the book. References 1. Sir M. Atiyah, Raoul Harry Bott, 24 September 1923 – 20 December 2005, Biographical Memoirs of the Fellows of the Royal Society 53 (2007), 63 – 76. 2. R. Bott, On topology and other things, Notices Amer. Math. Soc. 32 (1985), no. 2, 152 – 158. , Autobiographical sketch, Raoul Bott: Collected Papers. Vol. 1 (R. D. MacPherson, ed.), 3. Contemp. Mathematicians, Birkh¨ auser, Boston, MA, 1994, pp. 3 – 9. , The Dioszeger years (1923 – 1939), Raoul Bott: Collected Papers. Vol. 1 (R. D. 4. MacPherson, ed.), Contemp. Mathematicians, Birkh¨ auser, Boston, MA, 1994, pp. 11 – 26.

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5. A. Jackson, Interview with Raoul Bott, Notices Amer. Math. Soc. 48 (2001), no. 4, 374 – 382. 6. R. D. MacPherson (ed.), Raoul Bott: collected papers. Vol. 1: Topology and Lie groups, Contemp. Mathematicians, Birkh¨ auser, Boston, MA, 1994. 7. L. W. Tu, The life and works of Raoul Bott, Notices Amer. Math. Soc. 53 (2006), no. 5, 554 – 570. 8. S.-T. Yau (ed.), The founders of index theory: Reminiscences of and about Sir Michael Atiyah, Raoul Bott, Friedrich Hirzebruch, and I. M. Singer, 2nd ed., International Press, Somerville, MA, 2009. Department of Electrical and Computer Engineering, Boston University, Boston, MA 02215, USA E-mail address: [email protected]

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Part 1

Montr´ eal, the 1940s, and Mathematical Prehistory

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https://doi.org/10.1090/crmp/050/02

Centre de Recherches Math´ ematiques CRM Proceedings and Lecture Notes Volume 50, 2010

My Parents’ Montr´ eal Years and Growing Up with Raoul as my Father Candace Bott Abstract. These are some remarks that I made on behalf of the family at the CRM conference on the evening of June 10, 2008, at a welcome banquet that was held at the restaurant Chez Queux in Vieux Montr´ eal.

It’s a privilege and a delight to share in this celebration with so many of my father’s close colleagues and friends. And it’s special that the venue is Montr´eal, the place where my parents’ life together began and the place where my father achieved his dream of becoming a mathematician. My mother Phyllis, my brother Tony, my sister Jocelyn and my twin Renee, as well as the grandchildren, would have loved to join us here were it possible. And so, speaking on behalf of the entire family, I would like to express everyone’s appreciation. We are extremely pleased that this conference came together. Thanks to the vision and inspiration of Robert Kotiuga, who first proposed the idea, and thanks to the people who made it happen: Michael Atiyah, Fran¸cois Lalonde, Louis Pelletier and everyone at CRM. Were my mother present, she would regale you with stories of her first encounters with Raoul at McGill, where he was nicknamed “the Count” for his highbrow taste and dashing appearance. (He would typically don a fur hat and, with his Hungarian accent, was able to trick people into thinking that he came from nobility.) My mother remembers their first meeting at a bowling alley, where she was paired with my father, the inexperienced player, in a challenge against many people, only to lose and pay for it: The lowest-scoring team had to treat everyone to beer. She met him for the second time in 1945, after he had graduated from McGill and joined the Canadian army. He arrived in full uniform and jested that they should marry because it would increase his army stipend and vastly improve the size of his living quarters (at which point he pointed out the window to his onebedroom rental across the street). After many more encounters, she was intrigued, but not convinced that he was the man for her. But then there was the pivotal moment when my father proclaimed: “much to my surprise I find I like you” (which, despite his usual 2000 Mathematics Subject Classification. 01A60. This is the final form of the paper. c
2010
2010 American c American Mathematical Mathematical Society

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C. BOTT

chutzpah, began their courtship; only because my mother was usually good at reading innuendo). During their courtship, my mother remembers a time that my father showed up in his newly purchased 1926 Ford to show her around the city, when the car careened out of control, up onto a sidewalk and finally stalled on some trolley tracks. (Unbeknownst to my mother, it was Raoul’s first time behind the wheel, minus the benefits of Driver’s Ed! He’d purchased his license for 50 cents, which was possible in Montr´eal in the 1940s, without bothering to take driving lessons.) There was also the time when Raoul coaxed my mother into climbing over a wall to get into a McGill football game without purchasing tickets only to get caught and booted out; the irony being that my dad, who had perfected the art of slipping into events by passing himself off as someone important, had never dreamt of getting caught. So Raoul didn’t exactly get high marks for his courtship skills, only his playful sense of adventure. High marks, it turns out, were not a priority for him at McGill either. In a convocation address that he gave at McGill in 1987, he told the graduating class how he’d always managed, despite his mediocre grades, to “beat the Dean.” As my father explained it, this meant the students who’d scored high enough to pass a course were listed on a bulletin board in linear order of merit, and at the bottom of this terrifying document came the signature of the Dean. Hence the expression of “beating the Dean” if one got through. As Raoul’s daughter, I can tell you a bit about what it was like to have this Hemingwayesque man as my father. When I was a small child, I remember him presiding over family dinners, evaluating our conduct, refining our table manners, and seeing that we ate everything on our plates — little did he know that some of us were feeding the dog under the table. After dinner, he would get lost in his own world of mathematics or music and either retreat to his attic study to prepare his lectures or to the piano where he would play Bach, Beethoven, or Schubert into the wee hours of the night. As an old-world European, he believed in discipline and wanted to instill good values in his children. This often meant, at least from a small child’s perspective, renouncing pleasure for punishment. I remember him encouraging us to explore the vast galleries of the Louvre and the Oxford Ashmolean until we knew the true meaning of museum fatigue, or asking that we put in countless hours of yard work as a way of earning our keep, or demanding that we forfeit summer beach time in California for lessons in long division. Aside from wanting his children to master the basics of math to get a leg up in school, he wanted us to become enraptured with art, music, poetry and literature. He also encouraged us to be active in nature and to take on feats of physical endurance. Later in life, I remember relaxed times on Martha’s Vineyard when the family would gather at my parents’ summer home in Chilmark to take in the beauty and wonder of nature and to mingle as a family. These are among my most treasured memories; maybe because Chilmark was a place where my father’s boundless appreciation for life would come to the fore, and, with no constraints on his or my time, we could let our conversations explore deeper, more personal terrain.

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´ UPYEARS ´ MYYEARS PARENTS’ MONTREAL MY PARENTS’ MONTREAL & GROWING WITH RAOUL AS MY FATHER 11 13

I regard my father’s warm and adventurous spirit combined with his lust for life as the most treasured aspects of growing up with him. Life with Raoul was one big adventure. He brought joy, surprise and gravitas into my life, for which I am grateful, and it’s wonderful to see his influence as a person and a thinker reach into the greater, more universal, realm of mathematics. Clay Mathematics Institute, Cambridge, MA 02138, USA E-mail address: [email protected]

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https://doi.org/10.1090/crmp/050/03

Centre de Recherches Math´ ematiques CRM Proceedings and Lecture Notes Volume 50, 2010

Raoul Bott, McGill, the 1940s Joachim Lambek

Let me begin by congratulating Kotiuga for assembling a spectacular collection of mathematicians in one place and thank him for proposing the title of this talk. But why am I here? Although I never collaborated with Raoul, I knew him longer than anyone else here, having met him 65 years ago, when most of you were not even born. I was inscribed in an Honours Mathematics and Physics program at McGill, as were a few others destined to become professional mathematicians: the number theorist Raymond Ayoub, later a Professor at Penn State, the algebraist Martin Burrow and the analyst Louis Nirenberg, both at N.Y.U.. However, Raoul was studying Electrical Engineering, after having spent a year at a prestigious private school. I first met Raoul at the McGill students’ union, now the McCord museum. McGill was very much smaller then, so students from different faculties got together. Raoul sat down at a piano and gave a magnificent performance of Bach fugues, apparently having memorized the score. When I congratulated him on his phenomenal memory, he laughed and said he was just improvising. Evidently he had internalized Bach’s underlying principles. Raoul had come from Bratislava, formerly in Austro-Hungary, but now the capital of Slovakia. I was surprised to learn only here that he spoke both Slovak and Hungarian. With me he occasionally conversed in German. His stepfather owned a Delikatessen shop on St. Lawrence Boulevard, called “Sepp’s.” He had tried to enroll Raoul in one of the McGill dormitories, which was presided over by MacLennan, the head of the Philosophy department. MacLennan later told me that he was not sufficiently impressed to accept his stepson, but later changed his mind after listening to a talk by Raoul at a student affair. Having completed his Engineering program, Raoul enrolled in a graduate Mathematics and Physics program. We both took a couple of courses from Gillson, who had studied Physics at Cambridge. Gillson also became Vice Principal in charge of Dawson College, a temporary McGill campus in St. Jean, to accomodate returning veterans. Both Raoul and I taught elementary mathematics courses to huge classes: calculus, analytic geometry and spherical trigonometry, which was then required for 2000 Mathematics Subject Classification. 01A30. This is the final form of the paper. c
2010 c
2010 American American Mathematical Mathematical Society

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14 16

J. LAMBEK

navigation. We often travelled together on the bus between Montreal and St. Jean and discussed mathematics and physics. I recall Raoul once asking: “Did I make a mistake when I told my students that sine and cosine are called ‘orthogonal’ functions, because their graphs intersect at right angles?” We also made a serious attempt to study a book on quantum mechanics (I believe by Sommerfeld), but we didn’t get very far. I remember trying to persuade Raoul that this should be done with the help of quaternions, a subject I then was enamoured with, but only got around to publishing an article on half a century later. I was to learn that Raoul would exploit quaternions more seriously in his own research. I had not seen my family in England since before the Blitz, and Raoul too wanted to revisit Europe. So we decided to cross the Atlantic by working on a steamer. From adventure stories we both had read, we inferred that this could be arranged in a tavern at the port. We found no tavern in the Montreal harbour, but did manage to make contact in a home for sailors, only to learn that to get a job on a boat one had to be a paid up member of the seaman’s union. We reluctantly booked passage as paying passengers on a freighter. Raoul later changed his mind and I ultimately went alone. I returned to Montreal to complete my graduate studies, but Raoul moved to the United States, where he wrote a spectacular thesis on electrical networks. This led to two years at the Institute for Advanced Study in Princeton, where he learned Morse theory, and the rest is history. Years later, on Raoul’s advice I too spent a sabbatical at the Institute, but made no immortal contributions to mathematics. Still, a memoir, written in collaboration with Nathan Fine and Leonard Gillman, has recently been reprinted. On a later sabbatical in Z¨ urich, I met Raoul again. One day he told me that his car had disappeared, after he had parked it illegally at the cathedral in the old town. Having looked for it on all four sides of the church, he could not find it. The police denied having impounded the car, which only turned up when Raoul realized that the church had more than four sides. I guess the number of sides is not important for a topologist. After the sixties, I only saw Raoul on rare occasions, when he visited Montreal. Unfortunately, I was not present when he received his honorary doctorate at McGill. In the forties, the McGill mathematics department was very much smaller than it is now, with only six or seven members on the teaching staff, one of whom, Charley Sullivan, taught all the honours courses. Lloyd Williams, one of the founders of the Canadian Mathematical Congress (now called “Society”) shared an office with the renowned number theorist Gordon Pall, with whom I wrote my Master’s thesis. The Mathematics department had no secretaries; in fact, the entire Faculty of Arts and Sciences had just one secretary, who typed my thesis. Today the Mathematics department has seven secretaries, but mathematicians are expected to type their own manuscripts. So much for progress! I was the first to obtain a Ph.D. at McGill, under the direction of Hans Zassenhaus in 1950. The second was Jean Maranda, a distinguished algebraist, who became a professor at the Universit´e de Montr´eal and a fellow of the Royal Society

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RAOUL BOTT, MCGILL, THE 1940S

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of Canada. Sadly, he died in a car crash on his way to a meeting of the Royal Society in Ottawa. Department of Mathematics and Statistics, McGill University, 805 Sherbrooke Street West, Montr´ eal, QC H3A 2K6, Canada E-mail address: [email protected]

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https://doi.org/10.1090/crmp/050/04 Centre de Recherches Math´ ematiques CRM Proceedings and Lecture Notes Volume 50, 2010

Iron Rings, Doctor Honoris Causa Raoul Bott, Carl Herz, and a Hidden Hand P. Robert Kotiuga Abstract. The degree of Doctor of Sciences, honoris causa, was conferred on Raoul Bott by McGill University in 1987. Much of the work to make this happen was done by Carl Herz. Some of the author’s personal recollections of both professors are included, along with some context for the awarding of this degree and ample historical tangents. Some cultural aspects occurring in the addresses are elaborated on, primarily, the Canadian engineer’s iron ring. This paper also reprints both the convocation address of Raoul Bott and the presentation of Carl Herz on that occasion.

Introduction Raoul Bott needs no introduction in this volume. However, reprinting his address at the 1987 McGill convocation both gives some insight into the effort to award him an Honorary Doctorate in Mathematics from McGill, and a context to develop some less than mathematical themes, unashamedly from the point of view of an electrical engineer who enjoys the historical aspects of his discipline. Early on I was tipped off that Carl Herz was behind the effort, and the memory of Prof. Herz made me realize that I had to follow the trail like a hound. The result complements more technical presentations and anecdotes pertaining to Montr´eal1 . I am grateful to Candace Bott for digging up her father’s commencement2 address, and to Dominique Papineau of McGill University who showed up in Boston with a complete file pertaining to the awarding of the honorary degree from McGill’s archives. The support of various McGill faculty who listened to me think aloud is much appreciated, namely Peter Caines, Jacques Hurtubise, Joachim Lambek, and Peter Russell.

2000 Mathematics Subject Classification. Primary 01A60; Secondary 01A65. This is the final form of the paper. 1 For anecdotes from Bott’s years in Montr´ eal see [25] and Candace Bott’s remarks in this volume. 2 Although “convocation” and “commencement” have different meanings in general, in the context of a university graduation ceremony they have the same meaning in Canada and in the USA respectively. For the purposes of this article, they are used interchangeably. c 2010 American Mathematical Society

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P. R. KOTIUGA

How I got to know Raoul Bott I can’t remember who first connected Raoul Bott to McGill Engineering in my mind; most likely it was Peter Caines, Robert Hermann or Carl Herz. However, I do remember attending his talk in the Physics department at McGill in, I believe, 1982. These days, if the fact that I dragged my wife to be to the talk is mentioned, and that she coped with my enthusiasm with a sense of humor, my kids will kindly remind me of all my thrifty ideas for a good time! During those years I spent many hours “teething” on the book of Bott and Tu, and as a NSERC post-doc in the MIT Mathematics department in 1985 I finally got the opportunity to audit a course given by Raoul Bott. Not only was his mathematics entrancing, but we were both McGill engineers! During the first lecture he spotted my iron ring. I introduced myself afterwards, and we talked about a variety of things. Before long, I made a habit of auditing every course he taught. About a year later, he told me that he would be the recipient of an honorary doctorate from McGill — forty years after McGill wouldn’t have him as a graduate student in the mathematics department (he was unwilling to complete a second undergraduate degree in mathematics). Past history aside, he seemed genuinely honored, but he didn’t quite know who to share this news with. Reading the convocation speech, I now see it as his way to make peace with history and a means to repackage it constructively for graduates forty years younger than himself. Stepping back from the ceremony from decades ago, the reader is invited to read his “Autobiographical Sketch” [6]. I clearly enjoyed his lecturing style as well as the attention and discipline he demanded in the classroom. It certainly contrasted with the convocation speech. In the first lecture, he encouraged students to ask questions and claimed that he liked “stupid questions” because he could answer them. He was very serious about this and if we didn’t realize it initially, we eventually learned that he put the bar very high, and even higher for himself: His answers to questions were always more profound than the original questions and he once walked out of his own lecture in frustration because he didn’t like it! That was dramatic, baffling, and unexpected — especially since we knew he liked to think on his feet. In this particular case, he reappeared the next class, his arguments were impeccably clear and elegant, and his credibility was not only restored, it soared! Going beyond the classroom, Bott’s mentoring of graduate students is the source of legend. This volume has plenty of testimony from his students, and the reminiscences by Robert MacPherson [19] testify to his high expectations. The convocation speech is very different in that it demonstrates his ability to connect with an audience that has never seen him and most likely will never see him again. Before attempting to advise graduates, he warms up the audience by telling of the pivotal event in his professional life and saying: “I tell you all this only in part as a jest.” Only when the stage is set does he give the essence of his address credibly and in a few sentences. The message appears in a flash after saying “But my time is up!” The style mirrors his approach to giving a colloquium talk. Bott enjoyed cultivating certain habits which were best left unmentioned in the convocation speech. For instance, he lectured at 8:30 a.m. in order to have a flexible day after 10:00 a.m.! Graduate students felt this cramped both their style and sleeping habits, and this is where some of Bott’s Old World sensibilities kicked in when they dared to doze off in class. He could toss chalk and have it land on the table inches from the sleeping student’s face, startling them. He clearly relished

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IRON RINGS, DOCTOR HONORIS CAUSA RAOUL BOTT,. . .

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doing so. The memorable line which would meet the startled face changed with every successive offense. From “didn’t want you to miss anything” to “how much is tuition at Harvard?” to “who pays your tuition?” In Old World style, this was all for the benefit of the student and there was little room for self-preservation. These days it isn’t easy for a professor to be respected for doing this in a private university where students can feel like paying customers once the tuition bill is settled. Somehow Bott was consistently more mischievous than the students, and got away with it. Clearly he had extensive experience testing his teachers and this experience always gave him the upper hand in the classroom. As for the impact Bott’s lectures made on me, I’d be treading on thin ice if I tried to say why they were fantastic. Loring Tu says that3 Victor Guillemin, at a conference celebrating his 60th birthday, proudly announced that he took twelve courses from Bott, and to Loring’s chagrin, he could only list eleven. Clearly, I am in no position to speak with authority about Bott’s lectures! In my case, I loved his lecture style, the lecture material, and I felt a definite kinship since I could ask tangentially related questions after class and consistently get profound answers. There is perhaps one personal anecdote I can add to the many that I’ve heard. One day in class, after Bott explained the set-up of the Lefschetz fixed point theorem in terms of the transverse intersection of the graph of a map from a manifold to itself with the graph of the identity map, he claimed that, by duality, the Lefschetz number could be easily computed by picking a basis for integral cohomology, pulling back by the appropriate maps, taking wedge products with Poincar´e duals and integrating. When he claimed that it reduced to basic matrix algebra involving the induced automorphisms on cohomology groups which an engineer could do, the class just didn’t make the type of eye contact he was hoping for. At that point he called me up to the board and told me to fill in the details of the calculation! As I (methodically) wrapped up the calculation, he identified me as an engineer, emphasized that budding topologists shouldn’t shy away from such concrete calculations, and took satisfaction in the fact that he made his point. In my mind he reinforced the fact that Daniel Quillen’s thesis advisor could make us think functorially while, as a student of Richard Duffin, he could encourage us to “think with our fingers” and to always maintain a balance between the conceptual and the computational. Obviously, I’m enamored with Raoul Bott, and thrilled that he was the recipient of an honorary degree from McGill. However, my purpose here must be more focused. Specifically, in my mind, a few key points need elaboration: • Who was the driving force behind getting McGill to award Bott an honorary degree in Mathematics forty years after he left McGill as an engineer? Clearly, Bott was deserving, but it takes a kindred spirit to overcome the inertia of a bureaucracy and Carl Herz was such a kindred spirit. • In his convocation address, Bott vividly describes the pivotal moment at McGill when he decided to become a mathematician. However, how he was going to do it was not at all clear at the time. The wonderful and profound connections between topology and physics have been studied intensely in recent decades, but what is needed is a hint of the path from “engineering mathematics” to the mathematics Bott is known for [4, 17]. In retrospect, it almost seems that this path could have been more clear to Gauss and Maxwell than to modern specialists. We’ll soon 3 See

“Reminiscences of Working with Raoul Bott” in [32]

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P. R. KOTIUGA

see, that the career of Hermann Weyl provides us with a perspective and some key insights. • I’m fascinated with Bott’s struggle to reconcile old and new world sensibilities. I saw this encoded in my interactions with him and in the convocation address. I use the word “encoded” in reference to the address, because his references to the uniquely Canadian iron ring have their roots in various Quebec City bridge disasters, Rudyard Kipling’s poem “Sons of Martha,” and associated Biblical references. These details are required to fully decode the message. Carl Herz as a gateway to history I can distinctly remember the day Carl Herz knocked me off my feet. At the time I was a graduate student in Electrical Engineering at McGill and he was a feisty and famous professor of Mathematics. We began to chat after some seminar in the EE department, and he asked me what I was doing for a thesis. He listened as I told him how I felt that the reformulation of Maxwell’s equations in terms of differential forms was essential for the resolution of some key problems in computational electromagnetics. Specifically, most of the boundary value problems in low frequency electromagnetics amounted to Hodge theory on manifolds with boundary, with the periods of harmonic forms identified with the variables found in Kirchhoff’s laws. Furthermore, I told him that the whole framework has a variational setting which can be discretized by appealing to “Whitney forms” in order to obtain a finite element discretization with desirable properties. To me it was all obvious if one read the papers written by Donald Spencer and his students in the 1950s and interfaced them with Whitney’s “Geometric Integration Theory.” In retrospect, this was a natural connection given the work of Jozef Dodziuk and Werner M¨ uller’s proof of the Ray-Singer conjecture, but it was not apparent at the time. Carl listened, started pacing back and forth and I was beginning to worry that he was going into a trance! I don’t know what was going on in his mind, but I braced myself for what could come out of his mouth. I think I was standing in stunned silence when Carl stopped and asked me if I ever read Maxwell. Sheepishly, I told him that I read a good deal of Maxwell and that everyone in my field swears by Maxwell. He then asked me if I knew what a periphractic number was. When I expressed my ignorance, he went on to point out that Betti’s paper was written a decade after Maxwell’s treatise, and that in Maxwell’s treatise the first Betti number was called a “cyclomatic number” — a term introduced by Kirchhoff, and still used in graph theory. He went on to tell me that the second Betti number was called a “periphractic number”. . . . I later found out that Maxwell borrowed the term from Listing4 and that Listing was the person who coined the term topology. In one swoop Carl convinced me that Maxwell was often quoted but never read, and that if I wanted to get to the origin of these topological ideas the origin would be in some language other than English. Clearly, I was humbled — but I felt better when I looked up “periphractic” in the unabridged Oxford dictionary and found that Maxwell’s treatise is the first and last use of the word in the English language!5 4 See

Breitenberger [8] in James [16] for an article on Johann Benedikt Listing and his book. is irresistible to point out the connection between Maxwell and Morse theory in this article about Bott. Listing [18] is credited as being the first to systematically obtain a cell decompositions of 3-manifolds by tracking the change in topology as level sets cross a critical point. Maxwell [20] 5 It

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IRON RINGS, DOCTOR HONORIS CAUSA RAOUL BOTT,. . .

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Besides being awed by Carl’s encyclopedic knowledge, there are two big lessons I have learned over the years and which were initiated by my encounter with Carl and other mathematicians from his generation who “read the masters.” The first was that Maxwell had a profound experimental and theoretical knowledge, and that much of the inspiration for his theoretical work came from reading and corresponding with Germans (Gauss, Riemann, Kirchhoff, Clausius, Helmholtz, Listing,. . . ). Furthermore, it was the Germans who took Maxwell seriously when no one else did — from Helmholtz’ student Hertz demonstrating radio waves, to Boltzmann developing statistical mechanics, and to Einstein developing the logical physical conclusions of Maxwell’s theory. Contrast this with the situation in England where Oliver Heaviside was considered a self-educated eccentric who died in poverty despite making brilliant contributions to Maxwell’s theory and being awarded an honorary doctorate from G¨ottingen University in 1905. The other “Maxwellians” didn’t make it into the limelight either. The second big lesson I learned from my encounter with Carl is to never ignore the Institute for Advanced Study (IAS) in Princeton or the profound influence of its two first founding permanent members: Hermann Weyl and Albert Einstein. Carl was a student of Salomon Bochner and thrived on all the mathematics emanating from the IAS. In retrospect, the world seems quite small. It was Weyl who in 1948 invited Bott to the IAS, it was Weyl who earlier got de Rham, Kodaira, and Spencer to put Hodge theory on a rigorous footing, and it was Weyl [29, 31] and his close colleague Einstein who were the true curators of the developments arising from Maxwell’s theory. It turns out that the “Whitney forms” that I was so fond of have their origins in a 1952 paper of Andr´e Weil called Sur les th´eor`emes de de Rham. Clearly, every part of the novel mathematics I was using could be traced back to the IAS; even if my application of these ideas to computational electromagnetics was unforeseen. If ever I was in denial about details, I could check in with Donald Spencer’s student, Robert Hermann, to verify facts6 . If details were scarce in the literature, contemplating the influence of Hermann Weyl could help bring things into focus. Enough said about my interactions with Carl Herz. To appreciate Carl Herz’ contributions to harmonic analysis and other fields of mathematics, as well as the feisty character himself, through the eyes of his colleagues, the reader is referred to other sources [10,11]. Needless to say, when I realized that Carl Herz was behind the effort to award Bott an honorary doctorate from McGill, it seemed like a big piece of the puzzle fell into place. He plays a central role on almost all correspondence with the university administration on the matter and in the end, he’s the one who presented Bott for the degree in June of 1987. then wrote a paper citing Cayley and Listing. In his treatise [21], Maxwell uses the rudiments of Morse theory with the fact that a harmonic function cannot achieve a maximum or minimum in the interior of a region in order to make topological deductions. 6 The Advanced Calculus text Nickerson, Spencer, and Steenrod [22] was Princeton-inspired but was never published. However, it initiated a wave of differential-form based multi-variable calculus texts in the 1960s. Although it is a very natural way to bring multivariable calculus to its roots in Physics, this wave of texts didn’t catch on. Bott never wrote a text for such an undergraduate audience and so one can only hypothesize about how he would have integrated Kirchhoff’s laws with Hodge theory and Maxwell’s equations. In retrospect, it took a couple of decades to get things right and ultimately, the books that reached out to engineers and physicists most effectively were written by Bott’s close colleagues [9, 15].

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22

P. R. KOTIUGA

The convocation presentation of Carl Herz Mr. Chancellor, I have the honour of presenting to you, in order that you may confer on him the degree of Doctor of Sciences, honoris causa, Professor Raoul Bott. Raoul Bott was born in Hungary, but his university education up to the M.Eng. was at McGill. He received his B.Eng. from McGill in 1945. After a short stint in the infantry, he continued his studies in electrical engineering at McGill. The immediate postwar period saw a great demand for mathematics teachers, and Bott taught calculus here while studying for his master ’s degree. In addition he took some courses from Professor Gilson, then Chair of the Department of Mathematics. Nevertheless, he remained a student of electrical engineering until he left McGill to go to Carnegie Tech for his doctorate. Electrical engineering has a close affiliation with what might be viewed as an abstruse branch of mathematics, algebraic topology, Professor Bott’s specialty. One has only to recall that “Betti numbers,” the fundamental numerical invariants of topology, are named for an Italian electrical engineer, and one can read James Clerk Maxwell for profound insights into the subject. At an even more primitive level, circuit theory has always been a source of good problems for topologists. Bott’s earliest work was rather algebraic. The Bott – Duffin theorem (1949) on circuit synthesis was described by a reviewer thus: “This proof of the realizability of the driving point impedance without the use of transformers is one of the most interesting developments in network theory in recent years.” It continues to be a much-cited result. This work came shortly after Bott had obtained a D.Sc. in mathematics. After the doctorate, Raoul Bott went to the Institute for Advanced Study in Princeton. He was at the Institute during 1949 – 1951 and returned in 1955 – 1957. He joined the faculty of the University of Michigan in 1951 where he remained until 1959 when he was invited to his present academic home, Harvard, where he is William Caspar Graustein Professor of Mathematics. Professor Bott’s seminal contributions to mathematics are too extensive for me to do justice to them here. Most of his early ideas seem to have drawn their inspiration from the Calculus of Variations in its global version known as “Morse Theory.” Bott applied Morse Theory in an unexpected and striking way. Over a long period he, together with his various collaborators, worked out the topology of Lie groups and symmetric spaces. One must mention the Bott Periodicity Theorem which brought some order to the chaos of homotopy theory. He went on to study fixed point theorems and their application to other branches of mathematics including differential equations. Most recently Bott has been working on applications of topology and geometry to the Yang – Mills equations in quantum field theory. For his achievements, Bott was awarded the Veblen Prize of the American Mathematical Society in 1964. In addition to his purely scientific accomplishment, Raoul Bott stimulates all those who are about him. He is one of the best and most exciting expositors of mathematics I have had the privilege to listen to. Mr. Chancellor, McGill can take great pride in honoring this year, as it did last,7 another of its graduates who stand in the forefront of mathematics of the twentieth century. 7 Raoul Bott, Joachim Lambek and Louis Nirenberg all graduated from McGill in 1945 and Nirenberg was awarded an honorary doctorate from McGill in 1986.

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IRON RINGS, DOCTOR HONORIS CAUSA RAOUL BOTT,. . .

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The Eleventh Day of June, Nineteen Hundred and Eighty-seven Carl Herz Professor of Mathematics and Statistics. Given my encounters with Professor Herz, his correspondence with the McGill administration, and his encyclopedic breadth, it is clear that he played a central role in the case for the honorary degree. The masterful presentation of Carl Herz shows how a broad perspective can lead to a reorganization of knowledge that lets the likes of Paul Dirac and Eugene Wigner move from Engineering to Physics, and the likes of Raoul Bott, Solomon Lefschetz, John Milnor, and Donald Spencer move from Engineering to Mathematics. On the other hand, since Professor Herz always enjoyed an argument (in the very best sense of the word!), I’ll take the liberty to make a qualification and perhaps an elaboration. The qualification I might add is that Enrico Betti was clearly not an electrical engineer but a mathematician. Indeed, Betti made contributions to both Elasticity theory and Electromagnetism, and Maxwell does indeed cite Betti’s work in his treatise, but he was a mathematician. Betti, like the entire school of Italian Algebraic Geometry, was highly influenced by Riemann and topological ideas. However, the level of rigor in 19th century Italy was lax by modern standards and so his influence on current mathematical research may seem far removed. I revere Carl’s respect for historical detail, and I’ll refrain from calling his labeling Betti as an Electrical Engineer as a mistake. Rather I’d say Betti, like Gauss, Riemann, and Vito Volterra, had broad interests, and that Professor Herz suppressed the pedant in himself and took some license in his interpretation of history. The hidden hand of Hermann Weyl The role of Hermann Weyl in getting mathematics off the ground in the earliest days of the IAS is now well documented [1]. What I find fascinating is the first and fateful encounter between Hermann Weyl and Raoul Bott. The encounter has a lot to do with the interplay between electrical circuit theory, the early days of algebraic topology, and the perception of topology. The presentation of Carl Herz leaves out a lot of detail, much as a movie based on a book has to forgo a lot of detail. Given the encyclopedic knowledge of Carl Herz, it is tempting to speculate on what he could have put into a longer presentation. Bott told the story of his first encounter with Hermann Weyl many times, emphasizing different aspects and different amounts of detail. See for example [5]. I like the following rendition of the basic facts. During his grad student days as a student of Richard Duffin at Carnegie Tech, Bott played a large role in organizing the department colloquium. Being fluent in German, Hungarian, and Slovakian he would have an edge over other grad students in terms of “chatting up” foreign-born visitors. When Hermann Weyl visited, they were introduced, and Bott immediately began to tell Weyl of his thesis work [3]. There are interesting aspects of his thesis which predate both the Bott – Duffin Synthesis procedure and Wang algebras[12]. One key aspect is the “impedance potential” and how it defines a generalized inverse of a matrix. Of course the Moore – Penrose axioms for a generalized inverse were only formulated in the 1950s and so Bott does not use the term.8 Early on, he 8 See Chapter 2 and Appendix A of Ben-Israel and Greville’s book [2] for an exposition that puts the Bott – Duffin constrained inverse in the context of generalized inverses, and for putting the “Moore” of “Moore – Penrose” in historical perspective.

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and Duffin called it a “constrained inverse”[3], and in expository talks Bott later described it in terms of orthogonal projections in a complex (i.e., Hodge theory). It turns out that his impedance potential is a determinant which is intimately related to what graph theorists call a “matrix-tree formula” — a result that goes back to Kirchhoff and was used in Maxwell’s treatise. The logarithmic derivative of the impedance potential with respect to branch impedances gives a generalized inverse. When Bott explained the formalism and associated results to Hermann Weyl, Weyl grasped that the Bott – Duffin synthesis was indeed a contribution to network synthesis, but that the connection between Hodge theory and Kirchhoff’s laws was not. He pointed Bott to some papers connecting Kirchhoff’s laws to topology which he wrote in the early 1920s.9 Needless to say, Bott was invited to the IAS, but Bott felt a bit deflated about the Hodge theoretic aspect and that Weyl saw it concretely in Kirchhoff’s work. To be fair to Bott, we have to ask why these papers of Hermann Weyl were so obscure. G¨ ottingen was very closely tied to the technological aspects of Maxwell’s theory, and so why were these two papers as obscure as Maxwell’s periphractic numbers? What was the point Weyl was trying to make? To give some insight, a digression is in order. In the winter of 2005 I spent a month in the math department at the ETH in Z¨ urich while on sabbatical. When I arrived, my host gave me a choice of offices: a huge office with a stunning view of Z¨ urich belonging to a colleague on Sabbatical, or a very small empty office in the back of the building where “pure mathematicians hide their guests.” I told my host that I wanted the freedom to “spread out,” and that I felt more comfortable in the small back office. He was perplexed but obliged. It turns out that my cozy office was next to that of Beno Eckmann. On the centenary of Einstein’s golden year, Z¨ urich celebrated Eckmann as the last person in the city who had personal contact with Einstein! Since Hermann Weyl was the head of the ETH mathematics department in the 1920s, I naturally wanted to pick Beno’s brain for anecdotes. Not wanting to mess with his work habits, I planned to chat him up while he was a sitting target. In the hallway outside our offices was a high-tech espresso machine and every morning Beno would take a break to sit and enjoy an espresso outside our offices. The first day, I “coincidentally” joined him and he related wonderful anecdotes from 1950 – 1955, after Hermann Weyl retired from the IAS, resettled in Z¨ urich, and frequented the department. (I was sufficiently impressed that when I returned to Boston, I contacted the editors of the Notices of the AMS and a year later the anecdotes appeared in print[14]). The next day I resolved to ask Beno a question which I didn’t think any living person could answer. Little did I know that he had written a paper on the subject [13] and had a definite opinion on every nuance I could ask him to elaborate on! The conversation went something like this: RK: Beno, there is something I really don’t understand about Hermann Weyl. BE: What is it? RK: Well, in his collected works, there are are two papers about electrical circuit theory and topology dating from 1922/3. They are written in Spanish and published in an obscure Mexican mathematics journal. They are also 9 More precisely, Weyl’s papers dealt with Kirchhoff’s laws [26] and combinatorial topology [27].

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the only papers he ever wrote in Spanish, the only papers published in a relatively obscure place, and just about the only expository papers he ever wrote on algebraic topology. It would seem that he didn’t want his colleagues to read these papers. BE: Exactly! RK: What do you mean? BE: Because topology was not respectable! RK: Why was topology not respectable? BE: Hilbert! RK: Hilbert? BE: Just look at his 23 problems from 1900. Do you see anything to do with combinatorial group theory or topology? No! RK: Why? BE: Poincar´e !10 RK: What did Hilbert think of Poincar´e’s work on toplogy? BE: Poincar´e would write a huge paper on Analysis Situs. Half of it would be completely wrong! So, he’d write another huge paper trying to correct the first, but it would be half wrong! And so he’d write a third paper, but it would be half wrong. And so on. . . deuxi`eme compl´ement, troisi`eme, quatri`eme, cinqui`eme,. . . and in the end what did we get? Dubious results and conjectures! Hilbert didn’t think this was mathematics! RK: So why did Hermann Weyl write these papers? BE: He wanted to take stock of the honest results and reorganize them using a more modern abstract algebraic approach. Emmy Noether and others were doing interesting things in algebra and he had a need to write these papers for himself. These papers also contain some new results like the signature of a 4-d manifold. Beno went on to portray Hilbert as a bit of a reactionary figure, around which Hermann Weyl had to tip-toe. However, if Weyl wanted an opportunity to move things forward, it came in 1930 when Weyl succeeded Hilbert upon his retirement from G¨ottingen. Although he was only at the helm from 1930 until he fled the Nazis in 1933, during this very brief time German topology flowered in the hands of Emil Artin, Kurt Reidemeister, and others. According to Beno Eckmann, Weyl made a historic decision in 1930 which was highly controversial at the time, but ultimately vindicated: he appointed Heinz Hopf, a young researcher and relatively unseasoned, as his successor at ETH. If Beno’s historical perspective is taken superficially, there is a temptation to suspect there was some lasting disagreement between Weyl and Hilbert. However, one only needs to read Weyl’s masterful summary of Hilbert’s work [28] to realize that both men held themselves to the highest standards. In a sense, every time Hilbert or Poincar´e dug their heels in, Weyl found an opportunity to move mathematics forward. Algebraic topology may be one example and the continuum hypothesis may be another; perhaps the best example is the fact that Kurt G¨odel was among the first four hires at the IAS, but unemployable in Europe. 10 Sarkaria

[24] has given a modern executive summary of Poincar´e’s work in Topology.

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What did Carl know? We can only speculate. I can only say that he is one of the many people who impressed upon me the importance of having a historical perspective when reconciling algebraic topology with its applications. From History to Bott’s reconciliation with it The historical details discussed so far predate 1952. The next eight years would usher in the revolution in homotopy theory brought on by Serre’s thesis, CW complexes which tie Morse theory to homotopy theory, Bott periodicity, generalized cohomology theory such as K-theory and Rene Thom’s cobordism theory, and the reformulation of generalized cohomology theories in terms of spectra. Beno Eckmann pointed out to me that in the 1950s, those in Z¨ urich who dismissed Hermann Weyl as an old man were favorably stunned by his summary of the work of Kunihiko Kodaira and Jean-Pierre Serre on the occasion of their being awarded the Fields Medal in 1954 [30]. Nonetheless, contrasting Weyl’s presentation of the work of the two Fields medalists, it is apparent that he was challenged by the homotopy theoretic world Bott had entered into, even if he did a lot to unleash the homotopy theoretic perspective. Enough said; it is time to leave threads of mathematical history and experience another view of history: The convocation address of Raoul Bott Mr. Chancellor, Mr. Chairman of the Board – my dear fellow graduates: Congratulations to you — class of ’87 ! You look splendid ! I think you wash more behind the ears than your American cousins at Harvard do. It is nice to get a degree, isn’t it? Of course you only had to work hard for four years or so to get yours, while it took me over forty years to get mine. And presumably you have paid for yours, while I am paying for mine at this moment by being here on this platform, making a fool of myself. But, there is really nothing like one’s first degree. And what I loved especially about my Bachelor of Engineering was that an iron ring (from a fallen bridge) came with it. I hope this tradition continues, so that at least you engineers, can contrive — as I did — to display it on every occasion. It is a marvelous way of starting a conversation and at the same time lets one know that you have “graduated.” So my first admonition to you is: “Flaunt your degree in front of the whole world !” For a few weeks enjoy it to the hilt! The real world will rein you in soon enough. Of course the people who enjoy your degree most are your parents. So by all means — here comes my second admonition — Get yourselves some children, in time for degree-harvesting when you are still in your forties! (That way you might also have time to repay the loans before you die.) But let me tell you now a little bit about the good old days, just to keep some sort of historical perspective in a society, whose customs change at such a rate that the last forty years most probably represent two hundred uninflated ones. First of all I must tell you that, beautiful as your Campus is today, it used to be even more so in 1941. There were lawns to stretch out on, there was even a tennis court by the Redpath library! There was so much space and such a fine line of proportion! And there were no skyscrapers! (On the other hand, the area around McGill was very rundown. And I see that our “greasy spoon” has now flowered into a pizza joint.)

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Figure 1. Raoul Bott with Principal Johnson at Place des Arts, receiving his honorary degree from McGill in 1987.

Classes were small and some of my professors wore robes, as we are now, to teach in. They billowed and flowed delightfully with each step. These gowns were usually torn and completely covered with dust; still they added to the performance. I remember that later when I had my own calculus class to teach — the veterans had returned in huge numbers in the fall of 1945 and the Math. Department had pressed a lowly engineer into service to meet the demand — my dear friend and mentor, Professor McLennan — the Socrates of our campus — lent me his well-weathered gown. “Try it in your class,” he said, with a twinkle in his eye. Well, the class of course guffawed at first, but then actually settled down to work in a more businesslike manner than usual. Possibly it was this ballet-like aspect of the lectures that kept me going to classes very diligently in the beginning. However, this epoch of my life came to an end in short order after one of my roommates in our boarding house on Durocher called me in for a serious talk. Elwood Henneman was his name and he continues as a dear friend and colleague at Harvard. Elwood, with the full authority of a first year medical student and a Harvard B.A. warned me of the danger of being addicted to classes. “Never become a slave to them,” he declared ; “do the bulk of your thinking on your own!” This point of view made immediate sense; and thereafter it was safest to look for me in the Music room. Usually in the company of my dear friend, Walter Odze, and much later on also with my wife to be. How much this had to do with my falling grades I don’t know, but in any case I did manage to always “beat the Dean” as we used to say. Do you know what I mean? (But it sounds like good fun in any case, doesn’t it?) Well, one of the endearing procedures of our Alma Mater at that time was that they didn’t divulge

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our final grades until August — I think. Then suddenly your name was printed in the Gazette — if you passed, that is — and with an asterisk if you flunked one course, etc. On the same day the names of all the passing students were listed on a billboard in linear order of merit. Those who had failed were not on the list, and at the bottom of this terrifying document came the signature of the dean! Hence the expression of beating the Dean if one got through. But speaking of Deans and advice, let me tell you about one McGill Dean who in his own inimitable way gave me the best advice of my life. These were the war years and in ’45 right after graduation, I joined up in the Canadian infantry and was being trained for combat in Japan. After three months in basic training the atomic bomb was dropped on Hiroshima and Nagasaki, the war ended abruptly and my fellow recruits and I were thereby suddenly and miraculously reprieved — in this unbelievable and terrible manner. Of course, the one great advantage of being in the army is that one has no career problems whatsoever ! Hence the doubts I had about my vocation in engineering were completely submerged by my efforts to keep out of the Sergeant ’s hair. But when in October I found myself back in the Engineering Department, where they had very kindly let me return for a Master ’s Degree on — as you can imagine — very short notice, the old doubts flared up again and I was in a quandary about what to do. It was sometime in ’46 then that I presented myself at Dean Thompson’s office and asked him whether he could see his way to putting me through medical school. (On the Jewish side of my family they always did say: “chutzpah he does not lack.”) And Dean Thompson was quite encouraging at first. “We need scientifically trained doctors,” he said. “But,” he continued, “first tell me a little about yourself.” It was at this point that our interview started to go sour. No, I never enjoyed Biology much. No I hated dissecting frogs. Alas, Botany bored me and I had little use for Chemistry! After this sorry litany, Dr. Thompson surveyed me and the situation for a while, pipe in hand, and lost in thought. “Is it maybe that you want to do good for humanity,” he said at last. I hemmed and hawed in my seat, but before I had time to say anything he came out with: “Because they make the lousiest doctors!” Well, that was it for me — and you must admit that it explains a lot of things, doesn’t it? In any case, I got up and as I went to the door, I thought to myself : well you (expletive deleted) if that is how the land lies, then I will simply do what I like best: “I will become a Mathematician. Put that in your pipe and smoke it!” I tell you all this only in part as a jest. I would also like it to be a word of encouragement to those of you who, degree in hand, still are not quite certain of your path. May you also be blessed with a counselor with such diagnostic skills and such a knack for putting you on the right course. But my time is up! Still I cannot resist a serious word. Over my McGill days the War hung like an everpresent black cloud, subtly affecting every aspect of our lives. For you in the nuclear age the cloud is, thank God, farther away, but potentially much, much darker. These things you will have to live with and somehow hope to conquer. But for this road I know of no better advice than my friend Elwood ’s — “Do your own thinking.” In our more immediate lives we are also beset today more than ever before, with show, with image, with jargon; and here again — to pick one’s way through this quagmire, there is no better exhortation than: Be your own man; be your own woman. For then, I am confident, you will never confuse fashion with substance,

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heroes with the people who depict them on the tube, computers with people, Science with virtue, or wealth with happiness. Yes, may God bless you and may you be joyfully and productively yourselves — but may you also be ultimately servants of a larger and an all-encompassing benign world view. And that is really no more than what I take to be the correct reading of Dr. Thompson’s advice to me forty years ago. Only remember that the concerns of your generation must even transcend those off our human family. They must embrace every aspect of life itself on this deeply troubled, but magnificent and magical planet of ours. Raoul Bott On iron rings and other aspects of the convocation address Bott’s convocation address sets the stage for my fascination with his struggle to reconcile old and new world sensibilities. He was clearly the same age as my parents and like my parents, the disruption of his adolescence by Hitler, Stalin, and the events in the Europe of his youth was a traumatic experience and a profound education even if they didn’t view it that way at the time [7]. His refugee experience was a stark contrast to the way kids grew up in North America during the decades after WWII. This was evident in his sense of humor and in the way he handled those who did not choose their words correctly. The testimonies of his students in this volume attest to this. The war years and the economic turmoil that preceded it left him with little tolerance for the dangerous comforts of self-preservation. The Engineer’s iron ring along with its uniquely Canadian origin seem to frame some of his advice on responsibility and independent thinking. Reading the convocation speech, and recalling our interaction after the first class I audited at Harvard, it is apparent that Bott had a much more profound appreciation of, and respect for, the iron ring than did the students he was addressing. One cannot do justice to the topic here11 , but it is useful to connect a few key ideas to the events and sensibilities of other times. Engineering is full of trade-offs, and the story of the iron ring is about the interface between technical trade-offs, ethics and ambition. One engineering trade-off is between the theoretical effort that goes into designing something without making a physical model, and the willingness to build prototypes and make mistakes. If one were to design a paper clip, one would make many prototypes in order to see “what works.” On the other hand, if one were building a bridge, one would like to avoid disasters, and the development of a theoretical model with predictive properties is in order. In the case of bridge building, especially new designs, one is very cautious because public confidence is paramount. However, there have been many bridge disasters and contrary to what one might naively expect, they usually do not involve new designs! They usually involve refined designs which take into account that earlier designs were overly cautious, too costly, and less than ambitious. The temptations involved are quite universal and are not restricted to bridges; one can send up space shuttles routinely without an accident, but when one decides that the rules for launch are overly cautious in light of an opportunity to make some “State of the Union Address” spectacular, strange things can happen — just like when the 11 These days one has to look for stainless steel if one wants to spot an “Iron Ring” — the iron of the early rings used to be eaten away by sweat and was soon replaced.

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detailed properties of O-rings were purposely ignored in the lead-up to the Challenger disaster. Similarly, the dismissing of foam impacts during launch as routine in the lead-up to the Columbia Shuttle disaster underlines the vigilance required to make complicated things work. These days we have “financial engineering” and computer models so predictive that there is a temptation to lose track of the underlying assumptions and to consider the regulation of investors as unimaginative and cumbersome — here again, strange things can happen when ambition trumps regulation. Engineering disasters create teachable moments and they are very well documented when the stakes are high. In the case of bridges, the spectacular disasters have been studied and categorized, and scholarly books such as the one by Petroski [23] have been written. Chapter three of Henry Petroski’s book details the Qu´ebec City bridge disaster(s) that led to the iron ring worn by Canadian Engineers. It has a lot of detail on the New York based construction firm, the details of the bridge, the ignoring of warning signs and the 75 people killed in the first disaster. A key ingredient in this August 1907 disaster was an attempt to redesign the bridge during construction in order to ensure it broke a world record. There was a second disaster in 1916 during the construction of the redesigned bridge which killed 13 workers. In all, a total of 89 workers were killed in the construction of the bridge. The completion of the 1800 foot span of the Qu´ebec City bridge in 1917 made it the largest cantilever bridge in the world and vindicated the concept of the cantilever bridge for a mix of rail and automobile traffic. However, worldwide, no other major cantilever bridge was completed until the 1930s. To this day the Qu´ebec City bridge has the longest span of any cantilever bridge — other bridge designs are used for longer spans. The original iron rings were made of the collapsed bridge’s iron as a reminder of the stupidities engineers are capable of, and as a reminder of the engineer’s responsibility to society — soon after the rings were made of stainless steel. Rudyard Kipling was the recipient of the 1907 Nobel Prize in literature and lived in Brattleboro Vermont for a few years in the 1890s. It was in Vermont, between Qu´ebec City and the home of the bridge’s architect in New York, that he wrote The Jungle Book. It was also in 1907, following the first Qu´ebec City bridge disaster, that he wrote a poem called The sons of Martha.12 It is inspired by the Gospel of Luke (10:38 – 42) and forms the basis of the original iron ring ceremony which Kipling was commissioned to write. The original Canadian ceremony was called The ritual of the calling of an engineer and it was first performed in 1922. These days the original ceremony with its Biblical references is considered noninclusive. The iron ring ceremony, first performed in the United States in 1970, is centered around The obligation of the engineer, which is devoid of Biblical references.13 In Bott’s commencement address he is clearly referring to the the original ceremony and I recommend a read through Kipling’s poem to appreciate the iron ring as Bott would have been introduced to it. 12 See, for instance: John Beecroft, Kipling: A selection of his stories and poems, Doubleday, 1956 (in two volumes). In Vol. II, The Sons of Martha appears on p .451. 13 Kipling lived after “the days of wooden ships and iron men,” and in the peak of the English Empire. Of the 89 workers killed in the two Qu´ebec City bridge disasters, it appears that 33 were Mohawk steel workers from the Kahnawake reserve just outside of Montr´ eal, creating 24 widows and numerous fatherless children. The Mohawk workers were well adapted to heights and were the “high tech” workers of their time. I have yet to see this aspect arise in the context of a modern iron ring ceremony, or in the world view of Kipling’s time.

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This concludes my musings on a refugee as an engineering student in Montr´eal, his metamorphosis into a topologist, and a new world success story who was ultimately awarded an honorary doctorate by his alma mater. References 1. S. Batterson, Pursuit of genius: Flexner, Einstein, and the early faculty at the Institute for Advanced Study, A K Peters Ltd., Wellesley, MA, 2006. 2. A. Ben-Israel and T. N. E. Greville, Generalized inverses: Theory and applications, 2nd ed., CMS Books Math./Ouvrages Math. SMC, vol. 15, Springer, New York, 2003. 3. R. Bott and R. J. Duffin, On the algebra of networks, Trans. Amer. Math. Soc. 74 (1953), 99 – 109. 4. R. Bott, On topology and other things, Notices Amer. Math. Soc. 32 (1985), no. 2, 152 – 158. , On induced representations, The Mathematical Heritage of Hermann Weyl (Durham, 5. NC, 1987), Proc. Sympos. Pure Math., vol. 48, Amer. Math. Soc., Providence, RI, 1988, pp. 1 – 13. , Autobiographical sketch, Raoul Bott: Collected Papers. Vol. 1: Topology and Lie 6. Groups (MacPherson. R. D., ed.), Contemp. Mathematicians, Birkh¨ auser, Boston, MA, 1994, pp. 3 – 9. , The Dioszeger years (1923 – 1939), Raoul Bott: Collected Papers. Vol. 1: Topology 7. and Lie Groups (MacPherson. R. D., ed.), Contemp. Mathematicians, Birkh¨ auser, Boston, MA, 1994, pp. 11 – 26. 8. E. Breitenberger, Johann Benedikt Listing, History of Topology (I. M. James, ed.), NorthHolland, Amsterdam, 1999, pp. 909 – 924. 9. P. Bamberg and S. Sternberg, A course in mathematics for students of physics. 1, 2, Cambridge Univ. Press, Cambridge, 1991. 10. J. R. Choksi, W. H. J. Fuchs, and N. Th. Varopoulos, Carl Herz (1930 – 1995), Notices Amer. Math. Soc. 43 (1996), no. 7, 768 – 771. 11. S. W. Drury, The mathematical work of Carl S. Herz, Harmonic Analysis and Number Theory (Montr´ eal, QC, 1996) (S. W. Drury and M. R. Murty, eds.), CMS Conf. Proc., vol. 21, Amer. Math. Soc., Providence, RI, 1997, pp. 1 – 10. 12. R. J. Duffin, An analysis of the Wang algebra of networks, Trans. Amer. Math. Soc. 93 (1959), 114 – 131. 13. B. Eckmann, Is algebraic topology a respectable field?, Mathematical Survey Lectures 1943 – 2004, Springer, Berlin, 2006, pp. 245 – 255. , Hermann Weyl in Z¨ urich 1950 – 1955, Notices Amer. Math. Soc. 53 (2006), no. 10, 14. 1222 – 1223. 15. T. Frankel, The geometry of physics: An introduction, 2nd ed., Cambridge Univ. Press, Cambridge, 2004. 16. I. M. James (ed.), History of topology, North-Holland, Amsterdam, 1999. 17. A. Jackson, Interview with Raoul Bott, Notices Amer. Math. Soc. 48 (2001), no. 4, 374 – 382. 18. J. B. Listing, Der Census r¨ aumlicher Complexe oder Verallgemeinerung des Eulerschen Satzes von den Poly¨ edern, Abhandlungen der K¨ oniglichen Gesellschaft der Wissenschaften in G¨ ottingen (1861), 97 – 182. 19. R. D. MacPherson, Introduction to Volume 2, Raoul Bott: Collected Papers. Vol. 2: Differential Operators (MacPherson. R. D., ed.), Contemp. Mathematicians, Birkh¨ auser, Boston, MA, 1994, pp. vii – xvi. 20. J. C. Maxwell, On hills and dales, Philos. Mag. 40 (1870), 421 – 427. , A treatise on electricity and magnetism, 3rd ed., Dover, New York, 1954. 21. 22. H. K. Nickerson, D. C. Spencer, and N. E. Steenrod, Advanced calculus, D. Van Nostrand Co., Inc., Toronto – Princeton, NJ – New York – London, 1959. 23. H. Petroski, Engineers of dreams: Great bridge builders and the spanning of America, Alfred A. Knopf, New York, 1995. 24. K. S. Sarkaria, The topological work of Henri Poincar´ e, History of Topology (I. M. James, ed.), North-Holland, Amsterdam, 1999, pp. 123 – 167. 25. L. W. Tu, The life and works of Raoul Bott, Notices Amer. Math. Soc. 53 (2006), no. 5, 554 – 570.

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26. H. Weyl, Reparticion de corriente en una red conductora: Introducccion al analisis combinatorio, Rev. Mat. Hisp.-Amer. 5 (1923), 153 – 164. , Analisis situs combinatorio., Rev. Mat. Hisp.-Amer. 5 (1923), 43; Analisis situs 27. combinatorio (cotinuacion), Rev. Mat. Hisp.-Amer. 6 (1924), 1 – 9, 33 – 41. , David Hilbert and his mathematical work, Bull. Amer. Math. Soc. 50 (1944), 612 – 28. 654. , Space, time, matter, 4th ed., Dover, New York, 1952. 29. , Address of the President of the Fields Committee 1954, Proceedings of the Interna30. tional Congress of Mathematicians (Amsterdam, 1954). 31. C. N. Yang, Hermann Weyl’s contribution to physics, Hermann Weyl, 1885 – 1985: Centenary lectures (K. Chandrasekharan, ed.), ETHZ, Z¨ urich and Springer, Berlin, 1986, pp. 7 – 21. 32. S.-T. Yau (ed.), The founders of index theory: Reminiscences of Atiyah, Bott, Hirzebruch, and Singer, International Press, Somerville, MA, 2003. Department of Electrical and Computer Engineering, Boston University, Boston, MA 02215, USA E-mail address: [email protected]

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https://doi.org/10.1090/crmp/050/05

Centre de Recherches Math´ ematiques CRM Proceedings and Lecture Notes Volume 50, 2010

The Bott – Duffin Synthesis of Electrical Circuits John H. Hubbard Raoul Bott started his career as an electrical engineer, his adviser at CarnegieMellon was Richard Duffin. In a joint paper from 1949 [1], they published a proof of the following basic result in circuit theory, which in some sense tells us exactly what a passive circuit can do; the proof, completely constructive, tells us how to synthesize a circuit to do whatever is possible. A rational function is the impedance of a passive circuit if and only if it has real coefficients and maps the right half-plane to itself. The original paper is two pages long and crystal clear; it can scarcely be improved. My excuse for the present paper is that the original is written for electrical engineers, and includes (of course) no background. The vocabulary will be unfamiliar to many mathematicians (and even engineers who do not design circuits). Also, at one point I can improve the original presentation: it uses a result called Richards’ theorem [3] in the engineering literature. Although this was evidently not known to Bott or Duffin, I will show that Richards’ theorem is really Schwarz’s lemma in light disguise. Bott and Duffin were not working in a vacuum: there had been extensive work on the construction of circuits with a given impedance. In particular, Otto Brune [2] had shown how to construct a circuit using ideal transformers whose impedance is an arbitrary real rational function mapping the right half-plane to itself. In the words of Bott and Duffin, this use of ideal transformers was “found to be objectionable.” Bott and Duffin’s contribution was to remove the ideal transformers. For more extensive background in the spirit of the present paper I recommend Chapter 10 in [4]. 1. The synthesis theorem An electrical circuit is a graph with each edge carrying a circuit element, i.e., one of a resistor, a capacitor or an inductor, to which one should add current and voltage sources. Transformers, gyrators, memristors, etc, are often included among passive circuit elements, but they will not be allowed here. The circuits that Bott and Duffin consider are passive 1-ports, i.e., they are circuits as above with no sources, and with two wires sticking out, labeled + and −, 2000 Mathematics Subject Classification. 94C05. This is the final form of the paper. c c JohnJohn
2010 Hamal
2010 H. Hubbard

35 33

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together called the port, which will be connected to either a voltage source or a current source. If you use a current source and impose a current I(t) = I0 est , through the port, then the voltage across the port is (as we will see) V (t) = I0 Z(s)est , for some rational function Z(s), called the impedance of the circuit. The variable s, evidently the variable of the Laplace transform, has units inverse time. The values Z(s) have units ohms = voltage/current. The physically meaningful values of s are the purely imaginary1 ones s = jω, at least when studying steady-state phenomena; for transients, damped sinusoids are also important. It is convenient to have a notation Y (s) = 1/Z(s) for the inverse of the impedance, called the admittance of the circuit. An analytic function f mapping the half-plane Re s > 0 to itself and mapping the positive real axis to itself is called by the (rather regrettable) name Positive Real Function, abbreviated PRF. We will be interested only in PRFs that are rational functions; in this paper a PRF is a rational function with real coefficients that maps the right half-plane to itself. The main result about passive circuits can now be stated. Theorem 1. (a) The rational function Z(s) is a PRF. (b) Conversely, every PRF is the impedance of a passive 1-port. The Bott – Duffin contribution is part (b); part (a) was known long before. 2. Circuits and Kirchhoff ’s laws To set up the equations describing the time-evolution of an electrical circuit Γ, we need to orient the edges of the circuit. To each oriented edge e is associated a current ie and a voltage drop ve . The orientation of the edges has no physical meaning: if you reverse the orientation of edge e and change the sign of ie and ve , you describe the same behavior of the circuit. The currents and voltage drops are subject to Kirchhoff ’s current law : For each node x of Γ, we have

ie = ie edges e ending at x

edges e starting at x

and to Kirchhoff ’s voltage law : For every oriented loop L in Γ we have
ve = edge e ⊂ L oriented with L




ve .

edges e ⊂ L oriented against L

Let E be the set of edges of Γ. Set I ⊂ RE to be the subspace of currents allowed by the Kirchhoff current law, and V ⊂ RE to be the space of voltage drops allowed by the Kirchhoff voltage law. Theorem 2 (Tellegen’s theorem). The spaces I and V are orthogonal complements in RE . 1Bott once told me that the first step to becoming an electrical engineer is to call j = √−1.

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THE BOTT – DUFFIN SYNTHESIS OF ELECTRICAL CIRCUITS

35 37

Proof. They are respectively the kernel of M and the image of M  , where M is the matrix with rows labeled by nodes and columns labeled by edges, given by ⎧ ⎪ if node i is the end of edge j ⎨1 mi,j = −1 if node i is the origin of edge j
⎪ ⎩ 0 otherwise. 3. The equations of the circuit elements The circuit elements are described by equations, algebraic for resistors, differential for capacitors and inductors. Resistance (Ohm’s law). There is a number R, called the resistance, and measured in ohms, such that the current i in a resistor and the voltage drop v across it satisfy R

v = Ri.

Capacitance. There is a number C, called the capacitance (or capacity), and measured in farads, such that the current i in a capacitor and the voltage drop v across it satisfy C

dv = i. dt

C

Inductance (Lenz’s law). There is a number L, called the inductance, and measured in henrys, such that the current i in an inductor and the voltage drop v across it satisfy L

di = v. dt

L

These laws describe ideal circuit elements; real circuit elements obey these laws only approximately, and even then only over appropriate ranges of the variables. Resistors melt if the current in them becomes too high, and at high frequencies even wires behave like inductors, creating a magnetic field that affects the current. The equations for capacitors and inductors are fairly easy to justify in some approximation from Maxwell’s equations, but although resistors are closer to the intuition, their equation is harder to justify from first principles. 4. The impedance of a passive 1-port is a PRF Choose some minimal set of currents and voltages such that all the others can be expressed in terms of these using Kirchhoff’s laws and Ohm’s law. Form the vector x from these variables; the differential equations for the inductors and capacitors will then lead to a differential equation of the form x (t) = Ax(t) + f (t) for some square matrix A, where f is the current imposed by the current generator, expressed in the variables of x.

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36 38

J. H. HUBBARD

Using undetermined coefficients, we can find the solution of the differential equation x = Ax + f (t) describing the circuit with the source current iS given by iS = I0 est with I0 real, as vS = −I0 Z(s)est iλ = Iλ est

so that vλ = Lλ Iλ sest

vγ = Vγ est

so that iγ = Cγ Vγ sest

iρ = Iρ est

so that vρ = Rρ Iρ est

Tellegen’s theorem gives −¯ıS vS =




iλ vλ +




¯ıγ vγ +




γ

λ

¯ıρ vρ .

ρ

Cancel the common factor |est |2 to get


I02 Z(s) = s Lλ |Iλ |2 + s¯ Cγ |Vγ |2 + Rρ |Iρ |2 . γ

λ

Take real parts: I02 Re Z(s) = Re s




Lλ |Iλ |2 +

λ

ρ




 Cγ |Vγ |2

+




γ

Rρ |Iρ |2 .

ρ

Everything on the right is ≥ 0 when Re s ≥ 0. 5. Some preliminaries to the synthesis theorem The rank of a rational function is the maximum of the degrees of the numerator and the denominator. If a circuit consists of two circuits with impedances ZA and ZB in series, the resulting impedance is Z Z(s) = ZA (s) + ZB (s).

ZA

ZB

If a circuit consists of two circuits with acceptances YA and YB in parallel, the resulting acceptance is

Y (s) = YA (s) + YB (s).

Y YA YB

In terms of impedances,, this becomes 1 1 1 = + . Z(s) ZA (s) ZB (s) Lemma 3. All the poles and zeroes of a rational PRF in the right half-plane Re s ≥ 0 are on the imaginary axis or at infinity. Such poles and zeroes must be simple, with positive derivative or positive residue.

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THE BOTT – DUFFIN SYNTHESIS OF ELECTRICAL CIRCUITS

37 39

Proof. A nonconstant analytic function is open, so if a rational function maps some point of the right half-plane to 0 or to ∞, it must map some nearby points to points with negative real part. If at a (purely imaginary) zero of a PRF the derivative is not real and positive, again the PRF will map some points of the right half-plane to the left half-plane. The same argument holds for the residues of poles.
6. The synthesis theorem A PRF must satisfy one of the following three conditions. (1) (a) inf Re s>0 Re Z(s) = R > 0 or (b) inf Re s>0 Re Y (s) = 1/R > 0 (2) (a) The function Z(s) has a pole on the imaginary axis or at ∞ or (b) The function Y (s) has a pole on the imaginary axis or at ∞ (3) (a) There exists ω > 0 and L > 0 with Z(jω) = Ljω or (b) There exists ω > 0 and C > 0 with Y (jω) = Cjω. In each case we will see how to produce a “simpler” PRF Z1 or Y1 , in the sense that if we can synthesize Z1 , then we can synthesize Z. • If Z or Y satisfies (1), then Z1 or Y1 will satisfy (2) or (3) without increasing the rank. • if Z or Y satisfies (2), then Z1 or Y1 will have strictly smaller rank, and • if Z or Y satisfies (3), then Z1 or Y1 will satisfy (2) without increasing the rank. Case 1. We can write Z(s) = R + Z1 (s) or Y (s) = 1/R + Y1 (s). Put a resistor of resistance R in series with a circuit realizing Z1 to realize Z in the first case, and put the resistor in parallel with a circuit realizing Y1 in the second case. Case 2. If W is a PRF with at least one pole on the imaginary axis or at infinity, write W (s) = PW (s) + W1 (s) with

k0
2ki s + + k∞ s s s2 + ωi2 i=0 m

PW (s) =

with k0 ≥ 0, all ki ≥ 0 and k∞ ≥ 0, and at least one of these numbers is not 0. Then W1 is a PRF which is bounded on the right half-plane of strictly smaller rank than W . Thus if we can realize W1 as either an impedance or an admittance, and if we can realize the various terms of PW , then we can realize W . Realizing the various terms of PW is accomplished by Proposition 4. Proposition 4. (1) The PRF k0 /s is • the impedance of a capacitor of capacitance 1/k0 , and • the admittance of an inductor of inductance 1/k0 . (2) The PRF k∞ s is • the impedance of an inductor of inductance k∞ , and • the admittance of a capacitor of capacity k∞ . (3) The PRF 2ks/(s2 + ω 2 ) is • the impedance of a capacitor of capacitance C = 1/2k in parallel with an inductor of inductance L = 2k/ω 2 , and

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38 40

J. H. HUBBARD

• the admittance of a capacitor of capacitance C = 1/2k in series with an inductor of inductance L = 2k/ω 2 . To end Case 2, put the circuit elements given by Proposition 4 in series if PW is an impedance (Case 2(a)) and in parallel if PW is an admittance (Case 2(b)). Case 3. This case requires Richards’ theorem. Theorem 5 (Richards’ theorem). Let W (s) be a PRF with no zeros or poles on the imaginary axis or at infinity. (a) There then exists a unique s0 ∈ R+ with W (s0 ) = s0 . (b) The function sW (s) − s20 W1 (s) = s − W (s) is a PRF with rank W1 (s) ≤ rank W (s). (c) If W (jω) = jω, then W1 has a pole at jω. Remark. You should not think of W as either an impedance or an admittance: the units are wrong. We are interested in a fixed point of W , which makes sense only if the domain and the range have the same units. When we come to apply Richards’ theorem, we will modify the impedance or admittance so that this condition is satisfied. Proof. Recall Schwarz’s lemma: If D ⊂ C is the unit disc and f : D → D is analytic with f (0) = 0, then |f (z)| ≤ |z| for all z ∈ D, and |f  (0)| ≤ 1. If |f (z)| = |z| for a single z ∈ D, or if |f  (0)| = 1, then f (z) = az for some a ∈ C with |a| = 1. (a) The existence of s0 follows from the intermediate value theorem. The uniqueness follows from Schwarz’s lemma: a map from the right halfplane to itself that has more than one fixed point is the identity. (b) The map s − s0 ϕ(s) = s + s0 maps the right half-plane to the unit disc, and ϕ(s0 ) = 0. So f = ϕ ◦ W ◦ ϕ−1 maps the unit disk to itself with f (0) = 0. By Schwarz’s lemma, so does g(z) = f (z)/z. Thus W1 = ϕ−1 ◦ g ◦ ϕ is a PRF. If you work it out, you will find W1 (s) =

sW (s) − s20 . s − W (s)

(c) If W (jω) = jω, then W1 has a pole at jω. Suppose that Z(jω) = Ljω for some L > 0. Set Z(s) , Z1 (s) = L

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THE BOTT – DUFFIN SYNTHESIS OF ELECTRICAL CIRCUITS

and R(s) =

39 41

sZ1 (s) − s20 . s − Z1 (s)

Remark. The impedance Z has units voltage/current, whereas s has units 1/time. But L has units voltage · time/current, so Z1 has units inverse time in both the domain and range, and it makes sense to apply Richards theorem to it. The function R has units inverse time. A bit of computation shows that 1 1 + . Z(s) = 2 1/(LR(s)) + 1/(Ls) R(s)/(Ls0 ) + s/(Ls20 ) Suppose that the 1-ports Γ1 , Γ2 , Γ3 , Γ4 have admittances 1 , LR(s)

1 , Ls

R(s) , Ls20

s . Ls20

Then the circuit below has impedance Z(s). 1/LR(s)

R(s)/Ls20

1/Ls

s/Ls20

An example of the Bott – Duffin synthesis. Let us find a circuit with impedance s4 + 2s3 + 6s2 + 2s + 4 . s3 + s2 + 4s This function has a pole at ∞ and at 0, which we can remove by partial fractions to find 1 s2 + s + 1 1 Z(s) = s + + 2 = s + + Z1 (s). s s +s+4 s If we can find a 1-port Γ1 with impedance Z1 (s), then the circuit formed by Γ1 in series with a capacitor of 1F and an inductor of 1H will have impedance Z(s).

(1)

Z(s) =

Remark. The PRF under consideration is not physically realistic. A 1F capacitor or a 1H inductor is a massive thing, at least if it handles any substantial current. At the electronics store, capacities are measured in micro- or picofarads and inductances in micro- or picohenrys. The function Z1 (s) has no zeroes or poles on the imaginary axis, but setting s = jω and solving Re Z(jω) = 0 yields √ √ 2 Z1 (j 2) = j . 2 So we are in case 3(a), with L = 12 , and must find s0 > 0 with Z1 (s0 ) = s0 /2. Computed out, this equation becomes s30 − s20 + 2s0 − 2 = (s20 + 2)(s0 − 1) = 0, and we see that the solution is s0 = 1, unique in the right half-plane as it should be.

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40 42

J. H. HUBBARD

Set Z2 (s) = Z1 (s)/L = 2Z1 (s). Then the rational function R(s) =

s · Z2 (s) − 12 2s3 + s2 + s − 4 2s2 + 3s + 4 = 3 = s − Z2 (s) s − s2 + 2s − 2 s2 + 2

is a PRF (evidently with poles on the imaginary axis), and solving back for Z1 (s) we find 1 1 1 1 sR(s) + 1 = + := + . Z1 (s) = 2s + 2R(s) 2s + 2R(s) 2/R(s) + 2/s Y1 (s) Y2 (s) Now the admittance 6s Y1 (s) = 2s + 2R(s) = 4 + 2s + 2 s +2 is realized by a resistor of 14 Ω, a capacitor of 2F , and a circuit with admittance 6s 1 = s2 + 2 s/6 + 1/3s in parallel. This last is realized by an inductor of 16 H and a capacitor of 3F in series. Similarly the admittance 2 2 1 Y2 (s) = + 2R(s) = + s s 1 + 32 s/(s2 + 2) can be realized by putting in parallel an inductor of 12 H with a circuit consisting of a resistor of 1Ω in series with a circuit with impedance 3s/(2s2 + 4). This last can be realized by a capacitor of 23 F in parallel with an inductor of 34 H. Thus a circuit with impedance given by (1) is given by the following figure. 1 4Ω

1F

1H

1 2H

2F 1 6H

2 3F

3F

1Ω 3 4H

References 1. R. Bott and R. J. Duffin, Impedance synthesis without use of transformers, J. Appl. Phys. 20 (1949), 816. 2. O. Brune, Synthesis of a finite two terminal network whose driving-point impedance is a prescribed function of frequency, Journal of Mathematics and Physics 10 (1931), 191 – 236. 3. P. I. Richards, A special class of functions with positive real part in a half-plane, Duke Math. J. 14 (1947), 777 – 786. 4. M. W. Hirsch and S. Smale, Differential equations, dynamical systems, and linear algebra, Pure Appl. Math, vol. 60, Academic Press, New York – London, 1974. Department of Mathematics, Cornell University, Ithaca NY 14853, USA E-mail address: [email protected]

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Part 2

Early Students and Colleagues

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https://doi.org/10.1090/crmp/050/06

Centre de Recherches Math´ ematiques CRM Proceedings and Lecture Notes Volume 50, 2010

Raoul Bott as We Knew Him Raoul’s stepmother and my mother were friends. At some time one of them mentioned her son, a brilliant mathematician, to which the other replied by describing her son as another brilliant mathematician. This is how we first heard of each other. When Harvard cast its eye on Raoul, then at Michigan, they invited him to visit Harvard for a week. At the end of the week the dean, McGeorge Bundy, had a talk with Raoul about what he liked and what he didn’t like about Harvard. Raoul replied that he liked almost everything, but that there was one thing that bothered him, that he met so many stuffy people. Bundy replied: “You see, Professor Bott, the policy of Harvard is to invite the best people in each field, and let them do as they please. Many of them please to be stuffy.” Peter D. Lax We were friends for 53 of his life span of 82 years. Inge and I met Raoul and Phyllis when we visited Ann Arbor during our “honeymoon trip” in January 1953. Raoul, Michael Atiyah and I spent the year 1955/56 in Princeton, Raoul and Michael at the Institute, I at the University. He greatly improved and generalized the work of Armand Borel and me on homogeneous vector bundles. In the summer of 1956 Raoul, Michael with Lily, and I attended the famous Mexico conference where the three men lectured. During one of the weekends, Raoul and I went to Acapulco. For the first night we stayed in a hotel where our room was a dirty hole. Under protest we went to the best hotel and had a splendid room with a balcony overlooking the ocean, and, of course, there were always so many things to discuss. In 1958 the Botts came to Bonn where Raoul gave lectures on Morse theory, and during the Arbeitstagung — it was the second one — he presented his famous brand new periodicity theorems. In his comments to his Collected Papers, he writes, “These lectures, delivered in Bonn during the summer of 1958, marked my first return to Central Europe since 1939. Our family had now grown to four and my wife, Phyllis, was expecting. The trip was a high adventure for all of us.” Further Raoul writes, “Nevertheless, the ideas of Atiyah and Hirzebruch, interpreting the periodicity theorems as ‘K¨ unneth’ formulae in an ‘extraordinary cohomology theory’ came as a complete surprise to me! In one swoop my special computations had become a potential tool in all aspects of topology.” 2000 Mathematics Subject Classification. 01A60. This is the final fom of the paper. c
2010 c
2010 American American Mathematical Mathematical Society

45 43

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44 46

RAOUL BOTT AS WE KNEW HIM

Remembering our encounters I could go on and on. Raoul lectured for my 60th birthday in Bonn, I lectured for his 70th birthday at Harvard, and he came to my University retirement in 1993 and to my retirement as director of the Max Planck Institute in 1995. For my University retirement, there was a dinner given by the rector of Bonn University. The president of the German National Science Foundation attended and later told me that his image of mathematicians totally changed, because this was one of the best dinners he ever had, mathematicians are not cool, dry, and withdrawn, they are wonderful, full of spirit, charming and humorous and know how to laugh. No wonder, if one sits next to Raoul. I believe the chances of mathematics in the German National Science Foundation greatly increased. The last time Inge and I met Raoul was in Princeton for the 75th anniversary of the School of Mathematics of the Institute for Advanced Study. During a discussion we explained how our stays at the Institute had been formative for our scientific careers. Raoul had not lost his humour when he said, “The doctors tell me I am sick, but I don’t believe them.” In German, he called this “Galgenhumor” (gallows humour). Michael, Inge and I had wonderful days with Raoul. For Raoul’s 80th birthday, the president of the “Deutsche Akademie der Naturforscher Leopoldina” wrote a letter of congratulations in German as he always does when a member of the Academy reaches the age of eighty. I quote a few sentences from the presidential letter in English translation. After speaking about Raoul’s “Dioszeger years” (1923 – 1939) and his voyage to Canada, the president continues: “Hermann Weyl visited the Carnegie Institute and was so impressed by your results on networks (Theorem of Bott – Duffin which solved a 20 year old problem) that he invited you to the Institute for Advanced Study beginning in the fall of 1949. . . Soon you published path breaking papers on Lie groups. . . A group of German students set to music the famous Bott period Z 2, Z 2, Zero, Z, Zero, Zero, Zero, Z for the orthogonal groups and sang it full of enthusiasm. . . In 1990, you mention your happiness that your two home countries [Hungary and Slovakia] are finally liberated. The president concludes his letter with cordial greetings, “also to your wife Phyllis with whom you could celebrate the fiftieth anniversary of your marriage in 1997.” Raoul wrote an e-mail to me on October 9, 2003: “A few days ago I received a long letter of congratulations on my 80th from the president of the Leopoldina! I must admit that I was rather moved by this bit of European courtesy, so alien to the informal ways of this country! (I certainly did not get such a letter from The American Academy.) Of course I know that the president did not do all that research himself, and I have a strong hunch, who the ghostwriter was. Should you have a chance to meet him, (say during your morning shave), please express my thanks to him!” Friedrich Hirzebruch “Friends come first, mathematics after.” Raoul once said, as he smiled and wheeled his bicycle out of his office, and down the corridor to meet a visiting friend of his in a local coffee shop, his worn golden-brown portfolio balanced on his bicycle seat.

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RAOUL BOTT AS WE KNEW HIM

45 47

Raoul Bott was a wonderful colleague whose presence in any gathering made it joyous, and he was — as we all know — also one of the great mathematicians of the twentieth century. What a great range of profound insights and interests: from his early applied work in circuit theory, to his celebrated Periodicity Theorem which, in turn, introduced the power of Morse theory and loop-space arguments to differential topology; and which fed into the emergence of K-theory in the hands of Atiyah, Hirzebruch and Bott himself; and which connects with the majestic development of the Riemann – Roch theme in algebraic geometry and elliptic partial differential equations; and, in more recent times, to involvement with some of the marvellous issues in string theory. Raoul’s smile — his friendly quick laugh after saying something — meant that conversations with him often gleamed with a kind of humor, and sunshine. Raoul often presented himself as a skeptic. He said, for example, “I was always a little skeptical of Bourbaki. The subject is just too big. It doesn’t just have one main road.” But Raoul seemed to have developed utterly his own brand of skepticism that came along with such an extraordinary amount of humor and optimism and openness, that he could also claim, as he did: “I can’t say that there is any mathematics that I don’t like.” Often, in a mathematical seminar in a subject far from what he was working on at the time, Raoul would be the person to ask “the generous question.” That is, a question that simply wells up in full simplicity, something that is, perhaps, second nature to the cognoscenti, but a question coming from an ardent desire to follow the thread, the type of question that most in the audience are thankful for its having been raised. Splendid generosity pervaded much of Raoul’s involvement in mathematics: his work, his interchanges, and his teachings. How wonderful was his wise way of recommending that each of us refashion our mathematical understanding so that it sits well in our mind, rather than make do with formulations that might sit well in the mind of some disembodied Lagrangian deity; how marvellous his tip about how to make the best use of your time in a lecture where you understand
of the material (your exercise, sitting there, is to try to shape in your mind something, anything however small that is alive to you and that can be “taken home” from the lecture with enjoyment, and possibly, with understanding). Whatever a “mathematical samurai” is, Raoul claimed to be one. I suppose — in saying that — Raoul was simply acknowledging the freedom — the essential autonomy — of his intellectual spirit. “It’s the problem you go after,” Raoul once said, “rather than the field. You have to trust your instincts and hope that sometimes you will hit upon a subject to which you can maybe make a contribution.” But to be successful here, it is is useful to have the extraordinary range of mathematical experience that Raoul had. And his good judgment. Judgment is key, as is his breadth of vision, his humility, his openness, and his humanity. But — on top of all that — it helps to be, as Raoul was, one of the great mathematicians of our times. To remember Raoul is to savor all that I’ve just said, and beyond that, to remember the aura of Raoul’s presence: as he smiled, made his way down a corridor, looked for his lost gloves, walked his bicycle. And to recall how he had the uncanny gift of being able to radiate appreciation. . . of people, of ideas. . . in such a way that appreciation itself became infectious. Barry Mazur

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46 48

RAOUL BOTT AS WE KNEW HIM

I had the great good fortune to be a second year graduate student when Raoul burst onto the Harvard scene. He presented a mind-blowing course on algebraic topology, simultaneously thrilling and terrifying most of us. I recall him arriving at each class with no notes, puffing on a cigarette right under the No Smoking sign, and simply living the mathematics in our presence. At the end of that year, I mustered my courage to ask him to direct my thesis and I’ll never forget what he told me. He said, “Well, Larry, you’re a good student, but what we have to find out now is whether you can dream!” The second time Raoul profoundly affected my mathematical career was about eight years after my Ph.D. He came to Mexico City to deliver his famous lectures on Foliations and Characteristic Classes. Spotting me in the audience, he said “and Larry, you will write up the notes for these lectures.” Had he asked me if I would do so, I might have weaseled out, since I hardly knew what a foliation was; but who could refuse his royal command? Paul Schweitzer and I organized an informal seminar to explore his lectures. The seminar mostly consisted of Paul explaining to me the intricacies of these new ideas. Subsequent to the actual lectures, Raoul discovered the whole array of exotic characteristic classes and invited me to visit him at IAS for a couple of days so that he could bring me up to speed and get this new theory included in the lecture notes. As it turned out, he was summoned away by a family emergency and could only manage to give me a two-hour lecture on these classes. It’s a tremendous testament to his teaching skills that he managed to communicate this wonderful mathematics in such a short time and in such a way that I could actually understand it (for the most part). In this way, Raoul introduced me to a wonderfully rich and then relatively young area of mathematics that has excited me to this day. Lawrence Conlon Professor Bott was a very gracious person. Here is an incident from 50 years ago. While I was a graduate student, about to start on research for a thesis, Bott spent about two hours with me in his office sketching an approach to the solution of the vector fields on spheres problem. I was too inexperienced (in K-theory) to take this up, and in any case it looked too fanciful to work. About a year later, Bott himself spoke in his weekly seminar. He had received a manuscript from J. F. Adams containing the solution to the vector fields on spheres problem, which followed ALMOST EXACTLY the sketch I had seen on the blackboard a year earlier. Bott generously credited the proof entirely to Adams, without the slightest suggestion that he had been so close himself. Professor Bott was more interested in the mathematics than he was in claiming any credit for himself. I think he was appreciative rather than jealous of what Adams had done. I was Bott’s Ph.D. student (the third after Smale and Edwards, I believe). Of course Bott knew Eilenberg’s semisimplicial theory. In 1961, in addition to Bott’s seminar at Harvard, I began to attend Kan’s seminar at MIT, where I learned some special semisimplicial techniques (which I relayed to Bott). The group theory (Tietze transformations, etc.) I learned from Milnor. So they all had a hand in my thesis (and even more in my Annals paper of 1965). Edward B. Curtis

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RAOUL BOTT AS WE KNEW HIM

47 49

Raoul Bott’s arrival at Harvard in the fall of 1959 came at a fortunate time for me. The thesis advisor I had been working with was on leave that year, and as I began my third year as a graduate student at Harvard I still felt far from finding a thesis topic, let alone writing a thesis. But by the end of that academic year I had the beginnings of a thesis, by the end of the next year I had the Ph.D., and the following fall I began as a Peirce Instructor at Harvard. Fifty years later, I don’t remember exactly how it all happened, but I know that Bott played a crucial role in launching my academic career. The phrase I associate with Bott’s style of mathematics is “no nonsense.” I seem to recall that he loved to quote Sammy Eilenberg’s self-mocking phrase “generalized abstract nonsense” to describe not just Eilenberg and Cartan’s homological algebra, but all sorts of mathematics that reached for the upper limit of abstraction. The word “humility” certainly doesn’t seem right for Bott — he was not a humble man — but “modesty” might do. When he was first at Harvard I remember he would often point out that he would never have been able to get into Harvard as an undergraduate. He had plenty of confidence in his own abilities and his own judgement, but at the same time he was modest about how much he knew and he was always ready to admit his ignorance and work though a new idea starting at the most elementary level. One moment that I have always remembered and always thanked him for came in a seminar on algebraic topology in which an overly self-confident graduate student — or at any rate a graduate student who was successfully feigning self-confidence — ran into trouble doing what he said was a simple computational verification of something. “It’s the sort of thing that at home at your desk takes you five minutes, but I can’t seem to get it right just now,” he said. Bott muttered in a voice loud enough for all to hear, “It’s the sort of thing that at home at your desk takes you a day and a half, but you don’t let on!” He didn’t mean you the speaker, he meant you the mathematician — he included himself among those who struggled at home at their desks and didn’t let on. The graduate student’s only sin was in pretending otherwise. A similar moment was the time he mentioned to me something about a paper he was writing. I can’t reconstruct the context, but I remember him holding his flattened palm about a foot over his desk (he was standing at the time — somehow in my memory of him he is always standing) and said, “The drafts are stacked that high on my desk at home, and they just keep saying the same thing over and over and over again.” His willingness to show students that even his own work called for large amounts of persistence, revision, and patience had a lot to do with his success as a teacher and role model. Another feature of the Bott style that I admired was his habit of always going back to first principles. My memory can’t be accurate, but it tells me that every lecture began with “Let E be a vector bundle over a base space X,” and he would meticulously make an X on the clean blackboard, then make an arrow pointing down to it, and finally make a triumphant E on top of the arrow. But whatever the topic, he would carefully set the stage and review the known ideas and examples that would be needed to make the next step. Razzle dazzle was not his style. Rabbits were not pulled from hats. His goal was to leave us convinced that we could have thought of it ourselves. (Of course he didn’t convince us, but we appreciated his effort and did our best to encourage him.)

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And, perhaps most important of all, his evident joy in his work, and his evident belief that it was a joy we would all be able to share, made him the most inspiring of teachers. At the joint mathematics meetings in January 2008, I gave the talk on pages 65–70 about Riemann’s mathematics, which, although it is not directly related to Bott’s work, expresses an attitude that I believe Bott fostered, namely, that grand abstractions of the sort for which Riemann is famous must always spring from concrete examples and plenty of computational spadework. Certainly this is a lesson I feel I learned from his successful efforts fifty years ago to introduce me to the forbidding abstractions of modern mathematics. Harold M. Edwards The first time in my life I learnt about simplicial manifolds and the like was in a talk of R. Bott at the B. Eckmann festival in April 1977 at the ETH Z¨ urich when B. Eckmann’s 60th birthday was celebrated. It was in R. Bott and G. Segal, ‘The cohomology of the vector fields on a manifold’ that I understood the notion of cosimplicial space, which I used to develop a rigorous approach to lattice gauge theory, including a conceptual approach to extended moduli spaces and a combinatorial calculation of the Chern – Simons invariant for flat SU(2)-connections over lens spaces. A version of these extended moduli spaces are the quasi-Hamiltonian G-spaces developed by Alekseev, Meinrenken, etc. Curiously enough, Bott also contributed to quasi-Hamiltonian G-spaces: ‘Surjectivity for Hamiltonian Loop Group Spaces’. Raoul Bott, Susan Tolman, Jonathan Weitsman. Bott’s paper ‘On the Chern – Weil homomorphism and the continuous cohomology of Lie groups’ intrigued me for many years. In particular, a desire to understand the significance of his decomposition lemma for equivariant cohomology was a driving force for me and, having been professionally and scientifically close to Dold and Puppe for several years in Heidelberg, I also wanted to understand how the Dold – Puppe derived functors came into play here. Johannes Huebschmann Bott’s vanishing theorem for integrable bundles came just as I was learning de Rham’s theorem and characteristic classes, and I was excited by it. Raoul used to use the word “seduced” by mathematics. Sometimes he would say “inveigled”, which I had to look up the meaning of. In giving a lecture, sometimes he would come across a part of a proof that needed a computation and he would say he needed to do the calculation “in the privacy of his boudoir”. He taught many of us to think functorially, like thinking of a group as a category with one object and a morphism for each element, a manifold as a category of pairs (open set, point in the open set), and a bundle as an equivalence class of functors. When someone asked him who invented functors, he said “functors are prehistoric!”. He talked about “folk” theorems. . . theorems everyone knew, but were never written down. I remember playing tennis doubles at a conference with Raoul, Jim Heitsch, Joel Pasternack. He held his own with all of us, younger than him by many years.

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RAOUL BOTT AS WE KNEW HIM

We called him “The Great Man” and idolized him. We were shocked when he used to complain about how slow his mind was working or that he didn’t understand something. Later I realized he always wanted to translate the idea into his own internal way of thinking about it. One time he referred to himself as a “semi-expert” in one area and suggested they contact the real expert, Michael Atiyah, for the answer. One time he invited me to his house to work together, and in the middle of discussing classifying spaces for Haefliger structures, the phone rang. He picked it up and talked a bit, then hung up, made some derogatory comment, and said it was his stock broker. He also said something about his heating bill. As a young researcher, it seemed incongruous for me to hear the great man dealing with mundane things. He wrote a paper with Hochschild on the continuous cohomology of k-forms on a Lie group. He complained that Hochschild had forced him to be a joint author even though Hochschild had done all the work. Herbert Shulman Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, NY 10012-1185, USA E-mail address: [email protected] ¨r Mathematik, Vivatsgasse 7, 53111 Bonn, Germany Max-Planck-Institut fu E-mail address: [email protected] Department of Mathematics, Harvard University, One Oxford Street, Cambridge, MA 02138, USA E-mail address: [email protected] Department of Mathematics, Washington University, St. Louis, MO 63130, USA E-mail address: [email protected] Department of Mathematics, University of Washington, Seattle, WA 98195-4350, USA E-mail address: [email protected] Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, NY 10012-1185, USA E-mail address: [email protected] ´matiques, Universit´ UFR de Mathe e des Sciences et Technologies de Lille, CNRSUMR 8524, 59655 Villeneuve d’Ascq C´ edex, France E-mail address: [email protected] AT&T Laboratories, 200 Laurel Avenue, Middletown, NJ 07748, USA E-mail address: [email protected]

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https://doi.org/10.1090/crmp/050/07 Centre de Recherches Math´ ematiques CRM Proceedings and Lecture Notes Volume 50, 2010

Working with Raoul Bott: From Geometry to Physics Michael Atiyah

1. Introduction I was a close friend and collaborator of Raoul and our association lasted for 50 years from our first meeting at the Institute for Advanced Study in Princeton in 1955 to our last conference at Santa Barbara in 2005, shortly before his death. We wrote 12 joint papers together, published over the 20 years between 1964 – 1984, but this represents only the visible part of our joint collaboration. Our mathematical interaction started soon after we first met and continued wherever and whenever we met. We spent lengthy periods together at Princeton, Harvard, Cambridge, Oxford, Bonn while we also shared shorter conferences in India, China, Hungary, Italy,. . . Raoul and I were both geometers in the broad sense, with interests spanning topology, differential and algebraic geometry and eventually theoretical physics. We had different backgrounds but over time we learnt from each other and our interests fused. What was important was that we shared a common view in which global topological ideas provided a general framework to understand complex problems. Analysis was often at the deep core but the beauty lay in the way in which global results emerged in an elegant fashion, exemplified in the 1950s by the dominant role of sheaf cohomology in the hands of Cartan and Serre. About half-way through our 50 years Raoul and I, together with Is Singer and a few others, were suddenly confronted with new ideas in the theoretical physics of gauge theories. The wide gap between mathematics and physics of the time suddenly acquired a new connecting link, a bridge over which ideas and techniques flowed in both directions. Over the next few years this developed into a flood with spectacular results. Our teacher in all this was Edward Witten whom we encountered in the mid-seventies when he was a Junior Fellow at Harvard. Young enough to absorb new mathematical ideas, but old enough to have all physics at his fingertips, he was the ideal tutor for the two of us, together with our collaborators and students. A whole new generation has now grown up, in which physicists can manipulate spectral sequences and mathematicians can talk glibly about quantization. 2000 Mathematics Subject Classification. Primary 01A60; Secondary 58-03. This is the final form of the paper. c 2010 American Mathematical Society

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In retrospect I feel we are all fortunate to have been there at the right time — a great period in the history of mathematics and physics. I have said many times in the past that I always felt that I was following in the footsteps of Hermann Weyl, that I was traversing territory that Hermann Weyl had previously explored. Certainly, as the pioneer of gauge theory, he would have been delighted with recent developments. I know that Raoul also felt he was a disciple of Weyl. It was indeed Weyl who plucked Raoul from obscurity at Carnegie Tech. and brought him to Princeton; and Lie groups, which lay at the heart of much of Weyl’s work, were also central for Raoul. I have written elsewhere [2] a biographical memoir of Raoul, in which I describe both his personality and his work, but our friendship is beautifully encapsulated by Raoul in the inscription he wrote for me in Volume 1 of his collected works (January 1995): To Michael — comrade in arms and so many exploits, both in the wonderful world of mathematics and in the less perfect corporeal one, with admiration, gratitude and deep affection. In surveying the corpus of Raoul’s work I would like to pick out five themes which are at the heart of his mathematics. These are • • • • •

Morse Theory Lie Groups Loop Spaces Equivariant Cohomology Explicit Formulae

The role of the first four will become clear in the detailed descriptions that are in subsequent sections, but I should say something about the last theme, which is on a rather different footing. It refers not to content but to style. For Raoul mathematics was a balance between general structure, which gave shape and cohesion, and fine detail, where the beauty lay in precision and elegance. Like architecture, mathematics has beauty at both large and small scales. For Raoul a beautiful explicit formula was the ultimate objective. It showed that you really understood and could tame the general theory. This attention to minute detail was a source of “creative tension” between us. Frequently I thought explicit formulae an unnecessary distraction, but Raoul’s preference often paid off and I was partially converted. An explicit formula, as for example in a representative differential form for a cohomology class, can be informative and ultimately useful, as in Chern – Simons theory. In the subsequent sections I have selected a number of the papers I wrote with Raoul which represent our best joint work and which illustrate the five themes above. While they deal with separate issues they are not totally independent and there are links between many of them.

2. Bott periodicity and K-theory There is little doubt that the periodicity theorems for the homotopy of the classical groups [13] are Raoul’s greatest discovery. They are beautifully simple, have widespread consequences, were totally unexpected and were proved by methods from analysis (Morse theory).

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Figure 1. Raoul Bott Let me first recall the Bott periodicity theorem for the unitary groups. Let U = lim U(N ) N →∞

be the stable unitary group, the limit being taken with respect to the standard inclusions U(N ) → U(N + 1). Then the Bott periodicity theorem asserts that there is a homotopy equivalence Ω2 U ∼ U where Ω2 denotes the double loop space. In particular the homotopy groups of U are periodic with period 2:  Z q odd Πq (U) = 0 q even There is a similar periodicity for the stable orthogonal group O, but here the period is 8. Bott’s original proof came from Morse theory and was an outcome of his work with Hans Samelson. As with all important and beautiful theorems, there are many different proofs of Bott periodicity, each reflecting some particular aspect and

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employing a variety of techniques. It is said that Gauss had eight proofs of quadratic reciprocity and there may be a similar number of proofs of Bott periodicity. Bott periodicity takes an elegant form in the framework of K-theory as developed in topology by Hirzebruch and myself, following Grothendieck’s revolutionary work in algebraic geometry. In K-theory, for any compact space X, the tensor product gives a natural homomorphism (1)

K(X) ⊗ K(S 2 ) → K(X × S 2 )

and Bott periodicity essentially asserts that this is an isomorphism. The history of this version is actually quite interesting. When Hirzebruch and I were developing the foundations of K-theory we stumbled across (1) and, in analogy with algebraic geometry, we automatically assumed this must be the Bott isomorphism. When preparing a Bourbaki seminar on K-theory I suddenly realized there was a gap at this point. I filled this by a roundabout route employing cohomology but felt this was unsatisfactory, especially as it would fail for the orthogonal group. In desperation I wrote to Raoul appealing for help. By looking carefully at his proof he was indeed able to show that it implied the isomorphism (1), and the gap was filled. This was all explained in [14], the only paper of Bott’s that appeared in French, not a language that he was comfortable with. Later he regretted not knowing his translator so that he could be properly thanked! Entirely different proofs of the periodicity theorem emerged many years later from the index theorem for elliptic operators, as I will describe in the next section. 3. Index theory In algebraic geometry Grothendieck had introduced K-theory in order to give a new proof of the Hirzebruch – Riemann – Roch theorem, and to generalize it. When we move away from algebraic geometry to topology and differential equations, Ktheory is again fundamental and plays a key role in the index theory of elliptic differential operators on manifolds. The main theorem, developed by Singer and myself in the early sixties [12] gives an explicit formula for index D = dim(ker D) − dim(coker D) of an elliptic differential operator D on a compact manifold X. The answer is given by an integral over X of a differential form (or cohomology class) constructed from the symbol σD of D. This depends only on the highest-order terms of D and gives a map from the cotangent sphere bundle of X into GL(N, C). This is for D operating on CN -valued functions and there is a natural generalization when D acts on sections of vector bundles. In one direction the index theorem generalizes to families of elliptic operators Dy parametrized by y ∈ Y . The index is then an element of K(Y ). This is close to Grothendieck’s generalization and can be used to establish the periodicity theorem in the form (1) [1]. However there is another generalization of the index theorem which leads to an entirely different proof of (1). This involves the index theorem for manifolds X with boundary Z. Here one needs a good notion of a boundary condition so that the index is still well defined. Such a notion had been identified earlier by analysts and, examining this carefully, Raoul and I found how to use the boundary condition to extend the symbol from the cotangent sphere bundle S(X) to its interior over Z.

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Figure 2. Raoul Bott during his McGill years at Parc du MontRoyal, 1942 This meant that the integral formula for the index of D could now be extended suitably over Z, and the formula for index D still held [4]. The details of how the boundary condition for D connects to the topology over Z are quite delicate and very beautiful. They depend in fact on a new elementary proof of Bott periodicity which is explained in [3]. The starting point here is that loops in GL(N, C) should be approximated by finite Fourier series f (Z) =

k 

An Z n

−k

where the coefficients An are N × N complex matrices. All this shows that there is a deep connection between K-theory and the index theory of elliptic operators. Much of this has now been absorbed in the more general theories developed by Alain Connes and others. In looking back over the past, one can sometimes find problems that have been overlooked and might be re-examined. One such problem is to extend the elementary proof of periodicity to the orthogonal group. Here Fourier series, which are appropriate for the circle, have to be replaced by the representation theory of the orthogonal group and spherical harmonics. I am sure Raoul would have liked to see such a programme developed. It would involve explicit formulae of the type he loved. 4. The heat equation One approach to the index theorem which has become very popular, particularly because it fits in well with physics, is that of the heat equation. This is restricted to the Dirac type operators associated to Riemannian geometry, but in

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that context it has significant advantages. Essentially it has a local form and this is required for the APS boundary conditions studied in [11] and the η-invariant. The basic analytic idea is extremely simple. Since the self-adjoint operators DD∗ and D∗ D have non-negative discrete spectra we can write  ∗ Trace(etD D ) = etλ λ≥0

where λ runs over the discrete spectrum (with finite multiplicity). It is easy to see that the terms for λ = 0 are the same for D∗ D and DD∗ , while the difference for λ = 0 gives index D. Thus we have the formula (2)

index D = Trace(etD

∗

D

∗

) − Trace(etDD ).

This is an identity for all t > 0, but for t → 0 there are explicit asymptotic ∗ ∗ expansions for the kernel of the heat operators etD D and etDD . In particular (2) gives an explicit integral formula for index D. Unfortunately the asymptotic expansion of the heat kernel is extremely complicated involving high derivatives — how high depends on the dimension of the manifold. However, quite miraculously, the difference of the heat kernel traces in (2) involves remarkable cancellations, all the higher derivatives drop out, and one is left with a much simpler formula which can then be identified as an algebraic expression in the curvature. This represents a certain characteristic class and so we end up with a simple cohomological formula for index D. This algebraic reduction process was discovered first by Patodi [20] and subsequently generalized by Gilkey [16]. These methods appeared complicated to Raoul and me, so together with Patodi we finally [9] explained the whole process in terms of invariant theory. This involved both the classical results of Weyl on invariants of the orthogonal group and the results of Riemannian geometry, reducing everything to the Riemann tensor and the Bianchi identities. Patodi’s original proof involved some slick algebra which Raoul and I could not fathom. I believe this has now been understood by physicists using supersymmetry. There are two questions arising from all this which I would like to pose. (1) What is the relation between the supersymmetry proof and the invariant theory proof? (2) Can the invariant proof, plus the Bianchi identities, be understood as invariant theory of Diff(X)? This is, I believe, related to a question raised by Gelfand many years ago. 5. Fixed-point theorems At the Woods Hole conference of 1964 Raoul and I defined a generalization of the index theorem for a map f : X → X commuting with an elliptic operator D. This involves the Lefschetz number L(f, D) = Trace(f | ker D) − Trace(f | coker D) Motivated by a problem of Shimura we proposed that, when f has isolated (and transverse) fixed points, this analytic Lefschetz number should be given by a formula of the type  νP (f, D) L(f, D) = P

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Figure 3. Bott and the author

the sum being over the fixed points P of f and νP (f, D) being given by an explicit local formula at P . Having formulated our conjecture, we asked the assembled experts at Woods Hole to compute some simple examples as a check on it. Their verdict was that our conjectured formula was wrong. Fortunately we did not believe them; our formula looked too beautiful! We were in fact justified and our conjecture was eventually proved [5]. This is my version of the events at Woods Hole. Interestingly, none of the experts I consulted could remember the story. As Freud understood, we tend to forget our mistakes and remember our successes! This fixed-point formula was one of Raoul’s favourites. He liked the elegance of the result and its breadth of applications. On the one hand, as we quickly discovered, it implies the famous Hermann Weyl formula for the characters of the irreducible representations of compact semi-simple Lie groups. In a quite different direction it implies the truth of an old conjecture of P. A. Smith for actions of cyclic groups on spheres. Another surprising consequence of the fixed point theorem was a result discovered by Hirzebruch and myself [10] which showed that, if a spin manifold admits a nontrivial circle action, then the index of the Dirac operator (the Aˆ genus) is zero (actually we needed an extension of the fixed-point theorem dealing also with higher-dimensional fixed points). The idea of the proof was simple. One used the explicit formula for the fixedpoint contributions of L(z, D) where |z| = 1 gives the circle action. Analytically continuing to all complex numbers z (including ∞) we found there were no poles, so L(z, D) was independent of z (and actually zero).

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Many years later Witten applied the fixed-point theorem formally in infinite dimensions. He discovered [22] generalizations of my theorem with Hirzebruch. For a whole sequence of representations Rn of the orthogonal group the index of the Dirac operator, coupled to Rn , was rigid, i.e., independent of the circle action. A rigorous mathematical proof of Witten’s result was subsequently obtained by Bott and Taubes [15]. This was modelled on the proof that Hirzebruch and I had given but using modular forms rather than just rational functions. This is closely related to later work of M. Hopkins on elliptic cohomology, and may point to further links between physics and number theory in the future. 6. Yang – Mills and algebraic curves Whenever Raoul and I met we would exchange ideas and frequently a new collaboration would emerge. This happened in Oxford one time when Raoul was returning from a trip to the Tata Institute in Bombay. He was much taken with the study of vector bundles on algebraic curves, the first subject I had studied for my Ph.D. At this time, Yang – Mills theory was flavour of the month for 4-dimensional geometry. Walking across the park one day, on the way to lunch in my college, we began to consider what would come out of Yang – Mills in two dimensions, i.e., for Riemann surfaces or algebraic curves. At first sight we thought the theory would be trivial, but in fact it turned out to be unexpectedly fruitful and resulted in our longest joint paper [6]. The key idea, quite natural to Raoul, was to study the Yang – Mills functional FA 2 on the infinite-dimensional space A of G-connections on a compact Riemann surface X. From the physicists we had learnt of the infinite-dimensional group G = Map(X, G) of gauge transformations. This acted on A preserving the Yang – Mills functional, so naively speaking one should pass to the quotient. However Raoul knew from experience that, because the actions were not free, it was better to work with G-equivariant cohomology. We also discovered the simple but beautiful fact that A has a natural symplectic structure and that the action of G defines a moment map μ : A → Lie(G)∗ which turns out to be just the curvature A → FA . These two facts led us to a new and more direct way of computing the Betti numbers of moduli spaces of flat G-bundles over X (or equivalently of moduli spaces of holomorphic GC -bundles). These Betti numbers had been calculated earlier by Harder and Narasimhan [17] by totally different methods. They had worked over finite fields, counting rational points and used the Weil conjectures (subsequently established by Deligne). Although these two methods are quite different the actual structures of the proofs have remarkable similarities. In both cases one starts with an infinitedimensional situation and reduces by automorphisms. It is a very intriguing question to search for an underlying connection. Two ideas suggest themselves: (1) Is there an infinite-dimensional version of the Weil conjectures? (2) Can we find a quantum field theory that is closely related and makes the Feynman integral analogous to the Tamagawa measure?

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These questions have been around for over twenty years. Perhaps the time is now right to re-examine them? Let me end this section with a number of notes. (3) The method used in [6] turns out to work in general for finite-dimensional geometric invariant theory. This was shown by F. Kirwan [19]. (4) For genus zero (i.e., when X = S 2 ) [6] is close to the work of Bott on ΩG. (5) The ideas in [6] have proved useful in physics [23]. (6) In a subsequent paper [7] Raoul and I pursued further the relation between the moment map and equivariant cohomology. 7. Lacunas Although much of my joint work with Raoul centred round elliptic PDE there was one major work on hyperbolic PDE. This was in tripartite collaboration with Lars G˚ arding who initiated the project and instructed us in the relevant analysis. We were essentially a contract team brought in by Lars because of our expertise in topology and algebraic geometry. The problem we were brought in to study was that of lacunas for hyperbolic PDE, the generalization of the famous Huygens principle for the propagation of light. Igor Petrowsky [21] had made a detailed study of the problem and established many key results, but the topological techniques of the time were inadequate and the geometrical arguments difficult to understand. Our role was to modernize Petrowsky’s presentation making it more intelligible and easier to generalize. The basic idea is that the fundamental solution of a hyperbolic linear PDE (with constant coefficients) is given, through Fourier transforms, by an explicit integral over a cycle on the complexified characteristic hypersurface. This “period” integral depends on the region of space where we compute the fundamental solution. If it turns out to be zero (as it does trivially outside the entire “light-cone”) then the region is a lacuna. There are indeed nontrivial lacunas and determining them is a subtle problem. Fortunately, once Raoul and I had understood the problem, the modern techniques of algebraic geometry gave us the solution [8]. A key ingredient was a result attributed to Grothendieck which asserted that the cohomology of an affine algebraic variety was given by the complex of rational differential forms. In fact (though I did not realize it at the time) this result is actually implicit in the early paper [18] I wrote with Hodge. Not only did these modern methods clear up the problems in Petrowsky’s work but, under the guidance of Lars G˚ arding, they led to substantial generalizations. References 1. M. F. Atiyah, Bott periodicity and the index of elliptic operators, Quart. J. Math. Oxford Ser. (2) 19 (1968), 113–140. , Raoul Harry Bott, Biographical Memoirs of Fellows — Royal Society 53 (2007), 63–76. 2. 3. M. F. Atiyah and R. Bott, On the periodicity theorem for complex vector bundles, Acta Math. 112 (1964), 229–247. , The index problem for manifolds with boundary, Differential Analysis, Bombay Col4. loq., 1964, Oxford Univ. Press, London, 1964, pp. 175–186. , A Lefschetz fixed point formula for elliptic differential operators, Bull. Amer. Math. 5. Soc. 72 (1966), 245–250. , The Yang – Mills equations over Riemann surfaces, Philos. Trans. Roy. Soc. London 6. Ser. A 308 (1983), no. 1505, 523–615.

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Figure 4. Raoul Bott lecturing at the Universit¨ at Bonn. c Wolfgang Vollrath

7. , The moment map and equivariant cohomology, Topology 23 (1984), no. 1, 1–28. 8. M. F. Atiyah, R. Bott, and L. G˚ arding, Lacunas for hyperbolic differential operators with constant coefficients. I, Acta Math. 124 (1970), 109–189. 9. M. F. Atiyah, R. Bott, and V. K. Patodi, On the heat equation and the index theorem, Invent. Math. 19 (1973), 279–330. 10. M. F. Atiyah and F. Hirzebruch, Spin-manifolds and group actions, Essays on Topology and Related Topics (M´ emoires d´ edi´ es a ` Georges de Rham), Springer, New York, 1970, pp. 18–28. 11. M. F. Atiyah, V. K. Patodi, and I. M. Singer, Spectral asymmetry and Riemannian geometry. I, Math. Proc. Cambridge Philos. Soc. 77 (1975), 43–69. 12. M. F. Atiyah and I. M. Singer, The index of elliptic operators on compact manifolds, Bull. Amer. Math. Soc. 69 (1963), 422–433. 13. R. Bott, The stable homotopy of the classical groups, Ann. of Math. (2) 70 (1959), 313–337. , Quelques remarques sur les th´ eor` emes de p´ eriodicit´ e, Bull. Soc. Math. France 87 14. (1959), 293–310. 15. R. Bott and C. Taubes, On the rigidity theorems of Witten, J. Amer. Math. Soc. 2 (1989), no. 1, 137–186. 16. P. B. Gilkey, Curvature and the eigenvalues of the Laplacian for elliptic complexes, Advances in Math. 10 (1973), 344–382. 17. G. Harder and M. S. Narasimhan, On the cohomology groups of moduli spaces of vector bundles on curves, Math. Ann. 212 (1974/75), 215–248. 18. W. V. D. Hodge and M. F. Atiyah, Integrals of the second kind on an algebraic variety, Ann. of Math. (2) 62 (1955), 56–91.

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19. F. C. Kirwan, Cohomology of quotients in symplectic and algebraic geometry, Math. Notes, vol. 31, Princeton Univ. Press, Princeton, NJ, 1984. 20. V. K. Patodi, An analytic proof of Riemann – Roch – Hirzebruch theorem for K¨ ahler manifolds, J. Differential Geometry 5 (1971), 251–283. 21. I. Petrowsky, On the diffusion of waves and the lacunas for hyperbolic equations, Rec. Math. [Mat. Sbornik] N. S. 17(59) (1945), 289–370. 22. E. Witten, Elliptic genera and quantum field theory, Comm. Math. Phys. 109 (1987), no. 4, 525–536. , Two-dimensional gauge theories revisited, J. Geom. Phys. 9 (1992), no. 4, 303–368. 23. The School of Mathematics, The University of Edinburgh, James Clerk Maxwell Building, The King’s Buildings, Mayfield Road, Edinburgh, EH9 3JZ, UK E-mail address: [email protected]

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https://doi.org/10.1090/crmp/050/08

Centre de Recherches Math´ ematiques CRM Proceedings and Lecture Notes Volume 50, 2010

The Algorithmic Side of Riemann’s Mathematics Harold M. Edwards

Richard Dedekind once compared his own work to Bernhard Riemann’s in the following way: “My efforts in number theory have been directed toward basing the work not on arbitrary representations [Darstellungsformen] or expressions but on simple foundational concepts and thereby — although the comparison may sound a bit grandiose — to achieve in number theory something analogous to what Riemann achieved in function theory, in which connection I cannot suppress the passing remark that in my opinion Riemann’s principles are not being adhered to in a significant way by most writers — for example, even in the newest works on elliptic functions; almost always they disfigure the theory by unnecessarily bringing in forms of representation [Darstellungsformen again] which should be results, not tools, of the theory.” I believe that many historians of mathematics endorse this view of Riemann as a forerunner of the 20th-century style that marched under the banner of “structural mathematics” and that regarded formulas as clutter in the path of true understanding. Dedekind was a personal friend of Riemann in G¨ottingen, and could testify on the basis of face-to-face conversations with Riemann about Riemann’s views on the philosophy of mathematics. Nonetheless, the phrase I just quoted was written many years after Riemann’s death and is so contrary to my reading of Riemann’s works that I venture today to contradict him. Surely a mathematician of Riemann’s greatness would want to simplify and organize his formulas in the clearest possible way, but to say that Riemann would insist that Darstellungformen should always be results, not tools, of the theory is, I believe, a serious misrepresentation. (No pun intended.) Now the set-theoretic formulation of mathematical ideas was just being born at the time Dedekind was writing — he was of course one of the foremost pioneers of this conception of mathematics — so it is not certain that when he indicated that Riemann’s approach was 2000 Mathematics Subject Classification. Primary 01A55; Secondary 11-03. Presented at Joint AMS – MAA Meetings, San Diego, January 8, 2008. This is the final form of the paper. c
2010 c
2010 American American Mathematical Mathematical Society

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based on simple foundational concepts that he had in mind anything like the settheoretic formulations that we automatically imagine today. Still, I can’t accept that Riemann in any way repressed his use of various Darstellungsformen as tools in his theories. He was, rather, a virtuoso of Darstellungsformen. I will present my case in four exhibits. Exhibit A. The Riemann – Siegel asymptotic formula for the zeta function in the critical strip. In a letter quoted in his Collected Works, Riemann said that some of the statements in his paper containing the Riemann hypothesis were based on “a development of the function that I have not simplified enough to make it suitable for communication.” This suggests, as I would expect, that developments of functions — Darstellungsformen — and their simplification and interpretation, played a central role in Riemann’s work. The formula he was speaking of here is generally thought to be what is now called the Riemann – Siegel formula. It is based on a sophisticated technique for the asymptotic evaluation of definite integrals called the “saddle point method.” Carl Ludwig Siegel exhumed the formula from Riemann’s chaotic Nachlass, many decades after Riemann’s death, and it was a substantial contribution to the theory of the zeta function when it was published in 1932, surpassing some work that had been done in the intervening 70 years. It is the basis for most of the modern computer verifications of the Riemann hypothesis. (As you may know, these verifications have now reached into the billions of zeros. You may not know the vital role of the Riemann – Siegel formula in them.) One of Siegel’s most amazing discoveries was that Riemann himself, without the aid of a computer, had used his technique to find numerically the first two zeros of zeta in the critical strip. Perhaps he was primarily interested in grand general abstract concepts, but it appears that, at least on this occasion, he did not venture into these higher realms without doing a lot of serious computation to lay the groundwork for his flights. It is impossible to convey much of the substance of this highly sophisticated formula in the brief time that I have. You will get the flavor of the formula if you see its rather lengthy statement on page 154 of my book on Riemann’s zeta function. (In that statement, Z(t) is a multiple of ζ( 21 + it) in which the multiplier is an easily determined nonzero number. While you are at it, you should have a look at page 156, which reproduces the page of Riemann’s notes from which Siegel gleaned the Riemann – Siegel formula. As this page shows, if Riemann shunned formulas, it was not for any lack of ability to generate them.) Exhibit B. The functional equation of the ζ-function. When the basic ideas of Riemann’s paper on the number of primes less than a given magnitude are summarized, it is usually said that Riemann proved that the function ζ(s) defined by the convergent series 1 + 1/2s + 1/3s + 1/4s + · · · for real numbers s > 1 (or for complex numbers s with real part greater than 1) has an analytic continuation to the entire complex plane except for a simple pole at s = 1. The term “analytic continuation” suggests an image of disks of varying radii lined up along a curve in the complex plane, in which the function is defined by a

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power series convergent in each disk and the power series coefficients in each disk are determined by the ones in the preceding disk. But of course this image of successive disks is one of those non-constructive constructions that began to prevail in the decades after Riemann’s death and that dominated mathematics in the 20th century. Riemann’s analytic continuation of ζ(s) was not just a truly constructive construction, it was, if Dedekind will pardon the expression, a formula. Here it is:
∞ (−x)s−1 dx . 2 sin(πs) · Π(s − 1)ζ(s) = i ex − 1 ∞ The function Π(s − 1) is what is called Γ(s) today. The integral on the right-hand side of the equation requires a fair amount of explanation. The term (−x)s−1 in the numerator of the integrand means, of course, e(s−1) log(−x) , so its definition requires choosing a branch of the function log(−x) along the path of integration. Riemann describes the path of integration as “from +∞ to +∞ in the positive sense around the boundary of a domain which contains the value 0 but no other singularity of the integrand in its interior” (the singularities of the integrand are at the places x = ±2πni where the denominator is zero) and stipulates that “the logarithm of −x is determined in such a way that it is real for negative values of x.” There is no point in working through the details of this formula — the convergence of the integral and the truth of the formula for s > 1 — because the only point I want to make is that Riemann achieves his result not by eschewing formulas and staying on an abstract, general plane, but by deploying formulas with great technical ability. His next step in the paper is an even more impressive manipulation of his description of the zeta function in which he modifies the definite integral in such a way as to symmetrize the correspondence between s and 1 − s and to state the functional equation of the zeta function in a simple form, but I will not go into that at all. He did not avoid Darstellungsformen but rather manipulated them and chose among them masterfully. Riemann says nothing about “analytic continuation.” What he says is that the equation above “gives the value of ζ(s) for all complex s and shows that it is single-valued and finite for all values of s other than 1, and that it vanishes when s is a negative even integer.” (This last follows from the fact that Π(s − 1) has poles when s is a negative even integer, so ζ(s) must have zeros to cancel them because the integral on the right-hand side has no poles and sin πn is not zero when n is even.) (Parenthetically, I have to relate this choice of the right representation of a function to my paper on Euler’s definition of the derivative in the Fall ’07 issue of the Bulletin of the AMS. There I say that Euler defined the derivative to be the value of ∆y/∆x when ∆x = 0, the trick being to find a representation of ∆y/∆x that is meaningful when ∆x = 0. Transformations of functions into different forms was an important part of Euler’s mathematics, and, I do not doubt, of Riemann’s too. In short, the manipulation of Darstellungsformen is a fundamental activity, not an act that disfigures — Dedekind’s word was verunzieren — mathematical theories.) Exhibit C. Hypergeometric functions. I think it is very probable that Dedekind had in mind Riemann’s work on hypergeometric functions when he said Riemann was after the essence of things

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and sought to transcend particular Darstellungsformen. Lars Ahlfors in his wellknown book Complex Analysis says, “Riemann was a strong proponent of the idea that an analytic function can be defined by its singularities and general properties just as well as or perhaps better than through an explicit expression.” But my reading of Riemann’s paper leads to a different interpretation. It appears to me to be almost entirely about Darstellungsformen. In his summary Anzeige to the paper, which appears after the paper in the collected works, Riemann writes primarily about the work of his predecessors on hypergeometric functions, making the most extended reference to the work of Kummer, about which he says, “Kummer succeeded in making Euler’s method [a certain transformation of hypergeometric series] into an algorithm for finding the totality of transformations; however, to actually carry it out required such an extensive discussion that he left aside the transformations of the third degree and contented himself with the full derivation of the transformations of the first and second degrees and those composed of them.” He then says of his own work — without undue modesty — that it “gives all of the previous results almost without calculation.” Two remarks on this description. First, he has just described Kummer’s great contribution to the subject as a method of finding transformations of hypergeometric functions. I think that a reading of Riemann’s paper bears out the impression this leaves, that the results he is finding “almost without calculation” deal largely with the theory of transformations of hypergeometric functions. In other words, his method gives all those Darstellungsformen of a hypergeometric function that Kummer’s method produced only in a form that was too complicated to be carried out. Second, as the most cursory inspection of Riemann’s paper [page 76 of Riemann’s Werke] will show, Riemann’s idea of “almost without calculation” is not today’s notion of “almost without calculation.” Exhibit D. Riemann surfaces. Except for the Riemann hypothesis — perhaps even without exception — the most widely known of Riemann’s ideas among mathematicians today is the idea of a Riemann surface. Is there any sense in which Riemann surfaces can be connected to an algorithmic view of mathematics? In answering that question, it is essential to distinguish Riemann’s work itself from Hermann Weyl’s depiction of it in his 1913 book “Die Idee der Riemannschen Fl¨ache.” Weyl’s preface acknowledges that in his view Riemann’s presentation “veils” the “true relation of the functions to the Riemann surface” — he conjectures that Riemann did not want his presentation to present too great a challenge to his contemporaries — in that he only described the surfaces as many-sheeted coverings of the complex plane. He says that the general conception of the surfaces was first developed with transparent clarity by Felix Klein. In other words, what he presents in his book is what he imagines Riemann would have wanted to say had he not worried about shocking his readers too much. This is the sort of view historians must guard against. The work of great mathematicians of the past is not a series of partially successful efforts to put mathematics in the form deemed best by present-day mathematicians. Weyl himself expressed such an objection very differently in 1955 in the preface to the English language third edition of the book when he said of the original edition that its “enthusiastic preface betrayed the youth of the author.” He says Klein “had been

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the first to develop the freer conception of a Riemann surface, in which the surface is no longer a covering of the complex plane; thereby he endowed Riemann’s basic ideas with their full power. It was my fortune to discuss this thoroughly with Klein in divers conversations. I shared his conviction that Riemann surfaces are not merely a device for visualizing the many-valuedness of analytic functions, but rather an indispensable essential component of the theory; not a supplement, more or less artificially distilled from the functions, but their native land, the only soil in which the functions grow and thrive.” That’s very lyrical, but note that he has not reiterated his theory that Riemann veiled the full force of his theory from his readers, and in fact he makes no conjecture at all about the relation between Riemann’s original thought and the reformulation of it by Klein and himself. It would be foolhardy for me to attempt to give any analysis of Riemann’s works — and of the extensive notes of Riemann’s that were published in later editions of his works — in an attempt to confirm that his goals were in some way algorithmic. I will simply say that I see indications that his conception of manysheeted coverings of the complex plane did serve algorithmic ends. First, there is the encyclopedic description of the transformations of hypergeometric functions that I have already mentioned — and I should add that supplementary material in later editions of Riemann’s works gives many more indications of the role of the Riemann surface picture in the analysis of these transformations. See, for example, page 101 of the Nachtr¨ age that was added to the Riemann Collected Works by Noether and Wirtinger. Second, there is the point mentioned by Ahlfors in the passage I read, about Riemann describing analytic functions in terms of their singularities and their general properties. This is a very prominent feature of Riemann’s 1851 dissertation. But note that such a description is algorithmic too. To describe a polynomial by its roots or a rational function by its zeros and poles often serves very concrete algorithmic purposes. Such a method also gives insight into the number of arbitrary parameters in an algebraic function of a certain type — which is the number evaluated by the Riemann – Roch theorem. Finally, I will mention Riemann’s treatise on Abelian functions. He clearly explains that it is divided into two parts, the first part containing that part of the theory that he can cover without the use of his multi-variate theta function (and I can’t resist pointing out that the author of this paper does not appear to be a man worried about presenting too great a challenge to his readers) and the second part using that function. To the best of my very poor ability to understand that work, I would judge it to be quite distinctly algorithmic. It deals with Riemann surfaces of finite genus and deals with Abelian functions, which are integrals of algebraic differentials on such surfaces. Judging from the many explicit formulas it contains, it would seem to me to be hard to call it anything but algorithmic. In conclusion, I would like to quote a remark Carl Ludwig Siegel made in the introduction to his publication of the Riemann – Siegel formula. “The legend,” he wrote, “according to which Riemann found his mathematical results through grand general ideas without requiring the formal tools of analysis, is not as widely believed today as it was during Felix Klein’s lifetime. Just how strong Riemann’s analytic technique was is especially clearly shown by the derivation and transformation of his asymptotic series for ζ(s).” (My translation.)

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What I hear Siegel saying is that Felix Klein — and I would add Dedekind — conveyed a mistaken impression of Riemann’s contribution by emphasizing the generality of his concepts and methods. His more technical achievements are at least as impressive, at least as important a legacy, and, in any case, were the indispensable basis of all his works. References 1. L. V. Ahlfors, Complex analysis, 2nd ed., McGraw-Hill, New York – Toronto – London, 1966. 2. R. Dedekind, Gesammelte mathematische Werke, Vol. III, Vieweg, Braunschweig, 1932, pp. 468 – 469. Letter to R. Lipschitz, June 6, 1876. 3. H. M. Edwards, Riemann’s zeta function, Pure Appl. Math., vol. 58, Academic Press, New York – London, 1974; Dover, Mineola, NY, 2001. 4. B. Riemann, Gesammelte mathematische Werke, Teubner, Leipzig, 1876. ¨ 5. C. L. Siegel, Uber Riemanns Nachlaß zur analytischen Zahlentheorie, Gesammelte Abhandlungen, Vol. I, Springer, Berlin, 1966, pp. 275 – 310. 6. H. Weyl, Die Idee der Riemannschen Fl¨ ache, 2nd ed., Teubner, Leipzig, 1923. , The concept of a Riemann surface, ADIWES International Series in Mathematics, 7. Addison-Wesley, Reading, MA – London, 1964. Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, N.Y. 10012-1185, USA E-mail address: [email protected]

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https://doi.org/10.1090/crmp/050/09 Centre de Recherches Math´ ematiques CRM Proceedings and Lecture Notes Volume 50, 2010

Actions of Lie Groups and Lie Algebras on Manifolds M. W. Hirsch Dedicated to the memory of Raoul Bott

Abstract. Questions of the following sort are addressed: Does a given Lie group or Lie algebra act effectively on a given manifold? How smooth can such actions be? What fixed point sets are possible? What happens under perturbations? Old results are summarized, and new ones presented, including: For every integer n there are solvable (in some cases, nilpotent) Lie algebras g that have effective C ∞ actions on all n-manifolds, but on some (in many cases, all) n-manifolds, g does not have effective analytic actions.

Introduction Lie algebras were introduced by Sophus Lie under the name “infinitesimal group,” meaning the germ of a finite dimensional, locally transitive Lie algebra of analytic vector fields in Rn . In his 1880 paper Theorie der Transformationsgruppen [19, 20] and his later book with F. Engel [21], Lie classified infinitesimal groups acting in dimensions 1 and 2 up to analytic coordinate changes. This work stimulated much research, but attention soon shifted to the classification and representation of abstract Lie algebras and Lie groups. Later the topology of Lie groups was studied, with fundamental contributions by Bott. In 1950 G. D. Mostow [23] completed Lie’s program of classifying effective transitive surface actions.1 One of his major results is: Theorem 1 (Mostow). A surface M without boundary admits a transitive Lie group action if and only if M is a plane, sphere, cylinder, torus, projective plane, M¨ obius strip or Klein bottle. 2000 Mathematics Subject Classification. Primary 57S20; Secondary 22E25, 57S25. Key words and phrases. Transformation groups, Lie groups, Lie algebras. I thank M. Belliart, K. DeKimpe, W. Goldman, G. Mostow, J. Robbin, D. Stowe, F.-J. Turiel and J. Wolf for invaluable help. This is the final form of the paper. 1 For each equivalence class of transitive surface actions, Mostow describes a representative Lie algebra by formulas for a basis of vector fields. Determining whether one of these representatives is isomorphic to a given Lie algebra can be nontrivial. Here the succinct summary of the classification in M. Belliart [3] is useful. c 2010 American Mathematical Society

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By a curious coincidence these are the only surfaces without boundary admitting nontrivial compact Lie group actions (folk theorem). The following nontrivial extension of Theorem 1 deserves to be better known: Theorem 2. Let G be a Lie group and H a closed subgroup such that the manifold M = G/H is compact. Then χ(M ) ≥ 0, and if χ(M ) > 0 then M has finite fundamental group. This is due to Gorbatsevich et al. [11, Corollary 1, p. 174]. See also F´elix et al. [10, Proposition 32.10], Halperin [12], Mostow [24]. While much is known about the topology of compact group actions, there has been comparatively little progress on classification of actions of Lie algebras and noncompact groups, an exception being D. Stowe’s classification [28] of analytic actions of SL(2, R) on compact surfaces. The present article addresses the easier tasks of deciding whether a group or algebra acts nontrivially on a given manifold, determining the possible smoothness of such actions, and investigating their orbit structure. Most proofs are omitted or merely outlined, with details to appear elsewhere. The low state of current knowledge is illustrated by the lack of both counterexamples and proofs for the following Conjectures. Let g denote a real, finite-dimensional Lie algebra. (C1) If g has effective actions on M n , then g also has smooth effective actions on M n . (C2) If g is semisimple and has effective smooth actions on M n , n ≥ 2, then g also has effective analytic actions on M n . But however plausible these statements may appear, they can’t both be true: (C1) or (C2) is false for g = sl(2, R). For sl(2, R) has effective actions on every M 2 (Theorem 7), but no effective analytic action if M 2 is compact with Euler characteristic χ(M 2 ) < 0 (Corollary 17(b)). It is unknown whether such a surface can support a smooth effective action β of sl(2, R). If it does, Theorem 16(ii) implies that the vector fields X β are infinitely flat at the fixed points of so(2, R)β . The analog of (C2) for nilpotent algebras is false. If n denotes the Lie algebra of 3 × 3 niltriangular real matrices, by Theorem 3 and Example 13: On every connected surface n × n has effective C ∞ actions, but no effective analytic actions. Further conjectures and questions are given below. Terminology. F stands for the real field R, or the complex field C. We write z = a + ıb to indicate that ¡ z ∈ C has real part a = z and ¡ imaginary part ¯ := a − ıb. The sets of integers, b := z. The complex conjugate of λ := a + ıb is λ positive integers and natural numbers are Z, N+ = {1, 2, . . . } and N = 0 ∪ N+ respectively. i, j, k, l, m, n, r denote natural numbers, assumed positive unless the contrary is indicated. s denotes the largest integer ≤ s. M or M n denotes an n-dimensional analytic manifold, perhaps with boundary; its tangent space at p is Tp M . vs (M ) denotes the vector space of C s vector fields on M , with the weak C s topology (1 ≤ s ≤ ∞). The Lie bracket makes v∞ a Lie

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algebra, with analytic vector fields forming a subalgebra. The value of Y ∈ v1 (M ) at p ∈ M is Yp . The derivative of Y at p is a linear operator on dYp on Tp M . Except as otherwise indicated, manifolds, Lie groups and Lie algebras are real and finite dimensional; manifolds and Lie groups are connected, and maps between manifolds are C ∞ .
The G denotes a Lie group with Lie algebra g and universal covering group G. k-fold direct product G × · · · × G is Gk and similarly for g. SL(m, F) is the group of m × m matrices over F of determinant 1, and ST(m, F) is the subgroup of upper triangular matrices. The corresponding identity components and Lie algebras are denoted by SL0 (m, F), st(m, F) and so forth. An action α of G on M , indicated by (α, G, M ), is a homomorphism g → g α from G to the group of homeomorphisms of M with a continuous evaluation map evα : G × M → M , (g, x) → g α (x). We call α smooth, or analytic, when evα has the corresponding property.2 Small gothic letters denote linear subspaces of Lie algebras, with g and h reserved for Lie algebras. Recursively define g(0) = g and g(j+1) = g(j) = [g(j) , g(j) ] = commutator ideal of g(j) . Recall that g (and also G) is solvable of derived length l = l(g) = l(G) if l ∈ N+ is the smallest number satisfying g(l) = 0. For example, l(st(m, F)) = m. g is nilpotent if there exists k ∈ N such that g(k) = {0}, where g(0) = g and g(j+1) := [g, g(j) ]. It is known that g is solvable if and only g is nilpotent. g is supersoluble if the spectrum of ad X is real for all X ∈ g, where ad := adg denotes the adjoint representation of g on itself defined by (ad X)Y = [X, Y ]. Equivalently: g is solvable and faithfully represented by upper triangular real matrices. An action β of g on M , recorded as (β, g, M ), is a continuous homomorphism X → X β from g to v∞ (M ). An n-action means an action on an n-dimensional manifold. A smooth action (α, G, M ) determines a smooth action (α, ˆ g, M ). Conversely, if G is simply connected and (β, g, M ) is such that each vector field X β is complete (as when M is compact), then there exists (α, G, M ) such that β = α. ˆ The orbit of p ∈ M under (α, G, M ) is {g α (p) : g ∈ G}, and the orbit of p under a Lie algebra action (β, g, M ) is the union over X ∈ g of the integral curves of p for X β . An action is transitive if it has only one orbit. The fixed point set of (α, G, M ) is Fix(α) = {x ∈ M : g α (x) = x, g ∈ G}, and that of (β, g, M ) is Fix(β) := {p ∈ M : Xpβ = 0, X ∈ g} The support of any action γ on M is the closure of M \ Fix(γ). An action is effective if its kernel is trivial, and nondegenerate if the fixed point set of every nontrivial element has empty interior. Effective analytic actions are nondegenerate. A group action is almost effective if its kernel is discrete.

2 Most

of the results here can be adapted to C r actions and local actions.

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Construction of actions Every G acts effectively and analytically on itself by translation. Every g admits a faithful finite-dimensional representation R : g → gl(n, R) by Ado’s theorem (Jacobson [18]). If R(g) has trivial center, it induces an effective analytic action by g on the projective space Pn−1 and the sphere Sn−1 . An action gives rise to actions on other manifolds by blowing up invariant submanifolds in various ways; this preserves effectiveness and analyticity. Blowing up fixed points of standard actions of ST0 (3, R) on P2 , S2 and D2 yields: Theorem 3 (F. Turiel [30]). ST0 (3, R) has effective analytic actions on all compact surfaces. Conjecture. ST0 (3, R) has effective analytic actions on all surfaces. Analytic approximation theory is used to prove: Theorem 4 (M. Hirsch & J. Robbin [15]). The vector group Rm has effective analytic actions on M n if m ≥ 1, n ≥ 2. On open manifolds it is comparatively easy to produce effective Lie algebra actions: Theorem 5. Assume there is an effective action (α, g, W n ). Then a noncompact M n admits an effective action (β, g, M n ) in the following cases: (a) M n is parallelizable (b) n = 2 and W 2 is nonorientable. Moreover β can be chosen nondegenerate, analytic, transitive or fixed-point-free provided α has the same property. Proof. Define β as the pullback of α through an immersion M n → W n (for immersion theory see Hirsch [14], Poenaru [26], Adachi [1]).
Corollary 6. Every noncompact M 2 supports effective analytic actions by sl(3, R) and sl(2, C). Every parallelizable noncompact M n has effective analytic actions by sl(n + 1, R), by sl(n/2, C) if n is even, and by sl(n/2 + 1, C) if n is odd.
on R, and by compactification Actions of G on the circle S1 lift to actions of G to actions on [0, 1]. Such actions can be concatenated to get effective actions of 1 × · · · × G  G m on [0, 1]. Further topological constructions lead to effective actions on closed n-disks, trivial on the boundary. Embedding such disks disjointly into an n-manifold leads to:  0 (2, R)j × ST0 (2, R)k × Rm acts effectively on every manifold Theorem 7. SL of positive dimension (j, k, m ≥ 0). In many cases such actions cannot be analytic and their smoothness is unknown; but see Theorem 9. Algebraically contractible groups The actions constructed above are either analytic or merely continuous. Next we exhibit a large class of solvable groups having effective actions — often smooth —

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on manifolds of moderately low dimensions. In many case these are smooth but cannot be analytic. Let E(G) denote the space of endomorphisms of G, topologized as a subset of the continuous maps G → G. We call G and g algebraically contractible (AC) if there is a path φ = {φt } in E(G) joining the identity endomorphism φ0 of G to the trivial endomorphism φ1 . Equivalently: G is solvable and simply connected, and the identity and trivial endomorphisms of g are joined by a path ψ = {ψt } in the affine variety E(g) of Lie algebra endomorphisms of g. Every path ψ comes from a unique path φ. The class of AC groups contains the vector group Rn , the matrix groups 0 (n, C), and many of their subgroups and quotient groups. It is ST0 (n, R), ST closed under direct products. If g is AC and an ideal h is mapped into itself by every endomorphism of g, then h and g/h are AC. However, some nilpotent Lie algebras are not AC (Dekimpe [7]): The derivation algebra of an AC Lie algebra cannot be unipotent, but there are 8-dimensional nilpotent Lie algebras having unipotent derivation algebras (Dixmier – Lister [8], Ancochea – Campoamor [2]). Proposition 8. Assume G is algebraically contractible and (α, G, M ) is almost effective. There is an effective action (β, G, M × R) with the following properties: (a) g β (x, 0) = (g α (x), 0). (b) g β (x, t) = (x, t) if |t| ≥ 1. (c) If α is smooth so is β. Proof. We can choose the path ψ : [0, 1] → E(g) in the definition of AC to be C ∞ and constant in a neighborhood of {0, 1}. The corresponding path φ : [0, 1] → E(G) has the same properties. Extend φ over R by setting φt = φ1 (= the trivial endomorphism) for t ≥ 1, and φt = φ−t for t ≤ 0. Now define β by g β (x, t) := φt (g)α (x),

(g ∈ G, (x, t) ∈ M × R).




Theorem 9. Assume Gi is AC and (αi , G, Sn−1 ) is almost effective (i = 1, . . . , k). For every M n there exists an effective action (δ, G1 × · · · × Gk , M n ) that is smooth provided the αi are smooth. Proof. Let (βi , Gi , Sn−1 × R) obtained from αi as in Proposition 8. Through an identification Sn−1 ×R = Dn \(Sn−1 ∪0), extend βi to an action (γi , Gi , Dn ) with compact support in Dn \ Sn−1 . (Here Dn is the unit n-disk with boundary §n − 1.) n n Transfer the γi to actions δi in k disjoint coordinate  ndisks Di ⊂ M . Define δ to n
coincide with δi in Di and to be trivial outside i Di . Corollary 10. Assume Gi ⊂ GL(n, R) is algebraically contractible and contains no scalar multiple of the identity matrix, (i = 1, . . . , k). Then G1 × · · · × Gk has effective smooth actions on all n-manifolds. Proof. The natural actions of Gi on Pn−1 and Sn−1 are smooth and effective. Apply Theorems 9 and 5.
The Epstein – Thurston theorem D. B. A. Epstein and W. P. Thurston [9, Theorem 1.1] discovered fundamental lower bounds on the dimensions in which solvable Lie algebras can act effectively:

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Theorem 11 (Epstein – Thurston). Assume g is solvable and has an effective n-action. Then n ≥ l(g) − 1, and n ≥ l(g) if g is nilpotent. In the critical dimensions there is further information on orbit structure: Theorem 12. Let α be an effective n-action of a solvable Lie algebra g. Assume n = l(g) − 1, or g is nilpotent and n = l(g). (i) There is an open orbit. If α is nondegenerate the union of the open orbits is dense. (ii) Assume g(n−1) ⊂ c = the center of g. Then: (a) each nontrivial orbit of g(n−1) lies in an open orbit of g and has dimension 1, (b) the number of open orbits is ≥ dim g(n−1) (c) if α is nondegenerate then dim c = 1. Proof. The union of orbits of dimensions < n is a closed set L in which g(l(g)−1) acts trivially by Epstein – Thurston. Therefore M n \ L, the union of the open orbits, is nonempty because α is effective, and dense if α is nondegenerate. This proves (i). Next we prove (ii). (a) Let L be a nontrivial orbit of g(n−1) and let O be the orbit of g containing L. Then O is an open set because dim(O) = n by Epstein – Thurston. This proves the first assertion of (a). To prove the second we can assume the action is transitive. Fix a 1-dimensional subspace z ⊂ c having a 1-dimensional orbit L1 ⊂ L. After replacing O by a suitably small open subset, we can assume the domain of the action is O = Rn−1 × R with the slices x × R being the orbits of z. The induced action of g on the n-dimensional space of z-orbits kills g(n−1) by Epstein – Thurston. This implies L1 = L, which implies (a). (b) Suppose dim g(n−1) = s ≥ 1 and there are exactly r open orbits Oi , i = 1, . . . r. As g acts transitively in Oi and g(n−1) is central, there is acodimension-one subalgebra ki ⊆ g(n−1) acting trivially in Oi . If 1 ≤ r < s then i ki has positive dimension and acts trivially in each open orbit, and also in all other orbits by Epstein – Thurston. This implies (b). (c) Assume α is nondegenerate. By (a) there is an open orbit O, which we can assume is the only orbit. Let O, L, z be as in the proof of (a). If (c) is false we choose z so that the central ideal j := g(n−1) + z has dimension ≥ 2. In the proof of (a) we saw that every nontrivial orbit of j is 1-dimensional, hence every orbit of j is 1-dimensional because α is transitive and j is central. Therefore for every p ∈ O there is a maximal nontrivial linear subspace kp ⊂ j annihilated by α. As α is transitive and j is central, all the kp coincides with an ideal that acts trivially in O. This contradicts the assumption that α is nondegenerate.
Example 13. The nilpotent algebra n = st(n + 1, R) × R has derived length n and 2-dimensional center st(n + 1, R)(n−1) × R. Being algebraically contractible, n acts effectively on all n-manifolds by Corollary 10. On the other hand, Theorem 12 implies: • Every n-action of n is degenerate and therefore nonanalytic. Weight spaces and spectral rank Let T : g → g be linear. For λ in the spectrum spec(T ) ⊂ C define the (generalized) weight space w(T, λ) ⊂ g to be the largest T -invariant subspace on which T

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¯ The largest subspace of w(T, λ) on which T acts semisimply is has spectrum {λ, λ}.  kernel of T − λI if λ ∈ R m(T, λ) := 2 2 kernel of T − 2(Re λ)T + |λ| I if λ ∈ /R For any set S ⊂ C let Γ(S) denote the additive free abelian subgroup of C generated by S. The rank of Γ(spec(T )) is the spectral rank r(T ). The rank of Γ(spec(T ) \ R) is the nonreal spectral rank r NR (T ). For a Lie algebra g define r(g) = max r(ad X), X∈g

r NR (g) = max r NR (ad X) X∈g

I

Thus r (g) = 0 if and only if g is supersoluble. Example 14. The following can be shown: • A semisimple algebra g of rank r has spectral rank r (compare Jacobson [18, p. 117, XII]), and imaginary spectral rank 2r if g admits a complex structure. • If X ∈ st(m, R) is a sufficiently irrational diagonal matrix then   r st(m, R) = r(ad X) = m − 1. • Set k = m/2. If X ∈ so(k) and r I (X) = k then r I (st(m, R)) = r I (X) = k. Call Y ∈ v∞ (M ) is flat at p ∈ M when its Taylor series vanishes in local coordinates centered at p. If such a Y is analytic it is trivial. Given (α, g, M ) and p ∈ M , define fp (α) ⊂ g as the set of Y ∈ g such that Y α is flat at p. This is an ideal. Proposition 15. Assume (α, g, M n ) is smooth, X ∈ g and p ∈ Fix(X α ). Suppose m(ad X, λ) ∩ fp (α) = 0 for all λ ∈ spec(ad X) \ 0. Then spec(ad X) ⊂ Γ(spec(dXpα )) and therefore n ≥ max{r(X), 2rI (ad X)}. n n ∞ n Proof. We can assume M = R , p = 0. Write every Z ∈ v (R ) as the formal sum r∈N Z(r) where the components of the vector field Z(r) are homogeneous α α polynomial functions of degree r. Then X(0) = 0, X(1) = dXpα . The order of Z is the smallest r for which Z(r) = 0 if Z is not flat at 0, otherwise the order is ∞. Suppose Y ∈ k(ad X, λ) is not flat at 0 and has finite order r. Then (adC⊗g X − λI)Y = 0, α α α , Y(r) ] = λY(r) . Hence λ ∈ spec(adv∞ (Rn ) dXpα ). A calculation shows implying [X(1) that spec(adv∞ (Rn ) Z) ⊂ Γ(spec(Z)) for every linear vector field Z : Rn → Rn .
Apply this to Z := dXpα .

The following result is derived from Proposition 15: Theorem 16. Suppose (α, g, M n ) is smooth, X ∈ g and p ∈ Fix(X α ). (i) Assume r(ad X) = n + k > n. Then ad X has k different eigenvalues λ = 0 such that w(ad X, λ) ⊂ fp (α). (ii) Assume 2r I (ad X) = n, α is effective, and m(ad X, λ) ∩ fp (α) = 0 for all λ ∈ spec(ad X) \ R. Then dXpα has only nonreal eigenvalues, X α has index 1 at p, and if M n is compact then χ(M n ) = # Fix(X α ) ≥ 1.

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This has powerful consequences for analytic actions: Corollary 17. Assume (α, g, M n ) is effective and analytic and X ∈ g. (a) If Fix(X α ) = ∅ then n ≥ max{r(ad X), 2rI (ad X)}. (b) Suppose M n is compact and n = 2r I (ad X). Then χ(M n ) = # Fix(X α ) ≥ # Fix(α). Therefore χ(M n ) ≥ 0, and Fix(α) = ∅ if χ(M n ) = 0. For surface actions, (b) is due to Turiel [30]. Corollary 18. Assume M n is compact and χ(M n ) = 0. If (α, g, M n ) is analytic with kernel k, then dim k ≥ max{r(g) − n, r I (g) − n/2}. Example 19. Assume (α, st(2k, R), M n ) is effective and analytic. Corollary 17 implies that almost all X in the subalgebra so(2k, R) have the following properties: • If n ≤ 2k then Fix(X α ) = ∅. • If n = 2k and M n is compact then # Fix(X α ) = χ(M n ) > 0. Example 20. Assume m, n, l ∈ N+ with m ≤ n. Theorem 9 shows that every n-manifold supports a smooth effective action of st(m + 1, R)l . Because r(st(m + 1, R)) = ml, Corollary 18 has the following consequence: • Assume M n is compact and χ(M n ) = 0. If (α, st(m + 1, R)l , M n ) is analytic and effective then l ≤ n/m. For instance: • st(n + 1, R)2 does not have an effective analytic action on any compact n-manifold. Nevertheless, there are effective smooth actions of ST(n+1, R)2 on every n-manifold by Corollary 10. Fixed points For actions of G on compact surfaces M 2 the following results are known: Proposition 21. (a) (Lima [22], Plante [25], Belliart – Liousse [4], Turiel [29, 31]) ST0 (2, R) has effective, fixed-point free C ∞ actions on all compact surfaces. (b) (Belliart [3]) If G acts without fixed point and χ(M 2 ) < 0 then ST0 (2, R) is a quotient group of G. (c) (Turiel [30]) If G acts analytically without fixed point, χ(M 2 ) ≥ 0. (d) (Lima [22], Plante [25]) If G is nilpotent and acts without fixed point, χ(M 2 ) = 0. (e) (Hirsch – Weinstein [16]) If G is supersoluble and acts analytically without fixed point, χ(M 2 ) = 0. Careful use of the blowup construction shows that some supersoluble groups have effective analytic surface actions with arbitrarily large numbers of fixed points: Theorem 22. Let Mg2 denote a closed surface of genus g ≥ 0. For every k ∈ N there is an effective analytic action (β, ST0 (3, R), Mg2 ) such that  2(g + k + 1) if Mg2 is orientable, # Fix(β) = g+k if Mg2 is nonorientable and g ≥ 1.

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On the other hand: Suppose G is not supersoluble. If M 2 is compact and (α, G, M 2 ) is effective and analytic, then 0 ≤ # Fix(α) ≤ χ(M 2 ) ≤ 2. This follow from Corollary 17(b), because r I (G) ≥ 1 . Questions. Is the analog of Proposition 21(a) true for ST0 (3, R)? Does this group have an effective analytic action with a unique fixed point on some orientable closed surface? Can ST(3, R) act effectively on S2 with a unique fixed point? Can a smooth effective action of SL(2, R) on S1 × S1 have a fixed point? For noncompact group actions in higher dimensions the following are known: (Poincar´e [27], Hopf [17]) R acts effectively without fixed point on a compact M n ⇐⇒ χ(M n ) = 0. (Borel [5]) An algebraic action of a solvable complex algebraic group on a complete complex algebraic variety has a fixed point. (Bonatti [6]) If M n is compact, n = 3 or 4, and χ(M n ) = 0, then every analytic action of R2 on M n has a fixed point. Spectral rigidity A1 (g, M ) denotes the space of C ∞ actions of g on M under the smallest topology making the maps A1 (g, M ) → v1 (M ), α → X α , continuous for all X ∈ g. An action (α, g, M ) is spectrally rigid at (X, p) if X ∈ g, p ∈ Fix(X α ), and there exist arbitrarily small neighborhoods N ⊂ A1 (g, M n ) of α and W ⊂ M of p such that for all β ∈ N : (SR1) Fix(X β ) ∩ W = ∅ (SR2) q ∈ Fix(X β ) ∩ W =⇒ dXqβ and dXpα have the same nonzero eigenvalues. While spectral rigidity is impossible for nontrivial abelian algebras and dubious for nilpotent algebras, many solvable and semisimple algebras exhibit it: Theorem 23. Assume (α, g, M n ) is effective and analytic, X ∈ g and r(ad X) = n. Then α is spectrally rigid at (X, p) for all p ∈ Fix(X α ). The proof is based on Proposition 15. Conjecture. An analytic action α of a semisimple Lie algebra s is spectrally rigid at (X, p) for all X ∈ s, p ∈ Fix(α). References 1. M. Adachi, Embeddings and immersions, Transl. Math. Monogr., vol. 124, Amer. Math. Soc., Providence, RI, 1993. 2. J. M. Ancochea and R. Campoamor, Characteristically nilpotent Lie algebras: a survey, Extracta Math. 16 (2001), no. 2, 153–210. 3. M. Belliart, Actions sans points fixes sur les surfaces compactes, Math. Z. 225 (1997), no. 3, 453–465. 4. M. Belliart and I. Liousse, Actions affines sur les surfaces, 1996. 5. A. Borel, Groupes lin´ eaires alg´ ebriques, Ann. of Math. (2) 64 (1956), 20–82. 6. C. Bonatti, Champs de vecteurs analytiques commutants, en dimension 3 ou 4: existence de z´ eros communs, Bol. Soc. Brasil. Mat. (N.S.) 22 (1992), no. 2, 215–247. 7. K. Dekimpe, 2005, personal communication.

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8. J. Dixmier and W. G. Lister, Derivations of nilpotent Lie algebras, Proc. Amer. Math. Soc. 8 (1957), 155–158. 9. D. B. A. Epstein and W. P. Thurston, Transformation groups and natural bundles, Proc. London Math. Soc. (3) 38 (1979), no. 2, 219–236. 10. Y. F´ elix, S. Halperin, and J.-C. Thomas, Rational homotopy theory, Grad. Texts in Math., vol. 205, Springer, New York, 2001. 11. V. V. Gorbatsevich, A. L. Onishchik, and E. B. Vinberg, Foundations of Lie theory and Lie transformation groups, Springer, Berlin, 1997. 12. Stephen Halperin, Finiteness in the minimal models of Sullivan, Trans. Amer. Math. Soc. 230 (1977), 173–199. 13. S. Helgason, Differential geometry and symmetric spaces, Pure Appl. Math., vol. 12, Academic Press, New York, 1962. 14. M. W. Hirsch, The imbedding of bounding manifolds in Euclidean space, Ann. of Math. (2) 74 (1961), 494–497. 15. M. W. Hirsch and J. W. Robbin (2003), manuscript. 16. M. W. Hirsch and A. Weinstein, Fixed points of analytic actions of supersoluble Lie groups on compact surfaces, Ergodic Theory Dynam. Systems 21 (2001), no. 6, 1783–1787. ¨ 17. H. Hopf, Uber die Curvatura integra geschlossener Hyperfl¨ achen, Math. Ann. 95 (1926), no. 1, 340–367. 18. N. Jacobson, Lie algebras, Interscience Tracts in Pure and Applied Mathematics, vol. 10, Interscience Publishers, New York, 1962. 19. S. Lie, Theorie der Transformationsgruppen I, Math. Ann. 16 (1880), no. 4, 441–528. , Sophus Lie’s 1880 transformation group paper, translated by M. Ackerman, with 20. comments by R. Hermann, Lie Groups: History, Frontiers and Applications, vol. 1, Math. Sci. Press, Brookline, MA, 1975. 21. S. Lie and F. Engel, Theorie der Transformationsgruppen, Vol. 3, Teubner, Leipzig, 1893. 22. E. L. Lima, Common singularities of commuting vector fields on 2-manifolds, Comment. Math. Helv. 39 (1964), 97–110. 23. G. D. Mostow, The extensibility of local Lie groups of transformations and groups on surfaces, Ann. of Math. (2) 52 (1950), 606–636. , A structure theorem for homogeneous spaces, Geom. Dedicata 114 (2005), 87–102. 24. 25. J. F. Plante, Fixed points of Lie group actions on surfaces, Ergodic Theory Dynam. Systems 6 (1986), no. 1, 149–161. 26. V. Poenaru, Sur la th´ eorie des immersions, Topology 1 (1962), 81–100. 27. H. Poincar´e, Sur les courbes d´ efinies par une ´ equation diff´ erentielle, J. Math. Pures Appl. 1 (1885), 167–244. 28. D. C. Stowe, Real analytic actions of SL(2, R) on a surface, Ergodic Theory Dynam. Systems 3 (1983), no. 3, 447–499. 29. F.-J. Turiel, An elementary proof of a Lima’s theorem for surfaces, Publ. Mat. 33 (1989), no. 3, 555–557. , Analytic actions on compact surfaces and fixed points, Manuscripta Math. 110 30. (2003), no. 2, 195–201. , 2006, personal communication. 31. University of California at Berkeley and University of Wisconsin at Madison Current address: 7926 Hill Point Road, Cross Plains, WI 53528, USA E-mail address: [email protected]

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https://doi.org/10.1090/crmp/050/10

Centre de Recherches Math´ ematiques CRM Proceedings and Lecture Notes Volume 50, 2010

PDE from the Point of View of Multiplier Ideals Joseph J. Kohn

Introduction I have always admired Raoul Bott both as a mathematician and as a human being. One of the things that has greatly inspired me is his ability for clear and lucid exposition. In this lecture I have tried to present the main ideas and to avoid technical detail in an attempt to emulate Bott’s style. My early recollections of Bott date back to the 1950s when I was a graduate student at Princeton. In those days Solomon Lefschetz and Marston Morse were dominant figures in mathematics. Bott often said that it is fruitful to combine the approaches to topology that Lefschetz and Morse had even though they seemed far apart. Several graduate students, including me, had desks in Lefschetz’s office and Lefschetz was actively interested in our work. One day Lefschetz asked me what I was working on and I told him that I was studying fiber bundles. He said that, in his opinion, too much emphasis was being placed on fiber bundles, they were a useful first step but they were based on the condition of constant rank whereas in actual practice the objects of greatest interest had singularities so that the constant rank condition was too restrictive. At the time I took this statement, along with many of Lefschetz’s pronouncements, with a grain of salt. Only much later did I appreciate the insight that it carries. ¯ I was working on the ∂-Neumann problem on pseudoconvex domains. In this problem the ellipticity breaks down and one cannot prove an elliptic estimate but sometimes, depending on the geometry of the domain, one can prove subelliptic estimates for the corresponding energy form. An elliptic estimate is a special case of a subelliptic estimate with ε = 1; subelliptic estimates with ε > 0 imply local regularity as will be explained below . When the domain is strongly pseudoconvex then the energy form satisfies a subelliptic estimate with ε = 12 . This estimate can be analyzed using the fiber bundles (i.e., the constant rank condition). The natural question is what happens if the domain is weakly pseudoconvex, that is, pseudoconvex but not strongly pseudoconvex. Then the constant rank condition breaks down and one is faced with the analysis of singularities, the 12 -estimate can no longer hold but but under some conditions one can prove an ε-estimate with ε > 0 which still implies the desired smoothness. It is the study of the weakly pseudoconvex case that leads to algebraic geometric constructions such as the work 2000 Mathematics Subject Classification. Primary 32W05; Secondary 32W10, 35N15. This is the final form of the paper. c
2010 c
2010 American American Mathematical Mathematical Society

81 79

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of D’Angelo on finite type (see [7]), Catlin on multitypes (see [4]), and mine on multiplier ideals (see [13]). The analysis on weakly pseudoconvex domains in C2 is comparatively simple; it depends in the behavior of a single vector field on the boundary of the domain (see [12]). The analysis in C2 shows that the crucial invariant is the maximum order of contact of a regular complex analytic curve to the boundary of the domain (in two dimensions this is equal to the maximum order of contact of all complex analytic curves). An example of Bloom and Graham (see [1]) shows that in C3 there is a pseudoconvex domain such that through a point of the boundary there is a complex analytic curve with infinite order of contact but that the order of contact of regular complex analytic curves through that point is at most six. To understand subellipticity in domains in Cn+1 , with n ≥ 3, one must take into account the order of contact of singular curves. This problem was addressed indirectly in [13] by introducing ideals of multipliers. This approach combined with the results of Diederich and Fornaess (see [9]) settled the case when the boundary of the domain is real analytic. A few years later Catlin, in his remarkable papers [3, 5], completely settled the problem using the the fundamental work of D’Angelo (see [7]). The method of using multipliers to analyze partial differential equations has been applied with great success in various contexts. Here we will discuss the uses of this method in proving local smoothness. This method has also been especially effective in proving global existence (see [20, 21]). In this introduction we will give a description of multiplier ideals, which are involved in the proof of local smoothness, in general terms. Then we will discuss the ¯ case of the ∂-Neumann problem on pseudoconvex domains and other applications in greater detail. For a valuable guide to the relevant geometry underlying the ¯ ∂-Neumann problem as well as further references see [8]. We are concerned with the study of partial differential operators of the form F u = (F1 u, . . . , Fp u), where 
u(x) = u1 (x), . . . , uq (x) q  with x ∈ Rn and Fj u = ajk Dk ui for j = 1, . . . , p, i=1

with ajk ∈ C ∞ (Rn ). In particular, we are interested in Sobolev norm estimates that deal with existence and regularity. The operator F is hypoelliptic if given U ⊂ Rn and s ∈ R there exists an  s
s ∈ R such that if u ∈ H s0 (Rn ) is a distribution such that F (u)|U ∈ Hloc (U ) then s (U ). Thus if F (u)|U ∈ C ∞ (U ) then u|U ∈ C ∞ (U ). The corresponding a u|U ∈ Hloc priori estimate is the following. Let ζ, ζ  ∈ C0∞ (U ) with ζ  = 1 in a neighborhood of the support of ζ and let s0 < s ∈ R; then there exist constants s , C ∈ R such that (1)

ζus ≤ C(ζ  F (u)s
+ us0 ),

for all u ∈ C ∞ (Rn ) ∩ H s0 (Rn ). Whenever this estimate can be applied to distributions u ∈ H s0 (Rn ) then it implies hypoellipticity. It is not always possible to pass from C ∞ (Rn ) to u ∈ H s0 (Rn ). In the situations discussed below it can be accomplished with the use of smoothing operators. To study (1) we use hypoelliptic multipliers, defined as follows. Definition. A hypoelliptic multiplier at x0 ∈ Rn for an operator F is a zero order pseudodifferential operator P on C0∞ (U ), where U is a neighborhood of x0 , such that for ζ, ζ  ∈ C0∞ (U ) with ζ  = 1 in a neighborhood of the support of ζ and

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s0 , s ∈ R there exist s = s (ζ, ζ  , s0 , s) and C = C(ζ, ζ  , s0 , s, s ) such that P ζus ≤ C(ζ  F (u)s
+ us0 ),

(2)

for all u ∈ C ∞ (Rn )∩H s0 (Rn ). We denote by P the set of all hypoelliptic multipliers at x0 . The following general properties of P will be used to find conditions under which 1 ∈ P so that (1) holds. • If P ∈ P then P ∗ ∈ P. • If σ(P ) and σ(P  ) denote the principal symbols of P and P  , further suppose that P ∈ P and that σ(P ) = σ(P  ) then P  ∈ P. • P is an ideal and the set of principal symbols of elements of P is an ideal. • Suppose P ∈ P satisfies (2) and suppose that P  is a pseudodifferential operator whose principal symbol satisfies |σ(P  )|m ≤ |σ(P )|; then P  ζus/m ≤ C(P ζus + us0 ) and hence P  ∈ P. • Suppose that (Pij ) is a q ×q matrix of zero-order pseudodifferential operators satisfying     q    P ζu (3) ij j  ≤ C(ζ F (u)s
+ us0 ),  s

j=1

∞

for all u ∈ C (R ) ∩ H (R ). Then det(Pij ) ∈ P. • Suppose that X is a real vector field such that there exists an open set U ⊂ Rn and C > 0 so that n q

n

Xu ≤ C(F (u) + us0 ),

(4) for all u ∈

s0

(C0∞ (U )q .

Suppose further that P ∈ P; then [X, P ] = XP − P X ∈ P. ¯ 1. The ∂-Neumann problem

Let Ω ⊂ Cn+1 be a bounded domain with a smooth boundary bΩ. Consider the equation ¯ = α, (5) ∂u  where u ∈ C ∞ (Ω) and α = αi d¯ zi with αi ∈ C ∞ (Ω). Suppose that αi ∈ L2 (Ω) and  ¯ = (6) ∂α (αi¯z − αj z¯ ) d¯ zi ∧ d¯ zj = 0. j

i

i 0 such that   2 2 ¯ Li ϕj  + ϕn+1 1 + cij ϕi ϕ¯j dS ≤ CQ(ϕ, ϕ), (11) i,j

i,j

bΩ

for all ϕ ∈ C0∞ (U ∩ Ω) ∩ Dom(∂¯∗ ). Remark. Using a partition of unity and appropriate weight functions it can be shown that the local estimate (11) implies the global estimate (12)

ϕ2 ≤ CQ(ϕ, ϕ),

for all ϕ ∈ C ∞ (Ω) ∩ Dom(∂¯∗ ). Then, using smoothing operators, one can show that ¯ this estimate also holds for all ϕ ∈ Dom(∂)∩Dom( ∂¯∗ ). This then implies that, given ¯ ∩ Dom(∂¯∗ ) a (0, 1)-form α ∈ L2 (Ω) there exists a unique (0, 1)-form ϕ ∈ Dom(∂) such that Q(ϕ, ψ) = (α, ψ), for all ψ ∈ Dom(∂¯∗ ). It then follows that ϕ ∈ Dom(
) and that
ϕ = α. Given P ∈ bΩ our goal is to find conditions such that for some neighborhood U of P there exist constants ε and C such that ϕ2ε ≤ CQ(ϕ, ϕ), holds for all ϕ ∈ C0∞ (U ∩ Ω) ∩ Dom(∂¯∗ ). The use of this estimate is given in the following result (see [17]). (13)

Theorem 3. Suppose that (13) holds on U ∩ Ω and that α ∈ L2 (Ω) is a (0, 1)s form such that the restriction of α to U ∩ Ω is in Hloc (U ∩ Ω) and further suppose s+2ε that
ϕ = α; then the restriction of ϕ to U ∩ Ω is in Hloc (U ∩ Ω). Corollary. Thus if, in the above, the restriction of α to U ∩Ω is in C ∞ (U ∩Ω) ¯ = α and then the restriction of ϕ to U ∩ Ω is in C ∞ (U ∩ Ω). Furthermore, if ∂u ∞ if u ⊥ H(Ω) then the restriction of u to U ∩ Ω is in C (U ∩ Ω).

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Next we examine (13) in the cases Ω ⊂ Cn+1 with n = 0, 1, and n ≥ 2. Dimension one. In this case n = 0 so that ϕ1 = 0 on U ∩ bΩ. Hence in this case we see that (13) implies that ϕ21 = ϕ1 21 ≤ CQ(ϕ, ϕ) so that ε = 1 and the problem in this case is just the Dirichlet problem, as used by Riemann to study ∂¯ in one complex variable. Dimension two. Here we have ϕ = ϕ1 ω 1 + ϕ2 ω 2 and ∂¯∗ ϕ2 = L1 ϕ1 2 + O(ϕ2 2 + ϕ2 ) ≤ CQ(ϕ, ϕ) 1

combining this with (13) we obtain ¯ 1 ϕ1 2 ≤ CQ(ϕ, ϕ). L1 ϕ1 2 + L Let X = Re(L1 ) and Y = Im(L1 ); then ¯ 1 ϕ1 2 ∼ Xϕ1 2 + Y ϕ1 2 . L1 ϕ1 2 + L Then we can use H¨ ormander’s result on the subellipticity of X 2 + Y 2 (see [11], this result will be discussed in Section 3) and we obtain the following (see [10, 12]). Theorem 4. If Ω ⊂ C2 and if P ∈ bΩ then (13) holds if and only if the Lie ¯ 1 restricted to P equals CTP (bΩ). This condition is algebra generated by L1 and L ¯ 1 ) such that p(L1 , L ¯ 1 )c11 (P ) = 0. equivalent to the existence of a monomial p(L1 , L Remark. Note that the above condition implies that there there is no nonsingular complex analytic curve through P which is contained in bΩ. In general if Ω ⊂ Cn+1 and if W ⊂ bΩ is an n-dimensional complex variety then W is nonsingular. Thus, the condition in the above theorem implies that there are complex curves that lie in bΩ. In case the dimension of W is less than n the situation is quite different. Consider the defining function r = Re(z3 ) + |z12 + z23 |2 in C3 . The singular complex curve given by z3 = z12 +z23 = 0 is contained in the real hypersurface r = 0. This curve cannot be detected by evaluating the Lie algebra of the vector fields in T 1,0 (U ∩ bΩ) + T 0,1 (U ∩ bΩ) on P . Dimension greater than 2. The general case is studied by means of multi¯ plier ideals. For the ∂-Neumann problem we will use only functions for multipliers. In the subsequent sections we will need general pseudodifferential operators. It follows from (13) that 2    n  cij ϕj  ≤ CQ(ϕ, ϕ),   j=1

1/2

for i = 1, . . . , n. Hence, as in the introduction (3), we have that det(cij ) is a multiplier:  det(cij )ϕ21/2 ≤ CQ(ϕ, ϕ), for all ϕ ∈ C ∞ (U ∩ bΩ) ∩ Dom(∂¯∗ ). 0

Definition 5. A germ f of a C ∞ function at P ∈ bΩ is a subelliptic multiplier if there exists a neighborhood U of P and constants ε and C such that f ϕ2ε ≤ CQ(ϕ, ϕ), for all ϕ ∈ C0∞ (U ∩ bΩ) ∩ Dom(∂¯∗ ). The set of all subelliptic multipliers at P is denoted by I(P ). (14)

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PDE FROM THE POINT OF VIEW OF MULTIPLIER IDEALS

Lemma 6. If f ∈ I(P ) then  2  n    (15) L (f )ϕ i i 

85 87

≤ CQ(ϕ, ϕ),

ε/2

i=1

for all ϕ ∈ C0∞ (U ∩ bΩ) ∩ Dom(∂¯∗ ). Definition 7. If I is an ideal of germs of C ∞ functions at P then √the real radical of I is the ideal of germs of C ∞ functions g at P , denoted by R I, and defined by √ R I = {g | ∃ m ∈ Z+ and f ∈ I such that |g|m ≤ |f |}. Definition 8. If f1 , . . . , fk are germs of C ∞ functions at P then we define the n × (n + k) matrix M(f1 , . . . , fk ) by ⎛ ⎞ ... cn1 c11 ⎜ .. .. ⎟ .. ⎜ . . . ⎟ ⎜ ⎟ ⎜ cn1 ⎟ . . . c nn ⎜ ⎟ M(f1 , . . . , fk ) = ⎜ ⎟ L (f ) . . . L (f ) 1 1 n 1 ⎜ ⎟ ⎜ .. . . .. .. ⎟ ⎝ . ⎠ L1 (fk )

. . . Ln (fk )

We denote by det M(f1 , . . . , fk ) the set of determinants of n × n submatrices of M(f1 , . . . , fk ). If J is a set of germs of C ∞ functions at P then we denote by det M(J) the union of all det M(f1 , . . . , fk ) with f1 , . . . , fk ∈ J. The set I(P ) has the following properties: • I(P

) is an ideal. • R I(P ) = I(P ). • r, det(cij ) ∈ I(P ). • If f1 , . . . , fk ∈ I(P ) then det M(f1 , . . . , fk ) ⊂ I(P ). Definition 9. The ideals Im (P ) are defined inductively as follows: I1 (P ) = √  R
r, det(cij ) and

 R
Im (P ), det M(Im (P ) . Im+1 = P ∈ bΩ is defined to be of finite ideal type if there exists m ∈ Z+ such that 1 ∈ Im (P ). It follows that Im (P ) ⊂ I(P ); hence we have the following result. Theorem 10. The subelliptic estimate (13) holds if P is of finite ideal type. Remark. The above implies, in particular, that there is no complex analytic curve V with P ∈ V ∈ bΩ. To see directly that finite ideal type implies this we argue by contradiction. Suppose that P ∈ V ∈ bΩ. Let P  ∈ V be a regular point of V close to P . Let L be a vector field in a neighborhood of P  such that L ∈ T 1,0 (bΩ), the restriction of L to V is in T 1,0 (V ), and LP
= 0. We write ¯ P
∈ CTP
(V ) and since CT (V ) is spanned by L and L ¯ L = ni ζi Li . Then [L, L]  in a neighborhood of P we conclude that n  cij ζi = 0, i

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for j = 1, . . . , n. Then, by Cramer’s rule, we have det(cij ) = 0 on V near P  . Hence if f ∈ I1 (P ) then f = 0 on V near P so that I1 (P ) ⊂ IP (V ), where IP (V ) denotes the ideal of germs of holomorphic functions that vanish on V . Hence L(f ) ∈ IP (V ) so that n  Li (f )ζi = 0 i

and, again using Cramer’s rule, we conclude that I2 (P ) ⊂ IP (V ). Proceeding inductively we conclude that Im (P ) ⊂ IP (V ) for all m, which is a contradiction. ¯ Next we consider the ∂-Neumann problem on (p, q)-forms. We choose P ∈ bΩ, U a neighborhood of P , and the bases for (0, 1) vector fields and forms as above. The operator
on (p, q)-forms is defined by (9). The problem is to study the equation ¯ = α, where θ and α are (p, q − 1)- and (p, q)-forms, respectively. Again, if ∂θ ¯ =α ¯ = 0 then the solution of
ϕ = α has the property that θ = ∂¯∗ ϕ satisfies ∂θ ∂α and that θ is orthogonal to the space of square integrable (p, q)-forms that are ¯ Proceeding as above we give orthogonal to the nullspace of ∂. Definition 11. If f1 , . . . , fk are germs of C ∞ functions at P then we define det M(f1 , . . . , fk ) to be the set of determinants of (n−q+1)×(n−q+1) submatrices of M(f1 , . . . , fk ). If J is a set of germs of C ∞ functions at P then we denote by detq M(J) the union of all detq M(f1 , . . . , fk ) with f1 , . . . , fk ∈ J. q

Definition 12. A germ f of a C ∞ function at P ∈ bΩ is a subelliptic multiplier for (p, q)-forms if the subelliptic estimate (14) holds for (p, q)-forms in C0∞ (U ∩ Ω) ∩ Dom(∂¯∗ ). The set of all subelliptic multipliers for (p, q)-forms at P is denoted by I q (P ). Note that the subelliptic estimate (14) holds for (p, q)-forms at P if and only if it holds for (0, q)-forms at P , and hence I q (P ) is independent of p. As above, the set I q (P ) has the following properties: • I q (P ) is an ideal. • R I q (P ) = I q (P ). • If f1 , . . . , fk ∈ I q (P ) then detq M(f1 , . . . , fk ) ⊂ I q (P ). Note that, in particular, detq M(0) ⊂ I q (P ) and that M(0) = (cij ). q (P ) are defined inductively as follows: Definition 13. The ideals Im

 R
detq M(0) I1q (P ) =

and q = Im+1

R


 q Im (P ), detq M(Im (P ) .

P ∈ bΩ is defined to be of finite ideal type for (p, q)-forms if there exists m ∈ Z+ q such that 1 ∈ Im (P ). q (P ) ⊂ I q (P ); hence we have the following result. It follows that Im

Theorem 14. The subelliptic estimate for (p, q)-forms (13) holds whenever P is of finite ideal type for (p, q)-forms. The above gives a sufficient condition for (13). In case the the defining function r is real analytic this condition is also sufficient (this is proved in [13] using a result of Diederich and Fornaess (see [9]).

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Domains with real analytic boundaries. The following result (see [13]) is proved using results of Diederich and Fornaess (see [9]). Theorem 15. If P ∈ bΩ and if a defining function of Ω is real analytic in a neighborhood of P then there exists a  germ of a complex q-dimensional analytic q (P ). variety V ⊂ bΩ at P if and only if 1 ∈ m Im Under the above hypothesis, using the implicit function theorem, we conclude that there exist holomorphic coordinates z1 , . . . , zn+1 with origin at P such that in a neighborhood of P there exists a real analytic function F , with dF (0) = 0, so that r(z1 , . . . , zn+1 ) = Re(zn+1 ) + F (z1 , . . . , zn+1 ). Proposition 16. If r is a real analytic defining function given by the above formula and if V is a germ of a complex analytic variety such that 0 ∈ V ⊂ bΩ then zn+1 ∈ I(V ). Proof. Let D = {t ∈ C | |t| < 1} and z : D → V with z(0) = 0. Expanding in a power series in t and t¯ we have ∞ 
 1 r z1 (t), . . . , zn+1 (t) = zn+1 (t) + aij ti t¯j = 0. 2 i≥0 j>0

Hence (d/ dt)m zn+1 (0) = 0 for all m so that zn+1 (t) ≡ 0,




Special domains. In these domains the subellipticity is determined by holomorphic multipliers. Let h1 , . . . , hN be germs of holomorphic functions at 0 ∈ Cn . Let Ω ⊂ Cn+1 be a domain whose defining function r near the origin is given by (16)

r(z1 , . . . , zn+1 ) = Re(zn+1 ) +

N 

|hk (z1 , . . . , zn )|2 .

k=1

Note that, from the above we can conclude that if V is a germ of a complex analytic variety at 0 with V ⊂ bΩ then V ⊂ W ⊂ bΩ, where W = {z ∈ Cn+1 | q zn+1 = h1 (z) = · · · = hN (z) = 0}. In this case the ideals Im (0) contain the ideals q of germs of holomorphic functions Jm (0) which are defined below. These ideals q q (0) if and only if 1 ∈ Im (0). We have: have the property that 1 ∈ Jm Definition 17. Given germs of holomorphic functions g1 , . . . , gk at the origin in Cn we define the n × (N + k) matrix N (g1 , . . . , gk ) by ⎛ ⎞ h1z1 . . . h1zn ⎜ .. .. ⎟ .. ⎜ . . . ⎟ ⎜ ⎟ ⎜hN z1 . . . hN zn ⎟ ⎜ ⎟ ⎟ N (g1 , . . . , gk ) = ⎜ ⎜ g1z1 . . . g1zn ⎟ ⎜ . ⎟ . .. .. ⎟ ⎜ .. . ⎜ ⎟ ⎝ gkz . . . gkzn ⎠ 1 . Denote by detq N (g1 , . . . , gk ) the set of determinants of all (n − q + 1) × (n − q + 1) submatrices of N (g1 , . . . , gk ). If G is an ideal of germs of holomorphic functions at

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0 then we denote by detq N (G) the union of all detq N (g1 , . . . , gk ) with gj ∈ G. The q (0) are defined inductively by ideals Jm


 detq N (0) J1q (0) = and q (0) = Jm+1


 q q Jm (0), detq N (Jm (0)) .

Theorem 18. The variety W = {z ∈ Cn+1 | zn+1 = h1 (z) = · · · = hN (z) = 0} q (0) for some m. has dimension less than q if and only if 1 ∈ Jm 2. CR manifolds A real hypersurface M ⊂ Cn+1 is a manifold of real dimension 2n + 1. The subbundle of the complexified tangent bundle of M consisting of (1, 0) vectors determines a CR structure on M . This gives rise to the tangential Cauchy-Riemann equations and the corresponding operator ∂¯b . To the operator ∂¯b we associate the ¯ Laplacian
b and we carry out an analysis which is analogous to the ∂-Neumann problem. Such an analysis was initiated for the case ε = 12 in [18]. Definition 19. Let M be a manifold of real dimension 2n + 1. A CR structure on M is given by a subbundle T 1,0 (M ) ⊂ CT (M ) satisfying the following properties. (i) The fiber dimension of T 1,0 (M ) is n. (ii) For each P ∈ M we have T 1,0 (M ) ∩ T 1,0 (M ) = {0}. (iii) If U ⊂ M is an open set and if L and L are vector fields on U with values in T 1,0 (M ) then [L, L ] = LL − L L has values in T 1,0 (M ). Note that if Ω ⊂ Cn+1 then bΩ is a CR manifold whose CR structure is given by T 1,0 (bΩ) = T 1,0 (Cn+1 ) ∩ CT (bΩ). Definition 20. Suppose that P ∈ M and γ = 0 is a 1-form on a neighborhood U of P , such that γ = −γ, which annihilates the sections over U of T 1,0 (M ) + T 0,1 (M ), where T 0,1 (M ) = T 1,0 (M ). Then the quadratic form on TP0,1 (M ) defined by ¯

(L, L ) → dγP , L ∧ L is the Levi form with respect to γ. If there exists a γ so that the Levi form is nonnegative then M is pseudoconvex at P and if it is positive then M is strongly pseudoconvex at P . From now on we will assume that M is compact and pseudoconvex and we will ¯  ≥ 0. choose γ so that dγP , L ∧ L Let A(M ) denote differential forms on M , let C(M ) = γ ∧ A(M ), and B(M ) =

A(M ) . C(M )

Then B(M ) are exterior forms on the space T 1,0 (M ) ⊕ T 0,1 (M ), which induces a direct sum decomposition B(M ) = B p,q (M ). The elements of B p,q (M ) are called the (p, q)-forms on M . The operator ∂¯b : B p,q (M ) → B p,q+1 (M ) is defined as follows. Denote the projection operators that correspond to the above direct sum by Πp,q : A(M ) → B p,q (M ). Then, if ϕ ∈ B p,q (M ), let ϕ ∈ A(M ) with ϕ ≡ ϕ mod C(M ) and define ∂¯b ϕ = Πp,q+1 dϕ . This definition is independent of the choice of ϕ .

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PDE FROM THE POINT OF VIEW OF MULTIPLIER IDEALS

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We fix a Riemannian metric on M which induces a Hermitian inner product on T 1,0 (M ) such that for each x ∈ M we have Tx1,0 (M ) ⊥ Tx0,1 (M ). Thus we get a volume element on M and an L2 inner product on the spaces of forms defined above. We denote by L2p,q (M ) the Hilbert space induced on B p,q (M ) and let ∂¯b also denote the maximal L2 extension of ∂¯b ; the domain of this extension will be denoted by Domp,q (∂¯b ). Further we denote by ∂¯∗ : B p,q (M ) → B p,q−1 (M ) the L2 adjoint of ∂¯ : B p,q−1 (M ) → B p,q (M ) whose domain is denoted by Domp,q (∂¯b ), and by
b : B p,q (M ) → B p,q (M ) the operator defined by
b = ∂¯∂¯∗ + ∂¯∗ ∂¯ with domain Domp,q (
b ). We set Hbp,q (M ) = {ϕ ∈ B p,q (M ) |
b ϕ = 0. Given α ∈ B p,q (M ) with α ⊥ Hbp,q (M ) we will study the equation
b ϕ = α,

(17) Hbp,q (M ),

with α ∈ L2p,q (M ), α ⊥ ϕ ∈ Domp,q (
b ), and ϕ ⊥ Hbp,q (M ). For this we need a priori estimates on Qb (ϕ, ϕ) + ϕ2 , where Qb (ϕ, ϕ) = ∂¯b ϕ2 + ∂¯b∗ ϕ2 since the solution ϕ of (17) is characterized by ϕ ⊥ Hbp,q (M ) and Qb (ϕ, ψ) = (α, ψ), for all {ψ ∈ Domp,q (∂¯b ) ∩ Domp,q (∂¯b∗ )}. The a priori estimate we will study is ϕ2ε ≤ C(Qb (ϕ, ϕ) + ϕ2 ).

(18)

Theorem 21. Suppose that there exist ε and C such that (18) holds for all ϕ ∈ B p,q (M ) then Hp,q (M ) ⊂ B p,q and is finite dimensional. Furthermore, if α ∈ L2 (M ) and α ⊥ Hp,q (M ) then there exists {ϕ ∈ Domp,q (∂¯b ) ∩ Domp,q (∂¯b∗ )} satisfying (17). Local regularity is given by the following result. Theorem 22. Suppose that U ⊂ M is open and that there exist ε and C such that (18) holds for all ϕ ∈ C0∞ (U ) ∩ B p,q (M ). Suppose further that α ∈ L2p,q (M ), α ⊥ Hbp,q (M ), α|U ∈ C ∞ (U ), and that ϕ satisfies (18). Then ϕ|U ∈ C ∞ (U ). If P ∈ M and U is a small neighborhood of P let L1 , . . . , Ln be a basis of the vector fields on U with values in T 1,0 (M ) which is orthonormal at each point. We choose γ so that |γ| = 1; then there is a unique vector field T on U such that T = ¯ i = 0, |T | = 1, and γ, T = 1. Then L1 , . . . , Ln , L ¯1, . . . , L ¯ n, T −T , T, Li = T, L is an orthonormal basis of CTx (M ) at each point of M . Let ω1 , . . . , ωn , ω 1 , . . . , ω n , γ be the dual basis. Then we can identify ω1 , . . . , ωn and ω 1 , . . . , ω n with bases of B 1,0 (M ) and B 0,1 (M ), respectively. The Levi form is then given by  dγ ≡ cij ωi ∧ ω j mod γ  ¯j ] = ¯1, . . . , L ¯ n ). If u ∈ C ∞ (M ) = and also by [Li , L cij T mod (L1 , . . . , Ln , L B 0,0 (M ) then on U we have n  ¯ ¯ i uω i , L ∂b u = 1

the L adjoint, ∂¯b∗ : B 0,1 (M ) → B 0,0 (M ) on U is of the form 2

∂¯b∗ ϕ = −

n 

Li ϕi + O(ϕ).

i

By integration by parts we obtain the identity:   ¯ i ϕj 2 + L (cij T ϕi , ϕj ) = Qb (ϕ, ϕ) + O(ϕ2 ), (19) i,j

i,j

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for all ϕ ∈ C0∞ (U ) ∩ B 0,1 (M ). Set S 2n = {ξ ∈ R2n+1 | |ξ| = 1}, let ψ + ∈ C ∞ (S 2n ) with supp(ψ + ) ⊂ {ξ ∈ R2n+1 | ξ2n+1 > 0, and ψ + = 1 in a neighborhood of (0, . . . , 0, 1). We also denote by ψ + an extension such that ψ + ∈ C ∞ (R2n+1 ) with ψ + (ξ) = ψ + (ξ/|ξ|) when |ξ| ≥ 1. Let M+ denote the set of all ψ + ∈ C ∞ (R2n+1 ) and let M− denote the set of all ψ − ∈ C ∞ (R2n+1 ), where ψ − ∈ C ∞ (R2n+1 ) is a function such that ψ − (ξ) = ψ + (−ξ) for some ψ + ∈ M− . Let ψ 0 ∈ C ∞ (S n ) with supp(ψ 0 ) ⊂ {ξ ∈ R2n+1 | −1 < −c < ξ2n+1 < c < 1 for some c ∈ (0, 1). Again extend to ψ 0 ∈ C ∞ (Rn+1 ) with ψ 0 (ξ) = ψ 0 (ξ/|ξ|) when |ξ| ≥ 1. The set of all such ψ 0 is denoted by M0 . Definition 23. Let P + denote the set of pseudodifferential operators on Rn+1 whose symbols are the elements of M+ , that is, Ψ+ ∈ P + if and only if  Ψ+ u(x) = eix
ξ ψ + (ξ)ˆ u(ξ) dξ. Rn+1

C0∞ (U ) +

then the (+)-microlocalization of u, denoted by u+ is a function of the If u ∈ form ζΨ u for some ζ ∈ C0∞ (U ) with ζ = 1 on a neighborhood of supp(u) and some Ψ+ ∈ P + . Analogously we define (−)-microlocalization and (0)-microlocalization. Let x1 , . . . , x2n+1 be local coordinates with origin P ∈ M such that at P we have   ∂ = Re(Li |P ), ∂xi P for i = 1, . . . , n,   ∂ = Im(Li |P ), ∂xi P for i = n + 1, . . . , 2n, and   √ ∂ = −1 T |P . ∂x2n+1 P On the support of ψ + we have ξ2n+1 ≥ const.(1 + |ξ|2 )1/2 , when |ξ| ≥ 1. Hence using G˚ arding’s inequality we obtain from (19)  1/2 + (cij Λ1/2 ϕ+ ϕj ) ≤ C(Qb (ϕ, ϕ) + ϕ2 ), (20) i ,Λ i,j

for all ϕ ∈ C0∞ (U ) ∩ B p,q (M ). Here the constant C depends on the support of ψ + and the diameter of U . Then we obtain the (+)-microlocalization estimate  det(cij )ϕ+ 21/2 ≤ C(Qb (ϕ, ϕ) + ϕ2 ), for all ϕ ∈ C0∞ (U ) ∩ B p,q (M ). To derive the (−)-microlocalization estimate we integrate by parts and obtain from (19) the identity   Li ϕj 2 − (bij T ϕi , ϕj ) = Qb (ϕ, ϕ) + O(ϕ2 ), (21) i,j

i,j

n where bij = δij k=1 ckk − cij . Note that since (cij ) for all ϕ ∈ is positive semidefine then (bij ) is also positive semidefinite. Furthermore, when n = 1 then (bij ) = b11 = 0 and when n > 1 then (bij ) ≥ (cij ). On the support of C0∞ (U ) ∩ B 0,1 (M ),

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ψ − we have ξ2n+1 ≤ −const.(1 + |ξ|2 )1/2 . Hence again using G˚ arding’s inequality we obtain from (21)  1/2 − (22) (bij Λ1/2 ϕ− ϕj ) ≤ C(Qb (ϕ, ϕ) + ϕ2 ), i ,Λ i,j

C0∞ (U )

∩ B p,q (M ). Here the constant C depends on the support of ψ − for all ϕ ∈ and the diameter of U . Then we obtain the (−)-microlocalization estimate  det(bij )ϕ− 21/2 ≤ C(Qb (ϕ, ϕ) + ϕ2 ), for all ϕ ∈ C0∞ (U ) ∩ B p,q (M ). To analyze the (0)-microlocalization we note that in the support of ψ 0 we have 2n |ξ| ≤ const. k=1 |ξk | and that  2n+1   √ 1 ∂ ∂ ∂ ¯ Li = + −1 ak (x) ), + 2 ∂xi ∂xi+n ∂xk k=1

where ak (0) = 0. Hence if u ∈ C0∞ (U ) and if the diameter of U is small we obtain the elliptic estimate    2n   ∂u 2 0 2 2   u 1 ≤ C  ∂xk  + u . k=1

Hence substituting this in (19) we obtain the following estimate for the (0)-microlocalization: ϕ0 21 ≤ C(Q(ϕ, ϕ) + ϕ2 ),

(23)

for all ϕ ∈ C0∞ (U ) ∩ B p,q (M ). ¯ Next, as in our treatment of the ∂-Neumann problem we define the n × (n + k) − matrices M+ (f , . . . , f ) and M (f , . . . , f ), where f1 , . . . , fk are germs of C ∞ 1 k 1 k b b functions at P , as follows. Definition 24.

⎛

c11 ⎜ .. ⎜ . ⎜ ⎜ cn1 + Mb (f1 , . . . , fk ) = ⎜ ⎜L1 (f1 ) ⎜ ⎜ . ⎝ ..

... .. . ... ... .. .

⎞ cn1 .. ⎟ . ⎟ ⎟ cnn ⎟ ⎟ Ln (f1 )⎟ ⎟ .. ⎟ . ⎠

L1 (fk ) . . . Ln (fk ) and

⎛

b11 ⎜ .. ⎜ . ⎜ ⎜ bn1 − Mb (f1 , . . . , fk ) = ⎜ ⎜L1 (f1 ) ⎜ ⎜ . ⎝ ..

... .. . ... ... .. .

⎞ bn1 .. ⎟ . ⎟ ⎟ bnn ⎟ ⎟ Ln (f1 )⎟ ⎟ .. ⎟ . ⎠

L1 (fk ) . . . Ln (fk ) + We denote by det Mb (f1 , . . . , fk ) and by detq M− b (f1 , . . . , fk ) the sets of determinants of q × q submatrices of M+ (f , . . . , f ) and M− 1 k b b (f1 , . . . , fk ), respectively. If ∞ J is a set of germs of C functions at P then we denote by detq M+ b (J) the union q

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q − of all detq M+ b (f1 , . . . , fk ) with f1 , . . . , fk ∈ J. The set det Mb (J) is defined analogously.

The multiplier ideals corresponding to microlocal subelliptic estimates are defined as follows. q+ Definition 25. The ideals Im (P ) are defined inductively as follows:

 R
detq M+ I1q+ (P ) = b (0)

 R
q+ q+ q+ Im (P ), detq M+ = Im+1 b (Im (P ) . P ∈ bΩ is, by definition, of finite (+) ideal type for (p, q)-forms if there exists q+ q− (P ). Similarly we define Im (P ) and P ∈ bΩ is, by m ∈ Z+ such that 1 ∈ Im definition, of finite (−) ideal type for (p, q)-forms if there exists m ∈ Z+ such that q− 1 ∈ Im (P ). and

Then if P is of finite (+) ideal type for (p, q)-forms then there exist a neighborhood U of P and constants ε and C such that ϕ+ 2ε ≤ C(Qb (ϕ, ϕ) + ϕ2 ), for all ϕ ∈ C0∞ (U ) ∩ B p,q . Similarly if P is of finite (−) ideal type for (p, q)-forms then there exist a neighborhood U of P and constants ε and C such that ϕ− 2ε ≤ C(Qb (ϕ, ϕ) + ϕ2 ), for all ϕ ∈ C0∞ (U ) ∩ B p,q . Then, combining this with (23), we conclude that if both of the above microlocal ideal types are finite then ϕ2ε ≤ C(Qb (ϕ, ϕ) + ϕ2 ), q+ (P ) ⊂ Im (P ) for all ϕ ∈ C0∞ (U )∩B p,q . Observe that if n > 1 and 1 ≤ q then Im so that if n = 1 the (−) ideal type is not finite and if n > 1 and the (+) ideal type is finite then the (−) ideal type is finite and thus the above subelliptic estimate holds. (n−q)−

Remark. When n = 1 the subelliptic estimate does not hold. Nevertheless if 1+ 1 ∈ Im (P ) and if the range in L2 (M ) of ∂¯b is closed then the solution of ∂¯b u = α with u ⊥ Hb0,0 (M ) has the property that u|U ∈ C ∞ (U ) whenever α|U ∈ C ∞ (U ) (see [14]). Further, if M is the boundary of a bounded pseudoconvex domain Ω ⊂ Cn then the range in L2 (M ) of ∂¯b is closed (see [15]). 3. Second-order PDE First we will indicate how multipliers can be used to analyze the H¨ ormander sum of squares operators. Let X1 , . . . , Xm be real vector fields in a neighborhood of the origin in Rn . Assume that these vector fields satisfy the bracket condition at 0. That is, suppose that the Lie algebra generated by the vector fields evaluated at 0 spans the tangent space. Theorem 26. Given real vector fields X1 , . . . , Xm satisfying the bracket condition at 0 then the operator m  (24) E= Xi2 i=1

is hypoelliptic in a neighborhood of 0.

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To prove this theorem in suffices to prove the following a priori estimate. There exists a neighborhood U of 0 and positive constants ε and C such that u2ε

(25)

≤C

m 

Xi u2 ,

i=1

for all u ∈ C0∞ (U ). Theorem 27. Suppose that (25) holds. Let f and u be distributions such that Eu = f ; it follows that if f |U ∈ C ∞ (U ) then u|U ∈ C ∞ (U ). More precisely if s+2ε s (U ) then u|U ∈ Hloc (U ). f |U ∈ Hloc The estimate (25) is proved by using multipliers as follows. Definition 28. A pseudodifferential operator P of order zero is a subelliptic multiplier corresponding to ε if there exists a neighborhood U of 0 an ε > 0 and a C > 0 such that   m P u2ε ≤ C Xi u2 + u2 , i=1

for all u ∈ to ε.

C0∞ (U ).

Denote by Pε the set of all subelliptic multipliers corresponding

The subelliptic multipliers have the following properties. • Pε is an ideal. • If P ∈ Pε then P ∗ ∈ Pε . • If ε ≤ 1 and P ∈ Pε then [Xi , P ], [Xi∗ , P ] ∈ Pε/2 . Proof. [Xi , P u2ε/2 = (Xi P u, Rε u) − (P Xi u, Rε u), where Rε is the pseudodifferential operator of order ε defined by Rε = Λε [Xi , P ]. Then |(Xi P u, Rε u)| = |(P u, Xi∗ Rε u)| ≤ |(P u, Rε Xi∗ u)| + |(P u, [Xi∗ , Rε ]u)| ≤ |(P u, Rε Xi u)| + |(P u, S ε u)| + O(u2 ) ≤ C(P uε Xi u + P uε u + u2 )   m 2 2 Xi u + u , ≤C i=1 ε

where S =

[Xi∗ , Rε ].

Similarly we show that   m ε 2 2 |(P Xi u, R u)| ≤ C Xi u + u . i=1

Finally note that

Xi∗ u

= Xi u + O(u), which concludes the proof.




• If R−1 is a pseudodifferential operator of order −1 then R−1 Xi ∈ P1 . • If A is a first-order differential operator such that R−1 A ∈ Pε then R−1 [Xi , A] ∈ Pε/2 .

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J. J. KOHN

The bracket condition implies that there exists a number N such that on a small neighborhood U of 0 we have  J ∂ = ai k FJk , ∂xi k≤N,Jk

where Jk = jk , jk−1 , . . . j1 with jh = 1, . . . , n, and the FJk are defined inductively as follows. FJ1 = Xj1 and FJk = [Xjk , FJk−1 ]. It then follows that Λ−1 ∂/∂xi ∈ Pε  with ε = 2−N +1 . Hence, since I = Λ−2 ( ∂ 2 /∂x2i + I), we have I ∈ P2−N +1 which implies (25). Suppose that X1 , . . . , Xm are complex-valued vector fields defined in a neighborhood of 0 ∈ Rn satisfying the bracket condition at 0, that is, the Lie algebra evaluated at 0 spans the complexified tangent space. We will study the hypoellipticity of the operator m  Xi∗ Xi , F =C i=1

where Xi∗ denotes the adjoint of Xi . We cannot proceed as in the real case because if P is a subelliptic multiplier it is not necessarily true that [Xi∗ , P ] is a subelliptic multiplier. First, if the Xi span the complexified tangent space at 0 ∈ Rn then F is elliptic. Next, in case that only brackets of order one are needed to span we have the following result. Theorem 29. If the Xi together with the [Xi , Xj ] span the complexified tangent space then there exits a neighborhood U of 0 and C > 0 such that n  (26) u21/2 ≤ Xi u2 , i=1

for all u ∈ C0∞ (U ). Hence if u and f are distributions such that F u = f and if the restriction of f to U is C ∞ then the restriction of u to U is C ∞ . More precisely, s+1 s (U ) then u|U ∈ Hloc (U ). if f |U ∈ Hloc To prove (26) one first proceeds as in the real case. If P = R−1 Xj then [Xi , P ]u21/2 ≤ |(Xi P u, R0 u)| + |(P Xi u, R0 u)| ¯ i u + Xi uP ∗ u1 + u2 ) ≤ C(P u1 X ≤ large const.

m  k=1

Xk u2 + small const.

m 

¯ k u2 + Cu2 X

k=1

and ¯ k ]u, u) + O(X ¯ k uu + uXk u + u2 ) ¯ k u2 = ([Xk , X X ¯ k u2 + Xk u2 ) + l.c.u2 . ≤ C(u21 + s.c.(X 2

Combining the above we deduce the estimate (26). When brackets of order two or higher are needed in the bracket condition the situation is quite different than in the real case. In fact, the operator F might not be hypoelliptic. When it is hypoelliptic and two or more brackets are needed, then typically F will “lose” derivatives. This means that there are distributions s (U ) and there exists s < s with u and f such that F (u) = f with f ∈ Hloc s
3 u ∈ / Hloc (U ). The following operators on R give examples of this phenomenon. Consider the pseudoconvex hypersurface in Mm ⊂ C2 defined by Re(z2 ) = |z1 |2m

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PDE FROM THE POINT OF VIEW OF MULTIPLIER IDEALS

95 97

then we identify Mm with R3 ≈ C × R so that the vector field Lm which spans T 1,0 (M ) is given by ∂ ∂ + i¯ z |z|2(m−1) . Lm = ∂z ∂t The following theorem was proved for m = 1 in [16] and for general m in [2]. Theorem 30. The operator Fm,k on R3 defined by ¯m + L ¯ m |z|2k Lm Fm,k = Lm L s is hypoelliptic and if for U ⊂ M open we have Fm,k u ∈ Hloc (U ) then u ∈ s−(k−1)/m s (U ) and further there exists a u such that Fm,k u ∈ Hloc (U ) and u ∈ / H
H s (U ) if s > s − (k − 1)/m.

¯ m and X2 = z k Lm then Fm,k = X1∗ X1 + X2∗ X2 and Note that if we set X1 = L that X1 , X2 satisfy the bracket condition of order k + m − 1. Further, note that when k = 0 then Fm,k “gains” 1/m derivatives, when k ≥ 1 it “loses” (k − 1)/m ¯ m , when derivatives. When k → ∞ we the limit of the Fm,k is the operator Lm L m = 1 this gives the example of Lewy (see [19]), that is, there are smooth functions ¯ m u = f is not solvable. If m → ∞ the limiting operator f for which the equation L is independent of t and thus not hypoelliptic. The following theorem, proved in [6], shows that the bracket condition does not imply hypoellipticity. Theorem 31. The operators Gm,k on R4 defined by Gm,k = Fm,k +

∂2 ∂s2

are not hypoelliptic. References 1. T. Bloom and I. Graham, On “type” conditions for generic real submanifolds of Cn , Invent. Math. 40 (1977), no. 3, 217–243. 2. A. Bove, M. Derridj, J. J. Kohn, and D. S. Tartakoff, Sums of squares of complex vector fields and (analytic-) hypoellipticity, Math. Res. Lett. 13 (2006), no. 5-6, 683–701. ¯ 3. D. Catlin, Necessary conditions for subellipticity of the ∂-Neumann problem, Ann. of Math. (2) 117 (1983), no. 1, 147–171. , Boundary invariants of pseudoconvex domains, Ann. of Math. (2) 120 (1984), no. 3, 4. 529–586. ¯ , Subelliptic estimates for the ∂-Neumann problem on pseudoconvex domains, Ann. of 5. Math. (2) 126 (1987), no. 1, 131–191. 6. M. Christ, A remark on sums of squares of complex vector fields, available at arXiv:math/ 0503506. 7. J. P. D’Angelo, Real hypersurfaces, orders of contact, and applications, Ann. of Math. (2) 115 (1982), no. 3, 615–637. , Several complex variables and the geometry of real hypersurfaces, Stud. Adv. Math., 8. CRC Press, Boca Raton, FL, 1993. 9. K. Diederich and J. E. Fornaess, Pseudoconvex domains with real-analytic boundary, Ann. Math. (2) 107 (1978), no. 2, 371–384. ¯ 10. P. Greiner, Subelliptic estimates for the ∂-Neumann problem in C2 , J. Differential Geometry 9 (1974), 239–250. 11. L. H¨ ormander, Hypoelliptic second order differential equations, Acta Math. 119 (1967), 147– 171. 12. J. J. Kohn, Boundary behavior of ∂ on weakly pseudo-convex manifolds of dimension two, J. Differential Geometry 6 (1972), 523–542.

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96 98

13. 14.

15. 16. 17. 18. 19. 20. 21. 22.

J. J. KOHN

¯ , Subellipticity of the ∂-Neumann problem on pseudo-convex domains: sufficient conditions, Acta Math. 142 (1979), no. 1-2, 79–122. , Estimates for ∂¯b on pseudoconvex CR manifolds, Pseudodifferential Operators and Applications (Notre Dame, IN, 1984), Proc. Sympos. Pure Math., vol. 43, Amer. Math. Soc., Providence, RI, 1985, pp. 207–217. , The range of the tangential Cauchy – Riemann operator, Duke Math. J. 53 (1986), no. 2, 525–545. , Hypoellipticity and loss of derivatives, Ann. of Math. (2) 162 (2005), no. 2, 943–986. J. J. Kohn and L. Nirenberg, Non-coercive boundary value problems, Comm. Pure Appl. Math. 18 (1965), 443–492. J. J. Kohn and H. Rossi, On the extension of holomorphic functions from the boundary of a complex manifold, Ann. of Math. (2) 81 (1965), 451–472. ? Lewy. A. M. Nadel, Multiplier ideal sheaves and K¨ ahler – Einstein metrics of positive scalar curvature, Ann. of Math. (2) 132 (1990), no. 3, 549–596. Y.-T. Siu, Effective very ampleness, Invent. Math. 124 (1996), no. 1-3, 563–571. , Effective termination of Kohn’s algorithm for subelliptic multipliers, available at arXiv:0706.4113.

Department of Mathematics, Princeton University, Fine Hall, Washington Road, Princeton, NJ 08544-1000, USA E-mail address: [email protected]

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https://doi.org/10.1090/crmp/050/11

Centre de Recherches Math´ ematiques CRM Proceedings and Lecture Notes Volume 50, 2010

Dirac Operator and K-Theory for Discrete Groups Paul Baum Dedicated with admiration and affection to the memory of Raoul Bott

1. Introduction K-theory and K-homology are (respectively) the cohomology theory and the homology theory associated to the Bott spectrum. In each case Bott Periodicity is the foundation of the theory. If X is a compact Hausdorff topological space, then K 0 X is the Grothendieck group of C-vector bundles on X. If X is a finite CW complex then K∗ X is the group of K-cycles. A precise definition of K-cycle will be given below. For the case of twisted K-homology, K-cycles are the D-branes of string theory. In a certain sense K-theory and K-homology are more direct and elementary than classical cohomology and homology. In particular if E is a vector bundle on X, then the inclusion of X in E via the zero section is a normally nonsingular map. Thus K 0 X is exhausted by normally nonsingular cocycles. Similarly, K∗ X is exhausted by tangentially nonsingular cycles. For classical cohomology and homology it is not true that every element can be obtained from a normally nonsingular (respectively, tangentially nonsingular) cocycle (respectively) cycle. A careful study of this issue led M. Goresky and R. MacPherson to the definition of intersection homology. Another feature of K-theory and K-homology is that equivariant versions of these two theories are immediate and canonical. In classical (co)homology there is some ambiguity about what is the “correct” definition of equivariant (co)homology. The aim of this note is to state the BC (Baum – Connes) conjecture in a way that is very close to the Atiyah – Singer Index Theorem. This will be done by using equivariant K-homology. The overall point of view is that for a discrete countable group Γ, tangentially nonsingular equivariant K-cycles are normally nonsingular for Cr∗ Γ and so give elements in the K-theory of Cr∗ Γ. Acknowledgment. It is a pleasure to thank M. F. Atiyah for very interesting conversations about K-homology. Also, there were many enjoyable and enlightening discussions with Raoul Bott. 2000 Mathematics Subject Classification. Primary 19K33; Secondary 19K35, 58J22. This is the final form of the paper. c
2010 c
2010 American American Mathematical Mathematical Society

99 97

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PAUL BAUM

2. K∗top (Γ) Let Γ be a countable discrete group. “Countable” means that as a set Γ is finite or countable. Let (M, E) be a pair such that: (1) M is a C ∞ manifold (without boundary) on which Γ is acting by a given smooth proper and co-compact action. Γ×M →M (2) For M , a Γ-equivariant Spinc -structure is given. (3) E is a Γ-equivariant C vector bundle on M . Remarks. “Smooth” in (1) means that each γ ∈ Γ acts on M by a diffeomorphism. “Proper” means that the map Γ×M →M ×M (λ, p) → (λp, p) is proper (i.e., the pre-image in Γ × M of any compact set in M × M is compact). “Co-compact” is that the quotient space Γ\M , with the quotient topology, is compact. (2) asserts that the structure group of T M , the tangent bundle of M , has been changed from GL(n, R) to Spinc (n) (n = dimR (M )). This has been done in an equivariant way so that the resulting spinor vector bundle on M is a Γ-equivariant C-vector bundle on M . For n ≥ 3 π1 (SOn ) = Z/2Z. Spin(n) is the unique nontrivial two-fold cover of SOn , i.e., Spin(n) is the universal covering group of SOn . Spinc (n) := S 1 ×Z/2Z Spin(n). In (3) E is a Γ-equivariant vector bundle in the usual sense [2]. In particular, each fiber of E is a finite dimensional C-vector space. (1) implies that the quotient space Γ\M is a compact Hausdorff orbifold. If Γ is not a finite group, then M is not compact. In any case, M is a finite dimensional paracompact Hausdorff and second countable C ∞ manifold. Two such pairs (M, E), (M  , E  ) are isomorphic if there exists a Γ-equivariant diffeomorphism f : M → M mapping M onto M  such that f preserves the given Γ-equivariant Spinc structures for M and M  , and as Γ-equivariant C-vector bundles on M , E and f ∗ (E  ) are isomorphic, where f ∗ (E  ) is the pull-back via f of E  . Two such pairs (M0 , E0 ), (M1 , E1 ) are bordant if there exists a pair (W, E) such that (1) W is a C ∞ manifold with boundary on which Γ is acting by a given smooth proper and co-compact action. Γ×W →W (2) For W a Γ-equivariant Spinc structure is given. (3) E is a Γ-equivariant C-vector bundle on W . (4) (∂W, E|∂W ) is isomorphic to the disjoint union (M0 , E0 ) ∪ (−M1 , E1 ).

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Remarks. In (4) ∂W is given the Spinc structure it receives from W . E|∂W is the restrction of E to ∂W . −M1 is M1 with the Spinc structure of M1 reversed. Consider the collection of all pairs (M, E) satisfying (1) – (3). On this collection let ∼ be the equivalence relation generated by three elementary steps: • bordism • direct sum – disjoint union • vector bundle modification Bordism. If (M0 , E0 ) is bordant to (M1 , E1 ), then (M0 , E0 ) ∼ (M1 , E1 ). Direct sum – disjoint union. If E and E  are two Γ-equivariant C vector bundles on M , then the disjoint union (M, E)∪(M, E  ) is equivalent to (M, E ⊕E  ). (M, E) ∪ (M, E  ) ∼ (M, E ⊕ E  ) Vector bundle modification. Suppose that F is a Γ-equivariant C ∞ Spinc vector bundle on M . Thus F is a C ∞ Γ-equivariant R-vector bundle on M whose structure group has been changed from GL(m, R) to Spinc (m) (m = dimR (Fp ), p ∈ M ). This change of structure group has been done in a C ∞ and Γ-equivariant way so that the resulting spinor bundle H is a C ∞ Γ-equivariant C-vector bundle on M . Assume that each fiber Fp of F is an even-dimensional R vector space. That is, dimR (Fp ) is even. Θ1 denotes the trivial R-line bundle on M Θ1 = M × R. In an evident way, Θ1 is a C ∞ Γ-equivariant R-vector bundle on M . The action of Γ on Θ1 is γ(p, t) = (γp, t) γ ∈ Γ, p ∈ M, t ∈ R F ⊕ Θ1 is then a Γ-equivariant C ∞ Spinc vector bundle on M with odddimensional fibers. S(F ⊕ Θ1 ) denotes the unit sphere bundle of F ⊕ Θ1 . Note that the projection ρ : S(F ⊕ Θ1 ) → M has even-dimensional spheres as fibers. The spinor bundle H for F is also the spinor bundle for F ⊕ Θ1 . Each fiber Hp of H is a module over the complexified Clifford algebra of Fp ⊕ R. Cliff C (Fp ⊕ R) ⊗ Hp → Hp This Clifford module structure determines an endomorphism α with α ◦ α = I of the pull-back via ρ, ρ∗ H, of H. If p ∈ M and ξ ∈ ρ−1 (p) then the fiber at ξ of ρ∗ (H) is Hp . α maps this fiber to itself by v → iξv

√ where ξv is the Clifford multiplication by ξ and i = −1. Since α ◦ α = I, ρ∗ H decomposes into the eigenspaces of 1 and −1. Denote this direct-sum decomposition of ρ∗ H by ρ∗ H = V + ⊕ V − . Let β be the dual vector bundle to V + . Thus for ξ ∈ S(F ⊕ Θ1 ) βξ = HomC (Vξ+ , C). Then: (8) (M, E) ∼ (S(F ⊕ Θ1 ), β ⊗ ρ∗ E).

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PAUL BAUM

Remarks. The restriction of β to each fiber of ρ : S(F ⊕ Θ1 ) → M is the Bott generator vector bundle of that even-dimensional sphere. Hence vector bundle modification is very closely connected to Bott periodicity and to the Thom isomorphsim in K-theory. In (8) the Spinc structure on S(F ⊕ Θ1 ) is determined by the Spinc structure of F plus the Spinc structure of M . To define K∗top (Γ) let {(M, E)} denote the collection of all pairs (M, E) satisfying (1) – (3). Then K∗top (Γ) := {(M, E)}/∼. K∗top (Γ) is an abelian group with respect to disjoint union: (M, E) + (M  , E  ) = (M ∪ M  , E ∪ E  ). The negative of (M, E) is (−M, E) where −M is M with the Spinc structure for M reversed. The zero element of K∗top (Γ) is given by any (M, E) which bounds. With j = 0, 1 let Kjtop (Γ) be the subgroup of K∗top (Γ) determined by all (M, E) such that every connected component of M has its dimension congruent to j modulo 2. Hence: K∗top (Γ) = K0top (Γ) ⊕ K1top (Γ) 3. IndΓ : Kjtop (Γ) → Kj Cr∗ Γ Cr∗ Γ is the reduce C ∗ algebra of Γ and K∗ Cr∗ Γ its K theory. A homomorpism of abelian groups IndΓ : Kjtop (Γ) → Kj Cr∗ Γ is defined as follows. With (M, E) as above let DE be the Dirac operator of M tensored with E. DE is a Γ-equivariant elliptic first order differential operator. DE is formally self-adjoint and is essentially self-adjoint. If S is the spinor bundle of M , then DE maps the C ∞ compactly supported sections of S ⊗ E to itself. DE : Cc∞ (S ⊗ E) → Cc∞ (S ⊗ E) If M is even-dimensional, then S splits into the even and odd half-spinor bundles. S = S0 ⊕ S1 The map ∞ ∞ (S 0 ⊗ E ⊕ S 1 ⊗ E) → CC (S 0 ⊗ E ⊕ S 1 ⊗ E) DE : CC

is off-diagonal with respect to this splitting DE =

0

− DE

+ DE

0

+ − where DE maps Cc∞ (S 0 ⊗E) to Cc∞ (S 1 ⊗E) and DE maps Cc∞ (S 1 ⊗E) to Cc∞ (S 0 ⊗ − + E). DE is the formal adjoint of DE . IndΓ (M, E) is then the index (taken as an element of K∗ Cr∗ Γ) of DE . To make this precise, let C[Γ] be the purely algebraic ∗ group algebra of Γ.
C[Γ] is a dense subalgebra of Cr Γ. An element of∞C[Γ] is a finite formal sum γ∈Γ λγ [γ] where λγ ∈ C. Suppose that s1 , s2 ∈ Cc (S ⊗ E). For γ ∈ Γ let γs2 be s2 translated by γ:

(γs2 )(p) = γ(s2 (γ −1 p)).

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Set

s1 (p), (γs2 )(p) dp

λγ = M

where  , is the Hermitian structure on S ⊗ E and dp is the measure on M determined by the Riemannian metric of M . Since s1 , s2 have compact support and the action
of Γ on M is proper, λγ is zero except for at most finitely many γ ∈ Γ. Thus γ∈Γ λγ [γ] ∈ C[Γ] ⊂ Cr∗ Γ. Cc∞ (S⊗E) can now be completed with respect to this Cr∗ Γ-valued inner product and the completion is a Hilbert Cr∗ Γ-module, denoted H(S ⊗ E). Converting DE to a bounded operator then produces an endomorphism T : H(S ⊗ E) → H(S ⊗ E) Cr∗ Γ

module H(S ⊗ E). Hence from DE an element of KK j (C, Cr∗ Γ) of the Hilbert has been constructed, where j is the dimension modulo 2 of M . For a C ∗ -algebra A there is the standard isomorphism [10] KK j (C, A) ∼ = Kj A. So from DE an element of Kj Cr∗ Γ has been obtained, and this element of Kj Cj∗ Γ is denoted IndΓ (M, E). The BC (Baum – Connes) conjecture for Γ can now be stated. Conjecture ([5, 6, 14]). If Γ is any countable discrete group, then IndΓ : Kjtop (Γ) → Kj Cr∗ Γ is an isomorphism for j = 0, 1. Remarks. If Γ is any countable discrete group for which BC is true, then the Novikov conjecture, the stable Gromov – Lawson – Rosenberg conjecture and the Kadison – Kaplansky conjecture [5, 6, 14] are true for Γ. Note that if BC is valid for Γ, then: (i) Every element in K∗ Cr∗ Γ is of the form IndΓ (M, E) for some (M, E) (surjectivity). (ii) Two pairs (M, E), (M  , E  ) have IndΓ (M, E) = IndΓ (M  , E  ) if and only if it is possible to pass from (M, E) to (M  , E  ) by a finite sequence of the three elementary moves: • bordism • direct sum – disjoint union • vector bundle modification In other words, IndΓ (M, E) is a complete invariant for the equivalence relation generated by these three elementary moves (injectivity). This assertion is nonobvious and nontrivial even when Γ is the trivial one-element group. Compare [7]. When Γ is the trivial one-element group, this statement is implied by Bott periodicity and has as an immediate corollary the Atiyah – Singer formula [3, 4] for the index of DE . index(DE ) = (ch(E) ∪ Td(M ))[M ] Observe that this formulation of BC is easily proved to be equivalent to the statement of BC in [5]. However, is this formulation of BC equivalent to BC as stated in [6] using EΓ, the universal example for proper actions of Γ? The answer to this question is provided by an isomorphism of abelian groups : K∗top (Γ) → K∗Γ (EΓ).

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PAUL BAUM

4.
: K∗top (Γ) → K∗Γ (EΓ) EΓ is the universal example for proper actions of Γ. From the homotopy theory point of view, EΓ is the universal space associated to (F) where (F) is the family of all finite subgroups of Γ. If X is a paracompact Hausdorff topological space on which Γ is acting Γ×X →X by a proper action with a paracompact Hausdorff quotient space Γ\X, then there exists a Γ-map (i.e., a continuous Γ-equivariant map) X → EΓ, and any two such maps are homotopic through Γ-maps. K∗Γ (EΓ) is the Kasparov Γ-equivariant K homology [10] of EΓ with Γ-compact supports. Recall [6] that a closed subset ∆ ⊂ EΓ is Γ-compact if (i) γp ∈ ∆ for all (γ, p) ∈ Γ × ∆, and (ii) the quotient space Γ\∆ is compact. Any such ∆ is locally compact Hausdorff. If ∆1 , ∆2 are two Γ-compact subsets of EΓ with ∆1 ⊂ ∆2 , then the evident map of C ∗ -algebras C0 (∆1 ) ← C0 (∆2 ) induces a homomorphism of abelian groups KKΓj (C0 (∆1 ), C) → KKΓj (C0 (∆2 ), C). KjΓ (EΓ) is, by definition, the direct limit: KjΓ (EΓ) :=

lim

∆⊂EΓ ∆ Γ-compact

KKΓj (C0 (∆), C)

Let (M, E) and DE be as above. Then DE determines an element, denoted [DE ], in KKΓj (C0 (M ), C) where j is the dimension modulo 2 of M . By the universal property of EΓ, there exists a Γ-map f : M → EΓ. Set ∆ = f (M ) ⊂ EΓ. Since Γ\M is compact, ∆ is Γ-compact. The map f: M →∆ ∗

gives a map of C algebras C0 (M ) ← C0 (∆) which induces a homomorphism of abelian groups KKΓj (C0 (M ), C) → KKΓj (C0 (∆), C). Consider the composition KKΓj (C0 (M ), C) → KKΓj (C0 (∆), C) → KjΓ (EΓ) where the second map is due to the inclusion ∆ ⊂ EΓ plus the definition of KjΓ (EΓ). Denote by (D, E) the element of KjΓ (EΓ) obtained by applying this composition to [DE ] ∈ KKΓj (C0 (M ), C). Theorem (P. Baum, N. Higson, and T. Schick [8]). Let Γ be a countable discrete group. Then : K∗top (Γ) → K∗Γ (EΓ) is an isomorphism.

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DIRAC OPERATOR AND K-THEORY FOR DISCRETE GROUPS

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Granted this theorem, one then observes that the diagram Kjtop (Γ) 0 (recall λ0 = 0),  k k 
∞

i −t∆i i −tλj (−1) tr e = (−1) e dim Ei (λj ) i=0

i=0 j=0

=

∞


e

−tλj

j=0


k

 i

(−1) dim Ei (λj ) =

i=0

k


(−1)i dim Ei (λ0 ).

i=0

Now Hodge Theory tells us that Ei (λ0 )  H (E, d), so i

k


(−1)i dim Ei (λ0 ) =

i=0

k





(−1)i dim H i (E, d) = Ind(E, d).

i=0

The heat operator e−t∆i is much more than trace class. In fact it is a smoothing operator, so there is a smooth section kti (x, y) of Hom(π2∗ Ei , π1∗ Ei ) over M × M , (where πj : M × M → M are the projections), so that for s ∈ C ∞ (Ei ),  −t∆i (s)(x) = kti (x, y)s(y) dy. e M i In particular, if ξj,l is an orthonormal basis of Ei (λj ), we have
i i kti (x, y) = e−tλj ξj,l (x) ⊗ ξj,l (y), j,l i i (x) ⊗ ξj,l (y) on s(y) is where the action of ξj,l i i i i (x) ⊗ ξj,l (y)(s(y)) = ξj,l (y), s(y)ξj,l (x), ξj,l

and ·, · is the inner product on Ei,y . It follows fairly easily that    −t∆i = tr kti (x, x) dx, tr e M

so we have that Ind(E, d) =

k


 (−1)

  tr kti (x, x) dx =

i M

i=0



k
  (−1)i tr kti (x, x) dx, M i=0

which is independent of t. For t near zero, the heat operator is essentially a local operator and so is subject to local analysis. It is a classical result, see for instance [10, 17], that it has an asymptotic expansion as t → 0. In particular, for t near 0, k



  (−1)i tr kti (x, x) ∼ tj/2 aj (x),

i=0

j≥−n

where the aj (x) can be computed locally (that is, in any coordinate system and relative to any local framings) from the ∆i . Each aj (x) is a complicated expression in the derivatives of the ∆i , up to a finite order which depends on j. Now we have 
  k
 i  j/2 Ind(E, d) = (−1) tr kt (x, x) dx = lim t aj (x) = a0 (x), M i=0

t→0

j≥−n

M

M

  since the quantity M i=0 (−1)i tr kti (x, x) dx is independent of t. It was the hope, first raised explicitly by McKean and Singer [23], that there might be some “miraculous” cancellations in the complicated expression for a0 (x) that would 

k

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THE LEFSCHETZ PRINCIPLE, FIXED POINT THEORY, AND INDEX THEORY

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yield the Atiyah – Singer integrand, that is, there would be a local index theorem. Atiyah – Bott – Patodi and Gilkey showed that this was indeed the case, at least for Dirac operators twisted by Hermitian bundles. For a particularly succinct proof, which shows that the cancellations are not at all “miraculous,” but rather natural, see [15]. Standard arguments in K-theory then lead directly to the full Atiyah – Singer index theorem.
In [8], Atiyah and Singer proved the index theorem for families of compact manifolds. For a heat equation proof of this result see [11], and for a heat equation proof in the case of foliations, (a theorem due to Connes [14]), see [21]. We end this section with an outline of this extension to foliations. One major problem is that, in general, a foliation F of a compact Riemannian manifold M will have both compact and noncompact leaves. This introduces a number of difficulties, for example noncompact leaves can limit on compact ones, causing fearsome problems with the transverse smoothness of the heat operators. Some of these difficulties can be solved by working on the graph G of F instead of F itself. G is constructed by associating to each point in M the holonomy cover of the leaf through that point, so G has a natural foliation Fs , and there is a natural covering map G → M which takes leaves of Fs to leaves of F . The possible noncompactness of the leaves of Fs causes problems with the spectra of the leafwise Laplacians, since on even the simplest noncompact manifold, namely R, the spectrum of the usual Laplacian is the interval [0, ∞). Thus we cannot think of the heat operators as nice infinite-dimensional diagonal matrices with entries going quickly to zero. However, these heat operators are still smoothing, so have nice smooth Schwartz kernels when restricted to any leaf. If G is Hausdorff, then it is almost (but not quite) a fiber bundle, and this implies that Duhamel’s formula for the derivative of a family of heat kernels extends to heat kernels defined on the leaves of Fs ; see [19]. The heat kernel 2 we are interested in is e−Bt , where Bt is the Bismut superconnection obtained using the metric on M scaled by the factor 1/t. Suppose that D : Cc∞ (E) → Cc∞ (E) is a generalized Dirac operator defined along the leaves of √F , and ∇ is a connection on the bundle E over M . Then, in simple cases, Bt = tD + ∇ pulled back to G by the natural map G → M . 2 One of the major results of [19] is that the Schwartz kernel of e−Bt is smooth in all its variables, both leafwise and in directions transverse to the leaves. This allows us to define a Chern character which takes values in the “de Rham cohomology of the space of leaves of F .” This is in quotes because the space of leaves is usually a badly behaved space, so has no de Rham cohomology in the usual sense. Fortunately, Haefliger [18] has defined a de Rham theory for foliations which plays this role rather well. The Chern character is then defined using a (super) trace Trs on Schwartz kernels of leafwise smoothing operators, and this trace takes values in the Haefliger forms. The proof of the families index theorem for foliations then has three steps. The 2 first is to show that Trs (e−Bt ) is a closed Haefliger form and its cohomology class is independent of t, that is, the Lefschetz principle still holds. This is the main result of [19]. The fact that it is closed relies heavily on Duhamel’s formula and the trace property of Trs , while the independence from the metric is a fairly standard 2 argument. The second step is to compute the limit as t → 0 of Trs (e−Bt ). The calculation for families of compact manifolds in [11] works just as well for foliations

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J. L. HEITSCH

because the operator e−Bt becomes very Gaussian along the diagonal as t → 0, so the result is purely local, and locally the foliation case looks just like the compact 2 families case. Of course the final step is to compute the limit as t → ∞ of Trs (e−Bt ), which is the main result of [21]. To do this, we adapt an argument of [10], and split the spectrum of D into three pieces, namely 0 and the intervals (0, t−a ) and [t−a , ∞), for judicious choice of a > 0. In [10], they are dealing with the compact families case, and so for t large enough, the spectrum in the interval (0, t−a ) is the empty set. We are not so lucky. To handle this interval, we must make some assumptions. The first is that the spectral projections P0 onto the kernel of D, and Pt associated to the interval (0, t−a ) are transversely smooth, that is, have Schwartz kernels which are differentiable in all directions, both leafwise and transversely to the leaves of Fs . The second is that the “density” of the spectrum of D in the interval (0, t−a ) is not too great, in particular, we assume that Tr(Pt ) is O(t−β ) for sufficiently large β. The interval [t−a , ∞) is somewhat easier to handle as here 2 Trs (e−Bt ) is decaying very rapidly as t → ∞. Then a rather lengthy and quite complicated argument shows that 2

lim Trs (e−Bt ) = Trs (e−(P0 ∇P0 ) ). 2

2

t→∞

The proof is finished by noting that P0 ∇P0 is a “connection” on the “index bundle,” 2 that is, on the kernel of D minus the cokernel of D, and that Trs (e−(P0 ∇P0 ) ) is just the Chern character of this index bundle. For an extension of this result, which significantly reduces the assumptions needed by using a more complicated operator of heat type, see [9]. 3. Atiyah – Bott fixed point theorem Our final application of the Lefschetz principle is to the proof of the very general Atiyah – Bott fixed point theorem, [1], for elliptic complexes. Let (E, d) be an elliptic complex over a compact manifold M . An endomorphism T of (E, d) is a collection of maps Ti : C ∞ (Ei ) → C ∞ (Ei ), such that Ti+1 ◦ di = di ◦ Ti . Then each Ti induces Ti∗ : H i (E, d) → H i (E, d), and we set L(T ) =

k


(−1)i tr(Ti∗ ).

i=0

We will be concerned only with the so-called geometric endomorphisms associated with a smooth map f : M → M . Now the problem with f is that, in general, it does not induce a map from sections of Ei to sections of Ei , but rather from Ei to the pull-back f ∗ Ei of Ei . To correct for this, we assume that we have bundle maps Ai : f ∗ Ei → Ei so that if we define Ti : C ∞ (Ei ) → C ∞ (Ei ) to be the composition f∗

i C ∞ (Ei ), C ∞ (Ei ) −→ C ∞ (f ∗ Ei ) −→

A

then Ti+1 ◦ di = di ◦ Ti . This is not a very strong restriction, as the examples below will show. At a fixed point x of f , the fibers of f ∗ Ei and Ei agree, so Ai,x : f ∗ Ei,x = Ei,x → Ei,x has a trace. Theorem 11 (Atiyah – Bott). Let (E, d) be an elliptic complex over the compact manifold M . Suppose that the graph of f : M → M is transversal to the

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diagonal ∆M ⊂ M × M . Let T be a geometric endomorphism associated to f , derived from bundle maps Ai : f ∗ Ei → Ei . Then
k (−1)i tr(Ai,x ) i=0 . L(T ) = | det(1 − f∗,x )| f (x)=x

Proof. Recall the spectral decompositions L2 (Ei ) = j Ei (λj ) associated to the heat operators of the elliptic complex (E, d), and denote by Pij the projection of L2 (Ei ) onto the λj eigenspace Ei (λj ). As λ0 = 0, Ei (λ0 )  H i (E, d), and we have the commutative diagram Ei (λ0 )

Pi0 Ti Pi0

/ Ei (λ0 )

  Ti∗ / H i (E, d). H i (E, d)   Thus, for each i, tr(Ti∗ ) = tr Pi0 Ti Pi0 , so L(T ) =

k


(−1)i tr(Ti∗ ) =

i=0

k


  (−1)i tr Pi0 Ti Pi0 .

i=0

Once again we bring the Lefschetz principle to bear in the form Proposition 12 (general fixed point Lefschetz principle). For all t > 0, L(T ) =

k


  (−1)i tr e−t∆i Ti e−t∆i .

i=0 j Proof. The spectral projections Pij : L2 (Ei ) → Ei (λj ) satisfy Pi+1 di = di Pij , so for each positive eigenvalue λj , we have the commutative diagram, with exact rows /0 / E0 (λj ) d0 / E1 (λj ) d1 / · · · dk−1 / Ek (λj ) 0 P0j T0 P0j

0

 / E0 (λj )

Pkj Tk Pkj

P1j T1 P1j d0

 / E1 (λj )

/ ···

d1

dk−1

 / Ek (λj )

/ 0.

Thus, for each λj > 0, k


  (−1)i tr Pij Ti Pij = 0.

i=0

As above, we then have k



 k ∞  −t∆i −t∆i 
 j j i −2tλj = (−1) tr e Ti e (−1) e tr Pi Ti Pi i

i=0

i=0

=

∞
j=0

=

k


j=0

e

−2tλj


k

  (−1) tr Pij Ti Pij



i

i=0

  (−1)i tr Pi0 Ti Pi0 = L(T ).

i=0

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J. L. HEITSCH

    Now tr e−t∆i Ti e−t∆i = tr Ti e−2t∆i , so L(T ) = lim

t→0

= lim

t→0

k


  (−1)i tr e−t∆i Ti e−t∆i

i=0 k


k
    (−1)i tr Ti e−2t∆i = lim (−1)i tr Ti e−t∆i . t→0

i=0

i=0

−t∆i

As t → 0, the Schwartz kernel of e becomes very Gaussian shaped along the diagonal ∆M ⊂ M × M . This means that given any neighborhood of ∆M , we may choose t so small that kti (x, y) is essentially supported inside that neighborhood. This is what we meant in the previous section when we said that for small t, e−t∆i is essentially a local operator. The operator Ti e−t∆i also has smooth Schwartz kernel ktT,i (x, y), given by kti (x, y)

ktT,i (x, y) = Ai,x kti (f (x), y). Thus, as t → 0, ktT,i (x, y) becomes very Gaussian shaped along the graph of f , Gr(f ) ⊂ M × M . Now,         tr Ti e−t∆i = tr ktT,i (x, x) dx = tr ktT,i (x, x) dx. ∆M ⊂M ×M

M

  By taking t sufficiently small we may force the support of tr ktT,i (x, y) to be essentially contained in any neighborhood of the graph of f we choose. Thus, in order to compute    k
  (−1)i lim tr ktT,i (x, x) dx , L(T ) = i=0

t→0

∆M ⊂M ×M

we may restrict to any neighborhood of Gr(f ) intersected with ∆M , provided that t is sufficiently small. But a neighborhood of Gr(f ) intersected with ∆M is just a neighborhood of the fixed points of f . The theorem now follows by a direct local computation.
This result has a large number of interesting and deep applications. For the classical complexes, it has the following beautiful specializations. This material is taken from [2], and for details the reader should consult that paper. • The de Rham complex. In this case, we immediately recover Theorem 7, the classical Hopf theorem. T ∗ This result extends to the tensor product of the de Rham complex (  M, d) with any flat vector bundle E over M . This yields the elliptic complex (E ⊗ ∗ T ∗ M, 1⊗ d). If f is a transversal map, and A : f ∗ E → E is a bundle map preserving the flat structure, then the Lefschetz number of the resulting endomorphism T is


det(1 − f∗,x ) L(T ) = tr(Ai,x ) . | det(1 − f∗,x )| f (x)=x

• The signature complex. Suppose that f is an isometry of a compact, oriented, Riemannian, 2n-dimensional manifold M , and that f is transversal to ∆M . Then at each fixed point

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x ∈ M , f∗,x : T Mx → T Mx is an isometry, so T Mx decomposes into an orthogonal direct sum of 2-dimensional subspaces, T Mx = E1 ⊕ E2 ⊕ · · · ⊕ En , which are preserved by f∗,x . The action of f∗,x on Ei,x is given by rotation through the angle θix , and the collection θ1x , . . . , θnx is called a coherent system of angles for f∗,x . The Atiyah – Bott theorem in this case takes the form L(f ) =


f (x)=x

i−n

n 

cot(θkx /2).

k=1

As interesting applications of this result we have the following. Theorem 13 (Atiyah – Bott). Let M be a compact, connected, oriented manifold (of positive dimension), and let f : M → M be an automorphism of prime power pl with p odd. Then f cannot have just one fixed point. Theorem 14 (Atiyah – Bott, Milnor). Let G be a compact Lie group of diffeomorphisms of a homology sphere M with fixed points x and y. Assume that the action is free except at x and y. Then the induced representations of G on T Mx and T My are isomorphic. • The Spin complex. Suppose that f is an isometry of a compact, oriented, Riemannian, 2n-dimensional manifold M , and that f is transversal to ∆M . For each fixed point x of f , denote by θ1x , . . . , θnx a coherent system of angles for f∗,x . Suppose further that M admits a Spin structure, and that f admits a lifting fˆ to this Spin structure. The Spin number Spin(fˆ, M ) is then given by
 Spin(fˆ, M ) = (fˆ, x)(i/2)n csc(θkx /2), f (x)=x

k

where (fˆ, x) = ±1, depending on the particular lifting fˆ. • The Dolbeault complex. For a compact, complex analytic manifold M , we actually have a family of elliptic complexes. The complexified cotangent bundle T ∗ M ⊗R C splits naturally into two complex sub-bundles, T ∗ M ⊗R C = T 1,0 ⊕ T 0,1 . In local holomorphic coordinates, T 1,0 is spanned by the dzi , while T 0,1 is spanned by the d¯ zi , so T 1,0 has a holomorphic structure, while T 0,1 has an anti-holomorphic structure. Set p,q   = p T 1,0 ⊕ q T 0,1 .   The operator d ⊗ 1 on C ∞ ( ∗ T ∗ M ⊗R C) = C ∞ ( ∗,∗ ) splits naturally as ¯ d ⊗ 1 = ∂ + ∂,     where ∂ maps C ∞ ( p,q ) to C ∞ ( p+1,q ), and ∂¯ maps C ∞ ( p,q ) to C ∞ ( p,q+1 ). For each p = 1, . . . , n = dimC M ,    ∂¯ ∂¯ ∂¯ 0 → C ∞ ( p,0 ) − → C ∞ ( p,1 ) − → ··· − → C ∞ ( p,n ) → 0, is an elliptic complex. The cohomology groups of this complex are denoted H p,∗ (M ). ¯ so it induces If f : M → M is a holomorphic map, then f ∗ commutes with ∂,

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122 124

J. L. HEITSCH

f p,∗ : H p,∗ (M ) → H p,∗ (M ), and for each p = 1, . . . , n, we have the Lefschetz number L(f p,∗ ). As a real vector space, T ∗ Mx  T 1,0 Mx , so it has a complex structure. If A is any C-linear map of T ∗ Mx , we may compute its C-trace, trC (A), and C determinant, detC (A). Suppose that f is transversal to ∆M . At a fixed point x of f , ∗ ∗ ∗ f x → T Mx is x: T M pjust∗ such a C-linear map (in fact an isomorphism), as is p ∗ p ∗ fx : T Mx → T Mx . With this in mind, the Atiyah – Bott fixed point formula now takes the form
trC ( p f ∗ ) x p,∗ . L(f ) = detC (1 − fx∗ ) f (x)=x

More generally, if E is any holomorphic vector bundle over M , there is the associated elliptic complex (note that p = 0 here) 0,0 1⊗∂¯ ∞ 0,1 1⊗∂¯ 0,n 1⊗∂¯ 0 → C ∞ (E ⊗C ) −−−→ C (E ⊗C ) −−−→ · · · −−−→ C ∞ (E ⊗C ) → 0. If f : M → M is a holomorphic map and A : f ∗ E → E is a holomorphic bundle map, then there is the associated endomorphism T of this complex, and we have the Lefschetz number L(T ). If f is transversal to ∆M , then
trC (Ax ) , L(T ) = detC (1 − fx∗ ) f (x)=x

∗

where for a fixed point x, Ax : f Ex = Ex → Ex . As interesting applications of the Atiyah – Bott Dolbeault fixed point formula we have: any holomorphic self map of a rational algebraic manifold must have a fixed point; when applied to S 1 actions, it implies the Weyl character formula; if ˆ ) = 0. This last is due to S 1 acts nontrivially on a Spin manifold M , then the A(M Atiyah and Hirzebruch, [4], and uses the extension mentioned below. Note that Theorem 11 extends to more general fixed point sets N . The map f∗,N : T M/T N → T M/T N is required to satisfy det(I − f∗,N ) = 0. The identity map satisfies this (vacuously), so this result contains the Atiyah – Singer index theorem as a special case. For a discussion of the history of this result, see [17], and for its extension to foliations, see [20]. 4. Afterword When I originally wrote this talk, my intention was to give the audience a feeling for some of the wonderful mathematics of Raoul Bott in a way that was both informative and entertaining (and as close to his style as I could). This necessitated a bit of loose play, sometimes called “fictionalized history,” but nothing too egregious, I hoped. I was honored to have Sir Michael Atiyah in the audience. At the end of the talk, he murmured, “Very nice. But it didn’t happen quite that way.” My only defense was to reply, “But it makes for such a good story this way.” References 1. M. F. Atiyah and R. Bott, A Lefschetz fixed point formula for elliptic complexes. I, Ann. of Math. (2) 86 (1967), 374–407. , A Lefschetz fixed point formula for elliptic complexes. II: Applications, Ann. of Math. 2. (2) 88 (1968), 451–491. 3. M. F. Atiyah, R. Bott, and V. K. Patodi, On the heat equation and the index theorem, Invent. Math. 19 (1973), 279–330.

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4. M. F. Atiyah and F. Hirzebruch, Spin-manifolds and group actions, Essays on Topology and Related Topics (Geneva, 1969) (A. Haefliger and R. Narasimhan, eds.), Springer, New York, 1970, pp. 18–28. 5. M. F. Atiyah and I. M. Singer, The index of elliptic operators on compact manifolds, Bull. Amer. Math. Soc. 69 (1963), 422–433. , The index of elliptic operators. I, Ann. of Math. (2) 87 (1968), 484–530. 6. , The index of elliptic operators. III, Ann. of Math. (2) 87 (1968), 546–604. 7. , The index of elliptic operators. IV, Ann. of Math. (2) 93 (1971), 119–138. 8. 9. M.-T. Benameur and J. L. Heitsch, Index theory and non-commutative geometry. II: Dirac operators and index bundles, J. K-Theory 1 (2008), no. 2, 305–356. 10. N. Berline, E. Getzler, and M. Vergne, Heat kernels and Dirac operators, Grundlehren Math. Wiss., vol. 298, Springer, Berlin, 1992. 11. J.-M. Bismut, The Atiyah – Singer index theorem for families of Dirac operators: two heat equation proofs, Invent. Math. 83 (1985), no. 1, 91–151. 12. R. Bott, Vector fields and characteristic numbers, Michigan Math. J. 14 (1967), 231–244. , A residue formula for holomorphic vector-fields, J. Differential Geometry 1 (1967), 13. 311–330. 14. A. Connes, Noncommutative geometry, Academic Press Inc., San Diego, CA, 1994. 15. E. Getzler, A short proof of the local Atiyah – Singer index theorem, Topology 25 (1986), no. 1, 111–117. 16. P. B. Gilkey, Curvature and the eigenvalues of the Laplacian for elliptic complexes, Advances in Math. 10 (1973), 344–382. , Invariance theory, the heat equation, and the Atiyah – Singer index theorem, Math. 17. Lecture Ser., vol. 11, Publish or Perish, Wilmington, DE, 1984. 18. A. Haefliger, Some remarks on foliations with minimal leaves, J. Differential Geom. 15 (1980), no. 2, 269–284. 19. J. L. Heitsch, Bismut superconnections and the Chern character for Dirac operators on foliated manifolds, K-Theory 9 (1995), no. 6, 507–528. 20. J. L. Heitsch and C. Lazarov, A Lefschetz theorem for foliated manifolds, Topology 29 (1990), no. 2, 127–162. , A general families index theorem, K-Theory 18 (1999), no. 2, 181–202. 21. 22. H. B. Lawson Jr. and M.-L. Michelsohn, Spin geometry, Princeton Math. Ser., vol. 38, Princeton Univ. Press, Princeton, NJ, 1989. 23. H. P. McKean Jr. and I. M. Singer, Curvature and the eigenvalues of the Laplacian, J. Differential Geometry 1 (1967), no. 1, 43–69. 24. J. W. Milnor and J. D. Stasheff, Characteristic classes, Ann. of Math. Stud., vol. 76, Princeton Univ. Press, Princeton, NJ, 1974. 25. R. S. Palais, Seminar on the Atiyah – Singer index theorem, with contributions by M. F. Atiyah, A. Borel, E. E. Floyd, R. T. Seeley, W. Shih, and R. Solovay, Ann. of Math. Stud., vol. 57, Princeton University Press, Princeton, NJ, 1965. 26. F. W. Warner, Foundations of differentiable manifolds and Lie groups, Grad. Texts in Math., vol. 94, Springer, New York, 1983. Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, 851 S. Morgan Street, Chicago, IL 60607-7045, USA and Mathematics Department, Northwestern University, 2033 Sheridan Road, Evanston, IL 60208-2730, USA E-mail address: [email protected] and [email protected]

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https://doi.org/10.1090/crmp/050/13

Centre de Recherches Math´ ematiques CRM Proceedings and Lecture Notes Volume 50, 2010

A New Look at the Theory of Levels John Cantwell and Lawrence Conlon

1. Introductory remarks This note is an initial step in what may prove to be a rather interesting project. It was inspired by a question, posed by B. Deroin, V. Kleptsyn and A. Navas, as to whether the smoothness hypothesis in the classical theory of levels [1, Chapter 8; 2; 3; 6; 13] might be weakened. This theory concerns foliations of codimension one, assumed to be smooth of class at least C 2 . One easily observes that the proofs go through for class C 1+Lipschitz and even for C 1+bounded variation (S. Egashira [10]). Taking a new look at the published proofs, we noticed that the smoothness hypothesis was used a great deal more than was strictly necessary and narrowed down its use to the proof of one key property: the existence of local minimal sets. Assuming this property, we use strictly C 0 arguments to develop an interesting and useful theory of levels, albeit with some surprising new properties (which, as an array of examples shows, do occur). A striking example of the elimination of smoothness requirements is our C 0 proof of a key result, Theorem 5, which seems to be interesting in its own right. We characterize our results as “a theory looking for a hypothesis.” That is, one seeks a sharp condition, whether analytic or topological, guaranteeing the Local Minimal Set Property (Definition 2). A personal note by the second author. I would like to express my gratitude to Raoul for sparking my interest in foliations. I was one of his early students, before his foliation period, and finished graduate school barely knowing what a foliation was. Years later, I attended his Mexico City lectures on “foliations and characteristic classes.” He spotted me in the audience and announced, to my dismay, that “Larry will write up the lecture notes.” But how could I refuse his Royal Decree? Happily, my friend Paul Schweitzer was also in attendance and suggested that we organize an informal seminar after each of Raoul’s lectures. Paul was already an expert in foliation theory (I believe he even understood the proof of Novikov’s theorem by that time) and the seminar mainly consisted of him explaining to a few of us both the basics and the finer points of what was going on. After returning to the States and inspired by the Godbillon – Vey note [11] that had just 2000 Mathematics Subject Classification. Primary 57R32; Secondary 57R30. This is the final form of the paper. c
2010 c
2010 American American Mathematical Mathematical Society

127 125

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appeared, Bott teamed up with Milnor for a seminar in which they constructed the whole panoply of exotic characteristic classes of foliations. Raoul wanted this new material included in the notes and invited me to visit him for a couple of days at the Institute so that he could explain it. Unfortunately, he had a family emergency that limited our visit to a couple of hours, which amazingly turned out to be enough (along with a few frantic phone calls). I believe this new material made the notes truly foundational and it was thrilling to be introduced to it almost on the ground floor. Subsequently, my interests diverged from the quantitative to the qualitative theory of foliations in which there was a worldwide burst of activity for the next three decades or so (and there is still a lot of life in this subject). My serious participation in this activity began when I joined up with John Cantwell for what has turned out to be nearly four decades of exciting collaboration. We are both profoundly grateful for these many years of personal and professional friendship and we both thank Raoul for inspiring it all. 2. Levels and smoothness Let M be a compact m-manifold with a transversely oriented, C 0 foliation F of codimension one (if ∂M
= ∅ we assume that the boundary components are leaves). It is known that there exists a transverse, one-dimensional foliation L to F. This is guaranteed by a deep theorem of L. Siebenmann [19] (for more details, see [14, Theorem 1.1.2]). Of course, if F is assumed to be tangent to a C 0 hyperplane field, the existence of the transverse foliation is trivial. One has a finite foliated atlas of M that is biregular for F and L [1, Proposition 5.1.4]. The differentiability class of the foliated manifold is the highest differentiability class of such an atlas. In fact, one is usually concerned only with the transverse differentiability class given by the local transformations of the L-coordinate. Definition 1. A local minimal set (LMS) is an F-saturated subset X ⊆ M which has an open, saturated neighborhood W such that X is a minimal set of F|W . Remark. Local minimal sets come in three flavors. A proper leaf is a LMS, an open, saturated set in which each of the leaves is dense is a LMS, and there remains the exceptional type, the nonempty intersections of which with open, transverse arcs are open subsets of a Cantor set. Definition 2. The foliated manifold has the LMS property if, for every open, ¯ ∩ W contains a saturated set W and every leaf L ⊂ W , the relative closure L minimal set of F|W . Theorem 3 (Duminy – Hector, Cantwell – Conlon). If the foliated manifold (M, F) is transversely of class C 1+Lipschitz , it has the LMS property. For a proof, see [1, Theorem 8.1.8]. It has been pointed out by Egashira [10] that this proof goes through if (M, F) is transversely piecewise C 1+bounded variation (C 1 with first derivatives of bounded variation). By a well-known example of G. Hector [1, Example 8.1.13], the LMS property the fails in general for C 0 foliations. According to Navas (private communication) √ example can be modified to be smooth of class C 1+α , 0 ≤ α < ( 5 − 1)/2. It is quite possible that one can improve this to 0 ≤ α < 1. But there are infinitely

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many classes smoother than these that do not imply C 1+Lipschitz and one should investigate the possibility that Theorem 3 might extend to some of these. The basic theorem in the classical theory of levels is the following. Theorem 4 (Duminy – Hector, Cantwell – Conlon). Suppose that the foliated manifold (M, F) as above is transversely of class C 1+Lipschitz . Then there is a filtration ∅ = M−1 ⊂ M0 ⊂ M1 ⊂ · · · ⊂ Mk ⊂ · · · ⊆ Mγ = M by compact, F-saturated sets indexed by the ordinals −1 ≤ k ≤ γ ≤ ω, where ω is the first infinite ordinal and, for each finite index k ≥ 0, M
k is the union of Mk−1 and the minimal sets of F|(M \ Mk−1 ). If γ = ω, then ω>k≥0 Mk is dense in M. Any
union of leaves in Mk \ Mk−1 is said to lie at level k. Leaves (if any) in Mω \ ω>k≥0 Mk are said to lie at level ω or simply at infinite level. If there are leaves at infinite level, there are uncountably many and they are rather inaccessible to intuition. A great deal is known about them, however. For instance, (1) such a leaf L does not have a proper side and its holonomy (generally nontrivial) has fixed points clustering on L from both sides, and so L has trivial infinitesimal holonomy; ¯ contains a finite number (nonzero) of LMS at level k; (2) for each integer k ≥ 0, L (3) every open, orientable surface is diffeomorphic to a leaf at infinite level in a smooth, transversely orientable foliation of each compact 3-manifold [4]. In foliations with the LMS property we will deduce an analogous theory of levels, although the highest level γ might be any ordinal strictly less than the first uncountable ordinal Ω. This difference is due to the fact that we do not assume that the theorem of R. Sacksteder on the holonomy of transversely Cantor F-saturated sets [17] holds for our foliations. There are other appeals to Sacksteder’s theorem in the published literature on levels, but in this note we replace them with more elementary arguments using only the LMS property. We sketch briefly two striking applications of the classical theory: (1) The Godbillon – Vey class gv(F) has fascinated foliators since the publication of [11], and has proven to be very reluctant to give up its secrets. Many researchers found special classes of foliations for which the invariant vanishes, working toward a conjecture of R. Moussu and F. Pelletier [16] and D. Sullivan [18] that the class vanishes if no leaf has exponential growth. G. Duminy proved this conjecture and then some, proving that the class vanishes if no leaf is resilient (a resilient leaf is one that captures itself by a holonomy contraction). Resilient leaves have exponential growth, but not vice-versa. Duminy isolates the “Godbillon” part of the class, localizing it to a σ-additive measure on the ring of F-saturated Borel sets, taking its values in the vector space Hom(H m−1 (M, F), R). The vanishing of this measure on M is equivalent to gv(F) = 0. Using the level hierarchy and the fact that no leaf is resilient, he decomposes M into a countable family of saturated Borel sets on each of which the Godbillon measure vanishes. (2) Let F be a foliation of class C 1+Lipschitz having all leaves proper. Since leaves at level ω cannot be proper, all leaves of F lie at finite levels, although there may be no finite upper bound to these levels. In each of these leaves, there is a compact submanifold, called a “juncture,” related to the holonomy of the leaf. The compactness of the junctures is a consequence of C 1+Lipschitz smoothness [1, Theorem 8.1.26]. Using this, the level hierarchy and the Epstein – Millett filtration

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[15], one proves that the foliated manifold (M, F) is homeomorphic to a C ∞ -foliated manifold [5, 7]. Compactness of the junctures is proven in a very similar way to Theorem 3, so these theorems may both extend to some differentiability classes strictly less than C 1+Lipschitz . If so, the results of this paper would also prove that such proper foliations are C ∞ -smoothable. 3. Local minimal sets in C 0 foliations The purpose of this section is to remove the smoothness hypothesis in [1, Theorem 8.3.2 and Corollary 8.3.3]. Specifically, we will prove the following. Theorem 5. For (M, F) of class C 0 and W an open, F-saturated set, the (possibly empty) union of minimal sets of F|W is relatively closed in W . What is more, if W is connected, either W is minimal in itself or there are at most finitely many exceptional minimal sets of F|W and the relatively closed leaves of F|W (if any) fall into finitely many homeomorphism types, the union of leaves of each homeomorphism type being relatively closed in W . For W = M , this is well known. Indeed, the assertion about exceptional sets appears in P. Dippolito’s paper [9] and the assertion about closed (compact) leaves is a celebrated theorem of A. Haefliger [12] (see also [1, Theorem 6.1.1]). 3.1. The exceptional sets. We continue using the notation introduced above, assuming that W is connected. Recall also that the transverse metric com of W is a (generally noncompact) manifold with finitely many boundary pletion W  border leaves of W . They are components. We will call the components of ∂ W  leaves of F, although two components of ∂ W might be identified as a single leaf in F. We often blur this distinction. Open, connected saturated sets fall into two classes. The foliated products have  ∼ W = L × I, where L is a border leaf and the interval fibers {x} × I, x ∈ L, are  subarcs of leaves of L. The other types will be called “fat pieces,” for which W has a compact, connected nucleus that is not a product and finitely many “arms” that are products. For this “octopus decomposition” and other details about the structure of open, saturated sets, see [1, Section 5.2]. Theorem 6. There are at most finitely many exceptional minimal sets of F|W . The proof falls into some lemmas. Lemma 7. If M1 , M2 are minimal sets of F|W and U is a component of W \ M1 that is a foliated product with M2 ⊂ U , then M2 is homeomorphic to each border leaf of U and is a proper leaf. Proof. Indeed, at least one border leaf L of U lies in M1 . Since M2 cannot accumulate on this leaf, it meets any fiber {x} × I in a point y closest to L. Thus,  and the assertion is proven.
y is a fixed point of the total holonomy group of U We will assume that there are infinitely many exceptional minimal sets of F|W and deduce a contradiction. Choose one and call it M0 . Set W0 = W . The complement W0 \ M0 consists of foliated products and finitely many fat pieces [1, Corollary 5.2.9]. By Lemma 7, at least one fat piece, call it W1 , will contain infinitely many of the exceptional minimal sets. Choose one exceptional minimal

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set M1 ⊂ W1 . The complement W1 \ M1 consists of foliated products and finitely many fat pieces. Again, by Lemma 7, at least one of the fat pieces, call it W2 , will contain infinitely many of the exceptional minimal sets. Choose one exceptional minimal set M2 ⊂ W2 . Continue in this way to obtain exceptional minimal sets M0 , M1 , M2 , . . ., and fat pieces Wk ⊃ Mk where Wk is a component of Wk−1 \Mk−1 and W1 ⊃ W2 ⊃ · · ·. k → ∞ as k → ∞, The strategy is to show that the number of components of ∂ W which contradicts the following. Lemma 8. If the number of biregular charts in the foliated atlas is n, then an arbitrary open, connected, F-saturated set W has at most 4n border leaves. Proof. This is implicit in the proof of [1, Lemma 5.2.5]. Indeed, if W is a foliated product, the assertion is trivial and, otherwise, each border leaf contains a “special” border plaque [1, Definition 5.2.4]. It is clear that there are at most 4n such plaques (2n if ∂M = ∅).
Fix a biregular cover {Ui }ni=1 for the foliation. Local coordinates are chosen so that the closure of each biregular neighborhood is of the form U i = Dm−1 × [0, 1] (where Dm−1 denotes the closed unit disk in Rm−1 ). Here, of course, the disks Dm−1 × {t} are closed plaques of F and the intervals {x} × [0, 1] are closed plaques of L. Let ck be the number of biregular neighborhoods Ui with U i ⊂ Wk , 1 ≤ k < ∞. Then c1 ≥ c2 ≥ c3 ≥ · · ·. In fact, since none of the Wk are products, all ck > 0 by the following lemma. Lemma 9. If an open, connected F-saturated set V contains no U i , then V is a foliated product. Proof. Let L denote the one-dimensional foliation of V induced by L. It is clear that every leaf of L is an arc with endpoints in distinct leaves of ∂ V . Thus, V is fibered by intervals and V must be a foliated product.
Choose K so large that ck = c∗ > 0 is constant, for all K ≤ k < ∞. Lemma 10. For all k > K, Wk−1 \ Mk−1 has only one component that is a fat piece, namely Wk . Proof. The component Wk was chosen to be a fat piece. Since k > K, k − 1 ≥ K and ck = ck−1 = c∗ . Then, Lemma 9 and the fact that no component of Wk−1 \ Mk−1 other than Wk contains one of the biregular neighborhoods Ui with U i ⊂ Wk−1 implies that every component of Wk−1 \ Mk−1 other than Wk is a product.
K+r has at least r components. Thus, the number Lemma 11. If r ≥ 1, ∂ W of border leaves of Wk becomes unbounded as k → ∞. Proof. Note that the border leaves of a foliated product accumulate on exactly the same leaves of F. If a component of Wk−1 \ Mk−1 which is a product has one k−1 , then L would accumulate on border leaf in Mk−1 and one (call it L) in ∂ W Mk−1 , clearly a contradiction. Thus, if k > K, the only component of Wk−1 \ Mk−1 that can have a border leaf F in common with Wk−1 is Wk . Clearly this will happen k−1 . if and only if Mk−1 does not accumulate on F from within W

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Now WK+1 must have at least one border leaf L1 ⊂ Mk and MK+1 cannot accumulate on L1 . Thus, L1 survives as a border leaf of WK+2 which also has a border leaf L2 ⊂ MK+1 . Proceeding inductively, we see that WK+r has at least r
distinct border leaves L1 , L2 , . . . , Lr . As remarked above, this establishes Theorem 6. 3.2. Proper leaves. Recall that a leaf L is proper if and only if there is an open, connected F-saturated set W such that L is a relatively closed leaf of F|W [1, Proposition 8.1.19]. We turn to the proof of the following. Theorem 12. The (relatively) closed leaves of F|W fall into finitely many homeomorphism types, the union of leaves of each homeomorphism type being relatively closed in W . Hereafter, the relatively closed leaves will simply be called the closed leaves of F|W . The following is a special case of Theorem 12 and is fundamental to the proof of that theorem. Lemma 13. If W is a foliated product, then all closed leaves of F|W are homeomorphic and the union of these leaves is relatively closed in W . Proof. Let L ⊂ W be a relatively closed leaf. There are two cases to consider:  or it does. either L does not accumulate on a component of ∂ W In the first case, let y ∈ L and note that y is a fixed point of the total holonomy  , acting on the fiber I = {x}×I containing y. The group G of the foliated bundle W set of all such fixed points in int I is relatively closed in that open interval and so  and the leaves through those points are homeomorphic to the components of ∂ W unite to form a relatively closed subset X ⊆ W . If there were also a closed leaf L accumulating on a boundary component F , then X would have a closest leaf to F and L would accumulate also on that leaf, contradicting that L is relatively closed in W .  but, of course, In the second case, L accumulates on both components of ∂ W  on nothing else in W . Indeed, if L does not accumulate on one of the components , L ¯ ∩ W has a closest leaf to F and we are in the situation of the first case. F of ∂ W  Then W = W \ L is a foliated product and contains all the closed leaves of F|W   are homeomorphic to L and we are reduced other than L. The components of ∂ W to the first case.
Suppose that W is a fat piece. Let F1 , F2 , . . . , Fr be closed leaves of F|W and let V be a component of W \ F1 \ F2 \ · · · \ Fr . Recall that the inclusion i : V → M extends canonically to a continuous map ˆı : V → M which is one-to-one on each component of ∂ V , but may identify some of these components pairwise. Let X ⊆ W denote the union of all closed leaves of F|W . Lemma 14. If X ∩ V is relatively closed in V , then X ∩ ˆı(V ) is relatively closed in W . Proof. The set ˆı(∂ V ) is the union of the border leaves of V . The set-theoretic boundary of V in M is the closure of this set. These border leaves will consist of some of the border leaves of W and some of the leaves Fj . Thus, the set-theoretic

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boundary of V in W is the union F∗ of some of the relatively closed leaves Fj . Since X ∩ V is relatively closed in V and ˆı(V ) ∩ W = V ∪ F∗ is relatively closed in W ,
we conclude that X ∩ ˆı(V ) = (X ∩ V ) ∪ F∗ is relatively closed in W . Lemma 15. If V as above is a foliated product and if X ∩ V
= ∅, then this set is relatively closed in V and is exactly the union of all closed leaves of F|V . Proof. Clearly every leaf of X ∩ V is a closed leaf of F|V . In particular, since at least one of the border leaves of V is a closed leaf of F|W , if X ∩ V
= ∅ we are in the first case of the proof of Lemma 13. Thus, each closed leaf of F|V cobounds foliated products with leaves of X ∩ V and so must be a closed leaf of F|W . The final assertion of Lemma 13 completes the proof.
If W has at most finitely many relatively closed leaves, we are done, so assume that there are infinitely many. If one of them, say L0 , does not separate W0 = W , set W1 = W0 \ L0 , an open saturated set having two copies of L0 among the 1 . Consider the closed leaves of F|W0 other than L0 , noting components of ∂ W that they are also closed leaves of F|W1 . If one of them, say L1 , does not separate W1 , set W2 = W1 \ L1 , noticing that, since L1 cannot accumulate on either side 2 contains two copies of L0 as well as two copies of L1 . Continuing in of L0 , ∂ W this way, we obtain open, connected, saturated sets W0 ⊃ W1 ⊃ · · · ⊃ Wk such k is at least 2k. By Lemma 8, we obtain the that the number of components of ∂ W following. Lemma 16. The above process terminates. That is, for some K ≥ 0, every closed leaf of F|W in WK separates WK . If WK is a foliated product, then K = 1 and the closed leaves of F|W in W1 1 . Further, by Lemma 15, the union of closed do not individually cluster on ∂ W leaves of F|W1 is exactly X ∩ W1 . We are in the first case of Lemma 13 which, with Lemma 14, implies that that the closed leaves of F|W are mutually homeomorphic and that their union is relatively closed in W . We assume that WK is fat and note the following. Lemma 17. If U is a fat piece and if L is a closed leaf of F|U that separates U , then at least one of the two components of U \L, say U  , is fat. Also, if L ⊂ U  is a closed leaf of F|U that separates U , it is relatively closed in U  and separates that set. Let F ⊂ WK be a closed leaf of F|W and set WK+1 = WK \ F . In each fat component of that set, select a closed leaf of F|W (if there is one) and set WK+2 equal to the complement in WK+1 of the union of those leaves. If there is no such leaf, the process terminates. Otherwise, select closed leaves of F|W in each fat component of WK+2 (if there are any) and continue in this way. If one runs out of fat pieces that can be further divided, the process terminates and all the remaining closed leaves of F|W lie in product components. Theorem 12 then follows from Lemma 13, Lemma 14 and Lemma 15. If the process does not terminate, set 1 = n0 ≤ n2 ≤ · · · ≤ nk ≤ · · ·, where nk is the number of fat components of WK+k . Lemma 18. For some integer l ≥ 0, nk is constant, l ≤ k < ∞. Proof. Indeed, the fat components of WK+k are disjoint and, by Lemma 9, each contains at least one of the closed charts U i , 1 ≤ i ≤ n.


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Let V be one of the fat components of WK+l and suppose that V contains a leaf F0 that is relatively closed in W . By Lemmas 17 and 18, one of the components of 0 is F0 and the other V \ F0 is a foliated product U0 . One of the components of ∂ U is a leaf F in ∂ V . Let U = {Uα }α∈A be the family of all foliated products obtained in this way and having F as a border leaf. We linearly order U by inclusion.  a Lemma 19. The ordered set U contains a maximal element U . We call U maximal collar of F in V . (We also call the completions of product components of WK+l maximal collars of their border leaves). Proof. If the index set A is finite, we are done. Otherwise, for each α ∈ A, let α other than F . Set U =
Fα be the component of ∂ U α∈A Uα , an open, connected    F-saturated set. Let F ⊂ V be a component of ∂ U other than F . If some Fα accumulates on F  from within V , then F must also accumulate on F  from within  with one end x ∈ F  , V , also, a clear impossibility. Now, if J is an arc of L in U then points xα ∈ J ∩ Fα must cluster at x as α ranges over A and, by the previous remark, we can take xα to be the closest point of J ∩ Fα to x. Thus, the Dippolito semistability theorem [1, Theorem 5.3.4] implies that, for suitable α ∈ A, Fα and F  cobound a foliated product. It follows that F  and F cobound a foliated product. Since V is not a product, F  is not a border leaf of V and so F  ⊂ V ⊂ W . Since F  cobounds foliated products with closed leaves Fα of F|W , F  must be such a closed leaf and our assertion is proven.
The proof of Theorem 12 is now easy. Indeed, the closed leaves of F|W fall into finitely many of the border leaves of the components of WK+l , together with the K+l . These maximal closed leaves in maximal collars of some of the leaves of ∂ W collars conform to the first case in the proof of Lemma 13, and the theorem follows from Lemmas 14 and 15. Proof of Theorem 5. For the case that W is connected, the theorem reduces to Theorems 6 and 12. For the nonconnected case, let X ⊆ W be the union of minimal sets, {xk }∞ k=1 ⊂ X a sequence that converges to a point x ∈ W . Applying what we have already proven to the component of W containing x, we see that x ∈ X.
4. Levels We continue to assume all previous properties of the foliated manifold (M, F), except for the compactness of M . For this, we substitute the hypothesis that M is an open, connected, saturated subset of a compact, foliated manifold (N, G) and F = G|M . It is clear that Theorem 5 continues to hold. Also, the LMS property continues to make sense and will, in fact, be a consequence of the LMS property for (N, G). This moderately generalized context is useful for the proof of Theorem 31, stated but not proven in the next section. A useful theory of levels will be founded on a level filtration in the following sense. Definition 20. A level filtration for (M, F) is a filtration ∅ = M−1 ⊂ M0 ⊂ M1 ⊂ · · · ⊂ Mα ⊂ · · · ⊆ Mγ = M by closed, F-saturated sets, indexed by the ordinals −1 ≤ α ≤ γ < Ω, where Ω is the first uncountable ordinal. For each nonlimit ordinal −1 < α ≤ γ, Mα is the

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union of Mα−1 and the
minimal sets of F|(M \ Mα−1 ). If α ≤ γ is a limit ordinal, Mα is the closure of 0≤β 0, let Ωµ := Ωµ M = {γ ∈ ΩM : γ|[−2µ,2µ] ⊂ Bε }. If µ is sufficiently small, the support of α will be contained in Ωµ . Let η : (Ωµ × [0, 1], Ωµ × {0, 1}) → (Ωµ × [0, 1], Ωµ × [0, 1] ∩ Uµ ) be the inclusion map. We next exhibit a smooth family of maps {Ψr }, 0 ≤ r ≤ 1: Ψr : (Ωµ × [0, 1], Ωµ × [0, 1] ∩ Uµ ) → (Ωµ × [0, 1], Ωµ × [0, 1] ∩ Uµ ). with the following properties:

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(i) Ψ0 is the identity map. (ii) The image of Ψ1 lies in (Ωµ × [0, 1], Ωµ × {0, 1}). (iii) The image of F ∩ Ωµ × [0, 1] under each Ψr lies in F ∩ Ωµ × [0, 1]. (iv) Ψ1 ◦ η is homotopic to the identity map. Thus Ψ1 is homotopy inverse to the inclusion map η, and (v) η∗ ◦ Υ = (∩τ ) ◦ η∗ . The map Ψr is constructed on Ωµ M × [0, 1], starting with the identity map Ψ0 as follows: First replace (γ, s) by (h◦γ, s), where we have used geodesic coordinates in a neighborhood of x1 to get a map h : M → M that is the identity outside of B2ε , and takes Bε to x1 . Next replace (h ◦ γ, s) by (h ◦ γ, s), where in h ◦ γ the restriction of h ◦ γ to [0, min{s, 2µ}] and to [max{s, 1 − 2µ}] have been replaced by minimal geodesics. Finally replace (h ◦ γ, s) with (h ◦ γ, g(s)) where g : [0, 1] × [0, 1] is the (simplest piecewise linear map with: 0 → 0, µ → 0, 2µ → 2µ, 1−2µ → 1−2µ, 1 − µ → 1, 1 → 1. To see (iii), note that if s ∈ [2µ, 1 − 2µ], (h ◦ γ, g(s)) = (h ◦ γ, s), so γ(s) = x1 implies h ◦ γ(s) = x1 ; if s ≤ 2µ, then h ◦ γ is geodesic on [0, s], so γ(s) = x1 implies h ◦ γ ≡ x1 on [0, s] and h ◦ γ(g(s)) = x1 since g(s) ≤ s. We leave it to the reader to fill in the homotopy from Ψ0 to Ψ1 , and to check properties (ii), (iv). For each k with 2 ≤ k ≤ m, there is a smooth family φkr , 0 ≤ r ≤ 1, of diffeomorphisms φkr : Uk → M with φk0 : Uk → Uk the identity map, and φk1 : Uk → U1 covering the identity map on N . The family of maps φkr can be extended to a smooth family of diffeomorphims k φr : M → M , with φk0 : M → M the identity, and so that each φkr is the identity ouside Bε . Let κ := κµ : [0, 1] → [0, 1] be piecewise linear with κ(r) = 0 on [0, 12 − µ] ∪ 1 [ 2 + µ, 1] , and with κ(r) = 1 in a neighborhood of { 12 }. Let θ1/2→s : R/Z → R/Z be the (simplest) piecewise linear map [0, 1] → [0, 1] taking 0 → 0, 12 → s, and 1 → 1. For s ∈ (0, 1) define κs : [0, 1] → [0, 1] by κs = θ1/2→s ◦ κ ◦ θs→1/2 so that κs (r) = 1 for r in a neighborhood of {s}. For s ∈ (0, 1), r ∈ [0, 1], and 1 < k ≤ m define Φkr : ΩM × (0, 1) → ΩM by Φkr (γ, s)(t) = φkrκs (t) ◦ γ(t). The loop Φkr (γ, s) has the same endpoints as γ, and coincides with γ except when t is near s. When r = 0 the loop coincides with γ for all t, but when r = 1, k Φk1 (γ, s)(s) = φk1 ◦ γ(s). Note that if γ(0) = φk1 ◦ γ(s), i.e., if (γ, s) ∈ F(0,1) , k k then (Φ1 (γ, s), s) ∈ F(0,1) . Moreover (γ, s) → (Φ0 (γ, s), s) is the identity map ΩM × (0, 1) → ΩM × (0, 1): Φk0 (γ, s) = (γ, s); For each fixed r, the map Φkr : ΩM × (0, 1) → ΩM × (0, 1) by (γ, s) → (Φkr (γ, s), s)

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has a continuous inverse   (γ, s) → (Φkr )−1 (γ, s), s . and (since φkr is the identity ouside Bε ) takes (Ωµ × (0, 1), Ωµ × (0, 1) ∩ Uµ ) to itself. Because the following diagram commutes: 

F(0,1) M  O

/ ΩM × (0, 1) O

(Φk 1 ,id)

 k F(0,1) M

(Φk 1 ,id)

/ ΩM × (0, 1).

and Φk1 takes the normal bundle to Fk M to the normal bundle to FM (preserving orientation), if α ∈ H∗ (Ωµ M ), and η∗ α is the image of α in Hi+1 (Ωµ M × [0, 1], (Ωµ M × [0, 1]) ∩ Uδ ), then (using (v) for the first equality) η∗ Υ(α) = (η∗ α) ∩ τ = (Φk1 )∗ (((Φk1 )−1 ∗ η∗ α) ∩ τk ).   k −1  But because (Φ1 ) , id is homotopic, through the family (Φkr )−1 , id) of maps (Ωµ × (0, 1), Ωµ × (0, 1) ∩ Uδ ) → (Ωµ × (0, 1), Ωµ × (0, 1) ∩ Uδ ), to the identity map, we have

(18)

(19)

(Φk1 )−1 ∗ η∗ α = η∗ α;

we thus (using (18), (19)) have     η∗ Υ(α) = (Φk1 )∗ (η∗ α) ∩ τk = (Φk1 )∗ η∗ (α ∩ τk ) = η∗ (Φk1 )∗ (α ∩ τk ) Since η∗ is injective, this implies that (20)

Υ(α) = (Φk1 )∗ (α ∩ τk ).

k It is easy to see that the map c ◦ Φk1 on F(0,1) is (up to homotopy) the map k k  F(0,1) → ΩM × ΩM defined as follows: if (β, s) ∈ F(0,1) M , and if β|[0,s] = χ, and β|[s,1] = ς (in which case χ(0) = x1 , χ(1) = xk , ς(0) = xk , ς(1) = x1 , then c ◦ ((Φk1 )(β, s)) = (χ · ν, ν −1 · ς), where ν is the path φkr |(x1 ),(0 ≤ r ≤ 1) from xk to x1 . The hypothesis that π1 (N ) = 0 implies that the loop f ◦ ν at y is trivial. Thus f∗ (χ · ν) is homotopic to f∗ χ, and f∗ (ν −1 · ς) is homotopic to f∗ ς; thus

(21)

c∗ f∗ (Φk1 )∗ (α ∩ τk ) = c∗ f∗ (α ∩ τk )

using (20) and (21) we now have (17): c∗ f∗ (α ∩ τk ) = c∗ f∗ Υ(α) and Lemma 3 and thus the theorem are proved.

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LOOP PRODUCTS ON CONNECTED SUMS OF PROJECTIVE SPACES

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Boundary of the lift of the cycle Γ to the free loop space of the blowup . There is a There is not a lift of the space Γ of circles on M to a cycle on ΛM   natural chain Γ on ΛM lifting Γ which we will describe. We will need the following facts, which are well-known and/or easily verified: Let z ∈ KP d . The projective space P (Tz (KP d )) ≈ KP d−1 is naturally diffeomorphic to the set F(z ) of points in KP d at maximal distance from z ; the diffeomorphism is induced by the map that takes a vector V ∈ Tz (KP d ) to the point of intersection of F(z ) with a geodesic with initial vector V . If y = z and y ∈ / F(z ), then there is a unique arclength-parameterized minimal geodesic  (0) from the complement of γy from z to y ; thus we get a map y → γy d d d−1 F(z ) ∪ {z } in KP to P (Tz (KP )) ≈ KP . This map extends smoothly to a map φ : KP d − {z } → F(z ) ≈ PK (Tz (KP d )) ≈ KP d−1 that fixes F(z ). The image under φ of a circle in KP d − {z } is a circle in KP d−1 . This follows from the fact that φ lifts to a map Φ : S q(d+1)−1 − h−1 (z ) → S qd−1 that takes geodesics to geodesics and commutes with the Hopf maps. The map Φ is the composition of orthogonal projection from Kd+1 to the subspace Kd orthogonal to Kz, followed by radial projection onto the sphere S qd−1 ⊂ Kd . The image under φ of a circle in KP d meeting z is a point in KP d−1 (since every nontrivial circle in KP d meeting z lies on a unique KP 1 in KP d ). However φ does not give a continuous map from circles on KP d to circles in KP d−1 . Let v, w ∈ Tz (S q(d+1)−1 ) be unit vectors. Since T is a subbundle of the trivial bundle T Kd+1 we can also view tangent vectors to the sphere S q(d+1)−1 as points in Kd+1 , and unit tangent vectors as elements of S q(d+1)−1 ⊂ Kd+1 . Thus u, v ∈ S q(d+1)−1 and u, v ⊥ z. Assume that w ⊥Kz, v ∈ / Kz, and w ⊥ v. Let 0 ≤ ρ ≤ε with q(d+1)−1 starting at the point (z+ρw)/ 1 + ρ2 ε small, fixed. The geodesic on S with initial vector v descends by the Hopf map to a circle γ(v, ρw) on KP d . When ρ = 0, γ(v, ρw) is a circle beginning at z with initial vector V := Hopf ∗ (v), and φ(γ(v, 0) − {z } = {Φ(v) } in KP d−1 , but when ρ = 0 the circle φ(γ(v, ρw)) is qd−1 that is the intersection the image under the Hopf map of the great circle  on S qd−1 2 with the plane through Φ((z+ρw)/ 1 + ρ ) = Φ(w) and Φ(v ), of S independent of ρ. It is not hard to see that every circle on KP d sufficiently close to γ(v, 0) is an (orientation-preserving) reparameterization of some γ(v, ρw), and that v and  ρw are unique. Here ρw is determined by the condition that the point (z+ρw)/ 1 + ρ2 is the point on the inverse image of the given circle under the Hopf map closest to the point z on the sphere. If v ∈ / Kz, and if ε = ε(v) is sufficiently small, then the circle γ(v, ρw) is closest to z at t = 0. Given a nontrivial circle through z ∈ KP d , say that it is the image under Hopf of the great circle starting at z with initial vector v. (Then v ⊥ z and v∈ / Kz; fix ρ > 0, z and v.) Since circles on KP d−1 starting at Φ(v) are in 1 – 1 correspondence with oriented real 2-dimensional subspaces of Kd containing Φ(v), every circle on KP d−1 starting at Φ(v) is of the form φ(γ(v, ρw)) for a unique w ∈ Kd+1 with the properties w ⊥ Kz, and w ⊥ v. Note that φ(γ(v, ρw)) is independent of ρ so long as ρ > 0.  carry the metric induced by the product metric on M × F(z ). Let Let M ∗ Γ be the nontrivial circles on M not meeting z . There is a unique continuous

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N. HINGSTON

∗ of Γ∗ to ΛM . As mentioned earlier, the energy is unbounded on this lift. lift Γ However, because of the fact that the image under φ of a circle on the projective space M = KP d is a circle on F(z = KP d−1 , the energy will be bounded if we ∗ proportional to arclength. Assume this is done. reparameterize all the loops in Γ Let  Z = Γ z ∗E ΓE ⊂ ΛM where ΓE is the space of circles on E, and “∗E ” indicates that we take the Pontrjagin product of all composable pairs. We will assume that the loops in Z are parameterized proportional to arclength. It should be clear that the limit points of ∗ all lie in S 1 × Z (where S 1 acts by reparameterization), since each limit point Γ is a continuous loop whose projection onto each factor is a circle. It is not hard to see (using the parameterization (v, w, ρ) (w ⊥ Kz, w ⊥ v, and ρ < ε(v)) near  has the structure of a manifold with  of Γ ∗ in ΛM the boundary) that the closure Γ boundary  = S 1 · Z. ∂Γ ∗ ; we do not know in general It is not clear to us how to “fill in” the rift in Γ if the nonnilpotent homology class Θ = [Γ] (representing the highest-dimensional ). In the example below it is not. homology at level l) is in the image of H∗ (ΛM The case M = S n The theorem is nontrivial when M is a sphere, and gives information about the loop cohomology product on H ∗ (ΛKP d , Λ=0 KP d ), and also information about the map Π∗ : H ∗ (ΛS n , Λ=0 S n ) → H ∗ (ΛKP d , Λ=0 KP d ). For spheres and complex or quaternionic projective spaces, it is easy to keep track of the homology/cohomology because the rank is at most 1 in each dimension. For  = P = CP d (d = n/2), it is easy to check that the example when M = S n and M ) class Θm ∈ H3n−2 (ΛM ) is not in the image of Π∗ ; and that Π∗ Θp = 0 ∈ H2n (ΛM (since Θp has a representative cycle whose image factors through the based loop space ΩM , which has trivial homology in dimension 2n). However the cohomology class ) , Λ=0 M ω  := Π∗ (ωm ) ∈ H n−1 (ΛM is nonnilpotent (by the theorem), but level nilpotent, since (as follows from a simple dimension argument) in the standard metric on CP d cr( ω ) = l;

cr( ω
ω  ) = 3l.

Thus ω 
ω  is a nontrivial product that would not be detected by the methods of [7], where the authors computed the associated graded ring for the energy filtration. References 1. A. L. Besse, Manifolds all of whose geodesics are closed, Ergeb. Math. Grenzgeb., vol. 93, Springer, Berlin, 1978. 2. R. Bott, On the iteration of closed geodesics and the Sturm intersection theory, Comm. Pure Appl. Math. 9 (1956), 171–206. 3. G. E. Bredon, Topology and geometry, Grad. Texts in Math., vol. 139, Springer, New York, 1993. 4. M. Chas and D. Sullivan, String topology, available at arXiv:math/9911159.

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5. R. L. Cohen, J. D. S. Jones, and J. Yan, The loop homology algebra of spheres and projective spaces, Categorical Decomposition Techniques in Algebraic Topology (Isle of Skye, 2001), Progr. Math., vol. 215, Birkh¨ auser, Basel, 2004, pp. 77–92. 6. R. L. Cohen, J. R. Klein, and D. Sullivan, The homotopy invariance of the string topology loop product and string bracket, J. Topol. 1 (2008), no. 2, 391–408. 7. M. Goresky and N. Hingston, Loop products and closed geodesics, available at arXiv:0707. 3486. 8. M. Morse, The calculus of variations in the large, Amer. Math. Soc. Colloq. Publ, vol. 18, Amer. Math. Soc., Providence, RI, 1996. Reprint of the 1932 original. 9. D. Sullivan, Open and closed string field theory interpreted in classical algebraic topology, Topology, Geometry and Quantum Field Theory (Oxford, 2002), London Math. Soc. Lecture Note Ser., vol. 308, Cambridge Univ. Press, Cambridge, 2004, pp. 344–357. 10. W. Ziller, The free loop space of globally symmetric spaces, Invent. Math. 41 (1977), no. 1, 1–22. Mathematics & Statistics, The College of New Jersey, PO Box 7718, Ewing, NJ 08628-0718, USA E-mail address: [email protected]

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https://doi.org/10.1090/crmp/050/17

Centre de Recherches Math´ ematiques CRM Proceedings and Lecture Notes Volume 50, 2010

Equivariant Cohomology and Reflections James A. Bernhard Abstract. In this paper, we investigate and reflect upon an observation by Raoul Bott that certain stationary phase expansions on even-dimensional spheres have only finitely many terms. In order to explain this phenomenon, we describe a generalization of the Duistermaat – Heckman theorem to functions arising from some particular higher-dimensional equivariantly closed forms.

Of the many mathematical lessons I learned from Raoul Bott, perhaps my favorite is contained in his gentle admonition, “Do not hide your ignorance.” Throughout my time as a graduate student under Bott, he demonstrated to me the unspoken parts of this precept: instead of hiding your ignorance, explore it, probe it, learn from it. Those familiar with Bott either personally or through his work can certainly see evidence of his adherence to this principle. He had a keen ability to ask simple yet deep and insightful questions, questions that would focus attention on just the right unknown aspect of a problem. I experienced this myself time and again when I would discuss with him something new that I had learned or figured out. He would often respond by asking some simple question such as, “What does this look like on S 2 ?”, upon which I would have to rethink my entire understanding (or lack thereof) of the topic, and which would frequently lead to new insight into the problem at hand. My dissertation [2] and the open problems related to it, which I now describe, stemmed from one of these seemingly simple questions posed by Bott. For a discussion of these results in further detail, see [3]. The question from Bott that gave rise to my dissertation is: Why is the stationary phase expansion of the “height function” on an even-dimensional sphere finite? More specifically, suppose we denote the unit sphere in R2n+1 by S 2n = {(x1 , x2 , . . . , x2n , z) | x21 + x22 + · · · + x22n + z 2 = 1}, so that the last coordinate z can be thought of as “height” on the sphere. A direct computation shows that if dV is the standard volume form on S 2n , then

1  (1) etz dV = c etz (1 − z 2 )n−1 dz = etz Pz (t−1 ), S 2n

−1

z=±1

2000 Mathematics Subject Classification. Primary 55N91; Secondary 53D20. This is the final form of the paper. c
2010
2010 American c American Mathematical Mathematical Society

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J. A. BERNHARD

where c ∈ R and P−1 and P1 are polynomials. Now (1) gives exactly what would be expected as the stationary phase expansion of the integral on the left, except that P−1 and P1 would in general be only power series, not polynomials. In cases such as this one where the stationary phase expansion has only finitely many terms, we refer to the expansion as finite. Bott then asked, what is it about the height function that causes the finiteness of this stationary phase expansion? Put otherwise, where does the height function come from? As a clue to the origin of this property of the height function, there is a wellknown explanation on S 2 . The Duistermaat – Heckman theorem asserts that on a 2n-dimensional symplectic manifold M with symplectic form ω having moment map f under a Hamiltonian action by S 1 with isolated fixed points, the stationary phase expansion of
etω dV (2) M k

is finite, where dV = ω /k! is the volume form associated with ω. Now S 2 is symplectic with a Hamiltonian action by S 1 given by rotation around the z axis, and (up to constants) its symplectic volume form is the standard volume form and the associated moment map is the height function. The answer to the question of where the height function comes from on S 2 is therefore provided by the Duistermaat – Heckman theorem: it is a moment map of the standard symplectic form on S 2 , and from this it follows that the stationary phase expansion in (1) is finite. But S 2n is not symplectic for n > 1, so how does this generalize? For this question, a closer look at the Atiyah – Bott proof (given in [1]) of the Duistermaat – Heckman theorem is helpful. In this proof, suppose that M is as above, the symplectic form on M is denoted by ω, and u ∈ s1∗ is dual to a nonzero X ∈ s1 . Then the assertion that a function f is a moment map translates to the assertion that the equivariant differential form ω  = ω + fu is equivariantly closed. To investigate the finiteness of the stationary phase expansion of interest, we exponentiate ω :  


ωn ω2 + ··· + eω
= ef u eω = ef u 1 + ω + . 2! n! M M M The left side can be shown to have the desired finite stationary phase expansion using the equivariant localization theorem, so the remaining key to the proof is that all the terms on the right except the ω n /n! = dV term contribute nothing to the integral because their dimensions are too small. This leaves precisely the integral (2) whose stationary phase expansion we wanted to compute, which completes the proof of the Duistermaat – Heckman theorem. The Atiyah – Bott proof of the Duistermaat – Heckman theorem suggests an approach to the question of finite stationary phase expansions on S 2n : we might try replacing the equivariantly closed form ω +f u with a more general equivariantly closed form α  = αk + αk−1 u + · · · + α1 uk−1 + f uk , where each αj ∈ Ω2j (M ) and f : M → R. When such an equivariantly closed form is exponentiated, the results will be similar to those from exponentiating ω  , but

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EQUIVARIANT COHOMOLOGY AND REFLECTIONS

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now there will be additional forms of de Rham degree 2n in the integral obtained, and these additional terms complicate matters considerably. Also, exactly which equivariantly closed form α  to try is not so obvious. The  is form α  should have the property that αk is non-degenerate, so that when α exponentiated, the 2n-dimensional piece arising from αk is a volume form. The form α  should also have the property that f is the height function on S 2n under the action of rotations about the z axis (although other choices of actions are possible). What other properties it should have is not clear. An additional complication is that there are many different functions f that could be part of an equivariantly closed form that begins with any particular αk . Continuing to explore the original question, we consider the case of S 2n with the Levi-Civita connection of the metric induced by restriction of the Euclidean metric on R2n+1 , and with an S 1 action induced by rotations with arbitrary nonzero weights λ1 , . . . , λn of the first n coordinate planes in R2n+1 (meaning {x1 , x2 }, {x3 , x4 }, etc.). It turns out that in this case the equivariant Euler class, defined as the Pfaffian of 1/2π times the equivariant curvature form, has as its 0th degree de Rham form term a constant times the height function z. This makes it a good candidate for a suitable α  to which a generalized version of the Duistermaat – Heckman theorem might apply. Straight exponentiation does not work as in the original Duistermaat – Heckman theorem though, since there are many 2ndimensional terms, but it turns out that with the assumption of constant curvature these terms can be dealt with. Full details are given in [3, Sections 3 and 4], but in brief the behavior of the additional terms can be tracked by noticing in the constant curvature case certain relationships among the different de Rham degree terms in the equivariant Euler class. To derive these relationships, first we choose an orthonormal basis for the tangent space at an arbitrary point p, to give a convenient structure for the matrix of the 0th degree de Rham form part of the curvature. The constant curvature condition dictates how the matrix for the curvature R looks relative to an orthonormal basis. This facilitates the derivation of the desired relationships among the terms in the equivariant Euler class. In the S 2n case, as described in the previous paragraph, the curvature is constant and the height function z is proportional to the 0th degree de Rham form term of the equivariant Euler class. We deduce that the stationary phase expansion of the height function z in (1) is finite. This answers the original question posed by Bott. However, it also begs many more questions. Is the assumption of constant curvature necessary? Whether it is or not, do other equivariant characteristic classes give rise to functions with similar behavior? Do other higher-dimensional nondegenerate equivariantly closed forms have similar properties? More generally, exactly which equivariantly closed forms give rise to functions with the finite stationary phase expansion property? Similarly, is there a version of the Duistermaat – Heckman theorem that holds for a more general class of higher-dimensional equivariantly closed forms? More generally still, are there other properties of symplectic forms that generalize to higher dimensions in this manner? As an additional problem of interest, there is another function on S 2n that also gives rise to a finite stationary phase expansion (see [3]), namely:


z

(1 − ζ 2 )n−1 dζ.

f (z) = 0

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J. A. BERNHARD

This function f actually mimics the properties of moment maps of symplectic forms even more closely than the height function z does. Also, this function f appears in an equivariantly closed form whose highest degree de Rham form is the Euler class (or the volume form) on S 2n but which is not the equivariant Euler class. Where does this function come from? Is it the more natural generalization of the moment map of a symplectic form in the Duistermaat – Heckman theorem, or is the height function more natural, or are both of these manifestations of a more general phenomenon? I invite the reader to explore these unsolved problems stemming from one of Raoul Bott’s many interesting and remarkable mathematical questions, just as he so often and so kindly invited me to do. Acknowledgments. I would like to thank the anonymous referee of this paper for providing helpful corrections and suggestions, and to extend many thanks to Robert Kotiuga for all his work in organizing A Celebration of Raoul Bott’s Legacy in Mathematics. References 1. M. F. Atiyah and R. Bott, The moment map and equivariant cohomology, Topology 23 (1984), no. 1, 1–28. 2. J. A. Bernhard, Equivariant cohomology and stationary phase expansions, Ph.D. dissertation, Harvard University, 2000. , Finite stationary phase expansions, Asian J. Math. 9 (2005), no. 2, 187–198. 3. Department of Mathematics and Computer Science, University of Puget Sound, Tacoma, WA 98416-1043, USA E-mail address: [email protected]

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https://doi.org/10.1090/crmp/050/18

Centre de Recherches Math´ ematiques CRM Proceedings and Lecture Notes Volume 50, 2010

Connectedness of Level Sets of the Moment Map for Torus Actions on the Based Loop Group Lisa Jeffrey Abstract. This article describes results in [4] where the authors show that under certain conditions the level sets of the moment map for a torus action on the based loop group are connected.

1. Introduction: the based loop group This article is dedicated to Raoul Bott, a master of Morse theory whose work on the based loop group provided the foundations for the understanding of its topology and geometry. Let G be a compact, simple and simply connected Lie group. The loop group L1 G is defined as the set of maps from S 1 to G of Sobolev class H 1 , meaning that the Sobolev norm |f |1 is finite, where (|f |1 )2 = |f |2 + |df |2 in terms of the sup norm. Strictly speaking the Sobolev norm is defined on the Lie algebra of L1 G and the exponential map is used pointwise to transfer this definition to L1 G. L1 G is an infinite-dimensional Hilbert manifold. It contains the set C ∞ (G) of C ∞ maps from S 1 to G, which however is not a Hilbert manifold (although it is a Fr´echet manifold). The subset Ω1 G of L1 G (the based loop group) consists of those loops f : S 1 → G for which f (1) is the identity element. There is a surjective map from L1 G to Ω1 G defined as follows: F : h → h(1)−1 h This map sends the submanifold of constant loops (which may be identified with the group G) to the identity element in Ω1 G, so it identifies Ω1 G with the homogeneous space L1 G/G. The space Ω1 G is symplectic. The symplectic form at the identity element e is  
1 dY (s) ds (1) ωe (X, Y ) = X(s), 2π s∈S 1 ds 2000 Mathematics Subject Classification. Primary 53D20; Secondary 22E67. LCJ was partially supported by a grant from NSERC.. This is the final form of the paper. c
2010 c
2010 American American Mathematical Mathematical Society

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L. JEFFREY

for X, Y ∈ L1 (g). 2. Torus actions on the based loop group The rotation group S 1 acts on Ω1 G as follows: (2)

(eiθ f )(s) = f (s + θ)f (θ)−1 .

The maximal torus T acts on Ω1 G by conjugation: for t ∈ T , (3)

(tf )(s) = tf (s)t−1 .

These actions commute. It can be shown that these actions are Hamiltonian and the moment maps are as follows. The moment map for the rotation action (the “energy” E) is
2π 1 |f (s)−1 f  (s)|2 ds E(f ) = 4π 0 The moment map for the conjugation action of T is
2π 1 pr f (s)−1 f  (s) ds µ(f ) = 2π Lie(T ) 0 (where the projection is onto the Lie algebra of the maximal torus T ). 3. Convexity theorem Theorem 1 (Atiyah [1]; Guillemin – Sternberg [3]). Let M be a compact symplectic manifold with a Hamiltonian action of a torus T . Then the image of the moment map is the convex hull of the images of the components of the fixed point set under the moment map. The image of the moment map is a convex polytope (the Newton polytope). Atiyah and Pressley proved an analogue for the based loop group: Theorem 2 (Atiyah – Pressley [2]). Let R := T × S 1 act on Ω1 G as above. Then the image of the moment map is convex and it is the convex hull of the images of the fixed points. Note that the fixed points of the action of R are those loops which are homomorphisms f (s) = exp(sΛ) for some Λ ∈ Lie(G) with exp Λ = e.

Figure 1. The shaded region is the image of the moment map for the T 2 = T 1 × S 1 action on Ω(SU(2)). (We have also drawn in the critical values of µ, which are the line segments.)

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4. Connectedness of level sets A key step in the proofs of convexity by Atiyah [1] and Guillemin – Sternberg [3] is proving that the level sets of the moment map are connected. Atiyah and Pressley do not prove this en route to their convexity theorem. This motivated us to investigate whether it was in fact true in the based loop group example. Definition 3. Ωalg G is the subset of Ω1 G consisting of loops which have a finite Fourier series (viewed as N × N matrices for some positive integer N ). Theorem A ([4]). Any level set of the moment map µ for the T × S 1 action restricted to Ωalg G is connected (for regular or singular values of the moment map). Theorem B ([4]). Let µ be the moment map for the T × S 1 action on Ω1 G. The level set µ−1 (a) of the moment map is connected, provided a is a regular value. Sketch proof of Theorem A. This involves Bruhat cells [14] (analogous to the Bruhat decomposition of flag manifolds). We first prove connectedness of the intersection of any level set with any closed Bruhat cell. Next we prove that if C1 and C2 are two Bruhat cells and µ : Ωalg G → t ⊕ R is the moment map for the T × S 1 action, then (µ−1 () ∩ C1 ) ∪ (µ−1 () ∩ C2 ) is connected. This uses the fact that the union of any two Bruhat cells is contained in the closure of a larger Bruhat cell. The proof of connectedness of level sets for Bruhat cells uses some geometric invariant theory and uses the Grassmannian model for loop groups.
Sketch proof of Theorem B. Ωalg G is dense in Ω1 G (in the H 1 topology), To show that level sets in Ω1 G are connected, it suffices to show that their intersection with Ωalg G is dense in the level set. We use Morse theory for infinitedimensional manifolds. To prove denseness, for any point x ∈ µ−1 (a) we construct a sequence in µ−1 (a) ∩ Ωalg (G) converging to it. We do this by first using the fact that Ωalg (G) is dense in Ω1 (G) to find a sequence of points in Ωalg G (but not necessarily in µ−1 (a)) converging to it. We then use the Morse flow with respect to independent components of the moment map µ, restricted to Ωalg (G), to produce a new sequence of points in µ−1 (a) ∩ Ωalg (G) which converges to x. This construction — which uses Morse flows with respect to components of the moment map — requires the use of the Palais – Smale condition C: Every sequence {un } with a function f such that f (un ) is bounded and f  (un ) → 0 has a convergent subsequence [12]. But Chuu-Lian Terng has proved in [15] that the components of the moment map for the torus action on Ωalg G satisfy the Palais – Smale condition C. This requires the hypothesis that a is a regular value of the moment map µ.
Remark 4. We have also given a proof that all level sets (regular and singular) of the energy function E are connected. Remark 5. Augustin-Liviu Mare [8] was able to eliminate the regular value hypothesis for the momentum map µ. His argument works for the space of C ∞ loops and also for the space of loops of Sobolev class H s for any s ≥ 1 (including the case s = 1).

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L. JEFFREY

References 1. M. F. Atiyah, Convexity and commuting Hamiltonians, Bull. London Math. Soc. 14 (1982), no. 1, 1 – 15. 2. M. F. Atiyah and A. N. Pressley, Convexity and loop groups, Arithmetic and Geometry, Vol. II, Progr. Math., vol. 36, Birkh¨ auser, Boston, MA, 1983, pp. 33 – 63. 3. V. Guillemin and S. Sternberg, Convexity properties of the moment mapping, Invent. Math. 67 (1982), no. 3, 491 – 513. 4. M. Harada, T. S. Holm, L. C. Jeffrey, and A.-L. Mare, Connectivity properties of moment maps on based loop groups, Geom. Topol. 10 (2006), 1607 – 1634. 5. F. C. Kirwan, Cohomology of quotients in symplectic and algebraic geometry, Math. Notes, vol. 31, Princeton Univ. Press, Princeton, NJ, 1984. , Rational intersection cohomology of quotient varieties. II, Invent. Math. 90 (1987), 6. no. 1, 153 – 167. 7. R. R. Kocherlakota, Integral homology of real flag manifolds and loop spaces of symmetric spaces, Adv. Math. 110 (1995), no. 1, 1 – 46. 8. A.-L. Mare, Connectivity of pre-images for moment maps on various classes of loop groups, Osaka J. Math, to appear, available at arXiv:math/0702792. 9. S. A. Mitchell, A filtration of the loops on SU(n) by Schubert varieties, Math. Z. 193 (1986), no. 3, 347 – 362. 10. D. Mumford, J. Fogarty, and F. Kirwan, Geometric invariant theory, 3rd ed., Ergeb. Math. Grenzgeb. (2), vol. 34, Springer, Berlin, 1994. 11. R. S. Palais, Morse theory on Hilbert manifolds, Topology 2 (1963), 299 – 340. 12. R. S. Palais and C.-L. Terng, Critical point theory and submanifold geometry, Lecture Notes in Math., vol. 1353, Springer, Berlin, 1988. 13. A. N. Pressley, The energy flow on the loop space of a compact Lie group, J. London Math. Soc. (2) 26 (1982), no. 3, 557 – 566. 14. A. N. Pressley and G. B. Segal, Loop groups, Oxford Math. Monogr., Oxford Univ. Press, New York, 1986. 15. C.-L. Terng, Polar actions on Hilbert space, J. Geom. Anal. 5 (1995), no. 1, 129 – 150. Department of Mathematics, University of Toronto, 40 St. George Street, Toronto, ON M5S 2E4, Canada E-mail address: [email protected]

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https://doi.org/10.1090/crmp/050/19

Centre de Recherches Math´ ematiques CRM Proceedings and Lecture Notes Volume 50, 2010

Computing Characteristic Numbers Using Fixed Points Loring W. Tu This article has its genesis in a course that Raoul Bott gave at Harvard in the fall of 1996 shortly before his official retirement. The topic of the course was equivariant cohomology, which is simply the cohomology of a group action. The course was to culminate in the equivariant localization formula, discovered by Berline and Vergne, and independently by Atiyah and Bott, around 1982. When a manifold has a torus action, the equivariant localization formula, while formulated in equivariant cohomology, is a powerful tool for doing calculations in the ordinary cohomology of the manifold. It extends and simplifies Bott’s work several decades earlier on the relationship between characteristic numbers and the zeros of a vector field ([5]). In one of the lectures during the course, Raoul Bott considered an action of a circle on a projective space and computed its equivariant cohomology from the fixed points using the Borel localization theorem. After class, he asked me if I could do the same for a homogeneous space G/H of a compact, connected Lie group G by a closed subgroup H of maximal rank, under the natural action of a maximal torus. Unbeknownst to me at the time, and perhaps to him also, this problem had been solved earlier and is in retrospect not so difficult (see [1] and [9], which contain much more than this). As was his wont, Raoul liked to understand everything ab initio and in his own way. Using Bott’s method, I worked out the equivariant cohomology ring of G/H from the fixed points of the torus action. I saw then that some of the lemmas I developed for this calculation may be used, in conjunction with the equivariant localization formula, to calculate the ordinary (as well as the equivariant) characteristic numbers of G/H. The idea of relating integration of ordinary differential forms to integration of equivariant differential forms is folklore. It is implicit in Atiyah – Bott ([2, Section 8]) and explicitly stated for the Chow ring in Edidin – Graham ([11, Proposition 5, p. 627]). When the fixed points are isolated and the manifold is compact, by converting ordinary integration to equivariant integration, one can hope to obtain the original integral as a finite sum over the fixed points of the action. While the idea is simple, its execution in specific examples is not necessarily so. The explicit formulas for the characteristic numbers of G/H obtained here are involved but 2000 Mathematics Subject Classification. Primary 57R20; Secondary 14C17, 14M17, 55N25. Key words and phrases. equivariant localization formula, equivariant cohomology, equivariant characteristic numbers, homogeneous spaces. This is the final form of the paper. c
2010 c
2010 American American Mathematical Mathematical Society

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beautiful, and serve as an affirmation of the power and versatility of the equivariant localization formula. Throughout this project, I met with Raoul many times over a period of several years. This article is a testimony to his generosity with his time, ideas, and friendship, and so I think it is particularly appropriate as a contribution to a volume on his mathematical legacy. To make the article more self-contained and in an effort to emulate Raoul Bott’s style, about half of the article is exposition of known results, for example, the computation from scratch of the ordinary cohomology rings of G/T and G/H. Although the results are known, I have also included the computation of the equivariant cohomology rings of G/T and G/H, partly because Bott’s method of using the Borel localization theorem may be new — at least I have not seen it in the literature — and may be applicable to other situations. I think of this article as an application of the equivariant localization formula to one nontrivial example. It is nonetheless a key example, since every orbit of every action is a homogeneous space. In the hope that the article could be understood by a graduate student with a modicum of knowledge of equivariant cohomology, perhaps someone who has read Bott’s short introduction to equivariant cohomology [7], I have allowed myself the liberty of being more expository than in a typical research article. Throughout this paper, H ∗ ( ) stands for singular cohomology with rational coefficients. The main technical results are the restriction formulas (Proposition 10 and Theorem 15) and the Euler class formulas (Proposition 13 and Theorem 19), which allow us to apply the Borel localization theorem and the equivariant localization formula to compact homogeneous spaces. Using techniques of symplectic quotients, Shaun Martin ([14, Proposition 7.2]) has obtained a formula for the characteristic numbers of a Grassmannian similar to our Proposition 23. Some of the ideas of this paper, in particular that of looking at the fixed points of the action of T on G/T , have parallels in the Atiyah – Bott proof of the Weyl character formula ([6]). It is a pleasure to thank Aaron W. Brown for patiently listening to me and giving me feedback, Jeffrey D. Carlson for his careful proofreading and comments, and Jonathan Weitsman for helpful advice. 1. Line bundles on G/T and on BT Let G be a compact, connected Lie group and T a maximal torus in G. The cohomology classes on the homogeneous space G/T and on the classifying space BT all come from the first Chern class of complex line bundles on these spaces. Both G/T and BT are base spaces of principal T -bundles: G/T is the base space of the principal T -bundle G → G/T and BT is the base space of a universal principal T -bundle ET → BT . For this reason, we will first construct complex line bundles on the base space of an arbitrary principal T -bundle. A character of the torus T is a multiplicative Lie group homomorphism of T into C∗ . Let T
be the group of characters of T , with the multiplication of characters written additively: for α, β ∈ T
and t ∈ T , tα+β := tα tβ = α(t)β(t). Denote by Cγ the complex vector space C with an action of T given by the character γ : T → C∗ . A character α : T → C∗ has the form α(t1 , . . . , tl ) = tn1 1 · · · tnl l , where

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COMPUTING CHARACTERISTIC NUMBERS USING FIXED POINTS

187 189

ti ∈ S 1 and ni ∈ Z ([10, Proposition 8.1, p. 107]). So the character group T
is isomorphic to Zl . Suppose a torus T of dimension l acts freely on the right on a topological space X so that X → X/T is a principal T -bundle. By the mixing construction, a character γ on T associates to the principal bundle X → X/T a complex line bundle L(X/T, γ) over X/T : Lγ := L(X/T, γ) := X ×T Cγ := (X × Cγ )/T, where T acts on X × Cγ by (x, v)t = (xt, γ(t−1 )v). The equivalence class of (x, v) is denoted [x, v]. It is easy to check that as complex line bundles over X/T , Lα+β  Lα ⊗ Lβ . ∨

Hence, L−α  Lα , the dual bundle of L. The first Chern class of an associated complex line bundle defines a homomorphism of abelian groups   γ → c1 L(X/T, γ) . (1) c : T
→ H 2 (X/T ), Let Sym(T
) be the symmetric algebra with rational coefficients generated by the additive group T
. The group homomorphism (1) extends to a ring homomorphism c : Sym(T
) → H ∗ (X/T ), called the characteristic map of X/T , sometimes denoted cX/T . We apply the construction of the associated bundle of a character γ ∈ T
to two situations: (i) The classifying space BT . Applied to the universal bundle ET → BT , this construction yields line bundles Sγ := L(ET /T, γ) = L(BT, γ) over BT and cohomology classes c(γ) = c1 (Sγ ) in H 2 (BT ). The characteristic map c : Sym(T
) → H ∗ (BT ) is a ring isomorphism, since both Sym(T
) and H ∗ (BT ) are polynomial rings in l generators and the generators correspond. If χ1 , . . . , χl is a basis for the character group T
and ui = c1 (Sχi ), then Sym(T
) is the polynomial ring Q[χ1 , . . . , χl ], and H ∗ (BT ) = H ∗ (BS 1 × · · · × BS 1 )  H ∗ (BS 1 ) ⊗Q · · · ⊗Q H ∗ (BS 1 )  Q[u1 , . . . , ul ], because BS 1  CP ∞ . (ii) The homogeneous space G/T . Applied to the principal T -bundle G → G/T , this construction yields line bundles Lγ := L(G/T, γ) over G/T . For each character γ on the torus T , the relationship of the line bundle Sγ over BT and the line bundle Lγ over G/T is as follows. The universal G-bundle EG → BG factors through EG → EG/T → BG. The total space EG is a contractible space on which G acts freely. A fortiori, EG is also a contractible space on which

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the subgroup T acts freely. Hence, EG = ET . It follows that (EG)/T = (ET )/T = BT , so there is a commutative diagram  / ET G   G/T 

 / BT

  pt 

 / BG,

representing G/T as a fiber of the fiber bundle BT → BG and the principal T bundle G → G/T as the restriction of the principal T -bundle ET → BT from BT to G/T . Hence, the associated bundle Lγ = G ×γ C is the restriction of the associated bundle Sγ = ET ×γ C from BT to G/T . 2. The actions of the Weyl group The Weyl group of a maximal torus T in the compact, connected Lie group G is W = NG (T )/T , where NG (T ) is the normalizer of T in G. The Weyl group is a finite reflection group. We use w to denote both an element of W and a lift of the element to the normalizer NG (T ). The Weyl group W acts on the character group T
of T by (w · γ)(t) = γ(w−1 tw). This action induces an action on Sym(T
) as ring isomorphisms. Let R be the polynomial ring R = Sym(T
) = Q[χ1 , . . . , χl ] W and R the subring of W -invariant polynomials. If the Lie group G acts on the right on a space X, then the Weyl group W acts on the right on X/T by rw (xT ) = (xT )w = xwT. This action of W on X/T induces by the pullback an action of W on the line bundles over X/T and also on the cohomology ring H ∗ (X/T ): if L is a line bundle on X/T and a ∈ H ∗ (X/T ), then ∗ L, w · L = rw

∗ w · a = rw a,

∗ for the pullback of a line bundle and for the where we use the same notation rw pullback of a cohomology class.

Proposition 1. The action of the Weyl group W on the associated line bundles over X/T is compatible with its action on the characters of T ; more precisely, for w ∈ W and γ ∈ T
, ∗ L(X/T, γ)  L(X/T, w · γ). rw ∗ L(X/T, γ) → L(X/T, w · γ) by Indication of proof. Define ϕ : rw

ϕ(xT, [xw, v]) = [x, v]. If we use a different representative xt to represent xT , then (xT, [xw, v]) = (xtT, [xtw, v  ])

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COMPUTING CHARACTERISTIC NUMBERS USING FIXED POINTS

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and it is easily verified that v  = γ(w−1 t−1 w)v. Hence, ϕ(xtT, [xtw, v  ]) = [xt, v  ] = [xt, (w · γ)(t−1 )v] = [x, v]. This shows that ϕ is well defined. It has the obvious inverse map ψ([x, v]) = (xT, [xw, v]).




Corollary 2. The characteristic map c : Sym(T
) → H ∗ (X/T ) is W -equivariant. Proof. Let γ be a character on the torus T . Writing Lγ instead of L(X/T, γ), it follows from the proposition that ∗ ∗ ∗ rw c(γ) = rw c1 (Lγ ) = c1 (rw Lγ ) = c1 (Lw·γ ) = c(w · γ).




Corollary 3. For γ a character on T and w an element of the Weyl group W , w · c1 (Sγ ) = c1 (Sw·γ ). Proof. This is a special case of the preceding corollary with X = ET .




3. Fiber bundles with fiber G/T Let G be a compact, connected Lie group, T a maximal torus, and W = NG (T )/T the Weyl group of T in G. Suppose G acts freely on the right on a topological space X so that X → X/G is a principal G-bundle. Then the natural projection X/T → X/G is a fiber bundle with fiber G/T . We will have frequent occasion to call on the following topological lemma. Lemma 4. The rational cohomology of X/G is the subspace of W -invariants of the rational cohomology of X/T : H ∗ (X/G)  H ∗ (X/T )W . The proof is based on the following two facts from [13]: Fact 1. The compact homogeneous space G/T has a cellular decomposition into even-dimensional cells indexed by the Weyl group. This is a consequence of the well-known Bruhat decomposition (see [13, p. 35])1, using the fact that a compact homogeneous space G/T has a complex description GC /B = G/T , where GC is the complexification of G and B is a Borel subgroup containing T . It implies that H ∗ (G/T ) vanishes in odd degrees and that the Euler characteristic of G/T is χ(G/T ) = |W |. Fact 2. If N = NG (T ) is the normalizer of T in G, then G/N is acyclic. Proof. The projection G/T → G/N is a regular covering map with group W = N/T . Hence, H ∗ (G/N ) = H ∗ (G/T )W . It follows that like G/T , G/N also has nonzero cohomology classes only in even degrees. Since χ(G/N ) =

1 1 χ(G/T ) = |W | = 1, |W | |W |

H 0 (G/N ) = Q and H k (G/N ) = 0 for k > 0.




1Fact 1 may also be obtained from Morse theory; there is a proof in [4] for G compact, connected, and simply connected.

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Proof of Lemma 4. Factor X/T → X/G into X/T → X/N → X/G. Because G/N is acyclic, the constant function 1 defines a global cohomology class on X/N that restricts to a generator of cohomology on each fiber G/N of X/N → X/G. By the Leray – Hirsch theorem, H ∗ (X/N )  H ∗ (X/G). Since X/T → X/N is a regular covering with group W = N/T , H ∗ (X/N )  ∗
H (X/T )W . Thus, H ∗ (X/G)  H ∗ (X/T )W . 4. Cohomology rings of G/T and G/H Let EG → BG and ET → BT be universal bundles for the compact, connected Lie group G and its maximal torus T . Since ET = EG, BG = (EG)/G and BT = (EG)/T . So the natural projection BT → BG is a fiber bundle with fiber G/T . Choose a basis χ1 , . . . , χl for the character group T
, let Sχi be the associated complex line bundles over BT , and set ui = c1 (Sχi ). As noted earlier, if R = Sym(T
), then H ∗ (BT ) = Q[u1 , . . . , ul ]  Q[χ1 , . . . , χl ]  R. By Lemma 4, the cohomology of BG is the subring of W -invariants in H ∗ (BT ): H ∗ (BG) = H ∗ (BT )W = RW . W be the submodule of RW generated by all homogeneous Theorem 5. Let R+ W W elements of positive degree, and (R+ ) the ideal in R generated by R+ . Then ∗ W H (G/T )  R/(R+ ).

Proof. Consider the spectral sequence of the fiber bundle BT → BG with fiber G/T . Since both the base BG and the fiber BT have only even-dimensional cohomology, all the differentials dr vanish for r ≥ 2 and the spectral sequence degenerates at the E2 term ([8, Theorems 14.14 and 14.18]). Therefore, H ∗ (BT ) = E∞  E2  H ∗ (BG) ⊗Q H ∗ (G/T ). In the picture of E2 , H ∗ (G/T ) is the zeroth column and H ∗ (BG) is the bottom row. From the picture of E2 , one sees that the kernel of the restriction to the fiber, H ∗ (BT ) → H ∗ (G/T ), is the shaded area, which is the ideal generated by  p W p>0 H (BG) = R+ . Hence, (2)

H ∗ (BT ) R = . W) p (R+ p>0 H (BG)

H ∗ (G/T )  

This isomorphism is a priori a module isomorphism, but because the restriction map H ∗ (BT ) → H ∗ (G/T ) is a ring homomorphism, the module isomorphism (2) is in fact a ring isomorphism.
In terms of the basis χ1 , . . . , χl for T
, let Lχi be the line bundle over G/T associated to the character χi and let yi = c1 (Lχi ) ∈ H 2 (G/T ). If i : G/T → BT is the inclusion map as a fiber of BT → BG, then as noted in Section 1, Lχi = i∗ Sχi and yi = c1 (Lχi ) = c1 (i∗ Sχi ) = i∗ c1 (Sχi ) = i∗ ui . Let H be a closed, connected subgroup of maximal rank in the compact, connected Lie group G, and T ⊂ H a maximal torus. Denote by WG and WH the Weyl groups of T in G and in H respectively.

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COMPUTING CHARACTERISTIC NUMBERS USING FIXED POINTS

q

H ∗ (G/T ) 0 0

1 2
 p>0

3

p 

W H p (BG) = (R+ )

Figure 1 Theorem 6. If R = H ∗ (BT ), RWH the subring of WH -invariant elements, WG and (R+ ) the ideal generated by WG -invariant elements of positive degree in RWH , then R WH H ∗ (G/H)   WG  . R+ Proof. The natural projection G/T → G/H is a fiber bundle with fiber H/T . By Lemma 4 and Theorem 5,   WH R R WH ∗ ∗ WH H (G/H)  H (G/T )   WG  =  WG  .
R+ R+ 5. Equivariant cohomology and equivariant characteristic classes Suppose a topological group G acts on the left on a topological space M , and EG → BG is the universal principal G-bundle. Since G acts freely on EG, the diagonal action of G on EG × M , (e, x)g = (eg, g −1 x), is also free. The space MG := EG ×G M := (EG × M )/G is called the homotopy quotient of M by G, and its cohomology H ∗ (MG ) the equivariant cohomology of ∗ (M ). For the basics of equivariant cohomology, M under the G-action, denoted HG we refer to [7] or [12]. A G-equivariant vector bundle E → M induces a vector bundle EG → MG . An equivariant characteristic class cG (E) of E → M is defined to be the corresponding ordinary characteristic class c(EG ) of EG → MG . By the definition of homotopy quotient, MG → BG is a fiber bundle with fiber M . Let i : M → MG be the inclusion map as a fiber. We say that a cohomology ∗ class α  ∈ HG (M ) is an equivariant extension of α ∈ H ∗ (M ) if i∗ α  = α. For example, if E → M is a G-equivariant vector bundle, then any characteristic class c(E) has an equivariant extension cG (E), since (3)

i∗ cG (E) = i∗ c(EG ) = c(i∗ EG ) = c(E).

Now suppose G is a compact, connected Lie group with maximal torus T . We associate to a character γ : T → C∗ the complex line bundle Lγ on G/T : Lγ = G ×γ C.

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There is an action of the torus T on Lγ : t · [g, v] = [tg, v],

for [g, v] ∈ G ×γ C,

where [g, v] is the equivalence class of (g, v) ∈ G × C. This action is compatible with the action of T on G/T in the sense that the diagram Lγ  G/T

t

t

/ Lγ  / G/T

commutes for all t ∈ T . Therefore, Lγ → G/T is a T -equivariant line bundle and induces a line bundle (Lγ )T over the homotopy (G/T )T . Fix a basis  quotient  χ1 , . . . , χl for the characters of T and let yi = c1 (Lχi )T ∈ HT2 (G/T ). These are equivariant extensions of the cohomology classes yi := c1 (Lχi ) ∈ H 2 (G/T ). The T -equivariant cohomology of a point is (4)

HT∗ (pt) = H ∗ (BT ) = Q[u1 , . . . , ul ],

ui = c1 (Sχi ).

For any T -space M , let π : G/T → pt be the constant map. The pullback map π ∗ : HT∗ (pt) → HT∗ (M ) makes HT∗ (M ) into an algebra over the polynomial ring Q[u1 , . . . , ul ]. 6. Restriction to a fixed point in G/T Although it is possible to give a shorter derivation of the equivariant cohomology ring HT∗ (G/T ) (see [9]), we will determine the ring structure of HT∗ (G/T ) from the fixed points of the action of T on G/T using the following localization theorem of Borel. This approach leads to a restriction formula (Proposition 10) that will be useful in our subsequent computation of characteristic numbers. Theorem (Borel localization theorem, [13, Proposition 2, p. 39]). Suppose a torus T acts on a manifold M with fixed point set F . Let iF : F → M be the inclusion map. Then both the kernel and cokernel of the restriction homomorphism i∗F : HT∗ (M ) → HT∗ (F ) are torsion H ∗ (BT )-modules. To apply this theorem to the action of T on G/T by left multiplication, it is necessary to know the fixed point set F of the action as well as the restriction homomorphism i∗F : HT∗ (G/T ) → HT∗ (F ). Proposition 7. For a compact Lie group G with maximal torus T , let NG (T ) be the normalizer of T in G and let T act on G/T by left multiplication. Then the fixed point set F of the action is WG = NG (T )/T , the Weyl group of T in G. Proof. The coset xT is a fixed point ⇐⇒ txT = xT ⇐⇒ x

−1

⇐⇒ x

−1

txT = T

for all t ∈ T for all t ∈ T

tx ∈ T

⇐⇒ x ∈ NG (T ).

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COMPUTING CHARACTERISTIC NUMBERS USING FIXED POINTS

193 195

At a fixed point w in G/T , the group T acts on the fiber (Lγ )w of the line bundle Lγ . Therefore, the fiber (Lγ )w is a complex representation of T . Lemma 8. At the fixed point w = xT ∈ WG , the torus T acts on the fiber of the line bundle Lγ as the representation w · γ, i.e., (Lγ )w = Cw·γ . Proof. The fiber of Lγ at a fixed point xT consists of elements of the form [x, v] ∈ G ×γ C. If t ∈ T , then t · [x, v] = [tx, v] = [x(x−1 tx), v] = [x, γ(x−1 tx)v] = [x, (w · γ)(t)v]. Hence, under the identification [x, v] ↔ v, the torus T acts on the fiber (Lγ )w as the representation w · γ.
Lemma 9. Let w be a point in the Weyl group WG ⊂ G/T and iw : {w} → G/T the inclusion map. For a character γ of T , the restriction of the line bundle (Lγ )T from (G/T )T to {w}T  BT is given by (iw )∗T (Lγ )T  Sw·γ , where Sw·γ is the complex line bundle on BT associated to the character w · γ. Proof. By Lemma 8 the restriction of the line bundle Lγ to the fixed point w gives rise to a commutative diagram of T -equivariant maps  / Lγ Cw·γ    {w} 

iw

 / G/T .

Taking the homotopy quotients results in the diagram  / (Lγ )T (Cw·γ )T    BT

(iw )T

 / (G/T )T .

But (Cw·γ )T = ET ×T Cw·γ is precisely the line bundle Sw·γ over BT associated to the character w · γ of T .
To avoid a plethora of subscripts, we write (iw )∗T also as i∗w . To describe the yi ) for each restriction i∗F : HT∗ (G/T ) → HT∗ (F ), we need to describe i∗w (ui ) and i∗w (˜ w ∈ W. Proposition 10 (Restriction formula for G/T ). At a fixed point w ∈ W , let i∗w : HT∗ (G/T ) → HT∗ ({w}) be the restriction map in equivariant cohomology. Then (i) i∗w ui = ui . (ii) For any character γ ∈ T
, i∗w cT1 (Lγ ) = w · c1 (Sγ ). In particular, if yi = cT1 (Lχi ) and ui = c1 (Sχi ), then i∗w yi = w · ui .

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Proof. (i) Let π : G/T → {w} be the constant map. Since π ◦ iw = 1w , the identity map on {w}, i∗w π ∗ = 1HT∗ (G/T ) . The elements ui in HT∗ (G/T ) are really π ∗ ui , so i∗w ui = i∗w π ∗ ui = ui . (ii) By the naturality of c1 and by Lemma 9,   i∗w c1 (Lγ )T = c1 (i∗w (Lγ )T ) = c1 (Sw·γ ) = w · c1 (Sγ ) (by Corollary 3).


Now take γ to be χi . 7. Additive structure of HT∗ (G/T )

Consider the action of T on G/T by left multiplication. Write W = WG = WG (T ) for the Weyl group of T in G. By Theorem 5, the rational cohomology ring of G/T is W W ) = R/(R+ ), H ∗ (G/T ) = Q[y1 , . . . , yl ]/(R+ W where R = Q[y1 , . . . , yl ]  H ∗ (BT ) and (R+ ) is the ideal generated by the W -invariant polynomials of positive degree in R. In particular, the cohomology H ∗ (G/T ) has only even-dimensional cohomology classes. For dimensional reasons, viz., H odd (G/T ) = H odd (BT ) = 0, all the differentials d2 , d3 , . . . in the spectral sequence of the fiber bundle

(5)

G/T → (G/T )T → BT

vanish, and additively   H ∗ (G/T )T = E2 -term of the spectral sequence

  W = H ∗ (BT ) ⊗Q H ∗ (G/T )  R ⊗Q H ∗ (G/T )  R ⊗Q R/(R+ )   W ) . = Q[u1 , . . . , ul ] ⊗Q Q[y1 , . . . , yl ]/(R+

This shows that HT∗ (G/T ) is a free R-module of rank equal to dim H ∗ (G/T ). Moreover, because the differentials dr , r ≥ 2, in the spectral sequence all vanish, the classes y1 , . . . , yl extend to global classes on (G/T )T . Indeed, since yi = c1 (Lχi ) is the first Chern class of a T -equivariant line bundle on G/T , by (3) its global extension is   yi = c1 (Lχi )T in HT∗ (G/T ). Thus, HT∗ (G/T ) is generated as a Q-algebra by u1 , . . . , ul , y1 , . . . , yl , and it remains only to determine the relations they satisfy. 8. The equivariant cohomology rings of G/T and G/H Suppose G is a compact, connected Lie group with maximal torus T . Choose a basis χ1 , . . . , χl of the character group T
. Let Lχi be the associated line bundles over G/T , and yi = c1 (Lχi ) ∈ H 2 (G/T ) and yi = cT1 (Lχi ) ∈ HT2 (G/T ) be their ordinary and equivariant first Chern classes. Similarly, let Sχi be the associated line bundles over BT , and ui = c1 (Sχi ) ∈ H 2 (BT ) their first Chern classes. Recall that R = H ∗ (BT ) = Q[u1 , . . . , ul ]. Theorem 11. (i) The equivariant cohomology ring of G/T under the action of T on G/T by left multiplication is HT∗ (G/T ) 

Q[u1 , . . . , ul , y1 , . . . , yl ] , J

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where J is the ideal in Q[u1 , . . . , ul , y1 , . . . , yl ] generated by b( y ) − b(u) for all polyWG , the WG -invariant polynomials of positive degree. nomials b ∈ R+ (ii) If H is a closed subgroup of G containing the maximal torus T , and T acts on G/H by left multiplication, then y1 , . . . , yl ]WH ) Q[u1 , . . . , ul ] ⊗Q (Q[ , J where J is the ideal in Q[u1 , . . . , ul ] ⊗Q (Q[ y1 , . . . , yl ]WH ) generated by b( y ) − b(u) WG for all polynomials b ∈ R+ . HT∗ (G/H) 

Proof. By Proposition 7, the fixed point set F of the action of T on G/T is the Weyl group W = WG . For each w ∈ W , by the restriction formula (Proposition 10), i∗w yi = w · ui . Hence, if b(u) = b(u1 , . . . , ul ) is a W -invariant polynomial with coefficients in Q, then i∗w b( y1 , . . . , yl ) = b(w · u1 , . . . , w · ul ) = b(u1 , . . . , ul ). With π : G/T → {w} being the constant map and iw : {w} → G/T the inclusion map, π ◦ iw = 1. Thus, b(u) = i∗w π ∗ b(u). It follows that i∗w b( y ) = b(u) = i∗w π ∗ b(u), or   y ) − π ∗ b(u) = 0. i∗w b( As is customary, in HT∗ (G/T ), we identify π ∗ b(u) with b(u), so we can write   i∗w b( y ) − b(u) = 0. Since this is true at any fixed point w ∈ F ,   i∗F b( y ) − b(u) = 0, where iF : F → G/T is the inclusion map. Let R = H ∗ (BT ). By the Borel localization theorem, the kernel of the restriction map i∗F : HT∗ (G/T ) → HT∗ (F ) is a torsion R-module. Since HT∗ (G/T ) is a free R-module, the kernel must be 0. Therefore, i∗F is injective. So we obtain the relations b( y ) − b(u) = 0

(6) in

HT∗ (G/T )

for all W -invariant polynomials b(u1 , . . . , ul ).   y) − Let J be the ideal b( y ) − b(u) in Q[u1 , . . . , ul , y1 , . . . , yl ] generated by b( W b(u) for all polynomials b ∈ R+ . Then there is a surjective ring homomorphism

R ⊗Q R Q[u1 , . . . , ul , y1 , . . . , yl ]  → HT∗ (G/T ). J J From the spectral sequence of the fibering (5), we know that HT∗ (G/T ) is a free R-module of rank equal to dim H ∗ (G/T ). To prove that φ is an isomorphism, it suffices to show that (R ⊗Q R)/J is a free R-module of the same rank, for in that case the kernel of φ, being of rank 0, is a torsion submodule of a free module and is therefore the zero module. Note that R ⊗Q R  R ⊗RW R, J (7)

φ:

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since the tensor product over RW is obtained from the tensor product over Q by dividing out by the ideal generated by elements of the form b ⊗ 1 − 1 ⊗ b for b ∈ RW . Moreover,  RW R ⊗RW R  (R ⊗Q Q) ⊗RW R  R ⊗Q ⊗RW R W) (R+  W R R ⊗RW R  R ⊗Q  R ⊗Q W W) (R+ ) (R+ ∗  R ⊗Q H (G/T ) (Theorem 5). So (R ⊗Q R)/J is a free R-module of rank equal to dim H ∗ (G/T ). This proves that φ is an isomorphism. (ii) Since the fiber bundle G/H → G/T with fiber H/T is T -equivariant, it induces a map on homotopy quotients (G/H)T → (G/T )T , which is also a fiber bundle with fiber H/T . By Lemma 4, (8)

HT∗ (G/H) = HT∗ (G/T )WH =

y1 , . . . , yl ]WH ) Q[u1 , . . . , ul ] ⊗Q (Q[ , J

WG where as before, J is the ideal generated by b( y ) − b(u) for all b ∈ R+ .




9. The tangent bundle of G/H Let G be a Lie group with Lie algebra g, and H a closed subgroup with Lie algebra h. Proposition 12 (See also [16, p. 130]). The tangent bundle of G/H is diffeomorphic to the vector bundle G ×H (g/h) associated to the principal H-bundle G → G/H via the adjoint representation of H on g/h. Proof. For g ∈ G, let lg : G/H → G/H denote left multiplication by g. Then lg is a diffeomorphism and its differential (lg )∗ : TeH (G/H) → TgH (G/H) is an isomorphism. Now define ϕ : G ×H (g/h) → T (G/H) by ϕ([g, v]) = (lg )∗ (v). To show that ϕ is well defined, pick another representative of [g, v], say [gh, Ad(h−1 )v] for some h ∈ H. Then (lgh )∗ (Ad(h−1 )v) = (lgh )∗ (lh−1 )∗ (rh )∗ v = (lg )∗ (lh )∗ (lh−1 )∗ (rh )∗ v = (lg )∗ v, since right multiplication rh on G/H is the identity map. Since dim G/H = dim g/h, ϕ is a surjective bundle map of two vector bundles of the same rank, and so it is an isomorphism.


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10. Equivariant Euler class of the normal bundle at a fixed point of G/T In this section G is a compact, connected Lie group, T a maximal torus in G, g and t their respective Lie algebras, and W = NG (T )/T the Weyl group of T in G. The adjoint representation of T on g decomposes g into a direct sum 

Cα , g=t⊕ α∈∆+

where ∆+ is a choice of positive roots. Let w ∈ W be a fixed point of the left multiplication action of T on G/T , and νw the normal bundle of {w} in G/T . The normal bundle at a point is simply the tangent space. Let l(w) be the length of w and (−1)w := (−1)l(w) the sign of w. Proposition 13 (Euler class formula for G/T ). The equivariant Euler class of the normal bundle νw at a fixed point w ∈ W of the left action of T on G/T is    c1 (Sα ) = (−1)w c1 (Sα ) ∈ H ∗ (BT ). eT (νw ) = w · α∈∆+

α∈∆+

Proof. By Proposition 12 the tangent bundle of G/T is the homogeneous  vector bundle associated to the adjoint representation of T on g/t = α∈∆+ Cα . In the notation of Section 1, with Lα being the complex line bundle over G/T associated to the character α of T ,



Cα  Lα . (9) T (G/T )  G ×T (g/t)  G ×T α∈∆+

α∈∆+

By (9) and Lemma 8, at a fixed point w the normal bundle νw is



νw = Tw (G/T ) = (Lα )w  Cw·α . α∈∆+

α∈∆+

It follows that the equivariant Euler class of νw is 



Cw·α = e (Cw·α )T eT (νw ) = eT α∈∆+

α∈∆+



=e Sw·α

(by the definition of Sα )

α∈∆+

=



α∈∆+



=w·



c1 (Sw·α ) =

w · c1 (Sα )

(Corollary 3)

α∈∆+



c1 (Sα )

(w is a ring homomorphism)

α∈∆+

Each simple reflection s in the Weyl group W = NG (T )/T carries exactly one ∨ positive root to a negative root. Note that c1 (S−α ) = c1 (Sα ) = −c1 (Sα ). Hence,     c1 (Sα ) = c1 (Ss·α ) = − c1 (Sα ). s· α∈∆+

α∈∆+

α∈∆+

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Since w is the product of l(w) reflections,    c1 (Sα ) = (−1)l(w) c1 (Sα ). w· α∈∆+




α∈∆+

11. Fixed points of T acting on G/H The quotient space WG /WH can be viewed as a subset of G/H via NG (T ) G NG (T ) WG = → .  WH NH (T ) NG (T ) ∩ H H Proposition 14. Under the action of T on G/H by left multiplication, the fixed point set F is WG /WH . Proof. The coset xH is a fixed point ⇐⇒ txH = xH

for all t ∈ T

⇐⇒ x

−1

txH = H

for all t ∈ T

⇐⇒ x

−1

tx ∈ H

for all t ∈ T

⇐⇒ x

−1

T x ⊂ H.

Since any two maximal tori in H are conjugate by an element of H, there is an element h ∈ H such that h−1 x−1 T xh = T. Therefore, xh ∈ NG (T ), and xH = xhH. Thus any fixed point can be represented as yH for some y ∈ NG (T ). Conversely, if y ∈ NG (T ), then yH is a fixed point of the action of T on G/H, since T yH = yT H = yH. It follows that there is a surjective map NG (T )  F ⊂ G/H, y → yH with fiber NG (T ) ∩ H and hence a bijection NG (T )  F. NG (T ) ∩ H




12. Restriction to a fixed point of G/H Theorem 15 (Restriction formula for G/H). With HT∗ (G/H) as in Theorem 11(ii) and w a fixed point in WG /WH , the restriction map i∗w in equivariant cohomology i∗w : HT∗ (G/H) → HT∗ ({w}) = Q[u1 , . . . , ul ] is given by i∗w ui = ui ,

i∗w f ( y ) = w · f (u)

for any WH -invariant polynomial f ( y ) ∈ Q[ y1 , . . . , yl ]WH .

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Proof. Write w = xH ∈ NG (T )/(NG (T ) ∩ H) ⊂ G/H, with x ∈ NG (T ). The commutative diagram of T -equivariant maps  ixT / {xT }  G/T σx

  {xH} 

σ

ixH

 / G/H

induces a commutative diagram in equivariant cohomology H ∗ (BT ) o O

i∗ xT

HT∗ (G/T ) O

∗ σx

σ∗

H ∗ (BT ) o

i∗ xH

HT∗ (G/H).

Since σ ∗ : HT∗ (G/H) → HT∗ (G/T ) is an injection and σx∗ : HT∗ ({xH}) = H ∗ (BT ) → H ∗ (BT ) = HT∗ ({xT }) is the identity map, the restriction i∗xH is given by the same formula as the restriction i∗xT . The theorem then follows from the restriction formula for G/T (Proposition 10).
13. Pullback of an associated bundle Assume that ρ : H → GL(V ) is a representation of the group H on a vector space V over any field. By restriction, one obtains a representation of the maximal torus T . Proposition 16. Let σ : G/T → G/H be the projection map. Under σ the associated bundle G ×H V pulls back to G ×T V : σ ∗ (G ×H V )  G ×T V. Proof. Let [g, v]T and [g, v]H denote the equivalence classes of (g, v) in G×T V and G ×H V respectively. Define ϕ : G ×T V → σ ∗ (G ×H V ) by [g, v]T → (gT, [g, v]H ). 



If (gT, [g , v ]H ) is any element of σ ∗ (G ×H V ), then gH = g  H. Hence, g  = gh for some h ∈ H and (gT, [g  , v  ]H ) = (gT, [gh, v  ]H ) = (gT, [g, hv  ]H ) = ϕ([g, hv  ]T ), which shows that ϕ is surjective. A surjective bundle map between two vector bundles of the same rank is an isomorphism.
14. Pulling back the tangent bundle of G/H to G/T Let g, h, and t be the Lie algebras of G, H, and T respectively. Under the adjoint representation of H the Lie algebra g decomposes into a direct sum of H-modules g = h ⊕ m. By Proposition 12, the tangent bundle of G/H is the associated bundle (10)

T (G/H)  G ×H m.

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Restricting the adjoint representation to the maximal torus T , the Lie algebras of H and G decompose further into a sum of T -modules 

h=t⊕ Cα α∈∆+ (H)

and







g=t⊕

Cα





⊕

Cα ,

α∈∆+ \∆+ (H)

α∈∆+ (H) +

where ∆ (H) denotes a choice of positive roots for H, ∆+ a choice of positive roots for G containing ∆+ (H), and ∆+ \ ∆+ (H) the complement of ∆+ (H) in ∆+ . Hence, as a T -module,

(11) m= Cα . α∈∆+ \∆+ (H)

Proposition 17. Under the natural projection σ : G/T → G/H, the tangent bundle T (G/H) pulls back to a sum of associated line bundles:

Lα . σ ∗ T (G/H)  α∈∆+ \∆+ (H)

Proof. σ ∗ T (G/H)  σ ∗ (G ×H m)  G ×T m 

= G ×T Cα

(by (10) and Proposition 16) (by (11))

α∈∆+ \∆+ (H)

=



Lα

(definition of Lα )




α∈∆+ \∆+ (H)

15. The normal bundle at a fixed point of G/H At a fixed point w = xH of G/H, with x ∈ NG (T ), the action of T on G/H induces an action of T on the tangent space Tw (G/H). Proposition 18. At a fixed point w = xH of G/H, the tangent space Tw (G/H) decomposes as a representation of T into

Tw (G/H)  Cw·α . α∈∆+ \∆+ (H)

Proof. Let σ : G/T → G/H be the projection map. Then   TxH (G/H) = T (G/H) xH (fiber of tangent bundle at xH)  ∗  = σ T (G/H) xT 

= (Lα )xT (Proposition 17) α∈∆+ \∆+ (H)





Cw·α

(Lemma 8)

α∈∆+ \∆+ (H)

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Theorem 19 (Euler class formula for G/H). At a fixed point w ∈ WG /WH of the left action of T on G/H, the equivariant Euler class of the normal bundle νw is given by   c1 (Sα ) . eT (νw ) = w · α∈∆+ \∆+ (H)

Proof. The normal bundle νw at the point w is the tangent space Tw (G/H). By the multiplicativity of the Euler class and the fact that the Euler class of a complex line bundle is its first Chern class, the equivariant Euler class of νw is 

  Cw·α (Proposition 18) eT (νw ) = eT Tw (G/H) = eT =

 α∈∆+ \∆+ (H)

=



α∈∆+ \∆+ (H)



cT1 (Cw·α ) =

  c1 (Cw·α )T

α∈∆+ \∆+ (H)



c1 (Sw·α ) = w ·

α∈∆+ \∆+ (H)



c1 (Sα ) .




α∈∆+ \∆+ (H)

16. Equivariant characteristic numbers of G/H and G/T Suppose a torus T acts on a compact oriented manifold M . Let π : M → pt be the constant map and π∗ : HT∗ (M ; R) → HT∗ (pt; R) = H ∗ (BT ; R) the push-forward or integration map in equivariant cohomology. For any class η˜ ∈ HT∗ (M ; R), the equivariant localization formula computes π∗ η˜ in terms of an integral over the fixed point set F ([2, 3]). In case the fixed points are isolated, the equivariant localization formula states that  i∗p η˜ . (12) π∗ η˜ = eT (νp ) p∈F

On the right-hand side the calculation is performed in the fraction field of H ∗(BT ; R) and so is a priori a rational function of u1 , . . . , ul , but it is part of the theorem that the sum will be a polynomial in u1 , . . . , ul since the left-hand side π∗ η˜ ∈ H ∗ (BT ; R) is a polynomial in u1 , . . . , ul . Although the equivariant localization formula is stated for real cohomology, by viewing rational cohomology as a subset of real cohomology, we can apply the equivariant localization formula to rational cohomology classes. Now suppose G is a compact, connected Lie group, T a maximal torus in G, and H a closed subgroup of G containing T . For η˜ ∈ HT∗ (G/H; Q), π∗ η˜ is a priori a real class in H ∗ (BT ; R). However, the explicit formula (13) below shows that it is in fact a rational class in H ∗ (BT ; Q). For M = G/H with the standard torus action, by Theorem 11(ii), an element of HT∗ (G/H; Q) is of the form   π ∗ ai (u) fi ( y ), η˜ = where ai (u) ∈ Q[u1 , . . . , ul ] and fi ( y ) ∈ Q[ y1 , . . . , yl ]WH . By the projection formula,  ai (u)π∗ fi ( y ). π∗ η˜ =

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Thus, to calculate π∗ : HT∗ (G/H; Q) ⊂ HT∗ (G/H; R) → H ∗ (BT ; R), it suffices to y ) for f ( y ) = f ( y1 , . . . , yl ) a WH -invariant polynomial with coefficalculate π∗ f ( cients in Q. Since y1 , . . . , yl are all equivariant characteristic classes, π∗ f ( y ) is called an equivariant characteristic number of G/H. With the aid of the restriction formula (Theorem 15) and the Euler class formula (Theorem 19), the equivariant localization formula (12) gives for any f ( y ) ∈ Q[ y1 , . . . , yl ]WH ,   f (u) (13) π∗ f ( y) = w·  . α∈∆+ \∆+ (H) c1 (Sα ) w∈WG /WH

y ) by replacing yi by ui . In this formula, f (u) = f (u1 , . . . , ul ) is obtained from f ( y ) ∈ H ∗ (BT ; R) is a polynomial in u1 , . . . , ul with real Since the left-hand side π∗ f ( coefficients, so is the right-hand side. But the right-hand side clearly has rational y ) ∈ H ∗ (BT ; Q). coefficients. Hence, π∗ f ( For G/T , Q[u1 , . . . , ul , y1 , . . . , yl ] HT∗ (G/T ) = , WG (b( y ) − b(u) | b ∈ R+ ) and the fixed point set is WG . For f ( y ) ∈ Q[ y1 , . . . , yl ], by the restriction formula (Proposition 10) and the Euler class formula (Proposition 13) for G/T ,    (−1)w w · f (u) f (u) G . (14) π∗ f ( y) = w·  = w∈W α∈∆+ c1 (Sα ) α∈∆+ c1 (Sα ) w∈WG

17. Ordinary integration and equivariant integration We state a general principle (Proposition 20), well known to the experts, relating ordinary integration and equivariant integration. Proposition 20. Let M be a compact oriented manifold of dimension n on ∗ which a compact, connected Lie group G acts and let π∗ : HG (M ; R) → H ∗ (BG; R) be equivariant integration. Suppose a cohomology class η ∈ H n (M ; R) has an equin (M ; R). Then variant extension η˜ ∈ HG  η = π∗ η˜. (15) M

Remark 21. For a torus action the right-hand side π∗ η˜ of (15) can be computed using the equivariant localization formula in terms of the fixed point set F of T on M . In case the fixed points are isolated, this gives   i∗p η˜ . η= eT (νp ) M p∈F

Proof of Proposition 20. The commutative diagram M



i

τ

  pt 

/ MG π

j

 / BG

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induces by the push-pull formula [12, p. 158] a commutative diagram in cohomology H n (M ; R) o

i∗

τ∗

(16)

 H 0 (pt; R) o

n HG (M ; R) π∗

j∗

 H 0 (BG; R).

In degree 0, the restriction j ∗ : H 0 (BG; R) → H 0 (pt; R) = R is an isomorphism. Hence, if η has degree n in H n (M ; R), then by the commutative diagram (16) and the push-pull formula  η = τ∗ η = τ∗ i∗ η˜ = j ∗ π∗ η˜ = π∗ η˜.
M

Since all ordinary characteristic classes of G-vector bundles have equivariant extensions, all ordinary characteristic numbers of G-vector bundles can be computed from the equivariant localization formula. Fix a basis χ1 , . . . , χl of the character group T
of the maximal torus T in the compact, connected Lie group G. Let H be a closed subgroup containing T in G. On G/T , we have associated bundles Lχi := G×T Cχi . Let yi = c1 (Lχi ) ∈ H 2 (G/T ; Q). Let R be the polynomial ring Q[y1 , . . . , yl ]. By Theorem 6, H ∗ (G/H; Q) =

R WH Q[y1 , . . . , yl ]WH = . WG WG (R+ ) (R+ )

Theorem 22. Let f (y) ∈ Q[y1 , . . . , yl ]WH be a WH -invariant polynomial of degree dim G/H, where each yi has degree 2. Then the characteristic number f (y) of G/H is given by G/H    f (u)  f (y) = π∗ f ( y) = w· . G/H α∈∆+ \∆+ (H) c1 (Sα ) w∈WG /WH

is an equivariant extension of yi , the cohomology Proof. Since yi = class f ( y ) ∈ HT∗ (G/H; Q) is an equivariant extension of f (y). Combining Proposition 20 and (13), the formula for the ordinary characteristic numbers of G/H follows. As noted earlier, (13) shows that if f ( y ) is a rational class, then so is y ).
π∗ f ( cT1 (Lχi )

18. Example: the complex Grassmannian In this example, we work out the T -equivariant cohomology ring as well as the characteristic numbers of the complex Grassmannian G(k, n) of k-planes in Cn . As a homogeneous space, G(k, n) can be represented as G/H, where G is the unitary group U(n) and H is the closed subgroup U(k) × U(n − k). A maximal torus contained in H is ⎧ ⎫ ⎡ ⎤! ! t1 ⎪ ⎪ ⎨ ⎬ ! ⎢ ⎥ ! . .. T = U(1) × · · · × U(1) = t = ⎣ ⎦ ! ti ∈ U(1) . ! ⎪ ⎪   ⎩  ⎭ n tn ! A basis for the characters of T is χ1 , . . . , χn , with χi (t) = ti . The characters χi define line bundles Sχi over the classifying space BT . We let ui = c1 (Sχi ) ∈

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H 2 (BT ). A choice of positive roots for G and for H is ∆+ = {χi − χj | 1 ≤ i < j ≤ n}, ∆+ (H) = {χi − χj | 1 ≤ i < j ≤ k} ∪ {χi − χj | k + 1 ≤ i < j ≤ n}. Therefore, ∆+ \ ∆+ (H) = {χi − χj | 1 ≤ i ≤ k, k + 1 ≤ j ≤ n}. If α = χi − χj , then c1 (Sα ) = c1 (Sχi ⊗ Sχ∨j ) = c1 (Sχi ) − c1 (Sχj ) = ui − uj .

(17)

The Weyl groups of T in G and H are WG = Sn , the symmetric group on n letters, WH = Sk × Sn−k . (Notation: Sα , Sχ are line bundles over BT associated to the characters α and χ, but Sk , Sn are symmetric groups.) A permutation in the symmetric group Sn is a bijection w : {1, . . . , n} → {1, . . . , n}, (18)

w(1) = i1 , . . . , w(k) = ik , w(k + 1) = j1 , . . . , w(n) = jn−k ,

where I = (i1 , . . . , ik ) and J = (j1 , . . . , jn−k ) are two complementary multi-indices, i.e., I ∪ J = {1, . . . , n}. The equivalence class of w in Sn /(Sk × Sn−k ) has a unique representative with I = (i1 < · · · < ik ) and J = (j1 < · · · < jn−k ) both strictly increasing. By Theorem 11(ii), the rational equivariant cohomology ring of the Grassmannian G(k, n) under the torus action is   Q[u1 , . . . , un ] ⊗Q (Q[ y1 , . . . , yk , yk+1 , . . . , yn ]Sk ×Sn−k ) , HT∗ G(k, n) = J where J is the ideal in Q[u1 , . . . , un ] ⊗Q (Q[ y1 , . . . , yk , yk+1 , . . . , yn ]Sk ×Sn−k ) generated by b( y ) − b(u) for all symmetric polynomials b( y ) ∈ Q[ y1 , . . . , yn ]. Let σr be the rth elementary symmetric polynomial. Since every symmetric polynomial is a polynomial in the elementary symmetric polynomials, J is also the ideal generated y ) − σr (u) for r = 1, . . . , n. Thus, we may write by σr ( (19)

  Q[u1 , . . . , un ] ⊗Q (Q[ y1 , . . . , yk , yk+1 , . . . , yn ]Sk ×Sn−k )    HT∗ G(k, n) = (1 + yi ) − (1 + ui )   In this formula, the notation p(u, y) means the ideal generated by the homogeneous terms of the polynomial p(u, y). If S and Q are the universal sub- and quotient bundles over G(k, n), then setting

(20)

s˜r = cTr (S) = σr ( y1 , . . . , yk ),

q˜r = cTr (Q) = σr ( yk+1 , . . . , yn ),

we have 1 + s˜1 + · · · + s˜k =

k 

(1 + yi ),

and

1 + q˜1 + · · · + q˜n−k =

i=1

n 

(1 + yi ).

i=k+1

Thus (20) can be rewritten in the form   Q[u1 , . . . , un , s˜1 , . . . , s˜k , q˜1 , . . . , q˜n−k ] .  HT∗ G(k, n) =  (1 + s˜1 + · · · + s˜k )(1 + q˜1 + · · · + q˜n−k ) − (1 + ui )

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COMPUTING CHARACTERISTIC NUMBERS USING FIXED POINTS

205 207

Using the relation  (1 + ui )  , q˜i = (21) 1+ 1 + s˜i i=1     one can eliminate all the q˜i from HT∗ G(k, n) ; in other words, HT∗ G(k, n) is generated as an algebra over Q[u1 , . . . , un ] by s˜1 , . . . , s˜k with relations given by terms of degree > 2(n − k) in (21). In computing degrees, keep in mind that deg ui = 2 and deg s˜i = deg q˜i = 2i. Similarly, in this notation, the rational cohomology ring of the Grassmannian G(k, n) is n−k 

  Q[y1 , . . . , yk , yk+1 , . . . , yn ]Sk ×Sn−k   H ∗ G(k, n) = (1 + yi ) Q[s1 , . . . , sk , q1 , . . . , qn−k ] , = (1 + s1 + · · · + sk )(1 + q1 + · · · + qn−k ) where sr = cr (S) = σr (y1 , . . . , yk ) and qr = cr (Q) = σr (yk+1 , . . . , yn ). Proposition 23. The characteristic numbers of G(k, n) are   k σr (ui , . . . , ui )mr k m1 mk r=1  1  , c1 (S) · · · ck (S) = (22) (ui − uj ) G(k,n) i∈I j∈J I  where mr = k(n − k), I runs over all multi-indices 1 ≤ i1 < · · · < ik ≤ n and J is its complementary multi-index.   Proof. In Theorem 22, take f (y) to be kr=1 cr (S)mr = kr=1 σr (y1 , . . . , yk )mr and w = (i1 , . . . , ik , j1 , . . . , jn−k ) as in (18). Because w · u1 = ui1 , . . . , w · uk = uik , w · f (u) =

k 

σr (ui1 , . . . , uik )mr .

r=1

By (17), 

c1 (Sα ) =

α∈∆+ \∆+ (H)

k n  

(ui − uj ).

i=1 j=k+1

By (18),  w·



α∈∆+ \∆+ (H)

c1 (Sα )

=w·

 k n 

(ui − uj )

i=1 j=k+1

=



(ui − uj ).




i∈I j∈J

One of the surprising features of the localization formula is that although the right-hand side of (22) is apparently a sum of rational functions of u1 , . . . , un , the sum is in fact an integer. Example. As an example, we compute the characteristic numbers of CP 2 = G(1, 3). The rational cohomology of CP 2 is H ∗ (CP 2 ) = Q[x]/(x3 ), generated by

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206 208

L. W. TU ∨

x = c1 (S ) = −c1 (S). By Proposition 23,   3  u2i  x2 = c1 (S)2 = CP 2 G(1,3) j=i (ui − uj ) i=1 u21 u22 u23 + + , (u1 − u2 )(u1 − u3 ) (u2 − u1 )(u2 − u3 ) (u3 − u1 )(u3 − u2 ) which simplifies to 1, as expected. =

References 1. A. Arabia, Cohomologie T -´ equivariante de la vari´ et´ e de drapeaux d’un groupe de Kac-Moody, Bull. Soc. Math. France 117 (1989), no. 2, 129–165. 2. M. F. Atiyah and R. Bott, The moment map and equivariant cohomology, Topology 23 (1984), no. 1, 1–28. 3. N. Berline and M. Vergne, Classes caract´ eristiques ´ equivariantes. Formule de localisation en cohomologie ´ equivariante, C. R. Acad. Sci. Paris S´er. I Math. 295 (1982), no. 9, 539–541. 4. R. Bott, An application of the Morse theory to the topology of Lie-groups, Bull. Soc. Math. France 84 (1956), 251–281. , Vector fields and characteristic numbers, Michigan Math. J. 14 (1967), 231–244. 5. , On induced representations, The Mathematical Heritage of Hermann Weyl (Durham, 6. NC, 1987), Proc. Sympos. Pure Math., vol. 48, Amer. Math. Soc., Providence, RI, 1988, pp. 1–13. , An introduction to equivariant cohomology, Quantum Field Theory: Perspective and 7. Prospective (Les Houches, 1998), NATO Sci. Ser. C Math. Phys. Sci., vol. 530, Kluwer Acad. Publ., Dordrecht, 1999, pp. 35–56. 8. R. Bott and L. W. Tu, Differential forms in algebraic topology, Corr. 3rd printing, Grad. Texts in Math., vol. 82, Springer, New York, 1995. 9. M. Brion, Equivariant cohomology and equivariant intersection theory, Representation Theories and Algebraic Geometry (Montr´ eal, QC, 1997), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 514, Kluwer Acad. Publ., Dordrecht, 1998, pp. 1–37. 10. T. Br¨ ocker and T. tom Dieck, Representations of compact Lie groups, Grad. Texts in Math., vol. 98, Springer, New York, 1985. 11. D. Edidin and W. Graham, Localization in equivariant intersection theory and the Bott residue formula, Amer. J. Math. 120 (1998), no. 3, 619–636. 12. V. W. Guillemin and S. Sternberg, Supersymmetry and equivariant de Rham theory, Math. Past Present, Springer, Berlin, 1999. 13. W.-y. Hsiang, Cohomology theory of topological transformation groups, Vol. 85, Springer, New York, 1975. Ergeb. Math. Grenzgeb. 14. S. Martin, Symplectic quotients by a nonabelian group and by its maximal torus, available at arXiv:math.SG/0001002. 15. A. Pedroza and L. W. Tu, On the localization formula in equivariant cohomology, Topology Appl. 154 (2007), no. 7, 1493–1501. ´ 16. G. Segal, Equivariant K-theory, Inst. Hautes Etudes Sci. Publ. Math. 34 (1968), 129–151. Department of Mathematics, Tufts University, Medford, MA 02155-7049, USA E-mail address: [email protected]

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https://doi.org/10.1090/crmp/050/20

Centre de Recherches Math´ ematiques CRM Proceedings and Lecture Notes Volume 50, 2010

From Minimal Geodesics to Supersymmetric Field Theories Henning Hohnhold, Stephan Stolz, and Peter Teichner In memory of Raoul Bott, friend and mentor.

Abstract. There are many models for the K-theory spectrum known today, each one having its own history and applications. The purpose of this note is to give an elementary description of eight such models (and certain completions of them) and to relate all of them by canonical maps, some of which are homeomorphisms (rather than just homotopy equivalences). Our survey begins with Raoul Bott’s iterated spaces of minimal geodesics in orthogonal groups, which he used to prove his famous periodicity theorem, and includes Milnor’s spaces of Clifford module structures as well as the Atyiah – Singer spaces of Fredholm operators. From these classical descriptions we move via spaces of unbounded operators and super-semigroups of operators to our most recent model, which is given by certain spaces of supersymmetric (1|1)-dimensional field theories. These spaces were introduced by the second two authors for the purpose of generalizing them to spaces of certain supersymmetric (2|1)dimensional Euclidean field theories that are conjectured to be related to the Hopkins – Miller spectrum TMF of topological modular forms.

Introduction At the first Arbeitstagung 1957 in Bonn, Alexander Grothendieck presented his version of the Riemann – Roch theorem in terms of a group (now known as the Grothendieck group) constructed from (isomorphism classes of) algebraic vector bundles over algebraic manifolds. Some people say that he used the letter K to abbreviate ‘Klassen,’ the German word for (isomorphism) classes. Michael Atiyah and Friedrich Hirzebruch instantly realized that the same construction can be applied to all complex vector bundles over a topological space X, yielding a commutative ring K(X), where addition and multiplication come from direct sum, respectively tensor product, of vector bundles. For example, every complex vector bundle over 2000 Mathematics Subject Classification. Primary 19L99; Secondary 18D10, 47A53, 57R56. Key words and phrases. Differential geometry, algebraic geometry. The second and third author were supported by the Max-Planck Society and grants from the NSF. This is the final form of the paper. c c
2010 American
2010 Mathematical Society by the authors

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H. HOHNHOLD, S. STOLZ, AND P. TEICHNER

the circle is trivial and hence K(S 1 ) = Z. Moreover, K(S 2 ) = Z[L]/(L − 1)2 , is generated by Hopf’s line bundle L over the 2-sphere. At the second Arbeitstagung in 1958, Raoul Bott explained his celebrated periodicity theorem, which can be expressed as the computations for all n: K(S 2n−1 ) ∼ = K(S 1 ) ∼ = Z and K(S 2n ) ∼ = K(S 2 ) ∼ = Z2 Bott also proved a real periodicity theorem involving the Grothendieck group of (isomorphisms classes of) real vector bundles over X. After dividing by the subgroup generated by trivial bundles (the quotient is denoted by a tilde over the K-groups) one obtains ⎧ ⎪ for n ≡ 0 mod 4 ⎨Z n ∼
KO(S ) = Z/2 for n ≡ 1, 2 mod 8 ⎪ ⎩ 0 else It was again Atiyah and Hirzebruch who realized that Bott’s periodicity theorem could be used to define generalized cohomology theories K n (X) and KO n (X) that are 2-periodic, respectively 8-periodic, and satisfy K 0 = K, KO 0 = KO. They satisfy the same Eilenberg – Steenrod axioms (functoriality, homotopy invariance and Mayer – Vietoris principle) as the ordinary cohomology groups H n (X) but if one takes X to be a point one obtains nontrivial groups for some n = 0. In fact, the above computation yields by the suspension isomorphism  Z for n even −n −n 0 ∼ 0 n ∼ ∼  K (pt) = K (S ) = K (S ) = 0 for n odd and similarly for real K-theory. Several classical problems in topology were solved using this new cohomology theory. For example, the maximal number of independent vector fields on the n-sphere was determined explicitly. A modern way to express any generalized cohomology theory is to write down a spectrum, i.e., a sequence of pointed CW-complexes En with structure maps ΣEn → En+1 . For ordinary cohomology, these En would be the Eilenberg – Mac Lane spaces K(Z, n). The purpose of this note is to give an elementary description of eight models (and their completions) of the spaces En in the K-theory spectrum and relate all of them by canonical maps, most of which are homeomorphisms (rather then just homotopy equivalences). We will exclusively work with real K-theory KO but all statements and proofs carry over to the complex case, and, with a little more care, also to Atiyah’s Real K-theory. Before we start a more precise discussion, we’ll give a list of the models that will be described in this paper. Recall that a pointed CW-complex En is said to represent the functor KO n if there are natural isomorphisms of pointed sets [X, En ] ∼ = KO n (X) for all CW-complexes X. By Brown’s representation theorem, such En exist and n
(En ) ∼ are unique up to homotopy equivalence. The suspension isomorphism KO =
n+1 (ΣEn ) in real K-theory then takes the identity map on En to a map ΣEn → KO En+1 whose adjoint must be a homotopy equivalence



n : En − → ΩEn+1 .

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The sequence {En ,
n }n∈Z of spaces and structure maps is a special type of spectrum, namely an Ω-spectrum, that represents the cohomology theory KO. The homotopy groups of the Ω-spectrum are then given by the following connected components: πn KO := KO n (pt) = KO −n (pt) ∼ = π0 E−n which explains partially why we describe these negatively indexed spaces below. Note that for n ≥ 0, we also have π0 E−n ∼ = π 0 Ωn E 0 ∼ = πn E0 but since we are considering periodic KO-theory, the spectrum has nontrivial negative homotopy groups. To fix notation, let Cn be the Clifford algebra associated to the positive definite inner product on Rn . It has generators e1 , . . . , en satisfying the relations e2i = −½,

ei ej + ej ei = 0 for i = j

and it turns into a C ∗ -algebra via e∗i = −ei . For n ≥ 0, we define C−n to be the Clifford C ∗ -algebra for the negative definite inner product, so the operators ei are self-adjoint and e2i = ½. We also fix a separable real Hilbert space Hn with a ∗-representation of Cn−1 such that all (i.e., one or two) irreducible Cn−1 modules appear infinitely often. Note that Cn is a Z/2-graded algebra such that the even part Cnev is isomorphic to Cn−1 . Thus we get a Z/2-graded Cn -module Hn := Hn ⊗Cnev Cn . We denote the grading involution by α. All gradings in this paper are Z/2-gradings. Main Theorem. The following spaces are all homotopy equivalent and represent the (−n)th space in the periodic real KO-theory Ω-spectrum. The spaces in (2) to (5) are homeomorphic and so are their “completed ” partners (written in parentheses). (1) The Bott space Bn . Here B1 is the union of all orthogonal groups O(k) where k ∈ N. For n > 1, the space Bn is the space of minimal geodesics in Bn−1 . (2) The Milnor space Mfin n (respectively Mn ) of “Cn−1 -module structures” on Hn . More precisely, for n > 1 these are unitary structures J on Hn , such that J − en−1 has finite rank (is compact) and Jei = −ei J for 1 ≤ i ≤ n − 2. (3) The space Inf fin n (respectively Inf n ) of “infinitesimal generators”, i.e., odd, self-adjoint unbounded Cn -linear operators on Hn with finite rank (compact) resolvent. (4) The configuration space Conf fin n (respectively Conf n ) of finite-dimensional ungraded mutually perpendicular Cn -submodules Vλ of Hn , labelled by finitely (discretely) many λ ∈ R satisfying V−λ = α(Vλ ). (5) The space SGOfin n (respectively SGOn ) of super-semigroups of self-adjoint Cn -linear finite rank (compact) operators on Hn . (6) The classifying space of the internal groupoid 1|1-PEFT−n of supersymmetric, positive Euclidean Field Theories of dimension (1|1) and degree (−n). (7) The classifying space Qn of a category that arises from a Cn -module category by a topological version of Quillen’s S −1 S-construction. (8) The Atiyah – Singer space Fn of certain skew-adjoint Fredholm operators on Hn , anticommuting with the Cn−1 -action. The above theorem only gives very rough descriptions of the spaces involved. Detailed definitions for each item (k), k = 1, . . . , 8, can be found in Section k

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H. HOHNHOLD, S. STOLZ, AND P. TEICHNER

below. Section k also contains the proof that the spaces in (k) are homeomorphic (respectively homotopy equivalent) to spaces appearing previously. To the best of our knowledge, the homeomorphisms between the spaces in (2), (4), (5) are new, even though it was well known that the spaces are homotopy equivalent for abstract reasons (since they represent the same Ω-spectrum and have the homotopy type of a CW complex; in fact, our results imply that this is the case for all the spaces in our Main Theorem). Moreover, our maps relating the spaces in (4) and (7) respectively (3) and (8) seem to be new and slightly easier than the original ones. Another new aspect is the precise treatment of super-semigroups and the spaces SGOn . Finally, the main new result is the equivalence of the older descriptions of Ktheory with the one in (6). This paper does not contain the definitive treatment of this equivalence, we only outline the case for n = 0, see [12, 29] for more details. Studying supersymmetric Euclidean field theories is a long term project of the last two authors, attempting to give a geometric description of the Hopkins – Miller theory of topological modular forms, the “universal” elliptic cohomology theory, by raising the dimension of the worldsheet from (1|1) to (2|1). Remark 0.1. The spaces in (3) to (7) are defined for all n ∈ Z and the theorem holds for all integers n. The Bott and Milnor spaces only make sense for n ≥ 1 and the same seems to be true for the spaces in (8). This comes from the fact that the Atiyah – Singer spaces Fn are defined in terms of the ungraded Hilbert space Hn and for n ≤ 0 our translation to Hn doesn’t work well. However, this can be circumvented by never mentioning the Clifford algebra Cn−1 in the definitions and working with the ungraded algebra Cnev instead. Then the spaces Fn are defined for all n ∈ Z and our theorem holds. We chose the formulation above to better connect with the reference [2]. We should also mention some other descriptions of the spaces in the KOtheory spectrum; we apologize in advance for any omissions. In [26], Graeme Segal introduces the technology of Γ-spaces to construct a connective spectrum from a symmetric monoidal category. The 0th space E0 in the resulting Ω-spectrum is the group completion of the commutative H-space resulting from the classifying space of the symmetric monoidal category. If one applies his machine to finite-dimensional vector spaces (with direct sum operation), one obtains connective ko-theory. This means that for n ≥ 0 the spaces E−n = Ωn E0 are homotopy equivalent to the ones in our theorem but En is (n − 1)-connected and hence the negative homotopy groups of the spectrum vanish. In his book on K-theory [14, Theorem III.4.27], Max Karoubi describes the nth space for KO as the space of gradings of Hn . These are orthogonal involutions that anticommute with the ei (and have no relation with the given grading α). In [10], Guentner and Higson use for the (−n)th space for KO all Z/2-graded C ∗ -algebra homomorphisms from C0 (R) to the Cn -linear compact operators on Hn . Here the C ∗ -algebra of continuous functions on R that vanish at ∞ is graded via the map x → −x for x ∈ R. We showed in [28, Lemma 3.15] that this space is actually homeomorphic to SGOn in (5) above. This follows from the fact that C0 (R) is the 1|1 graded C ∗ -algebra generated by the super-semigroup R>0 used in the definition of SGOn . In Pokman Cheung’s thesis [5], another version of the (−n)th space in the real K-theory spectrum was introduced. The author starts with Conf n as a discrete set

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of objects, introduces a space of morphisms which gives a topological category and shows that its classifying space is homotopy equivalent to the Atiyah – Singer space Fn . We shall compare his results with ours in Remark 7.3. Survey of the spaces in our Main Theorem. We now give a rough outline of where the spaces come from and how they are related. The spaces Bn were defined by Raoul Bott in his classical paper [4] on “the stable homotopy of the classical groups” whose periodicity theorem is the heart and soul of K-theory. This model actually predates the invention of K-theory as a generalized cohomology theory, but was used by Bott to completely calculate the coefficients of this theory. Atiyah, Bott, and Shapiro [1] showed the significance of Clifford algebras in K-theory and they suggested to look for a proof of the periodicity theorem using Clifford algebras. A proof along these lines was then found by Wood in [31] and also by Milnor in his beautiful book [22] on Morse theory. We will recall in Section 1 how one can easily compute (iterated) spaces of minimal geodesics in the orthogonal group in terms of Clifford module structures on Hn . In Section 2 we’ll define the Milnor spaces Mn to be a certain completion of these spaces of Clifford module structures. These new spaces no longer depend on a basis of Hn but they have the same homotopy type as Bn by a theorem of Palais [23]. The configuration spaces Conf fin n are the easiest to work with because one can geometrically picture its points very well as a finite collection of real numbers λ, labelled by (finite-dimensional and pairwise orthogonal) Cn -submodules Vλ ⊂ Hn such that V−λ = α(Vλ ). In [27], Segal had introduced such configuration spaces F (X) in the case n = 0 (and ungraded) for any compact space X (with basepoint x0 ) instead of the real line, one-point compactified to S 1 (with ∞ as basepoint). The topology on F (X) is such that if two points of X collide then the corresponding labels (also known as subspaces of H = H0 ) add and moreover, a point can move to x0 and is then discarded. Segal proved that if X is connected then one has isomorphisms kon (X, x0 ) ∼ = πn F (X, x0 ) ∼ = πn+1 F (ΣX, x0 ) In particular, it follows that for n > 0 the space F (S n ) is the nth space of the connective ko-theory Ω-spectrum. However, for n = 0 this is clearly not the case of finite-dimensional subspaces of H and hence has since F (S 0 ) is the Grassmannian  the homotopy type of k BO(k). As a consequence, one needs to group complete this space (with respect to direct sum) to obtain the correct homotopy type BO × 0 Z. Our graded version Conf fin 0 of Segal’s configuration space F (S ) already has this homotopy type and hence it should be thought of as a “group completion of configurations”. We believe that more generally, if X is connected then F (X) = F (X; H) is homotopy equivalent to Conf(X × S 1 , (x0 , ∞); H). This is written in the notation of Section 4, where H = H ⊕ H and the involution on X × (R ∪ ∞) is given by (x, λ) → (x, −λ). It came as a surprise to us that (equipped with the correct topology) the configuration space Conf fin n is actually the classifying space Qn of a certain (internal) Quillen category appearing as (7) in our Main Theorem. In the light of the above considerations regarding group completion, this was expected only for the homotopy type of the classifying space for n = 0 and we shall outline in Section 7 how one can 0 use Quillen’s results to prove from this point of view that Conf fin 0 represents KO .

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Given a configuration {Vλ } in Conf fin n , one can interpret it as the eigenspaces fin and eigenvalues of an odd, self-adjoint, C n -linear operator D ∈ Inf n with domain λ Vλ that is given by D(v) := λ · v

∀ v ∈ Vλ .

We note that the domain of D is finite-dimensional; in particular, D is not densely defined, a common requirement for self-adjoint operators in textbooks. We expand the usual definition of a self-adjoint (unbounded) operator by just requiring that the operator is self-adjoint on the closure of its domain. As a consequence, it is very natural to study completions Conf n of the spaces Conf fin n where there is a discrete set of labels and hence the corresponding operator D ∈ Inf n may have dense domain (and it has compact resolvent). The resulting spaces Inf n are equipped with the generalized norm topology and the fact that one can retract the completed spaces back to their finite rank subspaces goes back to (at least) Segal [27] but we re-prove this fact here. The operator D can be used as the infinitesimal generator of the super-semigroup (t, θ) → e−tD

2

+θD

of finite rank (respectively compact) operators on Hn . These are the elements 1|1 of SGOfin n (respectively SGOn ). Here (t, θ) ∈ R>0 parametrize a certain supersemigroup whose super Lie algebra is free on one odd generator. Such supersemigroups of operators should be considered the ‘fermionic’ or ‘odd’ analogue of usual semigroups of operators. The homeomorphism from Conf fin n to Mn is basically given by applying the inverse of the Cayley transform to the operator D. If one applies this transformation to elements in Conf n one obtains interesting completions of the Milnor and Bott spaces. The spaces SGOn were introduced by two of us in [28] as super-semigroups of self-adjoint operators and at the time we thought of them as Euclidean field theories, without having a precise definition for the latter. In the meantime, we have developed a precise notion of supersymmetric Euclidean field theories of dimension d|δ, see [29]. These are certain fibred functors from a Euclidean bordism category to a category of topological vector spaces, both fibred over the site of complex super manifolds, see Section 6. Using invertible natural transformations between such functors leads to a groupoid of Euclidean field theories. Using the inner Hom in fibred categories, one naturally gets a groupoid internal in topological spaces and hence one has a classifying space. In Section 6 we will also assume that the Euclidean field theories are unitary and strongly positive and for d|δ = 1|1 we denote the resulting classifying space by 1|1-PEFT−n . Here the superscript refers to a degree −n twist that will not be explained in this paper but see [11]. If one starts with a closed n-dimensional spin manifold M then the Cn -linear Dirac operator DM (called Atiyah – Singer operator in [17, p. 140]) is an example of a (nonfinite) element in Inf n , where Hn are the L2 -sections of the Cn -linear spinor bundle on M . One can think of this operator as the infinitesimal generator of a supersymmetric (1|1)-dimensional quantum field theory, with Euclidean (rather than Minkowski) signature. This is the reason why we use the terminology Euclidean field theory. Actually, physicists would call it supersymmetric quantum mechanics on M , not a field theory, since space is 0-dimensional.

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The spaces Fn first appeared in the article [2] by Atiyah – Singer and are probably the most common model for K-theory. They make all the wonderful applications to analysis possible. From our point of view, the connection is easiest to make with the space Inf n : Starting with a skew-adjoint Fredholm operator T0 on Hn that anticommutes with e1 , . . . , en−1 , we can turn it into an odd, self-adjoint, Cn -linear Fredholm operator

 0 T0∗ T = T0 ⊗ en or equivalently T = T0 0 on Hn ∼ = Hn ⊕ Hn . It is easy to see that the map T0 → T is a homeomorphism and it is important to note that the skew-symmetry of T0 is equivalent to the relation T en = en T . This correspondence actually extends to the well known case n = 0 where one starts with all Fredholm operators on H0 and gets all odd, self-adjoint Fredholm operators on H0 ⊕ H0 . The essential spectrum of a Fredholm operator has a gap around zero and hence one can push the essential spectrum outside zero all the way to ±∞ by a homotopy. This turns a bounded operator into an unbounded one and is the basic step in the homotopy equivalence that takes a Fredholm operator T to an infinitesimal generator D. In the analytic literature, one can sometimes find concrete formulas in terms of functional calculus (which just describes the movement of the spectrum of T ) like D . T = 1 + D2 Such precise formulas are not important from our point of view but the following subtlety arises in the operator D: its eigenspace at ∞, by definition the orthogonal complement of the domain of D, is decomposed into the parts at +∞, respectively −∞. Such a datum is not present in general elements of Inf n and it reflects the fact that we started with a bounded operator. Roughly speaking, this represents no problem up to homotopy if both these parts at ±∞ are infinite-dimensional. This uses Kuiper’s theorem and is the only nonelementary aspect of this paper. Taking into account the Cn -action, this is related to the following well-known subtlety in the Atiyah – Singer spaces of Fredholm operators. If n ≡ 3 mod 4, the spaces Fn are given by operators T0 (or equivalently T ) as above. However, Atiyah – Singer showed that for n ≡ 3 mod 4 the space of Cn−1 -antilinear skew ± adjoint Fredholm operators on Hn has two boring, contractible components F n consisting of operators T0 such that e1 · e2 · · · en−1 · T0 is essentially positive (respectively negative). Recall that an operator is essentially positive if it is positive on a closed invariant subspace of finite codimension. So in ± are disregarded. It turns the precise version for Fn in [2], the two components F n out that the above functional calculus leads to a map of all Cn−1 -antilinear skewadjoint Fredholm operators to our spaces Inf n but this map is a quasifibration (with contractible fibres) only on the component Fn . Hence our spaces automatically remove the need for thinking about the above subtleties that arise from bounded operators and are interestingly only visible in the presence of special Clifford actions. A symmetric ring spectrum for K-theory. We end this introduction by explaining the easiest description (that we know) of a symmetric ring spectrum

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that represents K-theory. Let H−1 be a C−1 -module as in the paragraph preceding our Main Theorem, in particular it contains a submodule V that is isomorphic ⊗n has the properties required in our to C−1 as a C−1 -module. Then H−n := H−1 Main Theorem and hence there are corresponding spaces En := Inf fin −n (H−n ) of infinitesimal generators as in (3) of that theorem. One can also use the completed version Inf −n instead. We point out that En contains a canonical base point, namely the operator whose domain is zero (and thus all eigenvalues are at ∞). Theorem 0.2. For n ≥ 0, the spaces En form a symmetric ring spectrum representing real K-theory. The relevant structures are given as follows. • The symmetric group Σn acts by permuting the n tensor factors of H−n . • The multiplication maps En ∧ Em → En+m are given by the formula (Dn , Dm ) → Dn ⊗ ½ + ½ ⊗ Dm • The Σn -equivariant structure maps Rn → En are given by sending v ∈ Rn to its Clifford action on the C−n -module V ⊗n (and ∞ on the orthogonal complement in H−n ). As v → ∞, Clifford multiplication also goes to ∞ and hence the structure maps can be extended to S n , sending the point at ∞ to the base point in En . These operators are odd and self-adjoint which explains the negative sign of −n. This result is a reformulation of a theorem of Michael Joachim [13], so we shall not give a proof. By using complex Hilbert spaces and Clifford algebras, all our results translate to complex K-theory. In fact, keeping track of the involution of complex conjugation, one also gets Atiyah’s Real K-theory which contains both real K-theory (via taking fixed points) and complex K-theory (by forgetting the conjugation map). 1. Bott spaces of minimal geodesics The origin of topological K-theory is Raoul Bott’s classical paper [4] on “The stable homotopy of the classical groups”. For a Riemannian manifold M , let v = (P, Q, h) denote a ‘base point in M ’ which is actually a pair of points P, Q ∈ M , together with a fixed homotopy class h of paths connecting P and Q. If P = Q then h is just an element in π1 (M, P ). Bott considers the space M v of minimal geodesics from P to Q in the homotopy class h, a subspace of Ωv , the space of all such paths. Let |v| be the first positive integer which occurs as the index of some geodesic with base point v. Then Bott proved the following theorem in [4]: Theorem 1.1 (Bott). If M is a symmetric space, so is M v . Moreover, the based loop space Ωv can be built, up to homotopy, by starting with M v and attaching cells of dimension ≥ |v|: Ωv  M v ∪ e|v| ∪ (higher-dimensional cells)

|v|

written as M v −→ M .

For example, if M = S n , n > 1, with the round metric and P, Q are not antipodal, then there is a unique minimal geodesic from P to Q. The second shortest geodesic from P to Q reaches the point −Q that has an (n−1)-dimensional variation of geodesics to Q and hence |v| = n − 1. In the notation above, one gets (n−1)

pt = M v −−−−→ M = S n

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which implies that ΩS n is (n − 2)-connected, or equivalently, that S n is (n − 1)connected, not such a great result. However, if one considers all geodesics, one can say much more. In fact, the indices of geodesics from P to Q are k(n − 1) for k = 0, 1, 2, . . . . This is a case of Morse’s original application of his theory to infinite-dimensional manifolds: the energy functional

1 E : Ωv M → R, E(γ) := |γ  (t)|2 dt 0

is a Morse function with critical points the geodesics and indices given by the number of conjugate points (counted with multiplicity) along the given geodesic. Morse shows that infinite-dimensionality is not an issue, because the space of paths with bounded energy has the homotopy type of a finite-dimensional space, namely the piecewise geodesics (where the number of corners is related to the energy bound and the injectivity radius of M ). As a consequence of our example above, Ωv S n  S n−1 ∪ e2(n−1) ∪ e3(n−1) ∪ · · · If P and Q are antipodal points on S n , then the energy is not a Morse function. For example, the minimal geodesics form an (n − 1)-sphere, parametrized by the equator in S n . However, Bott developed a theory for such cases, now known as Morse-Bott theory, where the critical points form a submanifold whose tangent space equals the null space of the Hessian of the given function, the Morse – Bott condition. Applied to the case at hand, we can derive the same cell decomposition of Ωv S n as above but this time the bottom cell consists of the minimal geodesics: 2(n−1)

S n−1 = M v −−−−→ M = S n So Freudenthal’s suspension theorem is a direct consequence of this result, using only the index of the first nonminimal geodesics (not the first two, as in the generic case studied by Morse). More generally, for any symmetric space, this is Bott’s proof of Theorem 1.1 above. His approach to study the homotopy types of the classical groups was to apply this method to compact Lie groups which are symmetric spaces in their bi-invariant metric. For example, consider M = O(2m) and P = ½, Q = −½. Then every geodesic γ from P to Q is of the form γ(t) = exp(πt · A),

t ∈ [0, 1]

where A is skew-adjoint. Thus we can ‘diagonalize’ A by an orthogonal matrix T , i.e., T AT −1 is a block sum B of matrices 

0 ai −ai 0 with a1 , . . . , am ≥ 0 after normalization. Since γ(1) = −½, we see that the ai are odd integers. It is not hard to see that the energy of γ is given by the formula E(γ) = 2(a21 + · · · + a2m ) so that minimal energy (or equivalently, minimal length) means that all ai = 1. We note that for m > 1, the energy determines the homotopy class h of such a path so that we don’t need to mention it for minimal energy (or length) paths (this stays true in all considerations below as well). We conclude that A2 = T −1 B 2 T = T −1 (−½)T = −½ and

A∗ = −A = A−1

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so that A is a complex structure on R2m . Just like for the standard complex structure we have A = exp((π/2) · A) = γ(1/2) and we obtain the following result. Proposition 1.2 (Bott). The space B2 (2m) of minimal geodesics in O(2m) with basepoint v as above is isometric to the space M2 (2m) of unitary structures on R2m (consisting of J ∈ O(2m) with J 2 = −½). Moreover, this is a totally geodesic submanifold of O(2m) and the homeomorphism is given by sending a geodesic γ to its midpoint γ( 12 ). We are introducing a notation that is consistent with B1 (m) = O(m) = M1 (m) and will lead to the Bott and Milnor spaces in the limit when m → ∞. Now recall that B2 (2m) is again a symmetric space by Bott’s theorem so that we can iterate the construction: Pick a complex structure J1 and study the space B3 (4m) of minimal geodesics in B2 (4m) from J1 to −J1 (they automatically lie in a fixed homotopy class). By a very similar discussion as above, it turns out that the midpoint map gives an isometry B3 (4m) ∼ = M3 (4m) := {J ∈ O(4m) | J 2 = −½, JJ1 = −J1 J}. Note that the right-hand side is the space of (orthogonal) quaternion structures on R4m that are compatible with the given unitary (or orthogonal complex) structure J1 . The set of such structures form a totally geodesic submanifold of O(4m). More generally, we make the following Definition 1.3. Assume that Rm is a Cn−2 -module for some n > 1. More precisely, there is a ∗-homomorphism Cn−2 → End(Rm ) sending ei to Ji for i = 1, . . . , n. Note that the compatibility with ∗ implies that Ji ∈ O(m). Then we define Milnor spaces Mn (m) := {J ∈ O(m) | J 2 = −½, JJi = −Ji J∀i = 1, . . . , n − 2} to be “the space of all Cn−1 -structures on Rm ”, compatible with the given Cn−2 structure. Note that these spaces can be empty. Proposition 1.4 (Bott). All Mn (m) are totally geodesic submanifolds of O(m). The space of minimal geodesics in Mn (m) from Jn−1 to −Jn−1 is isometric to Mn+1 (m) via the midpoint map. Proof. Let’s assume the first sentence and show the second assertion. Any geodesic γ from Jn−1 to −Jn−1 is of the form γ(t) = Jn−1 · exp(πt · A),

t ∈ [0, 1]

for some skew-adjoint matrix A. One checks that γ( 12 ) = Jn−1 ·A has square −½ and anticommutes with J1 , . . . Jn−2 if and only if γ lies in the submanifold Mn (m).
Definition 1.5. Inductively, let the Bott spaces Bn+1 (m) be the space of minimal geodesics in Bn (m) from Jn−1 to −Jn−1 (for those m where such a path exists). Then the previous discussion shows that the midpoint map gives an isometry ∼ Mn+1 (m) Bn+1 (m) =

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1.1. Bott periodicity. This leads to Bott’s original proof of the celebrated periodicity theorems as follows. Theorem 1.6 (Bott). Let dn be the minimal dimension of a Cn -module. Recall Bott’s notation for the cell decomposition of the space of loops. Then m d −1

Bn+1 (m) −−n−−→ Bn (m) This implies in particular that for m → ∞, the smallest dimension of a cell needed to get the loop space from the space of minimal geodesics also goes to infinity. Thus in the limit, one gets a homotopy equivalence Ωv Bn  Bn+1 . To make this precise, we define the spaces Mn (∞), which were studied by Milnor in [22], as the union of all Mn (m) inside O(∞) (which is the union of all O(m)). In fact, we only take the union over those m that are divisible by dn . With a similar definition of Bn = Bn (∞), the midpoint maps give homeomorphisms Bn (m) ≈ Mn (m) ∀ m = 1, 2, . . . (including m = ∞). between these Bott and Milnor spaces. Now by Morita equivalence Mn (∞) ≈ Mn+8 (∞) because Cn+8 is a real matrix ring over Cn . As a consequence, Corollary 1.7 (Bott). There are homeomorphisms and homotopy equivalences Bn ≈ Bn+8  Ω8 Bn and the homotopy groups of O(∞) are 8-periodic. These groups are known as the ‘stable’ homotopy groups of the orthogonal group because πi O(m) ∼ = πi O(∞) ∀i < m − 1. 2. Milnor spaces of Clifford module structures For each n ≥ 1, let Hn be a separable Hilbert space that is a Cn−1 -module such that each irreducible representation of Cn−1 appears with infinite multiplicity. Definition 2.1. For n = 1 we define Mfin 1 = {A ∈ O(H1 ) | A ≡ 1 modulo finite rank operators}. For n ≥ 2 the (finite rank) Milnor space Mfin n is the space of orthogonal operators J on Hn satisfying • J 2 = −½, or, equivalently, J is skew-adjoint, • J anticommutes with e1 , . . . , en−2 , • J − en−1 has finite rank. If we replace finite rank operators by compact operators in the above definition, we get the Milnor spaces Mn for n ≥ 1. The main result from the previous section is that the Bott spaces Bn are homeomorphic to the filtered union (with the direct limit topology) Mn (∞) =

∞ 

Mn (k · dn )

k=1

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i.e., they can be calculated in terms of spaces of Clifford algebra structures on R∞ , see Definition 1.3. Let M be the unique irreducible Cn−1 -module if n is not divisible by 4, respectively the sum of the two irreducible Cn−1 -modules if n is divisible by 4. We may choose embeddings m  Hn (m) := M ⊂ Hn ∀ m = 1, 2, . . . (including m = ∞). k=1

This is just the choice of an orthonormal basis in the case n = 1. We get an embedding of Hn (∞) (= R∞ for n = 1) into Hn and an induced inclusion of Mn (∞) into Mn . Theorem 2.2. For all n ≥ 1, this inclusion is a homotopy equivalence


Mn (∞) − → Mn . It will follow from Proposition 4.6 that the inclusions Mfin n → Mn are homotopy equivalences (and hence so are the inclusions Mn (∞)→Mfin n ). In fact, in the definition of the Milnor spaces, one can use any space of operators in between finite rank and compact operators (with the norm topology) to make this result true. Proof. We shall use Palais’ Theorem (A) from [23] which states the following. Let E be a Banach space and π(m) continuous projection operators onto finitedimensional subspaces E(m) ⊂ E(m + 1) which tend strongly to the identity as m → ∞. Then for any open subspace O of E, the inclusion map O(∞) → O is a homotopy equivalence. Here O(∞) is the direct limit of all O(m) := O ∩ E(m). In our setting, we have an extra parameter n ≥ 2, where we leave the easiest case n = 1 to the reader. We define En := {A ∈ K(Hn ) | A∗ = −A, Aei = −ei A for i = 1, . . . , n − 2} where K(Hn ) is the Banach space of compact operators on Hn with the norm topology. Consider the Cn−1 -linear orthogonal projections Hn  Hn (m) which by pre- and post-composition induce projections π(m) : En  En (m) := En ∩ K(Hn (m)). We may assume that Hn is the closure of Hn (∞) and hence that the π(m) tend strongly to the identity. For the open subset in Palais’ theorem we choose On := {A ∈ En | A + en−1 is invertible} and On (m) := On ∩ En (m). Then the map A → A + en−1 gives homeomorphisms of On , respectively On (m), with Gn := {B ∈ GL(Hn ) | B − en−1 ∈ En }   respectively Gn (m) := Gn ∩ GL Hn (m) . Palais’ theorem says that the inclusion induces a homotopy equivalence


Gn (∞) − → Gn . To prove our Theorem 2.2, we need to replace the general linear groups by the orthogonal groups in all of the above. This can be done by the polar decomposition of the invertible skew-adjoint operators B above. We may write B =U ·P =P ·U

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√ √ where P := B ∗ B = BB ∗ is positive and U is an orthogonal operator. Note that U is actually also skew-adjoint: U ∗ = (BP −1 )∗ = P −1 B ∗ = P −1 (−B) = −U Moreover, B anticommutes with ei for i = 1, . . . , n − 2 and therefore P commutes with these ei and hence U again anticommutes with them. Finally, it is easy to check that U − en−1 is a compact operator and hence U has the same properties as B and therefore lies in Mn ⊂ Gn . This implies that there is a commutative diagram Gn (∞)




/ Gn





 Mn (∞)

 / Mn

where the vertical maps are given by polar decomposition B → U . Since these are well known to be the homotopy inverses to the inclusion maps, the proof of Theorem 2.2 is completed.
3. Infinitesimal generators In this section we will review some basic facts about self-adjoint (unbounded) operators, reminding the reader of a nice topology on this space. Let H be a separable complex Hilbert space. Denote by Inf the set of all self-adjoint operators on H with compact resolvent. Note that we do not require an element D ∈ Inf to be densely defined. By ‘self-adjoint’ we mean that D defines a self-adjoint operator on the closure of its domain. The compact resolvent condition means that the spectrum of D consists of eigenvalues of finite multiplicity that do not have an accumulation point in R. Hence, if the domain of D is infinite-dimensional, the operator D on dom(D) is necessarily unbounded. Because of this, we will think of dom(D)⊥ as the eigenspace of D associated with the ‘eigenvalue’ ∞. Functional calculus gives a bijection, see, e.g., [10], Inf ↔ Hom(C0 (R), K) where the right-hand side is the space of all C ∗ -homomorphisms from (complexvalued) continuous functions on R that vanish at ∞ to the compact operators K on H. Note that both of these C ∗ -algebras do not have a unit. Below, we will also deal with C ∗ -algebras that do have a unit and in this case Hom will denote those C ∗ -homomorphisms that preserve the unit.  to be just as above, except that we do not require We define the space Inf the spectrum to be discrete (and the eigenspaces can be infinite-dimensional). The “Cayley transform” is defined for such operators by functional calculus using the M¨ obius transformation x+i c(x) := x−i which takes R ∪ {∞} to the unit circle S 1 . It defines the mapping from the very left to the very right in the following theorem. Theorem 3.1. There are bijections c

a b  ← → Hom(C0 (R), B) ∼ →U Inf = Hom(C(S 1 ), B) ←

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H. HOHNHOLD, S. STOLZ, AND P. TEICHNER

where B and U are the bounded, respectively unitary, operators on H. Moreover, the bijection b on the right, given by functional calculus, is a homeomorphism from the pointwise norm topology on Hom(C(S 1 ), B) to the operator norm topology on U.  the topology coming from the above bijections. Definition 3.2. We give Inf This is sometimes referred to as the generalized norm topology because of Lemma 3.4 below. Remark 3.3. Just like Inf has an interpretation in terms of configuration  spaces, by using the pattern of eigenvalues and their eigenspaces, the space Inf can be interpreted as the space of all projection-valued measures on R, see [24, Theorem VIII.6]. The fact that the operators may not be densely defined is reflected in the fact that the projection corresponding to all of R is not necessarily the identity but projects onto the domain. Thus the result becomes cleaner than in [24] where the map b is not onto. Theorem 3.1 is well known; we just need to collect various bits and pieces of the argument, for example from Rudin [25] or Reed – Simon [24]. These authors only define the adjoint of a densely defined operator D because otherwise the adjoint is not determined by the formula Dv, w = v, D∗ w In particular, self-adjoint operators are assumed to have dense domain. As a consequence, [25, Theorem 13.19] proves that the Cayley transform gives an inclusion of all densely defined self-adjoint operators onto the space of unitary operators without eigenvalue 1. If one allows nondense domains, i.e., eigenvalue ∞ (defining the adjoint also to be ∞ on that subspace), then the Cayley transform takes the eigenspace of ∞ to the eigenspace of 1 and therefore becomes onto all unitary  ↔ U. operators, i.e., gives the desired bijection Inf Proof of Theorem 3.1. The map a is given by functional calculus, which is well defined on self-adjoint operators that are densely defined. Since the functions  by defining f (D) to be zero on the f vanish at ∞ one can extend this for all D ∈ Inf orthogonal complement of the domain of D. For the second map, note that C(S 1 ) is obtained from C0 (R) by adding a unit ½ (and using the above M¨obius transformation c). We get an isomorphism between the two spaces of C ∗ -homomorphisms since we require that ½ maps to ½ (if the algebras have units). Finally, the map b is given by evaluating a homomorphism at the identity map z : S 1 → S 1 . It is clear that the composition from left to right is therefore the Cayley transform and hence a bijection. Recall that by Fourier decomposition, there is an isomorphism of complex C ∗ -algebras C(S 1 ) ∼ = C ∗ (Z) where Z is the infinite cyclic group, freely generated by an element z (which corresponds to the above identity z on S 1 ). It follows that C ∗ (Z) is free as a C ∗ -algebra on one unitary element z and hence C ∗ -homomorphisms out of it are just unitary elements in the target. Moreover, the bijection is given by evaluating functions on this unitary z which is our map b above. To show that b is a homeomorphism, we need to show that a sequence ϕn of C ∗ -homomorphisms converges if and only if ϕn (z) converges (in norm). By

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definition, the ϕn converge if ϕn (f ) converges (in norm) for all f , so one direction is obvious. For the other, assume that un := ϕn (z) converges to u ∈ U and note that ϕn (f ) = f (ϕn (z)) = f (un ). We want to show that f (un ) converges to f (u) and we claim that this is easy to check in the case when f is a Laurent polynomial. For the general case, pick
> 0 and choose a Laurent polynomial p = p(z) such that f − psup <
/3 and an N  0 such that p(un ) − p(u) <
/3 for n > N . Then for n > N we have       f (un ) − f (u) =  f (un ) − p(un ) + p(un ) − p(u) + p(u) − f (u)  <
and hence f (un ) converges to f (u). This argument is very similar to the one in [24, Theorem VIII.20(a)].
Lemma 3.4. The Cayley transform on bounded operators  ↔U Bsa ⊂ Inf  extends the operator is an open embedding, i.e., the generalized norm topology on Inf sa norm topology on B , the bounded self-adjoint operators on H. Again this result is well known, see for example [24, Theorem VIII.18]. Reed and Simon use the resolvent instead of the Cayley transform but this is just a different choice of M¨obius transformation, using x → (x + i)−1 instead of c. This has the effect that the image of R is not the unit circle but a circle of radius 12 inside the unit circle. Therefore, one does not get unitary operators but there is certainly no difference for the induced topology. Unfortunately, in the above Theorem VIII.18, Reed and Simon assume an additional property on the sequence considered, namely that it is uniformly bounded. It turns out, however, that this assumption is unnecessary, which is an easy consequence of Theorem VIII.23(b) in [24]. Remark 3.5. It is interesting to recall from [25, Theorem 13.19] that the Cayley transform can also be applied to symmetric operators, i.e., those that are formally self-adjoint and with Dom(D) ⊆ Dom(D∗ ). The result is an isometry U with Dom(U ) = Range(D + i · ½) and

Range(U ) = Range(D − i · ½)

D is closed if and only if U is closed and D is self-adjoint if and only if U is unitary. Using the Cayley transform and its inverse one sees that the self-adjoint extensions of D are in 1-1 correspondence with unitary isomorphisms between the orthogonal complements of Dom(U ) and Range(U ). In particular, self-adjoint extensions exist if and only if these complements have the same dimensions, usually referred to as the deficiency indices. An example to keep in mind is the infamous right shift, which is an isometry with deficiency indices 0 and 1. Thus its inverse Cayley transform has no self-adjoint extension. Let Inf fin be the space of all self-adjoint operators on H, with finite spectrum and multiplicity (not necessarily densely defined). Proposition 3.6. The Cayley transform induces the following bijections c

a b Inf ← → Hom(C0 (R), K) ∼ → U ∩ (K + C · ½) = Hom(C(S 1 ), K + C · ½) ← c

a b Inf fin ← → Hom(C0 (R), FR) ∼ → U ∩ (FR + C · ½) = Hom(C(S 1 ), FR + C · ½) ←

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where K and FR are the compact, respectively finite rank, operators on H. Moreover, the bijections b on the right, given by functional calculus, are homeomorphisms from the pointwise norm topology on the spaces of C ∗ -homomorphisms to the operator norm topology on U. Proof. The Cayley transforms give the bijections from the very left to the very right because one can read off the conditions of being compact, respectively finite rank, from the spectrum and multiplicities of the operators. These conditions are mapped into each other by definition of the spaces. The fact that the maps b are homeomorphisms is proved exactly as in Theorem 3.1.
We now have complete control over the topology on our various spaces. The  is homeomorphic to U and hence contractible by Kuiper’s theorem, largest space Inf whereas the subspaces Inf fin and Inf are homotopy equivalent (see Proposition 4.6) and have a very interesting topology. We shall now add some bells and whistles, like grading, real structure and Clifford action to make these spaces even more interesting. In a first step, assume that our complex Hilbert space H has a real structure, i.e., that H = HR ⊗R C for some real Hilbert space HR . If we think of the real structure (also known as complex conjugation) on H as a grading involution α (which has the property that the even and odd parts are isomorphic) then the above Proposition 3.6 leads to the following result. Proposition 3.7. The Cayley transform induces homeomorphisms c

Inf odd (H) ≈ O(HR ) ∩ (K + C · ½)

and

c

Inf fin odd (H) ≈ O(HR ) ∩ (FR + C · ½)

Here Inf odd (H) denotes the subspace of odd operators in Inf(H) (which are still C-linear ) and O(HR ) is the usual orthogonal group, thought of as the subgroup of real operators in the unitary group U(H). Proof. Since both sides have the subspace topology, it suffices to show that the Cayley transform is a bijection between the spaces given in the Proposition. An operator D in Inf(H) is odd if and only if Dα := αDα = −D. Since our grading involution α is C-antilinear, we also have iα = −i and therefore1

α D+i D+i −D − i α = = c(D) c(D) = c(D) = = D−i −D + i D−i Since the operators in U(H) that commute with complex conjugation are clearly those in the real orthogonal group O(HR ), we get c(D) ∈ O(HR ) and as before, c(D) − ½ is compact (respectively finite rank). Conversely, a similar calculation
shows that if c(D) is real then Dα = −D. Let Hn be as in the introduction, a graded real Hilbert space with a ∗-action of the real Clifford algebra Cn . For example, the above discussion is the case n = 1 if we define H1 = H with the grading and C1 = C-action as above. Note that in this case one can think of C-linear operators, say in K(H), as R-linear operators that commute with the C1 -action. This motivates the following definition. 1The notation (D±i)−1 best interpreted on each pair of eigenspaces V ⊕V λ −λ of D separately; there it definitely makes sense and that is all we care about here.

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Definition 3.8. We denote by Kn (respectively FRn ) the space of all Cn linear self-adjoint compact (respectively finite rank) operators on Hn , and by Inf n fin (respectively Inf fin n ) the subspace of Inf (respectively Inf ) that consists of all Cn -linear and odd operators. In order to extend Proposition 3.7 to Inf n we have to identify the image of these spaces under the Cayley transform in O(Hn ). We now assume the following model for our graded Hilbert space Hn . For n ≥ 1, let Hn be a real Hilbert space that is a Cnev -module and consider the graded Cn -module Hn := Hn ⊗C ev Cn ∼ = Hn ⊗R C. n

The last isomorphism should be interpreted as saying that the complex structure on Hn is given by the last basis element en ∈ Cn and that the grading can be thought of as corresponding to complex conjugation, just like in our previous discussion. Proposition 3.9. For all n ≥ 1 there are homeomorphisms fin Inf fin n ≈ Mn

and

Inf n ≈ Mn .

For n = 1 they are given by the Cayley transform D → c(D) and for n > 1 by D → en−1 c(D). Proof. We will show that there is a homeomorphism Inf n ≈ Mn which restricts to the desired homeomorphism on Inf fin n . The case n = 1 was discussed above because in this case we have by definition O(H1 ) ∩ (½ + K) = M1

O(H1 ) ∩ (½ + FR) = Mfin 1 .

and

Now, let n ≥ 2. Recall that the complex structure on Hn is given by en , hence the relation Den = en D gives the C-linearity of c(D). We claim that the relations Dei = ei D for the remaining n − 1 generators ei of Cn imply that the generators ei of Cn−1 satisfy ei c(D) = c(D)−1 ei . To see this, note that we have the relations ei (D ± i) = (D ∓ i)ei

and

ei (D ± i)−1 = (D ∓ i)−1 ei

which together yield ei c(D) = ei (D − i)(D + i)−1 = (D + i)(D − i)−1 ei = c(D)−1 ei . Note that all these are operators on Hn but that our odd operator D gives an action of c(D) on Hn . We assert that the same relation holds for this operator c(D) on Hn . First, note that since c(D) is C-linear, i.e., it commutes with en , we have en ei c(D) = c(D)−1 en ei . Next, one checks that under the isomorphism Cnev ∼ = Cn−1 the action of en ei ∈ Cn , i = 1, . . . , n − 1, corresponds to the automorphism ei ⊗ id of Hn ⊗ C. This together with the relation we computed for c(D) ⊗ id implies ei c(D) = c(D)−1 ei for i = 1, . . . , n − 1. Hence we see that the Cayley transform c gives a homeomorphism Inf n ≈ {A ∈ O(Hn ) | A ≡ 1 mod K(Hn ) and ei A = A−1 ei for i = 1, . . . , n − 1}. The space on the right-hand side is not quite Mn yet. However, we claim that it can be identified with Mn by associating to an operator A the complex structure J := en−1 A ∈ Mn .

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It is clear that J ≡ en−1 mod K(Hn ). Furthermore, J is indeed a complex structure: J 2 = en−1 Aen−1 A = en−1 AA−1 en−1 = −½. It remains to check that J anticommutes with the generators of Cn−2 . The following computation shows this claim using e˜i = en−1 ei : (en−1 ei )(en−1 A) = (en−1 ei )(A−1 en−1 ) = en−1 Aei en−1 = −(en−1 A)(en−1 ei ) where we have interpreted Hn as a module over Cn−2 via ∼ =

ev Cn−2 − → Cn−1 ,

e˜i → en−1 ei ,

with e˜1 , . . . , e˜n−2 denoting the standard generators of Cn−2 .




For later use, we define the real graded C ∗ -algebra S to be given by real-valued functions in C0 (R) with trival ∗ and with grading involution induced by x → −x (leading to the usual decomposition into even and odd functions). To motivate the use of self-adjoint operators, we make the following easy observation that comes from the above case n = 1. Lemma 3.10. Restriction to self-adjoint elements defines homeomorphisms Homgr (C0 (R), K(H)) ↔ Homgr (S, Ksa (H)) where H is a complex Hilbert space with grading involution as above and Homgr denotes grading preserving ∗-homomorphisms. The analogous statement holds for FR in place of K. Proof. Recall that S are just the real valued functions in C0 (R) (also known as the self-adjoint elements in this complex C ∗ -algebra) and that the grading involutions agree. Moreover, there is an isomorphism ∼ C0 (R) S ⊗R C = The same statements apply to K (respectively FR) and therefore the complexification map gives an inverse to the restriction map in the lemma.
The following result follows from the observation that for an odd operator D ∈ Inf, functional calculus leads to a grading preserving ∗-homomorphism f → f (D) from S to K. Vice versa, if this ∗-homomorphism is grading preserving then D must have been odd to start with. Proposition 3.11. Functional calculus induces the homeomorphisms Inf n ≈ Homgr (S, Kn )

and

Inf fin n ≈ Homgr (S, FRn ).

4. Configuration spaces The unbounded operators of the previous section can be visualized as configurations on the real line: an operator D ∈ Inf is completely determined by its eigenvalues and eigenspaces and hence by the map V that associates to λ ∈ R the subspace V (λ) on which D = λ. We call V a ‘configuration on R’, since V (λ) may be thought of as a label attached at λ ∈ R. Since slightly different spaces of configurations will appear in Section 8, we give a general definition that also covers the case considered there. Let Λ be a topological space equipped with an involution s and H a separable graded Hilbert space with grading involution α. A configuration on Λ indexed

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by orthogonal subspaces of H is a map V from Λ to the set of closed (ungraded) subspaces of H such that • the subspaces V (λ) are pairwise orthogonal • H is the Hilbert sum of the V (λ)’s • V is compatible with s and α, i.e., V (s(λ)) = α(V (λ)) for all λ ∈ Λ. Recall that closed subspaces of H correspond precisely to continuous self-adjoint projection operators on H. Hence we may interpret V as a map V : X → Proj(H) ⊂ B(H). To save space, we write Vλ := V (λ). Define supp(V ) := {λ ∈ Λ | Vλ = 0}. Definition 4.1. The space Conf(Λ; H) of configurations on Λ indexed by orthogonal subspaces of H is the set of all configurations V : Λ → Proj(H) equipped with the topology generated by the subbasis consisting of the sets       B(U, L) := V ∈ Conf(Λ; H)  VU := Vλ ∈ L, supp(V ) ∩ ∂U = ∅ , λ∈U

where U and L range over all open subsets U ⊂ Λ and L ⊂ Proj(H). We will need the following variations. Let Θ ⊂ Λ be a subspace that is preserved under the involution s. Define Conf(Λ, Θ; H) ⊂ Conf(Λ; H) to be the subspace of configurations V such that Vλ has finite rank for all λ ∈ Θc := Λ \ Θ and such that the subset of all λ ∈ Θc with V (λ) = 0 is discrete in Θc .2 Replacing the discreteness condition by requiring that there should be only finitely many λ ∈ Θc with Vλ = 0 we obtain the space Conf fin (Λ, Θ; H) of configurations that are ‘finite away from Θ’. Finally, if C is an R-algebra and H is a C-module, we can replace subspaces of H by C-submodules in order to obtain spaces Conf C (Λ, Θ; H). If C is graded then we assume that H is a graded C-module, but the subspaces V (λ) are still ungraded; only those for which s(λ) = λ are graded modules over C. Our main examples will be the Clifford algebras C = Cn . Examples 4.2. Consider the one-point compactification R of R equipped with the involution s(x) := −x. Define Conf n := Conf Cn (R, {∞}; Hn ), where Hn is the graded Cn -module from the previous section. We will see in Proposition 4.4 that Conf n gives a different model for the space Inf n of unbounded operators introduced above. The homeomorphism Inf n → Conf n is given by mapping D ∈ Inf n to the configuration defined by associating to λ ∈ R the λ-eigenspace Vλ of D. Here we let V∞ := dom(D)⊥ . Since D has compact resolvent, the set of λ ∈ R with Vλ = 0 is indeed discrete in R and each eigenspace Vλ , λ ∈ R, is finite-dimensional. The relation V (s(λ)) = α(V (λ)) corresponds to D being odd. In order to get a better feeling for the topology on Conf n , let us describe a neighborhood basis for each configuration in Conf n . This will also be useful for the proof that the map Inf n → Conf n is a homeomorphism. We begin by pointing out that the topology on Conf n is generated by the sets B(U, L), where U ⊂ R is bounded. To see this, note that, by definition of Conf n , ∞ ∈ supp(V ) for all V ∈ Conf n . Hence B(U, L) = ∅ whenever ∞ ∈ ∂U so that 2This terminology will be convenient for our purposes. However, we should point out that with our notation Conf(Λ; H) = Conf(Λ, Λ; H) and not Conf(Λ; H) = Conf(Λ, ∅; H).

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the case of unbounded U ⊂ R is irrelevant. Furthermore, if ∞ ∈ U we can use B(U, L) = B(U c , 1 − L) to describe B(U, L) in terms of U c := R \ U . Thus it is sufficient to consider B(U, L) for U ⊂ R bounded. Definition 4.3. Let V ∈ Conf n and let K be a (large) positive real number such that VK = 0. Let BK (0) be the ball of radius K around 0 and denote by λ1 , . . . , λlK the numbers in BK (0) such that Vλi = 0. Let δ > 0 and ε > 0 be (small) real numbers; we may choose δ so small that Bδ (λi ) ∩ Bδ (λj ) = ∅ for i = j. Denote by VK,δ,ε the set of all configurations W such that VBδ (λi ) − WBδ (λi )  < ε for all i and such that Wλ = 0 for all λ ∈ BK (0) that do not lie in one of the balls Bδ (λi ). Thus, an element W ∈ VK,δ,ε almost looks like V on BK (0): the only thing that can happen is that a label Vλ ‘splits’ into labels Wλj with |λ − λj | small  W (< δ) and λj close to Vλ (< ε). The VK,δ,ε indeed form a neighborhood j n basis of V : assume V ∈ k=1 B(Uk , Lk ), with Uk ⊂ R bounded. Choose K as n n above with k=1 Uk ⊂ BK (0). Picking δ > 0 so small that Bδ (λi ) ⊂ k=1 Uk for all i it follows easily using the triangle inequality that for ε > 0 sufficiently small  VK,δ,ε ⊂ ni=1 B(Ui , Li ). In particular, we see that the topology on Conf n controls configurations well on compact subsets of R but not near infinity. The discussion also shows that Conf n is first countable since we may choose K, δ, and ε in Q. ¯ {∞}; Hn ) and any function f ∈ S, we can Given any V ∈ Conf n = Conf Cn (R, define a Cn -linear operator f (V ) on Hn by requiring that f (V ) has eigenvalue f (λ) exactly on Vλ . This operator is always compact and it is of finite rank if and only if fin V ∈ Conf fin n := Conf Cn (R, {∞}; Hn ) ⊂ Conf n , the subspace of configurations that are finite away from {∞}. Moreover, the relation V (s(λ)) = α(V (λ)) corresponds to f → f (V ) being grading preserving. Proposition 4.4. Functional calculus F (V )(f ) := f (V ) gives homeomorphisms ≈

→ Homgr (S, Kn ) F : Conf n −

and

≈

Conf fin → Homgr (S, FRn ). n −

Combining this result with Proposition 3.11 we obtain as a corollary Inf n ≈ Conf n

and

fin Inf fin n ≈ Conf n .

Proof. It is clear that F is a bijection because the map that identifies operators with the eigenspaces and eigenvalues is obviously a bijection and it is the composition of F with a homeomorphism. Since Conf n is first countable, we can check the continuity of F on sequences. To do so, assume Vn → V and fix f ∈ S. We have to prove f (Vn ) → f (V ). Given ε > 0, choose K > 0 such that |f (x)| < ε if |x| > K. Since the continuous map f is automatically uniformly continuous on compact sets, we can find a δ > 0 such that for all x ∈ BK (0) we have |f (x) − f (y)| < ε provided |x − y| < δ. The assumption Vn → V tells us that Vn ∈ VK,δ,ε for large n.

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The claim now follows from the following estimate that holds for all W ∈ VK,δ,ε :       f (λ)V − f (µ)W f (V ) − f (W ) =  λ µ  µ∈R λ∈R    

    ≤ f (µ)Wµ  + 2ε f (λ)Vλ −   λ∈BK (0) Vλ =0

µ∈Bδ (λ)

≤ #{λ ∈ BK (0) | Vλ = 0} ·



 max f (λ) · ε + ε + 2ε

λ∈BK (0)

≤ C · ε, where the constant C only depends on f and V . The first inequality follows by rearranging the terms and using the triangle inequality together with |f (x)| < ε for |x| > K. The second inequality follows from      f (λ)Vλ −  f (µ)W µ     µ∈Bδ (λ)    ≤ f (λ)(Vλ − WBδ (λ) ) +  (f (λ) − f (µ))Wµ   µ∈Bδ (λ)

≤ max f (λ) · ε + ε. λ∈BK (0)

The space Homgr (S, Kn ) is also first countable. This follows since f (Vn ) → 2 2 f (V ) for all f ∈ S if and only if this is the case for f (x) = e−x and f (x) = xe−x (see −1 on sequences the proof of Lemma 5.12). Thus we can check the continuity of F as well. Assume f (Vn ) → f (V ) for all f ∈ S and V ∈ B(U, L). We have to show Vn ∈ B(U, L) for n sufficiently large. More explicitly: supp(Vn ) ∩ ∂U = ∅ for n large and limn→∞ (Vn )U − VU  = 0.  Note that for an accumulation point γ ∈ R of the set n supp(Vn ) we must have Vγ = 0, because otherwise we would also have f (Vn ) − f (V ) ≥ 12 for infinitely many n if we choose f to be a bump function with f (γ) = 1 that is concentrated near γ. This together with supp(V ) ∩ ∂U = ∅ implies that there is a neighborhood v(∂U ) of ∂U such that (Vn )λ = 0 for λ ∈ v(∂U ) occurs only for finitely many n. In particular, supp(Vn ) ∩ ∂U = ∅ for n large. Now, choose f ∈ S such that f |R\U = 0 and f |U\v(∂U) = 1. By construction, f (Vn ) = χU (Vn ) for n large, where χU denotes the indicator function for U . The same identity holds for V and hence we can conclude lim (Vn )U − VU  = lim χU (Vn ) − χU (V ) = lim f (Vn ) − f (V ) = 0.

n→∞

n→∞

n→∞




This completes the proof.

Remark 4.5. A continuous map f : Λ → Λ that commutes with the involutions on Λ and Λ induces  Vλ . f∗ : Conf(Λ; H) → Conf(Λ ; H), (f∗ (V ))λ
:= λ∈f −1 (λ
)

We show that the map f∗ is continuous under the assumption that the space Λ is normal. Let V ∈ f∗−1 (B(U, L)). From the definition of B(U, L) we find supp(f∗ V )∩

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H. HOHNHOLD, S. STOLZ, AND P. TEICHNER

∂U = ∅. Since Λ is normal, there is an open neighborhood NV of supp(f∗ V ) such that NV ∩ ∂U = ∅. Unravelling the definitions one finds V ∈ B(f −1 (U ), L) ∩ B(f −1 (NV ), {idH }) ⊂ f∗−1 (B(U, L)) so that f∗−1 (B(U, L)) is a neighborhood of V . Thus f∗−1 (B(U, L)) is open. The additional properties that we required in the definition of the spaces Conf(Λ, Θ; H) are not stable under pushforward. In order to get an induced map we have to require f to be ‘nice’. For example, if f : (Λ, Θ) → (Λ, Θ ) is a proper map between locally compact Hausdorff spaces, we get an induced map f∗ : Conf(Λ, Θ; H) → Conf(Λ, Θ ; H). A sightly more complicated argument along the same lines can be used to show that a homotopy h : Λ × [0, 1] → Λ induces a homotopy H : Conf(Λ, H) × [0, 1] → Conf(Λ, H), at least if Λ is compact. The following result implies that the ‘finite’ and ‘nonfinite’ versions of the spaces considered in the previous chapters are homotopy equivalent. Proposition 4.6. The inclusion Conf fin n → Conf n is a homotopy equivalence. Proof. Consider the family of maps ht : R → R defined by  x/(1 − t|x|) if x ∈ (−1/t, 1/t) ht (x) := ∞ else. These induce a homotopy Ht := (ht )∗ : Conf n → Conf n from the identity on Conf n to H1 . Note that the image of H1 equals Conf fin n . → Conf is a homotopy equivalence with Thus, we see that the inclusion Conf fin n n
homotopy inverse H1 : Conf n → Conf fin n . Remark 4.7. We will later consider the space colimk→∞ Conf (k) n , where Conf (k) is the subspace of configurations V ∈ Conf such that V =  0 for at n λ n most k numbers λ with 0 < λ < ∞. As a set, colimk→∞ Conf (k) is just Conf fin n n , fin but the topology has more open sets than the topology of Conf n . We claim that colim Conf (k) n → Conf n k→∞

is also a homotopy equivalence. The same homotopy as in the proof of Proposition 4.6 can be used. This works because the map H1 : Conf n → colimk→∞ Conf (k) n is still continuous. This follows from the observation that  C : Conf n → N, V → C(V ) := dim Vλ −1≤λ≤1

is locally bounded so that for every V ∈ Conf n we can find an open neighborhood N of V such that H1 (N ) ⊂ Conf (k) n for some k (since H1 moves all labels outside (−1, 1) to ∞). Since continuity can be checked locally and since on Conf (k) n the topology induced from Conf n and the colimit topology coincide, it follows that H1 : Conf n → colimk→∞ Conf (k) n is continuous. fin In particular, we see that the identity map id : colimk→∞ Conf (k) n → Conf n is a homotopy equivalence.

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The same argument applies in the case of the filtration Xk of Conf fin n given by fin ⊂ Conf is the subspace of configurations the dimension of a configuration: if X k n V with dim(V ) := dimCn ( λ∈R Vλ ) ≤ 2k, then the identity map colimk→∞ Xk → Conf fin n is a homotopy equivalence. 5. Super-semigroups of operators In this section we will define super-semigroups of operators (SGOs) using as little supermathematics as possible. We will only need basic definitions and results from the theory of super manifolds, as can be found in Chapter 2 of [6]. Supermanifolds are particular ringed spaces, i.e., topological spaces together with a sheaf of rings, and morphisms are maps of ringed spaces. The local model for a super manifold of dimension (p|q) is Euclidean space Rp equipped with the sheaf of commutative super R-algebras U → C ∞ (U ) ⊗ Λ∗ (Rq ). This ringed space is the super manifold Rp|q . Definition 5.1. A super manifold M of dimension (p|q) is a pair (|M |, OM ) consisting of a (Hausdorff and second countable) topological space |M | together with a sheaf of commutative super R-algebras OM that is locally isomorphic to Rp|q . To every super manifold M there is an associated reduced manifold M red := (|M |, OM /nil) obtained by dividing out nilpotent functions. By construction, this gives a smooth manifold structure on the underlying topological space |M | and there is an inclusion of super manifolds M red → M . For example, (Rp|q )red = Rp . The main invariant of a super manifold M is its ring of functions C ∞ (M ), defined as the global sections of the sheaf OM . For example, C ∞ (Rp|q ) = C ∞ (Rp )⊗ Λ∗ (Rq ). It turns out that the maps between super manifolds M and N are just given by grading preserving algebra homomorphisms between the rings of functions: Hom(M, N ) ∼ = HomAlg (C ∞ (N ), C ∞ (M )) Example 5.2. Let E → M be a real vector bundle of fiber dimension q over the smooth manifold M p . Then (M, Γ(Λ∗ E)) is a super manifold of dimension (p|q). Batchelor’s theorem [3] says that every super manifold is isomorphic (but not canonically) to one of this type. This result does not hold in analytic categories, and it shows that, in the smooth category, super manifolds are only interesting if one takes their morphisms seriously and doesn’t just consider isomorphism classes. Definition 5.3. Define the ‘twisted’ super Lie group structure on R1|1 by m : R1|1 × R1|1 → R1|1 ,

(t, θ), (s, η) → (t + s + θη, θ + η).

This super Lie group plays a special role in supergeometry, the reason being the particular structure of its super Lie algebra: Lie(R1|1 ) ∼ = R[D] is the super Lie algebra generated freely by one odd generator D. Thus, R1|1 may be considered the odd analogue of the Lie group R. For example, integrating an odd vector field on a super manifold M leads to a flow M × R1|1 → M , and formulating the flow property involves the ‘twisted’ group structure.

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H. HOHNHOLD, S. STOLZ, AND P. TEICHNER 1|1

From the definition of m it is clear that the open sub-super manifold R>0 defined by the inclusion R>0 ⊂ R inherits the structure of a super-semigroup.3 Now we can already guess what a SGO should be: just as an ordinary semigroup of operators is a homomorphism from R>0 to an algebra of operators, a super1|1 semigroup of operators will be a homomorphism from the super-semigroup R>0 to a (Z2 -graded) operator algebra. In order to make sense of such a homomorphism, we will consider the latter to be a generalized super-semigroup using the ‘functor of points’ formalism (see [6, Sections 2.8 – 2.9]). Note that we, implicitly, already used the ‘functor of points’ language when writing down the group law m. The formula above tells us what the product of two elements in the group Hom(S, R1|1 ) is. Since the rule holds functorially for all super manifolds S, this defines the map m by the Yoneda lemma. Finally, we would like to remark that the structure of Lie(R1|1 ) and the existence of an odd infinitesimal generator D for a SGO Φ that we will prove below are closely related: D is nothing but the image of D under the derivative of Φ. However, making this precise requires some work (note that Φ maps to an infinitedimensional space!). We will avoid such problems altogether: the super Lie algebras do not appear in our argument. 5.1. Generalized super manifolds and super Lie groups. We will use the following, somewhat primitive, extension of the notion of super manifolds: Definition 5.4. A generalized super manifold M is a contravariant functor from super manifolds to sets.4 Similarly, if M takes values in the category of (semi)groups, we call it a generalized super-(semi )group. Morphisms in all these categories are natural transformations. Examples 5.5. (1) The Yoneda lemma implies that super manifolds are embedded as a full subcategory in generalized super manifolds by associating to a super manifold M the functor S → M (S) := Hom(S, M ). The analogous statement holds for super-(semi)groups. For example, we will con1|1 sider R>0 as a generalized super-semigroup by identifying it with the contravariant functor 1|1 S → Hom(S, R>0 ) from super manifolds to semigroups. (2) Every Z2 -graded real Banach space B = B0 ⊕ B1 may be considered as a generalized super manifold as follows. We define the value of the functor B on a superdomain U = (|U |, C ∞ ( )[θ1 , . . . , θq ]) ⊂ Rp|q to be B(U ) := (C ∞ (|U |, B)[θ1 , . . . , θq ])ev . The superscript ev indicates that we pick out the even elements, so that an element f ∈ B(U ) is of the form  fI θ I f= I 3A super (Lie) semigroup is a super manifold M together with an associative multiplication

M × M → M . In terms of the functor of points language: the morphism sets Hom(S, M ) carry semigroup structures, functorially in S. 4We use this simple notion here in order to avoid dealing with infinite-dimensional super manifolds.

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 where I ⊂ {1, . . . , q} and θ I := j∈I θj and each fI is a smooth map |U | → B|I| . For a map ϕ : U  → U between superdomains, the map B(ϕ) is defined using the formal Taylor expansion, just as in the case of usual super manifolds. This functor on superdomains may be extended to the whole category of super manifolds by gluing. (3) If B is a Z2 -graded Banach algebra, B(U ) is an algebra and thus B is a generalized super-semigroup. Again, B may be extended to all super manifolds by gluing. Remark 5.6. Giving a morphism from an ordinary super manifold T to a generalized super manifold B amounts to prescribing the image of the universal element id ∈ Hom(T, T ) in B(T ). Hence B(T ) is exactly the set of morphisms from T to B. Now assume that, in addition, T and B carry super-(semi)group structures. A map Φ : T → B is a homomorphism if / Hom(S, T )

Hom(S, T ) × Hom(S, T ) Φ×Φ

Φ

 B(S) × B(S)

 / B(S).

commutes for all super manifolds S. Again, it suffices to check the commutativity for the universal element pr1 × pr2 ∈ Hom(T × T, T ) × Hom(T × T, T ). Definition 5.7. Let H be a Z2 -graded Hilbert space, and denote by B(H) the Banach algebra of bounded operators on H equipped with the Z2 -grading inherited from H. (1) A super-semigroup of operators on H is a morphism of generalized supersemigroups 1|1 Φ : R>0 → B(H). As explained in the previous remark, Φ is of the form A + θB, where A : R>0 → B ev (H) and

B : R>0 → B odd (H)

are smooth maps. The homomorphism property amounts to certain relations between A and B (cf. the proof of Proposition 5.9). (2) If K ⊂ B(H) is a subset, we say Φ is a super-semigroup of operators with values in K if the images of A and B are contained in K. (3) If H is a module over the Clifford algebra Cn , we say Φ is Cn -linear if it takes values in Cn -linear operators. Examples 5.8. SGOs arise in a natural way from Dirac operators. We give two examples of that type and then extract their characteristic properties to describe a more general class of examples. The verification of the SGO properties for these more general examples also includes the case of Dirac operators. (1) Let D be the Dirac operator on a closed spin manifold X. There is a corresponding SGO on the Hilbert space of L2 -sections of the spinor bundle S over X. It is given by the super-semigroup of operators 1|1

R>0 → B(L2 (S)),

(t, θ) → e−tD + θDe−tD (= e−tD 2

2

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2

+θD

)

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and takes values in the compact, self-adjoint operators K sa (L2 (S)) ⊂ B(L2 (S)). (2) If dim X = n, one can consider the Cn -linear spinor bundle and the associated Cn -linear Dirac operator (see [17, Chapter 2, §7]). Using the same formula as in the previous example one obtains a Cn -linear SGO. (3) Now, let H be any Z2 -graded Hilbert space. For any closed subspace V∞ ⊂ H invariant under the grading involution and any odd, self-adjoint operator D ⊥ with compact resolvent, there is a unique super-semigroup of self-adjoint, on V∞ compact operators Φ = A + θB defined (using functional calculus) by A(t) = e−tD

2

B(t) = De−tD

2

and

⊥ on V∞

and A(t) = B(t) = 0 on V∞ . The first thing to check is that the maps A and B are indeed smooth; this follows easily using the fact that the map R>0 → C0 (R), 2 t → e−tx , is smooth. Since D is self-adjoint, the same holds for A and B. Finally, we have to show that Φ is a homomorphism. Let t, θ, s, η be the usual coordinates 1|1 1|1 on R>0 ×R>0 . It suffices to consider the universal element pr1 × pr2 = (t, θ)×(s, η). The computation, which, of course, heavily uses that odd coordinates θ and η square to zero, goes as follows (cf. [28, p. 38]): Φ(t, θ)Φ(s, η) = (e−tD + θDe−tD )(e−sD + ηDe−sD ) 2

2

2

2

= e−tD e−sD + e−tD ηDe−sD + θDe−tD e−sD + θDe−tD ηDe−sD 2

2

2

2

2

2

2

2

= e−(t+s)D + (θ + η)De−(t+s)D + θDηDe−(t+s)D 2

2

2

= (1 − θηD2 )e−(t+s)D + (θ + η)De−(t+s)D 2

2

= e−(t+s+θη)D + (θ + η)De−(t+s+θη)D 2

2

= Φ(t + s + θη, θ + η) The second to last equality uses the typical Taylor expansion in supergeometry. We call D the infinitesimal generator of Φ. We will see presently that every super-semigroup of self-adjoint, compact operators has a unique infinitesimal generator and is hence one of our examples. Note that if V∞ is a Cn -submodule and if D is Cn -linear, then A and B will also be Cn -linear. Next, we will construct infinitesimal generators for super-semigroups of operators. We restrict ourselves to the compact, self-adjoint case, which makes the proof an easy application of the spectral theorem for compact, self-adjoint operators. However, invoking the usual theory of semigroups of operators it should not be too difficult to prove the result for more general SGOs. Proposition 5.9. Every super-semigroup Φ of compact, self-adjoint operators on a Z2 -graded Hilbert space H has a unique infinitesimal generator D as in Example 3 above and is hence of the form Φ(t, θ) = e−tD + θDe−tD . 2

2

If Φ is Cn -linear, so is D. We need the following technical lemma:

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Lemma 5.10. Let A, B : R>0 → Ksa (H) be smooth families of self-adjoint, compact operators on the Hilbert space H, and assume that the following relations hold for all s, t > 0: (1)

A(s + t) = A(s)A(t)

(2)

B(s + t) = A(s)B(t) = B(s)A(t)

(3)

A (s + t) = −B(s)B(t).

Then H decomposes uniquely into orthogonal subspaces Hλ , λ ∈ R := R ∪ {∞}, such that on Hλ 2 2 A(t) = e−tλ and B(t) = λe−tλ (where we set e−∞ = 0, ∞ · e−∞ = 0). For λ ∈ R, the dimension of Hλ is finite. Furthermore, the subset of λ in R with Hλ = 0 is discrete. Proof. The identities (1) – (3) above show that all operators A(s), B(t) commute. We apply the spectral theorem for self-adjoint, compact operators to obtain a decomposition of H into simultaneous eigenspaces Hλ of the A(s) and B(t); the label λ takes values in R and will be explained presently. We define functions Aλ , Bλ : R>0 → R by A(t)x = Aλ (t)x and

B(t)x = Bλ (t)x

for all x ∈ Hλ .

Clearly, Aλ and Bλ are smooth and satisfy the same relations as A and B. Aλ ≤ 0, hence Aλ is From (1) we see that Aλ is nonnegative, and (3) shows  n decreasing. On the other hand, (1) implies Aλ (1/n) = Aλ (1), so that Aλ (0) := lim Aλ (t) exists and equals 0 or 1. t→0

In the first case we conclude Aλ ≡ 0 and thus also Bλ ≡ 0; the label of the corresponding subspace is λ = ∞. In the second case, we have Aλ (1) = 0 and using (1) again we compute Aλ (s + t) − Aλ (s) Aλ (1) lim Aλ (1) t→0 t Aλ (1 + t) − Aλ (1) Aλ (s) lim = −λ2 Aλ (s), = Aλ (1) t→0 t

Aλ (s) =

where λ2 := −Aλ (1)/Aλ (1) defines the label λ up to choice of a sign. By uniqueness of solutions of ODEs, we must have Aλ (t) = e−tλ . 2

Finally, (3) gives Bλ (t) = λe−tλ , picking the appropriate sign for λ. 2




Proof of Proposition 5.9. Let Φ = A + θB be a super-semigroup of com1|1 1|1 pact, self-adjoint operators. As before, we consider U = R>0 ×R>0 with coordinates t, θ, s, η. For the universal element pr1 × pr2 = (t, θ) × (s, η) the homomorphism property of Φ gives that Φ(t + s + θη, θ + η) = A(t + s + θη) + (θ + η)B(t + s + θη) = A(t + s) + A (t + s)θη + (θ + η)(B(t + s) + B  (t + s)θη) = A(t + s) + θB(t + s) + ηB(t + s) + θηA (t + s)

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equals Φ(t, θ)Φ(s, η) = (A(t) + θB(t))(A(s) + ηB(s)) = A(t)A(s) + θB(t)A(s) + ηA(t)B(s) − θηB(t)B(s). Comparing the coefficients5 yields exactly the relations in Lemma 5.10. Using the corresponding decompostion of H into subspaces Hλ we define the operator D by letting D = λ on Hλ . From the construction it is clear that D is the desired infinitesimal generator. Since A is even and B is odd, it follows that D is an odd
operator. If Φ is Cn -linear, so is D. We can finally give the definition promised in part (5) of our Main Theorem. Definition 5.11. Let H be a Z2 -graded Hilbert space and C ⊂ B(H) a subspace of the algebra of bounded operators on H. We denote by SGO(C) the set of super-semigroups of operators with values in C and in particular SGOn := SGO(Kn ) and

SGOfin n := SGO(FRn )

(the subspaces Kn and FRn were defined in Definition 3.8). We endow SGO(C) with the topology of uniform convergence on compact subsets, i.e., Φn = An + θBn → Φ = A + θB if and only if for all compact K ⊂ R>0 we have An (t) → A(t) and

Bn (t) → B(t)

uniformly on K

with respect to the operator norm on B(H). We will now relate the spaces SGOn with our configuration spaces Conf n . We have a triangle (and an analogous one for finite rank operators) Homgr (S, Kn ) SGO9n o 99 x< xx 99 x x x I 99 xx F  Conf n , R

where I maps a super-semigroup of operators to its infinitesimal generator, F is given by functional calculus, F (D)(f ) := f (D), and R is given by R(ϕ) := ϕ(e−tx ) + θϕ(xe−tx ). 2

2

Proposition 5.12. The maps I, F , and R are homeomorphisms, and similarly for finite rank operators. Proof. From the previous discussion it is clear that the composition of the three arrows is the identity no matter where in the triangle we start. We already know from Proposition 4.4 that F is a homeomorphism. We complete the proof by showing that R is a homeomorphism. 5Just to make the formal aspect of this computation clearer, we would like to point 1|1 1|1 out that the considered identity is an equation in the algebra K sa (H)(R>0 × R>0 ) = ∞ sa ev C (R>0 × R>0 , K (H))[θ, η] .

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The continuity of R−1 follows from the following assertion. We claim that we have convergences of operators f (Dn ) → f (D) for all f ∈ C0 (R) if and only if the following two sequences converge: e−Dn → e−D 2

2

and

Dn e−Dn → De−D . 2

2

The first obviously implies the second condition. To see the converse, note that the assumption implies that f (Dn ) → f (D) for all f that can be written as a 2 2 polynomial in the functions e−x and xe−x . Furthermore, the Stone – Weierstraß 2 2 Theorem implies that e−x and xe−x generate C0 (R) as a C ∗ -algebra so that the set of such f is dense in C0 (R). Using that from f (D) ≤ f  for all D and the triangle inequality we can deduce that f (Dn ) → f (D) holds for all f ∈ C0 (R). The continuity of R amounts to showing that if f (Dn ) → f (D) for all f , then 2 2 2 −tD2n e → e−tD and Dn e−tDn → De−tD uniformly for all t in a compact subset K ⊂ R>0 . As before, we can use f (D) ≤ f  and the triangle inequality to see that for a given ε > 0 we can find N such that not only do we have f (Dn )−f (D) ≤ ε for all n ≥ N , but that this estimate also holds for all g in a small neighborhood of f . This together with the compactness of K and the continuity of the maps 2 2
t → e−tx and t → xe−tx implies the claim. Remark 5.13. The arguments in the last parts of the proof can be used to show that we could also have equipped SGOn with the topology that controls all derivatives of a super-semigroup map Φ and still would have obtained the same topological space. We find this interesting, because this is the topology that one usually considers on spaces of smooth maps. 5.2. Super-semigroups of operators and Euclidean field theories. In the context of supersymmetric Euclidean field theories of dimension (1|1) slight variations of the spaces SGOn appear. We conclude this chapter by describing these spaces and showing that they have the same homotopy type as the spaces SGOn . More information about the axiomatic definition of supersymmetric EFTs will be given in the next chapter. Definition 5.14. Let TC(Hn ) be the Banach algebra of trace class operators on Hn , equipped with the trace or nuclear norm. Define SGOTC n to be the space of 1|1 super-semigroup homomorphisms R>0 → TC(Hn ) with values in self-adjoint Cn linear operators. As before, we consider the topology given by uniform convergence on compact subsets K ⊂ R>0 . In the same way, we define SGOHS n , where TC(Hn ) is replaced by HS(Hn ), the Banach algebra of Hilbert-Schmidt operators on Hn , equipped with the Hilbert – Schmidt norm. Examples 5.15. An SGO defined by an infinitesimal generator D lies in SGOTC respectively SGOHS if the eigenvalues of D converge to infinity sufficiently fast. This is, for example, the case for Dirac operators on closed spin manifolds, cf. [17, Chapter 3, §5]. Proposition 5.16. The injections SGOTC n → SGOn

and

SGOHS n → SGOn

are homotopy equivalences.

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HS Proof. We give the proof for SGOTC n , the argument for SGOn being very similar. The basic observation we use is that on operators of rank at most k the operator norm and the nuclear norm are equivalent. More precisely,

T  ≤ T nuc ≤ k · T  k ≥ · · · ≥ sk ≥ 0 for if rank(T ) ≤ k. Namely, if T = i=1 si ·, ei fi with s1 orthonormal systems ei and fi , then T  = s1 and T nuc = ki=1 si ≤ ks1 . Denote by S(k) ⊂ SGOTC n the subspace of SGOs whose infinitesimal generator has domain of dimension ≤ 2k. By our basic observation, the topology on S(k) is the same as the one we get by considering it as a subspace of SGOn . Using the identification I : SGOn ≈ Conf n and Remark 4.7 we see that S := colimk→∞ S(k)  SGOn . Hence, if we can show that i : S → SGOTC n is a homotopy equivalence, then the same is true  and H TC by the commutative diagram for SGOTC → SGO . Define homotopies H n n  / SGOn × [0, 1] ≈ / Conf n ×[0, 1] SGOTC n × [0, 1]
H

H TC

  SGOTC n

H

 / SGOn

≈

 / Conf n ,

where H is the homotopy used in the proof of Proposition 4.6. It is easy to see that the map i−1 ◦ H1TC : SGOTC → S is continuous. We claim that it is a homotopy n inverse for i. The argument works exactly as in Remark 4.7. The only thing to check is that the homotopy H TC is continuous (note that now we are using the topology that comes from the nuclear norm). For t = 0, the continuity follows from  and the fact that H TC locally maps to S(k) , for some k, as long the continuity of H as t = 0. Let us now consider the case t = 0. Assume Φm → Φ and tm → 0. For C ⊂ R, we denote by ΦC the SGO defined by the infinitesimal generator obtained from I(Φ) by omitting all labels in R \ C. Let K ⊂ R>0 compact, ε > 0. Write Φ := sups∈K (A(s)nuc + B(s)nuc ). Since Φ ∈ SGOTC , we can choose a big symmetric interval C = [−κ, κ] ⊂ R with I(Φ)κ = 0 such that Φ − ΦC  < ε. C  Note that Φm → Φ also implies ΦC m → Φ . The continuity of H implies that the TC (k) restriction of H to S × [0, 1] is continuous for all k. In particular, we have C H TC (ΦC m , tm ) → Φ . Now, choose N ∈ N such that for all m > N Φ − Φm  < ε,

ΦC − ΦC m  < ε,

C and H TC (ΦC m , tm ) − Φ  < ε.

It then follows that for all m > N TC |Φ − H TC (Φm , tm ) ≤ Φ − ΦC  + ΦC − H TC (ΦC (ΦR\C m , tm ) + H m , tm )

≤ 2ε + Φm − Φ + Φ − ΦC  + ΦC − ΦC m ≤ 5ε R\C

R\C

where we used that H TC (Φm , tm ) ≤ Φm  = Φm − ΦC m . This inequality √ follows directly from the definition of H TC (provided we choose κ ≥ mins∈K 1/ 2s), so that H TC is which we can certainly do). Hence, H TC (Φm , tm ) → Φ in SGOTC n continuous.
Remark 5.17. In [28] the notation 1|1-EFT−n was used for the space SGOHS n . In this paper we want to reserve the notation 1|1-EFT for the “correct” notion of

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Euclidean field theories: we give a geometric definition of supersymmetric positive Euclidean field theories of dimension (1|1) in the next section. It turns out that the resulting classifying spaces are homotopy equivalent to SGOTC n . 6. Supersymmetric Euclidean field theories In this section we outline how the spaces SGOTC of super-semigroups of selfn adjoint trace-class operators, studied in the previous section, are related to supersymmetric (1|1)-dimensional positive Euclidean field theories. Here we shall only treat the case n = 0, so there is no Clifford algebra Cn in the game. If n = 0, we would have to explain twisted Euclidean field theories that are defined in terms of certain 2-functors between 2-categories. This would lead too far afield in this paper and in the end, it would only give a different way in which the Clifford algebras Cn arise. The Atiyah – Segal definition of a d-dimensional topological field theory (TFT) is in terms of a symmetric monoidal functor E : d-Bord → Vect, or shorter, E ∈ Fun⊗ (d-Bord, Vect) The target of the functor E is the category Vect of vector spaces with tensor product. The domain category d-Bord is the bordism category whose objects are closed (d − 1)-manifolds and whose morphisms are homeomorphism classes of compact d-dimensional bordisms with boundary decomposed into an incoming and an outgoing part. Composition is given by gluing bordism along the specified parts of the boundary. The symmetric monoidal structures is given by disjoint union of manifolds. Our notion of a Euclidean field theory is based on such functors but we need to vary the source and target categories, mainly putting in geometry in the source, topology in the target and working in families. The new target category is easiest to describe and so we do this in the next subsection, where we shall recall some basic facts about Fr´echet spaces and their (projective) tensor product. In the following subsections we shall add geometry, families and supersymmetry into the definition of various bordism categories. In the last subsection we shall put all this information together and discuss the spaces 1|1-PEFT mentioned in our Main Theorem. 6.1. The symmetric monoidal category of Fr´ echet spaces. We start with a review of some basic facts on Fr´echet spaces, see for example [15] or [20]. These are the complete topological vector spaces whose topology can be defined by an increasing sequence of semi-norms ρ1 ≤ ρ2 ≤ · · · . Typical examples are Banach spaces (defined by a single norm) or spaces of continuous functions C 0 (X), for X a union of an increasing sequence of compact sets Kn . In this case, the ρn are given by the supremum norms on Kn . For a smooth manifold M , the Sobolev norms on C ∞ (M ) qualify for the ρs . For a Banach space, the dual space again has a complete norm and hence a preferred topology. For a Fr´echet space V there are many interesting topologies on the continuous dual V  . For example, one can define a strong topology on V  that agrees with the norm topology if V is Banach. Moreover, a Fr´echet space is Banach

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238 240

H. HOHNHOLD, S. STOLZ, AND P. TEICHNER

if and only if its strong dual is a Fr´echet space and in general there is no topology on V  that always ends up with a Fr´echet space! The two basic theorems for Fr´echet spaces are the closed graph theorem and the open mapping theorem: For a linear map A : V → W between Fr´echet spaces the former says that A is continuous if and only if its graph is closed and the latter says that A is open if it is continuous and surjective. It follows that there is at most one Fr´echet topology, finer than the subspace topology, on a subspace of a linear Hausdorff space and that a continuous bijective linear map between Fr´echet spaces is an isomorphism. There are many topologies on the algebraic tensor product of two topological vector spaces V and W . We will only study the projective topology which is the finest topology on the algebraic tensor product V ⊗alg W for which the canonical bilinear map V × W → V ⊗alg W is continuous. This means that a linear map V ⊗alg W → G is continuous if and only if the corresponding bilinear map V × W → G is continuous. Lemma 6.1. If V, W are Fr´echet spaces then the completion V ⊗ W of the algebraic tensor product in the projective topology is again a Fr´echet space, [16, 178]]. This tensor product is associative and commutative in the sense that SFr, the category of Fr´echet spaces, becomes a symmetric monoidal category under this projective tensor product. Examples 6.2. If V is Fr´echet then C ∞ (M ; V ) ∼ = C ∞ (M ) ⊗ V . In particular, C ∞ (M × N ) ∼ = C ∞ (M ) ⊗ C ∞ (N ) for any smooth manifolds M and N . In fact, in the graded category these statements continue to hold for super manifolds. If V, W happen to be Hilbert spaces then V ⊗W can be identified with the space of trace-class operators from V to W by the canonical map. Note that the Hilbert tensor product of V and W leads to the space of Hilbert-Schmidt operators. If V = W is infinite-dimensional, the inner product extends to a continuous homomorphism on the former but not the latter space, which is why we cannot use the monoidal category of Hilbert spaces as the target for our field theories. The following lemma will be used in the Main Theorem of this section in order to pick out one Fr´echet space from a zoo of possibilities. Lemma 6.3. Let V be a Fr´echet space and h : V ⊗ V → C a continuous linear map that is Hermitian, positive definite and induces an (algebraic) isomorphism V → V  . Then (V, h) is a Hilbert space whose norm topology agrees with the original Fr´echet topology. Before we give the proof of this lemma, we point out the case V = C ∞ (M ) for a compact Riemannian manifold M . The inclusion C ∞ (M ) → L2 (M ) is continuous but not open. It induces a positive definite inner product h on V which satisfies all the conditions except that V → V  is only 1 − 1 and not onto (V  is a space of distributions). So the following proof of Lemma 6.3 has to decisively use that V → V  is an isomorphism. Proof. It is clear that the composition ∆

h

V −→ V × V − →C

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is continuous and therefore the identity is a continuous map V → H, where V denotes the original Fr´echet space and H denotes V with the norm topology given by v2 := h(v, v). We will see in the next paragraph that the pre-Hilbert space H is actually complete. Hence we can apply the open mapping theorem to the identity map V → H and conclude that it is an isomorphism, as asserted. Since the identity map V → H is continuous, we have an inclusion ι : H  → V  . The composition H → V  of ι with the canonical map c : H → H  is surjective since it is equal to the composition H = V → V  which is surjective by assumption. It follows that the canonical map H → H  is surjective and hence the pre-Hilbert space H is complete.
6.2. Riemannian bordism categories. In [29] we defined the Riemannian bordism category d-RB as an internal category over the 2-category of symmetric monoidal categories. This means that we have objects: closed piecewise smooth (d−1)-manifolds Y c with a d-dimensional Riemannian bicollar Y . (1v ) vertical 1-morphisms: germs of isometries between objects (Y, Y c ).

(0)

(1h ) horizontal 1-morphisms: d-dimensional Riemannian bordisms Σ between such objects. (2) 2-morphisms: germs of isometries of horizontal 1-morphisms Σ. This 2-categorical language is very important if one wants to make twisted field theories precise, for example for degree n = 0. In this paper we decided to skip this complication and hence we collapse the above structure to that of a symmetric monoidal category. We hope that using the same notation d-RB as in [29] will not be confusing for the readers. Here are the detailed definitions used in this paper. One important aspect is that all our manifolds are without boundary, making it easier to deal with geometric structures. Definition 6.4. The objects of the category d-RB are quadruples (Y, Y c , Y ± ), where Y is a Riemannian d-manifold (usually noncompact) and Y c ⊂ Y is a compact codimension 1 submanifold which we call the core of Y . Moreover, we require that Y Y c = Y + Y − , where Y ± ⊂ Y are disjoint open subsets whose closures contain Y c . An isomorphism in d-RB from (Y0 , Y0c , Y0± ) to (Y1 , Y1c , Y1± ) is the germ of an invertible isometry f : W0 → W1 . Here Wj ⊂ Yj are open neighborhoods of Yjc and f is required to send Y0c to Y1c and W0± to W1± where Wj± := Wj ∩ Yj± . As usual for germs, two such isometries represent the same isomorphism if they agree on some smaller open neighborhood of Y0c in Y0 . Disjoint union makes this a symmetric monoidal groupoid, the invertible part of d-RB. We shall now introduce the general morphisms of d-RB as (isometry classes of) certain bordisms and we shall rediscover the above isomorphisms as very special types of bordisms. Definition 6.5. A Riemannian bordism from Y0 = (Y0 , Y0c , Y0± ) to Y1 = (Y1 , Y1c , Y1± ) is a triple (Σ, i0 , i1 ) consisting of a Riemannian d-manifold Σ and smooth maps ij : Wj → Σ. Here Wj ⊂ Yj are open neighborhoods of the cores Yjc . ± ± ± Letting i± j : Wj → Σ be the restrictions of ij to Wj := Wj ∩ Yj , we require that (+) (c)

− c i+ j are isometric embeddings into Σ \ i1 (W1 ∪ W1 ) and   the core Σc := Σ \ i0 (W0+ ) ∪ i1 (W1− ) is compact.

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H. HOHNHOLD, S. STOLZ, AND P. TEICHNER

Yc

Y−

Y+

Figure 1. An object (Y, Y c , Y ± ) of 2-RB Particular bordisms are given by isometries f : W0 → W1 as above, namely by using + Σ = W1 , i1 = idW1 and i0 = f . Note that in this case the images of i+ 0 and i1 are not disjoint but we didn’t require this condition. Below is a picture of a Riemannian bordism; we usually draw the domain of the bordism to the right of its range, since we want to read compositions of bordisms, like compositions of maps, from right to left. Roughly speaking, a Riemannian bordism between objects Y0 and Y1 of d-RB is just an ordinary bordism Σc from Y0c to Y1c equipped with a Riemannian metric, thickened up a little bit near its boundary to make gluing possible. The composition of two Riemannian bordisms Σ : Y0 → Y1 and Σ : Y1 → Y2 is defined as follows. Consider the maps i1 : W1 → Σ and i1 : W1 → Σ that are part of the data for our Riemannian bordisms. Here W1 , W1 ⊂ Y1 are open neighborhoods of Y1c and we set W1 := W1 ∩ W1 . Our conditions guarantee that i1 and i1 restrict to isometric embeddings of (W1 )+ := W1 ∩ Y1+ . We use these isometries to glue Σ and Σ along W1 to obtain Σ defined as follows:     Σ := Σ \ i1 ((W1 )+ \ (W1 )+ ) ∪(W1

)+ Σ \ i1 (W1− ∪ W1c ) The maps i0 : W0 → Σ and i2 : W2 → Σ can be restricted to maps (on smaller open neighborhoods) into Σ and they induce isometric embeddings as required i1 (Y1c )

i0 (Y0c )

Σ i1 (W1− )

i1 (W1+ )



i0 (W0− )

!

Σc

i0 (W0+ )

"

Figure 2. A 2-dimensional Riemannian bordism (Σ, i0 , i1 )

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by our conditions. This makes Σ a Riemannian bordism from Y0 to Y2 using the following remark. Remark 6.6. We point out that the conditions (+) and (c) in the above definition of a Riemannian bordism also make sure that the composed bordism is again a Hausdorff space. In other words, gluing two topological spaces along open subsets preserves conditions like “locally homeomorphic to Rn ” and structures like Riemannian metrics. However, it can happen that the glued up space is not Hausdorff, for example if one glues two copies of R along the interval (0, 1). The reader is invited to check that our claim follows from the following easy lemma. Lemma 6.7. Let X, X  be manifolds and let U be an open subset of X and X . Then X ∪U X  is a manifold if and only if the natural map U → X × X  sends U to a closed set. 

An isometry between Riemannian bordisms Σ, Σ : Y0 → Y1 is a germ of a triple of isometries F : X → X

f0 : V0 → V0

f1 : V1 → V1 .

Here X ⊂ Σ (respectively Vj ⊂ Wj ∩ i−1 j (X) ⊂ Yj ) are open neighborhoods of Σc (respectively Yjc ) and similarly for X, V0 , V1 . We require the conditions for fj to be an isomorphism in d-RB as in Definition 6.4 and that the following diagram commutes: i1 / X o i0 V 0 V1 f1

 V1

f0

F i
1

 / X o

i
0

 V0

Two such triples (F, f0 , f1 ) and (G, g0 , g1 ) represent the same germ if there there  are smaller open neighborhoods X  of Σc ⊂ X and Vj of Yj ⊂ Vj ∩ i−1 j (X ) such   that F and G agree on X , and fj and gj agree on Vj . Isometries with a germ of the form (F, idV0 , idV1 ) are referred to as being rel boundary. Definition 6.8. We define the morphisms in d-RB to be isometry classes (rel boundary) of Riemannian bordisms as defined above. The composition is welldefined and associative on such isometry classes. It is not hard to check that the isomorphisms in d-RB all come from isometries as explained above. Disjoint union makes d-RB into a symmetric monoidal category. Remark 6.9. It is extremely important to observe that there are no orientations used to distinguish between incoming and outgoing boundaries. From our perspective, it is an unfortunate historical coincidence that in many papers orientations are mixed up with categorical source and target information. In the above approach the convention is that incoming boundaries of Σ have the −-side on the inside whereas outgoing boundaries have the −-side on the outside. In the figures for d = 1 below, one can also remember this by noting that the •, representing the core of a point, is on the outside of the pictured bordism if and only if it is an outgoing boundary component. It will also become clear later that this is the correct convention when studying functors into infinite-dimensional vector spaces.

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The following object pt ∈ 1-RB is clearly the unique connected nonempty object up to isomorphism, implying that π0 (1-RB) = N0 : pt := (pt, ptc , pt± ) := (R, {0}, R± ) In fact, pt is the germ around {0} and a good picture is given by drawing the core ptc as a •, the +collar pt+ as a dashed line and ignoring the −collar pt− altogether: •_

_ _

pt

This choice comes from the fact that only the +collar is assumed to be embedded when considering Riemannian bordisms in and out of pt. We draw the interval of length t ≥ 0, a morphism It ∈ 1-RB(pt, pt) as follows: •_

_ _ 

pt

i1

/

•

It

•_

_ _

o

i0

? _ •_ pt _ _

The underlying Riemannian manifold of It is R, the incoming embedding i0 is given by translation by t and the outgoing embedding i1 is the identity. As for the object pt we only draw the +collars embedded in It (and not the images of the −collars that aren’t assumed to be embedded). The corresponding topological bordism Itc is the usual compact interval [0, t], obtained by removing the dashed line on the right. Here we only needed to remove the image of the +collar under the embedding i+ 0 since the image of the −collar under i1 isn’t even drawn. Note that I0 represents the identity morphism for the object pt. Two pictures of important morphisms in 1-RB are as follows: (4)

sl L0 • KR

• Rt •

We draw such bordisms so that their source is on the right and their target is on the left, to make it easier to compare pictures with algebraic formulas. We call Rt : ∅ → pt  pt the right elbow and Lt : pt  pt → ∅ the left elbow. It is very important to notice that Rt is only defined for t > 0, for t = 0 condition (+) in the definition of a Riemannian bordism is violated. However, it is satisfied for L0 which is an example of a noninvertible Riemannian bordism for which the core is a single point (just like for isometries of pt). Remark 6.10. Originally we thought that the intervals It are all one needs to understand 1-manifolds: just pick a triangulation that decomposes any 1-manifold into intervals. However, one should be more careful and rather think of manifolds decomposed into handles, then 0- and 1-handles are relevant for 1-manifolds. Categorically, this can also be seen by observing that there is no way to recover Rt just from intervals. To deal with this problem, we introduced the ‘adjunction transformations’ on the bordism categories and on Vect in our original approach [28]. We required the functor E to preserve these extra structures, leading to a more awkward definition. However, if one defines the category 1-RB as above, these additional properties are automatically satisfied as one can see by playing with the morphism L0 . We now prefer this approach because it introduces the concept of handle decomposition of 1-manifolds in a precise fashion: In the following Theorem, think of L0 as associated to a 0-handle (of length 0) and Rt as associated to 1-handles (of length t > 0). The

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relations come from Morse cancelling a pair of such handles and the orientation reversing isometry of an interval. Theorem 6.11. The symmetric monoidal category 1-RB is generated by the object pt and the morphisms L0 and Rt , t ∈ R>0 , subject to the following relations • L0 and Rt are symmetric, i.e., they interact as follows with the symmetry braiding σ of 1-RB: L0 = L0 ◦ σ and Rt = σ ◦ Rt . • Rt1 +t2 equals the composition Rt1 ◦L0 Rt2 := (I0  L0  I0 ) ◦ (Rt1  Rt2 ): • Rt 1

(5)

sl L0 • KR

•

• Rt 2 •

Proof. Having a set of generators for a symmetric monoidal category B means that the symmetric monoidal category freely generated by this set comes with an essentially surjective functor to B. In our case, this is true since (1) any object in 1-RB is isometric to the disjoint union of points, (2) any morphism in 1-RB is isometric to the disjoint union of Riemannian circles and intervals, (3) the moduli space of intervals (and circles) is (0, ∞) given by the length t, (4) all ways of assigning source and target to an interval are given by It , Lt , Rt . Moreover, L0 and Rt can be composed to give It , Lt , (5) circles are obtained from composing Lt and Rt . To find the relations between the generators L0 and Rt we have to understand their isometries relative boundary: These are clearly given by reflections and the first set of relations arises. The second relation is a geometric version of Morse cancellation if one thinks of L0 as associated to a 0-handle (of length 0) and Rt as associated to 1-handles (of length t > 0). In this dimension, this is the only possible Morse cancellation. With some care, the usual statement of handlebody theory, namely that any two handle decompositions of a manifold differ by a sequence of Morse cancellations, leads to a proof of our theorem.
For those readers who are not familiar with the notion of generators and relations for symmetric monoidal categories, the following consequence of Theorem 6.11 can serve as a definition. Corollary 6.12. For any symmetric monoidal category C, the groupoid of symmetric monoidal functors from 1-RB to C (and natural isomorphisms) is equivalent to a certain groupoid C3 : Fun⊗ (1-RB, C)  C3 Here the objects of C3 are triples (V, µ, ρt ) where V ∈ C and µ : V ⊗ V → ½ respectively ρt :

½ → V ⊗ V, t ∈ R>0

are symmetric morphisms in C subject to the relations ρt1 +t2 = ρt1 ◦µ ρt2 := (½ ⊗ µ ⊗ ½) ◦ (ρt1 ⊗ ρt2 )

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shown in (5). Note that this figure explains how to use a fixed morphism µ to introduce a monoidal structure on C(½, V ⊗ V ), denoted by ◦µ above. This works in any symmetric monoidal category C and our relation says that ρ is a homomorphism of monoids R>0 → C(½, V ⊗ V ). The morphisms C3 ((V, µ, ρt ), (V, µ, ρt )) are isomorphisms in C(V, V  ) for which the obvious diagrams involving µ, µ, ρt , ρt commute. In particular, applying this result to C = Vect, we see that 1-dimensional Riemannian field theories form a groupoid equivalent to Vect3 . For example, given a Hilbert space V with inner product µ and an operator A on V , one can form the semigroup of operators ρˆt := exp(−tA) and hope that it comes from ρt ∈ V ⊗ V . There are a few problems with this example. (1) Since µ is a bilinear pairing, rather then Hermitian, it qualifies as an inner product only over the real numbers. To relate to Hermitian pairings over the complex numbers, one needs to introduce orientations on the Riemannian manifolds which we do in [12]. (2) If we are using the algebraic tensor product on Vect then V ⊗ V can be identified with the finite rank operators, putting severe restrictions on A. This problem is resolved by working in the category of Fr´echet space with projective tensor product as explained in Section 6.1. Note that the monoidal category of Hilbert spaces only works in finite dimensions because in infinite dimensions the pairing µ is not defined on the Hilbert tensor product of V with itself: this tensor product is isomorphic to the Hilbert-Schmidt operators on V whereas the projective tensor product gives trace-class operators on V . In the translation, the pairing µ turns into the trace which is defined on this projective tensor product but doesn’t make sense for all Hilbert – Schmidt operators. (3) Most importantly, if we want that all Riemannian field theories arise in this manner, we have to make sure that the elements ρt vary continuously, or even smoothly, in t. Then the semigroup ρˆt can be differentiated to give an infinitesimal generator A. This requirement leads naturally to include family versions of the categories d-RB, introduced in the next two subsections. 6.3. Families of rigid geometries. Let M denote one of the categories of manifolds Man, super manifolds SMan, or complex super manifolds csM. The latter are defined in [6] in terms of an ordinary p-manifold, together with a sheaf of commutative superalgebras over C that are locally isomorphic to C ∞ (Rp|q ) ⊗ C In this case, p|q is again referred to as the superdimension. One should think of these manifolds as having p real coordinates and q complex coordinates and they all arise from complex vector bundles over ordinary manifolds. Here and in the following, a manifold will stand for an object in M (which has no boundary). Let G be a group object in M, i.e., a Lie group, Lie supergroup or Lie cs-group. We want to think of a G-action on a fixed manifold M as the local model for rigid geometries with isometry group G. This idea is very well explained in [30] and goes back to Klein’s Erlangen program. Definition 6.13. A (G, M)-structure on a manifold Y is a maximal atlas consisting of charts which are diffeomorphisms ϕi

→ Vi ⊆ M Y ⊇ Ui −∼ =

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between open subsets of Y and open subsets of M such that the Ui ’s cover Y and for all i, j the transition function ϕj ◦ϕ−1

i M ⊇ ϕi (Ui ∩ Uj ) −−−−− → ϕj (Ui ∩ Uj ) ⊆ M

is given by an element g ∈ G, it is the restriction of the map g×id

M = pt ×M −−−→ G × M → M for some g : pt → G. Here G × M → M is the action map and this careful formulation is particularly relevant for super manifolds. In the category of super manifolds (or cs-manifolds), any morphism pt → G factors uniquely through Gred ⊂ G. As a consequence, a (G, M)-structure on a super manifold Y is the same thing as a (Gred , M)-structure on Y . Therefore, the relevant notion for super manifolds is that of families of (G, M)-manifolds which we define next. Definition 6.14. A family of (G, M)-manifolds is a morphism p ∈ M(Y, S) together with a maximal atlas consisting of charts which are diffeomorphisms ϕi between open subsets of Y and open subsets of S ×M making the following diagram commutative: ϕi / Vi ⊆ S × M Y ⊇U ∼ ??i = y ?? yy ? yyp1 y p ??  |yyy S We require that the open sets Ui cover Y and that for all i, j the transition function ϕj ◦ϕ−1

i S × M ⊇ ϕi (Ui ∩ Uj ) −−−−− → ϕj (Ui ∩ Uj ) ⊆ S × M

is of the form (s, m) → (s, gij (s)m), where gij ∈ M(p(Ui ∩ Uj ), G). We note that the conditions imply in particular that p is a submersion and p(Ui ∩ Uj ) ⊆ S are open. If Y → S and Y  → S  are two families of (G, M)-manifolds, an isometry between them is a pair of maps (f, fˆ) making the following diagram commutative: Y  S

fˆ

f

/ Y  / S

We require that fˆ preserves the fiberwise (G, M)-structure in the following sense: There are charts (Ui , ϕi ), respectively (Ui , ϕi ), of Y , respectively Y  , such that fˆ(Ui ) ⊆ Ui and ϕi ◦ fˆ|Ui ◦ ϕ−1 is of the form (s, m) → (f (s), gi (s)m) for some i gi ∈ M(p(Ui ), G). The result is a category M(G, M) of families of (G, M)-manifolds and their isometries. It is easy to see that the forgetful functor M(G, M) → M is a Grothendieck fibration, i.e., pullbacks exist. Example 6.15. In the category Man of manifolds, we can consider M := Ed , the d-dimensional Euclidean space. Let G := Iso(Ed ) be the isometry group of Ed , which is the Euclidean group of translations, reflections and rotations of Ed . A

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Euclidean structure on a smooth d-manifold Y is an (Iso(Ed ), Ed )-structure in the above sense. It is clear that such an atlas determines a flat Riemannian metric on Y by transporting the standard metric on Ed to Ui via the diffeomorphism ϕi . Conversely, a flat Riemannian metric can be used to construct such an atlas. We want to give a super version of this geometric structure, at least in dimension 1|1. We would like the local model to be M = R1|1 and the translational part of the isometry group to be R1|1 with the group structure from Definition 5.3. This is the right group structure for the current setting because its Lie algebra is free on one odd generator D with square ∂t , the infinitesimal generator of the translation group R. In Iso(E1 ) we have the reflection that takes t → −t, physically it’s the time reversal. If we want an automorphism of R1|1 that induces this reflection on the reduced group R then it has to induce an action on the Lie algebra that sends D → iD, where i is a square root of −1. Thus super-time R1|1 must be a cs-Lie group if we want to include a super-time reversal. Example 6.16. In the category csM of cs-manifolds, consider the following rigid geometry. Define Euclidean cs-space of dimension 1|1 to be 1|1 M := E1|1 := Rcs 1|1

and let G := Rcs  Z/4 be the isometry group. So by definition, Iso(E1|1 ) := G, 1|1 where the group structure on the translational part Rcs comes from Definition 5.3. The “rotational” part Z/4 is generated by the pin generator pg which is the auto1|1 morphism of the group Rcs given by pg(t, θ) := (−t, −iθ),

i2 = −1.

One checks that G acts on M by first rotating and then left translating. Then csM(Iso(E1|1 ), E1|1 ) is the category of families of Euclidean cs-manifolds. Note that the underlying reduced Lie group of Iso(E1|1 ) is R  Z/4, a central double covering of Iso(E1 ). Hence the underlying reduced manifold of a Euclidean 1|1-manifold comes equipped with a Riemannian metric and pin-structure (more precisely, a tangential Pin− (1)-structure). Remark 6.17. We used to think of a 1|1-dimensional Euclidean structure in terms of an odd complex vector field Q that can be locally written as ∂θ + θ∂t and that is well-defined up to multiplication by complex numbers of length 1. It 1|1 is actually not difficult to see that the above group G := Rcs  Z/4 is exactly the 1|1 group of isometries of this geometric structure on Rcs . We recently decided to use the approach via rigid geometries because in this setting one can easily define d|δ-dimensional Euclidean manifolds (and bordism categories), see [29]. For later use in the definition of real field theories, we shall introduce the concept of complex conjugation on categories M(G, M). Recall that for a cs-manifold S, the complex conjugate cs-manifold S has the same real sheaf of functions but the complex numbers act by precomposing with complex conjugation. If S is a manifold or super manifold, we define S := S.

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Definition 6.18. A real structure on a manifold S (in Man, SMan or csM) is a isomorphism rS : S → S such that r¯ ◦ r = idS For example, if O(S) is the real sheaf of functions on a super manifold S then there is a canonical cs-manifold Scs with (Scs )red = Sred and sheaf of functions O(Scs ) = O(S) ⊗ C. The usual real structure on C then induces a real structure on Scs which is the identity on the reduced manifold. A morphism f : S  → S is real if rS ◦ f = f ◦ rS
. The above functor S → Scs gives an equivalence of categories between SMan and the category of real csmanifolds (with real structures that are the identity on the reduced part) and real morphisms. In the following, we shall not require our real structures to be the identity on the reduced part. A real structure on a local model (G, M) for rigid geometry is a real structure on G and a real structure on M such that all structure maps (group multiplication, unit, inverse, group action) are real. Lemma 6.19. A real structure on (G, M) induces an involution on the category M(G, M) that extends the complex conjugation functor Y → Y on total spaces of families of (G, M)-manifolds. In particular, the forgetful functor M(G, M) → M respects these complex conjugations. Proof. Let ϕi : Ui → Vi be charts for a family p : Y → S as in Definition 6.14 above with transition functions coming from gij ∈ M(p(Ui ∩ Uj ), G). Then ϕ ¯i

id ×r

M Y ⊇ U i −∼ → V i −−S−∼−−→ Wi ⊆ S × M

=

=

are charts for Y → S. One can easily check that, as required for a family of (G, M)manifolds, the transition functions are of the form (x, m) → (x, (rG ◦ g¯ij )(x)m) where g¯ij ∈ M(p(U i ∩ U j ), G). If an isometry f : Y → Y  is locally given by gi ∈ M(p(Ui ), G) then one similarly checks that the isometry f¯: Y → Y  is locally given by rG ◦ g¯i ∈ M(p(U i ), G).
Example 6.20. In the case of 1|1-dimensional Euclidean cs-structures from 1|1 Example 6.16, we can define the real structure rM on M = E1|1 = Rcs as in Definition 6.18. This also defines the real structure on the translational part of 1|1  Z/4 G = Iso(E1|1 ) = Rcs

If we think of Z/4 as the 4th roots of unity inside C then the usual real structure on C induces one on Z/4. On the pin generator pg, this real structure is rG (pg) := pg−1 which means that it permutes the sheets, even of the reduced manifold R  Z/4. This permutation is necessary in order to make (rG , rM ) into a real structure on the local model (G, M) for Euclidean cs-manifolds. 6.4. Fibred Euclidean bordism categories. In this subsection we define the (family) Euclidean bordism category d-EB → Man and its generalizations. These are symmetric monoidal categories (Grothendieck) fibred over the site Man of smooth manifolds, compare [29] or [11]. Definition 6.21. An object in d-EB is given by (S, Y, Y c , Y ± ), where S is a smooth manifold, Y → S is a family of d-dimensional Euclidean manifolds and Y c ⊂ Y is a smooth codimension 1 submanifold such that the restriction of p to

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H. HOHNHOLD, S. STOLZ, AND P. TEICHNER

Y c is proper. This assumption is a family version of our previous assumption in Definition 6.4 that Y c is compact, since it reduces to that assumption for S = pt. Also part of the data is the decomposition of Y  Y c as the disjoint union of two open subsets Y ± , both of which contain Y c in their closure. A morphism Σ in d-EB, lying over a smooth map f : S → S  of manifolds, is defined as follows: First fix the source (S, Y0 ) and target (S, Y  ) of Σ which are objects as above. Then pull back the target to get another object (S, f ∗ (Y  )) over S which we denote by (S, Y1 ). Then Σ = (Σ, pΣ , i0 , i1 ) contains the following additional data: • a family of d-dimensional Euclidean manifolds pΣ : Σ → S • for = 0, 1, smooth maps ij : Wj → Σ over S, where Wj ⊂ Yj are open ± neighborhoods of the cores Yjc . Letting i± j : Wj → Σ be the restrictions ± ± of ij to Wj := Wj ∩ Yj , we require that (+) (c)

− c i+ j are isometric embeddings into Σ \ i1 (W1 ∪ W1 ) and   the restriction of pΣ to the core Σc := Σ \ i0 (W0+ ) ∪ i1 (W1− ) is proper.

Two such families of Euclidean bordisms Σ → S and Σ → S represent the same morphism in d-EB if they are isometric (rel. boundary) over the identity of S. Then d-EB → Man is a fibred category which has a fibrewise symmetric monoidal structure, simply given by disjoint union (of total spaces Y respectively Σ). It is customary to denote by d-EBS the fibre categories of this fibration: For fixed S ∈ Man, the objects are families Y → S and the morphisms are restricted to lie over the identity map of S. Recalling the categories M(G, M) of families of (G, M)-manifolds (and their isometries) from Section 6.3, it is not hard to use the same technique as above to define the family bordism categories B(G, M) → M. If M is one of the two possible sites of super manifolds then we note that the properness assumptions on p : Y c → S is only an assumption on the reduced (ordinary) manifolds. In particular, using Euclidean cs-manifolds of dimension 1|1 as defined in Example 6.16, we get a fibred symmetric monoidal category 1|1-EB → csM. Definition 6.22. In this paper we shall make an additional positivity assumption on the cores of our Euclidean families: If Y → S is a familiy of Euclidean manifolds (representing objects or morphisms in d-EB) then we require that the induced map of cores Y c → S is a locally trivial fibre bundle and refer to this as a positive family. Roughly speaking, this means that we only explore the topology on the open part of the moduli space of Euclidean bordisms. We shall denote this positive Euclidean bordism category by d-PEB. Using the same positivity assumption, we get a fibred category 1|1-PEB → csM of positive Euclidean cs-bordisms. For d = 1 positivity implies that there are no smooth families that shrink the intervals It or Lt as t → 0. Such families exist for t > 0 and the morphisms I0 = idpt and L0 also exist but we don’t consider the limits t → 0, motivating the use of the word positive. As a consequence, I0 and L0 represent isolated points in the moduli space of intervals. This assumption of positivity is related to our choice of the family version of the target category Vect to be that of (locally trivial) vector bundles. In future

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work, we plan to investigate more general family bordism categories (partial compactifications of our open moduli space) mapping to more general sheaves of vector spaces. The forgetful functor d-PEB → Man sending (S, Y, Y c , Y ± ) to S is a Grothendieck fibration and will be considered as part of the structure of the family bordism category. We will denote by d-PEBS the fibre category over the manifold S, describing S-families of positive Euclidean bordisms. The disjoint union of total spaces Y gives a (fibrewise) symmetric monoidal structure on this fibration as explained in [11] and [12]. We have the following family version of Theorem 6.11. Theorem 6.23. The fibred symmetric monoidal category 1-PEB → Man is generated by the object pt ∈ 1-PEBpt and the morphisms L0 ∈ 1-PEBpt (pt  pt, ∅) and R ∈ 1-PEBR>0 (∅, pt  pt) subject to the following relations • L0 and R are symmetric as before: L0 = L0 ◦ σ, R = σ ◦ R • The relation Rt1 +t2 = Rt1 ◦L0 Rt2 shown in (5) holds, interpreted as saying that using the three maps p1 , p2 , m : R>0 × R>0 → R>0 , where m is addition and R>0 , one has p∗1 (R) ◦L0 p∗2 (R) = m∗ (R) ∈ 1-PEBR>0 ×R>0 (∅, pt  pt) We note that Rt is obtained from R by pullback via the map pt → R>0 with image t > 0. Theorem 6.23 follows from Theorem 6.11 by an analysis of the topology of the moduli space of intervals, see [12]. As in Corollary 6.12, the theorem has the following important consequence (which can again be used as the definition of the notion of “generators and relations” for fibred symmetric monoidal categories). Corollary 6.24. For any symmetric monoidal category C → Man, fibred over smooth manifolds, the groupoid of fibred symmetric monoidal functors from 1-PEB to C (and natural isomorphisms) is equivalent to a certain groupoid C3 : 3 Fun⊗ Man (1-PEB, C)  C

Here the objects of C3 are triples (V, µ, ρ) where V ∈ Cpt and µ ∈ Cpt (V ⊗ V, ½) respectively ρ ∈ CR>0 (½, V ⊗ V ) are symmetric morphisms in C subject to the relations ρt1 +t2 = ρt1 ◦µ ρt2 as explained in the theorem above. The morphisms C3 ((V, µ, ρ), (V, µ, ρ )) are isomorphisms in Cpt (V, V  ) for which the obvious diagrams involving µ, µ, ρ, ρ commute. Note that the same result holds without the above restriction to isomorphisms but we chose this formulation because it is the groupoids whose classifying space later leads to the correct homotopy types. In particular, we can consider as target the fibred category FrR → Man of real Fr´echet vector bundles over smooth manifolds, equipped with fibrewise projective tensor product. This leads to the following corollary. Corollary 6.25. The groupoid Fun⊗ Man (1-PEB, FrR ) is equivalent to the groupoid of triples (V, µ, ρ) where V is a Fr´echet space with a symmetric pairing µ : V ⊗ V → R and a smooth symmetric semigroup ρ : R>0 → V ⊗ V . Morphisms are isomorphisms of Fr´echet spaces, compatible with the extra structure given by µ, ρ.

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The equivalence is given by assigning to a field theory E the triple (V, µ, ρ), where V = E(pt),

µ = E(L0 )

and

ρ = E(R)

Definition 6.26. A 1-dimensional field theory E ∈ Fun⊗ Man (1-PEB, FrR ) is nonnegative if µ(v, w) ≥ 0 for all v, w ∈ V . We call E positive if V is in addition infinite-dimensional and separable and µ gives an (algebraic) isomorphism from V to its continuous dual space V  . The full subgroupoid of Fun⊗ Man (1-PEB, FrR ) given by positive field theories is the denoted by 1-PEFT, the groupoid of positive Euclidean field theories of dimension 1. Remark 6.27. We point out that the notion of a positive Euclidean field theory has two aspects: (1) the functor E is defined on the positive Euclidean bordism category. (2) The inner product µ = E(L0 ) is positive in the above sense. Both these conditions have their own reasons to be called positive and as a (desirable) consequence, we only need one adjective to describe the resulting Euclidean field theory. In the positive case, Lemma 6.3 implies that V is a real Hilbert space with inner product µ. Therefore, V ⊗ V is the space of trace-class operators and we obtain the following final computation from the fact that any two infinite-dimensional, separable Hilbert spaces are isometric. Corollary 6.28. Let (H, µ) be an infinite-dimensional separable real Hilbert space. Then the groupoid 1-PEFT of positive Euclidean field theories of dimension 1 is equivalent to the groupoid 1-PEFT(H) of smooth semigroups ρt of self-adjoint trace-class operators on H. The morphisms are isometries of (H, µ) that are compatible with the semigroups. We note that in the absence of the positivity condition in Definition 6.22, 1EFTs satisfying the conditions of Definition 6.26 do not exist at all! This comes from the fact that if the semigroup ρt = exp(−tA) extends smoothly to t = 0 then its infinitesimal generator A needs to give a continuous operator on V . If V is a Hilbert space then A needs to be bounded and the trace-class condition implies that A and hence all ρt have finite rank. Taking the limit t → 0 shows that ρ0 = idV and hence V is finite-dimensional. Our assumption that the EFT is only defined on positive bordisms in 1-PEB implies in particular that ρ0 does not have to be the identity on V , it only has to be a projection operator. We shall refer to the kernel of ρ0 as the ∞-Eigenspace of A. In [12] we shall study general 1-EFTs, i.e.,. the groupoid Fun⊗ Man (1-EB, FrR ), so that we have to allow V to be a Fr´echet space like C ∞ (S 1 ). Then the pairing µ may only induce an injection V → V  and may even be indefinite. The advantage of using C ∞ , rather than L2 -functions, is that the infinitesimal generator A can be unbounded but continuous on V , for example if A comes from the differentiation operator. We believe that this general class of 1-EFTs still leads to a homotopy equivalent classifying space if one uses the construction from the next section, see Remark 6.30.

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6.5. Internal groupoids of smooth field theories. In this section we explain how groupoids of smooth field theories can be naturally equipped with a topology so that we may consider the corresponding classifying spaces. More precisely, if G is one of these groupoids, we shall naturally define a new groupoid Man(M, G) for every manifold M ∈ Man which serves as the groupoid of smooth maps from M to G. In fact, these groupoids will fit together to a category G → Man fibred in groupoids. In all cases at hand, we will show that G is equivalent to a fibred groupoid coming from a contravariant functor from manifolds to small groupoids: sG : Manop → sGrp In particular, this functor sG has a well defined classifying space, obtained from realizing the simplicial space sG

|.|

∆op → Manop −→ →sGrp −→ Top We shall denote this classifying space simply by |G| to emphasize the naturality of the construction of G and since the choice of sG cannot alter the homotopy type of this classifying space. We shall give this construction in great generality in order to (1) express its naturality, (2) be able to apply it to supersymmetric EFTs, (3) use it in dimensions d > 1 in future papers. The main goal of the current Section 6 is the following result: Theorem 6.29. In the notation of this and the following subsection, the groupoids 1-PEFT and 1|1-PEFT of positive Euclidean field theories of dimensions 1 respectively 1|1 have classifying spaces whose homotopy type is given by: |1-PEFT|  pt

and

|(1|1)-PEFT|  BO × Z

Remark 6.30. In [12] we ignore all positivity assumptions and work with general sheaves of Fr´echet spaces to obtain the groupoids 1-EFT and 1|1-EFT. The construction of this section applies to obtain the corresponding classifying spaces and we conjecture that this does not change homotopy types: |1-EFT|  |1-PEFT| and

|(1|1)-EFT|  |(1|1)-PEFT|

Note however, that applying our current construction in dimensions 0 and 0|1 leads to contractible classifying spaces |0-EFT|  |(0|1)-EFT|  pt This is the reason why we worked with field theories over a manifold X in [11]. We proved that this approach leads to the correct homotopy type for de Rham cohomology to appear (in the twisted case) by using simplices in place of X. We believe that this more difficult construction gives the correct homotopy type in all dimensions and view our current method as a shortcut for K-theory, i.e., for dimension 1|1. We are uncertain whether this shortcut also works in the most interesting dimension 2|1. The groupoids G for which we shall construct a classifying space |G| will be of the following form: Let S be a base category, in this paper S = Man or SMan. We require that S has finite products and we fix a functor Man → S. Our notation will not reflect this functor, i.e., we will just think of manifolds as objects in S (which is certainly correct in the two cases at hand). Let B and V be two categories fibred

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over S, where B will later be some bordism category and V some category of vector bundles. Then we consider the groupoid of fibred functors G = FunS (B, V) where the morphisms are natural isomorphisms over S. In our applications, B and V will actually be symmetric monoidal fibred categories and the functors will be assumed to be symmetric monoidal. Definition 6.31. The category G → S fibred in groupoids is defined as the inner Hom in fibred categories. Its fibre GS over an object S ∈ S is given by GS = FunS (B, V)S = FunS (B, V(S)) where V(S) = FunS (S, V) is the fibred category with fibres V(S)T := VS×T . Here all functor categories are considered as groupoids, i.e., we only allow natural isomorphisms, not all natural transformations, as morphisms. Remark 6.32. This definition is a certain version of inner Hom, where the variable S is built in on the right hand side. This implies good functoriality properties and applications of Corollary 6.24 in the case B = 1-PEB. We note that in general there are equivalences of categories FunS (B, FunS (S, V))  FunS (B × S, V)  FunS (B, V)S because V(S)T = VS×T  FunS (S × T , V)  FunS (S, V)T As an example, let’s take the relevant fibred categories from the previous section, i.e., S = Man, B = 1-PEB and V = FrR Then we can apply Corollary 6.24 to the fibred category C = FrR (S) for any manifold S. We obtain an equivalence of groupoids 3 Fun⊗ Man (1-PEB, FrR )S  FrR (S)

where the objects of the groupoid FrR (S)3 are triples (V, µ, ρ). Here V is a Fr´echet bundle over S with a symmetric pairing µ and symmetric semigroup ρ. We now restrict the functors to be positive, in the (family version) sense of Definition 6.26 and obtain (full) subgroupoids 1-PEFTS ⊂ Fun⊗ Man (1-PEB, FrR )S The above equivalence takes this subgroupoid to the category of triples (V, µ, ρ) with V is a Hilbert bundle over S, fibrewise inner product µ and self-adjoint semigroup ρ. By the contractibility of the orthogonal group O(H), this category is equivalent to the one where V is the trivial bundle H × S and µ is constant. Then ρ : S → SGOTC (H) is a smooth map into all self-adjoint semigroup homomorphisms R>0 → H ⊗H, where the right hand side denotes the projective tensor product, isomorphic to the space of trace-class operators TC(H). Smoothness means that the adjoint of ρ, the map S ×R>0 → H ⊗H, is smooth. It is self-adjoint and has the semigroup property for each fixed s ∈ S.

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Remark 6.33. The notation ‘semigroup of operators’ SGOTC is also used in Proposition 5.16 where there is an additional index n, expressing the fact that the Hilbert space in question is graded and has a Cn -action. Here we have no grading (until we get to supersymmetric field theories) and no Cn -action (until we discuss twisted field theories of degree n) but we prefer to keep the infinite-dimensional separable Hilbert space H in the notation. Summarizing, we get Corollary 6.34. The fibred category 1-PEFT → Man is equivalent to the one coming from the functor into small groupoids: SGOTC (H)/O(H) : Manop → sGrp This is actually a quotient fibration, i.e., the value of this functor at a manifold S is the transport groupoid for the action of C ∞ (S, O(H)) on C ∞ (S, SGOTC (H)). The last step in this discussion is the computation of the classifying space of this quotient fibration. As explained at the beginning of this subsection, it is defined by using (extended) k-simplices as manifolds S above to obtain a small simplicial groupoid that can be geometrically realized to get |1-PEFT|. This realization means that one forms a bisimplicial set by taking the nerves of the groupoids involved. Then one has the choice of realizing this bisimplicial set in the two possible orders. If we first realize the original simplicial direction we get a simplicial space that is the nerve of an internal groupoid in Top with objects respectively morphisms given by the realizations of the simplicial sets [k] → C ∞ (∆k , O(H)) respectively [k] → C ∞ (∆k , SGOTC (H)) By a version of the smooth approximation theorem, these simplicial sets have the same homotopy type as the singular simplicial sets of O(H), respectively SGOTC (H) (which are formed by replacing C ∞ by C 0 in the above equations). Both of these spaces have the homotopy type of CW-complexes and hence the realizations of this singular simplicial sets are homotopy equivalent to the original spaces O(H) respectively SGOTC (H). The nerve of this internal (transport) groupoid in Top then has realization |1-PEFT|, proving the first part of the following result. Theorem 6.35. The classifying space of the groupoid 1-PEFT of positive 1EFTs has the homotopy type |1-PEFT|  |SGOTC (H)/O(H)|  SGOTC (H)  pt Proof. The second homotopy equivalence follow from the contractibility of O(H). It remains to show that the space SGOTC (H) of self-adjoint trace-class semigroups in H is contractible. The graded analogue was discussed at length in Section 5, see particularly Propositions 5.12 and 5.16. Therefore we shall be fairly brief in this argument. First recall that the zero semigroup ρt ≡ 0 serves as the basepoint in SGOTC (H). It corresponds to all eigenspaces of the infinitesimal generator A, ρt = exp(−tA), being at ∞. A contraction H : SGOTC (H) × [1, ∞] → SGOTC (H) to this basepoint is then given by the formula H(ρt , s) := ρs·t . This has infinitesimal generator s · A and, looking at its eigenspaces, we see that they move to ∞ as s → ∞.


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Remark 6.36. By looking at the eigenspaces of infinitesimal generators, the set SGOfin (H) can be identified with Segal’s configuration space F ((−∞, +∞]) discussed in the introduction. This comes from the fact that eigenspaces can move exactly to +∞ in SGOfin (H) because exp(−sA) becomes zero as s → +∞ and unbounded as s → −∞. Our contraction of SGOfin (H) in the proof above is then just the configuration space version of the contraction of (−∞, +∞] to the point +∞. 6.6. Proof of part (6) of our Main Theorem. In the remaining parts of Section 6, we shall redo all of the above considerations for super manifolds to obtain the classifying space of positive 1|1-dimensional Euclidian field theories. It will turn out that it has an interesting homotopy type, rather than being contractible as above. We need the following steps: Step (1) Recall the Euclidean cs-bordism category 1|1-EB and its positive subcategory 1|1-PEB. Step (2) Find generators and relations for 1|1-PEB. Step (3) Identify the resulting category 1|1-PEFT → Man fibred in groupoids with the quotient fibration SGOTC (H)/O ev (H) for a fixed Z/2-graded separable Hilbert space H such that the even and odd parts are infinite-dimensional. Step (4) Compute the homotopy type of the classifying space for supersymmetric positive 1|1-EFTs |(1|1)-PEFT|  |SGOTC (H)/O ev (H)|  SGOTC (H)  BO × Z Similarly to the discussion around Theorem 6.35, step (4) follows from step (3) by Propositions 5.12 and 5.16. Step (1) was explained in Section 6.4, where we defined the family bordism categories B(G, M) → M for arbitrary (G, M)-structures on the site M. The case of Euclidean structures in arbitrary dimension d|δ is explained in [29], for dimension 1|1 see Example 6.16 and Definition 6.21. Step (2) is then a computation of the relevant super-moduli spaces of 0- and 1-handles and will be carried out in the next subsections. In the remainder of this subsection, we’ll use the following corollary to Theorem 6.48 of to finish step (3) of the current discussion. Corollary 6.37. Let H be a separable real Hilbert space, graded in a way such that the even and odd parts are both infinite-dimensional. Then the groupoid 1|1-PEFT of positive Euclidean field theories of dimension 1|1 is equivalent to the groupoid 1|1-PEFT(H) of super-semigroups t,θ of even self-adjoint trace-class operators on H. The morphisms are even isometries of H that are compatible with the semigroups. Corollary 6.38. The fibred category 1|1-PEFT → Man is equivalent to the one coming from the quotient fibration SGOTC (H)/O ev (H) : Manop → sGrp The same computation of the classifying space of this quotient fibration as in Theorem 6.35 shows that it has the homotopy type of SGOTC (H). This is the space in Proposition 5.16, where it was shown that its homotopy type denoted by SGOTC 0 is that of |(1|1)-PEFT|  SGO0 ≈ Conf 0  BO × Z

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This finishes the proof of part 6 of our Main Theorem in the introduction, modulo Theorem 6.48. 6.7. Generators and relations for 1|1-PEB. First we note that any super or cs-manifold has an involution that negates its odd coordinate. We shall refer to this as the pin flip, written fl. If Y has a (G, M)-structure, one may ask whether flY is an isometry. It turns out that this is indeed the case if flM comes from an element g ∈ G in the sense explained at the end of Definition 6.13. This is true in the example of 1|1-dimensional Euclidean cs-manifolds, where flM is the square of the pin generator pg, see Example 6.16. The following object sp ∈ 1|1-EB will be referred to as the super-point   1|1 sp := (sp, spc , sp± ) := E1|1 , R0|1 , R± = pt ×R0|1 Note that the core is again a codimension 1 submanifold and for the following lemma it is essential that an automorphism of the super-point is required to preserve it. Lemma 6.39. The (inner ) isometry group Iso(sp) of the super-point is Z/2, generated by fl. Proof. By definition, the isometry group of our super Euclidean space E1|1 = R is Iso(E1|1 ) = R1|1  Z/4 and we need to decide which of these isometries preserve the additional data spc and sp± . The core spc of sp is given by coordinates (0, η). Hence none of the translations Tt,θ (0, η) = (t + θη, θ + η) preserve the core. The pin generator pg(t, θ) = (−t, −iθ) does preserve the core but only its square preserves the parts sp± of the super point. This is exactly the pin flip fl.
1|1

Remark 6.40. There is another object in 1|1-EB that looks just like the above super-point, with the same ambient manifold and core but where the ±-parts are interchanged. The pin generator is an isometry from this new object to sp and hence we don’t need to consider this as an extra generator. In the oriented bordism category, these two objects are not isomorphic. To obtain interesting families of Euclidean cs-manifolds over a fixed cs-manifold S, we start with a function f ∈ C ∞ (S) ∼ = csM(S, R1|1 )

1|1

such that fred : Sred → Rred = R satisfies fred > 0.

Then we can use the super group structure m on R1|1 given in Definition 5.3 to define the translation map id ×f ×id

id ×m

Tf : S × R1|1 −−−−−−→ S × R1|1 × R1|1 −−−−→ S × R1|1 Let Σ := S × E1|1 with iout the identity map S × sp → Σ and iin be given by Tf . Definition 6.41. For any cs-manifold S, write f ∈ C ∞ (S) as f = t + θ where t and θ are even respectively odd functions on S and assume that fred = tred > 0. Then the above morphism (Σ, iin , iout ) is called the super-interval of length (t, θ), written as It,θ ∈ 1|1-PEBS (S × sp, S × sp).

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It lies over the identity map of S in csM. There are two close cousins also lying over the identity of S, the left, respectively right, elbow of length (t, θ), written as     Rt,θ ∈ 1|1-PEBS ∅, S×(sp  sp) respectively Lt,θ ∈ 1|1-PEBS S × (sp  sp, ∅) . These are defined like It,θ with one exception, namely the pin generator pg has to enter the discussion (it does not depend on the parameters in S). For the left elbow, iout is empty and iin is the disjoint union of pg and the translated embedding Tf . For the right elbow, iin is empty and iout is the disjoint union of the identity and Tf precomposed by pg. We note that as before, the left elbow L0 also exists, except that it’s not related smoothly to the above family. The above definitions are direct generalizations of bosonic families which we didn’t explain in such detail. In particular, the use of the pin generator gives on the reduced part a reflection that needed to be used to parametrize one of the boundary points of Rt in (5). 1|1 Let’s spell out the last definition in the universal case S = R>0 and f = idS where the right elbow is just denoted by R = Rt,θ . Then iout is the disjoint union of the identity id : S × sp → S × E1|1 and the isometry S × sp → S × E1|1 ,

pg

(t, θ, y, η) −→ (t, θ, −y, −iη) → (t − y − iθη, θ − iη)

With this convention, one can easily check the super-semigroup relation (4) for R in Theorem 6.42 below. The symmetry condition for L0 turns into relation (2): L0 ◦ σ = L0 ◦ (fl  idsp ) where the right-hand side is L0 pre-composed by an isometry of two super-points. This follows from the fact that one has to use the pin generator as the isometry that exchanges the two boundary points of L0 which has the claimed consequence for the parametrization of the boundaries. The complete list of generators and relations is as follows. Here σ denotes the braiding symmetry of the symmetric monoidal category 1|1-PEB. Theorem 6.42. The fibred symmetric monoidal category 1|1-PEB → csM is generated by the super point sp ∈ 1|1-PEBpt , its pin flip fl and morphisms L0 ∈ 1|1-PEBpt (sp  sp, ∅) respectively R ∈ 1|1-PEBR1|1 (∅, sp  sp). These are subject >0 to the following relations (1) fl2 = idsp (2) L0 ◦ (fl  fl) = L0 and L0 ◦ (fl  id) = L0 ◦ σ (3) (fl  fl) ◦ Rt,θ = Rt,−θ and (fl  id) ◦ Rt,iθ = σ ◦ Rt,θ (4) R(t1 +t2 +θ1 θ2 ,θ1 +θ2 ) = R(t1 ,θ1 ) ◦L0 R(t2 ,θ2 ) The last relation is pictured in (5) and says that R is a super-semigroup. This 1|1 1|1 1|1 is made precise by using the three maps p1 , p2 , m : R>0 × R>0 → R>0 . Here m 1|1 is multiplication from Definition 5.3, restricted to R>0 := (0, ∞) × R0|1 , and the precise relation is p∗1 (R) ◦L0 p∗2 (R) = m∗ (R) ∈ 1|1-PEBR1|1 ×R1|1 (∅, sp  sp) >0

>0

The proof of Theorem 6.42 will be given in [12], the verification of the above relations (1) – (4) being the first (easy) step. The other steps are the same as in Theorem 6.11, except that at the end of the argument one now needs to compute the relevant cs-moduli spaces of cs-intervals. This is done by a cs-version of the

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developing map, compare [30]. It is used to construct global (G, M)-charts for rigid geometries on simply connected manifolds. It then follows that cs-intervals can be embedded isometrically into E1|1 and that the whole information is contained in the parametrization of one boundary point. As in Corollary 6.24, this theorem has the following important consequence (which can again be used as the definition of the notion of “generators and relations” for fibred symmetric monoidal categories over csM). Corollary 6.43. For any fibred symmetric monoidal category C → csM, the groupoid of fibred symmetric monoidal functors from 1|1-PEB to C (and natural isomorphisms) is equivalent to a certain groupoid C4 : 4 Fun⊗ csM (1|1-PEB, C)  C

Here the objects of C4 are quadruples (V, α, µ0 , ρ) where V ∈ Cpt and α ∈ Cpt (V, V ),

µ0 ∈ Cpt (V ⊗ V, ½)

and

ρ ∈ CR1|1 (½, V ⊗ V ) >0

are morphisms in C subject to the relations (1) – (4) given in the theorem above. The morphisms in C4 are isomorphisms in Cpt (V, V  ) for which the obvious diagrams commute. Note that we didn’t need to restrict to isomorphisms but we chose to do so in order to later get the correct homotopy type for the relevant classifying space. In particular, we can consider as target the fibred category SFr → csM of complex Fr´echet super vector bundles over cs-manifolds, equipped with fibrewise graded projective tensor product. The braiding symmetry comes with the usual sign in this setting. Note also that the pin flip fl is taken by any functor E to a grading E(fl) of the (already graded) Fr´echet space V = E(sp). Definition 6.44. We shall say that E satisfies the spin-statistics relation if E(fl) equals the given grading on V . We shall write E ∈ Fun⊗,ss for such functors. Remark 6.45. We initially used the pin flip to induce the Z/2-grading on the vector spaces E(sp) for any EFT E. This leads to a different category of (positive) 1|1-EFTs whose classifying space we believe also has the correct homotopy type BO × Z. Moreover, the tensor product of EFTs gives an H-space structure on this classifying space. However, the induced braiding symmetry is not the one that extends to degree n twists because it comes from the unsigned flip on the category of Z/2-graded vector spaces. The reason for this is that the Clifford algebras Cn only allow graded algebra isomorphisms Cn ⊗ Cm ∼ = Cn+m if one uses the signed flip on the category of graded vector spaces, in fact this symmetric monoidal category is by definition that of super vector spaces. Recall that a braiding of a monoidal category is needed in order to give the tensor product of two monoidal objects the structure of a monoidal object. Different braidings, like the unsigned flip versus the signed flip in the monoidal category of graded vector spaces, lead to different notions of monoidal objects. As a consequence, EFTs can only lead to the ring spectrum KO if they naturally lead to the signed flip. We do not see a better way than to introduce this braiding on the target category of super Fr´echet spaces SFr to begin with. Using the spin-statistics relation leads to the following corollary.

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Corollary 6.46. The groupoid Fun⊗,ss csM (1|1-PEB, SFr) is equivalent to the groupoid of triples (V, µ0 , ρ) where V is a graded Fr´echet space with an even pairing µ0 : V ⊗ V → C that is symmetric in the naive sense: µ0 (v, w) = µ0 (w, v)

for all v, w ∈ V .

1|1

Moreover, ρ : R>0 → V ⊗ V is a morphism of generalized cs-semigroups, as in Definition 5.7, with respect to the composition on V ⊗V induced by µ0 . In addition, ρ satisfies the following symmetry condition with respect to the decomposition of V ⊗ V into the four blocks V ev ⊗ V ev , etc. given by the grading of V :

0  At iθτ Bt ρt,θ = θBt A1t where A0 : R+ → V ev ⊗ V ev , A1 : R+ → V odd ⊗ V odd and B : R+ → V ev ⊗ V odd are smooth maps such that A0 and A1 are symmetric in the naive sense. Moreover, τ : V ev ⊗ V odd → V odd ⊗ V ev is the isomorphism given by u ⊗ v → v ⊗ u. Morphisms in this groupoid are isomorphisms of Fr´echet spaces, compatible with the extra structure given by µ0 , ρ. It is important to observe that µ0 is C-bilinear and hence we cannot relate this result directly to Hilbert spaces. In the next subsection we shall introduce a notion of real Euclidean field theories for which we obtain an inner product on a real Hilbert space. Proof of Corollary 6.46. Let’s define V := E(pt), ρ := E(R) and µ0 := E(L0 ). By the spin-statistics relation, the grading on V is given by α = E(fl). Therefore, the relations in (2) of Theorem 6.42 say that µ0 is even and satisfies for all homogeneous vectors v, w ∈ V µ0 (α(v), w) = (−1)|vw| µ0 (w, v) If v and w have different parity, both sides of this equation vanish since µ0 is even. If both vectors are even then it just gives naive symmetry. If both vectors are odd, then both sides pick up a minus sign, and hence again one gets the naive symmetry as claimed. The next step is to observe that the first relation is (3) of Theorem 6.42 says 1|1 that ρ : R>0 → V ⊗ V is a morphism of generalized cs-semigroups. This follows 1|1 from the fact that the grading on R>0 is given by (t, θ) → (t, −θ) whereas it is given by two copies of α on V ⊗ V . As explained in Definition 5.7, this implies that ρ is of the form A + θB, where A : R>0 → (V ⊗ V )ev

and

B : R>0 → (V ⊗ V )odd

are smooth maps. It is easy to check that the second relation in (3) of Theorem 6.42 is equivalent to the matrix decomposition of ρ as in the statement of our corollary.
6.8. Real Euclidian field theories. We recall from Section 6.3 the notion of real structures on cs-manifolds and Euclidean cs-manifolds, see Example 6.20. For 1|1 G = Iso(E1|1 ) = Rcs  Z/4

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the real structure on the 4th roots of unity Z/4 is obtained by restriction from the usual real structure on C. In particular, rG (pg) = pg−1 which implies the following effect on the generators of the bordism category 1|1-PEB from Theorem 6.42: ∼ sp, fl = fl, L0 = L0 ◦ (fl  id) and I t,θ = It,θ sp = Here we used Lemma 6.19 to get these complex conjugate Euclidean cs-manifolds. This actually induces an involution of complex conjugation on the whole Euclidean cs-bordism category 1|1-PEB → csM, compatible with the usual one on csM. Similarly, complex conjugation of complex vector bundles induces such an involution on the fibred category SFr → csM. Definition 6.47. Consider the resulting conjugation action on the groupoid Fun⊗,ss csM (1|1-PEB, SFr), written as E → E. Then the groupoid of real Euclidean field theories is defined to be the homotopy fixed point groupoid of this Z/2-action and is denoted by Fun⊗,ss csM (1|1-PEB, SFr)R The objects are pairs (E, rE ) where E ∈ Fun⊗,ss csM (1|1-PEB, SFr) and rE : E → E is a natural isomorphism such that r¯E ◦rE = idE . Morphisms are natural isomorphisms of EFTs that are compatible with the real structures rE . Using Corollary 6.46 and the above relations, we can spell out the data involved in such a real EFT. Because the super-intervals It,θ are real (also know as invariant under complex conjugation), the discussion simplifies in the case where the pairing µ0 = E(L0 ) is nondegenerate on V := E(sp), in the sense that the induced map V ⊗V → End(V ) is injective. We refer to the image of this map simply as the traceclass operators. In this nondegenerate case one does not lose any information when going from Rt,θ to It,θ = Rt,θ ◦sp L0 . We simply call such EFTs nondegenerate, adding a superscript nd into the notation. nd Theorem 6.48. The groupoid Fun⊗,ss csM (1|1-PEB, SFr)R of nondegenerate real 1|1-dimensional EFTs is equivalent to the groupoid of triples (W, µ, ) where W is a graded real Fr´echet space with an even pairing µ : W ⊗ W → R that is symmetric in the naive sense:

µ(v, w) = µ(w, v)

for all v, w ∈ W .

1|1

Moreover,  : R>0 → End(W ) is an even super-semigroup of trace-class operators that is self-adjoint with respect to µ: µ(t,θ v, w) = µ(v, t,θ w)

for all v, w ∈ W .

Morphisms in this groupoid are even isomorphisms of Fr´echet spaces, compatible with the extra structure given by µ, . Proof. We start with a real EFT (E, rE ) and first collect the data given by E as in Corollary 6.46: These are V = E(sp), a complex Fr´echet space graded by α = E(fl) with pairing µ0 = E(L0 ) and semigroup ρ = E(R). The real structure rE induces a real structure rV on V that is compatible with the grading α because fl = fl. Thus the fixed point set W of rV is a graded real Fr´echet space. The relation L0 = L0 ◦ (fl  id) implies that µ0 (rV (v), rV (w)) = µ0 (α(v), w)

for all

v, w ∈ V.

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This means that µ0 is real on W ev but purely imaginary on W odd . That’s why we can’t use µ0 as the required pairing. However, there is a simple fix to this problem: Define µ by setting it equal to µ0 on V ev and to i · µ0 on V odd . Then µ is a real pairing in the sense that it takes real values on W . Also recall that µ0 , and by definition µ, are even, i.e., they vanish on vectors of distinct parity. This factor of i does not only give a real pairing but it also cancels the annoying factor of i in relation (3) of Theorem 6.42: Let t,θ := E(It,θ ) : V → V , an even super-semigroup of trace-class operators. Then this relation (3) implies that µ0 (t,θ v, w) = µ0 (v, t,iθ w)

for all

v, w ∈ V.

We can write  in the form A + θB as in the proof of Corollary 6.46. Here A : R>0 → End(V )ev and B : R>0 → End(V )odd are smooth maps. Then we check the self-adjointness with respect to µ instead of µ0 , using the fact that both these pairings are even. If v, w ∈ V ev or v, w ∈ V odd then µ(t,θ v, w) = µ(At v, w) = µ0 (At v, w) = µ0 (v, At w) = µ(v, t,θ w) In the case where v ∈ V ev , w ∈ V odd (or vice versa), we have µ(t,θ v, w) = µ(θBt v, w) = iµ0 (θBt v, w) = µ0 (v, θBt w) = µ(v, θBt w) = µ(v, t,θ w) implying the self-adjointness of t,θ with respect to µ. Since It,θ is real, the same holds for t,θ and we have proven that the triple (W, µ, ) satisfies all the conditions stated in our corollary. Conversely, given such a triple, we need to reconstruct the real EFT. This is completely straightforward, except that we have to use the nondegeneracy of µ (or 1|1 equivalently, of µ0 ) to reconstruct the super-semigroup ρ : R>0 → V ⊗ V from the semigroup . Then the conditions line up exactly.
The last step we need is to nail down the Fr´echet space by the following positivity condition that’s the super analogue of Definition 6.26. It implies the nondegeneracy required above and we use the notation from the previous theorem. Definition 6.49. A 1|1-dimensional real Euclidean field theory E ∈ Fun⊗,ss csM (1|1-PEB, Fr)R is nonnegative if µ(v, w) ≥ 0 for all v, w ∈ W . We call E positive if W is in addition separable and its even and odd parts are both infinite-dimensional and µ gives an (algebraic) isomorphism from W to its continuous dual space W  . The full subgroupoid of Fun⊗,ss csM (1|1-PEB, Fr)R given by positive field theories is denoted by 1|1-PEFT, the groupoid of positive Euclidean field theories of dimension 1|1. In the positive case, Lemma 6.3 implies that W is a graded real Hilbert space with even inner product µ. Therefore, W ⊗ W is the space of trace-class operators and we obtain the final computation already used as Corollary 6.37 from the fact that any two such Hilbert spaces are isometric.

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7. Quillen categories and their classifying spaces In this section we shall first survey some constructions going back to Quillen and then relate certain Cn -linear generalizations to our configuration spaces Conf fin n . Recall from [9] that for any symmetric monoidal category (C, ⊕), Quillen defined the S −1 S-construction of C, usually denoted by C−1 C, as follows: Objects are pairs (C + , C − ) of objects of C and a morphism from (C + , C − ) to (D+ , D− ) is a triple (A, α+ , α− )

where A ∈ C and α± : C ± ⊕ A → D±

modulo the relation that two triples (A, α+ , α− ) and (B, β + , β − ) represent the same morphism if there is an isomorphism γ : A → B such that α± = β ± ◦(id ⊕γ). Quillen showed that if C is a groupoid and adding objects A ∈ C defines faithful endofunctors of C then the classifying space of C−1 C is the topological group completion of the classifying space of C (in the world of homotopy commutative and associative H-spaces). For example, let C be the groupoid of finite-dimensional real inner product spaces (and isometries), with orthogonal sum as monoidal structure. The existence of orthogonal complements simplifies the above definitions and it is easy to see that one obtains the following description of the topological category D := C−1 C: • objects of D are graded inner product spaces and • the morphism spaces D(W1 , W2 ) consist of pairs (f, R) where f : W1 → W2 is an isometric embedding and R is an odd orthogonal involution on the cokernel W2 − f (W1 ). Here and in the following we use the notation W2 − W1 for the orthogonal complement of W1 in W2 . In this case, a topological version of Quillen’s theorem implies that the classifying space |N• (D)|  BO × Z  because the right-hand side is the group completion of |N• (C)|  n BO(n). To relate these categories to our configurations, let’s fix a separable real Hilbert space H of infinite dimension and denote by Cδ (H) the same topological groupoid as above, except that the objects are required to be subspaces of H. Moreover, let C(H) be the internal groupoid in Top that has the same objects and morphisms as Cδ (H) but where the the topology on the objects isn’t discrete but has the usual topology of the Grassmannian of H. It is well known that the two continuous functors C ← Cδ (H) → C(H) induce homotopy equivalences on classifying spaces. Note that the two new categories are not monoidal since the direct sum is only partially defined: One needs that two given subspaces of H are orthogonal before one can take their sum (in a way that agrees with the direct sum in C). As a consequence, we cannot directly apply Quillen’s S −1 S-construction (even though there is a version for partially defined symmetric monoidal structures). However, there are obvious ambient analogues of the category D above: Consider H = H ⊕ H as a graded Hilbert space and denote by Dδ (H), respectively D(H), the categories in Top having as objects graded subspaces of H (with the discrete, respectively natural, topology) and with morphism spaces just like for D. Again there are two continuous functors D ← Dδ (H) → D(H)

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and one can show that they give homotopy equivalences on classifying spaces. In the ambient setting, there is one more simplification one can make. Namely, let D0 be the subcategory of D(H) with the same space of objects (graded subspaces of H) but with morphisms restricted to those pairs (f, R) where f is an inclusion of subspaces. In particular, there can only be a morphism from W1 to W2 in D0 if W1 ⊆ W2 . Theorem 7.1 below says that the classifying space of this category in Top has the homotopy type of Conf fin 0 . Using the contractibility of O(H) one can show that the continuous functor D0 → D(H) also induces a homotopy equivalence on classifying spaces, hence providing an alternative way to see that Conf fin 0 and BO × Z have the same homotopy type. For any graded algebra A, there are A-linear analogues of the above category D. These are not directly related to the Quillen construction, at least not to our eyes. In the following, we shall make the ambient versions of these categories precise for real Clifford algebras A = Cn . Fix a Z/2-graded real Cn -module Hn as in the introduction. Using graded Cn -submodules of Hn as objects, one can define categories Dδ (Hn ), D(Hn ) and Dn in Top as follows: The morphism spaces consist of pairs (f, R) as above such that both f and R are Cn -linear (and f is restricted to inclusions for Dn ). As before one can show that the two continuous functors Dn → D(Hn ) ← Dδ (Hn ) give homotopy equivalences on classifying spaces; the end of Remark 7.3 contains a very good reference. We shall be mostly concerned with Dn since its classifying space is homeomorphic to a configuration space. So let’s make that case completely precise: There is no morphism from W1 to W2 in Dn unless W1 ⊆ W2 and in this case MorDn (W1 , W2 ) := {R ∈ OCn (W2 − W1 ) | Rα = −R, R2 = 1}. The operators R are Cn -linear, orthogonal, odd involutions on this complement. The ±1-eigenspaces V± of R provide an orthogonal decomposition W2 − W1 = V+ ⊥ V−

with V− = α(V+ ).

In fact, mapping an operator R as above to its +1-Eigenspace gives a bijection with the set of subspaces V+ ⊂ W2 − W1 satisfying the above property. As in Section 4 we think of subspaces of Hn as orthogonal projection operators, hence identifying the set of objects ObjDn of Dn with a subspace of B(Hn ). In order to topologize the set of morphisms MorDn of Dn we identify it with the set of triples (W1 , W2 , A), where W1 , W2 ∈ ObjDn and A ∈ B(Hn ) such that W2 −W1 is an invariant subspace for A, the kernel of A is (W2 − W1 )⊥ and A defines an odd, orthogonal involution on W2 − W1 . We make MorDn into a topological space, in fact, a metric space, by considering it as a subspace of the product of three copies of B(Hn ). The nerve N• (Dn ) is a simplicial space, whose geometric realization Qn := |N• (Dn )| is the classifying space of Dn and it is directly related to configuration spaces:
 n : Qn − → Conf fin Theorem 7.1. There is a bijective continuous map G n that is also a homotopy equivalence.

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Proof. It suffices to consider the case n = 0. In this case, we suppress the index n altogether. In order to get the claim for general n, apply the n = 0 case to Hn (merely considered as a Z2 -graded real Hilbert space) to obtain a bijective  (see below)  : BD ≈ Conf fin . It is clear from the definition of G continuous map G that the classifiying space BDn ⊂ BD of the subcategory Dn of D corresponds fin  Since the homotopy used to prove that under G. precisely to Conf fin n ⊂ Conf  G is a homotopy equivalence preserves the subspace Conf fin n , the claim follows for general n. Recall that the k-simplices x ∈ Nk D of the nerve of our internal space category D are chains of Z/2-graded finite-dimensional subspaces W0 ⊆ W1 ⊆ · · · ⊆ Wk ⊂ H together with odd, orthogonal involutions Ri on Wi − Wi−1 for i = 1, . . . , k. We abbreviate this to x = (Wi , Ri ). The classifying space BD is the quotient space $

# Nk D × ∆k (β ∗ (x), t) ∼ (x, β∗ (t)) ∀ β : [m] → [n] k≥0

In our context, it is convenient to replace the usual standard simplex by k

∆k := {t = (t1 , . . . , tk ) ∈ R | 0 ≤ t1 ≤ · · · ≤ tk ≤ ∞}. The face map di : [k − 1] → [k] induces the map (di )∗ : ∆k−1 → ∆k , given by repeating ti , for i = 1, . . . , k. Moreover, (d0 )∗ adds a first coordinate equal to 0 and (dk )∗ adds a last coordinate equal to ∞. For i = 0, . . . , k − 1, the degeneracy maps si : [k] → [k − 1] induce (si )∗ : ∆k  ∆k−1 , given by skipping ti+1 . Now, every (x, t) ∈ Nk C × ∆k defines a configuration G(x, t) ∈ Conf fin as follows. The label G(x, t)0 at zero is Wi , where i is the largest index with ti = 0. For 0 < λ < ∞, G(x, t)±λ is the sum of the ±1-eigenspaces of all operators Ri with indices i with ti = λ. The label G(x, t)∞ is the orthogonal complement of all the other G(x, t)λ ’s. We claim that the map  : BD → Conf fin , G

 t] := G(x, t) G[x,

is well defined. We have to check that for all (x, t) and face and degeneracy maps β we have G(β ∗ (x), t) = G(x, β∗ (t)). We start with the face maps. In these cases we write t = (t1 , . . . , tk−1 ). If β = d0 : [k − 1] → [k] then β ∗ (x) is the chain of subspaces where W0 and R1 have been removed (and the indices of the other Wi and Ri are shifted to the left). Since β∗ (t) = (0, t1 , . . . , tk−1 ) it is clear from the definition of G that the labels of G(β ∗ (x), t) and G(x, β∗ (t)) coincide. For i = 1, . . . , k − 1 and β = di , the chain β ∗ (x) is obtained by composing the morphisms Ri and Ri+1 . This means that on Wi+1 − Wi−1 we get an orthogonal sum of these two operators. Since the ±1-eigenspaces of this orthogonal sum equals the direct sum of the ±1-eigenspaces of Ri and Ri+1 and since β∗ (t) just repeats ti , it is clear that G(β ∗ (x), t)ti = G(x, β∗ (t))ti . All the other labels are clearly unchanged so that G(β ∗ (x), t) = G(x, β∗ (t)). If β = dk : [k − 1] → [k] then β ∗ (x) is the chain of subspaces where Wk and Rk have been removed. However, since β∗ (t) = (t1 , . . . , tk−1 , ∞), we have G(β ∗ (x), t)λ = G(x, β∗ (t))λ for all λ ∈ R and so G(β ∗ (x), t) = G(x, β∗ (t)). For a degeneracy map β = si : [k] → [k − 1], where i = 0, . . . , k − 1, the argument is even easier. Then for a (k − 1)-simplex x, we get a chain β ∗ (x) of

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length k by inserting the identity at the ith subspace. This operation does not alter the operators Rj (the identity corresponds to R = 0 on a 0-space), it only shifts the indices > i to the right. Similarly, β∗ (t1 , . . . , tk ) = (t1 , . . . , tˆi , . . . , tk ), so that again a shifting of indices > i to the right occurs and G(β ∗ (x), t) = G(x, β∗ (t)) follows.  is well defined. It is bijective, since it has an inverse Hence the map G fin Conf → BD defined by mapping a configuration {Vλ } with exactly k nontrivial labels Vλ1 , . . . , Vλk with 0 < λi < ∞ to the equivalence class [x, t], where x is defined by the chain W0 ⊆ · · · ⊆ Wk , where W0 = V0 and Wi := Wi−1 ⊕ Vλi ⊕ V−λi for i > 0 and the operator Ri on Vλi ⊕ V−λi is defined to be the one with the  ±1-eigenspaces V±λi . It is clear that this defines an inverse for G.  It is easy to see that G is continuous: the description of the neighborhood basis for Conf in Definition 4.3 and the definition of the topology on MorD imply that G  is continuous. Hence the same is true for G.  It remains to show that G is a homotopy equivalence. Let Conf (k) ⊂ Conf fin be the subspace defined in Remark 4.7 and denote by BD(k) the image of Nk C × ∆k  restricts to a homeomorphism in BD. According to Lemma 7.2 below, G  (k) : BD(k) → Conf (k) G for all k ≥ 0. This together with the fact that id : colimk→∞ Conf (k) → Conf fin  is a homotopy equivalence. This completes the (see Remark 4.7) implies that G proof of Theorem 7.1.
Lemma 7.2. For all k ≥ 0, the map  (k) : BD(k) → Conf (k) G is a homeomorphism.  is continuous, so is G  (k) . The proof that G  (k) is open is based Proof. Since G on the following fact. Fact. There is a function εk : (0, 12 ) → R>0 satisfying εk (δ) → 0 as δ → 0 with the following property. If l ≤ k and V, W1 , W2 , . . . , Wl , are finite-dimensional subspaces of a Hilbert space H such that PV − Pl Wi  < δ < 12 and the Wi ’s i=1 are mutually orthogonal, then there exists a splitting of V into mutually orthogonal subspaces V1 , . . . , Vk such that PVi − PWi  < εk (δ) for all i. ˜ i := PV (Wi ) ⊂ V give a The idea of the proof is simple. The subspaces W ˜ decomposition of V and the Wi are almost orthogonal. Then use Gram – Schimdt ˜ i . Standard estimates plus induction on k yield the desired to orthogonalize the W −(k−1) works). function εk (e.g., εk (δ) = 3k−1 δ 2 Now, given a neighborhood N of [x, t] ∈ BD(k) we have to find K > 0 and  (k) (N ), where V := G  (k) [x, t] and VK,δ,δ is the δ > 0 such that VK,δ,δ ∩ Conf (k) ⊂ G neighborhood of V described in Definition 4.3. We may choose K to be any number such that (−K − 1, K + 1) contains all nontrivial labels λ ∈ R of V . In order to find k δ, observe that the preimage of N in i=0 Ni D × ∆i contains a full ε-neighborhood of the equivalence class [x, t] (recall that Ni D × ∆i is a metric space for all i) for

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some ε > 0. The existence of ε follows from the Lebesgue covering lemma and the fact that [x, t] is compact. Now choose δ > 0 such that εk (δ) < ε. In order to show that VK,δ,δ ∩ Conf (k) ⊂  (k) (N ), consider any W ∈ VK,δ,δ ∩ Conf (k) . Let j be minimal with W ∈ Conf (j) . G Using the Fact it is easy to see that there is (y, s) ∈ [x, t] ∩ Nj D × ∆j that lies within ε-range of the unique preimage of W in Nj D × ∆j under G. In other words,  (k) lies in (y, s) lies within ε-distance from [x, t]. Hence the preimage of W under G (k)  is open.
N , as desired. This completes the proof that G Remark 7.3. In his thesis [5], Pokman Cheung considers Conf n as a discrete set of objects and introduces a space of morphisms which gives a topological category that we shall denote by Cδ (Hn ). He refers to this as the “category of 1-dimensional super Euclidean field theories,” even though the morphisms are not defined in terms of field theories but rather in terms of configurations in Hn , i.e., points in Conf n . This is reflected by the fact that in the 2-dimensional case, Cheung can only define an analogous category for field theories based solely on annuli where one again can express the data in terms of configurations of Eigenspaces of an infinitesimal generator. The space of morphisms in Cδ (Hn ) is closely related to the path-space of Conf n . Via the Moore method, the space of paths actually gives a more direct way to get a topological category with classifying space Conf n . However, Cδ (Hn ) has the advantage that it connects very well to the Quillen type category Dδ (Hn ) discussed at the beginning of this section. There are two canonical maps on objects from Cδ (Hn ) to Dδ (Hn ) and back: One map takes a configuration {Vλ } ∈ Conf n and maps it to V0 ⊂ Hn , the index or Z/2-graded kernel of the odd infinitesimal generator corresponding to the configuration. This map is not continuous in the natural topologies and hence the use of discrete topologies is essential. The other canonical map (which is continuous in the natural topologies) starts with a finite-dimensional Cn -submodule W ⊂ Hn and associates to it the configuration with V0 := W and no other nontrivial eigenspaces. The new idea in [5] is to define the morphisms in Cδ (Hn ) in a way such that these two maps extend to continuous functors between topological categories. In fact, a main result [5, Proposition 1.4.1] says that the classifying spaces of Cδ (Hn ) and Dδ (Hn ) are homotopy equivalent via these two inverse functors. Moreover, it is shown that the latter classifying space is homotopy equivalent to the AtiyahSinger space Fn studied in the next section. The first steps of the argument [5, Lemmas 1.4.3, 1.4.4] actually show that the classifying spaces of Dδ (Hn ) and Dn are homotopy equivalent: |N• D(Hn )|  |N• Dn | = Qn The rest of Cheung’s argument is thus an alternative to our proof, in this and the next section, where we exhibit canonical homotopy equivalences Qn  Conf n  Fn . 8. Spaces of Fredholm operators In this chapter we relate the spaces Conf n to the spaces of skew-adjoint Fredholm operators considered by Atiyah and Singer in [2]. See the introduction for a quick survey of this section.

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8.1. Fredholm operators. Recall that a Fredholm operator is a bounded operator whose kernel and cokernel are finite-dimensional. Let Fred(H) ⊂ B(H) be the subspace of Fredholm operators on the infinite-dimensional separable real Hilbert space H. Denote by C(H) := B(H)/K(H) the C ∗ -algebra of bounded operators modulo compact operators (also known as Calkin algebra) and by π : B(H) → C(H) the projection. Then Fred(H) is the preimage of the group of units in C(H) under π, i.e., we have T ∈ B(H) is Fredholm ⇐⇒ π(T ) ∈ C(H) is invertible. We will need the following facts about the spectrum σ(T ) of a self-adjoint bounded operator T . Let σess (T ) := σ(π(T )) be the essential spectrum of T , i.e., the spectrum of π(T ) in C(H). Then there is a decomposition σ(T ) = σess (T )  σdiscrete (T ), where σdiscrete (T ) consists precisely of the isolated points in σ(T ) such that the corresponding eigenspace has finite dimension. From the definition of the essential spectrum it is clear that for a Fredholm operator T (∗)

σess (T ) ∩ (−ε(T ), ε(T )) = ∅ for ε(T ) := π(T )−1 −1 C(H) ,

where  · C(H) is the C ∗ -norm on the Calkin algebra. In other words: the essential spectrum of T has a gap of size ε(T ) around 0. Note that the map ε : Fred(H) → R>0 is continuous. 8.2. K-theory and Fredholm operators. The most important invariant of a Fredholm operator T is its index index(T ) := dim(ker T ) − dim(coker T ). It turns out that the index is invariant under deformations, i.e., it is a locally constant function on Fred(H). In fact, it defines an isomorphism ∼ =

→ Z, π0 Fred(H) −

[T ] → index(T ).

This is a special case of the well-known result that Fred(H) is a classifying space for the real K-theory functor KO 0 . More explicitly, for compact spaces X there are natural isomorphisms KO 0 (X) ∼ = [X, Fred(H)]. This isomorphism is defined as follows. Consider [f ] ∈ [X, Fred(H)]. Changing f by a homotopy one can achieve that the dimensions of the kernel and the cokernel of f (x) are locally constant. This implies that they define vector bundles ker f and coker f over X. The image of [f ] is defined to be [ker f ] − [coker f ] ∈ KO 0 (X). For X = pt this reduces to the above isomorphism ∼ =

π0 Fred(H) ∼ → KO 0 (pt) ∼ = [pt, Fred(H)] − =Z Atiyah and Singer showed that the other spaces in the Ω-spectrum representing real K-theory can also be realized as spaces of Fredholm operators.

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8.3. The Atiyah – Singer spaces Fn . Let n ≥ 1 and let Hn be a real Hilbert space with an action of Cn−1 , just as before. Define  n := {T0 ∈ Fred(Hn ) | T ∗ = −T0 and T0 ei = −ei T0 for i = 1 . . . , n − 1}. F 0  n if n ≡ 3 (mod 4). In the case n ≡ 3 (mod 4) define Furthermore, let Fn := F  Fn ⊂ Fn to be the subspace of operators T0 satisfying the following additional condition (AS): the essential spectrum of the self-adjoint operator e1 · · · en−1 T0 contains positive and negative values (we say that e1 · · · en−1 T0 is neither essentially positive nor negative). Atiyah and Singer introduce this condition, because it turns  n has three connected components, two out that for n ≡ 3 (mod 4) the space F of which are contractible. However, for the relation with K-theory only the third component, whose elements are characterized by the above requirement on the essential spectrum of e1 · · · gen−1 T0 , is interesting. In fact, the main result of [2] is that for all n ≥ 1 the space Fn represents the functor KO −n . We shall re-prove this result in terms of our configuration spaces.  n can also be interpreted as odd operators on the Z2 -graded The elements in F Hilbert space Hn = Hn ⊗Cnev Cn . If we define  gr := {T ∈ Fred(Hn ) | T is odd, Cn -linear, and self-adjoint} F n  gr using the homeomorphism  n and F we can identify F n ≈  gr n − → Fn , ψ⊗en : F

T0 → T := T0 ⊗ en .

The operator T has the matrix representation

 0 T0∗ T = T0 0 ∼ with respect to the decomposition Hn = Hn ⊕ Hn . It is important to note that the skew-symmetry of T0 is equivalent to the relation T en = en T .  gr is homotopy equivalent to a configIn the next lemma, we will show that F n  := [−∞, ∞] be the two-point compactification of R equipped uration space. Let R with the involution s(x) := −x.  gr of all operators T with T  = 1 and Lemma 8.1. The subspace A ⊂ F n  gr . ε(T ) = 1, see (∗) for the definition of ε(T ), is a strong deformation retract of F n  {±∞}; Hn ) Furthermore, A is homeomorphic to the configuration space Conf Cn (R, as defined in Chapter 4.  gr by  gr × [0, 1] → F Proof. Define a homotopy H : F n n   −1 (T, t) → Ht (T ) := (t + (1 − t)T ) · φ (tε(T ) + (1 − t)T −1 ) · T . Here φ : R → [−1, 1] is defined by φ|[−1,1] = id, φ|[1,∞) ≡ 1 and φ|(∞,−1] ≡ −1, and φ(·) denotes the functional calculus with φ. The continuity of H follows from the continuity of  ·  and ε and from the usual continuity properties of functional calculus, see [24, Theorem VIII.20]. Also, Cn -linearity and parity of T are preserved under functional calculus. Furthermore, H0 = idF
ngr ,

 gr ) ⊂ A. Ht = idA for all t, and H1 (F n

 gr . Hence A is a strong deformation retract of F n

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Now, for all T ∈ A we have σ(T ) ⊂ [−1, 1] and all λ ∈ σ(T ) ∩ (−1, 1) are eigenvalues of finite multiplicity. The spectral theorem for self-adjoint operators implies that the eigenspaces V (T )λ of T are pairwise orthogonal and span all of Hn . Since T is odd, V (T )−λ = α(V (T )λ ), where α is the grading involution on Hn . We thus obtain a map A → Conf Cn ([−1, 1], {±1}; Hn ),

T → V (T )

by associating to T the configuration λ → V (T )λ on [−1, 1]. Here the involution on [−1, 1] is x → −x. It is easy to see that this map is a homeomorphism. Finally, ≈  → R, x → x/(1 − |x|) induces a homeomorphism of configuration spaces [−1, 1] −  {±∞}; Hn ), thus completing the proof of Conf Cn ([−1, 1], {±1}; Hn ) ≈ Conf Cn (R, the second statement in the lemma.
Now we can formulate the relationship between the Atiyah-Singer spaces Fn  → R that is the and our configuration spaces Conf n . Consider the map p : R identity on R and that maps ±∞ to ∞. It induces a continuous map  {±∞}; Hn ) → Conf C (R, {∞}; Hn ) = Conf n . p∗ : Conf C (R, n

n

Let H be the homotopy equivalence defined as the composition ≈  gr
n −  {±∞}; Hn ). →F − → Conf C (R, H: F n

n

The main result of this section is: Theorem 8.2. For all n ≥ 1, p∗ H restricts to a homotopy equivalence


→ Conf n . p∗ H|Fn : Fn − Proof. Since H is a homotopy equivalence, the same is true for H|Fn . It remains to show that the restriction of p∗ to the path component H(Fn ) of  {±∞}; Hn ) is a homotopy equivalence. In order to do this, it will be Conf Cn (R, convenient to work with subspaces consisting of certain ‘finite’ elements. More precisely, if we define  {±∞}; Hn ), Conf  := H(Fn ) ∩ Conf fin (R, n

Cn

then we have a commutative diagram H(Fn ) O

p∗




? Conf n

/ Conf n O


p∗

? / Conf fin n

whose vertical arrows are homotopy equivalences (for the right arrow this was done in Proposition 4.6; the same argument works for the arrow on the left). Hence p∗ |H(Fn ) is a weak homotopy equivalence exactly if this is the case for p∗ : Conf n → Conf fin n . This will be proved in Theorem 8.5 below. It follows that p∗ H|Fn is a weak homotopy equivalence. Since Fn and Conf n both have the homotopy type of a CW
complex, the map p∗ H|Fn is a homotopy equivalence, cf. [21]. As a first step towards Theorem 8.5, let us give a characterization of the configurations contained the subspace Conf n . Since the map H is surjective, we have   Conf n = Conf fin Cn (R, {±∞}; Hn ) for n ≡ 3 (mod 4) (recall that Fn = Fn in this case).

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The interesting case is n ≡ 3 (mod 4). Our task is to understand what the Atiyah – Singer condition (AS) means for the corresponding configurations. Before we proceed, we need to collect some representation theoretic facts about graded Clifford modules. Recall from [17] that for n ≡ 3 (mod 4) the Clifford algebra Cn is simple, whereas it is the product of two simple algebras in the other cases. Therefore, there is a unique irreducible Cn -module (and a unique graded irreducible Cn+1 -module) for n ≡ 3 (mod 4), otherwise there are exactly two such modules. Remark 8.3. Let n ≡ 3 (mod 4) and recall from [17, Chapter I, Proposition 5.9] that in this case the action of the volume element e := e1 · · · en ∈ Cn distinguishes the two distinct (ungraded) irreducible Cn -modules. (Since e is a central orthogonal involution, it acts as ± id on these modules.)  gr . The diagonal entries of the even operator eT are given Now, let T ∈ F n by e1 · · · en−1 T0 : Hn → Hn and therefore the Atiyah – Singer condition (AS) that the operator e1 · · · en−1 T0 : Hn → Hn is neither essentially positive nor negative is equivalent to the same condition on eT : Hn → Hn . Denote by Afin ⊂ A the subspace of operators with finite spectrum. Lemma 8.4. Assume n ≡ 3 (mod 4). Let T ∈ Afin and denote by W± the (±1)-eigenspaces of T . (1) eT is essentially positive (respectively negative) if and only if the volume element e has a finite-dimensional (−1)-eigenspace on W+ (respectively W− ). (2) The (AS) condition is equivalent to the (±1)-eigenspaces of e, restricted to W+ , both being infinite-dimensional. (3) W+ is a Cn -module and the (AS) condition is equivalent to W+ containing both irreducible Cn -modules infinitely often. Proof. For part (1) we observe that eT = T e and hence we can find simultanous eigenspace decompositions for these self-adjoint operators. Note that the eigenvalues (±1) are the only possible accumulation points in the spectrum of T and hence such eigenspace decompositions exist. Since e2 = 1, the operator e has spectrum inside {±1}. A vector v ∈ Hn is in an essentially positive eigenspace of eT if and only if either v ∈ W+ and e(v) = +1, or v ∈ W− and e(v) = −1. Since T is odd, its spectrum is symmetric and, in particular, the grading involution α takes W+ to W− . Furthermore, α anticommutes with e and hence e|W+ = e ◦ α|W− = −α ◦ e|W− so that α takes the (+1)-eigenspace of e|W+ to the (−1)-eigenspace of e|W− . In particular, these vector spaces have the same dimension. This finishes the proof of part (1) as well as part (2). To prove part (3) notice that T is Cn -linear and therefore W+ is a Cn -module (which is not graded since α takes it to W− ). The claim follows from the well known algebraic fact stated at the beginning of the lemma.
Note that Conf n is exactly the image of Afin ∩ Fngr under the identification  {±∞}; Hn ) from Lemma 8.1. Moreover, the (±1)-eigenspaces W± A ≈ Conf Cn (R,

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of an operator T ∈ Afin turn into the (±∞)-eigenspaces W±∞ of the corresponding  configuration W ∈ Conf fin Cn (R, {±∞}; Hn ). Hence part (3) of Lemma 8.4 tells us that for n ≡ 3 (mod 4) the subspace  Conf n ⊂ Conf fin Cn (R, {±∞}; Hn ) consists precisely of the configurations {Wλ } such that W±∞ contains both irreducible Cn -modules infinitely often. We will now see why this is a very natural condition in terms of our configuration spaces: Theorem 8.5. The restriction p := p∗ |Conf
n : Conf n → Conf fin n is a quasi-fibration with contractible fibers (see Definition 8.8). Remark 8.6. In the case n ≡ 3 (mod 4) the space Conf n is the unique con nected component of Conf fin Cn (R, {±∞}; Hn ) that is not contractible. On the two remaining components the map p∗ is not a quasi-fibration as we shall see below (the fibres have distinct homotopy groups). Lemma 8.7. Let M be a graded Cn -module. (1) If M contains all (one or two) irreducible Cn -modules infinitely often then the Cn -action on M extends to a graded Cn+1 -action. (2) Let M0 be a graded irreducible Cn -module. Then there is a graded vector space R, the multiplicity space, such that M is isomorphic to the graded tensor product M0 ⊗ R. In the case n ≡ 3 (mod 4) we may, and shall, assume that R is concentrated in even degree. (3) With this notation, the grading preserving Clifford linear orthogonal group OCn (M ) is isomorphic to ∼ O(R) = ∼ O(Rev ) × O(Rodd ) OC (M ) = n

In particular, this group is contractible (by Kuiper ’s theorem) if and only if the multiplicity spaces Rev and Rodd are both either zero or infinitedimensional. This is equivalent to M containing either only one type of graded irreducible Cn -module, or containing both infinitely often. Proof. There are two cases to consider for proving (1): If only one graded irreducible Cn -module exists, then take any irreducible Cn+1 -module M0 and restrict it to Cn . It is clear that over Cn , M must be given by infinitely many copies of M0 . Let’s say there are two graded irreducible Cn -modules M0 , M1 and hence n is divisible by 4. By assumption, M is the sum of infinitely many copies of M0 ⊥ M1 . It then suffices to show that M0 ⊥ M1 has an Cn+1 -action. We first claim that op M1 ∼ = M0 , i.e., M1 is obtained from M0 by flipping the grading. Using the characterization of ungraded Cn−1 -modules given in Lemma 8.4, it suffices to show that the volume element e˜1 · · · e˜n−1 acts with different sign on M0ev and M0odd . Here e˜i = ei en are the usual generators of Cn−1 ∼ = Cnev . Since n is divisible by 4, it follows that e˜1 · · · e˜n−1 = (e1 en ) · · · (en−1 en ) = e1 · · · en =: e is also the volume element in Cn . Writing M0odd = ei · M0ev for some i, our claim follow from eei = −ei e. Finally, the module M0 ⊥ M0op has a Cn+1 -action given by the element en+1 = f α, where f flips the two summands and α is the grading involution.

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For part (2) one again needs to know that in the case that there are two graded irreducible Cn -modules M0 , M1 , they differ from each other by flipping the grading. This was proven above. Part (3) is obvious.
Proof of Theorem 8.5. We begin by proving that the fibers of p are con−1 (V ) consists of all W ∈ Conf n such tractible. Fix V ∈ Conf fin n . The fiber p that Vλ = Wλ for λ ∈ R and V∞ is the orthogonal sum of W∞ and W−∞ . Since W−∞ = α(W∞ ), where α is the grading involution on Hn , we may identify p−1 (V ) with the space of decompositions of the graded Cn -module V∞ of the form V∞ = W∞ ⊥ α(W∞ ), where W∞ is an ungraded Cn -submodule of V∞ that for n ≡ 3 (mod 4) satisfies the (AS) condition: both irreducible Cn -modules appear infinitely often in W∞ . Without the (AS) condition, it is straightforward to show that this space of decompositions of V∞ is homeomorphic to the following space of Cn+1 -structures on V∞ :  n+1 (V∞ ) := {en+1 ∈ O(V∞ ) | e2 = −½, en+1 ei = −ei en+1 , i = 1, . . . , n} C n+1

Namely, given en+1 , one can define W±∞ to be the (±1)-eigenspaces of en+1 α (and vice versa). Under this correspondence, the (AS) condition translates into the requirement that en+1 defines a Cn+1 module structure on V∞ which contains n+1 (V∞ ) both graded irreducibles infinitely often. We denote this subspace of C simply by Cn+1 (V∞ ) and observe that all these module structures en+1 on V∞ are isomorphic. We show in the following 4 steps that the fibre p−1 (V ) ≈ Cn+1 (V∞ ) is contractible under our assumptions. Step 1. By our basic assumption, the ambient Hilbert space Hn contains all graded irreducible Cn -modules infinitely often. Since V was a finite configuration to start with, it follows that V∞ has the same property and by part (1) of Lemma 8.7 it follows that Cn+1 (V∞ ) is not empty. Step 2. Since any two points in Cn+1 (V∞ ) lead to Cn+1 -module structures on V∞ that are isomorphic, the orthogonal group OCn (V∞ ) acts transitively (by conjugation) on Cn+1 (V∞ ). The stabilizer of a particular Cn+1 -structure is OCn+1 (V∞ ) and hence Cn+1 (V∞ ) ≈ OCn (V∞ )/OCn+1 (V∞ ) We shall show that this space is contractible, as a quotient of two contractible groups. Step 3. As a Cn -module, V∞ contains both graded irreducible Cn -modules infinitely often, that’s what we need by part (3) of Lemma 8.7 for the contractibility of the larger group OCn (V∞ ). Step 4. For the smaller group OCn+1 (V∞ ), the (AS) condition tells us again that the assumptions of part (3) of Lemma 8.7 are satisfied. To finish the proof of Theorem 8.5, it remains to show that p is indeed a quasifibration. We will use the criterion in Theorem 8.10 but first we give the relevant definitions. Definition 8.8. A map p : E  B is a quasi-fibration if for all b ∈ B, i ∈ N and e ∈ p−1 (b), p induces an isomorphism ∼ =

πi (E, p−1 (b), e) − → πi (B, b).

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From the long exact sequence of homotopy groups for a pair it follows that p is a quasi-fibration exactly if there is a long exact homotopy sequence connecting fibre, total space and base space of p, just like for a fibration. However, p does not need to have any (path) lifting properties as the following example shows. Example 8.9. The prototypical example of a quasi-fibration that is not a fibration is the projection of a ‘step’ (−∞, 0] × {0} ∪ {0} × [0, 1] ∪ [0, ∞) × {1} ⊂ R2 onto the x-axis. Even though all fibers have the same homotopy type (they are contractible), the map doesn’t have the lifting property of a fibration, since it is impossible to lift a path that passes through the origin. The following sufficient condition for a map to be a quasi-fibration is an easy consequence of the results of [7]: Theorem 8.10. Let p : Y → X be a continuous map between Hausdorff spaces and X0 ⊂ X1 ⊂ X2 ⊂ · · · an increasing sequence of closed subsets of X s.t. X = colimi→∞ Xi . Assume further that for all i ≥ 0 the map p|Yi+1 \Yi is a Serre fibration, where Yi := p−1 (Xi ), and that there exists an open neighborhood Ni of Xi (i) (i) in Xi+1 and homotopies dt = dt : Ni → Ni and Dt = Dt : p−1 (Ni ) → p−1 (Ni ) s.t. (1) D covers d, i.e., p ◦ Dt = dt ◦ p for all t. (2) D0 = id, Dt (Yi ) ⊂ Yi for all t, and D1 (p−1 (Ni )) ⊂ Yi (3) For every x ∈ Ni , the map D1 : p−1 (x) → p−1 (d1 (x)) is a weak homotopy equivalence. Then p is a quasi-fibration. Proof. According to Satz 2.15 in [7] p is a quasi-fibration provided that p|Yi is a quasi-fibration for all i ≥ 0. To see this, we proceed by induction on i. It is clear that p|Y0 is a quasi-fibration, since, by assumption, it is a Serre fibration. Now assume that we already know that p|Yi is a quasi-fibration. Applying Hilfssatz 2.10 in [7] with B = Ni , B  = Xi , q = p|p−1 (Ni ) , D = D, and d = d, implies that p|p−1 (Ni ) is a quasi-fibration. Now, applying the Korollar of Satz 2.2 in [7] with U = Ni+1 and V = Yi+1 \ Yi we see that p|Yi+1 is a quasi-fibration. (Note that p is a quasi-fibration over U ∩ V , since it is even a Serre fibration.)
Now, in order to apply Theorem 8.10 we filter Conf fin n by the closed subspaces  

    fin  Vλ ≤ 2i . Xi := V ∈ Conf n  dim(V ) := dimCn λ∈R

For each i consider the continuous function Li : Xi+1 → [0, ∞],

 

  V → Li (V ) := sup r  dimCn

 Vλ

 ≤ 2i .

λ∈(−r,r)

L−1 i (∞)

L−1 i ((1, ∞]).

Clearly, = Xi . Define Ni := The homotopies d(i) and D(i) are  now easy to find. Namely, consider the homotopies on Conf fin n and Conf n induced by the family of maps ht defined in the proof of Proposition 4.6 (in the case of  Conf n , note that the formula defining ht indeed also determines a homotopy on R  and hence on Conf n ). It is easy to check that almost all assumptions of the DoldThom theorem are hold: (1) and (2) obviously satisfied. Condition (3) is trivial,

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since all fibers are contractible. The map p|Xi+1 \Xi is a fiber bundle, in particular, it is a Serre fibration. The crucial point in the proof of this is that dimension dim(V ) of configurations in Xi+1 \ Xi is constant. This makes it possible to choose an open neighborhood N of V in Xi+1 \ Xi such that the orthogonal projection PV∞ : W∞ → V∞ is an isomorphism for all W ∈ N . It is easy to write down a local trivialization of p over such an N ; this shows that p|Xi+1 \Xi is a fiber bundle. The only condition in the Dold – Thom theorem that is violated is that Conf fin n is not the colimit over the subspaces Xi (cf. Remark 4.7). However, this is not a  serious issue: we can endow Conf fin n and Conf n with the colimit topologies with −1 respect to the filtrations Xi and p (Xi ) and apply the Dold – Thom theorem to see that p is a quasi-fibration in this case. It follows directly from the definition of a quasi-fibration that the same also holds if we consider the original topologies, since  the identity maps colimi→∞ Xi → Conf fin n and colimi→∞ Yi → Conf n are homotopy fin equivalences (see Remark 4.7 for the Conf n case, the same argument works in the
case Conf n ). This completes the proof of Theorem 8.5.

References 1. M. F. Atiyah, R. Bott, and A. Shapiro, Clifford modules, Topology 3 (1964), suppl 1, 3 – 38. 2. M. F. Atiyah and I. M. Singer, Index theory for skew-adjoint Fredholm operators, Inst. Hautes ´ Etudes Sci. Publ. Math. 37 (1969), 5 – 26. 3. M. Batchelor, The structure of super manifolds, Trans. Amer. Math. Soc. 253 (1979), 329 – 338. 4. R. Bott, The stable homotopy of the classical groups, Ann. of Math. (2) 70 (1959), 313 – 337. 5. P. Cheung, Supersymmetric field theories and cohomology, Ph.D. dissertation, Stanford University, 2006, arXiv:0811.2267. 6. P. Deligne and J. W. Morgan, Notes on supersymmetry (following Joseph Bernstein), Quantum Fields and Strings: A Course for Mathematicians, Vol. 1, 2 (Princeton, NJ, 1996/1997), Amer. Math. Soc., Providence, RI, 1999, pp. 41 – 97. 7. A. Dold and R. Thom, Quasifaserungen und unendliche symmetrische Produkte, Ann. of Math. (2) 67 (1958), 239 – 281. 8. D. S. Freed, Five lectures on supersymmetry, Amer. Math. Soc., Providence, RI, 1999. 9. D. Grayson, Higher algebraic K-theory. II (after Daniel Quillen), Algebraic K-theory (Evanston, IL, 1976), Lecture Notes in Math., vol. 551, Springer, Berlin, 1976, pp. 217 – 240. 10. N. Higson and E. Guentner, Group C ∗ -algebras and K-theory, Noncommutative Geometry, Lecture Notes in Math., vol. 1831, Springer, Berlin, 2004, pp. 137 – 251. 11. H. Hohnhold, M. Kreck, S. Stolz, and P. Teichner, Differential forms and 0-dimensional supersymmetric field theories, preprint. 12. H. Hohnhold, S. Stolz, and P. Teichner, K-theory and 1-dimensional supersymmetric field theories, in progress. 13. M. Joachim, K-homology of C ∗ -categories and symmetric spectra representing K-homology, Math. Ann. 327 (2003), no. 4, 641 – 670. 14. M. Karoubi, K-theory: An introduction, Grundlehren Math. Wiss., vol. 226, Springer, Berlin, 1978. 15. G. K¨ othe, Topological vector spaces. I, Grundlehren Math. Wiss., vol. 159, Springer, New York, 1969. , Topological vector spaces. II, Grundlehren Math. Wiss., vol. 237, Springer, New York, 16. 1979. 17. H. B. Lawson Jr. and M.-L. Michelsohn, Spin geometry, Princeton Math. Ser., vol. 38, Princeton Univ. Press, Princeton, NJ, 1989. 18. A. Libman, Orbit spaces, the normaliser decomposition and Minami’s formula for compact Lie groups, preprint. 19. E. Markert, Connective 1-dimensional Euclidean field theories, Ph.D. thesis, University of Notre Dame, 2005.

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20. R. Meise and D. Vogt, Introduction to functional analysis, Oxford Grad. Texts Math., vol. 2, Oxford Univ. Press, New York, 1997. 21. J. Milnor, On spaces having the homotopy type of CW-complex, Trans. Amer. Math. Soc. 90 (1959), 272 – 280. , Morse theory, Annals of Math. Stud., vol. 51, Princeton Univ. Press, Princeton, NJ, 22. 1963. 23. R. S. Palais, On the homotopy type of certain groups of operators, Topology 3 (1965), 271 – 279. 24. M. Reed and B. Simon, Methods of modern mathematical physics. I: Functional analysis, 2nd ed., Academic Press Inc., New York, 1980. . 25. W. Rudin, Functional analysis, 2nd ed., Internat. Ser. Pure Appl. Math., McGraw-Hill Inc., New York, 1991. 26. G. Segal, Categories and cohomology theories, Topology 13 (1974), 293 – 312. , K-homology theory and algebraic K-theory, K-Theory and Operator Algebras 27. (Athens, GA, 1975), Lecture Notes in Math., vol. 575, Springer, Berlin, 1977, pp. 113 – 127. 28. S. Stolz and P. Teichner, What is an elliptic object?, Topology, Geometry and Quantum Field Theory, London Math. Soc. Lecture Note Ser., vol. 308, Cambridge Univ. Press, Cambridge, 2004, pp. 247 – 343. , Supersymmetric Euclidean field theories and generalized cohomology, preprint. 29. 30. W. P. Thurston, Three-dimensional geometry and topology. Vol. 1, Princeton Math. Ser., vol. 35, Princeton Univ. Press, Princeton, NJ, 1997. 31. R. Wood, Banach algebras and Bott periodicity, Topology 4 (1965/1966), 371 – 389. Statistisches Bundesamt, 65180 Wiesbaden, Germany E-mail address: [email protected] Department of Mathematics, University of Notre Dame, 255 Hurley Hall, Notre Dame, IN 46556-4618, USA E-mail address: [email protected] Department of Mathematics, University of California, Berkeley, CA, 94720-3840, USA Current address: Max-Planck Institut f¨ ur Mathematik, P.O.Box 7280, 53072 Bonn, Germany E-mail address: [email protected]

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Part 4

Dualities and Interactions with Quantum Field Theory

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https://doi.org/10.1090/crmp/050/21

Centre de Recherches Math´ ematiques CRM Proceedings and Lecture Notes Volume 50, 2010

Raoul Bott as My Math Teacher Cumrun Vafa

1 It is a great pleasure to write this short note in memory of a brilliant mathematician, a generous human being, a dedicated teacher, and a good friend, Raoul Bott. Even though I have benefited from numerous interactions with him as a colleague throughout many years at Harvard, here I will focus on my view of him as my mathematics teacher before I joined the Harvard Faculty. I was an undergraduate student at MIT, double majoring in mathematics and physics. I got to know Raoul when I cross registered to take the introductory course on differential geometry that he was teaching at Harvard University in the spring of 1981. Perhaps I should first give some background to place this in proper context. I grew up in Iran and attended an excellent high school, the Alborz High School. Mathematics was my very favorite subject at high school. Iran has a rich tradition of mathematics dating back to great mathematicians such as Omar Khayyam, Khwarazmi and Biruni. A characteristic of this old Iranian school of mathematics was its emphasis on Euclidean geometry. Thus, it is not surprising that at my high school Euclidean geometry was perhaps the best taught subject. I loved Euclidean geometry and spent many hours trying to solve difficult geometry problems. What struck me most about it was the intuitive nature of the material, the elegance of geometric insights, and the power of proofs and constructions. I found the subject very beautiful. That one could see the proof by simply drawing clever auxillary lines and circles was very exciting to me. After high school I came to the US, where I was admitted to the undergraduate program at MIT in 1977. I double majored in mathematics and physics. After taking a few courses in modern mathematics, I began to feel a bit disillusioned with the abstract nature of modern math. I did not see the analogs of the beautiful geometric pictures I was used to from high school, which had so attracted me to mathematics in the first place. I was very lucky, however, to have Dan Quillen as my math advisor at MIT. Very kind and generous with his time, he went out of his way to help me get a well grounded education in mathematics (for example, he introduced me to the subject of instantons!). Having noticed my disillusionment with my math courses, 2000 Mathematics Subject Classification. 01A60. c
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C. VAFA

he suggested that I sign up for the differential geometry course that Raoul Bott was teaching at Harvard, and I am glad to say that I obliged. Raoul Bott’s course completely changed my perception of modern math. I learned that the methodology of modern math is not so different from what I was used to in the context of Euclidean geometry. His lectures were enjoyable to listen to, and he knew how to convey the core ideas so that the students would understand the main points without getting lost in irrelevant technical details. He instilled in me the concept of naturalness in mathematics and would often refer to natural objects and constructions as ‘God given’. He made modern geometry look as intuitive and exciting as I had perceived Euclidean geometry to be. Raoul reignited my interest in mathematics, profoundly influencing my later studies. Raoul was more than a teacher: He treated students almost as friends and was very easy to talk with. For example, after I took the final exam in his class he asked me where I was planning to go after my graduation. I replied that I would be going to Princeton University, where I had been admitted to the graduate program in physics. He immediately exclaimed: ‘Oh that is where Ed Witten is! You know, he is very good! I am willing to sign under anything he says!’ This also indicated his openness to ideas coming from physics. In fact, he contributed to many of the bridges that connect the two fields. Raoul Bott was a prominent figure in the twentieth century mathematics. I hope my brief recounting of my interactions with him gives a glimpse of his personality and influence beyond his mathematical accomplishments. He will be fondly remembered by his colleagues, students and friends. Department of Physics, Harvard University, 17 Oxford St., Cambridge, MA 02138, USA E-mail address: [email protected]

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https://doi.org/10.1090/crmp/050/22

Centre de Recherches Math´ ematiques CRM Proceedings and Lecture Notes Volume 50, 2010

A Physics Colloquium at McGill that Changed My Life Steven Lu

In the early 80s I was studying for a double bachelor’s degree at McGill University. Undecided about whether to pursue a career in physics or pure math, I went to this physics colloquium with a sense of anticipation since it would be given by a pure mathematician whose first two university degrees were earned in Montreal. His name was Raoul Bott. I had been fascinated by high energy physics and I recall one physicist in the audience questioned the Bott’s use of the term high energy in this mathematical talk intended for a wide audience. Bott readily apologized and jokingly said something to the effect that its understanding requires high energy of the brain. Needless to say, that talk by Bott lead me to pursue a Ph.D. in mathematics at Harvard and a life long pursuit of mathematics that is still filled with awe at his inspiration. ` Montr´ D´ epartement de math´ ematiques, Universit´ e du Qu´ ebec a eal, CP 8888, succ Centre-ville, Montr´ eal, QC H3C 3P8, Canada E-mail address: [email protected]

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https://doi.org/10.1090/crmp/050/23

Centre de Recherches Math´ ematiques CRM Proceedings and Lecture Notes Volume 50, 2010

Geometric Langlands from Six Dimensions Edward Witten Abstract. Geometric Langlands duality is usually formulated as a statement about Riemann surfaces, but it can be naturally understood as a consequence of electric-magnetic duality of four-dimensional gauge theory. This duality in turn is naturally understood as a consequence of the existence of a certain exotic supersymmetric conformal field theory in six dimensions. The same six-dimensional theory also gives a useful framework for understanding some recent mathematical results involving a counterpart of geometric Langlands duality for complex surfaces.

1. Introduction A d-dimensional quantum field theory (QFT) associates a number, known as the partition function Z(Xd ), to a closed d-manifold Xd endowed with appropriate structure.1 Depending on the type of QFT considered, the requisite structure may be a smooth structure, a conformal structure, or a Riemannian metric, possibly together with an orientation or a spin structure, etc. In physical language, the partition function can usually be calculated via a path integral over fields on X. However, this lecture will be partly based on an exception to that statement. To a closed (d − 1)-dimensional manifold Xd−1 (again with some suitable structure), a d-dimensional QFT associates a vector space H(Xd−1 ), usually called the space of physical states. In the case of a unitary QFT (such as the one associated with the Standard Model of particle physics), H is actually a Hilbert space, not just a vector space. The quantum field theories considered in this lecture are not necessarily unitary. The partition function associated to the empty d-manifold is Z(∅) = 1, and the vector space associated to the empty (d − 1)-manifold is H(∅) = C. There is a natural link between these structures. To a d-manifold Xd with boundary Xd−1 , a d-dimensional QFT associates a vector ψXd ∈ H(Xd−1 ). (In physical terminology, ψXd can usually be computed by performing a path integral for fields on Xd that have prescribed behavior along its boundary.) This generalizes 2000 Mathematics Subject Classification. Primary 81T60; Secondary 14D21, 81T45. Supported in part by NSF Grant Phy-0503584. This is the final form of the paper. 1We will slightly relax the usual axioms in Section 4.1. c
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the partition function, since if Xd−1 = ∅, then ψXd ∈ H(∅) = C is simply a complex number, which is the partition function Z(Xd ). What I have said so far is essentially rather familiar to physicists. (The reason for the word “essentially” in the last sentence is that for most physical applications, a less abstract formulation is adequate.) Less familiar is that it is possible to continue the above discussion to lower dimensions. The next step in the hierarchy is that to a closed (d−2)-manifold Xd−2 (with appropriate structure) one associates a category C(Xd−2 ). Then, for example, to a (d − 1)-manifold Xd−1 with boundary Xd−2 , one associates an object P(Xd−1 ) in the category C(Xd−2 ). (For relatively informal accounts of these matters from different points of view, see [2, 8]; for some recent developments, see [21] as well as [25].) 1.1. Categories and physics. In practice, physicists do not usually specify what should be associated to Xd−2 . This is not necessary for most purposes — certainly not in standard applications of QFT to particle physics or condensed matter physics. However, before getting to the main subject of this talk, I will briefly explain a few cases in which that language is or might be useful for physicists. So far, the most striking physical application of the “third tier,” that is, the extension of QFT to codimension two, is in string theory, where one uses twodimensional QFT to describe the propagation of a string. In this case, since d = 2, a (d − 2)-manifold is just a point. So the extra layer of structure is just that the theory is endowed with a category C, which is the category of what physicists call boundary conditions in the quantum field theory, or D-branes. For d = 2, a connected (d−1)-manifold with boundary is simply a closed interval I, whose boundary consists of two points. To define a space H(I) of physical states of the open string, one needs boundary conditions B and B
at the two ends of I. To emphasize the dependence on the boundary conditions, the space of physical states is better denoted as H(I; B, B
). In category language, this space of physical states is called the space of morphisms in the category, HomC (B, B
). (This construction has two variants that differ by whether the manifolds considered are oriented; they are both relevant to string theory.) Another case in which the third tier can be usefully invoked, in practice, is three-dimensional Chern – Simons gauge theory. This is a quantum field theory for d = 3 with a compact gauge group G and a Lagrangian that is, roughly speaking,2 an integer k times the Chern – Simons functional. A closed (d − 2)-manifold is now a circle, and again, the extra layer of structure is that a category C is associated to the theory; it is the category of positive energy representations of the loop group of G at level k. Finally, the state of the Universe in the presence of a black hole or a cosmological horizon is sometimes described in terms of a density matrix rather than an ordinary quantum state, to account for one’s ignorance of what lies beyond the horizon. This point of view (which notably has been advocated by Stephen Hawking) can possibly be usefully reformulated or refined in terms of categories. The idea here would be that, in d-dimensional spacetime, the horizon of a black hole (or a cosmological horizon) is a closed (d − 2)-manifold. Indeed, suppose that Xd is a d-dimensional Lorentz signature spacetime with an “initial time” hypersurface Xd−1 . Suppose further that a black hole is present; its horizon intersects Xd−1 on a codimension two 2This formulation suffices if G is simple, connected, and simply connected. In general, k is an element of H 4 (BG, Z).

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submanifold Xd−2 . It is plausible that to Xd−2 , we should associate a category C, and then to Xd−1 we would associate not — as we would in the absence of the black hole — a physical Hilbert space H(Xd−1 ) — but rather an object P in that category. To make this more concrete, suppose for example that C is the category of representations of an algebra S. Then P is an S-module, which in this context would mean a Hilbert space H(Xd−1 ; Xd−2 ) with an action of S. Physical operators would be operators on this Hilbert space that commute with S. Intuitively, S is generated by operators that act behind the horizon of the black hole. (That cannot be a precise description in quantum gravity, where the position of the horizon can fluctuate.) This point of view is most interesting if the algebra S is not of type I, so that it does not have irreducible modules and the category of S-modules is not equivalent to the category of vector spaces. At any rate, even if the categorical language is relevant to quantum black holes, it may be an oversimplification to suppose that C is the category of representations of some algebra. 1.2. Geometric Langlands. Our aim here, however, is to understand not black holes but the geometric Langlands correspondence. In this subject, one studies a Riemann surface C, but the basic statements that one makes are about categories associated to C. Indeed, the basic statement is that two categories associated to C are equivalent to each other. For G a simple complex Lie group, let YG (C) = Hom(π1 (C), G) be the moduli stack of flat G-bundles over C. And let ZG (C) be the moduli stack of holomorphic G-bundles over C. To the group G, we associate its Langlands [18] or GNO [11] dual group G∨ . (The root lattice of G is the coroot lattice of G∨ , and vice-versa.) Then the basic assertion of the geometric Langlands correspondence [3] is that the category of coherent sheaves on YG∨ (C) is naturally equivalent to the category of D-modules on ZG (C). If we are going to interpret this statement in the context of quantum field theory, we should start with a theory in dimension d = 4, so that it will associate a category to a manifold of dimension d − 2 = 2, in this case the two-manifold C. We need then an equivalence between a quantum field theory defined using G and a quantum field theory defined using G∨ , both in four dimensions. In fact, there is a completely canonical theory with the right properties. It is the maximally supersymmetric Yang – Mills theory in four dimensions. This theory, which has N = 4 supersymmetry, depends on the choice of a compact3 gauge group G. It also depends on the choice of a complex-valued quadratic form on the Lie algebra g of G; the imaginary part of this quadratic form is required to be positive definite. If G is simple, then Lie theory lets us define a natural invariant quadratic form on g (short coroots have length squared 2), and any such form is a complex multiple of this one. We write the multiple as 4πi θ + 2 , 2π e where e and θ (known as the gauge coupling constant and theta-angle) are real. We call τ the coupling parameter.

(1)

τ=

3In the formulation via gauge theory, we begin with a compact gauge group, whose complexification then naturally appears by the time one makes contact with the usual statements about geometric Langlands. Geometric Langlands is usually described in terms of this complexification.

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The classical statement (which evolved from early ideas of Montonen and Olive [22]) is that N = 4 super Yang – Mills theory with gauge group G and coupling parameter τ is equivalent to the same theory with dual gauge group G∨ and coupling parameter (2)

τ ∨ = −1/ng τ.

(Here ng is the ratio of length squared of long and short roots of G or G∨ .) The equivalence between the two theories exchanges electric and magnetic fields, in a suitable sense, and is known as electric-magnetic duality. There are also equivalences under (3)

τ → τ + 1,

τ ∨ → τ ∨ + 1,

that can be seen semiclassically (as a reflection of the fact that the instanton number of a classical gauge field is integer-valued). The nonclassical equivalence (2) combines with the semiclassical equivalences (3) to form an infinite discrete structure. For instance, if G is simply laced, then ng = 1, G and G∨ have the same Lie algebra, for many purposes one can ignore the distinction between τ and τ ∨ , and the symmetries (2) and (3) generate an action of the infinite discrete group SL(2, Z) on τ . There is a “twisting procedure” to construct topological quantum field theories (TQFTs) from physical ones. Applied to N = 2 super Yang – Mills theory, this procedure leads to Donaldson theory of smooth four-manifolds. Applied to N = 4 super Yang – Mills theory, the twisting procedure leads to three possible constructions. Two of these are quite similar to Donaldson theory in their content, while the third is related to geometric Langlands [17]. The equivalence between this third twisting for the two groups G and G∨ (and with an inversion of the coupling parameter) leads precisely at the level of “categories,” that is, for two-manifolds, to the geometric Langlands correspondence. (The underlying electric-magnetic duality treats G and G∨ symmetrically. But the twisting depends on a complex parameter; the choice of this parameter breaks the symmetry between G and G∨ . That is why the usual statement of the geometric Langlands correspondence treats G and G∨ asymmetrically.) So this is the basic reason that geometric Langlands duality, most commonly understood as a statement about Riemann surfaces, arises from a quantum field theory in four dimensions. Remark 1. For another explanation of why four dimensions is a natural starting point for geometric Langlands, see [37]. This explanation uses the fact that the mathematical theory as usually developed is based on moduli stacks rather than moduli spaces; but a two-dimensional sigma model whose target is the moduli stack of bundles is best understood as a four-dimensional gauge theory. This relies on the gauge theory interpretation of the moduli stack, introduced in a well-known paper by Atiyah and Bott [1]. 2. Defects of various dimension In the title of this talk, I promised to get up to six dimensions, not just four. Eventually we will, but first we will survey the role of structures of different dimension in a four-manifold.

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Suppose that a quantum field theory on a manifold M is defined by some sort of path integral, schematically 

(4) DA . . . exp − L , M

where L is a Lagrangian density that depends on some fields A (and perhaps on additional fields that are not written). “Inserting a local operator O(p) at a point p ∈ M ” means modifying the path integral at that point. This may be done by including a factor in the path integral that depends on the fields and their derivatives only at p. It may also be done in some more exotic way, such as by prescribing a singularity that the fields should have near p. In addition to local operators, we can also consider modifications of the theory that are supported on a p-dimensional submanifold N ⊂ M . We give some examples shortly. A local operator is the case p = 0. The general case we call a p-manifold operator. In much of physics, the important operators are local operators. This is also the case in Donaldson theory. The local operators that are important in Donaldson theory are related to characteristic classes of the universal bundle. I should point out that geometrically, a local operator may be a tensor field of some sort on M ; it may be, for example, a q-form for some q. If Oq is a local operator valued in q-forms, we can integrate it over a q-cycle Wq ⊂ M to get Oq . The most important operators in Donaldson theory are of this kind, with Wq q = 2. For our purposes, we need not distinguish a local operator from such an integral of one. (What we call a p-manifold operator cannot be expressed as an integral of q-manifold operators with q < p.) Local operators also play a role in geometric Langlands. Indeed, a construction analogous to that of Donaldson is relevant. Imitating the construction of Donaldson theory and then applying electric-magnetic duality, one arrives at results, many of which are known in the mathematical literature, comparing the group theory of G to the cohomology of certain orbits in the affine Grassmannian of G∨ . But local operators are not the whole story. In gauge theory, for example, given an oriented circle S ⊂ M , and a representation R of G, we can form the trace of the holonomy of the connection A around S in the given representation. Physicists denote this as    (5) WR (S) = TrR P exp − A . S

When included as a factor in a quantum path integral, WR (S) is known as a Wilson operator. Wilson operators were introduced over thirty years ago in formulating a criterion for quark confinement in the theory of the strong interactions. WR (S) cannot be expressed as the integral over S of a local operator. We call it a one-manifold operator. Electric-magnetic duality inevitably converts WR (S) to another one-manifold operator, which was described by ’t Hooft in the late 1970s. The ’t Hooft operator is defined by prescribing a singularity that the fields should have along S. (See [17] for a review. Operators defined in this way are often called disorder operators, while operators like the Wilson operator that are defined by interpreting a classical expression in quantum mechanics are called order operators.) The relevant singularities in G gauge theory are in natural correspondence with representations

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R∨ of the dual group G∨ . Electric-magnetic duality maps a Wilson operator in G∨ gauge theory associated with a representation R∨ to an ’t Hooft operator in G gauge theory that is also associated with R∨ . If one specializes to the situation usually studied in the geometric Langlands correspondence, the ’t Hooft operators correspond to the usual geometric Hecke operators of that subject. The electric-magnetic duality between Wilson and ’t Hooft operators leads to the usual statement that a coherent sheaf on YG∨ (C) that is supported at a point is dual to a Hecke eigensheaf on ZG (C). (Saying that a D-module on ZG (C) is a Hecke eigensheaf is the geometric analog of saying that a classical modular form is a Hecke eigenform.) Moving up the chain, the next step is a two-manifold operator. In general, in d-dimensional gauge theory, one can define a (d − 2)-manifold operator as follows. One omits from M a codimension two submanifold L. Then, fixing a conjugacy class in G, one considers gauge fields on M \L with holonomy around L in the prescribed conjugacy class. For d = 4, we have d − 2 = 2, so L is a two-manifold. Classical gauge theory in the presence of a singularity of this kind has been studied in the context of Donaldson theory by Kronheimer and Mrowka. In geometric Langlands, to get a class of two-manifold operators that is invariant under electric-magnetic duality, one must incorporate certain quantum parameters in addition to the holonomy [12]. Once one does this, one gets a natural quantum field theory framework for understanding “ramification,” i.e., the geometric Langlands analog of ramification in number theory. The next case is composed of operators supported on a three-manifold W ⊂ M . With M being of dimension four, W is of codimension one and locally divides M into two pieces. The theory of such three-manifold operators is extremely rich and [9,10] there are many interesting constructions, even if one requires that they should preserve the maximum possible amount of supersymmetry (half of the supersymmetry). For example, the gauge group can jump in crossing W . We may have G gauge theory on one side and H gauge theory on the other. If H is a subgroup of G, a construction is possible that is related to what Langlands calls functoriality. Other universal constructions of geometric Langlands — including the universal kernel that implements the duality — are similarly related to supersymmetric three-manifold operators. As long as we are in four dimensions, this is the end of the road for modifying a theory on a submanifold. A modification in four dimensions would just mean studying a different theory. So to continue the lecture, we will, as promised in the title, try to relate geometric Langlands to a phenomenon above four dimensions.

3. Self-dual gerbe theory in six dimensions Until relatively recently, it was believed that four was the maximum dimension for nontrivial (nonlinear or non-Gaussian) quantum field theory. One of the surprising developments coming from string theory is that nontrivial quantum field theories exist up to (at least) six dimensions. To set the stage, I will begin by sketching a linear, but subtle, quantum field theory in six dimensions. The nonlinear case is discussed in Section 4.

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In six dimensions, with Lorentz signature − + + + ++, a real three-form H can be self-dual, obeying H = H, where  is the Hodge star operator.4 Let us consider such an H and endow it with a hyperbolic equation of motion (6)

dH = 0.

That equation is analogous to the Bianchi identity dF = 0 for the curvature twoform F of a line bundle. It means that (in a mathematical language that physicists generally do not use) H can be interpreted as the curvature of a U(1) gerbe with connection. In contrast to gauge theory, there is no way to derive this system from an action. The natural candidate for an action, on a six-manifold M6 , would seem to   be M6 H ∧ H, but if H is self-dual this is the same as M6 H ∧ H = 0. Nevertheless, there is a quantum field theory of the closed, self-dual H field. To explain how one part of the structure of quantum field theory emerges, suppose that the Lorentz signature six-manifold M6 admits a global Cauchy hypersurface M5 . M5 is thus a five-dimensional Riemannian manifold. Fixing the topological type of a U(1) gerbe in a neighborhood of M5 , the space of gerbe connections with self-dual curvature, modulo gauge transformations, is an (infinite-dimensional) symplectic manifold in a natural way. (Roughly speaking, if B is the gerbe connection, then the symplectic form is defined by the formula ω = M5 δB ∧ dδB.) Quantizing this space, we get a Hilbert space associated to M5 . This association of a Hilbert space to a five-manifold is part of the usual data of a six-dimensional quantum field theory. The rest of the structure can also be found, with some effort. (For a little more detail, see [14, 15, 36].) An important fact is that the quantum field theory of the H field is conformally invariant. Classically, the equations H = H, dH = 0, are conformally invariant. The passage to quantum mechanics preserves this property, because the theory is linear. Now let us consider the special case that our six-manifold5 takes the form M6 = M4 × T 2 , where M4 is a four-manifold and T 2 is a two-torus. We assume a product conformal structure on M4 × T 2 . After making a conformal rescaling to put the metric on T 2 in a standard form (say a flat metric of unit area), we are left with a Riemannian metric on M 4 . The conformal structure of T 2 is determined by the choice of a point τ in the upper half of the complex plane — modulo the action of SL(2, Z). Next, in M4 ×T 2 , let us keep fixed the second factor, with a definite metric, and let the first factor vary. We let M4 be an arbitrary four-manifold with boundaries, corners, etc. Starting with a conformal field theory on M6 , this process gives us a four-dimensional quantum field theory (not conformally invariant) that depends on τ as a parameter. Clearly, the induced four-dimensional theory depends on the conformal structure of T 2 only up to isomorphism. So if we parametrize the 4 The quantum theory of a real self-dual three-form in six dimensions can be analytically continued to Euclidean signature, whereupon H is still self-dual but is no longer real. Such a continuation will be made shortly. In general, analytic continuation from Lorentz to Euclidean signature and back is an important tool in quantum field theory; the basic reason that it is possible is that in Lorentz signature the energy is nonnegative. 5Henceforth, and until Section 5.3, we generally work in Euclidean signature, using the analytic continuation mentioned in footnote 4.

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induced four-dimensional theory by τ , we will have a symmetry under the action of SL(2, Z) on τ . The induced four-dimensional quantum field theory is actually closely related to U(1) gauge theory, which is its “infrared limit.” Let us think of T 2 as C/Λ, where C is the complex plane parametrized by z = x + iy and Λ is the lattice generated by the complex numbers 1 and τ . Further, make an ansatz (7)

H = F ∧ dx + F ∧ dy,

where F is a two-form on M4 (pulled back to M6 = M4 × T 2 ), and  is the fourdimensional Hodge star operator. Then the equations dH = 0 become Maxwell’s equations (8)

dF = d  F = 0.

This gives an embedding of four-dimensional U(1) gauge theory in the sixdimensional theory. To be more precise, we should think of H as the curvature of a U(1) gerbe connection; then F is the curvature of a U(1) connection. Of course, we have described the embedding classically, but it also works quantum mechanically. This construction is more than an embedding of four-dimensional U(1) gauge theory in a six-dimensional theory. The four-dimensional U(1) gauge theory is the infrared limit of the six-dimensional theory in the following sense. We have endowed M6 with a product metric g6 that we can write schematically as g6 = g4 ⊕ g2 , where g4 and g2 are metrics on M4 and T 2 , respectively. Now we modify g6 to g6 (t) = t2 g4 ⊕ g2 , where t is a real parameter. The claim is that for t → ∞, the theory on M6 converges to U(1) gauge theory on M4 . (This theory is conformally invariant, so the t2 factor in the metric of M4 can be dropped.) This is usually described more briefly by saying that U(1) gauge theory on M4 is the long distance or infrared limit of the underlying theory on M6 . Even though U(1) gauge theory on M4 gives an effective and useful description of the large t limit of the six-dimensional theory on M6 , something is obscured in this description. The process of compactifying on T 2 and taking the large t limit is canonical in that it depends only on the geometry of T 2 and not on a choice of coordinates. But to go to a description by U(1) gauge theory, we used the ansatz (7), which depended on a choice of coordinates x and y. As a result, some of the underlying symmetry is hidden in the description by U(1) gauge theory. Concretely, though the six-dimensional theory does not have a Lagrangian, the four-dimensional U(1) gauge theory does have one:

θ 1 F ∧ F. F ∧ F + 2 (9) I= 2 4e M4 8π The coupling parameter θ 4πi + 2 . 2π e of the abelian gauge theory is simply the τ -parameter of the T 2 in the underlying six-dimensional description. The six-dimensional theory depends on τ only modulo the usual SL(2, Z) equivalence τ → (aτ + b)/(cτ + d), with integers a, b, c, d obeying ad − bc = 1, since values of τ that differ by the action of SL(2, Z) correspond to equivalent tori. Therefore, the limiting four-dimensional U(1) gauge theory must also have SL(2, Z) symmetry. However, there is no such classical symmetry. Manifest SL(2, Z) symmetry was lost

(10)

τ=

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in the reduction from six to four dimensions, because the ansatz (7), which was the key step in reducing to four dimensions, is not SL(2, Z)-invariant. Hence this ansatz leads to a four-dimensional theory with a “hidden” SL(2, Z) symmetry, one which relates the description by a U(1) gauge field with curvature F to a different description by a different U(1) gauge field with another curvature form (which, roughly speaking, is related to F by the action of SL(2, Z)). What we get in this way is an SL(2, Z) symmetry of quantum U(1) gauge theory that does not arise from a symmetry of the classical theory. To physicists, this symmetry is known as electric-magnetic duality. The name is motivated by the fact that an exchange (x, y) → (y, −x) in (7), which is a special case of SL(2, Z), would exchange F and F , and thus in nonrelativistic terminology would exchange electric and magnetic fields. So we have seen that electric-magnetic duality in U(1) gauge theory in four dimensions follows from the existence of a suitable conformal field theory in six dimensions [32]. The starting point in this particularly nice explanation is the existence in six dimensions of a quantum theory of a gerbe with self-dual curvature. (It is also possible to demonstrate the four-dimensional duality by a direct calculation, involving a sort of Fourier transform in field space; see [34].) 4. The nonabelian case Since there is not a good notion classically of a gerbe whose structure group is a simple nonabelian Lie group, one might think that it is too optimistic to look for an analogous explanation of electric-magnetic duality for nonabelian groups. However, it turns out that such an explanation does exist — in the maximally supersymmetric case. The picture is simplest to describe if G is simply laced, in which case G and G∨ have the same Lie algebra (and to begin with, we will ignore the difference between them, though this is precisely correct only if G = E8 ; a more complete picture can be found in Section 4.1). For G to be simply laced is equivalent to the condition that ng = 1 in (2). For many purposes, we can ignore the difference between τ and τ ∨ , and then the quantum duality (2) and the semiclassical equivalence (3) combine to an action of SL(2, Z) on τ . For every simply laced Lie group G, there is a six-dimensional conformal field theory that in some sense is associated with gerbes of type G. The theory is highly supersymmetric, so supersymmetry is essential in what follows. The existence of this theory was discovered in string theory in the mid-1990s. (The first hint [35] came by considering type IIB superstring theory at an ADE singularity.) Its existence is probably our best explanation of electric-magnetic duality — and therefore, in particular, of geometric Langlands duality. It is, in the jargon of quantum field theory, an isolated, non-Gaussian conformal field theory. This means among other things that it cannot be properly described in terms of classical notions such as partial differential equations. However, it has two basic properties which in a sense justify thinking of it as a quantum theory of nonabelian gerbes. Each property involves a perturbation of some kind that causes a simplification to a theory that can be given a classical description. The two perturbations are as follows: (1) After a perturbation in the vacuum expectation values of certain fields (which are analogous to the conjectured Higgs field of particle physics), the theory

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reduces at low energies to a theory of gerbes, with self-dual curvature, and structure group the maximal torus T of G. This notion does make sense classically, since T is abelian. In fact, the self-dual gerbe theory of T is much like the U(1) theory described in Section 3, with U(1) replaced by T . (Supersymmetry plays a fairly minor role in the abelian case.) (2) Let M6 = M5 × S 1 be the product of a five-manifold M5 with a circle; we endow it with a product metric g6 = g5 ⊕ g1 . The six-dimensional theory on M6 has a description (valid at long wavelengths) in terms of G gauge fields (and other fields related to them by supersymmetry) on M5 , but this description involves a highly nonclassical trick. If the circle factor of M6 = M5 × S 1 has radius R, then the effective action for the gauge fields in five dimensions is inversely proportional to R:
1 (11) I5 = Tr F ∧ F. 8πR M5 The factor of R−1 multiplying the action is a simple consequence of conformal invariance in six dimensions. (Under multiplication of the metric of M6 by a positive constant t2 , the Hodge operator  mapping two-forms to three-forms in five dimensions is multiplied by t, while R is also multiplied by t, so the action in (11) is invariant.) Though easily understood, this result is highly nonclassical. Equation (11) is a classical Lagrangian for gauge fields in five dimensions. Can it arise from a classical Lagrangian for gauge fields on M6 = M5 × S 1 ? Given a six-dimensional Lagrangian for gauge fields, we would reduce to a five-dimensional Lagrangian (for fields that are pulled back from M5 ) by integrating over the fibers of the projection M5 × S 1 → M5 . This would give a factor of R multiplying the five-dimensional action, not R−1 . So a theory that leads to the effective action (11) cannot arise in this way. The theory in six dimensions should be, in some sense, not a gauge theory but a gerbe theory instead, but this does not exist classically in the nonabelian case. What I have said so far is that the same six-dimensional quantum field theory can be simplified to either (i) a six-dimensional theory of abelian gerbes, or (ii) a five-dimensional theory with a simple nonabelian gauge group. The two statements together show that one cannot do justice to this theory in terms of either gauge fields (as opposed to gerbes) or abelian groups (as opposed to nonabelian ones). Now let us look more closely at the implications of the peculiar factor of 1/R in (11). We will study what happens for M6 = M4 × T 2 , the same decomposition that we used in studying the abelian gerbe theory in Section 3. However, for simplicity we will take T 2 = S 1 × S1 to be the orthogonal product of a circle S 1 of radius R and a second circle S1 of radius S. The tau parameter of such a torus (which is made by identifying the sides of a rectangle of height and width 2πR and 2πS) is R S or τ = i , (12) τ =i R S depending on how one identifies the rectangle with a standard one. The two values of τ differ by 1 (13) τ →− . τ We first view the six-manifold M6 as M6 = M5 × S 1 , where M5 = M4 × S1 . The six-dimensional theory on M6 reduces at long distances to a supersymmetric

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gauge theory on M5 . According to (11), the action for the gauge fields is
1 Tr F ∧ F. (14) I5 = 8πR M4 ×S
1 Now if M4 is much larger than S1 , then at long distances we can assume that the fields are invariant under rotation of S1 and we can deduce an effective action in four dimensions by integration over the fiber of the projection M4 × S1 → M4 . This second step is purely classical, so it gives a factor of S. The effective action in four dimensions is thus
S (15) I4 = Tr F ∧ F. 8πR M4 The important point is that this formula is not symmetric in S and R, even though they enter symmetrically in the starting point M6 = M4 × S 1 × S1 . Had we exchanged the two circles before beginning this procedure, we would have arrived at the same formula for the four-dimensional effective action, but with S/R replaced by R/S. Looking back to (12) and (13), we see that the two formulas differ by τ → −1/τ . Thus, we have deduced6 that for simply laced G, the four-dimensional gauge theory that corresponds to the maximally supersymmetric completion of (15) has a quantum symmetry that acts on the coupling parameter by τ → −1/τ . This is the electric-magnetic duality that has many applications in physics and also underlies geometric Langlands duality. What we have gained is a better understanding of why it is true in the nonabelian case. Remark 2. Unfortunately, despite its importance, there is no illuminating and widely used name for the six-dimensional QFT whose existence underlies duality in this way. According to Nahm’s theorem [23], the superconformal symmetry group of a superconformal field theory in six dimensions, when formulated in Minkowski spacetime, is OSp(2, 6|2r) for some r. The known examples have r = 1 or 2, and the theory with the properties that I have just described is the “maximally symmetric” one with r = 2. This theory is rather inelegantly called the six-dimensional (0, 2) model of type G, where 2 is the value of r (and the redundant-looking number 0 involves a comparison to six-dimensional models that are supersymmetric but not conformal). Remark 3. The bosonic subgroup of OSp(2, 6|2r) is SO(2, 6) × Sp(2r), where SO(2, 6) is the conformal group in six dimensions, and Sp(2r) is an “internal symmetry group” (it acts trivially on spacetime) and is known as the R-symmetry group. Thus, the R-symmetry group of the (0, 2) model is Sp(4). This is the group (sometimes called Sp(2)) of 2 × 2 unitary matrices of quaternions; its fundamental representation is of quaternionic dimension 2, complex dimension 4, or real dimension 8. Sp(4) is also the group that acts on the cohomology of a hyper-K¨ ahler manifold; this is no coincidence, as we will see later. 4.1. The space of conformal blocks. Among the simple and simply laced Lie groups, only E8 is simply connected and has a trivial center. Equivalently, its root lattice Γ endowed with the usual quadratic form is unimodular, that is, equal 6To keep the derivation simple, we considered only a rectangular torus. It is possible by using (24) to similarly analyze the case of a general torus.

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to its dual Γ∨ . In general, if G is simple, simply laced, and simply connected, its center is Z = Γ∨ /Γ, and the quadratic form on Γ leads to a perfect pairing (16)

Z × Z → R/Z = U(1).

(The pairing actually takes values in the subgroup Zn of U(1), where n is the smallest integer that annihilates Z.) What has been said so far is sufficient for E8 , but more generally, a refinement is necessary. (Most of this article does not depend on the following details.) For orientation, consider two-dimensional current algebra (that is, the holomorphic part of the WZW model7) of the simply connected and simply laced group G at level 1. This theory, formulated on a closed Riemann surface W , does not have a unique partition function (which is required in the usual axioms of quantum field theory, as indicated in the introduction to this article). Rather, it has a vector space of possible partition functions, known as the space of conformal blocks. This vector space (in the particular case of a simply laced group G at level 1) can be constructed as follows. The pairing (16) together with the intersection pairing on the cohomology of W leads to a perfect pairing (17)

H 1 (W, Z) × H 1 (W, Z) → U(1).

This pairing enables us to define a Heisenberg group extension (18)

1 → U(1) → F → H 1 (W, Z) → 0.

Up to isomorphism, the group F has a single faithful irreducible representation R in which U(1) acts in the natural way; it is obtained by “quantizing” the finite group H 1 (W, Z). One picks a decomposition of H 1 (W, Z) as A × B, where A and B (which can be constructed using a system of A-cycles and B-cycles on W ) are maximal subgroups on which the extension (18) is trivial. One then lets B act by multiplication — in the sense that R is the direct sum of all one-dimensional characters of B. Since (17) restricts to a perfect pairing A × B → U(1), characters of B correspond to elements of A. Thus, R has a unitary basis consisting of of A is a(ψa ) = ψaa , while B acts by b(ψa ) = elements ψa ,a ∈ A; the action   exp 2πi(b, a) ψa (where exp 2πi(b, a) denotes the pairing between A and B). The dimension of R is thus (#Z)g , where g is the genus of W and #Z is the order of Z. The space of conformal blocks of the level 1 holomorphic WZW model on a Riemann surface W with a simple and simply laced symmetry group G is isomorphic to R. Thus, for G = E8 , the space of conformal blocks has dimension bigger than 1. That means that this theory does not have a distinguished partition function and so does not quite obey the full axioms of quantum field theory. One may either relax the axioms slightly, study the ordinary (nonholomorphic) WZW model, or in some other way include holomorphic or nonholomorphic degrees of freedom so as to be able to define a distinguished partition function. The situation in the six-dimensional (0, 2) theory is similar, with the finite group H 3 (M6 , Z) playing the role of H 1 (W, Z) in two dimensions. From (16) and 7 There is no satisfactory terminology in general use. The WZW model is really [33] a two-dimensional quantum field theory that is modular-invariant but neither holomorphic nor antiholomorphic. Its holomorphic part corresponds to what physicists know as two-dimensional current algebra (which is a much older construction than the WZW model). But the phrase “two-dimensional current algebra” is not well known to mathematicians, and may even be unclear nowadays to physicists.

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Poincar´e duality, we again have a perfect pairing H 3 (M6 , Z) × H 3 (M6 , Z) → U(1), leading to a Heisenberg group extension (19)

1 → U(1) → F → H 3 (M6 , Z) → 0.

Again, up to isomorphism, F has a unique faithful irreducible module T with natural action of U(1). The theory on a general six-manifold has a space of conformal blocks that is isomorphic to T . For G a simple and simply laced Lie group that is not of type E8 , this again represents a slight departure from the usual axioms of quantum field theory. Our options are analogous to what they were in the twodimensional case: live with it (which will be our choice in the present paper) or consider various more elaborate constructions in which one can avoid the problem. Now let us consider an illuminating example. We take M6 = M5 × S 1 . We have a decomposition H 3 (M6 , Z) = H 2 (M5 , Z) ⊕ H 3 (M5 , Z). Calling the summands A and B, we can as above construct the space T of conformal blocks as the direct sum of characters of B. Hence, as in the two-dimensional case, T has a basis consisting of elements ψa , a ∈ A = H 2 (M5 , Z). On the other hand, the (0, 2) model on M6 = M5 × S 1 is supposed to be related to gauge theory on M5 . So in gauge theory on M5 , we should find a way to define a partition function for every a ∈ H 2 (M5 , Z). This is easily done once one appreciates that one should use the adjoint form of the group, which we will call Gad . A Gad bundle over any space X has a characteristic class a ∈ H 2 (X, Z) (where Z is the center of the simply connected group G or equivalently the fundamental group of Gad ). In Gad gauge theory on M5 , we define for every a ∈ H 2 (M5 , Z) a corresponding partition function Za by summing the path integral of the theory over all bundles whose characteristic class equals a. In defining the Za , we are relaxing the usual axioms of quantum field theory a little bit. If the gauge group is supposed to be G, the characteristic class must vanish and the partition function is essentially Z0 . (I will omit some elementary factors involving the order of Z.) If the gauge group is supposed to be Gad , all values of the characteristic class are allowed and the partition function is a Za . For groups intermediate between G and Gad , certain formulas intermediate between those two will arise. But for no choice of gauge group is the partition function precisely Za , for some fixed and nonzero a. Clearly, on the other hand, it is natural to permit ourselves to study these functions. So this is a situation in which we probably want to be willing to slightly generalize the usual axioms of quantum field theory. Now as before let us consider the case M5 = M4 × S1 , where S1 is another circle, so that M6 = M4 × S 1 × S1 can be viewed in more than one way as the product of a circle and a five-manifold. For simplicity, let us assume that H 1 (M4 , Z) = H 3 (M4 , Z) = 0. Then H 3 (M6 , Z) = A ⊕ B, where (20)

A = H 2 (M4 , Z) ⊗ H 1 (S 1 , Z),

B = H 2 (M4 , Z) ⊗ H 1 (S1 , Z).

The extension is trivial on both A and B. Reasoning as above, the space T of conformal blocks has a basis ψa , a ∈ A. On the other hand, exchanging the roles of A and B, it has a second basis ψb , b ∈ B. As is usual in quantization, the relation between these two bases (which are analogous to “position space” and “momentum space”) is given by a Fourier transform. In the present case, both A and B can be identified with H 2 (M4 , Z) and the Fourier transform is a finite sum:

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ψb = C



  exp 2πi(a, b) ψa .

a∈H 2 (M4 ,Z)

  Here C is a constant and we write exp 2πi(a, b) for the perfect pairing H 2 (M4 , Z)× H 2 (M4 , Z) → U(1). Let us interpret this formula in four-dimensional gauge theory. In Gad gauge theory on M4 , we can as before define a partition function Za by summing over bundles with a fixed characteristic class a ∈ H 2 (M4 , Z). Identifying these with the ψa , we find that under electric-magnetic duality the Za must transform by

  (22) Zb (−1/τ ) = C exp 2πi(b, a) Za (τ ). a

We have incorporated the fact that (because it exchanges the last two factors in M6 = M4 × S 1 × S1 ) electric-magnetic duality inverts τ , in addition to its action on the label a. This formula was first obtained in purely four-dimensional terms in [31], where more detail can be found. Here we have given a six-dimensional context for this result. If G is a simply laced and simply connected Lie group, then its GNO or Langlands dual group G∨ is precisely the adjoint group Gad . Apart from elementary constant factors that are considered in [31], the partition function of the theory with gauge group G is Z0 (since a must vanish if the gauge group is the simply connected form G), and the partition function of the theory with gauge group Gad is a Za (since all choices of a are equally allowed if the gauge group is the adjoint form). As noted in [31], a special case of (22) is that Z0 transforms under τ → −1/τ into a constant multiple of a Za . This assertion means that in this particular case the G and G∨ theories are dual. Other specializations of (22) correspond to duality for forms intermediate between G and Gad , but in general (22) contains more information than can be extracted from such special cases. Remark 4. The close analogy between the conformal blocks of the six-dimensional (0, 2) model and those of the the level 1 WZW model in two dimensions make one wonder if there might be an analog in six dimensions of the WZW models at higher level. All one can say here is that the usual (0, 2) model has appeared in string theory in many ways and as of yet there is no sign of a hypothetical higher level analog. 4.2. What is next? In view of what we have said, if we specialize to sixmanifolds of the form M6 = M4 × T 2 , where we keep the two-torus T 2 fixed and let only M4 vary, the six-dimensional (0,2) theory gives a good framework for understanding geometric Langlands. We can do other things with this theory, since we are free to consider more general six-manifolds. This will be our topic in Section 5. But perhaps we should first address the following question. Is this the end? Or will physicists come back next year and say that geometric Langlands should be derived from a theory above six dimensions? There is a precise sense in which six dimensions is the end. It is the maximum dimension for superconformal field theory, according to an old result of Nahm [23]. To get farther, one needs a different kind of theory. If one wishes to go beyond six dimensions, the next stop is presumably string theory (dimension ten). Indeed, the existence and most of the essential properties of

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the six-dimensional QFT that underlies four-dimensional electric-magnetic duality are known primarily from the multiple relations this theory has with string theory. 5. Geometric Langlands duality for surfaces 5.1. Circle fibrations. As we have discussed, one of the most basic properties of the six-dimensional (0, 2) theory is that when formulated on M6 = M5 × S 1 , it gives rise at long distances to five-dimensional gauge theory on M5 . The simplest generalization8 of this is to consider not a product M5 × S 1 , but a fibration over M5 with S 1 fibers: / M6 S1 (23)

 M5 .

(For simplicity, we assume that the fibers are oriented.) In this situation, the long distance limit is still gauge theory on M5 , with gauge group G. But there is an important modification. We pick on M6 a Riemannian metric that is invariant under rotation of the fibers of the U(1) bundle M6 → M5 . Such a metric determines a connection on this U(1) bundle, and therefore a curvature two-form f ∈ Ω2 (M5 ). Let A be the gauge field on M5 (so A is a connection on a G-bundle over M5 ), and let CS(A) be the associated Chern – Simons three-form. (As is customary among physicists, we will normalize this form so that its periods take values in R/2πZ.) Then the twisting of the fibration M6 → M5 results in the presence in the long distance effective action of an additional term ∆I that roughly speaking is
i f ∧ CS(A). (24) ∆I = 2π M5 To be more precise, one should define −i∆I as the integral of a certain Chern – Simons five-form for the group U(1) × G. This Chern – Simons five-form is associated to an invariant cubic form on the Lie algebra of U(1) × G that is linear on the first factor of this Lie algebra and quadratic on the second. Since ∆I is i times the integral of a Chern – Simons form, ∆I is well defined and gauge-invariant mod 2πiZ assuming that M5 is a compact manifold with boundary. This ensures that exp(−∆I) is well defined as a complex number, so that it is possible to include a factor of exp(−∆I) in the integrand of the path integral of five-dimensional supersymmetric gauge theory on M5 . (Saying that ∆I appears as a term in the effective action means precisely that the integrand of the path integral has such a factor.) 5.2. Allowing singularities. However, it is natural to relax the conditions that we have imposed so far. Describing M6 as a U(1) bundle over some base M5 amounts to exhibiting a free action of the group U(1) on M6 ; if such an action is given, one simply defines M5 = M6 / U(1) and then M6 is a U(1) bundle over M5 . Clearly, a more general situation is to consider a six-manifold M6 together with a nontrivial action of the group U(1). After possibly replacing U(1) by a 8The material in this section was presented in more detail in lectures at the IAS in the spring of 2008. Notes by D. Ben-Zvi can be found at http://www.math.utexas.edu/users/benzvi/ GRASP/lectures/IASterm.html.

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finite quotient of itself (to eliminate a possible finite subgroup that acts trivially), we can assume that U(1) acts freely on a dense open set in M6 . The quotient M5 = M6 / U(1) is a five-manifold possibly with singularities where the U(1) action is nonfree. The above description, with the term (24) in the effective action, is applicable away from the nonfree locus in M5 (which consists of the points in M5 that correspond to nonfree orbits in M6 ). Along the nonfree locus, one should expect the gauge theory description to require some kind of modification. What sort of modification is needed depends on how the U(1) action fails to be free. U(1) may have nonfree orbits in codimension 2, 4, or 6, and these nonfree orbits may be either fixed points of the whole group, or semi-free orbits whose stabilizer is a finite subgroup of U(1). (To characterize the local behavior, one also needs to specify the action of U(1) in the normal space to the nonfree locus. Further, though this will not be important for our purposes, in general one wishes to allow the possibility of a U(1) symmetry that acts via a homomorphism to the R-symmetry group Sp(4), in addition to acting geometrically on M6 .) Thus, for a full analysis of this problem, there are many interesting cases to consider, most of which have not been analyzed yet. A simple example is that U(1) may act on M6 with a fixed point set of codimension 2, in which case M5 is a manifold with boundary. Thus a natural boundary condition in five-dimensional supersymmetric gauge theory will have to appear. For our purposes, we will consider just one situation, in which one knows the appropriate modification of the effective field theory that occurs near the exceptional set in M5 . This is the case of a codimension 4 fixed point locus W such that the action of U(1) on the normal space to W can be modeled by the natural action of U(1) on C2 ∼ = R4 . Thus, focussing on the normal space to W , we take U(1) to act on C2 by (z1 , z2 ) → (eiθ z1 , eiθ z2 ), for eiθ ∈ U(1). Clearly, this gives an action of U(1) on C2 that is free except for an isolated fixed point at the origin. Somewhat less obvious — but elementary to prove — is that the quotient C2 / U(1) is actually a smooth manifold. In fact, it is a copy of R3 : (25) C2 / U(1) ∼ = R3 . We can get this statement by taking a cone over the Hopf fibration. The Hopf fibration is the U(1) bundle S 3 → S 2 . A cone over S 3 is R4 ∼ = C2 , while a cone over 2 3 4 3 4 S is R . So, writing 0 for the origin in R or R , R \{0} is a U(1) bundle over R3 \{0}. Gluing back in the origin on both sides, we arrive at the assertion (25). It follows from (25) that if U(1) acts on M6 freely except for a codimension 4 fixed point set W as just described, then M5 = M6 / U(1) is actually a smooth manifold. A few simple facts about the geometry of M5 deserve attention. One obvious fact is that W is naturally embedded as a codimension 3 submanifold of M5 . Moreover, it is only away from W that the natural projection M6 → M5 = M6 / U(1) is a U(1) fibration. This projection thus gives a U(1) bundle over M5 \W , which topologically cannot be extended over M5 . The obstruction to extending the U(1) bundle can be measured as follows. Let S be a small two-sphere in M5 \W that has linking number 1 with W . (One can construct a suitable S by choosing a normal three-plane N to W at some chosen point p ∈ W and letting S consist of points in N a distance  from p, for some small .) Then the U(1) bundle over M5 \W , restricted to S, has first Chern class 1.

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This fact can be expressed as an equation for the curvature two-form f of the U(1) bundle over M5 \W . As a form on M5 \W , f is closed, obeying df = 0. But f has a singularity along W which can be characterized by the statement (26)

df = 2πδW ,

where δW is the Poincar´e dual to W . 5.2.1. Role of W in the quantum theory. In light of this information, let us consider now the (0, 2) theory of type G formulated on M6 , and its reduction to an effective description on M5 . Away from W , as we have already discussed, the effective theory on M5 is simply supersymmetric gauge theory with gauge group G, and with the additional interaction (24) that reflects the twisting of the fibration M6 → M5 . The gauge field is a connection on a G-bundle E → M5 . However, there is an important and very interesting modification along W . This modification results from the fact that the interaction ∆I is not well defined in the usual sense. We can define a Chern – Simons five-form on M5 \W for the group U(1) × G, but as M5 \W is not compact, the integral of this form is not gauge-invariant, even modulo 2π. Consequently, exp(−∆I), the corresponding factor in the path integral, is not well-defined as a complex number, but as a section of a certain complex line bundle L. L is a line bundle over the space of all G-valued gauge fields, modulo gauge transformations, on W . More exactly, L is a line bundle over the space of all connections on E|W modulo gauge transformations (E|W is simply the restriction to W of the G bundle E → M5 ). We write A for the space of connections on E|W and G for the group of gauge transformations; then L is a line bundle over the quotient A/G (or equivalently, a G-invariant line bundle over A). In fact, L is the fundamental line bundle over A/G, often loosely called the determinant line bundle. (The motivation for this terminology is that if G = SU(n) or U(n) for some n, then L can be defined as the determinant line bundle of a ∂¯ operator. It can also be defined as the Pfaffian line bundle of a Dirac operator if G = SO(n) or Sp(2n).) The characterization of L can be justified as follows. The interaction ∆I as defined in (24) does not depend on a choice of gauge for the U(1) bundle M6 \W → M5 \W , as it is written explicitly in terms of the U(1) curvature f . On the other hand, under an infinitesimal G gauge transformation A → A − dA , the Chern – Simons three-form CS(A) transforms by CS(A) → CS(A) + dX2 , where X2 is known to physicists as the anomaly two-form (explicitly, X2 = (1/4π) Tr  dA). Substituting this gauge transformation law in (24), integrating by parts, and using (26), we see that under such a gauge transformation, ∆I transforms by
X2 , (27) ∆I → ∆I − i W

which is equivalent to saying that exp(−∆I) should be understood as a section of the line bundle L. Physicists would describe this situation by saying that the factor exp(−∆I) in the path integral has an anomaly under gauge transformations that are nontrivial along W . The anomaly must be canceled by incorporating in the theory another ingredient with an equal and opposite anomaly. This additional ingredient must be supported on W (since away from W we already know what is the right effective field theory). The theory that does the job is the two-dimensional (holomorphic)

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WZW model (or in other words, current algebra, as explained in footnote 7) on W , at level 1. This then is the secret of W : it supports this particular two-dimensional quantum field theory. This is the main fact that we will use in interpreting recent mathematical results [4, 19, 24] about instantons and geometric Langlands for surfaces. 5.2.2. More concrete argument. This somewhat abstract argument can be replaced by a much more concrete one if G is a group of classical type, rather than an exceptional group. (A similar analysis has been made independently for somewhat related reasons in [7]. See also [28]. The following discussion requires more detailed input from string theory than the rest of the present article, and the reader may wish to jump to Section 5.3.) The simplest case is that G is SU(n) or, even better, U(n). We use the fact [27] that the (0, 2) model of U(n) describes the low energy behavior of a system of n parallel M5-branes. We consider M-theory on ahler four-manifold R7 × TN, where TN is the Taub-NUT space, a certain hyper-K¨ that topologically is R4 . (It is described in detail in Section 5.4.) TN has a U(1) symmetry with TN/ U(1) = R3 , as suggested by (25); we denote by 0 the point in R3 that corresponds to the U(1) fixed point in TN. Inside R7 × TN, we consider n M5-branes supported on R2 × TN (for some choice of embedding R2 ⊂ R7 ); this gives a realization of the (0, 2) theory of type U(n) on R2 × TN. We want to divide by the U(1) symmetry of TN to reduce the six-dimensional (0, 2) model supported on the M5-branes to a five-dimensional description. This may be done straightforwardly. For any seven-manifold Q7 , M-theory on Q7 × TN is equivalent [29] to type IIA superstring theory on Q7 × R3 with a D6-brane supported on Q7 × {0}. So M-theory on R7 × TN is equivalent to type IIA on R7 × R3 with a D6-brane supported at R7 × {0}. In this reduction, the n M5-branes on R2 × TN turn into n D4-branes supported on R2 × R3 . The low energy theory on the D4-branes is N = 4 super Yang – Mills theory with gauge group U(n). The D4-branes intersect the D6-brane on the Riemann surface W = R2 × {0}, and a standard calculation (which uses the fact that the D4-branes and the D6-brane intersect transversely on W ) shows the appearance on W of U(n) current algebra at level 1. The behavior of the (0, 2) model of type Dn can be analyzed similarly by replacing R7 in the starting point with R5 /Z2 × R2 . 5.3. Compactification on a hyper-K¨ ahler manifold. We are going to consider the (0, 2) theory in a very special situation. We take M6 = R × S 1 × X, where X will be a hyper-K¨ ahler four-manifold. We think of R as parametrizing the “time” direction. On M6 , we take the obvious sort of product metric, giving circumference 2π to S 1 . We could take the metric on M6 to be of Euclidean signature (which would agree well with some of our earlier formulas), but it is actually more elegant in what follows to use a Lorentz signature metric, that is, a metric of signature − + + + ++, with the negative eigenvalue corresponding to the R direction.9 9One of the important general facts about quantum field theory, as remarked in footnote 4, is that in the world of unitary, physically sensible quantum field theories with positive energy — such as the six-dimensional (0, 2) model considered here — it is possible in a natural way to formulate the “same” quantum field theory on a space of Euclidean or Lorentzian signature. In the following analysis, the main thing that we gain by using Lorentz signature is that the supersymmetry generators are Hermitian and the energy is bounded below.

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The most obvious ordinary or bosonic conserved quantities in this situation are the ones that act geometrically: the Hamiltonian H, which generates translations in the R direction, the momentum P , which generates rotations of S 1 , and possible additional conserved quantities associated with symmetries of X. The (0, 2) model also has a less obvious bosonic symmetry group; this is the R-symmetry group ahler, Sp(4), mentioned in Remark 3. Because R × S 1 is flat and X is hyper-K¨ so that M6 = R × S 1 × X admits covariantly constant spinor fields, there are also unbroken supersymmetries. In fact, there are eight unbroken supersymmetries Qα , α = 1, . . . , 8; they are Hermitian operators that transform in the fundamental representation of the R-symmetry Sp(4) (which has real dimension 8). They commute with H and P , and obey a Clifford-like algebra. With a suitable choice of normalizations and orientations, this algebra reads (28)

{Qα , Qβ } = 2δαβ (H − P ).

Accordingly, the operator H − P is positive semi-definite; it can be written in many different ways as the square of a Hermitian operator. States that are annihilated by H − P are known as BPS states and play a special role in the quantum theory [39]. We write V for the space of BPS states. V admits an action of U(1) × Sp(4) (or possibly a central extension thereof), where U(1) is the group of rotations of S 1 and Sp(4) is the R-symmetry group. The center of Sp(4) is generated by an element of order 2 that we denote by (−1)F ; it acts as +1 or −1 on bosonic or fermionic states, respectively. So in particular, V is Z2 -graded by the eigenvalue of (−1)F . We refer to V, with its action of U(1) × Sp(4), as the spectrum of BPS states. One important general fact is that P is bounded below as an operator on V; indeed, on general grounds, H is bounded below in the full Hilbert space of the (0, 2) theory, while H = P when restricted to V. Certain features of the spectrum of BPS states are “topological invariants,” that is, invariant under continuous deformations of parameters. (In the present problem, the relevant parameters are the moduli of the hyper-K¨ahler metric of X.) The most obvious such invariant is the “elliptic genus,” F (q) = TrV (−1)F q P , where q is a complex number with |q| < 1. (It has modular properties, since it can be represented by the partition function of the (0, 2) model on Σ × X, where Σ is an elliptic curve whose modular parameter is τ = ln q/2πi.) F (q) is invariant under smooth deformation of the spectrum by virtue of the same arguments that are usually used to show that the index of a Fredholm operator is invariant under deformation. In the present problem, the whole spectrum of BPS states, and not only the index, is invariant under deformation of the hyper-K¨ ahler metric of X. One approach to proving this uses the fact that V can be characterized as the cohomology of Q, where Q is any complex linear combination of the Hermitian operators Qα that squares to zero. Picking any one complex structure on X (from among the complex structures that make up the hyper-K¨ ahler structure of X), one makes a judicious choice of Q to show that the spectrum of BPS states is invariant under deformations of the K¨ahler metric of X (keeping the chosen complex structure fixed). Repeated moves of this kind (specializing at each stage to a different complex structure and therefore a different choice of Q) can bring about arbitrary changes of the hyperK¨ ahler metric of X, so the spectrum of BPS states is independent of the moduli of X.

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In our discussion in Section 5.4, we compare two computations of V in two different regions of the moduli space of hyper-K¨ ahler metrics on X. The results must be equivalent in view of what has just been described. Remark 5. There is some sleight of hand here, as the arguments above have assumed X to be compact, and we will use the results for noncompact X. So some refinement of the arguments is actually needed. 5.4. Taub-NUT spaces. Now the question arises of what sort of hyperK¨ ahler four-manifold X we will select in the above construction. We will choose X to admit a triholomorphic U(1) symmetry, that is, a U(1) symmetry that preserves the hyper-K¨ ahler structure of X. (Among other things, this ensures that this U(1) also commutes with the unbroken supersymmetries Qα of eqn. (28).) Hyper-K¨ahler four-manifolds with triholomorphic U(1) symmetry are highly constrained [20]. The general form of the metric is 1 (29) ds2 = U dx · dx + (dθ + ω · dx)2 , U where x parametrizes R3 , U is a harmonic function on R3 , and (away from singularities of U ) θ is an angular variable that parametrizes the U(1) orbits. This form of the metric shows that the quotient space X/ U(1) (assuming X is complete) is equal to R3 . Indeed, the natural projection X → X/ U(1), which was considered in Section 5.2, has a special interpretation in this situation. It is the hyper-K¨ ahler moment map µ  and it is a surjective map to R3 : (30)

µ  : X → R3 .

The most obvious hyper-K¨ ahler four-manifold with a triholomorphic U(1) symmetry is R4 . This corresponds to the choice U = 1/2|x|. The U(1) action on R4 has a fixed point at the origin (where U has a pole and the radius of the U(1) orbits vanishes, according to (29)). This fixed point is precisely of the sort considered in Section 5.2. To verify this, begin with the fact that the rotation group of R4 has SU(2)L × SU(2)R for a double cover; SU(2)L and SU(2)R are two copies of SU(2). We can pick a hyper-K¨ ahler structure on X compatible with its flat metric such that SU(2)L rotates the three complex structures and SU(2)R preserves them. We simply take U(1) to be a subgroup of SU(2)R . Then, upon picking a complex structure on R4 that is invariant under SU(2)L × U(1) (this complex structure is not part of its U(1)-invariant hyper-K¨ ahler structure), we can identify R4 with C2 and U(1) acts in the natural way (z1 , z2 ) → (eiθ z1 , eiθ z2 ). This then is the situation that was considered in Section 5.2, and the statement (30) gives a hyper-K¨ahler perspective on the fact that the quotient R4 / U(1) is R3 , as was asserted in (25). Although R4 has the properties we need from a topological point of view, there is a different hyper-K¨ ahler metric on R4 that will be more useful for our application in Section 5.5. This is the Taub-NUT space, which we will call TN. To describe TN explicitly, we simply choose U to be 1 1 , (31) U= 2+ R 2|x| where R is a constant. Looking at (29), the interpretation of R is easy to under√ stand: the U(1) orbits have circumference 2π/ U , which at infinity approaches 2πR. The flat metric on R4 is recovered in the limit R → ∞; in R4 , of course, the circumference of an orbit diverges at infinity.

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Accordingly, the hyper-K¨ ahler metric on TN is quite different at infinity from the usual flat hyper-K¨ ahler metric on R4 . However, in one sense the difference is subtle. If we pick any one of the complex structures that make up the hyperK¨ahler structure, then it can be shown that, as a complex symplectic manifold in this complex structure, TN is equivalent to R4 ∼ = C2 . A more general choice of X is also important. First of all, naively we could pick an integer k > 1 and take X = R4 /Zk , where Zk is the subgroup of U(1) consisting ahler metric with of points of order k. Certainly R4 /Zk has a (singular) hyper-K¨ a triholomorphic U(1) symmetry. The singularity at the origin of R4 /Zk is known ahler resolution of this as an Ak−1 singularity. It is possible to make a hyper-K¨ singularity, still with a triholomorphic U(1) symmetry. This is accomplished by picking k points x1 , . . . , xk in R3 and setting U = 12 kj=1 1/|x − xj |. This gives a complete hyper-K¨ ahler manifold which is smooth if the xj are distinct. As a complex symplectic manifold in one complex structure, it can be described by an equation (32)

uv = f (w),

where f (w) is a kth order monic polynomial. This is the usual complex resolution of the Ak−1 singularity. In this description, the holomorphic symplectic form is du ∧ dv/f
(w), and the triholomorphic U(1) symmetry is u → λu, v → λ−1 v. However, again, a generalization is more convenient for our application in Section 5.5. We simply add a constant to U and take 1 1

1 . + 2 R 2 j=1 |x − xj | k

(33)

U=

This gives a complete hyper-K¨ ahler manifold, originally constructed in [13], that we call the multi-Taub-NUT space and denote as TNk . As a complex symplectic manifold in any one complex structure, TNk is independent of the parameter R and coincides with the usual resolution (32) of the Ak−1 singularity. However, the addition of a constant to U markedly changes the behavior of the hyper-K¨ahler metric at infinity. Just as in the k = 1 case that was considered earlier, the asymptotic value at infinity of the circumference of the fibers of the fibration TNk → R3 is 2πR. The space TNk is smooth as long as the xj are distinct. When r of them coincide, an Ar−1 singularity develops, that is, an orbifold singularity of type R4 /Zr . In general, for x → xj , we have U → ∞. So at those points, and only there, the radius of the U(1) fibers vanishes. The k points x = xj are, accordingly, the fixed points of the triholomorphic U(1) action. 5.4.1. A note on the second cohomology. We conclude this subsection with some technical remarks that will be useful in Section 5.5 (but which the reader may choose to omit). Topologically, TNk is, as we have noted, the same as the resolution of the Ak−1 singularity. A classical result therefore identifies H 2 (TNk , Z) with the root lattice of the group Ak−1 = SU(k). However, TNk is not compact and one should be careful with what sort of cohomology one wants to use. It turns out that another natural definition is useful. We define an abelian group Γk as follows: an element of Γk is a unitary line bundle L → TNk with anti-self-dual and square-integrable curvature and whose connection has trivial holonomy when restricted to a fiber at infinity of µ  : TNk → R3 . Γk is

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a discrete abelian group with a natural and integer-valued quadratic form, defined as follows; if L is a line bundle with anti-self-dual curvature F , we define (L, L) = − TNk F ∧ F/4π 2 . It turns out that Γk ∼ = Zk , with the quadratic form corresponding to the quadratic function of k variables y12 + y22 + · · · + yk2 . (A basis of Γk is described in Section 5.4.2.) Thus, Γ corresponds to the weight lattice of the group U(k). In many string theory problems involving Taub-NUT spaces, one must use Γk as a substitute for H 2 (TNk , Z), which does not properly take into account the behavior at infinity. This is notably true if one considers the (0, 2) model on M6 = W × TNk , for W a Riemann surface. In Section 4.1, we explained that the (0, 2) model on a compact six-manifold M6 has a space of conformal blocks that is obtained by quantizing, in a certain sense, the finite abelian group H 3 (M6 , Z). For M6 = W × TNk , the appropriate substitute for this group is  3 (W × TNk , Z) = H 1 (W, Z) ⊗ Γk . (34) H 5.4.2. Basis of Γk . It is furthermore true that Γk has a natural basis corresponding to the U(1) fixed points xj , j = 1, . . . , k. To show this, we first describe a dual basis of noncompact two-cycles. For j = 1, . . . , k, we let lj be a path in R3 from xj to ∞, not passing through any xr for r = j. Then we set Cj = µ  −1 (lj ). Cj 2 is a noncompact two-cycle that is topologically R . A line bundle L that represents a point in Γk is trivialized at infinity on Cj because  its connection is trivial on the fibers of µ  at infinity. So we can define an integer Cj c1 (L). One can pick a basis of  Γk consisting of line bundles Lr such that Cj c1 (Lr ) = δjr . (The Lr are described explicitly in [38].)  3 (W × TNk , Z) in (34). From what Now let us reconsider the definition of H 1 we have just said, H (W, Z) ⊗ Γk has a natural decomposition as the direct sum of 1 (W, Z) of H 1 (W, Z) associated with the fixed points: copies H(j) (35)

 3 (W × TNk ) = H

k 

1 H(j) (W, Z).

j=1

Upon quantization, this means that the space of conformal blocks of the (0, 2) model on W × TNk is the tensor product of k factors, each of them isomorphic to the space of conformal blocks in the level 1 WZW model (associated with the group G) on W . The factors are naturally associated to the U(1) fixed points. Remark 6. Similarly, we can enrich the definition of the two-dimensional characteristic class a of a Gad bundle over TNk . Normally, a takes values in H 2 (TNk , Z). However, suppose E → TNk is a Gad bundle that is trivialized over each fiber at infinity of  µ : TNk → R3 . Then E is trivialized at infinity on each Cj , so one can define a pairing aj = a, Cj for each j; the aj take values in Z. Equivalently, we  2 (TNk , Z) = Γk ⊗ Z Z. This also has an anacan consider a as an element of H log if we are given a conjugacy class C ⊂ Gad and the monodromy of E on each fiber at infinity lies in C. Then one can define a C-dependent torsor for the group  2 (TNk , Z), and one can regard a as taking values in this torsor. Concretely, this H means that, once we pick a path in Gad from C to the identity (an operation that trivializes the torsor), we can define the elements aj ∈ Z as before. Two different paths from C to the identity would differ by a closed loop in Gad , corresponding to

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an element b ∈ Z; if we change the trivialization of the torsor by changing the path by b, then the aj are shifted to aj + b. (b is the same for all j, since the regions at infinity in the two-cycles Cj can be identified, by taking the paths lj to coincide at infinity.) 5.5. Two ways to compute the space of BPS states. Now we are going to study in two different ways the space of BPS states of the (0, 2) model formulated on (36)

M6 = R × S 1 × TNk .

Though computed with different metrics, the results will automatically be equivalent, as explained at the end of Section 5.3. M6 admits an action of U(1)×U(1)
(the product of two factors of U(1)), where U(1) acts by rotation of S 1 , and U(1)
is the triholomorphic symmetry of TNk . We choose a product metric on M6 , such that S 1 has circumference 2πS, and TNk has a hyper-K¨ ahler metric in which the U(1)
orbit has asymptotic circumference 2πR. In Section 5.3, we took S = 1; in any event, because the (0, 2) model is conformally invariant, only the ratio R/S is relevant. U(1) and U(1)
play very different roles in the formalism because of the structure of the unbroken supersymmetry algebra, which we repeat for convenience: (37)

{Qα , Qβ } = 2δαβ (H − P ).

Here P is the generator of the U(1) symmetry. It appears in the definition of the elliptic genus F (q) = TrV q P (−1)F , where V is the space of BPS states. The function F (q) has modular properties, so if it is nonzero (as will turn out to be the case), there are BPS states with arbitrarily large eigenvalues of P . By contrast, it turns out that U(1)
acts trivially on V. One of our two descriptions of V will be good for S → 0 or equivalently R → ∞; the other description will be good for R → 0 or equivalently S → ∞. Comparing them will give a new perspective on the results of [4, 19, 24]. 5.5.1. Description I. For S → 0, the low energy description is by gauge theory on M6 / U(1) = R × TNk . As U(1) acts freely, we need not be concerned here with the behavior at fixed points. As the metric of M6 is a simple product S 1 × M5 (with M5 = R × TNk = M6 / U(1)), we also need not worry about the interaction described in (24). So we simply get maximally supersymmetric Yang – Mills theory on R × TNk , with gauge group G. In formulating gauge theory on R × TNk , we specify up to conjugacy the holonomy U of the gauge field over a fiber at infinity of µ  : TNk → R3 . This choice (which has a six-dimensional interpretation) leads to an important bigrading of the physical Hilbert space H of the theory and in particular of the space V of BPS states. First of all, let H be the subgroup of G that commutes with U . Classically, one can make a gauge transformation that approaches at infinity a constant element of H; quantum mechanically, to avoid infrared problems, the constant should lie in the center of H. So the center of H acts on H and V. We call this the electric grading. (The center of H is, of course, abelian, and the eigenvalues of its generators are called electric charges.) A second “magnetic” grading arises for topological reasons. When U = 1, the topological classification of finite energy gauge fields on TNk becomes more elaborate. Near infinity on TNk , the monodromy around S 1 reduces the structure group from G to H, and the bundle can be pulled back from an H-bundle over the

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region near infinity on R3 . Infinity on R3 is homotopic to S 2 , so we get an H-bundle over S 2 . The Hilbert space of the theory is then graded by the topological type of the H-bundle. We call this the magnetic grading. (Its components corresponding to U(1) subgroups of H are called magnetic charges.) According to [11], electric-magnetic duality exchanges the electric and magnetic gradings. In our context, this will mean that the electric grading in Description I matches the magnetic grading in Description II, and vice versa. In the simplest situation, if U is generic, then H is a maximal torus T of G; the electric and magnetic gradings correspond to an action of T and T ∨ , respectively. Actually, to extract the maximum amount of information from the theory, we want to allow an arbitrarily specified value of the two-dimensional characteristic  2 (TNk , Z), class a. As described in Remark 6, a takes values in a certain torsor for H which means, modulo a trivialization of the torsor, that a assigns an element of Z to each fixed point. (The origin of a in six dimensions was discussed in Section 4.1.) Roughly speaking, allowing arbitrary a means that we do Gad gauge theory, but there is a small twist: to extract the most information, we divide by only those gauge transformations that can be lifted to the simply connected form G. This means that the monodromy U can be regarded as an element of G (up to conjugacy), and similarly that in Description II, we meet representations of the Kac – Moody group of G (not Gad ). In gauge theory on R × TNk , U(1)
acts geometrically, generating the triholomorphic symmetry of TNk . But how does U(1) act? The answer to this question is that in this description, the generator P of U(1) is equal to the instanton number I. (This fact is deduced using string theory.) The instanton number is defined via a familiar curvature integral, normalized so that on a compact four-manifold and with a simply connected gauge group, it takes integer values. In the present context, the values of the instanton number are not necessarily integers, because TNk is not compact. The analog of integrality in this situation is the following. First, one should add to the instanton number I a certain linear combination of the magnetic charges (with coefficients given by the logarithms of the monodromies). Then there is a fixed real number r, depending only on the Let us call the sum I. monodromy at infinity and the characteristic class a, such that I takes values in r + Z. So in this description, eigenvalues of P are not necessary integers, but (for bundles with a fixed a and U ) a certain linear combination of the eigenvalues of P and the magnetic charges are congruent to each other modulo the integers. Since P
generates the U(1)
symmetry of TNk , one might expect its eigenvalues to be integers, but here we run into the electric charges. There is an operator that generates the triholomorphic symmetry and whose eigenvalues are integers; it is not simply P
but the sum of P
and a central generator of H (this generator is the logarithm of the monodromy at infinity), or in other words the sum of P
and a linear combination of electric charges. What are BPS states in this description? Classically, the minimum energy fields of given instanton number are the instantons — that is, the gauge fields that are independent of time and are anti-self-dual connections on TNk . Instantons on TNk have recently been studied by D-brane methods [5, 6, 38]. In particular [5], certain components of the moduli space M of instantons on TNk , when regarded as complex symplectic manifolds in one complex structure, coincide with components of the moduli space of instantons on the corresponding ALE space (the resolution

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of R4 /Zk ). All components of instanton moduli space on the ALE space arise in this way, but there are also components of instanton moduli space on TNk that have no analogs for the ALE space. (According to [5] and as explained to me by the author of that paper, these are the components of nonzero magnetic charge, corresponding to nonzero electric charge in Description II.) An instanton is a classical BPS configuration, but to construct quantum BPS states, we must, roughly speaking, take the cohomology of the instanton moduli space M. Actually, M is not compact and by “cohomology,” we mean in this context the space of L2 harmonic forms on M. (These are relevant for essentially the same reasons that they entered in one of the early tests of electric-magnetic duality [26].) So V is the space of L2 harmonic forms on M. Of course, to construct V we have to include contributions from all components Mn of M:  HL∗ 2 harm (Mn ), (38) V= n

HL∗ 2 harm

where we write for the space of L2 harmonic forms. The action of P on V is multiplication by the instanton number, and similarly the magnetic grading is determined by the topological invariants of the bundles parametrized by a given Mn . P
and the electric charges act trivially on V because they correspond to continuous symmetries of Mn that act trivially on its cohomology. ahler manifold, and accordingly the group Sp(4) — which Each Mn is a hyper-K¨ in the present context is the R-symmetry group (as explained in Remark 12) — acts on the space of L2 harmonic forms on Mn and hence on V. However, as in similar problems [16], it seems likely that Sp(4) acts trivially on these spaces. (This is equivalent to saying that L2 harmonic forms exist only in the middle dimension and are of type (p, p) for every complex structure.) This would agree with what one sees on the other side of the duality, which we consider next. Remark 7. If we simply replace TNk by R4 (with its usual metric) in this analysis, we learn in the same way that BPS states of the (0, 2) model on R×S 1 ×R4 correspond to L2 harmonic forms on instanton moduli space on R4 , with its usual metric. The same holds with an ALE space instead of R4 . The advantage of TNk over R4 or an ALE space is that there is an alternative second description. 5.5.2. Description II. The other option is to take R → 0. In this case, the fibers of µ  : TNk → R3 collapse, so to go over to a gauge theory description, we replace TNk by R3 , with special behavior at the U(1) fixed points xj , j = 1, . . . , k, where holomorphic WZW models will appear. We get a second description, then, in terms of maximally supersymmetric gauge theory on M5 = R × S 1 × R3 , with level 1 holomorphic WZW models of type G supported on the k two-manifolds Wj = R × S 1 × xj , j = 1, . . . , k. Once again, we must specify the holonomy U at infinity of the gauge field around S 1 . This is simply the same as the corresponding holonomy at infinity in Description I. Suppose for a moment that U is trivial. Then we also must pick, for each xj , j = 1, . . . , k, an integrable representation of the affine Kac – Moody algebra of G at level 1. For a simply laced and simply connected group, the integrable representations are classified by characters of the center Z of G, or, as there is a perfect pairing Z × Z → U(1), simply by Z. So for each j, we must give an element aj ∈ Z. This is precisely the data that we obtained in Description I  2 (TNk , Z). Since the second homology group of from the characteristic class a ∈ H

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R × S 1 × R3 vanishes, there is no two-dimensional characteristic class to be chosen in Description II (matching the fact that there was no Kac – Moody representation in Description I). More generally, for any U , we can canonically pick up to isomorphism a Gbundle on R × S 1 × R3 with that monodromy at infinity, namely a flat bundle with holonomy U around S 1 . In the presence of this flat bundle, the Kac – Moody algebra on each S 1 × xj is twisted; if θ is an angular parameter on S 1 , then instead of the currents obeying J(θ + 2π) = J(θ), they obey J(θ + 2π) = U J(θ)U −1 . The representations of this twisted Kac – Moody algebra at level 1 are a torsor for Z — the same torsor that we met in Remark 6. The torsor is the same for each j since each Kac – Moody algebra is twisted by the same U . (The torsor property means concretely that the representations of the Kac – Moody algebra are permuted if U undergoes monodromy around a noncontractible loop in Gad .) In Description II, P generates the rotations of S 1 . For reasons that will become apparent, what is important is how P acts on the representations of the Kac – Moody algebra. In the Kac – Moody algebra, P corresponds to the operator — usually called L0 — that generates a rotation of the circle. First set U = 1. Then L0 has integer eigenvalues in the vacuum representation of the Kac – Moody algebra (that is, the representation whose highest weight is G-invariant). In a more general representation (but still at U = 1), L0 has eigenvalues that are congruent mod Z to a fixed constant r that depends only on the highest weight. This matches the fact that, in Description I (at U = 1) the instanton number takes values in r + Z where r depends only on the characteristic class a. In the Kac – Moody theory, when the twisting parameter U is varied away from 1, the eigenvalues of L0 shift. However (recalling that H is the commutant of U in G), one can add to L0 a linear 0 with the property that combination of the generators of H to make an operator L in a given representation of the twisted Kac – Moody algebra, its eigenvalues are congruent mod Z. Thus, electric charges play precisely the role in Description II that magnetic charges play in Description I. On the other hand, in Description II, P
is the instanton number of a G-bundle on the initial value surface S 1 × R3 . If the monodromy U at infinity is trivial, then P
is integer-valued, just as in Description I. In general, for any U , a certain linear combination of P
and the magnetic charges (with coefficients given as usual by the logarithms of the monodromies) takes integer values. This mirrors the fact that in Description I, a linear combination of P
and the electric charges takes integer values. What is the space of BPS states in Description II? Supported on R×S 1 ×xj for each j = 1, . . . , k, there is a level 1 Kac – Moody module Wj . This module has H = P for all states (mathematically, the representation theory of affine Kac – Moody algebras is usually developed with a single L0 operator, not two of them), and

consists entirely of BPS states. The space of BPS states is simply V = kj=1 Wj . In particular, as the R-symmetry group Sp(4) acts trivially on the Wj , it acts trivially on V. The analogous statement in Description I was explained at the end of Section 5.5.1. Comparing the results of the two descriptions, we learn that (39)

k  j=1

Wj =



HL∗ 2

harm

(Mn ).

n

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The right-hand side is graded by instanton number and magnetic charge, and the left-hand side by L0 and electric charge. This equivalence closely parallels a central claim in [4, 19, 24]. A couple of differences may be worthy of note. Our description uses L2 harmonic forms; different versions of cohomology are used in recent mathematical ahler metric, not on the papers. Also, our instantons live on TNk with its hyper-K¨ resolution of the Ak−1 singularity. This does not affect Mn as a complex symplectic manifold (as long as one considers on the TNk side judiciously chosen components of the moduli space [5]), but it certainly affects the hyper-K¨ahler metric of Mn and therefore the condition for an L2 harmonic form. The components Mn of instanton moduli space on TNk that do not have analogs on the resolution of the Ak−1 singularity are also presumably important. One may wonder why we do not get additional BPS states from quantizing the moduli space of instantons, as we did in Description I. This can be understood as follows. Generically, curvature breaks all supersymmetry. In Description I, because the curvature of TNk is anti-self-dual, it leaves unbroken half of the supersymmetry. The half that survives is precisely the supersymmetry that is preserved by an instanton (since an instanton bundle also has anti-self-dual curvature). Hence instantons are supersymmetric and must be considered in constructing the space of BPS states. By contrast, in Description II, there is no curvature to break supersymmetry. Instead, there is a coupling (24), which (when extended to include fields and terms that we have omitted) leaves unbroken half the supersymmetry, but a different half from what is left unbroken by anti-self-dual curvature. The result is that in Description II, instantons are not supersymmetric. So in Description II, the instanton number and similarly the magnetic charges annihilate any BPS state. This implies that P
and the magnetic charges annihilate V in Description II, just as P
and the electric charges do in Description I. 5.5.3. A note on the dual group. The reader may be puzzled by the fact that in this analysis of two ways to describe the space of BPS states, we have not mentioned the dual group G∨ . The reason for this is that for simplicity, we have limited ourselves to the case that G is simply laced. When this is so, G and G∨ have the same Lie algebra. Instead of merely comparing G and G∨ theories, we can learn more, as explained in Section 4.1, by considering Gad bundles with an arbitrary two-dimensional characteristic class a. This is what we have done. For groups that are not simply laced, a slight variant of the construction that we have used is available [30]. The basic idea is that outer automorphisms of a simply laced group G can appear as symmetries of the (0, 2) model of type G in six dimensions. To a simple but not simply laced Lie group H, one associates a pair (G, λ), where G is simply laced and λ is an outer automorphism of G. The (0, 2) model of type G, when formulated on M6 = M5 × S 1 , can be “twisted” by λ in going around S 1 , in which case the low energy limit on M5 is maximally supersymmetric gauge theory of type H ∨ . Now consider M6 = M4 × S 1 × S1 , with circles of radius S and R and a twist by λ around S 1 . Repeating the analysis of Section 4, we get at long distances on M4 × S1 a description by H ∨ gauge theory; this further reduces on M4 to a description by H ∨ gauge theory with coupling parameter τ ∨ = iS/R. Alternatively, we get a description by G gauge theory on M4 × S 1 with a twist around S 1 that reduces G to H; this further reduces on M4

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to H gauge theory with coupling parameter τ = iR/S. The comparison of these two descriptions is essentially the perspective offered in [30] on electric-magnetic duality in four dimensions for non-simply-laced groups. We can learn more if we do not reduce all the way to four dimensions. We take M6 = R × S 1 × TNk , where S 1 has circumference 2πS and the fiber at infinity of µ  : TNk → R3 has circumference 2πR. For small S, we get Description I, which involves H ∨ instantons on TNk . For small R, we get Description II, which now involves level 1 modules of the affine Kac – Moody algebra of G, twisted by λ. This is equivalent to the claim of [4] (although the roles of H and H ∨ are reversed in our presentation here). To make contact with the formulation given in [4], one must know that the Langlands or GNO dual of the Kac – Moody group of H ∨ is not the Kac – Moody group of H but the λ-twisted Kac – Moody group of G. Remark 8. The construction in the last paragraph does not appear to have an analog with a twist around the circle at infinity in TNk . Precisely because this circle is contractible in the interior of TNk , it is not possible to twist by a discrete symmetry of the (0, 2) model in going around this circle. 5.5.4. Points with multiplicity. So far we have considered the points xj to be distinct, so that TNk is smooth. It is important, however, to consider the behavior as some of the xj become coincident. In general, suppose that the xj become coincident for j = i1 , . . . , ir . Without any essential change in the following remarks, we could allow several subsets of xj to simultaneously become coincident. In Description I, when this happens, TNk develops an Ar−1 singularity, which is an orbifold singularity, locally modeled by R4 /Zr . Gauge theory on R4 /Zr is defined as Zr -invariant gauge theory on R4 , but the notion of Zr invariance depends on the choice of how Zr acts on the fiber of a G-bundle at the fixed point (the origin in R4 ). Such a choice is a homomorphism φ : Zr → G. When a Zr orbifold singularity develops, the space V of BPS states becomes graded by the choice of φ. This is in addition to the grading by the part of the characteristic class a that can be defined on the complement of the singularity. The dual in Description II is that r of the points xj that support level 1 holomorphic WZW models become coincident, say at a point y ∈ R3 . Then the submanifold R × S 1 × y of M6 supports a level r holomorphic WZW model of type G. The affine Kac – Moody group of G supports several inequivalent integrable highest weight modules of level r, and when the points xi1 , . . . , xir become coincident, the Hilbert space of the theory decomposes as a direct sum of subspaces transforming in different such representations. This also gives a decomposition of the space V of BPS states. So duality must establish a correspondence between two types of data (φ and the relevant part of a on one side; a choice of level r integrable representation on the other side). Such a correspondence is used in [4] and can be described as follows in physical terms. For simplicity, we set G = E8 , so that we can dispense with a. Consider three-dimensional Chern – Simons theory with gauge group G at level r. A Wilson loop can be considered in any irreducible representation R that is the highest weight of a level r integrable module of the affine Kac – Moody algebra. On the other hand, around the Wilson loop, the gauge field acquires a monodromy that is an element of G of order r. This gives a correspondence between level r integrable modules and conjugacy classes of order r. It should be possible to use

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string theory arguments to show that this is the correspondence that enters in comparing Descriptions I and II, but this will not be attempted here. The general story is to consider points y1 , . . . , ys ∈ R3 with multiplicities s r1 , . . . , rs . In our presentation, we obtain this case starting with k = i=1 ri points, all of multiplicity 1, and letting them coalesce in clumps of the appropriate sizes. Then in Description I, we consider a TNk space that is constrained to have singularities of type Ari −1 for i = 1, . . . , s. Gauge theory at the ith singularity is defined by a choice of homomorphisms φi : Zri → G. Description II is based on supersymmetric gauge theory on R × S 1 × R3 , with each R × S 1 × yi supporting a holomorphic level ri WZW model, associated with the level ri integrable representation that corresponds to φi . This correspondence, moreover, is compatible with further coalescences of points, in close parallel to the picture of [4]. Any further coalescence of points leads to a further decomposition of V on both sides. Acknowledgments. I thank A. Braverman for explanations of the ideas of [4]. I also thank S. Cherkis for careful comments on the manuscript and G. Segal for a discussion of the sense in which the (0, 2) theory does not have a Lagrangian. References 1. M. F. Atiyah and R. Bott, The Yang-Mills equations over Riemann surfaces, Philos. Trans. Roy. Soc. London Ser. A 308 (1983), no. 1505, 523 – 615. 2. J. C. Baez and J. Dolan, Higher-dimensional algebra and topological quantum field theory, J. Math. Phys. 36 (1995), no. 11, 6073 – 6105. 3. A. Beilinson and V. Drinfeld, Quantization of Hitchin’s integrable system and Hecke eigensheaves, available at www.math.uchicago.edu/~mitya/langlands.html. 4. A. Braverman and M. Finkelberg, Pursuing the double affine Grassmannian. I: Transversal slices via instantons on Ak singularities, available at arXiv:0711.2083. 5. S. A. Cherkis, Moduli spaces of instantons on the Taub-NUT space, available at arXiv:0805. 1245. , Instantons on the Taub-NUT space, available at arXiv:0902.4724. 6. 7. R. Dijkgraaf, L. Hollands, P. Sulkowski, and Cumrun Vafa, Supersymmetric gauge theories, intersecting branes and free fermions, J. High Energy Phys. 2 (2008), 106. 8. D. S. Freed, Higher algebraic structures and quantization, Comm. Math. Phys. 159 (1994), no. 2, 343 – 398. 9. D. Gaiotto and E. Witten, Supersymmetric boundary conditions in N = 4 super Yang – Mills theory, available at arXiv:0804.2902. , S-Duality of boundary conditions In N = 4 super Yang – Mills theory, available at 10. arXiv:0807.3720. 11. P. Goddard, J. Nuyts, and D. Olive, Gauge theories and magnetic charge, Nuclear Phys. B 125 (1977), no. 1, 1 – 28. 12. S. Gukov and E. Witten, Gauge theory, ramification, and the geometric Langlands program, Current Developments in Mathematics, 2006, Int. Press, Somerville, MA, 2008, pp. 35 – 180. 13. S. W. Hawking, Gravitational instantons, Phys. Lett. A 60 (1977), no. 2, 81 – 83. 14. M. Henningson, The quantum Hilbert space of a chiral two-form in d = 5 + 1 dimensions, J. High Energy Phys. 3 (2002), 021. 15. M. Henningson, B. E. W. Nilsson, and P. Salomonson, Holomorphic factorization of correlation functions in (4k +2)-dimensional (2k)-form gauge theory, J. High Energy Phys. 9 (1999), 008. ahler quotients, Comm. Math. Phys. 211 (2000), no. 1, 16. N. Hitchin, L2 -cohomology of hyperk¨ 153 – 165. 17. A. Kapustin and E. Witten, Electric-magnetic duality and the geometric Langlands program, Commun. Number Theory Phys. 1 (2007), no. 1, 1 – 236.

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18. R. P. Langlands, Problems in the theory of automorphic forms, Lectures in Modern Analysis and Applications. III, Lecture Notes in Math., vol. 170, Springer, Berlin, 1970, pp. 18 – 61. 19. A. Licata, Framed rank r torsion-free sheaves on CP 2 and representations of the affine Lie
available at arXiv:math/0607690. algebra gl(r), 20. U. Lindstr¨ om and M. Roˇ cek, Scalar tensor duality and N = 1, 2 nonlinear σ-models, Nuclear Phys. B 222 (1983), no. 2, 285 – 308. 21. J. Lurie, On the classification of topological field theories, available at math.mit.edu/~lurie/. 22. C. Montonen and D. I. Olive, Magnetic monopoles as gauge particles?, Phys. Lett. B 72 (1977), no. 1, 117 – 120. 23. W. Nahm, Supersymmetries and their representations, Nuclear Phys. B 135 (1978), no. 1, 149 – 166. 24. H. Nakajima, Quiver varieties and branching, SIGMA Symmetry Integrability Geom. Methods Appl. 5 (2009), 003. 25. G. Segal, to appear. 26. A. Sen, Dyon-monopole bound states, self-dual harmonic forms on the multi-monopole moduli space, and SL(2, Z) invariance in string theory, Phys. Lett. B 329 (1994), no. 2-3, 217 – 221. 27. A. Strominger, Open p-branes, Phys. Lett. B 383 (1996), no. 1, 44 – 47. 28. M. C. Tan, Five-branes in M-theory and a two-dimensional geometric Langlands duality, available at arXiv:0807.1107. 29. P. K. Townsend, The eleven-dimensional supermembrane revisited, Phys. Lett. B 350 (1995), no. 2, 184 – 188. 30. C. Vafa, Geometric origin of Montonen – Olive duality, Adv. Theor. Math. Phys. 1 (1997), no. 1, 158 – 166. 31. C. Vafa and E. Witten, A strong coupling test of S-duality, Nuclear Phys. B 431 (1994), no. 1-2, 3 – 77. 32. E. Verlinde, Global aspects of electric-magnetic duality, Nuclear Phys. B 455 (1995), no. 1-2, 211 – 225. 33. E. Witten, Nonabelian bosonization in two dimensions, Comm. Math. Phys. 92 (1984), no. 4, 455 – 472. , On S-duality in abelian gauge theory, Selecta Math. (N.S.) 1 (1995), no. 2, 383 – 410. 34. , Some comments on string dynamics, Strings ’95 (Los Angeles, CA, 1995), World Sci. 35. Publ, River Edge, NJ, 1996, pp. 501 – 523. , Conformal field theory in four and six dimensions, Topology, Geometry and Quan36. tum Field Theory, London Math. Soc. Lecture Note Ser., vol. 308, Cambridge Univ. Press, Cambridge, 2004, pp. 405 – 419. , Mirror symmetry, Hitchin’s equations, and Langlands duality, available at arXiv: 37. 0802:0999. , Branes, instantons, and Taub-NUT spaces, available at arXiv:0902.0948. 38. 39. E. Witten and D. I. Olive, Supersymmetry algebras that include topological charges, Phys. Lett. B 78 (1978), 97 – 101. School of Natural Sciences, Institute for Advanced Study, Princeton, NJ 08540, USA Current address: Theory Group, CERN, Geneva Switzerland E-mail address: [email protected]

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https://doi.org/10.1090/crmp/050/24

Centre de Recherches Math´ ematiques CRM Proceedings and Lecture Notes Volume 50, 2010

Duality and Equivalence of Module Categories in Noncommutative Geometry Jonathan Block In memory of Raoul

Abstract. We develop a general framework to describe dualities from algebraic, differential, and noncommutative geometry, as well as physics. We pursue a relationship between the Baum – Connes conjecture in operator Ktheory and derived equivalence statements in algebraic geometry and physics. We associate to certain data, reminiscent of spectral triple data, a differential graded category in such a way that we can recover the derived category of coherent sheaves on a complex manifold.

Introduction In various geometric contexts, there are duality statements that are expressed in terms of appropriate categories of modules. We have in mind, for example, the Baum – Connes conjecture from noncommutative geometry, T-duality and Mirror symmetry from complex geometry and mathematical physics. This is the first in a series of papers that sets up a framework to study and unify these dualities from a noncommutative geometric point of view. We also view this project as an attempt to connect the noncommutative geometry of Connes, [6] with the categorical approach to noncommutative geometry, represented for example by Manin and Kontsevich. Traditionally, the complex structure is encoded in the sheaf of holomorphic functions. However, for situations we have in mind coming from noncommutative geometry, one can not use local types of constructions, and we are left only with global differential geometric ones. A convenient setting to talk about integrability of geometric structures and the integrability of geometric structures on their modules is that of a differential graded algebra and more generally, a curved differential graded algebra. Thus, for example, a complex structure on a manifold is encoded ¯ and a holomorphic vector bundle can be in its Dolbeault algebra A = (A0,• (X), ∂), viewed as the data of a finitely generated projective module over A0,0 together with 2000 Mathematics Subject Classification. Primary 58B34; Secondary 18E30, 19K35, 46L87, 58J42. J.B. partially supported by NSF grant DMS02-04558. This is the final form of the paper. c
2010 c
2010 American American Mathematical Mathematical Society

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¯ a flat ∂-connection. Similarly, holomorphic gerbes can be encoded in terms of a curved differential graded algebra with non-trivial curvature. Curved dgas appear naturally in the context of matrix factorizations and Laundau – Ginzburg models, [11, 21]. Indeed, these fit very easily into our framework. Of course, one is interested in more modules than just the finitely generated projective ones. In algebraic geometry, the notion of coherent module is fundamental. In contrast to projective algebraic geometry however, not every coherent sheaf has a resolution by vector bundles; they only locally have such resolutions. Toledo and Tong, [28,29], handled this issue by introducing twisted complexes. Our construction is a global differential geometric version of theirs. We have found the language of differential graded categories to be useful, [5, 10, 18]. In particular, for a curved dga A we construct a very natural differential graded category PA which can then be derived. The desiderata of such a category are • it should be large enough to contain in a natural way the coherent holomorphic sheaves (in the case of the Dolbeault algebra), and • it should be flexible enough to allow for some of Grothendieck’s six operations, so that we can prove Mukai type duality statements. The reason for introducing PA is that the ordinary category of dg-modules over the Dolbeault dga has the wrong homological algebra; it has the wrong notion of quasi-isomorphism. A morphism between complexes of holomorphic vector bundles considered as dg-modules over the Dolbeault algebra is a quasi-isomorphism if it induces an isomorphism on the total complex formed by the gloabal sections of the Dolbeault algebra with values in the complexes of holomorphic vector bundles, which is isomorphic to their hypercohomology. On the other hand, PA and the modules over it have the correct notion of quasi-isomorphism. In particular, PA is not an invariant of quasi-isomorphism of dga’s. To be sure, we would not want this. For example, the dga which is C in degree 0 and 0 otherwise is quasi-isomorphic to the Dolbeault algebra of CPn . But CPn has a much richer module category than anything C could provide. We show that the homotopy category of PA where A is the Dolbeault algebra of a compact complex manifold X is equivalent to the derived category of sheaves of OX -modules with coherent cohomology. Our description of the coherent derived category has recently been used by Bergman, [2] as models for B-model D-branes. To some extent, what we do is a synthesis of Kasparov’s KK-theory, [17] and of Toledo and Tong’s twisted complexes, [20, 28, 29]. In appreciation of Raoul Bott. I am always amazed by the profound impact that he had, and still has, on my life. During the time I was his student, I learned much more from him than mere mathematics. It was his huge personality, his magnanimous heart, his joy in life and his keen aesthetic that has had such a lasting effect. I miss him. Acknowledgements. We would like to thank Oren Ben-Bassat, Andre Caldararu, Calder Daenzer, Nigel Higson, Anton Kapustin, the referee, Steve Shnider, Betrand Toen and especially Tony Pantev for many conversations and much guidance regarding this project.

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1. Baum – Connes and Fourier – Mukai There are two major motivations for our project. The first is to have a general framework that will be useful in dealing with categories of modules that arise in geometry and physics. For example, we will apply our framework to construct categories of modules over symplectic manifolds. Second, as mentioned earlier in the introduction, this series of papers is meant to pursue a relationship between (1) the Baum – Connes conjecture in operator K-theory and (2) derived equivalence statements in algebraic geometry and physics. In particular, we plan to refine, in certain cases, the Baum – Connes conjecture from a statement about isomorphism of two topological K-groups to a derived equivalence of categories consisting of modules with geometric structures, for example, coherent sheaves on complex manifolds. We will see that there are natural noncommutative geometric spaces that are derived equivalent to classical algebraic geometric objects. Let us explain the obvious formal analogies between (1) and (2). For simplicity let Γ be a discrete torsion-free group with compact BΓ. In this situation, the Baum – Connes conjecture says that an explicit map, called the assembly map, µ : K∗ (BΓ) → K∗ (Cr∗ Γ)

(1.1)

is an isomorphism. Here Cr∗ Γ denotes the reduced group C ∗ -algebra of Γ. The assembly map can be described in the following way. On C(BΓ) ⊗ Cr∗ Γ there is a finitely generated projective right module P which can be defined as the sections of the bundle of Cr∗ Γ-modules EΓ ×Γ Cr∗ Γ. This projective module is a “line bundle” over C(BΓ) ⊗ Cr∗ Γ. Here, C(X) denote the complex-valued continuous functions on a compact space X. The assembly map is the map defined by taking the Kasparov product with P over C(BΓ). This is some sort of index map. µ : x ∈ KK(C(BΓ), C) → x ∪ P ∈ KK(C, Cr∗ Γ) where P ∈ KK(C, C(BΓ) ⊗ Cr∗ Γ) We now describe Mukai duality in a way that makes it clear that it refines Baum – Connes. Now let X be a complex torus. Thus X = V /Λ where V is a g-dimensional complex vector space and Λ ∼ = Z2g is a lattice in V . Let X ∨ denote the dual complex torus. This can be described in a number of ways: • as Pic0 (X), the manifold of holomorphic line bundles on X with first Chern class 0 (i.e., they are topologically trivial); • as the moduli space of flat unitary line bundles on X. This is the same as the space of irreducible unitary representations of π1 (X), but it has a complex structure that depends on that of X; ∨ • and most explicitly as V /Λ∨ where Λ∨ is the dual lattice, Λ∨ = {v ∈ V

∨

| Imv, λ ∈ Z ∀λ ∈ Λ}.

∨

Here V consists of conjugate linear homomorphisms from V to C. We note that X = BΛ and that C(X ∨ ) is canonically Cr∗ Λ. Hence Baum – Connes predicts (and in fact it is classical in this case) that K∗ (X) ∼ = K∗ (Cr∗ Λ) ∼ = ∗ ∨ K (X ).

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On X × X ∨ there is a canonical line bundle, P, the Poincar´e bundle, which is uniquely determined by the following universal properties: • P|X × {p} ∼ = p where p ∈ X ∨ and is therefore a line bundle on X. ∨ • P|{0} × X is trivial. Now Mukai duality says that there is an equivalence of derived categories of coherent sheaves Db (X) → Db (X ∨ ) induced by the functor F → p2∗ (p∗1 F ⊗ P) where pi are the two obvious projections. The induced map at the level of K0 is an isomorphism and is clearly a holomorphic version of the Baum – Connes Conjecture for the group Λ. 2. The dg-category PA of a curved dga 2.1. dg-categories. Definition 2.1. For complete definitions and facts regarding dg-categories, see [5, 10, 18, 19]. Fix a field k. A differential graded category (dg-category) is a category enriched over Z-graded complexes (over k) with differentials increasing degree. That is, a category C is a dg-category if for x and y in Ob C the hom set C(x, y) forms a Z-graded complex of k-vector spaces. Write (C • (x, y), d) for this complex, if we need to reference the degree or differential in the complex. In addition, the composition, for x, y, z ∈ Ob C C(y, z) ⊗ C(x, y) → C(x, z) is a morphism of complexes. Furthermore, there are obvious associativity and unit axioms. 2.2. Curved dgas. In many situations the integrability conditions are not expressed in terms of flatness but are defined in terms of other curvature conditions. This leads us to set up everything in the more general setting of curved dga’s. These are dga’s where d2 is not necessarily zero. Definition 2.2. A curved dga [23] (Schwarz [25] calls them Q-algebras) is a triple A = (A• , d, c) where A• is a (nonnegatively) graded algebra over a field k of characteristic 0, with a derivation d : A• → A•+1 which satisfies the usual graded Leibniz relation but d2 (a) = [c, a] where c ∈ A2 is a fixed element (the curvature). Furthermore we require the Bianchi identity dc = 0. Let us write A for the degree 0 part of A• , the “functions” of A.

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A dga is the special case where c = 0. Note that c is part of the data and even example of a curved dga. The if d2 = 0, that c might not be 0, and gives
a non-dga
  prototypical example of a curved dga is A• M, End(E) , Ad ∇, F of differential forms on a manifold with values in the endomorphisms of a vector bundle E with connection ∇ and curvature F . 2.3. The dg-category PA . Our category PA consists of special types of Amodules. We start with a Z-graded right module E • over A. Definition 2.3. A Z-connection E is a k-linear map E : E • ⊗A A• → E • ⊗A A• of total degree one, which satisfies the usual Leibniz condition
 E(eω) = E(e ⊗ 1) ω + (−1)e edω Such a connection is determined by its value on E • . Let Ek be the component of E such that Ek : E • → E •−k+1 ⊗A Ak ; thus E = E0 + E1 + E2 + · · · . It is clear that E1 is a connection on each component E n in the ordinary sense (or the negative of a connection if n is odd) and that Ek is A-linear for k = 1. Note that for a Z-connection E on E • over a curved dga A = (A• , d, c), the usual curvature E ◦ E is not A-linear. Rather, we define the relative curvature to be the operator FE (e) = E ◦ E(e) + e · c and this is A-linear. Definition 2.4. For a curved dga A = (A• , d, c), we define the dg-category PA : (1) An object E = (E • , E) in PA , which we call a cohesive module, is a Zgraded (but bounded in both directions) right module E • over A which is finitely generated and projective, together with a Z-connection E : E • ⊗A A• → E • ⊗A A• that satisfies the integrability condition that the relative curvature vanishes FE (e) = E ◦ E(e) + e · c = 0 •

for all e ∈ E . (2) The morphisms of degree k, PAk (E1 , E2 ) between two cohesive modules E1 = (E1• , E1 ) and E2 = (E2• , E2 ) of degree k are {φ : E1• ⊗A A• → E2• ⊗A A• | of degree k and φ(ea) = φ(e)a ∀a ∈ A• } with differential defined in the standard way

 d(φ)(e) = E2 φ(e) − (−1)|φ| φ E1 (e) Again, such a φ is determined by its restriction to E1• and if necessary we denote the component of φ that maps (2.1)

E1• → E2•+k−j ⊗A Aj

by φj . Thus PAk (E1 , E2 ) = HomkA (E1• , E2• ⊗A A• ). Proposition 2.5. For A = (A• , d, c) a curved dga, the category PA is a dgcategory.

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This is clear from the following lemma. Lemma 2.6. Let E1 , E2 be cohesive modules over the curved dga A = (A• , d, c). Then the differential defined above d : PA• (E1 , E2 ) → PA•+1 (E1 , E2 ) satisfies d2 = 0. 2.4. The homotopy category and triangulated structure. Given a dgcategory C, one can form the subcategory Z 0 C which has the same objects as C and whose morphisms from an object x ∈ C to an object y ∈ C are the degree 0 closed morphisms in C(x, y). We also form the homotopy category Ho C which has the same objects as C and whose morphisms are the 0th cohomology
 Ho C(x, y) = H 0 C(x, y) . We define a shift functor on the category PA . For E = (E • , E) set E[1] = (E[1]• , E[1]) where E[1]• = E •+1 and E[1] = −E. It is easy to verify that E[1] ∈ PA . Next for E, F ∈ PA and φ ∈ Z 0 PA (E, F ), define the cone of φ, Cone(φ) = (Cone(φ)• , Cφ ) by ⎛ • ⎞ F Cone(φ)• = ⎝ ⊕ ⎠ E[1]• and   F φ Cφ = 0 E[1] We then have a triangle of degree 0 closed morphisms φ

E− → F → Cone(φ) → E[1]

(2.2)

Proposition 2.7. Let A be a curved dga. Then the dg-category PA is pretriangulated in the sense of Bondal and Kapranov, [5]. Therefore, the category Ho PA is triangulated with the collection of distinguished triangles being isomorphic to those of the form (2.2). Proof. The proof of this is the same as that of Propositions 1 and 2 of [5].




2.5. Homotopy equivalences. As described above, a degree 0 closed morphism φ between cohesive modules Ei = (Ei• , Ei ), i = 1, 2, over A is a homotopy equivalence if it induces an isomorphism in Ho PA . We want to give a simple criterion for φ to define such a homotopy equivalence. On the complex PA (E1 , E2 ) define a decreasing filtration by F k PAj (E1 , E2 ) = {φ ∈ PAj (E1 , E2 ) | φi = 0 for i < k} where φi is defined as in (2.1). Proposition 2.8. There is a spectral sequence
 (2.3) E0pq =⇒ H p+q PA• (E1 , E2 ) where E0pq = gr PA• (E1 , E2 ) = {φp ∈ PAp+q (E1 , E2 ) : E1• → E2•+q ⊗A Ap } with differential d0 (φp ) = E02 ◦ φp − (−1)p+q φp ◦ E01

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Proposition 2.9. A closed morphism φ ∈ PA0 (E1 , E2 ) is a homotopy equivalence if and only if φ0 : (E1• , E01 ) → (E2• , E02 ) is a quasi-isomorphism of complexes of A-modules. Proof. Let E = (E • , E) be any object in PA . Then φ induces a map of complexes (2.4)

φ : PA• (E, E1 ) → PA• (E, E2 )

We show that the induced map on E1 -terms of the spectral sequences are isomorphisms. Indeed, the quasi-isomorphism of (E1• , E01 ) → (E2• , E02 ) implies that they are actually chain homotopy equivalent since E1• and E2• are projective, hence for each p that φ0 ⊗ I : (E1• ⊗A Ap , E01 ⊗ I) → (E2• ⊗A Ap , E02 ⊗ I) is a quasi-isomorphism and then •+q pq •+q gr(φ) = φ0 : E0pq ∼ = HomA (E • , E1 ⊗A Ap ) → E0 ∼ = HomA (E • , E2 ⊗A Ap )

is a quasi-isomorphism after one last double complex argument since the modules E • are projective over A. Thus (2.4) is a quasi-isomorphism for all E and this implies φ is an isomorphism in Ho PA . The other direction follows easily.
2.6. The dual of a cohesive module. We define a duality functor which will be of use in future sections. Let A = (A• , d, c) be a curved dga. Its opposite is A◦ = (A◦• , d, −c) where A◦• is the graded algebra whose product is given by a ·◦ b = (−1)|a||b| ba We will not use the notation ·◦ for the product any longer. We can now define the category of left cohesive modules over A as PA◦ . We define the duality dg functor ∨

: PA → PA◦

by E = (E • , E) → E ∨ = (E ∨• , E∨ ) where E ∨k = HomA (E −k , A) and for φ ∈ E ∨•

 (E∨ φ)(e) = d φ(e) − (−1)|φ| φ E(e) It is straightforward that E ∨ is indeed cohesive over A◦ . There is a natural pairing between E and E ∨ . And moreover the connection was defined so that the relation E∨ (φ), e + (−1)|φ| φ, E(e) = dφ, e holds. Note that the complex of morphisms PA• (E1 , E2 ) between cohesive modules can be identified with (E2• ⊗A A• ⊗A E1∨• , 1 ⊗ 1 ⊗ E∨ 1 + 1 ⊗ d ⊗ 1 + E2 ⊗ 1 ⊗ 1)

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2.7. Functoriality. We now discuss a construction of functors between categories of the form PA . Given two curved dgas, A1 = (A•1 , d1 , c1 ) and A2 = (A•2 , d2 , c2 ) a homomorphism from A1 to A2 is a pair (f, ω) where f : A•1 → A•2 is a morphism of graded algebras, ω ∈ A12 and they satisfy (1) f (d1 a1 ) = d2 f (a1 ) + [ω, f (a1 )] and (2) f (c1 ) = c2 + d2 ω + ω 2 . Given a homomorphism of curved dgas (f, ω) we define a dg functor f∗ : PA1 → PA2 •

as follows. Given E = (E , E) a cohesive module over A1 , set f ∗ (E) to be the cohesive module over A2 (E • ⊗A1 A2 , E2 ) where E2 (e⊗b) = E(e)b+e⊗(d2 b+ωb). One checks that E2 is still an E-connection and satisfies (E2 )2 (e ⊗ b) = −(e ⊗ b)c2 . This is a special case of the following construction. Consider the following data, X = (X • , X) where (1) X • is a graded finitely generated projective right-A2 -module, (2) X : X • → X • ⊗A2 A•2 is a Z-connection, (3) A•1 acts on the left of X • ⊗A2 A•2 satisfying a · (x · b) = (a · x) · b and


 X a · (x ⊗ b) = da · (x ⊗ b) + a · X(x ⊗ b)

for a ∈ A•1 , x ∈ X • and b ∈ A•2 , (4) X satisfies the following condition: X ◦ X(x ⊗ b) = c1 · (x ⊗ b) − (x ⊗ b) · c2 •

on the complex X ⊗A2 A•2 . Let us call such a pair X = (X • , X) an A1 -A2 -cohesive bimodule. Given an A1 -A2 -cohesive bimodule X = (X • , X), we can then define a dgfunctor (see the next section for the definition) X ∗ : PA1 → PA2 by X ∗ (E • , E) = (E • ⊗A1 X • , E2 ) where E2 (e ⊗ x) = E(e) · x + e ⊗ X(x), where the · denotes the action of A•1 on X • ⊗A1 A•2 . One easily checks that X ∗ (E) is an object of PA2 . We will write E ⊗ X for E2 . Remark 2.10. (1) The previous case of a homomorphism between curved dgas occurs by setting X • = A2 in degree 0. A•1 acts by the morphism f and the Z-connection is X(a2 ) = d2 (a2 ) + ω · a2 .

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(2) To give another example of an A1 -A2 -cohesive bimodule, consider a manifold M with two vector bundles with connection (E1 , ∇2 ) and (E2 , ∇2 ). Let ci be the curvature of ∇i . Set Ai = (A•i , di , ci ) = (A• (M ; End(Ei ), Ad ∇i ). Then we define a cohesive bimodule between them by setting
 X • = Γ M ; Hom(E2 , E1 ) in degree 0. X • has a Z-connection


 X(φ)(e2 ) = ∇1 φ(e2 ) − φ(∇2 e2 )

and maps X • → X • ⊗A2 A•2 . Then (X)2 (φ) = c1 · φ − φ · c2 as is required. This cohesive bimodule implements a dg-quasi-equivalence between PA1 and PA2 . (See the next section for the definition of a dg-quasi-equivalence.) 3. Modules over PA It will be important for us to work with modules over PA and not just with the objects of PA itself. 3.1. Modules over a dg-category. We first collect some general definitions, see [18] for more details. Definition 3.1. A functor F : C1 → C2 between two dg-categories is a dgfunctor if the map on hom sets (3.1)

F : C1 (x, y) → C2 (F x, F y)

is a chain map of complexes. A dg-functor F as above is a quasi-equivalence if the maps in (3.1) are quasi-isomorphisms and Ho(F ) : Ho C1 → Ho C2 is an equivalence of categories. Given a dg-category C , one can define the category of (right) dg-modules over C, Mod-C. This consists of dg-functors from the opposite dg-category C ◦ to the dg-category C(k) of complexes over k. More explicitly, a right C-module M is an assignment, to each x ∈ C, a complex M (x) and chain maps for any x, y ∈ C (3.2)

M (x) ⊗ C(y, x) → M (y)

satisfying the obvious associativity and unit conditions. A morphism f ∈ Mod-C(M, N ) between right C-modules M and N is an assignment of a map of complexes (3.3)

fx : M (x) → N (x)

for each object x ∈ C compatible with the maps in (3.2). Such a map is called a quasi-isomorphism if fx in (3.3) is a quasi-isomorphism of complexes for each x ∈ C. One can make modules over a dg-category into a dg-category itself. The morphisms we have defined in Mod-C are the degree 0 closed morphisms of this dg-category. The category of left modules C-Mod is defined in an analogous way. The category Mod-C has a model structure used by Keller to define its derived category, [18, 19]. The quasi-isomorphisms in Mod-C are those we just defined. The fibrations are the componentwise surjections and the cofibrations are defined by the usual lifting property. Using this model structure we may form the homotopy category of Mod-C, obtained by inverting all the quasi-isomorphisms in Mod-C. This is what Keller calls the derived category of C, and we will denote it by D(Mod-C).

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There is the standard fully faithful Yoneda embedding Z 0 C → Mod-C

where x ∈ C → hx = C(·, x).

Moreover, the Yoneda embedding induces a fully faithful functor Ho C → D(Mod-C) This is simply because for an object x ∈ C, the module hx is trivially cofibrant. Definition 3.2. (1) A module M ∈ Mod-C is called representable if it is isomorphic in Mod-C to an object of the form hx for some x ∈ C. (2) A module M ∈ Mod-C is called quasi-representable if it is isomorphic in D(Mod-C) to an object of the form hx for some x ∈ C. Definition 3.3. Let M ∈ Mod-C and N ∈ C-Mod. Their tensor product is defined to be the complex



  α M ⊗C N = cok M (c) ⊗ C(c , c) ⊗ N (c ) − → M (c) ⊗ N (c) c,c ∈C

c∈C





where for m ∈ M (c), φ ∈ C(c , c) and n ∈ N (c ) α(m ⊗ φ ⊗ n) = mφ ⊗ n − m ⊗ φn Bimodules are the main mechanism to construct functors between module categories over rings. They play the same role for modules over dg-categories. Definition 3.4. Let C and D denote two dg-categories. A bimodule X ∈ D-Mod-C is a dg-functor X : C ◦ ⊗ D → C(k) More explicitly, for objects c, c ∈ C and d, d ∈ D there are maps of complexes D(d, d ) ⊗ X(c, d) ⊗ C(c , c) → X(c , d ) satisfying the obvious conditions. Definition 3.5. For a bimodule X ∈ D-Mod-C and d ∈ D, we get an object X d ∈ Mod-C

where X d (c) = X(c, d).

Similarly, for c ∈ C, we get an object c

X ∈ D-Mod

where c X(d) = X(c, d).

Therefore, we may define for M ∈ Mod-D the complex M ⊗D c X Furthermore the assignment c → c X defines a functor C ◦ → D-Mod and so c → M ⊗D c X defines an object in Mod-C. Thus · ⊗D X defines a functor from Mod-D → Mod-C. Moreover, by deriving this functor, we get a functor L

M → M ⊗D X from D(Mod-D) → D(Mod-C). Definition 3.6 (Keller, [18]). A bimodule X ∈ D-Mod-C is called a quasifunctor if for all d ∈ D, the object X d ∈ Mod-C is quasi-representable. Such a bimodule therefore defines a functor Ho D → Ho C.

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Toen [27] calls quasi-functors right quasi-representable bimodules and it is a deep theorem of his that they form the correct morphisms in the localization of the category of dg-categories by inverting dg-quasi-equivalences. 3.2. Construction and properties of modules over PA . We now define a class of modules over the curved dga A that will define modules over the dg-category PA . Definition 3.7. For a curved dga A = (A• , d, c), we define a quasi-cohesive module to be the data of X = (X • , X) where X • is a Z-graded right module X • over A together with a Z-connection X : X • ⊗A A• → X • ⊗A A• that satisfies the integrability condition that the relative curvature FX (x) = X ◦ X(x) + x · c = 0 •

for all x ∈ X . Thus, they differ from cohesive modules by having possibly infinitely many nonzero graded components as well as not being projective or finitely generated over A. Definition 3.8. To a quasi-cohesive A-module X = (X • , X) we associate the ˜ X , by PA -module, denoted h ˜ X (E) = {φ : E • ⊗A A• → X • ⊗A A• | of degree k and φ(xa) = φ(x)a ∀a ∈ A• } h with differential defined in the standard way

 d(φ)(ex) = X φ(x) − (−1)|φ| φ E(x) ˜ X because of its similarity to the Yoneda for all E = (E • , E) ∈ PA . We use h embedding h, but beware that X is not an object in PA . However, in the same way ˜ X is shown to be a module over PA . For two as PA is shown to be a dg-category, h quasi-cohesive A-modules X and Y , and f : X • ⊗A A• → Y • ⊗A A• of degree 0 and satisfying f X = Yf , we get a morphism of PA -modules ˜X → h ˜Y ˜f : h h The point of a quasi-cohesive A-module X = (X • , X) is that the differential and morphisms decompose just the same as they do for cohesive modules. For example, X = k Xk where Xk : E • → X •−k+1 ⊗A Ak and similarly for morphisms. Proposition 3.9. Let X and Y be quasi-cohesive A-modules and f a morphism. Suppose f 0 : (X • , X0 ) → (Y • , Y0 ) is a quasi-isomorphism of complexes. ˜ f is a quasi-isomorphism in Mod-PA . The converse is not true. Then h It will be important for us to have a criterion for when a quasi-cohesive Amodule X induces a quasi-representable PA -module. Definition 3.10. Define a map φ : C → D between A-modules to be algebraically A-nuclear, [24], if there are finite sets of elements φk ∈ HomA (C, A) and yk ∈ D, k = 1, . . . , N such that

φ(x) = yk · φk (x) k

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Proposition 3.11 (See Quillen, [24, Proposition 1.1]). For C • a complex of A-modules, the following are equivalent: (1) C • is homotopy equivalent to a bounded complex of finitely generated projective A-modules. (2) For any other complex of A-modules D• , the homomorphism HomA (C • , A) ⊗A D• → HomA (C • , D• ) is a homotopy equivalence of complexes (over k). (3) The endomorphism 1C of C • is homotopic to an algebraically nuclear endomorphism. Definition 3.12. Suppose A = (A• , d, c) is a curved dga. Let X = (X • , X) be a quasi-cohesive module over A. Suppose there exist A-linear morphisms h0 : X • → X •−1 of degree −1 and T 0 : X • → X • of degree 0 satisfying (1) T 0 is algebraically A-nuclear, (2) [X0 , h0 ] = 1 − T 0 Then we will call X a quasi-finite quasi-cohesive module. Our criterion is the following. Theorem 3.13. Suppose A = (A• , d, c) is a curved dga. Let X = (X • , X) be a quasi-cohesive module over A. Then there is an object E = (E • , E) ∈ PA such ˜ X is quasi-representable, under either ˜ X is quasi-isomorphic to hE ; that is, h that h of the two following conditions: (1) X is a quasi-finite quasi-cohesive module. (2) A• is flat over A and there is a bounded complex (E • , E0 ) of finitely generated projective right A-modules and an A-linear quasi-isomorphism e0 : (E • , E0 ) → (X • , X0 ). Proof. In either case (1) or (2) of the theorem, there exists a bounded complex of finitely generated projective right A-modules (E • , E0 ) and a quasi-isomorphism e0 : (E • , E0 ) → (X • , X0 ). In case (1), X is quasi-finite-cohesive, and Proposition 3.11 implies that e0 is in fact a homotopy equivalence. In case (2) it is simply the hypothesis. In particular, e0 E0 −X0 e0 = 0. Now we construct a Z-connection term by term. The Z-connection X on X • induces a connection H : H k (X • , X0 ) → H k (X • , X0 ) ⊗A A1 for each k. We use the quasi-morphism e0 to transport this connection to a connection on H k (E • ; E0 ) / H k (E • , E0 ) ⊗A A1

H k (E • ; E0 ) (3.4)

e0 ⊗1

e0

 H k (X • , X0 )

H

 / H k (X • , X0 ) ⊗A A1

The right vertical arrow above e0 ⊗ 1 is a quasi-isomorphism; in case (1) this is because e0 is a homotopy equivalence and in case (2) because A• is flat. The first step is handled by the following lemma.

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Lemma 3.14. Given a bounded complex of finitely generated projective A modules (E • , E0 ) with connections H : H k (E • ; E0 ) → H k (E • , E0 ) ⊗A A1 , for each k, there exist connections  : E k → E k ⊗A A1 H lifting H. That is,   0 = (E0 ⊗ 1)H HE and the connection induced on the cohomology is H. Proof (of lemma). Since E • is a bounded complex of A-modules it lives in some bounded range of degrees k ∈ [N, M ]. Pick an arbitrary connection on E M , ∇. Consider the diagram with exact rows / H M (E • , E0 ) E M RRR RRR RRRθ RRR ∇ RRR H (   j⊗1 / H M (E • , E0 ) ⊗A A1 E M ⊗A A1

/0

j

(3.5)

/ 0.

In the diagram, θ = H ◦ j − (j ⊗ 1) ◦ ∇ is easily checked to be A-linear and j ⊗ 1 is surjective by the right exactness of tensor product. By the projectivity of E M , θ lifts to θ˜ : E M → E M ⊗A A1 ˜ With H  = ∇ + θ.  in place of ∇, the diagram above so that (j ⊗ 1)θ˜ = θ. Set H commutes. Now choose on E M −1 any connection ∇M −1 . But ∇M −1 does not necessar 0 = 0. So we correct it as follows. Set µ = HE  0− ily satisfy E0 ∇M −1 = HE 0 0 (E ⊗ 1)∇M −1 . Then µ is A-linear. Furthermore, Im µ ⊂ Im E ⊗ 1; this is because  ∈ Im E⊗1 since H  lifts H. So by projectivity it lifts to θ: E M −1 → E M −1 ⊗A A1 HE  Then  : E M −1 → E M −1 ⊗A A1 to be ∇M −1 + θ. such that (E0 ⊗ 1) ◦ θ = θ. Set H 0 0  E H = HE in the right most square below.

(3.6)

E0

/ E N +1

E0

E0

E / E M −1 / EM MMM MMMµ ∇M −1 MMM H
M&   E0 ⊗1 E0 ⊗1 E0 ⊗1 E0 ⊗1 / E N +1 ⊗A A1 / ... / E M −1 ⊗A A1 / E M ⊗A A1 E N ⊗A A1

EN

/ ...

0

/0 / 0.

 : E • → E • ⊗A A1 satisfying (E0 ⊗ 1)H  Now we continue backwards to construct all H 0 
= HE = 0. This completes the proof of the lemma.  1 = (−1)k H  on E k . Then (Proof of the theorem, continued). Set E  1 E0 = 0 1 + E E0 E  1 − X1 e0 = 0. We correct this as follows. but it is not necessarily true that e0 E 0 1 1 0 • • Consider ψ = e E − X e : E → X ⊗A A1 . Check that ψ is A-linear and a map of complexes.

(3.7)

(E • ⊗A A1 , E0 ⊗ 1) mm6 mmm e0 ⊗1 m m mmm  mmmψ m / (X • ⊗A A1 , X0 ⊗ 1). E• ˜ ψ

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In the above diagram, e0 ⊗ 1 is a quasi-isomorphism since e0 is a homotopy equivalence. So by [20, Lemma 1.2.5] there is a lift ψ˜ of ψ and a homotopy e1 : E • → X •−1 ⊗A A1 between (e0 ⊗ 1)ψ˜ and ψ, ψ − (e0 ⊗ 1)ψ = (e1 E0 + X0 e1 )  Then  1 − ψ. So let E1 = E E0 E1 + E1 E0 = 0 and

(3.8)

e0 E1 − X1 e0 = e1 E0 + X0 e1 .

So we have constructed the first two components E0 and E1 of the Z-connection and the first components e0 and e1 of the quasi-isomorphism E • ⊗A A• → X • ⊗A A• . To construct the rest, consider the mapping cone L• of e0 . Thus L• = E[1]• ⊕ X • Let L0 be defined as the matrix L0 =

(3.9) Define L1 as the matrix

 0  E [1] 0 e0 [1] X0

 1  E [1] 0 L = e1 [1] X1 1

(3.10)

Now L0 L0 = 0 and [L0 , L1 ] = 0 express the identities (3.8). Let   0 0 (3.11) D = L1 L1 + + rc X2 e0 [X0 , X2 ] where rc denotes right multiplication by c. Then, as is easily checked, D is A-linear and (1) [L0 , D] = 0 and (2) D|0⊕X • = 0. Since (L• , L0 ) is the mapping cone of a quasi-isomorphism, it is acyclic and since A• is flat over A, (L• ⊗A A2 , L0 ⊗ 1) is acyclic too. Since E • is projective, we have that
 Hom•A (E • , E0 ), (L• ⊗A A2 , L0 ) is acyclic. Moreover

  Hom•A (E • , E0 ), (L• ⊗A A2 , L0 ) ⊂ Hom•A L• , (L• ⊗A A2 , [L0 , ·]) is a subcomplex. Now we have that D ∈ Hom•A (E • , L• ⊗A A2 ) is a cycle and so  2 ]. Define L2 on L• by  2 ∈ Hom• (E • , L• ⊗A A2 ) such that −D = [L0 , L there is L A    2 + 0 02 . (3.12) L2 = L 0 X Then





0 0 (3.13) [L , L ] = L , L + 0 X2 0

2

0

2







0 0 = −D+ L , L + 0 X2 0

2

So L0 L2 + L1 L1 + L2 L0 + rc = 0. We continue by setting   0 0 1 2 2 1 (3.14) D =L L +L L + . X3 e0 [X0 , X3 ]

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 = −L1 L1 −rc .

DUALITY AND EQUIVALENCE

325 327

Then D : L• → L• ⊗A A3 is A-linear, D|0⊕X • = 0 and [L0 , D] = L1 ◦ rc − rc ◦ L1 = 0 by the Bianchi identity d(c) = 0. Hence, by the same reasoning as above, there is  3 ]. Define  3 ∈ Hom• (E • , L• ⊗A A3 ) such that −D = [L0 , L L A    3 + 0 03 . (3.15) L3 = L 0 X 3 Then one can compute that i=0 Li L3−i = 0. Now suppose we have defined L0 , . . . , Ln satisfying for k = 0, 1, . . . , n k

Li Lk−i = 0

for k = 2

i=0

and 2

Li L2−i + rc = 0

for k = 2.

i=0

Then define (3.16)

D=

n

i=1

 LL i

n+1−i

+

0

Xn+1 e0

 0 . [X0 , Xn+1 ]

D|0⊕X • = 0 and we may continue the inductive construction of L to finally arrive at a Z-connection satisfying LL + rc = 0. The components of L construct both the Z-connection on E • as well as the morphism from (E • , E) to (X • , X).
4. Complex manifolds We justify our framework in this section by showing that, for a complex manifold, the derived category of sheaves on X with coherent cohomology is equivalent to the homotopy category PA for the Dolbeault algebra. Throughout this section ¯ 0) the Dollet X be a compact complex manifold and A = (A• , d, 0) = (A0,• (X), ∂, 0,• ¯ , ∂, 0). beault dga. This is the global sections of the sheaf of dgas (A•X , d, 0) = (AX Let OX denote the sheaf of holomorphic functions on X. Koszul and Malgrange have shown that a holomorphic vector bundle ξ on a complex manifold X is the ¯ i.e., an operator same thing as a C ∞ vector bundle with a flat ∂-connection, ∂¯ξ : Eξ → Eξ ⊗A A1 ¯ )φ + f ∂¯ξ (φ) for f ∈ A, φ ∈ Γ(X; ξ) and satisfying the such that ∂¯ξ (f φ) = ∂(f integrability condition that ∂¯ξ ◦ ∂¯ξ = 0. Here Eξ denotes the global C ∞ sections of ξ. The notion of a cohesive module over A clearly generalizes this notion but in fact will also include coherent analytic sheaves on X and even more generally, bounded complexes of OX -modules with coherent cohomology as well. For example, if (ξ • , δ) denotes a complex of holomorphic vector bundles, with ¯ ∂¯ξ : E i → E i ⊗A A1 then the corresponding global C ∞ -sections E • and ∂-operator ¯ ¯ holomorphic condition on δ is that δ ∂ξ = ∂ξ δ. Thus E = (E • , E), where E0 = δ and E1 = (−1)• ∂¯ξ defines the cohesive module corresponding to (ξ • , δ). So we see that, for coherent sheaves with locally free resolutions, there is nothing new here.

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4.1. The derived category of sheaves of OX -modules with coherent cohomology. Pali, [22] was the first to give a characterization of general coherent ¯ analytic sheaves in terms of sheaves over (A•X , d) equipped with flat ∂-connections. ¯ He defines a ∂-coherent analytic sheaf F to be a sheaf of modules over the sheaf of C ∞ -functions AX satisfying two conditions: Finiteness: locally on X, F has a finite resolution by finitely generated free modules, and ¯ Holomorphic: F is equipped with a ∂-connection, i.e., an operator (at the level of sheaves) ∂¯ : F → F ⊗AX A1X and satisfying ∂¯2 = 0. Theorem 4.1 (Pali, [22]). The category of coherent analytic sheaves on X is ¯ equivalent to the category of ∂-coherent sheaves. We prove our theorem independently of his proof. We use the following proposition of Illusie, [3]. Proposition 4.2. Suppose (X, AX ) is a ringed space, where X is compact and AX is a soft sheaf of rings. Then (1) The global sections functor Γ : Mod-AX → Mod-AX (X) is exact and establishes an equivalence of categories between the category of sheaves of right AX -modules and the category of right modules over the global sections AX (X). (2) If M ∈ Mod-AX locally has finite resolutions by finitely generated free AX -modules, then Γ(X; M ) has a finite resolution by finitely generated projectives. (3) The derived category of perfect complexes of sheaves Dperf (Mod-AX ) is equivalent to the derived category of perfect complexes of modules
Dperf Mod-AX (X) . Proof. See [3, Proposition 2.3.2, Expos´e II].




Our goal is to derive the following description of the bounded derived category of sheaves of OX -modules with coherent cohomology on a complex manifold. Note that this is equivalent to the category of perfect complexes, since we are on a smooth ¯ 0), the Dolbeault dga, is the manifold. Recall that A = (A• , d, 0) = (A0,• (X), ∂, 0,• ¯ • global sections of the sheaf of dgas (AX , d, 0) = (AX , ∂, 0) Theorem 4.3. Let X be a compact complex manifold and A = (A• , d, 0) = ¯ 0) the Dolbeault dga. Then the category Ho PA is equivalent to the (A (X), ∂, bounded derived category of complexes of sheaves of OX -modules with coherent cob homology Dcoh (X). 0,•

Remark 4.4. This theorem is stated only for X compact. This is because Proposition 4.2 is stated only for X compact. A version of Theorem 4.3 will be true once one is able to characterize the perfect AX -modules in terms of modules over the global sections for X which are not compact. A module M over A naturally localizes to a sheaf MX of AX -modules, where MX (U ) = M ⊗A AX (U )

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p,q For an object E = (E • , E) of PA , define the sheaves EX by p,q (U ) = E p ⊗A AqX (U ). EX p,q • , E) = ( p+q=• EX , E). This is a complex We define a complex of sheaves by (EX ¯ of soft sheaves of OX -modules, since E is a ∂-connection. The theorem above will be broken up into several lemmas. • Lemma 4.5. The complex EX has coherent cohomology and • , E) E = (E • , E) → α(E) = (EX b (X). defines a fully faithful functor α : Ho PA → Dperf (X)  Dcoh

Proof. Let U be a polydisc in X. We show that on a possibly smaller polydisc V , there is gauge transformation φ : E • |V → E • |V of degree zero such that ¯ Thus E • |V is gauge equivalent to a complex of holomorphic φ ◦ E ◦ φ−1 = F0 + ∂.
 ¯ vector bundles. Or, in other words, for each p the sheaf H p E •,0 , E0 is ∂-coherent, ¯ with ∂-connection E1 . Since U is Stein there is no higher cohomology (with respect to E1 ) and we are left with the holomorphic sections over U of each of these ¯ ∂-coherent sheaves, which are thus coherent. The construction of the gauge transformation follows the proof of the integrability theorem for complex structures on vector bundles, [9, Section 2.2.2, p. 50]. Thus we may assume we are in a polydisc U = {(z1 , . . . , zn ) | |z|i < ri }. In these coordinates we may write the Z-connection E as E = E0 + ∂¯ + J, where  J : E p,q (U ) → E i,q+(p−i)+1 (U ) i≤p 

is AX (U )-linear. Now write J = J ∧d¯ z1 +J  where ι∂/∂ z¯1 J  = ι∂/∂ z¯1 J  = 0. Write ¯ ∂¯i for d¯ zi ∧(∂/∂ z¯i ). As in [9, p. 51], we find a φ1 such that φ1 (∂¯1 +J  ∧d¯ z1 )φ−1 1 = ∂1 , −1 ¯  by solving φ1 ∂1 (φ1 ) = J ∧ d¯ z1 for φ1 , possibly having to shrink the polydisc. Here, we are treating the variables z2 , . . . , zn as parameters. Then we set E1 = φ1 (E0 + ∂¯ + J  + J  )φ−1 1 . Then E1 ◦ E1 = 0 and we can write E1 = E01 + ∂¯1 + ∂¯≥2 + J1 where ι∂/∂ z¯1 J1 = 0 and we can check that both E01 and J1 are holomorphic in z1 . For 0 = E1 ◦ E1 and therefore (4.1)

0 = ι∂/∂ z¯1 (E1 ◦ E1 ) = ι∂/∂ z¯1 (E01 ◦ ∂¯1 + ∂¯1 ◦ E10 + J1 ◦ ∂¯1 + ∂¯1 ◦ J1 )
 = ι∂/∂ z¯1 ∂¯1 (E01 ) + ∂¯1 (J1 )

Now each of the two summands in the last line must individually be zero since

 ι∂/∂ z¯1 ∂¯1 (E01 ) increases the p-degree by one and ι∂/∂ z¯1 ∂¯1 (J1 ) preserves or decreases the p-degree by one. So we have arrived at the following situation: (1) E01 ◦ E01 = 0, (2) E01 and J1 are holomorphic in z1 , and (3) ι∂/∂ z¯1 J1 = 0. z2 +J1 where ι∂/∂ z¯1 J1 = ι∂/∂ z¯2 J1 = We now iterate this procedure. Write J1 = J1 ∧d¯   ι∂/∂ z¯1 J1 = ι∂/∂ z¯2 J1 = 0. Now solve  ¯ z2 φ−1 2 ∂2 (φ2 ) = J1 ∧ d¯

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for φ2 . Since J1 is holomorphic in z1 and smooth in z2 , . . . , zn , so will φ2 . Then as before we have ¯ φ2 (∂¯2 + J1 ∧ d¯ z2 )φ−1 2 = ∂2 as well as ¯ φ2 (∂¯1 )φ−1 2 = ∂1 since φ2 is holomorphic in z1 . Setting E2 = φ2 ◦ E1 ◦ φ−1 2 , we see that E2 = E02 + ∂¯1 + ∂¯2 + ∂¯≥3 + J2 where ι∂/∂ z¯1 J2 = ι∂/∂ z¯2 J2 = 0 and we can check as before that both E02 and J2 are ¯ holomorphic in z1 and z2 . We continue until we arrive at F = En = E0n + ∂.
• , d) on X with Lemma 4.6. To any complex of sheaves of OX -modules (EX coherent cohomology there corresponds a cohesive A-module E = (E • , E), unique up to quasi-isomorphism in PA and a quasi-isomorphism

α(E) → (E • , d) This correspondence has the property that, for any two such complexes E1• and E2• , the corresponding twisted complexes (E1• , E1 ) and (E2• , E2 ) satisfy
 ExtkOX (E1• , E2• ) ∼ = H k PA (E1 , E2 ) Proof. Since we are on a manifold we may assume that (E • , d) is a perfect • • ¯ = E • ⊗OX AX . Now the map (E • , d) → (E∞ ⊗A A•X , d ⊗ 1 + 1 ⊗ ∂) complex. Set E∞ is a quasi-isomorphism of sheaves of OX -modules by the flatness of AX over OX . • , d) is a perfect comAgain, by the flatness of AX over OX , it follows that (E∞ plex of AX -modules. By Proposition 4.2, there is a (strictly) perfect complex • ), d). More(E • , E0 ) of A-modules and quasi-isomorphism e0 : (E • , E0 ) → (Γ(X, E∞ • ¯ over (Γ(X, E∞ ), d ⊗ 1 + 1 ⊗ ∂) defines a quasi-cohesive module over A. So the hypotheses of Theorem 3.13(2) are satisfied. The lemma is proved.
4.2. Gerbes on complex manifolds. The theorem above has an analogue for gerbes over compact manifolds. X is still a compact complex manifold. A × × ) defines an OX -gerbe on X. From the exponential sequence of class b ∈ H 2 (X, OX sheaves exp 2πı · × 0 → ZX → OX −−−−−→ OX →0 there is a long exact sequence × · · · → H 2 (X; OX ) → H 2 (X; OX ) → H 3 (X; ZX ) → · · ·

If b maps to 0 ∈ H 3 (X; Z) (that is, the gerbe is topologically trivializable) then b pulls back to a class represented by a (0, 2)-form B ∈ A0,2 (X). Consider the curved ¯ B) — the same Dolbeault algebra as before but dga A = (A• , d, B) = (A0,• (X), ∂, with a curvature. Then we have a theorem [4], corresponding to (4.3), Theorem 4.7. The category Ho PA is equivalent to the bounded derived category of complexes of sheaves on the gerbe b over X of OX -modules with coherent b (X)(1) . cohomology and weight one Dcoh Sheaves on a gerbe are often called twisted sheaves. One can deal with gerbes which are not necessarily topologically trivial, but the curved dga is slightly more complicated, [4].

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5. Examples 5.1. Elliptic curved dgas. In this section we define a class of curved dga’s A such that the corresponding dg-category PA is proper, that is, the cohomology of the hom sets are finite dimensional. It is often useful to equip a manifold with a Riemannian metric so that one can use Hilbert space methods. We introduce a relative of the notion spectral triple in the sense of Connes, [6], so that we can use Hilbert space methods to guarantee the properness of the dg-category. Again, our basic data is a curved dga A = (A• , d, c). Definition 5.1. We say that A is equipped with a Hilbert structure if there is a positive definite Hermitian inner product on A• ·, · : Ak × Ak → C satisfying the following conditions: Let H• be the completion of A• . (1) For a ∈ A• , the operator la (respectively ra ) of left (respectively, right) multiplication by a extends to H• as a bounded operator. Furthermore, the operators la∗ and ra∗ map A• ⊂ H• to itself. (2) A has an anti-linear involution ∗ : A → A such that for a ∈ A, there is (la )∗ = la∗ and (ra )∗ = ra∗ . (3) The differential d is required to be closable in H• . Its adjoint satisfies d∗ (A• ) ⊂ A• and the operator D = d + d∗ is essentially self-adjoint with core A• . (4) For a ∈ A• , [D, la ], [D, ra ], [D, la∗ ] and [D, ra∗ ] are bounded operators on H• . Definition 5.2. An elliptic curved dga A = (A• , d, c) is a curved dga with a Hilbert structure which in addition satisfies 2 (1) The operator e−tD is trace class for all t > 0.  (2) A• = n Dom(Dn ) The following proposition follows from very standard arguments. Proposition 5.3. Given an elliptic curved dga A then for E = (E • , E) and F = (F • , F) in PA one has that the cohomology of PA (E, F ) is finite dimensional. Bondal and Kapranov have given a very beautiful formulation of Serre duality purely in the derived category. We adapt their definitions to our situation of dgcategories. Definition 5.4. For a dg-category C, such that all Hom complexes have finite dimensional cohomology, a Serre functor is a dg-functor S: C → C which is a dg-equivalence and so that there are pairings of degree zero, functorial in both E and F ·, · : C • (E, F ) × C • (F, SE) → C[0] satisfying dφ, ψ + (−1)|φ| φ, dψ = 0 which are perfect on cohomology for any E and F in C. Motivated by the case of Lie algebroids below, we make the following definition, which will guarantee the existence of a Serre functor.

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Definition 5.5. Let A = (A• , d, c) be an  elliptic curved dga. A dualizing module (of dimension g) is a triple ((D, D), ¯∗, ) where (1) (D, D) is an A-A cohesive bimodule, (2) ¯ ∗ : Ak → D ⊗A Ag−k is a conjugate linear isomorphism and satisfies ¯ ∗(aω) = ¯∗(ω)a∗

and

¯∗(ωa) = a∗ ¯∗(ω)

for a ∈ A and ω ∈ A• .   (3) There is a C-linear map : D ⊗A Ag → C such that D(x) = 0 for all x ∈ D ⊗A A• and   |ω||x| ω · x = (−1) x·ω for all ω ∈ A• and x ∈ D ⊗A A• , and  ω, η = ¯∗(ω)η Proposition 5.6. Given an elliptic curved dga A = (A• , d, c) with a dualizing  module ((D, D), ¯ ∗, ), the category PA has a Serre functor given by the cohesive bimodule (D[g], D). That is, S(E • , E) = (E ⊗A D[g], E # D) is a dg-equivalence for which there are functorial pairings ·, · : PA• (E, F ) × PA• (F, SE) → C satisfying dφ, ψ + (−1)|φ| φ, dψ = 0 is perfect on cohomology for any E and F in PA . 5.2. Lie algebroids. Lie algebroids provide a natural source of dga’s and thus, by passing to their cohesive modules, interesting dg-categories. Let X be a C ∞ -manifold and let a be a complex Lie algebroid over X. Thus a is ∞ a C vector bundle on X with a bracket operation on Γ(X; a) making Γ(X; a) into a Lie algebra and such that the induced map into vector fields ρ : Γ(X; a) → V(X) is a Lie algebra homomorphism and for f ∈ C ∞ (X) and x, y ∈ Γ(X; a) we have [x, f y] = f [x, y] + (ρ(x)f )y. Let g be the rank of a and n for the dimension of X. There is a dga corresponding to any Lie algebroid a over X as follows. Let • A•a = Γ(X; a∨ ) denote the space of smooth a-differential forms. It has a differential d of degree one, with d = 0 given by the usual formula,


 (5.1) (dη)(x1 , . . . , xk ) = (−1)i+1 ρ(xi ) η(x1 , . . . , x ˆ i , . . . , xk ) i

+

(−1)i+j η([xi , xj ], . . . , x ˆi , . . . , x ˆj , . . . , xk ).

i 1 the class [˜ We illustrate this phenomenon in a concrete example. Example 2.5 (Bunke – Schick [6]). Let M be a principal 2-torus bundle over B with H = 0. Choose an identification t ∼ = R2 and a connection (θ1 , θ2 ) for M so that ci = dθi represent the Chern classes of M . Following the construction from Proposition 2.1, we see that the trivial 2-torus bundle B × S 1 × S 1 with 3-form

= c1 ∧ θ˜1 + c2 ∧ θ˜2 . H is T-dual to M . over B with Chern classes [c1 ] and −[c2 ] A less obvious T-dual is the bundle M (equivalent to changing H = 0 by the exact 3-form d(θ1 ∧ θ2 ) and then using the

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is T-dual to M by choosing a construction of Proposition 2.1). We can see that M 1 2 , R ) so that dθ˜ = (c1 , −c2 ) and choosing connection θ˜ ∈ Ω (M

= d(θ˜1 ∧ θ˜2 ) = c1 ∧ θ˜2 + c2 ∧ θ˜1 . H

we have For this H

= d(θ1 ) ∧ θ˜2 + (dθ2 ) ∧ θ˜1 p∗ (H) − p˜∗ (H) = d(θ1 ∧ θ˜2 + θ2 ∧ θ˜1 ) − (θ1 ∧ c2 − θ2 ∧ c1 ) = d(θ1 ∧ θ˜2 + θ2 ∧ θ˜1 + θ1 ∧ θ2 ). ) a T-dual pair. So, F = θ1 ∧ θ˜2 + θ2 ∧ θ˜1 + θ1 ∧ θ2 is unimodular and makes (M, M Having described the basic behaviour of the T-duality relation between principal torus bundles with 3-form flux, we proceed to the consequence of this relation which principally concerns us. , H)

Theorem 2.6 (Bouwknegt – Evslin – Mathai [4]). If (M, H) and (M ∗ ∗

are T-dual, with p H − p˜ H = dF , then the following map

• • (10) τ : ΩT k (M ) → ΩT k (M ) τ (ρ) = eF ∧ ρ, Tk

), d
), is an isomorphism of the differential complexes and (Ω•T k (M H →M . where the integration above is along the fibers of M ×B M (Ω•T k (M ), dH )

Sketch of the proof. The proof consists of two parts. The first, which is just linear algebra, consists of proving that τ is invertible. This boils down to the requirement that F satisfies (6). The second part is to show that τ is compatible with the differentials, which is a consequence of (5):

dH
(eF ∧ ρ) dH
τ (ρ) = Tk

∧ eF ∧ ρ + eF d ∧ ρ + H

∧ eF ∧ ρ = (H − H) Tk

= H ∧ eF ∧ ρ + eF ∧ dρ Tk

= τ (dH ρ).




3. T-duality as a map of Courant algebroids In this section we rephrase the T-duality relation as an isomorphism of Courant algebroids. This will allow us to transport any T k -invariant generalized geometrical , and vice versa. structures on M to similar structures on M The map τ introduced in Theorem 2.6 may be described as a composition of pullback, B-field transform and pushforward, all operations on the Clifford module of T k -invariant differential forms. To make τ into an isomorphism of Clifford modules, we need to specify an isomorphism of T k -invariant sections ϕ : (T M ⊕T ∗ M )/T k )/T k such that ⊕ T ∗M → (T M τ (v · ρ) = ϕ(v) · τ (ρ),

(11) ∗

for all v ∈ Γ(T M ⊕ T M )/T and ρ ∈ Ω•T k (M ). We now define such a map ϕ. k

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GENERALIZED COMPLEX GEOMETRY AND T-DUALITY

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, H),

consider the diagram Given T-dual spaces (M, H) and (M , p∗ H − p˜∗ H)

(M ×B M TTTTp˜ p jjj TTTT j j T) j u jjj , H)

(M, H) U (M UUUU ii i i UUUU i UUUU iiiiπ˜ π * tiiii B Given X + ξ ∈ (T M ⊕ T ∗ M )/T k , we can try and pull it back to the correspondence . While p∗ ξ is well defined, the same is not true about the vector space M ×B M ), any other lift differs from X  by  ∈ T (M ×B M part. If we pick a lift of X, X k a vector tangent to TM , the fiber of the projection p. Then we form the B-field  + p∗ ξ by F to obtain transform of X (12)

  + p∗ ξ − F (X). X

We want to define ϕ as the pushforward of the element above to an element of )/T k ; however, there are two problems. First, p˜∗ (X)  is not well defined, ⊕T ∗ M (T M  of X. Second, we can only pushforward the as it depends of the choice of lift X ∗  form p ξ − F (X) if it is basic, i.e., only if (13)

 Y ) = 0, ξ(Y ) − F (X,

∀Y ∈ tkM .

As is often said, two problems are better than one: the nondegeneracy of F on k k  TM × TM  means that there is only one lift X for which (13) holds and hence we define ϕ as the pushforward of (12) for that choice of lift of X:  + p∗ ξ − F (X).  ϕ(X + ξ) = p˜∗ (X) If one traces the steps above in the definition of ϕ, it is clear that it satisfies (11). Further, the compatibility of τ with the differentials translates to a compatibility , as we now explain. of ϕ with the differential structure of M and M , H)

be T-dual spaces. The map ϕ defined Theorem 3.1. Let (M, H) and (M above is an isomorphism of Courant algebroids, i.e., for v1 , v2 ∈ (T M ⊕ T ∗ M )/T k v1 , v2  = ϕ(v1 ), ϕ(v2 )

and

ϕ([v1 , v2 ]H ) = [ϕ(v1 ), ϕ(v2 )]H
.

Proof. Both of these properties follow from (11). To show that ϕ is orthogonal, let v ∈ T M ⊕ T ∗ M and ρ ∈ ΩT k (M ) and compute v, vτ (ρ) = τ (v, vρ) = τ (v · (v · ρ)) = ϕ(v) · (ϕ(v) · τ (ρ)) = ϕ(v), ϕ(v)τ (ρ). To prove compatibility with the brackets we use (1) together with Theorem 2.6: ϕ([v1 , v2 ]H ) · τ (ρ) = τ ([v1 , v2 ]H · ρ) = τ ([[dH , v1 ], v2 ] · ρ) = [[dH
, ϕ(v1 )], ϕ(v2 )] · τ (ρ) = [ϕ(v1 ), ϕ(v2 )]H
· τ (ρ).




Example 3.2. In this example we give a concrete expression for the map ϕ , H)

conintroduced above in the case of the T-dual S 1 -bundles (M, H) and (M structed in Example 2.2. Recall that M and M were endowed with connections θ ˜ and θ˜ for which F = −θ ∧ θ.

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The presence of the connections θ and θ˜ provides us with a splitting T M/S 1 ∼ = T B ⊕ ∂θ , where T B is the space of invariant horizontal vector fields and ∂θ is an invariant period-1 generator of the circle action. Similarly T ∗ M/S 1 ∼ = T ∗ B ⊕ θ ∗ 1 . So, an element in (T M ⊕ T M )/S can be written as and the same holds for M X + f ∂θ + ξ + gθ, is given by with X horizontal and ξ basic. The pullback of this element to M ×B M X + f ∂θ + k∂θ˜ + ξ + gθ, where k will be determined later. Then the B-field transform by F is X + f ∂θ + k∂θ˜ + ξ + gθ + f θ˜ − kθ. →M is equivalent to The requirement that ξ + gθ + f θ˜ − kθ is basic for M ×B M , yielding k = g, and ϕ is defined as the pushforward of this element to M (14)

˜ ϕ(X + f ∂θ + ξ + gθ) = X + g∂θ˜ + ξ + f θ,

In the final expression, we recognize the exchange of tangent and cotangent ‘directions’ described by physicists in the context of T-duality. be T-dual spaces as constructed in Proposition 2.1. Remark. Let M and M allows us to split the invariant tangent and The choice of connections for M and M cotangent bundles: (T M ⊕ T ∗ M )/T k ∼ = T B ⊕ T ∗ B ⊕ tk ⊕ tk∗ , )/T k ∼ ⊕ T ∗M (T M = T B ⊕ T ∗ B ⊕ tk ⊕ tk∗ . In this light, the map ϕ is the permutation of the terms tk and tk∗ . This is BenBassat’s starting point for the study of T-duality and generalized complex structures in [3], where he deals with the case of linear torus bundles with vanishing background 3-form H. 4. T-duality and generalized structures , H)

form a T-dual pair, then we obtain a Courant isomorIf (M, H) and (M . phism ϕ from Theorem 3.1 between the T k -invariant sections of T ⊕T ∗ of M and M k This immediately implies that any T -invariant generalized geometrical structure, since it is defined purely in terms of the structure of T ⊕ T ∗ , may be transported from one side of the T-duality to the other. What is interesting is that the resulting structure may have very different behaviour from the original one, stemming from the fact that tangent and cotangent directions have been exchanged in the T-duality. We explore this phenomenon in detail in this section. , H)

be T-dual spaces. Then any Dirac, Theorem 4.1. Let (M, H) and (M generalized complex, generalized K¨ ahler or SKT structure on M which is invariant . under the torus action is transformed via ϕ into a structure of the same kind on M Note also that the Courant isomorphism ϕ came together with a Clifford module isomorphism τ from Theorem 2.6; it follows similarly that any decomposition of forms induced by generalized geometrical structures is preserved by τ :

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GENERALIZED COMPLEX GEOMETRY AND T-DUALITY

353 355

, H)

be T-dual spaces, and let J and J be Corollary 4.2. Let (M, H) and (M a pair of T-dual generalized complex structures on these spaces, as obtained from k k and UM Theorem 4.1. Let UM  be the decompositions of forms induced by J and J . Then τ (U k ) = U k and also M

 M

τ (∂M ψ) = ∂M τ (ψ)

τ (∂¯M ψ) = ∂¯M τ (ψ),

˜ for ∂M , ∂M  the generalized Dolbeault operators associated to J , J . A special case of the above occurs when the generalized complex structure J has a holomorphically trivial canonical bundle, in the sense that it has a non-vanishing dH -closed section ρ ∈ Γ(K). This additional structure is called a generalized Calabi – Yau structure [19]. In this case, any T-dual generalized complex structure J˜ also has this property, with preferred generalized Calabi – Yau structure ρ˜ = τ (ρ). In the following example we see how the behaviour of generalized complex structures may change under T-duality. This change of geometrical type is at the heart of mirror symmetry, where complex and symplectic structures are exchanged between mirror Calabi-Yau manifolds. Example 4.3 (Change of type of generalized complex structures). Recall that the type of a generalized complex structure ρ = eB+iω ∧ Ω is the degree of the decomposable form Ω. Using this description, we determine how the types of T-dual generalized complex structures are related. If ρ = eB+iω ∧ Ω is a locally-defined invariant form defining a T k invariant generalized complex structure on M , then the correspond is determined by the pure spinor ing generalized complex structure on the T-dual M τ (ρ), where τ is the map defined in (10). Expanding the exponential, we see that the lowest degree form in τ (ρ) is determined by the smallest j for which

(F + B + iω)j ∧ Ω = 0 (15) Tk

and hence the type of the T-dual structure J is type(J ) = type(J ) + 2j − k, where j is the smallest natural number for which (15) holds. For example, when performing T-duality along a circle, j is either 1 or 0, depending on whether Ω is basic or not, respectively. Hence, the type of J is either type(J ) + 1 or type(J ) − 1, depending on whether Ω is basic or not. In order to obtain more concrete expressions for the relation between the type of T-dual structures, we make some assumptions about the algebraic form of one ) of the generalized complex structures involved and of the form F ∈ Ω2 (M ×B M which defines the T-duality. ˜ is the form If M 2n is a principal T n bundle with connection and F = −θ, θ used in Proposition 2.1, then we can determine the change in type for certain generalized complex structures under T-duality, assuming the fibers are n-tori with special geometric constraints: Example 4.4 (Hopf surfaces). Given two complex numbers a1 and a2 , with |a1 |, |a2 | > 1, the quotient of C2 by the action (z1 , z2 ) → (a1 z1 , a2 z2 ) is a primary Hopf surface (with the induced complex structure). Of all primary Hopf surfaces,

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Table 1. Change of type of generalized complex structures under T-duality according to the type of fiber F . Structure on M Fibers of M

l

Fibers of M r Structure on M

Complex Complex

Complex n/2 0 Complex Real (T F ∩ I(T F ) = {0}) n 0 Symplectic

Complex Lagrangian

Symplectic Symplectic

Symplectic Lagrangian

Symplectic Real

0 0

n Symplectic 0 Complex

these are the only ones admitting a T 2 action preserving the complex structure (see [2]). If a1 = a2 , the orbits of the 2-torus action are elliptic surfaces and hence, according to Example 4.3, the T-dual will still be a complex manifold. If a1 = a2 , then the orbits of the torus action are real except for the orbits passing through (1, 0) and (0, 1), which are elliptic. In this case, the T-dual will be generically symplectic except for the two special fibers corresponding to the elliptic curves, where there is type change. This example also shows that even if the initial structure on M has constant type, the same does not need to be true in the T-dual. Example 4.5 (Mirror symmetry of Betti numbers). Consider the case of the mirror of a Calabi – Yau manifold along a special Lagrangian fibration. We have k induced by both the complex and symplectic structure seen that the bundles Uω,I are preserved by T-duality. Hence U p,q = Uωp ∩ UIq is also preserved, but U p,q will be associated in the mirror to UI˜p ∩ Uω˜q , as complex and symplectic structures get swapped. Finally, as remarked in the previous section (equation (2)), we have an isomorphism between Ωp,q and U n−p−q,p−q . Making these identifications, we have ∼ U n−p−q,p−q (M ) = ∼ U n−p−q,p−q (M ∼ Ωn−p,q (M ) = ). Ωp,q (M ) = This, in cohomology, gives the usual ‘mirror symmetry’ of the Hodge diamond. Since the map ϕ from Theorem 3.1 is an orthogonal isomorphism, we may transport an invariant generalized metric G on (T M ⊕ T ∗ M )/T k to a generalized )/T k . If C± are the ±1-eigenspaces of G ⊕ T ∗M metric G = ϕ ◦ G ◦ ϕ−1 on (T M

± = ϕ(C± ) are the ±1-eigenspaces of G.

This is a complete description of then C the T-dual generalized metric from the point of view of T M ⊕ T ∗ M . However, we saw that a generalized metric may also be described in terms of a Riemannian metric g on M and a real 2-form b ∈ Ω2 (M, R), so that C+ is the graph of b + g. It is natural to ask how g and b change under T-duality. We shall describe the dual metric g˜ and 2-form ˜b in the case of principal circle bundles, as in Example 2.2. , H)

Example 4.6 (T-duality of generalized metrics). Let (M, H) and (M ˜ Let be T-dual principal circle bundles as in Example 2.2, with connections θ and θ. G be a generalized metric on M invariant with respect to the circle action, so that the induced metric g on T M and 2-form b are both S 1 -invariant and hence can be written as g = g0 θ  θ + g1  θ + g2 b = b1 ∧ θ + b2 ,

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where the gi and bi are basic forms of degree i. So, the elements of C+ , the +1eigenspace of G, are of the form
 X + f ∂θ + (iX g2 + f g1 + iX b2 − f b1 ) + g1 (X) + f g0 + b1 (X) θ. Applying ϕ, the generic element of ϕ(C+ ) = C˜+ is given by 
˜ X + g1 (X) + f g0 + b1 (X) ∂θ˜ + (iX g2 + f g1 + iX b2 − f b1 ) + f θ. This is the graph of ˜b + g˜ for ˜b, g˜ given by b1  b1 − g1  g1 1 ˜ ˜ b1 ˜ θθ−  θ + g2 + g0 g0 g0 ∧ b g g 1 ˜b = − 1 ∧ θ˜ + b2 + 1 g0 g0

g˜ = (16)

These equations are certainly not new: they were encountered by physicists in their computations of the dual Riemannian metric for T-dual sigma-models [8, 9] and carry the name Buscher rules. The metric along the S 1 fiber, g0 , is sent to g0−1 , considered the hallmark of a T-duality transformation. As we saw in Theorem 1.3, a generalized K¨ ahler structure on M induces a bi-Hermitian geometry (g, I± ). We wish to understand how this bi-Hermitian structure varies under T-duality. Again, we consider the simple case of T-dual S 1 -bundles. Example 4.7 (T-duality of bi-Hermitian structure). The choice of a generalized metric G gives us two orthogonal spaces C± = {X + b(X, ·) ± g(X, ·) : X ∈ T M }, and the projections π± : C± → T M are isomorphisms. Hence, any endomorphism A ∈ End(T M ) induces endomorphisms A± on C± . Using the map ϕ we can transport this structure to a T-dual: A+ ∈ End(C+ )

ϕ

π

/ A˜+ ∈ End(C˜+ ) π

 ). A˜± ∈ End(T M O

 A ∈ End(T M ) O π

π

A− ∈ End(C− )

ϕ

/ A˜− ∈ End(C˜− )

As we are using the generalized metric to transport A and the maps π± and ϕ are orthogonal, the properties shared by A and A± will be metric properties, e.g., self-adjointness, skew-adjointness and orthogonality. In the generalized K¨ahler case, it is clear that if we transport I± via C± , we obtain the corresponding bi-Hermitian structure of the induced generalized K¨ ahler structure on the T-dual: −1 −1 −1 I˜± = π ˜± ϕπ± I± (˜ π± ϕπ± ) .

In the case of a metric connection, θ = g(∂θ , ·)/g(∂θ , ∂θ ), we can give a concrete −1 description of I˜± . We start by describing the maps π ˜± ϕπ± . If V is orthogonal to

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∂θ , then g1 (V ) = 0 and −1 π ˜± ϕπ± (V ) = π ˜± ϕ(V + b1 (V )θ + b2 (V ) ± g2 (V, ·))   ∂ =π ˜± V + b1 (V ) + b2 (V ) ± g2 (V, ·) ∂ θ˜ ∂ = V + b1 (V ) , ∂ θ˜

and for ∂θ we have −1 (∂θ ) π ˜± ϕπ±



∂ + b1 ± =π ˜± ϕ ∂θ



1 θ + g1 g0



 =π ˜±

 1 ∂ 1 + θ˜ = ± ∂θ˜. ˜ g0 ∂ θ g0

If V± is the orthogonal complement to span{∂θ , I± ∂θ }, we may describe I˜± as follows: ⎧ ⎪ if w ∈ V± ⎨I± w, (17) I˜± w = ± g10 I± ∂θ if w = ∂θ˜ ⎪ ⎩ ∓g0 ∂θ˜ if w = I± ∂θ . Therefore, if we identify ∂θ with ∂θ˜ and their orthogonal complements with each other via T B, I˜+ is essentially the same as I+ , but stretched in the directions of ∂θ and I+ ∂θ by g0 , while I˜− is I− conjugated and stretched in those directions. In particular, I+ and I˜+ induce the same orientation while I˜− and I− induce opposite orientations. Example 4.8 (T-duality and the generalized K¨ ahler structure of Lie groups). Any compact semi-simple Lie group G, together with its Cartan 3-form H, admits a generalized K¨ ahler structure ([16, Example 6.39]). These structures are obtained using the bi-Hermitian point of view: any pair of left and right invariant complex structures on the Lie group Il and Ir , orthogonal with respect to the Killing form, satisfy the hypotheses of Theorem 1.3. Any generalized K¨ ahler structure obtained this way will be neither left nor right invariant since at any point it depends on both Il and Ir . However one can also show that Il and Ir can be chosen to be bi-invariant under the action of a maximal torus, and hence the corresponding generalized K¨ ahler structure will also have this invariance. In this case, according to Theorem 4.1 and Example 2.4, T-duality furnishes other generalized K¨ ahler structures on the Lie group. If we chose I+ = Ir and I− = Il , the computation above shows that T-duality will furnish a new structure on the Lie group coming from Ir and I˜l , where I˜l is still left invariant but induces the opposite orientation of Il . Of course we may also swap the roles of I± , changing the right invariant complex structure while the left invariant structure is fixed. Example 4.9. An SKT structure is normally defined as a Hermitian structure (g, I) on a manifold M for which ddc ω = 0. According to Theorem 4.1, if we endow M with the closed 3-form dc ω, then any T-dual to (M, H) obtains an SKT structure. ˜ is given by combining the Buscher rules for g˜ and The T-dual SKT structure (˜ g, I) the transformation law given in the previous example for the complex structure I− .

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5. Explicit examples In this section we study some instructive explicit examples of T-duality. Example 5.1 (The symplectic 2-sphere). Consider the standard circle action on the 2-sphere S 2 fixing the north and south poles N, S. we may view S 2 \{N, S} as a trivial circle bundle over the interval (−1, 1). Using coordinates (t, θ) ∈ (−1, 1) × (0, 2π), the round metric is given by ds2 = (1 − t2 ) dθ 2 +

1 dt2 . 1 − t2

Using the Buscher rules with b = 0, the T-dual metric (introducing a coordinate θ˜ along the T-dual fiber) is 1 1 dθ˜2 + dt2 . 1 − t2 1 − t2 Observe that the fixed points give rise to circles of infinite radius at a finite distance. This metric is not complete. Given an invariant symplectic structure on the sphere ω = w(t) dt∧ dθ, and any invariant B-field B = b(t) dt ∧ dθ, we T-dualize the generalized complex structure
 defined by the differential form eB+iω = 1+ b(t)+iw(t) dt∧ dθ. The dual structure is given by
 τ (eB+iω ) = dθ˜ + b(t) + iw(t) dt. d˜ s2 =

Note that this defines a complex structure, with complex coordinate ˜

˜ = eiθ− z(t, θ)

t −1

(w(t
)−ib(t
)) dt


Therefore the T-dual is biholomorphic to an annulus with interior radius 1 and exterior radius exp(− S 2 ω). While this is an accurate picture of what T-duality does to a (twice-punctured) symplectic sphere, from the physical point of view this is incomplete. In order for the symplectic sphere and the complex annulus to describe the same physics, one must deform the complex structure to C∗ and endow the resulting space with a complex-valued function called the superpotential, making it a Landau – Ginzburg model [20]. Example 5.2 (Odd 4-dimensional structures and the Gibbons – Hawking Ansatz). A generalized Calabi – Yau metric structure on (M, H) is a generalized K¨ ahler structure (J1 , J2 ) such that the canonical bundles of J1 and J2 both admit nowhere-vanishing dH -closed sections ρ1 , ρ2 , with the volume normalization (18)

(ρ1 , ρ1 ) = (ρ2 , ρ2 ).

This last condition is the generalization of the usual Monge – Amp`ere equation for Calabi – Yau metrics. The description of generalized Calabi – Yau geometry in real dimension 4 can be divided into two cases, according to whether the bi-Hermitian induced complex structures J± determine the same orientation or not. If they determine different orientations, the corresponding differential forms ρ1 and ρ2 are of odd degree and J± commute (see [16, Remark 6.14]). The real distributions S± = {v ∈ T M : J+ v = ±J− v} are integrable [1], yielding a pair of transverse foliations for M . If

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we choose holomorphic coordinates (z1 , z2 ) for J+ respecting this decomposition, then (z1 , z2 ) furnish holomorphic coordinates for J− . As the metric g is of type (1, 1) with respect to both J± , it is of the form g = g1¯1 dz1 dz1 + g2¯2 dz2 dz2 . The graphs of the i-eigenspaces of J± via b ± g coincide with the intersections L1 ∩ L2 and L1 ∩ L2 of the +i-eigenspaces L1 , L2 of J1 , J2 respectively. Hence we can recover L1 and L2 from J± , g and b. In this case, the differential forms annihilating L1 , L2 and hence generating the canonical bundles for J1 , J2 are ρ1 = eb+g22

dz2 ∧ dz2

∧ f1 dz1 ,

ρ2 = eb+g11

dz1 ∧ dz1

∧ f2 dz2 .

The generalized Calabi – Yau condition dρ1 = dρ2 = 0 implies that f1 = f1 (z1 ) is a holomorphic function of z1 and f2 = f2 (z2 ) a holomorphic function of z2 , and hence, with a holomorphic change of coordinates, we have ρ1 = eb+g2¯2

dz2 ∧ dz2

∧ dz1 ,

ρ2 = eb+g1¯1

dz1 ∧ dz1

∧ dz2 .

After rescaling ρ2 , if necessary, the compatibility condition (18) becomes g11 = g22 , showing that the metric is conformally flat. Call this conformal factor V . Finally, the integrability conditions, dρ1 = dρ2 = 0, impose db ∧ dzi = dV ∧ ∗ dzi = (∗ dV ) ∧ dzi ,

i = 1, 2,

where ∗ is the Euclidean Hodge star. Therefore, (19)

db = ∗ dV,

showing that the conformal factor V is harmonic with respect to the flat metric. Suppose the structure described above is realized on the 4-manifold S 1 × R3 in a way which is invariant by the obvious S 1 action, and remove a collection of points in R3 so as to allow poles of V . The invariance of V implies it is a harmonic function on R3 . Choosing the flat connection 1-form θ, we may write b = b1 ∧θ +b2 . Then, (19) implies that db1 = ∗3 dV and db2 = 0. According to the Buscher rules, the T-dual metric is given by 1 ˜b = b2 , g˜ = V ( dx21 + dx22 + dx23 ) + (θ˜ − b1 )2 ; V with db1 = ∗3 dV and b2 closed. This is nothing but a B-field transform of the hyper-K¨ ahler metric given by the Gibbons – Hawking ansatz. 6. Reinterpretations of T-duality In this section we show that it is possible to rephrase the definition of Tduality in two different ways: as a double quotient and as a submanifold. While the first point of view makes clear that the Courant algebroids (T M ⊕ T ∗ M )/T k )/T k are isomorphic, the second likens T-duality to a Fourier – ⊕ T ∗M and (T M Mukai transform, much in the spirit of [12]. 6.1. T-duality as a quotient. In this section we review the process of reduction of Courant algebroids from [7], and interpret T-duality in this light. We refer to [7] for more details on reduction. Given a manifold with closed 3-form (M, H), any section v ∈ Γ(T ⊕T ∗ ) defines a natural infinitesimal symmetry of the orthogonal and Courant structures on T ⊕T ∗ , via the Courant adjoint action adv (w) = [v, w]H .

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The fact that this is an infinitesimal symmetry is a consequence of the following properties of the Courant bracket: Lπ(v) w1 , w2  = [v, w1 ]H , w2  + w1 , [v, w2 ]H    [v, [w1 , w2 ]H ]H = [v, w1 ]H , w2 H + [w1 , [v, w2 ]H ]H . Using the decomposition T ⊕ T ∗ , we may write adX+ξ in block matrix form   LX 0
 Yη . (20) adX+ξ (Y + η) = dξ − iX H LX It is clear from this expression that adX+ξ consists of an infinitesimal symmetry of M (the Lie derivative LX ) together with a B-field transform. Definition 9. Let G be a connected Lie group acting on a manifold M and ψ : g → Γ(T M) be the corresponding Lie algebra map. A lift of the action of G to T M ⊕ T ∗ M is a bracket-preserving map Ψ : g → T M ⊕ T ∗ M such that the following diagram commutes: /g

Id

g

ψ

Ψ

 Γ(T M ⊕ T ∗ M)

πT

 / Γ(T M)

and the infinitesimal g action induced by Ψ(g) integrates to a G action on T M ⊕ T ∗ M. Under these circumstances, T M⊕T ∗ M is an equivariant G-bundle. According to (20), the action of Ψ(γ) = Xγ + ξγ preserves the splitting of T M ⊕ T ∗ M if and only if (21)

iXγ H = dξγ ,

in which case, according to (20), the g action integrates to the standard G action on T M and T ∗ M obtained by differentiation. In particular, if Ψ preserves the splitting T M ⊕ T ∗ M, it must be a lifted action, and hence (21) provides an effective way to check whether a bracket preserving map Ψ is a lifted action. Conversely, given a lift of an action of a compact Lie group, one can always average T M for that lifted action to obtain a new splitting of T M ⊕ T ∗ M which is preserved by the action [7]. So, for compact groups, the requirement that the splitting is preserved is not restrictive. Remark. As an aside, given a lifted action Ψ : g → Γ(T M ⊕ T ∗ M) for which (21) holds, define Ψ = X + ξ with X ∈ Γ(T M; g∗ ) and ξ ∈ Γ(T ∗ M; g∗ ). Then the combined form H + ξ is an equivariant 3-form in the Cartan complex, and (22)

dg (H + ξ) = Ψ(·), Ψ(·) ∈ Sym2 g∗ ,

showing that the pairing Ψ(g), Ψ(g) is constant over M. Hence, a lift of an action gives rise to a symmetric 2-form on g. If the G-action on M is free and proper and Ψ : g → Γ(T M ⊕ T ∗ M) is a lift of ψ, then the distribution K ⊂ T M ⊕ T ∗ M generated by Ψ(g) is actually a smooth subbundle. Furthermore, K is G-invariant, and its orthogonal complement (with

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360 362

G. R. CAVALCANTI AND M. GUALTIERI

respect to the natural pairing), K ⊥ , is also G-invariant. Then ΓG (K ⊥ ) is closed under the bracket and ΓG (K ∩ K ⊥ ) is an ideal in ΓG (K ⊥ ). Therefore  K⊥ Ered = G, K ∩ K⊥ as a bundle over M/G, inherits a bracket as well as a nondegenerate pairing. This is the main argument in the following theorem which is a particular case of [7, Theorem 3.3]. Theorem 6.1. Let G act freely and propertly on M, and let Ψ : g → T M ⊕ T ∗ M be a lift of the G-action satisfying (21). Letting K = Ψ(g), the distribution  K⊥ Ered = G K ∩ K⊥ is a bundle over M/G which inherits a bracket and a nondegenerate pairing, making it into a Courant algebroid over M/G. This Courant algebroid is exact1 if and only if K is isotropic. Definition 10. The bundle Ered defined over Mred = M/G, equipped with its induced bracket and pairing, is the reduced Courant algebroid and Mred is the reduced manifold. Observe that the condition that K is isotropic is equivalent to the requirement that the symmetric pairing (22) vanishes. This would imply that H + ξ defines an equivariantly closed extension of H. We now specialize to two examples of this quotient construction which will be relevant for T-duality. We distinguish them according to the symmetric form induced on the Lie algebra. Example 6.2 (Isotropic actions). As we have just seen, to obtain exact reduced Courant algebroids, one must require that the lifted action, Ψ = X + ξ, is isotropic, i.e. H + ξ must be equivariantly closed. In this example we study this setting in detail and provide an explicit isomorphism between the reduced algebroid and T M/G ⊕ T ∗ M/G. In order to fully describe the reduced algebroid over M/G, we choose a connection θ ∈ Ω1 (M; g) for M, viewed as a principal G bundle. The extended action is given by Ψ = X + ξ with X ∈ Γ(T M; g∗ ) and ξ ∈ Γ(T ∗ M; g∗ ). We then apply a B-field transform by B = θ, ξ ∈ Ω2 (M, R) to T M ⊕ T ∗ M, so that the generators of the action become X + ξ − iX θ, ξ = X + ξ − ξ = X and the 3-form curvature of T M ⊕ T ∗ M becomes the basic 3-form Hred = H − dθ, ξ. After this B-field transform, the lifted action lies in T M, and hence K ⊥ = T M + Ann(ψ(g)), and 
 K⊥ G∼ = (T M/ψ(g) ⊕ Ann ψ(g) G ∼ = T M/G ⊕ T ∗ M/G, K where M/G is endowed with the 3-form Hred defined above. 1An exact Courant algebroid over N is one which is locally isomorphic to T N ⊕ T ∗ N .

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Example 6.3. In this example, consider a lifted action of a Lie group G of dimension 2k, with Lie algebra G, for which the pairing induced on G is nondegenerate with split signature. As we will show shortly, this is the case which arises in T-duality. As before, let K = Ψ(G). The nondegeneracy of the pairing on K implies that K⊥ ∩ K = {0}, hence the reduced Courant algebroid is not exact. Also, it is given by the G-invariant sections of K⊥ :  K⊥ Ered = G = K⊥ /G. K ∩ K⊥ Since the natural pairing has split signature on K, we can choose a polarization

We show that of K, expressing it as a sum of two isotropic subspaces, K = K ⊕ K. such a decomposition provides us with alternative descriptions of Ered . Namely, it

hence, as vector bundles with a is clear that K ⊥ = K⊥ + K and similarly for K symmetric pairing, 

⊥  K⊥ K ∼ ∼ (23) Ered = G. G=

K K

to be Courant integrable, Note, however, that since we did not require K and K the resulting descriptions of Ered do not inherit natural Courant brackets.

k , and each We remedy this in the case where G is a product, G = Gk × G of g, ˜ g is isotropic in G = g ⊕ ˜g for the induced pairing. In this case, we are

presented with a natural choice of isotropic splitting K = Ψ(g) ⊕ Ψ(˜g) = K ⊕ K,

and since both K and K are closed with respect to the Courant bracket, the spaces

and (K

⊥ /K)/(G

inherit a Courant bracket which agrees (K ⊥ /K)/(G × G) × G) ⊥ with that on Ered = K . In this way, we obtain two alternative descriptions of Ered .

G×G

isotropic

⊥ /K)/ In light of Example 6.2, the description Ered ∼ with K = (K

G

is precisely the reduction of T M⊕

⊥ /K)/ is very suggestive. Indeed, the space (K

to M = M/G,

obtaining an exact Courant T ∗ M by the isotropic action of G

G

×G

⊥ /K)/ algebroid over that space. Hence the reduced algebroid Ered ∼ = (K ⊥

G)/G,

/K)/ which is isomorphic to corresponds to the quotient vector bundle ((K the G-invariant sections of T M ⊕ T ∗ M , viewed as a bundle over B = M/G.

we see that the same Courant algebroid can Reversing the roles of G and G,

⊥ /K)/G)/

which corresponds to be described as the quotient vector bundle ((K G ⊥ ∼ over

⊕ T ∗M

/K)/G the quotient bundle of the exact Courant algebroid (K = TM M = M/G, so we have the following diagram: T M ⊕ T ∗M RRR K RRR R) ulll ⊥

/K)/G

(K ⊥ /K)/G. (K QQQ m QQQQ mmm ( vmmm/G
/G

K⊥ /G × G
lll K l

(24)

Theorem 6.4. Let (M, H) be the total space of a principal T k × T k -torus bundle and let Ψ : tk × ˜tk → Γ(T M ⊕ T ∗ M) be a lift of the 2k-torus action for which the natural pairing on K = Ψ(tk × ˜tk ) is nondegenerate, of split signature, obtained by and such that Ψ(tk ) and Ψ(˜tk ) are isotropic. Then the spaces M, M

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reducing M by the action of T k and T k are T-dual. Conversely, any pair of T-dual spaces arises in this fashion. Proof. By the argument from Example 6.2, one can transform T M ⊕ T ∗ M by a B-field so that the lift of the action of T k lies in T M. From now on, we assume that the action is of this form and denote by H the 3-form curvature of T M ⊕ T ∗ M associated to this splitting. Therefore, it follows from Example 6.2 that the reduced space for the T k is M = M/T k and that H is the pull back of H, the 3-form on M . By the same argument, one can transform T M ⊕ T ∗ M by a B-field F so that the lift of the action of T k lies in T M. The reduction of M by this action gives = M/T k as a reduced space, with 3-form given by H

= H + dF . So, M is the M

= dF . correspondence space for M and M , and there we have H − H Finally, since the pairing on K is nondegenerate, the 2-form F defined above gives rise to a nondegenerate pairing between ψ(tk ) and ψ(˜tk ), proving that (M, H) , H)

are T-dual. and (M This procedure can be reversed to prove the converse. Namely, given T-dual , H),

let M be the correspondence space and define a lifted spaces (M, H) and (M T k × T k -action infinitesimally by Ψ(t, t˜) = Xt − iXt F + Xt˜, where Xt and Xt˜ are the vector fields associated to the Lie algebra elements t and t˜. The individual lifts of the actions of T k and T k are isotropic, but nondegeneracy of F means that the natural pairing restricted to K = Ψ(T k × T k ) is nondegenerate, with split signature.
Remark. From this point of view, Theorem 3.1 may be viewed as the observation that the Courant algebroid of T k -invariant sections of T M ⊕ T ∗ M is because they are ⊕ T ∗M isomorphic to the algebroid of T -invariant sections of T M by both isomorphic to the reduction of the correspondence space M = M ×B M k k

the full T × T action. The approach to T-duality via quotients has also been described by Hu in [21], where he drives the theory much further, using it to study nonabelian Poisson group duality, essentially using this theorem as a definition of the duality relation. There is also a strong similarity between this approach and the gauged sigma model approach in [25]. 6.2. T-duality as a generalized submanifold. In this section we rerinterpret the T-duality relation between generalized complex manifolds (M, H, J ) and , H,

J˜) as a special kind of submanifold in the product M × M . (M Given a manifold with 3-form (N , H), a generalized submanifold is a submanifold ι : M → N together with a 2-form F ∈ Ω2 (M) such that dF = ι∗ H. For any such generalized submanifold, one can form the generalized tangent bundle τF = {X + ξ ∈ T M ⊕ T ∗ N : ξ|M = F (X)}. If (N , H) is endowed with a generalized complex structure J , we say that (M, F ) is a generalized complex submanifold if τF is invariant under J . When N is a usual complex manifold, this condition specializes to the usual notion of

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complex submanifold, and when N is symplectic, Lagrangian submanifolds provide examples. , H)

be two principal k-torus bundles over Theorem 6.5. Let (M, H) and (M , H − H)

be their product and M = M ×B M a manifold B. Let (N , H) = (M × M their fiber product, so that we have an inclusion ι : M → N . Then M and M are Tdual if and only if there is a 2-form F ∈ Ω2 (M) making (M, F ) into a generalized submanifold and such that τF is everywhere transversal to τM = T M ⊕ T ∗ M ⊂ T N ⊕ T ∗ N , . and similarly for M Proof. The requirement that M is a generalized submanifold is nothing but the condition

= dF p∗ H − p˜∗ H from the definition of T-duality. Now consider τF ∩ τM = {X + ξ ∈ T M ∩ T M ⊕ T ∗ M : ξ|M = iX F }. has Let X +ξ be an element of the above intersection. Since the projection M → M k k and p˜∗ X = 0, we see that X must be tangent to TM . Therefore, kernel the fibers TM requiring that τF ∩τM = {0} is equivalent to requiring that F (X) ∈ Ω1 (M) is not a k k k , i.e., that F : TM ×TM pullback from M for each X ∈ TM  → R is nondegenerate.
As we now show, this viewpoint allows us to express the T-duality relation , J˜) as the existence of a genbetween generalized complex manifolds (M, J ), (M . In this way, the T-duality eralized complex submanifold in the product M × M relation for generalized complex structures may be viewed as a generalization of the Fourier – Mukai transform for complex manifolds. , H)

be a T-dual pair and let J , J be generTheorem 6.6. Let (M, H) and (M alized complex structures on these spaces. Endow the product (N , H) = , H − H)

with the generalized complex structure (J , cJ c−1 ), where (M × M → TM ⊕ T ∗M is given in matrix form by ⊕ T ∗M c: TM   1 0 . 0 −1 Then J and J are T-dual if and only if the correspondence space (M, F ) is a generalized complex submanifold of (N, H). Proof. Observe that the generalized tangent space of (M, F ) is given by )}. ⊕ T ∗M τF = {(v, cϕ(v)) ∈ (T M ⊕ T ∗ M ) ⊕ (T M Hence this space is invariant under (J , cJ˜c−1 ) if and only if cJ (v) = cϕJ ϕ−1 (v),

∀v ∈ T M ⊕ T ∗ M,

that is, if and only if J = ϕJ ϕ−1 . But this is precisely the T-duality relation for (J , J ).


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27. R. Zucchini, A sigma model field theoretic realization of Hitchin’s generalized complex geometry, J. High Energy Phys. 11 (2004), 045. , Generalized complex geometry, generalized branes and the Hitchin sigma model, J. 28. High Energy Phys. 3 (2005), 022. Department of Mathematics, Utrecht Univesity, 3584 CD Utrecht, The Netherlands E-mail address: [email protected] Department of Mathematics, University of Toronto, 40 St. George Street, Toronto, ON M5S 2E4, Canada E-mail address: [email protected]

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https://doi.org/10.1090/crmp/050/26

Centre de Recherches Math´ ematiques CRM Proceedings and Lecture Notes Volume 50, 2010

Topological Quantum Field Theories from Compact Lie Groups Daniel S. Freed, Michael J. Hopkins, Jacob Lurie, and Constantin Teleman

Let G be a compact Lie group and BG a classifying space for G. Then a class in H n+1 (BG; Z) leads to an n-dimensional topological quantum field theory (TQFT), at least for n = 1, 2, 3. The theory for n = 1 is trivial, but we include it for completeness. The theory for n = 2 has some infinities if G is not a finite group; it is a topological limit of 2-dimensional Yang – Mills theory. The most direct analog for n = 3 is an L2 version of the topological quantum field theory based on the classical Chern – Simons invariant, which is only partially defined. The TQFT constructed by Witten and Reshetikhin – Turaev which goes by the name ‘Chern – Simons theory’ (sometimes ‘holomorphic Chern – Simons theory’ to distinguish it from the L2 theory) is completely finite. The theories we construct here are extended, or multi-tiered, TQFTs which go all the way down to points. For the n = 3 Chern – Simons theory, which we term a ‘0-1-2-3 theory’ to emphasize the extension down to points, we only treat the cases where G is finite or G is a torus, the latter being one of the main novelties in this paper. In other words, for toral theories we provide an answer to the longstanding question: What does Chern – Simons theory attach to a point? The answer is a bit subtle as Chern – Simons is an anomalous field theory of oriented manifolds.1 This framing anomaly was already flagged in Witten’s seminal paper [30]. Here we interpret the anomaly as an invertible 4-dimensional topological field theory A , defined on oriented manifolds. The Chern – Simons theory is a “truncated morphism” Z : 1 → A from the trivial theory to the anomaly theory. For example, on a closed oriented 3-manifold X the anomaly theory produces a complex line A (X) and the Chern – Simons invariant Z(X) is a (possibly zero) element of that line. This is the standard vision of an anomalous quantum field theory in general; 2000 Mathematics Subject Classification. Primary 57R56; Secondary 18D05, 18D10. The work of D. S. F. is supported by NSF grant DMS-0603964. The work of M. J. H. is supported by NSF grants DMS-0306519 and DMS-0757293 as well as the DARPA grant FA9550-07-1-0555. The work of J. L. is supported by the American Institute of Mathematics. The work of C. T. is supported by NSF grant DMS-0709448. DSF, MJH, and CT would like to acknowledge the Hausdorff Research Institute for Mathematics for its hospitality while some of this work was carried out. This is the final form of the paper. 1It is not anomalous as a theory of framed manifolds. c
2010 c
2010 American American Mathematical Mathematical Society

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D. S. FREED, M. J. HOPKINS, J. LURIE, AND C. TELEMAN

here we use this description down to points. The invariant of a 4-manifold in the theory A involves its signature and Euler characteristic. It was first discovered in a combinatorial description [10], and Walker [29] also uses A in his description of Chern – Simons (for a more general class of gauge groups). Since a torus is an abelian group, the classical Chern – Simons action is quadratic in the connection and so the theory is in some sense “free”. Indeed, one expects that the semi-classical approximation is exact. This is the point of view taken by Manoliu [22], who constructs Chern – Simons for circle groups as a 2-3 theory. The invariant this theory assigns to a closed oriented 3-manifold is made from classical invariants of 3-manifold topology: it is the integral over the space of flat connections of the square root of the Reidemeister torsion times a spectral flow phase. (There is an overall “volume” factor as well.) The extension to a 1-2-3 theory is determined by a modular tensor category by a theorem of Reshetikhin and Turaev [28]; it is assigned to the circle by the theory. For toral Chern – Simons this is a well-known category, and its relation to Chern – Simons theory was explored recently by Stirling [27], inspired by earlier work by Belov – Moore [6]. In a different direction the Verlinde ring, which encodes the 2-dimensional reduction of Chern – Simons theory, was recognized by three of the authors as a twisted form of equivariant K-cohomology [15, 16]. That description, or rather its equivalent in K-homology, inspires the categories of skyscraper sheaves which we use to extend toral Chern – Simons to a 0-1-2-3 theory. The gauge theories discussed in this paper have a classical description as pure gauge theories whose only field is a G-bundle with connection. On any fixed manifold M the G-bundles with connection form a groupoid, and it is the underlying stack which should be regarded as the fields in the theory. When M is a point, there is up to isomorphism a unique G-bundle with connection (the trivial one), but it has a group of automorphisms isomorphic to G. Thus the stack of G-bundles with connections is ∗
G = BG. A cohomology class in H n+1 (BG; Z), or rather the geometric manifestation which we describe (esp. §§4.1 and 5.1), defines the classical gauge theory on M  pt. We directly “quantize” this classical data over a point. This may be regarded, as in [14], as the “path integral” over the fields on a point. Integration over I
G, which is implemented as a categorical limit, amounts to taking G-invariants, which is a higher version of the Gauss Law in canonical quantization. It is also worth noting that the space of connections on M = S 1 is the finite-dimensional groupoid G
G, where G acts on G by conjugation. Properly interpreted, this is the loop space of the fields on a point. On the other hand, for any manifold of dimension at least two, the stack of connections is infinite dimensional unless G is finite. In the latter case — for a finite group G — the path integral in all dimensions reduces to a finite sum, so is manifestly well-defined and satisfies the axioms of a field theory. This idea was initiated in [14], where the notion of a classical action in an n-dimensional field theory is extended to manifolds of all dimensions < n and a higher categorical version of the path integral is used to heuristically define the quantum invariants.2 The developments in higher category theory in the intervening years, and the new definitions and structure theory for TQFT, make it possible to give a rigorous treatment of these finite path integrals and to generalize 2The application in [14] is to derive the quantum group which appears in 3-dimensional Chern – Simons theory from the classical Chern – Simons action.

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them. We give some indications in §3 and §8; we hope to develop these ideas in detail elsewhere. That structure theorem is the Baez – Dolan cobordism hypothesis, for which we include an exposition in §2.3. A much more detailed account may be found in [21], which also contains a detailed sketch of the proof. This theorem asserts that a fully extended TQFT — that is, a TQFT extended down to points — is determined by its value on a point. Furthermore, it characterizes the possible values on a point. This theorem provides one possible construction of many of the theories in this paper.3 The anomaly theory A which appears in Chern – Simons theory has a classical description as a 4-dimensional field theory. In §5.2 we sketch two such classical theories, each based on a 2-groupoid of gerbes (rather than a 1-groupoid of connections, which is what appears in gauge theory). These are free theories, so the path integral on a 4-manifold is Gaussian and can be carried out explicitly. Presumably this is a classical description of the theories put forward by Crane and Yetter [10], though we do not attempt to make the connection. One of these classical theories is based on a finite group, and for that we can apply the finite path integrals of §3 (in the generalized form for 2-groupoids) to construct the quantum theory. In §6 we start at the other end of the theory, that is, with points rather than 4-manifolds. We describe two braided tensor categories which may be attached to a point in the anomaly theory A ; they are Morita equivalent, so both define A . In §6.2 we introduce a finite (lattice) approximation of the continuous version of the anomaly theory, and so obtain many more Morita equivalent braided tensor categories which define that theory (Proposition 6.5). Finally, in §9.1 we spell out what the anomaly theory assigns to manifolds of various dimensions from 0 to 4. The Chern – Simons theory proper is described in §9.2; see Theorem 9.3. The tables in §9.3 are helpful in organizing the motley characters who play a part in our story. Our account here emphasizes 0-manifolds as that is the new element. We begin in §1 with a discussion of the 1-dimensional case to set down some basic notions. Here the gauge group is an arbitrary compact Lie group G, and the path integral reduces to an integral over G. The 2-dimensional case ( §2) exhibits more features. In §4 we discuss the 3-dimensional gauge theory with finite gauge group. On the one hand this is an application of the finite path integral and on the other a warm-up for the treatment in later sections of torus groups. Section 7 is a set of variations on ‘algebra’ of increasing (categorical) complexity. It provides some underpinnings for our discussion of the anomaly theory A and Chern – Simons theory. Section 8 places the discussion of our finite gauge theories in the general framework of higher finite groupoids: we give an algebraic model for quantizing finite homotopy types, leading to TQFTs in dimensions 0 through n, for any n. In the top two dimensions, these TQFTs may have been first considered by Kontsevich [18, Section 7]; see also [19] for a recent application. As befits a conference proceedings, our presentation here emphasizes the big picture and our speculation is uninhibited; many details are not worked out. The recent paper [7] generalizes the 3-dimensional untwisted Chern – Simons theory for finite groups in a different direction and gives applications to representation theory. Also, Bartels, Douglas, and Henriques have recently announced 3A corollary of the cobordism hypothesis characterizes theories on oriented and spin manifolds: for these the value on a point is endowed with certain equivariance data. For the 3- and 4-dimensional theories of most interest we do not pursue this equivariance data here.

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a construction of Chern – Simons theory for 1-connected compact Lie groups [11] which uses conformal nets and the cobordism hypothesis. Raoul Bott was an inspiration to each of us, both personally and through his mathematics. We offer this paper as a tribute. 1. The cohomology group H 2 (BG; Z) and one-dimensional theories To begin, we recall the definition of a topological quantum field theory. Let (BordSO n , ) be the bordism category whose morphisms are oriented n-manifolds; it carries a symmetric monoidal structure given by disjoint union. Let (VectC , ⊗) denote the symmetric monoidal category of finite-dimensional complex vector spaces under tensor product. Definition 1.1. An n-dimensional topological quantum field theory F on oriented manifolds is a symmetric monoidal functor F : (BordSO n , ) → (VectC , ⊗). Thus a field theory assigns to each closed oriented (n−1)-manifold Y a complex vector space F (Y ) and to each closed oriented n-manifold X a number F (X) ∈ C. If X : Y0 → Y1 is a compact manifold with boundary4 −Y0  Y1 , then F (X) : F (Y0 ) → F (Y1 ) is a linear map. The functor F maps the gluing operation of bordisms to the composition of linear maps. We refer the reader to [1, 21, 26] for more details and exposition about this definition. There are many possible variations on Definition 1.1. For the domain we may take a bordism category of manifolds with other topological structures: oriented manifolds, spin manifolds, framed manifolds, etc. There are also other choices for the codomain. This definition describes “two-tiered theories”; we will see more tiers in §2. If, say, n = 3, then we designate this a ‘2-3 theory’ to emphasize the two tiers. Let n = 1. Then a theory F assigns a vector space F (pt+ ) to the positively oriented point. We use the oriented interval with all possibilities for decomposing the boundary into incoming and outgoing components to deduce that F (pt− ) is the dual space of F (pt+ ) and the intervals give the duality pairing. Finally, F (S 1 ) = dim F (pt), and the entire theory is determined up to isomorphism by this nonnegative integer. We now describe 1-dimensional pure gauge theory with compact gauge group G and action given by a class in H 2 (BG; Z). Let T ⊂ C be the circle group. ∼ =

Proposition 1.2. There is an canonical isomorphism Hom(G,T) − → H 2 (BG; Z). In purely topological terms the classifying space BG carries a universal principal G-bundle, and the isomorphism assigns to an abelian character G → T the Chern class of the associated principal T-bundle. A more rigid viewpoint: abelian characters are in 1:1 correspondence with isomorphism classes of principal T-bundles on the groupoid ∗
G. A proof that the latter are classified by H 2 (BG; Z) may be found in several papers, e.g., [2, Proposition 6.3; 9, Appendix]. Let λ : G → T be an abelian character. Then the classical 1-dimensional gauge theory associated to λ assigns to each G-bundle with connection on the circle the complex number λ(g), where g is the holonomy. This is what physicists term the 4Here ‘−Y ’ denotes the oppositely oriented manifold. 0

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‘exponentiated action’ of the theory. Notice that holonomy depends on an orientation on the circle, so this is a theory of oriented 0- and 1-manifolds. To the unique connection on a point is attached the trivial complex line C; the automorphism group G of the unique connection acts on C via λ. More precisely, this is correct if the point is positively oriented; for the negatively oriented point the action is via λ−1 . Now the standard quantization procedure constructs a theory F with F (pt+ ) the elements in C invariant under the action of the automorphisms. (This is called the ‘Gauss law’ in physics.) The result is the zero vector space if λ = 1 is nontrivial and is C if λ = 1 is trivial. Then F (S 1 ) = 0 in the first case and F (S 1 ) = 1 in the second, as F (S 1 ) = dim F (pt+ ). These values may be understood as the result of the path integral over the groupoid G
G of connections on S 1 with respect to Haar measure: 
0, λ = 1; 1 λ(g) dg = vol G G 1, λ = 1. Remark 1.3. There are cohomology theories h with the property that h2 (BG) is isomorphic to the set of Z/2Z-graded abelian characters (λ, ), i.e., λ : G → T is an abelian character and  = 0, 1. For example, we can take cohomology theory h determined by a spectrum with two nontrivial homotopy groups: Z in degree 0 and Z/2Z in degree −2 connected by the nontrivial k-invariant. A less efficient alternative is the Anderson dual of the sphere [17, Definition B.2]. In either case spin structures are required to define pushforward, so the corresponding field theory is defined on spin manifolds. It is then natural to let the values of the theory lie in Z/2Z-graded complex vector spaces. If  = 0, the theory factors through oriented manifolds and reduces to the previous one. If  = 1, then the classical theory assigns the odd line C to the unique connection on a point. For λ = 1 we obtain the trivial theory as above, but now if λ = 1, then F (Sn1 ) = −1 and F (Sb1 ) = 1, where the subscript indicates whether we consider the circle as endowed with the nonbounding (n) or bounding (b) spin structure. There are similar generalizations of the higher-dimensional theories discussed below. Remark 1.4. All of the classical theories alluded to in this paper are purely topological and are most systematically defined by refining the class in hn+1 (BG) to an object in the corresponding differential theory and then transgressing; see [13] for the case n = 3. The quantum theories, which are our focus, do not depend on the differential refinement. 2. The cohomology group H 3 (BG) and two-dimensional theories 2.1. H 3 (BG; Z) for arbitrary G. Analogous to Proposition 1.2 we have the following. Proposition 2.1. Let G be a compact Lie group. The cohomology group H 3 (BG; Z) can be identified with the set of isomorphism classes of central extensions T → Gτ → G. In purely topological terms the class in H 3 (BG; Z) attached to a central extension T → Gτ → G is the obstruction to lifting the universal G-bundle to a

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Gτ -bundle.5 A central extension is equivalently a smooth Hermitian line bundle K = K τ → G together with isomorphisms (2.1)

θx,y : Kx ⊗ Ky → Kxy ,

x, y ∈ G,

for each pair in G, which satisfy an associativity constraint (2.2)

θxy,z (θx,y ⊗ idKz ) = θx,yz (idKx ⊗ θy,z ),

for each triple in G. Recall the simplicial model oo o ∗ oo (2.3) G oo G2 oo G3 oo

oo

x, y, z ∈ G,

o

G4 · · ·

for BG, which is quite useful for computing cohomology [8]. Write Sp = Gp for the pth space in (2.3). In these terms a central extension is given by a line bundle K τ → S1 , a trivialization (2.1) of the alternating tensor product of its pullback to S2 , the latter constrained so that the alternating product of its pullbacks to S3 is trivial (2.2). A proof of Proposition 2.1 may be found in [2, Proposition 6.3; 9, Appendix; 16, Part 1, §2.2.1]. Recall that the complex group algebra C[G] of G is the abelian group of complex functions on G with multiplication given by convolution with respect to Haar measure, that is, pushforward under multiplication m : G × G → G. If dim G > 0 then we take the continuous functions on G, which form a topological ring using the sup norm. If dim G = 0 (G is finite) then the center of C[G] is the commutative algebra with natural basis the δ-functions on the conjugacy classes. Alternatively, it is the subalgebra of central functions, a description which persists for all G. Associated to a central extension T → Gτ → G is a twisted complex group algebra Cτ [G], the algebra of sections of K τ → G under convolution. Its center is again the commutative algebra of central functions. 2.2. Gauge theory with finite gauge group. In an n-dimensional gauge theory with finite gauge group G the path integral over a closed n-manifold X reduces to a finite sum, and this sum defines the quantum invariant F (X). The starting point of [14] is an expansion of this standard picture: the classical action can be extended to manifolds of dimension < n and a higher categorical version of the path integral determines the quantum invariants. For any manifold M of dimension ≤ n, the groupoid of G-bundles may be identified with the fundamental groupoid of the mapping space Map(M, BG). One can obtain a larger class of theories by replacing BG with a topological space which has a finite number of nonzero homotopy groups, each of which is a finite group [25]. At least heuristically, the path integrals on manifolds of dimension ≤ n define an extended field theory down to points which is manifestly local and functorial. We write these finite sums explicitly in this section. In §3 we indicate one possible path which should put these ideas on a rigorous footing. For the 2-dimensional finite gauge theory based on the central extension T → Gτ → G the classical invariant attached to a G-bundle P → X over a closed oriented 2-manifold is I(X, P ) = e2πiσX (P ) , 5One familiar case is T → Spinc → SO in which case the obstruction is the third Stiefel – n n Whitney class W3 .

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where σ ∈ H 2 (BG; Q/Z) ∼ = H 3 (BG; Z) is the characteristic class corresponding to the central extension (see Proposition 2.1) and σX (P ) ∈ Q/Z is the characteristic number of the bundle. Physicists call I(X, P ) the ‘exponentiated action’ of the field P on X. The quantum invariant of X is then (2.4)

F (X) =

 P

1 I(X, P ), # Aut P

where the sum is over a set of representative G-bundles on X, one in each equivalence class. For the trivial central extension I(X, P ) = 1 for all P and (2.4) counts the number of representations of π1 X into the finite group G. (There is an overall factor of 1/ # G.) In that case we can extend to a theory of unoriented manifolds. Now consider this 2-dimensional theory over the circle S 1 . Recall from the introduction that the groupoid of G-bundles over S 1 is equivalent to G
G, where G acts on itself by conjugation. The value of the classical theory I is given by the central extension Gτ → G, viewed as an equivariant principal G-bundle. In other words, the value I(S 1 , P ) of this theory at a bundle with holonomy x ∈ G is the circle torsor Gτx (together with the action of the centralizer of x, the automorphisms of P , by conjugation in the central extension.) In the quantum theory we make a sum of the corresponding Hermitian lines KP analogous to (2.4) and so compute that F (S 1 ) is the vector space of central sections of K τ → G: (2.5)

F (S 1 ) =

 P

1 (KP )Aut P # Aut P

In this expression the metric on the Hermitian line Kx is scaled by the prefactor. The quantum “path integrals” (2.4) and (2.5) may be expressed in categorical language. Let BordSO 2 (G) denote the bordism category whose objects are closed oriented 1-manifolds equipped with a principal G-bundle and whose morphisms are compact oriented 2-manifolds equipped with a principal G-bundle. Then the classical topological theory I is a 2-dimensional topological quantum field theory in the sense of Definition 1.1 with a special feature: its values are invertible. Thus the numbers attached to closed oriented surfaces with G-bundle are nonzero and the vector spaces attached to G-bundles over the circle are lines (which are invertible in the collection of vector spaces under tensor product). Now there is an obvious SO which omits the G-bundle. Then the forgetful functor π : BordSO 2 (G) → Bord2 quantum field theory F , obtained by summing over G-bundles, can be viewed as a kind of pushforward of I along π. The relevant pushforward procedure has an analog in classical topology. Let π : E → S be a proper fiber bundle of topological manifolds whose fibers carry a suitable n-dimensional orientation. Then there is a map π∗ : h• (E) → h•−n (S) variously termed the ‘pushforward’ or ‘direct image’ or ‘wrong-way map’ or ‘umkehr map’ or ‘Gysin map’ or ‘transfer’. The analogy with our situation is tighter if we think of this pushforward on the level of cochains, or geometric representatives, rather than cohomology classes. Formulas (2.4) and (2.5), together with a similar formula for compact surfaces with boundary, can be used to define the functor F . This constructs a 1-2 theory. We would like, however, to continue down to points, i.e., to a 0-1-2 theory. To express the higher gluing laws encoded in an n-dimensional TQFT which extends down to points we use the language of higher category theory; see [21] for

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an introduction and for much more exposition about the n-categories we now in6 troduce. Let BordSO n denote the n-category whose objects are finite unions of oriented points, 1-morphisms are oriented bordisms of objects, 2-morphisms are oriented bordisms of 1-morphisms, and so forth.7 Note BordSO n carries a symmetric monoidal structure given by disjoint union. Let C be any symmetric monoidal n-category. Definition 2.2. An extended n-dimensional TQFT with values in C is a symmetric monoidal functor F : BordSO n → C. The 0-1-2 finite gauge theory can be constructed by hand. 2.3. 0-1-2 theories for general G. Let F be a fully extended topological field theory of dimension n. We can regard F as a prescription for assigning invariants to manifolds with corners of all dimensions ≤ n, together with set of rules for how these invariants behave when we glue manifolds together. Since every smooth manifold M can be assembled by gluing together very simple pieces (for example, by choosing a triangulation of M ), we might imagine that the value of F on arbitrary manifolds is determined by its values on a very small class of manifolds. In order to formulate this idea more precisely, we need to introduce a bit of terminology. Definition 2.3. Let M be a manifold of dimension m ≤ n. An n-framing of M is a trivialization of the vector bundle TM ⊕ Rn−m , where Rn−m denotes the trivial bundle (on M ) of rank n − m. We let Bordfr n denote the bordism n-category whose k-morphisms are given by n-framed k-manifolds for k ≤ n. Theorem 2.4 (Baez – Dolan Cobordism Hypothesis). Let C be a symmetric monoidal n-category. Then the construction F → F (∗) induces an injection from the collection of isomorphism classes of symmetric monoidal functors F : Bordfr n → C to the collection of isomorphism classes of objects of C. A version of Theorem 2.4 was originally conjectured by Baez and Dolan in [3]. We refer the reader to [21] for a more extensive discussion and a sketch of the proof. Let C be a symmetric monoidal n-category. We will say that an object C ∈ C is fully dualizable if there exists an extended TQFT F : Bordfr n → C and an isomorphism C  F (∗). Theorem 2.4 asserts that if C is a fully dualizable object of C, then the field theory F is uniquely determined by C. Remark 2.5. It is possible to state a more precise version of Theorem 2.4 by describing the class of fully dualizable objects C ∈ C without mentioning the n-category Bordfr n . If n = 1, this condition is easy to state: the object C should admit a dual C ∨ , so that there exist evaluation and coevaluation maps C ⊗ C∨ − →1 e

1− → C∨ ⊗ C c

6BordSO should be regarded as an (∞, n)-category. This means roughly that we consider n families of manifolds parametrized by a topological space rather than simply single manifolds. 7 An (∞, n)-category has r-morphisms for all r; they are invertible for r > n. The r-morphisms in BordSO n for r > n are given by (r − n − 1)-parameter families of diffeomorphisms of manifolds which preserve all boundaries (and corners).

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(here 1 denotes the unit object of C) which are compatible in the sense that the compositions C → C ⊗ C∨ ⊗ C → C C∨ → C∨ ⊗ C ⊗ C∨ → C∨ both coincide with the identity. For n > 1, we need to assume that analogous finiteness assumptions are satisfied not only by the object C, but by the 1-morphisms e and c. We refer the reader to [21] for a more complete discussion. Theorem 2.4 has a curious consequence: since the orthogonal group O(n) acts on the n-category Bordfr n (by change of framing), we get an induced action of O(n) on the collection of fully dualizable objects of any symmetric monoidal n-category C (more precisely, the orthogonal group O(n) can be made to act on the classifying space for the underlying n-groupoid of the fully dualizable objects in C). When n = 1, this action is simply given by the involution that takes a dualizable object C ∈ C to its dual C ∨ . We can use the action of the orthogonal group to formulate an analog of Theorem 2.4 for more general types of manifolds: Theorem 2.6. Let C be a symmetric monoidal n-category. The construction F → F (∗) establishes a bijection between the set of isomorphism classes of symmetric monoidal functors BordSO → C with the set of isomorphism classes of n (homotopy) fixed points for the action of the group SO(n) on the fully dualizable objects of C. Remark 2.7. Theorem 2.6 has an obvious analog for other types of manifolds: unoriented manifolds, spin manifolds, and so forth. Example 2.8. Let n = 2, and let C denote the 2-category Alg of (complex) algebras, bimodules, and intertwiners. Every object of C admits a dual: the dual of an algebra A is the opposite algebra Aop , where both the evaluation and coevaluation maps are given by A (regarded as a (C, A ⊗ Aop )-bimodule. An algebra A is fully dualizable if and only if A is dualizable both as a C-module and as an A ⊗ Aop -module. The first condition amounts to the requirement that A be finite dimensional over C, while the second condition requires that the algebra A be semisimple. The circle group SO(2) acts on the classifying space of the 2-groupoid of fully dualizable objects of Alg. In more concrete terms, this means that every fully dualizable object A ∈ Alg determines a functor from the fundamental 2-groupoid of SO(2) into Alg which carries the identity element of SO(2) to A. Applying this to a generator of the fundamental group π1 SO(2), we get an automorphism of A in Alg: this automorphism is given by the vector space dual A∨ , regarded as an (A, A)-bimodule. To realize A as a fixed point for the action of SO(2), we need to choose an identification of A with A∨ as (A, A)-bimodules. In other words, we need to choose a nondegenerate bilinear form b : A ⊗ A → C which satisfies the relations b(xa, a ) = b(a, a x)

b(ax, a ) = b(a, xa ).

If we set tr(a) = b(1, a), then the first condition shows that b(a, a ) = tr(a a), while the second condition shows that b(a, a ) = tr(aa ). It follows that tr is a trace on the algebra A: that is, it vanishes on all commutators [a, a ] = aa − a a. Conversely, given any linear map tr : A → C which vanishes on all commutators, the

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formula b(a, a ) = tr(aa ) defines a bilinear form b giving a map of (A, A)-bimodules A → A∨ . We say that tr is nondegenerate if this map is an isomorphism. A pair (A, tr) where A is a finite-dimensional algebra over C and tr is a nondegenerate trace on A is called a Frobenius algebra. We can summarize the above discussion as follows: giving a fully dualizable object of Alg which is fixed under the action of the group SO(2) is equivalent to giving a semisimple Frobenius algebra (A, tr). Theorem 2.6 implies that every such pair (A, tr) determines an extended TQFT F : BordSO 2 → Alg such that F (∗)  A. The value of F on the circle S 1 can be identified with the (C, C)-vector space given by the tensor product A⊗A⊗Aop A: in other words, the quotient of A by the subspace [A, A] generated by all commutators. Using the self-duality of F (S 1 ) provided by the field theory F and the self-duality of A provided by the bilinear form b, we can also identify F (S 1 ) with the orthogonal [A, A]⊥ of [A, A] with respect to b: that is, with the center of the algebra A. In particular, if G is a finite group and η ∈ H 3 (G; Z) is a cohomology class, then the twisted group algebra Cτ [G] admits a canonical trace (obtained by taking the coefficient of the unit in G and dividing by the order of G). Applying Theorem 2.6 in this case, we obtain another construction of the topological field theory described in §2.2. 3. Finite path integrals We sketch an idea to construct extended field theories via finite sums. Remark 3.1. Integration over a finite field, such as a gauge field with finite gauge group, sometimes occurs in a quantum field theory with other fields. These cases also fit into this framework. Examples include orbifolds and orientifolds in string theory. As in [21, §3.2] let Famn denote the n-category whose objects are finite groupoids X. (A groupoid X is finite if there is a finite number of inequivalent objects and each object has a finite automorphism group.) A 1-morphism C : X → Y between finite groupoids is a correspondence C JJ p JJ2 tt tt JJ t yt % p1

(3.1) X

Y

of finite groupoids. A 2-morphism in Famn is a correspondence of 1-morphisms, and so forth until the level n; we regard two n-morphisms in Famn as identical if they are equivalent. Composition is homotopy fiber product. Cartesian product of groupoids endows Famn with a symmetric monoidal structure. There is a symmetric monoidal functor (3.2)

BunG : BordSO n → Famn

which attaches to each manifold M the finite groupoid of G-bundles on M . This is the space of classical fields on M . (The functor (3.2) replaces the category BordSO n (G) which appears in our previous formulation in the paragraph following (2.5).) Let C be a symmetric monoidal n-category and Famn (C) the symmetric monoidal n-category of correspondences equipped with local systems valued in C: for example, an object of Famn (C) is a finite groupoid X and a functor X → C, and morphisms are also equipped with functors to C (as before, C denotes the codomain

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of our field theories). Then the classical theory is encoded by a functor I which fits into a commutative diagram / Famn (C) BordSO nD DD yy DD yy y D y BunG DD |yy " Famn I

where the right arrow is the obvious forgetful functor. Remark 3.2. For a classical theory the values of I lie in the invertible objects and morphisms of C. The finite sum construction does not depend on the invertibility, so will apply in the situations envisioned in Remark 3.1. Example 3.3. We recast the discussion in §2.1 in these terms. As a preliminary suppose B, C are 2-categories. We describe the data of a functor I : B → C. Each 2-category consists of objects (ob), 1-morphisms (1-mor), and 2-morphisms (2mor) and the data which defines I includes a map between these corresponding collections in B and C. For strict 2-categories these maps are required to respect the composition strictly. But for more general 2-categories there are two additional pieces of data. First, there is a map u : ob(B) → 2-mor(C) which for each object X ∈ ob(B) gives a 2-morphism u(X) : idI(X) → I(idX ). Second, there is a map a : 1-mor(B) ×ob(B) 1-mor(B) → 2-mor(C) which expresses the failure of I on 1morphisms to be a strict homomorphism, namely a(x, y) : I(y) ◦ I(x) → I(y ◦ x) for y x every pair · − →·− → · of composable 1-morphisms in B. These data are required to y x z →·− →·− → · is a triple of composable obey a variety of axioms. For example, if · − morphisms, then (3.3)

a(y ◦ x, z) ◦ {a(x, y) ∗ idI(z) } = a(x, z ◦ y) ◦ {idI(x) ∗ a(y, z)}.

As in Example 2.8, let C = Alg be the 2-category whose objects are complex algebras A. A morphism A → A is an (A , A)-bimodule and a 2-morphism is a homomorphism of bimodules. The symmetric monoidal structure is given by tensor product over C. Note that the unit object in C is the algebra C and a (C, C)bimodule is simply a vector space. Thus the category of 1-morphisms C → C is the category VectC of complex vector spaces. Now for the finite 2-dimensional gauge theory based on the central extension T → Gτ → G, the functor BunG assigns to the point pt+ the groupoid ∗
G. The lift I includes the functor (3.4)

χ : ∗
G → Alg

which assigns the (invertible) algebra C to the unique object ∗, the complex line Kx to the 1-morphism x ∈ G, and the identity map to the identity 2-morphisms in ∗
G. (There are only identity 2-morphisms in ∗
G.) The map u in the previous paragraph is then the identity. The map a in the previous paragraph assigns to each pair x, y of group elements the isomorphism (2.1), viewed as a 2-morphism in C. The associativity constraint (3.3) is (2.2). The quantization via a finite sum will be implemented by a symmetric monoidal functor of n-categories (see §8) (3.5)

Sumn : Famn (C) → C

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Given (3.5) we simply define the quantum theory as the composition (3.6)

Sum

I

n F : BordSO → Famn (C) − −−−→ C n −

The functor Sumn depicted in (3.5) is given by a purely categorical procedure: if X is a finite groupoid — an object in Famn — and χ : X → C is a C-valued local system on X, then Sumn (X, χ) is given by the colimit limx∈X χ(x). To guarantee ←− that this formula describes a well-defined functor from Famn (C) to C, we need to make certain assumptions on C: namely, that it is additive in a strong sense which guarantees that the colimit limx∈X χ(x) exists and coincides with the limit −→ limx∈X χ(x). ←− Remark 3.4. This discussion goes through if we replace Famn by the ncategory whose objects are finite n-groupoids (see §8). We will use the generalization to n = 2 in §5.2. Example 3.5. To illustrate the idea consider the 1-dimensional gauge theory of §1 for a finite gauge group G. Recall that it is specified by an abelian character λ : G → T. In this case let C be the symmetric monoidal category VectC with tensor product. Then the “classical” functor I : BordSO 1 → Fam1 (VectC ) sends the point pt+ to the functor ∗
G → VectC which sends ∗ to C and is the homomorphism λ on morphisms. The path integral (3.5) is defined as follows. If X ∈ Fam1 (VectC ) is a finite groupoid equipped with a functor χ : X → VectC then Sum1 (X, χ) ∈ VectC is the limit (3.7)

Sum1 (X, χ) = lim χ(x) ←− x∈X

If the finite groupoid X = X0 oo X1 is finitely presented (X0 and X1 are finite sets), then χ determines an equivariant vector bundle over X0 and the limit (3.7) is the vector space of invariant sections. On the other hand, a morphism in Fam1 (VectC ) is given by a correspondence (3.1) of finite groupoids; functors  χ: X  → and δ : Y → Vect ; and for each c ∈ C a linear map ϕ(c) : χ p (c) → Vect C C 1   δ p2 (c) . We define a map Sum1 (C, ϕ) : Sum1 (X, χ) → Sum1 (Y, δ) Assume X, Y, C are finitely presented. Given x ∈ X let Cx be the sub-groupoid of C consisting of c ∈ C such that p1 (c) = x. Then  ϕ(c) (3.8) Sum1 (C, ϕ) = # Aut(c) [c]∈π0 Cx

where the sum is over equivalence classes of objects in Cx . (Compare (2.4).) The reader can check that (3.7) and (3.8) reproduce the results in the paragraph preceding Remark 1.3 in the 1-dimensional finite gauge theory. Example 3.6. We continue Example 3.3 and content ourselves with the computation of F (pt+ ) as a limit in the 2-category C = Alg: F (pt+ ) = lim χ, ←− ∗
G

where χ is the functor (3.4). Unraveling the definitions, this limit is given by an algebra A with the following universal property: for any algebra B, category of

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(A, B)-bimodules is equivalent to the category of right B-modules M equipped with a compatible family of B-module isomorphisms {Kx ⊗ M  M }x∈G . This limit can be represented by the twisted group algebra  Kx . A = Cτ [G] = x∈G

We leave as an exercise to the reader the computation of F (S 1 ) as a (1categorical) limit over the groupoid G
G of G-bundles on S 1 . The argument is similar to Example 3.5. 4. Three-dimensional theories with finite gauge group 4.1. H 4 (BG; Z) for finite G. Let G be a finite group. Definition 4.1. A 2-cocycle on G with values in Hermitian lines is a pair (K τ , θ τ ) consisting of a Hermitian line bundle K τ → G × G, for each triple x, y, z ∈ G an isometry (4.1)

−1 −1 θx,y,z : Ky,z ⊗ Kxy,z ⊗ Kx,yz ⊗ Kx,y →C

and a cocycle condition (4.2)

−1 −1 θy,z,w θxy,z,w θx,yz,w θx,y,zw θx,y,z = 1,

x, y, z, w ∈ G.

for each quadruple of elements of G. Proposition 4.2. For G finite the cohomology group H 4 (BG; Z) is the set of isomorphism classes of 2-cocycles (K τ , θ τ ) on G with values in Hermitian lines. Proof. Since H 4 (BG; R) = 0, the Bockstein homomorphism H 3 (BG; T) → H (BG; Z) from the exponential sequence of coefficients is an isomorphism.8 Given (K τ , θ τ ) choose kx,y ∈ Kx,y of unit norm. Then 4

−1 −1 ωx,y,z = θx,y,z (ky,z kxy,z kx,yz kx,y )

is a 3-cocycle with values in T. A routine check shows that the resulting element of H 3 (BG; T) is independent of {kx,y } and of the representative (K τ , θ τ ) in an equivalence class. Also, the resulting map from equivalence classes of 2-cocycles to H 3 (BG; T) is an injective homomorphism. Surjectivity is also immediate: if ωx,y,z is a cocycle, set Kx,y = C and θx,y,z = ωx,y,z .
Suppose (K τ , θ τ ) is given. Define the line bundle L → G × G by (4.3)

∗ Lx,y = Kyxy −1 ,y ⊗ Ky,x .

The cocycle isomorphism (4.1) leads to an isomorphism Lyxy−1 ,y
⊗ Lx,y → Lx,y
y which is summarized by the statement that L is a line bundle over the groupoid G
G formed by the G action on itself by conjugation. In §4.2 the line bundle K enters into the quantization of a point and the line bundle L enters into the quantization of the circle. 8Here the circle T has either the discrete topology or the continuous topology.

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D. S. FREED, M. J. HOPKINS, J. LURIE, AND C. TELEMAN

4.2. Three-dimensional finite gauge theory. Let G be finite and fix a 2-cocycle (K τ , θ τ ) with values in Hermitian lines. Let Vectτ [G] be the category whose objects are complex vector bundles over G and morphisms are linear vector bundle maps. Define a monoidal structure on Vectτ [G] by twisted convolution: if W, W  → G are vector bundles set  Kx,x
⊗ Wx ⊗ Wx
. (W ∗ W  )y = xx
=y τ

Then Vect [G] is a linear tensor category. Remark 4.3. One way to regard this category is as the “crossed product” T
Vect, with the tensor category Vect replacing the customary algebra with T action. (Actions of T on Vect, as a tensor category, assign to each t ∈ T an invertible (Vect, Vect)-bimodule, which must of course be isomorphic to Vect. The associator for the action is a 2-cocycle on T with values in Pic, the group of units in Vect; and associators for equivalent actions differ by a coboundary. Our action is classified by τ , as per the discussion in §4.1.) There is a TQFT F τ such that F τ (pt) = Vectτ [G] and it can be constructed in several ways. First, and most directly, we can construct it as a 0-1-2-3 theory directly using the finite sum path integral in §3. Secondly, following Reshetikhin and Turaev [28], as a 1-2-3 theory it may be constructed by specifying the modular tensor category which is attached to a circle; it appears in Proposition 4.6 below. A third approach is to realize Vectτ [G] as a fully dualizable object of a symmetric monoidal 3-category C. The Baez-Dolan cobordism hypothesis then implies that Vectτ [G] determines a 0-1-2-3 theory which is defined on framed 3-manifolds. To remove the dependence on a choice of framing, we should go further and exhibit Vectτ [G] as an SO(3)-equivariant fully dualizable object. For present purposes, we will be content to sketch a definition of the relevant 3-category C and to give some hints at what the relevant finiteness conditions correspond to. Definition 4.4. The 3-category C can be described informally as follows: (a) The objects of C are tensor categories over C: in other words, C-linear categories equipped with a tensor product operation which is associative up to coherent isomorphism. (b) Given a pair of tensor categories A and A  , a 1-morphism from A to A  in C is an A -A  bimodule category: that is, a C-linear category D equipped with a left action A × D → D and a right action D × A  → D which commute with one another, up to coherent isomorphism. (c) Given a pair of tensor categories A and A  and a pair of bimodule categories D and D , a 2-morphism from D to D in C is a functor between bimodule categories: that is, a functor F : D → D which commutes with the actions of A and A  up to coherent isomorphism. (d) Given a pair of tensor categories A and A  , a pair of bimodule categories D and D , and a pair of bimodule category functors F, F  : D → D , a 3-morphism from F to F  in C is a natural transformation α : F → F  which is compatible with the coherence isomorphisms of (c). The category Vectτ [G] is a fully dualizable object of C. Roughly speaking, the verification of this takes place in three (successively more difficult) steps. First, we verify that Vectτ [G] is dualizable: in other words, that it is a fully dualizable object

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of the underlying 1-category of C (and therefore gives rise to a 1-dimensional field theory). This is completely formal: every tensor category A is a dualizable object of C, the dual being the same category with the opposite tensor product. The next step is to verify that Vectτ [G] is a fully dualizable object of the underlying 2category of C (and therefore gives rise to a 2-dimensional field theory, which assigns vector spaces to surfaces). This is a consequence of the fact that Vectτ [G] is a rigid tensor category: that is, every object of X ∈ Vectτ [G] has a dual (given by taking the dual vector bundle X ∨ of X and pulling back under the inversion map g → g −1 from G to itself). Finally, to get a 3-dimensional field theory, we need to check that Vectτ [G] satisfies some additional 3-categorical finiteness conditions which we will not spell out here. (However, we should remark that these 3-categorical finiteness conditions are in some sense the most concrete, and often amount to the finite dimensionality of various vector spaces associated to Vectτ [G]: for example, the vector spaces of morphisms between objects of Vectτ [G].) The paper [5] is presumably relevant to the full dualizability of Vectτ [G]. Definition 4.5. Let A be a monoidal category with product ∗. Its (Drinfeld ) center Z(A ) is the category whose objects are pairs (X, X ) consisting of an object X in A and a natural isomorphism X (–) : X ∗ – → – ∗ X. The isomorphism X is compatible with the monoidal structure in that for all objects Y, Z in A we require     X (Y ∗ Z) = idY ∗ X (Z) ◦ X (Y ) ∗ idZ . The center Z(A ) of any monoidal category A is a braided monoidal category. M¨ uger [23] proves that if A is a linear tensor category over an algebraically closed field which satisfies certain conditions, then Z(A ) is a modular tensor category. That applies to part (ii) of the following result. Proposition 4.6. (i) The value F τ (S 1 ) of the field theory F τ on the circle is the center of the monoidal category Vectτ [G]. (ii) The center of Vectτ [G] consists of twisted equivariant vector bundles W → G, that is, vector bundles with a twisted lift (4.4)

Lx,y ⊗ Wx → Wyxy−1

of the G-action on G by conjugation, where L → G × G is defined in (4.3). This center is the well-known modular tensor category attached to a circle in the twisted finite group Chern – Simons theory [14]. Proof. We compute F τ (S 1 ) by decomposing the circle into two intervals IL and IR . We regard IL as a morphism in the oriented bordism category from the empty set to the disjoint union of two oppositely oriented points; IR is a bordism in the other direction. Then S 1 is the composition IR ◦ IL . Let A = Vectτ [G], which is the object in C — a tensor category — attached to a positively oriented point. Then the tensor category Aop with the opposite monoidal structure is attached to the negatively oriented point. Now F τ (IL ) is A viewed as a left module for A ⊗ Aop and F τ (IR ) is A viewed as a right module for A ⊗ Aop . Thus F τ (S 1 ) ∼ = F τ (IR ◦ IL ) ∼ = A ⊗A⊗Aop A (This is by definition the Hochschild homology of A.) But now we use additional structure on Vectτ [G] which gives an isomorphism of A with its linear dual A∨ =

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Hom(A, Vect). Namely, there is a trace θ : A → Vect which maps a vector bundle over G to the fiber over the identity element, and the corresponding bilinear form A ⊗ A → Vect W ⊗ W  → θ(W ∗ W  ) induces the desired identification. Therefore, F τ (S 1 ) ∼ = A ⊗A⊗Aop A∨ ∼ = HomA⊗Aop (A, A) which we may identify with the center of A (the Hochschild cohomology). For (ii) suppose W → G is in the center. Let W  → G be the vector bundle which is the trivial line Cy at some y ∈ G and zero elsewhere. Then the braiding gives, for every x ∈ G, an isomorphism Ky,x ⊗ Cy ⊗ Wx → Kyxy−1 ,y ⊗ Wyxy−1 ⊗ Cy


which, by (4.3), is the desired isomorphism (4.4). 5. 2-cocycles on tori

5.1. H (BG; Z) for torus groups. Let G = T be a compact connected abelian Lie group, i.e., a torus. Associated to it are the dual lattices (finitely generated free abelian groups) Π = Hom(T, T ) ∼ = H2 (BT ; Z) = H1 (T ; Z) ∼ 4

Λ = Hom(T, T) ∼ = H 2 (BT ; Z) = H 1 (T ; Z) ∼ Let t be the Lie algebra of T . Then Π ⊂ t by differentiation of a homomorphism, and dually Λ ⊂ t∗ , also by differentiation. The cohomology ring of BT is the symmetric ring on H 2 (BT ; Z) = Λ, so in particular ∼ Sym2 Λ (5.1) H 4 (BT ; Z) = We identify Sym2 Λ with the group of homogeneous quadratic functions q : Π → Z. For there is a natural quotient map Λ⊗2 → Sym2 Λ with kernel the alternating tensors, and the value q(π) of the lift of an element of Sym2 Λ on π⊗π is independent of the lift. Then the symmetric bi-additive homomorphism (form) (5.2)

π1 , π2  = q(π1 + π2 ) − q(π1 ) − q(π2 ),

π1 , π2 ∈ Π,

is even (π, π ∈ 2Z) and q(π) = 12 π, π. Therefore, the group Sym2 Λ in (5.1) is also isomorphic to the group of even forms Π × Π → Z. Definition 5.1. A class in H 4 (BT ; Z) is nondegenerate if the corresponding form –, – is nondegenerate over Q. The form induces a homomorphism τ : Π → Λ; it is nondegenerate over Q if the map τQ : Π⊗Q → Λ⊗Q on rational vector spaces is an isomorphism, or equivalently if τ is injective. Remark 5.2. The group (Sym2 Π)∗ of all symmetric bi-additive homomorphisms Π × Π → Z fits into the exact sequence 0 → Sym2 Λ → (Sym2 Π)∗ → Hom(Π, Z/2Z) → 0 where the quotient map takes a form –, – to π → π, π (mod 2). The quotient is the cohomology group H 2 (BT ; Z/2Z), the kernel is the cohomology group H 4 (BT ; Z), and we can identify the middle term with the cohomology group h4 (BT ), where h is the first cohomology theory mentioned in Remark 1.3.

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Fix a nondegenerate class in H 4 (BT ; Z) with corresponding form –, – and homomorphism τ : Π → Λ. Applying ⊗ R we extend the form to t × t and obtain a linear map τR : t → t∗ . The nondegeneracy implies that τR is invertible. Let L → t × t be the trivial line bundle and lift the action of Π × Π on t × t to L by setting (5.3)

(π, π  ) : Lξ,ξ
→ Lξ+π,ξ
+π
,

to act as multiplication by (5.4)

π, π  ∈ Π,

ξ, ξ  ∈ t,

  π, ξ   − ξ, π   + π, π   e , 2

where e(a) = e2πia for a real number a. Also, define the correspondence (5.5)

N v C gN NNNNNpN2 vv N N' v s N zvv Λ/τ (Π) T p1

by C = (t ⊕ Λ)/Π, where the action by π ∈ Π on (ξ, λ) ∈ t × Λ is   (5.6) π · (ξ, λ) = ξ + π, λ + τ (π) . Proposition 5.3. (i) The expression (5.4) does lift the action of Π × Π, whence there is a quotient Hermitian line bundle L → T × T . There are natural isomorphisms (5.7)

Lx,y
⊗ Lx,y → Lx,y
y ,

x, y, y  ∈ T,

which lift to the identity map on t×3 and which satisfy an associativity condition. Hence for each x ∈ T the bundle Lx,– → T determines a central extension T → T x → T . (ii) The fiber of p1 over x ∈ T may be identified with the Λ-torsor of splittings of T → T x → T . The splitting χ(ξ,λ) corresponding to (ξ, λ) ∈ t × Λ is determined by 

 ξ  −1  ξ  ∈ t. (5.8) χ(ξ,λ) (ξ ) = e τR (λ) − , ξ ∈ Lξ,ξ
, 2 (iii) C is a group and p2 is split by the homomorphism s : Λ/τ (Π) → C defined by (5.9)

  s(λ) = τR−1 (λ), λ ∈ t ⊕ Λ,

λ ∈ Λ.

The proof is a series of straightforward verifications from (5.4), (5.6), (5.8), and (5.9). Notation 5.4. Let F ⊂ C denote the image of s and F ⊂ T the image p1 (F ). Both F and F are finite groups isomorphic to Λ/τ (Π). The abelian groups F and Λ/τ (Π) are in Pontrjagin duality by pairing characters in Λ with elements of F ⊂ T . The map p2 has as fibers affine spaces for t. The subgroup F contains a unique element in each affine space. Also, for each x ∈ F the lift x ˆ ∈ F ⊂ C defines a distinguished projective character of Tx . Observe that t ⊂ C is the fiber of p2 over 0, and C is isomorphic to the direct sum t ⊕ F .

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A class in H 4 (BT ) may also be represented by a 2-cocycle,9 as in Proposition 4.2, and it has an explicit construction analogous to that of L in (5.3). It depends on a choice of a (nonsymmetric) bilinear form B : Π × Π → Z which whose symmetrization is the form (5.2): B(π1 , π2 ) + B(π2 , π1 ) = π1 , π2 ,

π1 , π2 ∈ Π.

Namely, let K → t × t be the trivial line bundle and lift the action of Π × Π on t × t to K by setting (5.10)

(π, π  ) : Kξ,ξ
→ Kξ+π,ξ
+π
,

π, π  ∈ Π,

ξ, ξ  ∈ t,

to act as multiplication by   B(π, ξ  ) − B(ξ, π  ) + B(π, π  ) (5.11) e , 2 Proposition 5.5. (i) The expression (5.11) does lift the action of Π × Π, whence there is a quotient Hermitian line bundle K → T × T . There are natural isomorphisms θx,y,z as in (4.1) which lift to the identity map on t×3 and which satisfy the cocycle condition (4.2). (ii) There is an isomorphism ∼ =

−1 → Kx,y ⊗ Ky,x Lx,y −

such that (5.7) is θx,y
,y θy
,y,x θy−1
,x,y . The proof is a series of straightforward verifications. 5.2. Classical descriptions. We continue with the notation of §5.1. Recall that the level is a class in H 4 (BT ; Z) and is represented by a homogeneous quadratic map q : Π → Z. It determines a homomorphism τ : Π → Λ. The finite subgroup F ⊂ T , defined in Notation 5.4, may be identified as F ∼ = τ −1 (Λ)/Π ⊂ (Π ⊗ Q)/Π. Q

Then q induces a homogeneous quadratic map qQ/Z : F → Q/Z. We give a topological interpretation. For an abelian group A and nonnegative integer n, let K(A, n) be the corresponding Eilenberg – Mac Lane space. Lemma 5.6 ([12, Theorem 26.1]). Let A, B be discrete abelian groups. Then the set of homogeneous quadratic forms q : A → B is isomorphic to the homotopy classes of maps q˜: K(A, 2) → K(B, 4). Applying the lemma we may represent qQ/Z by a continuous map (5.12)

q˜Q/Z : K(F, 2) → K(Q/Z, 4).

Now for any manifold X let FF (X) denote the 2-groupoid of “F -gerbes” on X. Recall that an F -gerbe10 is a geometric representative of a class in H 2 (X; F ). One possible topological model is that an F -gerbe on X is a map X → K(F, 2). In this  model FF (X) is the fundamental 2-groupoid of the mapping space Map X, K(F, 2) . There are three nonzero homotopy groups: π0 FF (X) ∼ = H 2 (X; F ), π1 FF (X) ∼ = H 1 (X; F ), π2 FF (X) ∼ = H 0 (X; F ). 9A 2-cocycle on T with values in Hermitian lines is defined as in Definition 4.1 with the additional requirement that the line bundle K → T × T and isometry (4.1) be smooth. 10Physicists are familiar with gerbes as fields in a field theory in string theories, where they are known as “B-fields”.

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Composition with (5.12) gives the Lagrangian of the field theory. Suppose X is a closed oriented 4-manifold. Then the action is defined by integrating the Lagrangian, which in this case means pairing with the fundamental class given by the orientation. The result only depends on the equivalence class of the gerbe and gives a homogeneous quadratic map qX : H 2 (X; F ) → Q/Z, which is defined using the cup square and the quadratic form q. The quantum invariant is then a finite path integral ( §3) over the stack of F -gerbes: AF (X) =

(5.13)

 # H 0 (X; F ) G

# H 1 (X; F )

e2πiqX (G) ,

where the sum is over a set of representative F -gerbes on X. (Compare (2.4).) This Gauss sum may be evaluated explicitly [20]: (5.14)

 # H 0 (X; F ) # H 2 (X; F ) exp 2πi(sign b)(sign X)/8 # H 1 (X; F ) = ( # F )Euler X µ(sign b)(sign X) ,

AF (X) =

where sign X is the signature and Euler X the Euler characteristic of the 4-manifold X; sign b is the signature of the bilinear form b = –, – in (5.2) associated to q (after tensoring with Q); and µ = exp(2πi/8) is a primitive 8th root of unity. We claim that the finite path integral procedure of §3, applied to the F -gerbes and the quadratic form q, defines an invertible 4-dimensional TQFT AF . For this we need to first specify a target symmetric monoidal 4-category C. The relevant category can be described informally as a “de-looping” of the 3-category of Definition 4.4. Roughly speaking, the objects of C are braided tensor categories A (which we can think of as associative algebras in the setting of tensor categories). A 1-morphism from A to A  is an A -A  bimodule in the setting of tensor categories: that is, a tensor category D equipped with commuting central actions of the braided monoidal categories A and A  . The 2-morphisms in C are given by linear categories, the 3-morphisms by functors, and the 4-morphisms by natural transformations. (See §7.2 for further discussion.) Here we give a second classical description of the anomaly theory which leads to a finite dimensional but not finite path integral, and we use some heuristics in its evaluation on a 4-manifold. In this theory the finite group F is replaced by the Lie algebra t with the discrete topology. Now we use the real homogeneous quadratic form qR : t → R and apply Lemma 5.6 to obtain (5.15)

K(t, 2) → K(R, 4) → K(R/Z, 4).

Let Ft (X) be the 2-groupoid of t-gerbes on a closed oriented 4-manifold X. Then (5.15) determines a homogeneous quadratic map (5.16)

qX : H 2 (X; t) → R/Z.

The finite sum (5.13) is now replaced by a sum over an uncountable set, which we interpret as an integral:
vol H 0 (X; t) 2πiqX (G) e . At (X) = 1 G∈H 2 (X;t) vol H (X; t)

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Here ‘vol’ denotes a formal volume which we regularize below. This is a Gaussian integral, and we evaluate it as11

 vol H 0 (X; t) (5.17) At (X) = vol H 2 (X; t) exp 2πi(sign b)(sign X)/8 1 vol H (X; t) = λEuler X µ(sign b)(sign X) , √ where λ is a constant we choose equal to # F to match (5.14). Before leaving these classical descriptions we indicate a classical coupling of the usual toral Chern – Simons to the classical gerbe theory At on a compact oriented 4-manifold X with boundary a closed oriented 3-manifold Y . We freely use generalized differential cohomology [17].12 Let G be a t-gerbe on X; since t has the discrete topology G is flat. Its restriction ∂G to Y has an exponential exp ∂G which is a topologically trivial flat T -gerbe. The field in toral Chern – Simons is a ‘nonflat trivialization’ P of exp ∂G. More precisely, there is a groupoid whose objects are t-gerbes on Y and whose morphisms are torsors for the differential cohomology } 2 (Y ); these morphisms are equivalence classes of nonflat trivializations of group HΠ the exponentials of the t-gerbes. (The actual nonflat trivializations are morphisms in a 2-groupoid.) Now P extends to a nonflat trivialization of exp G on X. Let ω ∈ Ω2 (X; t) be its curvature. Work over a base manifold S. Then the quadratic form (5.15), applied to ∂G and integrated over the fibers of Y → S, yields a flat R/Z-bundle over S whose equivalence class in H 1 (S; R/Z) is computed by a relative version of (5.16) for the relative 3-manifold Y → S. Because ∂G is extended to a t-gerbe on X, this R/Z-bundle comes with a trivialization. The Chern – Simons action is a section of this circle bundle, and using the trivialization may be identified with the function
ω ∧ ω (mod Z) (5.18) X/Z

on S. In the quantum Chern – Simons theory we integrate the exponential of (5.18) over the stack of nonflat trivializations P for fixed G. The result lives in the complex line bundle L(∂G) → S which is the exponential of the R/Z-bundle in the previous paragraph. Automorphisms of ∂G act on L(∂G) and the result of the path integral is invariant. Therefore, if these automorphisms act nontrivially the path integral vanishes. (This is called the ‘Gauss law’ in physics.) Now an automorphism α of ∂G acts through its equivalence class [α] ∈ H 1 (Y ; t), and the action only depends on 2 the equivalence class [∂G]   ∈ H (Y ; t) of the gerbe: namely, it acts as multiplication by exp 2πi[α]  [∂G] . This shows that the path integral vanishes unless ∂G is trivializable as a flat t-gerbe. Relative to a trivialization the Chern – Simons field 11If Q is a symmetric bilinear form on a finite-dimensional real vector space V , it induces

a map V → V ∗ whose determinant is a map Det Q : Det V → Det V ∗ , so an element Det Q ∈ (Det V ∗ )⊗2 . The integral of eiQ(x,x)/2 over V has an algebraic evaluation as e2πi(sign Q)/8 ∈ | Det V |,
| det Q| where | Det V | is the real line associated to V by the absolute value character of R=0 . A translationinvariant volume form on V may be viewed as an element of the dual real line | Det V ∗ |, which then gives a numerical answer which matches the usual Gaussian integral. This explains the signature factors in (5.17). 12We use differential theories based on the Eilenberg – Mac Lane spectrum HΠ.

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P is a usual T -bundle with connection and we recover the standard description of classical Chern – Simons. 6. The basic tensor category and its center 6.1. Drinfeld centers. Let T be a torus, τ ∈ H 4 (BT ; Z) a nondegenerate twisting. Recall from Proposition 5.5 the line bundle K → T × T , which is a 2-cocycle. Define a convolution on the category Sky[T ] of skyscraper sheaves of finite-dimensional vector spaces on T , with finite support, by setting Cx ∗ Cy = Kx,y ⊗ Cxy for the skyscrapers at x, y, xy ∈ T . The cocycle property of K ensures that this defines a tensor category, which we denote by Skyτ [T ]. This is an analogue of the “twisted group ring” Vectτ [G] discussed in §4.2 for finite groups G. The tensor structure lifts to the universal cover t of T , but it is trivializable there. This is because we can trivialize the pullback of K as a 2-cocycle valued in Pic, compatibly with our chosen trivialization of the pullback of L, as in (5.10) and (5.3). However, we shall see that Skyτ [t] carries a higher structure, a braiding. This ∼ is specified by a family of automorphism Cξ ∗ Cξ
− → Cξ
∗ Cξ , ξ, ξ  ∈ t forming a bi-multiplicative section of the line bundle L of (5.3). In the defining trivialization, the distinguished section is the function   σ(ξ, ξ  ) = exp −πiξ, ξ   . The structure extends in fact to the larger category Skyτ [C] of skyscrapers on the correspondence space C = (t × Λ)/Π in (5.5) by a reformulation of Proposition 5.3(ii). Alternatively, there is a natural quadratic function θ : C → T (Remark 6.2(ii) below), which determines the braided tensor structure on Skyτ [C] by a general construction (cf. the end of §8.1). The conceptual meaning for this structure is given by the following. Proposition 6.1. (i) The braided tensor category Skyτ [C] is the “continuous” Drinfeld center of Skyτ [T ]. The natural functor from Skyτ [C] to Skyτ [T ] is induced by projection. (ii) As braided tensor categories, Skyτ [C] ≡ Skyτ [t] ⊗ Skyτ [F ], sitting in C by the obvious inclusions, and lifting the splitting C = t × F of abelian groups. Moreover, the two factors are mutual commutants. The last statement means that any Skyτ [C]-object braiding trivially with all of Skyτ [F ] is in Skyτ [t], and similarly with F and t interchanged. Regarding the notion of “continuous Drinfeld center,” we will limit ourselves to the following observation. All our categories are semisimple, with the simple isomorphism classes corresponding to the points on the underlying spaces. They are ‘categorifications’ of the underlying abelian groups. We then ask that the half-braiding in Definition 4.5 should be continuous on irreducible objects, in their natural topology. Developing this in more detail would be a distraction here for two reasons: first, the key ingredient for us is not quite the Drinfeld center, which is a Hochschild cohomology, but its dual notion, a Hochschild homology, which is a tensor product. Second, as we briefly indicate in the next section, the easiest way to justify our story rigorously is via approximations by finite abelian groups, in the time-tested “lattice approximation” of quantum field theory; and continuity plays no rˆole in that setting.

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Sketch of proof. An alternative description of C will be useful. Note that τ defines an isogeny τT : T → T ∗ , the Langlands dual torus. Its kernel is the group F described in Notation 5.4. Interpreting T ∗ as the moduli space of flat line bundles on T , the map τT classifies the line bundle L. Then, C∼ = t∗ ×T ∗ T. To identify this fiber product over T ∗ with the standard description (t ⊕ Λ)/Π, send (ξ, λ) in the latter space to (dτ (ξ) − λ, eξ ) in the former. In the second description of C, we interpret a point x ∈ T as the skyscraper object Cx and an element in the fiber of τT (x) as a continuous character of the central extension of the group T defined by Lx,– ; see Proposition 5.3(ii). But this is precisely a continuous half-braiding of Cx with the simple objects of Skyτ [T ]. It is easy to see a priori that simple objects of the center must be supported at single points, and then that they have rank one, so we have just found all of them. If we start at the identity with the trivial braiding, and keep a continuous choice of character as we move in t, we sweep out the identity copy of t in C. The projective characters of T thus swept out have the property that they are trivial on F ⊂ T (where every central extension in the family is naturally trivialized). On the other hand, Lx,– has trivial holonomy at the points x ∈ F , and there we can choose the trivial character in t∗ . This defines the copy F of F in C. Clearly, this braiding commutes with the braiding by the copy of t just described. Nondegeneracy of the quadratic form on t implies that its commutant is no larger than Skyτ [F ]. That the two in fact commute, and the analogous statement for F , follow from the formula for the ribbon element below. One can also argue directly that the braiding is defined by the perfect bi-character on C described in Remark 6.2.
Remark 6.2. (i) The two descriptions of C make clear its remarkable property of being Pontrjagin self-dual: the groups t∗ × T and t × Λ are Pontrjagin dual, and the kernel of the addition map of the first group to T ∗ is dual to the quotient of the second by the dual inclusion of Π. This self-duality is symmetric, and is induced by the quadratic “ribbon” map θ : C → T below. (ii) The braided tensor category Skyτ [C] and its subcategories Skyτ [F ], τ Sky [t] are in fact ribbon categories with ribbon function on the simple object (ξ, λ)   θ(ξ, λ) = exp πi τ −1 (λ)2 − ξ − τ −1 (λ)2 , using the norm associated to the quadratic form (5.2). The square of the braiding is given by the standard formula θ(XY )θ −1 (X)θ −1 (Y ). (iii) The equivalence in Proposition 6.1(ii) is not one of ribbon categories, because the ribbon function is quadratic and not linear on objects. (iv) One should mind that the tensor structure defined by the restriction τ ∈ H 4 (BF ; Z) is not necessarily trivial, that is, Skyτ [F ] may differ from Sky[F ] as a tensor category. For instance, this always happens when T = S 1 . But τ is always 2-torsion on F . (v) The category Skyτ [F ] is in fact a modular tensor category [4,28] and defines the 1-2-3-dimensional Chern – Simons theory (of framed manifolds) associated to the torus T at level τ via the Reshetikhin – Turaev theorem.

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6.2. Finite approximation of T and t. Let us now describe the finite (“lattice”) approximations of C, t, T and develop the finite version of Chern – Simons theory in the next section. Let n be a positive integer, which will become infinitely ∗ ⊂ T ∗ the subgroup of n-torsion points and divisible in the limit13; denote by T(n) by T (n) → T the dual covering torus of T , with Galois group Π(n) ∼ = Π/nΠ, Pon∗ ∗ . Finally, let T(nF ) denote the inverse image of T(n) in T under trjagin dual to T(n) (n) (n) τT and t that of T(nF ) in T . There is also the Pontrjagin dual t∗(nF ) of t(n) , ∗ which is a Galois cover of T(n) (restricted from T ∗ ) with group Λ(nF ) , Pontrjagin ∗ dual to T(nF ) . It is an exercise to check that the restriction τT : T(nF ) → T(n) lifts (n) ∼ ∗(nF ) naturally to an isomorphism t = t . As n → ∞, the finite group T(n) will be our approximation for T , t(n) will ∗ approximate the Lie algebra, t∗(nF ) its dual, T(n) will play the rˆole of T ∗ , Π(n) that of Π and Λ(nF ) that of Λ. ∗ = nd , # T(nF ) = # F · nd , Remark 6.3. If d is the dimension of T , then # T(n) # t(n) = # F · n2d .

The twisting τ restricts to H 4 (BT(n) ; Z) and defines a tensor category Vectτ (T(n) ) as discussed in §4.2, which is a full subcategory of Skyτ [T ]. Consider   ∗ C (n) := t(n) ⊕ Λ(nF ) /Π(n) ∼ T(nF ) . = t∗(nF ) ×T(n) It is easy to check that the ribbon function θ of Remark 6.2(ii) descends to a nondegenerate quadratic function on C (n) , and gives a braided tensor structure on Skyτ [C (n] ), with the restricted twisting τ ). The projection t∗(nF ) ∼ = t(n) → T(n) defines a splitting C (n) ∼ = t(n) × F . The proof of the following, discrete analogue of Proposition 6.1 is left to the reader. Proposition 6.4. (1) The braided tensor category Skyτ [C (n) ] is the Drinτ feld center of Sky [T(nF ) ]. The natural functor from Skyτ [C (n) ] to Skyτ [T(n) ] is induced by projection. (2) As braided tensor categories, Skyτ [C (n) ] ≡ Skyτ [t(n) ] ⊗ Skyτ [F ], sitting in (n) C by the obvious inclusions, and lifting the splitting C (n) = t(n) × F ∗ of abelian groups. Moreover, the two factors are mutual commutants. 6.3. Morita relations between our categories. The categories discussed in this section are closely related. Just as the right notion of quasi-isomorphism for algebras in their natural world is Morita equivalence, there is a corresponding notion for tensor categories and even braided tensor categories. In the next section, we sketch a minimal background for these notions; but let us now state a key result and its significance for TQFTs. Proposition 6.5. (i) Skyτ [t] and Skyτ [F ] are Morita equivalent by means τ of Sky [T ]. (ii) Skyτ [t(n] ) and Skyτ [F ] are Morita equivalent by means of Skyτ [T(n) ]. (iii) All these braided tensor categories are quasi-invertible, more precisely, Skyτ [S] ⊗ Sky-τ [S] is Morita equivalent to Vect in all cases S = t, F , t(n) (with (−τ ) indicating the opposite braiding). 13That is, we’ll take the limit over N ordered by divisibility.

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Thus, Skyτ [t] and Skyτ [t(n) ] are quasi-invertible in their world. Moreover, they are equivalent to the unit braided tensor category Vect in the following cases: • When the signature of –, – on t is divisible by 8; • When Z/2-graded vector spaces are used. In the first case, we apply Proposition 6.5(i) to a product T of copies of the E8 maximal torus, with its generating class τ ; unimodularity of the E8 lattice ensures that F = {1}. The second case uses a variant of our categories and twistings with graded vector spaces, which we have  notdiscussed. In this case, with T = U(1), the graded tensor category Gr - Vect U(1) can be twisted by a class which is half 14 the generator of H 4 (B U(1); Z), and leads again to a trivial group F . (The group F corresponding to the generator of H 4 (B U(1); Z) has two elements.) The braided tensor categories above lead to invertible 4-dimensional TQFTs for oriented manifolds: these are the theories described in §5.2. Invertibility has the obvious meaning, in the tensor structure on TQFTs, and follows from Proposition 6.5(iii). Graded vector spaces, however, require spin structures on the manifold. We conclude that the (isomorphic) theories defined by Skyτ [t] and Skyτ [t(n) ] are trivial on oriented manifolds when the signature of τ is a multiple of 8, and always trivial on spin manifolds.15 7. Higher algebra: tensor and bimodule categories 7.1. m-Algebras. Our picture of Chern – Simons theory requires an ascent to categorical altitudes which exceed the safe limit without special equipment: we will need linear 3-categories describing (very simple) 4-dimensional theories. Fortunately, there is a recursive procedure to produce the higher categorical objects needed and, even more fortunately, up to the range of dimensions we study there are user-friendly models for these. We hope to return to an extensive discussion of these structures elsewhere; here we sketch their basic features. Recall first that vector spaces and linear maps form a symmetric tensor category, and that algebra objects in a symmetric tensor category form in turn a symmetric tensor category (of one level higher, but who’s counting?). Definition 7.1. A 0-algebra is a complex vector space, and a morphism of 0-algebras is a linear map. For m > 0, an m-algebra is an algebra object in the (symmetric tensor m-) category of (m−1)-algebras. Morphism between m-algebras are bimodule objects in the category of (m − 1)-algebras. (Higher morphisms are defined recursively.) The 2-category of (1-)algebras, modules and intertwiners is familiar enough — see Example 2.8 — and here we just give analogous pictures (or adequate substitutes) for the next two levels. A 2-algebra is an algebra A together with an (A, A⊗2 )-bimodule M defining the 2-multiplication and a left A-module E defining the identity; there must be the associativity and unit intertwiners (2-morphisms in the category of algebras) satisfying the ‘obvious’ compatibility rules. A morphism 14This half-generator is defined in the generalized cohomology theory h of Remark 1.3; this leads to a TQFT for spin, rather than oriented manifolds. 15Some structure has been swept under the carpet here. Defining the theory for oriented or spin manifolds, rather than framed ones, requires us to specify an action of SO(4) or Spin(4), which adds some parameters. One of those is the constant λ coupled to the Euler class in (5.17).

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of 2-algebras, from A to B, is a (B, A) bimodule in the world of algebras: an algebra N , plus an (N, B ⊗ N )-bimodule P and an (N, N ⊗ A)-bimodule Q, plus compatibility isomorphisms defining the bialgebra structure. In general, a 2-algebra structure on A defines a tensor structure on the category of left A-modules; sending A → (A-Mod) is a fully faithful functor from the 3category of 2-algebras to that of tensor categories. Modulo the problem of realizing linear categories as module categories, one can use tensor categories instead of 2-algebras. This, of course, is what we have been doing. For example, any commutative algebra becomes a 2-algebra via its own multiplication map; a morphism between commutative algebras A and B, viewed as 2-algebras, is simply a B ⊗ A-algebra. However, commutative algebras have a more obvious and economical embedding into tensor categories, sending A to the tensor category with one object and endomorphism ring A. We will write A[2] when thinking of A as a 2-algebra. More relevant is the example of a Hopf algebra H, which becomes a 2-algebra by means of the multiplication M := H ⊗ H, which is a left H-module via the Hopf structure. For the Hopf algebra A = C(G) of a finite group with the point-wise multiplication, but comultiplication induced by the Hopf structure, the category (A-Mod) is that of vector bundles over G; and the tensor category defined by the Hopf 2-algebra structure is precisely the earlier Vect[G] in §4.2. A twisting class τ ∈ H 4 (BG; Z) changes the associator of the 2-multiplication. One level up, a 3-algebra structure on A can be interpreted as follows: the 2algebra structure turns (A-Mod) into a tensor category, having a 2-category of modules; and the 3-algebra structure gives a tensor structure on the latter 2-category. (This, in turn, has a linear 3-category of modules, affirming the 3-categorical nature of a 3-algebra.) For example, if A is commutative, two (A-Mod)-modules can be tensored over (A-Mod) to produce a new (A-Mod)-module, since the latter is a symmetric tensor category. Similarly, in the case of the group ring, Vect[G]modules can be tensored over Vect[G], using the Hopf structure on the latter. This is usually not possible for the twisted versions Vectτ [G], where the Hopf structure is broken by the twisting; this is much like the case of twisted group rings over C defined from T-central extensions of the group, which do not usually have 2-algebra promotions. 7.2. Braided tensor categories. Manipulating tensor 2-categories can be rather daunting, and no doubt the most general 3-algebras are no friendlier; but in some cases they can be captured by a more concise structure, namely a braided tensor category (BTC). These are special kinds of algebra objects in the category of all tensor categories, and represent the simplest type of structure on the category (A-Mod) which promotes A to a 3-algebra. For example, each braided tensor category Skyτ [S] of the previous section promotes the algebra of functions on S, with pointwise multiplication, into a 3-algebra.16 As we will see, these higher structures are the natural result of quantization. Even if we restrict to BTC among all algebra objects in the symmetric 3category of tensor categories, the description of all module objects — hence the description of bimodule objects, which are the morphisms in the 4-category of 16The cases of S = t and S = C require a priori a grain of salt, because we are considering

very special modules, but in fact the structure supplied does give a 3-algebra in each case, albeit with poor finiteness conditions.

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BTCs — can be awkward; see the case of 2-algebras above. However, there is a nice class of special module objects M , those for which the action of B on M is defined by a functor B ⊗ M → M . This must be a tensor functor, which forces the action of B to half-braid with the multiplication on M ; in other words, the algebra map B → M induced by tensoring with 1 ∈ M must lift to a braided tensor functor into the Drinfeld center Z(M ). We call these ‘B-modules’ half-braided algebras over B. The reader should think of the analogy with a commutative 1-algebra A and the 2-category of central algebras over it; here, the braiding can be thought of as a homotopy between left and right multiplications. In spite of these difficulties, the following simplified four-category C will be a suitable replacement the 4-category of 3-algebras. Objects of C will be braided tensor categories; morphisms from A to B will be half-braided (B, A) bialgebra categories, that is, tensor categories M with a braided tensor functor B ⊗ Aop to Z(M ). 2-morphisms will be bimodules between bialgebras (with compatible B ⊗ Aop -action), and then functors and natural transformations round this up. 7.3. Quasi-isomorphisms. The key moral of the story is that the correct notion of equivalence becomes increasingly obscure for higher algebras, although not less precise. The correct notion is always a pair of functors f, g with f ◦ g and g ◦ f both equivalent to the respective identities.17 Thus, for 1-algebras we have Morita equivalence. For 2-algebras, the familiar Morita conditions P ⊗R Q ≡ S,

Q ⊗S P ≡ R

become Morita equivalences themselves, as P and Q are now bialgebras. This continues, as we now illustrate in the proof of Proposition 6.5. Recall for this purpose the braided tensor categories Skyτ [F ], Skyτ [t], Skyτ [C] from the previous section, the latter two accompanied by their finite approximations Skyτ [t(n) ] and Skyτ [C (n) ]. In addition, the tensor categories Skyτ [T ] and Skyτ [T(n) ] are halfbraided bialgebras over the first three, respectively their finite versions. This makes them into 1−morphisms in the 4-category C. Sketch of proof of Proposition 6.5. We prove part (ii), part (i) is similar. We must produce Morita equivalences  Skyτ [T(n) ] ∼ Skyτ [F ], Skyτ [T(n) ] (7.1)

Skyτ [t(n) ]

Skyτ [T(n) ]



Skyτ [T(n) ] ∼ Skyτ [t(n) ],


] Skyτ [F

and similarly for t and T . The Morita objects realizing (7.1) are Skyτ [T(n) ] in all cases (respectively, Skyτ [T ] for part (i)). Indeed, the desired identity is  Skyτ [T(n) ] Skyτ [T(n) ] ≡ Skyτ [F ], (Skyτ [T(n) ]⊗Skyτ [t(n) ] Skyτ [T(n) ])

and the obvious permutations. Were we to tensor over the braided tensor category Skyτ [T(n) ] ⊗ Skyτ [T(n) ] instead, the left side would be the Hochschild homology A ⊗A⊗A A. By semisimplicity (or rather, self-duality of Skyτ [T(n) ] as a 17This is not circular, since it relies on the lower-algebra definition of equivalence.

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self-bimodule), this would give (a linear category equivalent to) the Drinfeld center Skyτ [C (n) ] of Skyτ [T(n) ]. Instead, the effect of working over the tensor product Skyτ [T(n) ] ⊗Skyτ [t(n) ] Skyτ [T(n) ] is to pick the relative Drinfeld center of Skyτ [T(n) ] over Skyτ [t(n) ], the commutant of Skyτ [t(n) ] in the full center Skyτ [C (n) ]. By Proposition 6.4, this is Skyτ [F ]. Exchanging F and tn and repeating the argument we obtain the second identity in (7.1). Regarding Part (iii), this is a similar argument. Noting that Sky-τ [S] can be identified with the opposite braided tensor category of Skyτ [S], we will use the tensor category Skyτ [S] with its left-right action on itself as Morita bimodule. Furthermore, we use Skyτ [S] again to produce the Morita equivalence of  Skyτ [S] Skyτ [S] Skyτ [S]⊗Sky-τ [S]

with Vect. Indeed, after identifying one copy of Skyτ [S] with the dual category, the tensor square of Skyτ [S] over the left category is the relative Drinfeld center of Skyτ [S] over Skyτ [S] ⊗ Sky-τ [S]. However, the center is the double Skyτ [S] ⊗ Skyτ [S ∗ ], and because τ gives an isomorphism S ∼ = S ∗ , this double τ -τ is nothing but Sky [S] ⊗ Sky [S] again, with its half-braided tensor action on Skyτ [S]. Nondegeneracy of τ again ensures that the identity is the only simple object braiding trivially with everything; therefore the relative center is Vect, as desired.
8. Quantization of groupoids In this section, we outline a “canonical quantization” procedure starting from “classical” topological quantum field theories with target higher groupoids (satisfying some finiteness conditions). In other words, we elaborate on the map Sumn in (3.6). For our purposes, “higher groupoids” are spaces, with 0-groupoids being discrete sets and ordinary, or 1-groupoids, being homotopy 1-types. The case n = 2 is used in §5.2. The constraint is finiteness of the homotopy groups. We hope to develop the full story elsewhere; here we outline the construction while flagging the cases of immediate interest. There is a quantization procedure for each n ≥ 0, leading to a TQFT in the respective dimension. At its root is a linearization functor from spaces to higher algebras. (This works without finiteness.) This “higher groupoid ring” is an malgebra, which is what a TQFT in dimension n = m + 1 assigns to a point. The construction enhances the group algebra of a finite group (m = 1), or even more basically, of the vector space of functions on a finite set (m = 0). The discrepancy between m (the algebra level) and n (the TQFT dimension) — for which we apologize — is caused by the fact that the term n-vector space has been used for a much more restrictive notion than our (n − 1)-algebras. The m-algebra associated to a (connected, pointed) space X has as its category of modules the m-category of representations of the based loop space ΩX on (m−1)algebras; for disconnected spaces, we must sum over components. The dependence on base points can be removed by viewing its representations as the category of local systems of (m − 1)-algebras over X. We will write Rm (X) for the groupoid algebra when not relying on basepoint choices.

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There is an untwisted construction, leading to a TQFT of unoriented manifolds, and a construction with twists by a phase, which requires orientations. Notation 8.1. Let G be a group and A an algebra. Then A[G] denotes the group algebra of G under convolution and A(G) the A-valued functions on G under pointwise multiplication. 8.1. Low m examples. Let X be a space, which for our application may be assumed to have finitely many nonzero homotopy groups, each of which is finite. We now produce a candidate for the m-algebra Rm (X). For m = −1, our construction produces the number of isomorphism classes of points of the groupoid, weighted by their automorphisms;   (−1)i # πi (X, x) (8.1) x∈π0 X i≥1

A number counts as a “(−1)-algebra”; for natural numbers at least, which are cardinalities of sets, this matches our intuition. The weight of each isomorphism class is the alternating product of the orders of the homotopy groups, which places an obvious finiteness condition. Without it, we must abandon the top layer of the TQFT. For m = 0, we get the vector space of functions on π0 X. For m = 1, we produce the usual groupoid algebra. After a choice of basepoints, this is the direct sum of the group algebras of the π1 ’s of the components, but a Morita equivalent basepoint free construction is the path algebra of X.18 Things become more interesting for m = 2: the groupoid X = K(π2 , 2) quantizes to the 2-algebra C[π2 ][2] associated to the commutative group algebra C[π2 ]. If, in addition, a π1 is present, we get a crossed product 2-algebra π1
C[π2 ][2]. This is easiest to describe by means of the tensor structure on the category of modules of the underlying 1-algebra. As a 1-algebra, π1
C[π2 ][2] consists of the functions on π1 with values in the algebra C[π2 ], and pointwise multiplication. Its linear category of modules is Rep(π2 )(π1 ), consisting of bundles of π2 -representations over π1 . Now, the tensor category of C[π2 ][2]-modules is equivalent to Rep(π2 ) as a linear category, but carries the nonstandard tensor structure corresponding to convolution of characters. In the Fourier transformed picture, this is the category Vect(π2∗ ) of vector bundles on the Pontryagin dual group, but with the pointwise tensor structure.19 The group π1 acts by automorphisms of the tensor category Vect(π2∗ ): the action comprises the obvious automorphisms of π2 , as well as the k-invariant k ∈ H 3 (Bπ1 ; π2 ), if present. Indeed, we may interpret k as a crossed homomorphism π1 → B 2 π2 , and π2 is a group of central 2-automorphisms of Vect(π2∗ ) as a tensor category: namely, elements of π2 give 1-automorphisms (multiplications by Fourier modes) of Vect(π2∗ ) as a bimodule category over itself, which as such represents the identity morphism on the tensor category Vect(π2∗ ). Thus, we can form the desired crossed product tensor category20 π1
Vect(π2∗ ) of modules for π1
C[π2 ][2]. More succinctly but loosely, when π1 acts trivially and the k-invariant 18We are assuming a discrete model for spaces, such as simplicial sets. 19The standard tensor structure on Rep(π ) would correspond to convolution on Vect(π ∗ ). 2 2 20An alternative view: k classifies an extension of π by Bπ , which twists the obvious

crossed product tensor structure on π1
Vect(π2∗ ).

1

2

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is null, H = C[π2 ](π1 ) is a Hopf algebra over C[π2 ] and this gives a 2-algebra structure as explained in §7.1. A nontrivial action of π1 is incorporated by twisting the left H-module structure over H, as is the k-invariant. The case m = 3 gives rise to 3-algebras. Each connected component Xx is BΩx X, and R3 (Xx ) is the promotion of R2 (Ωx X) to a 3-algebra using the extra multiplication in the loop space. However, for simply connected X, a braided tensor category arises naturally. Namely, the tensor category π2
Ωk Vect(π3∗ ) was defined from the second loop space Ω2 X, which has a homotopy-commutative multiplication, homotopy-commuting with the third multiplication on R3 (Xx ). (The action of π2 is classified by the looping Ωk ∈ H 3 (Bπ2 ; π3 ) of the unique Postnikov invariant of X, k ∈ H 4 (K(π2 ; 2); π3 ).) Otherwise put, the category π2
Ωk Vect(π3∗ ) has a second multiplication compatible with the first: this structure is equivalent to a braiding. A π1 , of course, would spoil the requisite commutativity. Remark 8.2. Recall that k is equivalent to the datum of a quadratic map π2 → π3 (Lemma 5.6), so this quadratic map is all that is needed for the construction of the braiding on the category π2
Ωk Rep(π3 ). When π3 is a subgroup of T, B 2 π3 acts by automorphisms of the 2-algebra C[2] (equivalently, automorphisms of the tensor category Vect), and we see from the same construction that a braiding on Vect[π2 ] is determined by a T-valued quadratic form on π2 . 8.2. Outline of the general construction. By now, the reader may have imagined the inductive procedure for constructing the m-algebra Rm (X). The underlying vector space, or 0-algebra, comprises the (finitely supported) functions on the m-truncated homotopy  πm (X, x) × · · · × π1 (X, x). x∈π0 X

As a 1-algebra, we see the C[πm ]-valued functions on the union of the (πm−1 ×· · ·×π1 ), with pointwise multiplication. The full m-algebra structure can be described recursively. First, as an (m − 1)-algebra,  Rm (X) = Rm−1 (Ωx X). x∈π0 X

Now, the loop spaces Ωx X carry a multiplication, and the induced multiplication on each Rm−1 allows its promotion to an m-algebra. Finally, Rm is the direct sum of the resulting m-algebras. Remark 8.3. Let us go one step further in unraveling this description: for each basepoint x, π1 (X, x) acts on B 2 Ω2x X; this defines an action on the m-algebra Rm (B 2 Ω2x X), and we have Rm (Xx ) := π1 (X, x)
Rm (B 2 Ω2x X). This procedure has the advantage of producing finite-dimensional objects whenever the homotopy groups of X are finite, but the choice of basepoints in the induction step breaks hopes of functoriality. (This is similar to the construction of a minimal model for X from its Postnikov tower in rational homotopy theory.) So our notation Rm (X) is somewhat abusive: we produce something akin to the group ring of the based loop space ΩX. A remedy would be to cross with the Poincar´e groupoids, instead of π1 . This would produce a Morita equivalent algebra; but even with a finite model for a higher groupoid, there is then little hope of a finite-dimensional answer.

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8.3. Twisting by cohomology classes in T. The construction of the malgebra (and in fact of the entire n = (m + 1)-dimensional TQFT) can be twisted by a class τ ∈ H n (X; T) of our space X. This gives a projective cocycle for actions of ΩX on (m − 1)-algebras, and the twisted group ring can now again be defined as the m-algebra with the ‘same’ representation category. Some examples: for m = −1, the points are now weighted by their phase in H 0 (X; T) before counting. For m = 0, we get a nontrivial action of π1 on C for each component of X, and only the invariant lines are summed up to produce the vector space. For m = 1, the class gives a central extension of each π1 , which we quantize to the sum of the corresponding twisted group algebras. For m = 2 and connected X, H 3 (X; T) classifies the crossed homomorphisms π1 → π2∗ , and τ twists the action of π1 on Vect(π2∗ ), resulting in a different crossed product π1
τ,k C[π2 ][2]. Finally, for pure gerbes K(π, 2), a 4-class τ defines a quadratic map π → T, which defines a braided tensor structure on Vect[π]. The categories Skyτ [t], Skyτ [F ] from §7 are of this form. The construction of a general twisted groupoid m-algebra follows the inductive procedure sketched earlier. Note that, in the absence of twistings, Rm (X) = Rm (X)op ; this is related to the fact that the associated TQFT does not require orientations. 8.4. Construction of the quantization map Sumn . The category F H of spaces with finitely many, finite homotopy groups has disjoint unions, products and fiber products (homotopy fiber products). As in §3, define the n-category F Hn of correspondences in F H (truncated at level n). We will enhance the assignment X → Rn−1 (X) into a symmetric tensor21 functor Sumn : F Hn → Alg[n − 1]

(8.2)

into the n-category Alg[n − 1] of (n − 1)-algebras. Having fixed an X ∈ Ob(F H), we pre-compose with the mapping space functor I : Bordn → F Hn to produce the ‘TQFT with target X,’ a theory with values in Alg[n − 1]. 22

Remark 8.4. There is a similar functor F τ on the category F Hnτ , whose objects are spaces with an n-cocycle τ valued in T, 1-morphisms are correspondences equipped with a homotopy between pulled-back cocycles, and so forth. This leads to the twisted TQFTs for oriented manifolds; we will not spell out its details here. Whereas on objects Sumn (X) = Rn−1 (X) the formula for morphisms is increasingly complex we pause for a moment to explain why. If Rn−1 was a contravariant functor with “good” tensor properties, we would just apply it to morphisms of all levels. The reader might think of the algebra of cochains on a space, which converts correspondences to bimodules. (As a technical aside, this does not always have the required tensor properties, but a closely related functor does: the coalgebra of chains, with cotensor products.) However, our Rn−1 is the Koszul dual functor of chains on the based loop space. We thus use Koszul duality as a recipe to define Sumn . The general rule is that a correspondence at level k, given by a k-storied diagram of spaces, gets sent to the colimit of the same diagram of group rings. 21Sum

n

will take disjoint unions to direct sums and fiber products to tensor products.

22This is a particular choice of ‘C’ in §3.

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We illustrate this for 1- and 2-morphisms. We may assume connectivity of our spaces (separate components are handled separately). For a correspondence c : C → X × Y , call F the homotopy fiber of c. It may be viewed as the antidiagonal quotient (ΩX ×ΩY )/ΩC; the obvious action of ΩX ×ΩY on this last space is the structural one on F . To C, we assign the (Rn−1 (X), Rn−1 (Y ))-bimodule  Sumn (C) := Rn−1 (X) Rn−1 (Y ) ∼ = Rn−2 (F ). Rn−1 (C)

This is an (n − 2)-algebra, as the top multiplication layer has been used up to build the tensor product. The geometric ΩX × ΩY actions on F induce the bimodule structure on Rn−2 (F ) Now let D → C × C  be a correspondence of correspondences; the compositions D → X × Y must agree. With H denoting the homotopy fiber of the last map, there is induced a map h : H → F × F  with compatible ΩX × ΩY action. The homotopy fiber G of h carries ΩF × ΩF  actions and intertwining ΩX × ΩY actions; in fact, (8.3)

G∼

Ω2 X × Ω2 Y ΩF × ΩF  ∼ 2 , ΩH Ω C ×Ω2 D Ω2 C 

with the left × right action of Ω2 C × Ω2 C  on each factor in the second numerator; we have just enough commutativity to mod out by the antidiagonal Ω2 D. The desired multi-module is  Rn−2 (F  ) Sumn (D) := Rn−3 (G) = Rn−2 (F ) Rn−2 (H)

= Rn−2 (X)



Rn−2 (Y );

Rn−2 (C)⊗Rn−2 (D) Rn−2 (C
)

in the first presentation, tensoring in the (n−2)-multiplications ensures that we stay in the realm of “(Rn−1 (X), Rn−1 (Y )) bialgebra (Rn−2 (F ), R(n−2) (F  ))-bialgebras,” but uses up the top two level products on Sumn (D) and leaves an (n − 3)-algebra. In the second presentation, we view D as a (C, C  )-correspondence in a lowerdimensional field theory and tensor over the (n − 2)-algebra Sumn−1 (D), acting left × right on each factor. The fun continues, but we will stop. Remark 8.5. Our description of Sumn involves a crude simplification, relying on the fact that the morphisms of algebras were represented by maps of underlying vector spaces, rather than bi-modules. The tensor product B ⊗A C must generally be replaced by M ⊗A N , where M and N are the bi-modules realizing morphisms C → A, A → B. We hope to return to a full treatment of this topic in a future paper. Remark 8.6. Recall the finite group F from §5.1. Consider the gerbe B 2 F , equipped with the class τ ∈ H 4 (B 2 F ; T) described in Lemma 5.6. The space C of maps from a closed 4-manifold X to B 2 F can be viewed as a top-level correspondence from the point to itself, and (8.1) is the Gauss sum computed in §5.2. 9. The quantum theories The braided tensor categories Skyτ [F ], Skyτ [t(n) ] and Skyτ [t] generate 4-dimensional TQFTs with target space the corresponding (τ -twisted) gerbes. Their

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4-manifold invariants were computed as a path integral in §5.2; we have seen their agreement with the numbers provided by the general quantization procedure. The invariants for t need to be renormalized by a power of the volume of t; this is explained by the increasing number of points in its finite approximation t(n) . We discuss these theories further in §9.1. 9.1. The quantum gerbe theories on F, t and t(n) . We now describe the theories AS on closed manifolds of dimension below 4; At can be justified either as the limit of At(n) ’s, or in its own right by judicious use of the word ‘continuous’. The Lie algebra theory also has a few interesting variations, which tie in with positive energy representations of Lt and their fusion. The point. We already know the braided categories Skyτ [F ], Skyτ [t], and τ (n) Sky [t ] assigned by A to a positively oriented point; A sends the negative point to the opposite object, the category with opposite braiding. Recall that, in all cases, the twisting τ ∈ H 4 (−; T) which defines the braided structure is given by a nondegenerate quadratic map q to T (see (5.12), (5.16), §6.1). The circle. Here, AF assigns the 2-algebra quantization of the groupoid LB 2 F = BF × B 2 F , with twisting Ωτ ∈ H 3 (LB 2 F ; T) transgressed from τ ; this class Ωτ represents the bihomomorphism b : F × F → T derived from q. According to §8.3, the quantization is the crossed product R := F
Ωτ C[F ][2] with action twisted by the transgression of τ . We have several pictures for this. (1) The tensor category of R-modules is F
b Vect(F ∗ ), after identifying C[F ]modules with vector bundles on F ∗ . The group F acts on F ∗ by translation, via b. Note that this is equivalent to the matrix algebra MF ∗ ×F ∗ (Vect) on Vect(F ∗ )! (2) R-modules in algebras are C[F ]-algebras with an action of the group F by automorphisms, which twisted commutes with the central C[F ]: af (f  ) = f  ·b(f, f  ) for f, f  ∈ F , with af denoting the automorphism and f  ∈ F embedded in the central C[F ]. (3) Related to this is the 2-category of R-linear categories: these are C-linear categories with a projective action of F × BF with cocycle b. This is an action of F by linear functors plus a second action of F by automorphisms of the identity functor (central automorphisms of all objects), which must be related by af (fx ) = fa f (x) · b(f, f  ), for each object x. (4) The same 2-category has a different presentation which emphasizes its loopy nature: the 2-category Br τ (F ) of (fully) braided bimodule categories over the braided category Skyτ [F ]. The full braiding gives the BF -action. We will see that its counterpart Br τ (t) relates nicely to positive energy representations of Lt. The algebra C[F ] is a module object over R: this is because the linear category Rep(F ) of C[F ]-modules is naturally a module over the tensor category F
b Vect(F ∗ ) in (i); more precisely, it is the standard simple module of MF ∗ ×F ∗ (Vect). This implies the following. Proposition 9.1. For nondegenerate q, R is Morita equivalent to C[2], with bimodule C[F ]. Closed surfaces. The space of maps from a closed surface Σ to B 2 F factors as B 2 F × H 1 (Σ; F ) × F . The transgression τΣ of the twisting cocycle has two components: one of them defines the Heisenberg central extension of H 1 (Σ; F ) constructed from q and the Poincar´e duality pairing, the other, on B 2 F , gives the

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character f → b(f, f0 ), in the component labeled by f0 . Quantization produces the   group ring, the Weyl algebra Wτ H 1 (Σ; F ) for the component f0 = 0, and kills the other components. At this stage, invertibility of the theory has become obvious, because we get a matrix algebra. Despite this, the theory is not trivial on surfaces. Indeed, while the mapping class group of Σ acts by automorphisms on the Weyl algebra, this conceals a central extension by eighth roots of unity, given (when t has rank one) by the reduction mod 8 of the Maslov index. This appears when attempting to lift the odinger representations action on Lagrangian subspaces in H 1 to the associated Schr¨ [24, §I.4]. Remark 9.2. Even when this extension is trivial, such as for spin surfaces, two trivializations may differ by a (half-integral) power of the determinant line. This stems from an invertible 3-dimensional theory, and the usual framing anomaly in Chern – Simons theory can be concealed therein. 3-manifolds with boundary. A bounding 3-manifold M gives a Lagrangian subodinger representation S of W (∂M ) space in H1 (Σ); we can then form the Schr¨ [24]. This is the morphism AF (M ) from C to the Weyl algebra. Gluing two such manifolds into a closed one produces the line HomW (∂M ) (S− , S+ ). This line has a preferred trivialization on any oriented 3-manifold, but also carries a natural action of the group Z of global frame changes. 9.2. Chern – Simons theory. Consider the 3-dimensional gauge theory Z(n) with finite group T(nF ) and twisting class τ ∈ H 3 (BT(nF ) ; T). Such theories were described in §4.2. From the perspective of Theorem 2.4, the theory Zn is generated by the tensor category23 Skyτ [T(nF ) ]. Similarly, we should view Skyτ [T ] as the generating tensor category for a theory Z, which is an L2 version of Chern – Simons theory with gauge group T . For example, the vector space associated to a closed surface is the space of L2 sections of the Theta-line bundle Θ(τ ) on the T -Jacobian JT of flat bundles (whereas the usual, holomorphic Chern – Simons theory would supply the holomorphic sections). It is easiest to justify the relation of Skyτ [T ] to L2 gauge theory by using the finite approximations Z(n) : sections of Θ(τ ) on T(nF ) -bundles approximate L2 (JT ) as n → ∞. We will see in §9.1 that this goes for 1- and 3-dimensional outputs as well. (The limit must be regularized, so that, for instance, summation over torsion points becomes integration in the limit. L2 Chern – Simons itself requires regularization: vectors of bounding 3-manifolds are δ-sections on the respective Lagrangians in JT , and the 3-manifold invariant is only ‘obviously’ finite for for rational homology spheres.) One picture of a 3-dimensional TQFT, as a stand-alone theory, is as an endomorphism of the trivial 4-dimensional theory, with the top level truncated.24 If, however, a 3D theory is a module over another, nontrivial 4D TQFT A — meaning that the algebras arising various dimensions is are module objects over their 4D counterparts, in a consistent way — then it can be viewed as a truncated morphism 1 → A . If A is invertible, this is the notion of an ‘anomalous field theory’ with 23In 2-algebra language, Skyτ [T (nF ) ] is the tensor category of modules for the algebra of

functions on T(nF ) , with τ -twisted Hopf 2-algebra structure. 24Without truncation, all morphisms between TQFTs are isomorphisms. At any rate, a 3D TQFT does not supply anything in dimension 4.

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anomaly A . Similarly, a bimodule for theories (At , AF ) yields a truncated morphism AF → At . Of course, concerning Z, we have the gerbe theories for F and t in mind. The following summarizes our results. Theorem 9.3. (i) The L2 -Chern – Simons theory Z generated by Skyτ [T ] gives an isomorphism AF → At of oriented 4D theories. (ii) The finite gauge theory Z(n) generated by Skyτ [T(nF ) ] gives an isomorphism AF → At(n) of oriented 4D theories. (iii) Z gives a truncated morphism 1 → At of oriented 4D theories. (iv) A Morita equivalence Vect ∼ Skyτ [t] induces an isomorphism At → 1; after a composition with such an isomorphism, Z becomes isomorphic to 3D (holomorphic) Chern – Simons theory. (v) Z(n) gives a truncated morphism 1 → At(n) of oriented 4D theories. A Morita equivalence Vect ∼ Skyτ [t(n) ] induces an isomorphism At(n) → 1; after a composition with such an isomorphism, Z(n) becomes isomorphic to 3D (holomorphic) Chern – Simons theory. Remark 9.4. Recall from §6.3 that a Morita isomorphism as in (iii) and (v) can be found whenever 8 divides the signature of τ , or anytime we work with graded objects and spin manifolds. Items (ii) and (v) are perhaps not of much interest, but in the n → ∞ limit they serve to justify rigorously the claims (i), (iii) and (iv); without that, we would need to delve into topological categories and their continuous centers. The truncated morphism Z in (iii) is (holomorphic) Chern – Simons as an anomalous theory. The theorem is really a corollary of Proposition 6.5 — or at least it would be so if we supplied the information needed to make the braided tensor categories into fully dualizable objects with SO(4)-actions in the world of braided tensor categories.25 We shall not do this; instead let us explain what the theories assign in each dimension. In particular, we will recover the usual modular tensor category [27] for holomorphic Chern – Simons for T as the relative Drinfeld center of Skyτ [T ] over Skyτ [t]. 9.3. Chern – Simons as an anomalous theory. We condense the relations between our theories on closed manifolds X in Table 1, in which the third column gives a Morita isomorphism between the second and fourth columns. We have written Z(X) for the L2 Chern – Simons invariant of a 3-manifold, a renormalized count of the flat T -bundles on X [22], while AF,t (X) refers to the 4-manifold invariants computed in §5.2. There is a matching equivalence with the finite theories for T(n) , t(n) ; see Table 2. Experts may have recognized in the right column the double of Chern – Simons theory for the torus T at level τ . This is explained as follows. The second and third columns in each table give our advertised description of (holomorphic) Chern – Simons theory as an anomalous theory. We claim that Z is ‘finite as a module over At , with holomorphic Chern – Simons theory as a basis’ (and similarly for Z(n) and At(n) ). The most obvious instance is that of the vector 25Something is missing, as can be seen from the invariant computation in §5.2, where an Euler characteristic-coupled parameter must be supplied.

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Table 1. Theories on closed manifolds At (X)

dim X 0 1 2 3 4

Z(X)

AF (X)

Sky [t] Sky [T ] Skyτ [F ] Skyτ [C] F
b Sky[F ∗ ] t
b Sky[t∗ ]  1      Wτ H (X; t) L2 JT (X); Θ(τ ) Wτ H 1 (X; F ) C Z(X) C – AF (X) At (X) τ

τ

Table 2. Finite theories on closed manifolds At(n) (X)

dim X 0 1 2 3 4 space (9.1)

Z(n) (X)

AF (X)

Skyτ [t(n) ] Skyτ [T(nF ) ] Skyτ [F ] ∗(n) τ (n) ] Sky [C ] F
b Sky[F ∗ ] t
b Sky[t  1      Wτ H (X; t(n) ) L2 JT(nF ) (X); Θ(τ ) Wτ H 1 (X; F ) C C Z(n) (X) – AF (X) At(n) (X) (n)

  Z(Σ) = L2 JT (Σ); Θ(τ ) ,

for a closed surface Σ. The theory of Theta-functions tells us that, after a choice of complex structure, (9.1) factors into holomorphic and anti-holomorphic sectors; the former is the space of holomorphic Theta-functions, an irreducible representation of the Heisenberg group on H ( Σ; F ), and the latter the anti-holomorphic Fock representation of the Weyl algebra W H 1 (Σ; t) . One dimension down, the center Skyτ [C] of Skyτ [T ] plays the role of the modular tensor category for L2 Chern – Simons. The braided structure on Skyτ [C] makes this into a braided bimodule over Skyτ [t], thus a t
b Sky[t∗ ]-module. It is free with basis Skyτ [F ], in the sense that it converts into the latter, after the Morita equivalence of t
b Skyτ [t∗ ] with 1 defined by Sky[t∗ ]. Such a Morita equivalence can be induced from a trivialization of At by means of Skyτ ’ [T ] at a level τ  giving a trivial group F  ;26 for indeed, Skyτ ’ [C  ] is then just Sky[t∗ ], as a braided bimodule. (The change to another level of the same signature can be accommodated by scaling the Lie algebra.) This is also related to the computation of the relative center of Skyτ [T ] over the braided category Skyτ [t]. In dimension 0, our data for Chern – Simons theory is new. 9.4. Surfaces with boundary in another model for t-gerbe theory. In this model Rep(Lt) for Skyτ [t], closely related to loop groups, the objects are semisimple, projective, positive energy modules of the smooth loop Lie algebra Lt, with projective cocycle  (ξ, η) →

ξ, dη.

26Using graded vector spaces, see §6.3.

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D. S. FREED, M. J. HOPKINS, J. LURIE, AND C. TELEMAN

These representations are invariant under diffeomorphisms of the circle; noninvariantly, they factor into a semisimple representation of t and the Fock representation of Lt/t, the latter being a Morita factor in the equivalence. The fusion of two representations is defined using a pair of pants P : the Weyl algebra Wτ (P ) of the pair of pants (see below) accepts maps from three commuting copies of the Weyl algebra Wτ (Lt) of Lt. The fusion of two boundary representations is the induced module from their tensor product, as a Wτ (P )-module, to the third boundary. Fusion gives a braided tensor structure on Rep(Lt), which makes it equivalent to Skyτ [t]. Wτ (Lt) itself is the underlying 3-algebra, when equipped with the fusion product via the braided tri-module Wτ (P ). Associated to a surface Σ with boundary is the symplectic vector space SΣ := Ω1 (Σ)/dΩ0 (Σ, ∂Σ) of closed forms modulo differentials of functions vanishing on ∂Σ; the symplectic form is Σ ϕ ∧ ψ. The Weyl algebra Wτ (SΣ ⊗ t) is a braided bialgebra for the Wτ (Lt)’s at each boundary circle. This promotes Wτ (Σ) to an object of Br τ , the 2-category (equivalent to that) of t
b Sky[t∗ ]-modules, and the thus promoted Wτ (Σ) is At (Σ).  for SΣ that removes the Morita factors Fock(Lt/t) from W uses A model SΣ only differentials whose restriction to each boundary circle is constant (in a fixed parameterization). We lose the connection to loop groups, but this is convenient for describing Z(Σ) (without having to Morita-modify C). Now, Z(Σ) is a functor between products of copies of Skyτ [C], one for each boundary circle: in effect, a vector bundle over a product C n . The moduli space JT (Σ, ∂Σ) of flat T -bundles on Σ equipped with constant connections on ∂Σ projects to tn by the boundary holonomies; call this map p. Z(Σ) is the bundle over tn of fiber-wise L2 sections of Θ(τ ) along p; it is naturally a finite module over Wτ (S  ⊗ t) under its translation of the Jacobian. The F -components of Z(Σ) are determined by the weight space decomposition under the actions of F at the boundaries. References ´ 1. M. Atiyah, Topological quantum field theories, Inst. Hautes Etudes Sci. Publ. Math. 68 (1988), 175 – 186. 2. M. Atiyah and G. Segal, Twisted K-theory, Ukr. Mat. Visn. 1 (2004), no. 3, 287 – 330; English transl., Ukr. Math. Bull. 1 (2004), no. 3, 291 – 334. 3. J. C. Baez and J. Dolan, Higher-dimensional algebra and topological quantum field theory, J. Math. Phys. 36 (1995), no. 11, 6073 – 6105. 4. B. Bakalov and A. Kirillov Jr., Lectures on tensor categories and modular functors, Univ. Lecture Ser., vol. 21, Amer. Math. Soc., Providence, RI, 2001. 5. J. W. Barrett and B. W. Westbury, Spherical categories, Adv. Math. 143 (1999), no. 2, 357 – 375. 6. D. Belov and G. W. Moore, Classification of abelian spin Chern – Simons theories, available at arXiv:hep-th/0505235. 7. D. Ben-Zvi and D. Nadler, The character theory of a complex group, available at arXiv: 0904.1247. 8. R. Bott, Lectures on characteristic classes and foliations, Lectures on Algebraic and Differential Topology (Mexico City, 1971), Lecture Notes in Math., vol. 279, Springer, Berlin, 1972, pp. 1 – 94. 9. J.-L. Brylinski, Gerbes on complex reductive Lie groups, available at arXiv:math/0002158. 10. L. Crane and D. Yetter, A categorical construction of 4D topological quantum field theories, Quantum Topology, Ser. Knots Everything, vol. 3, World Sci. Publ., River Edge, NJ, 1993, pp. 120 – 130. 11. C. Douglas, Two-dimensional algebra and quantum Chern – Simons theory, Topological Field Theory (Evanston, IL, 2009).

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TQFTS FROM COMPACT LIE GROUPS

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12. S. Eilenberg and S. Mac Lane, On the groups H(Π, n). II: Methods of computation, Ann. of Math. (2) 60 (1954), 49 – 139. 13. D. S. Freed, Classical Chern – Simons theory. II, Houston J. Math. 28 (2002), no. 2, 293 – 310. , Higher algebraic structures and quantization, Comm. Math. Phys. 159 (1994), no. 2, 14. 343 – 398. 15. D. S. Freed, M. J. Hopkins, and C. Teleman, Twisted K-theory and loop group representations, available at arXiv:math/0312155. , Loop groups and twisted K-theory. I, available at arXiv:0711.1906; II, available at 16. arXiv:math/0511232. 17. M. J. Hopkins and I. M. Singer, Quadratic functions in geometry, topology, and M-theory, J. Differential Geom. 70 (2005), no. 3, 329 – 452. 18. M. Kontsevich, Rational conformal field theory and invariants of 3-manifolds, preprint, available at www.ihes.fr/~maxim/TEXTS/3TQFT_and_RCFT.pdf. 19. M. Kreck and P. Teichner, Positivity of topological field theories in dimension at least 5, J. Topol. 1 (2008), no. 3, 663–670. 20. J. Milnor and D. Husemoller, Symmetric bilinear forms, Ergeb. Math. Grenzgeb., vol. 73, Springer, New York, 1973. 21. J. Lurie, On the classification of topological field theories, available at www-math.mit.edu/ ~lurie/. 22. M. Manoliu, Abelian Chern – Simons theory. I: A topological quantum field theory, J. Math. Phys. 39 (1998), no. 1, 170 – 206; II: A functional integral approach, 207 – 217. 23. M. M¨ uger, From subfactors to categories and topology. II: The quantum double of tensor categories and subfactors, J. Pure Appl. Algebra 180 (2003), no. 1-2, 159 – 219. 24. A. Polishchuk, Abelian varieties, theta functions and the Fourier transform, Cambridge Tracts in Math., vol. 153, Cambridge Univ. Press, Cambridge, 2003. 25. F. Quinn, Lectures on axiomatic topological quantum field theory, Geometry and Quantum Field Theory (Park City, UT, 1991), IAS/Park City Math. Ser., vol. 1, Amer. Math. Soc., Providence, RI, 1995, pp. 323 – 453. 26. G. D. Segal, Stanford notes. Lecture 1: Topological field theories, available at www.cgtp.duke. edu/ITP99/segal/stanford/lect1.pdf. 27. S. D. Stirling, Abelian Chern – Simons theory with toral gauge group, modular tensor categories, and group categories, available at arXiv:0807.2857. 28. V. G. Turaev, Quantum invariants of knots and 3-manifolds, de Gruyter Stud. in Math., vol. 18, de Gruyter, Berlin, 1994. 29. K. Walker, Fields, blobs, and TQFTs, available at canyon23.net/math/talks/. 30. E. Witten, Quantum field theory and the Jones polynomial, Comm. Math. Phys. 121 (1989), no. 3, 351 – 399. Department of Mathematics, University of Texas, 1 University Station C1200, Austin, TX 78712-0257, USA E-mail address: [email protected] Department of Mathematics, Harvard University, 1 Oxford Street, Cambridge, MA 02138, USA E-mail address: [email protected] Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139-4307, USA E-mail address: [email protected] Department of Mathematics, University of California, Berkeley, 970 Evans Hall #3840, Berkeley, CA 94720-3840, USA E-mail address: [email protected]

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Titles in This Series 50 P. Robert Kotiuga, Editor, A celebration of the mathematical legacy of Raoul Bott, 2010 49 Miguel Abreu, Fran¸ cois Lalonde, and Leonid Polterovich, Editors, New perspectives and challenges in symplectic field theory, 2009 48 David Avis, David Bremner, and Antoine Deza, Editors, Polyhedral computation, 2009 47 John Harnad and Pavel Winternitz, Editors, Groups and symmetries: From Neolithic Scots to John McKay, 2009 46 Jean-Marie De Koninck, Andrew Granville, and Florian Luca, Editors, Anatomy of integers, 2008 45 Panos M. Pardalos and Pierre Hansen, Editors, Data mining and mathematical programming, 2008 44 Stanley Alama, Lia Bronsard, and Peter J. Sternberg, Editors, Singularities in PDE and the calculus of variationa, 2007 43 Andrew Granville, Melvyn B. Nathanson, and Jozsef Solymosi, Editors, Additive combinatorics, 2007 42 Donald A. Dawson, Vojkan Jakˇ si´ c, and Boris Vainberg, Editors, Probability and mathematical physics: a volume in honor of Stanislav Molchanov, 2007 41 Andr´ e Bandrauk, Michel C. Delfour, and Claude Le Bris, Editors, High-dimensional partial differential equations in science and engineering, 2007 40 V. Apostolov, A. Dancer, N. Hitchin, and M. Wang, Editors, Perspectives in Riemannian geometry, 2006 39 P. Winternitz, D. Gomez-Ullate, A. Iserles, D. Levi, P. J. Olver, R. Quispel, and P. Tempesta, Editors, Group theory and numerical analysis, 2005 38 Jacques Hurtubise and Eyal Markman, Editors, Algebraic structures and moduli spaces, 2004 37 P. Tempesta, P. Winternitz, J. Harnad, W. Miller, Jr., G. Pogosyan, and M. Rodriguez, Editors, Superintegrability in classical and quantum systems, 2004 36 Hershy Kisilevsky and Eyal Z. Goren, Editors, Number theory, 2004 35 H. E. A. Eddy Campbell and David L. Wehlau, Editors, Invariant theory in all characteristics, 2004 34 P. Winternitz, J. Harnad, C. S. Lam, and J. Patera, Editors, Symmetry in physics, 2004 33 Andr´ e D. Bandrauk, Michel C. Delfour, and Claude Le Bris, Editors, Quantum control: Mathematical and numerical challenges, 2003 32 Vadim B. Kuznetsov, Editor, The Kowalevski property, 2002 31 John Harnad and Alexander Its, Editors, Isomonodromic deformations and applications in physics, 2002 30 John McKay and Abdellah Sebbar, Editors, Proceedings on moonshine and related topics, 2001 29 Alan Coley, Decio Levi, Robert Milson, Colin Rogers, and Pavel Winternitz, Editors, B¨ acklund and Darboux transformations. The geometry of solitons, 2001 28 J. C. Taylor, Editor, Topics in probability and Lie groups: Boundary theory, 2001 27 I. M. Sigal and C. Sulem, Editors, Nonlinear dynamics and renormalization group, 2001 26 J. Harnad, G. Sabidussi, and P. Winternitz, Editors, Integrable systems: From classical to quantum, 2000 25 Decio Levi and Orlando Ragnisco, Editors, SIDE III—Symmetries and integrability of difference equations, 2000 24 B. Brent Gordon, James D. Lewis, Stefan M¨ uller-Stach, Shuji Saito, and Noriko Yui, Editors, The arithmetic and geometry of algebraic cycles, 2000 23 Pierre Hansen and Odile Marcotte, Editors, Graph colouring and applications, 1999

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A five-day conference celebrating the legacy of Raoul Bott was held at the CRM on June 9–13, 2008. The conference focused on the extraordinary impact Bott had on both topology and interactions between mathematics, physics and technology. The conference was co-organized by the Clay Mathematics Institute and had support from the National Science Foundation (Award 0805925). Montréal was a natural venue for such an event since Raoul Bott obtained two degrees in Electrical Engineering at McGill University in the 1940s and an Honorary Doctorate from McGill in 1987. The fact that Bott’s presence is still fresh in the minds of all those involved made for a tremendous amount of enthusiasm and every attempt has been made to channel this energy into this book. The contributions to this book come from three generations of Bott’s students, coauthors, and fellow kindred spirits in order to cover six decades of Bott’s research, identify his enduring mathematical legacy, and the consequences for emerging fields. The contributions can be read independently. In order to help a whole to emerge from the parts, the book is broken into four sections and to make the book accessible to a wide audience, each section starts with easier-to-read reminiscences and works its way into more involved papers.

CRMP/50

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