Quantum theory and symmetries with Lie theory... Vol. 1 9789811327148, 9789811327155

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Springer Proceedings in Mathematics & Statistics

Vladimir Dobrev Editor

Quantum Theory and Symmetries with Lie Theory and Its Applications in Physics Volume 1 QTS-X/LT-XII, Varna, Bulgaria, June 2017

Springer Proceedings in Mathematics & Statistics Volume 263

Springer Proceedings in Mathematics & Statistics This book series features volumes composed of selected contributions from workshops and conferences in all areas of current research in mathematics and statistics, including operation research and optimization. In addition to an overall evaluation of the interest, scientific quality, and timeliness of each proposal at the hands of the publisher, individual contributions are all refereed to the high quality standards of leading journals in the field. Thus, this series provides the research community with well-edited, authoritative reports on developments in the most exciting areas of mathematical and statistical research today.

More information about this series at http://www.springer.com/series/10533

Vladimir Dobrev Editor

Quantum Theory and Symmetries with Lie Theory and Its Applications in Physics Volume 1 QTS-X/LT-XII, Varna, Bulgaria, June 2017

123

Editor Vladimir Dobrev Institute for Nuclear Research and Nuclear Energy Bulgarian Academy of Sciences Sofia, Bulgaria

ISSN 2194-1009 ISSN 2194-1017 (electronic) Springer Proceedings in Mathematics & Statistics ISBN 978-981-13-2714-8 ISBN 978-981-13-2715-5 (eBook) https://doi.org/10.1007/978-981-13-2715-5 Library of Congress Control Number: 2018955909 Mathematics Subject Classification (2010): 11M32, 11R42, 19F27, 35Q53, 17B37, 37J35, 70H06, 17A70, 20G42, 33D80, 81R50, 58B34, 81R60, 83C65, 17B35, 17B65, 22E65, 81R10, 17B80, 37K10, 91B80, 11S40 © Springer Nature Singapore Pte Ltd. 2018 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Preface of Volume 1

This is the first volume of the Proceedings of the joint conference X. International Symposium “Quantum Theory and Symmetries” (QTS-10) and XII. International Workshop “Lie Theory and Its Applications in Physics” (LT-12), 19–25 June 2017, Varna, Bulgaria. The first symposium of the QTS series was held in Goslar, Germany, in 1999; then, it was held in Cracow (2001), Cincinnati (2003), Varna (2005), Valladolid (2007), Lexington (2009), Prague (2011), Mexico City (2013), Yerevan (2015). The series started around the core concept that symmetries underlie all descriptions of quantum systems. It has since evolved to a symposium on the frontiers of theoretical and mathematical physics (for more details on this series, see here http://theo.inrne.bas.bg/*dobrev/QTS-homepage.htm). The LT series covers the whole field of Lie theory in its widest sense together with its applications in many facets of physics. As an interface between mathematics and physics, the workshop serves as a meeting place for mathematicians and theoretical and mathematical physicists. The first three workshops of the LT series were organised in Clausthal (1995, 1997, 1999), the fourth was part of the 2nd Symposium ‘Quantum Theory and Symmetries’ in Cracow (2001), the fifth was organised in Varna (2003), the sixth was part of the 4th Symposium ‘Quantum Theory and Symmetries’ in Varna (2005), but has its own volume of Proceedings, and the seventh, eighth, ninth, tenth were organised in Varna (2007, 2009, 2011, 2013); see: http://theo.inrne.bas.bg/*dobrev/. In the division of the material between the two volumes, we have tried to select for the first, respectively, second, volume more mathematics, respectively, physics, oriented papers. However, this division is relative since many papers could have been placed in either volume. The scientific level was very high as can be judged by the speakers. The plenary speakers contributing to Volume 1 are: Branko Dragovich (University of Belgrade and Serbian Academy of Sciences and Arts), Tamar Friedmann (Smith College, Northampton, MA), Malte Henkel (Université de Lorraine, Nancy), Alexey Isaev (JINR, Dubna), Anthony Joseph (Weizmann Institute of Science, Israel), Toshiyuki Kobayashi (University of Tokyo and Kavli IPMU), Takeo Kojima (Yamagata v

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Preface of Volume 1

University), Ivan Todorov (INRNE, BAS, Bulgaria), J. Van der Jeugt (Ghent University). The topics covered the most modern trends in the field of the joint conferences: Representation Theory, Integrability, Entanglement, Quantum Groups and Noncommutative Geometry, Algebraic Geometry and Number Theory, Various Mathematical Results. The joint meeting of QTS-10 and LT-12 was organised by the Institute of Nuclear Research and Nuclear Energy of the Bulgarian Academy of Sciences (BAS) in June 2017 at the International House of Scientists “Frederic Joliot-Curie” (IHS) on the Black Sea Coast near Varna. The overall number of participants in the 2017 joint conference was 130, and they came from 33 countries. (The list is given at the end of the volume.) The organising committee was: V. K. Dobrev (Chairman), L. K. Anguelova, V. I. Doseva, V. G. Filev, A. Ch. Ganchev, K. K. Marinov, D. T. Nedanovski, T. V. Popov, D. R. Staicova, M. N. Stoilov, N. I. Stoilova, S. T. Stoimenov.

Acknowledgements We express our gratitude to the – Institute of Nuclear Research and Nuclear Energy – Abdus Salam International Centre for Theoretical Physics – International Association of Mathematical Physics for financial help. We thank the International Advisory Committee and the QTS Conference Board (see the web-page) for the invaluable help. We thank the Publisher, Springer Japan, represented by Ms. Chino Hasebe (Executive Editor in Mathematics, Statistics, Business, Economics, Computer Science) and Mr. Masayuki Nakamura (Editorial Department), for assistance in the publication. Last but not least, I thank the members of the organising committee who, through their efforts, made the workshop run smoothly and efficiently. Sofia, Bulgaria May 2018

Vladimir Dobrev

Contents

Part I

Plenary Talks

Image of Conformally Covariant, Symmetry Breaking Operators for Rp;q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Toshiyuki Kobayashi and Alex Leontiev

3

Trails for Minuscule Modules and Dual Kashiwara Functions for the Bð‘Þ Crystal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Anthony Joseph

37

From Euler’s Play with Infinite Series to the Anomalous Magnetic Moment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ivan Todorov

55

On the Derivation of the Wallis Formula for p in the 17th and 21st Centuries . . . . . . . . . . . . . . . . . . . . . . . . . . . Tamar Friedmann

75

Variations of Infinite Derivative Modified Gravity . . . . . . . . . . . . . . . . . Ivan Dimitrijevic, Branko Dragovich, Zoran Rakic and Jelena Stankovic

91

Infinite-Dimensional Metaconformal Symmetries: 1D Diffusion-Limited Erosion and Ballistic Transport in ð1 þ 2Þ Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 Malte Henkel and Stoimen Stoimenov Behrends–Fronsdal Spin Projection Operator in Space-Time with Arbitrary Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 A. P. Isaev and M. A. Podoinitsyn Wakimoto Realization of the Quantum Affine Superalgebra b Uq ð slðMjNÞÞ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 Takeo Kojima

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viii

Contents

On Superdimensions of Some Infinite-Dimensional Irreducible Representations of ospðmjnÞ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 N. I. Stoilova, J. Thierry-Mieg and J. Van der Jeugt Non-commutativity in Unified Theories and Gravity . . . . . . . . . . . . . . . 177 G. Manolakos and G. Zoupanos Part II

Representation Theory

Webs of Quantum Algebra Representations in 5d N ¼ 1 Super Yang–Mills . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 Jean-Emile Bourgine Parallelizations for Induced Representations of SO0 ð2; qÞ . . . . . . . . . . . 219 Patrick Moylan Jordan Algebra and Hydrogen Atom . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 Todor Popov Screening Operators for the Lattice Vertex Operator Algebras of Type A1 at Positive Rational Level . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 Takuya Matsumoto Symmetries of the S3 Dirac–Dunkl Operator . . . . . . . . . . . . . . . . . . . . . 255 Hendrik De Bie, Roy Oste and Joris Van der Jeugt Part III

Integrability

The Superintegrable Zernike System . . . . . . . . . . . . . . . . . . . . . . . . . . . 263 Natig M. Atakishiyev, George S. Pogosyan, Cristina Salto-Alegre, Kurt Bernardo Wolf and Alexander Yakhno The Two Bosonizations of the CKP Hierarchy: Bicharacter Construction and Vacuum Expectation Values . . . . . . . . . . . . . . . . . . . . 275 Iana I. Anguelova Recursion Operator and Bäcklund Transformation for Super mKdV Hierarchy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293 A. R. Aguirre, J. F. Gomes, A. L. Retore, N. I. Spano and A. H. Zimerman Lie Symmetry Analysis of a Third-Order Equation Arising from a General Class of Lotka–Volterra Chains . . . . . . . . . . . . . . . . . . 311 Kyriakos Charalambous and Christodoulos Sophocleous Part IV

Entanglement

Probing Anderson Localization Using the Dynamics of a Qubit . . . . . . . 321 Hichem Eleuch, Michael Hilke and Richard MacKenzie

Contents

ix

Pure Spinors, Impure Spinors and Quantum Mechanics . . . . . . . . . . . . 331 Mike Hewitt Higher-Derivative Oscillators in AdS5  S5 T-Dual Penrose Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337 H. Dimov, S. Mladenov, R. Rashkov and T. Vetsov Part V

Quantum Groups and Related Structures

A Unified Approach to Poisson–Hopf Deformations of Lie–Hamilton Systems Based on sl(2) . . . . . . . . . . . . . . . . . . . . . . . . 347 Ángel Ballesteros, Rutwig Campoamor-Stursberg, Eduardo Fernández-Saiz, Francisco J. Herranz and Javier de Lucas Two-Parameter Quantum General Linear Supergroups . . . . . . . . . . . . . 367 Huafeng Zhang Noncommutative Geometry and An Index Theory of Infinite-Dimensional Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377 Doman Takata Part VI

Various Mathematical Results

A Stable Version of Terao Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . 385 Cristian Anghel Multiplication of Distributions and Nonperturbative Calculations of Transition Probabilities . . . . . . . . . . . . . . . . . . . . . . . . . 393 J. F. Colombeau, J. Aragona, P. Catuogno, S. O. Juriaans and C. Olivera About Markov, Gibbs, ... Gauge Theory ... Finance . . . . . . . . . . . . . . . . 403 Alexander Ganchev A Method for Classifying Filiform Lie Superalgebras . . . . . . . . . . . . . . 413 Rosa M. Navarro and José M. Sánchez Equivalence of Vector Field Realizations of Lie Algebras from the Lie Group Point of View . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421 Maryna Nesterenko and Severin Pošta

List of Participants

Cristian Anghel, Institute of Mathematics of the Romanian Academy, Bucharest, Romania Iana Anguelova, College of Charleston, USA Lilia Anguelova, INRNE, BAS, Bulgaria Constantin Arcus, Secondary School Cornelus Radu, Romania Roberto Auzzi, Università Cattolica del Sacro Cuore, Brescia, Italy Mirela Babalic, IBS Center for Geometry and Physics, Pohang, Republic of Korea Sigiswald Barbier, Ghent University, Belgium Thomas Basile, University of Tours, France Benjamin Basso, École Normale Supérieure, Paris, France Nadir Bizi, Université Pierre et Marie Curie, Paris, France Alexander Rivera Bonilla, Universidade Federal de Juiz de Fora, Brazil Loriano Bonora, SISSA, Trieste, Italy Jean-Emile Bourgine, Korea Institute for Advanced Study, Seoul, Republic of Korea Alex Buchel, University of Western Ontario/Perimeter Institute for Theoretical Physics, Canada Cestmir Burdik, Department of Mathematics, FNSP CTU in Prague, Czech Republic Junpeng Cao, Institute of Physics, Chinese Academy of Sciences, Beijing, China Jose Carrasco, Universidad Complutense de Madrid, Spain Martin Cederwall, Department of Physics, Chalmers University of Technology, Gothenburg, Sweden Kyriakos Charalambous, University of Nicosia, Cyprus Qingtao Chen, ETH Zurich, Switzerland Rene Chipot, University of Basel, Switzerland Jeong Ryeol Choi, Daegu Health College, Republic of Korea Jean Francois Colombeau, UNICAMP, Campinas, Brazil

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xii

List of Participants

Ben Craps, Vrije Universiteit Brussel, Belgium Sumit Das, University of Kentucky, Lexington, USA Ivan Dimitrijevic, Faculty of Mathematics, University of Belgrade, Serbia Vladimir Dobrev, INRNE, BAS, Bulgaria Heinz-D. Doebner, TU Clausthal, Germany Branko Dragovich, Institute of Physics, University of Belgrade, Serbia Valeriy Dvoeglazov, Universidad de Zacatecas, Mexico Tresor Ekanga, Institut de mathématiques de Jussieu, Paris Rive Gauche, France Eduardo Fernandez-Saiz, Complutense University of Madrid, Spain Veselin Filev, Institute of Mathematics and Informatics, BAS, Bulgaria Davide Fioravanti, INFN Bologna, Italy Tamar Friedmann, Smith College and University of Rochester, USA Cohl Furey, DAMTP; Cavendish, University of Cambridge, UK Alexander Ganchev, AUBG, Blagoevgrad, Bulgaria Alexandre Gavrilik, Bogolyubov Institute for Theoretical Physics of NASU, Kiev, Ukraine Jose Francisco Gomes, Instituto de Fisica Teorica (IFT-UNESP), Sao Paulo, Brazil John Haas, University of Missouri, Columbia, USA Ludmil Hadjiivanov, INRNE, BAS, Bulgaria Malte Henkel, Institute for Building Materials—ETH Zurich, Switzerland Francisco J. Herranz, University of Burgos, Spain Mike Hewitt, Canterbury Christ Church University, UK Jiri Hrivnak, Czech Technical University in Prague, Czech Republic Baran Hynek, Mathematical Institute in Opava, Czech Republic Alexey Isaev, BLTPh, JINR, Dubna, Russia Evgeny Ivanov, BLTPh, JINR, Dubna, Russia Elchin Jafarov, Institute of Physics, Azerbaijan National Academy of Sciences, Baku, Azerbaijan Josef Janyska, Masaryk University at Brno, Czech Republic Antal Jevicki, Brown University, Providence, USA Anthony Joseph, Weizmann Institute of Science, Rehovot, Israel Toshiyuki Kobayashi, University of Tokyo, Japan Takeo Kojima, Yamagata University, Japan Peter Koroteev, Perimeter Institute for Theoretical Physics, Waterloo, Canada Ivan Kostov, IPhT–Saclay, France Zhanna Kuznetsova, Federal University of ABC (UFABC), Sao Paulo, Brazil Andras Laszlo, Wigner Research Centre for Physics of the Hungarian Academy of Sciences, Budapest, Hungary Calin Lazaroiu, IBS-CGP, Pohang, Republic of Korea Helge Oystein Maakestad, NAV, Norway Tomasz Maciazek, Center for Theoretical Physics, PAS, Warsaw, Poland Richard Mackenzie, Université de Montréal, Canada

List of Participants

xiii

Stefano Mancini, University of Camerino, Italy Kalin Marinov, INRNE, BAS, Bulgaria Takuya Matsumoto, Nagoya University, Japan Anna Melnikov, University of Haifa, Israel Stefan Mladenov, Department of Physics, Sofia University, Bulgaria Marco Modugno, Department of Mathematics and Informatics Florence, Italy Lenka Motlochova, Czech Technical University in Prague, Czech Republic Patrick Moylan, the Pennsylvania State University, Abington, USA Rosa Navarro, University of Extremadura, Spain Dimitar Nedanovski, INRNE, BAS, Bulgaria Stam Nicolis, CNRS, Laboratoire de Mathématiques et Physique Théorique, Tours, France Nikolay Nikolov, INRNE, BAS, Bulgaria Emil Nissimov, INRNE, BAS, Bulgaria Petr Novotny, FNSPE Czech Technical University in Prague, Czech Republic Satoshi Ohya, Nihon University, Japan Mariano del Olmo, Universidad de Valladolid, Spain Marco Oppio, University of Trento, Italy Roy Oste, Ghent University, Belgium Svetlana Pacheva, INRNE, BAS, Bulgaria Tchavdar Palev, INRNE, BAS, Bulgaria Valentina Petkova, INRNE, BAS, Bulgaria George Pogosyan, Yerevan State University, Armenia Todor Popov, INRNE, BAS, Bulgaria Erich Poppitz, University of Toronto, Canada Severin Posta, Czech Technical University in Prague, Czech Republic Ana Retore, Instituto de Fisica Teórica (IFT-UNESP), Sao Paulo, Brazil Dario Rosa, Korea Institute for Advanced Study, Seoul, Republic of Korea Christopher Sadowski, Ursinus College, Collegeville, USA Pavle Saksida, Faculty of Mathematics and Physics, University of Ljubljana, Slovenia Hadi Salmasian, University of Ottawa, Canada Igor Salom, Institute of Physics, Belgrade, Serbia Al Shapere, University of Kentucky, Lexington, USA Kuldip Singh, National University of Singapore, Singapore Walter Smilga, Germany Emeri Sokatchev, LAPTh, France Nathaly Spano, IFT-UNESP, Sao Paulo, Brazil Denitsa Staicova, INRNE, BAS, Bulgaria Marian Stanishkov, INRNE, BAS, Bulgaria Jelena Stankovic, Teacher Education Faculty, University of Belgrade, Serbia Anastasia Stavrova, St. Petersburg State University, Russia

xiv

List of Participants

Ovidiu Cristinel Stoica, Horia Hulubei National Institute for R&D in Physics and Nuclear Engineering, Magurele, Romania Michail Stoilov, INRNE, BAS, Bulgaria Neli Stoilova, INRNE, BAS, Bulgaria Stoimen Stoimenov, INRNE, BAS, Bulgaria Fumihiko Sugino, Center for Theoretical Physics of the Universe, Institute for Basic Science, Seoul, Republic of Korea Peter Suranyi, University of Cincinnati, USA Franciszek Szafraniec, Academic Computer Centre, Krakow, Poland Doman Takata, Kyoto University, Japan Meng-Chwan Tan, National University of Singapore, Singapore Elena Tobisch, Institute for Analysis, J. Kepler University, Linz, Austria Ivan Todorov, INRNE, BAS, Bulgaria Francesco Toppan, Centro Brasileiro de Pesquisas Fisicas, Brazil Amalia Torre, ENEA, Fusion and Nuclear Safety Department, C. R. Frascati, Italy Valdemar Tsanov, Mathematisches Institut, Universität Gottingen, Germany Benoit Vallet, IPhT, CEA Saclay, France Joris Van Der Jeugt, Ghent University, Belgium Tsvetan Vetsov, Sofia University, Bulgaria Petr Vojcak, Mathematical Institute, Silesian University in Opava, Czech Republic Apostolos Vourdas, University of Bradford, UK Dinh Long Vu, IPhT, CEA Saclay, France Mirjana Vuletic, University of Massachusetts, Boston, USA L.c.rohana Wijewardhana, University of Cincinnati, Department of Physics, USA Kurt Bernardo Wolf, Instituto de Ciencias Fisícas, Universidad Nacional Autónoma de México, Morelos, Mexico Michael Wright, the Archive Trust for Research in Mathematical Sciences, Oxford, UK Asher Yahalom, Ariel University, Kiryat Hamada, Israel Yi Yang, National Chiao Tung University, Hsinchu, Taiwan Huafeng Zhang, ETH Zurich, Switzerland George Zoupanos, Physics Department, National Technical University in Athens, Greece

List of Participants

xv

Summary

Countries (33)

Number

Bulgaria USA France Czech Republic Brazil Italy S. Korea Japan Spain Canada Serbia Belgium UK Switzerland Romania Russia Germany Israel Mexico Singapore Poland Armenia Austria Azerbaijan China Cyprus Greece Hungary Norway Slovenia Sweden Ukraine Taiwan Total

21 11 9 8 7 7 6 5 5 5 4 4 4 4 3 3 3 3 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 130

Part I

Plenary Talks

Image of Conformally Covariant, Symmetry Breaking Operators for R p,q Toshiyuki Kobayashi and Alex Leontiev

Abstract We consider the meromorphic continuation of an integral transform that gives rise to a conformally covariant, symmetry breaking operator Aλ,ν between the natural family of representations I (λ) and J (ν) of the indefinite orthogonal group G = O( p + 1, q + 1) and its subgroup G  = O( p, q + 1), respectively, realized in function spaces on the conformal compactifications of flat pseudo-Riemannian manifolds R p,q ⊃ R p−1,q . In this article, we determine explicitly the image of the renormalized operator Aλ,ν for all (λ, ν) ∈ C2 . In particular, the complex parameters (λ, ν) for which the image of Aλ,ν coincides with {0}, C, finite-dimensional representations, the minimal representation, or discrete series representations for pseudo-Riemannian space forms are explicitly classified. A graphic description of the K -types of the image is also provided. Our results extend a part of the prior results of Kobayashi and Speh [Memoirs of Amer. Math. Soc. 2015] in the Riemannian case where q = 0. Keywords Representation theory · Reductive group · Branching law · Broken symmetry · Conformal geometry · Symmetry breaking operator Pseudo-Riemannian manifold 2010 MSC Primary 22E46 · Secondary 33C45 · 53C35

1 Introduction Let (π, Hπ ) be a representation of a group G, and (π  , Hπ  ) the one of a subgroup G  . A symmetry breaking operator is a linear map T. Kobayashi (B) Graduate School of Mathematical Sciences, The University of Tokyo and Kavli IPMU, Tokyo, Japan e-mail: [email protected] A. Leontiev Graduate School of Mathematical Sciences, The University of Tokyo, Tokyo, Japan e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2018 V. Dobrev (ed.), Quantum Theory and Symmetries with Lie Theory and Its Applications in Physics Volume 1, Springer Proceedings in Mathematics & Statistics 263, https://doi.org/10.1007/978-981-13-2715-5_1

3

4

T. Kobayashi and A. Leontiev

T : Hπ → Hπ  that intertwines the actions of the subgroup G  . Then the image of T is a G  submodule of π  . In the last decade, symmetry breaking operators for infinite-dimensional representations of reductive groups G ⊃ G  have been actively studied as a new line of investigation on branching problems of representation theory [10, 12, 22, 23] and also interacted with some other areas such as automorphic form theory or conformal geometry among others, see [4, 11].

1.1 Conformal Representations I (λ) and J(ν) Associated with Pseudo-Riemannian Manifolds R p,q ⊃ R p−1,q In this article we discuss symmetry breaking operators motivated from conformal geometry. Let G = O( p + 1, q + 1) be the automorphism group of the quadratic form on R p+q+2 of signature ( p + 1, q + 1) defined by Q p+1,q+1 (x) = x02 + · · · + x 2p − x 2p+1 − · · · − x 2p+q+1 . Let R p,q be the ( p + q)-dimensional vector space R p+q endowed with flat pseudoRiemannian structure ds 2 = d x12 + · · · + d x 2p − d x 2p+1 − · · · − d x 2p+q of signature ( p, q). Then, the group G acts isometrically on R p+1,q+1 , and conformally on the conformal compactification X := (S p × S q )/{±1} of R p,q , which is the direct product of p- and q-spheres equipped with pseudoRiemannian structure g S p ⊕ (−g Sq ), modulo the direct product of antipodal maps, see Segal [24, Chap. II]. By the general theory of conformal groups [16, Sect. 2], one has a natural family of representations I (λ) of G on C ∞ (X ) with parameter λ ∈ C. We normalize I (λ) such that I (0) is the space of sections, and I (dim X ) is the space of densities. Via the twisted pull-back ι∗λ : C ∞ (X ) → C ∞ (R p,q ) of the conformal embedding ι : R p,q → X , we may realize I (λ) on the subspace ι∗λ (C ∞ (X )) of C ∞ (R p,q ), see [18, (2.8.6)]. Similarly, another group G  := O( p, q + 1) acts on the conformal compactification Y := (S p−1 × S q )/{±1}

Image of Conformally Covariant, Symmetry Breaking Operators for R p,q

5

of the flat pseudo-Riemannian manifold R p−1,q , and one has a natural family of representations J (ν) of G  on C ∞ (Y ) for ν ∈ C. Thus we have a G-module I (λ) and a module J (ν) of the subgroup G  with complex parameters λ and ν. The object of our study is symmetry breaking operators I (λ) → J (ν) with focus on their images.

1.2 Degenerate Principal Series Representations The representation I (λ) of G = O( p + 1, q + 1) defined in Sect. 1.1 by using conformal geometry may be interpreted as a degenerate principal series representation of the real reductive Lie group G as follows. Let P = M AN+ be a maximal parabolic subgroup of G with Levi part M A O( p, q) × {±1} × R. For a character Cλ of A R, we regard it as that of P via the quotient map P → P/M N+ A, and form a G-equivariant line bundle Lλ = G × P Cλ → G/P. Then the (unnormalized) induced representation Ind GP (Cλ ) is realized in the Fréchet space of smooth sections for the line bundle Lλ → G/P. Our parametrization is chosen in a way that Ind GP (Cλ ) contains a finite-dimensional submodule if −λ ∈ 2N and a finite-dimensional quotient if λ − ( p + q) ∈ 2N. Then we have an isomorphism of G-modules I (λ) Ind GP (Cλ ). The realization on ι∗λ (C ∞ (X )) (⊂ C ∞ (R p+q )) is referred to as the N -picture of I (λ). Similarly to I (λ), we have an isomorphism as G  -modules 

J (ν) Ind GP  (Cν ) , 

where Ind GP  (Cν ) is the (unnormalized) induced representation of G  from a character Cν of a maximal parabolic subgroup P  with Levi part O( p − 1, q) × {±1} × R.

1.3 Construction of Symmetry Breaking Operators We realize R p−1,q as a submanifold of R p,q by letting x p = 0. This determines the embeddings Y → X between their conformal compactifications, and G  = O( p, q + 1) → G = O( p + 1, q + 1) between conformal groups. Applying the general results proven in Kobayashi–Speh [22, Chap. 3] to our specific setting, we see

6

T. Kobayashi and A. Leontiev

that, for any breaking operator T : I (λ) → J (ν), there exists a distribution  symmetry  K T ∈ D  R p+q such that for all f ∈ I (λ) ι∗ν (T





f )(x ) = Rest x p =0 ◦

R p+q

K T (x − y)(ι∗λ f )(y)dy,

(1)

where x  = (x1 , . . . , xp , . . . , x p+q ) ∈ R p−1,q and x = (x1 , · · · , x p+q ) ∈ R p,q . The distribution kernel K T satisfies certain covariance properties, which characterize that T is a G  -intertwining operator (i.e., T is a symmetry breaking operator), see [22, Thm. 3.16]. By [21, Lem. 2.22], T is a differential symmetry breaking operator if Supp(K T ) = {0}. In contrast, T is called a regular symmetry breaking operator ([22, Def. 3.3]) if Supp(K T ) contains an interior point, or equivalently, if Supp(K T ) = R p+q in our setting. Differential symmetry breaking operators I (λ) → J (ν) in our setting were classified in [19], by using the F-method [7, 8], see also [13] for a generalization. On the other hand, there exists a unique holomorphic family of symmetry breaking operators, to be denoted by Aλ,ν , up to scalar multiplication, such that Aλ,ν is a regular symmetry breaking operator for an open dense subset of (λ, ν) ∈ C2 , see Remark 12 below. It is constructed as follows. We set Q p,q (x) := x12 + · · · + x 2p − x 2p+1 − · · · − x 2p+q . Theorem 1 (regular symmetry breaking operator) Suppose that p, q ≥ 1. We let (G, G  ) := (O( p + 1, q + 1), O( p, q + 1)) as before. The linear operator Aλ,ν : I (λ) → J (ν), initially defined as the integral operator (1) with locally integrable kernel function Aλ,ν :=

Γ

 λ−ν  2

Γ



1 λ+ν− p−q+1 2



Γ

 λ+ν− p−q   |Q p,q (x)|−ν  1−ν  x p

(2)

2

on R p,q for Re ν 0 and Re(λ + ν)  0, intertwines the action of the subgroup G  , and extends to a family of symmetry breaking operators that depend holomorphically on (λ, ν) in the entire C2 . Remark 2 The Gamma factor in (2) is chosen in an optimal way in the sense that • Aλ,ν depends holomorphically on (λ, ν) ∈ C2 ; • the set of the zeros of Aλ,ν is a discrete subset in C2 (see Theorem 4) below. Remark 3 Theorem 1 gives a generalization of Kobayashi–Speh [22, Thm. 1.5] which treated the q = 0 case. We note that the normalizing Gamma factor is different in the q = 0 case.

Image of Conformally Covariant, Symmetry Breaking Operators for R p,q

7

1.4 Image of the Symmetry Breaking Operators Aλ,ν The goal of this article is to determine the image of the holomorphic continuation of the regular symmetry breaking operator Aλ,ν : I (λ) → J (ν) for all (λ, ν) ∈ C2 given in Theorem 1. We note that I (λ) is a G-module and J (ν) is a G  -module and that G  G  . This sort of problem was first studied in Kobayashi– Speh [22, Chap. 13], and a complete solution was given in the Riemannian case where q = 0. In this article, we shall consider a more general case where q > 0. This means that R p,q is of indefinite metric and the conformal group G = O( p + 1, q + 1) has real rank greater than one. For simplicity of the exposition, we confine ourselves to the case p > 1 in this article. Our main theorem (Theorem 15) will be formulated in Sect. 3 after preparing some combinatorial notation. As an introduction, we avoid complicated definitions in the general case, and focus on specific features of Theorem 15 instead, giving explicit criteria for the parameter (λ, ν) to fulfill the following conditions: (1) (2) (3) (4) (5)

Image(Aλ,ν ) = {0}, i.e., Aλ,ν vanishes (Theorem 4); Image(Aλ,ν ) is finite-dimensional (Theorem 5); Image(Aλ,ν ) is the trivial one-dimensional representation (Corollary 7); Image(Aλ,ν ) is the minimal representation (Theorem 8); Image(Aλ,ν ) is a discrete series for the pseudo-Riemannian space form (Theorem 9).

(1) Vanishing condition of Aλ,ν . In the theory of symmetry breaking operators, it is an important question to determine the zeros and poles of the meromorphic continuation of the regular symmetry breaking operators. Once it is normalized as in Remark 2, of particular importance is to find precisely the parameters for which the holomorphic continuation of the normalized regular symmetry breaking operator vanishes. In those places, we expect that the representations are reducible and that the dimension of symmetry breaking operators jumps up, see [22, Thm. 11.4] for instance. In the case q = 0, it is proved in [22, Thm. 8.1] that the zeros of the normalized regular symmetry breaking operator are given by the following discrete set: L even := {(λ, ν) ∈ Z2 : λ ≤ ν ≤ 0, λ ≡ ν

mod 2}.

(3)

In the case q ≥ 1, the zeros of Aλ,ν are given as follows. For simplicity, we assume p = 1. Theorem 4 Suppose p ≥ 2 and q ≥ 1. Then the following two conditions on (λ, ν) ∈ C2 are equivalent: (i) Aλ,ν = 0.

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T. Kobayashi and A. Leontiev

(ii) (λ, ν) ∈ //∩ |||. The definition of the subsets // and ||| will be given C2 in Sect. 3.1. For here we present a more concrete formula of the intersection //∩ ||| by comparing it with L even . For this, we consider the following discrete set in C2 . Γ := {(λ, ν) ∈ Z2 : 0 < ν and λ ≤ ν}. Then by (12) and (14) below, one sees //∩ |||=

(L even ∩ (2Z)2 )  (Γ ∩ (2Z)2 ) if q is odd, if q is even. L even  (Γ ∩ (2Z + 1)2 )

(4)

(2) When is Image(Aλ,ν ) finite-dimensional? Whereas the vanishing condition of the symmetry breaking operator Aλ,ν depends on the parity of q in the previous theorem, see (4), it turns out that the condition on (λ, ν) ∈ C2 for Image(Aλ,ν ) to be a (nonzero) finite-dimensional vector space is independent of the parities of p and q as below. Theorem 5 (finite-dimensional image). Suppose p ≥ 2 and q ≥ 1. Then the following two conditions on (λ, ν) ∈ C2 are equivalent. (i) Image(Aλ,ν : I (λ) → J (ν)) is nonzero and finite-dimensional. (ii) ν ∈ 2Z, ν ≤ 0 and λ satisfies one of the following: • λ ∈ 2Z and ν < λ; • λ ∈ C − 2Z. Graphically, Theorem 5 corresponds to the colored red left corner bounded by the “barrier A++ ” in Case A of Theorems 20, 26, 31 and 35 in later sections. We note that if one of (therefore both of) the equivalent conditions (i) and (ii) in Theorem 5 are fulfilled, then Image(Aλ,ν ) is an irreducible representation of the subgroup G  . Remark 6 See [22, Thm. 13.1] for an analogous theorem in the case q = 0 and [22, Thm. 14.9] for some application. (3) When is Image(Aλ,ν ) isomorphic to the trivial one-dimensional representation? The trivial one-dimensional representation of G  occurs as a subrepresentation of the degenerate principal series representation J (ν) with ν = 0. Then the equivalence (i) ⇔ (iii) in the following corollary is an immediate consequence of Theorem 5 with ν = 0. The equivalence (i) ⇔ (ii) follows from a trick in Kobayashi–Speh [22, Chap. 14]. Corollary 7 Suppose p ≥ 2 and q ≥ 1. Then the following three conditions on λ ∈ C are equivalent: (i) Image(Aλ,0 : I (λ) → J (0)) is the trivial one-dimensional representation of G  .

Image of Conformally Covariant, Symmetry Breaking Operators for R p,q

9

(ii) The regular symmetry breaking operator Aλ,0 induces a nonzero G-intertwining operator (5) I (λ) → C ∞ (G/G  ). (iii) λ ∈ C − {−2, −4, −6, . . .}. In Corollary 7, the resulting G-intertwining operator (5) is nothing but a (generalized) Poisson transform to the semisimple symmetric space G/G  , which is the pseudo-Riemannian space form M + p,q+1 of signature ( p, q + 1) with positive constant curvature. (4) When is Image(Aλ,ν ) isomorphic to the minimal representation? For p + q odd (≥ 7), there exists an irreducible unitary representation  of G  = O( p, q + 1), referred to as the minimal representation. Here by minimal representation we mean that the annihilator of the smooth representation  ∞ in the enveloping algebra U (gC ) is the Joseph ideal [3]. This is the unique irreducible unitary representation of G  whose Gelfand–Kirillov dimension is equal to p + q − 2, which is smaller than that of any other infinite-dimensional unitary representation of G  . It is remarkable that our symmetry breaking operator Aλ,ν constructs the minimal representation as its image when p + q ≡ 1 mod 4 by a specific choice of (λ, ν) as follows. Theorem 8 (minimal representation). Suppose that p + q = 4k + 1 for some k ∈ Z with k ≥ 2. If we take (λ, ν) := (2k + 1, 2k − 1), then the underlying (g , K  )-module of Image(Aλ,ν : I (λ) → J (ν)) is isomorphic to that of the minimal representation  of the subgroup G  = O( p, q + 1). In particular, Aλ,ν f is a Yamabe harmonic on the pseudo-Riemannian manifold Y = (S p × S q )/{±1} for any f ∈ I (λ). The last statement of Theorem 8 follows from the geometric construction of the minimal representation proved in [16, Thms. 3.4.2 and 3.6.1]. See Case E in Theorem 26 for a graphic interpretation of Theorem 8. (5) When is Image(Aλ,ν ) isomorphic to a discrete series representation for a generalized hyperboloid? Let M εp,q be the ( p + q)-dimensional pseudo-Riemannian space form of signature ( p, q) of constant sectional curvature +1 (ε = +) and −1 (ε = −), referred also to as a generalized hyperboloid from its realization as hypersurfaces in R p+q+1 : p+1,q : Q p+1,q (x) = 1}, M+ p,q = {x ∈ R p,q+1 M− : Q p,q+1 (x) = −1}. p,q = {x ∈ R

For simplicity, we treat only the case ε = − here. Then the group G  = O( p, q + 1) acts isometrically and transitively on M − p,q . As a homogeneous space, we have the following diffeomorphism

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T. Kobayashi and A. Leontiev

M− p,q O( p, q + 1)/O( p, q). Let  p,q be the Laplacian of the pseudo-Riemannian manifold M − p,q . For p > 0 and q > 0, the Laplacian  p,q is not an elliptic operator. In this case, there exist countably many L 2 -eigenvalues of  p,q on M − p,q , given explicitly by {ν(ν − 2ρ) : ν ∈ Z, ρ < ν},

(6)

see Faraut [1] or Strichartz [25]. Here we set ρ :=

1 ( p + q − 1). 2

(7)

The isometry group G  acts irreducibly on the Hilbert space of L 2 -eigenfunctions of  p,q for each eigenvalue ν(ν − 2ρ). Following the same notation as in [17, Sect. 5], p,q+1 we write π−,ν−ρ for the resulting irreducible unitary representation. The represenp,q+1 tation π−,ν−ρ is referred to as a discrete series representation for the generalized hyperboloid M − p,q . Theorem 9 (discrete series for pseudo-Riemannian space form). Suppose p ≥ 2 and q ≥ 1. Let ν ∈ Z satisfy ρ < ν. Then the underlying (g, K )-module of Image(Aλ,ν : I (λ) → J (ν)) is isomorphic to that of a discrete series representation for the pseudoRiemannian space form M − p,q if and only if ν ≡ q + 1 mod 2 and λ satisfies the following conditions. Case 1. q is even. (λ ∈ 2Z and ν < λ) or (λ ∈ C − 2Z). Case 2. q is odd. (λ ∈ 2Z + 1 and ν < λ) or (λ ∈ C − (2Z + 1)). p,q+1

To see Theorem 9, we use a realization of π−,ν−ρ as a subrepresentation of J (ν) with K -types E ν+− given by the barrier A+− p,q+1,−ν , see Example 14. Then Theorem 9 follows from a graphic description of Image(Aλ,ν ) in Cases E, E , and E bis Cases G and Gbis Cases C and Cbis Cases B, B , and Bbis

in Theorem 20; in Theorem 25 with ν > ρ; in Theorem 29 with ν > ρ; in Theorem 33.

The paper is organized as follows. In Sect. 2 we give brief comments on our problem from a perspective on the general problem of restrictions of representations, in particular, for pairs of reductive groups G ⊃ G  . In Sect. 3, we determine Image(Aλ,ν : I (λ) → J (ν)) for all (λ, ν) ∈ C2 in Theorem 15 when (G, G  ) = (O( p + 1, q + 1), O( p, q + 1)) with p ≥ 2 and q ≥ 1. A graphic description of Theorem 15 is given in Sects. 4–7 depending on the parities of p and q, from which theorems in Introduction follow. A detailed proof of Theorem 15 will appear elsewhere.

Image of Conformally Covariant, Symmetry Breaking Operators for R p,q

11

Notation. N := {0, 1, 2, . . .}. For two subsets A and B of a set, we write A − B := {a ∈ A : a ∈ / B} rather than the usual notation A \ B.

2 Branching Program ABC for Restriction of Representations In this section, we provide some brief comments on the topic treated here from a perspective on the general problem of restriction of (infinite-dimensional) representations of real reductive Lie groups. See [10, 11] for more details.

2.1 Finiteness Criterion for Multiplicities in Branching of Representations Suppose G ⊃ G  are a pair of reductive groups and π is an irreducible representation of G. The restriction of π to the subgroup G  is no more irreducible in general as a representation of G  . If G is compact, then any irreducible π is finite-dimensional and splits into a finite direct sum π |G  =



m(π, π  )π 

 π  ∈G

of irreducibles π  of G  with multiplicities m(π, π  ). In this case, the multiplicity m(π, π  ) is given by dim Hom G  (π  , π |G  ) = dim Hom G  (π |G  , π  ).

(8)

However, for noncompact G  and for infinite-dimensional π , the restriction π |G  is not always a direct sum of irreducible representations, even if π is a unitary representation of G. In general, we need the notion of direct integral of Hilbert spaces to give an irreducible decomposition of the restriction π |G  . Sometimes there is no continuous spectrum in the irreducible decomposition of the restriction π |G  , even when G  is noncompact. See [5, 6] for the condition that the restriction π |G  is discretely decomposable. For the more general case where π is nonunitary, the equality (8) does not hold: both of the spaces Hom G  (·, ·) depend on the underlying topologies on the representation spaces of π and π  . To clarify our formulation, we recall that, associated to a continuous representation π of a Lie group on a Banach space H , a continuous representation π ∞ is defined on the Fréchet space H ∞ of C ∞ -vectors of the Banach representation on H . Given another representation π  of the subgroup G  , we consider the space of continuous

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T. Kobayashi and A. Leontiev

G  -intertwining operators (symmetry breaking operators)   ∞  . Hom G  π ∞ |G  , π 

(9)

If both π and π  are admissible representations of finite length of reductive Lie groups G and G  , respectively, then the dimension of the space (9) is determined by the underlying (g, K )-module π K of π and the (g , K  )-module π K  of π  , and is independent of the choice of Banach globalizations because π ∞ and (π  )∞ are determined uniquely by π K and π K  , respectively, by the Casselman–Wallach theory [27, Chap. 11]. We denote by m(π, π  ) the dimension of (9), and call it the multiplicity of π  in the restriction π |G  . In general, the multiplicity m(π, π  ) may be infinite, even when G  is a maximal reductive subgroup of G and π is irreducible. This happens even when (G, G  ) is a symmetric pair. By using the theory of real spherical spaces initiated in Kobayashi– Oshima [20], the criterion for finite-multiplicities is discovered in [9, 20] as follows. Fact 10 Let (G, G  ) be a pair of real reductive Lie groups, and (G C , G C ) its complexification. (1) The multiplicity m(π, π  ) is finite for all irreducible representations π of G and all irreducible representations π  of G  if and only if a minimal parabolic subgroup of G  has an open orbit on the real flag variety of G. (2) The multiplicity m(π, π  ) is uniformly bounded if and only if a Borel subgroup of G C has an open orbit on the complex flag variety of G C . The complete classification of symmetric pairs (G, G  ) satisfying the above geometric criteria was accomplished in Kobayashi–Matsuki [15]. The (G, G) = (O( p + 1, q + 1), O( p, q + 1)) satisfies the criterion in (2) (and in particular, the criterion in (1), too), and therefore, m(π, π  ) is uniformly bounded. Furthermore, Sun–Zhu [26] proved that m(π, π  ) ≤ 1. In the theory of symmetry breaking operators, we consider “quotient map” from a representation of a group G to that of the subgroup G  . On the other hand, one may reverse arrows and consider “embedding map” from a representation of a subgroup G  to that of G, e.g., consider the following spaces: Hom G 

  ∞ ∞    π , π |G  or Homg ,K  π K  , π K |g ,K  .

We observe that there are canonical injective maps:   ∞     ∨ ∞ ∞ π , π ∨ |G  ⊂ Hom G  π ∞ |G  , π  ,     Homg ,K  (π  )∨K  , (π ∨ ) K |g ,K  ⊂ Homg ,K  π K |g ,K  , π K  ,

Hom G 

where the symbol ∨ stands for the contragredient representation. The study of these objects in the left-hand sides is closely related to the theory of discretely decomposable restrictions, [5, 6], which we do not discuss here. Concerning the right-hand

Image of Conformally Covariant, Symmetry Breaking Operators for R p,q

13

sides for symmetry breaking operators in the category of (g, K )-modules and in the category of admissible smooth representations of moderate growth, we raised a question in [9, Sect. 10] about automatic continuity property for symmetry breaking operators as a generalization of the theory of Casselman–Wallach (G = G  case): it is plausible that if (G, G  ) satisfies one of (therefore any of) the equivalent conditions in Fact 10 (1), then the natural injection map below is surjective     Hom G  π ∞ |G  , π ∞ → Homg ,K  π K |g ,K  , π K  , see [9, Rem. 10.2 (4)].

2.2 Program ABC for Branching The study of restriction of representations (branching problem) is an important but involves different types of difficult problems even in very special cases. The first author analysed various (in fact, wild) features and phenomena about restrictions for reductive Lie groups, and proposed in [10] a program for studying the restriction of representations of reductive groups, which may be summarized as follows: Stage A. Abstract features of the restriction; Stage B. Branching law of π |G  ; Stage C. Construction of symmetry breaking operators. Fact 10 is an example for Stage A in branching problems. Stage A aims for developing the general theory of the restrictions π |G  (e.g., spectrum, multiplicity), which would  single out the good triples G, G  , π . In turn, we could expect concrete and detailed study of those restrictions π |G  in Stages B and C. For instance, Fact 10 assures the following a priori estimate: m(π, π  ) is uniformly bounded if the pair of Lie algebras (g, g ) is a real form of (sl(n + 1, C), gl(n, C)) or (o(n + 1, C), o(n, C)), in particular, if (G, G  ) is of the form (G, G  ) = (O( p + 1, q + 1), O( p, q + 1)) .

(10)

“Stage B” is a traditional question, however, it is often very difficult to compute explicitly branching laws of infinite-dimensional representations of (noncompact) reductive groups. The first systematic study of “Stage C” is given by a monograph by Kobayashi–Speh [22], which corresponds to the case q = 0, more precisely, the case

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T. Kobayashi and A. Leontiev

π : spherical principal series representations of G = O(n + 1, 1), π  : spherical principal series representations of G  = O(n, 1), Stage C includes the following subproblems. (C1) (C2) (C3) (C4) (C5)

construct symmetry breaking operators explicitly; classify all symmetry breaking operators; find residue formulæ for symmetry breaking operators; study functional equations among symmetry breaking operators; determine the images of subquotients by symmetry breaking operators.

The subprogram (C1)–(C5) was considered by Kobayashi–Speh [22] with a complete answer for the pair (G, G  ) = (O(n + 1, 1), O(n, 1)) of real rank one groups (10). In [14], we discussed the subprograms (C1)–(C4) for degenerate spherical principal series representations π = I (λ) of G and π  = J (ν) of G  for the pair of higher real rank groups. The (C5) is the main issue of this article.

3 Main Theorem In this section we determine the image of the meromorphic continuation of the regular symmetry breaking operator Aλ,ν : I (λ) → J (ν) for all (λ, ν) ∈ C2 . The statement of the main results uses the following notation: (1) subsets //, \\, |||, X and X+ (see Sect. 3.1), (2) description of G  -submodules of the target space J (ν) (see Sect. 3.2). Theorems 4–9 are special cases of the main theorem of this section (Theorem 15).

3.1 Subsets //, |||, \\ and X in C2 We introduce some subsets of C2 . It should be noted that the symbols //, \\, ||, and ||| below are defined as subsets of C2 , and are not as binary relations. Definition 11 We let \\ := {(λ, ν) ∈ C2 : p + q − 1 − λ − ν ∈ 2N},

(11)

// := {(λ, ν) ∈ C : ν − λ ∈ 2N},

(12)

X := // ∩ \\,

(13)

2

||| := {(λ, ν) ∈ C : ν ∈ −2N ∪ (q + 1 + 2Z)}. 2

(14)

Image of Conformally Covariant, Symmetry Breaking Operators for R p,q

15

For the sets //, \\, and X, we have adopted the same notation with the one introduced in [22] which dealt with the q = 0 case. It is easy to see X ∩ {(λ, ν) ∈ C2 : ν ∈ Z} = ∅ if and only if p + q is even.

(15)

As in [22], we define  ∈ N and and k ∈ N by 2 = ν − λ for (λ, ν) ∈ //, 2k = p + q − 1 − λ − ν for (λ, ν) ∈ \\.

(16) (17)

We define two subsets of X = \\ ∩ // by X+ := {(λ, ν) ∈ X : ρ ≤ ν} = {(ρ −  − k, ρ +  − k) : , k ∈ N,  ≥ k} , X− := {(λ, ν) ∈ X : ρ > ν} = {(ρ −  − k, ρ +  − k) : , k ∈ N,  < k} . Here we recall from (7) that ρ = 21 ( p + q − 1). We decompose the set // ∪ \\ into a disjoint union // ∪ \\ = //  (\\ − X) , and further decompose the set // into three subsets // = (// − (X+ ∪ |||))  (//∩ |||)  (X+ − |||) , where we have used X+ ⊂ //. Combining these decompositions together, we have a decomposition of the parameter set C2 of (λ, ν) into a disjoint union of five subsets as follows C2 = (// ∪ \\)c  (// − (X+ ∪ |||))  (//∩ |||)  (X+ − |||)  (\\ − X) .

(18)

Then the image of the regular symmetry breaking operator is described according to which subset the parameter (λ, ν) belongs to. Remark 12 The support of the distribution kernel Aλ,ν of the symmetry breaking operator Aλ,ν does not contain an interior point, if (λ, ν) ∈ //, \\, or if ν ∈ 1 + 2N, respectively, when p, q ≥ 1 ([14, Thm. 6.3]).

3.2 Description of Submodules of the Principal Series J(ν) The degenerate spherical principal series representation J (ν) of the group G  = O( p, q + 1) has at most four irreducible subquotients. The number of irreducible subquotients depends on ν ∈ C and on the parities of p and q. In this section, we

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T. Kobayashi and A. Leontiev

give a quick review of the socle filtration of J (ν) from Howe and Tan [2]. We note that our group is G  = O( p, q + 1) whereas their group in [2] is O( p, q). Let K  = O( p) × O(q + 1). Then K  is a maximal compact subgroup G  = O( p, q + 1), and the G  -module J (ν) is multiplicity-free as K  -modules for any ν ∈ C. To describe its K  -types, it is convenient to use the notion of spherical harmonics, which we recall now. The space of spherical harmonics of degree a ∈ N is defined by H a (R p ) :=

⎧ ⎨ ⎩

F ∈ Pol[x1 , · · · , x p ] :

⎫ p p ⎬   ∂2 F ∂F = 0, x = a F j 2 ⎭ ∂x j ∂x j j=1 j=1

{ f ∈ C ∞ (S p−1 ) : Δ S p−1 f = −a(a + p − 2) f },

where the second isomorphism is induced by the restriction map F → f = F| S p−1 . Then H a (R p ) = {0} if p ≥ 2 or if p = 1 and a ∈ {0, 1}, and the orthogonal group O( p) acts irreducibly on H a (R p ). With this notation, the space J (ν) K  of K  -finite vectors is decomposed into the multiplicity-free direct sum of irreducible K  -modules as follows. J (ν) K 



H a (R p )  H b (Rq+1 ),

(19)

(a,b)∈N2even

where we set N2even := {(a, b) ∈ N2 : a ≡ b

mod 2}.

Therefore, any G  -submodule is characterized by its K  -types, which, in turn, is parametrized as a subset of N2even via (19). We introduce the following notation. Definition 13 We set  {(a, b) ∈ N2even ++ E ν := N2even  {(a, b) ∈ N2even E ν+− := N2even  {(a, b) ∈ N2even E ν−+ := N2even  {(a, b) ∈ N2even E ν−− := N2even

: a + b ≤ −ν}

if ν ∈ −2N, if ν ∈ / −2N,

: a − b ≤ −ν + q − 1}

if 1 − ν + q ∈ 2Z, if 1 − ν + q ∈ / 2Z,

: a − b ≥ ν − p + 2}

if ν − p ∈ 2Z, if ν − p ∈ / 2Z,

: a + b ≥ ν + 3 − p − q} if p + q − 1 − ν ∈ −2N, if p + q − 1 − ν ∈ / −2N.

Then the K  -types of any nonzero G  -submodules of J (ν) are given by the intersection of some E νδ,ε (δ, ε = ±). p,q+1

Example 14 In Sect. 1.4, we discussed discrete series representations π−,ν−ρ (ν ∈ Z and ν > ρ) for the pseudo-Riemannian space form M − p,q = O( p, q + 1)/O( p, q).

Image of Conformally Covariant, Symmetry Breaking Operators for R p,q

17

Then, if ν ≡ q + 1 mod 2, then the smooth representation (π−,ν−ρ )∞ of π−,ν−ρ is isomorphic to the subrepresentation of J (ν) with K -types given by E ν+− , see [17, Sect. 5]. p,q+1

p,q+1

As in [2], we define functions of R2 by A++ p,q,c (a, b) A+− p,q,c (a, b) A−+ p,q,c (a, b) −− A p,q,c (a, b)

:= := := :=

c − a − b, c − a + b + q − 2, c + a − b + p − 2, c + a + b + p + q − 4.

Then we may characterize E ν±± by the “barriers” as follows: • when ν ∈ 2Z, E ν++ = {(a, b) ∈ N2even : A++ p,q+1,−ν (a, b)  0}; • when 1 − ν + q ∈ 2Z, E ν+− = {(a, b) ∈ N2even : A+− p,q+1,−ν (a, b)  0}; • when ν − p ∈ 2Z, E ν−+ = {(a, b) ∈ N2even : A−+ p,q+1,−ν (a, b)  0}; • when p + q − 1 − ν ∈ 2Z, E ν−− = {(a, b) ∈ N2even : A−− p,q+1,−ν (a, b)  0}. In later sections, we use the symbol A±± to refer to the line (or “barrier”) defined by the zero locus of the function A±± (x, y), indicating that the submodules graphically given by the barrier.

3.3 Main Theorem: Image of Aλ,ν As we mentioned, since J (ν) is multiplicity-free as a K  -module, any G  -submodule of J (ν) is characterized by its K  -types, or equivalently, the corresponding subset of N2even via (19).   Theorem 15 Suppose p ≥ 2 and q ≥ 1. Then Image Aλ,ν : I (λ) → J (ν) is a G  submodule of J (ν) which is characterized by its K  -types according to the decomposition (18) of the parameter space as follows:

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T. Kobayashi and A. Leontiev

⎧ ++ +− ⎪ ⎪ E ν++ ∩ E ν+− ⎪ ⎪ ⎨ Eν ∩ Eν ∅ ⎪ ⎪ E ++ ∩ E ν+− ∩ E ν−+ ∩ E ν−− ⎪ ⎪ ⎩ ν++ E ν ∩ E ν+− ∩ E ν−+ ∩ E ν−−

if (λ, ν) ∈ C2 − (// ∪ \\) , if (λ, ν) ∈ // − (X+ ∪ |||) , if (λ, ν) ∈ //∩ |||, if (λ, ν) ∈ X+ − |||, if (λ, ν) ∈ \\ − X.

3.4 Restatement of Theorem 15 The conditions on the parameter (λ, ν) in Theorem 15 may look somewhat complicated, however, the subsets of C2 given in (18) are of simpler forms when we specify the parities of p, q and ν as follows. Proposition 16 Let ρ = 21 ( p + q − 1) as in (7). Suppose ν ∈ Z. Then the subsets in (18) in Table 1 reduce to the following sets when we impose conditions on p, q and ν in the left two columns in the table. To be precise about Table 1, we use the following convention. Under the conditions on p, q and ν described in the left two columns, the symbol in each box gives the same subset of (λ, ν) with that of the 0th row above the box. For example, // − X+ Table 1 Decomposition of C2 − (// ∩ \\) in (18) // − (X+ ∪ |||) p even q even

p odd q even

p even q odd

p odd q odd

//∩ |||

X+ − |||

\\ − X

∅

//

∅

\\

//

∅

∅

\\

∅

//

∅

\\

∅

//

∅

\\ − X

//

∅

X+

\\ − X

// − X+

∅

X+

∅

∅

//

∅

\\ − X

∅

//

∅

\\ − X

∅

//

∅

∅

ν odd

//

∅

∅

\\

ν even

∅

//

∅

\\

ν even ν≤0 ν even ν>0 ν odd ν even ν≤0 ν even 00 ν odd

//− ||| //∩ ||| ∅ // // ∅ ∅ //

Thus Lemma 19 is proved.



4.3 Description of the Image of Symmetry Breaking Operators (p Even, q Even) For p and q both even, the critical cases are when (λ, ν) ∈ Z2 . We divide the parameter space Z2 into the following regions (see Theorem 20 below for the precise definition). Here, we follow the convention of Kobayashi–Speh [22] that ν is for the x-axis and λ is for the y-axis.

We are ready to describe graphically the image of the regular symmetry breaking operators for p and q both even as follows. Theorem 20 Let p and q be both even. (1) Suppose ν ∈ / Z. Then the regular symmetry breaking operator Aλ,ν : I (λ) → J (ν) is surjective for any λ ∈ C. (2) Suppose ν ∈ 2Z.

22

T. Kobayashi and A. Leontiev

(2-a) For λ ∈ 2Z, the K  -types of the image of Aλ,ν are given by the subsets of N2even in the following colored red regions via (19).

(2-b) For λ ∈ 2Z + 1, we divide the parameter set (λ, ν) ∈ (2Z + 1) × (2Z) into the following three cases. Case A : ν ≤ 0, Case B : 0 < ν, λ + ν > p + q − 1, Case D: 0 < ν, λ + ν ≤ p + q − 1. Then the image of Aλ,ν in Cases A or B is described graphically by the same diagram with the one in Cases A or B, respectively, whereas the one in Case D is given as follows.

(2-c) For λ ∈ / Z, we divide the parameter space (C − Z) × 2Z into two cases.

Image of Conformally Covariant, Symmetry Breaking Operators for R p,q

23

Case Abis : ν ≤ 0, Case Bbis : 0 < ν. Then the image of the regular symmetry breaking operator Aλ,ν in Cases Abis or Bbis is described graphically by the same diagram with the one in Cases A or B, respectively. (3) Suppose ν ∈ 2Z + 1. We divide the parameter space C × (2Z + 1) into the following four cases: Case E: λ ∈ 2Z, Case E : λ ∈ 2Z + 1, ν < λ, Case F: λ ∈ 2Z + 1, ν ≥ λ, Case Ebis : λ ∈ C − Z. Then the image of Aλ,ν is described graphically by the following diagram.

Cases E, E′ and Ebis

Case F

Remark 21 In each case, the arrangement of the barriers A±± may vary. Assuming Theorem 15, we complete the proof of Theorem 20. Proof of Theorem 20. By Lemmas 18 and 19, Theorem 20 follows readily from Theorem 15. 

5 Graphic Description of the Image of the Regular Symmetry Breaking Operator: Case p Odd (≥3) and q Even In this section we give a graphic description of Theorem 15 in the case where p is odd (≥3) and q is even.

24

T. Kobayashi and A. Leontiev

5.1 Socle Filtration of the Target Space J(ν) In this section we give a graphic description of Theorem 15 in the case where p is odd (≥3) and q is even. We review from [2] the socle filtration of the principal series representations J (ν) of G  = O( p, q + 1) with p odd (≥3) and q even as in Fact 17. We keep the notation from (7) that ρ = 21 ( p + q − 1). Fact 22 Suppose p is odd (≥3) and q is even. (1) The G  -module J (ν) is irreducible if and only if ν ∈ C − Z or ν is an even integer satisfying 0 < ν < 2ρ. (2) For ν ∈ Z, G  -submodules of J (ν) are classified by their K  -types as follows: • For ν even,

{0}  E ν++  J (ν)

if ν ≤ 0,

{0}  J (ν)

if 0 < ν < 2ρ,

{0} 

E ν−−

 J (ν)

if 2ρ ≤ ν.

• For ν odd,

{0}  E ν+− ∩ E ν−+  E ν+− , E ν−+  J (ν) {0}  {0} 

E ν+− , E ν+− ,

E ν−+ E ν−+

 J (ν) 

E ν+−

if ν < ρ, if ν = ρ,

⊕

E ν−+

 J (ν)

if ρ < ν.

The following lemma formulated with the same convention as in Lemma 18 is readily seen from Definition 13 set theoretically, and fits well with Fact 22. Lemma 23 Suppose that p is odd (≥3) and q is even. We retain the notation that ρ = 21 ( p + q − 1). For ν ∈ Z, the G  -modules with the K  -types ε∈{±} E ν+ε or  δε δ,ε∈{±} E ν are given in the following table. E ν++ ∩ E ν+− E ν++ ∩ E ν+− ∩ E ν−+ ∩ E ν−− ν even ν≤0 E ν++ E ν++ 0 < ν < 2ρ J (ν) J (ν) 2ρ ≤ ν J (ν) E ν−− ν odd ν 1. Now set u = (s, k), v = (s, k − 1) viewed as elements of J (as above). Note that this implies that i u = i v = s. Set Fk γj s

 :=

αs , if j ∈]v, u], 0, otherwise.

(5)

We call Fsk : k > 1 the closed face of type s at position k > 1. This terminology was motivated by the geometry of wiring diagrams (see [6, Figures 1, 2]).

2.2 Adjoining Faces to Trails Given K ∈ Kt B Z we define K := K + Fsk by the rule

Trails for Minuscule Modules and Dual Kashiwara Functions …

Fk

γ Kj = γ Kj + γ j s .

43

(6)

However it is far from obvious that K ∈ Kt B Z . When this does hold we say that K (resp. K ) is obtained by adjoining (resp. removing) the face Fsk to K (resp. from K ). These operations are presented diagrammatically in [6, Figures 1, 2]. We may truncate a trail K at any point after it trivializes. Then the resulting truncated trail is given by a monomial e K in the simple root vectors ei : i ∈ I such that e K applied to the lowest weight vector v−t gives the extremal vector v−wϕ(K ) t of weight −wϕ(K ) t , up to a non-zero scalar. Then adjoining the closed face Fsk+1 to K to obtain K just means shifting a factor of es in e K to the right from position (s, k + 1) to position (s, k) giving a new monomial e K . Here we can assume that (s, k + 1) ≤ ϕ(K ) without loss of generality. Then as a result of (B)(ii) and a relatively easy property of Demazure modules, one may show a slight generalization of the fact (not needed here) that e K v−t is always [7, 4.2–4.4] a multiple of v−wϕ(K ) t . Then K is a trail exactly when e K v−t is a non-zero multiple of v−wϕ(K ) t . In [7, 7.6] we suggested exactly when vanishing occurs and this we expressed as the “absence of false trails”.

2.3 The Minuscule Case In [7] a detailed analysis was made of when a face can be adjoined to a trail and the structure which results. It involves a manipulation of the Chevalley-Serre relations and the theory of finite dimensional sl(2) modules. We showed [7, Corollary 8.3] that the absence of false trails (see Sect. 2.2) allows one to compute Kt B Z via the “canonical” S-graphs (mentioned in 1.7 and studied in detail in [8, Sects. 5–7]). These describe exactly how Kt B Z can be recovered by adjoining faces to the driving trail K t1 . This theory brought out a remarkable and entirely unexpected relationship between the Chevalley-Serre relation and the canonical S-graphs. This relationship is expressed through [7, Thm. 7.6]. Unfortunately we were not quite able to prove the absence of false trails by the induction argument that this theorem suggests. Here we shall restrict ourselves to the minuscule case when trails are combinatorially defined. Then a main point is to describe when faces may be added to trails. Besides this, our proof that the resulting dual Kashiwara functions satisfy (1) recovers the special case of [1, Thm. 3.9] for minuscule modules in a completely elementary fashion.

2.4 The Driving Function In view of the comments in Sect. 1.4, the Kashiwara functions described in [4, 2.3.2] are replaced by those obtained by interchanging roots and coroots. In the present notation [4, 2.3.2] becomes

44

A. Joseph

rsk = m ks +



αi∨j (αs )m j , ∀(s, k) ∈ I × N+ ,

(7)

j>(s,k)

where m ks := m (s,k) . By (7), the difference of successive Kashiwara functions is given by the sum 

+ m ks + rsk−1 − rsk = m k−1 s

αi∨j (αs )m j = z Fs , k

(8)

j∈](s,k−1),(s,k)[

where for the second equality taken K = Fsk in (3).  we have ∨ 0 One may define rt = j∈J αi j (αt )m j and check that the driving function z t1 defined in [6, 4.7], on again interchanging roots and coroots, becomes z t1 = rt0 − rt1 = m 1t +

(t,1)−1 

αi∨j (αt )m j = z Ft . 1

(9)

j=1

2.5 Extremal Weight Trails Drop condition (P) on a trail, that is on an element of Kt B Z . (In the minuscule case condition (P) will result automatically - see Sect. 3.9.) An extremal weight trail is a trail K ∈ Kt B K for which γ1K = −st t , γ Kj+1 ∈ {si j γ Kj , γ Kj }, ∀ j ∈ J.

(10)

Here it is important to note that an arbitrary sequence γ Kj : j ∈ J satisfying (10) forms a trail if and only if the γ Kj are increasing and (B) is satisfied. In the above we may view K as the subset of J defined by { j ∈ J |γ Kj+1 = γ Kj }. Indeed such a subset uniquely defines a sequence of weights satisfying (10). Note that if si j γ Kj = γ Kj , then we could omit j from K and obtain the same trail. Thus we shall always assume that K is maximal with respect to a given trail and in this case we say that K is well-chosen (with respect to t ). Of course the above definition of K gives K as a well-chosen subset of J . Not all subsets of J are well-chosen.

2.6 The Final Function The driving function z t1 can be viewed as the initial function of Z t from which one hopes to be able to obtain the whole of Z t by adding suitable differences of successive Kashiwara functions.

Trails for Minuscule Modules and Dual Kashiwara Functions …

45

Now suppose that the Weyl group is finite and set r = (w0 ). Set γ1 = −st t . For all j ∈ [1, r − 1], set γ j+1 = si j γ j = −w j st t , where w j = si j . . . si1 , which we recall is a reduced expression. It follows that the γ j are increasing. +1 form a trail. Then Observe that γr = sir w0 γ1 and set γr +1 = γr . Thus the {γ j }rj=1 the above calculations give  1 αi∨j (γ j+1 + γ j )m j = αi∨r (γr )m r . 2 +

(11)

j∈N

For all j ∈ J set β j = si1 . . . si j−1 αi j = w −1 j−1 αi j . As is well-known {β j } j∈ j is just the set of positive roots. Moreover αi∨j (γ j ) = −β ∨j (t ) = β ∨j (γ1 ). Let θ(t) is the unique element of {1, 2, . . . r } such that βθ(t) = αt . f

f

f

Lemma 2 There exists a trail K t ∈ Kt B Z such that z t := z K t = m θ(t) . Proof Set θ(t) = k. Suppose k = r . Then αi∨r (γr ) = βr∨ (γ1 ) = αt∨ (t ) = 1. Then the right hand side of (11) equals m r , as required. For the general case we take a new sequence γr +1 , . . . , γ1 , given by γ j = γ j : = γk and γ j+1 = si j γ j : j > k. j ≤ k whilst γk+1 Write w0 = w1 sik w2 using the given reduced expression for w0 . Then deleting sik in the reduced expression for w0 means replacing w0 by w1 w2 . Yet αt = βk = w2−1 αi1 , so we obtain w0 = w1 w2 sαt . Thus w1 w2 has reduced length r − 1 and is hence a reduced expression. Consequently the γ j are increasing. On the other hand +1 γr +1 = w1 w2 γ1 = −w0 sαt t = −w0 t and so (B) is satisfied. Thus the {γ j }rj=1 form a trail. On the other hand for this trail the right hand side of (3) becomes αi∨k (δk )m k = αi∨k (γk )m k = βk∨ (γ1 )m k = αt∨ (t )m k = m k , as required. Remark. The set Kt B Z resembles a crystal in the sense that there is a lowest f weight element z t1 and (in the finite case) a highest weight element z t . Then the j independence of ∪ j∈N+ E t,s on s is the analogue of a crystal being a union of sstrings for any s ∈ I . One can therefore anticipate that its proof will be difficult.

2.7 The S-set of a Totally Degenerate S-graph We verify the truth of the Preparation Theorem [5, 8.6] in the special case we need here. Recall 1.7 and assume that the coefficients ci : i ∈ N are all equal and strictly positive. For the minuscule case it is enough to consider the case ci = 1 for all i ∈ N . Then the S graph degenerates to its pointed chain [8, Lemma 5.8.2]. Let z i ; i ∈ Nˆ denote the function assigned to the vertex with label i in the pointed chain. Then by P3 of [4, 6.2] one obtains z i − z j = r i − r j . Here r k is a Kashiwara function

46

A. Joseph

of the type s ∈ I determined by the type of the driving function z K assigned to the distinguished vertex (see Sect. 1.7), that is to say r k = rsk , with the latter given by (7). Recall that [4, 2.3.2] asserts that f s enters into the (s, k)th place of b ∈ B J (∞), j that is to say it increases m ks by 1, exactly when rsk (b) − rs (b) > 0 for k > j and j j rsk (b) − rs (b) ≥ 0 for k < j. Substitution into (7) then gives rsk ( f s b) ≥ rs (b). The computation which is a tiny-winy bit tricky is given in detail in [6, 5.5]. The final piece of magic is that the coefficient of m ks (viewed as a linear function on B J ) in z k , is zero by property P8 of [4, 6.7]. This follows from the way the coefficients K ) ci are deduced from the driving function z K (of type s). Indeed ci := −αs∨ (δ(s,i) i K which by (3) is just the coefficient −m s in z . In the minuscule case we can suppose cn+1 = 0 and ci = 1 : i = 1, 2, . . . , n, up to choice of n and a shift of indices (see + · · · + 2m ns + m n+1 . 3.7). Yet by (8) one has z k − z n+1 = rsk − rsn+1 = m ks + 2m k+1 s s k Thus the coefficient of m s in z k is indeed zero. We conclude that maxz∈Z z(b) = z k (b) = z k ( f s b) = maxz∈Z z( f s b), as required for (2).

3 The Minuscule Case 3.1 Announcement of Main Thorem Recall that V (−t ) is minuscule if every weight λ of V (−t ) is of the form −wt : w ∈ W . This forces αi∨ (λ) ∈ {−1, 0, 1}, for all i ∈ I and W to be finite. It also means that if αi∨ (λ) ≤ 0, then λ and λ − αi cannot both be weights of V (−t ). This was is used below. All the fundamental modules in type A are minuscule; but this is exceptional and in general very few are minuscule, for example none are minuscule in types G 2 , E 8 . Set Z t = {z K } K ∈K t B Z . Theorem : Suppose V (−t ) is minuscule. Then Z t is t-semi-invariant. In particular Z t satisfies max z(b) = εt (b), ∀b ∈ B J (∞). z∈Z t

The theorem is proved in the subsections which follow. In this we assume throughout that V (−t ) is minuscule. It implies that all trails are extremal weight trails. Recall that r = (w0 ) and that J = {1, 2, . . . , r }. It is convenient to define Jˆ = {1, 2, . . . , r + 1}.

Trails for Minuscule Modules and Dual Kashiwara Functions …

47

3.2 Well-chosen Subsets Fix a reduced decomposition w0 = sir . . . si1 , of the unique longest element of W . In this case we may identify J with R := {r, r − 1, . . . , 1} viewed as an ordered set. Then Kt B Z may be identified with the set of well-chosen subsets of R. Here one may remark that being well-chosen depends on the fundamental weight t and on the reduced sequence J , which here are both fixed.

3.3 Adjoining Faces Take K ∈ Kt B Z . Let us write u = (s, k), v = (s, k − 1), as in (5). One has r ≥ u > v ≥ 1 and i u = i v = s. Since the γ Kj are increasing and V (−t ) is minuscule, one has K K γu+1 = aαs + γuK , γv+1 = bαs + γvK with a, b ∈ {0, 1}.

(12)

Take K ∈ Kt B Z . Recall (6) and assume that K = K + Fsk for some (s, k) ∈ I × N+ , with k > 1. In this case we say that K is obtained from K by adjoining the face Fsk . Set γ Kj = γ j , γ Kj = γ j , for all j ∈ Jˆ. Observe that (5) and (6) give γk

 − γk =

αs , if k ∈]v, u], 0, otherwise.

(13)

In the above conventions and notation we have the / K . Lemma 3 As well-chosen subsets of R, one has u ∈ / K , v ∈ K and u ∈ K , v ∈ ∨ ∨ ∨ ∨ Moreover αs (γu ) = αs (γv+1 ) = −1, αs (γu ) = αs (γv+1 ) = 1. Proof Indeed by (12) there exists a ∈ {0, 1} such that γu+1 = γu+1 = γu + aαs = = γu . Again by γu + (a − 1)αs , so (12) forces a = 1. Thus γu+1 = γu + αs , γu+1 (12) there exists b ∈ {0, 1} such that γv+1 = γv+1 + αs = (γv + bαs ) + αs = γv + = γv + αs . This proves the (b + 1)αs , so (12) forces b = 0. Thus γv+1 = γv , γv+1 first part. The second part follows because V (−t ) is minuscule and using (13).

3.4 Moving es Factors Retain the above notation, assume (13) holds and take k ∈]u, v[. = γk and so k ∈ K . Suppose k ∈ K , then γk+1 = γk , so by (13) one has γk+1 = γk + Suppose that k ∈ / K . Then γk+1 = γk + αik . Then by (13) we obtain γk+1 αik , so k ∈ /K.

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A. Joseph

Proposition : Let K be a well-chosen subset of R. Fix (s, k) ∈ I × N+ with k > 1. Set u = (s, k), v = (s, k − 1). There exists a well-chosen subset K of R such that K = K + Fsk , when the latter are identified as elements of Kt B Z , if and only if u∈ / K , v ∈ K and for all k ∈]u, v[ with k ∈ / K one has αs∨ (αik ) = 0. Moreover in this case K = K − {v} ∪ {u}. Proof By Sect. 3.4 “only if” obtains on noting that αs∨ (αik ) ≤ 0 for all k ∈]u, v[, so then αs∨ (γk ) can only decrease as k increases from v + 1 to u and so by the second part of Sect. 3.3 it cannot change at all. Conversely suppose for all k ∈]u, v[∩(R − K ) one has αs∨ (αik ) = 0. Yet sik and / K , v ∈ K means that sαs commute when αs∨ (αik ) = 0, and on the other hand u ∈ γu+1 = sαs γu , γv+1 = γv . Thus the factor sαs may be taken rightwards through the sik : k ∈]u, v[ from position u to position v in the trail defined by K to obtain the = γu , γv+1 = sαs γv . Moreover it is then obvious that trail defined by K so that γu+1 condition (B) being satisfied for K implies that condition (B) is satisfied for K . The very last part is clear. Remark. The above operation can also be viewed as moving a factor of es rightwards from position u to position v. When V (−t ) is not minuscule this is much more delicate. Here one may not have commutativity and one has to deal with the general Chevalley-Serre relations - see [7, Sect. 7].

3.5 Sequences for Fixed s ∈ I 3.5.1

Definition of Sequences

Fix a well-chosen subset K ⊂ R and s ∈ I . Set u k = (s, k) : k ∈ N+ . Recall the definition of δ Kj given in Sect. 1.5. Consider the behaviour of the sequence αs∨ (δuKk ) : k ∈ N+ . Suppose u k ∈ / K . Since K is well-chosen γuKk +1 = γuKk . In addition since V (−t ) is minuscule, one has αs∨ (γuKk ) = −1. Suppose u k ∈ K , then αs∨ (δuKk ) = αs∨ (γuKk ) ∈ {−1, 0, 1}. On the other hand by Sect. 1.5(i) we have αs∨ (δuK1 )

3.5.2

=

αs∨ (γuK1 )

 ∈

{1}, if s = t, K = K t1 , {0, −1}, otherwise.

(14)

Non-negativity at Maximal Element

The following holds whenever W is finite. It is a natural complement to Sect. 1.5(i). Lemma 4 Fix s ∈ I and let kmax be the largest value of j such that i j = s. Then for all K ∈ Kt B Z one has

Trails for Minuscule Modules and Dual Kashiwara Functions …

αs∨ (δuKkmax ) ≥ 0.

49

(15)

Proof By definition of kmax and since the γ Kj are increasing we obtain γrK+1 ∈ γuKkmax +1 + N(π − {αs }). Yet γrK+1 = −w0 t by (B). Since this is the highest weight / Ω, so αs∨ (γuKkmax +1 ) ≥ 0. Then by the first of V (−t ) it follows that γuKkmax +1 + αs ∈ ∨ K ∨ K part αs (γu kmax ) ≥ αs (γr +1 ) ≥ 0. Hence (15).

3.6 Non-positivity Retain the notation of 3.5.1. Lemma 5 Fix s ∈ I . Suppose αs∨ (δuKk ) = −1, then / K one has αs∨ (αi ) = 0. (i) For all  ∈ ]u k , u k+1 [ with  ∈ (ii) αs∨ (δuKk+1 ) = 1. Proof Under the hypothesis of the lemma, it follows that u k ∈ K . In particular γuKk = γuKk +1 . Since off-diagonal entries of the Cartan matrix are ≤ we obtain αs∨ (γuKk+1 ) ≤ αs∨ (γuKk +1 ) = αs∨ (γuKk ) = −1. Moreover equality holds since V (−t ) is minuscule. This forces (i). Moreover αs∨ (δuKk+1 )

 =

−1, if u k+1 ∈ K, / K. 0, if u k+1 ∈

(16)

This gives (ii).

3.7 Strategy Fix s ∈ I . Take K ∈ Kt B Z and assume that either s = t or K = K t1 . Let z K be the linear function on B J given by (3). In ([6, 5.2] we defined the notion of a driving function of type s. All we need to know here is that if z K is a driving function of type s ∈ I then the “coefficient set” {ck := −αs∨ (δuKk )}k∈N+ , consists of non-negative integers and determines an S-graph, as noted in Sect. 1.7. Furthermore the latter determines an S-set which satisfies (2), as noted in 2.7. In the minuscule case these integers are all in {0, 1} which is very special and makes the S-graph particularly simple, as noted in 1.7. (It also means that conversely z K is a driving function if the coefficient set consists of non-negative integers. This can fail in general [6, 5.2].) For a maximal subset in which these integers form the sequence (0, 1, 1, . . . , 1) the S-graph is reduced to its pointed chain 1.7. The result for an arbitrary sequence of zeros and ones (starting with a zero) obtains

50

A. Joseph

by the obvious joining of the S-graphs coming from the sequences (0, 1, 1, . . . , 1). A precise formulation can be found in [8, Lemma 5.8.1], in any case it is what results from the construction below. Suppose that K is a well-chosen subset of R such that αs∨ (δuKk ) ∈ {−1, 0} for all k ∈ N+ . By (15) one must have αs∨ (δuKkmax ) = 0. Let k1 ∈ N+ be maximal such that αs∨ (δuKk ) = −1. Then k1 < kmax and 1 ∨ K αs (δu k +1 ) = 0, so u k1 +1 ∈ / K , u k1 ∈ K . Let 1 ≤ k1 be maximal such that αs∨ (δuK −1 ) = 1 1 0, or failing this take 1 = 1. By Sect. 3.5, we may successively adjoin the faces Fsk1 +1 , Fsk1 +1 , . . . , Fs1 +1 to obtain a subset Z 1 ⊂ Kt B Z in which each element is given by a well-chosen subset R. This subset is exactly that obtained from the pointed chain to which the S-graph degenerates 1.7. If 1 > 1, then this process may be repeated and we obtain a union Z of such subsets, which is precisely the S-set obtained from the driving function z K using [8, Lemma 5.8.1]. We call Z so constructed, the completion of z K with respect to s and Z an scomplete set. j

It follows from the Preparation Theorem [5, Thm. 8.6] that Z t,s := Z satisfies (2). This relatively simple case is verified directly in 2.7. Indeed it was this easy case from which the Preparation Theorem originated.

3.8 Removing Faces Consider the contrary case when αs∨ (δuKk+1 ) = 1, for some k ∈ N. We can assume that k is minimal with this property. Assume that s = t or K = K t1 . Then by Sect. 1.5(i) we obtain k > 0. Then αs∨ (δuKk ) = −1, otherwise we would obtain a contradiction with (ii) of Sect. 3.6. Hence αs∨ (δuKk ) = 0, by the minimality of k. Set u = u k+1 , v = u k . Lemma 6 Suppose αs∨ (δuK ) = 1 and αs∨ (δvK ) = 0. Then the face Fsk+1 can be removed from K . More precisely u ∈ K , v ∈ / K and K := K − {u} ∪ {v} is a wellchosen subset of R such that K + Fsk+1 = K . Proof Through the hypothesis that αs∨ (δuK ) = 1, we obtain u ∈ K and moreover K + ∈]v,u[ n  αi , for some n  ∈ N. Hence αs∨ (γuK ) = αs∨ (δuK ) = 1. Now γuK = γv+1 K ) ≥ 1. Since V (−t ) is minuscule, equality holds and then for all  ∈]u, v[ αs∨ (γv+1 with  ∈ / K , one has αs∨ (αi ) = 0. K ) = 1, it follows that v ∈ / K and αs∨ (γvK ) = −1. Since αs∨ (δvK ) = 0 and αs∨ (γv+1 Moreover by the last assertion of the previous paragraph it follows as in Sect. 3.4 that

Trails for Minuscule Modules and Dual Kashiwara Functions …

51

K := K − {u} ∪ {v} is a well-chosen subset of R. Finally K + Fsk+1 = K obtains by interchanging K , K in Sect. 3.4.

3.9 Positivity of the Leading Coefficient Let K ∈ Kt B Z be an extremal weight trail. Suppose that γ Kj+1 = si j γ Kj , for all j ∈ K = −w0 t , so γ1K = −t , which contradicts (B)(i). J . By (B)(ii) one has γ(w 0 )+1 Thus let j ∈ J be maximal such that γ Kj+1 = si j γ Kj . Then γ Kj+1 = −w j t = γ Kj and moreover αi∨j (δ Kj ) = αi∨j (γ Kj+1 ) whilst αi∨j (γ Kj+1 ) > 0 by the choice of j. We conclude that the leading coefficient of z K is positive (and integer). (One may expect this to be a general property of any trail in Kt B Z .) The above result may be expressed in the notation of Sect. 3.8 as follows. Lemma 7 Let K be an extremal weight trail, then there exists (s, k) ∈ I × N such that αs∨ (δuKk+1 ) = 1.

3.10 Conclusion of Proof of Main Theorem Fix s ∈ I . Take K ∈ Kt B Z − {K t1 } and assume s = t or K = K t1 . If z K is not a driving function of type s, then by Sect. 3.8 we may remove finitely many faces of type s from K to obtain a new element K ∈ Kt B Z such that z K is a driving function of type s. Moreover a comparison of the conclusions of Sects. 3.4 and 3.8, shows that K lies in the subset Z obtained from K as defined in Sect. 3.7. By Sect. 2.2 the initial driving function z t1 is given by a well-chosen subset K t1 of K1

R. Set It := {s ∈ I − {t}|αs∨ (δ j t ) = 0| j ∈ [1, (t, 1)[}. If It = φ, then (as one easily checks) Z t := {z t1 } is already t-semi-invariant. One f may also check that z t1 = z t in this case. K1

Given s ∈ It , then by Sect. 1.5(i), one has αs∨ (δ j t ) ∈ {0, −1} for all j ∈ J . Thus 1 z t is a driving function of type s. Let Z denote the completion of z t1 with respect to s. For all s ∈ I , every function in Z besides z t1 lies in an s -complete set by the first paragraph above. (This point is a stumbling block outside the minuscule case.) We conclude that this process gives a finite set Z t which for each s ∈ I − {t} is a union of complete sets with respect to s whilst Z t − {t} is a union of complete sets with respect to t. Conversely given any well-chosen subset K of R, except that corresponding to K = K t1 , we can remove, via Sects. 3.8 and 3.9, faces from K . Since there are only finitely many well-chosen subsets of R this process must terminate at K t1 . We conclude that Z t is in bijection with the well-chosen subsets of R and that it is t-semi-invariant.

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This proves Theorem 3.1. Remark. We have shown that in the minuscule case that every K ∈ Kt B Z is obtained by adjoining faces to K t1 . Thus (P) is automatically satisfied in this case.

3.11 Giant S-sets In the minuscule case, Kt B Z is a giant S-set in the sense of [7, 8.2]. This means that for all s ∈ I , it is a disjoint union of S-sets of type s. We had thought that a giant S-set always exists and would be precisely the set of extremal elements of Kt B Z , viewed via (3), as functions on B J . A counter-example [6, 5.7] was found in type F4 , via computer computations of Zelikson. In general all one can hope for is that the independence on s ∈ I obtains on taking maxima as described in the last part of [7, Thm. 8.7]. Outside the minuscule case and type A, the set of functions defined by Kt B Z through (3) is highly redundant from the point of view of (1). Our construction gives a much smaller set though not always just the set of extremal elements.

4 Index of Notation Symbols occurring frequently are given below in the subsection in which they are first defined. 1.1 g, h, I, αi , αi∨ , W, (w), si , J, w j , B J , m j , f s , b∞ , B J (∞), B(∞). 1.2 Z t , εt . 1.3 z t1 .

Fk

1.5 Kt B Z , et , t , Ω, v Kj , γ Kj , (K ), δ Kj , z K . 2.1 , j < (s, k), ]v, u[, K t1 , Ftk , γ j s . 2.4 rsk , m ks . f 2.6 z t . Acknowledgements I would like to thank Polyxeni Lamprou and Shmuel Zelikson who worked together with me on this project stretching over several papers and several years. A key breakthrough on constructing S graphs developed from the computations of Lamprou. Despite this and her many other earlier excellent works she not succeed to stay in Academia. At least one can hope that the novelties in [8] will be one day fully appreciated.

Trails for Minuscule Modules and Dual Kashiwara Functions …

53

References 1. A. Berenstein and A. Zelevinsky, Tensor product multiplicities, canonical bases and totally positive varieties. Invent. Math. 143 (2001), no. 1, 77–128. 2. O. Gleizer and A. Postnikov, Littlewood-Richardson coefficients via Yang-Baxter equation. Internat. Math. Res. Notices 2000, no. 14, 741–774. 3. A. Joseph, Quantum groups and their primitive ideals. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 29. Springer-Verlag, Berlin, 1995. 4. A. Joseph, Consequences of the Littelmann path theory for the structure of the Kashiwara B(∞) crystal. Highlights in Lie algebraic methods, 25–64, Progr. Math., 295, Birkäuser/Springer, New York, 2012. 5. A. Joseph, A Preparation Theorem for the Kashiwara B(∞) Crystal, Selecta Math. (N.S.) 23 (2017), no. 2, 1309–1353. 6. A. Joseph, Dual Kashiwara functions for the B(∞) crystal, Transf. Groups (to appear). 7. A. Joseph, Trails S-graphs and identities in Demazure modules, arXiv: 1702.00243. 8. A. Joseph and P. Lamprou, A new interpretation of the Catalan numbers, J. Algebra 504 (2018), 85–128. 9. M. Kashiwara, Global crystal bases of quantum groups. Duke Math. J. 69 (1993), no. 2, 455– 485. 10. M. Kashiwara, The crystal base and Littelmann’s refined Demazure character formula. Duke Math. J. 71 (1993), no. 3, 839–858. 11. P. Littelmann, Paths and root operators in representation theory, Annals of Math. 142 (1995), 499–525.

From Euler’s Play with Infinite Series to the Anomalous Magnetic Moment Ivan Todorov

Abstract During a first St. Petersburg period Leonhard Euler, in his early twenties, became interested in the Basel problem: summing the series of inverse squares. In the words of André Weil [W] “as with most questions that ever attracted his attention, he never abandoned it”. Euler introduced on the way the alternating “phiseries”, the better converging companion of the zeta function, the first example of a polylogarithm at a root of unity. He realized - empirically! - that odd zeta values appear to be new (transcendental?) numbers. It is amazing to see how, a quarter of a millennium later, the numbers Euler played with, “however repugnant” this game might have seemed to his contemporary lovers of the “higher kind of calculus”, reappeared in the first analytic calculation (by Laporta and Remiddi) of g − 2 the anomalous magnetic moment of the electron, the most precisely calculated and measured physical quantity [K]. Mathematicians, on the other hand, are reviving the dream of Galois of uncovering a group structure of the periods, including the same multiple zeta values - the mixed Tate motives, inspired by ideas of Grothendieck and appearing in a variety of subjects - from algebraic geometry to Feynman amplitudes.

Introduction “The usefulness of useless knowledge” was the provocative title of a 1939 essay of Abraham Flexner, the founding director of the Institute for Advanced Study in Princeton, recalled - and vindicated - in 2017 by the Institute’s current director Robert Dijkgraaf. The present story is another illustration of the value of curiosity driven

I. Todorov (B) Institut des Hautes Etudes Scientifiques, 91440 Bures-sur-Yvette, France e-mail: [email protected] I. Todorov Institute for Nuclear Research and Nuclear Energy, Tsarigradsko Chaussee 72, 1784 Sofia, Bulgaria © Springer Nature Singapore Pte Ltd. 2018 V. Dobrev (ed.), Quantum Theory and Symmetries with Lie Theory and Its Applications in Physics Volume 1, Springer Proceedings in Mathematics & Statistics 263, https://doi.org/10.1007/978-981-13-2715-5_3

55

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research and of long term thinking in a time full of short term distractions. What could have seemed in mid eighteenth century as Euler’s idle play with numbers given by infinite series became a central topic in mathematics and quantum field theory in the last decades. The paper consists of three distinct parts. The first (Sect. 1) begins with Euler’s early encounter of Mangoli’s “Basel problem” and follows his repeated assaults on the zeta series and their alternating cousins.1 The second (Sect. 2) is concerned with what came to be called the g − 2 saga (from 1947 to 2017). Some related recent developments in number theory, algebraic geometry and perturbative quantum field theory are surveyed in Sect. 3. It should be noted that the stories in Sects. 2 and 3 are both open ended. The beginning of a theory of Feynman periods concerns the primitively divergent graphs in a (massless) scalar (φ 4 -)theory. It has been only recently realized that g − 2 calculations (in which infrared divergences are only resolved for sums of Feynman amplitudes) display properties under Galois coaction, similar to those discovered in the φ 4 theory. If it should become clear from our exposé that the study of the Galois coaction on quantum periods is in its infancy, we are not even alluding to the physical problem with the anomalous magnetic moment of the muon whose understanding does not match the success story of g − 2 for the electron.

1 Leonhard Euler (1707–1783): Zeta Values and Their Alternating Companions 1.1 The Basel Problem Having obtained a kind of master degree in Basel at the age of 17, Euler, not quite twenty, got an offer to join the Saint Petersburg Academy. There he was commissioned (fortunately, not for long) into the Russian navy (as he had won, just before leaving Basel, a prize for an essay on ship-building, never having seen a sea-going ship [62], Chapt. 3, Sect. II). Along with his major work on mechanics (in two volumes), on music theory and naval architecture Euler wrote during his first Petersburg period (1727–1741) some 70 memoirs on a great variety of topics.2 It was early in this period, around 1729, that Euler became first interested in the Basel problem - the problem of finding the sum of inverse squares or what we would 1A

first hand review of Euler’s work and an elementary introduction to multple zeta values is contained in Sects. 1–2 of Cartier’s Bourbaki lecture [24]. 2 A definitive collection of Euler’s works, Opera Omnia, has been published since 1911 by the Euler Commission of the Swiss Academy of Sciences. By the time of the appearance of his first full scale biography [21], at the end of 2015 the edition is nearing completion with over 80 large volumes published. The Eneström index of Euler’s papers counts 866 entries. A concise (30-page) biography of Euler with color illustrations is contained in [34]; shorter biographical sketches can be found in [5, 62].

From Euler’s Play with Infinite Series to the Anomalous Magnetic Moment

57

now call (after Riemann) ζ (2). It was the dawn of infinite series. One knew how to sum a similar series of inverse rectangles which allowed to prove that ζ (2) is a real number between one and two: ∞  1 , Re(s) > 1, ζ (s) = s n 1  ∞ ∞  ∞    1 1 1 1 = − = 1 < ζ (2) < 1 + = 2. (1) n(n + 1) n n+1 n(n + 1) 1 1 1

But what exactly is ζ (2)? (It is actually a problem which the young Pietro Mengoli (1625–1686), successor of Cavalieri in Bologna, posed in 1644 [5] and which excited the brothers-rivals Jacob and Johann Bernoulli in Basel.) Euler first tried to obtain a good numerical estimate for ζ (2). The series (1) is slowly convergent. To get from it a six digit accuracy (by 1731 Euler had ζ (2)  1.644934 [3]), one would have needed a million terms. On the way of obtaining a faster converging expression Euler first introduced the alternating phi-series, φ(s) =

∞  k=1

(−1)k−1

1 = (1 − 21−s )ζ (s), ks

φ(1) = ln(2),

(2)

a special case of the Dirichlet L-functions [57], introduced over hundred years later. (The series for φ(s) is convergent for Re(s) > 0 while the harmonic series ζ (1) diverges.) More importantly, Euler viewed ζ (2) as a special value of a power series, the dilogarithm [64], which also has an integral representation (noted by Leibniz in 1696 in a letter to Euler’s teacher, Johann Bernoulli): Li 2 (x) ==

 x t  x ∞  xn ln(1 − t) dtdt1 dt = . = − 2 n t t (1 − t1 ) 0 0 0 n=1

(3)

ζ (2) and φ(2) appear as the values of this function at the two square roots of unity: ζ (2) = Li 2 (1), φ(2) = Li 2 (−1). Furthermore, Euler derived the elementary functional equation (see [61] Eq. (3)): Li 2 (x) + Li 2 (1 − x) + ln(x)ln(1 − x) = Li 2 (1). Setting in it x = 1/2 he obtained a much faster converging series for ζ (2). In 1734 Euler had the bold idea to extrapolate Newton’s formula for the coefficient of a polynomial in terms of its roots to infinite series (after having obtained an approximation for ζ (2) with twenty decimal places). He announced the beautiful result, ζ (2) = π 2 /6 in letters to friends (including Daniel Bernoulli) and did not hide his excitement in the introduction to the article (quoted in [61] Sect. 2). Here is a sketch of this ingenious direct calculation (see for more detail [9, 61]). Taking the logarithmic derivative of the infinite product expansion of the sine function

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one finds: ∞ 

1 ⇒ x − kπ −∞  2 n ∞ ∞   x2 x = 1 − 2 ζ (2n) . xcot (x) = 1 − 2 2 2 2 k π −x π2 n=1 k=1 cot (x) =

(4)

Compared with the power series expansion of cot(x) this allows to compute ζ (2n) (at least for small n). In a next assault on the problem Euler recognized the appearance of the numbers Bn encountered in the posthumous work of Jacob Bernoulli (1655– 1705). He called them Bernoulli numbers and arrived (in 1740) at the beautiful general formula: 1 1 1 B2n (2πi)2n , B2 = , B4 = − , B6 = , (−1)n−1 B2n ∈ Q>0 , 2(2n)! 6 30 42 (5) Euler writes ζ (n) = Nn π n noting that for even n Nn is rational; he has computed ζ (3) to ten significant figures and convinced himself that N3 is not a rational number with a small denominator (see [28] where the original Euler’s paper - in Latin - is cited). He conjectured that Nn for odd n might be a function of ln(2)(= φ(1)), [5], but this did not work either. Later, in 1749, in his Berlin period, he acknowledges: “for n odd all my efforts have been useless until now” (cited in [5]). In fact, this failure made mathematicians, and especially mathematical physicists, believe that Euler had discovered new transcendental numbers, designed to play an important role in both pure mathematics and quantum field theory. ζ (2n) = −

1.2 Memorable Mathematical Developments Trying to find polynomial relations among zeta values Euler was led by the stuffle product ζ (m) ζ (n) = ζ (m, n) + ζ (n, m) + ζ (n + m) (6) to the concept of multiple zeta values (MZVs): ζ (n 1 , . . . , n d ) =

 0 0 integral Jn is calculated by applying Lemma 4,  Jn =

√ 1 n Pδ Q −g d4 x + 2 l=0 M n−1



Sμν (l P, n−1−l Q)δg μν M

√ −g d4 x. (43)

Variations of Infinite Derivative Modified Gravity

99

Using (37) in the first term we obtain 



Jn =

 √ Rμν Q  n P − K μν (Q  n P) δg μν −g d4 x

M n−1

1 + 2 l=0



Sμν (l P, n−1−l Q)δg μν

√ −g d4 x.

(44)

M

Summation over n yields the final result 

∞

 √ Pδ(F()Q) −g d4 x = f n Jn

I = M

=

∞  n=0

n=0





fn

 √ Rμν Q  n P − K μν (Q  n P) δg μν −g d4 x

M

 ∞ n−1 √ 1  fn Sμν (l P, n−1−l Q)δg μν −g d4 x + 2 n=1 l=0 M    √ = Rμν Q  F()P − K μν (Q  F()P) δg μν −g d4 x M ∞ n−1

1  + fn 2 n=1 l=0



Sμν (l P, n−1−l Q)δg μν

(45)

√ −g d4 x.

M

3 Equations of Motion Let us return to action (1). In order to calculate δS we introduce the following auxiliary actions  S0 =



(R − 2Λ)

√

−g d4 x,

M

S1 =

P(R)F()Q(R)

√ −g d4 x.

(46) (47)

M

Action S0 is Einstein–Hilbert action and its variation is    √ δS0 = G μν + Λgμν δg μν −g d4 x. M

(48)

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Lemma 5 Variation of the action S1 is  √ 1 δS1 = − gμν P(R)F()Q(R)δg μν −g d4 x 2 M    √ + Rμν W − K μν W δg μν −g d4 x M ∞

1  + fn 2 n=1 l=0 n−1



√ Sμν (l P(R), n−1−l Q(R))δg μν −g d4 x,

(49)

M

where W = P  (R)F()Q(R) + Q  (R)F()P(R). Proof Variation δS1 is equal to  

√ √ P(R)F()Q(R)δ( −g) + δ P(R)F()Q(R) −g M √  + P(R)δ(F()Q(R)) −g d4 x.

δS1 =

(50)

All the terms in the previous formula are obtained by Theorem 1. In particular (36) yields  M

√ 1 P(R)F()Q(R)δ( −g) d4 x = − 2



√ gμν P(R)F()Q(R) δg μν −g d4 x. M

(51) Also, from equation (37) we get 

√



√ δ(P(R))F()Q(R) −g d x = P  (R)δ R F()Q(R) −g d4 x M M      √  = Rμν P (R)F()Q(R) − K μν P (R)F()Q(R) −g d4 x. (52) 4

M

The last term is calculated by (38). 

√ P(R)δ(F()Q(R)) −gd4 x M     √ = Rμν Q  (R)F()P(R) − K μν Q  (R)F()P(R) −g d4 x M

+

∞  n=1

fn

n−1   l=0

  √ Sμν l P(R), n−1−l Q(R) −g d4 x. M

Adding equations (51), (52) and (53) together proves the Lemma.

(53)

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101

Theorem 2 Variation of the action (1) is equal to zero iff   1 1 Gˆ μν = G μν + Λgμν − gμν P(R)F()Q(R) + Rμν W − K μν W + Ωμν = 0, 2 2 (54) where W = P  (R)F()Q(R) + Q  (R)F()P(R), Ωμν =

∞  n=1

fn

n−1 

  Sμν l P(R), n−1−l Q(R) .

(55) (56)

l=0

Proof The proof of Theorem 2 is evident from the Lemma 5 and Theorem 1.

4 Second Variation of the Action In this section we set h μν = δgμν . From Lemma 2 we see that h μν = −δg μν . Also let h = g μν h μν be the trace of h μν . Operator δ is defined by (δ)V = δ(V ) − δV . Then we can prove the following Lemma Lemma 6 Let U, V be scalar functions. Then 1 (δ)V = −h μν ∇μ ∇ν V − ∇ μ h λμ ∇λ V + ∇ λ h∇λ V, 2   √ √ 1 4 U (δ)V −g d x = Sμν (U, V )δg μν −g d4 x. 2 M M

(57) (58)

Proof For the first part, start with (δ)V = δ(V ) − δV = δ(g μν ∇μ ∇ν V ) − δV λ = −h μν ∇μ ∇ν V − g μν δΓμν ∇λ V 1 = −h μν ∇μ ∇ν V − g μν (∇μ h λν + ∇ν h λμ − ∇ λ h μν )∇λ V 2 1 = −h μν ∇μ ∇ν V − ∇ μ h λμ ∇λ V + ∇ λ h∇λ V. 2

The second part of the Lemma is proved by

(59) (60) (61) (62) (63)

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√ U (δ)V −g d4 x M  √ 1 = U (−h μν ∇μ ∇ν V − ∇ μ h λμ ∇λ V + ∇ λ h∇λ V ) −g d4 x 2 M  √ 1 = − (U ∇μ ∇ν V − ∇μ (U ∇ν V ) + gμν ∇ λ (U ∇λ V ))h μν −g d4 x 2 M √ 1 = (−2∇μ U ∇ν V + gμν ∇ λ U ∇λ V + gμν U V )δg μν −g d4 x 2 M  √ 1 = Sμν (U, V )δg μν −g d4 x. 2 M

(64) (65) (66) (67) (68)

In the next lemma we find the variation of F(). Lemma 7 Let U, V be scalar functions. Then, 

∞

n−1

n=1

l=0

  √ U δ(F())V −g d4 x = fn M



√ Sμν (l U, n−1−l V )δg μν −g d4 x. M

(69) n−1 l Proof Note that δn = l=0  (δ)n−1−l for n > 0 and δ0 = δId = 0. Therefore summation over n and integration yields 

∞

n−1

n=1

l=0

  √ U δ(F())V −g d4 x = fn M

=

∞ 

fn

n=1

=

n−1   l=0

∞ 

fn

n=1

√ U l (δ)n−1−l V −g d4 x

(70)

M

√ l U (δ)n−1−l V −g d4 x

(71)

√ Sμν (l U, n−1−l V )δg μν −g d4 x.

(72)

M

n−1   l=0



M

Lemma 8 Let U be scalar function. Then, 



√ 1 (Rμν Y − K μν Y + Ψμν )δg μν −g d4 x, (73) 2 M M Y = U (P  F()Q + Q  F()P) + (P  F()(Q  U ) + Q  F()(P  U )), (74) √

U δW −g d x =

Ψμν =

+∞  n=1

fn

n−1   l=0

4

 Sμν (l (P  U ), n−1−l Q) + Sμν (l (Q  U ), n−1−l P) .

(75)

Variations of Infinite Derivative Modified Gravity

103

Proof Since W = P  F()Q + Q  F()P the variation δW is written as δW = P  F()Qδ R + P  δ(F())Q + P  F()(Q  δ R) + Q  F()Pδ R + Q  δ(F())P + Q  F()(P  δ R).

(76)

Integration of the second and fifth term in this sum is done by using Lemma 7. The remaining four terms are obtained by Theorem 1. Lemma 9 Let A, B be scalar functions. Then,  M

√ Sμν (δ A, B)δg μν −g d4 x = √ Sμν (A, δ B)δg μν −g d4 x =

 

√ σ1 (B)δ A −g d4 x,

(77)

√ σ2 (A)δ B −g d4 x,

(78)

M

where σ1 (B) = ∇ λ h∇λ B − 2∇μ h μν ∇ν B − 2h μν ∇μ ∇ν B, λ

μν

μν

σ2 (A) = −∇ h∇λ A − Ah − 2∇ν h ∇μ A − 2h ∇μ ∇ν A.

(79) (80)

Proof To prove the first equation recall the definition of Sμν (A, B) 

√ Sμν (δ A, B)δg μν −g d4 x M    √ = gμν ∇ α δ A∇α B + gμν δ AB − 2∇μ δ A∇ν B δg μν −g d4 x M  √ = −h∇ α δ A∇α B − hδ AB + 2h μν ∇μ δ A∇ν B −g d4 x M  α  √ = ∇ (h∇α B) − hB − 2∇μ (h μν ∇ν B) δ A −g d4 x M  α  √ = ∇ h∇α B − 2∇μ h μν ∇ν B − 2h μν ∇μ ∇ν B δ A −g d4 x M √ = σ1 (B)δ A −g d4 x. M

The proof of the second equation is similar.  l  n−1 n−1−l Lemma 10 Let Ωμν = ∞ Q(R) . Then, n=1 f n l=0 Sμν  P(R),  

√ δΩμν δg μν −g d4 x M

=

  +∞ M n=1

fn

n−1   l=0

h μν ∇ λ l P∇λ n−1−l Q + h∇μ l P∇ν n−1−l Q

(81) (82) (83) (84) (85) (86)

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I. Dimitrijevic et al.

1 Sμν (hl P, n−1−l Q) 2 + Rμν P  l (σ1 (n−1−l Q)) − K μν (P  l (σ1 (n−1−l Q)))

+ h μν l Pn−l Q −

 √ + Rμν Q  l (σ2 (n−1−l P)) − K μν (Q  l (σ2 (n−1−l P))) δg μν −g d4 x

+

1 2

  +∞ M n=1

fn

l−1  n−1  

Sμν (m (σ1 (n−1−l Q)), l−m−1 P)

l=1 m=0

 √ + Sμν ( (σ2 (n−1−l P)), l−m−1 Q) δg μν −g d4 x. m

(87)

Proof Note that δΩμν =

∞  n=1

fn

n−1 

  δSμν l P(R), n−1−l Q(R) .

(88)

l=0

Moreover,     √ √ Sμν (δ A, B) + Sμν (A, δ B) δg μν −g d4 x δSμν (A, B)δg μν −g d4 x = M M   + h μν ∇ λ A∇λ B + h∇μ A∇ν B M

+ h μν AB −

 √ 1 Sμν (h A, B) δg μν −g d4 x. 2

(89)

Using this formula for each term in δΩμν yields the result of the Lemma. Theorem 3 The second variation of the action (1) is given by 1 δ S= 16πG 2

 

√ 1 1 Uμν + Rμν X − K μν X + χμν + μν δg μν −g d4 x, (90) 2 4 M

where 1 λ Uμν = − h μν (R − 2Λ + PF()Q) + δ Rμν (W + 1) + δΓμν ∇λ W 2 1 + h μν W − Sμν (h, W ), (91) 2  1 X = (h + P  hF()Q + Q  F()(Ph)) + δ R(P  F()Q + Q  F()P) 2  + (P  F()(Q  δ R) + Q  F()(P  δ R)) , +∞

χμν =

1  fn Sμν (l (Ph), n−l−1 Q) 2 n=1 l=0 n−1

(92)

Variations of Infinite Derivative Modified Gravity

− +

+∞ 

fn

n−1  

Sμν (l (P  M), n−1−l Q) + Sμν (l (Q  M), n−1−l P)

n=1

l=0

+∞ 

n−1  

n=1

fn

105



h μν ∇ λ l P∇λ n−1−l Q + h∇μ l P∇ν n−1−l Q

l=0

+ h μν l Pn−l Q −

1 Sμν (hl P, n−1−l Q) 2

 + (Rμν − K μν )(P  l (σ1 (n−1−l Q)) + Q  l (σ2 (n−1−l P))) ,

(93)

and μν =

+∞ 

fn

n=1

l−1  n−1  

Sμν (m (σ1 (n−1−l Q)), l−m−1 P)

l=1 m=0

 + Sμν ( (σ2 (n−1−l P)), l−m−1 Q) , m

(94)

σ1 (B) = ∇ λ h∇λ B − 2∇μ h μν ∇ν B − 2h μν ∇μ ∇ν B, λ

μν

(95)

μν

σ2 (A) = −∇ h∇λ A − Ah − 2∇ν h ∇μ A − 2h ∇μ ∇ν A.

(96)

Proof In the pervious section we calculated the first variation of the action (1) δS =

1 16πG



√ Gˆ μν δg μν −g d4 x.

(97)

M

Moreover the second variation δ 2 S is  

1 1 2 μν 2 μν αβ ˆ μν √ ˆ ˆ δ G μν δg + G μν δ g − gαβ δg G μν δg δ S= −g d4 x. 16πG M 2 (98) At the beginning note that 

  √ δ G μν + Λgμν δg μν −g d4 x M  

√ 1 1 1 δ Rμν − (R − 2Λ)h μν + Rμν h − K μν h δg μν −g d4 x. = 2 2 2 M

(99) (100)

The next term is calculated by using Lemma 7 

  √ δ gμν PF()Q δg μν −g d4 x (101) M   √ √ = h μν PF()Qδg μν −g d4 x + gμν Pδ(F())Qδg μν −g d4 x M

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 − =



M

 √ P  hF()Q + Q  F()(Ph )δ R −g d4 x

(102)

√ h μν PF()Qδg μν −g d4 x

M +∞ 

n−1   √ 1 − fn Sμν (l (Ph), n−l−1 Q)δg μν −g d4 x 2 n=1 l=0    √ − Rμν P  hF()Q + Q  F()(Ph )δg μν −g d4 x M   √ + K μν P  hF()Q + Q  F()(Ph )δg μν −g d4 x.

The third term is  M

 M

  √ δ Rμν W δg μν −g d4 x and it is equal to

  √ δ Rμν W δg μν −g d4 x 

 √ W δ Rμν + Rμν δW δg μν −g d4 x M    √ 1 =− h μν + ∇μ ∇ν h − 2∇λ ∇μ h λν W δg μν −g d4 x 2 M  √ + Rμν δW δg μν −g d4 x. =

(103)

(104) (105)

(106)

M

The last integral of the above formula is obtained by Lemma 8. Similarly, we obtain    √ δ K μν W δg μν −g d4 x (107) M √ = K μν δW δg μν −g d4 x M   λ  √ + δΓμν ∇λ W + h μν W + gμν (δ)W h μν −g d4 x (108)  √ = δW K μν δg μν −g d4 x M  

√ 1 λ δΓμν ∇λ W + h μν W − Sμν (h, W ) δg μν −g d4 x. (109) − 2 At the end the last term

 M

√ δΩμν δg μν −g d4 x is calculated in Lemma 10.

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107

5 Perturbations 5.1 Background In this section we start with F RW metric, which for k = 0 can be written as ds 2 = −dt 2 + a(t)2 (dx 2 + dy 2 + dz 2 ).

(110)

Some relevant background quantities are i = H δ ij ,  = −∂t2 − 3H ∂t . R = 12H 2 + 6 H˙ , Γi0j = H gi j , Γ j0

(111)

For perturbations it is useful to employ the canonical ADM (1 + 3) decomposition and introduce the conformal time τ such that adτ = dt. Then the flat FRW metric (110) transforms to ds 2 = a(τ )2 (−dτ 2 + dx 2 + dy 2 + dz 2 ).

(112)

5.2 Perturbations The metric perturbations (see [17]) can be divided into three types: scalar, vector and tensor perturbations. The component h 00 is invariant under spatial rotations and translations and therefore (113) h 00 = 2a(τ )2 φ. The components h 0i are the sum of a spatial gradient of a function B and divergence free vector Si . (114) h 0i = a(τ )2 (∂i B + Si ). Similarly, components h i j , which transform as a tensor under 3-rotations are written as (115) h i j = a(τ )2 (2ψδi j + 2∂i2j E + ∂i F j + ∂ j Fi + ϕi j ), where ψ and E are scalar functions, Fi is a vector with zero divergence and 3-tensor satisfies ∂i ϕij = 0. (116) ϕii = 0,

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Note that there are four scalar functions, two vectors with two independent components each and tensor ϕi j has two independent components. Therefore, as expected we have in total ten functions. The scalar perturbations are defined by scalar functions φ, ψ, B, E and perturbed metric around F RW background is

ds 2 = a(τ )2 −(1 − 2φ)dτ 2 + ∂i Bdτ d x i + ((1 + 2ψ)δi j + 2∂i ∂ j E)d x i d x j . (117) The vector perturbations are defined by vectors Si and Fi , i.e.

ds 2 = a(τ )2 −dτ 2 + Si Bdτ d x i + (δi j + ∂i F j + ∂ j Fi )d x i d x j .

(118)

Tensor perturbations are defined by ϕi j and describe gravitational waves, which have no analog in Newton gravity theory.

ds 2 = a(τ )2 −dτ 2 + (δi j + ϕi j )d x i d x j .

(119)

Each of the types of perturbations can be studied separately. In this form of perturbations we have

 2φ ∂i B + S i 2 h μν = a(τ ) , (120) ∂i B + S i 2ψId + 2 Hess E + ∂ j Fi + ∂i F j + ϕi j h = −2φ + 6ψ + 2E, a 2  a  δ R = 2 6 φ + (φ − 2ψ) + 3 (B + E  ) a a a  a     + 3 (φ + 3ψ ) + (B + E ) + 3ψ  . a

(121)

(122)

Moreover, let Aμ = ∇ν h μν , then a 1 a 2 (φ + ψ) + B + 2 E + 2 φ , 3 2 3 a a a a a 2 1 i i A = 4 3 (∂i B + S ) + 2 (∂i ψ + ∂i E) + 2 (∂i B  + S i + F i ). a a a

A0 = 6

(123) (124)

Out of 4 scalar modes only 2 are gauge invariant. The convenient gauge invariant variables (Bardeen potentials) are introduced as =φ−

1 (a(B − E  )) , a

Ψ =ψ+

a (B − E  ). a

(125)

The prime denotes the differentiation with respect to the conformal time τ and the dot as before w.r.t. the cosmic time t. The (1 + 3) structure suggests to represent the perturbation quantities (which can depend on all 4 coordinates) as

Variations of Infinite Derivative Modified Gravity

109

f (τ , x) = f (τ , k)Y (k, x),

(126)

where x = (x, y, z) and k = |k| comes from the definition of the Y -functions as spatial Fourier modes (127) δ i j ∂i ∂ j Y = −k 2 Y. Then

Y = Y0 e±ikx .

(128)

The relevant expressions for the d’Alembert operator are =−

δ i j ∂i ∂ j a k2 1 2 2 ∂ − 2 ∂ + = −∂ − 3H ∂ − , τ t τ t a2 a3 a2 a2

(129)

where k = |k|. All the expression in this subsection are valid for a generic scale factor a in flat space.

6 Concluding Remarks In this paper we have considered a class of nonlocal gravity models without matter given by the action in the form S=

1 16πG

 (R − 2Λ + P(R)F()Q(R))

√ −g d4 x.

(130)

M

We have derived the equations of motion for this action. We also have presented the second variation of action (130) and basics of metric perturbations. In many research papers there are equations of motion which are special cases of our equations. In the case P(R) = Q(R) = R one obtains 1 S= 16πG

 (R − 2Λ + RF()R)

√

−g d4 x.

M

This nonlocal model is further elaborated in the series of papers [16, 18–27]. The action (130) for P(R) = R −1 and Q(R) = R was introduced in [24] as a new approach to nonlocal gravity. This model one can also find in [28]. The case P(R) = R p and Q(R) = R q we analyzed in [25, 26]. For the case P(R) = (R + R0 )m and Q(R) = (R + R0 )m see [8, 29]. Studies of this model with R = const can be found in [30, 31]. It is worth noting that cosmology with nonlocality in the matter sector was also investigated, see e.g. [32, 33].

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For some very recent achievements in higher derivative modified gravities one can see [11, 34–36]. ∞ Note that there is the following formula −1 = 0 e−α dα, which could be used in investigation of models containing −n , n ∈ N, where −n =

1 (n − 1)!



∞

αn−1 e−α dα .

0

Namely, formalism presented in the previous sections can easily incorporate this case taking F() = e−α and at the end performing integration over α. Acknowledgements Work on this paper was partially supported by the Ministry of Education, Science and Technological Development of Republic of Serbia, grant No 174012. B.D. thanks Prof. Vladimir Dobrev for invitation to participate and give a talk on nonlocal gravity, as well as for hospitality, at the X International Symposium “Quantum Theory and Symmetries”, and XII International Workshop “Lie Theory and its Applications in Physics”, 19–25 June 2017, Varna, Bulgaria. B.D. also thanks a support of the ICTP - SEENET-MTP project NT-03 CosmologyClassical and Quantum Challenges during preparation of this article.

References 1. R. M. Wald, General Relativity (University of Chicago Press, 1984). 2. T. Clifton, P. G. Ferreira, A. Padilla, C. Skordis: Modified gravity and cosmology. Physics Reports 513 (2012) 1–189; [arXiv:1106.2476v2 [astro-ph.CO]]. 3. S. Nojiri, S. D. Odintsov: Unified cosmic history in modified gravity: from F(R) theory to Lorentz non-invariant models. Physics Reports 505 (2011) 59–144; [arXiv:1011.0544v4 [grqc]]. 4. S. Nojiri, S. D. Odintsov, V. K. Oikonomou: Modified Gravity Theories on a Nutshell: Inflation, Bounce and Late-time Evolution. Physics Reports 692 (2017) 1–104. [arXiv:1705.11098 [grqc]] . 5. Novello, M., Bergliaffa, S.E.P.: Bouncing cosmologies. Phys. Rep. 463, 127–213 (2008). [arXiv:0802.1634 [astro-ph]]. 6. T. P. Sotiriou, V. Faraoni: f (R) theories of gravity. Rev. Mod. Phys. 82 (2010) 451–497. [arXiv:0805.1726v4 [gr-qc]]. 7. E. Belgacem, Y. Dirian, S. Foffa and M. Maggiore: Nonlocal gravity. Conceptual aspects and cosmological predictions. Journal of Cosmology and Astroparticle Physics, Volume 2018, (2018) [arXiv:1712.07066 [hep-th]]. 8. I. Dimitrijevic, B. Dragovich, J. Stankovic, A. S. Koshelev and Z. Rakic: On Nonlocal Modified Gravity and its Cosmological Solutions. Springer Proceedings in Mathematics & Statistics 191 (2016) 35–51. arXiv:1701.02090 [hep-th]. 9. B. Dragovich: On Nonlocal modified gravity and cosmology. Lie Theory and Its Applications in Physics, Springer Proceedings in Mathematics and Statistics 111, 251–262, 2014. 10. R. P. Woodard: Nonlocal models of cosmic acceleration. [arXiv:1401.0254 [astro-ph.CO]] (2014). 11. A. S. Koshelev, K. S. Kumar, A. A. Starobinsky: R 2 inflation to probe non-perturbative quantum gravity. Journal of High Energy Physics 1803 (2018) 071. [arXiv:1711.08864 [hep-th]]. 12. Modesto, L.: Super-renormalizable quantum gravity. Phys. Rev. D 86, 044005 (2012). [arXiv:1107.2403 [hep-th]]

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13. Modesto, L., Rachwal, L.: Super-renormalizable and finite gravitational theories. Nucl. Phys. B 889, 228 (2014). [arXiv:1407.8036 [hep-th]]. 14. Stelle, K.S.: Renormalization of higher derivative quantum gravity. Phys. Rev. D 16, 953 (1977). 15. Dragovich, B., Khrennikov, A. Yu., Kozyrev, S. V., Volovich, I. V., Zelenov, E. I.: p-Adic mathematical physics: the first 30 years. p-Adic Numbers Ultrametric Anal. Appl. 9 (2), 87– 121 (2017). [arXiv:1705.04758 [math-ph]]. 16. Biswas, T., Conroy, A., Koshelev, A.S., Mazumdar, A.: Generalized gost-free quadratic curvature gravity. [arXiv:1308.2319 [hep-th]]. 17. V. Mukhanov, Physical Foundations of Cosmology, (Cambridge, 2005). 18. Biswas, T., Mazumdar, A., Siegel, W: Bouncing universes in string-inspired gravity. J. Cosmology Astropart. Phys. 0603, 009 (2006) [arXiv:hep-th/0508194]. 19. Biswas, T., Koivisto, T., Mazumdar, A.: Towards a resolution of the cosmological singularity in non-local higher derivative theories of gravity. J. Cosmology Astropart. Phys. 1011, 008 (2010) [arXiv:1005.0590v2 [hep-th]]. 20. Biswas, T., Gerwick, E., Koivisto, T., Mazumdar, A.: Towards singularity and ghost free theories of gravity. Phys. Rev. Lett. 108, 031101 (2012) [arXiv:1110.5249v2 [gr-qc]]. 21. T. Biswas, A. S. Koshelev, A. Mazumdar, S. Yu. Vernov, Stable bounce and inflation in nonlocal higher derivative cosmology, JCAP 08 (2012) 024, [arXiv:1206.6374v2 [astro-ph.CO]]. 22. I. Dimitrijevic, B. Dragovich, J. Grujic , Z. Rakic: On modified gravity. Springer Proceedings in Mathematics and Statistics 36, 251–259 (2013) [arXiv:1202.2352 [hep-th]]. 23. Dimitrijevic, I., Dragovich, B., Grujic J., Rakic, Z.: New cosmological solutions in nonlocal modified gravity. Rom. Journ. Phys. 58 (5–6), 550–559 (2013) [arXiv:1302.2794 [gr-qc]]. 24. I. Dimitrijevic, B. Dragovich, J. Grujic and Z. Rakic: A new model of nonlocal modified gravity. Publications de l’Institut Mathematique 94 (108) (2013), 187–196. 25. I. Dimitrijevic, B. Dragovich, J. Grujic and Z. Rakic: Some Cosmological Solutions of a Nonlocal Modified Gravity. Filomat 29 (3), (2015) 619–628, arXiv:1508.05583 [hep-th]. 26. I. Dimitrijevic: Cosmological solutions in modified gravity with monomial nonlocality. Applied Mathematics and Computation, 285 (3), (2016) 195–203. 27. A. S. Koshelev, S. Yu. Vernov: On bouncing solutions in non-local gravity. Phys. Part. Nuclei 43, 666–668 (2012) [arXiv:1202.1289v1 [hep-th]]. 28. I. Dimitrijevic, B. Dragovich, J. Grujic and Z. Rakic: Some power-law cosmological solutions in nonlocal modified gravity. in: Lie Theory and Its Applications in Physics, Springer Proceedings in Mathematics and Statistics, 111 2014, pp. 241–250. 29. Dimitrijevic, I., Dragovich, B., Grujic J., Koshelev A. S., Rakic, Z.: Cosmology of modified gravity with a non-local f (R). arXiv:1509.04254 [hep-th]. 30. I. Dimitrijevic, B. Dragovich, J. Grujic and Z. Rakic: Constant curvature cosmological solutions in nonlocal gravity. AIP Conference Proceedings 1634, (2014) 18–23. 31. I. Dimitrijevic, B. Dragovich, Z. Rakic and J.Stankovic: On Nonlocal Gravity with Constant Scalar Curvature. Publications de l’Institut Mathematique, Nouvelle série, 103 (117) (2018), 53–59. 32. Aref’eva, I.Ya., Joukovskaya, L.V., Vernov, S.Yu.: Bouncing and accelerating solutions in nonlocal stringy models. JHEP 0707, 087 (2007) arXiv:0701184 [hep-th/0701184]. 33. E. Elizalde, E. O. Pozdeeva, S. Yu. Vernov: Stability of de Sitter Solutions in Non-local Cosmological Models. PoS, QFTHEP2011:038, 2013, arXiv:1202.0178. 34. L. Buoninfante, A. S. Koshelev, G. Lambiase and A. Mazumdar: Classical properties of nonlocal, ghost- and singularity-free gravity. [arXiv:1802.00399 [gr-qc]]. 35. A. S. Koshelev, L. Modesto, L. Rachwal and A. A. Starobinsky: Occurrence of exact R 2 inflation in non-local UV-complete gravity. Journal of High Energy Physics, 2016(11), 1–41. [arXiv:1604.03127 [hep-th]]. 36. A. S. Koshelev, J. Marto, A. Mazumdar: Towards non-singular metric solution in infinite derivative gravity. [arXiv:1803.00309 [gr-qc]].

Infinite-Dimensional Metaconformal Symmetries: 1 D Diffusion-Limited Erosion and Ballistic Transport in (1 + 2) Dimensions Malte Henkel and Stoimen Stoimenov

Abstract Meta-conformal invariance is a novel class of dynamical symmetries, with dynamical exponent z = 1, distinct from the standard ortho-conformal invariance. A physical example in the context of fluctuating interfaces is provided by diffusionlimited erosion. In d = 1 space dimension, its meta-conformal symmetry, non-local in space, is isomorphic to the direct sum of three loop-Virasoro algebras. Co-variant two-time response functions are derived and agree with the exact solution of the Langevin equation of diffusion-limited erosion. In d = 2, two different local representation of the meta-conformal Lie algebra are found, and the two-point functions are derived. Both generalisations act as dynamical symmetries of a ballistic transport equation, but they are not isomorphic. One of them is isomorphic to the non-local symmetry of 1D diffusion-limited erosion.

1 Ortho-Conformal and Meta-Conformal Invariances Many brilliant applications of conformal invariance are known, ranging from string theory and high-energy physics [48], or to two-dimensional phase transitions [5, 19, 25, 50] or the quantum Hall effect [10, 22]. These applications are based on a geometric definition of conformal transformations, considered as local coordiM. Henkel (B) Laboratoire de Physique et Chimie Théoriques (CNRS UMR 7019), Université de Lorraine Nancy, B.P. 70239, F-54506 Vandœuvre-lès-Nancy Cedex, France e-mail: [email protected] M. Henkel Centro de Física Teórica e Computacional, Universidade de Lisboa, P-1749-016 Lisboa, Portugal S. Stoimenov Institute of Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, 72 Tsarigradsko chaussee, Blvd., BG-1784 Sofia, Bulgaria e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2018 V. Dobrev (ed.), Quantum Theory and Symmetries with Lie Theory and Its Applications in Physics Volume 1, Springer Proceedings in Mathematics & Statistics 263, https://doi.org/10.1007/978-981-13-2715-5_6

113

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nate transformations r → r  = f (r), of spatial coordinates r ∈ R2 such that angles are kept unchanged. The associated Lie algebra is called the ‘conformal Lie algebra’. Can one adapt at least some ideas of conformal invariance to dynamical problems ? Then, time and space variables should be distinguished through their global rescaling, according to t → t  = b z t and r → r  = br which defines the dynamical exponent z. Clearly, conformal invariance must have z = 1. In general, z has a non-trivial value [53]. Attempts of identifying dynamical conformal invariance goes back at least to critical dynamics of a two-dimensional statistical system [11]. Motivated by studies of dynamical symmetries in partial differential equations, it appears that besides the standard, ortho-conformal symmetries a related, different kind of symmetry merits attention. We define this in (1 + 1) dimensions. Definition 1 A set of meta-conformal transformations M is a set of maps (t, r ) → (t  , r  ) = M (t, r ), which form a Lie group. Its maximal finite-dimensional Lie subalgebra is semi-simple and contains at least a Lie algebra isomorphic to sl(2, R) ⊕ sl(2, R). A physical system is meta-conformally invariant if its n-point functions transform covariantly under meta-conformal transformations. Definition 2 A set of ortho-conformal transformations O is a set of meta-conformal transformations (t, r ) → (t  , r  ) = O(t, r ), such that (i) the maximal finitedimensional Lie sub-algebra is isomorphic to sl(2, R) ⊕ sl(2, R) and that (ii) angles in the coordinate space of the points (t, r ) are kept invariant. A physical system is ortho-conformally invariant if its n-point functions transform covariantly under ortho-conformal transformations. Example 1 Ortho-conformal transformations in two dimensions are analytic or antianalytic maps, z → f (z) or z¯ → f¯(¯z ), of the complex variables z = t + ir , z¯ = t − ir . The Lie algebra generators, acting on ‘quasi-primary’ scaling operators φ = φ(z, z¯ ) = ϕ(t, r ) [5] read n = −z n+1 ∂z − Δ(n + 1)z n , ¯n = −¯z n+1 ∂z¯ − Δ(n + 1)¯z n

(1)

where Δ, Δ ∈ R are the conformal weights of the scaling operator φ. The scaling dimension is x := xφ = Δ + Δ. We shall often use the basis X n := n + ¯n and Yn := n − ¯n . Consider the Laplace operator Sφ = 4∂z ∂z¯ φ = ∂t2 + ∂r2 ϕ and compute [S, n ] φ(z, z¯ ) = −(n + 1)z n Sφ(z, z¯ ) − 4Δn(n + 1)z n−1 ∂z¯ φ(z, z¯ ).

(2)

If Δ = Δ = 0, the space of solutions of the Laplace equation Sφ = 0 is conformally invariant. This means that every solution φ is mapped onto another solution n φ (or ¯n φ) in the transformed coordinates. The maximal finite-dimensional Lie sub-algebra of projective conformal transformations is sl(2, R) ⊕ sl(2, R). Two-point functions of quasi-primary scaling operators are defined as C12 (t1 , t2 ; r1 , r2 ) := φ1 (z 1 , z¯ 1 )φ2 (z 2 , z¯ 2 ) = ϕ1 (t1 , r1 )ϕ2 (t2 , r2 ).

(3)

Infinite-Dimensional Metaconformal Symmetries: …

115

Their ortho-conformal covariance implies the projective Ward identities X n C12 = Yn C12 = 0 for n = ±1, 0 [5]. For scalars, Δi = Δi = xi . Then, up to normalisation  −x1 . C12 (t1 , t2 ; r1 , r2 ) = δx1 ,x2 (t1 − t2 )2 + (r1 − r2 )2

(4)

Example 2 An example of meta-conformal transformations in (1 + 1)D reads [26] X n = −t n+1 ∂t − μ−1 [(t + μr )n+1 − t n+1 ]∂r − (n + 1)xt n − γ − (n + 1) [(t + μr )n − t n ], μ Yn = −(t + μr )n+1 ∂r − (n + 1)γ(t + μr )n

(5)

with n ∈ Z. Herein, x, γ are the scaling dimension and the ‘rapidity’ of the scaling operator ϕ = ϕ(t, r ) on which these generators act. The constant 1/μ has the dimensions of a velocity. The Lie algebra X n , Yn n∈Z is isomorphic to the conformal Lie algebra [30], see Table 1, where it is called meta-1 conformal algebra. If γ = μx, the generators (5) act as dynamical symmetries on the equation Sϕ = (−μ∂t + ∂r )ϕ = 0. This follows from the only non-vanishing commutators of the Lie algebra with S, namely [S, X 0 ] ϕ = −Sϕ and [S, X 1 ] ϕ = −2tSϕ + 2(μx − γ)ϕ. The formulation of the meta-1 conformal Ward identities does require some care, since already the two-point function turns out to be a non-analytic function of the time- and spacecoordinates. It can be shown that the covariant two-point correlator is [31] −2x1



C12 (t1 , t2 ; r1 , r2 ) = δx1 ,x2 δγ1 ,γ2 |t1 − t2 |

μ 1+ γ1

   r1 − r2  −2γ1 /μ γ1  .  t −t  1 2

(6)

Although both examples 1 and 2 have z = 1 and isomorphic Lie algebras, the explicit two-point functions (4), (6), as well as the invariant equations Sϕ = 0, are different. The form of two-point functions depends mainly on the representation and not so much on the Lie algebra, a known phenomenon not restricted to the conformal

Table 1 Comparison of local ortho-conformal, conformal galilean and meta-1 conformal invariance, in (1 + 1)D. The non-vanishing commutators of the Lie algebra g are [X n , X m ] = (n − m)X n+m , [X n , Ym ] = (n − m)Yn+m and [Yn , Ym ] = (n − m) (g1 X n+m + g2 Yn+m ), where only the constants (g1 , g2 ) are listed. The defining equation of the generators, the invariant differential operator S and the covariant two-point function are indicated, where applicable. Physically, the co-variant quasiprimary two-point function C12 = ϕ1 (t, r )ϕ2 (0, 0) is a correlator, with the constraints x1 = x2 and γ1 = γ2 Ortho

Galilean

Meta-1

Meta-2

(g1 , g2 )

(1, 0)

(0, 0)

(0, μ)

(0, −μ)

S

   γ r  exp −2  1t 

−μ∂t + ∂r     γ r  −2γ1 /μ 1 + γμ  1t 

μ∂t + υ∂r + φ0 r −3 ∂υ

t 2x1 C 12

∂t2 + ∂r2   2 −x1 1 + rt

Generators

(1)

(8)

(5)

1

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algebra. Similarly, for the so-called Schrödinger algebra at least three distinct representations with different forms of the two-point function are known [27]. Example 3 The Vlassov equation with an external force (ϕ0 is a constant)   μ∂t + υ∂r + ϕ0 r −3 ∂υ f (t, r, υ) = 0

(7)

has a meta-conformal dynamical symmetry, with dynamical exponent z = 2. This is but one special case of a larger set of such symmetries, which contain both spatial and ‘velocity’ coordinates r and υ [52]. Example 4 Taking the limit μ → 0 in the meta-conformal representation (5) produces the generators X n = −t n+1 ∂t − (n + 1)t n r ∂r − (n + 1)xt n − (n + 1)nγt n−1 r Yn = −t n+1 ∂r − (n + 1)γt n

(8)

of the conformal Galilean algebra (cga) in (1 + 1)D or bms3 -algebra [7, 23, 24, 41]. This Lie algebra is obtained by standard contraction of the conformal Lie algebra, see Table 1.1 Hence the CGA is not a meta-conformal algebra, although z = 1. The co-variant two-point correlator, obtained either from the generators (8) [30], or by letting μ → 0 in (6), is C12 (t1 , t2 ; r1 , r2 ) = δx1 ,x2 δγ1 ,γ2 |t1 − t2 |

−2x1

    r1 − r2    exp −2 γ1 t1 − t2 

(9)

Clearly, this form is different from both ortho- and meta-1 conformal invariance. The shape of the scaling functions of the two-point functions (4), (6), (9) is compared in Fig. 1. All two-point functions obey the symmetries C12 (t1 , t2 ; r, r ) = C21 (t2 , t1 ; r, r ) , C12 (t, t; r1 , r2 ) = C21 (t, t; r2 , r1 ),

(10)

under permutation ϕ1 ↔ ϕ2 of the two scaling operators, as physically required for a correlator. The distinctive non-analyticity of the meta-conformal and galilean conformal correlators (6), (9), at u = 0, is clearly seen, and is in contrast to analytic ortho-conformal correlators. For u → ∞, the slow algebraic decay of orthoand meta-conformal invariance is distinct from the exponential decay of conformal Galilean invariance. This illustrates the variety of possible forms already for z = 1.

1 There

are no cga-invariant scalar equations (in the classical Lie sense) [13]. Such equations exist for the Newton-Hooke extension of the cga on a curved de Sitter/anti-de Sitter space, related to the Pais-Uhlenbeck oscillator [14, 37]. Alternatively, invariant equations are found from the central extensions of the cga [1, 2].

Infinite-Dimensional Metaconformal Symmetries: …

0.8

0.8

0.4

f(u)

Fig. 1 Scaling function f (u) of the covariant two-point correlator C (t, r ) = t −2x1 f (r/t), over against the scaling variable u = r/t, from Eqs. (4), (6), (9) respectively. Here x1 = γ1 = 21 and μ = 1

117

0.0 -2 -1 0 1 2 10 10 10 10 10

0.4

ortho meta -1 galilean 0.0

0

u

5

Below, we shall give further distinct forms of (meta-)conformal invariance. Physical systems with dynamical exponent z = 1 include the Jeans-Vlassov equation [9, 17, 40, 45]. Also, the non-equilibrium dynamics of quantum quenches generically has z = 1, related to ballistic spreading of signals, see [8], either near to quantum criticality [15] or deep in the two-phase coexistence region [54]. Similar effective equations of motion arise in generalised hydrodynamics of quantum systems [6, 12, 16, 47]. Table 2 collects several examples of infinite-dimensional time-space transformation groups. Of those, only the Schrödinger-Virasoro group has a dynamical exponent z = 2 distinct from one. The 2D meta-conformal transformations are new [35]. Here, we shall summarise recent results on meta-conformal symmetries. First, a new variant of meta-conformal invariance, formulated in terms of spatially non-local generators, arises from the non-local equation of motion of the diffusion-limited erosion process (dle), in d = 1 dimensions. The dynamical symmetries are built around both local and non-local spatial translations, such that the full symmetry algebra is isomorphic to the direct sum of three Virasoro algebras. Second, we shall also consider extension of the 1D local meta-conformal invariance to two spatial dimensions. As we shall show, the Lie algebra of dynamical symmetries is again infinite-dimensional and isomorphic to the one of 1D dle. The results presented here on the dle are taken from [33, 34]. Full details on 2D meta-conformal symmetries will be reported elsewhere [35].

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Table 2 Infinite-dimensional groups of time-space transformations, with the defining coordinate changes. Herein, f, f¯, θ, β are arbitrary functions, and a an arbitrary vector-valued function, of their argument. In addition, z = t + ir, z¯ = t − ir are (complex) light-cone coordinates, ρ = t + μr and w = t + βz, w¯ = t + β z¯ . The physical interpretation of the co-variant n-point functions as either correlators or responses follows from the extension of the Cartan sub-algebra [30, 31] Group Coordinate changes Co-variance Ortho-conformal (1 + 1)D SchrödingerVirasoro

z  = f (z) z = z t  = β(t) t = t

Conformal galilean

t  = β(t) t = t

Meta-conformal 1D Meta-conformal 2D

t t t t t

= f (t) =t =t =t = θ(t)

z¯  = z¯ z¯  = f¯(¯z ) r = (dβ(t)/dt)1/2 r r  = r + a(t) r  = (dβ(t)/dt) r r  = r + a(t) ρ = f (ρ) ρ = a(ρ) z  = f (z) z¯  = z¯  z =z z¯  = f¯(¯z ) w = w w¯  = w¯

Correlator Response

Correlator Correlator

2 Diffusion-Limited Erosion 2.1 Physical Background Here, we shall inquire about dynamical symmetries of the following stochastic Langevin equation, to be called diffusion-limited erosion (dle) Langevin equation, which reads in momentum space [38] B(t, q) d h(t, q) = −ν|q| h(t, q)dt + j(t, q)dt + (2νT )1/2 d

(11)

and describes the Fourier-transformed height h(t, q) = (2π)−d/2



dr e−iq·r h(t, r) Rd

Because of the (Fourier-transformed) standard brownian motion B, with the variance  B(t, q) B(t  , q  ) = min(t, t  )δ(q + q  ), this is a stochastic process, called diffusionlimited erosion (dle) process. Herein, ν, T are non-negative constants and δ(q) is the Dirac distribution. An external source j(t, q) allows to derive linear responses. Inverting the Fourier transform in order to return to direct space, Eq. (11) implies spatially long-range interactions. The dle equation can be obtained from the description of an interface created by the corrosive action of diffusively moving particles, see Fig. 2. A stability analysis

Infinite-Dimensional Metaconformal Symmetries: … Fig. 2 The dle process. (a) At small time, a diffusing particle (green path) arrives on a surface (full black line) and erodes a small part (red) of it. (b) Analogous process at a later time

(a)

119

(b)

then leads to the dle Langevin equation (11) [38]. Numerical studies have confirmed the dynamical exponent z = 1, see [38]. Alternatively, the contours of vicinal surfaces are described by the same universality class [51]. The associated world lines can be interpreted as the (1 + 1)D  world lines of fermions such that on each site, the particle number n = 21 1 + σnz = 0, 1 is rewritten in terms of spin variables and the hamiltonian H =−

N   w  − + z 2vσn+ σn+1 + 2v −1 σn− σn+1 + Δ σnz σn+1 −1 2 n=1

(12)

√  √ √ √ where v = p/q eλ , w = pq eμ and Δ = 2 p/q + q/ p e−μ [36, 51]. In a continuum limit, n (t) → (t, r ) = ∂r h(t, r ) is related to the height h which in turn obeys (11), with a gaussian white noise η [51]. The low-energy behaviour of H yields the dynamical exponent z = 1 [36, 51]. If one conditions the system to an a-typically large current, the large-time, large-distance behaviour of (12) is described by a conformal field-theory with central charge c = 1 [36]. The physical background of Eq. (11) is the growth of interfaces [3, 21, 39], a paradigm of the emergence of non-equilibrium collective phenomena [28, 53]. Interfaces are described in terms of a time-space-dependent height profile h(t, r). This random profile also depends on fluctuations of the set of initial states and on the noise in the Langevin equation. The interface  width measures the fluctuations. The spatially averaged height is h(t) := L −d r∈L h(t, r). On a hyper-cubic lattice L ⊂ Zd of |L | = L d sites, the interface width is defined by    2β 2 t 1   ; if t L −z  1 t 2βz ∼ w (t; L) := d h(t, r) − h(t) = L f w 2α L ; if t L −z  1 L Lz r∈L (13) Herein, . denotes an average over many independent samples. The Family-Vicsek scaling form [18] holds for large times t → ∞ and lattice sizes L → ∞ and defines the growth exponent β, the roughness exponent α and the dynamical exponent z = α/β. The saturation regime occurs for t L −z  1 and the growth regime occurs for t L −z  1. We shall focus on the growth regime from now on. 2

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Definition 3 A spatially infinite interface with a width w(t)  ∞ for large times t → ∞ is called rough. If limt→∞ w(t) is finite, the interface is called smooth. In particular, if in (13) β > 0, the interface is rough. Dynamical properties of the interface can be studied through the two-time correlators and responses. In the growth regime (where effectively L → ∞), one considers the double scaling limit t, s → ∞ with y := t/s > 1 fixed and expects       C(t, s; r) := h(t, r) − h(t) h(s, 0) − h(s)  δ h(t, r) − h(t)  R(t, s; r) :=   δ j (s, 0) 

= s −b FC



t r ; s s 1/z

  = h(t, r) h(s, 0) = s −1−a FR j=0





(14a) 

t r ; s s 1/z

(14b) where j is an external field conjugate2 to the height h. Spatial translation-invariance was implicitly admitted in (14). This defines the ageing exponents a, b. The autocorrelation exponent λC and the autoresponse exponent λ R are defined from the asymptotics FC (y, 0) ∼ y −λC /z and FR (y, 0) ∼ y −λ R /z , respectively, as y → ∞. These non-equilibrium exponents obey b = −2β and the bound λC ≥ (d + zb)/2. For the dle process (11) of an initially flat interface one has exactly [33, 38] −(d−1)/2  2 −(d−1)/2  T C0  2 ν (t − s)2 + r 2 − ν (t + s)2 + r 2 d −1 (15a)   2 −(d+1)/2 R(t, s; r) = C0 (t − s) ν(t − s) ν (t − s)2 + r 2 (15b)

C(t, s; r) =

where C0 is known constant and the Heaviside function  expresses the causality condition t > s. In particular, in the growth regime, the interface width reads, with a known constant C1 (Λ) ⎧ ; if d > 1 ⎨T C0 C1 (Λ)/(d − 1) t→∞ ; if d = 1 w 2 (t) = C(t, t; 0)  T C0 ln(2νt) (16) ⎩ T C0 (2ν)1−d /(1 − d) · t 1−d ; if d < 1 Hence d ∗ = 1 is the upper critical dimension of dle process. It follows that the dle-interface is smooth for d > 1 and rough for d ≤ 1. In contrast to the interface width w(t), which shows a logarithmic growth at d = d ∗ = 1, logarithms cancel in the two-time correlator C and response R, up to additive logarithmic corrections to scaling. This is well-known in the physical ageing at d = d ∗ of simple magnets or of the Arcetri model [20, 29]. 2 In

the context of Janssen-de Dominicis theory,  h is the conjugate response field to h, see [53].

Infinite-Dimensional Metaconformal Symmetries: …

121

Definition 4 The deterministic part of the Langevin Eq. (11) is obtained when formally setting B = 0. Inspired by Niederer’s treatment [43] of the dynamical symmetries of the free diffusion equation, our reinterpretation of the notion of dynamic symmetries of Langevin equations [28, 46], and the succesful application of the technique to the Edwards-Wilkinson equation and the Arcetri model [32, 49], we seek dynamical symmetries of the deterministic part of the dle process. In other words, we look for dynamical symmetries of the non-local equation (μ∂t − ∇r )ϕ = 0, in terms of the non-local Riesz-Feller derivative ∇r to be defined below. Theorem 1 ([33, 34]) The dynamical symmetry of the 1D dle process is a metaconformal algebra isomorphic to the direct sum of three Virasoro algebras without central charge. The Lie algebra generators are non-local in space. The general form of the co-variant two-time response function is (with t > s) ν(t − s) + FB (t − s)1+ψ−2x × − s)2 + r 2      2  πψ r 2 2 −(ψ+1)/2 − × ν (t − s) + r cos (ψ + 1) arctan ν(t − s) 2

R(t, s; r ) = FA (t − s)1−2x

ν 2 (t

(17)

where x, ψ are real parameters and FA,B are normalisation constants. The exact solution (15b) is reproduced by (17) if one takes x = 21 , ν > 0, FA = C0 and FB = 0. The symmetries to be found are merely symmetries of the ‘deterministic part’ of Eq. (11), obtained by setting T = 0. This is consistent with the T independence of the response (14b), (15b). On the other hand, since the correlator (14a), (15a) vanishes for T = 0, its form does not follow from a covariance argument. Different techniques must be developed for that calculation, see [32, 46].

2.2 Impossibility of a Local Meta-Conformal Invariance The known variants of conformal invariance, all of which have z = 1, are no valid dynamical symmetries of the dle process in 1 + 1 dimensions, because: 1. The deterministic part of the dle-equation (11) is distinct from their invariant equations Sϕ = 0. 2. The explicit response function (15b) of the dle process is distinct from the predictions (4), (6), (9), see also Table 1. The analogy of Schrödinger-invariant systems is suggestive: consider a Langevin equation of the form Sϕ = (2νT )1/2 η, where η = dB is a white noise of unit varidt ance. If the deterministic part, Sϕ0 = 0, is Schrödinger-invariant, then the Bargman super-selection rules [4] allow to reduce all averages to quantities computable only from its deterministic part [46]. In particular, the two-time noisy response function

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R(t, s; r) = R0 (t, s; r), is independent of the noise and R0 can be computed from its covariance under Schrödinger algebra [28, 46]. Although still unproven, we conjecture that an analogous procedure applies to the dle Langevin equation as well. Indeed, the two-time response R (15b) of dle is independent of T .

2.3 Riesz-Feller Fractional Derivative Definition 5 For functions f (r ) of a single variable r ∈ R, the Riesz-Feller derivative, of order α, is [42] iα ∇rα f (r ) := √ 2π



iα dk |k|α eikr f (k) = 2π R

R2

dkdx |k|α eik(r −x) f (x)

(18)

where f (k) denotes the Fourier transform of f (r ). We often write ∇r = ∇r1 = ∂r . If 0 ≤ α < 2, then for elements f ∈ H α/2 (R) of a fractional Sobolev space, the fractional derivative ∇rα f (r ) exists [42, Prop. 3.6]. We also require ∂r ∇r−1

1



f (r ) = √ 2π

R

dk eikr sign (k) f (k)

(19)

 2 and have ∂r ∇r−1 f (r ) = f (r ). Practical calculations require the identities [28, 33] ∇rα ∇rβ f (r ) = ∇rα+β f (r ) ,

α ∇r , r f (r ) = α∂r ∇rα−2 f (r )

α ∇rα f (μr ) = |μ|α ∇μr f (μr ) , ∇rα eiqr = (i|q|)α eiqr   α 2 2 α ∇ r f (r ) (q) = (i|q|) f (q) , ∇r f (r ) = ∂r f (r )

(20)

and, with n ∈ N

∇r , r n = nr n−1 ∂r ∇r−1 , r 2 ∇r , r ∂r = −r 2 ∇r , r ∂r ∇r−1 , r ∇r = −r ,





[∇r , ∂r ] = [r ∂r , r ∇r ] = r, ∂r ∇r−1 = r n ∂r , ∂r ∇r−1 = r n ∇r , ∂r ∇r−1 = 0. (21)

2.4 Non-local Meta-Conformal Generators The deterministic part of (11) becomes Sϕ = (−μ∂r + ∇r ) ϕ = 0, where μ−1 = iν. Physical intuition suggests that a Lie algebra should at least contain time translations X −1 = −∂t , dilatations X 0 = −t∂t − rz ∂r − xz and spatial translations Y−1 = −∂r . However, a long-standing difficulty was to include a spatially local generator of

Infinite-Dimensional Metaconformal Symmetries: …

123

generalised Galilei transformations as a dynamical symmetry. This has now been solved by considering simultaneously local and non-local spatial translations. Theorem 2 ([34]) Construct the generators, for all n ∈ Z and x, ξ constants An = −t n+1 (∂t − ∇r ) − (n + 1) (x − ξ) t n   1 n+1 ξ (t ± r )n 1 ± ∂r ∇r−1 Bn± = − (t ± r )n+1 (∇r ± ∂r ) − 2 2

(22)

Their non-vanishing commutators are given by [An , Am ] = (n − m)An+m ,



± Bn± , Bm± = (n − m)Bn+m

(23)

for n, m ∈ Z. Their Lie algebra is isomorphic to the direct sum of three Virasoro algebras with vanishing central charges. They are also dynamic symmetries of the deterministic equation Sϕ = (−∂t + ∇r ) ϕ = 0 of the dle process, provided that x = ξ, because of the commutators [S, An ] = −(n + 1)t n S + (n + 1)n (x − ξ) t n−1 ,

S, Bn± = 0

(24)

Proof The essential step is to generalise the first identity (21), for all n ∈ N:

∇r , (α ± r )n = ±n (α ± r )n−1 ∂r ∇r−1 where α is a constant. The rest are straightforward calculations, using (20), (21).  A more conventional form of the generators is obtained by defining Yn = Bn+ + Bn− , Z n = Bn+ − Bn− , X n = An + Yn

(25)

This is the dle-analogue of the ortho- and meta-1 conformal invariances, respectively, of the Laplace equation and of simple ballistic transport, treated above. Both local and non-local spatial translations are needed for realising the full dynamical symmetry of the dle process, which we call erosion-Virasoro algebra and denote by ev. The infinite-dimensional Lie algebra ev is built from three commuting Virasoro algebras, and has the maximal finite-dimensional Lie sub-algebra sl(2, R) ⊕ sl(2, R) ⊕ sl(2, R). The scaling operators ϕ = ϕ(t, r ) on which these generators act are characterised by two independent ‘scaling dimensions’ x = xϕ and ξ = ξϕ . By analogy with conformal galilean invariance [44], one expects that three independent central charges of the Virasoro type should appear if the algebra (23) will be quantised. Additional physical constraints (e.g. unitarity) may reduce the number of independent central charges.

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2.5 Ward Identities We adapt a key definition from (ortho-)conformal invariance [5]: Definition 6 A scaling operator ϕ = ϕ(t, r ) is quasi-primary, if it transforms covariantly under the action of the generators of the maximal finite-dimensional subalgebra of ev. A primary scaling operator transforms co-variantly under the action of all generators of ev. For non-equilibrium dynamics, a two-point function of quasi-primary scaling operators is either a correlators ϕ(t, r )ϕ(t  , r  ), or a response  δϕ(t,r )    ϕ(t, r ) ϕ(t , 0) = δ j (t  ,0)  , rewritten as a correlator with a response operator ϕ j=0

conjugate to the scaling operator ϕ within Janssen-de Dominicis theory [53]. In analogy with the time-space transformations of Table 2, the quasi-primary evWard identities are obtained from the explicit form of the Lie algebra generators (22), (25), generalised to n-body generators. We must also assign a signature ε = ±1 to each scaling operator. By convention, εi = +1 for scaling operators ϕi and εi = −1 for response operators ϕ i . The n-body generators then read [33, 34] [n] = X −1 = X −1



[−∂i ] ,

X 0 = X 0[n] =



i

X 1 = X 1[n] = Y−1 =

=



[−εi ∇i ] , Y0 = Y0[n] =

i

Y1 = Y1[n] =

(26)



−ti2 ∂i − 2ti ri Di − μεi ri2 ∇i − 2xi ti − 2μξi εi ri Di ∇i−1 , i

[n] Y−1

[−ti ∂i − ri Di − xi ] ,

i



[−εi ti ∇i − μri Di − μξi ] ,

i

   −εi ti2 + μ2 ri2 ∇i − 2μti ri Di − i

[n] Z −1 = Z −1

− 2μξi ti − 2μ2 ξi εi ri Di ∇i−1 ,  = [−Di ] , i

Z0 =

Z 0[n]



−ti Di − εi ri ∇i − μξi Di ∇i−1 , =

Z1 =

Z 1[n]

   − ti2 + μ2 ri2 Di − 2εi μti ri ∇i − =

i

i

− 2μξi ri − 2μξi εi ti Di ∇i−1

with the short-hands ∂i = ∂t∂i , Di = ∂r∂ i and ∇i = ∇ri . The generators (27) do obey the meta-conformal Lie algebra of the dle process. Define the (n + m)-point function Cn,m = Cn,m (t1 , . . . , tn+m ; r1 , . . . , rn+m ) (27) = ϕ1 (t1 , r1 ) · · · ϕn (tn , rn ) ϕn+1 (tn+1 , rn+1 ) · · · ϕ n+m (tn+m , rn+m )

Infinite-Dimensional Metaconformal Symmetries: …

125

of quasi-primary scaling and response operators. Their co-variance is expressed through the quasi-primary Ward identities, for k = ±1, 0 X k[n+m] Cn,m = Yk[n+m] Cn,m = Z k[n+m] Cn,m = 0,

(28)

The solution of this set of (linear) differential equations gives the sought (n + m)point function Cn,m .

2.6 Co-variant Two-Time Correlator and Response Proposition 1 Any two-point correlator C2,0 (t1 , t2 ; r1 , r2 ) = ϕ1 (t1 , r1 )ϕ2 (t2 , r2 ) built from ev-quasi-primary scaling operators ϕi , vanishes. Proof Time-translation-invariance implies that C2,0 = C2,0 (t1 − t2 ; r1 , r2 ). Invariance under both non-local and local space-translations gives Y−1 C2,0 = Z −1 C2,0 = 0. In Fourier space, this becomes (ε1 |q1 | + ε2 |q2 |) C 2,0 (t; q1 , q2 ) = 0 , (q1 + q2 ) C 2,0 (t; q1 , q2 ) = 0 with positive signatures ε1 = ε2 = +1. The only solution is C 2,0 (t; q1 , q2 ) = 0.  The ev algebra acts as a dynamical symmetry of the deterministic part of the dle Langevin equation (11) only, which corresponds to T = 0. The vanishing of C2,0 for T → 0 is seen explicitly in the exact dle-correlator (14a), (15a). This behaviour is fully analogous to Schrödinger-invariant Edwards-Wilkinson or Arcetri models, see [32, 49], where it follows from a Bargman superselection rule [4]. Consider the two-time response function: R = R(t1 , t2 ; r1 , r2 ) = C1,1 (t1 , t2 ; r1 , r2 ). Time-translation-invariance, X −1 R = 0, implies that R = R(t1 − t2 ; r1 , r2 ). In Fourier space, invariance under non-local and local space-translations give (t = t1 − t2 ) q1 , q2 ) = 0 , (q1 + q2 ) R(t; q1 , q2 ) = 0 ε1 (|q1 | − |q2 |) R(t; since the signatures are now ε1 = −ε2 = +1. Here, a non-vanishing solution is possible and we can write R = F(t, r ), with r = r1 − r2 . Proposition 2 ([34]) One has for the two-point response function a scaling form R = C1,1 = ϕ1 (t, r ) ϕ2 (0, 0) = t −2x f (r/t). If x = 21 (x1 + x2 ), ξ = 21 (ξ1 + ξ2 ), and ξ1 − ξ2 = x1 − x2 and if the scaling function f (υ) obeys

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(ε1 ∇υ + μυ∂υ + 2μξ) f (υ) = 0 , (x1 − x2 ) (ε1 ∇υ + μv∂υ + μ) f (υ) = 0. (29) then all ev-quasi-primary Ward identities are satisfied. The conditions (29) come from the deterministic part of the dle Eq. (11) and do not contain T . This is consistent with the T -independence of the exact dle-response function (14b), (15b). Equations (29) are compatible in two distinct cases: A: 2ξ = 1. Then (ε1 ∇υ + μυ∂υ + μ) f (υ) = 0 and x1 = x2 is still possible. B: x1 = x2 . Then ξ1 = ξ2 and (ε1 ∇υ + μυ∂υ + 2μξ) f (υ) = 0. We must also compare the differential operator S = −μ∂t + ∇r with the dle Langevin equation (11). Taking into account the normalisation in the definition of the Riesz-Feller derivative, we find μ−1 = iν. Physically, one should require ν > 0 in order that the correlators and responses vanish for large momenta |q| → ∞. Proof of Theorem 1 Both cases can be treated in the same way. The first Eq. (29) becomes in Fourier space

iε1 |q| − μq∂q + μ(2ξ − 1) f (q) = 0 In case A, the constant term vanishes, while it is non-zero in case B. Hence f 0 q 2ξ−1 exp (−ε1 ν|q|) f (q) = f 0 q 2ξ−1 exp (iε1 |q|/μ) = where f 0 is a normalisation constant and we can now adopt ε1 = +1. The constant ν from the dle Langevin equation (11) illustrates that f (q) → 0 for |q| large when ν is positive. Both cases A and B produce valid solutions of the linear Eqs. (29), such that the general solution should be a linear superposition of both cases. The inverse Fourier transforms are carried out straightforwardly, with ψ = 2ξ − 1.  While no choice of x1 will make the ortho-conformal prediction (4) compatible with (15b), our main conceptual point is: covariance under the non-local representation (22) of the meta-conformal algebra ev reproduces the correct scaling behaviour of the non-stationary response of the dle process. The non-local ev-meta-conformal invariance produces the response R = C1,1 , whereas the local ortho-, galilean and meta-conformal invariances of Table 2 yielded a correlator C2,0 . The main result (17) on the shape of the ev-meta-conformal response can be cast into the scaling form t x1 +x2 R12 (t, r ) = f (r/t), with the explicit scaling function  f (u) =

1 + u2

−1

−ξ  + ρ 1 + u 2 1 sin (πξ1 − 2ξ1 arctan u) . 1 + ρ sin πξ1

(30)

We see that the first scaling dimensions x1 , x2 merely arrange the data collapse, while the form of the scaling functions only depends on the second scaling dimension ξ1 = ξ2 and the amplitude ratio ρ (the exact solution (15b) of the dle-process corresponds to ρ = 0). The normalisation is chosen such that f (0) = 1. For ξ1 = 21 , we simply

Infinite-Dimensional Metaconformal Symmetries: …

127 1.5

ξ=1 , ρ= 1/3 ξ=1 , ρ= 0 ξ=1 , ρ=-1/3 ξ=1 , ρ=-2/3 ξ=1 , ρ=-1

ξ=2.5 , ρ= 1 ξ=2.5 , ρ= 2/3 ξ=2.5 , ρ= 1/3 ξ=2.5 , ρ= 0 ξ=2.5 , ρ=-1/3

1.0

f(u)

f(u)

1.0

0.5

0.5

0.0

1

2

u

3

4

5

0.0

1

2

u

3

4

5

Fig. 3 Scaling function f (u) for the response R (t, r ) = t −x1 −x2 f (r/t) of 1D dle

have f (u) = (1 + u 2 )−1 . In Fig. 3, several examples of the shape of f (u) are shown (values for different pairs (x1 , ξ1 ) are displayed in [34]). Clearly, these are quite distinct from all the examples of ortho-, meta-1- and Galilean-conformal invariance, displayed above in Fig. 1. By analogy with Schrödinger-invariance, fluctuation-dominated correlators should be obtained from certain time-space integrals of higher n-point responses. Preliminary support of this idea comes from the quantum chain representation of the terracestep-kink model. The exact stationary density-density correlator3 is [36] C(t, r ) = C A t −2

∗ 1 − ξ2 −ψ cos[2(q r − ωt)] + C t B (1 + ξ 2 )2 (1 + ξ 2 )ψ

with the scaling variable ξ = (r − vc t)/(νt), where vc is the global velocity of the interface, ψ ≥ 21 is a real parameter and C A,B are known normalisation constants. Their result is qualitatively very close to the form (17). Work on the calculation of correlators from the ev-algebra is in progress.

3 Meta-Conformal Algebra in d = 2 Spatial Dimensions Generalisations of the one-dimensional meta-conformal case to d = 2 space dimensions (with points (t, x, y) ∈ R3 ) proceed as follows. The generators of translations and dilatations read y

x = −∂x , Y−1 = −∂ y X −1 = −∂t , Y−1 X 0 = −t∂t − x∂x − y∂ y − δ.

3 In

our notation, the density u(t, r ) = ∂r h(t, r ).

(31a) (31b)

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The main task is to find the the special generator X 1 . Here we shall consider [35]     X 1 = − t 2 + α(x 2 + y 2 ) ∂t − 2t x + βx x 2 + (1 − p)β y x y + pβx y 2 ∂x   − 2t y + pβ y x 2 + (1 − p)βx x y + β y y 2 ∂ y − 2δt − 2γx x − 2γ y y. (32) along with the generator of spatial rotations in the form (such that [X 1 , Rx y ] = 0)4 R¯ x y = x∂ y − y∂x + Rγx γ y + Rβx β y R γ x γ y = γ x ∂γ y − γ y ∂γ x , R β x β y = β x ∂β y − β y ∂β x .

(33)

The first step consists in finding the generators of the finite-dimensional Lie algey bra X n , Ynx , Yn n∈{±1,0} . Using the translations gives the next two generators Y0x = y y 1 x [X 1 , Y−1 ], Y0 = 21 [X 1 , Y−1 ]. In turn, their commutator reads 2 (3 p − 1)( p + 1) y × [Y0x , Y0 ] = 8  × ( p − 1)(βx2 + β y2 )(β y x − βx y)∂t + 2(βx β y x + βx2 y)∂x +  + 2(β y2 x + βx β y y)∂ y

(34)

where in the second row we have substituted α = ( p+1)(4 p−1) β 2 [35]. The only recogy nisable simple structure arises if [Y0x , Y0 ] = 0 which occurs for two cases: 1. p = −1, hence α = 0. 2. p = 1/3, hence α = − 29 (βx2 + β y2 ). We shall take up these two distinct cases separately.

3.1 The Case p = −1 3.1.1

Generators y

First, since α = 0, the generators X 1 , Y0x and Y0 reduce to   X 1 = −t 2 ∂t − 2t x + βx x 2 + 2β y x y − βx y 2 ∂x   − 2t y − β y x 2 + 2βx x y + β y y 2 ∂ y − 2δt − 2γx x − 2γ y y. Y0x = −(t + βx x + β y y)∂x − (βx y − β y x)∂ y − γx y

Y0 = −(β y x − βx y)∂x − (t + β y y + βx x)∂ y − γ y . 4 This

(35a) (35b) (35c)

modification shows that the corresponding representations of the algebra depends on two additional vectors β = (βx , β y ) and γ = (γx , γx ) such that their component become variables equivalent to x, y.

Infinite-Dimensional Metaconformal Symmetries: …

129 y

y

The last two generators Y1x := [X 1 , Y0x ] and Y1 := [X 1 , Y0 ] become   Y1x = − t 2 + 2tβx x + 2tβ y y + (βx2 − β y2 )x 2 + 4βx β y x y − (βx2 − β y2 )y 2 ∂x   − 2tβx y − 2tβ y x − 2βx β y x 2 + 2(βx2 − β y2 )x y + 2βx β y y 2 (36a) − 2γx (t + βx x + β y y) − 2γ y (βx y − β y x)   Y1x = − 2tβ y x − 2tβx y + 2βx β y x 2 − 2(βx2 − β y2 )x y − 2βx β y y 2 ∂x   − t 2 + 2tβx x + 2tβ y y + (βx2 − β y2 )x 2 + 4βx β y x y − (βx2 − β y2 )y 2 ∂ y − 2γ y (t + βx x + β y y) − 2γx (β y x − βx y).

(36b)

Finally, the non-vanishing commutators holds, with n, m ∈ {0, ±1} [X n , X m ] = (n − m)X n+m , y x [X n , Ymx ] = (n − m)Yn+m , [X n , Ymy ] = (n − m)Yn+m , y

x [Ynx , Ymy ] = [Yny , Ymx ] = (n − m)(β y Yn+m + βx Yn+m ), y x x y y x [Yn , Ym ] = −[Yn , Ym ] = (n − m)(βx Yn+m − β y Yn+m ), [Ymx , R¯ x y ] = Ymy , [Ymy , R¯ x y ] = −Ymx

(37)

In the right-hand side the components of vector β appear which means that in order to have closed algebraic structure we must eliminate them as variables. It is enough to set Rβx β y = 0. Hence, the admissible values of βx and β y are (i) βx = β = cste., β y = 0; (ii) βx = 0, β y = β = cste.; (iii) βx = ±β y = cste. These generators act as dynamical symmetries of the linear differential equation Bˆ f (t, x, y) = (∂t + cx ∂x + c y ∂ y ) f (t, x, y) = 0 cy cx γx cx + γ y c y + δ = 0 , βx = − 2 , βy = − 2 . cx + c2y cx + c2y

(38) (39)

Indeed, one has in general that x ˆ X −1 ] = [ B, ˆ Y−1 ˆ Y−1 ] = 0 , [ B, ˆ X 0 ] = − B. ˆ [ B, ] = [ B, y

(40)

In addition, under the conditions (39), we also have ˆ Y0 ] = [ B, ˆ Y1x ] = [ B, ˆ Y1 ] = 0 , [ B, ˆ X 1 ] = −2t B. ˆ ˆ Y0x ] = [ B, [ B, y

y

which implies the invariance of the solution space.

(41)

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Infinite-Dimensional Extension

For a better understanding of the algebra choose coordinate axes such that β = (β, 0). This means that the ballistic transport is along the x-axis. The generator of rotations now reads (42) R = Rx y = x∂ y − y∂x + γx ∂γ y − γ y ∂γx y

Considering the commutators between Ynx and Yn , the peculiar signs arising suggest to go over to new generators Yn± :=

 1 x Yn ± i Yny 2

(43)

The resulting Lie algebra turns out to be isomorphic to the one found of the 1D diffusion-limited erosion process studied above in Sect. 2. We go over to complex spatial coordinates z = x − iy and z¯ = x + iy. Define the new generators An := X n −

1 + 1 − Y − Yn β n β

(44)

In the basis An , Yn+ , Yn− n∈{±1,0} , the Lie algebra becomes the direct sum sl(2, R) ⊕ sl(2, R) ⊕ sl(2, R). The above definition can now be extended to an infinite-dimensional set of generators, with n ∈ Z An = −t

n+1

    γ 1 ¯ γ¯ 1 n ∂t − ∂ − ∂ − (n + 1)t δ − − β β β β

Yn+ = −(t + βz)n+1 ∂ − (n + 1)γ(t + βz)n Yn− = −(t + β z¯ )n+1 ∂¯ − (n + 1)γ(t ¯ + β z¯ )n

(45)

with the complex components γ := 21 (γx + iγ y ) and γ¯ := 21 (γx − iγ y ). The only non-vanishing commutators are [An , Am ] = (n − m)An+m ,



± Yn± , Ym± = β (n − m)Yn+m

(46)

The Lie algebra (46) is isomorphic to (23), hence to vect(S 1 ) ⊕ vect(S 1 ) ⊕ vect(S 1 ), the direct sum of three Virasoro algebras without central charge. This is remarkable since the generators (22) from (45). Finally, the ballistic operator (39)  are distinct  becomes Bˆ = −∂t + β1 ∂ + ∂¯ and obeys the commutators 

   An , Bˆ = (n + 1)t n Bˆ − (n + 1)nt n−1 δ , Yn± , Bˆ = 0

where  δ := δ −

γ β

− βγ¯ . Summarising, we have proven:

(47)

Infinite-Dimensional Metaconformal Symmetries: …

131

Theorem 3 ([35]) In two spatial dimensions r = (x,  y), the linear ballistic transport equation has the normal form Bˆ f (t, x, y) = −∂t + β −1 ∂x f (t, x, y) = 0, where β is a constant. Its maximal dynamical symmetry is infinite-dimensional, spanned by the generators (45), if only  δ = δ − βγ − βγ¯ = 0. The Lie algebra (46) of dynamical symmetries is isomorphic to the direct sum of three centre-less Virasoro algebras. Working with the coordinates w = t + βz and w¯ = t + β z¯ , we see that the sym¯ while the action metries generated by Yn± are ortho-conformal in the variables (w, w), of the genertors An are meta-conformal, see also Table 2. This appears to be the first known example which combines ortho- and meta-conformal transformations into a single symmetry algebra. If  δ = 0, we actually have a spectrum-generating algebra for Bˆ = A0 . In spite of the symmetric formulation, the equation of motion (39) contains a bias, since the transport goes along the axis x = 21 (z + z¯ ), if β = 0. 3.1.3

Two-Point Function

A simple application of dynamical symmetries is the computation of covariantly transforming two-point functions. By analogy with ortho-conformal invariance [5], quasi-primary scaling operators φ(t, z, z¯ ), which tranform co-variantly under the ± . Each scaling operator is characterised finite-dimensional sub-algebra A±1,0 , Y±1,0 by three constants ( δ, γ, γ). ¯ Standard calculations give F(t, z, z¯ ) = φ1 (t, z, z¯ )φ2 (0, 0, 0) = 

= δδ1 ,δ2 δγ1 ,γ2 δγ¯ 1 ,γ¯ 2 t −2δ1 (t + βz)−2γ1 (t + β z¯ )−2γ¯ 1

(48)

up to normalisation. This shows a cross-over between an ortho-conformal two-point function when t  z, z¯ and a non-trivial scaling form in the opposite case t  z, z¯ . Cardy [11] suggested a long time ago to look for dynamical symmetries which keep 2D conformal invariance in space. The present example appears to be the first one of this kind, and desscribes a relaxation towards a 2D conformally invariant equilibrium state, with dynamical exponent z = 1.

3.2 The Case p = 1/3 3.2.1

Generators y

For simplicity, let β = (β, 0). If α = −(2/9)β 2 , the generators X 1 , Y0x , Y0 become    2 2 2 1 2 2 2 X 1 = − t − β (x + y ) ∂t − 2t x + βx + β y ∂x 9 3   2 − 2t y + βx y ∂ y − 2δt − 2γx x − 2γ y y. 3 

2

(49a)

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2 2 1 β x∂t − (t + βx)∂x − β y∂ y − γx 9  3  2 2 1 1 y Y0 = β y∂t − β y∂x − t + βx ∂ y − γ y . 9 3 3

Y0x =

y

(49b) (49c)

y

The last two generators Y1x := [X 1 , Y0x ] and Y1 := [X 1 , Y0 ] read   2 2 1 2 2 = β 2t x + βx + β y ∂t − (50a) 9 3   7 1 − t 2 + 2βt x + β 2 x 2 + β 2 y 2 ∂x − 9 9     2 2 2 2 2 2 βt y + β x y ∂ y − 2γx t − 2 βγx − β δ x − βγ y y, − 3 9 9 3     2 2 2 2 y Y1 = β 2 2t y + βx y ∂t − βt y + β 2 x y ∂x − 9 3 3 9   1 2 2 − t 2 + βt x + β 2 (x 2 − y 2 ) ∂ y − 2γ y t − βγ y x − 3 9 3   2 1 βγx − β 2 δ y (50b) −2 3 9

Y1x

Indeed, for n, m ∈ {0, ±1}, we have the non-vanishing commutators [X n , X m ] = (n − m)X n+m , y

x [X n , Ymx ] = (n − m)Yn+m , [X n , Ymy ] = (n − m)Yn+m , n−m y βYn+m , [Ynx , Ymy ] = [Yny , Ymx ] = 3   2 2 x x y y x [Yn , Ym ] = [Yn , Ym ] = (n − m) − β X n+m + βYn+m , 9 [Ymx , R¯ x y ] = Ymy , [Ymy , R¯ x y ] = −Ymx

(51)

  In addition, it can be checked that the Lie algebra X n , Yna n∈{±1,0},a∈{x,y}  sl(2, R) ⊕ sl(2, R) ⊕ sl(2, R) is not isomorphic to the one found for p = −1. Furthermore, the generators thus constructed act as dynamical symmetry algebra of the linear differential equation, if δ = (3/β)γx   3 Bˆ p f (t, x, y) = ∂t − ∂x f (t, x, y) = 0 β

(52)

Infinite-Dimensional Metaconformal Symmetries: …

133

To see this, it suffices to compute x ˆ X 0 ] = − B. ˆ ] = [ Bˆ p , Y±1,0 ] = 0 , [ B, [ Bˆ p , X −1 ] = [ Bˆ p , Y−1 y

(53)

In addition, we obtain   4β 2β x Bˆ p , [ Bˆ p , Y0x ] = − Bˆ p , [ Bˆ p , X 1 ] = − 2t + 3 3   β 8β t + x Bˆ p . [ Bˆ p , Y1x ] = − 3 3 which proves the invariance of the solution space. Theorem 4 ([35]) In two spatial dimensions   r = (x, y), the ballistic equation in normal form Bˆ f (t, x, y) = ∂t − 3/β 2 ∂x f (t, x, y) = 0 has the 10-dimensional dynamical symmetry algebra (51), whose generators are given by Eqs. (31), (49), (50).

3.2.2

Two-Point Function

The two-point function is build by the quasi-primary fields φ(t, x, y, γ, λ), where we write γ = γx and λ = γ y . which transform covariantly under the representation of (51) we have constructed. Taking into the account the covariance of time- and space-translations we write F = φ(t1 , x1 , y1 , γ1 , λ1 )φ(t2 , x2 , y2 , γ2 , λ2 ) = F(t, x, y, γ1 , γ2 , λ1 , λ2 ), where t = t1 − t2 , x = x1 − x2 , y = y1 − y2 . Ward identities expresse the covariance under the other generators. This closed linear system leads to the constraints γ1 = γ2 = γ, λ1 = λ2 = λ, δ1 = δ2 = δ, and the following form, up to normalisation of F = F(t, x, y) and w = t + βx/3 3γ/β−2δ  2(δ−3γ/β)  2 β2 2 2 t + βx y × F= w + 9 3    β|y| 2λ arctan × exp − 3β 3|w|

(54)

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4 Conclusions In spite of its tremendous richness in applications, 2D ortho-conformal invariance is not the only infinite-dimensional Lie group of time-space transformations. Beyond the previously known examples of Schrödinger-, 1D meta-conformal and conformal galilean invariance, which are all doubly infinite, we have studied two distinct examples with isomorphic triply infinite Lie algebras. Remarkably, one of our examples is local in space while the other is not, but both have a dynamical exponent z = 1. The non-local example was extracted from the 1D dle process and we showed how to reproduce its exact non-stationary two-time response (15b) from the covariance under the representation of the ev-algebra. A surprising feature is that both local and non-local spatial translations play an important rôle. The local example arose by generalising the 1D meta-conformal to d = 2 spatial dimensions. The covariant two-point function allows to follow the relaxation from an initial state to a 2D ortho-conformally-invariant equilibrium state, for a preselected dynamical exponent z = 1. An algebraically non-isomorphic example of another 2D meta-conformal algebra was also constructed. This work having been focused on establishing these new and unexpected algebraic structures, the physical and mathematical content of their representations remains to be explored beyond the simple examples mentioned here. Acknowledgements Most of this work was done during the visits of S.S. at Université de Lorraine Nancy and of M.H. at the joint conference “Quantum Theory and Symmetries QTS10” and “Lie Theories and Its Applications in Physics LT12”. These visits were supported by PHC Rila and by Bulgarian National Science Fund Grant DFNI - T02/6.

References 1. N. Aizawa, Z. Kusnetsova, F. Toppan, Prog. Theor. Exp. Phys. 083A01 (2016) [arXiv:1506.08488]. 2. N. Aizawa, Z. Kusnetsova, F. Toppan, J. Math. Phys. 57, 041701 (2016) [arXiv:1512.02290]. 3. A.-L. Barabási, H.E. Stanley H.E., Fractal Concepts in Surface Growth, Cambridge University Press (Cambridge 1995). 4. V. Bargman, Ann. of Math. 56, 1 (1954). 5. A.A. Belavin, A.M. Polykaov, A.B. Zamolodchikov, Nucl. Phys. B241, 333 (1984). 6. L. Bertini, A. De Sole, D. Gabrielli, G. Jona-Lasinio, C. Landim, Rev. Mod. Phys. 87, 593 (2015) [arXiv:1404.6466]. 7. H. Bondi, M.G.J. van der Burg, A.W.K. Metzner, Proc. Roy. Soc. London, A269, 21 (1962). 8. P. Calabrese, J.L. Cardy, J. Stat. Mech. P064003 (2016), [arXiv:1603.02889]. 9. A. Campa, T. Dauxois, D. Fanelli, S. Ruffo, Physics of Long-Range Interacting Systems, Oxford Univ. Press (Oxford 2014). 10. A. Cappelli, G.V. Dunne, C.A. Trugenberger, G.R. Zemba, Nucl. Phys. B398, 531 (1993), [arXiv:hep-th/9211071]. 11. J.L. Cardy, J. Phys. A18, 2271 (1985). 12. J.-S. Caux, B. Doyon, J. Dubail, R. Konik, T. Yoshimura, [arXiv:1711.00873]. 13. R. Cherniha, M. Henkel, J. Math. Anal. Appl. 369, 120 (2010) [arXiv:0910.4822].

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Behrends–Fronsdal Spin Projection Operator in Space-Time with Arbitrary Dimension A. P. Isaev and M. A. Podoinitsyn

Abstract On the basis of the Wigner unitary representations of the covering group I S L(2, C) of the Poincar’e group we construct spin-tensor wave functions of free massive particles with arbitrary spin. These wave functions automatically satisfy the Dirac–Pauli–Fierz equations. Spin-tensors of polarizations and conditions that fix the corresponding relativistic spin projection operators (Behrends–Fronsdal projection operators which determine the numerators in the propagators of fields of relativistic particles) are obtained. With the help of these conditions we find the generalization for relativistic spin projection operators for particles of arbitrary spins and for arbitrary space-time dimensions D > 2.

1 Introduction This report is based on the results of paper [1]. Here we consider the Wigner unitary representations [2] (see also [3, 4]) of the group I S L(2, C), which covers the Poincaré group. We show that these irreducible representations can be reformulated such that they act in the space of spin-tensor wave functions of a special type. The construction of these functions is carried out with the help of Wigner operators (a similar construction was developed in [5]; see also [3]), which translate the unitary massive representation of the group I S L(2, C) (induced from the irreducible representation of the stability subgroup SU (2)) acting in the space of Wigner wave functions to a representation of the group I S L(2, C), acting in the space of special spin-tensor fields of massive particles. These spin-tensor fields automatically satisfy the Dirac–Pauli–Fierz wave equations [6–8] for free massive particles of arbitrary A. P. Isaev (B) · M. A. Podoinitsyn Bogoliubov Laboratory of Theoretical Physics, JINR, Dubna 141980, Russia e-mail: [email protected] M. A. Podoinitsyn e-mail: [email protected] A. P. Isaev · M. A. Podoinitsyn State University of Dubna, University street, 19, Dubna 141980, Russia © Springer Nature Singapore Pte Ltd. 2018 V. Dobrev (ed.), Quantum Theory and Symmetries with Lie Theory and Its Applications in Physics Volume 1, Springer Proceedings in Mathematics & Statistics 263, https://doi.org/10.1007/978-981-13-2715-5_7

137

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A. P. Isaev and M. A. Podoinitsyn

spin. Further, we consider special expansion of a completely symmetric Wigner wave function, which gives a natural recipe for describing the polarizations for massive particles with arbitrary spins. As an application of this formalism, a generalization of the Behrends–Fronsdal projection operator is constructed, which determines spintensor structures of a two-point Green function (propagator) of massive particles of any higher spins for arbitrary space-time dimension D > 2.

2 Polarization Tensors for the Field of Arbitrary Spin Based on the Wigner construction of massive unitary and irreducible representations of the covering Poincare group I S L(2, C), one can show (see [1]) that the space of the representation of the group I S L(2, C) with spin j is the space of the spin-tensor wave functions ( 2p , r2 ) - type depending on four-momentum k = (k0 , k1 , k2 , k3 ): p r    −1† β˙ j 1  δi n ξ˙ j δ p+ j × A (A ) · · (q σ ˜ ) (k) n α ˙ (k) i ξj mr i=1 j=1

(β˙ ...β˙ )

(r ) 1 r (k) = ψ(α 1 ...α p )

× φ(δ1 ...δ p δ p+1 ...δ p+r ) (k)

(1)

Here ( p + r ) = 2 j, φ(δ1 ...δ p δ p+1 ...δ p+r ) (k) – is an arbitrary symmetric tensor of rank 2 j (the Wigner wave function), q n – components of the test four-momentum q n qn = qk η kn qn = q02 − q12 − q22 − q32 = m2 , η kn = diag(+1, −1, −1, −1) , parameter m is a mass, σ˜ n = (σ0 , −σ1 , −σ2 , −σ3 ); where σ1 , σ2 , σ3 – Pauli matrices and σ0 – unit (2 × 2) matrix. Matrices A(k) , A†−1 (k) ∈ S L(2, C), used in (1), are solutions of the equations (A(k) )γα (q n σn )γ α˙ (A†(k) )αγ˙˙ = (k n σn )αγ˙ ,

(2)

and parametrize coset space S L(2, C)/SU (2). The upper index (r ) of the spinwith respect to the number of dotted tensors ψ (r ) in (1) distinguishesthese spin-tensors ⊗r ⊗p (q σ) ˜ , used in (1) to translate the Wigner wave indices. The operators A(k) ⊗ A†−1 (k) functions into spin-tensor functions of ( 2p , r2 )-type, are called the Wigner operators. (β˙ ...β˙ )

(r ) 1 r Proposition 1 The wave functions ψ(α (k), defined in (1), satisfy the Dirac– 1 ...α p ) Pauli–Fierz equations [6–8]:

(β˙ ...β˙ )

(γ˙ β˙ ...β˙r )

(r ) 1 r (r +1) 1 1 (k) = m ψ(α k m (σ˜ m )γ˙ 1 α1 ψ(α 1 ...α p ) 2 ...α p )

k

m

˙ ˙ (σm )γ1 β˙ 1 ψ(α1 ...α p )

(r ) (β1 ...βr )

(k) =

˙ ˙ m ψ(γ1 α1 ...α p )

(r −1) (β2 ...βr )

(k) ,

(k) ,

(r = 0, . . . , 2 j − 1) , (r = 1, . . . , 2 j) ,

(3)

Behrends–Fronsdal Spin Projection Operator in Space-Time with Arbitrary Dimension

139

which describe the dynamics of a massive relativistic particle with spin j = ( p + r )/2. The compatibility conditions for the system of equations (3) are given by the mass shell relations (k n kn − m 2 ) ψ (r ) (k) = 0. 

Proof See [1]

It is convenient to write symmetrized Wigner functions φ(α1 ...α2 j ) (k) (defined in (1)) in the form of generating function: φ(k; v) = φ(α1 ...α2 j ) (k)v α1 . . . v α2 j ,

(4)

where v α are the components of auxiliary Weyl spinor v = (v 1 , v 2 ). Introduce monoj mials Tm (v): (v 1 ) j+m (v 2 ) j−m Tmj (v) = √ , ( j + m)!( j − m)!

(m = − j, − j + 1, . . . , j) ,

(5)

which can be considered as (2 j + 1) basis elements in the space of polynomials (4). j Indeed, any polynomial (4) can be expanded in terms of Tm (v): j

φ(k, v) =  φm (k) Tmj (v) . m=− j

(6)

The relation between the coefficients φm (k) and φ(α1 ...α2 j ) (k) is given by relation: φm (k) = √

(2 j)! φ( 1...1 2...2 ) (k) . 

( j + m)!( j − m)! 

j+m

(7)

j−m

The generators of the lie algebra s(2, C) acts in the space of polynomials (4) and (6) as differential operators S− = v 2 ∂v1 , S+ = v 1 ∂v2 , S3 =

1 1 (v ∂v1 − v 2 ∂v2 ). 2

(8)

j

The monomials Tm (v) are the eigenvectors of the operator S3 (the third component of the spin vector) given in (8). In fact, we have S3 Tmj (v) = m Tmj (v) ;

(9)

therefore, the coefficients φm (k) in the expansion (6) correspond to the projections − → m of the third component S = (S+ , S− , S3 ) of the spin operator. Formula (1), which transforms the Wigner wave function φ(α1 ...α2 j ) (k) to the spin-tensor fields (β˙ ...β˙ )

(r ) 1 r (k), can be written in terms of the generating functions as follows: ψ(α 1 ...α p )

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ψ (r ) (k; u, u) =

p  1 (u αi (A(k) )αiρi ∂ρ(v)i ) × (2 j)! mr i=1 r     β˙ ρ × u β˙ A−1† ˜  p+ ∂ρ(v)p+ φ(k; v), (k) · (q σ)

(10)

=1

where ∂ρ(v)i = ∂v∂ρi . We fix as usual the test momentum q = (m, 0, 0, 0) and substitute expression (6) for the Wigner wave function to the formula (10). After that, expanding the left and right-hand sides of (10) over u and u, we obtain ˙ ˙ ψ(α1 ...α p )

(r ) (β1 ...βr )

j p r     −1† β˙ ρ p+ (m) 1 A(k) σ˜ 0 (k) = φm (k) (A(k) )αiρi

ρ1 ...ρ2 j , (11) (2 j)! m=− j i=1 =1

where

j (v) (v)

(m) ρ1 ...ρ2 j = ∂ρ1 . . . ∂ρ2 j Tm (v) ,

(12)

are components of constant tensor (m) . Using these tensors we define spin-tensors e with components

(m)

p r ˙ ˙   −1† β˙  ρ p+ (m) 1  A(k) σ˜ 0 e (α1 ...α p ) (k) = √ (A(k) )αiρi

ρ1 ...ρ2 j , (2 j)! i=1 =1

(m) (β1 ...βr )

(13)

where the normalization factor √(21 j)! is chosen for convenience. Then, in terms of spin-tensors (13), formula (11) can be written as: (β˙ ...β˙ )

(r ) 1 r ψ(α (k) = √ 1 ...α p )

j ˙ ˙  1 (m) (β1 ...βr ) φm (k) e (α1 ...α p ) (k) . (2 j)! m=− j ˙

(14)

˙

(m) (β1 ...β j )

Proposition 2 The spin-tensors e (α1 ...α j ) , defined in (13) satisfy the relations:  (m) (β˙1 ...β˙j ) γ˙ μγ ∂λ∂ γ − λ ∂μ∂ γ˙ e (α1 ...α j )  (m) (β˙1 ...β˙j ) e (α1 ...α j ) = √( j+m)(1 j−m+1) λγ ∂μ∂ γ − μγ˙ ∂ γ˙ e (α1 ...α j ) ∂λ ˙ ˙  (m) (β˙1 ...β˙j ) (m) (β1 ...β j ) γ˙ ∂ 1 ∂ ∂ e (α1 ...α j ) = 2·m μγ ∂μγ − λγ ∂λγ + λ γ˙ − μγ˙ ∂μ∂ γ˙ e (α1 ...α j ) , ˙

˙

(m+1) (β1 ...β j )

e

(α1 ...α j ) ˙ ˙ (m−1) (β1 ...β j )

=

1 √ ( j−m)( j+m+1)

(15)

∂λ

γ˙

where μγ , λγ , μγ˙ , λ - Weyl spinors. Proof The proof is based on the use of formulas (8) and representation of matrices A(k) , A−1† (k) ∈ S L(2, C) in terms of Weyl spinors μ, λ

Behrends–Fronsdal Spin Projection Operator in Space-Time with Arbitrary Dimension

141

(A(k) )αβ μ=

=

μ1 μ2

1 z



λ2˙ −μ2˙ μ1 λ1 †−1 α˙ 1 , (A(k) ) β˙ = z ∗ , μ2 λ2 −λ1˙ μ1˙ ρ˙ 2 ρ ∗ 2 (z) = μ λρ , (z ) = μ λρ˙ ,

(16)

μ1˙ λ1˙ λ1 , μ= , λ= , λ2 μ2˙ λ2˙

(17)

, λ=

λα˙ = (λα )∗ ,

μα˙ = (μα )∗ . 

proposed in the paper [1].

For integer j, the spin-tensor functions ( 2j , 2j )-type are related to the vector-tensors in Minkowski space by the following formula: f n 1 ...n j (k) =

˙ ˙ 1 ( j) (β1 ...β j ) (σn 1 )α1 β˙ 1 . . . (σn j )α j β˙ j εα1 γ1 . . . εα j γ j ψ(γ (k) . 1 ...γ j ) j 2

(18)

Here the components f n 1 ...n j (k) of vector-tensor have only vector indices {n 1 , . . . , n j }. (β˙ ...β˙ )

Now we convert the spin-tensor wave functions ψ(γ11...γ jj) (k) to the vector-tensor func(β˙ ...β˙ )

tions f n 1 ...n j (k) by means of relation (18) and then use expression (14) for ψ(γ11...γ jj) (k) in terms of Wigner’s coefficients φm (k). As a result, we obtain the expansion f n 1 ...n j (k) = 

j 

1 2 j (2 j)!

φm (k) e(m) n 1 ...n j (k) ,

(19)

m=− j

where ˙ ˙ 1 (m) (β1 ...β j ) e(m) (σn 1 )α1 β˙ 1 . . . (σn j )α j β˙ j εα1 γ1 . . . εα j γ j e (γ1 ...γ j ) (k) . n 1 ...n j (k) = √ 2j

(20)

(m)

Using the definition (13) of spin-tensor e (k), one can show that the set of vector tensors e(m) n 1 ...n j (k) is normalized as follows: ) j  e(m) n 1 ...n j (k)e(m n 1 ...n j (k) = (−1) δ (m)(m ) , 

(21)

where e(m) n 1 ...n j (k) = (e(m) n 1 ...n j (k))∗ . Remark From definition (13), in the case r = p (i.e. for integer spins j), we deduce the recurrence relation: √ 1 (+) √ ( j + m)( j + m − 1) e(m−1) e(m) n 1 ...n j = 2 j (2 j−1) n 1 ...n j−1 en j +  √ √ (−) (0) + ( j − m)( j − m − 1) e(m+1) , 2( j − m)( j + m) e(m) n 1 ...n j−1 en j + n 1 ...n j−1 en j (22)

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where e(m) n 1 ...n j−1 ≡ 0 if |m| > j − 1. Solution of Eq. (22) completely determines the components e(m) n 1 ...n j of the polarization tensor for any j via the components of the vectors of polarization e(a) n (a = 0, ±1) for j = 1. The spin projection operator Θ(k) is constructed as the sum of products e(m) (k) · e(m) (k) over all polarizations m: n ...n

j

Θr11...r j j (k) := (−1) j  er(m)1 ...r j (k)e(m) n 1 ...n j (k)

(23)

m=− j

This operator is sometimes called the density matrix for a massive particle with integer spin j, or the Behrends–Fronsdal projection operator [9, 10]. Proposition 3 The operator Θ(k), defined in (23), satisfies the following properties: (1) projective property and reality: Θ 2 = Θ, Θ † = Θ; n ...n n ...n i ...n  ...  ...n i ... = Θr...n ; (2) symmetry: Θ...r1 i ...rj ... = Θ...r1  ...rj i ... , Θr...n 1 ...r j 1 ...r j n ...n n ...n 1 j 1 j r1 (3) transversality: k Θr1 ...r j = 0, kn 1 Θr1 ...r j = 0; n ...n (4) tracelessness: ηr1 r2 Θr11r2 ...rj j = 0. 

Proof See [1].

3 Spin Projection Operator for Arbitrary Spin For the four-dimensional space-time D = 4, the Behrends–Fronsdal projection operator Θ(k) for any spin j was explicitly constructed in [9, 10]. Here we find a generalization of the Behrends–Fronsdal operator to the case of an arbitrary number of dimensions D > 2. Also in this Section we prove an important formula which connects the projection operators for half-integer spins j with the projection operators for integer spins j + 1/2. The construction will be based on the properties of this operator, which are listed in Proposition 3. n ...n Let j be integer spin. Instead of the tensor Θr11...r j j (k) symmetrized in the upper and lower indices, it is convenient to consider the generating function n ...n

Θ ( j) (x, y) = x r1 . . . x r j Θr11...r j j (k) yn 1 . . . yn j .

(1) n ...n

For concreteness, we assume that the tensor Θ(k) with components Θr11...r j j (k) is defined in the pseudo-Euclidean D-dimensional space Rs,t (s + t = D) with an arbitrary metric η = ||ηmn ||, having the signature (s, t). Indices n  and r in (1) run through values 0, 1, . . . , D − 1 and (x0 , . . . , x D−1 ), (y0 , . . . , y D−1 ) ∈ Rs,t . Proposition 4 The generating function (1) of the covariant projection operator n ...n Θr11...r j j (in D-dimensional space-time), satisfying properties (1)–(4), in Proposition 3, has the form

Behrends–Fronsdal Spin Projection Operator in Space-Time with Arbitrary Dimension j

Θ

( j)

(x, y) =

[2] 

( j)

aA

 (y) (x)  A  (y)  j−2 A Θ(y) Θ(x) Θ(x) ,

143

(2)

A=0

where [ 2j ] – integer part of j/2,  1 A j! ( j) , aA = − 2 ( j − 2 A)! A! (2 j + D − 5)(2 j + D − 7) . . . (2 j + D − 2 A − 3) (A ≥ 1) , (3) ( j)

(y)

a0 = 1, and the function Θ(x) is defined as follows (ηr n – the metric of space Rs,t ): (y)

Θ(x) ≡ Θ (1) (x, y) = x r yn Θrn ,

Θrn = ηrn −

kr k n . k2

(4)

The generating function (2) satisfies the differential equation ∂ ∂ ( j) j ( j + D − 4)(2 j + D − 3) ( j−1) Θ (x, y) = (x, y) . Θ r ∂x ∂ yr (2 j + D − 5)

(5)

Proof We recall that the matrix Θrn which is defined in (4) is a projection operator on the subspace of vector orthogonal to the D-dimensional momentum with the components k r : k r Θrn = 0 = Θrn kn ,

Θrr = ηr n Θ r n = D − 1 ,

Θrn Θmr = Θmn .

(6)

n ...n

Taking into account this fact, the most general covariant operator Θr11...r j j (k), satisfying properties (2) and (3) from Proposition 3, is written as follows: n ...n

Θr11...r j j (k) =

1  n σ(1) n  ... m 1 Θ . . . Θ jσ( j) Bm11 ...mj j (k) Θrmμ(1) . . . Θrμ(jj) , ( j!)2 σ,μ∈S 1

(7)

j

where σ, μ are permutations of the indices {1, 2, . . . , j} i.e. σ and μ are elements of  ... the permutation group S j , and the components Bm11 ...mj j (k) are any covariant combinations of the metric ηr m and coordinates kr of the D-dimensional momentum. Since the matrices Θrm used in the right-hand side of (7) are transverse to the D-momentum  ... k (see (6)), the external indices of the tensor Bm11 ...mj j (k) can be associated only with the indices of the metric η and, therefore, this tensor is expressed in the form  ...

( j)



( j)



Bm11 ...mj j (k) = a0 (k) ηm11 . . . ηmj j + a1 (k) η 1 2 ηm1 m2 ηm33 . . . ηmj j +  ( j) +a2 (k) η 1 2 ηm1 m2 η 3 4 ηm3 m4 ηm55 . . . ηmj j + . . . ,

(8)

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A. P. Isaev and M. A. Podoinitsyn ( j)

where the coefficients a A (k) in general could be function of the invariants (k)2 = k r kr . Substitution (8) into (7) gives j

r r ...r Θn11 n22 ...nj j

[2] j A   ( j)   1  r rσ(2−1) rσ(2) . (9) = a Θ Θ Θnσ(i) n μ(2−1) n μ(2) μ(i) A 2 ( j!) A=0 σ,μ∈S i=2 A+1 =1 j

Finally, using formula (9) in (1), we obtain representation (2) for the generating function of the projection operator. ( j) Now we show that coefficients a A in (2), do not depend on (k)2 and their explicit form is fixed by properties (1) and (4) from Proposition 3. Property (4) (tracelessness) in Proposition 3 for the tensor Θ is equivalent to the harmonic equation for the generating function (2): (10) (∂x )2 Θ ( j) (x, y) = 0 , where (∂x )2 = ∂x r ∂xr and ∂x r = ∂/∂x r . Let us substitute expression (2) into the ( j) Eq. (10) to find the conditions which fix the coefficients a A . ( j)  1 A j! a0 ( j) aA = − , (11) 2 ( j − 2 A)! A! [(2 j + D − 5)(2 j + D − 7) . . . (2 j + D − 3 − 2 A)] ( j)

i.e., the condition (10) determines the coefficients a A up to a single arbitrary factor ( j) a0 . For A = 0 the product of factors in square brackets in the denominator of (11) have to be equal to unity. We note that when we substitute the coefficients (11) into the sum (2), this sum is automatically terminated for A > j/2 in view of the infinite factor ( j − 2 A)! = ∞ in the denominator of (11). Now we verify for solution (2), (3) the condition (1) in Proposition 3. First of all, the property Θ(x, y) = Θ(y, x) for the function (2) and reality condition Θ ∗ = Θ are equivalent to Θ † = Θ for the matrices (9). The projector condition Θ 2 = Θ (the property 1 in Proposition 3) for the matrix (9) can be checked directly: j

r r ...r (Θ 2 )n11 n22 ...nj j

=

r r ...r j Θm11 m2 2 ...m j

[2]  ( j) 1  a × 2 ( j!) A=0 σ,μ∈S A j

×

A 

Θ

=1

=

m σ(2−1) m σ(2)

Θn μ(2−1) n μ(2)

j 

 m σ(i) = Θn μ(i)

i=2 A+1

r r ...r j Θm11 m2 2 ...m j

 1  ( j)  m 1 m r r ...r ( j) a0 Θn μ(1) . . . Θn μ(j j) = a0 Θn11 n22 ...nj j , j! μ∈S j

where in the second and third equalities we used obvious identities r r ...r

j mi m =0, Θm11 m2 2 ...m jΘ

r r ...r

m

r r ...r

m

r r ...r

2 j 1 2 j 1 2 j i i Θm11 ...m  ...m j Θm i = Θm 1 ...m  ...m j ηm i = Θm 1 m 2 ...m j . i

i

Behrends–Fronsdal Spin Projection Operator in Space-Time with Arbitrary Dimension

145

Which follow from conditions (3) and (4) of Proposition 3. Thus, the projector ( j) condition is fulfilled, if we fix the initial coefficient in the expansion (2) as a0 = 1. With this value of at the formula (11) turns into formula (3). Finally, we prove the identity (5). For this we calculate ∂x r ∂ yr Θ ( j) (x, y) = =

j [ 2]

A=0

  ( j) (y) (x) (y) a A ∂x r ∂ yr (Θ(x) Θ(y) ) A (Θ(x) ) j−2 A =

j [ 2] 

A=0

 ( j) ( j) (y) (x) (y) a A+1 4(A + 1)2 + a A ( j − 2 A)(D − 2 + j + 2 A) (Θ(x) Θ(y) ) A (Θ(x) ) j−2 A−1

(12) Taking into account the explicit formula for the coefficients (3), we obtain relation ( j)

( j)

a A+1 4(A + 1)2 + a A ( j − 2 A)(D + j + 2 A − 2) =

j ( j + D − 4)(2 j + D − 3) ( j−1) aA . 2j + D − 5

Substitution of this relation into (12), gives the identity (5).



Remark 1 Identity (5) for the generating functions (1) leads to the equality that connects the projection operators (9) for the spins j and ( j − 1): n n ...n

r n ...n

ηnr11 (Θ ( j) )r11r22...r j j = (Θ ( j) )r11 r22...r j j =

( j + D − 4)(2 j + D − 3) ( j−1) n 2 ...n j (Θ )r2 ...r j . j (2 j + D − 5) (13)

In other words the trace of the matrix Θ ( j) over the pair of indices is proportional to the matrix Θ ( j−1) . Using formula (13), we can calculate the complete trace of the Behrends–Fronsdal projector Θ ( j) in the case of D-dimensional space-time (D ≥ 3): r r ...r

(Θ ( j) )r11 r22 ...r jj =

(D − 4 + j)! (2 j + D − 3) . j! (D − 3)!

(14)

This trace is equal to the dimension of the subspace, which is extracted from the space of vector-tensor wave functions f n 1 ...n j (k) by the projector Θ ( j) . In other words, the trace (14) is equal to the number N of independent components of symmetric vectortensor wave functions f (n 1 ...n j ) (k) that satisfy conditions k n 1 f (n 1 ...n j ) (k) = 0 ,

η n 1 n 2 f (n 1 n 2 ...n j ) (k) = 0 .

On the space of these functions an irreducible massive representation of the Ddimensional rotation group is realized. Remark 2 From relation (2) the useful identity [10] immediately follows:

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A. P. Isaev and M. A. Podoinitsyn

( j)

j

j

A=0

A=0

[2] [2]  j   j  ( j) ( j) (x, x) = Θ (1) (x, x) a A = (k)−2 j (k)2 (x)2 − (k n xn )2 aA ,

(15) ( j) where the coefficients a A are defined in (5). ( j) The sum of the coefficients a A in the right-hand side of (15) can be calculated explicitly by using relations (5) and (15). Indeed, we put xr = yr in (5), take into account (10) and apply the operator ∂x r ∂xr to both sides of equality (15). After that, comparing the results obtained in both sides of (15), we deduce the recurrence equation: ( j + D − 4) ( j−1) ( j) S , SD = (2 j + D − 5) D [ 2j ] ( j) ( j) where S D = A=0 a A . Solving this equation with the initial condition S D(1) = ( j) a0 = 1, we find: ( j)

SD =

( j + D − 4)! , (D − 3)! (2 j + D − 5)(2 j + D − 7) . . . (D − 1)

j >1.

(16)

Remark 3 The following formula which relates the polarizations of the integer and half-integer spin is valid: ) = (er(m) 1 ...r j−1/2 A



 (−) + ( j−m) er(m+1/2) , 1 ...r j−1/2 e A 2j m = − j, − j + 1, . . . , j , ( j+m) (m−1/2) er1 ...r j−1/2 e(+) A 2j

(17)

where index A = 1, 2, 3, 4 labels the components of the Dirac bispinors (17); (−) e(+) A , e A describe polarizations for particles with the spin j = 1/2. n ...n j−1/2  B for a We define the Behrends-Fronsdal projection operator (Θ ( j) )r11...r j−1/2 A half-integer spin j as the sum over m of products of the corresponding polarizations: j  ( j) n 1 ...n j−1/2  B (−1) j−1/2  (m) (Θ )r1 ...r j−1/2 A = (er1 ...r j−1/2 ) A (e(m) n 1 ...n j−1/2 ) B . 2 m=− j

(18)

The following statement holds (see [1]). Proposition 5 For arbitrary space-time dimension D > 2 and any half-integer spin j the projection operator from (18), satisfies the conditions (1)–(4) of the Proposition 3, and the additional spinor condition n ...n

n ...n

j−1/2 j−1/2 (Θ ( j) )r11...r j−1/2 · γn 1 = 0 = γ r1 · (Θ ( j) )r11...r j−1/2 ,

(19)

and the following formula holds: n ...n

n n ...n

j−1/2 j−1/2 ) AB = c( j) (Θ (1/2) ) AG (γ r )GC (γn )CB (Θ ( j+ 2 ) )r r11...r j−1/2 , ((Θ ( j) )r11...r j−1/2 1

(20)

Behrends–Fronsdal Spin Projection Operator in Space-Time with Arbitrary Dimension

147

where Θ ( j+ 2 ) – operator from (23), for the integer spin ( j + 21 ), factor c( j) = j+1/2 1 and (Θ (1/2) ) = 2m (γ n kn + m I ) (here operator I is 2[D/2] ⊗ 2[D/2] unit (2 j+D−2) matrix and [a] denotes the integer part of a), matrices γ n (n = 0, 1, . . . , D − 1) represents generators of the Clifford algebra in dimensions D. 1

4 Conclusion We hope that the formalism considered in this paper for describing massive particles of arbitrary spin will be useful in the construction of scattering amplitudes of massive particles in a similar way to the construction of spinor-helicity scattering amplitudes for massless particles [11, 12]. Some steps in this direction have already been done in papers [13–15] where the analogous formalism and its special generalization were used. Acknowledgements This work was supported by Russian Science Foundation, grant 14-11-00598.

References 1. A.P. Isaev and M.A. Podoinitsyn, Two-spinor description of massive particles and relativistic spin projection operators, Nucl. Phys. B, 929 (2018) 452–484; arXiv:1712.00833 [hep-th]. 2. E.P. Wigner, On unitary representations of the inhomogeneous Lorentz group, Annals of Mathematics, 40 (1) (1939) 149–204; V. Bargmann, E.P. Wigner, Group theoretical discussion of relativistic wave equations, Proceedings of the National Academy of Sciences of the USA, 34(5) (1948) 211. 3. Yu.V. Novozhilov, Introduction to Elementary Particle Theory, Volume 78 in International Series in Natural Philosophy, Pergamon Press, Oxford (1975). 4. N.N. Bogoliubov, A.A. Logunov, A.I. Oksak and I. Todorov, General principles of quantum field theory (Vol. 10), Springer Science & Business Media (2012). 5. S. Weinberg, Feynman Rules for Any Spin, Phys. Rev. 133 (1964) B1318; Feynman Rules for Any Spin. 2. Massless Particles, Phys. Rev. 134 (1964) B882. 6. P.A.M. Dirac, Relativistic wave equations, Proc. Roy. Soc. A, 155 (1936) 447. 7. M. Fierz, ber den drehimpuls von teilichen mit ruhemasse null und beliebigem spin. 8. M. Fierz and W. Pauli, On Relativistic Wave Equations for Particles of Arbitrary Spin in an Electromagnetic Field, Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, Vol. 173, No. 953 (1939) 211–232. 9. C. Fronsdal, On the theory of higher spin fields, Il Nuovo Cimento (1955-1965) 9 (1958) 416–443. 10. R.E. Behrends, C. Fronsdal, Fermy decay for higher spin particles, Physical Review 106.2 (1957) 345. 11. E. Witten, Perturbative gauge theory as a string theory in twistor space, Comm. Math. Phys 252, 1-3 (2004) 189–258. 12. H. Elvang and Y. Huang, Scattering amplitudes in gauge theory and gravity, Cambridge University Press, 2015. 13. E. Conde and A. Marzolla, Lorentz constraints on massive three-point amplitudes, Journal of High Energy Physics 09 (2016) 041; arXiv:1601.08113 [hep-th].

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14. E. Conde, E. Joung and K. Mkrtchyan, Spinor-Helicity Three-Point Amplitudes from Local Cubic Interactions, Journal of High Energy Physics 08 (2016) 040; arXiv:1605.07402 [hepth]. 15. A. Marzolla, The 4D on-shell 3-point amplitude in spinor-helicity formalism and BCFW recursion relations, in Proceedings of 12th Modave Summer School in Mathematical Physics (11–17 Sep 2016, Modave, Belgium), (2017) 002; arXiv:1705.09678 [hep-th].

Wakimoto Realization of the Quantum  Affine Superalgebra Uq ( sl(M|N)) Takeo Kojima

 Abstract A bosonization of the quantum affine superalgebra Uq (sl(M|N )) is presented for an arbitrary level k ∈ C. The Wakimoto realization is given by using ξ − η  )) are presented for system. The screening operators that commute with Uq (sl(M|N  the level k = −M + N . New bosonization of the affine superalgebra sl(M|N ) is obtained in the limit q → 1.

1 Introduction Bosonization is a powerful method to study representation theory and its application to mathematical physics [1]. Wakimoto realization is the bosonization that provides a bridge between representation theory of affine algebras and the geometry of the semi-infinite flag manifold. The Wakimoto realizations have been constructed for the  ), osp(2|2)(2) , affine Lie algebra g = (AD E)(r ) (r = 1, 2), (BC F G)(1) and sl(M|N (1) D(2, 1, a) [2–12]. They have been used to construct correlation functions of WZW models, in the study of Drinfeld–Sokolov reduction and W -algebras. It’s nontrivial to give quantum deformation of Wakimoto realization as the same as quantum Drinfeld– Sokolov reduction and quantum W -algebras. The quantum Wakimoto realizations  )) and Uq (sl(2|1))  [13–17]. In this paper have been constructed only for Uq (sl(N we study a higher-rank generalization of the previous works for the quantum affine  We give a bosonization of the quantum affine superalgebra superalgebra Uq (sl(2|1)).  Uq (sl(M|N )) for an arbitrary level k ∈ C, and give the Wakimoto realization using  )) for ξ − η system. We give the screening operators that commute with Uq (sl(M|N the level k = −M + N . Taking the limit q → 1, we obtain new bosonization of the  affine superalgebra sl(M|N ). This paper is a shorter review of the papers [18–21].

T. Kojima (B) Faculty of Engineering, Yamagata University, Jonan 4-3-16, Yonezawa 992-8510, Japan e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2018 V. Dobrev (ed.), Quantum Theory and Symmetries with Lie Theory and Its Applications in Physics Volume 1, Springer Proceedings in Mathematics & Statistics 263, https://doi.org/10.1007/978-981-13-2715-5_8

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 2 Quantum Affine Superalgebra Uq ( sl(M|N)) In this section we recall the definition of the quantum affine superalgebra  )) for M, N = 1, 2, 3, . . .. Throughout this paper, q ∈ C is assumed to be Uq (sl(M|N n −q −n 0 < |q| < 1. For any integer n, define [n]q = qq−q −1 . We set νi = +1 (1 ≤ i ≤ M), νi = −1 (M + 1 ≤ i ≤ M + N ) and ν0 = −1. The Cartan matrix (Ai, j )0≤i, j≤M+N −1  of the affine Lie superalgebra sl(M|N ) is given by Ai, j = (νi + νi+1 )δi, j − νi δi, j+1 − νi+1 δi+1, j .  )) [22] is the associative algeThe quantum affine superalgebra Uq (sl(M|N ±,i bra over C with the generators X m (i = 1, 2, . . . , M + N − 1, m ∈ Z), Hni (i = 1, 2, . . . , M + N − 1, n ∈ Z=0 ), H i (i = 1, 2, . . . , M + N − 1), and c. The Z2 grading of the generators is given by p(X m±,M ) ≡ 1 (mod 2) for m ∈ Z and zero otherwise. The defining relations of the generators are given as follows: c : central element,

[Ai, j m]q [cm]q δm+n,0 , m i ±, j ±, j [H , X (z)] = ±Ai, j X (z), [Ai, j m]q ∓ c |m| m ±, j q 2 z X (z), [Hmi , X ±, j (z)] = ± m (z 1 − q ±Ai, j z 2 )X ±,i (z 1 )X ±, j (z 2 ) = (q ±A j,i z 1 − z 2 )X ±, j (z 2 )X ±,i (z 1 ), for |Ai, j | = 0, [H i , Hmj ] = 0, [Hmi , Hnj ] =

[X ±,i (z 1 ), X ±, j (z 2 )] = 0 for |Ai, j | = 0,  δi, j c δ(q c z 2 /z 1 )Ψ+i (q 2 z 2 ) − [X +,i (z 1 ), X −, j (z 2 )] = −1 (q − q )z 1 z 2  c − δ(q −c z 2 /z 1 )Ψ−i (q − 2 z 2 ) , [X ±,i (z 1 ), [X ±,i (z 2 ), X ±, j (z)]q −1 ]q + (z 1 ↔ z 2 ) = 0 for |Ai, j | = 1, i = M, [X ±,M (z 1 ), [X ±,M+1 (w1 ), [X ±,M (z 2 ), X ±,M−1 (w2 )]q −1 ]q ] + (z 1 ↔ z 2 ) = 0, where we use [X, Y ]a = X Y − (−1) p(X ) p(Y ) aY X,  for homogeneous elements X, Y ∈ Umq (sl(M|N )). For simplicity we write [X, Y ] = [X, Y ]1 . Here we set δ(z) = m∈Z z and the generating functions

 Wakimoto Realization of the Quantum Affine Superalgebra Uq (sl(M|N ))

X ±, j (z) =



X m±, j z −m−1 ,

m∈Z c Ψ±i (q ± 2 z)

=q

151

 ±h i

exp ±(q − q

−1

)



 i H±m z ∓m

.

m>0

The multiplication rule for the tensor product is Z2 -graded and is defined for  )) by (X 1 ⊗ Y1 )(X 2 ⊗ Y2 ) = homogeneous elements X 1 , X 2 , Y1 , Y2 ∈ Uq (sl(M|N (−1) p(Y1 ) p(X 2 ) (X 1 X 2 ⊗ Y1 Y2 ), which extends to inhomogeneous elements through linearity. Let α¯ i , Λ¯ i (1 ≤ i ≤ M + N − 1) be the classical simple roots, the classical fundamental weights, respectively. Let (·|·) be the symmetric bilinear form satisfying (α¯ i |α¯ j ) = Ai, j and (Λ¯ i |α¯ j ) = δi, j for 1 ≤ i, j ≤ M + N − 1. Let us introduce the affine weight Λ0 and the null root δ satisfying (Λ0 |Λ0 ) = (δ|δ) = 0, (Λ0 |δ) = 1, and (Λ0 |α¯ i ) = (Λ0 |Λ¯ i ) = 0 for 1 ≤ i ≤ M + N − 1. The other affine weights and the affine roots are given by Λi = Λ¯ i + Λ0 , αi = α¯ i for 1 ≤ i ≤ M + N − 1, and  M+N −1  αi . Let V (λ) be the highest-weight module over Uq (sl(M|N )) α0 = δ − i=1 generated by the highest weight vector |λ = 0 such that Hmi |λ = X m±,i |λ = 0 (m > 0), X 0+,i |λ = 0, H i |λ = li |λ , where the classical part of the highest weight is λ¯ =

 M+N −1 i=1

li Λ¯ i .

 3 Bosonization of Uq ( sl(M|N))  In this section we give a bosonization of Uq (sl(M|N )) for an arbitrary level k ∈ C.

3.1 Boson i, j

We introduce bosons ami (m ∈ Z, 1 ≤ i ≤ M + N − 1), bm (m ∈ Z, 1 ≤ i < j ≤ i, j M + N ), cm (m ∈ Z, 1 ≤ i < j ≤ M + N ), and zero mode operators Q ia (1 ≤ i ≤ i, j i, j M + N − 1), Q b (1 ≤ i < j ≤ M + N ), Q c (1 ≤ i < j ≤ M + N ). Their commutation relations are 1 [(k + g)m]q [Ai, j m]q δm+n,0 , [a0i , Q aj ] = (k + g)Ai, j , m 1   i, j i , j  [bmi, j , bni , j ] = −νi ν j [m]q2 δi,i  δ j, j  δm+n,0 , [b0 , Q b ] = −νi ν j δi,i  δ j, j  , m

[ami , anj ] =

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T. Kojima

1     i, j [cmi, j , cni , j ] = νi ν j [m]q2 δi,i  δ j, j  δm+n,0 , [c0 , Q ic , j ] = νi ν j δi,i  δ j, j  , m √ i, j i , j  [Q b , Q b ] = π −1 (νi ν j = νi  ν j  = −1). The remaining commutators vanish. Here g = M − N stands for the dual Coxeter i, j number. We define free boson fields b± (z), bi, j (z) as follows: b± (z) = ±(q − q −1 ) i, j



b±m z ∓m ± b0 logq, i, j

i, j

m>0

 bmi, j i, j i, j z −m + Q b + b0 logz. bi, j (z) = − [m] q m=0 i (z), ci, j (z) are defined in the same way. We define free boson Free boson fields a± i, j i, j fields (ΔεL b± )(z), (ΔεR b± )(z) (ε = ±, 0) as follows:

i. j (ΔεL b± )(z)

i. j (ΔεR b± )(z)

=

b± (q ε z) − b± (z) (ε = ±), i+1, j i, j b± (z) + b± (z) (ε = 0), i+1, j

=

i, j

b± (q ε z) − b± (z) (ε = ±), i, j+1 i, j b± (z) + b± (z) (ε = 0). i, j+1

i, j

We define free boson fields with parameters L 1 , . . . , L r , M1 , . . . , Mr , α as follows:

=− +

 L1 L2 Lr i ... a (z; α) = M1 M2 Mr  [L 1 m]q [L 2 m]q . . . [L r m]q

ami −α|m| −m q z + [M1 m]q [M2 m]q . . . [Mr m]q [m]q m=0 L1 L2 . . . Lr (Q i + a0i logz). M1 M2 . . . Mr a

Normal ordering rules are defined as follows: :

  bmi, j bni , j

i, j

:=:

i , j 

: Qb Qb

  bni , j bmi, j

i , j 

:=: Q b

:= i, j

i, j i  , j 

bm bn (m < 0), i  , j  i, j bn bm (m > 0), i, j

i , j 

Q b := Q b Q b i, j

i, j

(i > i  or i = i  , j > j  ).

Normal ordering rules of ami , cm and Q c are defined in the same way.

 Wakimoto Realization of the Quantum Affine Superalgebra Uq (sl(M|N ))

153

3.2 Bosonization We define bosonic operators Ψ±i (z) (1 ≤ i ≤ M + N − 1) as follows: Ψ±i (q ± 2 z) =: ea± (q k

×e−

i

±( k2 +2M+1−l) ∓ i,l z) l=M+1 (Δ L b± )(q

×e

g i a± (q ± 2

×

: , (1 ≤ i ≤ M − 1),

(1)

=

(2)

 M−1 0 l,M ±( k +l)  M+N k z)− l=1 (Δ R b± )(q 2 z)+ l=M+2 (Δ0L b±M,l )(q ±( 2 +2M+1−l) z)

g a±M (q ± 2

k Ψ±i (q ± 2 z)

=: e

i M k l,i i,l ±( k2 +l) z)+ l=1 (Δ∓R b± )(q ±( 2 +l) z)− l=i+1 (Δ∓ z) L b± )(q

 M+N

k Ψ±M (q ± 2 z)

=: e

g ±2

:,

= M i k k l,i l,i z)− l=1 (Δ±R b± )(q ±( 2 +l−1) z)− l=M+1 (Δ±R b± )(q ±( 2 +2M−l) z)

 M+N

k

±( 2 +2M−l) ± i,l z) l=i+1 (Δ L b± )(q

×

: , (M + 1 ≤ i ≤ M + N − 1).

(3)

We define bosonic operators X ±,i (z) (1 ≤ i ≤ M + N − 1) as follows: X +,i (z) =

i 

ci, j (E + (z) − E i,−j (z)) , (1 ≤ i ≤ M − 1), (q − q −1 )z i, j

(4)

c M, j E M, j (z),

(5)

j=1

X +,M (z) =

M  j=1

X +,i (z) =

M 

ci, j E i, j (z) +

j=1

i 

ci, j (E i,+j (z) − E i,−j (z)), −1 )z (q − q j=M+1

(M + 1 ≤ i ≤ M + N − 1), X −,i (z) =

i−1  j=1

(q

di,1 j (F 1,− (z) − q −1 )z i, j

(6)

− Fi,1,+ j (z)) +

2 di,i

2,+ (F 2,− (z) − Fi,i (z)) + (q − q −1 )z i,i M M+N   di,3 j 3,− 3,+ (F (z) − F (z)) + di,3 j Fi,3 j (z), + i, j i, j −1 (q − q )z j=i+2 j=M+1

+

(1 ≤ i ≤ M − 1), X −,M (z) =

M−1  j=1

+

(q

1 d M, j (F 1,− (z) − q −1 )z M, j

2 d M,M

(q − q −1 )z

(7) 1,+ − FM, j (z)) +

2,− 2,+ (FM,M (z) − FM,M (z)) +

154

T. Kojima

+ X −,i (z) =

M+N 

3 d M, j

(q − q −1 )z j=M+2

M 

di,1 j Fi,1 j (z) +

j=1

3,− 3,+ (FM, j (z) − FM, j (z)),

i−1 

di,1 j

(q − q −1 )z j=M+1

(8)

1,+ (Fi,1,− j (z) − Fi, j (z)) +

2 di,i

2,+ (z)) + (F 2,− (z) − Fi,i (q − q −1 )z i,i M+N  di,3 j 3,+ (Fi,3,− + j (z) − Fi, j (z)), −1 )z (q − q j=i+2

+

(M + 1 ≤ i ≤ M + N − 1).

(9)

We set E i,±j (z) as follows: j,i+1

E i,±j (z) = : e(b+c) (q z)+b± (q z)−(b+c) (1 ≤ j < i ≤ M − 1), j,i

i,i+1

± (z) = : eb± E i,i

j−1

j−1

j,i+1

 j−1 l,i (q j−1±1 z)+ l=1 (Δ−R b+ )(q l z)

i−1 − l,i l (q i−1 z)−(b+c)i,i+1 (q i−1±1 z)+ l=1 (Δ R b+ )(q z)

:,

(1 ≤ j < i ≤ M − 1), ± (z) E i,i

=:e × e−

i,i+1 2M+1−i −b± (q z)−(b+c)i,i+1 (q 2M+1∓1−i z)

M

×

i−1 l,i + l,i l−1 z)− l=M+1 (Δ+R b+ )(q 2M−l z) l=1 (Δ R b+ )(q

:,

(M + 1 ≤ i ≤ M + N − 1), E i,±j (z)

=:e × e−

j,i+1

(b+c) j,i (q 2M+1− j z)−b±

(q 2M+1− j z)−(b+c) j,i+1 (q 2M+1∓1− j z)

M

 j−1 l,i + l,i l−1 z)− l=M+1 (Δ+R b+ )(q 2M−l z) l=1 (Δ R b+ )(q

×

:,

(M + 1 ≤ j < i ≤ M + N − 1). We set E i, j (z) as follows: (q E M, j (z) = : e(b+c) (q z)+b (1 ≤ j ≤ M − 1), j,M

E M,M (z) = : eb

M,M+1

j−1

j,M+1

j−1

 j−1 l,M z)− l=1 (Δ0R b+ )(q l z)

 M−1 0 l,M l (q M−1 z)− l=1 (Δ R b+ )(q z)

:,

:,

(1 ≤ j ≤ M − 1),

E i, j (z) = : e

 j−1 j,i l,i b+ (q j−1 z)−b j,i (q j z)+b j,i+1 (q j−1 z)− l=1 (Δ+R b+ )(q l−1 z)

(M + 1 ≤ i ≤ M + N − 1, 1 ≤ j ≤ M). 1 We set Fi,1,± j (z), Fi, j (z) as follows:

:,

:,

 Wakimoto Realization of the Quantum Affine Superalgebra Uq (sl(M|N )) a− (q Fi,1,± j (z) =: e i

×e

k+g − 2

155

z)+(b+c) j,i+1 (q −k− j z)−b± (q −k− j z)−(b+c) j,i (q −k− j∓1 z) j,i

×

i

M  M+N i,l i,l + l,i −k−l −k−l −k−2M−1+l z)− l=i+1 (Δ+ z)− l=M+1 (Δ+ z) l= j+1 (Δ R b− )(q L b− )(q L b− )(q

:

(1 ≤ j < i ≤ M − 1), 1,± FM, j (z) = −

k+g

j,M

−k− j

−k− j∓1

j,M+1

−k− j

−k− j+1

z)−b− (q z)−b (q z) =: ea− (q 2 z)−b± (q z)−(b+c) (q  M−1  M+N 0 l,M 0 M,l −k−l −k−2M−1+l − l= (Δ b )(q z)+ (Δ b )(q z) j+1 l=M+2 R − L − ×e :, (1 ≤ j ≤ M − 1), 1 Fi, j (z) = M

j,M+1

M − l,i −k−l+1 (q −k− j z)−b j,i+1 (q −k− j+1 z)+b j,i (q −k− j z)− l= z) j+1 (Δ R b− )(q i  M+N − i,l −k−2M+l − l,i −k−2M+l − l=M+1 (Δ R b− )(q z)+ l=i+1 (Δ L b− )(q z)

=: ea− (q i

×e

j,M

k+g − 2

j,i+1

z)−b−

:,

(M + 1 ≤ i ≤ M + N − 1, 1 ≤ j ≤ M), Fi,1,± j (z)

=

k+g i a− (q − 2

−k−2M+ j

j,i

−k−2M+ j

z)+(b+c) (q z)+b± (q z)−(b+c) =: e i  M+N − i,l −k−2M+l − l,i −k−2M+l z)+ l=i+1 (Δ L b− )(q z) ×e− l= j+1 (Δ R b− )(q :, j,i+1

j,i

(q −k−2M±1+ j z)

×

(M + 1 ≤ j < i ≤ M + N − 1). 2,± (z) as follows: We set Fi,i 2,± Fi,i (z) =: ea± (q i

×e−

i,i+1 ±(k+i+1) z)+b± (q z)+(b+c)i,i+1 (q ±(k+i) z)

M

 M+N i,l ∓ i,l ±(k+l) ±(k+2M+1−l) z)− l=M+1 (Δ∓ z) l=i+2 (Δ L b± )(q L b± )(q

2,± FM,M (z)

=: e

2,± (z) =: e Fi,i

×e

k+g ± 2

a±M (q ±

k+g 2

k+g i a± (q ± 2

 M+N

×

:, (1 ≤ i ≤ M − 1),

 M+N z)−b M,M+1 (q ±(k+M−1) z)+ l=M+2 (Δ0L b±M,l )(q ±(k+2M+1−l) z)

i,i+1 ±(k+2M−1−i) z)−b± (q z)+(b+c)i,i+1 (q ±(k+2M−i) z)

± i,l ±(k+2M−l) z) l=i+2 (Δ L b± )(q

:,

×

:, (M + 1 ≤ i ≤ M + N − 1).

3 We set Fi,3,± j (z), Fi, j (z) as follows: a+ (q Fi,3,± j (z) =: e i

×e

−

k+g 2

i+1, j

z)+(b+c)i, j (q k+ j−1 z)+b±

M

 M+N i,l − i,l k+l k+2M+1−l z)− l=M+1 (Δ− z) l= j (Δ L b+ )(q L b+ )(q k+g

i+1, j

i i, j k+2M− j z)−b+ Fi,3 j (z) =: ea+ (q 2 z)−b (q  M+N − i,l k+2M+1−l − l= z) j+1 (Δ L b+ )(q

×e 3,± FM, j (z) = =: ea+ (q

k+g 2

×

:, (1 ≤ i < j ≤ M − 1),

(q k+2M− j z)+bi+1, j (q k+2M+1− j z)

×

:, (1 ≤ i ≤ M − 1, M + 1 ≤ j ≤ M + N ), M+1, j

z)−b M, j (q k+2M− j z)−b± (q k+2M+1− j z)−(b+c) M+1, j (q k+2M+1∓1− j z)  M+N M+1, j 0 M,l k+2M+1−l b+ (q k+2M+1− j z)+ l= z) j+1 (Δ L b+ )(q M

×e

(q k+ j−1 z)−(b+c)i+1, j (q k−1±1+ j z)

×

:, (M + 2 ≤ j ≤ M + N ),

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T. Kojima

Fi,3,± j (z) = =: ea+ (q i

×e

 M+N

k+g 2

i+1, j

z)+(b+c)i, j (q k+2M+1− j z)−b±

+ i,l k+2M−l z) l= j+1 (Δ L b+ )(q

(q k+2M+1− j z)−(b+c)i+1, j (q k+2M+1∓1− j z)

×

:, (M + 1 ≤ i < j − 1 ≤ M + N − 1).

2 , di,3 j ∈ C satisfy the following conditions. The coefficients ci, j ∈ C and di,1 j , di,i

di,1 j

2 di,i

⎧ ⎪ ⎪ ⎨

1 (1 ≤ i ≤ M − 1, 1 ≤ j ≤ i − 1), 1 q j−1 (i = M, 1 ≤ j ≤ M − 1), = νi+1 × −k−1 (M + 1 ≤ i ≤ M + N − 1, 1 ≤ j ≤ M), q ci, j ⎪ ⎪ ⎩ 1 (M + 1 ≤ i ≤ M + N − 1, M + 1 ≤ j ≤ i − 1),  1 1 (1 ≤ i = M ≤ M + N − 1), = νi+1 × (i = M), q M−1 ci,i

di,3 j = νi+1 ⎧ ⎪ ⎪ ⎨

j−i−1 1  ci+l,i+1 × ci,i l=1 ci+l,i

1 (1 ≤ i ≤ M − 1, i + 2 ≤ j ≤ M), q k+3M+1−2 j (1 ≤ i ≤ M − 1, M + 1 ≤ j ≤ M + N ), × q (M−1)( j−M) (i = M, M + 2 ≤ j ≤ M + N ), ⎪ ⎪ ⎩ 1 (M + 1 ≤ i ≤ M + N − 1, i + 2 ≤ j ≤ M + N ). Theorem 1 The bosonic operators Ψ±i (z) defined in (1)–(3), and X ±,i (z) defined in (4)–(6) and (7)–(9) satisfy the defining relations of the quantum affine superalgebra  )) with the central element c = k ∈ C. Uq (sl(M|N

3.3 Wakimoto Realization In this section we introduce the ξ − η system and give the Wakimoto realization. We set the boson Fock space F( pa , pb , pc ) as follows. The vacuum state |0 = 0 is i, j i, j defined by ami |0 = bm |0 = cm |0 = 0 (m ≥ 0). Let | pa , pb , pc be | pa , pb , pc

⎛ M+N  −1 (A−1 )i, j i, j i, j pai Q ia − = exp ⎝ νi ν j pb Q b + k + g i, j=1 1≤i< j≤M+N ⎞  ⎟ pci, j Q i,c j ⎠ |0 , + 1≤i< j≤M+N νi ν j =+1

 Wakimoto Realization of the Quantum Affine Superalgebra Uq (sl(M|N ))

157

then | pa , pb , pc is the highest weight state of the boson Fock space F( pa , pb , pc ). i, j i, j The boson Fock space F( pa , pb , pc ) is generated by the bosons ami , bm , cm on the highest weight state | pa , pb , pc . We set the space F( pa ) by 

F( pa ) =

F( pa , pb , pc ).

i, j i, j pb =− pc ∈Z (νi ν j =+) i, j pb ∈Z (νi ν j =−)

Here we impose the restriction pb = − pc (νi ν j = +), because X m±,i change Q b + i, j  Q c . F( pa ) is Uq (sl(M|N ))-module. Let |λ = | pa , 0, 0 where pai = li (1 ≤ i ≤ M + N − 1). The generators H i , Hmi , X m±,i act on |λ as follows: i, j

i, j

i, j

Hmi |λ = X m±,i |λ = 0 (m > 0), X 0+,i |λ = 0, H i |λ = li |λ .  )). We have the level-k highest weight module V (λ) of Uq (sl(M|N V (λ) ⊂ F( pa ).  M+N −1 li Λ¯ i . Here the classical part of the highest weight is λ¯ = i=1 i, j i, j We introduce the ξ − η system We set bosonic operators ξm , ηm (νi ν j = +1, 1 ≤ i < j ≤ M + N ) as follows: η i, j (z) =



ηmi, j z −m−1 =: ec

i, j

(z)

:, ξ i, j (z) =

m∈Z



ξmi, j z −m =: e−c

i, j

(z)

:.

m∈Z

Fourier components  ηmi, j =

dz √ z m η i, j (z), ξmi, j = 2π −1



dz √ z m−1 ξ i, j (z) 2π −1 i, j

are well-defined on the module F( pa ). The Z2 -grading is given by p(ξm ) = i, j p(ηm ) = +1. We have direct sum decomposition. i, j i, j

i, j i, j

F( pa ) = η0 ξ0 F( pa ) ⊕ ξ0 η0 F( pa ), i, j

i, j i, j

i, j

i, j i, j

where Ker(η0 ) = η0 ξ0 F( pa ), Coker(η0 ) = ξ0 η0 F( pa ). We set η0 =

 1≤i< j≤M+N νi ν j =+1

i, j

η0 , ξ0 =

 1≤i< j≤M+N νi ν j =+1

i, j

ξ0 .

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T. Kojima

We introduce the subspace F( pa ) by F( pa ) = η0 ξ0 F( pa ). The operators η0 , ξ0 commute with X ±,i (z), Ψ±i (z) up to sign ±1. i, j

i, j

 Proposition 1 F( pa ) is Uq (sl(M|N ))-module.  We call F( pa ) the Wakimoto realization of Uq (sl(M|N )).

4 Screening Operator In this section we give the screening operators Q i (1 ≤ i ≤ M + N − 1) that com )) for the level c = k = −g. We define bosonic operators mute with Uq (sl(M|N Si (z) (1 ≤ i ≤ M + N − 1) that we call the screening currents as follows: Si (z) =

M 

M+N  ei, j − + (S (z) − S (z)) + ei, j Si, j (z) i, j i, j −1 (q − q )z j=i+1 j=M+1

(1 ≤ i ≤ M − 1), S M (z) =

M+N 

(10)

e M, j S M, j (z),

(11)

j=M+1

Si (z) =

M+N 

ei, j (Si,−j (z) − Si,+j (z)) −1 )z (q − q j=i+1 (M + 1 ≤ i ≤ M + N − 1).

(12)

We set Si,±j (z) as follows: Si,±j (z) = =: e−( k+g a )(z; 1

i

M

i, j k+g i+1, j (q M−N − j z)−b± (q M−N − j z)−(b+c)i, j (q M−N − j∓1 z) 2 )+(b+c)

 M+N i,l + i,l M−N −l −M−N +l−1 z)+ l=M+1 (Δ+ z) l= j+1 (Δ L b− )(q L b− )(q

×e Si,±j (z) =

k+g

−M−N + j

i, j

−M−N + j

:, (1 ≤ i < j ≤ M), −M−N + j±1

z)+b± (q z)−(b+c) (q =: e−( k+g a )(z; 2 )+(b+c) (q  M+N − i,l −M−N +l z) ×e− l= j+1 (ΔL b− )(q :, (M + 1 ≤ i < j ≤ M + N ). 1

i

i+1, j

×

i, j

z)

 Wakimoto Realization of the Quantum Affine Superalgebra Uq (sl(M|N ))

159

We set Si, j (z) as follows: Si, j (z) = k+g

−M−N + j

i+1, j

−M−N + j

−M−N + j+1

z)+b+ (q z)−b (q z) =: e−( k+g a )(z; 2 )+b (q ×  M+N + i,l −M−N −1+l (Δ L b− )(q z) l= j+1 ×e :, (1 ≤ i ≤ M − 1, M + 1 ≤ j ≤ M + N ), 1

i

i, j

S M, j (z) =: e−( k+g a )(z; 1

×e−

i

i+1, j

k+g M+1, j (q −M−N + j z)+b M, j (q −M−N + j z) 2 )+(b+c)

 M+N

0 M,l −M−N −1+l z) l= j+1 (Δ L b− )(q

×

:, (M + 1 ≤ j ≤ M + N ).

Here we set ei, j as follows: ⎧ 2 1/di,i (1 ≤ i ≤ M − 1), ⎨ 2 −N +1 /d M,M (i = M), ei,i+1 = −q ⎩ 2 (M + 1 ≤ i ≤ M + N − 1), −1/di,i ⎧ 1/di,3 j (1 ≤ i ≤ M − 1, i + 2 ≤ j ≤ M), ⎪ ⎪ ⎨ q k+1+M−N /d 3 (1 ≤ i ≤ M − 1, M + 1 ≤ j ≤ M + N ), i, j ei, j = 3 j−M−N −q /d (i = M, M + 2 ≤ j ≤ M + N ), ⎪ M, j ⎪ ⎩ −1/di,3 j (M + 1 ≤ i ≤ M + N − 1, i + 2 ≤ j ≤ M + N ). The Z2 -grading of the screening current is given by p(S M, j (z)) ≡ 1 (mod 2) for M + 1 ≤ j ≤ M + N and zero otherwise. The Jackson integral with parameters q ∈ C and s ∈ C∗ is defined by  s∞  f (w)dq w = s(1 − q) f (sq n )q n . 0

n∈Z

We define the screening operators Q i (1 ≤ i ≤ M + N − 1) as follows, when the Jackson integrals are convergent. 

s∞

Qi =

Si (w)dq 2(k+g) w.

(13)

0

Theorem 2 The screening operators Q i (1 ≤ i ≤ M + N − 1) defined in (10)–  )). (13) commute with the quantum affine superalgebra Uq (sl(M|N  ))] = 0. [Q i , Uq (sl(M|N

5 Limit q → 1  Bosonization of the affine superalgebra sl(M|N ) for an arbitrary level k have been  studied in [9–11]. We obtain new bosonization of the affine superalgebra sl(M|N ) in the limit q → 1.

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In what follows we set 

H i (z) =

Hmi z −m−1 (1 ≤ i ≤ M + N − 1).

m∈Z

We set the parameters ci, j = 1 in (4)–(6), (7)–(9), (10)–(12) for simplicity. In the limit i, j (z), γi, j (z) q → 1 we introduce operators αi (z) (1 ≤ i ≤ M + N − 1), βi, j (z), β † (1 ≤ i < j ≤ M + N , νi ν j = +), and ψi, j (z), ψi, j (z) (1 ≤ i < j ≤ M + N , νi ν j = −) as follows:   i, j αi (z) = ∂z a i (z) , γi, j (z) =: e(b+c) (z) :,  i, j    i, j i, j (z) =: ∂z e−bi, j (z) e−ci, j (z) :, βi, j (z) =: ∂z e−c (z) e−b (z) :, β ψi, j (z) =: eb

i, j

(z)

:, ψi,† j (z) =: e−b

i, j

(z)

:.

They satisfy the following relations. (k + g)Ai, j + ..., (z − w)2 δi,i  δ j, j  δi,i  δ j, j  + . . . , γi, j (z)βi  , j  (w) = − + ..., βi, j (z)γi  , j  (w) = z−w z−w     i, j (z)γi  , j  (w) = − δi,i δ j, j + . . . , γi, j (z)β i  , j  (w) = δi,i δ j, j + . . . , β z−w z−w  δ j, j   δ j, j  δ δ i,i i,i + . . . , ψi,† j (z)ψi  , j  (w) = + .... ψi, j (z)ψi† , j  (w) = z−w z−w

αi (z)α j (w) =

i In the limit q → 1 the operators a± (z), b± (z), (ΔL b± )(z) and (ΔR b± )(z) disappear. We obtain the following: i, j

H i (z) = αi (z) +

i 

i, j

i, j

j,i (z)γ j,i (z) − β j,i+1 (z)γ j,i+1 (z)) : : (β

j=1

+

M 

i+1, j (z)γi+1, j (z) − β i, j (z)γi, j (z)) : : (β

j=i+1

+

M+N 

† † : ((∂z ψi+1, j )(z)ψi+1, j (z) − (∂z ψi, j )(z)ψi, j (z)) :,

j=M+1

(1 ≤ i ≤ M − 1),

 Wakimoto Realization of the Quantum Affine Superalgebra Uq (sl(M|N )) M−1 

H M (z) = α M (z) +

j,M (z)γ j,M (z)) : : ((∂z ψ j,M+1 )(z)ψ †j,M+1 (z) + β

j=1

−

M+N 

M+1, j (z)γ M+1, j (z) + (∂z ψ M, j )(z)ψ † (z)) :, : (β M, j

j=M+2

H i (z) = αi (z) +

M 

: ((∂z ψ j,i+1 )(z)ψ †j,i+1 (z) − (∂z ψ j,i )(z)ψ †j,i (z)) :,

j=1

+

i 

j,i+1 (z)γ j,i+1 (z) − β j,i (z)γ j,i (z)) : : (β

j=M+1

+

M+N 

i, j (z)γi, j (z) − β i+1, j (z)γi+1, j (z)) :, : (β

j=i+1

(M + 1 ≤ i ≤ M + N − 1). X +,i (z) =

i 

: β j,i+1 (z)γ j,i (z) :, (1 ≤ i ≤ M − 1),

j=1

X +,M (z) =

M 

: γ j,M (z)ψ j,M+1 (z) :,

j=1

X +,i (z) =

M 

: ψ j,i+1 (z)ψ †j,i (z) : −

j=1

i 

: β j,i+1 (z)γ j,i (z) :,

j=M+1

(M + 1 ≤ i ≤ M + N − 1). X −,i (z) = − : αi (z)γi,i+1 (z) : −κi : ∂z γi,i+1 (z) : +

i−1 

: β j,i (z)γ j,i+1 (z) : −

j=1

−

M 

: βi+1, j (z)γi, j (z) : −

j=i+2

M+N 

: ψi+1, j (z)ψi,† j (z) :

j=M+1

+

M 

i, j (z)γi, j (z) − β i+1, j (z)γi+1, j (z))γi,i+1 (z) : : (β

j=i+1

+

M+N 

† : ((∂z ψi, j )(z)ψi,† j (z) − (∂z ψi+1, j )(z)ψi+1, j (z))γi,i+1 (z) :,

j=M+1

(1 ≤ i ≤ M − 1), X

−,M

(z) =: α M (z)ψ †M,M+1 (z) : +κ M : ∂z ψ †M,M+1 (z) :

161

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T. Kojima

−

M−1  j=1

−

M+N 

: β j,M (z)ψ †j,M+1 (z) : −

: β M+1, j (z)ψ †M, j (z) :

j=M+2

M+N 

† † M+1, j (z)γ † : (β M+1, j (z) + (∂z ψ M, j )(z)ψ M, j (z))ψ M,M+1 (z) :,

j=M+2

X −

−,i

(z) =: αi (z)γi,i+1 (z) : +κi : ∂z γi,i+1 (z) :

M 

: ψ j,i (z)ψ †j,i+1 (z) : +

j=1

−

i−1 

: β j,i (z)γ j,i+1 (z) : −

j=M+1

M+N 

: βi+1, j (z)γi, j (z) :

j=i+2

+

M+N 

i, j (z)γi, j (z) − β i+1, j (z)γi+1, j (z))γi, j (z) :, : (β

j=i+1

(M + 1 ≤ i ≤ M + N − 1). Here we have set the coefficients κi by κi =

⎧ ⎨

k +i (1 ≤ i ≤ M − 1) k+ M −1 (i = M) . ⎩ k + 2M − i (M + 1 ≤ i ≤ M + N − 1)

In what follows we assume k = −g. In the limit q → 1 we have the following. Si (z) =

M 

: s˜i (z)βi, j (z)γi+1, j (z) : +

j=i+1

M+N 

† : s˜i (z)ψi, j (z)ψi+1, j (z) :

j=M+1

(1 ≤ i ≤ M − 1), M+N  S M (z) = : s˜M (z)γ M+1, j (z)ψ M, j (z) :, j=M+1

Si (z) =

M+N 

: s˜i (z)βi, j (z)γi+1, j (z) :

(M + 1 ≤ i ≤ M + N − 1).

j=i+1

Here we have set the boson operator s˜i (z) =: e

  1 − k+g a i (z;0)

Our bosonization is different from [9–11].

:.

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163

Acknowledgements This work is supported by the Grant-in-Aid for Scientific Research C (26400105) from Japan Society for Promotion of Science. The author would like to thank Professor Michio Jimbo and Professor Vladimir Dobrev for giving advice. The author would like to thank Professor Zengo Tsuboi, Professor Pascal Baseilhac, Professor Kouichi Takemura and Professor Kenji Iohara for discussion. The author is thankful for the kind hospitality by the organizing committee of the 10-th International Symposium “Quantum Theory and Symmetries” (QTS10) and 12-th International Workshop “Lie Theory and Its Applications in Physics” (LT12).

References 1. E.V. Frenkel, Adv. Math. 195 (2005) 297–404. 2. M. Wakimoto, Commun. Math. Phys. 104 (1986) 605–609. 3. B.L. Feigin and E.V. Frenkel, Physics and Mathematics of Strings (World Scientific, Singapore 1980) 271–316. 4. B.L. Feigin and E.V. Frenkel, Commun. Math. Phys. 128 (1990) 161–189. 5. K. Ito and S. Komata, Mod. Phys. Lett. A6 (1991) 581–589. 6. J. de Boer and L. Fehér, Commun. Math. Phys. 189 (1997) 759–793. 7. M. Szczesny, Math. Res. Lett. 9 (2002) 433–448. 8. L. Fehér and B.G. Pusztai, Nucl. Phys. B674 (2003) 509–532. 9. X.-M. Ding, M.D. Gould and Y.-Z. Zhang, Phys. Lett. 318 (2003) 354–363. 10. W.-L. Yang, Y.Z. Zhang and X. Liu, J. Math. Phys. 48 (2007) 053514 (pp. 11). 11. K. Iohara and Y. Koga, Math. Proc. Camb. Phil. Soc. 132 (2002) 419–433. 12. A. Shafiekhani and W.-S. Chung, Mod. Phys. Lett. A13 (1998) 47–57. 13. A. Matsuo, Commun. Math. Phys. 160 (1994) 33–48. 14. J. Shiraishi, Phys. Lett. A171 (1992) 243–248. 15. H. Awata, S. Odake and J. Shiraishi, Commun. Math. Phys. 162 (1994) 61–83. 16. H. Awata, S. Odake and J. Shiraishi, Lett. Math. Phys. 42 (1997) 271–279. 17. Y.-Z. Zhang and M.D. Gould, J. Math. Phys. 41 (2000) 5291–5577. 18. T. Kojima, J. Math. Phys. 53 (2012) 013515 (pp. 15). 19. T. Kojima, Springer Proceedings 111 (2013) 263–276. 20. T. Kojima, J. Math. Phys.53 (2012) 083503 (pp. 30). 21. T. Kojima, Commun. Math. Phys. 355 (2017) 603–644. 22. H. Yamane, Publ. Res. Inst. Math. Sci. 35 (1999) 321–390.

On Superdimensions of Some Infinite-Dimensional Irreducible Representations of osp(m|n) N. I. Stoilova, J. Thierry-Mieg and J. Van der Jeugt

Abstract In a recent paper characters and superdimension formulas were investigated for the class of representations with Dynkin labels [0, . . . , 0, p] of the Lie superalgebra osp(m|n). Such representations are infinite-dimensional, and of relevance in supergravity theories provided their superdimension is finite. We have shown that the superdimension of such representations coincides with the dimension of a so(m − n) representation. In the present contribution, we investigate how this osp(m|n) ∼ so(m − n) correspondence can be extended to the class of osp(2m|2n) representations with Dynkin labels [0, . . . , 0, q, p].

1 Introduction Chiral spinors and self dual tensors of the Lie superalgebra osp(m|n) play a prominent role in some models of supergravity theory [1, 13]. As representations, these spinors and self dual tensors are characterized by Dynkin labels [0, . . . , 0, p], where p = 1 for the chiral spinor and p = 2 for the self dual tensor. It will be interesting to consider the class of representations with arbitrary positive integer p. Although all Dynkin labels are nonnegative integers, the corresponding representations are infinite-dimensional (as they do not satisfy the extra condition in Kac’s list of finite-

N. I. Stoilova Institute for Nuclear Research and Nuclear Energy, Boul. Tsarigradsko Chaussee 72, 1784 Sofia, Bulgaria e-mail: [email protected] J. Thierry-Mieg NCBI, National Library of Medicine, National Institute of Health, 8600 Rockville Pike, Bethesda, MD 20894, USA e-mail: [email protected] J. Van der Jeugt (B) Department of Applied Mathematics, Computer Science and Statistics, Ghent University, Krijgslaan 281-S9, 9000 Gent, Belgium e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2018 V. Dobrev (ed.), Quantum Theory and Symmetries with Lie Theory and Its Applications in Physics Volume 1, Springer Proceedings in Mathematics & Statistics 263, https://doi.org/10.1007/978-981-13-2715-5_9

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dimensional irreducible representations [6, 7]). In [16], we showed that the superdimension of these representations coincides with the dimension of the corresponding so(m − n) representation. Herein, the algebra should be interpreted differently when m − n is negative: as sp(n − m) when n − m is even, and as osp(1|n − m − 1) when n − m is odd. The results of [16] rely on the knowledge of the character for such osp(m|n) representations. In particular, the expansion or formulation of this character in terms of supersymmetric Schur functions turned out to be the crucial ingredient in order to obtain the osp(m|n) ∼ so(m − n) correspondence. In the present paper, we shift our attention to the class of osp(2m|2n) representations with Dynkin labels [0, . . . , 0, q, p]. In the distinguished Dynkin diagram of osp(2m|2n), all nodes have zero labels and only the two nodes of the fork have a non-negative integer label. Such representations are again infinite-dimensional. Our idea to deal with these representations is as follows: we will first investigate the finite-dimensional so(2k) representations of type [0, . . . , 0, q, p], conjecture that the osp(m|n) ∼ so(m − n) correspondence still holds, and as such obtain interesting new characters of osp(2m|2n) representations.

2 Preliminaries and Definitions The character formulas used in this paper are expressed in terms of symmetric or supersymmetric Schur functions, which are labelled by partitions. So it will be useful to recall some notation for this. The standard reference is [12]. A partition λ = (λ1 , λ2 , . . . , λn ) of weight |λ| and length (λ) ≤ n is a sequence of non-negative integers satisfying the condition λ1 ≥ λ2 ≥ · · · ≥ λn , such that their sum is |λ|, and λi > 0 if and only if i ≤ (λ). It is common to represent (and sometimes identify) a partition by its Young diagram. For example, the Young diagram of λ = (6, 4, 4, 2) is given by the first figure in (1). × × ×

× (1)

The conjugate partition λ corresponds to the Young diagram of λ reflected about the main diagonal. For the above example, λ = (4, 4, 3, 3, 1, 1). If λ, μ are two partitions, one writes λ ⊃ μ if the diagram of λ contains that of μ. The difference λ − μ is called a skew diagram [12]. For example, if μ = (4, 4, 3, 1), then the boxes of the skew diagram λ − μ are crossed in the second picture of (1). A skew diagram is a horizontal strip if it has at most one box in each column. The number of boxes of the horizontal strip is its length. The above example is a horizontal strip of length 4.

On Superdimensions of Some Infinite-Dimensional …

167

Partitions are used to label symmetric and supersymmetric functions. When dealing with characters of Lie algebras or Lie superalgebras, the Schur functions [12] or S-functions are the most useful basis. In terms of a set of n independent variables x = (x1 , x2 , . . . , xn ), the Schur function sλ (x) (with λ a partition) is a symmetric polynomial that can be defined by means of determinants [12]. When dealing with two sets of variables x = (x1 , . . . , xm ) and y = (y1 , . . . , yn ), one can define the socalled supersymmetric Schur function sλ (x|y) [2, 9]. Here, sλ (x|y) is zero whenever λm+1 > n. Following this, it is common to denote by Hm,n the set of all partitions with λm+1 ≤ n, i.e. the partitions (with their Young diagram) inside the (m, n)-hook. For characters of simple Lie algebras, ordinary Schur functions play a prominent role. Characters of finite-dimensional irreducible representation (irreps) of gl(n) or sl(n) are directly given by a Schur function, and characters of irreps of other simple Lie algebras can be expanded in Schur functions [10]. An irrep of gl(n) is characterized by a partition λ with (λ) ≤ n. In terms of the standard basis 1 , . . . , n n of the weight space of gl(n), the highest weight of this representation is i=1 λi i , λ and the representation space will be denoted by Vgl(n) . Its character is given by λ = sλ (x), where xi = ei . char Vgl(n) For Lie superalgebras, this role is played by the supersymmetric Schur functions, at least for certain classes of representations. For a partition λ ∈ Hm,n , the corresponding covariant representation of the Lie superalgebra gl(m|n) will be denoted λ . In terms of the standard basis 1 , . . . , m , δ1 , . . . , δn of the weight space by Vgl(m|n)  m λi i + nj=1 max(λj − of gl(m|n), the highest weight of this representation is i=1 m, 0)δ j , and the main result of [2] is λ = sλ (x|y), char Vgl(m|n)

(2)

where xi = ei and y j = eδ j .

3 Dimension, Superdimension and t-Dimension As is well known, the character of a representation gives all information on the weight structure of the representation. Sometimes, it is useful to consider certain specializations of characters, because of specific information that is needed, or because of elegant formulas that hold for certain specializations. Let V be a highest weight representation of a simple Lie algebra or Lie superalgebra, with highest weight Λ and character char V . A well known specialization of the character of V is the socalled q-dimension [8, Chap. 10]. The q-dimension of V is nothing else than the specialization dimq (V ) = F(e−Λ char V ),

where

F(e−αi ) = q,

(3)

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and the αi ’s are the simple roots of the Lie (super)algebra. So this corresponds to the principal gradation of the Lie (super)algebra, and one counts the dimension of the “levels” of the representation space starting from the top level (corresponding to the highest weight) according to this gradation. Here, we will be dealing with a different specialization, referred to as the tdimension. For a (simple) Lie algebra, of which the simple roots are commonly expressed in terms of the standard basis 1 , . . . , n , one defines dimt (V ) = F0 (e−Λ char V ), where F0 (e−i ) = t.

(4)

For a Lie superalgebra of type sl, gl or osp, of which the simple roots are commonly expressed in terms of the standard basis 1 , . . . , m , δ1 , . . . , δn , we define the tdimension and the t-superdimension: dimt (V ) = F0 (e−Λ char V ), where F0 (e−i ) = t and F0 (e−δi ) = t; −Λ

sdimt (V ) = F1 (e

−i

char V ), where F1 (e

) = t and F1 (e

−δi

) = −t.

(5) (6)

Intuitively, the t-dimension again counts the dimension of levels of a representation starting from the top level, but according to a gradation different from the principal one. Similarly, the t-superdimension counts the dimension of the same levels, but with alternating signs. For finite-dimensional representations, putting t = 1 in dimt (V ) gives the dimension of V , and putting t = 1 in sdimt (V ) gives its so-called superdimension (i.e. dim V0¯ − dim V1¯ , when V = V0¯ ⊕ V1¯ is written as the direct sum of its even and odd subspace). Let us consider some examples. For the orthogonal Lie algebra so(2n + 1), with simple roots 1 − 2 , . . . , n−1 − n , n , we will focus on representations V with Dynkin labels [0, . . . , 0, p], for which the highest weight is ( 2p , . . . , 2p ) in the basis. For this representation, the character is [3, 14] char[0, . . . , 0, p]so(2n+1) = (x1 · · · xn )− p/2



sλ (x).

(7)

λ1 ≤ p, (λ)≤n

So the sum is over all partitions λ such that the Young diagram of λ fits inside the n × p rectangle, of width p and height n. Specializing this character according to F0 , one finds:  λ dim Vgl(n) t |λ| . (8) dimt [0, . . . , 0, p]so(2n+1) = λ1 ≤ p, (λ)≤n

When the character is expressed in terms of Schur functions, as in (7), it yields in fact the branching of the representation according to so(2n + 1) ⊃ gl(n). When the character is specialized as in (8), it is a polynomial in t (or, in case of an infinitedimensional representation, a formal power series in t) such that the coefficient of t k counts the dimension “at level k” according to the Z-gradation induced by the gl(n)

On Superdimensions of Some Infinite-Dimensional …

169

subalgebra of so(2n + 1). For example, for so(7), one has dimt [0, 0, 1]so(7) = 1 + 3t + 3t 2 + t 3 , dimt [0, 0, 2]so(7) = 1 + 3t + 9t 2 + 9t 3 + 9t 4 + 3t 5 + t 6 , dimt [0, 0, 2]so(7) = 1 + 3t + 9t 2 + 19t 3 + 24t 4 + 24t 5 + 19t 6 + 9t 7 + 3t 8 + t 9 . The q-dimension, on the other hand, is a character specialization with a very different nature. It is a character specialization closely related to Weyl’s dimension formula, for which an explicit formula exists [8, (10.10.1)]. For the representations considered in this example, this yields (replacing q by q 2 in order to avoid half-integer powers): (1 − q p+5 )(1 − q p+4 )(1 − q p+3 )2 (1 − q p+2 )(1 − q p+1 ) . (1 − q 5 )(1 − q 4 )(1 − q 3 )2 (1 − q 2 )(1 − q) (9) So the q-dimension is a character specialization for the principal gradation of a Lie (super)algebra, leading to classical formulas. The t-dimension is a character specialization related to the gradation coming from the gl(n) subalgebra (or gl(m|n) subalgebra), thus typically related to the branching g ⊃ gl(n) or g ⊃ gl(m|n). As a second example, let us consider the t-dimension for a class of representations of g = osp(1|2n). The notation is as follows [5–7]: δ j are the basis elements for the weight space of osp(1|2n); the odd roots are given by ±δ j ( j = 1, . . . , n), the even roots by δi − δ j (i = j) and ±(δi + δ j ), and the simple roots by δ1 − δ2 , δ2 − δ3 , . . . , δn−1 − δn , δn . The subalgebra gl(n) is spanned by the root vectors corresponding to δi − δ j . The embedding gl(n) ⊂ osp(1|2n) leads to a Z-gradation of osp(1|2n) [16]. We consider here a class of infinitedimensional representations of osp(1|2n), namely the ones with highest weight given by (− 2p , − 2p , . . . , − 2p ) in the δ-basis. For this representation, the Dynkin labels are [0, 0, . . . , 0, − p]. The structure and character of this representation have been determined in [11]. Using the notation xi = e−δi , one has: dimq 2 [0, 0, p]so(7) =



char[0, 0, . . . , 0, − p]osp(1|2n) = (x1 · · · xn ) p/2

sλ (x).

(10)

λ, (λ)≤ p

This is an infinite sum over all partitions of length at most p. Since sλ (x) = 0 if (λ) > n, the sum is actually over all partitions satisfying (λ) ≤ min(n, p). Applying the above specialization F0 , one finds: dimt [0, 0, . . . , 0, − p]osp(1|2n) =



λ dim Vgl(n) t |λ| .

(11)

λ, (λ)≤min(n, p)

This infinite sum can be rewritten in an alternative form, see [16]. Some examples for osp(1|6) are given by:

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1 − 3t 2 + 3t 4 − t 6 1 = (1 − t)3 (1 − t 2 )3 (1 − t)3 = 1 + 3t + 6t 2 + 10t 3 + 15t 4 + · · ·

dimt [0, 0, −1]osp(1|6) =

1 − t3 = 1 + 3t + 9t 2 + 18t 3 + 36t 4 + · · · (1 − t)3 (1 − t 2 )3 1 = = 1 + 3t + 9t 2 + 19t 3 + 39t 4 + · · · 3 (1 − t) (1 − t 2 )3

dimt [0, 0, −2]osp(1|6) = dimt [0, 0, −3]osp(1|6)

4 Superdimensions for osp(2m + 1|2n) and osp(2m|2n) In this section we mainly summarize some of the main results of [16]. For the Lie superalgebra B(m, n) = osp(2m + 1|2n), we work with the distinguished set of simple roots in the -δ-basis [5, 6] δ1 − δ2 , . . . , δn−1 − δn , δn − 1 , 1 − 2 , . . . , m−1 − m , m .

(12)

The relevant gl(m|n) subalgebra is spanned by the root vectors corresponding to δi − δ j , i −  j , ±(i − δ j ), and g = osp(2m + 1|2n) admits a Z-gradation g = g−2 ⊕ g−1 ⊕ g0 ⊕ g+1 ⊕ g+2 with g0 = gl(m|n). The class of representations to be considered are the irreducible highest weight representations with highest weight given by ( 2p , . . . , 2p ; − 2p , . . . , − 2p ) in the -δbasis. This representation has Dynkin labels [0, 0, . . . , 0, p]. Using xi = e−i , yi = e−δi , the following character formula holds [15, 16]: char[0, . . . , 0, p]osp(2m+1|2n) = (y1 · · · yn /x1 · · · xm ) p/2



sλ (x|y).

(13)

λ, λ1 ≤ p

Here the sum is over all partitions λ inside the (m, n)-hook (otherwise sλ (x|y) is zero anyway) with λ1 ≤ p, or equivalently (λ ) ≤ p. Applying F1 , one should (apart from the factor in front of the above sum) specify xi = t and y j = −t in the above character, and so one finds  sλ (t, . . . , t| − t, . . . , −t) sdimt [0, . . . , 0, p]osp(2m+1|2n) = λ, λ1 ≤ p

=



sλ (1, . . . , 1| − 1, . . . , −1) t |λ|

λ, λ1 ≤ p

=



λ sdim Vgl(m|n) t |λ| .

(14)

λ, λ1 ≤ p

But superdimension formulas for covariant representations of gl(m|n) are well known [9], and reduce to dimensions of gl(k) irreps:

On Superdimensions of Some Infinite-Dimensional … λ λ sdim Vgl(n+k|n) = dim Vgl(k) ,

171 

λ λ sdim Vgl(m|m+k) = (−1)|λ| dim Vgl(k) .

(15)

λ In particular, when m = n, sdim Vgl(n|n) = 0 unless λ is the zero partition (0). λ = 0; when λ1 > k then Note that (15) implies: when (λ) > k then sdim Vgl(n+k|n) λ = 0. Applying this to (14) leads to three cases. sdim Vgl(m|m+k)

Case 1: m = n, osp(2n + 1|2n). All superdimensions of covariant representations of gl(n|n) are zero, except when λ = (0). Hence: sdimt [0, . . . , 0, p]osp(2n+1|2n) = 1.

(16)

Case 2: m = n + k, osp(2n + 2k + 1|2n). This is the most interesting case. The infinite sum in (14) reduces to a finite sum: sdimt [0, . . . , 0, p]osp(2m+1|2n) =



λ dim Vgl(k) t |λ|

λ, λ1 ≤ p

=



λ dim Vgl(k) t |λ| .

(17)

λ, λ1 ≤ p, (λ)≤k

This coincides with example (8). Hence we can write sdimt [0, 0, . . . , 0, p]osp(2n+2k+1|2n) = dimt [0, . . . , 0, p]so(2k+1) .

(18)

Case 3: n = m + k, osp(2m + 1|2m + 2k). One finds: sdimt [0, . . . , 0, p]osp(2m+1|2n) =





λ (−1)|λ| dim Vgl(k) t |λ|

λ, λ1 ≤ p, λ1 ≤k



μ

dim Vgl(k) (−t)|μ| .

(19)

sdimt [0, 0, . . . , 0, p]osp(2m+1|2m+2k) = dim−t [0, . . . , 0, − p]osp(1|2k) .

(20)

=

μ, (μ)≤min( p,k)

The right hand side is the same expression as (11), so

So in all three cases, the superdimension for osp(2m + 1|2n) simplifies and reduces to a dimension of so(2m + 1 − 2n) or osp(1|2n − 2m). Let us now turn to the Lie superalgebra D(m, n) = osp(2m|2n). The distinguished set of simple roots in the -δ-basis is δ1 − δ2 , . . . , δn−1 − δn , δn − 1 , 1 − 2 , . . . , m−2 − m−1 , m−1 − m , m−1 + m .

(21) It will be helpful to see this superalgebra in the subalgebra chain osp(2m + 1|2n) ⊃ osp(2m|2n) ⊃ gl(m|n).

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For the irreducible highest weight representation of osp(2m|2n) with highest weight given by ( 2p , . . . , 2p ; − 2p , . . . , − 2p ), with Dynkin labels [0, 0, . . . , 0, p], the character was determined in [16]: 

char[0, . . . , 0, p]osp(2m|2n) = (y1 · · · yn /x1 · · · xm ) p/2

sλ (x|y).

(22)

λ∈B, λ1 ≤ p

Herein, B denotes the set of partitions for which each part appears twice (including the zero partition). Thus, one finds sdimt [0, . . . , 0, p]osp(2m|2n) =



λ sdim Vgl(m|n) t |λ| .

(23)

λ∈B, λ1 ≤ p

This expression allows once again to deduce superdimension formulas in three cases: m = n, m > n and m < n, see [16]. Let us give here the formula for m > n, i.e. m = n + k, or osp(2n + 2k|2n). From (23) one has: sdimt [0, . . . , 0, p]osp(2m|2n) =



λ dim Vgl(k) t |λ|

λ∈B, λ1 ≤ p

=



λ dim Vgl(k) t |λ| .

(24)

λ∈B, λ1 ≤ p, (λ)≤k

This is to be compared to known characters of so(2k) irreps [16], where a distinction should be made between k even and k odd. For k even, one has  sλ (x). (25) char[0, . . . , 0, p]so(2k) = (x1 · · · xk )− p/2 λ∈B; λ1 ≤ p, (λ)≤k

For k odd, char[0, . . . , p, 0]so(2k) = (x1 · · · xk )− p/2



sλ (x).

(26)

λ∈B: λ1 ≤ p, (λ)≤k−1

Comparing with (24), yields: 

dimt [0, . . . , 0, 0, p]so(2k) for k even, dimt [0, . . . , 0, p, 0]so(2k) for k odd. (27) Here, the convention for the order of the simple roots of so(2k) is 1 − 2 , . . . , k−1 − k , k−1 + k . sdimt [0, 0, . . . , 0, p]osp(2n+2k|2n) =

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5 Characters of “fork” Representations for so(2m) and osp(2m|2n) The characters of so(2k) and so(2k + 1), used in the previous section, should be seen in the context of the subalgebra chain so(2k + 1) ⊃ so(2k) ⊃ gl(k). In (7) we obtained  sλ (x). (28) char[0, . . . , 0, p]so(2k+1) = (x1 · · · xk )− p/2 λ1 ≤ p, (λ)≤k

Essentially, this is the branching so(2k + 1) ⊃ gl(k), since Schur functions are characters of gl(k) irreps. Considering the representation with respect to the branching so(2k + 1) ⊃ so(2k), one finds (using Weyl’s character formula): char[0, . . . , 0, p]so(2k+1) =

p 

char[0, . . . , r, p − r ]so(2k) .

(29)

r =0

The so(2k) representations with Dynkin labels [0, . . . , r, p − r ] are sometimes referred to as fork representations, since the only non-zero Dynkin labels appear at the fork nodes of the diagram, see Fig. 1. The so(2k) characters – in terms of Schur functions – that were used in the identification of the right hand side of (24) were for the representations [0, . . . , 0, p] and [0, . . . , 0, p, 0]. Given (29), the question is how to write the character of the other so(2k) fork representations [0, . . . , r, p − r ] as a sum of Schur functions? Or in other words, what is the branching so(2k) ⊃ gl(k) for these representations? The answer is given by: Theorem 1 For k even, one has char[0, . . . , 0, r, p − r ]so(2k) = (x1 · · · xk )− p/2



sλ (x).

(30)

λ1 ≤ p, (λ)≤k; λ∈Br

Herein, Br stands for the set of partitions of B to which a horizontal strip of length r is attached. (Recall that B is the set of partitions for which each part appears twice.) The first condition (λ1 ≤ p, (λ) ≤ k) means that (the Young diagram of) λ fits inside the k × p rectangle. Similarly, for k odd: Fig. 1 Dynkin diagram of the fork representation of so(2k)

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char[0, . . . , 0, r, p − r ]so(2k) = (x1 · · · xk )− p/2

sλ (x).

(31)

λ1 ≤ p, (λ)≤k; λ∈B p−r

The proof is technical and can be obtained using the branching rules for so(2k) ⊃ gl(k) described in [10]. Note that, in accordance with (29), the union of all partitions of Br in the k × p rectangle, for r = 0, 1, . . . , p, is equal to the set of all partitions in the rectangle. In order to illustrate the sets Br , let us give some examples for so(8). char[0, 0, 0, 1]so(8) = (x1 · · · x4 )−1/2 (1 + s(1,1) + s(1,1,1,1) ) char[0, 0, 1, 0]so(8) = (x1 · · · x4 )−1/2 (s(1) + s(1,1,1) ) char[0, 0, 0, 2]so(8) = (x1 · · · x4 )−1 (1 + s(1,1) + s(2,2) + s(1,1,1,1) + s(2,2,1,1) + s(2,2,2,2) ) char[0, 0, 1, 1]so(8) = (x1 · · · x4 )−1 (s(1) + s(2,1) + s(1,1,1) + s(2,2,1) + s(2,1,1,1) + s(2,2,2,1) ) char[0, 0, 2, 0]so(8) = (x1 · · · x4 )−1 (s(2) + s(2,1,1) + s(2,2,2) ) char[0, 0, 0, 3]so(8) = (x1 · · · x4 )−3/2 (1 + s(1,1) + s(2,2) + s(1,1,1,1) + s(3,3) + s(2,2,1,1) + s(3,3,1,1) + s(2,2,2,2) + s(3,3,2,2) + s(3,3,3,3) ) char[0, 0, 1, 2]so(8) = (x1 · · · x4 )−3/2 (s(1) + s(2,1) + s(1,1,1) + s(2,1,1,1) + s(2,2,1) + s(3,2) + s(2,2,2,1) + s(3,2,1,1) + s(3,3,1) + s(3,2,2,2) + s(3,3,2,1) + s(3,3,3,2) ) char[0, 0, 2, 1]so(8) = (x1 · · · x4 )−3/2 (s(2) + s(2,1,1) + s(3,1) + s(2,2,2) + s(3,1,1,1) + s(3,2,1) + s(3,2,2,1) + s(3,3,2) + s(3,3,3,1) ) char[0, 0, 3, 0]so(8) = (x1 · · · x4 )−3/2 (s(3) + s(3,1,1) + s(3,2,2) + s(3,3,3) ) From these examples, one can indeed see that for representations [0, 0, 0, p], only partitions appear for which each part is repeated twice (inside the 4 × p rectangle). The partitions appearing in, e.g., [0, 0, 2, 1] are obtained from those of [0, 0, 0, 3] by attaching a horizontal strip of length 2. Note that indeed the union of all partitions appearing in, e.g., [0, 0, 0, 3], [0, 0, 1, 2], [0, 0, 2, 1] and [0, 0, 3, 0] give indeed all partitions inside the 4 × 3 rectangle. But now we can extend the analogy that we observed between representations [0, . . . , 0, p] of osp(m|n) and the corresponding ones of so(m − n). For osp(2m + 1|2n), one should compare Eq. (13) with (8). For osp(2m|2n), one should compare (22) with (25). For all these cases, the character of the corresponding representation (expressed in terms of Schur functions) is the same, up to the extra condition (λ) ≤ k for so(2k). We conjecture that this correspondence also holds for the characters of fork representations of osp(2m|2n) (see Fig. 2), by dropping the condition (λ) ≤ k in (30).

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Fig. 2 Dynkin diagram of the fork representation of osp(2m|2n)

Conjecture 1 For |m − n| even, one has char[0, . . . , 0, r, p − r ]osp(2m|2n) = (y1 · · · yn /x1 · · · xm ) p/2



sλ (x/y).

λ1 ≤ p, λ∈Br

(32) So in this case we have an expansion as an infinite sum of supersymmetric Schur functions, labeled by partitions λ inside the (m, n)-hook, of width at most p, and belonging to Br . For |m − n| odd, the result is similar, with Br replaced by B p−r , following (31). Note that this conjecture also has some interesting consequences, and yields the equivalence of (29): char[0, . . . , 0, p]so(2m+1|2n) =

p 

char[0, . . . , r, p − r ]so(2m|2n) .

(33)

r =0

Indeed, the expansion of the left hand side is given by (13), and involves all partitions λ with λ1 ≤ p. The expansion of the terms in the right hand side is given by (32); each term involves the partitions of Br with λ1 ≤ p. Clearly, {λ | λ1 ≤ p} is the disjoint union of the sets {λ ∈ Br | λ1 ≤ p},

r = 0, 1, . . . , p.

(34)

Obviously, every element of (34) belongs to {λ | λ1 ≤ p}. The other way round, when λ is an arbitrary partition with λ1 ≤ p, one should make the following construction. For λ = (λ1 , λ2 , λ3 , λ4 , . . .), let μ1 = μ2 = λ2 , μ3 = μ4 = λ4 , etc.; thus μ ∈ B (all parts appear twice). And λ − μ is by construction a horizontal strip of length r = |λ| − |μ|, where r ≤ p since λ1 ≤ p. So λ belongs to a unique set of (34) for some r ∈ {0, 1, . . . , p}. Now (33) follows. To conclude, in the current paper we have first analyzed characters and superdimensions for representations of the form [0, . . . , 0, p] for osp(2m + 1|2n) and osp(2m|2n), and related them to characters and dimensions of so(2k + 1) and so(2k) (for k = m − n). Exploiting this correspondence, we conjecture that it also holds for fork representations of the form [0, . . . , 0, r, p − r ] for osp(2m|2n). For this purpose, we have deduced characters of the corresponding fork representations of so(2k). The formal proof of the conjecture might be difficult or technical. One way is to try and use characters of more general osp(m|n) tensors which were studied in [4]. Here, the character formulas correspond to alternating series of S-functions,

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which are not easy to handle. Another way is to make use of the explicit construction of the representation [0, . . . , 0, p]so(2m+1|2n) in [15]. This method is in principle straightforward, but might be difficult to perform because of the complicated matrix elements appearing for these representations. Acknowledgements NIS and JVdJ were supported by the Joint Research Project “Lie superalgebras - applications in quantum theory” in the framework of an international collaboration programme between the Research Foundation – Flanders (FWO) and the Bulgarian Academy of Sciences. NIS was partially supported by the Bulgarian National Science Fund, grant DN 18/1. This research (JTM) was supported in part by the Intramural Research Program of the NIH, U.S. National Library of Medicine.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.

L. Baulieu and J. Thierry-Mieg, Nucl. Phys. B 228 (1983) 259–284. A. Berele and A. Regev, Adv. Math. 64 (1987) 118–175. A.J. Bracken and H.S. Green, Nuovo Cim. 9 (1972) 349–365. S.J. Cheng and R.B. Zhang, Adv. Math. 182 (2004) 124–172. L. Frappat, A. Sciarrino and P. Sorba, Dictionary on Lie Algebras and Superalgebras (Academic Press, London, 2000). V.G. Kac, Adv. Math. 26 (1977) 8–96. V.G. Kac, Lect. Notes in Math. 626 (1978) 597–626. V.G. Kac, Infinite dimensional Lie algebras. 2nd edition (Cambridge University Press, Cambridge, 1985). R.C. King, Ars. Combin. 16A (1983) 269–287. R.C. King and B.G. Wybourne, J. Math. Phys. 41(2000) 5002–5019. S. Lievens, N.I. Stoilova and J. Van der Jeugt, Commun. Math. Phys. 281 (2008) 805–826. I.G. Macdonald, Symmetric Functions and Hall Polynomials. 2nd edition (Oxford University Press, Oxford, 1995). C.R. Preitschopf, T. Hurth, P. van Nieuwenhuizen and A. Waldron, Nucl. Phys. B 56B (1997) 310–317. N.I. Stoilova and J. Van der Jeugt, J. Phys. A 41 (2008) 075202. N.I. Stoilova and J. Van der Jeugt, J. Phys. A 48 (2015) 155202. N.I. Stoilova, J. Thierry-Mieg and J. Van der Jeugt, J. Phys. A 50 (2017) 155201.

Non-commutativity in Unified Theories and Gravity G. Manolakos and G. Zoupanos

Abstract First, we briefly review the Coset Space Dimensional Reduction scheme and the results of the best model so far. Then, we present the introduction of fuzzy coset spaces used as extra dimensions and perform a dimensional reduction. In turn, we describe a construction which mimics the results of a reduction, starting from a 4-dimensional theory and we present a successful example of a dynamical generation of fuzzy spheres. Finally, we propose a construction of the 3-d gravity as a gauge theory on specific non-commutative spaces.

1 Introduction During the last decades, the unification of all the fundamental interactions has attracted the interest of theoretical physicists. The aim of unification led to a number of approaches and among them those that elaborate the notion of extra dimensions are particularly appealing. A consistent framework employing the idea of extra dimensions is superstring theories [1] with the Heterotic String [2] (defined in ten dimensions) being the most promising, due to the possibility that in principle could lead to experimentally testable predictions. More specifically, the compactification of the 10-dimensional spacetime and the dimensional reduction of the E8 × E8 initial

G. Manolakos · G. Zoupanos (B) Physics Department, National Technical University, 15780 Zografou, Greece e-mail: [email protected] G. Manolakos e-mail: [email protected] G. Zoupanos Max-Planck Institut für Physik, Föhringer Ring 6, 80805 Munchen, Germany © Springer Nature Singapore Pte Ltd. 2018 V. Dobrev (ed.), Quantum Theory and Symmetries with Lie Theory and Its Applications in Physics Volume 1, Springer Proceedings in Mathematics & Statistics 263, https://doi.org/10.1007/978-981-13-2715-5_10

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gauge theory lead to phenomenologically interesting Grand Unified theories (GUTs), containing the SM gauge group. A few years before the development of the superstring theories, another important framework with similar aims was employed, namely the dimensional reduction of higher-dimensional gauge theories. Pioneers in this field were Forgacs-Manton and Scherk-Schwartz studying the Coset Space Dimensional Reduction (CSDR) [3–5] and the group manifold reduction [6], respectively. In both approaches, the higher-dimensional gauge fields are unifying the gauge and scalar fields, while the 4-dimensional theory contains the surviving components after the procedure of the dimensional reduction. Moreover, in the CSDR scheme, the inclusion of fermionic fields in the initial theory leads to Yukawa couplings in the 4-dimensional theory. Furthermore, upgrading the higher-dimensional gauge theory to N = 1 supersymmetric, i.e. grouping the gauge and fermionic fields of the theory into the same vector supermultiplet, is a way to unify further the fields of the initial theory, in certain dimensions. A very remarkable achievement of the CSDR scheme is the possibility of obtaining chiral theories in four dimensions [7, 8]. The above context of the CSDR adopted some very welcome suggestions coming from the superstring theories (specifically from the Heterotic String [2]), that is the dimensions of the space-time and the gauge group of the higher-dimensional supersymmetric theory. In addition, taking into account the fact that the superstring theories are consistent only in ten dimensions, the following important issues have to be addressed, (a) distinguish the extra dimensions from the four observable ones by considering an appropriate compactification of the metric and (b) determine the resulting 4-dimensional theory. In addition, a suitable choice of the compactification manifolds could result into N = 1 supersymmetric theories in four dimensions, aiming for a chance to lead to realistic GUTs. Requiring the preservation of N = 1 supersymmetry after the dimensional reduction, Calabi–Yau (CY) spaces serve as suitable compact, internal manifolds [9]. However, the emergence of the moduli stabilization problem, led to the study of flux compactification, in the context of which a wider class of internal spaces, called manifolds with SU (3)-structure, was suggested. In this class of manifolds, a non-vanishing, globally defined spinor is admitted. This spinor is covariantly constant with respect to a connection with torsion, versus the CY case, where the spinor is constant with respect to the Levi-Civita connection. Here, we consider the nearly-Kähler manifolds, that is an interesting class of SU (3)−structure manifolds [10–13]. The class of homogeneous nearly-Kähler manifolds in six dimensions consists of the non-symmetric coset spaces G 2 /SU (3), Sp(4)/(SU (2) × U (1))non−max , SU (3)/U (1) × U (1) and the group manifold SU (2) × SU (2) [13] (see also [10–12]). It is worth mentioning that 4-dimensional theories which are obtained after the dimensional reduction of a 10-dimensional N = 1 supersymmetric gauge theory over non-symmetric coset spaces, contain supersymmetry breaking terms [14, 15], contrary to CY spaces. Another very interesting framework which seems to be a natural arena for the description of physics at the Planck scale is the non-commutative geometry [16– 38]. Regularizing quantum field theories, or even better, building finite ones are the features that render this approach as a promising framework. On the other hand,

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the construction of quantum field theories on non-commutative spaces is a difficult task and, furthermore, problematic ultraviolet features have emerged [20] (see also [21, 22]). However, non-commutative geometry is an appropriate framework to accommodate particle models with non-commutative gauge theories [23] (see also [24–26]). It is remarkable that the two frameworks (superstring theories and noncommutative geometry) found contact, after the realization that, in M-theory and open String theory, the effective physics on D-branes can be described by a noncommutative gauge theory [27, 28], if a non-vanishing background antisymmetric field is present. Moreover, the type IIB superstring theory (and others related with type IIB with certain dualities) in its conjectured non-perturbative formulation as a matrix model [29], is a non-commutative theory. In the framework of non-commutative geometry, of particular importance is the contribution of Seiberg and Witten [28], which is a map between commutative and non-commutative gauge theories and has been the basis on which notable developments [30, 31] were achieved, including the construction of a non-commutative version of the SM [32]. Unfortunately, such extensions fail to solve the main problem of the SM, which is the presence of many free parameters. A very interesting development in the framework of the non-commutative geometry is the programme in which the extra dimensions of higher - dimensional theories are considered to be non-commutative (fuzzy) [33–38]. This programme overcomes the ultraviolet/infrared problematic behaviours of theories defined in noncommutative spaces. A very welcome feature of such theories is that they are renormalizable, versus all known higher-dimensional theories. This aspect of the theory was examined from the 4-dimensional point of view too, using spontaneous symmetry breakings which mimic the results of the dimensional reduction of a higherdimensional gauge theory with non-commutative (fuzzy) extra dimensions. In addition, another interesting feature is that in theories constructed in this programme, there is an option of choosing the initial higher-dimensional gauge theory to be abelian. Then, non-abelian gauge theories result in lower dimensions in the process of the dimensional reduction over fuzzy coset spaces. Finally, the important problem of chirality in this framework has been successfully attacked by applying an orbifold projection on an N = 4 SYM theory. After the orbifolding, the resulting theory is an N = 1 supersymmetric, chiral SU (3)3 . Another interesting aspect is the study of gravity as a gauge theory on noncommutative spaces. The first and strong motivation came from Witten’s work [39], that (classical) 3-d gravity with or without cosmological constant can be described as a (renormalizable) gauge theory of the isometry group of dS/AdS or Minkowski spacetime, respectively. Having already the know-how from previous works mentioned above, namely the construction of gauge theories on non-commutative spaces as extra dimensions, motivated us to study 3-d gravity as a gauge theory on noncommutative spaces. At first, one has to determine suitable manifolds and then gauge their isometry groups, resulting with the transformations of the gauge fields and the curvature tensors. Then, one should propose an action and eventually, end up with

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the equations of motion [40]. Our long-term goal is to obtain a 4-d theory of gravity, hopefully with improved ultraviolet properties.

2 Reduction of a D-dimensional Theory An obvious and naive way to dimensionally reduce a higher-dimensional gauge theory is to consider all fields of the theory to be independent of the extra coordinates (trivial reduction), meaning that the Lagrangian will be independent as well. In contrast, a much more elegant way is the consideration of non-trivial dependence, that is a symmetry transformation on the fields by an element that belongs to the isometry group, S, of the compact coset space, B = S/R, formed by the extra dimensions will be a gauge transformation (symmetric fields). Therefore, the axiomatic consideration of gauge invariance of the Lagrangian, renders it independent of the extra dimensions. The above method of getting rid of the extra dimensions consists the basic concept of the CSDR scheme [3–5].

2.1 CSDR of a D-dimensional Theory We consider the action of a D-dimensional YM theory of gauge group G, coupled to fermions defined on M D with metric g MN :  A=

  √ 1 i ¯ M d 4 xd d y −g − Tr(FMN FKΛ )g MK g N Λ + ψΓ DM ψ , 4 2

(1)

1 θMN Λ Σ N Λ the spin connection of M D 2 and FMN = ∂M AN − ∂N AM − [AM , AN ], where M , N , K, Λ = 1 . . . D and AM and ψ are D-dimensional symmetric fields. The fermions can be accommodated in any representation F of G, unless an additional symmetry, e.g. supersymmetry, is involved. Let ξAα , (A = 1, . . . , dimS and α = dimR + 1, . . . , dimS the curved index) be the Killing vectors generating the symmetries of S/R and WA , the gauge transformation associated with ξA . The following constraint equations for scalar φ, vector Aα and spinor ψ fields on S/R, derive from the definition of the symmetric fields: where DM = ∂M − θM − AM , with θM =

δA φ = ξAα ∂α φ = D(WA )φ, δA Aα =

β ξA ∂β Aα

β ∂α ξA Aβ

+ = ∂α WA − [WA , Aα ], 1 δA ψ = ξAα ∂α ψ − G Abc Σ bc ψ = D(WA )ψ , 2

(2) (3) (4)

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where WA depend only on internal coordinates y and D(WA ) is the gauge transformation in the corresponding representation, in which the fields are assigned. Solving the constraints (2)–(4), one is led to [3, 4] the unconstrained 4-dimensional fields, as well as to the 4-dimensional gauge symmetry. We proceed with the process of the constraints of the fields of the theory. Gauge field AM on MD splits into its components as (Aμ , Aα ) corresponding to M 4 and S/R, respectively. Solving the corresponding constraint, (3), we get informed as follows: The 4-dimensional gauge field, Aμ , does not depend on the coset space coordinates and the 4-dimensional gauge fields commute with the generators of the subgroup R ∈ G. This means that the remaining gauge symmetry, H , is the subgroup of G that commutes with R, that is the centralizer of R in G, i.e. H = CG (RG ). Aα (x, y) ≡ φα (x, y), transform as scalars in the 4-dimensional theory and φα (x, y) act as intertwining operators connecting induced representations of R acting on G and S/R. In order to find the representation of scalars in the 4-dimensional theory, one must decompose G according to the following embedding: G ⊃ RG × H ,

adjG = (adjR, 1) + (1, adjH ) +

and S under R: S ⊃ R,

adjS = adjR +



si .

 (ri , hi ) ,

(5)

(6)

We conclude that for every pair ri , si , where ri and si are identical irreducible representations of R, there exists a remaining scalar (Higgs) multiplet, transforming under the representation hi of H . The rest of the scalars vanish. As for the spinors [4, 7, 8, 41], the analysis of the corresponding constraint, (4), is similar. Solving the constraint, one finds that spinors in the 4-dimensional theory do not depend on the coset coordinates and act as intertwining operators connecting induced representations of R in SO(d ) and in G. To obtain the representation under H of the fermions in the 4-dimensional theory, one has to decompose the initial representation F of G under the RG × H : G ⊃ RG × H ,

F=

 (ri , hi ),

(7)

and the spinor of SO(d ) under R: SO(d ) ⊃ R ,

σd =



σj .

(8)

Concluding, for each pair ri and σi , with ri and σi being identical irreducible representations, there exists a multiplet, hi of spinor fields in the 4-dimensional theory. If one considers Dirac fermions in the higher-dimensional theory, it is impossible to result with chiral fermions in four dimensions. But, if one imposes the Weyl condition in the chiral representations of an even (in an odd higher-dimensional theory Weyl condition cannot be applied) higher-dimensional theory, eventually, one is led to a chiral 4-dimensional theory. The most interesting case is the D = 2n + 2 higher

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dimensional theory, in which fermions are in the adjoint representation and the Weyl condition leads to two sets of chiral fermions with the same quantum numbers under H of the 4-dimensional theory. Imposing the Majorana condition, the doubling of the fermionic spectrum is lifted. In the case of D = 4n + 2 -the case of our interest-, the two conditions are compatible.

2.2 The 4-dimensional Effective Action We proceed with determining the 4-dimensional effective action. The first thing to do is the compactification of the space M D to M 4 × S/R, with S/R a compact coset space. After the compactification, the metric of the M D will take the following form:  g MN =

η μν 0 0 −g ab

 ,

(9)

where η μν is the mostly negative metric of the 4-dimensional Minkowski spacetime and g ab is the metric of the coset space. Replacing (9) into the action, (1), and taking into account the constraints of the fields, we obtain:   1 t tμν 1 4 A = C d x − Fμν F + (Dμ φα )t (Dμ φα )t + V (φ) + (10) 4 2  i ¯ a i ¯ μ Dμ ψ − ψΓ Da ψ , + ψΓ 2 2 where Dμ = ∂μ − Aμ and Da = ∂a − θa − φa , with θa = 21 θabc Σ bc the connection of the space and C the volume of the space. The potential, V (φ), is given as follows: 1 V (φ) = − g ac g bd Tr(fabC φC − [φa , φb ])(fcdD φD − [φc , φd ]), 4

(11)

where, A = 1, . . . , dimS and f ’s are the structure constants of the Lie algebra of S. The constraints of the fields, (2)–(3), dictate that scalar fields, φa , have to satisfy the following equation: faiD φD − [φa , φi ] = 0 ,

(12)

where φi are the generators of RG . This means that some fields will be cut, while others will survive after the reduction scheme and will be identified as the genuine Higgs fields. The potential, V (φ), expressed in terms of the scalars that passed the filter of the constraints (the Higgs fields), is a quartic polynomial, invariant under the 4-dimensional gauge group, H . Then, one has to determine the vacuum (minimum of the potential) and find out the remaining gauge symmetry [42–44]. In general, this is a tough procedure. However, there is a specific case in which one may result with

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the remaining symmetry, after the spontaneous symmetry breaking of H , very easily, in case the following criterion is satisfied. Whenever S has an isomorphic image SG in G, then the 4-dimenisonal gauge group H breaks spontaneously to a subgroup K, where K is the centralizer of SG in the gauge group of the initial, higher-dimensional, theory, G [4, 42–44]. This is demonstrated in the following scheme: G ⊃ SG × K ∪ ∩ G ⊃ RG × H

(13)

The potential of the resulting 4-dimensional gauge theory is always of spontaneous symmetry breaking form, when the coset space is symmetric.1 It is rather unpleasant, that in this case, after the application of the reduction scheme, the fermions obtained are supermassive -as in the Kaluza-Klein theory-. Let us now demonstrate some results coming from the dimensional reduction of the N = 1, E8 SYM over the nearly-Kähler manifold SU (3)/U (1) × U (1). The 4dimensional gauge group is obtained by decomposing E8 under R = U (1) × U (1), as follows: (14) E8 ⊃ E6 × SU (3) ⊃ E6 × U (1)A × U (1)B . Satisfying the criterion mentioned above, the resulting 4-dimensional gauge group is: (15) H = CE8 (U (1)A × U (1)B ) = E6 × U (1)A × U (1)B . The decomposition of the adjoint representation of E8 , the 248, under U (1)A × U (1)B gives the surviving scalar and fermion fields. After the application of the CSDR rules, one obtains the resulting 4-dimensional theory, which is an N = 1, E6 GUT, with U (1)A , U (1)B global symmetries. The potential is determined after a lengthy calculation found in Ref. [15]. Apart from the F- and D-terms contributing to this potential, one can determine also scalar masses and trilinear scalar terms, identified with the scalar part of the soft supersymmetry breaking sector of the theory. In addition, the gaugino becomes massive, receiving a contribution from the torsion, unlike the rest soft supersymmetry breaking terms. It is worth-noting that the CSDR scheme leads straight to the soft supersymmetry breaking sector without any additional assumption. Further breaking of the E6 GUT is achieved by the Wilson flux mechanism. Details for the present case can be found in Ref. [12]. The theory derived is a softly broken N = 1, chiral SU (3)3 theory which can be further broken to an extension of the MSSM.

1A

c = 0. coset space is called symmetric when fab

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3 Fuzzy Spaces A particularly interesting framework, in which particle (and gravity) models can be built on, is non-commutative geometry. For now, we focus on the fuzzy spaces (noncommutative spaces defined as matrix approximations of continuous manifolds), which will be used as extra dimensions in a higher dimensional theory. In this section we give details about the definition of a specific fuzzy space, the fuzzy sphere and the differential calculus on it. Then, we briefly present how to do gauge theory on this matrix-approximated sphere, concluding all the necessary information for the applications of the next section.

3.1 The Fuzzy Sphere We will introduce the fuzzy sphere, SN2 [18], through a modification of the familiar, ordinary sphere S 2 , which is considered as a manifold embedded into the 3dimensional Euclidean space, R3 . This embedding allows the specification of the algebra of functions on S 2 through R3 , by imposing the constraint 3 

xa2 = R2 ,

(16)

a=1

where xa are the coordinates of R3 and R the radius of the sphere. The isometry group of S 2 is a global SO(3), generated by the three angular momentum operators, La = −i abc xb ∂c , due to the isomorphism SO(3)  SU (2). Writing the three generators, La , in terms of the spherical coordinates θ, φ, they are expressed as La = −iξaα ∂α , where the greek index, α, denotes the spherical coordinates and ξaα are the components of the Killing vector fields, which generate the isometries of the sphere.2 The operator defined as: 1 √ L2 = −R2 S 2 = −R2 √ ∂a (g ab g∂b ) , g

(17)

has the spherical harmonics, Ylm (θ, φ) as eigenfunctions. In order to calculate the eigenvalues of L2 , one has to act on Ylm (θ, φ): L2 Ylm = l(l + 1)Ylm ,

(18)

with l being a positive integer. The eigenfunctions Ylm (θ, φ) satisfy the orthogonality condition:

2 The

S 2 metric can be expressed in terms of the Killing vectors as g αβ =

1 α β ξ ξa . R2 a

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 † Yl m = δll δmm . sin θd θd φYlm

(19)

Ylm (θ, φ) form a complete and orthogonal set of functions, therefore any function on S 2 can be expanded on them: a(θ, φ) =

∞  l 

alm Ylm (θ, φ) ,

(20)

l=0 m=−l

where alm are complex coefficients. In an alternative way, spherical harmonics can be expressed in terms of the coordinates xa , as: Ylm (θ, φ) =



falm xa1 ...al , 1 ...al

(21)

a

where falm is an l-rank (traceless) symmetric tensor. 1 ...al Let us now modify the above, in order to obtain the fuzzy version of S 2 . Fuzzy sphere is a typical case of a non-commutative space, meaning that functions do not commute, contrary to the S 2 case, with l having an upper limit. Therefore, this truncation yields a finite dimensional (non-commutative) algebra, l 2 dimensional. Thus, it is natural to consider the truncated algebra as a matrix algebra and it is consistent to consider the fuzzy sphere as a matrix approximation of the S 2 . According to the above, N -dimensional matrices are expanded on a fuzzy sphere as: aˆ =

N −1 

l 

alm Yˆ lm ,

(22)

l=0 m=−l

where Yˆ lm are spherical harmonics of the fuzzy sphere, given by: Yˆ lm = R−l



falm Xˆ a1 · · · Xˆ al , 1 ...al

(23)

a

where:

2R ) λ(N Xˆ a = √ a , 2 N −1

(24)

) lm where λ(N a are the SU (2) generators in the N -dimensional representation and fa1 ...al is the same tensor, used in (21). The Yˆ lm also satisfy the orthonormality condition:

 † ˆ TrN Yˆ lm Yl m = δll δmm .

(25)

Moreover, there is a correspondence between the expansion of a function, (20), and that of a matrix, (22), on the ordinary and the fuzzy sphere, respectively:

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aˆ =

N −1 

l 

alm Yˆ lm

→

l=0 m=−l

a=

N −1 

l 

alm Ylm (θ, φ) .

(26)

l=0 m=−l

The above obviously maps matrices to functions. The introduction of the fuzzy sphere as a truncation of the algebra of functions on S 2 , suggests, as a natural choice (but not unique), the consideration of the same alm . The above is a 1 : 1 mapping given by ref [45]:  † a(θ, φ) = TrN (Yˆ lm aˆ )Ylm (θ, φ) , (27) lm

while the matrix trace is mapped to an integral over the sphere:  1 1 TrN → dΩ . N 4π

(28)

To sum up, the fuzzy sphere is a matrix approximation of S 2 . The price one has to pay for the truncation of the algebra of functions is the loss of commutativity, yielding the non-commutative algebra of matrices, Mat(N ; C). Therefore, the fuzzy sphere, SN2 , is the non-commutative manifold with Xˆ a being the coordinate functions. As given by (24), Xˆ a are N × N hermitian matrices produced by the generators of SU (2) in the N -dimensional representation. It is obvious that they have to obey both the condition: 3  (29) Xˆ a Xˆ a = R2 , a=1

which is the equivalent of (16) and the commutation relation: [Xˆ a , Xˆ b ] = iα abc Xˆ c , α = √

2R N2 − 1

.

(30)

It is equivalent to consider the description of the algebra on SN by the antihermitian matrices: Xˆ a , (31) Xa = iαR also satisfying a variation of (29), (30): 3  a=1

Xa Xa = −

1 , [Xa , Xb ] = Cabc Xc , α2

(32)

1 where Cabc = abc . R Let us proceed by giving a short description of the differential calculus on the fuzzy sphere, which is 3-dimensional and SU (2) covariant. The derivations of a function f , along Xa are:

Non-commutativity in Unified Theories and Gravity

ea (f ) = [Xa , f ] ,

187

(33)

and, consequently, the Lie derivative on f is: La f = [Xa , f ] ,

(34)

where La obeys both the Leibniz rule and the commutation relation of the SU (2) algebra: (35) [La , Lb ] = Cabc Lc . Working with differential forms, let θa be the 1-forms dual to the vector fields ea , namely ea , θb  = δab . Therefore, the action of the exterior derivative, d on a function f , gives: (36) df = [Xa , f ]θa , while the action of the Lie derivative on the 1-forms θb gives: La θb = Cabc θc .

(37)

The Lie derivative obeys the Leibniz rule, therefore action on any 1-form ω = ωa θa gives: a c θ , (38) Lb ω = Lb (ωa θa ) = [Xb , ωa ]θa − ωa Cbc where we have applied (34) and (37). Therefore, one obtains the result: c . (Lb ω)a = [Xb , ωa ] − ωc Cba

(39)

After the description of the differential geometry of the fuzzy sphere, one could move on to the study of the differential geometry of M 4 × SN2 , that is the product of Minkowski spacetime and fuzzy sphere with fuzziness level N − 1. For example, any 1-form A of M 4 × SN2 can be expressed in terms of M 4 and SN2 , that is: A = Aμ dxμ + Aa θa ,

(40)

where Aμ , Aa depend on xμ and Xa coordinates. In addition, instead of functions, one may consider spinors on the SN2 [33]. Moreover, there are studies of the differential geometry of various higher-dimensional fuzzy spaces, e.g. of the fuzzy CP M [33].

3.2 Gauge Theory on Fuzzy Sphere Let us consider a field φ(Xa ) on the SN2 , depending on the powers of Xa [46]. The infinitesimal transformation of φ(Xa ) is given by:

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δφ(X ) = λ(X )φ(X ) ,

(41)

where λ(X ) is the gauge parameter. If λ(X ) is an antihermitian function of Xa , the (41) is an infinitesimal (abelian) U (1) transformation, while if λ(X ) is valued in Lie(U (P)) (the algebra of P × P hermitian matrices), then (41) is the infinitesimal (non-abelian), U (P). Also, δXa = 0, that is a condition which ensures the invariance of the coordinates under a gauge transformation. Therefore, the left multiplication by a coordinate is not a covariant operation: δ(Xa φ) = Xa λ(X )φ ,

(42)

Xa λ(X )φ = λ(X )Xa φ .

(43)

and in general it holds that:

Inspired by the non-fuzzy gauge theory, one may proceed with the introduction of the covariant coordinates, φa , such that: δ(φa φ) = λφa φ ,

(44)

δ(φa ) = [λ, φa ] .

(45)

which holds if:

Usual (non-fuzzy) gauge theory also suggests the definition: φa ≡ Xa + Aa ,

(46)

with Aa being interpreted as the gauge potential of the non-commutative theory. Therefore, the covariant coordinates, φa , are the non-commutative analogue of the covariant derivative encountered in ordinary gauge theories. From (45), (46), one is led to the gauge transformation of Aa : δAa = −[Xa , λ] + [λ, Aa ] ,

(47)

a form that encourages the interpretation of Aa as a gauge field. Then, it is natural to define a field strength tensor, Fab , as: c c Ac = [φa , φb ] − Cab φc . Fab ≡ [Xa , Ab ] − [Xb , Aa ] + [Aa , Ab ] − Cab

(48)

It can be proven that the field strength tensor transforms covariantly: δFab = [λ, Fab ] .

(49)

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4 Dimensional Reduction of Higher-Dimensional Theory with Fuzzy Extra Dimensions In this section we present the ordinary (naive) dimensional reduction of a higherdimensional theory with fuzzy extra dimensions and the coset space dimensional reduction, adjusted to the non-commutative framework.

4.1 Ordinary Fuzzy Dimensional Reduction Let us now apply the structure of the previous section, considering a higherdimensional theory with fuzzy extra dimensions and then perform a simple (trivial) dimensional reduction. The higher-dimensional theory is defined on M 4 × (S/R)F , with (S/R)F a fuzzy coset, e.g. the fuzzy sphere, SN2 , with symmetry governed by the gauge group G = U (P). The Y-M action is: SYM

1 = 2 4g

 d 4 xkTrtrG FMN F MN ,

(50)

with trG the trace over the generators of the gauge group, G, and kTr3 the integration over (S/R)F , i.e. the fuzzy coset described by N × N matrices and FMN the higherdimensional field strength tensor, which is composed of both 4-dimensional spacetime and extra-dimensional parts, i.e. (Fμν , Fμa , Fab ). The fuzzy extra-dimensional components of FMN are expressed in terms of the covariant coordinates φa : Fμa = ∂μ φa + [Aμ , φa ] = Dμ φa c Fab = [Xa , Ab ] − [Xb , Aa ] + [Aa , Ab ] − Cba Aac . Replacing the above equations in (50), the action becomes: 

 SYM =

d 4 xTrtrG

k 2 k F + (Dμ φa )2 − V (φ) 4g 2 μν 2g 2

 ,

(51)

where V (φ) is the potential, derived from the kinetic term of Fab , that is V (φ) = −

 k Trtr Fab Fab G 4g 2 ab

 k = − 2 TrtrG [φa , φb ][φa , φb ] − 4Cabc φa φb φc + 2R−2 φ2 . 4g

3 In

(52)

general, k is a parameter related to the size of the fuzzy coset space. In the case of the fuzzy sphere, k is related to the radius of the sphere and the integer l.

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The (51) admits a natural interpretation as an action of a 4-dimensional theory. Let λ(xμ , X a ) be the gauge parameter of an infinitesimal gauge transformation of G. This transformation can be viewed as a M 4 gauge transformation: λ(xμ , X a ) = λI (xμ , X a )T I = λh,I (xμ )T h T I ,

(53)

where T I denote the hermitian generators of the gauge group U (P) and λI (xμ , X a ) are the N × N antihermitian matrices, which means that they can be expressed as λI ,h (xμ )T h , where T h are the antihermitian generators of U (N ) and λI ,h (xμ ), h = 1, . . . , N 2 , are the Kaluza-Klein modes of λI (xμ , X a ). In turn, we can assume that the fields on the right hand side of (53) could be considered as a field valued in the tensor product Lie algebra Lie (U (N )) ⊗ Lie (U (P)), that is the algebra Lie (U (NP)). Similar consideration applies for the the gauge field Aν , too: μ h I Aν (xμ , X a ) = AIν (xμ , X a )T I = Ah,I ν (x )T T ,

(54)

which can be regarded as a gauge field on M 4 taking values in the Lie (U (NP)) algebra. A similar consideration can be applied for the scalars, too.4 A very important feature of the above structure is the enhancement of the gauge symmetry of the 4-dimensional theory as compared to the symmetry of the starting, higher-dimensional theory. Specifically, one may choose to start with abelian gauge group in higher dimensions and result with a non-abelian gauge symmetry in four dimensions. An undesirable result is that the scalars are accommodated in the adjoint representation of the 4-dimensional gauge group, meaning that they cannot trigger the electroweak symmetry breaking. In order to overcome this drawback, one should try to employ an alternative dimensional reduction.

4.2 Fuzzy CSDR There is an alternative way to obtain a 4-dimensional gauge theory from a higherdimensional theory. This is realized by performing a non-trivial dimensional reduction, which, in our case, is the CSDR, modified as it must, since the extra dimensions are now fuzzy coset spaces [33].5 We begin by presenting the similarities of the two reduction schemes: CSDR and fuzzy CSDR. The first similarity is that fuzziness does not affect the isometries and the second one is that gauge couplings defined on both spaces have the same dimensionality. A major difference between fuzzy and ordinary CSDR is that the 4-dimensional gauge group appearing in the ordinary CSDR after the geometrical breaking and before the spontaneous symmetry breaking -due to the 4-dimensional Higgs TrtrG is interpreted as the trace of the U (NP) matrices. also [47].

4 Also, 5 See

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fields- does not appear in the fuzzy CSDR. In the latter, the spontaneous symmetry breaking takes already place by solving the fuzzy CSDR constraints and the 4-dimensional potential appears already shifted to a minimum. Therefore, in four dimensions, appears only the physical Higgs field that survives after a spontaneous symmetry breaking. Correspondingly, in the Yukawa sector of the theory we have results of the spontaneous symmetry breaking, i.e. massive fermions and Yukawa interactions among fermions and the physical Higgs field. We conclude that if one would like to describe the spontaneous symmetry breaking of the SM in the present framework, then one would be naturally led to large extra dimensions. Another fundamental difference between the two CSDR reductions is the fact that a non-Abelian gauge group, G, is not required in many dimensions. Indeed, it turns out that the presence of a U (1) in the higher-dimensional theory is enough to obtain non-Abelian gauge theories in four dimensions. Another fundamental difference as compared to all known theories defined in more than four dimensions is that the present ones are renormalizable. For technicalities, one should consult the original paper or some review papers [33].

5 Orbifolds and Fuzzy Extra Dimensions The involvement of the orbifold structure (similar to the one developed in [48]) in the framework of gauge theories with fuzzy coset spaces as extra dimensions, was suggested in order to obtain chiral low-energy theories. In order to support the renormalizability of the theories constructed so far using fuzzy extra dimensions the reverse procedure was considered, that is to start from a renormalizable theory in four dimensions and reproduce the results of a higher-dimensional theory reduced over fuzzy coset spaces [34–36]. This idea was realized as follows: one starts with a gauge theory in four dimensions with an appropriate set of scalar fields and a suitable potential, which leads to vacua that could be identified as -dynamically generatedfuzzy extra dimensions, including a finite Kaluza-Klein tower of massive modes. This reverse procedure is targeting at proving that an initial abelian gauge theory does not have to be considered in higher dimensions, with the non-abelian gauge theory structure emerging from fluctuations of the coordinates [49, 50]. The whole idea share some similarities with the idea of dimensional deconstruction, introduced earlier [51, 52]. Then, there was an attempt to include fermions, but the best one could achieve (for some time) contained mirror fermions in bifundamental representations of the low-energy gauge group [35, 36]. Although mirror fermions do not exclude the possibility to make contact with phenomenology [53], it is preferable to result with exactly chiral fermions. In the following, we are going to deal with the Z3 orbifold projection of the N = 4 Supersymmetric Yang Mills (SYM) theory [54], studying the action of the discrete group on the fields of the theory and the emerging superpotential in the projected theory [37].

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5.1 N = 4 SYM Field Theory and Z3 Orbifolds Let us consider an N = 4 supersymmetric SU (3N ) gauge theory defined on the Minkowski spacetime with a particle spectrum of the theory (in the N = 1 terminology) that consists of an SU (3N ) gauge supermultiplet and three adjoint chiral supermultiplets Φ i , i = 1, 2, 3. The component fields of the above supermultiplets are the gauge bosons, Aμ , μ = 1, . . . , 4, six adjoint real (or three complex) scalars φa , a = 1, . . . , 6 and four adjoint Weyl fermions ψ p , p = 1, . . . , 4. The scalars and Weyl fermions transform according to the 6 and 4 representations of the SU (4)R R-symmetry of the theory, respectively, while the gauge bosons are singlets. For the introduction of the orbifolds, the discrete group Z3 has to be considered as a subgroup of SU (4)R . The Z3 is not embedded into SU (4)R in a unique way with the options not being equivalent, since the choice of embedding affects the amount of the remnant supersymmetry [48]: • Maximal embedding of Z3 into SU (4)R would lead to non-supersymmetric models, therefore it is excluded. • Embedding Z3 in a subgroup of SU (4)R : – Embedding into an SU (2) subgroup would lead to N = 2 supersymmetric models with SU (2)R R-symmetry. – Embedding into an SU (3) subgroup would lead to N = 1 supersymmetric models with U (1)R R-symmetry. We focus on the last embedding, which gives the desired remnant supersymmetry. Let us consider a generator g ∈ Z3 , labeled (for convenience) by three integers a = (a1 , a2 , a3 ) [55] satisfying the relation a1 + a2 + a3 = 0 mod 3 .

(55)

The last equation implies that Z3 is embedded in the SU (3) subgroup, i.e. the remnant supersymmetry is the desired N = 1 [56]. Since the various fields of the theory transform differently under SU (4)R , Z3 will act non-trivially on them. Gauge and gaugino fields are singlets under SU (4)R , therefore the geometric action of the Z3 rotation is trivial. The action of Z3 on the complex scalar fields is given by the matrix 2π γ(g)ij = δij ω ai , where ω = e 3 and the action of Z3 on the fermions φi is given 1 by γ(g)ij = δij ω bi , where bi = − (ai+1 + ai+2 − ai ).6 In the present case, the three 2 integers of the generator g are (1, 1, −2), meaning that ai = bi . The matter fields are not gauge invariant, therefore Z3 acts on their gauge indices, too. The action of this rotation is given by the matrix ⎛

⎞ 1N 0 0 γ3 = ⎝ 0 ω1N 0 ⎠ . 0 0 ω 2 1N 6 Also

modulo 3.

(56)

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There is no specific reason for these blocks to have the same dimensionality (see e.g. [57–59]). However, it is the same, because the projected theory must be free of anomalies. After the orbifold projection, the spectrum of the theory consists of the fields that are invariant under the combined action of the discrete group, Z3 , on the “geometric”7 and gauge indices [55]. As for the gauge bosons, the projection is Aμ = γ3 Aμ γ3−1 . Therefore, taking into consideration (56), the gauge group of the initial theory breaks down to the group H = SU (N ) × SU (N ) × SU (N ) in the projected theory. The complex scalar fields transform non-trivially under the gauge and R-symmetry, so the projection is φiIJ = ω I −J +ai φiIJ , where I , J are gauge indices. Therefore, J = I + ai , meaning that the scalar fields surviving the orbifold projection have the form φI ,J +ai and transform under the gauge group, H , as:

 3 · (N , N¯ , 1) + (N¯ , 1, N ) + (1, N , N¯ ) .

(57)

Similarly, fermions transform non-trivially under both gauge group and R-symmetry, i i = ω I −J +bi ψIJ . Therefore, the fermions surviving the with the projection being ψIJ i projection have the form ψI ,I +bi accommodated in the same representation as the scalars, (57), demonstrating the N = 1 remnant supersymmetry. It is notable that the representations (57) of the resulting theory are anomaly free. So, in a nutshell, fermions are accommodated into chiral representations of H , divided into three generations since the particle spectrum contains three N = 1 chiral supermultiplets. The interactions of the projected model are included in the superpotential. To specify it, one has to start with the superpotential of the N = 4 SYM theory [54]: WN =4 = ijk Tr(Φ i Φ j Φ k ) ,

(58)

where, Φ i , Φ j , Φ k are the three chiral superfields of the theory. After the projection, the structure of the superpotential remains the same, encrypting only the interactions among the surviving fields of the N = 1 theory: (proj)

WN =1 =



j

ijk ΦIi ,I +ai ΦI +ai ,I +ai +aj ΦIk+ai +aj ,I .

(59)

I

5.2 Dynamical Generation of Twisted Fuzzy Spheres proj

The superpotential WN =1 , (59), produces the scalar potential:

= 1 SYM theory, one obtains an N = 4 SYM Yang-Mills theory in four dimensions having a global SU (4)R symmetry which is identified with the tangent space SO(6) of the extra dimensions [14, 15, 61].

7 In case of ordinary reduction of a 10-dimensional N

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VN =1 (φ) =

1 i j † i j  Tr [φ , φ ] [φ , φ ] , 4

(60) proj

where, φi are the scalar components of the superfield, Φ i . The potential VN =1 (φ) gets minimized by vanishing vevs of the fields, so, in order to result with solutions interpreted as vacua of a non-commutative geometry, some modifications have to take proj place. So, in order to result with minima of VN =1 (φ), soft N = 1 supersymmetric 8 terms of the form VSSB =

1  2 i† i 1  mi φ φ + hijk φi φj φk + h.c. 2 i 2

(61)

i,j,k

are included, with hijk = 0 unless i + j + k ≡ 0 mod3. The introduction of SSB terms does not cause embarrassment, since the presence of an SSB sector is necessary anyway for a model to have phenomenological viability, see e.g. [60]. The D-terms of the theory are given by 1 1 (62) VD = D2 = DI DI , 2 2 where DI = φ†i T I φi , where T I are the generators, accommodated in the representation of the corresponding chiral multiplets. Therefore, putting all potential terms together, the total potential of the theory is: proj

V = VN =1 + VSSB + VD .

(63)

An appropriate choice of the parameters m2i and hijk of (61) is m2i = 1 and hijk = ijk . Therefore, the scalar potential, (63), takes the form: V =

1 ij † ij (F ) F + VD , 4

(64)

where F ij is defined as: F ij = [φi , φj ] − i ijk (φk )† .

(65)

The first term of, (64), is positive, therefore, the global minimum of the potential is: [φi , φj ] = i ijk (φk )† , φi (φj )† = R2 ,

(66)

where (φi )† denotes the hermitian conjugate of φi and [R2 , φi ] = 0. It is clear that the above relations are related to a fuzzy sphere. This gets even more transparent, after the consideration of the untwisted fields, φ˜ i , defined by : proj

8 The SSB terms that will be inserted into V N =1 (φ), are purely scalar. Although this is enough for our purpose, it is obvious that more SSB terms have to be included too, in order to obtain the full SSB sector [60].

Non-commutativity in Unified Theories and Gravity

φi = Ω φ˜ i ,

195

(67)

where Ω = 1 satisfies the relations: Ω 3 = 1 , [Ω, φi ] = 0 , Ω † = Ω −1 , (φ˜ i )† = φ˜ i ⇔ (φi )† = Ωφi .

(68)

Therefore, (66) reproduces the fuzzy sphere relations, generated by φ˜ i [φ˜ i , φ˜ j ] = i ijk φ˜ k , φ˜ i φ˜ i = R2 ,

(69)

demonstrating the fact that non-commutative space generated by φi is actually a twisted fuzzy sphere, S˜ N2 . Next, configurations of the twisted fields, φi , can be found, i.e. fields satisfying (66). Such a configuration is: φi = Ω(13 ⊗ λi(N ) ) ,

(70)

where λi(N ) are the SU (2) generators in the N -dimensional irreducible representation and Ω is the matrix: ⎛ ⎞ 010 Ω = Ω3 ⊗ 1N , Ω3 = ⎝ 0 0 1 ⎠ , Ω 3 = 1 . (71) 100 According to the transformation (67), the “off-diagonal” orbifold sectors, (57), take the block-diagonal form: ⎛

⎞ ⎞ ⎛ i 0 (λi(N ) )(N ,N¯ ,1) 0 0 λ(N ) 0 0 0 (λi(N ) )(1,N ,N¯ ) ⎠ = Ω ⎝ 0 λi(N ) 0 ⎠ . (72) φi = ⎝ i 0 0 λi(N ) (λ(N ) )(N¯ ,1,N ) 0 0 The untwisted fields generating the ordinary fuzzy sphere, φ˜ i , are in block-diagonal form. Each block is considered as a fuzzy sphere, since each one satisfies the corresponding commutation relations (69). In turn, the above configuration in (72), that is the vacuum of the theory, has the form of three fuzzy spheres, with relative angles 2π/3. Concluding, the solution φi can be viewed as the twisted equivalent of three fuzzy spheres, in accordance to the orbifolding. Note that the F ij of (65), can be interpreted as the field strength tensor of the spontaneously generated fuzzy extra dimensions. The term VD of the potential induces a change on the radius of the sphere (in a similar way to the case of the ordinary fuzzy sphere [34, 36, 62]).

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5.3 Chiral Models After the Fuzzy Orbifold Projection - The SU(3)c × SU(3)L × SU(3)R Model The resulting groups after the orbifold projection are various because of the different ways the gauge group SU (3N ) is spontaneously broken. The minimal, anomaly free unified models are SU (4) × SU (2) × SU (2), SU (4)3 and SU (3)3 .9 Let us focus on the breaking of the latter, that is the trinification group SU (3)c × SU (3)L × SU (3)R [64, 65] (see also [66–70] and for a string theory approach see [50]). At first, the integer N has to be decomposed as N = n + 3. Then, for SU (N ), the considered embedding is: SU (N ) ⊃ SU (n) × SU (3) × U (1) . (73) Therefore, the embedding for the gauge group SU (N )3 is: SU (N )3 ⊃ SU (n) × SU (3) × SU (n) × SU (3) × SU (n) × SU (3) × U (1)3 . (74) The three U (1) factors are ignored10 and the representations are decomposed according to (74), as: SU (n) × SU (n) × SU (n) × SU (3) × SU (3) × SU (3) , ¯ 1) (n, n¯ , 1; 1, 1, 1) + (1, n, n¯ ; 1, 1, 1) + (¯n, 1, n; 1, 1, 1) + (1, 1, 1; 3, 3, ¯ + (1, 1, 1; 3, ¯ 1, 3) + (n, 1, 1; 1, 3, ¯ 1) + (1, n, 1; 1, 1, 3) ¯ + (1, 1, 1; 1, 3, 3) ¯ 1, 1) + (¯n, 1, 1; 1, 1, 3) + (1, n¯ , 1; 3, 1, 1) + (1, 1, n¯ ; 1, 3, 1) . (75) + (1, 1, n; 3, Taking into account the decomposition (73), the gauge group is broken to SU (3)3 . Under SU (3)3 , the surviving fields transform as: SU (3) × SU (3) × SU (3) ,

 ¯ 1) + (3, ¯ 1, 3) + (1, 3, 3) ¯ , (3, 3,

(76) (77)

which correspond to the desired chiral representations of the trinification group. Under SU (3)c × SU (3)L × SU (3)R , the quarks and leptons of the first family transform as:

9 Similar

approaches have been studied in the framework of YM matrix models [63], lacking phenomenological viability. 10 As anomalous gaining mass by the Green-Schwarz mechanism and therefore they decouple at the low energy sector of the theory [58].

Non-commutativity in Unified Theories and Gravity

⎞ ⎛ c c c⎞ d uh d d d ¯ 1) , ¯ 1, 3) , q = ⎝ d u h ⎠ ∼ (3, 3, qc = ⎝ uc uc uc ⎠ ∼ (3, c c c d uh h h h ⎛ ⎞ c N E v ¯ , λ = ⎝ E N c e ⎠ ∼ (1, 3, 3) c c v e S

197

⎛

(78)

respectively. Matrices for the other two families come in a similar way. It is worth noting that this theory can be upgraded to a two-loop finite theory (for reviews see [66, 71–73]) and give testable predictions [66], too. Additionally, fuzzy orbifolds can be used to break spontaneously the unification gauge group down to MSSM and then to the SU (3)c × U (1)em . Summarizing this section, we conclude that fuzzy extra dimensions can be used for constructing chiral, renormalizable and phenomenologically viable field-theoretical models. A natural extension of the above ideas and methods have been reported in ref [74] (see also [75]), realized in the context of Matrix Models (MM). At a fundamental level, the MMs introduced by Banks-Fischler-Shenker-Susskind (BFSS) and Ishibashi-Kawai-Kitazawa-Tsuchiya (IKKT), are supposed to provide a nonperturbative definition of M-theory and type IIB string theory respectively [29, 76]. On the other hand, MMs are also useful laboratories for the study of structures which could be relevant from a low-energy point of view. Indeed, they generate a plethora of interesting solutions, corresponding to strings, D-branes and their interactions [29, 77], as well as to non-commutative/fuzzy spaces, such as fuzzy tori and spheres [78]. Such backgrounds naturally give rise to non-abelian gauge theories. Therefore, it appears natural to pose the question whether it is possible to construct phenomenologically interesting particle physics models in this framework as well. In addition, an orbifold MM was proposed by Aoki-Iso-Suyama (AIS) in [79] as a particular projection of the IKKT model, and it is directly related to the construction described above in which fuzzy extra dimensions arise with trinification gauge theory [37]. By Z3 - orbifolding, the original symmetry of the IKKT matrix model with matrix size 3N × 3N is generally reduced from SO(9, 1) × U (3N ) to SO(3, 1) × U (N )3 . This model is chiral and has D = 4, N = 1 supersymmetry of Yang-Mills type as well as an inhomogeneous supersymmetry specific to matrix models. The Z3 - invariant fermion fields transform as bifundamental representations under the unbroken gauge symmetry exactly as in the constructions described above. In the future we plan to extend further the studies initiated in refs [74, 75] in the context of orbifolded IKKT models. Our current interest is to continue in two directions. Given that the two approaches discussed here led to the N = 1 trinification GUT SU (3)3 , one plan is to examine the phenomenological consequences of these models. The models are different in the details but certainly there exists a certain common ground. Among others we plan to determine in both cases the spectrum of the Dirac and Laplace operators in the extra dimensions and use them to study the behaviour of the various couplings, including

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the contributions of the massive Kaluza-Klein modes. These contributions are infinite or finite in number, depending on whether the extra dimensions are continuous or fuzzy, respectively. We should note that the spectrum of the Dirac operator at least in the case of SU (3)/U (1) × U (1) is not known. Another plan is to start with an abelian theory in ten dimensions and with a simple reduction to obtain an N = (1, 1) abelian theory in six dimensions. Finally, reducing the latter theory over a fuzzy sphere, possibly with Chern-Simons terms, to obtain a non-abelian gauge theory in four dimensions provided with soft supersymmetry breaking terms. Recall that the last feature was introduced by hand in the realistic models constructed in the fuzzy extra dimensions framework.

6 Gravity as a Gauge Theory In this section we recall the particularly interesting relation between gravity and gauge theories [80–83] with ultimate aim to transfer it to the non-commutative framework.

6.1 4-Dimensional Gravity as a Gauge Theory Employing the vielbein formulation of general relativity, we recall that it can be reproduced -at least at a kinematical level- if considered as a gauge theory of its isometries on the 4-dimensional Minkowski spacetime, i.e. as a gauge theory of the Poincaré algebra, iso(1, 3), which has ten generators: the four generators of local translations Pa , a = 1, 2, 3, 4, and the six Lorentz transformations Mab , satisfying the commutation relations: [Mab , Mcd ] = 4η[a[c Md ]b] ,

[Pa , Mbc ] = 2ηa[b Pc] ,

[Pa , Pb ] = 0 ,

(79)

where ηab is the (mostly plus) Minkowski metric. In order to proceed with the gauging, one has to introduce a gauge field for each generator: the vielbein eμ a for translations and the spin connection ωμ ab for Lorentz transformations. The gauge connection will be: 1 (80) Aμ = eμ a (x)Pa + ωμ ab (x)Mab , 2 transforming in the adjoint representation: δAμ = ∂μ + [Aμ , ] , where the gauge transformation parameter is:

(81)

Non-commutativity in Unified Theories and Gravity

199

1

= ξ a (x)Pa + λab (x)Mab . 2

(82)

Therefore, one can calculate the transformations of the gauge fields: δeμ a = ∂μ ξ a + ωμ ab ξb − λa b eμ b , δωμ ab = ∂μ λab − 2λ[a c ωμ cb] ,

(83) (84)

but also, using the standard formula: Rμν (A) = 2∂[μ Aν] + [Aμ , Aν ]

(85)

and expanding Rμν (A) = Rμν a (e)Pa + 21 Rμν ab (ω)Mab , one may result with the curvatures of the gauge fields: Rμν a (e) = 2∂[μ eν] a − 2ω[μ ab eν]b , Rμν (ω) = 2∂[μ ων] ab

ab

(86)

− 2ω[μ ων]c . ac

b

(87)

The condition for vanishing torsion gives the expression of the spin connection with respect to the vielbein. The dynamics follow from the Einstein-Hilbert action: SEH4 =

1 2



d4 x μνρσ abcd eμ a eν b Rρσ cd (ω) ,

(88)

which is not an action that results from gauge theory (Yang-Mills type). Thus, rigorously, only the kinematics of 4-dimensional gravity is obtained by gauge theory, not its dynamics.

6.2 3-Dimensional Gravity as a Gauge Theory In the 3-dimensional case, gravity can be completely described as a gauge theory of the corresponding Poincaré algebra, as for the kinematics as for the dynamics [39]. The 3-dimensional Einstein-Hilbert action is:  1 (89) d3 x μνρ abc eμ a Rνρ bc (ω) , SEH3 = 2 which, as Witten showed, is identical to a Chern-Simons gauge theory of the Poincaré algebra iso(1, 2). This algebra has six generators, three translations Pa and three Lorentz transformations M a = abc Mbc , with a = 1, 2, 3. Those generators satisfy the following commutation relations: [Ma , Mb ] = abc M c ,

[Pa , Mb ] = abc P c ,

[Pa , Pb ] = 0 .

(90)

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Following the same procedure for the gauging as in the 4-dimensional case, gauge connection and gauge parameters are written as: Aμ = eμ a Pa + ωμ a Ma , = ξ a Pa + λa Ma ,

(91)

and then one can calculate the gauge transformations and the curvature of the connection. After the appropriate choice of the quadratic form of the algebra, the resulting Chern-Simons action is identical to the Einstein-Hilbert action, (89). Furthermore, it was proved that the inclusion of a cosmological constant is also possible but then one has to gauge the dS or AdS algebra in three dimensions, so(3, 1) and so(2, 2) respectively. In this case the generators of the translations are not commutative any more, but they satisfy the relation: [Pa , Pb ] = λMab ,

(92)

where λ is the cosmological constant.

6.3 3-Dimensional Gravity as a Gauge Theory on Non-commutative Spaces Having already studied gauge theories on non-commutative spaces (Sect. 3), and given the strong relation between gravity and gauge theories in three dimensions, our purpose is to study gravity as a gauge theory on non-commutative spaces. In order to accomplish this goal, the first and important step is to identify the noncommutative space because it will be its isometry group the one that one would gauge in order to derive the kinematics and the action. A very interesting space is the foliation of the 3-dimensional Euclidean space by fuzzy spheres, first considered in Ref. [84] (see also [85]). Non-commutative coordinates obey the algebra of SU (2), but, unlike the fuzzy sphere case, one does not consider the matrices to be proportional to the SU (2) generators in irreducible representations, but in reducible ones. The consideration of reducible representations results in the construction of large, block-diagonal matrices, with each block being an irreducible representation. Therefore the Hilbert space is: H = ⊕[],  = 0, 1/2, 1, . . . .

(93)

This fuzzy space is known as R3λ and can be viewed as being given by three operators Xi which satisfy: (94) [Xi , Xj ] = iλ ijk Xk , living in reducible representations of su(2) (cf. [84]). Allowing Xi to live in a reducible representation is equivalent to considering a sum of fuzzy 2-spheres of different radii.

Non-commutativity in Unified Theories and Gravity

201

Thus, this can be seen as a discrete foliation of 3D Euclidean space by multiple fuzzy 2-spheres, each being a leaf of the foliation.11 (cf. [86]). The above space has an SO(4) symmetry [87], which one would gauge. In this procedure, the need of including additional generators emerges, typical in non-Abelian non-commutative gauge theories, for the anticommutators to close. Therefore, the gauge theory considered in this case is the U (2) × U (2) in a fixed representation. Then, the procedure followed is the same as in the continuous case, adjusted to the non-commutative framework: First one has to establish the commutation and anticommutation relations of the generators: [Pa , Pb ] = i abc Mc , [Pa , Mb ] = i abc Pc , [Ma , Mb ] = i abc Mc , 1 1 1 {Pa , Pb } = δab 1l , {Pa , Mb } = δab γ5 , {Ma , Mb } = δab 1l . 2 2 2

(95) (96)

Then, one has to introduce a gauge field for each generator and therefore, the gauge connection is obtained, which modifies the coordinates to their covariant form, that is: Aμ ⊗ γ5 , (97) Xμ = Xμ ⊗ i1l + eμ ⊗ Pa + ωμ ⊗ Ma + Aμ ⊗ i1l +  The gauge parameter is valued in the Lie algebra, therefore it is defined as:

0 ⊗ γ5 .

= ξ a ⊗ Pa + λa ⊗ Ma + 0 ⊗ i1l + 

(98)

Using the above relations, one may end up with the transformations of the gauge fields. Also, using the covariant coordinates in the following relation: Rμν = [Xμ , Xν ] − iλ μνρ Xρ ,

(99)

one obtains the corresponding curvatures. Finally, the action proposed is: S = Tri μνρ Xμ Rνρ .

(100)

Varying the above action one ends up with the equations of motion.

7 Conclusions Kaluza and Klein gave a very insightful suggestion, in which geometry would lead to 4-dimensional gauge theories. Due to some difficulties, one was led to start with gauge theories in higher-dimensional theories and using the CSDR scheme (with support from the heterotic string), obtain 4-dimensional particle models, making contact 11 In the Lorentzian case there is a similar construction, where the 3-dimensional spacetime with Lorentzian signature is foliated by fuzzy hyperboloids [88].

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with phenomenology. Going one step further, the employment of fuzzy coset spaces as extra dimensions gave the virtue of renormalizability to the higher-dimensional theories. Then, in order to find support for these scenarios from another direction, a 4-dimensional particle model that would incorporate the results of a dimensional reduction was built, in which, after suitable additions, a particular fuzzy space, i.e. the fuzzy sphere, was dynamically generated. Eventually, in order to get closer to a complete picture of all interactions, a 3-d gravity model was proposed, constructed on non-commutative spaces as a gauge theory of their symmetries. The ultimate aim is to obtain such models in four dimensions, with hopes for better u-v properties. Acknowledgements We acknowledge support by the COST action QSPACE MP1405. G.Z. thanks the MPI for Physics in Munich for hospitality and the A.v.Humboldt Foundation for support.

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Part II

Representation Theory

Webs of Quantum Algebra Representations in 5d N = 1 Super Yang–Mills Jean-Emile Bourgine

Abstract Instanton partition functions of 5d N = 1 Super Yang–Mills reduced on S 1 are engineered in type IIB string theory from webs of ( p, q)-branes. Branes intersections are associated to the (refined) topological vertex, while the web diagram provides gluing rules. These partition functions are covariant under the action of a quantum toroidal algebra, the Ding–Iohara–Miki algebra. In fact, a web of representations can be associated to the brane web diagram, where ( p, q)-branes correspond to representations of levels (q, p), and topological vertices to intertwiners. Using this correspondence, the T -operator of a new type of quantum integrable systems can be constructed. Its vacuum expectation value reproduces the Nekrasov instanton partition function, while further insertion of algebra elements provides the qq-characters.

1 Introduction Since the 1990’s, Super Yang–Mills (SYM) theories in four dimensions with N = 2 supersymmetries are known to exhibit a correspondence with classical integrable systems [26]. These theories can be IR-regularized by deformation of the Euclidean background into the Omega background R21 × R22 depending on two cut-offs (1 , 2 ), thus allowing the exact calculation of the non-perturbative (instanton) partition function by localization [29]. This deformation of the background induces a quantization of the system in the previous correspondence: in the limit 2 → 0, the theory is described by a quantum integrable system (qIS) with Planck constant given by 1 [31]. In this context, the interpretation of the extra deformation parameter 2 has remained mysterious, despite the presence of an action of a quantum algebra

J.-E. Bourgine (B) Korea Institute for Advanced Study (KIAS), Quantum Universe Center (QUC), 85 Hoegiro, Dongdaemun-gu, Seoul, South Korea e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2018 V. Dobrev (ed.), Quantum Theory and Symmetries with Lie Theory and Its Applications in Physics Volume 1, Springer Proceedings in Mathematics & Statistics 263, https://doi.org/10.1007/978-981-13-2715-5_11

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(Spherical Hecke central algebra, isomorphic to the affine Yangian of gl1 ) on the partition function [23, 33].1 In order to extend the correspondence to finite regulators 1 , 2 , it turns out easier to consider the q-deformed context. Indeed, N = 2 SYM theories can be lifted to a five-dimensional background compactified on S R1 , and the corresponding partition function is invariant under the action of the Ding–Iohara–Miki (DIM) algebra [16, 27], which is in fact the quantum toroidal algebra of gl1 . In this language, the instanton partition function is obtained as the vacuum expectation value (vev) of a Baxter T -operator defining a new qIS based on this quantum toroidal algebra. The construction rules for the T -operator are prescribed by the ( p, q)-branes web diagram engineering the gauge theories in type IIB string theory [7, 28]. This diagram defines a representations web describing the interconnection of representations through Awata–Feigin–Shiraishi (AFS) intertwiners [6], the algebraic counterpart of the topological vertex. Aside from the correspondence with qIS, this algebraic construction present several advantages compared to the standard topological string realization of the gauge theory. For instance, the effective calculations are greatly simplified since the cumbersome summations of symmetric polynomials is replaced by simpler algebraic manipulations (normal ordering). Furthermore, the AGT correspondence can be simply formulated as a kind of Schur–Weyl duality, i.e. as the action of q-Virasoro/q-Wm algebras on tensor products of DIM modules. Finally, S-duality (and in fact the whole S L(2, Z) invariance) can also be realized explicitly as automorphism of the DIM algebra. In the second section, we present the construction of the T -operator using a generalized version of AFS intertwiners defined in [14], and reproduce the partition functions of gauge theories with linear quiver (unitary gauge group at each node). The T -operator also supplies the qq-characters, a natural generalization of Baxter’s TQ-equations, their derivation will be given in section three.

2 Construction of a T -Operator In contrast with the standard construction of the T -operator using a Lax matrix, i.e. an R-matrix in the appropriate physical⊗auxiliary representations, here the T -operator is obtained by gluing AFS intertwiners (the equivalent of Jimbo and Miwa’s vertex operators [22]). After introducing the DIM algebra, we present the construction of the T -operator using the representations web, and discuss its interpretation in the integrability framework.

1 The

AGT-correspondence with Liouville/Toda 2d CFT [4, 34] is another facet of this problem, promoting this duality to a triality but giving only limited insight on the formulation of an 2 deformed integrable system.

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2.1 DIM Algebra and Representations 2 ) The DIM algebra can be presented as a one-parameter deformation of the Uq (sl algebra that is behind the integrable properties of the Heisenberg XXZ spin chain. 2 ) when one of Thus, it depends on two parameters q1 and q2 and reduces to Uq (sl them is sent to zero. This is clearly observed in the second Drinfeld presentation where both algebras are defined in terms of the currents x ± (z) =



z −k xk± , ψ ± (z) =



± z ∓k ψ±k ,

(1)

k≥0

k∈Z

and the central element γ, ˆ obeying the following algebraic relations, g(γw/z) ˆ ψ − (w)ψ + (z), g(γˆ −1 w/z) ψ + (z)x ± (w) = g(γˆ ±1/2 z/w)±1 x ± (w)ψ + (z), [ψ ± (z), ψ ± (w)] = 0, ψ + (z)ψ − (w) =

(2)

ψ − (z)x ± (w) = g(γˆ ∓1/2 z/w)±1 x ± (w)ψ − (z) x ± (z)x ± (w) = g(z/w)±1 x ± (w)x ± (z)   − −1/2 [x + (z), x − (w)] = κ δ(γˆ −1 z/w)ψ + (γˆ 1/2 w) − δ(γz/w)ψ ˆ (γˆ w) .

At this level, the only difference lies in the complex parameter κ and the function g(z) that, in the case of DIM algebra, depends on the parameters q1 , q2 and q3 (related through q1 q2 q3 = 1) as κ=

 1 − qα z (1 − q1 )(1 − q2 ) , g(z) = . (1 − q1 q2 ) 1 − qα−1 z α=1,2,3

(3)

2 ) and DIM algebras have the structure of a quasi-triangular Hopf algebra Both Uq (sl with the Drinfeld coproduct Δ, that makes them suitable for the definition of qIS. c/2 However, the DIM algebra exhibits an extra feature: in addition to γˆ = q3 , it has a c/2 ¯ − + −1 second central element given by ψ0 = (ψ0 ) = q3 . Correspondingly, two grading elements can be introduced, and the universal R-matrix depends on two spectral parameters. Here, we consider only representations with integer levels (c, c) ¯ ∈ Z2 . More precisely, our construction involves vertical representations, of levels (0, m), and horizontal representations of levels (1, n). Horizontal representations act on the physical space of the qIS. They are the equiv2 ) [18]. These Fock representations alent of the level one representations for Uq (sl are defined upon the modes ak of a free boson satisfying |k|

|k|

|k|/2

[ak , al ] = k(1 − q1 )(1 − q2 )q3

δk+l .

(4)

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Explicitly, ⎛

⎞  z −k ∓|k|/4 q ρ(1,n) (x ± (z)) = u ±1 z ∓n : exp ⎝∓ ak ⎠ : u k 3 k=0   z ∓k k/2 ∓n/2 −k/2 (1,n) ± ρu (ψ (z)) = q3 exp ± (q3 − q3 )a±k , k k>0

(5)

where the complex parameter u is the weight of the representation. By definition, the Fock vacuum state |∅ is annihilated by positive modes. On the other hand, the auxiliary space involves vertical representations (that are somehow similar to evaluation representations). These are highest weight representations, with states parameterized by a m-tuple Young diagram λ, a two-dimensional version of the index for the (infinite) spin representations of sl2 . In this representation, Cartan elements ψ ± (z) are diagonal while x ± (z) add/remove boxes to/from the Young diagrams: (x + (z))|v, λ = ρ(0,m) v

 x∈A(λ)

ρ(0,m) (x − (z))|v, λ v

−m/2

= q3

δ(z/χx ) Res z=χx

 x∈R(λ)

−m/2

ρ(0,m) (ψ ± (z))|v, λ = q3 v

1 |v, λ + x , zYλ (z)

δ(z/χx ) Res z −1 Yλ (zq3−1 )|v, λ − x , z=χx

[Ψλ (z)]± |v, λ ,

(6)

where A(λ) and R(λ) are the set of addable/removable boxes. In this setting, a comj−1 plex number (the instanton position) χx = vl q1i−1 q2 is associated to each box x ∈ λ with x = (l, i, j), (i, j) ∈ λl . The parameters v1 , . . . , vl are the roots of the Drinfeld polynomial, they encode the weights of the representations. The other functions are defined as

−1 Yλ (zq3−1 ) x∈A(λ) 1 − z χx , Yλ (z) = , (7) Ψλ (z) = −1 −1 Yλ (z) x∈R(λ) 1 − z q3 χx and [· · · ]± denotes an expansion in powers of z ∓1 .

2.2 From ( p, q)-Brane Webs to Representations Webs In type IIB string theory, ( p, q)-branes are bound states of p D5 branes (1, 0) and q NS5 branes (0, 1) [2, 20]. In the ten dimensional target space, they fill the dimensions 01234 (giving rise to the 5d Omega background R21 × R21 × S R1 ) together with a one dimensional object (line or segment) in the 56-plane. Their orientation in this plane

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depends on their charges as Δx6 /Δx5 = p/q. The resulting brane configuration in the 56-plane determines uniquely the field content of the gauge theory, and the partition function can be computed by assigning a topological vertex to each brane junction, and gluing them using appropriate rules [1, 21, 25]. The AFS intertwiner introduced in [6] provides a free field expression of the refined topological vertex based on representations of the DIM algebra. It follows from a correspondence between branes of charge ( p, q) at position u and representations of levels (q, p) and weight u. More specifically, a vertical representation (0, 1) is associated to each D5 brane, and an horizontal representation (1, n) to each NS5 brane dressed with n D5 branes [7, 9, 28]. The refined topological vertex, with preferred direction along the D5 brane, is identified with the intertwiner of the DIM algebra (or its dual, depending on the orientation) constructed in the appropriate representations (1, n) × (0, 1) → (1, n + 1) (resp. (1, n + 1) → (0, 1) × (1, n + 1)) as a solution to the equation   Δ(e), (e)Φ[u, v] = Φ[u, v] · ρ(0,1) ⊗ ρ(1,n) ρ(1,n+1) v u u

  (1,n+1) Δ (e) · Φ ∗ [u, v], (e) = ρ(0,1) ⊗ ρ(1,n) Φ ∗ [u, v]ρu

v u

(8)

where e ∈ DIM is any element of the algebra, and Δ is the dual coproduct obtained by permutation of the components of the tensor product. Although the brane web reflects the physical configuration of the branes, it can be slightly simplified while retaining the algebraic information needed to construct the T -operator. In the representations web, D5-branes are drawn vertically and dressed NS5-branes horizontally, while brane charges (or representations levels) are indicated explicitly, thus allowing a direct reading of the Chern–Simons charges (corresponding to differences in D5-dressing of the NS5 branes). In addition, the stacks of m D5 branes associated to a single U (m) gauge group are replaced by a single line associated to a vertical representation of levels (0, m). Accordingly, generalized intertwiners have been introduced in [14] using a fusion procedure. The non-trivial part of this construction is to verify that the newly defined intertwiners satisfy the relations (8) with the representation (0, 1) replaced with (0, m) (and (1, n + 1) with (1, n + m) by charge conservation).2 As an example, the representations web for the A2 quiver with gauge group U (m 2 ) × U (m 1 ) is represented on Fig. 1. In addition, the Table 1 summarizes the dictionary between algebra, string and gauge quantities. To build the T -operator, intertwiners are coupled either in the vertical (D5) direction, giving rise to a scalar product, or in the horizontal (NS5) direction, rendered by the usual product in the Fock space. As a result, the T -operator associated to a general Ar quiver with total gauge group U (m r ) × · · · × U (m 1 ) reads: T [Ar ] =



(n ∗ ,m ∗1 )∗

Φλ(n11 ,m 1 ) [u 1 , v1 ] ⊗ Φλ11

[u ∗1 , v∗1 ]Φλ(n22 ,m 2 ) [u 2 , v2 ] ⊗ · · ·

λ1 ,λ2 ,...,λr

··· ⊗

(n ∗ ,m ∗ )∗ Φλr r−1−1 r −1 [u r∗−1 , vr∗−1 ]Φλ(nr r ,m r ) [u r , vr ]

(9) ⊗

(n ∗ ,m ∗ )∗ Φλr r r [u r∗ , vr∗ ],

2 It is noted that the representation (0, m) is not simply obtained from the coproduct of (0, 1) representations in our conventions where all Young diagrams λl appear in a symmetric way.

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Fig. 1 Representations web corresponding to the U (m 2 ) × U (m 1 ) A2 quiver gauge theory. This diagram provides the gluing rules for the generalized AFS intertwiners, thus defining the T -operator

(1, n2 )u2

Φ(n2 ,m2 ) [u2 , v2 ]

(1, n2 + m2 )u2

(0, m2 )v2

(1, n∗2 + m1 )u2

(1, n∗2 )u∗2 (n∗ 2 ,m2 )∗

Φ

Φ(n1 ,m1 ) [u1 , v1 ] (1, n1 + m1 )u1

[u∗2 , v2 ]

(0, m1 )v1

(1, n∗1 + m2 )u1

∗

Φ(n1 ,m1 )∗ [u∗1 , v1 ]

(1, n∗1 )u∗1

Table 1 Correspondence between algebraic and string/gauge parameters String theory Gauge theory DIM algebra Radius R, Flux D5 charge (0, m) NS5 charge (n, 1) D5 positions vl NS5 position u

Background parameters R, 1 , 2 Gauge group rank m Chern–Simons coupling κ = n − n ∗ Coulomb branch vevs vl = e Ral Exp. gauge coupling q = u/u ∗

Parameters q1 = e R1 , q2 = e R2 Vertical repres. level (0, m) Horizontal repres. level (1, n) Weights vertical repres. v Weight horizontal repres. u

where the summation spans the configurations of r m i -tuples Young diagrams λi . By construction, this operator acts in the tensor product of r + 1 bosonic Fock spaces, seen as the physical spaces of a chain of length r + 1. A similar expression can be written for the affine quivers Aˆ r , and for quiver with fundamental matter fields [14]. The T -operator associated to D-type quivers has also been obtained recently in [15]. In all these cases, the partition function is equal to the Fock vev of this operator, for instance (10) Z[Ar ] = ∅| ⊗ · · · ⊗ ∅| T [Ar ] |∅ ⊗ · · · ⊗ |∅ .

2.3 Integrability As a consequence of the relations (8), the product of two intertwiners L = Φ (n,m)∗ Φ (n,m) satisfies the property     (0,m) ⊗ ρ(1,n) Δ (e) L = L ρ(0,m) ⊗ ρ(1,n) Δ(e), e ∈ DIM. ρ

(11)

where we have suppressed the weights indication for simplicity. As a consequence, the product L is identified with a Lax matrix [9], up to a constant factor dubbed

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215

anomaly in [8],3 and the T -operator defined in (9) takes the standard form T [Ar ] = Φ1 · L12 · L23 . . . Lr −1r · Φr∗

(13)

where · denotes the product in the auxiliary space (a matrix product in the vertical representation). The two extra intertwiners Φ1 and Φr∗ at the end of the chain corresponds to the product of a boundary state times the Lax matrix. Similarly, the T -operators of affine quivers Aˆ r are related to chains with periodic boundary conditions: T [ Aˆ r ] = tr [L12 · L23 . . . Lr1 ] .

(14)

Interestingly, various objects involved in the (q-deformed) AGT correspondence [4, 5, 34] also receive an interpretation in terms of the integrable structure [14]. For instance, the vev of the Lax matrix coincides with the q-Virasoro/q-W N vertex operator (after the appropriate change of basis, as in [3]),4 while the intertwiners’ vev reproduce the Gaiotto states [19] V12 = ∅|L|∅ , |G, v = ∅|Φ (n

∗

,m)∗

[u ∗ , v]|∅ , G, v| = ∅|Φ (n,m) [u, v]|∅ . (15)

3 Defining qq-Characters The qq-characters have been introduced in [32] as a generalization of the q-characters of Yangians and quantum groups that are naturally associated to the Baxter TQequations of qIS. On the gauge side, they define a double deformation of the Seiberg–Witten curve [11, 24] and encode an infinite set of constraints (called nonperturbative Schwinger-Dyson equations) between the correlators of the BPS sector [30]. As suggested in [9], qq-characters can be obtained from the T -operator constructed previously by insertion of an element of the DIM algebra inside the vev. In the simplest case of a single U (m) gauge group, the fundamental qq-character

3 The

presence of this extra factor is due to the fact that the Lax matrix comes from a wrongly normalized R-matrix that does not satisfy the two additional relations (Δ ⊗ 1)R = R13 R23 , (1 ⊗ Δ)R = R13 R12 .

4 The

(12)

covariance properties of this operator, sometimes called vertical intertwiner, were analyzed in [12, 13], where it has also been used to define the qq-characters.

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follows from the insertion of the coproduct of x + (z) in horizontal representations (omitted here): (16) χ(z) ∝ ∅| ⊗ ∅| Δ(x + (z))T [A1 ] |∅ ⊗ |∅ . This object is a polynomial in z as a consequence of the vanishing commutator [Δ(x + (z)), T ]. Higher qq-characters are obtained by insertion of the products Δ(x + (z 1 )) . . . Δ(x + (z k )). In quiver SYM theories, a different qq-character is attached to each node. For a linear quiver, the character attached to the first node is obtained from the insertion of x + (z) with repeated action of the coproduct Δ, Δ ⊗ 1,... in order to build an operator acting in (r + 1)th copies of the Fock space. Characters on the remaining nodes are harder to construct: the relevant operator is obtained from the action of the Weyl group on tensor products of the form − − (z)x + (γˆ (s−1)! z) ⊗ ψ[2] (z) x + (γˆ (s−2)! z) ⊗ · · · ψ[1] − · · · ⊗ ψ[s−1] (z)x + (γˆ (1) z) ⊗ x + (z)(⊗1)r +1−s

(17)

This quantum Weyl transformation has been defined in [14], and all the qq-characters of gauge theories with linear quiver have been computed.

4 Discussion The study of integrable systems based on quantum toroidal algebras is only at its initial stage, and there are still many interesting open questions. The T -operator can be expanded in both instanton counting parameters and coulomb branch vevs, since both type of weights can be associated to a different spectral parameter. The resulting Hamiltonians of these double series should be related under S-duality, implemented here by Miki’s automorphism [27]. Then, the standard problems of qIS can be formulated: spectrum, eigenvectors, form factors,... In addition, the qIS relevant to 4d N = 2 SYM should follows from the degenerate limit R → 0 (the limit of Fock representations might be more delicate here). Several generalizations of this qIS are also possible: higher ranks (ALE space) [10], elliptic (6d SYM) and more [17]. On the gauge side, the precise relation between the DIM algebra and the elementary strings degrees of freedom remains mysterious. Furthermore, the algebraic derivation of the qq-characters of DE-type, and even affine, quivers, is still missing. Acknowledgements I would like to thank my collaborators D. Fioravanti, M. Fukuda, K. Harada, Y. Matsuo, H. Zhang, R.-D. Zhu, with whom I had the pleasure to study various aspects of instanton partition functions and quantum algebras.

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References 1. Aganagic M, Klemm A, Marino M, Vafa C (2005) The topological vertex. Communications in mathematical physics 254(2):425–478 2. Aharony O, Hanany A, Kol B (1998) Webs of (p,q) 5-branes, Five Dimensional Field Theories and Grid Diagrams. JHEP 9801:002, hep-th/9710116 3. Alba VA, Fateev VA, Litvinov AV, Tarnopolskiy GM (2011) On combinatorial expansion of the conformal blocks arising from AGT conjecture. LettMathPhys 98:33–64, 1012.1312 4. Alday LF, Gaiotto D, Tachikawa Y (2010) Liouville Correlation Functions from Fourdimensional Gauge Theories. LettMathPhys 91:167–197, 0906.3219 5. Awata H, Yamada Y (2010) Five-dimensional AGT Conjecture and the Deformed Virasoro Algebra. JHEP 01:125, 0910.4431 6. Awata H, Feigin B, Shiraishi J (2012) Quantum Algebraic Approach to Refined Topological Vertex. JHEP 2012(03):041, 1112.6074 7. Awata H, Kanno H, Matsumoto T, Mironov A, Morozov A, Morozov A, Ohkubo Y, Zenkevich Y (2016a) Explicit examples of DIM constraints for network matrix models. JHEP 07:103, 1604.08366 8. Awata H, Kanno H, Mironov A, Morozov A, Morozov A, Ohkubo Y, Zenkevich Y (2016b) Anomaly in RTT relation for DIM algebra and network matrix models 1611.07304 9. Awata H, Kanno H, Mironov A, Morozov A, Morozov A, Ohkubo Y, Zenkevich Y (2016c) Toric Calabi-Yau threefolds as quantum integrable systems. R-matrix and RTT relations 1608.05351 10. Awata H, Kanno H, Mironov A, Morozov A, Suetake K, Zenkevich Y (2017) (q, t)-KZ equations for quantum toroidal algebra and Nekrasov partition functions on ALE spaces 1712.08016 11. Bourgine JE, Fioravanti D (2017) Seiberg-Witten period relations in Omega background 1711.07570 12. Bourgine JE, Fukuda M, Matsuo Y, Zhang H, Zhu RD (2016a) Coherent states in quantum W 1+∞ algebra and qq-character for 5d Super Yang-Mills. PTEP 2016(12):123B05, 1606.08020 13. Bourgine JE, Matsuo Y, Zhang H (2016b) Holomorphic field realization of SHc and quantum geometry of quiver gauge theories. JHEP 04:167, 1512.02492 14. Bourgine JE, Fukuda M, Harada K, Matsuo Y, Zhu RD (2017a) (p,q)-webs of DIM representations, 5d N=1 instanton partition functions and qq-characters 1703.10759 15. Bourgine JE, Fukuda M, Matsuo Y, Zhu RD (2017b) Reflection states in Ding-Iohara-Miki algebra and brane-web for D-type quiver 1709.01954 16. Ding J, Iohara K (1997) Generalization of Drinfeld quantum affine algebra. Lett Math Phys 41(2):181–193, q-alg/9608002v2 17. Foda O, Wu JF (2017) A Macdonald refined topological vertex. J Phys A50(29):294,003, 1701.08541 18. Frenkel I, Jing N (1988) Vertex representation of quantum affine algebras. Proc Natl Acad Sci USA 85:9373–9377 19. Gaiotto D (2009) Asymptotically free N=2 theories and irregular conformal blocks 0908.0307 20. Hanany A, Witten E (1997) Type IIB superstrings, BPS monopoles, and three-dimensional gauge dynamics. Nucl Phys B492:152–190, hep-th/9611230 21. Iqbal A, Kozcaz C, Vafa C (2009) The Refined Topological Vertex. JHEP 2009(0910):069, hep-th/0701156 22. Jimbo M, Miwa T (1994) Algebraic analysis of solvable lattice models, vol 85, cbms edn. AMS Society 23. Kanno S, Matsuo Y, Zhang H (2013) Extended Conformal Symmetry and Recursion Formulae for Nekrasov Partition Function. JHEP 1308:028 24. Kimura T (2016) Double quantization of Seiberg-Witten geometry and W-algebras. 1612.07590 25. Leung N, Vafa C (1998) Branes and Toric Geometry. Adv Theor Math Phys 2:91–118, hepth/9711013 26. Marshakov A (1999) Seiberg-Witten theory and integrable systems. World Scientific

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27. Miki K (2007) A (q, γ) analog of the W1+∞ algebra. Journal of Mathematical Physics 48(12):123520 28. Mironov A, Morozov A, Zenkevich Y (2016) Ding–Iohara–Miki symmetry of network matrix models. Phys Lett B762:196–208, 1603.05467 29. Nekrasov N (2004) Seiberg-Witten prepotential from instanton counting. Adv Theor Math Phys 7:831, hep-th/0306211 30. Nekrasov N (2016) BPS/CFT correspondence: non-perturbative Dyson-Schwinger equations and qq-characters. JHEP 03:181, 1512.05388 31. Nekrasov N, Shatashvili S (2009) Quantization of Integrable Systems and Four Dimensional Gauge Theories. World Scientific Publishing Co. Pte. Ltd., pp 265–289, 0908.4052 32. Nekrasov N, Pestun V, Shatashvili S (2013) Quantum geometry and quiver gauge theories 1312.6689 33. Schiffmann O, Vasserot E (2013) Cherednik algebras, W algebras and the equivariant cohomology of the moduli space of instantons on A2 . Pub Math de l’IHES 118(1):213–342, 1202.2756v2 34. Wyllard N (2009) A N −1 conformal Toda field theory correlation functions from conformal N=2 SU(N) quiver gauge theories. JHEP 11:002, 0907.2189

Parallelizations for Induced Representations of SO0 (2, q) Patrick Moylan

Dedicated to I.E. Segal in commemoration of the centenary of his birth in 2018

Abstract We consider generalizations to S O 0 (2, q) of results in [13, 15, 16, 21] dealing with parallelizations of associated induced bundles for S O0 (2, 4) = SU (2, 2)/Z2 representations which are induced from (non-unitary) finite dimensional representations of a maximal parabolic subgroup.

1 S O(2, q), so(2, q) and Subgroups of S O0 (2, q) Consider the real vector space Rq+2 of dimension n = q + 2 with quadratic 2 + x02 − x12 − . . . − xq2 . Let β0 = diag(1, 1, −1, −1, . . , −1). form Q(x) = x−1 Let G = S O(2, q) = {g ∈ S L(n, R)|g † β0 g = β0 } and S O0 (2, q) be the component connected to the identity. († denotes transpose.) The Lie algebra of G is so(2, q) = {X ∈ sl(n, R) | X † β0 + β0 X = 0}. A basis of g = so(2, q) is Li j with i, j = −1, 0, 1, 2, . . . , q and i < j. Let Li j = −L ji for i > j. The Li j satisfy the

P. Moylan—On leave of absence from the Pennsylvania State University. P. Moylan (B) Department of Mathematics, Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague, Trojanova 13, 120 00 Prague, Czech Republic e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2018 V. Dobrev (ed.), Quantum Theory and Symmetries with Lie Theory and Its Applications in Physics Volume 1, Springer Proceedings in Mathematics & Statistics 263, https://doi.org/10.1007/978-981-13-2715-5_12

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following commutation relations: [Lab , Lbc ] = − eb Lac

(1)

with e−1 = e0 = −e1 = 1. All other commutators vanish. Some important subgroups of G are the following ⎫ ⎤ cos τ sin τ 0 ⎬ K = k(u) = ⎣ − sin τ cos τ 0 ⎦ τ ∈ [0, 2π) , u ∈ S O(q) ⎭ ⎩ 0 0 u ⎧ ⎨

⎧ ⎪ ⎪ ⎪ ⎪ ⎨

⎡

⎡

es ⎢0 ⎢ A0 = Σ × ⎢ ⎢0 ⎪ ⎪ ⎣0 ⎪ ⎪ ⎩ 0

0 0 et 0 0 Iq−2 0 0 0 0

0 0 0 et 0

⎤ ⎡ 10 0 ⎢0 1 0⎥ ⎥ −1 1 ⎢ ⎢ 0⎥ ⎥ Σ Σ = √2 ⎢ 0 0 ⎣0 1 0⎦ 10 es

0 0 ... 0 0

0 1 0 −1 0

⎤⎫ 1 ⎪ ⎪ ⎪ ⎪ 0 ⎥ ⎥⎬ 0 ⎥ ⎥⎪ 0 ⎦⎪ ⎪ ⎪ ⎭ −1

with s, t ∈ R. (The number of 1’s and −1’s on the diagonal of Σ are equal and, for q odd, the middle row of Σ consists of all zeros except for a 1 in the center of the row.) ⎫ ⎧ ⎡ ⎤ 0 y0 y yq−1 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎢0 0 z ⎥ 0 −y ⎬ ⎨ q−1 ⎢ ⎥ n0 −1 † † ⎥ ⎢ N0 = e n0 = n0 (y0 , y, yq−1 , z) = Σ × ⎢ 0 0 0q−2 −z −y ⎥ × Σ ⎪ ⎪ ⎪ ⎪ ⎣0 0 0 0 −y0 ⎦ ⎪ ⎪ ⎪ ⎪ ⎭ ⎩ 0 0 0 0 0 with y0 , yq−1 ∈ R, y, z ∈ Rq−2 . ⎧ ⎪ ⎪ ⎪ ⎨

⎡

2 2 u2 − v2 1 − u2 + v2 u v 2 2 ⎢ −u I1 0 u ⎢ n(u,v) N = n(u, v) = e =⎢ v† 0 Iq−1 −v † ⎪ ⎣ ⎪ ⎪ 2 2 2 2 ⎩ − u2 + v2 u v 1 + u2 − v2

with n(u, v) = ..., q − 1).

q−1 i=0

⎫ ⎤ ⎪ ⎪ ⎪ ⎥ ⎬ ⎥ q−1 ⎥ u ∈ R, v ∈ R ⎪ ⎦ ⎪ ⎪ ⎭

yi Pˆ i , y0 = 2u, yi = 2vi and −2Pˆ i = −L−1i + Liq (i = 0, 1,

⎧ ⎡ 2 2 1 − t2 + s2 ⎪ ⎪ ⎨ ⎢ t ˜ ˜ s) = en(u,v) =⎢ N˜ = n(t, † ⎣ s ⎪ ⎪ ⎩ 2 2 t − s2 2

⎫ ⎤ 2 2 −t s − t2 + s2 ⎪ ⎪ ⎬ ⎥ I1 0 t q−1 ⎥ t ∈ R, s ∈ R † ⎦ 0 Iq−1 s ⎪ ⎪ ⎭ t2 s2 t −s 1 + 2 − 2

Parallelizations for Induced Representations …

with n˜ (t, s) = q − 1).

q−1 i=0

221

xi Pi x0 = 2t, xi = 2si and −2Pi = L−1i + Liq (i = 0, 1, ...,

⎫ ⎧ ⎡ ⎤ ch(t) 0 sh(t) ⎬ ⎨ A = a(t) = etL−1q = ⎣ 0 Iq+1 0 ⎦ t ∈ R ⎭ ⎩ sh(t) 0 ch(t) +

S O0 (2, q)|g † γ0+ g

−

S O0 (2, q)|g † γ0− g

H = {g ∈

H = {g ∈

=

γ0+ }

=

γ0− }



  ∗ 0 = h = ∈ S O0 (2, q) 0 ±1 



+



  ±1 0 = h = ∈ S O0 (2, q) 0 ∗ −

where ⎡

1 ⎢ 0 γ0+ = ⎢ ⎣0 0

0 0 1 0 0 −Iq−1 0 0

⎤ 0 0⎥ ⎥ 0⎦ 1

⎡

and

1 ⎢ 0 γ0− = ⎢ ⎣0 0

0 −1 0 0

⎤ 0 0 0 0⎥ ⎥ Iq−1 0 ⎦ 0 1

If we neglect the row and column containing ±1, we see that H + and H − as (q + 1) × (q + 1) matrices are precisely the orthochronous anti-de Sitter group and orthochorous de Sitter group, respectively [2, 18, 20]. We let H0± denote their respective connected components. Specifically, H0+ is S O0 (2, q − 1), the anti de Sitter subgroup, and H0− is S O0 (1, q), the de Sitter subgroup. A finite subgroup of order two is   W = Iq+2 ∪ {Q} with ⎛

Q = e−π(L−1,0 −Lq−1,q )

−1 ⎜ 0 ⎜ =⎜ ⎜ 0 ⎝ 0 0

0 −1 0 0 0

0 0

0 0 Iq−2 0 0 −1 0 0

⎞ 0 0 ⎟ ⎟ 0 ⎟ ⎟. 0 ⎠ −1

If M = C H − (A) be the centralizer of A in H − , then M = S O0 (1, q − 1)  W where W acts on S O0 (1, q − 1) by conjugation. The two Poincaré subgroups are Pq = M0  N ∼ = S O0 (1, q − 1)  N

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P˜ q = M0  N˜ ∼ = S O0 (1, q − 1)  N˜ The two scale extended Poincaré subgroups are Wq = NG (Pq ) = ((S O0 (1, q − 1)  W ) × A)  N = (M × A)  N q = NG (P˜ q ) = ((S O0 (1, q − 1)  W ) × A)  N˜ = (M × A)  N˜ W where NG (Pq ) (NG (P˜ q )) is the normalizer in G of the subgroup Pq (P˜ q ). G/Wq  (S 1 × S q−1 )/Z2 = M0 [1]. M0 is q-dimensional compactified Minkowski space and M0 denotes Minkowski space (q dimensional). Some important subgroups of the S O0 (1, q) subgroup which we shall make use of are the following: ⎧⎡ ⎪ ⎪ 1 ⎨ ⎢0 K2 = ⎢ ⎣0 ⎪ ⎪ ⎩ 0

0 1 0 0

0 0 ui j uq j

⎫ ⎤ 0  ⎪ ⎪  ⎬ u i j u iq 0 ⎥ ⎥ ; = u ∈ S O(q) u iq ⎦ u q j u qq ⎪ ⎪ ⎭ u qq

⎫ ⎧ ⎡ ⎤ 1 0 0 0 ⎪ ⎪ ⎪ ⎪ ⎬ ⎨ ⎢ 0 ch(t) 0 sh(t) ⎥ ⎢ ⎥ ; t ∈ R A2 = a2 (t) = ⎣ 0 0 Iq−1 0 ⎦ ⎪ ⎪ ⎪ ⎪ ⎭ ⎩ 0 sh(t) 0 ch(t) ⎧⎡ 1 ⎪ ⎪ ⎨⎢ 0 M2 = ⎢ ⎣0 ⎪ ⎪ ⎩ 0

0 1 0 0

0 0 ui j 0

⎫ ⎤ 0 ⎪ ⎪ ⎬ 0⎥ ⎥ (u i j ) ∈ S O(q − 1) ∼ = S O(q − 1); 0 ⎦ ⎪ ⎪ ⎭ 1

⎧ ⎡ 1 0 0 0 ⎪ ⎪ ⎪ r2 ⎨ ⎢ 0 1 + r2 r ⎢ 2 2 N˜ 2 = n˜ 2 (r ) = en˜ 2 (r ) = ⎢ † † ⎪ ⎣ 0 r I r q−1 ⎪ ⎪ 2 ⎩ r2 0 − 2 −r 1 − r2 with n˜ 2 (r ) =

q−1 i=1

N2 =

with n2 (ξ) =

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

⎤

⎫ ⎪ ⎪ ⎪ ⎬

⎥ ⎥ ⎥ | r ∈ Rq−1 ⎪ ⎦ ⎪ ⎪ ⎭

˜ i , wi = −2ri and −2G ˜ i = L0i + Liq (i = 1, ..., q − 1). wi G ⎫ ⎤ 1 0 0 0 ⎪ ⎪ 2 2 ⎬ ⎢0 1 + ξ ξ −ξ ⎥ q−1 2 2 ⎥|ξ ∈R =⎢ ⎣ 0 ξ † Iq−1 −ξ † ⎦ ⎪ ⎪ ⎭ 2 2 0 ξ2 ξ 1 − ξ2 ⎡

n 2 (ξ) = en2 (ξ)

q−1 i=1

z i Gi , z i = −2ξi and −2Gi = −L0i + Liq (i = 1, ..., q − 1).

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2 Group Decompositions We have the following decompositions of G = S O0 (2, q). The Iwasawa decomposition of S O0 (2, q) is [8, 18] S O0 (2, q) ∼ = K × A0 × N0

(2)

A maximal parabolic subgroup of G is Wq = (M × A)  N and a Bruhat-like decomposition relative to this subgroup is [18] D0 = N˜ × M × A × N

(3)

with D0 open, dense in S O0 (2, q). The decompositions of Sekiguchi for S O0 (2, q) are [18] (4) D+ = H + × A × N D− = H − × A × N

(5)

with D+ and D− being dense, open subsets of S O0 (2, q). Corresponding decompositions for S O0 (1, q) are: the Iwasawa decomposition [5] (6) S O0 (1, q) ∼ = K 2 × A2 × N2 ; the Bruhat decomposition [5] D02 = N˜ 2 × M2 × A2 × N2 (M2 = S O(q − 1));

(7)

and the Hannabuss decomposition [5] D2 = N S O0 (1,q) (M0 ) × A2 × N2 = (M0  W2 ) × A2 × N2

(8)

with D02 and D2 being dense, open subsets of S O0 (1, q). M0 = S O0 (1, q − 1) and N S O0 (1,q) (M0 ) = M0  W2 where W2 = {Iq+2 } ∪ {β0 } for q even (cf. [5]).

3 Geometrical Results Let ei be the vector in R2+q with one in the ith position and zeros elsewhere. Set e = e−1 + en . The stabilizer of e is Pq . Wq sends e into ±λe where λ = es ∈ R+ and n˜ 2 (r )a2 (t)e =

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⎡

1 0 0 0 r2 ⎢ 0 1+ r 2 r ⎢ 2 2 ⎢ ⎣ 0 r † Iq−1 r † 2 2 0 − r2 −r 1− r2

⎤⎡

⎤ ⎤⎡ ⎤ ⎡ 1 1 1 0 0 0 ⎥⎢ 1 2 1 2 ⎢ ⎥ ⎢ r cht ⎥ ⎥ ⎢ 0 ch(t) 0 sh(t) ⎥ 2 ⎥. ⎥ ⎢ 0 ⎥ = ⎢ (1+ 2 r )sht+ ⎥⎣ † t ⎣ ⎣ ⎦ ⎦ ⎦ 0 r e ⎦ 0 0 Iq−1 0 1 2 1 2 1 0 sh(t) 0 ch(t) (1− 2 r )cht − 2 r sht

Let Vq ∼ = S O0 (1, q)/S O0 (1, q−1) which is q dimensional de Sitter space. Then (t, r ) are horospherical coordinates on one half of Vq [5] and using the above result we readily establish that as a decomposition into the Cartesian product of sets N 2 × A 2 × W2 ∼ = Vq

(9)

with Vq a dense, open subset of Vq . (Compare Ref. [12] for a slightly different but more detailed treatment of the q = 4 case.) From Eq. (5) and H − = S O0 (1, q)  W we get S O0 (2, q) ⊃ D− = S O0 (1, q)  W × A × N dense

(10)

By taking transposes of both sides of Eq. (8) we get the decomposition 2 × A2 × W2  S O0 (1, q−1) , S O0 (1, q) ⊃ D†2 = N dense

(11)

2 . (10) and (11) ⇒ since Λ ∈ S O0 (1, q−1) ⇒ Λ† ∈ S O0 (1, q−1) and N2† = N 2 × A2 × W2 × (S O0 (1, q−1)  W × A × N ) . S O0 (2, q) ⊃ U ∼ =N !" # dense

(12)

Wq

Hence

2  A2 ) × W2 ⊂ G/Wq ∼ (N = M0

(13)

2  A2 ) × W2 as a dense, open subset. The explicit injective diffeomorphism of ( N into M0 is given in [13] for the q = 4 case, which is completely representative of the general case. It makes use of a mapping between (pseudo)-spherical coordinates [13] on Vq and spherical coordinates on S 1 × S q−1 . By combining Sekiguchi’s decomposition, Eq. (4), for S O0 (2, q) with his decomposition, Eq. (5), for S O0 (2, q − 1) we can repeat this analysis to obtain results for Ad Sq analogous to those for Vq described in this section.

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4 Parallelizations of Induced Bundles and Representations of S O0 (2, q) A fibre bundle is a triple (T, π, B) where π : T → B is a map. The space B is the base space, the space T is the total space and the map π is the projection of the bundle. For each b ∈ B the space π −1 (b) is called the fibre of the bundle over b ∈ B. We use the notation π : T → B, or just T → B, to denote the fibre bundle. A cross section of a fibre bundle π : T → B is a map γ : B → T such that π ◦ γ = I B where I B is the identity map on B. In other words, a cross section is a map γ : B → T such that γ(b) ∈ π −1 (b), the fibre over b, for each b ∈ B. For H a topological group, π : T → B is a principal H bundle over B if T is a locally trivial free H -space [7]. Let G be a Lie group and let H be a closed subgroup of G. Then G is a principal H -bundle over the (left) coset space G/H . For ρ a representation of a subgroup P of G on a vector space V define the space of covariance functions: Cρ∞ (G) = { f : G → V , f (x · p) = ρ( p −1 ) f (x) ∀ x ∈ G, h ∈ P}.

(14)

Consider the associated induced bundle G ×ρ V → G/P defined as the quotient space (G × V )/P where P acts on G × V via the action (x, v) · p = (x · p, ρ( p −1 )v). Set Γ ∞ (G ×ρ V ) → G/P equal to the space of C ∞ sections of G ×ρ V → G/P. We have the following result whose proof is straightforward: Proposition 1 Γ ∞ (G ×ρ V ) and Cρ∞ (G) are naturally isomorphic as vector spaces over C. Specifically, for f ∈ Cρ∞ (G) define φ : Cρ∞ (G) → Γ ∞ (G ×ρ V ) by setting φ( f ) =  ∈ Γ ∞ (G ×ρ V ) where (b) = [(g, f (g))] for any choice of g ∈ G such that π(g) = b ∈ B and [(g, f (g))] denotes equivalence class of (g, f (g)) in G ×ρ V . Thus, for a representation π ρ of G on Cρ∞ (G) defined by [(π ρ (g)( f )](x) = f (g −1 x) (x ∈ G),

(15)

there is an associated representation π ρ of G on Γ ∞ (G ×ρ V ) defined by [π ρ (g)()](y) = g ◦ (g −1 · y) ( ∈ Γ ∞ (G ×ρ V )) where g −1 · y ∈ G/P with

φ ◦ π ρ = π ρ ◦ φ.

(16)

If π : G → G/P is trivial then the associated vector bundle G ×ρ V → G/P is trivial. Thus C ∞ sections  of G ×ρ V → G/P can be identified with C ∞ functions

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ψ from G/P to V given a particular trivialization (or parallelization) of G ×ρ V → G/P [6]. Parallelizations of induced bundles are interesting from the point of view of physics mainly because they allow for a unified comparison of different models of the universe. This is made abundantly clear in Refs. [15, 16] for the case of S O0 (2, 4) and in Ref. [14] for the general S O(2, n) case where a concrete application for calculating Casimir energies for n − 1 spheres in n dimensions is given. In the general case of G = S O0 (2, q), we can relate, in a one-to-one fashion, certain C ∞ functions on M0 (q dimensional Minkowski space), transforming according to a given representation of G, to C ∞ functions on M0 (a quotient of the q dimensional Einstein universe) associated with an equivalent representation of G, provided the associated bundles are trivial. The explicit description of parallelizable induced bundles and explicit formulae for relating different parallelizations are given in Ref. [15]. The following is a partial summary of Theorem 4.1 of Ref. [15]: (i) let N be a Lie subgroup of G; (ii) let there be given a (possibly only local) C ∞ action φ of G on N which satisfies φ(x)y = x y for x, y ∈ N; (iii) we are given a finite dimensional representation ρ of the isotropy subgroup G x0 of a point x0 ∈ N; (iv) C ∞ (N)  ψ(x) := S(x x0−1 )(x) for x ∈ N and  a C ∞ section of G ×ρ V → G/G x0 and where S(g, x) is a C ∞ function on G × N such that S(g; x0 ) = ρ(g) for g ∈ G x0 , the inducing subgroup. S(g; x) takes the fibre over φ(g −1 )x to the fibre over x. Now to representations of S O0 (2, q) induced from finite dimensional representations of Wq . Let A∗ be the space of all characters i.e. the space of all homomorphism q χ : A → C. For σ ∈ C, let A∗  χσ (a(s)) = e−(σ+ 2 )s . Irreducible (unitary) representations of W = {I, Q} are all one dimensional and are specified by ρ (I ) = (+1) and ρe (Q) = (−1) where takes the values 0 or 1. Since so(1, q − 1) and so(q) are just different real forms of their complexification, so(q, C), it follows that the finite dimensional non-unitary representations of S O0 (1, q − 1) are in one-to-one correspondence with and can be constructed in a straightforward way out of the unitary representations of S O(q) described in [3] or [17]. Let D s denote such a representation of M0 = S O0 (1, q − 1) where s = {i } are a sequence of numbers indexing the representation. The representation space is V s and the action of m 0 ∈ M0 on it is denoted by D s (m 0 ). Now let D (s, ) = D s ⊗ ρ be the corresponding representation of M = M0 × W on V s ⊗ C  V s . Consider D (s, ) ⊗ χσ : M × A → End(V s ) and extend this mapping to a mapping from the parabolic subgroup Wq = (M × A)  N to End(V s ) by requiring that it act trivially on N i.e. D (s, ) ⊗ χσ ⊗ 1 : Wq → End(V s ) with (D (s, ) ⊗ χσ ⊗ 1)( p) = D (s, ) (m)χσ (a(s))

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where p = ma(s)n ∈ P. We now define, according to Eq. (15), a representation π ρ (G) on Cρ∞ (G) with ρ = ρ(s, , σ) = D (s, ) ⊗ χσ ⊗ 1.

5 A Generalization of the Oscillator Parallelization for S O0 (2, q) (q even, q > 3) Define r = with

q−2 2

mutually commuting Heisenberg pairs {2 (i) , 3 (i)} (i = 1, . . ., r ) 2 (i) = L0 2i+1 + L1 2i+1 − L−1 2i+2 + L2 2i+2 3 (i) = L0 2i+2 + L1 2i+2 − L−1 2i+1 + L2 2i+1 .

We have % $ 2 (i), 3 ( j) = 1 δi j where 1 = 2(L−10 + L−11 + L12 + L02 ) and all other commutators vanishing. The Lie algebra spanned by 2 (i), 3 (i) (i = 1, . . ., r ) and 1 is isomorphic to the Heisenberg(-Weyl) algebra hr of order r [4, 11]. It is a 2r + 1 = q − 1 dimensional Lie algebra. Next consider q = L−10 − L12 + L34 + L56 + · · · + Lq−1 q and let the Lie algebra generated by q be a0 . We have % $ q , 2 (i) = −3 (i), $ % q , 3 (i) = 2 (i) and % $ q , 1 = 0. We consider the semidirect sum a0 ⊕s hr

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with the action of a0 on hr being the adjoint action [11]. This is the generalization to S O(2, q) (q even) of the oscillator algebra for S O(2, 4) described in Ref. [21]. It is a q dimensional Lie algebra, which is called the q dimensional oscillator algebra. The associated Lie group, which we denote by ea0 ⊕s hr , is a solvable Lie group [11] called the q dimensional oscillator group. It consists of all real q × q matrices of the form [21] 

e

x 1 1 +

r 

i=1

αi 2 (i)+

r 

i=1

 βi 3 (i)

e xq q

with x1 , αi , βi , xq ∈ R. It is a straightforward generalization of the results in Ref. [21] to prove that ea0 ⊕s hr is diffeomorphic to its image in M0 and that it is dense in M0 .

6 Results on Various Parallelizations for the Representations π (s,,σ) (S O0 (2, q)) We now apply the method given in Theorem 4.1 of Ref. [15] and outlined in Sect. 4 to describe some parallelizations for the representations π (s, ,σ) (S O0 (2, q)). In all cases, the isotropy subgroup G x0 of x0 is Wq ∼ = (M × A)  N , the scale extended Poincaré group, and N refers to the subgroup N in Theorem 4.1 of Ref. [15] and whose properties were stated in Sect. 4. (1) Curved (or compact) parallelization: (q = 2, 4 or 8) the subgroup N is a quotient of the subgroup K /K 2  S O(2) × S O(q)/S O(q − 1)  S 1 × S q−1 . N is diffeomorphic to (S 1 × S q−1 )/Z2  M0 where the Z2 action is the product of antipodal maps on S 1 × S q−1 [13, 15]. This parallelization is associated with the Iwasawa decomposition of S O0 (2, q) described above in Sect. 2. Explicit descriptions of the representations π (s, ,σ) (S O0 (2, q)) in this parallelization for the q = 4 case are given in much detail in Refs. [15, 16]. (2) Flat (or Minkowski space) parallelization: N is the translation subgroup N˜ of the Poincaré subgroup, P˜ q . This parallelization is well-known in physics as the one associated with the action of the conformal group, S O0 (2, q)/Z2 , on q-dimensional Minkowski space, M0 . The explicit expression for the parallelization map S(x x0−1 ) and explicit formulae relating the curved and flat parallelizations in the q = 4 case is worked out in detail in Refs. [13] and [15]. Details about the properties of the representations π (s, ,σ) (S O0 (2, q)) in this parallelization for the q = 4 case can be obtained from results in [10, 15, 16]. 2  A2 of Eq. (13). The relationship between (3) De Sitter parallelization: N is N the flat (Minkowski space) parallelization and this de Sitter parallelization leads to a Lorentzian variant of sterographic projection from q-dimensional de Sitter space to q-dimensional Minkowski space (cf. Ref. [13]).

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(4) Generalization of oscillator parallelization: the subgroup N is eh⊕s Wr . The oscillator parallelization for SU (2, 2), the two-fold cover of S O(2, 4), plays an important role in the DL F theory of A. Levichev [9]. Nevertheless, there does not seem to be much in the literature, if at all, by way of explicit description of the representations of S O0 (2, q) in this parallelization.

References 1. E. Angelopolous, M. Laoues, Rev. Math. Phys. 10 (3) (1998) 271–299. 2. T.P. Branson, J. Funct. Anal. 74 (1987) 199–291. 3. V.K. Dobrev, G. Mack, V.B. Petkova, S.G. Petrova and I.T. Todorov, Harmonic Analysis on the n-Dimensional Lorentz Group and Its Applications to Conformal Quantum Field Theory, Lecture Notes in Physics, 63, (Springer-Verlag, Berlin-Heidelberg-New York, 1977), pp. 45– 56. 4. G. Folland, Harmonic Analysis in Phase Space (Princeton University Press, Princeton, 1989). 5. K.C. Hannabuss, Math. Proc. Camb. Phil. Soc. 70 (1971) 283–299. 6. J. Hebda, P. Moylan, Math. Proc. Camb. Phil. Soc. 103 (1988) 285–298. 7. Dale Husemoller, Fibre Bundles, 2nd edn. (Springer, New York, 1975), pp. 39–41. 8. K. Iwasawa, Ann. of Math. 50 (2) (1949) 507–558. 9. A. V. Levichev, Phys. Scr. 83 (2011), 1–9 (015101). 10. G. Mack, Commun. Math. Phys. 55 (1977) 1–28. 11. A. Medina, P. Revoy, Manuscripta Mathematica 52 (1985) 81–95. 12. M.V. Mensky, Method of Induced Representations, Space-time and the Particle Concept [in Russian] (Nauk, Moscow, 1978). 13. P. Moylan, J. Math. Phys. 36 (6) (1995) 2826–2879. 14. P. Moylan in Lie Theory and Its Applications in Physics ed. V.K. Dobrev. Springer Proceedings in Mathematics & Statistics, 36 (Springer, Tokyo, 2015) pp. 231–238. 15. S.M. Paneitz, I.E. Segal, J. Funct. Anal. 47 (1982) 78–142. 16. S.M. Paneitz, J. Funct. Anal. 47 (1982) 78–142. 17. F. Schwarz, J. Math. Phys. 12 (1) (1971) 131–139. 18. J. Sekiguchi, Nagoya Math. J. 79 (1980) 151–185. 19. P. Sawyer, Linear Algebra and its Applications 493 (2016) 573–579. 20. R.F. Streater, A. S. Wightman, PCT, Spin and Statistics and All That, (W. A. Benjamin, New York, 1964), p. 11. 21. O.S. Svidersky, Proc. Amer. Math. Soc. 125 (8) (1997) 2485–2491.

Jordan Algebra and Hydrogen Atom Todor Popov

Abstract We show that the causal automorphisms of the Minkowski space-time and the dynamical group of the hydrogen atom stem both from the Tits–Kantor–Koecher conformal construction for a Jordan algebra.

1 Introduction The hydrogen atom is one of the few quantum mechanical systems that allows for an exact solution. The quantum system of an electron (without spin) interacting with the proton through the Coulomb potential is the quantum counterpart of the classical Kepler problem in the celestial mechanics. On the top of the angular momentum L one has integrals of motion given by the Laplace-Runge-Lenz vector X. These six integrals of motion span S O(4) group commuting with the hamiltonian, the so called accidental symmetry explaining the degeneracy of energy levels of nonrelativistic hydrogen atom (depending only on the principal quantum number n). The group S O(2) × S O(4) can be further extended to a conformal S O(2, 4) dynamical group of a hydrogen system. The dynamical symmetry of the hydrogen-like systems has been revealed in the works of Barut and collaborators [2]. Gouwu Meng [8] considered the “spectrum generating” algebra so(2, 4) of the hydrogen system from the point of view the conformal symmetry of the Jordan algebra structure on the Pauli matrices. In fact Meng has gone much further, he has attached to every euclidean Jordan algebra its own generalized hydrogen-like dynamical system based on the Tits–Kantor–Koecher construction of its conformal algebra.

T. Popov (B) Institute for Nuclear Research and Nuclear Energy, Tsarigradsko Chaussee 72, 1784 Sofia, Bulgaria e-mail: [email protected] T. Popov American University in Bulgaria, Blagoevgrad, Bulgaria © Springer Nature Singapore Pte Ltd. 2018 V. Dobrev (ed.), Quantum Theory and Symmetries with Lie Theory and Its Applications in Physics Volume 1, Springer Proceedings in Mathematics & Statistics 263, https://doi.org/10.1007/978-981-13-2715-5_13

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In this note we elucidate the connection between the causal automorphisms of the Minkowski space-time and the conformal dynamical symmetry of the hydrogen atom. We show that the two isomorphic representations of the conformal algebra so(2, 4) are realized as differential operators on the Jordan algebra J and J∗ . In fact the connection is provided by the Fourier transform on the light cone given in the work for minimal U (2, 2)-representations of Mack and Todorov [6]. At first sight it is puzzling how a non-relativistic system can be matched to a problem for the relativistic cone. It turns out that the solutions of the relativistic KleinGordon equation are mapped to the solutions for the non-relativistic Schroödinger equation, the accidental S O(4) symmetry being broken in a controlled way [2]. Thus the states of both relativistic and non-relativistic hydrogen system carry a modified action of the dynamical symmetry so(2, 4) which is simply related to the conformal algebra of the causal automorphisms, i.e., to the transformations preserving future cone of Minkowski space-time.

2 Jordan Algebra and Its Structure Algebra Jordan layed the foundation stone of the axiomatic purely algebraic treatment of the observables in his paper [5] introducing a class of operator algebras with a commutative product x ◦ y = y ◦ x and relations (x 2 ◦ y) ◦ x = x 2 ◦ (y ◦ x)

∀x, y ∈ J .

These algebras were coined Jordan algebras. The Jordan product ◦ was modelled as the symmetric part x ◦ y = 21 (x y + yx) of an associative product of Hermitian matrices which is stable under the Hermitian conjugation. A Jordan algebra is commutative but is not associative. However the identity implies the power-associativity, x m+n = x m ◦ x n . The Jordan multiplication map is defined to be Lx y = x ◦ y A derivation d ∈ der(J) of a Jordan algebra J is an element of gl(J) such that d(a ◦ b) = (da) ◦ b + a ◦ db

d ∈ der(J) .

Hence the commutator of a Jordan multiplication and a derivation is again a Jordan multiplication a ∈J. [d, L a ] = L da

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Therefore there is a bigger Lie algebra merging the derivations and the Jordan multiplications: str(J) = der(J)  L J called the structure algebra of a Jordan algebra J. The Jordan identity [[L x , L y ], L z ] = L [L x ,L y ]z implies that the commutator [L a , L b ] is an inner derivation, [L a , L b ] ∈ der(J). Hence the structure algebra is Z2 -graded str(J)1 = L J

str(J)0 = der(J) .

We assume that J has a unit e, then the operator L e generates the dilatation in J and it is in the center of str(J). The reduced structure algebra str (J) is str(J) modulo its center, str(J) = str (J) ⊕ RL e . Jordan triple product. The structure algebra str(J) allows for a uniform description with the help of the triple Jordan product (abc) = a ◦ (b ◦ c) − b ◦ (a ◦ c) + (a ◦ b) ◦ c = [L a , L b ]c + L a◦b c .

(1)

The Jordan triple product defines the Jordan Triple System with the relations (abc) = (cba)

(2)

(ab(cd x)) − (cd(abx)) = (a(dcb)x) − ((cda)bx) .

(3)

y

For any pair of elements (x, y) ∈ J × J one defines a linear map Sx : J → J through Sxy (z) = (x yz) . Note that the Jordan multiplication operators L y z arise as L y z := S ye (z) = Sey (z) . Then the product (abc) takes values in (a subspace of) str(J) and the operators Sab transform between themselves d b − Sc(dab) = Sa(bcd) − S(cda) [Sab , Scd ] = S(abc)

which is an equivalent form of the relations (3).

(4)

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3 Tits–Kantor–Koecher Construction The construction of Tits, Kantor and Koecher associated to a Jordan algebra J its Möbius algebra co(J) which is the 3-graded Lie algebra co(J) = g+1 ⊕ g0 ⊕ g−1 ∼ = J ⊕ str(J) ⊕ J∗ that is, [gi , g j ] = gi+ j and gi = 0 for i = 0, ±1. The elements Ua ∈ g−1 (U a ∈ g+1 ) are in bijection with the elements of the Jordan algebra a ∈ J. Due to the 3-grading of co(J) the elements of g1 (g−1 ) commute between themselves [Ua , Ub ] = 0

[U a , U b ] = 0 .

(5)

The Möbius algebra co(J) is endowed with a conjugation † U a = Ua† ∈ g+1

(gi )† = g−i

The self-conjugated subalgebra g0 ⊂ co is the Jordan structure algebra str(J) spanned by the elements Sba := [U a , Ub ] ∈ g0

(Sba )† = Sab

(6)

whose bracket is closed in view of Eq. (4). The action of g0 = str on the g±1 is given by [U c , Sba ] = U (abc) (7) [Sab , Uc ] = U(abc) where (abc) stays for the Jordan Triple Product (1). In summary the Tits–Kantor–Koecher construction associates functorially to every Jordan algebra J the Lie algebra co(J) havins as relations Eqs. (4)–(7). The Möbeus algebra co(J) compound of a triple of embedded Lie algebras der(J) ⊂ str(J) ⊂ co(J) . The symmetric cone Ω(J) consists of the positive elements in J, Ω(J) = {x 2 |x ∈ J} and the tube domain is defined by TΩ (J) = J ⊕ iΩ(J), a subspace in the complexification of J. Exponentiating the Lie algebras g to Lie groups G = Lie(g) yields the inclusion of the automorphisms Aut (J), the structure group Str (J) and the Möbius group Co(J) Aut (J) ⊂ Str (J) ⊂ Co(J) .

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The structure group Str (J) is preserving the symmetric cone Ω while the conformal group Co(J) consists of the holomorphic automorphisms of the tube domain TΩ (J). Oscillator Realization of co(J) [4]. Let eα be a basis of the Euclidean Jordan algebra of J: x = x α eα . The structure constants of Jordan Triple Product reads βσ eσ . (eα eβ eγ ) = Σαγ

The canonical commutation algebra with generators Aα and Aβ [Aα , Aβ ] = δαβ

[Aα , Aβ ] = 0 = [Aα , Aβ ]

(8)

lead to realization of the structure algebra str(J) = g0 (J) ac d A Ac Sba = −Σbd

⇒

cs a ac s [Sba , Sdc ] = −Σdb Ss + Σds Sb

Then the subalgebras g−1 (J) and g+1 (J) are represented by the oscillator modes as Ua = −Aa

as b c U a = Σbc A A As .

Finally it is easy to verify the remaining relations (7) of the Lie algebra co(J) [Ua , U b ] = Sab

as [Sba , Uc ] = Σbc Us .

We conclude that given the canonical commutation algebra (8) with generators in bijection with the basis of the Jordan algebra J one obtains a realization of the conformal algebra co(J).

4 Causal Automorphisms of Space-Time The Jordan algebra JC 2 of hermitian 2 × 2 matrices over C is a particularly appealing and instructive since its Tits–Kantor–Koecher algebra co(J) has a direct interpretation as a space-time symmetry. A general element X in JC 2 is a hermitian matrix μ

X = X σμ =



X0 + X3 X1 + i X2 X1 − i X2 X0 − X3

 μ = 0, 1, 2, 3

where σμ stay for the Pauli matrices. An element X ∈ JC 2 is parametrized by a 4vector in a Minkowski space, 21 tr (X σμ ) = X μ . The Jordan product on the basis of JC 2 is simply 1 σμ ◦ σν = δμ,ν = {σμ , σν } . 2

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The norm of the algebra JC 2 is given by the determinant of the matrix det X = gμν X μ X ν = (X 0 )2 − (X 1 )2 − (X 2 )2 − (X 3 )2 which is nothing but the length of the 4-vector X in Minkowski space with metric gμν = (+, −, −, −). Automorphisms Aut (JC 2 ). The unitary group SU (2) is acting on an hermitian by spinorial rotation matrix X ∈ JC 2 U XU † = X 

U ∈ SU (2) .

(9)

It produces an orthogonal transformation in the group S O(3) = Aut (JC 2) X i = O ij X j i = 1, 2, 3

X 0 = X 0

O ∈ S O(3)

preserving the time component X 0 of the 4-vector X μ . The automorphism group C Aut (JC 2 ) = S O(3) is the exponentiation of derivations der(J2 ) = so(3) whereas the group SU (2) is the double covering of S O(3). The generators of the rotations are the angular momentum operators Mi j .   Structure group Str (JC 2 ). The reduced structure group Str (J) = Lie(str (J)) is the group of invariance of the natural norm of a Jordan algebra J given by the determinant. When the matrix U in the transformation (9) take values in the group S L(2, C) it maps hermitian matrix to an hermitian matrix without changing the determinant (the norm) det X  = det X

X μ = Λμν (U )X ν .

The Lorentz length is a transformation invariant and it results into a Lorentz transformation S O(1, 3) = Str  (J) of the Minkowski vector X with generators Mμν of both ordinary and hyperbolic rotations(Lorentz boosts). The group S L(2, C) is the complexification of SU (2) and a double covering of the Lorentz group S O(1, 3). When the matrix U = ρ11 for ρ ∈ R then the norm is rescaled by a real factor ρ2 , giving rise to the dilatation transformation D : X  = ρX . Conformal group Co(JC 2 ). A fractional linear transformation of the Jordan algebra is given by the Möbius transformation 

−1

X = (AX + B)(C X + D)

 U=

A B C B

 ∈ SU (2, 2)

which preserves the hermicity provided that the matrix U is a pseudounitary matrix in SU (2, 2). The group SU (2, 2) is a double covering of the conformal group Co(JC 2 ) = S O(2, 4). The group S O(2, 4) has as a proper subgroup the Poincaré group I S O(1, 3) of inhomogeneous Lorentz transformations

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X μ = Λμν (U )X ν + a μ a general element (Λ, a) of Poincaré group consists of Lorentz rotation and spacetime translations. The generators of the translations X μ = X μ + a μ are the momentum operators P μ . The special conformal transformations in S O(2, 4) are the transformations X μ =

X μ − X 2 cμ . 1 − 2(X · c) + X 2 c2

(10)

These are preserving the light cone in Minkowski space and therefore the causal structure of the space-time. The generators of the special conformal transformations are denoted by K μ . The Tits–Kantor–Koecher construction for JC 2 leads to the Lie groups embeddings Aut (J) ⊂ Str (J) ⊂ Co(J) S O(3) ⊂ S O(1, 3) × R+ ⊂ S O(2, 4) . ∼ The conformal algebra co(JC 2 ) = so(2, 4) commutators are concisely written as [J A,B , JC,D ] = i(g BC J AD − g B D J AC − g AC J B D + g AD J BC ) where the generators J AB ∈ so(2, 4) are indexed in the 6-dimensional space A, B = 0, 1, 2, 3, 5, 6 with metric g AB = (+, −, −, −, −, +). We adopt the convention that the greek letter indices always take values in {0, 1, 2, 3}. The identification of the so(2, 4) elements J AB with the generators of the symmetries given above Mμν = Jμ,ν Pμ = Jμ6 + Jμ5 D = J56 K μ = Jμ6 − Jμ5 yields the conformal algebra of the Minkowski space-time [D, Pμ ] [D, K μ ] [K μ , Pν ] [K α , Mμν ] [Pα , Mμν ]

= i Pμ [Pμ , Pν ] = 0 = −i K μ [K μ , K ν ] = 0 = 2i(gμν D − Mμν ) [D, Mμν ] = 0 = i(gαμ K ν − gαν K μ ) = i(gαμ Pν − gαν K μ )

[Mμν , Mαβ ] = i(gαν Mμβ − gβν Mαν − gαμ Mνβ + gμβ Mαν )

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The subalgebra structure of the conformal co(JC 2 ) is summarized in the diagram ∼ su(2) ∼ Span{Mi j } der(JC = = so(3) 2) = ∩ ∼ str(JC Span{Mμν , D} = sl(2, C) ⊕ R ∼ = so(1, 3) ⊕ R 2) = ∩ ∼ ∼ co(JC 2 ) = Span{Pμ , K μ , Mμν , D} = su(2, 2) = so(2, 4) The oscillator realization of co(JC 2 ), Ab = −

∂ ∂x b

Ab = x b

coincides with the so called ladder representation [6] of so(2, 4) ∼ = su(2, 2) of helicity (spin) λ = 0. The ladder representations are describe symmetries of the wave equation of the massless particles and also the dynamical symmetries of the hydrogen atom [6]. A scalar (of helicity λ = 0) massless field transforms under the conformal symmetries according to the rules [χ(x), Pμ ] = i∂μ χ(x) [χ(x), Mμν ] = i(xμ ∂ν − xν ∂μ )χ(x) . [χ(x), D] = −i(1 + x μ ∂μ )χ(x) [χ(x), K μ ] = i(2xμ + 2xμ xν ∂ ν − x 2 ∂μ )χ(x)

(11)

The conformal transformations being symmetries of the wave equation of a particle of mass “zero”, in the case of the hydrogen system this is the photon, the carrier of the electromagnetic interaction. We are going to switch our attention to another realization of the conformal symmetry so(2, 4).

5 Hydrogen Atom and Its Spectrum Generating Algebra The non-relativistic hydrogen system with spin 0 is governed by the Hamiltonian which we write in atomic units H=

Z p2 L2 Z p2 − = r + 2− . 2 r 2 2r r

(12)

For energy E < 0 the Hamiltonian H has a discrete spectrum E n , its eigenfunctions being the wave functions of bounded states r, θ, φ|nlm = ψnlm (r, θ, φ) = Nnl Rnl (r )Ylm (θ, φ)

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where Ylm (θ, φ) stays for the spherical harmonics. The spherical symmetry of the Coulomb-Kepler potential results in the conservation of the angular momentum hence the eigenfunctions ψnlm transform in so(3) multiplets L 2 ψnlm = l(l + 1)ψnlm

L 3 ψnlm = mψnlm

with orbital quantum number l and magnetic quantum number m ranging in m = −l, . . . , l which account for the eigenvalues of the square of the angular momentum L 2 and its projection L 3 . The stationary Schrödinger equation H ψnlm = E n ψnlm after separation of variables yields the Schrödinger radial equation Rnl (r ) 

pr2 l(l + 1) Z + − −E 2 2r 2 r

 R(r ) = 0 .

The isotropy of the potential implies that the energy does not depend on the magnetic quantum number m, thus for every l the energy should have 2l + 1 degeneracy. However the energy depends only on the principal quantum number n En = −

1 Z2 2 n2

n = 1, 2, . . .

and each level is stratified by the orbital number l = 0, 1, . . . , n − 1. The quantum n−1 (2l + 1). overall degeneracy of E n is thus n 2 = l=0 The accidental degeneracy of the spectrum of the hydrogen atom has been explained by the presence of an extra symmetry of the Kepler potential [3] the conservation of the Laplace–Runge–Lenz vector X X=

r 1 1 ( p × L − L × p) − Z = r p 2 − p(r · p) + r H . 2 r 2

(13)

Replacing in X the operator H by its discrete value E n < 0 one gets the modiX . The angular momentum L together fied Laplace–Runge–Lenz vector V = √−2E n

X with V = √−2E generates so(4) such that all states with fixed E n transform in one n irreducible n 2 -dimensional so(4)-representation. On the other hand states with different energy E n are interpolated by an additional radial so(1, 2) algebra with generators

T1 =

  1 L2 r pr2 + −r 2 r

T2 = r · p − i

T3 =

  1 L2 r pr2 + +r 2 r

(14)

The radial algebra so(1, 2) commutes with the angular momentum algebra so(3). The principle quantum number n stems from the spectrum of the operator T3 commuting with the action of so(4) in a way that we now explain following the pedagogical exposition [1].

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Dilatation operator. The operator T2 is a dilatation (called also tilting operator) and the group element eiωT2 is a scaling transformation on functions eiωT2 f (r ) = eω f (eω r )

eω > 0

On operators the scaling transformation acts by conjugation O˜ = eiωT2 Oe−iωT2 .

(15)

The effect on the coordinate and momentum operators is r˜ = eω r

p˜ = e−ω p .

(16)

It turns out that with the help of the scaling transformation (15) of T3 T˜3 = T3 cosh ω − T1 sinh ω

(17)

the radial Schrödinger equation can be solved entirely in terms of the so(1, 2) algebra [1]. Indeed the radial equation for the nonrelativistic atom has the form 

pr2 η ξ + 2 − −τ 2 2r r

 R(r ) = 0

(18)

where the coefficients are set to the values ξ = l(l + 1), η = Z and τ = E. Multiplying the radial equation on the left by the factor 2r put in the front of the scene the so(1, 2) generators (14) [T1 (1 + 2τ ) + T3 (1 − 2τ ) − 2η] R(r ) = 0 .

(19)

√ The astute choice of the hyperbolic parameters ω: −8τ sinh ω = 1 + 2τ and √ 3 (17) lead −8τ cosh ω = 1 − 2τ together with the scaling transformation of T to a scaled version of the radial equation 

   3 − 2η(−8τ )−1/2 R(r ) = 0 . T1 sinh ω + T3 cosh ω − 2η(−8τ )−1/2 R(r ) = T

The equation takes a simpler form 

 ˜ )=0 T3 − η(−2τ )−1/2 R(r

λ = eω = (−2τ )−1/2 .

˜ ) = e−ω eiωT2 R(r ) = R(λr ). for the transformed function R(r The T3 -eigenfunctions |k, q spanning an infinite dimensional representation of the so(1, 2) T 2 |k, q = k(k + 1)|k, q

T± |k, q = Ckl |k, q ± 1

Jordan Algebra and Hydrogen Atom

T3 |k, q = q|k, q

241

q =k+1+μ

μ = 0, 1, 2, . . .

which is bounded below. The quantization of the energy originates from eigenvalue q of the operator T3 , by its identification with the principal quantum number q = n n = ηλ

⇔

τ =−

1 η2 2 n2

⇔

E =−

1 Z2 . 2 n2

On the other hand the orbital quantum number l is identified with k = l via the Casimir of the “spectrum generating” so(1, 2), Eq. (14) T 2 = T32 − T12 − T22 = ξ = k(k + 1) . The eigenfunctions are expressible through associated Laguerre functions nl (r ) = |n, l = Nnl e−r (2r )l L (2l+1) R n−l−1 (2r ) . The solutions of the original nonrelativistic radial equation (19) are thus nl (r ) = cλ−1 R(λ  −1r ) Rnl (r ) = ce−iωT2 R

c = (λn)− 2

1

λ=

n Z

which retrieves the textbook result for the radial Schrödinger solution by an algebraic method of scaling transformation. Note that the angular momentum L is scale invariant,  L = L in view of Eq. (16) 1 whereas the scaling of the modified Laplace–Runge–Lenz operator V = (−2E n )− 2 X yields the new rescaled Laplace–Runge–Lenz vector A=

1 1 2 . r p − p(r · p) − r = V 2 2

The set {T3 , L, A} generates a scaled algebra so(2) ⊕ so(4) , {H, L, V } → ,   } = {T3 , L, A} whose factors are commutants to each other. {H L, V It has been found in the works of Barut and his collaborators (see [2] and the references there) that all bound states ψnlm of the (nonrelativistic) hydrogen atom belong to one irreducible representation of “spectrum generating algebra” so(2, 4) with generators L=r×p

T1 = 21 (r p2 − r ) A = 21 r p2 − p(r · p) − 21 r

Γ =rp

T3 = 21 (r p2 + r ) B = 21 r p2 − p(r · p) + 21 r T2 = r · p − i

(20)

The radial so(1, 2) algebra (14) spanned by {T1 , T2 , T3 } is a subalgebra of so(2, 4) commuting only with the angular momentum algebra so(3) of L. The so(2, 4) generators L AB = −L B A with A, B ∈ {0, 1, 2, 3, 5, 6} are identified with the operators L, A, B, Γ and T acting on the Hilbert space of the hydrogen

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system throughout the antisymmetric matrix ⎞ ⎛ 0 L 01 L 02 0 Γ1 Γ2 Γ3 T1 T3 ⎜ 0 L 3 −L 2 A1 B1 ⎟ ⎜ 0 L 12 ⎟ ⎜ ⎜ ⎟ ⎜ ⎜ 0 L A B 0 1 2 2 ⎟=⎜ ↔⎜ ⎟ ⎜ ⎜ 0 A B 3 3 ⎟ ⎜ ⎜ ⎝ 0 T2 ⎠ ⎝ 0 ⎛

L AB

L 03 L 13 L 23 0

A0 A1 A2 A3 0

⎞ B0 B1 ⎟ ⎟ B2 ⎟ ⎟ B3 ⎟ ⎟ T2 ⎠ 0

(21)

where it is convenient to introduce the 4-vectors Aμ := (A0 , A) = (T1 , A) and Bμ := (B0 , B) = (T3 , B). The 15 conformal algebra so(2, 4) generators L AB satisfy the relations [L AB , L C D ] = −i(g AC L B D + g B D L AC − g AD L BC − g BC L AD )

(22)

where g AB is the diagonal metric with the signature (+ − − − −+) for the set of indices {0, 1, 2, 3, 5, 6}.1 The new generators are the “current operator” Γ = r p and the partner of the scaled Laplace–Runge–Lenz vector A stemming from the commutation with T2 [T2 , A] = i B

B = A+r .

(23)

The adjoint action eigenvectors of the dilatation T2 are combinations of the generators L AB L = r × p T3 − T1 = r B−A= r . (24) Γ =rp T3 + T1 = r p2 B + A = r p2 − 2 p(r · p) The generators L and r = B − A close the Galilean group S O(3)  R3 . The generators {L, B} are closing one more subalgebra so(1, 3) ⊂ so(2, 4) and the vectors B can be seen as Lorentz boosts [7] of the relativistic wavefunction from the rest-frame to another frame.

6 Causal Automorphisms Versus Spectrum Generating Algebra Let us consider the creation and annihilation operators for the massless scalar field χ(x) (11) and the conformal invariant vacuum [a( p), a ∗ ( p)] = 2| p|δ( p − p )

a ∗ ( p)|0 = 0

J AB |0 = 0 .

index convention changes to the Barut’s notations by the prescription 0, 1, 2, 3, 5, 6 → 6, 1, 2, 3, 4, 5.

1 Our

Jordan Algebra and Hydrogen Atom

243

The Fock space H arises by applying polynomials of the operators a ∗ ( p) on the vacuum |0. The “one-particle” states span a subspace in H of vectors of the form  Ψ → |Ψ  =

d3 p Ψ ( p)a ∗ ( p)|0 2| p|

One construct a Lorentz covariant, massless scalar field as a combination of creation and annihilation operators by  χ(x) =

  d 4 p δ+ ( p 2 ) a( p)e−i p·x + a( p)∗ ei p·x .

(25)

The transformation law for the field χ(x) implies the transformation law [a( p), J AB ] = ∂ AB a( p) where the operators ∂ AB read [6] Li j Γi T2 T3 + T1 Bi + Ai T3 − T1 Bi − Ai

↔ ∂i j ↔ ∂0i ↔ ∂56 ↔ ∂06 − ∂05 ↔ ∂i6 − ∂i5 ↔ ∂05 + ∂06 ↔ ∂i5 + ∂i6

: : : : : : :

Li j L 0i D K0 Ki P0 Pi

= −i( pi ∂∂p j − p j ∂∂pi ) = i| p| ∂∂pi = (1 + pi ∂∂pi ) = −| p|Δ p = pi Δ p − 2 p j ∂∂p j ∂∂pi − 2 ∂∂pi = p0 = | p| = pi

The comparison between the generators of the dynamical conformal symmetry of the hydrogen system (24) and the Fourier image of the generators of the causal automorphisms (6) shows that the two so(2, 4)-representations coincide upon the identification x μ ↔ p μ for the isotropic vectors x 2 = 0 = p 2 . In summary, the Möbius algebra co(JC 2 ) of the Jordan algebra of hermitian 2 × 2 complex matrices has two different representations, the causal automorphisms of space-time and the spectrum generating symmetry of the hydrogen atom C C ∗ co(JC 2 ) x − rep on J2 p − rep on (J2 ) g0 Mμν , D L μν , T2 g−1 Pμ Bμ + Aμ g+1 Kμ Bμ − Aμ

These two representations are related by the Fourier transform (25). Acknowledgements It is my pleasure to thank Tekin Dereli, Michel Dubois-Violette, Ludmil Hadjiivanov, Nikolay Nikolov, Petko Nikolov and Ivan Todorov for their encouraging interest in that work and many enlightening discussions. This work has been supported in part by the Bulgarian National Science Fund research grant DN 18/3 and the TUBITAK 2221 program.

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References 1. B.G. Adams, J. Cizek, and J. Paldus, “Lie algebraic methods and their applications to simple quantum systems”, Advances in quantum chemistry. Vol. 19. Academic Press, 1988. 1–85. 2. A.O. Barut and G.L. Bornzin, “SO(4,2)Formulation of the Symmetry Breaking in Relativistic Kepler Problems with or without Magnetic Charges”, Journal of Mathematical Physics 12 (1971) 841–846. 3. V. Fock, Z. Phys. 98 (1935) 145. 4. M. Günaydin, “Generalized conformal and superconformal group actions and Jordan algebras”, Modern Physics Letters A8 (1993) 1407–1416. 5. P. Jordan, Z. Phys. 80 (1933) 285. 6. G. Mack and Ivan Todorov, “Irreducibility of the ladder representations of U (2,2) when restricted to the Poincaré subgroup”, Journal of Mathematical Physics 10 (1969) 2078–2085. 7. Brian G. Wybourne, Classical groups for physicists (Wiley, 1974). 8. G. Meng “Euclidean Jordan Algebras, hidden actions and J-Kepler problems” Journal of Mathematical Physics 52 (2011) 112104.

Screening Operators for the Lattice Vertex Operator Algebras of Type A1 at Positive Rational Level Takuya Matsumoto

Abstract In this short note, we propose an algebraic construction of the screening operators, which commute with the Virasoro generators, for the lattice vertex operator algebras of type A1 at positive rational level.

1 Introduction The two-dimensional conformal field theories are remarkable quantum field theories in the sense that their correlation functions are completely determined by the infinite dimensional symmetry, called the Virasoro algebra. These theories were proposed by A. A. Belavin, A. M. Polyakov, and A. B. Zamolodchikov in 1984 [1]. In particular, they have utilized the representation theory of the Virasoro algebras with the central charge c p+ , p− = 1 − 6

( p + − p − )2 , p+ p−

(1)

where ( p+ , p− ) are coprime non-negative integers bigger than one. We shall refer to this central charge as the positive rational level. As a characteristic feature of these models, called the BPZ minimal models, there are finite number of the Virasoro primary fields and their operator products are closed among them. In order to investigate the Virasoro modules, the Virasoro module maps commuting with the Virasoro generators play very important roles. Such operators are referred to as screening operators. In fact, G. Felder characterized the BPZ minimal models in terms of the cohomology defined by the screening operators [2]. To explain his idea, it is convenient to consider the lattice vertex operator algebra (VOA), which includes the BPZ minimal series as a sub quotient. The simple modules of the lattice T. Matsumoto (B) Graduate School of Mathematics, Nagoya University Furocho, Chikusaku Nagoya 464-8602, Japan e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2018 V. Dobrev (ed.), Quantum Theory and Symmetries with Lie Theory and Its Applications in Physics Volume 1, Springer Proceedings in Mathematics & Statistics 263, https://doi.org/10.1007/978-981-13-2715-5_14

245

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T. Matsumoto

VOA are given by a direct sum of the boson Fock modules Fβr,s over a rank one lattice, i.e.   0¯ 1¯ = Fβr,s +nα0 , V[r,s] = Fβr,s +(n+ 21 )α0 . (2) V[r,s] n∈Z

n∈Z

where 1 ≤ r ≤ p+ , 1 ≤ s ≤ p− and α0 is the generator of the rank one even lattice. The screening operators Q [r+] and Q [s] − are the Virasoro module maps and relate 0¯ 1¯ the modules V[r,s] and V[r,s] . More precisely, they define the following the Felder complex consisting of the Fock modules [2]; [ p −r ]

Q+ +

[ p −r ]

Q [r+]

Q+ +

Q [s] −

Q− −

Q [r+]

· · · −−−−→ Fβr,s −−−−→ Fβ p+ −r,s+ p− −−−−→ Fβr,s+2 p− −−−−→ · · · [ p −s]

Q− −

[ p −s]

Q [s] −

· · · −−−−→ Fβr,s −−−−→ Fβr + p+ , p− −s −−−−→ Fβr +2 p+ ,s −−−−→ · · · and the compositions of the maps [ p −r ]

◦ Q [r+] : Fβr,s → Fβr,s+2 p− ,

[ p −s]

◦ Q [s] − : Fβr,s → Fβr +2 p+ ,s

Q+ +

Q− −

(3)

are zero. The simple modules of the Virasoro minimal models L hr,s = L h p+ −r, p− −s with 1 ≤ r ≤ p+ − 1 and 1 ≤ s ≤ p− − 1 are characterized by the cohomology at Fβr,s ; L hr,s = =

Ker(Q [r+] : Fβr,s → Fβ p+ −r,s+ p− ) [ p −r ]

Im(Q + +

: Fβ p+ −r,s− p− → Fβr,s )

Ker(Q [s] − : [ p− −s] Im(Q −

Fβr,s → Fβr + p+ , p− −s ) : Fβr − p+ , p− −s → Fβr,s )

.

(4)

Clearly, screening operators are a powerful tool for the Virasoro representation theory. Our motivation is to construct some extension of the screening operators acting 0¯ 1¯ and V[r,s] themselves. The natural candidates for on the lattice VOA modules V[r,s] [ p −r ] such operators are the composition of the screening operators such as Q + + ◦ [ p −s] Q [r+] and Q − − ◦ Q [s] − , because they have the correct charge ( p+ − r )α+ + r α+ = ( p− − s)α− + sα− = p± α± = ±α0 , corresponding to the lattice unit. However, as we have seen in (3), these operators are zero due to the Felder complex. A solution to this problem is deforming the conformal field theory so that it preserves the conformal symmetry [3]. More concretely, we deform the central charge c p+ , p− at positive rational level (1) by a continuous parameter  such that c p+ , p− () reduces to c p+ , p− in the  → 0 limit. This deformation allows us to lift-up the Felder complex over the ring of the formal power series O = C[[]] rather than the complex

Screening Operators for the Lattice Vertex Operator …

247

field C. Eventually, the maps given in (3) do not vanish. The screening operators E, F for the lattice VOA are actually obtained as the -order of the maps (3). The conceptual idea in [3, 4] could be illustrated as follows; Renormalization

K −−−−−−−−−→  ⏐  -Deformation⏐ C

O ⏐ ⏐Reduction ( → 0)  C = O/O

We start with the conformal field theory over C. Then, we perform the -deformation. A priori, the deformed theory would be defined over the formal Laurent power series K = C(()) rather than O. Thus, we need to renormalize the physical quantities which we are interested in so that they take the values in O. After this procedure, we are able to take  → 0 limit. The key ingredients are the screening operators defined over the ring O; [r ] O Q+

: O Fβr,s → O Fβ−r,s

(5)

with r ∈ Z>0 and s ∈ Z, where O Fβr,s is the Fock module over the ring O. These renormalized screening operators are constructed in [3] in the framework of the twisted de Rham theory (for instance, see [5, 6]). By expanding the screening operator with respect to , we have [r ] O Q+

= Q [r+] +  Q [r+],(1) +  2 Q [r+],(2) · · · ,

(6)

where the leading term Q [r+] is nothing but the screening operator for the Fock modules over C. In addition to this, for our purpose, the derivation Q [r+],(1) plays important [r+],(1) , we define the roles. After suitable modifications for the first order operator Q 1 following operator with 1 ≤ r ≤ p+ − 1 ; [r+],(1) . [+p+ −r ],(1) ◦ Q [r+] + Q [+p+ −r ] ◦ Q E≡Q

(7)

Then, we have the following main theorem. Main theorem 1 ([4]) The operator E is a Virasoro module map; E : Fβr,s −→ Fβr,s+2 p− .

(8)

This short article is organized as follows. Section 2 is devoted to introduce the basics of the free field realization. In Sect. 3, we construct the screening operator for the lattice VOA.

1 The

existence of such operator (7) is also mentioned in [7].

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T. Matsumoto

2 Free Field Realization In order to construct the screening operators, it is convenient to realize the Virasoro generators in terms of bosons (or equivalently Heisenberg generators), which is the so called free field realization of the Virasoro algebras. Let us first introduce the boson fields ϕ(z) and the boson current a(z) by ϕ(z) = aˆ + a[n] log z + a(z) =





a[n]

n=0

a[n]z

−n−1

z −n , −n

= ∂z ϕ(z) .

(9)

n=0

These are the generating functions of the Heisenberg generators a[n] (n ∈ Z) and aˆ satisfying the relations, [a[n], a[m]] = 2nδm+n,0 id. [a, ˆ a[n]] = 2δn,0 id.

(10)

The energy-momentum tensor is defined by T (z) =

τ () 1 : a(z)2 : + ∂z a(z) , 4 2

(11)

where the symbol : : denotes normal ordering for the bosons and the parameter τ () ∈ O = C[[]] is given by τ () = τ+ () + τ− () , τ± () = τ±(0)

τ±(0)

+

τ±(1)

 = ± p∓ / p± .

+

τ+ ()τ− () = −1 ,  2 τ±(2)

+ · · · ∈ O , (τ±(i) ∈ C) ,

(12)

Here, ( p+ , p− ) are coprime positive integers bigger than one. It is noted that T (z) is the generating function of the Virasoro generators L n (n ∈ Z) satisfying the following relations, [L n , L m ] = (n − m)L n+m +

cτ () 3 (n − n)δn+m,0 id. , 12

(13)

with the central charge cτ () = 1 − 6τ ()2 = cτ(0) + cτ(1) +  2 cτ(2) · · · .

(14)

The leading term cτ(0) is nothing but the central charge c p+ , p− of the BPZ theory [1] given in (1) .

Screening Operators for the Lattice Vertex Operator …

249

Since the energy-momentum tensor (11) is written in terms of the bosons, we could construct Virasoro modules by Fock modules as follows. The Fock module O Fβ over O is generated by O Fβ |β with the relations a[n]|β = 0 for n > 0, a[0]|β = (α, β)|β ,

(15)

where β ∈ P = Zω and ω is the fundamental weight which is dual to the simple root α of A1 root system Q = Zα ⊃ P with respect to the symmetric bilinear form ( , ) : P × P → Q defined by (α, α) = 2 , (α, ω) = 1 , (ω, ω) =

1 . 2

(16)

By the PBW theorem for the Heisenberg algebra, we have the following isomorphism as the free O-modules, O Fβ

O[a[−1], a[−2], · · · ]|β

(17)

Next, let us introduce the vertex operator Vβ (z) with β ∈ P by Vβ (z) =: e

ϕβ (z)

aˆ β aβ [0]

:= e z

   zl  z −l aβ [−l] exp aβ [l] exp l −l l>0 l>0

(18)

ˆ where we have denoted ϕβ (z) = (ω, β)ϕ(z), aβ [l] = (ω, β)a[l] and aˆ β = (ω, β)a. The vertex operator is the Virasoro primary field with the conformal dimension h β = −(β, β ∨ )/2 where β ∨ = τ ()α − β, which means that the following operator product expansion (OPE) holds; T (z)Vβ (w) ∼

hβ 1 ∂w Vβ (w) . Vβ (w) + (z − w)2 z−w

(19)

The screening operators are a special class of vertex operators with conformal dimension one. Solving the quadratic equation h xα = x 2 − τ ()x = 1, we have the two solutions τ± () ∈ O such that τ () = τ+ () + τ− () ,

τ+ ()τ− () = −1 .

(20)

This is actually the parameterization given in (12). In this case, the OPE (19) becomes a total derivative;

 1 V± (w) , (21) T (z)V± (w) ∼ ∂w z−w

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T. Matsumoto

where we have denote V± (w) = Vτ± ()α (w). Hence, the zero modes of the operator V± (w) commute with the Virasoro generators. These are the simplest case of screening operators. More precisely, for r, s ∈ Z, they are defined by [1] O Q+ ≡ [1] O Q−

≡



dzV+ (z) : O Fβ1,s () → O Fβ−1,s () , dzV− (z) : O Fβr,1 () → O Fβr,−1 () ,

(22)

where we have introduced the notation ; βr,s () ≡ (1 − r )τ+ ()ω + (1 − s)τ− ()ω .

(23)

These operators are Virasoro module maps over O. Namely, it holds that [1] [O Q [1] + , L n ] = [O Q + , L n ] = 0 .

(24)

For the generic r ∈ Z>0 and s ∈ Z, we can also define long screening operators by [r ] O Q+

≡

[ r (τ+ )]

V+ (z 1 ) · · · V+ (zr )dz 1 · · · dzr : O Fβr,s () → O Fβ−r,s () ,

(25)

Here, the twisted cycle [ r (τ+ )] has been constructed in [3, 6] (see also [5]). This is similar for O Q [s] − with s > 0.

3 Screening Operators for the Lattice VOA Now, we are ready to construct the screening operators for the lattice VOA. For this purpose, it is convenient to introduce the following operator, Ur,s () ≡ e(ω,βr,s ()−βr,s (0))aˆ ,

(26)

which separates the -dependence of the Fock modules ; |βr,s () = Ur,s ()|βr,s (0) .

(27)

By using the shift operator (27), we shall introduce operators in the Heisenberg picture, namely,  L n ≡ Ur,s ()−1 L n Ur,s () ,

[r ] O Q+

≡ Ur,s ()−1 O Q [r+] Ur,s () .

(28)

Screening Operators for the Lattice Vertex Operator …

251

Note that the Virasoro generators  L n satisfy the same relations in (13) with the same central charge (14). Then, for any u ∈ O Fβr,s (0) , it holds that [r+] ]u = 0 . [ Ln, O Q

(29)

In particular, when the vector u does not depend on the parameter , u ∈ C[a[−1], a[−2], · · · ]|βr,s (0) = C Fβr,s (0) ⊂ O Fβr,s (0) ,

(30)

by expanding the operators as  L (1) L n = L n +  n + ··· , [r ] O Q+

[r+],(1) · · · , = Q [r+] +  Q

(31)

we obtain that, for any u ∈ Fβr,s (0) , 

 [r ] [r ],(1) ] u = 0 . [ L (1) n , Q + ] + [L n , Q +

(32)

[r+],(1) , we have the following theorem. Using the sub-leading term Q Theorem 1 For r ∈ Z>0 , s ∈ Z, define the operator [+p+ −r ] (1) ◦ Q [r+] + Q [+p+ −r ] ◦ Q [r+] (1) : Fβr,s (0) −→ Fβr,s+2 p (0) . E≡Q −

(33)

This is the Virasoro module map. Proof First, we show that the operator E has the correct a[0]-charge. Since [r+] ] = 2r τ+ ()O Q [r+] , [a[0], O Q

(34)

the -order of this relation is [r+] ,(1) + 2r τ+(1) Q [r+] . [r+] ,(1) ] = 2r τ+(0) Q [a[0], Q

(35)

Thus, due to the Felder complex (3), we have [ p −r ] [r ] ,(1) ] Q+

[a[0], Q + +

[ p −r ] [r ] ,(1) Q+

= 2r τ+(0) Q + +

.

(36)

This is indeed the desired a[0]-charge. By the same argument, it is easy to see the [ p −r ] [r+] (1) of E has the same charge. Therefore, the image of E second term Q + + ◦ Q is in Fβr,s (0) . Second, we prove that E actually commute with the Virasoro generators. For any u ∈ Fβr,s (0) , it holds that

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T. Matsumoto

[r+ [L n , E]u = [L n , Q

∨

] (1)

∨

[r ] (1) + ]Q [r+] u + Q [r+ ] [L n , Q ]u

∨

∨

[r ] [r ] [r ] [r ] (1) = −[L (1) n , Q + ]Q + u − Q + [L n , Q + ]u .

(37)

where we have denote r ∨ = p+ − r and used the relation (32). Furthermore, due to the Felder complex (3), the above expression reduces to ∨

[L n , E]u = Q [r+ ] Q [r+] L (1) n u.

(38)

It is, however, noted that L (1) n u ∈ Fβr,s (0) . Thus, this quantity vanishes because of the Fleder complex. This completes the proof.  Finally, we would like to give some comments on the operator E in (33). Schematically, the operator may be regarded as the derivative of the following well-defined operator, ∨

[r+ ] ◦ Vr () ◦ O Q [r+] : O Fβr,s (0) −→ O Fβr,s+2 p (0) Vr ∨ () ◦ O Q −

(39)

where the charge matching operator Vr () is defined through the relation −1 () , e−aˆ δ = Ur ∨ ,s+ p− ()Vr ()Ur,s

(40)

ˆ The whole picture might be summarized where aˆ δ ≡ ( p+ τ+ ()ω + p− τ− ()ω, ω)a. in the following diagram; Vr ∨ ()

∨ [r ] O Q+

Vr ()

[r ] O Q+

Vr ∨ ()

[r ∨ ] O Q+

e−aˆ δ

[r ] O Q+

Fβ(0) ←−−−− Fα() ←−−−− Fβ(0) ←−−−− Fα() ←−−−− −r,s r,s+2 p− −r ∨ ,s+ p− r ∨ ,s+ p− ⏐  ⏐  ⏐U ⏐U −1 () ⏐U ∨ ⏐U −1 ()  r,s+2 p− () ⏐ r ∨ ,s+ p−  r ,s+ p− () ⏐ r,s

Fβ(0) ⏐r,s ⏐U ()  r,s

Fβ() ←−−−− Fβ() ←−−−− Fβ() ←−−−− Fβ() ←−−−− Fβ() r,s+2 p− −r,s r,s −r ∨ ,s+ p r ∨ ,s+ p −

−

In the above diagram, all Fock modules F are considered over O and we have dropped the subscripts as F := O F for the simplicity. The values αr,s () are appropriately determined so that the charges are conserved. Acknowledgements We are grateful for our collaborators Prof. Akihiro Tsuchiya and Prof. Yoshitake Hashimoto. This contribution is based on the work [4]. This work is also supported by Grant-in-Aid for Young Scientists (B) 16K17567 and TOYOAKI SCHOLARSHIP FOUNDATION.

References 1. A.A. Belavin, A.M. Polyakov, and A.B. Zamolodchikov, “Infinite conformal symmetry in twodimensional quantum field theory,” Nucl. Phys. B 241 (1984) 333–380.

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2. G. Felder, “BRST approach to minimal models,” Nucl. Phys. B 317 (1989) 215–236. 3. A. Tsuchiya and S. Wood, “On the Extended W-Algebra of Type sl2 at Positive Rational Level,” International Mathematics Research Notices, Volume 2015, Issue 14, 1 January 2015, Pages 5357–5435, 4. Y. Hashimoto, T. Matsumoto and A. Tsuchiya, in preparation. 5. K. Aomoto and M. Kita, “Theory of Hypergeometric Functions.” Springer, 2011. 6. A. Tsuchiya and Y. Kanie, “Fock space representations of the Virasoro algebra – Intertwining operators,” Publ. RIMS, Kyoto Univ. 22 (1986) 259–327. 7. B.L. Feigin, A.M. Gainutdinov, A.M. Semikhatov, and I.Y. Tipunin, “Logarithmic extensions of minimal models: Characters and modular transformations,” Nucl. Phys. B757 (2006) 303–343.

Symmetries of the S3 Dirac–Dunkl Operator Hendrik De Bie, Roy Oste and Joris Van der Jeugt

Abstract We work in three-dimensional Euclidean space on which the symmetric group S3 acts in a natural way. Here, we consider the Dunkl operators, a generalization of partial derivatives in the form of differential-difference operators associated to a reflection group, S3 in our case. In this setting, the main object of study is the Dunkl version of the Dirac operator. We determine the classes of symmetries of the Dirac– Dunkl operator and present the algebra they generate.

1 Introduction We present a different view on a recently considered specific case of more general abstract results. In a first paper [3], we determined the symmetries, and the algebraic structure they generate, for a class of Laplace–like and Dirac-like operators in the framework of Wigner quantization. These operators include, in particular, the Dunkl version of Laplace and Dirac operators associated to an arbitrary reflection group or root system. The term Dunkl refers to the Dunkl operators [5, 8], a generalization of partial derivatives in the form of differential-difference operators associated to a reflection group. In a second article [4], we moved from the abstract setting to a concrete example being the S3 Dirac–Dunkl operator, which appears in the Dirac Hamiltonian for the S3 Dunkl Dirac equation. Here, S3 is the symmetric group on three elements which acts in a natural way on three-dimensional Euclidean space by H. De Bie Faculty of Engineering and Architecture, Department of Mathematical Analysis, Ghent University, Krijgslaan 281-S8, 9000 Gent, Belgium e-mail: [email protected] R. Oste (B) · J. Van der Jeugt Faculty of Sciences, Department of Applied Mathematics, Computer Science and Statistics, Ghent University, Krijgslaan 281-S9 9000 Gent, Belgium e-mail: [email protected] J. Van der Jeugt e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2018 V. Dobrev (ed.), Quantum Theory and Symmetries with Lie Theory and Its Applications in Physics Volume 1, Springer Proceedings in Mathematics & Statistics 263, https://doi.org/10.1007/978-981-13-2715-5_15

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coordinate permutation, and occurs as the reflection group associated to the root system A2 . The structure of the symmetries and symmetry algebra in this case followed by substituting the corresponding Dunkl operators in the expressions obtained for the abstract Dirac-like operator, void of references to Dunkl operators or reflection groups. In the current paper, we will show that, though requiring more tedious computations, the algebraic relations for the symmetries of the S3 Dirac–Dunkl operator can be obtained also in the concrete setting of Dunkl operators and reflection groups. We do this not only to further clarify these results, but also to validate their correctness and moreover highlight the power and beauty of approaching and dealing with a problem in a more abstract general framework. In fact, the calculations in the current paper served as an inspiration for, and predate the work on the generalized version. They were in turn inspired by results on Dirac–Dunkl operators for other classes of reflection groups and root systems. We mention in particular the (Z2 )n and B3 cases [1, 2, 7] where the symmetry algebra was shown to generalize the so-called Bannai-Ito algebra. In Sect. 2, we go over the definition and notions required to introduce our main object of study, the Dirac–Dunkl operator related to S3 . In Sect. 3, we present the symmetries of this Dirac–Dunkl operator with explicit expressions and also give the algebra generated by them.

2 The S3 Dirac–Dunkl Operator We consider three-dimensional space R3 with coordinates x1 , x2 , x3 . The symmetric group S3 is generated by the transpositions g12 , g23 , g31 which act on functions in a natural way, e.g. g12 f (x1 , x2 , x3 ) = f (x2 , x1 , x3 ). Denoting g123 = g12 g23 = g31 g12 = g23 g31 and g321 = g23 g12 = g12 g31 = g31 g23 , the six elements of S3 are {1, g12 , g23 , g31 , g123 , g321 }. For a parameter κ, usually assumed to be positive, the Dunkl operators [5, 8] associated to S3 are given by  D1 = ∂ x 1 + κ

1 − g13 1 − g12 + x1 − x2 x1 − x3



 D3 = ∂ x 3 + κ

 , D2 = ∂ x 2 + κ

1 − g31 1 − g23 + x3 − x1 x3 − x2

1 − g23 1 − g12 + x2 − x1 x2 − x3

 ,

 .

The property that makes these generalizations of partial derivatives so special is that they commute with one another, [Di , D j ] = 0 for i, j ∈ {1, 2, 3}. Furthermore, the action of S3 on the Dunkl operators is simply given by g12 D1 = D2 g12 , g12 D2 = D1 g12 , g12 D3 = D3 g12 ,

(1)

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and similarly for g23 and g31 . The commutation relations with the coordinate variables are readily shown to be

[Di , x j ] = Di x j − x j Di =

 ⎧ gik i = j ⎨1 + κ ⎩

(2)

k=i

−κgi j

i = j

for i, j, k ∈ {1, 2, 3}. Note that when κ = 0 these reduce to [Di , x j ] = δi j , as the Dunkl operators then reduce to ordinary partial derivatives. In this setting, the Laplace–Dunkl operator is given by Δκ = (D1 )2 + (D2 )2 + (D3 )2 . The Dirac–Dunkl operator D is defined as a square root of the Dunkl Laplacian as follows: D = e1 D1 + e2 D2 + e3 D3 , where e1 , e2 , e3 generate the three-dimensional Euclidean Clifford algebra and hence of the wellsatisfy {ei , e j } = 2δi j for i, j∈ {1,  canbe realized  by means  2, 3}. They  01 0 −i 1 0 known Pauli matrices: e1 = , e2 = , e3 = . 10 i 0 0 −1 Together with the vector variable x = e1 x1 + e2 x2 + e3 x3 , the operator D generates a realization of the osp(1|2) Lie superalgebra, governed by the relations [{D, x}, D] = −2D,

[{D, x}, x] = 2x.

Here, the operator {D, x} = D x + x D can be written as 2(E + 3κ) where E = x1 ∂x1 + x2 ∂x2 + x3 ∂x3 is the Euler operator, which measures the degree of a homogeneous polynomial in x1 , x2 , x3 .

3 Symmetries of the Dirac–Dunkl Operator We now turn to the subject of symmetries of the operator D. By the word symmetry, we have in mind an operator which “supercommutes” (commutes or anticommutes) with the operator in question. A first symmetry of the Dirac–Dunkl operator is the so-called Scasimir operator of the osp(1|2) realization above, which anticommutes with D. It is given explicitly by 1 [D, x] − 1 = 1 + κ(g12 + g23 + g31 ) + e1 e2 L 12 + e2 e3 L 23 + e3 e1 L 31 , 2

(3)

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where the right-hand side is obtained through the commutation relations (2). The Scasimir operator is usually denoted by  + 1, where the notation  refers to the “angular Dirac operator”, appearing when D is written in spherical coordinates. Moreover, in the right-hand side of (3) L 12 = x1 D2 − x2 D1 ,

L 23 = x2 D3 − x3 D2 ,

L 31 = x3 D1 − x1 D3

are the Dunkl versions of the angular momentum operators, which commute with Δκ . They are the main object of study in a related paper on the Dunkl angular momentum algebra [6]. The square of the Scasimir operator yields the osp(1|2) Casimir operator which commutes with D: ( + 1)2 =  + κ(g12 + g23 + g31 ) + 3κ 2 (1 + g123 + g321 ) − (L 212 + L 223 + L 213 ). Another symmetry of D inherent to the Clifford algebra, is the pseudo-scalar e1 e2 e3 . Because of the anti-commutation relations of e1 , e2 , e3 , one immediately sees that [D, e1 e2 e3 ] = 0 and moreover (e1 e2 e3 )2 = −1. In fact, in the realization by means of the Pauli matrices, e1 e2 e3 is just i times the identity matrix. While the Dunkl Laplacian is invariant under the action of S3 , following the interaction (1), the Dirac–Dunkl operator is not, as the group action leaves the Clifford elements e1 , e2 , e3 unchanged. To get symmetries of D, we extend the S3 action to affect also Clifford elements. We do this by appending a group element of S3 with an appropriate element in the Pin group of the Clifford algebra. In this way, we arrive at the symmetries 1 1 1 G 12 = √ g12 (e1 − e2 ), G 23 = √ g23 (e2 − e3 ), G 31 = √ g31 (e3 − e1 ). 2 2 2

(4)

One readily verifies that they satisfy G 12 e1 = −e2 G 12 , G 12 e2 = −e1 G 12 , G 12 e3 = −e3 G 12 , (G 12 )2 = 1, with analogous relations for G 23 and G 31 . Hence, they anti-commute with D. We also have symmetries which commute with the S3 Dirac–Dunkl operator, corresponding to the two even elements of S3 : G 123 = G 12 G 23 =

1 g123 (e1 e2 + e2 e3 + e3 e1 − 1) = G 23 G 31 = G 31 G 12 , 2

G 321 = G 23 G 12 =

1 g321 (e2 e1 + e3 e2 + e1 e3 − 1) = G 31 G 23 = G 12 G 31 . 2

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Finally, we present the most interesting class of symmetries of D. They extend the notion of Dunkl angular momentum operators as symmetries of the Dunkl Laplacian to Dunkl total angular momentum operators in the context of Dirac theory. Theorem 1 The operators 1 O12 = x1 D2 − x2 D1 + e1 e2 + 2 1 O23 = x2 D3 − x3 D2 + e2 e3 + 2 1 O31 = x3 D1 − x1 D3 + e3 e1 + 2

κ √ (G 12 e1 + G 12 e2 − G 23 e1 − G 31 e2 ), 2 κ √ (G 23 e2 + G 23 e3 − G 31 e2 − G 12 e3 ), 2 κ √ (G 31 e3 + G 31 e1 − G 12 e3 − G 23 e1 ) 2

(5)

commute with D and with x. Proof We show that [O12 , D] = 0, the other results are completely analogous. Using the commutation relations (2), we have  x1 , D D2 = −(1 + κg12 + κg13 )e1 D2 + κg12 e2 D2 + κg13 e3 D2 ,

 − x2 , D D1 = −κg12 e1 D1 + (1 + κg21 + κg23 )e2 D1 − κg23 e3 D1 ,  1

e1 e2 , D = −e2 D1 + e1 D2 . 2

Finally, for i = j and k elements of {1, 2, 3} we have [G i j ek , D] = G i j {ek , D} = 2G i j Dk ,

(6)

where we used DG i j = −G i j D and the anticommutation relations of the Clifford algebra. Using relation (6) to compute the final terms of [O12 , D] and plugging in  the definition (4), all components of [O12 , D] cancel out. Theorem 2 The algebra generated by the symmetries  + 1, e1 e2 e3 , G 12 , G 23 , G 31 and O12 , O23 , O31 is governed by the following relations: •  + 1 and e1 e2 e3 commute with the other symmetries, • G 12 , G 23 , G 31 generate a copy of S3 and act on the indices of O12 , O23 , O31 by an S3 action with minus sign, i.e. G 12 O12 = −O12 G 12 , G 12 O23 = −O31 G 12 , G 12 O31 = −O23 G 12 , and analogous actions of G 23 and G 31 , • the commutation relations

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√ 3 2κ( + 1)e1 e2 e3 (G 23 − G 12 ) + κ 2 (G 123 − G 321 ) 2 √ 3 [O31 , O23 ] = O12 + 2κ( + 1)e1 e2 e3 (G 31 − G 23 ) + κ 2 (G 123 − G 321 ) 2 √ 3 [O12 , O31 ] = O23 + 2κ( + 1)e1 e2 e3 (G 12 − G 31 ) + κ 2 (G 123 − G 321 ). 2

[O23 , O12 ] = O31 +

Proof We work out [O23 , O12 ] — the other relations follow in a similar manner — by going over the commutators of the different components of the operators (5). By means of the commutation relations (2) we have [L 23 , L 12 ] = L 31 (1 + κg12 + κg23 ) − κ L 12 g23 − κ L 23 g12 , which can also be found in Ref. [6]. Here, we recognize a first ingredient to make O31 , namely L 31 . Recalling the expression√(3) for  + 1, the terms accompanied with a factor κ will form part of the product 2κ( + 1)e1 e2 e3 (G 23 − G 12 ). Another part of this product is given by the commutators [L i j , G kl em ] = (L i j − L gkl (i)gkl ( j) )G kl em , where the indices in the second term are permuted by the S3 -action. Next, we have [L 23 , e1 e2 ] = 0 = [e2 e3 , L 12 ], while [ 21 e2 e3 , 21 e1 e2 ] = 21 e3 e1 . The final ingredients to make O31 follow from the terms in the commutators [ei e j , G kl em ] = G kl (egkl (i) egkl ( j) em − em ei e j ) — with suitable values for i, j, k, l, m — where two of the three indices of the Clifford elements are equal and thus cancel out. √ The remaining terms of the latter commutators serve as parts of the product 2κ( + 1)e1 e2 e3 (G 23 − G 12 ), whose final ingredient follows from the commutators [G i j ek , G lm en ] = G i j G lm eglm (k) en − G lm G i j egi j (n) ek — again with appropriate values for i, j, k, l, m, n — when glm (k) = n or gi j (n) = k. This leaves eight terms such as G 23 e2 G 12 e1 = −G 23 G 12 = −G 321 of which two cancel out and the remaining six  terms form 23 κ 2 (G 123 − G 321 ), so we arrive at the desired result. For a further analysis of the symmetry algebra of Theorem 2, and its representations, the reader is referred to Ref. [4].

References 1. 2. 3. 4. 5. 6. 7.

H. De Bie, V.X. Genest, L. Vinet, Commun. Math. Phys. 344 (2016) 447–464 H. De Bie, V.X. Genest, L. Vinet, Adv. Math. 303 (2016) 390–414 H. De Bie, R. Oste, J. Van der Jeugt, Lett. Math. Phys. 108 (2018) 1905–1953 H. De Bie, R. Oste, J. Van der Jeugt, Ann. Phys. 389 (2018) 192–218 C.F. Dunkl, Trans. Amer. Math. Soc. 311 (1989) 167–183 M. Feigin, T. Hakobyan, J. High Energy Phys. 11 (2015) 107 V.X. Genest, L. Lapointe, L. Vinet, osp(1, 2) and generalized Bannai-Ito algebras, arXiv:1705.03761 8. M. Rösler, in Orthogonal Polynomials and Special Functions, Lecture Notes in Mathematics vol 1817 (Springer, Berlin, 2003), pp. 93–135

Part III

Integrability

The Superintegrable Zernike System Natig M. Atakishiyev, George S. Pogosyan, Cristina Salto-Alegre, Kurt Bernardo Wolf and Alexander Yakhno

Abstract We present a résumé of this year’s work on what we call the Zernike system. It stems from a differential equation proposed by Frits Zernike in 1934 to describe wavefronts at circular optical pupils through a basis of polynomial solutions on the unit disk and free boundary conditions. This system entails a classical model and a quantum model. The classical model leads to closed elliptic orbits while the quantum model yields bases of polynomial wavefunctions that separate in a manifold of coordinate systems, where only the polar one is orthogonal. Special functions that appear in the solutions and interbasis expansions include the Legendre, Gegenbauer, Jacobi, Hahn and Racah polynomials, as well as special Clebsch–Gordan and 6 j coefficients. The underlying symmetry is a cubic Higgs superintegrable algebra.

N. M. Atakishiyev Instituto de Matemáticas, Universidad Nacional Autónoma de México, Cuernavaca, Mexico e-mail: [email protected] G. S. Pogosyan · A. Yakhno Departamento de Matemáticas, Centro Universitario de Ciencias Exactas e Ingenierías, Universidad de Guadalajara, Guadalajara, Mexico e-mail: [email protected] A. Yakhno e-mail: [email protected] G. S. Pogosyan Yerevan State University, Yerevan, Armenia G. S. Pogosyan Joint Institute for Nuclear Research, Dubna, Russia C. Salto-Alegre Posgrado en Ciencias Físicas, Instituto de Ciencias Físicas-UNAM, Cuernavaca, Mexico e-mail: [email protected] K. B. Wolf (B) Instituto de Ciencias Físicas, Universidad Nacional Autónoma de México, Cuernavaca, Mexico e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2018 V. Dobrev (ed.), Quantum Theory and Symmetries with Lie Theory and Its Applications in Physics Volume 1, Springer Proceedings in Mathematics & Statistics 263, https://doi.org/10.1007/978-981-13-2715-5_16

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1 Introduction The differential equation solved by Frits Zernike to find a basis of orthogonal polynomial functions over the closed unit disk, D := {r = (x, y) | |r| ≤ 1}, was [1]    Z Ψ (r) := ∇ 2 − (r · ∇)2 − 2r · ∇ Ψ (r) = −E Ψ (r).

(1)

This equation can be seen as defining a classical system when we build a Hamiltonian system with momentum ∇ → ip, as was done in Ref. [2]. It can also be interpreted  = −1  Z as a Schrödinger equation with a non-standard quantum Hamiltonian H 2 2 that is hermitian in the space of square-integrable functions L (D) with the inner product (Ψ1 , Ψ2 )D := D d2 r Ψ1 (r)∗ Ψ2 (r), where the ‘energy’ spectrum of E in (1) is E n = n(n + 2) for n ∈ {0, 1, . . .} =: Z0+ , as was seen in Ref. [3]. The only evi := −i(x∂ y − y∂x ), dent symmetry of (1) is rotational, generated by the operator M with a combined spectrum (n, m), |m | ≤ 21 n, which is the same as that of the twodimensional harmonic oscillator system. This has led researchers astray when trying to relate both systems; we now see that their symmetries are different and that of Zernike is rather in a class with that of Kerr. In Sect. 2 we present the classical Zernike system and the quantum one in Sect. 3 with the new polynomial solutions, whose interbasis expansions [4] are reviewed in Sect. 4. The extra constants of the motion and their corresponding quantum generators, which form the cubic Higgs superintegrable algebra are collected in Sect. 5. Some conclusions are offered in Sect. 6.

2 The Classical Zernike System The classical counterpart of the Zernike differential equation (1) is obtained by its de-quantization, i.e., the replacement of ∇ → ip, with r = (x, y) and r := |r|. Thus, in Cartesian and polar coordinates ∇ 2 → −( px2 + p 2y ) = −( pr2 + pφ2 /r 2 ), r · ∇ → i(x px + yp y ) = i r pr .

(2)

The Hamiltonian function H obtained from (1) as − Zˆ → H is then, on the unit disk r ≤ 1, H := ( px2 + p 2y ) − (x px + yp y )2 + 2i(x px + yp y ) = (1 − r

2

) pr2

+

pφ2 /r 2

+ 2ir pr ,

(3) (4)

and E becomes a dimensionless energy. This √ Zernike Hamiltonian is anomalous because it contains the imaginary factor i = (−1), but in the end it leads to a purely real classical system. Moreover,

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Fig. 1 Trajectories on the unit disk in classical Zernike system (3)–(4) with angular momentum pφ = 3 and energies E = 35, 20, and 15 (left to right). The dots mark the positions of a point mass for equidistant times t ∈ [0, T ]

Zernike’s original equation [1] was given with two free parameters, α, β, in front of the second and third term in (1) and (3), but the requirement of the ‘quantum’ system (next section) to have Zˆ hermitian in L2 (D), restricted them to the values α = 1 and β = 2 (as written in their above form). The classical Hamiltonian (3)–(4) is not bound to these restrictions, and in fact allows for closed orbits inside or outside the unit disk. For brevity we do not present here the results for all their possible allowed and forbidden values, but concentrate on the ‘traditional’ form of Zernike’s equation. To solve the classical system with Hamiltonian (4), we use the Hamilton-Jacobi method with the polar coordinates (r, φ) and momenta ( pr , pφ ). Since H is independent of time t and angular coordinates φ, its Hamilton’s principal or action function can be separated as (5) S(r, φ) = R (r ) + pφ φ − Et. The derivative of this function with respect to the radius r is the radial momentum, pr = ∂ R (r )/∂r , this leads to an integral expression which, after derivation with respect to pφ and solution of the integral, provides the geometric orbit as r (φ). On the other had, derivation (5) with respect to the energy, ∂ R (r )/∂ E − t = −to leads to the determination of its dynamical behavior r (t) in time [2]. The geometric and dynamical orbits can be given in terms of the parameters A := E + pφ2 ,

B := 2 pφ2 , C :=

 (E − pφ2 )2 − 4 pφ2 ,

(6)

and the results are [2] r 2 (φ) =

B , A − C cos φ

r 2 (t) =

A + C cos(2πt/T ) . 2(E + 1)

(7)

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This describes the motion of a mass point on closed√elliptical orbits, with eccentricities 0 ≤ C/B ≤ 1, traversed with period T = 21 π/ (E+1), as shown in Fig. 1. It is found that angular momentum restricts the energies to the range pφ2 − 2| pφ | ≤ E ≤ pφ2 + 2| pφ |. The orbit half-axes are given by μ2y = B/(A−C) and μ2x = B/(A + C). √ When C = 0 these trajectories are circular with radius (B/A), while for pφ → 0, they degenerate to lines falling to the center. Extra constants of the motion and the separability of Zernike’s equation (1) in coordinate systems beyond the polar one, are features common to the classical and quantum systems, so we defer their common analysis to Sect. 4. Let it only be said that the analysis of Eq. (1) can be made with factors α and β in other ranges beyond 1 and 2 made here, as detailed in [2]. The strategy is to map the full r-plane vertically on a conic (sphere or 1- and 2-sheeted hyperboloids [5–7]) and choose among their coordinate systems those where one coordinate line matches the circular boundary of the disk. The trajectories will be either completely inside or outside the disk.

3 The Quantum Zernike System The Zernike differential operator −  Z in (1) defines quite naturally a quantum system with wavefunctions Ψ (r) on the disk D, and energies E that will be quantized since Z Ψ2 )D [1]. Yet it is it is hermitian and the range is compact: (  Z Ψ1 , Ψ2 )D = (Ψ1 ,  a peculiar system because the optically relevant solutions are required to be finite throughout, including at the boundary |r| = 1; this is a proper subset of squareintegrable L2 (D) solutions. Also, the functions are not required to be zero, or have fixed logarithmic derivative at the boundary, as in circular-well quantum systems. That is, we are searching for polynomial solutions in any separating coordinate system. Thus we can only refer to  Z as a hermitian (but not self-adjoint) operator. Using the manifest SO(2) symmetry of the disk, Zernike’s solution [1] employed polar coordinates (r, φ) to separate (1) into an angular momentum factor ∼eimφ times a radial Jacobi polynomial ∼Pn(r|m |,0) (2r 2 −1), with a radial quantum number n r := 21 (n−|m |) ∈ Z0+ , angular momentum m ∈ {−n, −n+2, . . . , n} and the principal quantum number n ∈ Z0+ ; the energy spectrum being E n = n(n+2) (the same as a two-dimensional harmonic oscillator —which it is not). The key to find further coordinate systems in which Zernike’s partial differential equation (1) separates into two ordinary simultaneous differential equations, is to perform an orthogonal (i.e., vertical) projection on the upper half-sphere H+ , such that it matches the boundaries of D, as shown in Fig. 2. Spherical coordinates (ϑ, ϕ) on H+ allow any maximal circle on the sphere to match this boundary. As shown in the figure, System I has the equator ϑ = 21 π on the boundary, which leads to the polar coordinates used by Zernike; or we can choose any meridian ϕ = constant of the sphere, tilted so that its poles lie in any direction on the boundary, of which we consider separately only the cases where they lie on the x- or y-axes, calling them Systems II and III, respectively. We thus introduce the three coordinates ξ for unit

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Fig. 2 Top row: The three coordinate systems (9)–(11), denoted as Systems I, II and III, on the half-sphere H+ . Bottom row: The projection of these systems on the disk D; there, only System I is orthogonal.eps

vectors on H+ with components ξ1 := x, ξ2 := y, ξ3 :=

 1 − x 2 − y 2 ≥ 0,

(8)

so |ξ| = 1, in terms of which we define [2, 3], System I: ξ1 = sin ϑ cos ϕ, ξ2 = sin ϑ sin ϕ, ξ3 = cos ϑ,

π/2

ϑ|0 , ϕ|π−π ,

(9)

ϑ |π0 , ϕ |π0 ,

(10)

System II: ξ1 = cos ϑ , ξ2 = sin ϑ cos ϕ , ξ3 = sin ϑ sin ϕ , System III: ξ1 = sin ϑ sin ϕ , ξ2 = cos ϑ , ξ3 = sin ϑ cos ϕ ,

π/2

ϑ |π0 , ϕ |−π/2 , (11)

noting that only System I yields orthogonal coordinates on D, whereas the coordinates of Systems II and III are not orthogonal there. Elliptical coordinate systems on H+ have been considered in the classical case [2] and are under consideration for the quantum case.

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I Fig. 3 Left: Ψn,m (r, φ), Zernike’s original solutions [1] over the disk (15). Rows count down n = 0, 1, . . ., and m horizontally. Since they are complex, we show the real part for m ≥ 0 and the imaginary part for m < 0. Right: ΨnII1 ,n 2 (x, y) in (16). Rows n count down; n 1 down left and n 2 down right. In System III the functions are those of System II, only rotated by 21 π

In passing from the disk to the half-sphere, the measure d2 r for the inner product on D changes to sin ϑ◦ dϑ◦ dϕ◦ on H+ in the three spherical coordinates in (9)–(11), which is dξ1 dξ2 d2 r =√ . (12) d2 S(ξ ) = ξ3 1 − |r|2 Accordingly, the Zenike operator and functions on H+ will be   ξ2 + ξ2  := (1−|r|)1/4  W Z (1−|r|)−1/4 = ΔLB + 1 2 2 + 1 , 4ξ3 Υ ◦ (ϑ◦, ϕ◦ ) ≡ Υ (ξ ) := (1−|r|)1/4 Ψ (r),

(13) (14)

L 21 +  L 22 +  L 23 and cyclic with the Laplace–Beltrami operator on the sphere ΔLB =   L i := ξ j ∂ξk − ξk ∂ξ j . The polynomial solutions obtained for the Zernike equation, which is now of  Υ ◦ (ϑ◦, ϕ◦ ) = −EΥ ◦ (ϑ◦, ϕ◦ ) in the three coordinate systems, the standard form W projected back to the disk through Ψ ◦ (r) := (1−|r|)−1/4 Υ ◦ (ϑ◦, ϕ◦ ) with r = (r, φ) or (x, y), are normalized on D and labelled by angular momentum or Cartesian eigenvalues, (n, m) or (n 1 , n 2 ). For Systems I and II, they are shown in Fig. 3: Zernike’s original System I in (9),  Ψn,m (r, φ) = (−1) I

1 2 (n−|m |)

n+1 |m | (|m |,0) r P 1 (n−|m |) (1−r 2 ) eimφ ; 2 π

(15)

System II in (10),  y  1 ; ΨnII1 ,n 2 (x, y) = Cn 1 ,n 2 (1−x 2 ) 2 n 1 Cnn21 +1 (x)Pn 1 √ 1−x 2

(16)

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System III in (11),  x  1 (1−y 2 ) 2 n 1 Cnn21 +1 (y), ΨnIII1 ,n 2 (x, y) = Cn 1 ,n 2 Pn 1  2 1−y the last two with the normalization constant (2n 1 + 1)(n 1 + n 2 + 1) n 2 ! . Cn 1 ,n 2 := 2n 1 n 1 ! π(2n 1 + n 2 + 1)!

(17)

(18)

Here Jν(α,β) , Cνα and Pν are the Jacobi, Gegenbauer and Legendre polynomials, whose quantum numbers relate through n 1 + n 2 = n = 2n r + |m | ∈ Z0+ ,

n 1 , n 2 , n, n r ∈ Z0+ , m ∈ {−n, −n+2, . . . n},

(19)

and the energy eigenvalues are E n = n(n+2). In spite of the resemblance of the patterns with that of the harmonic oscillator, there are no two-term raising-lowering operators, only three-term differential and recursion relations can exist (see Refs. [8– 12]). It has been remarked to us that the Zernike two-variable polynomial solutions ΨnII1 ,n 2 (x, y) in (16) have been considered long ago by F. Didon [13], and more recently named Class II two-variable analogues of the Jacobi polynomials [14, Eq. (3.8)].

4 Interbasis Expansions Since systems I, II and III stem from rotations of the spherical coordinates in Fig. 2, one would expect that the separated wavefunctions (9)–(11) with the same principal quantum number n (lying in the same horizontal multiplet of n + 1 members in Fig. 3) would be related through representations of the SO(3) Wigner-D functions; certainly the latter two. That this assumption is not borne out we deem as an indication that the Zernike system, which is not described by a Lie but a Higgs algebra, may be an instructive case for deeper study of covariance. Relating the systems I and II, the unitary transformation is [4, 15], ΥnII1 ,n 2 (ϑ , ϕ ) =

n

I Wnn,m Υn,m (ϑ, ϕ), 1 ,n 2

(20)

m=−n (2)

 where nm=−n (2) indicates the sum over m in the range (19) as all other indices above, and where the angles in (9) and (10) relate as sin ϑ sin ϕ . cos ϑ = sin ϑ cos ϕ, cos ϕ =  1 − sin2 ϑ cos2 ϕ

(21)

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The linear combination coefficients Wnn,m in (20) are given by hypergeometric 3 F2 1 ,n 2 polynomials of unit argument, which are the three-parameter Hahn polynomials Q p (x; α, β, γ) of degree p [16], and also a special kind of Clebsch–Gordan coefficients [17] with that number of effective parameters, Wnn,1 ,nm2

1 2n 1 + 1 in 1 (−1) 2 (m+|m |) (n!)2    = 1 1 n ! (n + n 1 + 1)! 2 (n − m) ! 2 (n + m) ! 2   × Q n 2 21 (n + m); −n − 1, −n − 1, n

(22)

= in 1 (−1) 2 (m+|m |) C n1 1n,, 0− 1 m; 1 n, 1 m . 1

2

2

2

2

Between systems I and III a similar relation with phases holds [4]. The relation between systems II and III, ΥIII1 ,2 (ϑ , ϕ ) =

n

n 2 =0

Un11,,n22 ΥnII1 ,n 2 (ϑ , ϕ ),

(23)

where 1 + 2 = n = n 1 + n 2 , and where the angles relate as cos ϑ = sin ϑ cos ϕ , cos ϕ = 

sin ϑ sin ϕ 1 − sin2 ϑ cos2 ϕ

,

(24)

is more complicated than the I–II relation: the form taken by the coefficients Un11,,n22 in (23) depends on the parities of the four indices. The nonzero coefficients are those where the parity of n 1 n 2 is the same as that of 1 2 , and the coefficients adopt four different forms. Each of the four forms Un11,,n22 of involves 3 F4 hypergeometric polynomials of unit argument, which are the four-parameter Racah polynomials Rq (λ(x ); α, β, γ, δ) of degree q. For the case of n 1 n 2 and 1 2 both even, so n = n 1 + n 2 = 2 p1 + 2 p2 = 1 + 2 = 2q1 + 2q2 is even, it was found that [4] √

  ρ( p1 ) Rq1 λ( p1 ); 21 n, − 12 (n+1), − 12 (n+2), 21 (n+1) , dq1 (25) with the Racah weight and norm factors 2 p ,2 p U2q11,2q22

= (−1)

p1 +q2

(n + 1)(4 p1 + 1) (2 p2 )! [( p1 + 21 n)!]2 , (2 p1 + n + 1)! ( p2 !)2 (n + 1)(2q1 + n + 1)! q2 ! , := q 1 1 (4q1 + 1)(2q2 )! 4 (q1 + 2 n)!

ρ( p1 ) := 42 p1 dq1

(26) (27)

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which are orthogonal over the quadratic lattice λ( p1 ) = p1 ( p1 + 21 ) [16]. In fact, this belongs to a special family of self-dual Racah polynomials that are invariant under the interchange p1 ↔ q1 . One may also express the coefficients (25) given by 6 j symbols [17] 2 p ,2 p U2q11,2q22

1 2 12 , =c 3  23

(28)

where c is a constant and the parameters are 1 = − 41 n − 1, 2 = 21 ( p1 − q1 − 1), 12 = 21 (q2 − p1 − 1); 3 = 41 n −

1 2

 = 21 (q1 − p1 − 1), 23 = −1 − q1 − 21 ( p1 + q2 ),

(29)

since n is even and so all parameters are integer or half-integer. In the other three cases of nonzero Un11,,n22 ’s, one finds expressions with the same structure as (25), but whose expression with 6 j symbols involves some negative parameters. Finally, an expression of all cases in terms of a sum of products of two Clebsch–Gordan coefficients can be derived through concatenating the direct and inverse III ↔ I ↔ II transformations [3].

5 Algebraic Structure of Zernike’s System  on the half-sphere H+ in terms In Eq. (13) we expressed the Zernike Hamiltonian W of the Laplace-Beltrami operator ΔLB . This would be the square angular momentum on the full sphere; only L 3 = ξ1 ∂ξ2 − ξ2 ∂ξ1 is the unequivocal symmetry generator of rotations in the x-y plane of the disk. When one separates the Hamiltonian (13) in systems II or III, the separating L 3 , are operators, beside J3 :=  1 ξ2 + ξ2 = L 21 + 2 2 3 , 2  4ξ3 4 sin ϕ 1 ξ12 + ξ32 2  + = L + , 2 4 cos2 ϕ 4ξ32

J1 := ∂ϕ2  +

(30)

J2 := ∂ϕ2 

(31)

 = J12 + J22 + J32 + 1 . Yet, J1 and and sum to the Zernike Hamiltonian on H+ as W 2 J2 are only related by a rotation to each other, so we introduce the three new operators, one of them built with a commutator, ξ1 ξ2  S2 := J1 − J2 ,  S3 := [ S1 ,  S2 ] = 2{ L 1,  L 2 }+ − 2 , S1 := J3 ,  ξ3

(32)

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where {·, ·}+ is the anticommutator. These three operators close nonlinearly under commutation,  S2 ] =  S3 , [ S3 ,  S1 ] = 4 S2 , [ S3 ,  S2 ] = 8 S13 − 8W S1 , [ S1 , 

(33)

into what is known as the cubic Higgs algebra [18] on the half-sphere H+ . Their corresponding operators on the (x, y) plane of the disk D are obtained through the map inverse to (13), and are 1 := y∂x − x∂ y , K  K 2 := −(1 − x 2 − y 2 )(∂x x − ∂ yy ) + 2x∂x − 2y∂ y , 3 := −4(1 − x 2 − y 2 )∂x y + 4y∂x + 4x∂ y , K

(34) (35) (36)

with the same commutation relations (33). Through the process of de-quantization, ∂x → i px , ∂ y → i p y , (34)–(36) provide three constants of the motion for the classical Zernike system, which will satisfy the cubic algebra (33) under Poisson brackets [2].

6 Conclusions The Zernike system has been shown to contain interesting algebraic properties, both in its classical and quantum realizations. It provides constants of the motion and is built out of an underlying SO(3) algebra, into a Higgs cubic superintegrable algebra [18–21]. Also, it has provided new special function relations between the classical and the higher Askey–Wilson polynomials of discrete variable [16]. Finally, it pertains the physically relevant optical setup of a circular pupil where wavefronts are analysed to correct aberrations. A recent paper by A. Fordy [22] offers a different perspective into classical and quantum superintegrability that applies, among other, to the classical and quantum Zernike systems [2, 3]. In the Introduction we mentioned the affinity that the Zernike system on the disk D shares with the one-dimensional Kerr model of nonlinear optics. Both Hamiltonians are built with ∇ 2 , ∇ · r and r2 ; but while the Kerr medium involves the compact quadratic operator (−∇ 2 +r2 )2 , the Zernike system adds the noncompact quadratic term (∇ · r)2 . That is, both Hamiltonians involve the generators of an SO(2,1) = Sp(2,R) algebra, plus a sui generis quadratic term. This topic is under current research. Acknowledgements We are grateful to Prof. Tom Koornwinder for pointing out previous works on two-variable polynomials that were not referenced in our recent papers on this subject. We thank Guillermo Krötzsch (icf- unam) for indispensable help with the figures. G.S.P. and A.Y. thank the support of project pro- sni- 2017 (Universidad de Guadalajara). N.M.A., C. S.-A. and K.B.W. acknowledge the support of unam- dgapa Project Óptica Matemática papiit-IN101115.

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References 1. F. Zernike, Beugungstheorie des Schneidenverfahrens und Seiner Verbesserten Form der Phasenkontrastmethode, Physica 1 (1934) 689–704. 2. G.S. Pogosyan, K.B. Wolf, and A. Yakhno, Superintegrable classical Zernike system, J. Math. Phys. 58 (2017) 072901. 3. G.S. Pogosyan, C. Salto-Alegre, K.B. Wolf, and A. Yakhno, Quantum superintegrable Zernike system, J. Math. Phys. 58 (2017) 072101. 4. N.M. Atakishiyev, G.S. Pogosyan, K.B. Wolf, and A. Yakhno, Interbasis expansions in the Zernike system, J. Math. Phys. 58 (2017) 103505. 5. C. Grosche, G.S. Pogosyan, and A.N. Sissakian, Path integral discussion for SmorodinskyWinternitz potentials. II. The two- and three-dimensional Euclidean sphere, Fortschr. Phys. 43 (1995) 523–563. 6. L.G. Mardoyan, G.S. Pogosyan, A.N. Sissakian, and V.M. Ter-Antonyan, Quantum Systems with Hidden Symmetry. Interbasis Expansions (Nauka, Fizmatlit, Moscow 2006, in Russian). 7. W. Miller Jr., S. Post, and P. Winternitz, Classical and quantum superintegrability with applications, J. Phys. A 46 (2013) 423001. 8. A.B. Bhatia and E. Wolf, On the circle polynomials of Zernike and related orthogonal sets, Math. Proc. Cambridge Phil. Soc. 50 (1954) 40–48. 9. E.C. Kintner, On the mathematical properties of the Zernike polynomials, Opt. Acta 23 (1976) 679–680. 10. A. Wünsche, Generalized Zernike or disc polynomials, J. Comp. App. Math. 174 (2005) 135– 163. 11. B.H. Shakibaei and R. Paramesran, Recursive formula to compute Zernike radial polynomials, Opt. Lett. 38 (2013) 2487–2489. 12. M.E.H. Ismail and R. Zhang, Classes of bivariate orthogonal polynomials, SIGMA 12 (2016) 021, 37 pages. 13. F. Didon, Développements sur certaines séries de polynômes à un nombre quelconque de variables (1), Ann. Sci. École Norm. Sup. 7 (1870) 247–268. 14. T. Koornwinder, Two-variable analogues of the classical orthogonal polynomials. In: Theory and Application of Special Functions, Ed. by R. Askey (Academic Press, 1975). 15. G.S. Pogosyan, K.B. Wolf, and A. Yakhno, New separated polynomial solutions to the Zernike system on the unit disk and interbasis expansion. J. Opt. Soc. Am. A 34 (2017) 1844–1848. 16. R. Koekoek, P.A. Lesky, and R.F. Swarttouw, Hypergeometric Orthogonal Polynomials and Their q-Analogues (Springer, 2010). 17. D.A. Varshalovich, A.N. Moskalev and V.K. Khersonski˘ı, Quantum Theory of Angular Momentum (World Scientific, Singapore, 1988). 18. P. W. Higgs, Dynamical symmetries in a spherical geometry, J. Phys. A 12 (1979) 309–323. 19. E.G. Kalnins, W. Miller Jr., and S. Post, Wilson polynomials and the generic superintegrable system on the 2-sphere, J. Phys. A 40 (2007) 11525–11538. 20. E.G. Kalnins, W. Miller Jr., and S. Post, Contractions of 2D 2nd order quantum superintegrable systems and the Askey scheme for hypergeometric orthogonal polynomials, SIGMA 9 (2013) 057. 21. V.X. Genest, L. Vinet, and A. Zhedanov, Superintegrability in two dimensions and the RacahWilson algebra, Lett. Math. Phys. 104 (2014) 931–952. 22. A. Fordy, Classical and quantum super-integrability: From Lissajous figures to exact solvability. arXiv:1711.10583vl [nlin.SI] 28 Nov 2017.

The Two Bosonizations of the CKP Hierarchy: Bicharacter Construction and Vacuum Expectation Values Iana I. Anguelova

Abstract We discuss the twisted vertex algebras involved in the two bosonizations of the CKP hierarchy. We show that they can be realized through the bicharacter construction of twisted vertex algebras, by endowing their Fock spaces with additional Hopf module-algebra structure and selecting appropriate bicharacters. We use the bicharacter descriptions to derive certain vacuum expectation values and identities.

1 Introduction Vertex algebra is the mathematical concept axiomatizing the properties of some, simplest, “algebras” of vertex operators from conformal and string theory (see for instance [10, 15, 17, 19, 21]). Vertex algebras are closely associated with integrable systems through the works of Date, Jimbo, Kashiwara and Miwa (e.g., [12, 14, 22]), Igor Frenkel [16], Victor Kac [19, 20] and many others. The boson-fermion correspondence, which is a super vertex algebra isomorphism between the charged free fermions super vertex algebra and the lattice super vertex algebra of the rank one odd lattice (see e.g. [19]), is a phenomenon well known and applied in many areas of representation theory and mathematical physics. It is famously associated with the Kadomtsev–Petviashvili (KP) hierarchy and the algebraic Hirota approach, whereby the KP hierarchy is written as a single Hirota equation involving the two charged fermion fields. Bosonization, which is one of the directions of any boson-fermion correspondence, translates the fermionic fields in the Hirota equation into bosonic fields, a process necessary to translate the purely algebraic Hirota equation into a hierarchy of actual differential equations. Besides “the” boson-fermion correspondence (which we can think of as type A, as it is associated with the infinite-dimensional a∞ I. I. Anguelova (B) Department of Mathematics, College of Charleston, Charleston, SC, USA e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2018 V. Dobrev (ed.), Quantum Theory and Symmetries with Lie Theory and Its Applications in Physics Volume 1, Springer Proceedings in Mathematics & Statistics 263, https://doi.org/10.1007/978-981-13-2715-5_17

275

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Lie algebra), there are other boson-fermion correspondences. Date, Jimbo, Kashiwara and Miwa introduced the boson-fermion correspondence of type B, which is associated to the BKP hierarchy ([14], see also [24]). They also introduced the CKP hierarchy and suggested an algebraic Hirota equation for it [13], as well as an approach to its bosonization (which though they did not complete). These, as well as other examples of bosonizations and boson-fermion correspondences, cannot be described through super vertex algebras due to the more general types of singularities in the Operator Product Expansions (OPEs) of the fields involved. Hence in [4, 5, 9] we developed the notion of a twisted vertex algebra, which accommodated these examples of boson-fermion correspondences. Twisted vertex algebras allow poles at z = ±w in the OPEs, as needed for the descriptions of the boson-fermion correspondences of type B, C and D; we recall the definition in the next Sect. 2. (Twisted vertex algebras, which perhaps should have been named “twisted chiral algebras” to avoid the potential confusion with twisted modules for vertex algebras, in general allow for poles at roots of unity in the OPEs, [5, 9], but we won’t need the general case here). The case of the CKP hierarchy and its Hirota equation, as introduced in [13], is particularly interesting and was shown recently to provide a wealth of surprises [2, 8, 23]. The first surprise was, that unlike any of the other cases, there is not one, but two bosonizations attached to the Hirota equation of the CKP hierarchy: one via a twisted Heisenberg field (see (11) below), and one via an untwisted Heisenberg field (see (12)). The second surprise was that, unlike the boson-fermion correspondence of type A, the spaces of highest weight vectors for these two Heisenberg actions themselves have structures of nontrivial twisted vertex algebras. As a result, this one case of the CKP hierarchy provided at least 5 different examples of twisted vertex algebras, with more than one isomorphism between them (we will recall them briefly in Sect. 2), and we suspect that is not the end of this story. Any such twisted vertex algebra isomorphism provides a wealth of applications, for instance, in [2] we studied the several Virasoro conformal structures involved, and the resulting character identities. Such examples should be understood very well, and specifically the reason behind the existence of more than one isomorphism involving these twisted vertex algebras. In this paper we attempt to understand another aspect of these twisted vertex algebras, namely the Hopf module-algebra structure underlying these examples. The more categorical approach to vertex algebras (super, twisted or quantum) was first suggested by Borcherds in [11], through the bicharacter construction via imposing a Hopf module-algebra structure on the underlying spaces. The bicharacter construction is not well understood among the vertex algebra researchers, as there is a major difference between the bicharacter description of a vertex algebra and the operator-based description typically used. In the operator-based description the examples are presented in terms of generating fields (vertex operators) and their OPEs (or commutation relations). With the bicharacter construction one starts instead with a (supercommutative supercocommutative) Hopf algebra M and its free Leibnitz module (see Sect. 3). The vertex operators then need to be defined through the bicharacter, as well as two auxiliary maps (the projection and the exponential maps, see Sect. 3, (40)). OPEs then result from the choice of a bicharacter r on the underlying Hopf algebra M— different bicharacters r will dictate different OPEs

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even with the same underlying algebra M. In Sect. 3 we show what choices for the underlying algebra Hopf algebra M and a bicharacter r describe the examples of the twisted vertex algebras associated to the CKP hierarchy. Even though the bicharacter description is somewhat counterintuitive, it provides another aspect of the algebraic structure involved, as well as allows us to see some of the properties from a different point of view in a more general light. One such property is the structure of the vacuum expectation values— we show that the type of vacuum expectation values really depends only on the structure of the underlying Hopf algebra M, for any bicharacter r , and not on the exact nature of the resulting generating fields. For instance, the fact that the vacuum expectation values are expressed through Pfaffians in all three of the boson-fermion correspondences of types B, C and D follows directly from the fact that the underlying Hopf algebra M involved is generated by a single odd primitive element, even though the bicharacters for each of the three cases are different, producing in turn different generating OPEs. In this paper we show that when the underlying Hopf algebra M is generated by a single even primitive element, then the resulting vacuum expectation values are Hafnians for any choice of bicharacter (see Proposition 4.4). As an immediate Corollary 4.6, we obtain the Hafnian-PfaffianProduct identity of [18], which was obtained also in [23] by direct calculation. There are more identities, including Borchardt’s identity involving determinant and permanent, that follow from the underlying bicharacter description of the untwisted bosonization, but we will have to leave those for a lengthier paper.

2 Twisted Vertex Algebras Related to the Two Bosonizations of the CKP Hierarchy Throughout this paper we will use common concepts and technical tools from the areas of vertex algebras and conformal field theory, such as the notions of field, locality, Operator Product Expansions (OPEs), normal ordered products, etc., for which we refer the reader to any book on the topic (such as [17, 19, 21]). We will also use the extension of these technical tools to the case of N -point locality, as introduced in [9]. In what follows we assume the underlying vector spaces are super (Z2 graded) vector spaces over the field of complex numbers C. For any homogeneous element a in a super vector space we denote by a˜ its Z2 grading, called parity. In this category the flip map τ is defined by ˜

τ (a ⊗ b) = (−1)a·˜ b (b ⊗ a)

(1)

for any homogeneous elements a, b in the super vector space, and extended by linearity. In a Hopf superalgebra H we denote the and the counit by  and η, coproduct the antipode by S. We will write (a) = a  ⊗ a  for the coproduct of a ∈ H (Sweedler’s notation). That means we will usually omit the indexing in (a) =

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ak ⊗ ak , especially when it is clear from the context. We want to also note that by a super Hopf algebra we mean that the product on H ⊗ H is defined by k

˜

(a ⊗ b)(c ⊗ d) = (−1)b·c˜ (ac ⊗ bd)

(2)

for any a, b, c, d homogeneous elements in H , and extended by linearity. Denote by HT−1 the Hopf algebra with a primitive generator D ((D) = D ⊗ 1 + 1 ⊗ D) and a grouplike generator T−1 ((T−1 ) = T−1 ⊗ T−1 ) subject to the relations: DT−1 = −T−1 D, and (T−1 )2 = 1

(3)

Denote by H D the Hopf subalgebra H D = C[D] of HT−1 . 2 (z, w) the space of rational functions in the variables z, w ∈ C with Denote by F± only poles at z = 0, z = ±w. Note that we do not allow poles at w = 0, i.e., if 2 2 (z, w), then f (z, 0) is well defined. Similarly, F± (z 1 , z 2 , . . . , zl ) is the f (z, w) ∈ F± space of rational functions in variables z 1 , z 2 , . . . , zl with only poles at z 1 = 0, or 2 (z, w) is a HT−1 ⊗ HT−1 Hopf algebra module by z j = ±z k . F± Dz f (z, w) = ∂z f (z, w), (T−1 )z f (z, w) = f (−z, w)

(4)

Dw f (z, w) = ∂w f (z, w), (T−1 )w f (z, w) = f (z, −w)

(5)

2 (z, w) by h z ·, and We will denote the action of elements h ⊗ 1 ∈ HT−1 ⊗ HT−1 on F± similarly h w · will denote the action of elements 1 ⊗ h ∈ HT−1 ⊗ HT−1 . We can now proceed to the concept of a twisted vertex algebra:

Definition 2.1 (Twisted vertex algebra of order 2, [4, 5, 9]) Twisted vertex algebra of order 2 is a collection of the following data (V, W, π f , Y ): • the space of fields V : a vector super space V , which is an HT−1 module, graded as an H D -module; • the space of states W : a vector super space W , W ⊂ V ; • a linear surjective projection map π f : V → W , such that π f |W = I dW • a field-state correspondence Y : a linear map from V to the space of fields on W ; • a vacuum vector: a vector 1 = |0 ∈ W ⊂ V . This data should satisfy the following set of axioms: • • • •

vacuum axiom: Y (1, z) = I dW ; modified creation axiom: Y (a, z)|0 |z=0 = π f (a), for any a ∈ V ; transfer of action: Y (ha, z) = h z · Y (a, z) for any h ∈ HT−1 ; analytic continuation: For any a, b, c ∈ V exists 2 (z, w) such that X z,w,0 (a ⊗ b ⊗ c) ∈ W [[z, w]] ⊗ F± Y (a, z)Y (b, w)π f (c) = i z,w X z,w,0 (a ⊗ b ⊗ c)

• symmetry: X z,w,0 (a ⊗ b ⊗ c) = X w,z,0 (τ (a ⊗ b) ⊗ c);

(6)

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• Completeness with respect to Operator Product Expansions (OPE’s): For each k ∈ Z, any a, b, c ∈ V , a, b–homogeneous w.r.to the grading by D, exist lk ∈ Z such that Resz=±w X z,w,0 (a ⊗ b ⊗ c)(z ∓ w)k =

finite 

s

wlk Y (vks , w)π f (c)

(7)

s

for some homogeneous elements vks ∈ V , where lks ∈ Z. Remark 1 In [5] (and [9]) we defined the more general concept of a twisted vertex algebra of order N , where the singularities in the OPEs are at roots of unity of order N . If V is an (ordinary) super vertex algebra, then the data (V, V, π f = I dV , Y ) is a twisted vertex algebra of order 1, and the definition above is of a twisted vertex algebra of order 2. Examples of twisted vertex algebras of order 2 are provided by the corresponding sides of the boson-fermion correspondences of types B, C and D [2, 4–6, 8, 14, 23, 24]. In this paper we will discuss the twisted vertex algebras associated to the CKP hierarchy and its two bosonizations. We start by recalling their descriptions in terms of generating fields and Fock spaces. Note that in [9] we provided uniqueness and existence theorems for twisted vertex algebras in terms of generating fields, which allows us to claim that the various fields we describe below do indeed generate corresponding twisted vertex algebras. The CKP hierarchy is described by a Hirota equation (see [2, 13, 23]), defined via the twisted neutral boson field χ(z) χ(z) =



χn z −n−1/2 ,

(8)

n∈Z+1/2

with OPE χ(z)χ(w) ∼

1 . z+w

(9)

In terms of commutation relations for the modes χn , n ∈ Z + 1/2, this OPE is equivalent to 1 [χm , χn ] = (−1)m− 2 δm,−n 1. (10) The modes of the field χ(z) form a Lie algebra which we denote by L χ . Let Fχ be the Fock module of L χ with vacuum vector |0 , such that χn |0 = 0 for n > 0. By Proposition 2.9 of [7] (see also Theorem 7.12 of [9]) there is a two-point local twisted vertex algebra structure with a space of fields generated by χ(z) and its descendent field χ(−z), acting on the space of states W = Fχ . We will denote this twisted vertex algebra for short by just Fχ . To describe the bosonizations maps, we need to recall the Heisenberg fields initiating the bosonizations.

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Proposition 2.2 ([7]) I. Let h Z+1/2 (z) = χ

1 : χ(z)χ(−z) : . 2

(−z) = h Z+1/2 (z), and we index h Z+1/2 (z) as h Z+1/2 (z) = We have h Z+1/2 χ χ χ χ t −2n−1 Z+1/2 hn z . The field h χ (z) has OPE with itself given by: (z)h Z+1/2 (w) ∼ − h Z+1/2 χ χ

 n∈Z+1/2

1 1 z 2 + w2 1 1 ∼− − , 2 2 2 2 2(z − w ) 4 (z − w) 4 (z + w)2

(11)

and its modes, h tn , n ∈ Z + 1/2, generate a twisted Heisenberg algebra HZ+1/2 with relations [h tm , h tn ] = −mδm+n,0 1, m, n ∈ Z + 1/2. II. Let 1 h Zχ (z) = (: χ(z)χ(z) : − : χ(−z)χ(−z) :) . 4z  We have h Zχ (−z) = h Zχ (z), and we index h Zχ (z) as h Zχ (z) = n∈Z h Zn z −2n−2 . The field h Zχ (z) has OPE with itself given by: h Zχ (z)h Zχ (w) ∼ −

(z 2

1 , − w 2 )2

(12)

and its modes, h Zn , n ∈ Z, generate an untwisted Heisenberg algebra HZ with relations [h Zm , h Zn ] = −mδm+n,0 1, m, n ∈ Z. Denote by Fχhwv the vector space spanned by the highest weight vectors for the untwisted Heisenberg algebra representation on Fχ , and by Fχt−hwv the vector space spanned by the highest weight vectors for the twisted Heisenberg algebra representation on Fχ . These spaces themselves have structures of twisted vertex algebras. First, using the Heisenberg fields, we can define the exponentiated boson fields as follows: Let   1 h Zn z −2n ; V − (z) = exp − n n>0 Define βχ (z 2 ) =

V + (z) = exp

  1 h Z−n z 2n . n n>0

(13)

χ(z) − χ(−z) χ(z) + χ(−z) ; γχ (z 2 ) = . 2z 2

(14)

and Z

H β (z 2 ) = V + (z)−1 βχ (z 2 )z −2h 0 V − (z)−1 ,

Z

H γ (z 2 ) = V + (z)γχ (z 2 )z 2h 0 V − (z). (15) (V − (z) and V + (z) are actually functions of z 2 , so the notation is unambiguous).

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Similarly denote Vt− (z) = exp

 n>0

 2 h t2n−1 z −2n+1 ; 2n − 1 2

  Vt+ (z) = exp − n>0

 2 h t 2n−1 z 2n−1 ; 2n − 1 − 2

(16) and

H χ (z) = Vt+ (z)−1 χ(z)Vt− (z)−1 .

(17)

Theorem 2.3 I. [8] The vector space Fχhwv spanned by the highest weight vectors for the untwisted Heisenberg algebra HZ has a structure of a super vertex algebra, strongly generated by the fields H β (z) and H γ (z), with vacuum vector |0 , translation hwv , and vertex operator map induced by operator T = L −1 Y (χ−1/2 |0 , z) = H γ (z), Y (χ−3/2 |0 , z) = H β (z).

(18)

This vertex algebra structure is a realization of the symplectic fermion vertex algebra, indicated by the OPEs: 1 1 , H γ (z)H β (w) ∼ − ; (z − w)2 (z − w)2 H β (z)H β (w) ∼ 0; H γ (z)H γ (w) ∼ 0.

H β (z)H γ (w) ∼

(19) (20)

II. [2] The vector space Fχt−hwv spanned by the highest weight vectors for the twisted Heisenberg algebra HZ+1/2 has a structure of an N = 2 twisted vertex algebra, generated by the field H χ (z), with vacuum vector |0 , and vertex operator map induced by (21) Y (χ−1/2 |0 , z) = H χ (z). This twisted vertex algebra structure is twisted fermionic, indicated by the OPEs: H χ (z)H χ (w) ∼

z−w . (z + w)2

(22)

Corollary 2.4 I. [1] Define the symplectic fermion Fock space: β

β

β

γ

γ

γ

S F := {H(m ) . . . H(m ) H(m ) H(n ) . . . H(n ) H(n ) |0 | k

2

1

s

2

1

| m k < . . . m 2 < m 1 , n s < . . . n 2 < n 1 ; m i , n j ∈ Z |z 2 | > · · · > |z 2n |. Note that such is the case of Mφ−χ = C{φχ }, hence we immediately have  z − z 2n  φ−χ 2n i j 0 | H χ (z 1 )H χ (z 2 ) . . . H χ (z 2n )|0 = i z P f r zi ,z j (φχ ⊗ φχ ) i, j=1 = i z P f . (z i + z j )2 i, j=1

(51) Similarly, we derived a general vacuum expectation value formula that is valid for any twisted vertex algebra based on L 1 = C[Zα] and any choice of a bicharacter r . Namely, Proposition V.6 of [5] asserts that 0 | em 1 α (z 1 )em 2 α (z 2 ) . . . em n α (z n )|0 = i z δm 1 +m 2 +···+m n ,0

n 

r zi ,z j (em i α ⊗ em j α ).

i< j, i, j=1

Hence, since our twisted vertex algebra is based on the tensor product Mχ−tb = Mφ−χ ⊗ L 1 , we immediately obtain that 0 | χtb (z 1 )χtb (z 2 ) . . . χtb (z 2n )|0 = i z P f

2n  z − z 2n  zi + z j i j · . 2 (z i + z j ) i, j=1 i< j z i − z j

(52)

To obtain vacuum expectation values for the original twisted vertex algebra generated by χ(z) on Fχ , we need to modify the Proposition V.4 of [5] to accommodate a pair

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(C{χ}, r ) with χ being an even primitive variable (which in fact will make it easier, as there will be no minus signs to account): Proposition 4.4 Let V be a twisted vertex algebra based on M = C{χ}, where χ is an even primitive variable, and any supersymmetric bicharacter r (in particular, V = HT−1 (M) = H D (C{χ, T χ}) and W = H D (C{χ})). Denote by χ(z) the field Y (χ, z) produced by definition (40), via (39). The following formula for the vacuum expectation values holds:  2n 0 | χ(z 1 )χ(z 2 ) . . . χ(z 2n )|0 = i z H f r zi ,z j (χ ⊗ χ) i, j=1 .

(53)

Here H f denotes the Hafnian of a symmetric matrix of even size: for a symmetric matrix A of size 2n by 2n H f (A) =



Ai1 ,i2 Ai3 ,i4 . . . Ai2n−1 ,i2n .

P

The sum is over all permutations P, P(k) = i k , of {1, 2, . . . , 2n}, such that i 1 < i 2 , i 3 < i 4 , . . . , i 2n−1 < i 2n , i 1 < i 3 < · · · < i 2n−1 . Proof The proof is very similar to the proof of V.4 of [5], but we want to explain why the bicharacter of the primitive element determines that the summation is over these particular permutations only, thus resulting in a Hafnian. Since χ is a primitive element we have r z,w (χ ⊗ 1) = r z,w (1 ⊗ χ) = 0 for any bicharacter. To calculate the vacuum expectation value, we need the analytic continuation X z1 ,z2 ,...,z2n (χ ⊗ χ . . . χ), which has explicit formula in terms of the (2n)-character r z1 ,z2 ,...,z2n (χ ⊗ χ ⊗ . . . ⊗ χ) (see [5]). The only contributions in the (2n)-character r z1 ,z2 ,...,z2n (χ ⊗ χ ⊗ . . . ⊗ χ) will come from the following situation: a nonzero summand in the (2n)-character r z1 ,z2 ,...,z2n (χ ⊗ χ ⊗ . . . ⊗ χ) will be a product of nonzero bicharacter factors, and that happens only when we have a sequence of either (1, 1) pairs (non-contributing, but non-zero, as r z,w (1 ⊗ 1) = 1), or (χ, χ) pairs (nontrivial contributions). If there is a mixed pair (1, χ) or (χ, 1) appearing in a factor in a summand, that summand will be 0. So a nonzero summand will have exactly n such nontrivial contributing pairs (χ, χ), and each pair forms one bicharacter r zi2k−1 ,zi2k (χ ⊗ χ). Corollary 4.5 (of Propositions 4.1 and 4.4) For the generating field χ(z) of the twisted vertex algebra Fχ we have 0 | χ(z 1 )χ(z 2 ) . . . χ(z 2n )|0 = i z H f



1 2n . z i + z j i, j=1

(54)

We finish with the following identity (proved in [18], and re-proved in [23] by using direct calculations and Wick’t Theorem), which now follows directly from the twisted bosonization isomorphism and the bicharacter description of both sides of the bosonization:

The Two Bosonizations of the CKP Hierarchy …

291

Corollary 4.6 2n  z − z 2n  zi + z j 1 2n i j = Pf · . Hf z i + z j i, j=1 (z i + z j )2 i, j=1 i< j z i − z j



(55)

Acknowledgements We would like to express our gratitude to all the organizers of the International Workshop “Lie Theory and Its Applications in Physics”, and especially Vladimir Dobrev, for their long standing and continuing effort to provide such an excellent forum for the researchers in the areas of Lie theory, quantum symmetries, vertex algebras, and mathematical physics in general, to meet, interact and exchange ideas; thank you.

References 1. Toshiyuki Abe. A Z2 -orbifold model of the symplectic fermionic vertex operator superalgebra. Mathematische Zeitschrift, 255(4):755–792, 2007. 2. Iana I. Anguelova. The two bosonizations of the CKP hierarchy: overview and character identities. to appear in Contemporary Mathematics (proceedings), Naihuan Jing and Kailash Misra, editors. arXiv:1708.04992 [math-ph]. 3. Iana I. Anguelova. Super-bicharacter construction of H D -quantum vertex algebras. Rep. Math. Phys., 61(2):253–263, 2008. 4. Iana I. Anguelova. Boson-fermion correspondence of type B and twisted vertex algebras. In Vladimir Dobrev, editor, Lie Theory and Its Applications in Physics, volume 36 of Springer Proceedings in Mathematics and Statistics, pages 399–410. Springer Japan, 2013. 5. Iana I. Anguelova. Twisted vertex algebras, bicharacter construction and boson-fermion correspondences. Journal of Mathematical Physics, 54(12):38, 2013. 6. Iana I. Anguelova. Boson-fermion correspondence of type D-A and multi-local Virasoro rep1 resentations on the Fock space F ⊗ 2 . Journal of Mathematical Physics, 55(11):23, 2014. 7. Iana I. Anguelova. Multilocal bosonization. Journal of Mathematical Physics, 56(12):13, 2015. 8. Iana I. Anguelova. The second bosonization of the CKP hierarchy. Journal of Mathematical Physics, 58(7), 2017. 9. Iana I. Anguelova, Ben Cox, and Elizabeth Jurisich. N -point locality for vertex operators: normal ordered products, operator product expansions, twisted vertex algebras. J. Pure Appl. Algebra, 218(12):2165–2203, 2014. 10. Richard E. Borcherds. Vertex algebras, Kac-Moody algebras, and the Monster. Proc. Nat. Acad. Sci. U.S.A., 83(10):3068–3071, 1986. 11. Richard E. Borcherds. Quantum vertex algebras. In Taniguchi Conference on Mathematics Nara ’98, volume 31 of Adv. Stud. Pure Math., pages 51–74, Tokyo, 2001. Math. Soc. Japan. 12. Etsur¯o Date, Michio Jimbo, Masaki Kashiwara, and Tetsuji Miwa. Transformation groups for soliton equations. III. Operator approach to the Kadomtsev-Petviashvili equation. J. Phys. Soc. Japan, 50(11):3806–3812, 1981. 13. Etsur¯o Date, Michio Jimbo, Masaki Kashiwara, and Tetsuji Miwa. Transformation groups for soliton equations. VI. KP hierarchies of orthogonal and symplectic type. J. Phys. Soc. Japan, 50(11):3813–3818, 1981. 14. Etsur¯o Date, Michio Jimbo, Masaki Kashiwara, and Tetsuji Miwa. Transformation groups for soliton equations. IV. A new hierarchy of soliton equations of KP-type. Phys. D, 4(3):343–365, 1981/82. 15. Edward Frenkel and David Ben-Zvi. Vertex algebras and algebraic curves, volume 88 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, second edition, 2004.

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16. Igor B. Frenkel. Two constructions of affine Lie algebra representations and boson-fermion correspondence in quantum field theory. J. Funct. Anal., 44(3):259–327, 1981. 17. Igor Frenkel, James Lepowsky, and Arne Meurman. Vertex operator algebras and the Monster, volume 134 of Pure and Applied Mathematics. Academic Press Inc., Boston, MA, 1988. 18. Masao Ishikawa, Hiroyuki Kawamuko, and Soichi Okada. A Pfaffian-Hafnian analogue of Borchardt’s identity. The Electronic Journal of Combinatorics, 12:8, 2005. 19. Victor Kac. Vertex algebras for beginners, volume 10 of University Lecture Series. American Mathematical Society, Providence, RI, second edition, 1998. 20. V. G. Kac and A. K. Raina. Bombay lectures on highest weight representations of infinitedimensional Lie algebras, volume 2 of Advanced Series in Mathematical Physics. World Scientific Publishing Co. Inc., Teaneck, NJ, 1987. 21. James Lepowsky and Haisheng Li. Introduction to vertex operator algebras and their representations, volume 227 of Progress in Mathematics. Birkhäuser Boston Inc., Boston, MA, 2004. 22. T. Miwa, M. Jimbo, and E. Date. Solitons: differential equations, symmetries and infinite dimensional algebras. Cambridge tracts in mathematics. Cambridge University Press, 2000. 23. J. W. van de Leur, A. Y. Orlov, and T. Shiota. CKP Hierarchy, Bosonic Tau Function and Bosonization Formulae. SIGMA, 8, 2012. 28pp. 24. Yuching You. Polynomial solutions of the BKP hierarchy and projective representations of symmetric groups. In Infinite-dimensional Lie algebras and groups (Luminy-Marseille, 1988), volume 7 of Adv. Ser. Math. Phys., pages 449–464. World Sci. Publ., Teaneck, NJ, 1989.

Recursion Operator and Bäcklund Transformation for Super mKdV Hierarchy A. R. Aguirre, J. F. Gomes, A. L. Retore, N. I. Spano and A. H. Zimerman

Abstract In this paper we consider the N = 1 supersymmetric mKdV hierarchy composed of positive odd flows embedded within an affine sˆ l(2, 1) algebra. Its Bäcklund transformations are constructed in terms of a gauge transformation preserving the zero curvature representation. The recursion operator relating consecutive flows is derived and shown to relate their Backlund transformations.

1 Introduction The algebraic formulation for integrable hierarchies presents itself as a powerful framework in order to discuss its integrable properties, symmetries and soliton solutions. In particular the supersymmetric mKdV hierarchy consists of a set of time evolution (flows) equations obtained from a zero curvature representation involving A. R. Aguirre Instituto de Física e Química, Universidade Federal de Itajubá - IFQ/UNIFEI, Av. BPS 1303, Itajubá, Minas Gerais 37500-903, Brazil e-mail: [email protected] A. L. Retore Physics Department of the University of Miami, Coral Gables, FL 33124, USA e-mail: [email protected] N. I. Spano · J. F. Gomes (B) · A. H. Zimerman Instituto de Física Teórica - IFT/UNESP, Rua Dr. Bento Teobaldo Ferraz 271, Bloco II, São Paulo 01140-070, Brazil e-mail: [email protected] N. I. Spano e-mail: [email protected] A. H. Zimerman e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2018 V. Dobrev (ed.), Quantum Theory and Symmetries with Lie Theory and Its Applications in Physics Volume 1, Springer Proceedings in Mathematics & Statistics 263, https://doi.org/10.1007/978-981-13-2715-5_18

293

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a two dimensional gauge potential lying within an affine sˆ l(2, 1) Kac–Moody algebra and a common infinite set of conservation laws [1–3]. Moreover, Bäcklund transformation can be employed to construct an infinite sequence of solitons solutions by purely superposition principle and also to link nonlinear equations to canonical forms as discussed for many examples in [4]. For the supersymmetric mKdV hierarchy, the Bäcklund transformation was derived for the entire hierarchy by an universal gauge transformation that preserves the zero curvature representation and henceforth the equations of motion [1]. The results obtained in [2, 5, 6] for the super sinh-Gordon were extended to the entire smKdV hierarchy by the construction of a Bäcklund-gauge transformation which connects two field configurations of the same equations of motion [1]. Such structure was first introduced in [7, 8] for the bosonic sine-Gordon theory in order to describe integrable defects in the sense that two solitons solutions are interpolated by a defect, as a set of internal boundary conditions derived from a Lagrangian density located at certain spatial position connecting two types of solutions. The integrability of the model is guaranteed by the gauge invariance of the zero curvature representation. The N = 1 supersymmetric modified Korteweg de-Vries (smKdV) hierarchy in the presence of defects was investigated in [1] through the construction of gauge transformation in the form of a Bäcklund-defect matrix approach. Firstly, we employ the defect matrix associated to the hierarchy which turns out to be the same as for the super sinh-Gordon (sshG) model. The method is general for all flows and as an example we have derived explicitly the Bäcklund equations in components for the first few flows of the hierarchy, namely t1 , t3 and t5 . Finally, this super Bäcklund transformation is employed to introduce type I defects for the supersymmetric mKdV hierarchy. In this note we propose an alternative derivation for the Bäcklund transformation obtained in [1] by employing a recursion operator. For the bosonic case of the mKdV hierarchy the recursion operator was constructed in [9] and it relates equations of motion for two consecutive time evolutions. We show that the same philosophy can be applied to the supersymmetric mKdV hierarchy to relate Bäcklund transformations for two consecutive flows. In what follows, we first derive the recursion operator for the supersymmetric mKdV hierarchy directly from the zero curvature representation. For technical reasons we change variables u(x, tN ) of mKdV equation as u(x, tN ) = ∂x φ(x, tN ) which seems more suitable to deal with Bäcklund transformations. We next conjecture that the Bäcklund transformations for consecutive flows are also related by the same recursion operator. In fact we verify our conjecture for the first few flows generated by t1 , t5 and t3 .

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295

2 The smKdV Hierarchy An integrable hierarchy can be obtained from the zero curvature condition   ∂x + Ax , ∂tN + AtN = 0

(1)

 where, Ax and AtN are the Lax pair lying into an affine Kac–Moody superalgebra (G) and tN represents the time flow of an integrable equation. Another important key ingredient to construct an integrable hierarchy is a grading operator Q and a constant grade one element E (1) that decomposes the affine superalgebra into the following subspaces = ⊕ G m = K(E) ⊕ M(E) G

(2)

m according to Q, i.e., [Q, Gm ] = mGm , where mis the degree of the subspace G /[ x, E (1) ] = 0 is the kernel of E (1) and M(E) is its complement ( K(E) = x ∈ G image). Now we can define the Lax pair as Ax = E (1) + A0 + A1/2 , AtN =

DN(N )

+

(N −1/2) DN

+ ··· +

(3) (1/2) DN

+

DN(0) ,

(4)

0 ∩ M(E), A1/2 ∈ G 1/2 ∩ M(E) with their respective components where A0 ∈ G m . defining the bosonic and fermionic fields of the theory and DN(m) ∈ G The decomposition of the Eq. (1) into graded subspaces yields the following system   (N + 1) : E (1) , DN(N ) = 0,     (N −1/2) + A1/2 , DN(N ) = 0, (N + 1/2) : E (1) , DN       (N −1/2) (N ) : ∂x DNN + A0 , DN(N ) + E (1) , DN(N −1) + A1/2 , DN = 0, .. .

(1) : (1/2) : (0) :

      (1/2) ∂x DN(1) + A0 , DN(1) + E (1) , DN(0) + A1/2 , DN = 0,     (1/2) (1/2) ∂x DN + A0 , DN + A1/2 , DN(0) − ∂tN A1/2 = 0,   ∂x DN(0) + A0 , DN(0) − ∂tN A0 = 0.

(5)

The set of Eq. (5) can be solved recursively yielding the time evolution equations for the fields in A0 and A1/2 as the zero and 1/2 grade components, respectively. In particular, the construction of the supersymmetric mKdV hierarchy is based on =  the judicious construction of an affine subalgebra of G sl(2, 1), with the principal

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(1) gradation operator Q = 2d + 21 h(0) = K (1) + K (2) . Its 1 and the constant element E generators may be regrouped as

(n+ 1 ) (n+ 1 ) (2n+ 3 ) (n+1) + E−α − E−α2 2 , F1 2 = Eα1 +α2 2 − Eα(n+1) 1 −α2 2 (2n+ 1 ) (n+ 1 ) (n+ 21 ) (n) F2 2 = −Eα(n)1 +α2 + Eα2 2 + E−α1 −α − E−α , 2 2 1 1 1 2n+ n+ n+ ( ) ( ) ( 2) (n) G 1 2 = Eα(n)1 +α2 + Eα2 2 + E−α1 −α + E−α 2 2 (2n+ 3 ) (n+ 1 ) (n+ 1 ) (n+1) G 2 2 = −Eα1 +α2 2 − Eα(n+1) + E−α + E−α2 2 , 1 −α2 2 (n+1) K1(2n+1) = −E−α − Eα(n)1 , 1 (n+ 1 ) (n+ 1 ) K2(2n+1) = h1 2 − h2 2 , (n+1) − Eα(n)1 , M1(2n+1) = E−α 1

M2(2n) = 2h(2n) 1

(6)

and decomposed as follows (see [10] for details),

Mbos = M2(2n) , M1(2n+1) ,

Kbos = K1(2n+1) , K2(2n+1) ,

(2n+ 1 ) (2n+ 3 ) Mfer = G 1 2 , G 2 2 , (2n+ 3 ) (2n+ 1 )

Kfer = F1 2 , F2 2

(7)

Notice that the fermionic generators Fi and G i , i = 1, 2 lying in the Kernel and Image respectively display an explicit Z2 structure in their affine indices, in the sense that the semi integers indices N + 1/2 are decomposed according to 2n + 1/2 and 2n + 3/2 disjoints subsets. Another Z2 structure arises, now decomposing the integers N into odd (2n + 1) and even (2n) subsets. Assign to the bosonic generators {K1 , K2 , M1 } and {M2 } the grades 2n + 1 and 2n respectively. The affine algebra displayed in the appendix is shown to close consistently with the Z2 structures described above. The x part of the Lax pair is then constructed from A0 = u(x, t)M2(0) and A1/2 = ¯ t)G (1/2) . ψ(x, 1 The first equation in the system (5) implies that DN(N ) ∈ K(E) and hence N = 2n + 1. In order to solve equations in (5) we expand DN(m) according to its bosonic or fermionic character using latin or greek coefficients, respectively following the grading given in equation (7),1 i.e.,

1 Moreover we use α , β m m

for grades m = 2n + 1/2 and γm , δm for m = 2n + 3/2 while am , bm , cm for m = 2n + 1 and dm for m = 2n.

Recursion Operator and Bäcklund Transformation for Super mKdV Hierarchy (2n+ 23 )

DN

297

(2n+ 3 ) (2n+ 3 ) = γ2n+ 23 F1 2 + δ2n+ 23 G 2 2 ,

DN(2n+1) = a2n+1 K1(2n+1) + b2n+1 K2(2n+1) + c2n+1 M1(2n+1) , mber (2n+ 1 ) (2n+ 1 ) (2n+ 1 ) DN 2 = α2n+ 21 F2 2 + β2n+ 21 G 1 2 , DN(2n) = d2n M2(2n) , (2n− 21 )

DN

(2n− 1 ) (2n− 1 ) = γ2n− 21 F1 2 + δ2n− 21 G 2 2 ,

DN(2n−1) = a2n−1 K1(2n−1) + b2n−1 K2(2n−1) + c2n−1 M1(2n−1) , (2n− 3 ) (2n− 3 ) (2n− 3 ) DN 2 = α2n− 23 F2 2 + β2n− 23 G 1 2 , DN(2n−2) = d2n−2 M2(2n−2) , .. . (3) (3) (3) DN2 = γ 23 F1 2 + δ 23 G 2 2 , DN(1) = a1 K1(1) + b1 K2(1) + c1 M1(1) , (1) (1) (1) DN2 = α 21 F2 2 + β 21 G 1 2 DN(0) = d0 M2(0) .

(8)

¯ where the am , bm , cm , dm and αm , βm , γm , δm are functionals of the fields u and ψ. Substituting this parameterization in the Eq. (5), one solve recursively for all D(m) , m = 0, · · · N . Starting with the highest grade equation in (5) in which N = 2n + 1, [K1(1) + K2(1) , a2n+1 K1(2n+1) + b2n+1 K2(2n+1) + c2n+1 M1(2n+1) ] = 0

(9)

We obtain after using the comutation relations given in the appendix that c2n+1 = 0. Now substituting this result in the next equation in (5), i.e, the equation for degree N + 1/2 we get, 1 ¯ (10) β2n+ 21 = ψ(a 2n+1 + b2n+1 ) 2 From the equation for degree N we find that a2n+1 , b2n+1 are constants and d2n = ¯ 2n+ 1 . Proceeding in this way until we reach the equation for degree ua2n+1 + ψα 2 N − 2, we get (N − 1/2) : (N − 1) : (N − 3/2) :

¯ 2n = 0 ∂x α2n+ 21 − uβ2n+ 21 + ψd

∂x β2n+ 21 − uα2n+ 21 + 2δ2n− 21 = 0

(11)

¯ 2n− 1 = 0 ∂x d2n − 2c2n−1 + 2ψγ 2

(12)

¯ 2n−1 = 0 ∂x γ2n− 21 − uδ2n− 21 + ψc ¯ 2n−1 + b2n−1 ) = 0 (13) ∂x δ2n− 21 − uγ2n− 21 + 2β2n− 23 − ψ(a

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(N − 2) :

¯ 2n− 3 = 0 ∂x a2n−1 + 2uc2n−1 − 2ψβ 2 ¯ 2n− 3 = 0 ∂x b2n−1 + 2ψβ 2 ¯ 2n− 3 = 0 ∂x c2n−1 − 2d2n−2 + 2ua2n−1 + 2ψα 2

(14)

The subsequent equations are all similar to the set above, in the sense that the equations for even grade will correspond to (12), the odd ones will be similar to the set in (14). For the semi-integer degree equations the following combinations are allowed: if (N − 21 − 2m) then it corresponds to the set (11) and if the grade can be written as (N − 21 − 2m − 1) it seems like (13) where m ∈ Z+ . Then for a specific n ∈ Z+ these results can be written in the following way, √ c2n+1 = 0,

β2n+ 21 =

i ¯ ψ(a2n+1 + b2n+1 ),

2 a2n+1 = constant b2n+1 = constant √ ¯ 2n+ 1 d2n = ua2n+1 + iψα 2 √ ¯ 2n+1−j = 0 ∂x α2n+ 23 −j − uβ2n+ 23 −j + iψd (odd j) ∂x β2n+ 23 −j − uα2n+ 23 −j + 2δ2n+ 21 −j = 0 (odd j) √ ¯ 2n+ 1 −j = 0 ∂x d2n+1−j − 2c2n−j + 2 iψγ (odd j) 2 √ ¯ 2n+1−j = 0 ∂x γ2n+ 23 −j − uδ2n+ 23 −j + iψc (even j) √ ¯ 2n+1−j + b2n+1−j ) = 0 (even j) ∂x δ2n+ 23 −j − uγ2n+ 23 −j + 2β2n+ 21 −j − iψ(a √ ¯ 2n+ 1 −j = 0 ∂x a2n+1−j + 2uc2n+1−j − 2 iψβ (even j) 2 √ ¯ 2n+ 1 −j = 0 ∂x b2n+1−j + 2 iψβ (even j) 2 √ ¯ 2n+ 1 −j = 0 ∂x c2n+1−j − 2d2n−j + 2ua2n+1−j + 2 iψα (even j) (15) 2 where j = 1, . . . , 2n. We proceed in this way until the grade (1/2) equation in (5) to get ¯ 0 ∂x α 21 = uβ 12 − ψd

(16)

∂t2n+1 ψ¯ = ∂x β 21 − uα 21

(17)

and the zero grade equation to obtain ∂t2n+1 u = ∂x d0

(18)

Therefore the problem is to solve recursively this set of equations, finding the respective coefficients for a given value of n and then substitute them in (18) and ¯ For example, if n = 0 the equations (17) to obtain the time evolution of the fields u, ψ. of motion are

Recursion Operator and Bäcklund Transformation for Super mKdV Hierarchy

¯ ∂t1 ψ¯ = ∂x ψ,

∂t1 u = ∂x u

299

(19)

For n = 1 we have the supersymmetric mKdV equation  ¯ x uψ¯x , 4∂t3 u = u3x − 6u2 ux + 3iψ∂  4∂t3 ψ¯ = ψ¯ 3x − 3u∂x uψ¯ .

(20) (21)

Then, for n = 2 we have 16∂t5 u = u5x − 10(ux )3 − 40u(ux )(u2x ) − 10u2 (u3x ) + 30u4 (ux ) ¯ x (uψ¯ 3x − 4u3 ψ¯ x + ux ψ¯ 2x + u2x ψ¯ x ) + 5iψ¯ x ∂x (uψ¯2x ), + 5iψ∂ ¯ + 10u2 ∂x (u2 ψ) ¯ 16∂t5 ψ¯ = ψ¯ 5x − 5u∂x (uψ¯2x + 2ux ψ¯ x + u2x ψ) ¯ − 10(ux )∂x (ux ψ).

(22) (23)

3 Recursion Operator for smKdV Hierarchy We shall now consider the construction of a set of supersymmetric integrable equations by solving the system in (15). Since the solution of (15) is similar for all values of n it is expected that there exists a connection among the time flows. The recursion operator is the mathematical object responsible for such connection and will be constructed in this section. In order to see this we consider the equations for N = 2n + 1 and N = 2n + 3 2n + 1 c2n+1 = 0,

2n + 3 c2n+3 = 0

a2n+1 = b2n+1 = 1, β2n+ 21 = ψ¯

(24)

a2n+3 = b2n+3 = 1, β2n+ 5 = ψ¯

(25) (26)

2

¯ 2n+ 1 , d2n = u + ψα 2

¯ 2n+ 5 d2n+2 = u + ψα 2

¯ 2n = 0, ∂x α2n+ 21 − uβ2n+ 21 + ψd

(27)

¯ 2n+2 = 0 ∂x α2n+ 5 − uβ2n+ 5 + ψd

(28)

∂x β2n+ 21 − uα2n+ 21 + 2δ2n− 21 = 0, ∂x β2n+ 5 − uα2n+ 5 + 2δ2n+ 23 = 0

(29)

¯ 2n− 1 = 0, ∂x d2n − 2c2n−1 + 2ψγ 2

(30)

.. .

¯ 0 = 0, ∂x α1/2 − uβ1/2 + ψd

2

2

2

2

¯ 2n+ 3 = 0 ∂x d2n+2 − 2c2n+1 + 2ψγ 2

.. .

¯ 2=0 ∂x α5/2 − uβ5/2 + ψd ∂x β5/2 − uα5/2 + 2δ3/2 = 0 ¯ 3/2 = 0 ∂x d2 − 2c1 + 2ψγ

(31) (32)

(33) ¯ (34) ∂x γ3/2 − uδ3/2 + ψc1 = 0 ¯ 1 + b1 ) + 2β1/2 = 0 (35) ∂x δ3/2 − uγ3/2 − ψ(a

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¯ 1/2 + 2uc1 = 0 ∂x a1 − 2ψβ ¯ 1/2 = 0 ∂x b1 + 2ψβ

(36) (37) (38) (39)

¯ 1/2 = 0 ∂x c1 − 2d0 + 2ua1 + 2ψα ¯ 0=0 ∂x α1/2 − uβ1/2 + ψd ∂t2n+3 ψ¯ = ∂x β1/2 − uα1/2

∂t2n+1 ψ¯ = ∂x β1/2 − uα1/2 , ∂t2n+1 u = ∂x d0 ,

(40) (41)

∂t2n+3 u = ∂x d0

Notice that until the Eq. (32) the aligned equations have the same solution, in such a way that we can make the following useful identifications,

d2

2n+3

= d0

2n+1

, β5/2

2n+3

= β1/2

2n+1

, α5/2

2n+3

= α1/2

2n+1

.

(42)

The case for N = 2n + 3 has eight additional Eqs. (32)–(39), which can be solved in terms of the coefficients for N = 2n + 1 by the relations (42). Then we will be able to relate the time evolution equations for t2n+3 to the time evolution equations for t2n+1 . Starting with the Eq. (32) by using (42) we get

δ3/2

2n+3

1 ¯ = − ∂t2n+1 ψ. 2

(43)

Substituting in (33) and (34)

1

¯ 3/2

= ∂t2n+1 u + ψγ c1 2n+3 2n+3 2 

1

¯ =− γ3/2 dx ∂t2n+1 (uψ). 2n+3 2

(44) (45)

Solving recursively the Eqs. (35)–(39) we get the following coefficients 

1 u

¯ + 1 ψ(a ¯ ¯ 1 + b1 )

= ∂x ∂t2n+1 ψ − , (46) dx ∂t2n+1 (uψ) β1/2 2n+3 2n+3 4 4 2   

1

¯ a1 = − dx u∂t2n+1 u + dx uψ¯ dx ∂t2n+1 (uψ) 2n+3 2  1 ¯ ¯ x ∂t2n+1 ψ, + (47) dx ψ∂ 2   

1

¯ ¯ x ∂t2n+1 ψ¯ + 1 dx uψ¯ dx ∂t2n+1 (uψ), =− (48) b1 dx ψ∂ 2n+3 2 2 

1 1 ¯

¯ ¯ 1/2

d0 = ∂x ∂t2n+1 u − ψ∂ dx u∂t2n+1 u + ψα t2n+1 (uψ) − u 2n+3 2n+3 4 4    ¯ u ¯ − ∂x ψ dx ∂t2n+1 (uψ) ¯ + dx uψ¯ dx ∂t2n+1 (uψ) 2 4

Recursion Operator and Bäcklund Transformation for Super mKdV Hierarchy

 u 2  1 = 4  1 + 2  1 − 4 +

α1/2

2n+3

301

¯ ¯ x ∂t2n+1 ψ, dx ψ∂

(49)

¯ x ∂t2n+1 u) dx (u∂x ∂t2n+1 ψ¯ − ψ∂  ¯ x ∂t2n+1 ψ) ¯ dx uψ¯ dx (u∂t2n+1 u − ψ∂   ¯ x ψ¯ ¯ dx ∂t2n+1 (uψ). dx u2 − ψ∂

(50)

Finally putting these coefficients in the equations of motion (40) and (41) we obtain that the t2n+3 equation of the smKdV hierarchy is given by ∂u ∂u ∂ ψ¯ = R1 + R2 , ∂t2n+3 ∂t2n+1 ∂t2n+1

∂u ∂ ψ¯ ∂ ψ¯ = R3 + R4 ∂t2n+3 ∂t2n+1 ∂t2n+1

(51)

where {R1 , R4 }, {R2 , R3 } are the bosonic and fermionic components of the recursion operator, respectively, which are given by 1 2 i i ¯ −1 ¯ i ¯ −1 ψ¯ D − u2 − ux D−1 u + ψ¯ ψ¯ x + u2 ψD ψ + ux D−1 uψD 4 4 4 2 i i ¯ −1 u ¯ − i ψ¯ x D−1 u2 D−1 ψ¯ + i ψ¯ x D−1 uψD − − ψ¯2x D−1 ψ¯ − ψ¯ x D−1 ψD 4 4 4 2 1 ¯ − ψ¯ x D−1 ψ¯ ψ¯x D−1 ψ, (52) 4

R1 =

R2 = − − R3 = − R4 = −

i i ¯ i i i ¯ −1 ¯ −1 u ¯ + i ux D−1 uψD uψD − uψ¯x − ux ψ¯ + u2 ψD u + ux D−1 ψD 2 2 4 4 2 2 i ¯ −1 i i 1 ¯ −1 ψD ¯ ψ2x D u + ψ¯ x D−1 uD − ψ¯ x D−1 u2 D−1 u + ψ¯ x D−1 uψD 4 4 4 2 1 ¯ −1 ¯ ¯ −1 (53) ψx D ψ ψx D u, 4 1 3 i 1 ¯ −1 ψ¯ + 1 uD−1 ψD ¯ − uψ¯ − ux D−1 ψ¯ − ψ¯ x D−1 u + ψ¯ x D−1 uψD 4 4 2 2 4 1 −1 ¯ −1 i 1 ¯ uD uψD u − uD−1 ψ¯ ψ¯ x D−1 ψ¯ + uD−1 u2 D−1 ψ, (54) 2 4 4 1 2 1 2 1 1 1 D − u − ux D−1 u − uD−1 uD + uD−1 u2 D−1 u 4 4 4 4 4 i −1 ¯ ¯ −1 i ¯ −1 u + i uD−1 uψD ¯ −1 ψD. ¯ uD ψ ψx D u + ψ¯ x D−1 uψD (55) 4 2 2

where D = ∂x and D−1 is its inverse.

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A. R. Aguirre et al.

In terms of u = φx we get ∂φ ∂φ ∂ ψ¯ = R1 + R2 , ∂t2n+3 ∂t2n+1 ∂t2n+1

∂φ ∂ ψ¯ ∂ ψ¯ = R3 + R4 ∂t2n+3 ∂t2n+1 ∂t2n+1

(56)

where R1 = D−1 R1 D, R2 = D−1 R2 , R3 = R3 D, R4 = R4 , with R1 = − − R2 = + − R3 = −

1 2 i i ¯ −1 ψ¯ − i ψ¯ 2x D−1 ψ¯ D − φ2x − φ2x D−1 φx + ψ¯ ψ¯ x + φ2x ψD 4 4 4 4 i ¯ −1 ¯ i ¯ −1 2 −1 ¯ i ¯ −1 ¯ −1 ψx D ψD − ψx D φx D ψ + ψx D φx ψD φx 4 4 2 1 ¯ −1 ¯ ¯ −1 ¯ i ¯ −1 ψ, ¯ (57) ψx D ψ ψx D ψ + φ2x D−1 φx ψD 4 2 i ¯ − i φx ψ¯ x − i φ2x ψ¯ + i φ2 ψD ¯ −1 φx + i φ2x D−1 ψD ¯ − i ψ¯ 2x D−1 φx φx ψD x 2 2 4 4 2 4 i ¯ −1 i ¯ −1 2 −1 1 ¯ −1 ¯ −1 ¯ ψx D φx D − ψx D φx D φx + ψx D φx ψD ψD 4 4 2 i 1 ¯ −1 ¯ ¯ −1 −1 ¯ −1 φx , (58) ψx D ψ ψx D φx + φ2x D φx ψD 4 2 3 i 1 1 ¯ −1 ψ¯ + 1 φx D−1 ψD ¯ − φx ψ¯ − φ2x D−1 ψ¯ − ψ¯ x D−1 φx + ψ¯ x D−1 φx ψD 4 4 2 2 4 1 ¯ −1 φx − i φx D−1 ψ¯ ψ¯ x D−1 ψ¯ + 1 φx D−1 φ2 D−1 ψ, ¯ (59) φx D−1 φx ψD x 2 4 4

1 2 1 2 1 1 1 D − φx − φ2x D−1 φx − φx D−1 φx D + φx D−1 φ2x D−1 φx 4 4 4 4 4 i i ¯ −1 φx + i φx D−1 φx ψD ¯ −1 ψD. ¯ − φx D−1 ψ¯ ψ¯ x D−1 φx + ψ¯ x D−1 φx ψD (60) 4 2 2

R4 =

We have explicitly checked that by employing Eq. (51) for n = 0 we recover the smKdV equation (20), (21). Also it was verified that (51) for n = 1, yields the t5 flow of the hierarchy (22), (23) as predicted.

4 The Bäcklund Transformations for the smKdV Hierarchy In this section we will start by reviewing the systematic construction of the Bäcklund transformation for the smKdV hierarchy, based on the invariance of the zero curvature Eq. (1) under the gauge transformation, ∂μ K = KAμ (φ1 , ψ¯1 ) − Aμ (φ2 , ψ¯2 )K.

(61)

Recursion Operator and Bäcklund Transformation for Super mKdV Hierarchy

303

where A1 = Ax , A0 = At2n+1 , ∂1 = ∂x , ∂0 = ∂t2n+1 and assuming the existence of a defect matrix K(φ1 , ψ¯1 , φ2 , ψ¯2 ) which maps a field configuration {φ1 , ψ¯1 } into another {φ2 , ψ¯2 }. It is important to point out that the spatial Lax operator Ax is common to all members of the smKdV hierarchy and is given by ⎛

√

⎞

−1 i ψ¯ ⎟ ⎜ λ − φx ⎟ ⎜ ⎜ √ 1/2 ⎟ ⎜ 1/2 Ax = ⎜ −λ λ + φx i λ ψ¯ ⎟ ⎟, ⎟ ⎜ ⎠ ⎝√ √ 1/2 ¯ 1/2 iλ ψ i ψ¯ 2λ 1/2

(62)

Moreover, based on this fact it has been shown that the spatial component of the Bäcklund transformation, and consequently the associated defect matrix, are also common and henceforth universal within the entire bosonic hierarchy [11, 12]. More recently, in [1], this result has been extended to the supersymmetric mKdV hierarchy with the following defect matrix ⎛ λ ⎜ ⎜ ⎜ 2 −φ 1/2 K =⎜ ⎜ − ω2 e + λ ⎜ ⎝ √ φ+ 2 i − 2 e f1 λ1/2 ω 1/2

− ω22 eφ+ λ−1/2 λ1/2 √ 2 i φ2+ e f1 ω

√ φ + − 2ω i e 2 f1

⎞

⎟ ⎟ ⎟ 2 i − φ2+ 1/2 ⎟ − ω e f1 λ ⎟ ⎟ ⎠ 2 1/2 + λ 2 ω √

(63)

where φ± = φ1 ± φ2 , ω represents the Bäcklund parameter, and f1 is an auxiliary fermionic field. We can now substitute (62) and (63) in the x-part of the gauge transformation (61), to get   φ+ 4 2i f1 ψ¯ + , ∂x φ− = 2 sinh(φ+ ) − sinh ω ω 2   φ+ 4 ¯ f1 , ψ− = cosh ω 2   φ+ ¯ 1 ∂x f1 = cosh ψ+ . ω 2

(64) (65) (66)

which is the spatial part of the Bäcklund transformations, where we have denoted ψ¯ ± = ψ¯ 1 ± ψ¯ 2 . The corresponding temporal part of the Bäcklund is obtained from the gauge transformation in (61) for μ = 0, i.e., ∂t2n+1 K = KAt2n+1 (φ1 , ψ¯1 ) − At2n+1 (φ2 , ψ¯2 )K.

(67)

304

A. R. Aguirre et al.

where we consider the corresponding temporal part of the Lax pair At2n+1 . Now, we will consider some examples. For n = 0 we have that Ax = At1 so the temporal part of the Bäcklund is, ∂t1 φ+ = ∂x φ+ , ∂t1 f1 = ∂x f1 .

(68) (69)

∂t1 φ− = ∂x φ− .

(70)

This implies that

Then, for n = 0 the x and t components of the Bäcklund are the same. The next non-trivial example is the smKdV equation (n = 1), √ ⎞ p0 + λ1/2 p1/2 − λφx + λ3/2 p+ − λ μ+ + λ1/2 ν+ + λ i√ ψ¯ 2 1/2 3/2 1/2 3/2 −λp− − λ −p0 + λ p1/2 + λφx + λ λ μ− + λν− + λ iψ¯ ⎠ , At3 = ⎝ √ √ λ1/2 μ− − λν− + λ3/2 iψ¯ μ+ − λ1/2 ν+ + λ iψ¯ 2λ1/2 p1/2 + 2λ3/2 ⎛

(71) where 1 4 1 p± = 2 p0 =

√    ¯ x ψ¯ , p1/2 = − i ψ∂ ¯ ν± = i ∂x ψ¯ ± ψ∂ ¯ xφ ¯ x ψ, 2(φx )3 − φ3x − 3iφx ψ∂ 2 2 √    ¯ 2x − 2ψ(φ ¯ x ψ¯ , μ± = i ∂ 2 ψ¯ ± φx ∂x ψ¯ ∓ ψφ ¯ x )2 . φ2x ± (φx )2 ∓ iψ∂ 4 x

(72)

By substituting (71) and (63) in (67), we obtain 4∂t3 φ− = − + − 4∂t3 f1 = + .

 φ    φ  i 32 + + (+) 2 φ(+) − φ cosh sinh ψ¯ + f1 − 6 sinh3 φ+ x ω 2x 2 2 ω   φ  φ  i + ¯ (+) + ¯ (+) f1 φ(+) ψ ψ cosh − 2 sinh x 2x ω x 2 2   (+) 2 2  (+) ¯ + ψ¯ (+) sinh φ+ 2φ cosh φ − φ sinh φ + i ψ + + x x ω 2  2x       φ+ 96i 3 φ+ 5 φ+ sinh + 4 sinh + 3 sinh (73) ψ¯ + f1 , ω5 2 2 2 φ    12 φ  1 + + ¯ (+) 2 cosh 2ψ¯ 2x ψ+ + 5 sinh2 φ+ cosh − ψ¯ + (φ(+) x ) 2ω 2 ω 2 φ    12 φ  1 + + (+) ¯ (+) sinh φ(+) − 4 sinh φ+ cosh2 ψ¯ + φ(+) x f1 2x − φx ψx 2ω 2 ω 2 (74)

Recursion Operator and Bäcklund Transformation for Super mKdV Hierarchy

305

the corresponding temporal part of the Bäcklund transformation for the smKdV i i ¯ ¯ (±) equation, where φ(±) ix = ∂x φ± and ψix = ∂x ψ± . In [1] this procedure have been applied to obtain these transformations for the t5 member of the smKdV hierarchy.

5 Recursion Operator for the Bäcklund Transformations In this section we will extend the idea of recursion operator to generate the Bäcklund transformation for smKdV hierarchy as an alternative method. In order to construct the recursion operator for the Bäcklund transformations we consider two different solutions of the Eq. (56) as ¯1 ∂φ1 (1) ∂φ1 (1) ∂ ψ = R1 + R2 , ∂t2n+3 ∂t2n+1 ∂t2n+1 ¯2 ∂φ2 (2) ∂φ2 (2) ∂ ψ = R1 + R2 , ∂t2n+3 ∂t2n+1 ∂t2n+1

¯1 ∂ ψ¯ 1 (1) ∂φ1 (1) ∂ ψ = R3 + R4 ∂t2n+3 ∂t2n+1 ∂t2n+1 ¯2 ∂ ψ¯ 2 (2) ∂φ2 (2) ∂ ψ = R3 + R4 ∂t2n+3 ∂t2n+1 ∂t2n+1

(75) (76)

  (p) (p) (p) (p) (p) where Ri = Ri φx , φ2x , ψ¯p , ψ¯x , ψ¯2x , i = 1, . . . , 4, p = 1, 2. And take the following combination of these solutions     (2) (2) ∂t2n+1 φ− + R(1) ∂t2n+1 ψ¯ − 2∂t2n+3 φ− = R(1) 1 + R1 2 + R2     (2) (2) ∂t2n+1 φ+ + R(1) ∂t2n+1 ψ¯ + , + R(1) 1 − R1 2 − R2

(77)

    (2) (2) 2∂t2n+3 ψ¯ − = R(1) ∂t2n+1 φ− + R(1) ∂t2n+1 ψ¯ − 3 + R3 4 + R4     (2) (2) ∂t2n+1 φ+ + R(1) ∂t2n+1 ψ¯ + . + R(1) 3 − R3 4 − R4

(78)

At this point, we conjecture that the Eqs. (77) and (78) correspond to the temporal part of the super Bäcklund transformation for an super integrable equation specified by n. We note that as well as the consecutive equations of motion within the hierarchy are connected by the same recursion operator, here the same occurs to the Bäcklund transformations. In order to clarify this hypothesis we next consider some examples. For n = 0 we have     (2) (2) ∂t1 φ− + R(1) ∂t1 ψ¯ − 2∂t3 φ− = R(1) 1 + R1 2 + R2     (2) (2) ∂t1 φ+ + R(1) ∂t1 ψ¯ + , + R(1) (79) 1 − R1 2 − R2     (2) (2) 2∂t3 ψ¯ − = R(1) ∂t1 φ− + R(1) ∂t1 ψ¯ − 3 + R3 4 + R4     (2) (2) ∂t1 φ+ + R(1) ∂t1 ψ¯ + . + R(1) (80) 3 − R3 4 − R4

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By using (68)–(70) and the x-part of the Bäcklund (64)–(66) we recover Eqs. (73) and (74), i.e. the time component of the Bäcklund transformations for n = 2 (smKdV). Next we consider the case for n = 2 and using again (73)–(74) we obtain from (77),   3  (−) 5 5 (−)  (−) 2  (+) 2 (+) (+) φ2x − 5φ(−) φx − φx + φ2x 2x φ2x φx 8 2     (+) 3 2  (+) 2  5 (+) (+) 3 − φ(−) φ φ(−) 5φ(−) φ − φ + φx x x x x 3x 3x 8 2    (+) 2  5i  ¯ ¯ (−) (−) (−) 2 ψ− ψx + ψ¯ + ψ¯ x(+) φ(−) φ − φ + 3 φx x x 3x 4   2 2   5i  ¯ ¯ (+) (+) ψ− ψx + ψ¯ + ψ¯ x(−) φ(+) φ(+) + 3 φ(−) x x 3x − φx 4  5i   5i ¯  ¯ (−) (+) (+) (−) (−) (+) (+) (−) + ψ¯ + ψ¯ 2x φx φ2x + ψ¯2x φ2x ψ+ ψ3x φx + ψ¯ 3x 4 4  15  (−) 3  (+) 2 5i ¯  (−) ¯ (−) ¯ (+) φx φx + ψ− φ2x ψ2x + φ(+) ψ 2x 2x 4 4  5i ¯  (−) ¯ (−) ¯ (+) (81) ψ− φx ψ3x + φ(+) x ψ3x 4

16∂t5 φ− = φ(−) 5x + + + + + + +

And for the Eq. (78) we get   5   5 (−) (−) (+) (+) ¯ + φ(−) φ(+) + φ(+) φ(−) 16∂t5 ψ¯− = ψ¯ 5x − − ψ¯ − φ(−) φ + φ φ ψ x x x x 4x 4x 4x 4x 4 4     5  ¯ (+) 2 2 (+) − 2φ(+) φ(+) + 3 φ(−) ψ− φ2x + ψ¯ + φ(−) x x 2x 3x − φx 4   2 2   5  ¯ (−) (−) − ψ− φ2x + ψ¯ + φ(+) 2φ(−) φ(−) + 3 φ(+) x x 2x 3x − φx 4   2 2   5 (−) − ψ¯ x(−) φ(−) 6φ(−) φ(−) + 6 φ(+) x x x 3x − φx 8    (+) 3  5 (+)  (−) (+) 5 − ψ¯ x(−) φ(+) 6φ(+) − ψ¯ x 4φ2x φ2x + 3φ(+) x 3x − φx 3x φx 8 4  2   5 ¯ (+) (+)  (−) (−) 2 − ψx φx 3φ3x − 2 φx + φ(+) x 4  5    15 ¯ (+)  (+) (−) (+) ¯ (−) φ(−) 2 + φ(+) 2 − − φ ψ2x φ2x φx + φ(−) ψ x x 2x x 4 4 3x      2 2 5 (+) (−) (+) 5 ¯ (−) φ(−) − ψ¯ 3x φx φx − ψ x + φ(+) 2x 2x 2 2  15 ¯ (−)  (−) (−) (+) (82) − ψ2x φ2x φx + φ(+) 2x φx 4

Recursion Operator and Bäcklund Transformation for Super mKdV Hierarchy

307

Now, using the x-part of the Bäcklund transformation (64)–(66) in these two equations we end up with the corresponding Bäcklund transformation for n = 3, that was obtained in [1]. Conclusions In this note we have considered a hierarchy of supersymmetric equations of motion ˆ underlined by an affine construction of a Kac–Moody algebra sl(2, 1). These equations of motion were shown to be related by a recursion operator that maps consecutive time flows. Moreover, it was shown that the Bäcklund transformation follows the same relation generated by the recursion operator. Such framework provides a general and systematic method of constructing Bäcklund transformations for the entire hierarchy completing and clarifying the question raised in Ref. [1]. An interesting point we would like to point out and is still under investigation is about the construction of the underlying affine Kac–Moody algebra. The key ingredient in the affinization was the decomposition of the integers and semi integers numbers according to a Z2 structure assigned to bosonic and fermionic generators defined in (6). For higher rank algebras we expect to systematize such construction decomposing both integers and semi-integers in disjoint subsets compatible with the ˆ 1). closure of the algebra, e.g. Zk for sl(k, Acknowledgements ALR thanks the Sao Paulo Research Foundation FAPESP for financial support under the process 2015/00025-9. JFG, NIS and AHZ thank CNPq for financial support.

Appendix Here we resume the commutation and Anti-commutation relations for the  sl(2,1) affine Lie superalgebra [K1(2n+1) , K2(2m+1) ] = 0, [M1(2n+1) , K1(2m+1) ] = 2M22(n+m+1) + (n + m)δn+m+1,0 cˆ , [M1(2n+1) , K2(2m+1) ] = 0, [K2(2n+1) , K2(2m+1) ] = −(n − m)δn+m+1,0 cˆ , [M2(2n) , K1(2m+1) ] = 2M12(n+m)+1 , [M2(2n) , K2(2m+1) ] = 0, [M1(2n+1) , M2(2m) ] = −2K12(n+m)+1 , [M1(2n+1) , M1(2m+1) ] = −(n − m)δn+m+1,0 cˆ , [M2(2n) , M2(2m) ] = (n − m)δn+m,0 cˆ , [K1(2n+1) , K1(2m+1) ] = (n − m)δn+m+1,0 cˆ ,

308

A. R. Aguirre et al. (2n+3/2)

[F1

(2n+3/2)

, K1(2m+1) ] = −[F1

, K2(2m+1) ] = F2

2(n+m+1)+1/2

,

(2n+1/2) (2n+1/2) 2(n+m)+3/2 [F2 , K1(2m+1) ] = −[F2 , K2(2m+1) ] = F1 , (2m+3/2) 2(n+m+1)+1/2 (2n+1) [M1 , F1 ] = G1 , (2m+1/2) (2m+3/2) 2(n+m)+3/2 (2n+1) (2n) [M1 , F2 ] = −[M2 , F1 ] = G2 , (2m+1/2) 2(n+m)+1/2 (2n) [M2 , F2 ] = −G 1 , (2n+1/2) 2(n+m)+3/2 (2m+1) [G 1 , K1 ] = −G 2 , (2n+1/2) 2(n+m)+3/2 (2m+1) [G 1 , K2 ] = −G 2 , (2n+3/2) 2(n+m+1)+1/2 (2m+1) [G 2 , K1 ] = −G 1 , (2n+3/2) 2(n+m+1)+1/2 (2m+1) [G 2 , K2 ] = −G 1 , (2m+1/2) 2(n+m)+3/2 (2n+1) [M1 , G1 ] = −F1 , (2m+3/2) 2(n+m+1)+1/2 (2n+1) [M1 , G2 ] = −F2 , (2m+1/2) 2(n+m)+1/2 (2n) [M2 , G 1 ] = −F2 , (2m+3/2) 2(n+m)+3/2 (2n) [M2 , G 2 ] = −F1 , (2n+3/2) (2m+1/2) {F1 , F2 } = [(2n + 1) − 2m]δn+m+1,0 cˆ , (2n+3/2)

, F1

(2n+1/2)

, F2

(2n+1/2)

, G1

(2n+3/2)

, G2

(2n+3/2)

, G1

(2n+1/2)

, G2

{F1 {F2 {F2 {F1 {F1 {F2

(2n+1/2)

{G 1

(2m+3/2)

} = 2(K22(n+m+1)+1 + K12(n+m+1)+1 ),

(2m+1/2)

} = −2(K22(n+m)+1 + K12(n+m)+1 ),

(2m+1/2)

} = 2M12(n+m)+1 ,

(2m+3/2)

} = −2M12(n+m+1)+1 ,

(2m+1/2)

} = 2M22(n+m+1) + [(2n + 1) + 2m]δn+m+1,0 cˆ ,

(2m+3/2)

} = −2M22(n+m+1) − [2n + (2m + 1)]δn+m+1,0 cˆ ,

(2m+3/2)

, G2

} = [2n − (2m + 1)]δn+m+1,0 cˆ ,

(2n+1/2) (2m+1/2) {G 1 , G1 } (2n+3/2) (2m+3/2) {G 2 , G2 }

= 2(K22(n+m)+1 − K12(n+m)+1 ), = −2(K22(n+m+1)+1 − K12(n+m+1)+1 ).

(83)

References 1. A.R. Aguirre, A.L. Retore, J.F. Gomes, N.I. Spano, A.H. Zimerman, “Defects in the supersymmetric mKdV hierarchy via Backlund transformations”, J. High Energ. Phys. (2018) 2018: 18., https://doi.org/10.1007/JHEP01(2018)018, arXiv:1709.05568. 2. A.R. Aguirre, J.F. Gomes, N.I. Spano, A.H. Zimerman N=1 super sinh-Gordon model with defects revisited JHEP 02 (2015) 175 [arXiv:1412.2579]. 3. H. Aratyn, J. F. Gomes, and A.H. Zimerman, Supersymmetry and the KdV equations for Integrable Hierarchies with a Half-integer Gradation, Nucl. Phys. B 676 (2004) 537 [hepth/0309099].

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4. C. Rogers and W.F. Shadwick, Bäcklund transformations and their applications, New York, Academic Press, 1982. 5. A.R. Aguirre, T.R. Araujo, J.F. Gomes, and A.H. Zimerman, Type-II Bäcklund transformations via gauge transformations, JHEP 12 (2011) 056 [nlin/1110.1589]. 6. A. R. Aguirre, J. F. Gomes, N. I. Spano and A. H. Zimerman, Type-II Super-Bäcklund Transformation and Integrable Defects for the N = 1 super sinh-Gordon Model, JHEP 1506, 125 (2015) [arXiv:1504.07978 [math-ph]]. 7. P. Bowcock, E. Corrigan and C. Zambon, Classically integrable field theories with defects, Int. J. Mod. Phys. A19 (2004) 82 [hep-th/0305022]. 8. P. Bowcock, E. Corrigan and C. Zambon, Affine Toda field theories with defects, JHEP 01 (2004) 056 [hep-th/0401020]. 9. P.J. Olver, Evolution equations possessing infinitely many symmetries, J. Math. Phys. 18 (1977) pp. 1212–1215. 10. J.F. Gomes, L.H. Ymai, and A.H. Zimerman, Soliton Solutions for the Super mKdV and sinhGordon Hierarchy, Phys. Lett. A 359 (2006) 630–637 [hep-th/0607107]. 11. J.F. Gomes, A.L. Retore, N.I. Spano, and A.H. Zimerman, Bäcklund Transformation for Integrable Hierarchies: example - mKdV Hierarchy , J. Phys.: Conf. Ser. 597 (2015) 012039 [arXiv:1501.00865]. 12. J.F. Gomes, A.L. Retore, and A.H. Zimerman, Construction of type-II Bäcklund transformation for the mKdV hierarchy, J. Phys.: Math. Theor. 48 (2015) 405203 [arXiv:1505.01024].

Lie Symmetry Analysis of a Third-Order Equation Arising from a General Class of Lotka–Volterra Chains Kyriakos Charalambous and Christodoulos Sophocleous

Abstract In the literature has recently appeared a class of bi-cubic equations from the study of a general class of Lotka–Volterra chains. Lie symmetry analysis is performed for this non-linear partial differential equation and a list of similarity reductions is presented with the employment of the appropriate optimal system. This family of bi-cubic equations can be generalized by taking the parameters as functions of time or as functions of the space variable x. The corresponding symmetry analysis for these two general cases is also presented.

1 Introduction Recently, Zilburg and Rosenau [6] considered a general class of Lotka–Volterra chains with symmetric 2N-neighbors interactions and classified the types of solitons which propagate along the chain. This was achieved by using quasi-continuum approximations depending on the coupling between neighbors. As a result, they construct a number of interesting partial differential equations (pdes). Such equation is the bi-cubic pde (1) u t = [bu 2 + 2kuu x x + (u x x )2 ]x . √ We note that the mapping x → |k|x normalizes the constant k equal to 1 or −1 depending if it is positive or negative. If we assume that the constants k and b preserve their values, the equivalence transformations for Eq. (1) have the form t  = αt + β, x  = x + γ , u  =

u . α

K. Charalambous (B) Department of Mathematics, University of Nicosia, CY 1700 Nicosia, Cyprus e-mail: [email protected] C. Sophocleous Department of Mathematics and Statistics, University of Cyprus, CY 1678 Nicosia, Cyprus e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2018 V. Dobrev (ed.), Quantum Theory and Symmetries with Lie Theory and Its Applications in Physics Volume 1, Springer Proceedings in Mathematics & Statistics 263, https://doi.org/10.1007/978-981-13-2715-5_19

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Taking into account that parameters which appear in a physical model can change in time or/and in space, coefficients in a partial differential equation can be taken as functions of time or/and of space variables. In fact, such equations can describe physical phenomena with more accuracy. In the last years a large number of variable coefficient equations have appeared in the literature which are studied from various points of view. In the present study we consider two cases, b and k being functions of t and x, respectively. These two variable coefficient equations are studied form the Lie symmetry point of view.

2 Lie Symmetries for Eq. (1) We classify the Lie symmetries of Eq. (1). The classical method for finding Lie point symmetries is well known, see for example in [1]. We look for vector fields of the form, (2) Γ = T (t, x, u)∂t + X (t, x, u)∂x + U (t, x, u)∂u , which generate one-parameter groups of point symmetry transformations of Eq. (1). These vector fields form the maximal Lie invariance algebra of this equation. Any such vector field, Γ , satisfies the criterion of infinitesimal invariance, i.e., the action of the third extension Γ (3) of Γ on Eq. (1) results in the conditions being an identity for all solutions of this equation. Namely, we require that Γ (3) [u t − 2buu x − 2ku x u x x − 2kuu x x x − 2u x x u x x x ] = 0

(3)

identically, modulo equation (1). Equation (3) is a multivariable polynomial with variables the derivatives of u. Since the right hand side of evolution equation (1) is a polynomial in the derivatives of u with respect to the spatial variable x, two of the coefficient functions have the restricted forms T = T (t) and X = X (t, x) [3]. Coefficients of u x x x u x x , u x x x u x and the term independent of derivatives in (3) imply that the coefficient functions have the forms T = T (t), X = a1 (t)x + a2 (t), U = (5a1 (t) − T (t))u + g(x).

(4)

Now, the coefficients of u x x x , u x x , u x u x x , u x and the term independent of derivatives in (3) lead to the following determining system that provide the forms of the functions T (t), a1 (t), a2 (t) and g(x), gx x + kg = 0, gx x x + kgx = 0, a1 k = 0, a1 b = 0, a1t x + a2t + 2kgx x + 2bg = 0, 5a1t − Ttt − 2kgx x x − 2bgx = 0.

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Depending on the relation between the parameters k and b, the solution of the determining system lead to five cases. For each case we give the Lie symmetries admitted by Eq. (1). 1. b = k 2 : Γ1 = ∂x , Γ2 = ∂t , Γ3 = t∂t − u∂u ; √ 2. b = k 2 , k√> 0 : Γ1 = ∂x , Γ2 = ∂t , Γ3 = t∂t − u∂u , Γ4 = cos( kx)∂u , Γ5 = sin( kx)∂u ; < 0 : Γ1 = ∂x , Γ2 = ∂t√ , Γ3 = t∂t − u∂u , 3. b = k 2 , k √ Γ4 = exp( −kx)∂u , Γ5 = exp(− −kx)∂u ; 4. k = 0, b = 0 : Γ1 = ∂x , Γ2 = ∂t , Γ3 = t∂t − u∂u , Γ4 = −2bt∂x + ∂u ; 5. b = k = 0 : Γ1 = ∂x , Γ2 = ∂t , Γ3 = t∂t − u∂u , Γ4 = x∂x + 5u∂u , Γ5 = x∂u , Γ6 = ∂u . The results in cases 1, 2 and 3 agree with those in Ref. [2]. The primary use of Lie symmetries is to obtain a reduction of variables. Similarity variables appear as first integrals of the characteristic system dx du dt = = . T X U Here we can reduce a PDE in two independent variables into an ordinary differential equation (ODE) using a one-dimensional subalgebra of Lie symmetry algebra. Reductions s could be obtained from any symmetry which is an arbitrary linear combination i=1 ai Γi , where s is the number of basis operators of maximal Lie symmetry algebra of the given partial differential equation. To ensure that a minimal complete set of reductions is obtained from the Lie symmetries of Eq. (1), we construct the socalled optimal system of one-dimensional subalgebras. Ovsiannikov [5] proved that the optimal system of solutions consists of solutions that are invariant with respect to all proper inequivalent subalgebras of the symmetry algebra. More detail about construction of optimal sets of subalgebras can be found in [4, 5]. Firstly we construct the Lie symmetry table for the Lie algebra of the Γi and then using the Lie series we construct a table showing the separate adjoint actions for each element Γi acting on all the other elements. This table enables us to derive the optimal system of subalgebras that provides all possible invariant solutions (Lie ansatzes). In the first case the optimal system consists of the list of one-dimensional inequivalent subalgebras Γ3 + cΓ1 , Γ2 + cΓ1 , Γ1 , where c is an arbitrary constant. The subalgebra Γ1  produces the constant solution, while the other two subalgebras give the following results: • Γ3 + cΓ1  produces the ansatz u = t −1 φ(ξ ), ξ = x − c ln t that reduces (1) into ordinary differential equation (φξ ξ + kφ)φξ ξ ξ + kφξ φξ ξ

  1 1 + bφ + c φξ + φ = 0. 2 2

(5)

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• Γ2 + cΓ1  leads to the ansatz u = φ(ξ ), ξ = x − ct that reduces (1) into ordinary differential equation (φξ ξ + kφ)φξ ξ ξ + kφξ φξ ξ

  1 + bφ + c φξ = 0. 2

(6)

In the second case, in addition to the subalgebras of the first case, we have the following inequivalent subalgebras Γ5 + Γ1 + cΓ3 , Γ5 + Γ3 , where c is an arbitrary constant and = ±1. We have the following results: • Γ5 + Γ1 + cΓ3 : We obtain the reduction √ √ √ − k cos ( kx) + c sin ( kx) u= + e−c x φ(ξ ), ξ = te− cx k + c2 that maps (1) into ordinary differential equation  2 φξ + 2c Φ(ξ ) + c ξ Φξ = 0, Φ(ξ ) = c2 ξ 2 φξ ξ + 3c2 ξ φξ + (k + c2 )φ . • Γ5 + Γ3 : We find the reduction u = nary differential equation

φ(x) t

√ + sin( kx) that maps (1) into ordi-

  φ + (φx x + kφ)2 x = 0. The results in case 3 are similar to the ones of case 2. In case 4, we have the additional subalgebras with the corresponding reductions: ) • Γ4 + cΓ2 + Γ3 : We find u = t+φ(ξ , ξ = x − 2b[c log(c + t) − t] and the c+ t reduced ordinary differential equation has the form

1 1 φξ ξ φξ ξ ξ + bφφξ + φ − c = 0. 2 2 • Γ4 + Γ2 : We obtain the mapping u = t + φ(ξ ), ξ = x + bt 2 that reduces Eq. (1) into φξ ξ φξ ξ ξ + bφφξ − = 0. x + φ(t) and the reduced equation has the form • Γ4 : We find u = − 2bt

tφt + φ = 0. The solution of the above simple ordinary differential equation provides the sim, where κ is a constant of integration, of Eq. (1). ilarity solution u = − x+2bκ 2bt

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3 Symmetry Analysis for Equation with Time-Dependent Coefficients We consider the case where the parameters in Eq. (1) dependent on the time. In other words, the equation under study has the form u t = [b(t)u 2 + 2k(t)uu x x + (u x x )2 ]x .

(7)

Equation (7) admits the equivalence transformations t  = αt + β, x  = γ x + δ, u  =

γ5 k(t) b(t) u, k  (t  ) = 2 , b (t  ) = 4 , α γ γ

(8)

where α, β, γ , δ are arbitrary constants and αγ = 0. We classify the Lie symmetries admitted by the generalized equation (7). The appearance of the functions b(t) and k(t) makes the analysis more difficult than the case where b and k were constants. This change does not effect the initial calculations. In fact, we find that the coefficient functions of the symmetry generator Γ have the restricted forms given by Eq. (4) which lead to the following determining system: gx x + kg = 0, gx x x + kgx = 0, T kt + 2a1 k = 0, T bt + 4a1 b = 0, a1t x + a2t + 2kgx x + 2bg = 0, 5a1t − Ttt − 2kgx x x − 2bgx = 0. The solution of the determining system, without taking into account the cases where both b(t) and k(t) are constants, gives the following results: 1. k(t), b(t) arbitrary functions: Γ1 = ∂x ; 2. k(t) = t n , b(t) = t 2n : Γ1 = ∂x , Γ2 = 2t∂t − nx∂x − (5n + 2)u∂u ; (5n+2) n Lie symmetry Γ2 provides the similarity reduction u = t − 2 φ(ξ ), ξ = t 2 x which maps Eq. (7) into 4φξ ξ φξ ξ ξ + 4φφξ ξ ξ + 4φξ φξ ξ + 4φφξ − nξ φξ + (5n + 2)φ = 0. 3. k(t) = ent , b(t) = e2nt : Γ1 = ∂x , Γ2 = 2∂t − nx∂x − 5nu∂u ; 5nt nt Here Lie symmetry Γ2 leads to the reduction u = e− 2 φ(ξ ), ξ = xe 2 and the corresponding reduced ordinary differential equation has the form n(ξ φξ − 5φ) − 2

 2

φξ ξ + φ = 0. ξ

4. k(t) = 0, b(t) = t n : Γ1 = ∂x , Γ2 = 4t∂t − nx∂x − (5n + 4)u∂u , Γ3 = 2t n+1 ∂x − (n + 1)∂u ; The optimal system consists of the inequivalent subalgebras

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4 4 Γ3 + Γ2 , n = − , Γ3 + Γ2 , n = − , Γ3 , Γ2 , Γ1  5 5 which leads to the following reductions: • Γ3 + Γ2 , n = − 45 : We obtain the reduction u=−

2 (n + 1) 5n+4 n 5n+4 + t − 4 φ(ξ ), ξ = xt 4 − t 4 5n + 4 (5n + 4)

which maps (7) into: 8φξ ξ φξ ξ ξ − nξ φξ + 8φφξ + (5n + 4)φ = 0.

(9)

• Γ3 + Γ2 , n = − 45 : We find u = φ(ξ ) −

1 log t, ξ = xt − 5 − log t 20 2

and the corresponding reduced ordinary differential equation has the form 40φξ ξ φξ ξ ξ + 4ξ φξ + 40φφξ + 10 φξ + = 0.

(10)

xt −(n+1) and the reduced equa• Γ3 : We find the reduction u = φ(t) − (n+1) 2 tion has the form tφt + (n + 1)φ = 0. which can be solved to give the similarity solution of (7), u = where κ is a constant of integration. • Γ2 : We derive the similarity reduction u = t−

5n+4 4

−(n+1)x+κ , 2t n+1

n

φ(ξ ), ξ = xt 4

which maps (7) into 8φξ ξ φξ ξ ξ − nξ φξ + 8φφξ + (5n + 4)φ = 0. 5. k(t) = 0, b(t) = ent : Γ1 = ∂x , Γ2 = 4∂t − nx∂x − 5nu∂u , Γ3 = 2ent ∂x − n∂u ; The corresponding optimal system consists of the inequivalent subalgebras Γ3 + Γ2 , Γ3 , Γ2 , Γ1  which leads to the following similarity reductions:

(11)

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• Γ3 + Γ2 : We find the reduction: 2 5nt 5nt nt u = e− 4 φ(ξ ) − , ξ = xe 4 − e4 5 5n which reduces (7) into 8φξ ξ φξ ξ ξ − nξ φξ + 8φφξ + 5nφ = 0.

(12)

• Γ3 : Here we find the mapping u = φ(t) − n2 xe−nt and the corresponding reduced ordinary differential equation has the form φt + nφ = 0. where κ is a constant of integraWe obtain the similarity solution u = −nx+κ 2ent tion. 5nt nt • Γ2 : We find the reduction: u = e− 4 φ(ξ ), ξ = xe 4 that reduces (7) into 8φξ ξ φξ ξ ξ − nξ φξ + 8φφξ + 5nφ = 0.

(13)

6. k(t) = 0 and b(t) satisfies the ordinary differential equation 

b bt



 3 + b = 0. b(t)dt + μ 2 tt

We obtain the following  Lie symmetries:  ( b(t)dt + μ)∂t − 2x( b(t)dt + μ)∂x Γ1 = ∂x , Γ2 = 8b b

   t   4bbtt 2 u + x ∂u , Γ3 = 2 b(t)dt∂x − ∂u . + 2 − 9 ( b(t)dt + μ) − 4b 2 b t

Due to the complicated forms of Lie symmetries, we do not proceed the analysis any further.

4 Symmetry Analysis for Equation with x-Dependent Coefficients Here we consider the general class u t = [b(x)u 2 + 2k(x)uu x x + (u x x )2 ]x ,

(14)

where the functions b(x) and k(x) are not both constants. Equation (14) admits the equivalence transformations (8). Equation (14) admits Lie symmetries in the following cases:

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1. k(x), b(x) arbitrary functions: Γ1 = ∂t , Γ2 = t∂t − u∂u ; 2. b(x) = k 2 (x): Γ1 = ∂t , Γ2 = t∂t − u∂u , Γ3 = g1 (x)∂u , Γ4 = g2 (x)∂u , where g1 (x) and g2 (x) form a fundamental set of solutions of the second-order linear 2 ordinary differential equation dd xg2 + k(x)g = 0; √ √ 3. k(x) = x −2 , b(x) = x −4 : Γ1 = ∂t , Γ2 = t∂t − u∂u , Γ3 = x cos( 23 log x)∂u , √ √ Γ4 = x sin( 23 log x)∂u , Γ5 = x∂x + 5u∂u . In the spirit of the previous sections, the corresponding similarity reductions can be derived.

5 Conclusion We have considered a nonlinear equation that appears recently in the literature [6]. We have carried out the Lie symmetry analysis for three cases: when the two parameters in the equation are constants, when they are functions of time and when they are functions of the space variable. Lie symmetries have been used to construct similarity reductions. Most of the reduced ordinary differential equations have a complicated form and hence, numerical or approximation methods are needed in order to proceed to obtain solutions. The investigation for non-local (potential) symmetries can follow the present work. In order to achieve this goal, firstly, a classification of conservation laws is needed. Equation (1) is written in a conserved form and hence, a auxiliary (potential) system has the form vx = u, vt = bu 2 + 2kuu x x + (u x x )2 . Lie symmetry analysis of the above system does not produce potential symmetries for (1). The study of non-Lie reductions is also another possible problem to be considered.

References 1. G.W. Bluman, S. Kumei, Symmetries and differential equations, (Springer-Verlag, New York, 1989). 2. S. Dimas and I.L. Freire, Appl. Math. Lett. 69 (2017) 121–125. 3. J.G. Kingston, C. Sophocleous, J. Phys. A:Math. Gen. 31 (1998) 1597–1619. 4. P. Olver, Applications of Lie Groups to Differential Equations, (Springer-Verlag, New York 1986). 5. L. Ovsiannikov, Group Analysis of Differential Equations, (Academic Press, New York 1982). 6. A. Zilburg, P. Rosenau, J. Phys. A: Math. Theor. 49 (2016) 095101 (21pp).

Part IV

Entanglement

Probing Anderson Localization Using the Dynamics of a Qubit Hichem Eleuch, Michael Hilke and Richard MacKenzie

Abstract Anderson localization refers to the absence of extended states in disordered lattice systems. One implication is exponential suppression of transmission. In this paper we study a two-level system coupled to an environment modelled by a semi-infinite chain via an N -site channel with and without disorder. We show that the decay of transmission affects the dynamics of a qubit coupled to the disordered system and we express the relaxation rate of the qubit in terms of the localization properties. As expected, adding static disorder to a channel coupled to a qubit will reduce the decoherence rate of the qubit compared to the channel without disorder. This could have practical applications: when designing electrodes that couple to qubits, it is possible to improve their performance by adding impurities to the channel.

H. Eleuch Department of Applied Sciences and Mathematics, College of Arts and Sciences, Abu Dhabi University, Abu Dhabi, UAE e-mail: [email protected] H. Eleuch Institute for Quantum Science and Engineering, Texas A&M University, College Station, TX 77843, USA M. Hilke Department of Physics, McGill University, Montreal, QC H3A 2T8, Canada e-mail: [email protected] R. MacKenzie (B) Groupe de physique des particules, Université de Montréal, C. P. 6128, Succursale Centre-ville, Montreal, QC H3C 3J7, Canada e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2018 V. Dobrev (ed.), Quantum Theory and Symmetries with Lie Theory and Its Applications in Physics Volume 1, Springer Proceedings in Mathematics & Statistics 263, https://doi.org/10.1007/978-981-13-2715-5_20

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1 Introduction The two-level system (TLS) is arguably the most elementary nontrivial quantum system. Its simplicity belies its importance, as many important systems amount to a TLS or a combination of a number of them. Examples of TLSs include: • spin 1/2 system • photon polarization • two-level atoms (atoms where all but the ground state and first excited state can be ignored) • quantum dots (empty/full) • double well (or double quantum dot) • K K¯ and B B¯ systems • neutrino oscillations (2 flavours) The TLS has been around since the birth of quantum mechanics; it has taken on new life more recently since it forms the building block of quantum computers, in which context it is referred to as a qubit. Indeed, we have quantum computation, and more generally quantum information, in mind as our main underlying motivation for this work. An isolated TLS will of course evolve unitarily; in particular, a pure state remains pure. Assuming the two states are coupled to one another, the system oscillates between the two base states. In reality, the TLS interacts with its environment. Thus, if one starts with a product state |ΨTLS  |Ψenv , it evolves into an entangled state. This is still a pure state globally, but from the point of view of observables restricted to the TLS, it is indistinguishable from a mixed state given by the partial trace of the entangled state over the environment. Thus, as far as the TLS itself is concerned, interaction with the environment causes the state to lose purity. In addition, a particle initially in the TLS can escape to the environment, so (again from the point of view of the TLS) probability is not conserved. We will ultimately study a system composed of three subsystems (see Fig. 1): the TLS (or double dot), a finite one-dimensional channel and a semi-infinite lead which is also a one-dimensional chain.

Fig. 1 TLS attached to a semi-infinite chain representing the environment via a finite chain which can be ordered or disordered

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We begin in the next section with the familiar example of a TLS in isolation as a warm-up and to establish notation. We then progressively work towards the final system, first connecting the TLS directly to a semi-infinite chain, then connecting the two through an ordered finite chain, and finally through a finite chain with disorder [1].

2 Two-Level System The TLS is governed by the Hamiltonian  HDD =

1 τ τ 2

 =

  τ 0 + δ0 /2 , τ 0 − δ0 /2

(1)

i are the energies of the uncoupled basis states, 0 = (1 + 2 )/2 is the average energy and δ0 = 1 − 2 is the energy splitting. These are modified by the  coupling τ ; the energy splitting and the energies of HDD are, respectively, δ = δ0 2 + 4τ 2 and λ± = 0 ± δ/2. The Green’s function for the system is  G DD (E) =

E − 1 −τ −τ E − 2

−1

.

(2)

For such a simple system, it is easy to find the time-dependent Green’s function via Fourier transformation; one of its components, for instance, is G DD 12 (t) = −

 2πiτ  −iλ+ t e − e−iλ− t . δ

(3)

The frequencies are the energies of HDD , and also the poles of G DD (E). For a system as simple as this, one hardly needs the Green’s function. Nonetheless, interesting questions can be answered in terms of it. For instance, one could ask for the probability that the TLS, if prepared in the state |1 = (1, 0)T at t = 0, will be seen to be in the state |2 = (0, 1)T at a later time t. In terms of the Green’s function, it is 2 2 P1→2 (t) = |G DD 12 (t)| /4π . This is displayed in Fig. 2. We see that the system oscillates away from |1 and then back to it, with frequency given by the energy splitting δ.

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Fig. 2 Oscillating probability P1→2 for 1 = −2 = 1, τ = 0.19

3 TLS Coupled to Infinite Chain We now couple the TLS to an infinite chain via a hopping term. The combined system is described by the Hamiltonian  H=

HDD V V † H∞

 (4)

with HDD as above and V and H∞ given by ⎛



 V =

0 0 ··· , tc 0 · · ·

H∞

0 ⎜1 ⎜ = ⎜0 ⎝ .. .

1 0 1 .. .

0 1 0 .. .

⎞ ... . . .⎟ ⎟ . . . .⎟ ⎠ .. .

(5)

We have scaled all energies in the problem so that the nonzero elements of H∞ are 1; we assume tc is real and positive. Using standard techniques (see, e.g., [2, 3]), the effect of the chain on the TLS can be described by a self-energy, leading to a modified Green’s function  G DD ∞ (E)

=

where Σ∞ (E) =

−τ E − 1 −τ E − 2 − Σ∞ (E)

−1

  tc 2 E − i 4 − E 2 = tc2 e−ik 2

(6)

(7)

where we have defined a wave vector k which will be convenient in what follows. Note that the effective Hamiltonian is the original Hamiltonian HDD with the replacement 2 → 2 + Σ∞ (E). This effective Hamiltonian has two unusual properties: it is non-Hermitian and energy-dependent. Both are quite easy to understand.

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S . Right: Fig. 3 Left: real and imaginary parts of the energy-dependent surface Green’s function G ∞ magnitude of the imaginary part of the time-dependent surface Green’s function

Non-hermiticity, as mentioned above, simply reflects the fact that the TLS is coupled to another system and, while finding the particle somewhere in the combined system is of course fixed at unity, the probability of finding it in the TLS is not constant. The energy dependence is an indication that the effect on the TLS of the semi-infinite chain is dependent on the energy, which is not surprising since the properties of the chain (e.g. its density of states) are energy-dependent. The non-conservation of probability is most evident by computing the timedependent self-energy, which is proportional to the surface Green’s function (the (1, 1) component of the matrix Green’s function) of the semi-infinite chain. This is displayed in Fig. 3.

4 Tripartite System (Ordered Case) We now consider the system depicted in Fig. Hamiltonian is ⎛ HDD VN H = ⎝ VN† HN 0 W†

1 with an ordered finite chain. The ⎞ 0 (8) W ⎠ H∞

where HN is an N × N truncation of H∞ , VN is a 2 × N truncation of V and W is an N × ∞ version of V with tc → t L (again assumed real and positive). The net effect of the chains on the TLS can again be described by a self-energy; rather than (6), we have  G DD N ∞ (E)

=

−τ E − 1 −τ E − 2 − Σ N ∞ (E)

where, defining s N ≡ sin(N k), etc.,

−1

,

(9)

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Σ N ∞ = tc 2

s N − t L 2 e−ik s N −1 s N +1 − t L 2 e−ik s N

(10)

The time-dependent Green’s function can no longer be evaluated analytically without approximation. Numerical and approximate analytic evaluations of the decay constant (that is, the escape rate or decoherence time) are described in [1] and they agree when expected to do so. The approximate analytical expression is:  −1 τ2 ≈ − 2  {Σ N ∞ (λ+ )} τφ δ0

(11)

where λ+ is one of the TLS energy eigenvalues given above. The decay rate is displayed in Fig. 4, where the generally excellent agreement between the two methods used can be seen. Also clearly visible is the greater disagreement in the upper panel, as expected. The appearance of three separate curves in the right panels is a manifestation of the fact that the graph depicts a periodic function of N sampled at integer

Fig. 4 (τφ )−1 as a function of the chain length N for 1 = −2 = 1, τ = 0.19, t L = 0.65 and tc = 0.2 and 0.1 (top and bottom panels, respectively). The left graph displays a small range of N while the right side shows a wider range. The red circles are from the approximate expression (11), where the self-energy is evaluated at λ+ . The black dots and lines are obtained by evaluating (τφ )−1 numerically using the decay of the time-dependent Green’s function. The horizontal line is the rate obtained in the limit N → ∞ (or equivalently, if we set t L to 1)

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values; it turns out that the period of this function, if N is viewed as a continuous parameter, is very close to 3 for the parameter values used, so when N changes by 3 the function has only changed by a small amount, as can be seen clearly in the left panels of the figure.

5 Tripartite System (Disordered Case) We now come to the final case to be considered, which is the previous case described by (8) with disorder added. This is achieved by adding to HN diagonal entries v1 , v2 , . . . , v N which are chosen randomly with a Gaussian probability distribution with a variety of widths v. The time-dependent Green’s function is displayed in Fig. 5 for four values of the disorder parameter v and also in Fig. 6.

Fig. 5 Absolute value of the time-dependent Green’s function for N = 4 and four values of the disorder parameter v. Each panel shows three cases. The upper curve of each panel is for a system with no semi-infinite bath. Since the system is finite in size, the Green’s function fluctuates but does not decay. The middle curve is for the full system. The lower curve corresponds to N = 0 examined in Sect. 3

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Fig. 6 Effect of disorder on decay rate (averaged over many disorder configurations, reducing the fluctuations). Bottom curve (labeled “no chain”) refers to the case discussed in Sect. 3; top curve (labeled “finite chain”) refers to the case where the TLS is connected to the finite chain only (corresponding to the top curves in Fig. 5)

It is clear that as the disorder increases, the Green’s function decreases more slowly. Once v is greater than about 2, there is no perceptible difference between the disordered case and the case with no bath. Thus, the TLS is effectively decoupled from its environment. This is a nice illustration of the well-known fact that disorder (giving rise to Anderson localization [4]) slows down decoherence (and transmission) from the double dot, in a sense “protecting” it from the influence of the environment. As a final observation, we can relate the decay rate to the transmission coefficient T of the N -site channel connected to two semi-infinite leads. One can show that T ≈ 4{G S∞ }{Σ N ∞ }, giving

 −1 T (λ+ ) τ2 ≈− 2 τφ S 4{G δ0 ∞ (λ+ )}

(12)

6 Summary In conclusion, we have studied a model allowing us to examine how coupling to a chain affects the decoherence rate of a TLS. In the case of a clean channel and semi-infinite lead, periodic behaviour as a function of N was noted, with potentially wildly different behaviours for adjacent values of N . In the case of a noisy channel, we found that noise impedes decoherence and escape. We also noted the relation between the decoherence rate and the transmission coefficient of the channel. Acknowledgements This work was supported in part by the Natural Science and Engineering Research Council of Canada and by the Fonds de Recherche Nature et Technologies du Québec via the INTRIQ strategic cluster grant.

Probing Anderson Localization Using the Dynamics of a Qubit

References 1. 2. 3. 4.

H. Eleuch, M. Hilke, R. MacKenzie, Phys. Rev. B 95 (2017) 062114. J.L. d’Amato and H.M. Pastawski, Phys. Rev. B 41 (1990) 7411. S. Datta, Quantum transport: atom to transistor (Cambridge University Press, 2005). P.W. Anderson, Phys. Rev. 109 (1958) 1492.

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Pure Spinors, Impure Spinors and Quantum Mechanics Mike Hewitt

Abstract The geometry of spinors in higher dimensional spaces is used to elucidate a potential ambiguity in the concept of a pure quantum state, and a ‘toroidal entropy’ is introduced to provide a measure of the geometrical ‘impurity’ of spinors. The geometry of the sub-manifold of geometrically pure spinors is described. The relationship of toroidal entropy with the preparation of a pure quantum state is discussed. It is shown that the toroidal entropy is trivial in 3 dimensions or for a single qubit system, but may be relevant to the physics of general quantum computation. A generalization of these concepts to general Lie group representations is also presented.

1 Introduction In this paper we will introduce a construct, the ‘toroidal entropy’ which is interesting for mathematical reasons, as it can be used to define novel invariants and metrics for group representations, and may be interpreted as a thermodynamic cost in preparing certain quantum states. A charged spin 21 particle in three spatial dimensions (e.g. the electron) is a canonical example of a single qubit system. Spatial rotations act though the group SU (2) on the 2 component spinor wave-function, which together with U (1) phase or gauge transformations, comprise the U (2) group of symmetries of the < | > Dirac inner product on the quantum state space [1]. Spin eigenstate components for such a particle can be resolved using a magnetic field, as in the classic Stern–Gerlach experiment [2]. Furthermore, from a given pure quantum state, any other may be obtained adiabatically through rotations induced by passing through magnetic fields. This behavior does not generalize to more than 3 spatial dimensions, or to general Lie group representations for an internal symmetry group, and the deviation from the properties of the simple 1 qubit system may be quantified by the ‘toroidal entropy’ defined below. In these more general settings, not M. Hewitt (B) Canterbury Christ Church University, Canterbury CT1 1QU, UK e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2018 V. Dobrev (ed.), Quantum Theory and Symmetries with Lie Theory and Its Applications in Physics Volume 1, Springer Proceedings in Mathematics & Statistics 263, https://doi.org/10.1007/978-981-13-2715-5_21

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all pure quantum states are accessible from a given pure state via adiabatic manipulation using magnetic fields. It is conjectured that the ‘toroidal entropy’ represents the minimum entropy that must be generated (e.g. by thermodynamically irreversible filters) to prepare such magnetically inaccessible states. The problem of preparing such states with these restricted means could be seen as analogous to geometric constructions using only e.g. compass and straight edge. The toroidal entropy can be used to define a family of deformed Kahler metrics on the representation space which are symmetric under the represented group which may be of mathematical interest.

2 Rotation Groups We will consider next the quantum mechanics of a single qubit system equivalent to a single spin 21 particle in a space S of D spatial dimensions (i.e. D + 1 spacetime dimensions). A system of D/2 qubits can be modeled straightforwardly in this way for even D. There is a generalization to representations of arbitrary Lie groups, explained below, but for concreteness we will deal first with rotation groups. In S with odd and even dimensions these have Lie algebras of types B and D respectively. A Lie generator s for a rotation group G is equivalent to a 2 form σ . In 2n+1 dimensions consider the 1-form v = ∗σ n (where * is the Hodge duality operator). Now V is orthogonal to σ , so the corresponding generator Σ operates in the 2n dimensional space orthogonal to v. So that the action of σ reduces to the even dimensional case. In D = 2n, take Σ to act on S C , the complexification of S. In a matrix representation Σ is Hermitian and so will have n pairs of complex conjugate eigenvectors which can be used as a basis e for S C . The matrix representation of σ is diagonal in this basis, and S C splits into n planes on which Σ acts separately. The basis e can now be used to define a pure spinor ψ with n positive eigenvalues (+1/2) for the plane rotations comprising Σ. This construction gives us a mapping m : σ − > ψ where the inverse image of ψ under m is a maximal torus M(ψ) of G. In this way, the moduli space B of maximal tori is isomorphic to the projective space of pure spinors [3], and B is the base of a natural torus fibration of G with ψ acting as a coordinate system on B. B has natural complex structure, related to the Borel–Weil–Bott theorem [4], which agrees with that of H , the Hilbert space containing ψ (the spinor representation).However, although m(B) spans H (as this is an irreducible representation of G) it does not by itself fill the unit sphere of H in D > 3. This is because dim(H ) = 2 ∗ (n) whereas dim(B) = n (where n = D/2) as D increases. Recall that ψ is a simultaneous eigenvector of the n constituent planar rotations of Σ. These planar rotations form a maximal commuting set of operators, corresponding to the quantum mechanical concept of a maximal compatible set of observables. In D > 3, because only pure spinors correspond to these observational states, and most spinors are not pure, the corresponding quantum states (for a spin particle) cannot be prepared by rotations from a given pure state alone.

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The spin of charged particles can be manipulated adiabatically using magnetic fields. A magnetic field Bμν is a 2-form, with the same components as a Lie rotation generator. Using the construction above, magnetic flux Bμν maps (in a many to one way) to pure spinors ψ. The energy eigenstates for our charged spin 1/2 particle will be the orbit of m(B) under the discrete group generated by spin flips in the n planes corresponding to Bμν giving a total of 2n eigenstates. General states split into 2n beams on passing through Bμν in a D dimensional Stern–Gerlach experiment. The probability distribution for the 2n outcomes if the path of the particle is subsequently measured can be assigned an entropy S(b) from density matrix ρ corresponding to a Bμν for b in B for this Stern–Gerlach splitting by the conventional formula S = −T r (ρ ln ρ)

(1)

A general ψ (not necessarily a pure spinor) generates non-negative function S(ψ, b) or S(ψ, φ) where φ is the pure spinor corresponding to b in B. For a pure spinorψ, S(ψ, b) will attain the value 0 at m(ψ), and this property indeed characterizes pure spinors. In general,the minimum value of S(ψ, b) over B defines a toroidal entropy ST (ψ) giving a measure of the impurity of a state or spinor, which takes the minimum value 0 exactly at the pure spinors. ST is rotationally invariant, and so is an invariant for spinor representations and so can be thought of as an intrinsic property. Its relationship to other invariants constructed from the Clifford algebra of spinors [5] is a topic for further investigation.

3 General Lie Group Representations A more general formulation of toroidal entropy for general Lie group representations may be motivated by the following ‘experiment’. Consider a beam of spin 0 particles (this time we can be in ordinary 3 dimensional space) from a representation R of dimension r of a general Lie group G. Let the beam pass through a magnetic field i i from the (in general non-abelian) gauge group G, where the direction of Bμν Bμν along the G fibers is covariantly constant and so lies on some maximal torus T of G. In general, the beam will split into r components, corresponding to the weights of R with respect to T . Let U R be the set of orthonormal bases u i of R so that < u i |u j >= δi, j . The Dirac inner product is invariant under U (r ) and the group G is embedded in U (r ) by the representation. The collection U R forms a torsor for U (r ). Let B be the set of maximal abelian subgroups of G. G is a natural torus fibration over B. For each b ∈ B, there will be a basis of unit eigenvectors ei (b), and these will form a subset E R of U R . Now there will be a basis ei (T ) corresponding to T ,

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i the maximal torus associated to our magnetic field Bμν . As a particle passes through the field, its state is resolved as r < ei |ψ > |ei > |ψ >= Σi=1

(2)

and the probability distribution associated with the density matrix ρi, j (|ψ >, T ) =< ei |ψ >< ψ|e j >

(3)

can be assigned the formal entropy S(|ψ >, T ) = −T r (ρ(|ψ >, T ) ln ρ(|ψ >, T )) ≥ 0

(4)

and the maximal value for S(|ψ >, T ) as T ranges over B gives us our definition of ST (ψ >): (5) ST (|ψ >) = maxb∈B S(|ψ >, b) and ST (|ψ >) = 0 iff |ψ >= ei for some eigenstate ei . ST (|ψ >) is invariant as G acts on |ψ > and is a local gauge invariant. In cases where 2r > dim(B) = dim(G) − rank(G)

(6)

there will be states |ψ > for which ST (|ψ >) > 0, and again ST (|ψ >) will discriminate between states in a gauge invariant way. (The factor 2 in (6) is based on R having r complex dimensions.) This mathematical definition of ST (|ψ >) coincides with that in Sect. 2 when G is a rotation group and R is a spinor representation, and so is a generalization of the above. How may ST (|ψ >) be interpreted as an entry or measure of information? The pure states that can be prepared with a single filter are the ei . If we take one of these as our starting point (following Newton’s celebrated experiments with light), it is possible to synthesize an arbitrary state |ψ > rotating, splitting and recombining beams using magnetic fields. While rotations can be accomplished adiabatically, and thus all eigenstates e j are directly accessible, some states require splitting, filtering and recombination to prepare and therefore will be thermodynamically irreversible. We conjecture that ST (|ψ >) is the minimum average entropy per particle that must be generated by a ‘factory’ for |ψ > states. In particular, an algorithm for a quantum computer that utilizes such states would need to generate a minimum amount of entropy (and heat) in operation. For an arbitrary state |ψ >, ST (|ψ >) will define an entropy function on the moduli space of maximal tori, and the minimum will identify the optimal filter for preparing |ψ > in terms of thermodynamic efficiency. As a mathematical observation, the projective space P = R/C of complex dimension r − 1 is a Kahler manifold symmetric under R, and may be deformed to a family of R symmetric Kahler metrics using

Pure Spinors, Impure Spinors and Quantum Mechanics

δgz,z ∗ = α∂z ∗ ∂z ST

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(7)

and for each φ ∈ P there is a Kahler deformation of B given by δgz,z ∗ = α∂z ∗ ∂z S(φ, .)

(8)

4 Multi Qubit Systems The analysis above can be applied to an N qubit system consisting of n electrons (or equivalent) subject to separate magnetic manipulation. Here R is the tensor product of n copies of the Hilbert space in Sect. 2, so r = 2 N with G = SU (2) N , so dim(G) = 3N and rank(G) = n. Thus (6) implies that algorithms exploiting general states in R will necessarily generate heat in execution at finite temperature. There is still plenty of scope to design adiabatic algorithms, however.

5 Conclusions We have introduced a measure of the ‘impurity’ of spinors and more general Lie group representations, and outlined some of the most obvious properties, in the hope that it will find useful applications in mathematical physics. Acknowledgements The author would like to thank the organizers for their kind invitation and hospitality in Varna, and Dan Waldram for enlightening conversations on group representations.

References 1. P.A.M. Dirac, The Principles of Quantum Mechanics, OUP, Oxford (1930). 2. W. Gerlach, O.Stern, Der experimentelle Nachweis der Richtungsquantelung im Magnetfeld, Zeitschrift fur Physik (1922) 9 349. 3. E.Cartan, The Theory of Spinors, Dover (1966). 4. J.P. Serre, Representations lineaires et espaces homognes kahlriens des groupes de Lie compacts (d’apres Armand Borel et Andre Weil, Seminaire Bourbaki 2, 447. 5. C.Chevalley, The Algebraic Theory of Spinors and Clifford algebras, Springer (1996).

Higher-Derivative Oscillators in Ad S5 × S5 T-Dual Penrose Limits H. Dimov, S. Mladenov, R. Rashkov and T. Vetsov

Abstract In this report we comment on the footprints of higher-derivative oscillators (more concretely the Pais–Uhlenbeck oscillator), which appear in the Penrose limit of the (non-)Abelian T-dual of the Ad S5 × S 5 string theory background. We show that the existence of a Kalb–Ramond B-field in the closed string action couples two of the equations of motion leading to an effective description given by the Pais– Uhlenbeck oscillator of fourth order. Consequently, we find that the infrared and the ultraviolet limits of the non-Abelian T-dual solution have an effective description in terms of a higher-derivative oscillators.

1 Introduction The interest in theories described by equations of motion (or equivalently Lagrangian) containing higher derivatives is fed by the search for a quantum field theory that is ultraviolet (UV) finite [1]. This idea was examined in gravity by showing that terms of higher order in curvature, when included in the Einstein action, lead to a renormalisable quantum theory [2]. The Hamiltonian formulation of such theories could be realised by the Ostrogradsky’s approach [3], which unfortunately suffers H. Dimov · S. Mladenov (B) · R. Rashkov · T. Vetsov Faculty of Physics, Sofia University, 5 J. Bourchier Blvd., 1164 Sofia, Bulgaria e-mail: [email protected] H. Dimov e-mail: [email protected] R. Rashkov e-mail: [email protected]; [email protected] T. Vetsov e-mail: [email protected] R. Rashkov Institute for Theoretical Physics, Vienna University of Technology, Wiedner Hauptstr. 8–10, 1040 Vienna, Austria © Springer Nature Singapore Pte Ltd. 2018 V. Dobrev (ed.), Quantum Theory and Symmetries with Lie Theory and Its Applications in Physics Volume 1, Springer Proceedings in Mathematics & Statistics 263, https://doi.org/10.1007/978-981-13-2715-5_22

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from unbounded from below Hamiltonian and therefore negative energies. The last flaw causes instabilities at classical level and negative-norm states (or ghosts) after quantisation. A simple toy model featuring higher derivatives is provided by the so-called Pais– Uhlenbeck oscillator (PUO) [4] defined by the following EoM1 :   x (4) (t) + ω12 + ω22 x (2) (t) + ω12 ω22 x(t) = 0,

(1)

where ω1 and ω2 represent oscillator’s frequencies. The presence of higher derivatives implies constraints between the model’s dynamical degrees of freedom, hence the quantisation requires inclusion of Dirac constraints [5] or path integral formalism [6]. Furthermore, the drawbacks of the Ostrogradsky’s approach could be easily remedied and the PUO has a few well-defined Hamiltonian formulations [7–9]. The last one stands out with its simplicity by reformulating the PUO in terms of harmonic oscillators. This strongly facilitates the application of the framework of thermo-field dynamics (TFD) [10] for investigation of various quantum properties (for example the entanglement entropy) of a system of interacting PUOs at (non-)equilibrium and at finite temperatures [11, 12]. In this report we focus on the role of the B-field in the Penrose limit of the (non-)Abelian T-dual geometry [13–15] of the Ad S5 × S 5 string background. We aim to reveal that the presence of B-field in the closed string action is a messenger of the appearance of higher-derivative oscillators in the corresponding equations of motion. In Sect. 2 we show that closed strings in the pp-wave limit of the Abelian T-dual geometry have an effective description in terms of the fourth-order PUO. In Sect. 3 we do similar analysis for the non-Abelian case. Finally, in Sect. 4 we summarise our results and comment on the thermodynamic properties of the PUO.

2 Closed Strings in the Ad S5 × S5 T-Dual pp-Wave: The Abelian Case In this section we consider the Penrose limit of the Abelian T-dual solution of Ad S5 × S 5 /Zk . The T-duality is performed along a direction on S 5 acting upon Zk . In order to restore the common factor of L 2 in the resulting metric, the new coordinate ψ˜ ∈ 2 [0, 2πk] is rescaled as ψ˜ = Lα ψ (L is the “radius” of the solution) and ψ has small range ψ ∈ [0, 2πkα /L 2 ] for large L. The Penrose limit is subsequently taken for geodesic motion of a (massless) particle along the directions ξ and ψ on the sphere, which basically means that the whole geometry is zoomed in a region near the geodesic. The geodesic equations describing this motion imply that the coordinates α and χ in the spherical part of the T-dual geometry are α = 0 and χ = π/2. The r¯ x¯ , α = 2L , χ = π2 + Lz and expansion pp-wave limit is defined by the rescaling r = 2L 1 In

the case of PUO of fourth order. The EoM could contain derivatives of arbitrary high even or odd integer number.

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w.r.t. L. The obtained plane wave solution in Brinkmann form has metric [15]: ds 2 = 2dudv + dr¯ 2 + r¯ 2 dΩ32 + dz 2 + dx 2 + x 2 dβ 2 + dw 2   2 8J 2 − 1 2 r¯ 2 2 + x + J z du 2 − 16 16

(2)

and field content e2Φ = gs2

α ≡ g˜s2 , L2

B2 =

u dz ∧ dw, 2

F4 =

2J x du ∧ dz ∧ dx ∧ dβ, g˜s

(3)

where u is the affine parameter along the null geodesic (the “lightcone” time). The action describing the dynamics of closed strings in such geometry reads [15]:   1 2  2 2  3 2  4 2  + X + X + X X 1 i i S=− dτ dσ ∂ X · ∂ X + (4)  4πα 16   5 2  6 2  2      + X X 2 8J − 1 + J 2 X 7 − κ˜ 1 X 7 ∂σ X 8 − κ˜ 2 X 8 ∂σ X 7 , + 16 where J is the conserved quantity along the cyclic coordinate ξ, and κ˜ 1 and κ˜ 2 are constants determining the non-zero B-field components in a particular gauge choice. Requiring reality of the solution for ψ as well as further exclusion of tachyonic modes imply that J can only take values in the range 1 1 √ ≤J≤ . 2 2 2

(5)

The equations of motion for the scalars X i are straightforwardly obtained by varying the action (4) (with accompanying boundary conditions) [15]: 1 i X = 0, i = 1, . . . 4, 16 8J 2 − 1 i X = 0, i = 5, 6, X i − 16 1 X 7 − J 2 X 7 + ∂σ X 8 = 0, 2 1 X 8 − ∂σ X 7 = 0, 2

X i −

(6)

where  ≡ η μν ∂μ ∂ν with ητ τ = −1, ησσ = 1. From now on, the track on the B-field is lost since κ˜ 1 and κ˜ 2 are not present in the EoMs (6), which means that one cannot switch off the B-field by taking the limit B → 0. The exact solutions for the scalars X i are trivial and do not present interest, hence we only derive the oscillator

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frequencies by making an ansatz of the form X i ∼ eiωt+inσ : 1 , i = 1, . . . 4, 16 8J 2 − 1 2 , i = 5, 6, = n2 + ωn,i 16 1 2 J2 2 ± ωn,± = n2 + n + J 4. 2 2 2 = n2 + ωn,i

(7)

We observe that two of the EoMs (the last two equations in (6)) describing the dynamics in the directions, in which the B-field is switched on, are coupled in a specific way. As a consequence they can be represented as one ordinary differential equation of fourth order governing the dynamics of a PUO. Let us make the ansatz X i = eimσ xi (τ ) in the last two equations of (6). After some simple algebra we get: x8(4)

   (2)  2 1 2 2 2 2 x8 = 0. + 2m + J x8 + m m + J − 4

(8)

This is the familiar PUO of fourth order. By comparing Eqs. (1) and (8) we can calculate the corresponding frequencies ωm,+ = ω1 and ωm,− = ω2 of the effective higher-derivative oscillator: 2 = m2 + ωm,±

1 2 J2 ± m + J 4. 2 2

(9)

Two comments are in order. First, the existence of antisymmetric NS two form in the action of closed bosonic strings moving in the pp-wave background of the T-dual Ad S5 × S 5 geometry leads to description in term of the PUO of fourth order, which is essentially an effective theory of coupled oscillators. This fact gives strong motivation to study the entanglement of the system. Second, the oscillator frequencies ωn,± in Eq. (7) are exactly equal to the frequencies of the PUO ωm,± in Eq. (9).

3 Closed Strings in the Ad S5 × S5 T-Dual pp-Wave: The Non-abelian Case The analysis in the non-Abelian case [13] mostly resembles the Abelian one. The T-duality is performed along the SU (2) isometry of Ad S5 × S 5 /Zk . The coordinates after the T-duality are denoted by ρ˜ ∈ [0, 2πk] and two angles (χ, ξ). The pp-wave 2 limit is then taken in the directions of ρ and ξ after appropriate rescale of ρ˜ = Lα ρ 3/2 and of the string coupling g˜˜ s = gs αL 3 . The Penrose limit is defined by expansion r¯ x z , χ = π2 + 2L . The plane wave solution has around the null geodesic r = 2L , α = 2L metric [15]

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ds 2 = 2dudv + dr¯ 2 + r¯ 2 dΩ32 + dx 2 + x 2 dβ 2 + dz 2 + dw 2  2  r¯ (ρ2 + 1)2 2 2 x2 2 2 2 − J z − Fz z − Fw w du 2 , + (8J − 1) + 16 16 ρ4

(10)

where the pp-wave spectrum coefficients Fz and Fw have the following form: Fz =

4J 2 (4ρ2 + 1) + 3(4J 2 − 1)ρ4 , 4ρ4 (ρ2 + 1)2

Fw = −

3 . 4(ρ2 + 1)2

(11)

The dilaton, NS-NS, and R-R fields take the form: ρ2 + 1 1 ρ2 + 3 (κ1 zdu ∧ dw − κ2 wdu ∧ dz), , B2 = − 2 2ρ +1 g˜˜ s2 2J x ρ2 + 1 F4 = du ∧ dx ∧ dz ∧ dw. g˜˜ s

e−2Φ =

(12)

We consider closed bosonic strings moving in the pp-wave background of the nonAbelian T-dual of the Ad S5 × S 5 geometry. The action is slightly more complicated and again has B-field (manifested by the constants κ1 and κ2 ) switched on in the directions X 7 and X 8 [15]:

 2  2  2  2   X1 + X2 + X3 + X4 1 i i · ∂ X + S=− dτ dσ ∂ X 4πα 16  5 2  6 2  2  2  2  X + X + 8J − 1 − Fz X 7 − Fw X 8 (13) 16   2  2  7 2  ρ +1 ρ2 + 3  2 κ 1 X 7 ∂σ X 8 − κ 2 X 8 ∂σ X 7 , X J + − ρ4 ρ2 + 1 where the coordinate ρ is a direction of movement of (massless) particle. The variation of the action (13) with appropriate boundary conditions easily gives the equations of motion for the scalars X i , i = 1, . . . , 8: 1 i X = 0, i = 1, . . . 4, 16 8J 2 − 1 i X i − X = 0, i = 5, 6, 16   2 ρ2 + 1 1 ρ2 + 3 7 2 7 ∂σ X 8 = 0, J − F + X − X z ρ4 2 ρ2 + 1

X i −

X 8 + Fw X 8 −

1 ρ2 + 3 ∂σ X 7 = 0. 2 ρ2 + 1

(14)

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We readily notice that the EoMs for the scalar fields X 7 and X 8 are coupled again and there is no trace of the B-field coefficients κ1 and κ2 . In this case, however, the masses (frequencies) of the system depend on the light-cone time (in this case the affine parameter u over the geodesic) through the coordinate ρ. In order to overcome this complication one can take the limit u 1, in which ρ becomes constant, the first two equations do not change, but the last two equations take the following form: 4 − a2 7 1 X + (1 + a)∂σ X 8 = 0, 16 2 2 3a 8 1 X − (1 + a)∂σ X 7 = 0, X 8 − 16 2

X 7 −

(15)

where a ≡ 1 − 4J 2 . Skipping the exact solutions for the scalars X i , the closed string quantisation in this limit gives the following oscillator frequencies by making the ansatz X i ∼ eiωt+inσ [15]: 1 , i = 1, . . . , 4, 16 1 − 2a 2 , i = 5, 6, = n2 + ωn,i 16  1 1  2 2 ωn,± a + 2 ± (a + 1) 16n 2 + (a − 1)2 . = n2 + 16 8

2 ωn,i = n2 +

(16)

Noticing again the coupled nature of Eq. (15), we are tempted to make the same ansatz as before, X i = eimσ xi (τ ), m = 7, 8, in order to transform the two coupled differential equations of second order into one EoM of PUO of fourth order,  1 2 + 16m 2 + a 2 x8(2) 8 



 1 3a 2 4 − a2 m2 + − m 2 (1 + a)2 x8 = 0. + m2 + 16 16 4

x8(4) +

(17)

The two frequencies ω1 = ωm,+ and ω2 = ωm,− of the fourth-order PUO can be extracted by comparing Eqs. (1) and (17): 2 = m2 + ωm,±

 1 1  2 a + 2 ± (a + 1) 16m 2 + (a − 1)2 . 16 8

(18)

To conclude, in the non-Abelian case we found out the same behaviour as in the Abelian one. Namely, the presence of a Kalb–Ramond field in the action of closed bosonic strings moving in the pp-wave limit of the non-Abelian T-dual background of the Ad S5 × S 5 geometry couples the equations of motion resulting in an effective theory of PUO of fourth order. The frequencies of this higher-derivative oscillator coincide with those of the original system. The role of the non-Abelian nature of the T-duality is to complicate the EoMs by making the frequencies coordinate-dependent.

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In the infrared limit u 1, however, they become constants. On the other hand, the ultraviolet limit of the non-Abelian T-dual solution is equivalent to its Abelian counterpart [14]. Hence these two limits can be effectively described by the fourthorder PUO. The real reason for the effective description in terms of higher-derivative oscillators, however, is the existence of B-field in the action.

4 Conclusion In this report we considered an interesting property of the Penrose limit of the (non-)Abelian T-dual solution of the Ad S5 × S 5 geometry. Specifically, we showed that the presence of antisymmetric NS-NS two-form in the action of closed bosonic strings couples two of the EoMs in such way that they become equivalent to one EoM of Pais–Uhlenbeck oscillator of fourth order. Moreover, there exists a relation between the Abelian and non-Abelian T-duals of Ad S5 × S 5 [14]. Namely, in the limit r → ∞ the NS-NS sector of the non-Abelian T-dual solution reduces to that of the Abelian one. The R-R sectors do not exactly coincide, but they are physically equivalent. Therefore two limits of the non-Abelian pp-wave solution—the infrared limit (u 1) and the ultraviolet limit (Abelian Tdual)—have an effective description in terms of higher-derivative oscillator due to the non-trivial B-field. This behaviour was observed for two similar systems in the context of the Pilch–Warner supergravity solution—the Penrose limit of this solution and the quadratic fluctuations around classical solutions of rotating strings [11]. The reason for the appearance of higher-derivative oscillators in these systems was traced back to the presence of B-field too. Therefore one can argue that the inclusion of B-field is an indicator for appearance of higher-derivative oscillators. Furthermore, the higher-derivative systems are intrinsically interacting. In the case of PUO, this property follows as a consequence of the fact that every PUO of order 2n possesses a description in terms of Hamiltonian, which is equal to the sum (with alternating signs) of n Hamiltonians of harmonic oscillators. Therefore one can calculate the entanglement entropy of interacting PUOs or consider PUOs as a composite system of harmonic oscillators. The latter could be easily applied for the two systems considered here. The renormalised entanglement entropy (in the context of thermo-field dynamics) of one of the d.o.f. (say the one corresponding to ω1 = ωm,+ ) to the other could be written as [12]: 

   K1 K1 kB  coth K 1 1 + coth − 2 ln e K 1 − 1 , S1 (K 1 ) = 2 4 4

(19)

where K 1 = ω1 β depends on the inverse temperature β and kB is the Boltzmann constant. The entanglement entropy is a reliable quantity characterising the amount of entanglement between the d.o.f. of one system. It can be easily promoted to the Fisher information metric, which gives the distance between two infinitesimally close points on a statistical manifold. The problem of recovering the distribution

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(PDF) corresponding to a particular Fisher metric is notoriously hard to solve in its generality. However insights counting on particular examples (as these provided here) might prove to be useful. Acknowledgements The authors would like to thank Prof. Vladimir Dobrev for the opportunity to present their work on the 10-th International Symposium “Quantum Theory and Symmetries” (QTS10) with 12-th International Workshop “Lie Theory and Its Applications in Physics” (LT12), 19-25 June 2017, Varna, Bulgaria. This work is supported by grant No. DN18/1 with the Bulgarian Science Fund.

References 1. W. Thirring, “Regularization as a consequence of higher order equations,” Phys. Rev. 77 (Feb, 1950) 570–570. 2. K. S. Stelle, “Renormalization of higher-derivative quantum gravity,” Phys. Rev. D 16 (Aug, 1977) 953–969. 3. M. V. Ostrogradsky, “Mémoire sur les équations différentielles relatives au problème des isopérimètres,” Mem. Acad. Sci. St. Petersbourg VI 4 (1850) 385–517. 4. A. Pais and G. E. Uhlenbeck, “On Field theories with nonlocalized action,” Phys. Rev. 79 (1950) 145–165. 5. P. D. Mannheim and A. Davidson, “Dirac quantization of the Pais-Uhlenbeck fourth order oscillator,” Phys. Rev. A71 (2005) 042110. 6. S. W. Hawking and T. Hertog, “Living with ghosts,” Phys. Rev. D65 (2002) 103515. 7. K. Bolonek and P. Kosinski, “Hamiltonian Structures for Pais–Uhlenbeck Oscillator,” Acta Physica Polonica B 36 (June, 2005) 2115. 8. A. Mostafazadeh, “A Hamiltonian Formulation of the Pais-Uhlenbeck Oscillator that Yields a Stable and Unitary Quantum System,” Phys. Lett. A375 (2010) 93–98. 9. I. Masterov, “An alternative Hamiltonian formulation for the PaisUhlenbeck oscillator,” Nucl. Phys. B902 (2016) 95–114. 10. Y. Hashizume and M. Suzuki, “Understanding quantum entanglement by thermo field dynamics,” Physica A Statistical Mechanics and its Applications 392 (Sept., 2013) 3518–3530. 11. H. Dimov, S. Mladenov, R. C. Rashkov and T. Vetsov, “Entanglement of higher-derivative oscillators in holographic systems,” Nucl. Phys. B 918, 317 (2017). 12. H. Dimov, S. Mladenov, R. C. Rashkov and T. Vetsov, “Entanglement entropy and Fisher information metric for closed bosonic strings in homogeneous plane wave background,” Phys. Rev. D 96, no. 12, 126004 (2017). 13. K. Sfetsos and D. C. Thompson, “On non-abelian T-dual geometries with Ramond fluxes,” Nucl. Phys. B 846, 21 (2011). 14. Y. Lozano and C. Nez, “Field theory aspects of non-Abelian T-duality and N = 2 linear quivers,” JHEP 1605, 107 (2016). 15. G. Itsios, H. Nastase, C. Nez, K. Sfetsos and S. Zacaras, “Penrose limits of Abelian and nonAbelian T-duals of Ad S5 × S 5 and their field theory duals,” JHEP 1801, 071 (2018).

Part V

Quantum Groups and Related Structures

A Unified Approach to Poisson–Hopf Deformations of Lie–Hamilton Systems Based on sl(2) Ángel Ballesteros, Rutwig Campoamor-Stursberg, Eduardo Fernández-Saiz, Francisco J. Herranz and Javier de Lucas

Abstract Based on a recently developed procedure to construct Poisson–Hopf deformations of Lie–Hamilton systems [9], a novel unified approach to nonequivalent deformations of Lie–Hamilton systems on the real plane with a Vessiot–Guldberg Lie algebra isomorphic to sl(2) is proposed. This, in particular, allows us to define a notion of Poisson–Hopf system in dependence of a parameterized family of Poisson algebra representations. Such an approach is explicitly illustrated by applying it to the three non-diffeomorphic classes of sl(2) Lie–Hamilton systems. Our results cover deformations of the Ermakov system, Milne–Pinney, Kummer–Schwarz and several Riccati equations as well as of the harmonic oscillator (all of them with tdependent coefficients). Furthermore t-independent constants of motion are given as well. Our methods can be employed to generate other Lie–Hamilton systems and their deformations for other Vessiot–Guldberg Lie algebras and their deformations.

Á. Ballesteros · F. J. Herranz Departamento de Física, Universidad de Burgos, 09001 Burgos, Spain e-mail: [email protected] F. J. Herranz e-mail: [email protected] R. Campoamor-Stursberg (B) Instituto de Matemática Interdisciplinar I.M.I-U.C.M, Pza. Ciencias 3, 28040 Madrid, Spain e-mail: [email protected] E. Fernández-Saiz Departamento de Geometría y Topología, Universidad Complutense de Madrid, Pza. Ciencias 3, 28040 Madrid, Spain e-mail: [email protected] J. de Lucas Department of Mathematical Methods in Physics, University of Warsaw, ul. Pasteura 5, 02-093 Warszawa, Poland e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2018 V. Dobrev (ed.), Quantum Theory and Symmetries with Lie Theory and Its Applications in Physics Volume 1, Springer Proceedings in Mathematics & Statistics 263, https://doi.org/10.1007/978-981-13-2715-5_23

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1 Introduction Since its original formulation by Lie [23], nonautonomous first-order systems of ordinary differential equations admitting a nonlinear superposition rule, the so-called Lie systems, have been studied extensively (see [11–13, 20, 28, 31, 32] and references therein). The Lie theorem [14, 23] states that every system of first-order differential equations is a Lie system if and only if it can be described as a curve in a finite-dimensional Lie algebra of vector fields, a referred to as Vessiot–Guldberg Lie algebra. Although being a Lie system is rather an exception than a rule [15, 21, 22], Lie systems have been shown to be of great interest within physical and mathematical applications (see [12] and references therein). Surprisingly, Lie systems admitting a Vessiot–Guldberg Lie algebra of Hamiltonian vector fields relative to a Poisson structure, the Lie–Hamilton systems, have found even more applications than standard Lie systems with no associated geometric structure [7, 8, 10, 16]. Lie–Hamilton systems admit an additional finite-dimensional Lie algebra of Hamiltonian functions, a Lie–Hamilton algebra, that allows for the algebraic determination of superposition rules and constants of motion of the system [10]. Apart from the theory of quasi-Lie systems [15] and superposition rules for nonlinear operators [21, 22], most approaches to Lie systems rely strongly in the theory of Lie algebras and Lie groups [12, 13, 26]. However, the success of quantum groups [18, 24] and the coalgebra formalism within the analysis of superintegrable systems [3, 5, 6], and the fact that quantum algebras appear as deformations of Lie algebras suggested the possibility of extending the notion and techniques of Lie–Hamilton systems beyond the range of application of the Lie theory. An approach in this direction was recently proposed in [9], where a method to construct quantum deformed Lie–Hamilton systems (LH systems in short) by means of the coalgebra formalism and quantum algebras was given. The underlying idea is to use the theory of quantum groups to deform Lie systems and their associated structures. More exactly, the deformation transforms a LH system with its Vessiot–Guldberg Lie algebra into a Hamiltonian system whose dynamics is determined by a set of generators of a Steffan–Sussmann distribution. Meanwhile, the initial Lie–Hamilton algebra (LH algebra in short) is mapped into a Poisson–Hopf algebra. The deformed structures allow for the explicit construction of t-independent constants of the motion through quantum algebra techniques for the deformed system. This work aims to illustrate how the approach introduced in [9] to construct deformations of LH systems via Poisson–Hopf structures allows for a further systematization that encompasses the nonequivalent LH systems corresponding to isomorphic LH algebras. Specifically, we show that Poisson–Hopf deformations of LH systems based on a LH algebra isomorphic to sl(2) can be described generically, hence providing the deformed Hamiltonian functions and the corresponding deformed Hamiltonian vector fields, once the corresponding counterpart of the non-deformed system is known.

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Moreover this work also provides a new method to construct LH systems with a LH algebra isomorphic to a fixed Lie algebra g. Our approach relies on using the symplectic foliation in g∗ induced by the Kirillov–Kostant–Souriau bracket of g. As a particular case, it is explicitly shown how our procedure explains the existence of three types of LH systems on the plane related to a LH algebra isomorphic to sl(2). This is due to the fact that each one of the three different types corresponds to one of the three types of symplectic leaves in sl∗ (2). Analogously, one can generate the only type of LH systems on the plane admitting a Vessiot–Guldberg Lie algebra isomorphic to so(3). Our systematization permits us to give directly the Poisson–Hopf deformed system from the classification of LH systems [8, 10], further suggesting a notion of Poisson– Hopf Lie systems based on a z-parameterized family of Poisson algebra morphisms. Our methods seem to be extensible to study also LH systems and their deformations on other more general manifolds. The structure of the contribution goes as follows. Section 2 is devoted to introducing the main aspects of LH systems and Poisson–Hopf algebras. The general approach to construct Poisson–Hopf algebra deformations of LH systems [9] is summarized in Sect. 3. For our further purposes, the (non-standard) Poisson–Hopf algebra deformation of sl(2) is recalled in Sect. 4. The novel unifying approach to deformations of Poisson–Hopf Lie systems with a LH algebra isomorphic to a fixed Lie algebra g are treated in Sect. 5. Such a procedure is explicitly illustrated in Sect. 6 by applying it to the three non-diffeomorphic classes of sl(2)-LH systems on the plane, so obtaining in a straightforward way their corresponding deformation. Next, a new method to construct (non-deformed) LH systems is presented in Sect. 7. Finally, our results are summarised and the future work to be accomplished is briefly detailed in the last section.

2 Lie–Hamilton Systems and Poisson–Hopf Algebras This section recalls the main notions that will be used in the sequel. Let {x1 , . . . , xn } be global coordinates in Rn and consider a nonautonomous system of first-order ordinary differential equations dxk = f k (t, x1 , . . . , xn ), dt

1 ≤ k ≤ n,

(1)

where f k : Rn+1 → R are arbitrary functions. Geometrically, this system amounts to a t-dependent vector field Xt : R × Rn → TRn given by Xt : R × Rn  (t, x1 , . . . , xn ) →

n  k=1

f k (t, x1 , . . . , xn )

∂ ∈ TRn . ∂xk

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We say that (1) is a Lie system if its general solution, x(t), can be expressed in terms of a finite number m of generic particular solutions {y1 (t), . . . , ym (t)} and n constants {C1 , . . . , Cn } in the form x(t) = Ψ (y1 (t), . . . , ym (t), C1 , . . . , Cn ), for a certain function Ψ : (Rn )m × Rn → Rn , a so-called superposition rule of the system (1). The Lie–Scheffers Theorem [12, 13, 23, 31, 32] states that Xt is a Lie system if and only if there exist t-dependent functions b1 (t), . . . , br (t) and vector fields X1 , . . . , Xr on Rn spanning an r -dimensional real Lie algebra V such that Xt (x, y) =

r 

bi (t)Xi .

i=1

Then, V is called a Vessiot–Guldberg Lie algebra of Xt . A Lie system is said to be a Lie–Hamilton system [16] whenever it admits a Vessiot–Guldberg Lie algebra V of Hamiltonian vector fields with respect to a Poisson structure. In our work, we will focus on LH systems on the plane admitting a Vessiot–Guldberg Lie algebra of Hamiltonian vector fields relative to a symplectic structure. It can be proved that all LH systems on the plane can be studied around a generic point in this way [8]. Hence, the LH systems to be studied hereafter admit a symplectic structure ω on R2 that is invariant under Lie derivatives with respect to the elements of V , namely L Xi ω = 0,

1 ≤ i ≤ r.

Due to the non-degeneracy of ω, each function h determines uniquely a vector field Xh , the Hamiltonian vector field of h, such that ιXh ω = dh, enabling us to define a Poisson bracket {·, ·}ω : C ∞ (R2 ) × C ∞ (R2 ) → C ∞ (R2 ) through the prescription { f, g}ω → Xg f.

(2)

In particular, this implies that (C ∞ (R2 ), {·, ·}ω ) is a Lie algebra. Similarly, the space Ham(ω) of Hamiltonian vector fields on R2 relative to ω is a Lie algebra with respect to the commutator of vector fields. These two Lie algebras are known to be related through the exact sequence (see [29] for details): ϕ

π

0 → R → (C ∞ (R2 ), {·, ·}ω ) −→ (Ham(ω), [·, ·]) −→ 0, where ϕ maps every function h ∈ C ∞ (R2 ) into −Xh .

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Going back to the theory of LH systems, recall that every LH system admits a Vessiot–Guldberg Lie algebra V of Hamiltonian vector fields relative to an ω. In view of (2), there always exists a finite-dimensional Lie subalgebra Hω of (C ∞ (R2 ), {·, ·}ω ) containing the Hamiltonian functions of V : a so-called Lie– Hamilton algebra of the LH system Xt . Let g be a Lie algebra isomorphic to Hω . This induces the universal enveloping algebra U (g) and the symmetric algebra S(g) (see [30] for details). The second one is the associative commutative algebra of polynomials in the elements of g, whereas U (g) is defined to be the tensor algebra of g modulo the two-sided ideal generated by the elements {v ⊗ w − w ⊗ v − [v, w] : v, w ∈ g}. Relevantly, S(g) and U (g) are isomorphic as linear spaces [30]. They also share a special property: they are Hopf algebras. The Lie bracket of g can be extended to S(g) turning this space into a Poisson algebra. Since the elements of g can be considered as linear functions on g∗ , then the elements of S(g) can be considered as elements of C ∞ (g∗ ), which allows us to ensure that the space C ∞ (g∗ ) can be endowed with a Poisson–Hopf algebra structure. Let us finally recall in this introduction the main properties of Hopf algebras. We recall that an associative algebra A with a product m and a unit η is said to be a Hopf algebra over R [1, 18, 24] if there exist two algebra homomorphisms called coproduct (Δ : A −→ A ⊗ A) and counit ( : A −→ R) satisfying (Id ⊗ Δ)Δ = (Δ ⊗ Id)Δ,

(Id ⊗ )Δ = ( ⊗ Id)Δ = Id,

along with an antihomomorphism, the antipode γ : A −→ A, such that the following diagram is commutative: A⊗ A

Id ⊗ γ

A⊗ A

Δ

m 

A

R

Δ

A⊗ A

η

A m

γ ⊗ Id

A⊗ A

3 Poisson–Hopf Deformations of Lie–Hamilton Systems The coalgebra method employed in [7] to obtain superposition rules and constants of motion for LH systems on a manifold M relies almost uniquely in the Poisson–Hopf algebra structure related to C ∞ (g∗ ) and a Poisson map D : C ∞ (g∗ ) → C ∞ (M),

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where we recall that g is a Lie algebra isomorphic to a LH algebra, Hω , of the LH system. Relevantly, quantum deformations allow us to repeat this scheme by substituting the Poisson algebra C ∞ (g∗ ) with a quantum deformation C ∞ (g∗z ), where z ∈ R, and obtaining an adequate Poisson map Dz : C ∞ (g∗z ) → C ∞ (M). The above procedure enables us to deform the LH system into a z-parametric family of Hamiltonian systems whose dynamic is determined by a Steffan–Sussmann distribution and a family of Poisson algebras. If z tends to zero, then the properties of the (classical) LH system are recovered by a limiting process, hence enabling us to construct new deformations exhibiting physically relevant properties. In essence, the method for a LH system on an n-dimensional manifold M consists essentially of the following four steps (see [9] for details):  1. Consider a LH system Xt := ri=1 bi (t)Xi on M with respect to a symplectic form ω and possessing a LH algebra Hω spanned by the functions {h 1 , . . . , h r } ⊂ C ∞ (M) and structure constants Cikj , i.e. {h i , h j }ω =

r 

Cikj h k ,

1 ≤ i, j ≤ r.

k=1

2. Consider a Poisson–Hopf algebra deformation C ∞ (H∗z,ω ) with (quantum) deformation parameter z ∈ R (respectively q := ez ) as the space of smooth functions F(h z,1 , . . . , h z,r ) for a family of functions h z,1 , . . . , h z,r on M such that {h z,i , h z, j }ω = Fz,i j (h z,1 , . . . , h z,r ),

(3)

where the Fz,i j are smooth functions depending also on z satisfying the boundary conditions lim h z,i = h i ,

z→0

lim grad h z,i = grad h i ,

z→0

lim Fz,i j (h z,1 , . . . , h z,r ) =

z→0

r 

Cikj h k .

(4)

k=1

3. Define the deformed vector fields Xz,i on M according to the rule ιXz,i ω = dh z,i ,

(5)

lim Xz,i = Xi .

(6)

so that z→0

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4. Define the Poisson–Hopf deformation of the LH system Xt as Xz,t :=

r 

bi (t)Xz,i .

i=1

We stress that the deformed vector fiel Xz,1 , . . . , Xz,r do not generally close on a finite-dimensional Lie algebra. Instead, they span a Stefan–Sussman distribution (see [27, 29]). Their corresponding commutation relations can be written in terms of the functions Fz,i j as [9] [Xz,i , Xz, j ] = −

r  ∂ Fz,i j k=1

∂h z,k

Xz,k .

(7)

Next, to determine the t-independent constants of the motion and the superposition rules of a LH system with a LH algebra Hω , the coalgebra formalism developed in [7] is applied. Let us illustrate this point. Consider the symmetric algebra S (g) of g  Hω , that can be endowed with a Poisson algebra structure by means of the Lie algebra structure of g. The Hopf algebra structure with a (non-deformed trivial) coproduct map Δ is given by Δ : S (g) → S (g) ⊗ S (g) ,

Δ(v) := v ⊗ 1 + 1 ⊗ v,

∀v ∈ g.

This is easily seen to be a Poisson algebra homomorphism with respect to the Poisson structure on S(g) and the natural Poisson structure in S(g) ⊗ S(g) induced by S(g). Due to density of the functions S(g) in C ∞ (g∗ ), the coproduct Δ can be extended in a unique way to       Δ : C ∞ g∗ → C ∞ g∗ ⊗ C ∞ g∗ . The extension by continuity of the Poisson–Hopf structure in S(g) to C ∞ (g∗ ) endows the latter with a Poisson–Hopf algebra structure [7]. Let now C = C(v1 , . . . , vr ) be a Casimir function of the Poisson algebra C ∞ (g∗ ), where {v1 , . . . , vr } is a basis for g. We can define a Lie algebra morphism φ : g → C ∞ (M) such that h i := φ(vi ). The Poisson algebra morphisms   D : C ∞ g∗ → C ∞ (M),

    D (2) : C ∞ g∗ ⊗ C ∞ g∗ → C ∞ (M) ⊗ C ∞ (M),

defined by D(vi ) := h i (x1 ),

D (2) (Δ(vi )) := h i (x1 ) + h i (x2 ),

1 ≤ i ≤ r,

  where xs = xs,1 , . . . , xs,n (s = 1, 2) are global coordinates in M, lead to tindependent constants of the motion F (1) := F and F (2) of Xt having the form F := D(C),

F (2) := D (2) (Δ(C)) .

(8)

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The very same argument holds for any deformed Poisson–Hopf algebra C ∞ (g∗z ) with deformed coproduct Δz and Casimir invariant C z = C z (vz,1 , . . . , vz,r ), where {vz,1 , . . . , vz,r } satisfy the same formal commutation relations of the h z,i in (3), and such that lim vz,i = vi , lim C z = C. lim Δz = Δ, z→0

z→0

z→0

Therefore, the deformed Casimir C z will provide the t-independent constants of motion for the deformed LH system Xz,t through the coproduct Δz .

4 The Non-standard Poisson–Hopf Algebra Deformation of sl(2) Amongst the LH systems in the plane (see [7, 8, 10] for details and applications), those with a Vessiot–Guldberg Lie algebra isomorphic to sl(2) are of both mathematical and physical interest; they cover complex Riccati, Milne–Pinney and Kummer– Schwarz equations as well as the harmonic oscillator, all of them with t-dependent coefficients. Furthermore, sl(2)-LH systems are related to three non-diffeomorphic Vessiot–Guldberg Lie algebras on the plane [8, 10]. This gives rise to different nonequivalent Poisson–Hopf deformations. Let us consider sl(2) with the standard basis {J3 , J+ , J− } satisfying the commutation relations [J+ , J− ] = J3 . [J3 , J± ] = ±2J± , In this basis, the Casimir operator reads C=

1 2 J + (J+ J− + J− J+ ). 2 3

(9)

Considering the non-standard (triangular or Jordanian) quantum deformation Uz (sl(2)) of sl(2) [25] (see also [2, 4] and references therein), we are led to the following deformed coproduct Δz (J+ ) = J+ ⊗ 1 + 1 ⊗ J+ , Δz (Jl ) = Jl ⊗ e2z J+ + e−2z J+ ⊗ Jl ,

l ∈ {−, 3}

and the commutation rules  

J3 , J+ J3 , J−





J+ , J−



z

= 2 shc(2z J+ )J+ ,

z

= −J− ch(2z J+ ) − ch(2z J+ )J− .

z

= J3 ,

Here shc denotes the cardinal hyperbolic sinus function defined by

A Unified Approach to Poisson–Hopf Deformations …

shc(ξ) :=

sh(ξ) , ξ

1,

355

for ξ = 0, for ξ = 0.

It is known that every quantum algebra Uz (g) related to a semi-simple Lie algebra g admits an isomorphism of algebras Uz (g) → U (g) (see [18, Theorem 6.1.8]). This allows us to obtain a Casimir operator of Uz (sl(2)) out of one, e.g. C given by (9), of U (sl(2)) in the form (see [18] for details) Cz =

1 2 1 J + shc(2z J+ )J+ J− + J− J+ shc(2z J+ ) + ch2 (2z J+ ), 2 3 2

which, as expected, coincides with the expression formerly given in [4]. There exists a new z-parametrized family of deformed Poisson–Hopf structures in C ∞ (sl∗z (2)) denoted by (C ∞ (sl∗z (2)), {·, ·}z ) and given by the relations {v1 , v2 }z = − shc(2zv1 )v1 , {v1 , v3 }z = −2v2 , {v2 , v3 }z = − ch(2zv1 )v3 , (10) along with the coproduct Δz (v1 ) = v1 ⊗ 1 + 1 ⊗ v1 ,

Δz (vk ) = vk ⊗ e2zv1 + e−2zv1 ⊗ vk ,

The Poisson algebra C ∞ (slz (2)) admits a Casimir function C z = shc(2zv1 ) v1 v3 − v22 , where v1 = J+ ,

v2 =

1 J3 , 2

v3 = −J− .

k = 2, 3. (11)

(12)

(13)

In the limit z = 0, the Poisson–Hopf structure in C ∞ (sl∗z (2)) recovers the standard Poisson–Hopf algebra structure in C ∞ (sl∗ (2)) with non-deformed coproduct and Poisson bracket Δ(vi ) = vi ⊗ 1 + 1 ⊗ vi , {v1 , v2 } = −v1 ,

i = 1, 2, 3,

{v1 , v3 } = −2v2 ,

{v2 , v3 } = −v3 ,

(14)

as well as the Casimir function C = v1 v3 − v22 .

(15)

We shall make use of the above Poisson–Hopf algebra C ∞ (sl∗z (2)) in the next section in order to construct the corresponding deformed LH systems from a unified approach.

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5 Poisson–Hopf Deformations of sl(2) Lie–Hamilton Systems This section concerns the analysis of Poisson–Hopf deformations of LH systems on a manifold M with a Vessiot–Guldberg algebra isomorphic to sl(2). Our geometric analysis will allow us to introduce the notion of a Poisson–Hopf Lie system that, roughly speaking, is a family of nonautonomous Hamiltonian systems of first-order differential equations constructed as a deformation of a LH system by means of the representation of the deformation of a Poisson–Hopf algebra in a Poisson manifold. Let us endow a manifold M with a symplectic structure ω and consider a Hamiltonian Lie group action Φ : S L(2, R) × M → M. A basis of fundamental vector fields of Φ, let us say {X1 , X2 , X3 }, enable us to define a Lie system Xt =

3 

bi (t)Xi ,

i=1

for arbitrary t-dependent functions b1 (t), b2 (t), b3 (t), and {X1 , X2 , X3 } spanning a Lie algebra isomorphic to sl(2). As is well known, there are only three nondiffeomorphic classes of Lie algebras of Hamiltonian vector fields isomorphic to sl(2) on the plane [8, 19]. Since X1 , X2 , X3 admit Hamiltonian functions h 1 , h 2 , h 3 , the t-dependent vector field Xt admits a t-dependent Hamiltonian function h=

3 

bi (t)h i .

i=1

Due to the cohomological properties of sl(2) (see e.g. [29]), the Hamiltonian functions h 1 , h 2 , h 3 can always be chosen so that the space h 1 , h 2 , h 3  spans a Lie algebra isomorphic to sl(2) with respect to {·, ·}ω . Let {v1 , v2 , v3 } be the basis for sl(2) given in (13) and let M be a manifold where the functions h 1 , h 2 , h 3 are smooth. Further, the Poisson–Hopf algebra structure of C ∞ (sl∗ (2)) is given by (14). In these conditions, there exists a Poisson algebra morphism D : C ∞ (sl∗ (2)) → C ∞ (M) satisfying D( f (v1 , v2 , v3 )) = f (h 1 , h 2 , h 3 ), ∀ f ∈ C ∞ (sl∗ (2)). Recall that the deformation C ∞ (sl∗z (2)) of C ∞ (sl∗ (2)) is a Poisson–Hopf algebra with the new Poisson structure induced by the relations (10). Let us define the submanifold O =: {θ ∈ sl∗ (2) : v1 (θ) = 0} of sl∗ (2). Then, the Poisson structure on sl∗ (2) can be restricted to the space C ∞ (O). In turn, this enables us to restrict the Poisson–Hopf algebra structure in C ∞ (sl∗ (2)) to C ∞ (O). Within the latter space, the elements

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357

vz,1 := v1 , vz,3

vz,2 := shc(2zv1 )v2 , v2 c := shc(2zv1 ) 2 + , v1 4 shc(2zv1 )v1

(16)

are easily verified to satisfy the same commutation relations with respect to {·, ·} as the elements v1 , v2 , v3 in C ∞ (sl∗z (2)) with respect to {·, ·}z (10), i.e. {vz,1 , vz,2 } = − shc(2zvz,1 )vz,1 ,

{vz,1 , vz,3 } = −2vz,2 ,

{vz,2 , vz,3 } = − ch(2zvz,1 )vz,3 .

(17)

In particular, from (16) with z = 0 we find that 2{v0,1 , v0,2 }v0,2 = −2v0,2 , v0,1 2 2 v0,2 v0,2 c c {v0,2 , v0,3 } = − 2 {v0,2 , v0,1 } − 2 {v0,2 , v0,1 } = − − = −v0,3 . v0,1 4v0,1 v0,1 4v0,1

{v0,1 , v0,2 } = −v0,1 ,

{v0,1 , v0,3 } =

The functions vz,1 , vz,2 , vz,3 are not functionally independent, as they satisfy the constraint 2 = c/4. (18) shc(2zv1,z )v1,z v3,z − v2,z The existence of the functions vz,1 , vz,2 , vz,3 and the relation (18) with the Casimir of the deformed Poisson–Hopf algebra is by no means casual. Let us explain why vz,1 , vz,2 , vz,3 exist and how to obtain them easily. Around a generic point p ∈ sl∗ (2), there always exists an open U p containing p where both Poisson structures give a symplectic foliation by surfaces. Examples of symplectic leaves for {·, ·} and {·, ·}z are displayed in Fig. 1. The splitting theorem on Poisson manifolds [29] ensure that if U p is small enough, then there exist two different coordinate systems {x, y, C} and {x z , yz , C z } where the Poisson bivectors related to {·, ·} and {·, ·}z read Λ = ∂x ∧ ∂ y and Λz = ∂xz ∧ ∂ yz . Hence, C z and C are Casimir functions for Λz and Λ, respectively. Moreover, x z = x z (x, y, C), yz = yz (x, y, C), C z = C z (x, y, C). It follows from this that Φ : f (x z , yz , C z ) ∈ C z∞ (U p ) → f (x, y, C) ∈ C ∞ (U p ) is a Poisson algebra morphism. If {v1 , v2 , v3 } are the standard coordinates on sl∗ (2) and the relations (10) are satisfied, then vi = ξi (x z , yz , C z ) holds for certain functions ξ1 , ξ2 , ξ3 : R3 → R. Hence, the vˆ z,i = ξi (x, y, C) close the same commutation relations relative to {·, ·} as the vi do with respect to {·, ·}z . As C is a Casimir invariant, the functions vz,i := ξi (x, y, c), with a constant value of c, still close the same commutation relations among themselves as the vi . Moreover, the functions vz,i become functionally dependent. Indeed, C z = C z (v1 , v2 , v3 ) = C z (ξ1 (x z , yz , C z ), ξ2 (x z , yz , C z ), ξ3 (x z , yz , C z )).

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Fig. 1 Representatives of the submanifolds in sl∗ (2) given by the surfaces with constant value of the Casimir for the Poisson structure in sl∗ (2) (left) and its deformation (right). Such submanifolds are symplectic submanifolds where the Poisson bivectors Λ and Λz admit a canonical form

Hence, c = C z (ξ1 (x z , yz , c), ξ2 (x z , yz , c), ξ3 (x z , yz , c)) and we conclude that c = C z (vz,1 , vz,2 , vz,3 ). The previous argument allows us to recover the functions (16) in an algorithmic way. Actually, the functions x z , yz , C z and x, y, C can be easily chosen to be x z := v1 ,

yz := −

v2 , shc(2zv1 )v1

C z := shc(2zv1 )v1 v3 − v22 ,

as well as x = v1 ,

y = −v2 /v1 ,

C = v1 v3 − v22 .

Therefore, ξ2 (x z , yz , C z ) = −yz shc(2zx z )x z , ξ1 (x z , yz , C z ) = x z , C z + x z2 yz2 shc2 (2zx z ) ξ3 (x z , yz , C z ) = . shc(2zx z )x z By assuming that C z = c/4, replacing x z , yz by x = v1 , y = −v2 /v1 , respectively, and taking into account that vz,i := ξi (x, y, c), one retrieves (16). It is worth mentioning that due to the simple form of the Poisson bivectors in splitting form for three-dimensional Lie algebras, this method can be easily applied to such a type of Lie algebras. Next, the above relations enable us to construct a Poisson algebra morphism Dz : f (v1 , v2 , v3 ) ∈ C ∞ (sl∗z (2)) → D( f (v1,z , v2,z , v3,z )) ∈ C ∞ (M)

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for every value of z allowing us to pass the structure of the Poisson–Hopf algebra C ∞ (sl∗z (2)) to C ∞ (M). As a consequence, Dz (C z ) satisfies the relations {Dz (C z ), h z,i }ω = 0,

i = 1, 2, 3.

Using the symplectic structure on M and the functions h z,i written in terms of {h 1 , h 2 , h 3 }, we can easily obtain  the deformed vector fields Xz,i in terms of the 3 bi (t)Xz,i holds, it is straightforward to vervector fields Xi . Finally, as Xz,t = i=1 ify that the brackets Xz,i D(C z ) = {D(C z ), h z,i } = 0, imply that the function D(C z ) is a t-independent constant of the motion for each of the deformed LH system Xz,t . Consequently, deformations of LH-systems based on sl(2) can be treated simultaneously, starting from their classical LH counterpart. The final result is summarized in the following statement. Theorem 1 If φ : sl(2) → C ∞ (M) is a morphism of Lie algebras with respect to the Lie bracket in sl(2) and a Poisson bracket in C ∞ (M), then for each z ∈ R there exists a Poisson algebra morphism Dz : C ∞ (sl∗z (2)) → C ∞ (M) such that for a basis {v1 , v2 , v3 } satisfying the commutation relations (14) is given by Dz ( f (v1 , v2 , v3 ))  c φ2 (v2 ) + . = f φ(v1 ), shc(2zφ(v1 ))φ(v2 ), shc(2zφ(v1 )) φ(v1 ) 4 shc(2zφ(v1 ))φ(v1 ) Provided that h i := φ(vi ), the deformed Hamiltonian functions h z,i := Dz (vi ) adopt the form h z,1 = h 1 , h z,3

h z,2 = shc(2zh 1 )h 2 , h2 c = shc(2zh 1 ) 2 + , h1 4 shc(2zh 1 )h 1

which satisfy the commutation relations (17). The Hamiltonian vector fields Xz,i associated with h z,i through (5) turn out to be  h2  Xz,1 = X1 , ch(2zh 1 ) − shc(2zh 1 ) X1 + shc(2zh 1 )X2 , Xz,2 = h1 

2  h2  c ch(2zh 1 ) Xz,3 = X1 ch(2zh 1 ) − 2 shc(2zh 1 ) − 2 2 h 21 4h 1 shc (2zh 1 ) h2 +2 shc(2zh 1 )X2 , h1 and satisfy the following commutation relations coming from (7)

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[Xz,1 , Xz,2 ] = ch(2zh z,1 )Xz,1 ,

[Xz,1 , Xz,3 ] = 2 Xz,2 ,

[Xz,2 , Xz,3 ] = ch(2zh z,1 )Xz,3 + 4z 2 shc2 (2zh z,1 )h z,1 h z,3 Xz,1 . As a consequence, the deformed Poisson–Hopf system can be generically described in terms of the Vessiot–Guldberg Lie algebra corresponding to the nondeformed LH system as follows: Xz,t =

  h2  ch(2zh 1 ) − shc(2zh 1 ) X1 bi (t)Xz,i = b1 (t) + b2 (t) h1 i=1 

2  h  c ch(2zh 1 ) X1 +b3 (t) 22 ch(2zh 1 ) − 2 shc(2zh 1 ) − 2 2 h1 4h 1 shc (2zh 1 )  h2 X2 . + shc(2zh 1 ) b2 (t) + 2b3 (t) h1

3 

This unified approach to nonequivalent deformations of LH systems possessing a common underlying Lie algebra suggests the following definition. Definition 1 Let (C ∞ (M), {·, ·}) be a Poisson algebra. A Poisson–Hopf Lie system is a pair consisting of a Poisson–Hopf algebra C ∞ (g∗z ) and a z-parametrized family of Poisson algebra representations Dz : C ∞ (g∗z ) → C ∞ (M) with z ∈ R. Next, constants of the motion for Xz,t can be deduced by applying the coalgebra approach introduced in [7] in the way briefly described in Sect. 3. In the deformed case, we consider the Poisson algebra morphisms   Dz : C ∞ sl∗z (2) → C ∞ (M),     Dz(2) : C ∞ sl∗z (2) ⊗ C ∞ sl∗z (2) → C ∞ (M) ⊗ C ∞ (M), which by taking into account the coproduct (11) are defined by Dz (vi ) := h z,i (x1 ) ≡ h (1) z,i , i = 1, 2, 3, Dz(2) (Δz (v1 )) = h z,1 (x1 ) + h z,1 (x2 ) ≡ h (2) z,1 , Dz(2) (Δz (vk )) = h z,k (x1 )e2zh z,1 (x2 ) + e−2zh z,1 (x1 ) h z,k (x2 ) ≡ h (2) z,k , k = 2, 3, where xs (s = 1, 2) are global coordinates in M. We remark that, by construction, the functions h (2) z,i satisfy the same Poisson brackets (17). Then t-independent constants of motion are given by (see (8)) Fz ≡ Fz(1) := Dz (C z ),

Fz(2) := Dz(2) (Δz (C z )) ,

where C z is the deformed Casimir (12). Explicitly, they read

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 2   c (1) (1) (1) h Fz = shc 2zh (1) h − h = , z,1 z,1 z,3 z,2 4  2   (2) (2) (2) (2) (2) Fz = shc 2zh z,1 h z,1 h z,3 − h z,2 .

6 The Three Classes of sl(2) Lie–Hamilton Systems on the Plane and Their Deformation We now apply Theorem 1 to the three classes of LH systems in the plane with a Vessiot–Guldberg Lie algebra isomorphic to sl(2) according to the local classification performed in [8], which was based on the results formerly given in [19]. Thus we have the manifold M = R2 and the coordinates x = (x, y). According to [8, 10], these three classes are named P2 , I4 and I5 and they correspond to a positive, negative and zero value of the Casimir constant c, respectively. Recall that these are nondiffeomorphic, so that there does not exist any local t-independent change of variables mapping one into another. Table 1 summarizes the three cases, covering vector fields, Hamiltonian functions, symplectic structure and t-independent constants of motion. The particular LH systems which are diffeomorphic within each class are also mentioned [10]. Notice that for all of them the following commutation relations are satisfied for the vector fields and Hamiltonian functions (the latter with respect to the corresponding ω): [X1 , X3 ] = 2X2 , [X2 , X3 ] = X3 , [X1 , X2 ] = X1 , {h 1 , h 2 }ω = −h 1 , {h 1 , h 3 }ω = −2h 2 , {h 2 , h 3 }ω = −h 3 . By applying Theorem 1 with the results of Table 1 we obtain the corresponding deformations which are displayed in Table 2. It is straightforward to verify that the classical limit z → 0 in Table 2 recovers the corresponding starting LH systems and related structures of Table 1, in agreement with the relations (4) and (6).

7 A Method to Construct Lie–Hamilton Systems Section 5 showed that deformations of a LH system with a fixed LH algebra Hω  g can be obtained through a Poisson algebra C ∞ (g∗ ), a given deformation and a certain Poisson morphism D : C ∞ (g∗ ) → C ∞ (M). This section presents a simple method to obtain D from an arbitrary g∗ onto a symplectic manifold R2n . Theorem 2 Let g be a Lie algebra whose Kirillov–Kostant–Souriau Poisson bracket admits a symplectic foliation in g∗ with a 2n-dimensional S ⊂ g∗ . Then, there exists a LH algebra on the plane given by the image of the Lie algebra morphism.

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Table 1 The three classes of LH systems on the plane with underlying Vessiot–Guldberg Lie algebra isomorphic to sl(2). For each class, it is displayed, in this order, a basis of vector fields Xi , Hamiltonian functions h i , symplectic form ω, the constants of motion F and F (2) as well as the corresponding specific LH systems • Class P2 with c = 4 > 0 ∂ ∂x 1 h1 = − y

∂ ∂ ∂ ∂ +y X3 = (x 2 − y 2 ) + 2x y ∂x ∂y ∂x ∂y x x 2 + y2 dx ∧ dy h2 = − h3 = − ω= y y y2 2 + (y + y )2 − x ) (x 1 2 1 2 F =1 F (2) = y1 y2 – Complex Riccati equation – Ermakov system, Milne–Pinney and Kummer–Schwarz equations with c > 0 • Class I4 with c = −1 < 0 X1 =

X2 = x

∂ ∂ ∂ ∂ ∂ ∂ + X2 = x +y X3 = x 2 + y2 ∂x ∂y ∂x ∂y ∂x ∂y 1 x+y xy dx ∧ dy h1 = h2 = h3 = ω= x−y 2(x − y) x−y (x − y)2

X1 =

F =−

1 4

F (2) = −

(x2 − y1 )(x1 − y2 ) (x1 − y1 )(x2 − y2 )

– Split-complex Riccati equation – Ermakov system, Milne–Pinney and Kummer–Schwarz equations with c < 0 – Coupled Riccati equations • Class I5 with c = 0 X1 =

∂ ∂x

h1 = − F =0

X2 = x

1 2y 2

∂ y ∂ + ∂x 2 ∂y

h2 = −

F (2) =

x 2y 2

X3 = x 2

h3 = −

x2 2y 2

∂ ∂ + xy ∂x ∂y ω=

dx ∧ dy y3

(x1 − x2 )2 4y12 y22

– Dual-Study Riccati equation – Ermakov system, Milne–Pinney and Kummer–Schwarz equations with c = 0 – Harmonic oscillator – Planar diffusion Riccati system

  Φ : g → C ∞ R2n relative to the canonical Poisson bracket on the plane. Proof The Lie algebra g gives rise to a Poisson structure on g∗ through the Kostant– Kirillov–Souriau bracket {·, ·}. This induces a symplectic foliation on g∗ , whose leaves are symplectic manifolds relative to the restriction of the Poisson bracket. Such leaves are characterized by means of the Casimir functions of the Poisson bracket. By assumption, one of these leaves is 2n-dimensional. In such a case, the Darboux Theorem warrants that the Poisson bracket on each leave is locally symplectomorphic to the Poisson bracket of the canonical symplectic form on R2n  T ∗ Rn .

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Table 2 Poisson–Hopf deformations of the three classes of sl(2)-LH systems written in Table 1. The symplectic form ω is the same given in Table 1 and F ≡ Fz • Class P2 with c = 4 > 0 ∂ ∂ ∂ Xz,2 = x ch(2z/y) + y shc(2z/y) ∂x ∂x ∂y  ∂ y2 ∂ ch(2z/y) = x2 − + 2x y shc(2z/y) ∂x ∂y shc2 (2z/y)

Xz,1 = Xz,3

hz,1 = − Fz(2) =

1 y

hz,2 = −

x shc(2z/y) y

hz,3 = −

x 2 shc2 (2z/y) + y 2 y shc(2z/y)

(x1 − x2 )2 shc(2z/y1 ) shc(2z/y2 ) e2z/y1 e−2z/y2 y1 y2

(y1 + y2 )2 shc2 (2z/y1 + 2z/y2 ) 2z/y1 −2z/y2 e e y1 y2 shc(2z/y1 ) shc(2z/y2 ) • Class I4 with c = −1 < 0 +

∂ ∂ + ∂x ∂ y    2z ∂ 2z ∂ x+y ∂ x−y ∂ = ch + + shc − 2 x−y ∂x ∂y 2 x−y ∂x ∂y     1 2z 2z ∂ ∂ = ch (x + y)2 + (x − y)2 shc−2 + 4 x−y x−y ∂x ∂y   2z ∂ 1 ∂ + (x − y)2 shc − 2 x − y  ∂x ∂ y  

Xz,1 = Xz,2 Xz,3

2z 2z − (x − y)2 (x + y) shc x−y (x + y)2 shc2 x−y 1   hz,2 = hz,3 = 2z x−y 2(x − y) 4(x − y) shc x−y   2z 2z 2z (x1 − x2 + y1 − y2 )2 − 2z Fz(2) = shc e x1 −y1 e x2 −y2 shc 4(x1 − y1 )(x2 − y2 ) x1 − y1 x2 − y2 ⎡  ⎤ 2z 2z 2z − x 2z (x1 + x2 − y1 − y2 ) shc x1 −y1 + x2 −y2 x2 −y2 (x − y ) 1 −y1 (x 2 − y2 ) e e 1 1 ⎣ −    +  ⎦ 4(x1 − y1 )(x2 − y2 ) shc x 2z shc x 2z −y −y

hz,1 =

1

1

2

2

• Class I5 with c = 0

 ∂  ∂   y Xz,2 = x ch z/y 2 + shc z/y 2 ∂x 2 ∂y  ∂  ∂   2 2 2 Xz,3 = x ch z/y + x y shc z/y ∂x ∂y     1 x x2 2 hz,3 = − 2 shc z/y 2 hz,2 = − 2 shc z/y hz,1 = − 2 2y 2y 2y     2 (x1 − x2 )2 (2) 2 2 z/y1 −z/y22 Fz = shc z/y1 shc z/y2 e e 4y12 y22

Xz,1 =

∂ ∂x

In particular, there exists some Darboux coordinates mapping the Poisson bracket on such a leaf into the canonical symplectic bracket on T ∗ Rn . The corresponding change of variables into the canonical form in Darboux coordinates can be understood as

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a local diffeomorphism h : S → T ∗ Rn mapping the Poisson bracket δk on the leaf Sk into the canonical Poisson bracket on T ∗ Rn . Hence, h gives rise to a canonical Poisson algebra morphism φ : C ∞ (S) → C ∞ (T ∗ Rn ). As usual, a basis {v1 , . . . , vr } of g can be considered as a coordinate system on bracket, they span an g∗ . In view of the definition of the Kirillov–Kostant–Souriau  r -dimensional Lie algebra. In fact, if [vi , v j ] = rk=1 cikj vk for certain constants cikj ,  then {vi , v j } = rk=1 cikj vk . Since S is a symplectic submanifold, there is a local immersion ι : S → g∗ which is a Poisson manifold morphism. In consequence, ∗

∗

{ι vi , ι v j } =

r 

cikj ι∗ vk .

k=1

Hence, the functions ι∗ vi span a finite-dimensional Lie algebra of functions on S. Since S is 2n-dimensional, there exists a local diffeomorphism T ∗ Rn  R2n and   Φ : v ∈ g → φ ◦ ι∗ v ∈ C ∞ R2n is a Lie algebra morphism. Let us apply the above to explain the existence of three types of LH systems on the plane. We already know that the Lie algebra sl(2) gives rise to a Poisson algebra in C ∞ (g∗ ). In the standard basis v1 , v2 , v3 with commutation relations (14), the Casimir is (15). It turns out that the symplectic leaves of this Casimir are of three types: • A one-sheet hyperboloid when v1 v3 − v22 = k < 0. • A conical surface when v1 v3 − v22 = 0. • A two-sheet hyperboloid when v1 v3 − v22 = k > 0. In each of the three cases we have the Poisson bivector Λ = −v1

∂ ∂ ∂ ∂ ∂ ∂ ∧ − 2v2 ∧ − v3 ∧ . ∂v1 ∂v2 ∂v1 ∂v3 ∂v2 ∂v3

Then, we have a changes of variables passing from the above form into Darboux coordinates v¯1 = v1 , v¯2 = −v2 /v1 , C = v1 v3 − v22 . Then, v1 = v1 ,

v2 = −v¯1 v¯2 ,

v3 = (C + v¯12 v¯22 )/v¯1 .

On a symplectic leaf, the value of C is constant, say C = c/4, and the restrictions of the previous functions to the leaf read ι∗ v1 = v1 ,

ι∗ v2 = −v¯1 v¯2 ,

ι∗ v3 = c/(4v¯1 ) + v¯1 v¯22 .

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This can be viewed as a mapping Φ : sl(2) → C ∞ (R2 ) such that Φ(v1 ) = x,

Φ(v2 ) = −x y,

Φ(v3 ) = c/(4x) + x y 2 ,

which is obviously a Lie algebra morphism relative to the standard Poisson bracket in the plane. It is simple to proof that when c is positive, negative or zero, one obtains three different types of Lie algebras of functions and their associated vector fields span the Lie algebras P2 , I4 and I5 as enunciated in [8]. Observe that since Φ(v1 )Φ(v3 ) − Φ(v2 )2 = c/4, there exists no change of variables on R2 mapping one set of variables into another for different values of c. Hence, Theorem 2 ultimately explains the real origin of all the sl(2)-LH systems on the plane. It is known that su(2) admits a unique Casimir, up to a proportional constant, and the symplectic leaves induced in su∗ (2) are spheres. The application of the previous method originates a unique Lie algebra representation, which gives rise to the unique LH system on the plane related to so(3). All the remaining LH systems on the plane can be generated in a similar fashion. The deformations of such Lie algebras will generate all the possible deformations of LH systems on the plane.

8 Concluding Remarks It has been shown that Poisson–Hopf deformations of LH systems based on the simple Lie algebra sl(2) can be formulated simultaneously by means of a geometrical argument, hence providing a generic description for the deformed Hamiltonian functions and vectors fields, starting from the corresponding classical counterpart. This allows for a direct determination of the deformed Hamiltonian functions and vector fields, as well as their corresponding Poisson brackets and commutators, by mere insertion of the data corresponding to the non-deformed LH system. This procedure has been explicitly illustrated by obtaining the deformed results of Table 2 from the classical ones of Table 1 through the application of Theorem 1. Moreover we have explained a method to obtain (non-deformed) LH systems related to a LH algebra Hω by using the symplectic foliation in g∗ , where g is isomorphic to Hω , which has been stated in Theorem 2. This result could further be applied in order to obtain deformations of LH systems beyond sl(2). It is also left to accomplish the deformation of LH systems in other spaces of higher dimension. It seems that the techniques provided here are potentially sufficient to provide a solution to the above mentioned problems. These will be the subject of further work currently in progress. Acknowledgements A.B. and F.J.H. have been partially supported by Ministerio de Economía y Competitividad (MINECO, Spain) under grants MTM2013-43820-P and MTM2016-79639-P (AEI/FEDER, UE), and by Junta de Castilla y León (Spain) under grants BU278U14 and VA057U16. The research of R.C.S. was partially supported by grant MTM2016-79422-P (AEI/FEDER, EU). E.F.S. acknowledges a fellowship (grant CT45/15-CT46/15) supported by the Universidad Complutense de Madrid. J. de L. acknowledges funding from the Polish National Science Centre under grant HARMONIA 2016/22/M/ST1/00542.

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Two-Parameter Quantum General Linear Supergroups Huafeng Zhang

Abstract The universal R-matrix of two-parameter quantum general linear supergroups is computed explicitly based on the RTT realization of Faddeev–Reshetikhin– Takhtajan.

1 Introduction Fix r, s non-zero complex numbers whose ratio rs is not a root of unity. Let M, N be positive integers and g := gl(M, N ) be the general linear Lie superalgebra. The enveloping algebra U (g) as a Hopf superalgebra admits a two-parameter deformation Ur,s (g) which is neither commutative nor cocommutative. In this paper we compute its universal R-matrix, an invertible element in a completed tensor square 2 satisfying R ∈ Ur,s (g)⊗ Δcop (x) = RΔ(x)R−1 for x ∈ Ur,s (g), together with other favorable properties. In the non-graded case N = 0, Benkart– Witherspoon [2, 3] proved the existence of universal R-matrix, and derived from it a braided structure in the category of finite-dimensional representations; the exact formula of universal R-matrix was unknown. Recently it was shown [6] that Ur,s (gl(M)) can be recovered from a special R-matrix in the spirit of Faddeev–Reshetikhin– Takhtajan [5], the RTT realization. In this paper we define the two-parameter quantum supergroup Ur,s (g) by RTT realization, based on a suitable R-matrix on the vector superspace C M|N . Our main result, Eqs. (9)–(11), is a factorization formula for the universal R-matrix R in terms of RTT generators. (This idea was previously applied to the quantum affine superalgebra of gl(1, 1); see [11].)

H. Zhang (B) Laboratoire Paul Painlevé, Université de Lille, 59655 Villeneuve d’Ascq, France e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2018 V. Dobrev (ed.), Quantum Theory and Symmetries with Lie Theory and Its Applications in Physics Volume 1, Springer Proceedings in Mathematics & Statistics 263, https://doi.org/10.1007/978-981-13-2715-5_24

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Let us compare with earlier works on universal R-matrices: [8] for Uq (sl M ); [9] for Uq (gl(M, N )); [1] for (quantum doubles of) Nichols algebras, which are believed to include two-parameter quantum (super)groups. In these works a key step is to construct root vectors by Lusztig isomorphisms or q-brackets. In our approach the root vectors are already encoded in the definition of the algebra. There is another two-parameter quantum supergroup Uq1 ,q2 (sl(2, 1)) proposed by R.B. Zhang [10]: for q1 = q2 = q it is Uq (sl(2, 1)), while for q1 = q2 its comultiplication is not yet clear.

2 RTT Realization and Orthogonality We define Ur,s (gl(M, N )) following Faddeev–Reshetikhin–Takhtajan [5], and prove an orthogonality property for the associated Hopf pairing. Let V = C M|N be the vector superspace with basis (vi )1≤i≤M+N and parity: |vi | = |i| = 0 if i ≤ M and |vi | = |i| = 1 if i > M. Define the elementary matrices E i j ∈ EndV : vk → δ jk vi . Define the two-parameter Perk–Schultz matrix R ∈ End(V⊗2 ) by (r

 i≤M

+s



)E ii ⊗ E ii + (

i>M

 i> j

+r s



)E ii ⊗ E j j + (r − s)



i< j

(−1)|i| E ji ⊗ E i j .

i< j

(1) Recall the super tensor product. For V = V0 ⊕ V1 a vector superspace and p ∈ Z2 = {0, 1}, let (EndV ) p denote the set of linear endomorphisms g ∈ EndV such that g(Vq ) ⊆ V p+q for all q ∈ Z2 . This makes EndV a superalgebra. Let W be another vector superspace. For f ∈ EndW and g ∈ (EndV ) p the super tensor product f ⊗ g ∈ End(W ⊗ V ) is defined by f ⊗ g : w ⊗ v → (−1) pq f (w) ⊗ g(v) for w ∈ Wq and v ∈ V. If V, W are finite-dimensional, this identifies the tensor product superalgebra EndW ⊗ EndV with End(W ⊗ V ). Let us define three elements of End(V⊗3 ): R12 = R ⊗ 1,

R23 = 1 ⊗ R,

R13 = (cV,V ⊗ 1)R23 (cV,V ⊗ 1).

Here cV,V ∈ End(V⊗2 ) : vi ⊗ v j → (−1)|i|| j| v j ⊗ vi is the graded flip. Lemma 1 (Yang–Baxter Equation) R12 R13 R23 = R23 R13 R12 .  := cV,V R ∈ End(V⊗2 ). Define R 12 and R 23 ∈ End(V⊗3 ) in the obvious Proof Set R way. The Yang–Baxter equation is equivalent to the braid relation 23 R 23 R 12 = R 12 R 23 ∈ End(V⊗3 ). 12 R R

(2)

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 s, M, N ) denote R.  To indicate the dependence on r, s, M, N , we shall also let R(r,  is of even parity, the validity of Eq. (2) is independent of the Z2 -grading Since R  s, M, 0) = R(s,  r, 0, M). By [3, Proposition 5.5], on V = C M|N . Observe that R(r,  s, n, 0). So it holds for R(r,  s, M + N , 0) =: S Eq. (2) holds for the matrix R(r,  s, 0, M + N ) =: S . After ignoring the super structure, the vector spaces and R(r, C M+N |0 , C0|M+N and C M|N are the same. So we view S, S ∈ End(V⊗2 ). We prove that (2) applied to va ⊗ vb ⊗ vc is true if a, b, c ∈ {1, 2, . . . , M + N } ij are two-by-two distinct. Let Skl be the coefficient of vk ⊗ vl in the vector S(vi ⊗ v j ). ij Then Skl = 0 implies {i, j} = {k, l}, and for i = j we have  i ⊗ v j ) = Sii jj vi ⊗ v j + (−1)|i|| j| S ijij v j ⊗ vi . R(v

(3)

Apply S12 S23 S12 = S23 S12 S23 to va ⊗ vb ⊗ vc , and let Ci jk be the coefficient of vi ⊗ v j ⊗ vk . Then Ci jk = 0 only if i jk is a permutation of abc. Based on the relation  and S, one proves that (3) of R 12 R 23 R 12 (va ⊗ vb ⊗ vc ) = R



23 R 12 R 23 (va ⊗ vb ⊗ vc ) si jk Ci jk vi ⊗ v j ⊗ vk = R

i jk

where si jk = ± is a signature depending on the permutation i jk of abc. Based on the braid relations on S and S , one shows that (2) applied to va ⊗ vb ⊗ vc is true if abc is a permutation of ii j such that i ≤ M or i, j > M. We are reduced to the case M = N = 1 and to show that the braid relation applied  s, 1, 1). Consider to v1 ⊗ v2 ⊗ v2 , v2 ⊗ v1 ⊗ v2 , v2 ⊗ v2 ⊗ v1 holds. Set T := R(r, the second vector u := v2 ⊗ v1 ⊗ v2 as an example: T12 T23 T12 (u) = T12 T23 (v1 ⊗ v2 ⊗ v2 ) = −sT12 (v1 ⊗ v2 ⊗ v2 ) = −s(r − s)v1 ⊗ v2 ⊗ v2 − r s 2 v2 ⊗ v1 ⊗ v2 = T23 ((r − s)v1 ⊗ v2 ⊗ v2 − r s 2 v2 ⊗ v2 ⊗ v1 ) = T23 T12 ((r − s)v2 ⊗ v1 ⊗ v2 + r sv2 ⊗ v2 ⊗ v1 ) = T23 T12 T23 (u). The first and the third vectors can be checked in the same way.



Definition 1 U := superalgebra generated by the coefficients Ur,s (gl(M, N )) is the of matrices T = i≤ j t ji ⊗ E ji , S = i≤ j si j ⊗ E i j ∈ U ⊗ EndV of even parity (so that si j and t ji are of parity |i| + | j|) with relations R23 T12 T13 = T13 T12 R23 , R23 S12 S13 = S13 S12 R23 , R23 T12 S13 = S13 T12 R23 , and the sii , tii are invertible for 1 ≤ i ≤ M + N . U is a Hopf superalgebra with coproduct Δ and counit ε: Δ(si j ) =

 k

sik ⊗ sk j , Δ(t ji ) =

 k

t jk ⊗ tki , ε(si j ) = ε(t ji ) = δi j .

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The antipode S : U −→ U is an anti-automorphism of superalgebra defined by equations (S ⊗ Id)(S) = S −1 , (S ⊗ Id)(T ) = T −1 in U ⊗ EndV. Let U + (resp. U − ) be −1 −1 (resp. the t ji , tkk ) for i ≤ j; these are the subalgebra of U generated by the si j , skk M+N Zi ; sub-Hopf-superalgebras. Algebra U is graded by the weight lattice P := ⊕i=1 we set si j and t ji to be of weight ±(i −  j ) respectively. The weight grading restricts to subalgebras U ± . We interpret Definition 1 as a quantum double construction, following [12, Sect. 3.1.3]. There exists a unique bilinear form ϕ : U + × U − −→ C such that 

ϕ(si j , tkl )E kl ⊗ E i j = R ∈ (EndV)⊗2 ,

(4)

i jkl

and for a, a ∈ U + and b, b ∈ U − super homogeneous

ϕ(a, bb ) = ϕ2 (Δ(a), b ⊗ b ), ϕ(aa , b) = (−1)|a||a | ϕ2 (a ⊗ a, Δ(b)).

Here ϕ2 (a ⊗ a , b ⊗ b ) = (−1)|a ||b| ϕ(a, b)ϕ(a , b ). Such a form is called Hopf pairing. The quantum double U + ⊗ U − is isomorphic to U as Hopf superalgebras via the multiplication map. This implies that in U : ba = (−1)|a(1) ||b|+(|b(2) |+|b(3) |)|a(2) |+|a(3) ||b(3) | ϕ(a(1) , S(b(1) ))a(2) b(2) ϕ(a(3) , b(3) ).

(5)

Here a(1) ⊗ a(2) ⊗ a(3) = (Δ ⊗ Id)Δ(a) is the Sweedler notation. The Hopf pairing respects the weight grading: for x ∈ U + and y ∈ U − being of weight α and β respectively, ϕ(x, y) = 0 only if α + β = 0. Let τ : EndV −→ EndV be the transposition E i j −→ (−1)|i|+|i|| j| E ji . Lemma 1 affords a vector representation ρ of U on V: (ρ ⊗ 1)(S) = (τ ⊗ 1)(R), (ρ ⊗ 1)(T ) = r s(τ ⊗ 1)(cV,V R −1 cV,V ).

(6)

Lemma 2 Let 1 ≤ i, j, k ≤ M + N be such that j ≤ k. Then sii s jk = ϕ(sii , t j j )ϕ(sii , tkk )−1 s jk sii , tii s jk = ϕ(s j j , tii )−1 ϕ(skk , tii )s jk tii , tii tk j = ϕ(s j j , tii )ϕ(skk , tii )−1 tk j tii , sii tk j = ϕ(sii , t j j )−1 ϕ(sii , tkk )tk j sii . Proof For the second identity, by Eq. (5) tii s jk = ϕ(s j j , S(tii ))s jk tii ϕ(skk , tii ) = ϕ(s j j , tii )−1 ϕ(skk , tii )s jk tii . Here we have used the three-fold coproduct formula of tii , s jk , and the fact that ϕ(sab , tii ) = 0 if a < b. The fourth identity can be proved similarly. For the first identity, by comparing the coefficients of vi ⊗ v j in the identical vectors R23 S12 S13 (vi ⊗ vk ) = S13 S12 R23 (vi ⊗ vk ) ∈ U ⊗ V⊗2 we obtain

Two-Parameter Quantum General Linear Supergroups

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xsii s jk + ys ji sik = zs jk sii + ws ji sik for certain x, z ∈ {1, r s, r, s} and y, w ∈ {0, r − s, s − r }. Here we set s pq = 0 if p > q. We prove that sii s jk ∈ Cs jk sii . If not, then j < i < k, in which case y = (−1)|i| (r − s) = w and xsii s jk = zs jk sii , a contradiction. Now the first identity is obtained from the vector representation (6): ρ(sii ) =



ϕ(sii , tkk )E kk , ρ(s jk ) = ϕ(s jk , tk j )(−1)|k|+| j| E jk for j < k.

k



The third identity can be proved in the same way.  It follows that a vector x ∈ U is of weight i λi i if and only if sii xsii−1 = ϕ(sii , tii )λi (r s)

 j of U + . Similarly, the b ji form a subset Y and generate a subalgebra U < of U − . Let H + (resp. H − ) be the subalgebra of U > (resp. U < ) generated by the sii (resp. the tii ). X, Y are totally ordered sets with lexicographic ordering: ai j ≺ akl and b ji ≺ blk if either (i < k) or (i = k, j < l). Lemma 3 Fix 1 ≤ i < j ≤ M + N and p ∈ Z>0 . Let x1 , x2 , . . . , x p ∈ X and y1 , y2 , . . . , y p ∈ Y be such that xl ai j and yl b ji for all 1 ≤ l ≤ p. (A) We have ϕ(ai j , b ji ) = (−1)|i| (s −1 − r −1 ). (B) If ϕ(ai j , y1 y2 . . . y p ) = 0, then p = 1, y1 = b ji . (C) If ϕ(x1 x2 . . . x p , b ji ) = 0, then p = 1 and x1 = ai j . Proof Let us first prove an auxiliary result: (D) If a ∈ U > , b ∈ U < and x± ∈ H ± , then ϕ(x+ a, x− b) = ϕ(x+ , x− )ϕ(a, b). One may assume that x+ is a product of the sii±1 so that ϕ(x+ , 1) = 1 and Δ(x + ) = x ⊗ x + . By definition, Δ(x+ a) − x+ ⊗ x+ a is a sum of xi ⊗ yi where each xi is of non-zero weight and so ϕ(xi , x− ) = 0. By Eq. (5), +

ϕ(x+ a, x− b) = ϕ2 (x+ ⊗ x+ a, x− ⊗ b) = ϕ(x+ , x− )ϕ(x+ a, b). Δ(b) − b ⊗ 1 is a sum of xi

⊗ yi

where each yi

is of non-zero weight and so ϕ(x+ , yi

) = 0. This implies ϕ(x+ a, b) = ϕ2 (a ⊗ x+ , b ⊗ 1) = ϕ(a, b)ϕ(x+ , 1) = ϕ(a, b).

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H. Zhang

This proves (D). We are able to compute ϕ(ai j , b ji ): ϕ(si j , t ji ) = ϕ(sii ai j , b ji tii ) = ϕ(sii ai j , ϕ(sii , tii )−1 ϕ(s j j , tii )tii b ji ) = ϕ(s j j , tii )ϕ(ai j , b ji ) = r sϕ(ai j , b ji ). (A) follows from Eqs. (1) and (4). For (B), the first tensor factors in Δ(ai j ) − ai j ⊗ sii−1 s j j , being either 1 or x ∈ X with x ≺ ai j , are orthogonal to y1 b ji . So ϕ(ai j , y1 y2 . . . y p ) = ϕ(ai j , y1 )ϕ(sii−1 s j j , y2 . . . y p ). Now ϕ(ai j , y1 ) = 0 forces  p = 1 and y1 = b ji . (C) is proved similarly. Lemma 4 Fix 1 ≤ i < j ≤ M + N . Let x1 , x2 , . . . , x p ∈ {x ∈ X | x ai j } and y1 , y2 , . . . , yq ∈ {y ∈ Y | y b ji }. Let m, n ∈ Z≥0 . Then ϕ(x1 x2 . . . x p aimj , y1 y2 . . . yq bnji ) = ϕ2 (x1 x2 . . . x p ⊗ aimj , y1 y2 . . . yq ⊗ bnji ), ϕ(aimj , bnji ) = δmn (m)!τi j ϕ(ai j , b ji )m . Here (m)u :=

m

u k −1 k=1 u−1

and τi j := (−1)|i|+| j| (r s)−1 ϕ(sii , tii )ϕ(s j j , t j j ).

Proof By induction on max(m, n): the case m = n = 0 is trivial. Assume m > 0 (the case n > 0 can be treated similarly). The left hand side of the first formula becomes (we set θ1 := |ai j ||aim−1 x1 x2 . . . x p |) j n lhs1 = (−1)θ1 ϕ2 (ai j ⊗ x1 x2 . . . x p aim−1 j , Δ(y1 y2 . . . yq b ji )).

For 1 ≤ j ≤ q, there exists a unique z j ∈ H − such that ϕ(1, z j ) = 1 and each of the first tensor factor of Δ(y j ) − z j ⊗ y j is an element of Y strictly greater than y j multiplied by an element of H − . By Lemma 3 (B), the Δ(y j ) − z j ⊗ y j do not contribute to lhs1 . Similarly, for the n copies of Δ(b ji ), only one of them contributes b ji ⊗ 1 to lhs1 , and the rest of them z ⊗ b ji with z = t j j tii−1 . lhs1 = (−1)θ1 ϕ2 (ai j ⊗ x1 x2 . . . x p aim−1 j , q  k=1

(z k ⊗ yk )

n  (z ⊗ b ji )l−1 (b ji ⊗ 1)(z ⊗ b ji )n−l ). l=1

Note that ϕ(ai j , z 1 z 2 . . . z q b ji ) = ϕ(ai j , b ji ). Also, by Lemma 2, b ji z = zb ji ϕ(sii , t j j )−1 ϕ(s j j , tii )−1 ϕ(sii , tii )ϕ(s j j , t j j ) = zb ji τi j (−1)|b ji | . n−1 b ji ⊗ bn−1 Thus (z ⊗ b ji )l−1 (b ji ⊗ 1)(z ⊗ b ji )n−l = (−1)(n−1)|b ji | τin−l j z ji and n−1 lhs1 = (−1)θ1 +θ2 (n)τi j ϕ2 (ai j ⊗ x1 x2 . . . x p aim−1 j , b ji ⊗ y1 y2 . . . yq b ji )

= (n)τi j ϕ2 (x1 x2 . . . x p aim−1 ⊗ ai j , y1 y2 . . . yq bn−1 j ji ⊗ b ji ).

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Here θ2 = |b ji ||bn−1 ji y1 y2 . . . yq |. In the second identity observe that ϕ respects the parity: ϕ2 (a ⊗ b, c ⊗ d) = ϕ2 (b ⊗ a, d ⊗ c) × (−1)|a||b|+|c||d| . The rest is clear from the induction hypothesis.  Let  be the set of functions f : X −→ Z≥0 such that f (x) ≤ 1 if |x| = 1. Such an f induces, by abuse of language, another function f : Y −→ Z≥0 defined by f (b ji ) := f (ai j ). Set a f :=



x f (x) ∈ U > , b f :=

x∈X



y f (y) ∈ U < .

(8)

y∈Y

 Here means the product with descending order. If i ≤ M < j, then τi j = −1 and ϕ(aimj , bmji ) = 0 for m > 1, which is the reason for f (ai j ) ≤ 1.

Corollary 1 For f, g ∈  we have ϕ(a f , bg ) = 0 if and only if f = g. Moreover, the a f and the b f form bases of U > and U < respectively. Proof The first statement comes from Lemmas 3–4; notably the a f (resp. the b f ) are linearly independent. For the second statement, consider U > for example. A slight modification of the arguments in the proof of [7, Lemma 2.1] by using R23 S12 S13 = S13 S12 R23 shows that U > is spanned by ordered products of the ai j . It remains to prove si2j = 0 (and so ai2j = 0) if si j is odd; this comes from a comparison of coefficients of vi ⊗ vi in the equality R23 S12 S13 (v j ⊗ v j ) = S13 S12 R23 (v j ⊗ v j ) ∈ U ⊗ V⊗2 . 

3 Universal R-Matrix In this section we compute the universal R-matrix of Ur,s . For this purpose, we first work with a topological version of quantum supergroups and view r, s as formal variables: r = e ∈ C[[, ℘]], s = e℘ ∈ C[[, ℘]]. Step 1. Extend U ± , U to topological Hopf superalgebras over C[[, ℘]] based on the ∗ weight grading : first add commutative primitive elements  (i )1≤i≤M+N of even parity such that [i∗ , x] = λi x for x ∈ U ± , U of weight λ = i λi i ∈ P; then identify (for the indexes 1 ≤ i, j ≤ M + N ) sii = e

(+℘)

 j M),

tii = e

(+℘)

 j M).

± Denote by U,℘ , U,℘ the resulting topological Hopf superalgebras. Set

374

H. Zhang ± U,℘

 Hi := ( + ℘)



 ∗j

+

i∗

×

j M).

+ − × U,℘ −→ C((, ℘)) by ϕ(i∗ , H j ) = δi j . Extend ϕ to a Hopf pairing ϕ : U,℘ Observe that ϕ(sii , t j j ) = ϕ(sii , t j j ), which shows in turn that ϕ exists uniquely. The multiplication map induces a surjective morphism of topological Hopf superalgebras + − ⊗ U,℘ to U,℘ with kernel generated by the i∗ ⊗ from the quantum double U,℘ ∗ 1 − 1 ⊗ i . ± generated by the i∗ . Then Step 2. Let U 0 be the topological subalgebra of U,℘ + − 0 > 0 < U,℘ = U U and U,℘ = U U . Corollary 1 still holds true. We obtain orthonormal bases of ϕ and the universal R-matrix of U,℘ :



 af ⊗ bf = R := R R , R = (−1) Ri j , ϕ(a f , b f ) i< j f ∈  ∗  ∗ ∗  ∗ ∗  ∗ ∗ R0 = ei ⊗Hi = s i ⊗i × r  j ⊗ j × (r s)k ⊗l , 0

+

i

+

|a f |

i≤M

j>M

⎧∞  ainj ⊗bnji ⎪ ⎪ if (i < j ≤ M), ⎪ (n)r! s −1 (s −1 −r −1 )n ⎪ ⎨n=0 ∞ ainj ⊗bnji Ri j =  if (M < i < j), ⎪ (n)!sr −1 (r −1 −s −1 )n ⎪ ⎪ n=0 ⎪ ⎩ a ⊗b ji 1 − s −1i j −r −1 if (i ≤ M < j).

(9) (10)

ll

λk μl

the non-graded case R0V,W is exactly the operator s × f V,W in [2, Sect. 4].

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  for v ∈ Vλ and w ∈ Wμ where  λ = i λi i and μ = i μi i . Next, for f ∈ , the weight of a f v ∈ V is λ + i< j f (ai j )(i −  j ). By condition (iii), a f v = 0 for all but finitely many f . So R+ V,W ∈ End(V ⊗ W ) is indeed a finite sum. Let RV,W := R0V,W R+ . From the quantum double construction of U we obtain: Category O V,W together with the RV,W is braided. Consider the vector representation (6). From the proof of Lemma 2 we see that vi is of weight i and V is in category O. Similar to [9, Sect. 10.7]: RV,V = cV,V Rs−1 −1 ,r −1 cV,V . Following [4, 6], define Drinfeld–Jimbo generators for 1 ≤ i < M + N : ei := sii−1 si,i+1 ,

f i := ti+1,i tii−1 , ki := sii−1 si+1,i+1 , li := ti+1,i+1 tii−1 .

The following relations are proved in the same way as [6]: Δ(ei ) = 1 ⊗ ei + ei ⊗ ki , Δ( f j ) = l j ⊗ f j + f j ⊗ 1, ei2 ei+1 − (r + s)ei ei+1 ei + r sei+1 ei2 = 0 if (1 ≤ i < M + N − 1, i = M), ei−1 ei2 − (r + s)ei ei−1 ei + r sei2 ei−1 = 0 if (1 < i < M + N , i = M), r s f i2 f i+1 − (r + s) f i f i+1 f i + f i+1 f i2 = 0 if (1 ≤ i < M + N − 1, i = M), r s f i−1 f i2 − (r + s) f i f i−1 f i + f i2 f i−1 = 0 if (1 < i < M + N , i = M), ei e j = e j ei ,

f i f j = f j f i , e2M = f M2 = 0 if (|i − j| > 1),

[ei , f j ] = δi j (−1)|i| (s −1 − r −1 )(ki − li ), e M−1 e M e M+1 e M + r se M+1 e M e M−1 e M + e M e M−1 e M e M+1 + r se M e M+1 e M e M−1 − (r + s)e M e M−1 e M+2 e M = 0 if M, N > 1, r s f M−1 f M f M+1 f M + f M+1 f M f M−1 f M + r s f M f M−1 f M f M+1 + f M f M+1 f M f M−1 − (r + s) f M f M−1 f M+2 f M = 0 if M, N > 1. Let R := cV,V R −1 cV,V . Then r s R in the non-graded case is the R-matrix [6, Definition 3.1] defining the two-parameter quantum group. The generators li+j and l −ji therein correspond to our si j and t ji .

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Noncommutative Geometry and An Index Theory of Infinite-Dimensional Manifolds Doman Takata

Abstract The Atiyah–Singer index theorem is one of the monumental works in geometry and topology. My dream is to construct an infinite-dimensional version of it. Although this project is very hard, I constructed several core objects for an analytic index theory for infinite-dimensional manifolds. The problem is the following: For an infinite-dimensional “Spin c ”-manifold M on which a loop group of a circle acts, construct a C ∗ -algebra A which carries some information of M, a Hilbert space which can be regarded as an “L 2 -space consisting of sections of the Spinor bundle”, and an operator D which can be regarded as a Dirac operator on M, and define a spectral triple coming from them for A. The core idea for the construction comes from representation theory of loop groups and Higson–Kasparov–Trout’s algebra [4].

1 Main Results We will study infinite-dimensional manifolds which we call “proper L T -spaces”. Let T , Trot and U (1) be circle groups, and let L T be the loop group C ∞ (Trot , T ) which is an infinite-dimensional Fréchet Lie group with respect to the pointwise multiplication. It acts on its dual Lie algebra Lt∗ via the gauge action: l.A := A − l l −1 . A proper L T -space is an infinite-dimensional manifold which can be “compared” with Lt∗ . Definition 1 Let M be a Banach manifold equipped with an isometric smooth action of L T . M is called a proper L T -space, if M admits a proper equivariant smooth map Φ : M → Lt∗ . Example 1 The flag manifold of L T , M := L T /T = Ω T , is a proper L T -space with respect to the left translation and the canonical embedding Φ(l) := −l  l −1 . It will appear in the Borel–Weil theory of L T . D. Takata (B) Department of mathematics, Kyoto University, Kitashirakara, Oiwake-cho, Sakyo-ku, Kyoto-shi 606-8502, Japan e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2018 V. Dobrev (ed.), Quantum Theory and Symmetries with Lie Theory and Its Applications in Physics Volume 1, Springer Proceedings in Mathematics & Statistics 263, https://doi.org/10.1007/978-981-13-2715-5_25

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Example 2 More generally, a Hamiltonian L T -space is a proper L T -space. For details of Hamiltonian loop group spaces, see [1] for example. For an index problem for Hamiltonian LG-spaces for general G, see [7]. To study an index problem, we need a “Spinor bundle” and a “Dirac operator”. Let us explain why we need such object. A Dirac operator is a differential operator which is a “square root of a Laplacian”. To find  such an operator D on a Euclidean space Rn , we suppose that D is of the form γ i ∂x∂ i . Then, its square is computed  2 as γ i γ j ∂x∂i ∂x j . If it is a Laplacian, the equations γ i γ j = −δi, j must be satisfied, where we use the convention of geometer’s Laplacian. Although the equation has no scalar-valued solutions unless n = 1, there are solutions if we deal with matrices. A minimal rank vector space on which matrices of a solution to the above equation acts, is called a Spinor. We need a Spinor bundle associated to the tangent bundle. In order to admit such a bundle, the tangent bundle must satisfy a certain topological condition. For example, a tangent bundle equipped with a complex structure admits a Spinor bundle. In order to explain the condition, we need to see how simple proper L T -spaces are. In fact, they are product of a certain finite-dimensional manifold and an infinite-dimensional Lie group. Proposition 1 L T has a canonical decomposition L T = (T × Z) × U . U acts on Lt∗ freely, and the U -bundle U → Lt∗ → Lt∗ /U ∼ = t is trivial. By the pullback of the  := Φ −1 (t) can be trivialized. trivialization t → Lt∗ , the U -bundle U → M → M  × U. From now on, we always identify M with M For proper L T -spaces, we impose the following.  has a T × Z-equivariant Spinor bundle S M . Assumption 1 M For some purposes, it is natural to twist a Spinor bundle with a certain complex line bundle. For example, the Bott’s quantization of a Hamiltonian G-space is defined by the equivariant index of a Spinor bundle twisted by a line bundle whose curvature is the symplectic form (called the pre-quantum line bundle, see [7]). We deal with the line bundle of the following type. Assumption 2 L T has an admissible and positive definite U (1)-central extension L T τ of L T in the sense of [3]. M has a τ -twisted L T -equivariant complex line bundle L, that is, an L T τ -equivariant line bundle such that U (1) ⊆ L T τ acts as scalar multiplication. We can talk about the key observation. U can be written as the exponential of the following Lie subalgebra 

   f (θ)dθ = 0 . f : Trot → t 

Trot acts there, and the action induces a complex structure on U . Using it, we can define a Spinor space SU by the complex exterior algebra. Hence we can define a global Spinor bundle S := S M  SU .

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In order to explain the main result, we need several noncommutative geometrical objects. Noncommutative geometry is a study of invariants of C ∗ -algebras. A “classical” space corresponds to a commutative C ∗ -algebra as observed by Gelfand. Therefore, a noncommutative C ∗ -algebra corresponds to a “quantum” space [2]. A crossed product is one of such examples: For a group Γ equipped with an invariant measure, and a C ∗ -algebra A equipped with a ∗-action of Γ , we can define a new C ∗ -algebra by the twisted convolution on Cc (Γ, A). It plays a role of a “modified quotient space”. A quantum analogue of a Dirac operator on a C ∗ -algebra is a “spectral triple” defined as follows: For a C ∗ -algebra A, a triple (A, H, D) is called a spectral triple, if A is a “smooth subalgebra” of A, H is a ∗-representation space of A, and D is a closed operator on H such that A preserves the domain of D, [a, D] is bounded for any a ∈ A, and (1 + D2 )−1 a is compact for any a ∈ A. The following is the main theorem of this document. Theorem 1 We can define a Hilbert space H :=“L 2 (M, L ⊗ S)” without a measure on M, a “Dirac operator D on M”, a C ∗ -algebra A :=“L T τ C0 (M)” without a measure on L T or C0 (M) itself. Moreover, A has a smooth subalgebra A, and the triple (A, H, D) is a spectral triple for A.  × U , we can divide the problem into the M-part  Using the identification M = M and the U -part. Since the bundle SU → U is trivial, in order to construct the Hilbert space consisting of “L 2 -sections” of the bundle L ⊗ S, it suffices to construct a Hilbert space consisting of “L 2 -sections” of a line bundle L|U → U . Let us add several properties of our spectral triple. Proposition 2 (Proposition 6.11 in [8]) While a spectral triple coming from a finitedimensional manifold is finitely-summable, our triple is not finitely-summable. Thus we should understand that our triple comes from an infinite-dimensional object. Proposition 3 (Theorem 7.1 in [8]) Any irreducible positive energy representation can be written as the analytic index of the τ -twisted L T -equivariant Spinor bundle twisted by a certain line bundle over Ω T = L T /T . This result is completely parallel to the Borel–Weil theory.

2 Constructions The subgroup U has a good base cos θ sin θ cos 2θ sin 2θ , √ ,..., √ , √ , √ π π 2π 2π thanks to the Fourier series theory. It defines finite-dimensional approximations  U N := SpanR

 cos θ sin θ cos 2θ sin 2θ cos N θ sin N θ . , √ , √ ,..., √ √ , √ , √ π π Nπ Nπ 2π 2π

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The strategy is the following: Construct objects on U N , associate them, and take a limit. Firstly, we study the L 2 -side. For this purpose, we quote a representation theoretical fact: U N has the unique irreducible representation L 2 (R N ) at level 1 (the Stone–von Neumann theorem, see [6] for details). Moreover, the following isomorphism holds. Proposition 4 (Sect. 4.2 in [8]) L 2 (U N , L|U N ) ∼ = L 2 (R N ) ⊗ L 2 (R N )∗ . Thanks to the structure of the U N -representation, L 2 (R N ) has a specific unit vector 1b , where b comes from “boson”. By Ω = 1b ⊗ 1∗b , we define data on the orthogonal complement of U N in U N +1 . With this method, we can define a sequence of Hilbert spaces: L 2 (U1 , L|U1 ) → L 2 (U2 , L|U2 ) → · · · → L 2 (U N , L|U N ) → · · · . The promised Hilbert space “L 2 (U, L|U )” is defined by the limit of the above sequence. Then, with a technique of [3], we obtain the following “K K -cycle”. Theorem 2 (Sect. 4.3 in [8]) An operator ∂
on H := L 2 (U, L|U ) ⊗ SU can be defined as a strongly converging infinite sum on a dense subspace L 2 (U, L|U )fin ⊗alg SU,fin . Moreover, ∂
has a well-defined analytic index as a formal difference of representations of L T . Our construction is not just a pair of Hilbert space and an operator, but a spectral triple. Before introducing the “twisted crossed product U τ C0 (U )”, we briefly explain why a twisted crossed product is required, for a classical case. Suppose that a finite-dimensional Lie group G acts on a finite-dimensional manifold X . The space of continuous sections of a G-equivariant vector bundle turns out to be a module over the ordinary crossed product G  C0 (X ). However, we need a twisted version of it, because we want to study a τ -twisted L T -equivariant Spinor bundle. Needless to say, we need a measure on G and the algebra C0 (X ) itself, to construct the twisted crossed product. Nevertheless, we can prove the following, without a measure on L T or C0 (M) itself, with a similar method of the above theorem. Theorem 3 (Theorems 6.1 and 6.10 in [8]) We can define a C ∗ -algebra A which can be regarded as “L T τ C0 (M)”. Moreover, H admits an A-action, the pair (H, ∂
) defines a smooth subalgebra A, and the triple (A, H, D) turns out to be a spectral triple.

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3 After the Conference We found that the result is not satisfying, after the conference. The following should be the model of our theorem. Theorem 4 ([5]) For a second countable, locally compact and Hausdorff group Γ and a complete Spin c -manifold X equipped with an isometric, proper and cocompact action of Γ , we suppose that a Spinor bundle S over X and a Γ -equivariant Dirac operator D are given. Then, although D itself is not a Fredholm operator, it has an ∗ (Γ )). Moreover, it has a “K K -theoretical description” analytic index in K K (C, Cred called the assembly map. We have made a progress on the above problem from the viewpoint of K K -theory in [9]. We have constructed an analytic index not as a difference of representations but as an element of a K K -group, and a part of the assembly map. Then we prove the coincidence between the new analytic index and the image of the assembly map. Acknowledgements I sincerely thank to the organizers for the hospitality during the conference. I am supported by JSPS KAKENHI Grant Number 16J02214 and the Kyoto Top Global University Project (KTGU).

References 1. A. Alekseev, A. Malkin and E. Meinrenken, J. Differential Geom. 48 (1998), no. 3, 445–495. 2. A. Connes, Noncommutative geometry, Academic Press, Inc., San Diego, CA, 1994., http:// www.alainconnes.org/docs/book94bigpdf.pdf. 3. D. Freed, M. Hopkins and C. Teleman, Journal of AMS. 26 (2013) no. 3, 595–644. 4. N. Higson, G. Kasparov and J. Trout, Advances in Mathematics, 135 (1998), no. 1, 1–40. 5. G. Kasparov, J. Noncommut. Geom. 10 (2016), no. 4, 1303–1378. 6. A. Kirillov, Lectures on the orbit methods, Graduate Studies in Mathematics, 64. American Mathematical Society, Providence, RI, 2004. 7. Y. Song, J. Geom. Phys. 106 (2016), 70–86. 8. D. Takata, arXiv, https://arxiv.org/abs/1701.06055, to appear in J. Noncommut. Geom. 9. D. Takata, arXiv, https://arxiv.org/abs/1709.06205.

Part VI

Various Mathematical Results

A Stable Version of Terao Conjecture Cristian Anghel

Abstract The aim of this note is to explain a famous conjecture in hyperplane arrangements theory, Terao’s conjecture, in its natural algebraic-geometric framework of vector bundles on projective spaces. Also, using the explicit relation with the extendability property of bundles, we introduce a closely related conjecture using the notion of infinitely stably extendability of vector bundles on Pn , considered and characterized by I. Coanda in 2009.

1 Introduction Terao conjecture (1981) introduced in [17], asserts that the freeness of a hyperplane arrangement depends only of its combinatorics. The freeness is equivalent with the fact that the associated bundle splits completely as direct sum of line bundles. This last property, thanks to Horrocks criterion [10, 12], is equivalent with the vanishing of certain cohomology modules of the bundle in question. Also, using the famous Barth-Van de Ven-Sato-Tyurin result [2, 16, 18], the freeness of an arrangement is equivalent with the infinitely extendability of the associated bundle. In the first part of the paper, we shall describe the above circle of ideas. The second part will be devoted to the notion of stably extendability, introduced by Horrocks (1966) in [11], and its connection with the above results, thanks to a theorem of Coanda (2009) from [4], which gives a characterization of infinitely stably extendable vector bundles in terms of the vanishing of some cohomology modules of the bundle. Finally, we shall formulate a problem with the same flavour as Terao conjecture, using the Coanda notion of infinitely stably extendability.

C. Anghel (B) Institute of Mathematics of the Romanian Academy, Calea Grivitei nr. 21, Bucuresti, Romania e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2018 V. Dobrev (ed.), Quantum Theory and Symmetries with Lie Theory and Its Applications in Physics Volume 1, Springer Proceedings in Mathematics & Statistics 263, https://doi.org/10.1007/978-981-13-2715-5_26

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2 Arrangements and Their Lattices An arrangement in the complex projective space Pn (C) is a finite collection of hyperplanes A = {H1 , . . . , Hk }. For a fixed arrangement A, its intersection lattice LA is the poset with elements the finite intersections between the Hi s, ordered by reverse inclusion: for L1 , L2 ∈ LA , L1 ≤ L2 iff L1 ⊇ L2 . Using the interaction lattice, we have a first equivalence relation for arrangements: Definition 2.1 Two arrangements A1 , A2 have the same combinatorics if their lattices LA1 , LA2 are isomorphic. For example, if A1 is defined by three concurrent lines in P2 (C) and A2 by three lines without a common point, then A1 and A2 have different combinatorics. A fundamental question in the theory of hyperplanes arrangements is to find which properties of the arrangement depends only on its lattice i.e. only of its combinatorics. For example, concerning the cohomology algebra of the complement we have the following celebrated result due to Arnold, Brieskorn, Orlik and Solomon [1, 3, 13]: Theorem 2.2 The cohomology ring H ∗ (Pn (C) \ {H1 , . . . , Hk } is combinatorially determined by

k 

Hi ) of the complement of A =

i=1 LA .

Also, a negative result in this direction, concern the homotopy type of the complek  ment: for example π1 (Pn (C) \ Hi ) is not combinatorially determined. In fact, i=1

Rybnikov (1998) in [14] constructed two arrangements in P2 (C) with the same combinatorics but different π1 for the complements.

3 Bundles Associated with Arrangements Apart the lattice and homological or homotopical invariants associated to an arrangement A another interesting object is the sheaf TA of vector fields with logarithmic poles along A. It was introduced for the first time by Saito and Deligne in the ’80s [15] and used in the context of hyperplane arrangements by Dolgachev, Kapranov [7, 8], Terao and others. Its construction goes as follows: denote by fi an homogenous equation of the hyperplane Hi and by f the product ki=1 fi . Then TA is defined as the kernel of the map → OPn (k − 1), OP⊕(n+1) n defined by the partial derivatives of f : (∂x0 f , . . . , ∂xn f ). The sheaf TA will be the principal object of study in the sequel. In general it is a rank-n sheaf on Pn , but we

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will be interested mainly in the case where it is locally free. For example, due to a result of Dolgachev this is the case for the normal crossing arrangements. An important problem concerning TA , in the case when it is locally free, is its splitability. Definition 3.1 A vector bundle on Pn is splittable if it is direct sum of line bundles. With the definition above, an arrangement A is called free if the associated sheaf TA is splittable. Example 3.2 A first class of free arrangements is the following [8]: normal crossing arrangements with at most n + 1 hyperplanes in Pn . Of course if A is free, then TA is locally free and consequently, concerning the freeness one can consider only arrangements with locally free TA . In the above terms, one can enounce the Terao’s conjecture. Conjecture 3.3 The freeness of an arrangement is combinatorially determined. Namely, for two arrangements A1 , A2 with isomorphic lattices, if A1 is free then A2 is also free. Despite the simplicity of the statement, it was proved only in very few particular cases. For example, in [9] it proved that Terao conjecture holds true for arrangements in P2 with at most 12 lines.

4 Freeness Versus Infinitely Extendability As long as the freeness of A means the splittability of TA , a good starting point in the study of free arrangements could be a criterion which ensure the splittability of a vector bundle on Pn . In this direction, the fundamental result is Horrocks theorem. Let F a vector bundle on Pn . For any 1 ≤ i ≤ n − 1 we denote by H∗i (F) the cohomology module  H i (Pn , F(k)), k∈Z

where, as usual F(k) = F ⊗ OPn (k). With the above notations we have the following criterion of Horrocks. Theorem 4.1 A vector bundle F on Pn splits completely as direct sum of line bundles iff for any 1 ≤ i ≤ n − 1 the cohomology module H∗i (F) is zero. Consequently, the freeness of an arrangement A is equivalent with the vanishing of all intermediate cohomology modules H∗i (TA ) of its bundle of logarithmic vector fields. A second viewpoint concerning the splittability of bundles on Pn is connected with the following definition.

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Definition 4.2 A vector bundle F on Pn is infinitely extendable if for any m ≥ n there exist a bundle Fm on Pm such that Fm|Pn  F. Example 4.3 All the line bundles on Pn , and their direct sums, are infinitely extendable. The following result, due to Barth, Van de Ven, Sato and Tyurin, asserts that in fact the infinitely extendability is equivalent with the complete splittability of the bundle in question. Babylonian tower theorem 4.4 For a vector bundle F on Pn , the following are equivalent: 1. F splits completely as direct sum of line bundles, 2. F is infinitely extendable. As consequence, one obtain another characterization of the freeness of an arrangement A, namely the infinitely extendability of TA . The conclusion of the above results is that the freeness of an arrangement, which is the main property in the statement of the Terao conjecture, admits at least two equivalent formulations. vanishing of all the intermediate cohomology of TA

⇔

freeness ⇔

infinitely extendability of TA .

The main question we will discus in the sequel is the following. Question 4.5 Is there a weaker (than freeness) property with a similar cohomological and geometrical flavour which could be used in a modified form of Terao conjecture? The answer is yes and is connected with the notion of infinitely stably extendability, characterized by Coanda in 2009.

5 The Stably Freeness of Arrangements In analogy with the previous notion of extendability, Horrocks (1966) introduced the following weaker concept: Definition 5.1 A vector bundle F on Pn is stably extendable on a larger space Pm if there exists a bundle Fm on Pm whose restriction to Pn is the direct sum between F and certain line bundles. A first remark is that an extendable bundle is obviously stably extendable, but the converse is not true. For example the tangent bundle of Pn , TPn is stably extendable

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but not extendable. Also, one should note that the above notion is connected with the complete splittability of a bundle by the following result of Horrocks: Theorem 5.2 If the bundle F on Pn extends stably to P2n−3 and the cohomology modules H∗1 (F), H∗n−1 (F) vanishes, then F splits completely on Pn . The result above, show that the condition of stably extendability of a bundle F, has a subtle connection with the property of complete splittability and is also a good motivation for the following definition introduced (and as we shall see, characterized) by Coanda in 2009. Definition 5.3 A vector bundle F on Pn is infinitely stably extendable, if for any m ≥ n it extends stably on Pm . Example 5.4 The tangent and the cotangent bundles of Pn , together with their twist are infinitely stably extendable. As in the case of stably extendability, the above property is strictly weaker than infinitely extendability, as long as again, the example of the tangent bundle of Pn shows that there are bundles infinitely stably extendable which are not splittable and therefore (using the babylonian tower theorem of Barth-Van de Ven-Sato-Tyurin) are not infinitely extendable. The main point concerning the above property is that, like the complete splittability and therefore -via the babylonian tower theorem- like the infinitely extendability, it admits an analogous cohomological characterization in terms of some intermediate cohomology modules. This one, was obtained by Coanda in 2009: Theorem 5.5 A vector bundle F on Pn is infinitely stably extendable iff for any 2 ≤ i ≤ n − 2 the intermediate cohomology module H∗i (F) vanishes. On should remark that in fact, the original theorem of Coanda, contains also a third characterization of the infinitely stably extendability, namely as the condition for F of being the cohomology of a free monad. However we do not use this fact for the moment. Also, one should note the following. Remark 5.6 The condition in the theorem is empty for n ≤ 3 but it can be proved that any bundle on P≤3 is infinitely stably extendable [5].

6 A “Stable” Terao Conjecture Inspired by the above result we consider de following: Definition 6.1 An arrangement A is stably free if its associated bundle of vector fields with logarithmic poles TA is infinitely stably extendable.

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Example 6.2 As a first class of non-free but stably free arrangements we can take n + 2 normal crossing hyperplanes in Pn [8]. Obviously, a free arrangement is stably free, the converse is not true and we have, as in the case of freeness, a similar characterization of stably freeness: vanishing of all the intermediate cohomology of TA in the range 2≤i ≤n−2

⇔ stably freeness =

infinitely stably extendability of TA .

Consequently, we introduce the following: Stable Terao Conjecture 6.3 The stably freeness of an arrangement is combinatorially determined. Namely, for two arrangements A1 , A2 with isomorphic lattices, if A1 is stably free then A2 is also stably free. A first remark is that the two conjectures are not comparable: no one implies the other. Also, it could be interesting to compare this notion of stably freeness which can be obviously be extended from arrangements to arbitrary union of hyper-surfaces to other weaker notions of freeness existing in literature. For convenience we mention only two: – the nearly free and almost free divisors introduced by Dimca and Sticlaru [6], – the quasi free divisors studied by Castro-Jimenez [19]. Another point to note is the following. Due to Horrocks and Coanda criteria, both conjectures can be expressed as the combinatorial invariance of the vanishing in a certain range of the intermediate cohomology modules of TA . From this viewpoint, one can ask the following question which already appeared in [20] in relation with the lattice cohomolgy introduced by Yuzvinsky in [21–23]: Question 6.4 For an arrangement A in Pn , and a fixed 1 ≤ i ≤ n − 1, can be the cohomology module H∗i (TA ) expressed/computed only in terms of the lattice LA of A? In terms of the above Question, one should note that using the final Remark from Sect. 5, the first nontrivial case for the Stable Terao conjecture is on P4 : in this case it asserts the combinatorial invariance of the vanishing of H∗2 (TA ).

References 1. V. I. Arnold, The cohomology ring of the group of dyed braids, Math. Notes 5 (1969), 138–140. 2. W. Barth, A. Van de Ven, A decomposability criterion for algebraic 2-bundles on projective spaces, Invent. Math. 25, 91–106 (1974). 3. E. Brieskorn, Sur les groupes de tresses, in: Seminaire Bourbaki, 1971/72, Lect. Notes in Math. 317, Springer-Verlag, 1973, pp. 21–44.

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4. I. Coanda, Infinitely stably extendable vector bundles on projective spaces Arch. Math. (94) (2010), 539–545. 5. I. Coanda, Private communication. 6. A. Dimca, G. Sticlaru, Nearly free divisors and rational cuspidal curves, arXiv:1505.00666. 7. I. Dolgachev, Logarithmic sheaves attached to arrangements of hyperplanes, J. Math. Kyoto Univ. (JMKYAZ) 47-1 (2007) 35–64. 8. I. Dolgachev, M. Kapranov, Arrangements of hyperplanes and vector bundles on Pn , Duke Math. J. Volume 71, Number 3 (1993), 633–664. 9. D. Faenzi, J. Valles, Logarithmic bundles and line arrangements, an approach via the standard construction , J. Lond. Math. Soc. (2) 90 (2014), no. 3, 675–694. 10. G. Horrocks, Vector Bundles on the Punctured Spectrum of a Local Ring, Proc. London Math. Soc. (1964),(4), 689–713. 11. G. Horrocks, On extending vector bundles over projective space, Quart. J. Math. Oxford (2) 17, 14–18 (1966). 12. G. Horrocks, Construction of bundles on Pn , In: A. Douady and J.-L. Verdier (eds.), Les equations de Yang-Mills, Seminaire E. N. S. (1977–1978), Asterisque 71-72, Soc. Math. de France, 197–203 (1980). 13. P. Orlik, L. Solomon, Combinatorics and topology of complements of hyperplanes, Invent. Math. 56 (1980), 167–189. 14. G. Rybnikov, On the fundamental group of the complement of a complex hyperplane arrangement, arXiv:math/9805056, Functional Analysis and its Applications Volume 45 (2011), Number 2, 137–148. 15. K. Saito, Theory of logarithmic differential forms and logarithmic vector fields, J. Fac. Sci. Univ. Tokyo, Sci. IA, 27, (1980), 265–291. 16. E. Sato, The decomposability of an infinitely extendable vector bundle on the projective space, II, In: International Symposium on Algebraic Geometry, Kyoto University, Kinokuniya Book Store, Tokyo, 663–672 (1978). 17. H. Terao, The exponents of a free hypersurface, Singularities, Part 2 (Arcata, Calif., 1981), 561–566, Proc. Sympos. Pure Math., 40, Amer. Math. Soc., Providence, RI, 1983. 18. A. N. Tyurin, Finite dimensional vector spaces over infinite varieties, Math. USSR Izv. 10, 1187–1204 (1976). 19. J.M. Ucha, F.J. Castro-Jimenez, Interesting examples of Quasi Free divisors, preprint. 20. M. Yoshinaga, Freeness of hyperplane arrangements and related topics, Annales de la Faculte des Sciences de Toulouse, (6) 23 no. 2 (2014), 483–512. 21. S. Yuzvinsky, Cohomology of local sheaves on arrangement lattices. Proc. Amer. Math. Soc. 112 (1991), no. 4, 1207–1217. 22. S. Yuzvinsky, The first two obstructions to the freeness of arrangements. Trans. Amer. Math. Soc. 335 (1993), no. 1, 231–244. 23. S. Yuzvinsky, Free and locally free arrangements with a given intersection lattice. Proc. Amer. Math. Soc. 118 (1993), no. 3, 745–752.

Multiplication of Distributions and Nonperturbative Calculations of Transition Probabilities J. F. Colombeau, J. Aragona, P. Catuogno, S. O. Juriaans and C. Olivera

Abstract In a mathematical context in which one can multiply distributions the “formal” nonperturbative canonical Hamiltonian formalism in Quantum Field Theory makes sense mathematically, which can be understood a priori from the fact the so called “infinite quantities” make sense unambiguously (but are not classical real numbers). The perturbation series does not make sense. A novelty appears when one starts to compute the transition probabilities. The transition probabilities have to be computed in a nonperturbative way which, at least in simplified mathematical examples (even those looking like nonrenormalizable series), gives real values between 0 and 1 capable to represent probabilities. However these calculations should be done numerically and we have only been able to compute simplified mathematical examples due to the fact these calculations appear very demanding in the physically significant situation with an infinite dimensional Fock space and the QFT operators.

J. F. Colombeau (B) Institut Fourier, Grenoble, France e-mail: [email protected] J. F. Colombeau · P. Catuogno · C. Olivera IME, UNICAMP, Campinas, SP, Brazil e-mail: [email protected] C. Olivera e-mail: [email protected] J. Aragona · S. O. Juriaans IME, USP, Sao Paulo, SP, Brazil e-mail: [email protected] S. O. Juriaans e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2018 V. Dobrev (ed.), Quantum Theory and Symmetries with Lie Theory and Its Applications in Physics Volume 1, Springer Proceedings in Mathematics & Statistics 263, https://doi.org/10.1007/978-981-13-2715-5_27

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1 Mathematical Background In 1954 L. Schwartz published a note Impossibility of the multiplication of distributions [19] and he claimed [20] Multiplication of distributions is impossible in any mathematical context, possibly different of distribution theory. This means therefore for ever. The viewpoint of mathematical physicists was then that multiplications of distributions in physics emerged from an erroneous mathematical formulation of physics to be replaced by a correct mathematical formulation to be discovered: this was at the origin of the development of various “axiomatic” field theories in which physics was summed up in a list of “axioms” and the mathematical difficulty was shifted to the construction of mathematical objects satisfying the axioms. Thirty years later in 1983 L. Schwartz presented to the French Academy of Sciences the note [9]. A General Multiplication of Distributions whose title is exactly the converse of the one of his 1954 note [19]. The detailed proofs of the note were published in form of two books [10], 1984 and [11] 1985. A clarification is needed.

2 A Mathematical Context in Which One Can Multiply Distributions This clarification is obvious from the two formulas (where H denotes the Heaviside function H (x) = 0 if x < 0, H (x) = 1 if x > 0, H (0) unspecified):  

(H 2 − H )(x)ψ(x)d x = 0 ∀ψ ∈ Cc∞ ,

(H 2 − H )(x)H  (x)d x = [

H 2 +∞ H3 1 − ]−∞ = − , 3 2 6

(1)

(2)

which both should hold in a context in which one can multiply distributions (so that (H 2 − H )H  makes sense), see [3]. First we give a coarse incorrect proof that these two formulas are contradictory: formula (1) implies H 2 − H = 0. Formula (2) implies H 2 − H = 0, hence a contradiction which proves impossibility of existence of a mathematical context in which one could multiply distributions. Where is the mistake? The mistake is that we assume we are in a new unknown mathematical context and we ignore if in this context

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F(x)ψ(x)d x = 0 ∀ψ ∈ Cc∞

395

 ⇒F =0

(3)

implicitly used by habit to claim H 2 − H = 0 (in the 1954 Schwartz proof this follows from a property imposed to any reasonable multiplication of distributions). The correct conclusion is: The familiar implication (3) ceases to be valid in a context in which one can multiply distributions. This causes no trouble for a differential calculus similar to the classical one: in 1982 one of the authors published such a context [8], later developed in the books and surveys [2, 4, 5, 10–14, 17, 18, 21, 22] among other. The first novelty is simply the need to introduce a new notation:  F ≈ G iff by definition (F − G)(x)ψ(x)d x = 0 ∀ψ ∈ Cc∞ which is some kind of weak relation that generalizes exactly the equality of distributions. We call this relation “association”. It has the properties and the peculiarities of distributions: F ≈ G ⇒ F  ≈ G  but  F K ≈ G K if K is another generalized function. As a basic example H 2 ≈ H and H 2 = H. As another example one can define easily various natural positive square roots of the Dirac distribution δ and one has √ √ δ ≈ 0 and δ = ( δ)2 ≈ 0 showing again incoherence between ≈ and nonlinear operations. In short one has defined a basically new concept of equality, more restrictive than the classical one. Then the natural extension of the classical concept of equality is given the name of “association” and the notation ≈. In the applications of this new theory one has to play with = and ≈ at their right place. Basically this is very simple. Let U and V be two distributions from physics that cannot be multiplied within the distributions and for which the product U V is usually considered as ambiguous. What does this theory give? A priori nothing more: if Fi ≈ U and G j ≈ V (there are many Fi and G j ) then the various products Fi G j take different values: one recovers the ambiguity. But a novelty: In this theory one can state more precisely than usual, on physical ground, the equations: this can give some selection on Fi and G j which can resolve the ambiguity: then one can obtain new formulas to be compared with experimental results.

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This has been done for the equations modeling strong collisions used to design the armor of battle tanks: [12]. From a differential statement of Hooke’s law used by engineers these equations are in nonconservative form which in the case of shock waves gives rise to products of distributions not defined within the distributions. A suitable formulation of the equations in this new context, motivated on physical ground, consisting in stating some equations of the system with = and the other ones with ≈ permits to resolve the ambiguity. Then one obtains well defined formulas ruling the shock waves for this system, which have been checked to be in agreement with the experimental observations.

3 A Brief Recall on the Calculations in the Canonical Hamiltonian Formalism for a Neutral Massive Self-interacting Boson Field The Hilbert space of states called Fock space is the Hilbertian direct sum 2 3 n F = ⊕∞ n=0 L s ((R ) ),

where the subscript s means symmetric in the n arguments in R 3 . The basic linear operators used in F are the creation and annihilation operators: if ψ ∈ L 2 (R 3 ) the creation operator a + (ψ) is the unbounded operator on F defined by ( f n ) −→ (0, f 0 ψ, . . . ,

√

nSym( f n−1 ⊗ ψ), . . . )

where Sym is the symmetrization operator. The annihilation operator a − (ψ) is given by the formula  ( f n ) −→

f 1 (λ)ψ(λ)dλ, . . . ,

√



 n+1

f n+1 (λ)ψ(λ)dλ, . . .

.

The free field operator is given by the formula: Φ0 (x, t) := a + (k → 2− 2 (2π)− 2 (k 0 )− 2 eik t e−ikx )+ 1

3

1

0

a − (k → 2− 2 (2π)− 2 (k 0 )− 2 e−ik t e+ikx ) 1

3

1

0

√ where k 0 = k 2 + m 2 . If Φ(x, t) denotes the (unknown) interacting field operator the total Hamiltonian operator is given by the formula   H (t) :=

 3 1 1 m2 2 g 2 2 N +1 (∂i Φ) + (x, t)d x. (∂t Φ) + Φ + Φ 2 2 i=1 2 N +1

(4)

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Another related important operator denoted H0 (t) is obtained by inserting the free field operator Φ0 into (4) in place of the interacting field operator Φ. The Hamiltonian formalism consists in formal calculations on the free field operator that produce the interacting field operator and the scattering operator, then transition probabilities that are compared with experimental results. We consider two time values τ < t. The interacting field operator is given by the formula Φ(x, t) := ei(t−τ )H0 (τ ) Φ0 (x, τ )e−i(t−τ )H0 (τ ) . Formal calculations give that it satisfies the “ interacting field equation” ∂t2 Φ(x, t) =

3 

∂i2 Φ(x, t) − m 2 Φ(x, t) − gΦ N (x, t).

i=1

The scattering operator is given by the formula: Sτ (t) := ei(t−τ )P0 e−i(t−τ )H0 (τ ) ,

(5)

where P0 is the energy operator defined by the formula: if F = ( f n ) ∈ F (P0 )F = (0, k→k 0 f 1 (k), . . . , (k1 , . . . , kn )→(k10 + · · · + kn0 ) f n (k1 , . . . , kn ), . . . ). Formal calculations give that the scattering operator permits to obtain the interacting field operator Φ(x, t) from the free field operator at same time Φ0 (x, t) through the formula Φ(x, t) = (Sτ (t))−1 Φ0 (x, t)Sτ (t). Formal calculations give that the scattering operator satisfies the ODE g ∂t Sτ (t) = −i N +1



Φ0 (x, t) N +1 d x Sτ (t), Sτ (τ ) = id.

The transition probabilities are given by the formula | < F1 , Sτ (t)F2 > |, F1 , F2 ∈ F.

(6)

These formal calculations are recalled in detail in [10, 15].

4 Interpretation of QFT in this Mathematical Context In order to interpret the canonical Hamiltonian formalism in the context of nonlinear generalized functions we need an heuristic understanding of the new context:

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a real number, a distribution, a nonlinear generalized function, are all merely: (a regularization) modulo an imprecision (the imprecision appears mathematically as a mathematical quotient). In more detail: • A real number is a Cauchy sequence (xn ) of rational numbers modulo the set of {all sequences that tend to 0}. • A distribution is a sequence ( f n ) of C ∞ functions having the property that ∀ψ f n ψd x tends to a limit modulo the set of {all sequences (gn ) such that ∀ψ gn ψd x → 0}. • A nonlinear generalized function is a certain sequence ( f n ) of C ∞ functions modulo the set of {all sequences that tend to 0 in a certain sense}. This interpretation of real numbers is well known. This interpretation of distributions is given in [1]. We observe that this is no more than the repetition of a standard process in mathematics: regularization of a “mysterious” new object to define it by means of known objects and then imprecision to diminish the number of new objects so constructed by regularization. How to interpret QFT with these nonlinear generalized functions? The free field operator Φ0 (x, t) is a distribution in x for each fixed t. To produce the needed family of C ∞ functions that should represent it we use a regularization by convolution. We set 1 x

, Φ0 (x, t, ) = (Φ0 (., t) ∗ φ )(x), φ (x) = 3 φ  

 φd x = 1.

(7)

This depends on a choice of φ and one should check that the final result of the theory should be independent of this choice. The calculations in Sect. 3 make sense from Theorem 1 The symmetric operators that occur (such as the total Hamiltonian H0 (τ )) admit a self-adjoint extension. A proof is in [10, 16]. This permits to define the imaginary exponentials such as e−i(t−τ )H0 (τ ) . In particular one defines the scattering operator (5), then the transition probabilities (6), that both are equivalence classes, in the new mathematical context, of representatives depending on the choice of a function φ in φ in (7). Let F1 , F2 ∈ F, of norms = 1. What is the transition probability | < F1 , Sτ (t)F2 > |? The generalized operator Sτ (t) is the equivalence class of the family {Sτ (t, )} that depends on  and, for each , | < F1 , Sτ (t)()F2 > | is a real number between 0 and 1. Usually it oscillates infinitely between 0 and 1 when  → 0 like |cos 1 |: this is due to the presence of “infinities” such as 1 when  → 0. In the theory developed in this paper the infinities are transformed into infinite oscillations because we are in a nonperturbative formulation and the infinities appear inside imaginary exponentials such as ex p( i ). How to interpret these infinite oscillations? since we wish a well defined real number between 0 and 1 to be interpreted as a probability. One has to obtain the answer from the interpretation of nonlinear generalized functions in physics.

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5 Calculations of an Infinite Number of Oscillations In classical mechanics an experiment is represented by a value of  > 0 very small, denoted 0 , that can only slightly change from an experiment to another due to small differences in experimental conditions: one can eliminate the influence of parasites (except in situations of turbulence). Since, in the examples from classical mechanics, the result, depending on , has a limit when  → 0 this limit is a very good approximation of the result obtained from the value 0 . Here, in the calculations of QFT, we have no limit but infinite oscillations between 0 and 1. The origin of these oscillations can be attributed to the influence of the void on the interaction itself (fictitious particle-antiparticle pairs) that we cannot eliminate and is aleatory. Therefore different experiments can give very different results. Since the final experimental result is an average (a probability) then the same should be done at the level of the theory. This suggests that the result to be compared with experiments should be N 1  | < F1 , Sτ (t, i )F2 > |, N very large , i > 0 very small, at random . (8) N i=1

Does such a mean value exist? One expects YES. This expectation is supported by two kinds of arguments: numerical calculations in very simplified cases and a theoretical proof also in a simplified case. Numerical calculations, from the definition of mean value, in Hilbert spaces of dimension n = 2 and 3, can be done on a PC very easily: it suffices to compute a large number of times ( j = 1, . . . , N , N very large) the quantity | < F1 , ex p(i H ( j ))F2 > | if H () is a hermitian symmetric n × n matrix whose coefficients are functions of  that can tend to ∞ when  → 0. A theoretical result is as follows: Theorem 2 In a Hilbert space of finite dimension n let H be a n × n symmetric matrix whose coefficients are “reasonable” functions of  that can tend to ∞ when  → 0 then | < F1 , ex p(i H ())F2 > | has a mean value when  → 0. The word reasonable means: polynomials and exponentials in 1 , their quotients with nonzero denominator, etc. Our proof is long and difficult. It uses results from the theory of almost periodic functions [6, 7]. We have not been able to extend the proof to the Fock space and the operators of QFT presumably because the matter becomes very complicated. The case of QFT looks exactly similar at a heuristic level. Here are a few numerical tests that can be reproduced at once on any PC. √ √ √ √ Example: mean value of | < u, ex p(−i H )v > |, u = ( 22 , 22 ), v = ( 22 , − 22 ), a c c b H = first line ( n1 + 2n 1 , −22 ), second line ( −22 , n 2 ) a = 1, b = 1.3, c = 0.4, n 1 = 1, n 2 = 1,  < 10−7 at random. We obtained

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N = 105 0.223 ten times, 0.224 two times N = 107 0.2235 ten times, 0.2236 two times N = 108 0.22353 six times, 0.22352 five times, 0.22351 one time Different values: a = 1, b = 1, c = 2, n 1 = 0.5, n 2 = 1,  < 10−7 at random. We obtained N = 105 0.1539 one time, 0.154 ten times, 0.1551 one time N = 106 0.1544 eight times, 0.1543 four times. Example: mean value of √16 |ex p( i ) − 2ex p( i2 ) + (1 + )ex p( √i  )| with  < 10−10 at random N = 104 : 0.9313, 0.9188, 0.9212 = 0.9 for sure N = 105 : 0.9257, 0.9258, 0.9243 = 0.92 for sure N = 106 : 0.9249, 0.9248, 0.9247 = 0.924 for sure?: more trials needed N = 107 : 0.9249, 0.9251, 0.9250 N = 108 : 0.9251, 0.9251, 0.9250 = 0.925 for sure N = 109 : 0.9251, 0.9251, 0.9251 = 0.9251 for sure?: more trials needed The calculations from (8) show that one needs large values of N to reach the mean value even in such very simple cases. This method is not efficient enough for large dimension a fortiori for the Fock space. This mean value appears very robust presumably independent on the choice of φ in (7).

6

Conclusion

We have presented a work in evolution whose achievement faces serious difficulties because the extension of the theoretical proof or the numerical tests to the genuine case of QFT appears particularly difficult. We can work in the directions of improvements to get closer to QFT, in particular try to find a significantly better numerical method that could apply first in Hilbert spaces of large dimension then in QFT. One could also try to compare the result from the mean value with those from a method of renormalization. It would be interesting to investigate if there exist a mathematically similar physical theory in a finite dimensional Hilbert space that could serve for a validation of the method of mean values presented in this paper by comparison theory-experiments. Our conclusion is that the mathematical interpretation of QFT in this paper is likely a possible nonperturbative QFT. However it is unfinished and there remains great difficulties that appear of a technical nature.

References 1. P. Antosik, J. Mikusinski, R. Sikorski. Theory of Distributions: the sequential approach. (Elsevier, Amsterdam, 1973). 2. J. Aragona, H.A. Biagioni. An intrinsic definition of the Colombeau algebra of generalized functions. Analysis Mathematica 17, 1991, p. 75–132.

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3. J. Aragona, P. Catuogno, J.F. Colombeau, S.O. Juriaans, Ch. Olivera in Lie Theory and its Applications in Physics, V. Dobrev ed. (Springer, Berlin-Heidelberg-New York, 2016), pp. 583–589. 4. J. Aragona,R. Fernandez, S.O. Juriaans. A discontinuous Colombeau differential calculus. Monatsh. Math. 144, 2005, pp. 13–29. 5. H.A. Biagioni. A Nonlinear Theory of Generalized Functions. Lecture Notes in Math. 1421. Springer, 1990. 6. A.S. Besicovitch. Almost Periodic Functions. (Cambridge University Press, 1954). 7. H. Bohr. Almost Periodic Functions. (Chelsea Pub Company, New York, 1947). 8. J.F. Colombeau. J. Math. Anal. Appl. 94, 1, (1983) 96–115. 9. J.F. Colombeau. Comptes rendus Acad. Sci. Paris 296, (1983), 357–360. 10. J.F. Colombeau. New Generalized Functions and Multiplication of Distributions. (NothHolland-Elsevier, Amsterdam, 1984). 11. J.F. Colombeau. Elementary Introduction to New Generalized Functions. (Noth-HollandElsevier, Amsterdam, 1985). 12. J.F. Colombeau. Multiplication of Distributions (Lecture Notes in Math. 1532, SpringerVerlag, Berlin-Heidelberg-New York, 1992). 13. J.F. Colombeau. Bull. Amer. Math. Soc., 23, 2, (1990), 251–268. 14. J.F. Colombeau. Nonlinear generalized functions. Sao Paulo J. Math. Sci. 7, 2013, 2, pp; 201– 239. 15. J.F. Colombeau. Mathematical problems on generalized functions and the canonical Hamiltonian formalism. arXiv:0708.3425. 16. J.F. Colombeau, A. Gsponer. The Heisenberg-Pauli canonical Hamiltonian formalism of quantum field theory in the rigorous mathematical setting of nonlinear generalized functions (part 1). arXiv:0807.0289. 2008. 17. M. Grosser, M. Kunzinger, M. Oberguggenberger, R. Steinbauer. Geometric Theory of Generalized Functions with Applications to General Relativity. (Kluwer, Dordrecht, 2001). 18. A. Gsponer. A concise introduction to Colombeau generalized functions and their application in classical electrodynamics. European J. Phys. 30, 2009, 1, pp. 109–126. 19. L. Schwartz. Comptes Rendus Acad. Sci. Paris 239, 1954, 847–848. 20. L. Schwartz. Théorie des Distributions, (Hermann, Paris, 1966). 21. S. Steinbauer, J. Vickers. The use of Generalized Functions and Distributions in General Relativity. Class. Quant. Grav. 23, pp. R91–113, 2006. 22. J.A. Vickers. Distributional geometry in general relativity. J. Geom. Phys. 62, 3, 2012, p. 692–705.

About Markov, Gibbs, … Gauge Theory … Finance Alexander Ganchev

Abstract The Hammersley–Clifford theorem states the equivalence between Markov and Gibbs random fields. The Markov property is a kind of ‘locality’ while the Gibbs property is a kind of ‘factorization’. We speculate that a generalization to gauge fields on graphs is possible. Such a generalization could provide a justification for using Gibbs measures in the application of gauge theories to finance.

1 Introduction Gauge field theories provide the foundation for the theory of elementary particles. In order to do numerical simulations and to try to understand quark confinement lattice gauge theory was developed with the lattice serving as a discretization of space-time. For the sake of background independent gravity also gauge fields on more general graphs have been considered. Recently models based on gauge theories were proposed in various different contexts – from non equilibrium thermodynamics (e.g., [13] and references there) to social psychology … to finance. The propagation of gauge theory ideas and methods from physics to other areas can be seen as a manifestation of Macks pushing Einsteins principles to the extreme [10]. The fact that financial notions such as prices, exchange rates, and discount factors are related with gauge connections and no-arbitrage with flat connection (trivial holonomy) possibly has occurred to several people but it is clearly stated for the first time in the works of Ilinski (see for example [7]). Probably the first use of gauge theory in economics is the Malaney–Weinstein connection [12, 17] used to define a

A. Ganchev (B) AUBG, 2700 Blagoevgrad, Bulgaria e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2018 V. Dobrev (ed.), Quantum Theory and Symmetries with Lie Theory and Its Applications in Physics Volume 1, Springer Proceedings in Mathematics & Statistics 263, https://doi.org/10.1007/978-981-13-2715-5_28

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consumer index which correctly accounts for the change of preferences. A very brief overview of these topics is in [5]. The Black-Scholes model for pricing of options is the most successful mathematical model in finance. The key assumptions of the BS model are no-arbitrage and diffusion of prices. These are equilibrium assumptions. In fact all of neoclassical economics is based on equilibrium. How to get away from equilibrium? For a discussion and a call for a non equilibrium dynamical economics from the point of view of a mathematical physicist see the paper of Smolin [15]. In a lattice quantum gauge theory the graph is a regular lattice in d dimensions, e.g., Zd , the gauge fields or connections live on the edges, gauge invariant quantities are Wilson loops, and the probability measure is of Boltzmann-Gibbs type exp(−S) where the action is S is the sum over Wilson loops over plaquettes (elementary squares on the lattice). The choice of action is made so to reproduce the Yang-Mills action in the continuum limit. Promising simulations based on financial lattice QED of Ilinski have been performed in [3, 4]. The model of Ilinski has met also justified criticism [16]. The use of gauge invariant quantities (Wilson loops) to describe the dynamics is undisputable. On the other hand the use of a Boltzmann-Gibbs distribution is rather ad hoc in the financial context and needs justification. The latter is the content of this note. A possible way to motivate the use of Gibbs measures in financial gauge theories is to generalize the Hammersley–Clifford theorem. The latter establishes the equivalence of Markov and Gibbs fields on graphs. While asking for a Gibbs measure could seem ad hoc in the context of finance the Markov property is very natural. The Markov property is a property of locality – if two regions I and J are separated by a third K (think of I as ‘interior’, J – ‘exterior’, and K the ‘boundary’, or K ‘shields’ or ‘screens’ I from J ) then the fields in I are independent on J conditional on K . I.e., any ‘influence’ from a region and its complement has to be mediated by the boundary. The Gibbs property, on the other hand, is a property of factorization of the probability measure (factorization over cliques of the graph). Thus we can restate loosely the Hammersley–Clifford theorem as “locality is equivalent to factorization”. The problem is that the Hammersley–Clifford theorem is for fields living on the nodes and ‘interactions’ occurring on the edges of a graph while for gauge theories we need a generalization for fields living on the edges. In Sect. 2 we introduce gauge theories on trade networks. In Sect. 3 we review Mobius inversion. Section 4 is devoted to the Hammersley–Clifford theorem. In Sect. 5 we speculate on a generalization of the Hammersley–Clifford theorem for gauge field theories.

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2 Gauge Theory of Trade Networks Let us start with the simplest trade network – two ‘tradable things’1 a and b. To be traded one needs to compare a and b or an exchange rate: if β units of b are exchanged for α units of a, set Uba = β/α. Here Uba is a positive real number. Set Uba Uab = 1.2 We can picture this by a graph with two nodes and a directed edge from a to b decorated by the exchange rate Uab . The important observation is that the units in which each asset is measured are a matter of convention. So we have a ‘gauge’ freedom to rescale at every node, or a local action of G L + (1, R). The exchange rate Uab is the connection on our graph allowing for the “parallel transport” of one tradable thing for another. Under local gauge transformations ga , gb ∈ G L + (1, R) the connection transforms as Uab → ga Uab gb−1 . In the case of three tradable things we will have a triangular graph with the exchange rates or connections Uab , Ubc , Uca on the edges and the freedom to rescale at every vertex independently. Transporting around a loop, in this case the triangle, we obtain the holonomy, or curvature Uab Ubc Uca . When the holonomy is not equal to one we have arbitrage, or equivalently, no-arbitrage corresponds to a flat connection. If we have a complex network of financial securities in different countries the network will look something like a backbone consisting of nodes corresponding to the major currencies connected by edges labelled with the foreign exchange rates. Each currency node is the center of a star made of edges connecting the currency with the financial securities tradable in this currency with the current prices of the securities labelling the corresponding edges.3 This network describes “space” at a certain moment of time. To evolve the network in time we have to evolve every node. To compare the value of an asset at time t and t + 1 we need a connection in the time direction. This is exactly the discounting factor with which a future value is discounted to compare to present value. When a bank account at interest rate r compounded continuously is ‘transported’ from t to t + 1 the amount is changed by er . Hence discount factors are the gauge connections in the time direction. Starting with our simplest graph with just one edge evolving in time for one time period we obtain a quadrilateral with vertices (a, t), (a, t + 1), (b, t), and (b, t + 1) and ‘space-like’ edges labelled by exchange rates Uab;t and Uab;t+1 and ‘time-like’ edges labelled by discounting factors Ua;t,t+1 and Ub;t,t+1 . The product Uab;t Ub;t,t+1 Uba;t+1 Ua;t+1,t is the holonomy around this quadrilateral loop. If it is not one we have arbitrage in the “time direction”. We have described a trade network as a graph equipped with a principle G L + (1, R) bundle (the ability to rescale the unit of measure at every node) and a connection living on the edges allowing us to ‘parallel transport’, i.e., to exchange or trade and evolve in time. 1 To

avoid going into financial definitions, this is anything that can be traded, goods, services, financial instruments. 2 It is standard in finance to ignore in first approximation any ‘friction’ or ‘cost of trading’ like taxes, buy/sell spreads, etc. 3 The price of any tradable thing is the connection on the edge between this thing and the corresponding currency.

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3 Mobius Inversion on Posets Here we quickly recall the notions of posets, incidence algebras on posets and Mobius inversion on posets developed by G.C. Rotta. A good introductory source are the lecture slides of B. Sagan [14]. This material will be used in Sect. 4.2.

3.1 Posets and Incidence Algebras A partially ordered set (poset) (P, ≤) is a set P with a partial order, i.e., a relation ≤ ⊂ P × P (instead of (a, b) ∈ ≤ it is standard to write a ≤ b) which is reflexive: a ≤ a; antisymmetric: a ≤ b and b ≤ a implies a = b; transitive: a ≤ b and b ≤ c implies a ≤ c; for any a, b, c ∈ P. An example of a poset is the powerset P(A) = {B : B ⊂ A} of a set A consisting of all subsets of A. It is partially ordered by inclusion. Another example of a poset is the poset of natural numbers with order relation given by divisibility, i.e., we write m|n if m divides n. A poset is a totally ordered set if any two elements are comparable, i.e., for any a and b either a ≤ b or b ≤ a. An example of a totaly ordered set is Cn = 0 < 1 < · · · < n, also called a chain. The posets with monotone (ordered preserving) maps between them form a category POSet. The categorical product of two posets P × Q is the cartesian product of their underlying sets with the product order on them, i.e., ( p1 , q1 ) ≤ ( p2 , q2 ) iff p1 ≤ p2 and q1 ≤ q2 . Given a poset (P, ≤) the (closed) intervals in P are [a, c] = {b ∈ P : a ≤ b ≤ c} for any a ≤ c. Denote by Dn the interval [1, n] in the poset (N, |), i.e., the poset of divisors of n ordered by divisibility. Denote Bn the powerset of an n-element set (the Boolean algebra of 2n ). Then we have the following examples of isomorphisms in POSet: D20  D5 × D4  C1 × C2 and B3  C1 × C1 × C1 . Assume (P, ≤) is a locally finite poset, i.e., all intervals are finite. (Assume k is a field (take k = R). The incidence algebra k[P] is defined as follows. It consists of functions ξ : P × P → k such that ξ(a, b) = 0 if a is not larger or equal to b. Addition and scaling is defined pointwise while multiplication is given by (ξ ∗ ψ)(a, c) =



ξ(a, b) ψ(b, c)

(1)

b∈[a,c]

usually called convolution (or matrix multiplication of triangular matrices). With a poset P one can associate a vector space k P of functions f : P → k on which the  incidence algebra acts (on the right) ( f · ξ)(b) = f (a)ξ(a, b). Note that posets and groups (more generally monoids) are special cases of categories, i.e., a poset is a category with few morphisms (between any two objects there are either zero or one morphism) while a monoid (group) is a category with a single

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object and many (invertible) morphisms. The notions of an incidence algebra k[P] of a poset P and the group algebra k[G] of a group G are both special cases of the notion of a category algebra k[C] of a category C. For a set P let Δ = {(a, a)|a ∈ P} ⊂ P 2 be the diagonal relation. Any order relation ≤ on P has to be reflexive, i.e., Δ ⊂ ≤, hence the indicator function of Δ, denoted δ(a, b) which is one if a = b and zero otherwise, is in the incidence algebra k[P] and plays the role of unit for the convolution product, i.e., ξ ∗ δ = δ ∗ ξ = ξ. The following fact is easy: an element ξ of the incidence algebra is invertible iff ξ(a, a) = 0 for every a (i.e., triangular matrices with nonzero diagonal elements are invertible).

3.2 Mobius Inversion Let (P, ≤) be a poset and k[P] its incidence algebra. The characteristic/indicator function of the order relation ≤ is denoted ζ, i.e. ζ(a, b) is one if a ≤ b and zero otherwise. From the previous fact it follows that the ζ -function is invertible and its inverse is called the Mobius function μ. Let f, g ∈ k P then Mobius inversion is f ·ζ =g

⇔

f =g·μ

(2)

or equivalently g(t) =



f (s)

⇔

f (t) =

s≤t



g(s)μ(s, t) .

(3)

s≤t

Three examples of Mobius inversion follow. First consider the set of natural numbers N with the standard order. For sequences f : N → R ‘integration’ (Σ f )(n) = nk=1 f (k) is convolution with ζ, while ‘differentiation’ (Δf )(n) = f (n) − f (n − 1) (we are assuming f (0) = 0) is convolution with μ. The ‘fundamental theorem of (discrete) calculus’, i.e., that integration and differentiation are inverses (ΔΣ f )(n) = f (n), is exactly Mobius inversion. The next example is the original one due to August Ferdinand Mobius (1790–1868). The poset is the set of natural numbers ordered by divisibility. Let n = p1m 1 . . . pkm k be the decomposition of a positive integer into different prime factors (the m’s are the multiplicities). Define the Mobius function μ : N → R as follows: μ(n) = (−1)k if m 1 = · · · = m k = 1 and μ(n) = 0 otherwise. For two functions f, g : N → R the following is true f (n) =

 m|n

g(m)

⇔

g(n) =



μ(n/m) f (m) .

(4)

m|n

The last example is the Inclusion Exclusion Principle (IEP) that will be used in the proof of the Hammersley–Clifford theorem. The IEP is a manifestation of Mobius inversion on the powerset poset. First note that the incidence algebra over a field k

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of the product P × Q of two posets P and Q with the product order is the tensor product the two incidence algebras, i.e., k[P × Q]  k[P] ⊗k k[Q]. Hence in particular ζ P×Q = ζ P ζ Q and μ P×Q = μ P μ Q . One also notes that Bn  C1×n , i.e., the Boolean algebra of 2n elements is isomorphic to the n-ht power of the Boolean algebra with 2 elements or the chain C1 = {0 < 1}. For the chain C1 = {0 < 1} obviously μC1 (0, 1) = −1, μC1 (0, 0) = μC1 (1, 1) = 1. For a set A let |A| denote the cardinality of A. Combining the above we obtain that the Mobius function for the powerset poset (P(U ), ⊂) is given by (5) μ(X, Y ) = (−1)|Y |−|X | if X ⊂ Y and is zero otherwise.

4 Hammersley–Clifford Theorem After a few definitions from graph theory we introduce Markov random fields and Gibbs fields on graphs and sketch the proof of the Hammersley–Clifford Theorem [6] (or a multitude of other sources, e.g. [2]).

4.1 Markov and Gibbs Fields on Graphs Let G = (V, E) be a finite simple graph, i.e., a finite undirected graph without loops or multiple edges, i.e., the number of vertices is finite, |V | < ∞, and the edges, E ⊂ V × V , consitue a symmetric, anti-reflexive relation on the vertices. Two vertices (or nodes) are adjacent or neighbours if they are connected by an edge and no node is adjacent to itself. For a node a ∈ V denote N (a) = {b ∈ V |(a, b) ∈ E} the set of neighbours of a. A subset of nodes C ⊂ V is a clique if any two nodes in C are neighbours. Denote by C(G) the collection of all cliques in G. Let V be a finite set of nodes/sites/vertices and let S be a finite space of states (phase space) at every node. A field configuration is a map x : V → S, or a collection {xa ∈ S}a∈V . Denote X = S V the space of field configurations. As both S and V are finite we turn X = S V into a measurable space (X, A) by taking A = P(X) as the algebra of events on X. For a subset A ⊂ V denote with x A = {xa ∈ S}a∈A the restriction of x to A. Fix a field x ∈ S V and let a ∈ V be some node, and A, B ⊂ V subsets. The (cylinder set) event (X a = xa ) is {X ∈ X : X a = xa } ∈ P(X). The event (X A = x A ) = ∩a∈A (X a = xa ) consists of X ’s equal to x on A and arbitrary away from A. For two subset A and B (X A∪B = x A∪B ) = (X A = x A ) ∩ (X B = x B )

(6)

holds. The objective is to consider an appropriate probability distribution P on the measurable space of configurations (X, A) turning it into a probability space

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(X, A, P). We will need conditional probabilities P(X A = x A |X B = x B ) =

P(X A∪B = x A∪B ) P(X B = x B )

(7)

and we make the (positivity) assumption that all conditional probabilities make sense (we are not dividing by zero). To make the notation more readable adopt the shorthand (xa ) ≡ (X a = xa ) and (x A ) ≡ (X A = x A ) and then the above conditional probability will be written as P(x A |x B ) = P(x A∪B )/P(x B ). Now we turn to Markov random fields. A random field is Markov if dependence/influence/interaction is local, namely, if two regions I and J are separated by a third K , then I and J are independent given K , or any interaction between I and J has to pass through (be mediated by) K . To describe the Markov property we need a neighbourhood system N = {N (a) ⊂ V : a ∈ V } defined by the properties: a∈ / N (a) for every a ∈ V and if b ∈ N (a) then a ∈ N (b). This is the same as turning V into a graph (V, E) by connecting two nodes a and b by an edge exactly when b ∈ N (a) or equivalently a ∈ N (b). We say that (X, A, P) is a Markov random field or P satisfies the Markov property if for any a ∈ V we have P(xa |x V −{a} ) = P(xa |x N (a) )

(8)

or more generally for any partition of the nodes V = I ∪ K ∪ J so there is no edge connecting a node from I with a node from J P(x I |x K ∪J ) = P(x I |x K )

(9)

holds, i.e., the field at I is independent of what happens at non-neighbouring nodes J conditional on the separating region K (a locality property). The Markov relation can be rewritten as P(x I ∪J ∪K ) P(x K ) = P(x K ∪I ) P(x K ∪J )

(10)

log P(x I ∪J ∪K ) + log P(x K ) − log P(x K ∪I ) − log P(x K ∪J ) = 0 .

(11)

or taking logarithms

This last formula will be used in the proof of the HC Theorem. Now we turn to Gibbs fields. Let G = (V, E) be a graph, let S be a space of states at a node a ∈ V , let the space of field configurations X = S V be the maps from V to S, let C(G) the set of all cliques on the graph G = (V, E). For a subset of nodes K ⊂ V and a field X ∈ S V let X K be the restriction of X to K . A probability distribution P on the space of field configurations X is a Gibbs distribution if it can be factorized over cliques (as before we are assuming positivity)

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⎛ P(X ) = Z −1 exp ⎝−



⎞ UC (X C )⎠ .

(12)

C∈C(G)

An example of a Gibbs random field is the Ising model. Here the graph G = (V, E) is a square lattice (lattice in the physics sense as a regular discretization of flat space not in the sense of order theory) the cliques are the edges/links/bonds of the lattice and the space of states S at every node is a two element set.

4.2 Hammersley–Clifford Theorem The Hammersley–Clifford theorem says that a field is Markov iff it is Gibbs. The interesting direction is Markov ⇒ Gibbs. Its proof consists in first using Mobius inversion and next imposing the Markov property to produce the factorization over cliques. Here is a sketch of the proof. Since we imposed positivity, with−H (X ) where Z , the ‘partition out loss of generality, we can write P(X ) = Z −1 e U function’ is a normalization factor, and H (X ) = B (X ). More generally B⊂V  ) = U (X ) and applying Mobius inversion to it we get we can write H (X A B⊂A B  U A (X ) = B⊂A (−1)|A−B| H (X B ). The Markov property gives us that U A (X ) = 0 if A ∈ / C(G) thus the  above sum reduces to a sum over cliques, i.e., the distribution is Gibbs H (X ) = C∈C(G) UC (X ). Indeed, assume A ⊂ V is not a clique, hence we can find a pair of nodes i, j ∈ A such that there is no edge between i and j, hence we can partition A into A = I ∪ J ∪ K with I = {i} and J = { j}, then the sum over B ⊂ A can be written as  (−1)|K −L| (H (X I ∪J ∪L ) + H (X L ) − H (X I ∪L ) − H (X J ∪L )) (13) U A (X ) = L⊂K

and the Markov property (11) gives us that the sum of the four terms is zero.

5 Outlook and Speculations The Hammersley–Clifford theorem shows the equivalence of Markov and Gibbs fields. These fields live on the nodes a graph. The Markov property and the cliques entering the Gibbs measure depend on the notion of neighbouring nodes, i.e., nodes connected by an edge (the interactions between fields on different nodes are on edges connecting them). We can restate that two nodes are adjacent by saying that they are in the boundary of an edge. (The boundary operator is just the incidence matrix of the graph.) We need a generalization of the Hammersley–Clifford theorem for gauge theories on graphs. Now the fields live on the edges of a graph. To define a Markov property

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we need to define which edges are neighbouring. (This is like climbing the dimension ladder in the sense of n-categories.) Thus to define a Markov property and cliques for gauge fields we need a 2-graph4 (or the 2-skeleton of a CW complex) having 0-cells or nodes, 1-cells or edges, and 2-cells or plaquettes.5 We will say that two edges are adjacent or neighbours if they are in the boundary of some plaquette. We will say that a 2-clique is a collection of edges such that any pair of edges is adjacent. For example in the Zd if the plaquettes are the standard plaquettes from lattice field theory then the 2-cliques coincide with the plaquettes. On the other hand in a tetrahedral-octahedral honeycomb lattice if the plaquettes are triangles the maximal 2-cliques are tetrahedra. With this definitions a straightforward generalization of the Hammersley–Clifford theorem will follow. Thus a generalization of the Hammersley–Clifford theorem to gauge theories on arbitrary graphs needs additional information about which edges are ‘neighbours’. The study of Markov properties of quantum field theories has a long history. For older references see for example [11] or [1], while for newer ones – [8, 9]. These results cannot be generalized immediately to finance because the graph of a trade network has no underlying manifold of which it is a discretization or in which it is embedded. I could find nothing in the literature regarding a generalization of the Hammersley–Clifford theory to gauge theories on arbitrary graphs. Thus this questions deserves attention and this sketch could be a first step.

References 1. S. Albeverio, R. Hoegh-Krohn, H. Holden, in Stochastic methods and computer techniques in quantum dynamics ed. by H. Mitter and L. Pittner (Springer, Vienna, 1984) pp. 211–231. 2. P. Clifford, in Disorder in Physical Systems: A Volume in Honour of John M. Hammersley, eds. G. Grimmett, D. Welsh (Oxford University Press, 1990) pp. 19–32. 3. B. Dupoyet, H. Fiebig, D. Musgrove, Physica A 389 (2010) 107–116. 4. B. Dupoyet, H. Fiebig, D. Musgrove, Physica A 390 (2011) 3120–3135. 5. A. Ganchev, in: Lie Theory and Its Applications in Physics, ed. by V. Dobrev (Springer, 2016), pp. 591–597. 6. J. M. Hammersley and P. Clifford, ‘Markov fields on finite graphs and lattices’ (1971), unpublished. 7. K. Ilinski, Physics of Finance - Gauge Modelling in Non-equilibrium Pricing, (John Wiley & Sons, New York, 2001). 8. Y. Le Jan, Markov paths, loops and fields. arXiv preprint arXiv:0808.2303 (2008). 9. T. Levy, Bull. Sci. math. 135 (2011) 629–649. 10. G. Mack, Pushing Einstein’s principles to the extreme, arXiv preprint arXiv:gr-qc/9704034. 11. G. Mack, Osterwalder-Schrader positivity and Markov property in lattice gauge theories, in Proc. of III School of Elementary Particles and High Energy Physics, Primorsko, Bulgaria, 1977, pp. 183–192. 12. P.N. Malaney, The Index Number Problem: A Differential Geometric Approach, PhD Thesis 1996. 4 This term is not standard an could be in conflict with other usages but lets adopt it for the moment. 5 We

are borrowing the term form lattice gauge theory but now a plaquette is a 2-cell bounded by several edges and not necessarily a square bounded by four edges.

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13. M. Polettini, EPL 97 (2012) 30003. 14. B. Sagan, Partially Ordered Sets and their Mobius Functions, lecture slides 2014, http://users. math.msu.edu/users/sagan/Slides/MfpBog1h.pdf. 15. L. Smolin, Time and symmetry in models of economic markets, arXiv preprint arXiv:0902.4274 [q-fin]. 16. D. Sornette, Int. J. Mod. Phys. C 9 (1998) 505–508. 17. E. Weinstein, Gauge Theory and Inflation: Enlarging the Wu-Yang Dictionary to a unifying Rosetta Stone for Geometry in Application Perimeter Institute lecture, 2006.

A Method for Classifying Filiform Lie Superalgebras Rosa M. Navarro and José M. Sánchez

Abstract Throughout the present work the latest contributions in determining filiform Lie superalgebras are reviewed. Filiform Lie superalgebras constitute itselves a very important type of nilpotent Lie superalgebras and their classification is at present and open and unsolved problem. Furthermore, the aforementioned contributions are mainly based on an adequate use of infinitesimal deformations of the model filiform Lie superalgebra.

1 Introduction The notion of filiform Lie algebras was firstly introduced in [14] by Vergne, having this type of nilpotent Lie algebras important properties. In particular, every filiform Lie algebra can be obtained by means of a deformation of the model filiform algebra Ln. Filiform Lie superalgebras, on the other hand, are a natural generalization of filiform Lie algebras and an important type of nilpotent Lie superalgebras. It has been showed that in the same way as occurs for filiform Lie algebras, all filiform Lie superalgebras can be obtained by means of infinitesimal deformations of the corresponding model Lie superalgebra L n,m . Thus, and using this result, it has been obtained a complete classification (up to isomorphisms) of complex filiform Lie superalgebras in low dimensions. Nevertheless, this method could be also used in higher dimensions.

R. M. Navarro (B) Dpto. de Matemáticas, Universidad de Extremadura, Cáceres, Spain e-mail: [email protected] J. M. Sánchez Departamento de Matemáticas, Universidad de Cádiz, Campus de Puerto Real, Puerto Real, 11510 Cádiz, Spain e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2018 V. Dobrev (ed.), Quantum Theory and Symmetries with Lie Theory and Its Applications in Physics Volume 1, Springer Proceedings in Mathematics & Statistics 263, https://doi.org/10.1007/978-981-13-2715-5_29

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Therefore, there will be considered infinitesimal deformations of L n,m which are defined by even 2-cocycles in Z 02 (L n,m , L n,m ). These deformations were studied along the papers [1, 4, 8, 9]. Note that all the vector spaces that appear in this paper (and thus, all the algebras) are considered to be C-vector spaces of finite dimension.

2 Preliminaries Recall that a superspace is nothing but a vector space with a Z2 -grading: V = V0 ⊕ V1 , and the elements of the space V0 are usually called even elements, and the elements of the space V1 , odd elements; the indices 0 and 1 are modulo 2. In a similar way, any linear mapping φ : V → W between two super vector spaces is called even iff φ(V0 ) ⊂ W0 and φ(V1 ) ⊂ W1 and is called odd iff φ(V0 ) ⊂ W1 and φ(V1 ) ⊂ W0 . Consequently, we will have the following decomposition: Hom(V, W ) = Hom(V, W )0 ⊕ Hom(V, W )1 , where the first summand is composed by all the even linear mappings and the second summand by all the odd ones. A Lie superalgebra (see [2, 12]) is a superspace g = g0 ⊕ g1 , with an even bilinear commutation operation (or “supercommutation”) [ , ], which satisfies the conditions: ∀X ∈ gα , ∀Y ∈ gβ . 1. [X, Y ] = −(−1)α·β [Y, X ] 2. (−1)γ·α [X, [Y, Z ]] + (−1)α·β [Y, [Z , X ]] + (−1)β·γ [Z , [X, Y ]] = 0 for all X ∈ gα , Y ∈ gβ , Z ∈ gγ with α, β, γ ∈ Z2 . The latter condition is usually called Graded Jacobi Identity. As a consequence of the definition we have that g0 is an ordinary Lie algebra and g1 is a module over g0 ; note that the Lie superalgebra structure also contains the symmetric pairing S 2 g1 −→ g0 . The descending central sequence of a Lie superalgebra g = g0 ⊕ g1 is defined in the same way as for lie algebras, that is, by C 0 (g) = g, C k+1 (g) = [C k (g), g] for all k ≥ 0. The Lie superalgebra will be called nilpotent if C k (g) = {0} for some k. In that case the nilindex of g will be the smallest integer k verifying C k (g) = {0}. Note that the dimension of each term of the descending central sequence is an invariant of the superalgebra, in the same way as occurs for Lie algebras. Moreover, each term of this sequence has an even part and an odd part, that is C k (g) = (C k (g))0 ⊕ (C k (g))1 , whose dimensions are also invariants of the superalgebra. For Lie superalgebras, there are also defined two others descending sequences called C k (g0 ) and C k (g1 ): C 0 (gi ) = gi , C k+1 (gi ) = [g0 , C k (gi )], k ≥ 0, i ∈ {0, 1}. These two descending sequences allow us to define super-nilindex or s-nilindex. Thus, if g = g0 ⊕ g1 is a nilpotent Lie superalgebra, then g has super-nilindex or s-nilindex ( p, q) if the following conditions hold: (C p−1 (g0 )) = 0

(C q−1 (g1 )) = 0,

C p (g0 ) = C q (g1 ) = 0.

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Nevertheless, it can be remarked that nowadays there is not any explicit formula or clear connection relating the concepts nilindex and s-nilindex. On the other hand, it can be noted that a module A = A0 ⊕ A1 on the Lie superalgebra g is an even bilinear map g × A → A satisfying ∀ X ∈ gα , Y ∈ gβ , a ∈ A : X (Y a) − (−1)αβ Y (Xa) = [X, Y ]a Lie superalgebra cohomology is a well-known concept. Next we recall briefly its definition (for more details it can be consulted [2, 13]): the superspace of q-dimensional cocycles of the Lie superalgebra g = g0 ⊕ g1 with coefficients in the g-module A = A0 ⊕ A1 is defined by C q (g; A) =



  Hom ∧q0 g0 ⊗ S q1 g1 , A

q0 +q1 =q q

q

This space is graded by C q (g; A) = C0 (g; A) ⊕ C1 (g; A) with C qp (g; A) =



  Hom ∧q0 g0 ⊗ S q1 g1 , Ar

q0 + q1 = q q1 + r ≡ p mod 2

Therefore, we have the cohomology groups  H pq (g; A) = Z qp (g; A) B qp (g; A) q

where, in particular, the elements of Z 0 (g; A) are called even q-cocycles and the q elements of Z 1 (g; A) odd q-cocycles. In complete analogy to Lie algebras [5–7], on the other hand, it can be denoted by N n+1,m the variety of nilpotent Lie superalgebras g = g0 ⊕ g1 with dim g0 = n + 1 and dim g1 = m. Thus, in [3] the following definition and theorem can be consulted. Definition 1 Any nilpotent Lie superalgebra g = g0 ⊕ g1 ∈ N n+1,m with s-nilindex (n, m) is called filiform. It is denoted by F n+1,m the subset of N n+1,m consisting of all the filiform Lie superalgebras. Before studying in deep this family of Lie superalgebras it is convenient finding a suitable basis, a so-called adapted basis. Thus, we have the following theorem. Theorem 1 If g = g0 ⊕ g1 ∈ F n+1,m , then there exists an adapted basis of g, namely {X 0 , X 1 , . . . , X n , Y1 , . . . , Ym }, with {X 0 , X 1 , . . . , X n } a basis of g0 and {Y1 , . . . , Ym } a basis of g1 , such that: 

[X 0 , X i ] = X i+1 , 1 ≤ i ≤ n − 1, [X 0 , X n ] = 0, [X 0 , Y j ] = Y j+1 , 1 ≤ j ≤ m − 1, [X 0 , Ym ] = 0.

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X 0 is called the characteristic vector. Consequently, from now on all the filiform Lie superalgebras considered throughout the present paper will be expressed in an adapted basis. Thus, it can be seen that the simplest filiform Lie superalgebra, denoted by L n,m , will be defined by the following non-null brackets:  L

n,m

:

[X 0 , X i ] = X i+1 , 1 ≤ i ≤ n − 1, [X 0 , Y j ] = Y j+1 , 1 ≤ j ≤ m − 1,

with {X 0 , X 1 , . . . , X n , Y1 , . . . , Ym } a basis of L n,m and the omitted brackets are regarded as zero. We will note by μ0 the law of L n,m . In complete analogy to Lie algebras, L n,m is the most important filiform Lie superalgebra since all the other filiform Lie superalgebras can be obtained from it using deformations. Then, we are going to consider its infinitesimal deformations that will be given by the even 2-cocycles, Z 02 (L n,m , L n,m ). Therefore, an infinitesimal deformation of L n,m can be seen as an element of the following space Z 02 (L n,m , L n,m ) = Z 2 (L n,m , L n,m ) ∩ Hom(g0 ∧ g0 , g0 ) ⊕ Z 2 (L n,m , L n,m ) ∩ Hom(g0 ∧ g1 , g1 ) ⊕ Z 2 (L n,m , L n,m ) ∩ Hom(S 2 g1 , g0 ) = A⊕ B ⊕C and g1 = L n,m where g0 = L n,m 0 1 . Note that the third component has been studied in [1, 4], and the first and the second component has been studied in [8, 9] respectively.

3 Method for Classifying Filiform Lie Superalgebras Firstly, we are going to see the following theorem ([10], p. 14) related to filiform color Lie superalgebras which are a generalization of filiform Lie superalgebras. Theorem 2 (1) Any filiform (G, β)-color Lie superalgebra law μ is isomorphic to μ0 + ϕ where μ0 is the law of the model filiform (G, β)-color Lie superalgebra and ϕ is an infinitesimal deformation of μ0 verifying that ϕ(X 0 , X ) = 0 for all X ∈ L, with X 0 the characteristic vector of the model one. (2) Conversely, if ϕ is an infinitesimal deformation of a model filiform (G, β)color Lie superalgebra law μ0 with ϕ(X 0 , X ) = 0 for all X ∈ L, then the law μ0 + ϕ is a filiform (G, β)-color Lie superalgebra law iff ϕ ◦ ϕ = 0. In general, color Lie superalgebras are a generalization of Lie superalgebras and consequently filiform color Lie superalgebras are a generalization of filiform Lie

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superalgebras. Indeed, Lie superalgebras are defined through antisymmetric and symmetric products. However, in color Lie superalgebras the product is neither symmetric nor antisymmetric and it is defined by means of a commutation factor β. Moreover, the basic tool for defining color Lie superalgebras is a grading which is determined by an abelian group G (instead of Z2 associated with Lie superalgebras). Therefore, if we replace the generic abelian group G by Z2 and the commutator factor β by β(i, j) = (−1)i j ∀i, j ∈ Z2 in the above theorem, we will obtain that any filiform Lie superalgebra is a linear deformation of the corresponding model filiform Lie superalgebra. Thus, we have the main result that we are going to use in our method for classifying filiform Lie superalgebras, that is, every filiform Lie superalgebra can be expressed by μ0 + Ψ with μ0 the law of the model filiform Lie superalgebra and Ψ a linear (infinitesimal) deformation verifying Ψ ◦ Ψ = 0, with Ψ ◦ Ψ (x, y, z) = Ψ (Ψ (x, y), z) + Ψ (Ψ (z, x), y) + Ψ (Ψ (y, z), x) Then, firstly we are going to consider Ψ such that Ψ = aΨ A + bΨ B + cΨC with a, b, c ∈ C, Ψ A ∈ A, Ψ B ∈ B and ΨC ∈ C. Secondly we are going to impose the condition to be integrable, that is Ψ ◦ Ψ = 0. Finally we will classify the family of laws μ0 + Ψ obtained. There are two important facts to be considered: first we will always have c = 0 otherwise we have Lie superalgebras that are actually Lie algebras (degenerate cases); second if a = b = 0 then Ψ is always integrable.

4 Classification in Low Dimensions Thus, following the aforementioned method, the classification of filiform Lie superalgebras in low dimensions has been obtained. Such classification can be seen in the following theorem (for more details it can be consulted [11]). Theorem 2 (Classification’s Theorem) Let g = g0 ⊕ g1 be a non-degenerated filiform Lie superalgebra with dim(g0 ) = n + 1 and dim(g1 ) = m. If the total dimension is less or equal to seven then the law of g will be isomorphic to a law μi(n+1,m) of j the following list of laws. Note that the laws μi(n+1,m) and μ(n+1,m) are not isomorphic for i = j. Furthermore two laws of the same one-parametric family μi(n+1,m) (α) and μi(n+1,m) (α ) with α = α are also non-isomorphic.

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List of Laws Pair of Dimensions: n = 2, m = 1. μ1(3,1) : μ0 + ϕ1,2 Pair of Dimensions: n = 3, m = 1. μ1(4,1) : μ0 + ϕ1,3 Pair of Dimensions: n = 4, m = 1. μ1(5,1) : μ0 + ϕ1,4 μ2(5,1) : μ0 + Ψ1,4 + ϕ1,4 Pair of Dimensions: n = 5, m = 1. μ1(6,1) : μ0 + ϕ1,5 μ2(6,1) : μ0 + Ψ1,4 + ϕ1,5 μ3(6,1) : μ0 + Ψ1,5 + ϕ1,5 μ4(6,1) : μ0 + Ψ2,5 + ϕ1,5 μ5(6,1) : μ0 + Ψ1,4 + Ψ2,5 + ϕ1,5 Pair of Dimensions: n = 2, m = 2. μ1(3,2) : μ0 + ϕ1,1 μ2(3,2) : μ0 + ϕ1,2 Pair of Dimensions: n = 3, m = 2. μ1(4,2) : μ0 + ϕ2,3 μ2(4,2) : μ0 + ϕ1,2 2 μ3(4,2) : μ0 + Ψ1,1 + ϕ1,2 4 μ(4,2) : μ0 + ϕ1,3 2 μ5(4,2) : μ0 + Ψ1,1 + ϕ1,3 Pair of Dimensions: n = 4, m = 2. μ1(5,2) : μ0 + ϕ2,4 μ2(5,2) : μ0 + ϕ1,3 2 μ3(5,2) : μ0 + Ψ1,1 + ϕ1,3 4 μ(5,2) : μ0 + ϕ1,4 2 μ5(5,2) : μ0 + Ψ1,1 + ϕ1,4 6 μ(5,2) : μ0 + Ψ1,4 + ϕ1,3 2 μ7(5,2) : μ0 + Ψ1,4 + Ψ1,1 + ϕ1,3 8 μ(5,2) : μ0 + Ψ1,4 + ϕ1,4 2 μ9(5,2) : μ0 + Ψ1,4 + Ψ1,1 + ϕ1,4 Pair of Dimensions: n = 2, m = 3. 3 + ϕ2,2 μ1(3,3) : μ0 + Ψ1,1

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A Method for Classifying Filiform Lie Superalgebras

μ2(3,3) μ3(3,3) μ4(3,3) μ5(3,3) μ6(3,3) μ7(3,3)

: μ0 + ϕ2,2 : μ0 + ϕ1,1 + ϕ2,2 : μ0 + ϕ1,2 3 : μ0 + Ψ1,1 + ϕ1,2 : μ0 + ϕ1,1 3 : μ0 + Ψ2,1 + ϕ1,1

Pair of Dimensions: n = 3, m = 3. 3 + ϕ1,1 + 21 ϕ2,3 μ1(4,3) : μ0 + Ψ2,1 2 μ(4,3) (α) : μ0 + ϕ1,1 + αϕ2,3 3 μ3(4,3) : μ0 + Ψ1,1 + ϕ1,2 + ϕ2,3 4 μ(4,3) : μ0 + ϕ1,2 + ϕ2,3 2 3 μ5(4,3) : μ0 + Ψ1,1 + Ψ1,1 + ϕ1,2 6 2 μ(4,3) : μ0 + Ψ1,1 + ϕ1,2 3 μ7(4,3) : μ0 + Ψ1,1 + ϕ1,2 8 μ(4,3) : μ0 + ϕ1,2 2 3 3 μ9(4,3) : μ0 + Ψ1,1 + Ψ1,1 + 2Ψ2,1 + ϕ2,3 10 2 3 μ(4,3) : μ0 + Ψ1,1 + 2Ψ2,1 + ϕ2,3 3 μ11 (4,3) : μ0 + Ψ1,1 + ϕ2,3 12 μ(4,3) : μ0 + ϕ2,3 2 3 μ13 (4,3) (α) : μ0 + Ψ1,1 + αΨ2,1 + ϕ1,3 3 μ14 (4,3) : μ0 + Ψ2,1 + ϕ1,3 15 2 3 μ(4,3) : μ0 + Ψ1,1 + Ψ1,1 + ϕ1,3 16 3 μ(4,3) : μ0 + Ψ1,1 + ϕ1,3 μ17 (4,3) : μ0 + ϕ1,3 Pair of Dimensions: n = 2, m = 4. μ1(3,4) : μ0 + ϕ2,1 μ2(3,4) : μ0 + ϕ1,1 + ϕ2,2 μ3(3,4) : μ0 + ϕ1,1 4 μ4(3,4) : μ0 + Ψ2,1 + ϕ1,1 5 μ(3,4) : μ0 + ϕ2,2 3 μ6(3,4) : μ0 + Ψ1,1 + ϕ2,2 7 4 μ(3,4) : μ0 + Ψ1,1 + ϕ2,2 3 4 μ8(3,4) : μ0 + Ψ1,1 + Ψ1,1 + ϕ2,2 9 μ(3,4) : μ0 + ϕ1,2 3 μ10 (3,4) : μ0 + Ψ1,1 + ϕ1,2 4 μ11 (3,4) : μ0 + Ψ1,1 + ϕ1,2 12 3 4 μ(3,4) : μ0 + Ψ1,1 + Ψ1,1 + ϕ1,2

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Acknowledgements First author was supported by Agencia Estatal de Investigación (Spain), grant MTM2016-79661-P (European FEDER support included, UE). Second author was supported by the PCI of the UCA ‘Teoría de Lie y Teoría de Espacios de Banach’, by the PAI with project number FQM298.

References 1. M. Bordemann, J.R. Gómez, Yu. Khakimdjanov, R.M. Navarro, Some deformations of nilpotent Lie superalgebras. J. Geom. Phys. 57 (2007) 1391–1403. 2. D.B. Fuks, Cohomology of Infinite-Dimensional Lie algebras. Plenum publishing Copr. (1986). 3. J.R. Gómez, Yu. Khakimdjanov, R.M. Navarro, Some problems concerning to nilpotent Lie superalgebras. J. Geom. Phys. 51 (2004) 473–486. 4. J.R. Gómez, Yu. Khakimdjanov, R.M. Navarro, Infinitesimal deformations of the Lie superalgebra L n,m . J. Geom. Phys. 58 (2008) 849–859. 5. Yu. Khakimdjanov, Variété des lois d’algèbres de Lie nilpotentes. Geometriae Dedicata 40 (1991) 269–295. 6. Yu. Khakimdjanov, Varieties of Lie Algebra Laws. Handbook of Algebra, vol.2, Elsevier Science, 2000, 509–541. 7. Yu. Khakimdjanov, Affine Structures on Filiform Lie Algebras. Amer. Math. Soc. Transl. (2) Vol. 213, 2005. 8. Yu. Khakimdjanov, R.M. Navarro, Deformations of filiform Lie algebras and superalgebras. J. Geom. Phys. 60 (2010) 1156–1169. 9. Yu. Khakimdjanov, R.M. Navarro, A complete description of all the infinitesimal deformations of the Lie superalgebra L n,m . J. Geom. Phys. 60 (2010) 131–141. 10. Yu. Khakimdjanov, R.M. Navarro, Filiform color Lie superalgebra. J. Geom. Phys. 61 (2011) 8–17. 11. R.M. Navarro, Low-dimensional filiform Lie superalgebras. J. Geom. Phys. 108 (2016) 71–82. 12. M. Scheunert, The Theory of Lie Superalgebras. Lecture Notes in Math. 716 (1979). 13. M. Scheunert, R.B. Zhang, Cohomology of Lie superalgebras and of their generalizations. J. Math. Phys. 39 (1998) 5024–5061. 14. M. Vergne, Cohomologie des algèbres de Lie nilpotentes. Application à l’étude de la variété des algèbres de Lie nilpotentes. Bull. Soc. Math. France. 98 (1970) 81–116.

Equivalence of Vector Field Realizations of Lie Algebras from the Lie Group Point of View Maryna Nesterenko and Severin Pošta

Abstract The equivalence of vector field realizations of Lie algebras is considered from the viewpoint of the Lie algebra and also from the corresponding (local) Lie group. It is shown that for some stages in the establishment of the equivalence things are simpler in the case of the Lie group. A lemma about the equivalence of the sum of realizations is proposed.

1 Introduction Representation of Lie algebras by vector fields on manifolds is still of great interest and has a huge amount of applications, see, e.g. [5] and references therein. One of the key problems that appear in realization theory is to detect if two given sets of vector fields representing the same Lie algebra are equivalent or not. This question was recently considered in [4], where the notion of equivalence of Lie algebra realizations and the quantities stable under the equivalence transformations were discussed; and an algorithm establishing the equivalence between any two realizations of a Lie algebra was proposed. In particular notions of the realization rank, essential variables and realization subalgebra were introduced and applied to realization identification. The equivalence of the unfaithful realizations was considered as well. It turned out, that a part of the equivalence of Lie algebra realizations come from the properties

M. Nesterenko Institute of Mathematics of NAS of Ukraine, 3 Tereshchenkivska Str., Kyiv 01004, Ukraine e-mail: [email protected] S. Pošta (B) Department of Mathematics, Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague, Trojanova 13, 120 00 Prague, Czech Republic e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2018 V. Dobrev (ed.), Quantum Theory and Symmetries with Lie Theory and Its Applications in Physics Volume 1, Springer Proceedings in Mathematics & Statistics 263, https://doi.org/10.1007/978-981-13-2715-5_30

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of the corresponding group actions, which is what motivates the present work, for example, transitivity of the realization (which is one of the key invariant properties) is not obvious from the Lie algebra point of view, but it is shown in Sect. 4 how to check it explicitly for the corresponding group. In this paper we also propose the new statement concerning the equivalence of formal sum of realizations. The paper is arranged as follows. In Sect. 2 we give all necessary notions and definitions, then in Sect. 3 we introduce equivalence of realizations and corresponding local groups. Section 4 is devoted to transitive and intransitive Lie groups, to their properties, that are useful in establishment of the equivalence; and to the practical detection of the transitivity. After that we propose the statement about the equivalence of realizations constructed as a sum of two different realization with non-overlapping variables. And, finally, in Conclusions we discuss key steps of the practical algorithm that allows to detect the equivalence existence.

2 Preliminaries Let g be an n-dimensional Lie algebra spanned by a basis {e1 , e2 , . . . , en } with the structure constants Cikj ∈ R, where i, j, k = 1, 2, . . . , n. We denote an open domain of Rm as M and the Lie algebra of vector fields on it as Vect(M). The vector fields are taken in a form of linear first-order differential operators with analytical coefficients and the Lie product of vector fields is given by their commutator. The automorphism group of g is denoted by Aut(g). A realization of the Lie algebra g is a representation of g by vector fields on M (i. e., a homomorphism R : g → Vect(M)). Let x = (x1 , . . . , xm ) be the coordinates on M and X i ∈ Vect(M) be the images of the basis elements ei , i = 1, . . . , n of g under the realization R, i. e. X i = R(ei ) =

m  j=1

ξ ij (x)∂ j , where ∂ j =

∂ . ∂x j

(1)

Let G be the local n-parametric transformation group that corresponds to the vector fields X i , that is, X i are the infinitesimal generators of the action of G on M. Denote the set of group parameters as a = (a1 , . . . an ), ai ∈ R. Then the group transformations have the form x j = f j (x1 , . . . , xm , a1 , . . . , an ) = f j (x, a).

(2)

Let us fix a point x ∈ M and let Rx be a realization of g at this point. Consider the linear map Rx : g → V ect (M)(x) that transforms a vector v ∈ g to it’s image R(v(x)) at x. The matrix that corresponds to this linear map is the n by m matrix ξ formed by the coefficients of the realization (1):

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⎛

⎞ ξ11 (x) ξ12 (x) . . . ξ1m (x) ⎜ξ21 (x) ξ22 (x) . . . ξ2m (x)⎟ ⎜ ⎟ ξ(x) = ⎜ . .. . . .. ⎟ . ⎝ .. . . . ⎠ ξn1 (x) ξn2 (x) . . . ξnm (x)

(3)

The rank of the linear map Rx , or, equivalently, the rank of the matrix ξ(x) at a point x is called a rank of realization R at point x and is denoted rank Rx . The realization rank value possess the obvious inequality 0 ≤ rank Rx ≤ n. A point x ∈ M is called a regular point of a realization R if there exists a neighborhood Ux of x, such that rank R y is constant for all y ∈ Ux . A point that is not regular is called singular.

3 Equivalence Notion of equivalence of realizations has been formulated and revisited several times [2, 4, 5], on the basis of these papers we can conclude that two realizations are equivalent on a neighborhood Ux of a regular point x, if they can be transformed to the identical form by means of non-singular automorphic basis changes (ei → e˜i ) (that do preserve the structure constants tensor) and 1 to 1 changes of variables (diffeomorphism) (xl → yl = ϕl (x)) with non-zero Jacobi determinant. Let us have a diffeomorphism of M such that for the corresponding x, y ∈ M we have y1 = ϕ1 (x1 , . . . , xm ), y2 = ϕ2 (x1 , . . . , xm ), . . . , ym = ϕm (x1 , . . . , xm ). Then the realization of the form (1) transforms to the following: ˜ i) = R(e

m  l=1

⎛ ⎞ m m   (x) ∂ϕ l ⎝ ⎠ ∂ yl . ξ˜il (y)∂ yl = ξi j (x) ∂ x j l=1 j=1

(4)

Note, that the coefficients ξ˜il (y) are written in terms of y using the inverse transformation ϕ −1 . Unfortunately, direct application of this definition to real problems is rather cumbersome and complicated. This suggest additional study of the equivalence properties. Let us look at the equivalence of realizations from the view point of the corresponding local groups [1, 3]. Consider two n-parametric groups on the same space M: x j = f j (x, a) and y j = F j (y1 , . . . , ym , b1 , . . . , bn ) = F j (y, b). Equivalence (or similarity) of these groups means that there exists invertible transformations y j = Φ j (x) and bi = βi (a), such that one group transforms into the other. Let ξi j (x) and ξ˜i j (y) be the infinitesimal transformations of the above groups.

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Note that the change of group parameters (functions βi (a)) introduce a linear transformation of the infinitesimal vectors of the group, what in terms of realization means a Lie algebra isomorphism; and the functions f j (x) do provide a nonsingular change of coordinates. By the non-singular change of basis, the structure of the similar groups can be made coinciding, so the groups of the same structure are equivalent if and only if ∂ F (x) there exist y j = F j (x) such that ξ˜i j = ξil ∂jxl . Therefore the ranks of the matrices formed by infinitesimals are necessary the same for the equivalent groups. In terms of realizations this means that the equivalence requires the same realization ranks.

4 Transitive Groups A group G is transitive on M if each point of M can be transformed to any other point of M by at least one of transformations from G; otherwise, the group is called intransitive. Let G acts on Rm and decomposes it to a family of invariant d-dimensional manifolds (orbits). Then G is called primitive if there is no such a decomposition (one orbit only) and imprimitive otherwise. Note that all primitive groups are transitive and intransitivity is a particular case of imprimitivity. For a group to be transitive it is necessary that n ≥ m and rank Rx = m. When m = n, G is called simply transitive and the corresponding realization is called generic. The case of the simply transitive group corresponds to the zero subalgebra realization, see [4]. Transitive groups do not have invariants and their equivalence is established by the usual subgroups, but for the intransitive groups in the case rank Rx < m invariant varieties do exist and the intransitive group G induces the q-parametric subgroup G q acting on the invariant variety. In this case the infinitesimals of the induced group are of the form [1] ηik = ηik (x1 , . . . , xm−q , ψm−q+1 (x1 , . . . , xm−q ), . . . , ψm (x1 , . . . , xm−q )), where i = 1 . . . , n, k = 1, . . . , m − q and the functions ψm−q+1 , . . . , ψm define the parameter-free group of transformations for the last q variables x j = ψ j (x1 , . . . , xm−q ),

j = m − q + 1, . . . , m.

Transitivity of the given Lie group can be easily verified. It is enough to know it’s defining equations [3], then the group G is transitive if and only if for a regular point x0 ∈ M the group G contains exactly n − m independent infinitesimal transformations, such that their expansions into the power series with respect to x − x0 start from the summand of the order one or higher. If n < m and rank Rx = p then the coordinates x j and y j can be chosen in such form that ξi j = 0 and ξ˜i j = 0 for j = p + 1 . . . m. This completely coincides with

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the notion of the reduction of unessential variables introduced in [4]. Note that after this change of coordinates, the explicit formulas for the group transformation of unessential variables do not depend on the group parameters. This means that they are stable (invariant) under the group action and there exist the respective subgroup and corresponding subalgebra (or series of subalgebras for intransitive case) of the dimension m − p.

5 Formal Sum of Realizations Let R  = ei and R  = ei be two realizations of g in the non-overlapping essential variables x1 , . . . , xm 1 and xm 1 +1 , . . . , xm 2 respectively. Consider the realization R = ei + ei formed as a formal sum of R  and R  . Using the general form [ei , e j ] = Cikj ek of the commutation relations of g we can see that the set of vector fields obtained by summation do realize the same Lie algebra, indeed, [ei + ei , ej + ej ] = [ei , ej ] + 0 + 0 + [ei , ej ] = Cikj (ek + ek ). The commutators [ei , ej ] and [ei , ej ] are equal to zero since the variables do not overlap. Suppose that the realizations R  and R  are taken in the second canonical coordinates, this means that the subgroups corresponding to the subalgebras preserve the coordinates xm 1 +1 , . . . , xm in the first case and x1 , . . . , xm 1 , xm 2 +1 , . . . , xm in the second case. After the summation only the coordinates xm 2 +1 , . . . , xm are to be preserved. This results in the following lemma. Lemma 1 Let a realization R1 of a Lie algebra g corresponds to a subalgebra h1 and a realization R2 of the same Lie algebra corresponds to a subalgebra h2 . Then the formal sum R of these realizations taken in non-overlapping variables corresponds to the subalgebra h1 h2 . Example 1 Consider two inequivalent realizations of the two-dimensional abelian Lie algebra 2 A1 [5] e1 = ∂1 , e2 = ∂2 and e1 = ∂1 , e2 = x2 ∂1 . In these cases rank R1 = 2 and rank R2 = 1. The first realization corresponds to the zero subalgebra h1 = {0}, since K er (R1 (x = (0, 0))) = 0. The second realization corresponds to the subalgebra h2 = e2 = K er (R2 (x = (0, 0))). It is easy to see that the first realization is generic and the corresponding Lie group is simply transitive. Consider the formal sum of these realizations R3 = R1 + R2 (R1 for the variables (x1 , x2 ) and R2 for the variables (x3 , x4 )), namely R3 (e1 ) = ∂1 + ∂3 ,

R3 (e2 ) = ∂2 + x4 ∂3 .

As far as [∂1 + ∂3 , ∂2 + x4 ∂3 ] = 0, R3 realizes the Lie algebra 2 A1 in the space of four variables (x1 , x2 , x3 , x4 ) and rank R3 = 3, what means that we can reduce variables to the essential ones as it was indicated in the previous section.

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Indeed, the diffeomorphism ϕ given by the non-singular functions x1 = ϕ1 (x1 , . . . , x4 ) = x1 , x2 = ϕ2 (x1 , . . . , x4 ) = x2 , x3 = ϕ3 (x1 , . . . , x4 ) = x1 − x3 + x2 x4 , x4 = ϕ4 (x1 , . . . , x4 ) = x4 transforms the realization R3 to the equivalent realization that corresponds to the subalgebra h3 = h1 ∩ h2 = {0}.

6 Conclusions To conclude let us revisit the procedure allowing us to compare and match two realizations of a given Lie algebra [4] and emphasize the advantages coming from the group point of view. 1. Preliminary testing. At this stage Lie groups give us the same requirements, mainly the coincidence of the structure of both groups. The important thing is that the structure constant tensor can be changed by the invertible functions of group parameters instead of the matrix transformation in the case of Lie algebra. To detect unfaithful realizations one should test the corresponding subgroup, in the case of unfaithful realizations the subgroups are normal. 2. Reduction to essential variables. This step is similar to the case of Lie algebra: it is necessary to study the rank of the infinitesimal coefficients and to replace non-essential variables by the absolute invariants of the group. 3. Subalgebra detection. In contrast to the algebraic approach, where the subalgebra can be found as the kernel of linear transformation in the origin of coordinates, in the case of Lie groups we are obliged to study the invariant varieties of the group and to construct the induced subgroup, which is much more complicated then the description of the kernel. 4. Construction of diffeomorphism and automorphism. At this stage we are expected to solve a system of partial differential equations to construct the desired diffeomorphism in the case of Lie algebra and to detect (if necessary) an automorphism after that. In the case of Lie groups we will solve the equations from the First fundamental Lie theorem and then a system of functional equations, which gives us both the parameter and variable transformations. So it seems to be reasonable to apply both approaches to a given task in order to simplify calculations and crosscheck the results.

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Acknowledgements MN is grateful for the hospitality extended to her at the Department of mathematics, FNSPE, Czech Technical University in Prague, where part of this work was done. SP is grateful for the hospitality extended to him at the Institute of Mathematics of NAS of Ukraine.

References 1. L.P. Eisenhart, Continuous groups of transformations, (Dover Publications, New York, 1961), pp. 79–80. 2. D. Gromada, S. Pošta, On classification of Lie algebra realizations, math-ph/1703.00808v1, (2017), 20 pages. 3. S. Lie, Theory of Transformation groupes I, ed., transl. J. Merker, (Springer, 2015), pp. 341–378. 4. M. Nesterenko, S. Pošta, Comparison of realizations of Lie algebras, Procedings of the XXVth International Conference on Integrable Systems and Quantum symmetries, J. Phys.: Conf. Ser. (2018), to appear. 5. R.O. Popovych, V.M. Boyko, M.O. Nesterenko, M.W. Lutfullin, J. Phys. A36 (2003) 7337– 7360.