The Mathematical Sciences after the Year 2000: Proceedings of the International Conference (Beirut, Lebanon, 11–15 January 1999) 9810242239, 9789810242237

Citation preview

Proceedings of the International Conference on

Beirut, Lebanon

11 -15 January 1999

Editors

Khalil Bitar Ali Chamseddine Wafic Sabra American University of Beirut, Lebanon

,III» World Scientific

Singapore· New Jersey· London· Hong Kong

Published by

World Scientific Publishing Co. Pte. Ltd. POBox 128, Farrer Road, Singapore 912805 USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

THE MATHEMATICAL SCIENCES AFfER THE YEAR 2000 Copyright © 2000 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book. or parts thereof, may not be reproduced in any form or by any means. electronic or mechanical. including photocopying. recording or any information storage and retrieval system now known or to be invented. without written permission from the Publisher.

For photocopying of material in this volume. please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive. Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.

ISBN 981-02-4223-9

Printed in Singapore.

ACKNOWLEDGMENTS These papers are the proceedings of a conference that was organized to inaugurate the Center for Advanced Mathematical Sciences (CAMS) at the American University of Beirut (AUB). Many world renowned experts in various disciplines in mathematics and physics were present for the occasion from January 11 to 15, 1999 at Hotel AI-Bustan, Beit Mery, and at AUB, and gave reviews of their respective fields. One of the principal aims of the conference was to raise public awareness of the recent revolutionary advances in science and mathematics. CAMS is a research center hosted by AUB, the oldest and largest Americanchartered university in the Middle East. The aim of CAMS is to provide a focused environment for research in pure mathematics, numerical methods, applied mathematics, theoretical physics, theoretical computer science, financial mathematics and mathematical biology. Given the seminal historical role of the Arab Middle East in the development of mathematics and astronomy, it is appropriate for the region to have established such a center. The establishment of CAMS is especially timely, in view of the significant scientific talent in the region and among its diaspora. At present, CAMS provides a vehicle for promoting research and graduate studies in the mathematical sciences, as well as a focal point for collaborative research among scientists and mathematicians in Lebanon and in the region at large. The establishment of CAMS would not have been possible without the vision of Dr. Nicola Khuri, who played a key role in presenting the proposal of CAMS to AUB. The help of Dr. Richard Debs, the Chairman of the AUB Board of Trustees, in the choice of AUB to host CAMS, is greatly appreciated. Furthermore, the role of the International Advisory Committee of CAMS, consisting of distinguished mathematicians and physicists, and especially the contributions of its Chairman, Sir Michael Atiyah, in providing scientific directions to CAMS, has been very instrumental in promoting the image and the role of the Center. Special thanks are due to the support of the Lounsbery foundation for a twoyear grant. CAMS also acknowledges the Abdus Salam International Center for Theoretical Physics in Trieste, Italy, and to its director Miguel Virasoro, who provided CAMS with a grant in the form of seed money. CAMS is also fortunate to have the generous financial support of Dr. Richard Debs, Mrs. Geraldine Ford, Dr. Nicola Khuri, Mr. Dirrar Al Ghanem, Dr. A. M. AI Qattan, Dr. Kamal Shair and Mr. Munib Masri. The support of Mrs. Mirna Al Bustani through her generous offer to CAMS to host a conference at Hotel Al Bustan every January, is greatly appreciated as well. v

vi

I would also like to acknowledge the help of the organizing committee for this conference, Dean Khalil Bitar, Dr. Ahmad Nasri, Dr. Wafic Sabra and Dr. Jalal Shatah. The tireless efforts of my assistant, Miss Rim Jaber, in helping the participants in the conference and in preparing this volume also deserve recognition.

Ali Chamseddine Director Center for Advanced Mathematical Sciences American University of Beirut

PROGRAM Monday, January 11 (AI-Bustan Hotel, Beit-Mery) Chair: Nicola Khuri , Rockefeller University 1.

2.

3.

Opening Ceremony by : • President John Waterbury • H.E. Salim El Hoss Prime Minister • H.E. Mohammad Yousef Baydoun Minister of Education, Higher Education and Culture • Nicola Khuri Rockefeller University, USA Sir Michael Atiyah University of Edinburgh,UK "Geometry and Physics in the 20th Century" Roman Jackiw MIT, USA "Field Theory:Why Have Some Physicists Abandoned It?" Chair: Khalil Bitar, Dean A&S AUB

4.

5.

Ivar Ekeland University of Paris IX, France "Mathematical Models in Economics" John Ball Oxford University, UK "From the Calculus of Variations to the Discovery of New Materials"

Tuesday, January 12 ( Issam Faris Hall, AUB) Chair: Ali Chamseddine, Director CAMS, AUB 6. Alain Connes College de France "Trace Formula in Noncommutative Geometry and the Zeros of the Riemann Zeta function" 7. Sir Michael Berry University of Bristol, UK "Extreme Twinkling and its Opposite" vii

viii

8. Henry McKean Courant Institute NY, USA "Probability and its Many Applications"

Chair: Sir Michael Atiyah, University of Edinburgh 10 Michael Duff Texas A&M, USA "A Layman's Guide to M-Theory" 11 Jacob Palis IMPA, Brazil "A Global Perspective on Dynamical Systems" 12 Miguel Virasoro ICTP, Trieste, Italy "International Cooperation in Basic Sciences: the Abdus Salam ICTP Experience"

Chair: Nicola Khuri, Rockefeller University 13 Murray Gell-Mann Caltech, USA "Simplicity and Complexity"

Wednesday, January 13 • Excursion to Baalbeck (All day)

Thursday, January 14 assam Faris Hall, AUB)

Chair: Jalal Shatah, Courant Institute NY 14 Andre Martin CERN-Geneva, Switzerland "Spying on Nature: How general Principles Lead to Physical Results" 15 Jean Pierre Bourguignon Institut des Hautes Etudes Scientifiques, IHES, France "Spinors and Special Geometries" 16 Eduardo Vesentini Accademia Lincei Rome, Italy "Variations on a Theorem of Banach and Stone" 17 George Papanicolaou Stanford University, USA "Mathematical Problems in Geophysical Wave Propagation"

ix

18 S.R. Varadhan Courant Institute, USA "Does Size Make a Difference? From the Microscopic to Macroscopic Scales" 19 Louis Nirenberg Courant Institute NY, USA "Variational Methods for Nonlinear Problems"

Friday, January 15 (409 Nicely, AUB) Chair: Jihad Touma, CAMS, AUB

20 Spenta Wadia Tata Institute of Fundamental Research, India "Black Holes, Information Paradox and String Theory" 21 Hermann Nicolai Einstein Institute, Germany "Hidden Symmetries in Supergravity : a Window on M-Theory" 22 Kelly Stelle Imperial College, London "M-Theory and the Universe" 23 Ramzi Khuri City University of New York, USA "Black Holes, Thermodynamics and Polymers" Chair: Walic Sabra, CAMS, AUB

24 V. Mukhanov Maxmilians University Munich, Germany "The Origin of the Universe: Conjectures vs. Facts" 25 Chris Hull Queen Mary College, London "String Theory and Duality, Space and Time"

CONTENTS

v

Acknowledgments Program

vii

Geometry and Physics in the 20 th Century Sir Michael Atiyah

1

A Layman's Guide to M-Theory Michael Duff

10

Mathematical Models in Economics Ivar Ekeland

37

Duality and Strings, Space and Time Chris Hull

45

Field Theory - Why Have Some Physicists Abandoned It? Roman Jackiw

65

Black Holes, Thermodynamics and Polymers RamziKhuri

71

Spying on Nature: How General Principles Lead to Physical Results Andre Martin

77

Probability & its Applications Henry McKean

90

Inflation: Conjectures vs. Facts V Mukhanov

96

Variational Methods in Nonlinear Problems Louis Nirenberg xi

116

xii

Homoclinic Bifurcations: From Poincare to Present Time

123

Jacob Palis Brane Charge Cohomology and Tension Lattices

135

Kelly Stelle Scaling Limits of Large Interacting Systems

144

S.R.S. Varadhan The Microscopic Origin of Black Hole Thermodynamics in String Theory

159

Spenta Wadia Organizing Committee

169

List ofSpeakers

170

List ofParticipants

171

Photos

175

GEOMETRY AND PHYSICS IN THE 20TH CENTURY MICHAEL ATIY AH University of Edinburgh. Scotland.

1

Introduction

It is hard to predict what will happen to mathematics and physics in the next century, and the best indicator is to look back over the past century, particularly the last quarter. The most striking thing that emerges is that the major events have been unpredictable and so we should expect similar surprises in the future. The past twenty-five years have seen a spectacular and totally unexpected new interaction between geometry and quantum theory and I will try to survey this scene. To put the matter into some philosophical framework it is perhaps useful to consider the eternal triangle, not the two boys and a girl scenario of Hollywood movies, but the much deeper one depicted below

PHYSICAL WORLD

MATHEMATICS

I

The relation between the human mind and our physical environment is the staple diet of philosophers and greatly exercised Kant. Quantum mechanics, with its role for the observer, has given this old story a new slant. Similarly the question whether mathematics is an invention of the human mind or a reflection of physical reality goes back to Plato. But if we accept the Platonic view of mathematics then there remains the mystery of why it is so successful in dealing with the physical world, even at scales well outside our personal experience. Wigner was one of those who puzzled over this conundrum and the developments of recent years, which I

2

shall be surveying, only increase the difficulty. In the bizarre world being unravelled by Witten and other physicists it is hard to know where mathematics ends and physics begins.

2

The classical era

To set the scene for modern developments it is necessary to take a brief look at the earlier history, before the advent of quantum mechanics. From our present perspective we can say that a very comprehensive picture emerges when we think of the fundamental forces of nature as producing a distortion or curvature. The first clear instance of this occurs already in Newton's laws of motion. A particle, in the absence of forces, moves in a straight line with uniform velocity. Under the action of a force it deviates from this path and moves along some curve. In Newton's picture of the world Euclidean geometry provides a fixed background in which all the action takes place, including notably the motion of the planets under gravitational force. In Einstein's General Relativity on the other hand space, or rather 4-dimensional space-time, becomes curved and itself embodies the gravitational force. This Einstein picture of gravitational force as curvature has become the paradigm for theories of other forces. Thus Maxwell's theory of electro-magnetism can be interpreted, following ideas of Hermann Weyl as curvature in an enlarged space-time of 5 dimensions. The extra dimension is a circle or phase and the way this twists relative to the other 4 coordinates is interpreted as the electro-magnetic force. This process can be carried further by envisaging additional dimensions which are to be related to very short-range nuclear forces. This "Kaluza-Klein" programme rests on the assumption that these additional dimensions are compact and very small, like a tightly curved circle. The idea is that we cannot directly detect such additional small dimensions. They manifest themselves only indirectly through their effects as nuclear forces. 3

The quantum era The 1920's saw the advent of quantum mechanics. This appeared to be ungeometrical in character with its emphasis on operators in Hilbert space. Certainly functional analysis, notably spectral theory, received a big boost from quantum theory. The future of physics seemed to lie more with analysis than geometry. But it is of course a mistake to regard the branches of mathematics as self-contained entities. Analysis without a geometrical interpretation can be austere and unintuitive, Geometry without analysis lacks the tools to tackle deep problems. So we should not be too surprised, in retrospect, that by the end of the 201h century geometry was back in the picture.

3

One of the key points leading to the re-emergence of geometry is that, during the 20 th century, geometry itself underwent an internal revolution. I refer to the rise of topology and the emphasis on global problems. The prototype of this is Hodge's theory of harmonic forms which dates back to the 1930's, but the roots go back to Riemann and Poincare in the previous century. In a sense the classical link between geometry and physics through the motion of curvature has been extended, in this century, to include a "quantum link", between topology and quantum theory. That such a link might exist could perhaps have been predicted from the basic fact that the key characteristics of quantum theory are: 1. 2.

Discrete quantities (e.g. energy levels, angular momentum etc.). Global effects (wave-functions not localised).

These are also the characteristics of topology which studies the global properties of various spaces which are unchanged by continuous variation, and so are discrete. The first real indications that this putative "quantum link" really existed can be traced to Dirac's argument, using magnetic monopoles, to explain the "quantisation of electric charge", i.e. the fact that all particles in nature have an electric charge which is an integer multiple of the charge of the electron. Closely related is the Aharanov Bohm effect which shows that the phase of an electron can change even when it is moving in a force-free region of space, provided there is a magnetic field elsewhere. Moreover the Kaluza-Klein programme, invoking extra dimensions, provided ample room for interesting topology to manifest itself. The internal topology of the extra dimensions could affect the quantum oscillations, in the manner of Hodgetheory, and this would have physical effects in space-time, particularly at high energies. But the full development of the link between quantum theory had to await further advances in both fields. In physics the early quantum mechanics moved on to quantum field theory with its long struggle with infinities, renormalisation and internal symmetry groups. After 50 years a lfIature stage had been reached where the "standard theory" of elementary particles was in a reasonably satisfactory stage. However the ultimate goal of incorporating gravity remained elusive until string theory appeared. This has made considerable progress and, while the ultimate theory remains out of sight, there are hopeful indications. Meanwhile topology and geometry were also maturing. 50 years after Hodge theory and the early work of Lefschetz, a highly sophisticated theory had emerged. At some stage it was inevitable that a new link would be established and this began with the mutual interest of both physicists and geometers in the famous Dirac equation. Since then the flood-gates have opened and the tentative link of the early days has been replaced by a structure which can be compared with the Golden Gate

4

Bridge in San Francisco. The traffic goes in both directions though at certain tImes the flow is much more in one direction than the other. In this lecture I will focus on the direction leading from physics to geometry. I will list some, but not all, of the spectacular results in geometry that have come directly out of this link to quantum theory. It makes an impressive story. I will select examples based on increasing space-time dimensions. This time I refer not to adding extra dimensions but to the contrary process of forgetting some dimensions. In real physics some spatial dimensions are irrelevant, either because we are constrained to lie on a surface (e.g. dealing with a thin film of atoms) or because axial symmetry (e.g. a long wire) means that we work with a cross-section. So I will choose examples from dimensions 2,3,4 respectively. As will become clear the examples are from quite different areas of geometry and appear to have little in common. Only quantum theory provides a unifying framework

4

Quantum Cohomology

Let me recall that "intersection theory" in geometry is concerned with counting the number of common points of a number of curves or surfaces. For example the number of common points of two curves in the plane. This subject has its roots in algebraic geometry and this led Lefschetz to develop a systematic topological theory in terms of homology on manifolds. Later this was dualised to give the cohomology ring of more general spaces. Quantum Cohomology is concerned with counting the number of rational curves (on suitable algebraic varieties) satisfying certain constraints. For example we might want to find the number of rational curves of a given degree which 1.

lie in a plane and pass through a given number of points (so as to get a finite answer)

or 2.

lie on a hypersurface in complex projective 4-space given by a generic polynomial equation of degree 5.

For a physicist such problems arise from 2-dimensional quantum field theory with the algebraic variety being the target space for the fields. Alternatively they arise when we consider strings moving in an algebraic variety, in which this is viewed as the extra Kaluza-Klein internal manifold. The heuristic computations of such problems made by physicists produce explicit beautiful answers and geometers have, with some success, been working overtime to give mathematical proofs of these results.

5

The striking thing about such results is that they provide answers not just for curves of a given degree, but for curves of all degrees. The answer appears in the form of a generating function [1]. This function or series has a constant term which is the "classical contribution" and all the higher terms are "quantum corrections". It should be emphasised that these are not the perturbative corrections produced by Feynman diagram expansions. They are non-perturbative corrections which are hard to calculate and can only be done when one has some other control of the theory. In many important cases this comes from some "duality" principle, which here (example (2) [11] goes under the heading of "mirror symmetry". This is now under intensive mathematical investigation. If we focus only on the constant term in these calculations we find that we are just dealing with classical cohomology theory. Thus "quantum cohomology" is an extension or deformation of ordinary or classical cohomology. Interestingly, whereas Lefschetz developed intersection theory in a general manifold context (and ordinary cohomology works for general topological spaces), quantum cohomology works only in a much more restrictive context. It is now recognised that the right context is that of symplectic geometry [4], a branch of geometry which arises both in hamiltonian mechanics and in complex algebraic geometry (as in Hodge theory). This story is really quite remarkable. The problems addressed and solved by quantum cohomology include very explicit problems that would have been of interest to 19th century algebraic geometers, though they did not have techniques to solve them. Moreover such traditional algebraic geometry seems very far from physics. It seems little short of miraculous that deep ideas from quantum theory should have any relevance to this area.

5

Jones - Witten invariants

Perhaps the most typical topological problem is the study of knots in 3-dimensional space. For a topologist a knot is a closed "piece of string" or more formally an embedding of a circle into 3-space. Two knots are equivalent if they can be continuously deformed into one another, without introducing self-crossings. The systematic classification of knots is highly non-trivial. Oddly enough it was first undertaken in earnest by the physicist P.G. Tait in the late 19th century, when Kelvin put forward his theory of "vortex atoms". According to this theory an atom was to be thought of as a knotted tube of ether, and the classification of atoms would be reduced to the topological classification of knots. After this theory was discarded by physicists the subject of knots was left to the mathematicians. A major step was the introduction of a knot-invariant by J.W. Alexander in 1928. This is now called the Alexander polynomial. It is a polynomial with integer coefficients (in a variable t and t -J ), which can be easily written down from the diagram of a plane projection of the knot. It is however independent of the choice of projection and so is an invariant. If two knots have different Alexander polynomials then they are inequivalent, but the converse is not true. For example a

6

trefoil and its mirror image have the same Alexander polynomial but they are not equivalent. This is true more generally: the Alexander polynomial can never distinguish between a knot and its mirror image. It was therefore a big surprise when Vaughan Jones in 1985 [3] discovered another polynomial invariant of knots (now named after him) which can distinguish mirror images. Unlike the Alexander polynomial which can easily be understood in terms of conventional homology theory the Jones polynomial did not seem to connect up with standard topology. Instead its parentage was an odd mixture of von Neumann algebras and statistical mechanics. But in 1988 Witten [7] made a big break-through by showing that the Jones polynomial could be elegantly explained as arising from a quantum field theory. Beside its conceptual simplification, Witten's work had a number of more concrete consequences. First of all it led to generalisations of the Jones polynomial with the Lie group SU(2) being replaced by any other compact Lie group. Second it enabled Witten to extend the definition of the Jones polynomial to knots in general 3-manifolds. In particular, when the knot itself is absent, Witten was able to define a polynomial invariant for any compact oriented 3-mainfold. This whole area of the Jones-Witten invariants totally revitalised the subject of knot theory. One of the by-products was the proof of 100 year old conjectures made empirically by Tait during his systematic work on knots. We therefore have the odd situation that knot theory was initially stimulated by a physical idea ("vortex atoms") which turned out to be incorrect but the knot conjectures made at this time were finally as proved a result of a quite different physical theory. The explanation of this may be that Kelvin's basic idea that the topological stability of knots should underlie the physical stability of atoms, though incorrect at the atomic level, has been resurrected more successfully at the deep level of elementary particles. Certainly topology is now seen as the explanation for various conserved quantities in fundamental physics.

6

Donaldson invariants

The study of differentiable manifolds, which had been very successful in dimensions ~ 5, was revolutionised by Simon Donaldson [2] when he showed that totally new phenomena appeared in dimension 4. Moreover Donaldson's main tool (the Yang-Mills equations) came from the new physics. What Donaldson did was to produce new polynomial invariants of compact oriented 4-manifolds. At first the relation to physics was slight. Only the classical Yang-Mills equations were used, just as the Laplace equation has appeared in various branches of mathematics. All the hard work and details involved geometry and analysis. But eventually Witten [8] showed that Donaldson's theory could be interpreted in terms of quantum field theory. Moreover the quantum field theory,

7

although formally complicated, was a small variant of a standard theory much studied by physicists. Now, although this explanation pleased physicists because it enabled them to understand Donaldson theory in their own terms, it appeared to be a retrograde step mathematically. After all Donaldson's work was rigorous mathematics, whereas Wittens re-interpretation depended on heuristic physical arguments. The only advantage would come from any additional progress that might be based on physical intuition. After some first steps in this direction by Witten [9] a spectacular break-through came in the work of Seiberg and Witten [10]. This was a by-product of more general ideas about electro-magnetic duality in the context of non-abelian quantum field theories. The idea is that certain quantum field theories have a dual and that this duality can switch classical and quantum computations. This makes some quantum non-perturbative calculations possible (as for mirror manifolds). A particular example shows that Donaldson theory is equivalent to a theory based on the much simpler Seiberg - Witten equations. Although a full mathematical justification of this duality has not yet been established the Seiberg - Witten equations themselves have been extensively and very productively used by mathematicians. Many problems which were difficult to settle by Donaldson theory have now been rigorously established.

7

Topological QFT

The three examples outlined in the previous sections all came from quantum fields theories (in various dimensions). In fact, as shown by Witten, they all arise from topological quantum field theories. These theories are ones in which all the output is topological and discrete. There are no variable real quantities. In a sense they describe the study of a vacuum before any particles or dynamics are introduced. The examples show how rich these Topological QFT are, how much structure is encoded in the vacuum itself, in other words in the framework. At this topological level various symmetries and dualities have their clearest expression. We could say that Euclidean geometry is the study of the classical vacuum. At this level there is no significant topology. But the quantum vacuum of present-day physics is a much more complicated entity and it contains a lot of topological information. The first and simplest instance of these remarks can already be seen in Hodge theory, where harmonic forms (corresponding to the ground state) are related to homology. This is a finite-dimensional example of what is happening in topological QFT, and this is well explained by Witten [6]. All these exciting developments cast new light on the philosophical questions raised at the beginning. The emergence of quantum mathematics seems hard to reconcile with Plato's view. We seem to have here new mathematics which has been forced on us by quantum physics, and no one could pretend that quantum theory was purely a creation of the human mind.

8

The 21 5t century

8

th

After this survey of the interaction of mathematics and physics during the 20 century we might ask ourselves what we should expect in the 21 51 century. As I warned at the beginning, the biggest break-throughs cannot be predicted and there will inevitably be surprises. However we can identify several different philosophical strands, each with their own following. 1.

In the first place there is the main physics community led by Witten_which sees string theory (or its new off-shoot M-theory) as the way forward. According to Witten this will force the emergence of new mathematics which will dominate the 21 sl century. Events over the past 25 years provide much support for this view.

2.

A more mathematical framework is the non-commutative geometry of Alain Connes. This has many roots in geometry, analysis and physics. It has the advantage of intellectual coherence and it is just possible that such mathematical ideas will tame physics.

3.

Roger Penrose takes an unorthodox view of physics and he believes that, in the conflict between quantum theory and General Relativity it is the former that will have to give ground (unlike the string theorists who want to incorporate GR into a quantum framework). Penrose's view is that some new physical insight is required which will alter our fundamental approach. Perhaps this will be the big break-through. Penrose's twistor theory has already been very fruitful mathematically so his ideas should not be discounted.

Which of these three scenerios will be closest to the truth? It is possible that they will all merge in some way to give different aspects of the same reality. There is already* an interesting new link between (1) and (2). Which may be a sign of things to come [5]. 51

The 21 century will be an exciting time! *This comment added September 1999. References 1. P. Candelas, X.c. de la Ossa, P.S. Green and L. Parkes, Nucl. Phys B 359 (1991),21. 2. S.K. Donaldson, Polynomial invariants for smooth four-manifolds, Topology 29 (1990), 257.

9

3. V.P.R. Jones, Hecke algebra representations of braid groups and link polynomials, Ann. Math. 126 (1987), 335 4. D. McDuff and D. Salamon, "J-Holomorphic Curves and Quantum Cohomology". American Mathematical Society 1994. 5. N. Nekrasov and A. Schwarz, Instantons on non commutative R4 and (2,0) superconformal six dimensional theory, hep-thl9802068. 6. E. Witten, Supersymmetry and Morse theory, J. Diff. Geom., 17 (1982), 661. 7. E, Witten, Quantum Field Theory and the Jones polynomial, Comm. Math. Phys. 121 (1989),351. 8. E. Witten, Topological quantum field theory, Comm. Math. Phys. 117 (1988), 353. 9. E. Witten, Supersymmetric Yang-Mills theory on a four-manifold, J. Math. Phys. 35 (1994),5101. 10. E. Witten, Monopoles and four-manifolds, Math. Research Letters 1 (1994), 769. 11. S.T. Yan (ed), "Essays on mirror manifolds" International Press Hong Kong 1997.

A LAYMAN'S GUIDE TO M-THEORY* M. J. DUFFt Center for Theoretical Physics Texas A&M University, College Station, Texas 77843 The best candidate for a fundamental unified theory of all physical phenomena is no longer ten-dimensional superstring theory but rather eleven-dimensional Mtheory. In the words of Fields medalist Edward Witten, "M stands for 'Magical', 'Mystery' or 'Membrane', according to taste". New evidence in favor of this theory is appearing daily on the internet and represents the most exciting development in the subject since 1984 when the superstring revolution first burst on the scene.

1

Magical Mystery Membranes

Up until 1995, hopes for a final theory 1 that would reconcile gravity and quantum mechanics, and describe all physical phenomena, were pinned on superstrings: one-dimensional objects whose vibrational modes represent the elementary particles and which live in a ten-dimensional universe 2. All that has now changed. In the last few years ten-dimensional superstrings have been subsumed by a deeper, more profound, new theory: eleven-dimensional M -theory 3. The purpose of the present paper is to convey to the layman some of this excitement. According to the standard model of the strong nuclear, weak nuclear and electromagnetic forces, all matter is made up of certain building block particles called fermions which are held together by force-carrying particles called bosons. This standard model does not incorporate the gravitational force, however. A vital ingredient in the quest to go beyond this standard model and to find a unified theory embracing all physical phenomena is supersymmetry, a symmetry which (a) unites the bosons and fermions, (b) requires the existence of gravity and (c) places an upper limit of eleven on the dimension of spacetime. For these reasons, in the early 1980s, many physicists looked to eleven-dimensional supergravity in the hope that it might provide that elusive superunified theory 5. Then in 1984 superunification underwent a major paradigm-shift: eleven-dimensional supergravity was knocked off its pedestal by ten-dimensional superstrings. Unlike eleven-dimensional supergravity, su"RESEARCH SUPPORTED IN PART BY NSF GRANT PHY-9722090. t ADDRESS FROM 1 SEPTEMBER 1999: RANDALL LABORATORY, DEPARTMENT OF PHYSICS, UNIVERSITY OF MICHIGAN, ANN ARBOR, MI 48109-1120; [email protected]

10

11

perstrings appeared to provide a quantum consistent theory of gravity which also seemed capable, in principle, of explaining the standard model a . Despite these startling successes, however, nagging doubts persisted about superstrings. First, many of the most important questions in string theory, in particular how to confront it with experiment and how to accommodate quantum black holes, seemed incapable of being answered within the traditional framework called perturbation theory, according to which all quantities of interest are approximated by the first few terms in a power series expansion in some small parameter. They seemed to call for some new, non-perturbative, physics. Secondly, why did there appear to be five different mathematically consistent superstring theories: the Es x Es heterotic string, the 80(32) heterotic string, the 80(32) Type I string, the Type lIA and Type lIB strings? If one is looking for a unique Theory of Everything, this seems like an embarrassment of riches! Thirdly, if supersymmetry permits eleven dimensions, why do superstrings stop at ten? This question became more acute with the discoveries of the supermembrane in 1987 the superfivebrane in 1992. These are bubble-like supersymmetric extended objects with respectively two and five dimensions moving in an eleven-dimensional spacetime, which are related to one another by a duality reminiscent of the electric/magnetic duality that relates an electric monopole ( a particle carrying electric charge) to a magnetic monopole (a hypothetical particle carrying magnetic charge). Finally, therefore, if we are going to generalize zero-dimensional point particles to onedimensional strings, why stop there? Why not two-dimensional membranes or more generally p-dimensional objects (inevitably dubbed p-branes)? In the last decade, this latter possibility of spacetime bubbles was actively pursued by a small but dedicated group of theorists 6, largely ignored by the orthodox superstring community. Although it is still too early to claim that all the problems of string theory have now been resolved, M -theory seems a big step in the right direction. First, it is intrinsically non-perturbative and already suggests new avenues both for particle physics and black hole physics. Secondly, it is is an elevendimensional theory which, at sufficently low energies, looks, ironically enough, like eleven-dimensional supergravity. Thirdly, it subsumes all five consistent string theories and shows that the distinction we used to draw between them is just an artifact of perturbation theory. See Table 1. Finally, it incorporates supermembranes and that is why M stands for Membrane. However, it may well be that we are only just beginning to scratch the surface of the ultimate aFor an up-to-date non-technical account of string theory, the reader is referred to the forthcoming popular book by Brian Greene 4.

12 Table 1. The five apparently different string theories are really just different corners of M-theory.

Es X Es heterotic string ) 80(32) heterotic string M theory 80(32) Type I string Type IIA string Type lIB string

meaning of M-theory, and for the time being therefore, M stands for Magic and Mystery too. 2

Symmetry and supersyrnrnetry

Central to the understanding of modern theories of the fundamental forces is the idea of symmetry: under certain changes in the way we describe the basic quantities, the laws of physics are nevertheless s~en to remain unchanged. For example, the result of an experiment should be the same whether we perform it today or tomorrow; this symmetry is called time translation invariance. It should also be the same before and after rotating our experimental apparatus; this symmetry is called rotational invariance. Both of these are examples of spacetime symmetries. Indeed, Einstein's general theory of relativity is based on the requirement that the laws of physics should be invariant under any change in the way we describe the positions of events in spacetime. In the standard model of the strong, weak and electromagnetic forces there are other kinds of internal symmetries that allow us to change the roles played by different elementary particles such as electrons and neutrinos, for example. These statements are made precise using the branch of mathematics known as Group Theory. The standard model is based on the group 8U(3) x 8U(2) x U(l), where U(n) refers to unitary n x n matrices and 8 means unit determinant. Grand Unified Theories, which have not yet received the same empirical support as the standard model, are even more ambitious and use bigger groups, such as 8U(5), which contain 8U(3) x 8U(2) x U(l) as a subgroup. In this case, the laws remain unchanged even when we exchange the roles of the quarks and electrons. Thus it is that the greater the unification, the greater the symmetry required. The standard model symmetry replaces the three fundamental forces: strong, weak and electromagnetic, with just two: the strong and electroweak. Grand unified symmetries replace these two with just one

13

strong-electroweak force. In fact, it is not much of an exaggeration to say that the search for the ultimate unified theory is really a search for the right symmetry. At this stage, however, one might protest that some of these internal symmetries fly in the face of experience. After all, the electron is very different from a neutrino: the electron has a non-zero mass whereas the neutrino is masslessb• Similarly, the electrons which orbit the atomic nucleus are very different from the quarks out of which the protons and neutrons of the nucleus are built. Quarks feel the strong nuclear force which holds the nucleus together, whereas electrons do not. These feelings are, in a certain sense, justified: the world we live in does not exhibit the SU(2) x U(l) of the standard model nor the SU(5) of the grand unified theory. They are what physicists call "broken symmetries". The idea is that these theories may exist in several different phases, just as water can exist in solid, liquid and gaseous phases. In some of these phases the symmetries are broken but in other phases, they are exact. The world we inhabit today happens to correspond to the brokensymmetric phase, but in conditions of extremely high energies or extremely high temperatures, these symmetries may be restored to their pristine form. The early stages of our universe, shortly after the Big Bang, provide just such an environment. Looking back further into the history of the universe, therefore, is also a search for greater and greater symmetry. The ultimate symmetry we are looking for may well be the symmetry with which the Universe began. M-theory, like string theory before it, relies crucially on the idea, first put forward in the early 1970s, of a spacetime supersymmetry which exchanges bosons and fermions. Just as the earth rotates on its own axis as it orbits the sun, so electrons carry an intrinsic angular momentum called "spin" as they orbit the nucleus in an atom. Indeed all elementary particles carry a spin s which obeys a quantization rule s = nh/47f where n = 0, 1,2,3, ... and h is Planck's constant. Thus particles may be divided into bosons or fermions according as the spin, measured in units of h/27f, is integer 0, 1,2, ... or half-odd-integer 1/2,3/2,5/2... Fermions obey the Pauli exclusion principle, which states that no two fermions can occupy the same quantum state, whereas bosons do not. They are said to obey opposite statistics. According to the standard model, the quarks and leptons which are the building blocks of all matter, are spin 1/2 fermions; the gluons, W and Z particles and photons which are the mediators of the strong, weak and electromagnetic forces, are spin 1 bosons and the Higgs particle which is responsible for the breaking of symmetries and for giving masses to the other particles, is a spin 0 bobOr at most has a very tiny mass.

14

son. Unbroken supersymmetry would require that every elementary particle we know of would have an unknown super-partner with the same mass but obeying the opposite statistics: for each boson there is a fermion; for each fermion a boson. Spin 1/2 quarks partner spin 0 squarks, spin 1 photons partner spin 1/2 photinos, and so on. In the world we inhabit, of course, there are no such equal mass partners and bosons and fermions seem very different. Supersymmetry, if it exists at all, is clearly a broken symmetry and the new supersymmetric particles are so heavy that they have so far escaped detection. At sufficiently high energies, however, supersymmetry may be restored. Supersymmetry may also solve the so-called gauge hiemrchy problem: the energy scale at which the grand unified symmetries are broken is vastly higher from those at which the electroweak symmetries are broken. This raises the puzzle of why the electrons, quarks, and W-bosons have their relatively small masses and the extra particles required by grand unification have their enormous masses. Why do they not all slide to some common scale? In the absence of supersymmetry, there is no satisfactory answer to this question, but in a supersymmetric world this is all perfectly natural. The greatest challege currently facing high-energy experimentalists at Fermi National Laboratory (Fermilab) in Chicago and the European Centre for Particle Physics (CERN) in Geneva is the search for these new supersymmetric particles. The discovery of supersymmetry would be one of the greatest experimental achievements of the century and would completely revolutionize the way we view the physical world c . Symmetries are said to be global if the changes are the same throughout spacetime, and local if they differ from one point to another. The consequences of local supersymmetry are even more far-reaching: it predicts gravity. Thus if Einstein had not already discovered General Relativity, local supersymmetry would have forced us to invent it. In fact, we are forced to a supergmvity in which the graviton, a spin 2 boson that mediates the gravitational interactions, is partnered with a spin 3/2 gmvitino. This is a theorist's dream because it confronts the problem from which both general relativity and grand unified theories shy away: neither takes the other's symmetries into account. Consequently, neither is able to achieve the ultimate unification and roll all four forces into one. But local supersymmetry offers just such a possiblity, and it is this feature above all others which has fuelled the theorist's belief in supersymmetry in spite of twenty-five years without experimental support. "This will present an interesting dilemma for those pundits who are predicting the End oj Science on the grounds that all the important discoveries have already been made. Presumably, they will say "I told you so" if supersymmetry is not discovered, and "See, there's one thing less left to discover" if it is.

15

3

Eleven-dimensional supergravity

Supergravity has an even more bizarre feature, however, it places an upper limit on the dimension of spacetime! We are used to the idea that space has three dimensions: height, length and breadth; with time providing the fourth dimension of spacetime. Indeed this is the picture that Einstein had in mind in 1916 when he proposed general relativity. But in the early 1920's, in their attempts to unify Einstein's gravity and Maxwell's electromagnetism, Theodore Kaluza and Oskar Klein suggested that spacetime may have a hidden fifth dimesion. This idea was quite succesful: Einstein's equations in five dimensions not only yield the right equations for gravity in four dimensions but Maxwell's equations come for free. Conservation of electric charge is just conservation of momentum in the fifth direction. In order to explain why this extra dimension is not apparent in our everyday lives, however, it would have to have a different topology from the other four and be very small. Whereas the usual four coordinates stretch from minus infinity to plus infinity, the fifth cordinate would lie between 0 and 27f R. In other words, it describes a circle of radius R. To get the right value for the charge on the electron, moreover, the circle would have to be tiny, R '" 10- 35 meters, which satisfactorily explains why we are unaware of its existence. It is difficult to envisage a spacetime with such a topology but a nice analogy is provide by a garden hose: at large distances it looks like a line but closer inspection reveals that at each point of the line, there is a little circle. So it was that Kaluza and Klein suggested there is a little circle at each point of four-dimensional spacetime. Moreover, this explained for the first time the empirical fact that all particles come with an electric charge which is an integer multiple of the charge on the electron, in other words, why electric charge is quantized. The Kaluza-Klein idea was forgotten for many years but was revived in the early 1980s when it was realized by Eugene Cremmer, Bernard Julia and Joel Scherk from the Ecole Normale in Paris that supergravity not only permits up to seven extra dimensions, but in fact takes its simplest and most elegant form when written in its full eleven-dimensional glory. Moreover, the kind of four-dimensional picture we end up with depends on how we compactify these extra dimensions: maybe seven of them would allow us to derive, a la Kaluza-Klein, the strong and weak forces as well as the electromagnetic. In the end, however, eleven dimensional supergravity fell out of favor for several reasons. First, despite its extra dimensions and despite its supersymmetry, elevendimensional supergravity is still a quantum field theory and runs into the problem from which all such theories suffer:: the quantum mechanical probability

16

for certain processes yields the answer infinity. By itself, this is not necessarily a disaster. This problem was resolved in the late 1940s in the context of Quantum Electrodynamics (QED), the study of the electromagnetic interactions of photons and electrons, by showing that these infinities could be absorbed in to a redefinition or renormalization of the parameters in the theory such as the mass and charge of the electron. This renormalization resulted in predictions for physical observables which were not only finite but in spectacular agreement with experiment. Spurred on by the success of QED, physicists looked for renormalizable quantum field theories of the weak and strong nuclear interactions which in the 1970s culminated in the enormously successful standard model that we know today. One might be tempted, therefore, to conclude that renormalizability, namely the ability to absorb all infinities into a redefinition of the parameters in the theory, is a prerequisite for any sensible quantum field theory. However, the central quandary of all attempts to marry quantum theory and gravity, such as eleven-dimensional supergravity, is that Einstein's general theory of relativity turns out non-renormalizable! Does this mean that Einstein's theory should be thrown on the scrapheap? Actually, the modern view of renormalizability is a little more forgiving. Suppose we have a renormalizable quantum field theory describing both light particles and heavy particles of mass m. Even such a renormalizable theory can be made to look non-renormalizable if we eliminate the heavy particles by using their equations of motion. The resulting equations for the light particles are then non-renormalizable but perfectly adequate for describing processes at energies less than me?, where c is the velocity of light. We run into trouble only if we try to extrapolate them beyond this range of validity, at which point we should instead resort to the original version of the theory with the massive particles put back in. In this light, therefore, the modern view of Einstein's theory is that it is perfectly adequate to explain gravitational phenomena at low energies but that at high energies it must be replaced by some more fundamental theory containing massive particles. But what is this energy, what are these massive particles and what is this more fundamental theory? There is a natural energy scale associated with any quantum theory of gravity. Such a theory combines three ingredients each with their own fundamental constants: Planck's constant h (quantum mechanics), the velocity of light c (special relativity) and Newton's gravitational constant G (gravity). From these we can form the so-called Planck mass mp = VhcjG, equal to about 10-8 kilograms, and the Planck energy mpc?, equal to about 1019 GeV. (GeV is short for giga-electron-volts=109 electron-volts, and an electron-volt is the energy required to accelerate an electron through a potential difference of one volt. ) From this we conclude that the energy at which Einstein's

17

theory, and hence eleven-dimensional supergravity, breaks down is the Planck energy. On the scale of elementary particle physics, this energy is enormousd : the world's most powerful particle acclerators can currently reach energies of only 104 GeV. So it seemed in the early 1980s that we were looking for a fundamental theory which reduces to Einstein's gravity at low energies, which describes Planck mass particles and which is supersymmetric. Whatever it is, it cannot be a quantum field theory because we already know all the supersymmetric ones and they do not fit the bill. Equally puzzling was that an important feature of the real world which is incorporated into both the standard model and grand unified theories is that Nature is chiral: the weak nuclear force distinguishes between right and left. (As Salam had noted with his left-handed neutrino hypothesis). However, as emphasized by Witten among others, it is impossible via conventional Kaluza-Klein techniques to generate a chiral theory from a non-chiral one and unfortunately, eleven-dimensional supergravity, in common with any odd-dimensional theory, is itself non-chiral. 4

Ten-dimensional superstrings

For both these reasons, attention turned to ten-dimensional superstring theory. The idea that the fundamental stuff of the universe might not be pointlike elementary particles, but rather one-dimensional strings had been around from the early 1970s. Just like violin strings, these relativistic strings can vibrate and each elementary particle: graviton, gluon, quark and so on, is identified with a different mode of vibration. However, this means that there are infinitely many elementary particles. Fortunately, this does not contradict experiment because most of them, corresponding to the higher modes of vibration, will have masses of the order of the Planck mass and above and will be unobservable in the direct sense that we observe the lighter ones. Indeed, an infinite tower of Planck mass states is just what the doctor ordered for curing the non-renormalizability disease. In fact, because strings are extended, rather than pointlike, objects, the quantum mechanical probabilities involved in string processes are actually finite. Moreover, when we take the low-energy limit by eliminating these massive particles through their equations of motion, we recover a ten-dimensional version of supergravity which incorporates Einstein's gravity. Now ten-dimensional quantum field theories, dFor this reason, incidentally, the End of Science brigade like to claim that, even if we find the right theory of quantum gravity, we will never be able to test it experimentally! As I will argue shortly, however, this view is erroneous.

18

as opposed to eleven-dimensional ones, also admit the possibility of chirality. The reason that everyone had still not abandoned eleven-dimensional supergravity in favor of string theory, however, was that the realistic-looking Type I string, which incorporated internal symmetry groups containing the SU(3) x SU(2) x U(l) of the standard model, seemed to suffer from inconsistencies or anomalies, whereas the consistent non-chiral Type I I A and chiral Type lIB strings did not seem realistic. Then came the September 1984 superstring revolution. First, Michael Green from QueenMary and Westfield College, London, and John Schwarz from the California Institute of Technology showed that the Type I string was free of anomalies provide the group was uniquely SO(32) where O(n) stands for orthogonal n x n matrices. They suggested that a string theory based on the exceptional group Es x Es would also have this property. Next, David Gross, Jeffrey Harvey, Emil Martinec and Ryan Rohm from Princeton University discovered a new kind of heterotic (hybrid) string theory based on just these two groups: the Es x Es heterotic string and the SO(32) heterotic string, thus bringing to five the number of consistent string theories. Thirdly, Philip Candelas from the University of Texas, Austin, Gary Horowitz and Andrew Strominger from the University of California, Santa Barbara and Witten showed that these heterotic string theories admitted a Kaluza-Klein compactification from ten dimensions down to four. The six-dimensional compact spaces belonged to a class of spaces known to the pure mathematicians as Calabi- Yau manifolds. The resulting four-dimensional theories resembled quasi-realistic grand unfied theories with chiral representations for the quarks and leptons! Everyone dropped eleven-dimensional supergravity like a hot brick. The mood of the times was encapsulated by Nobel Laureate Murray Gell-Mann in his closing address at the 1984 Santa Fe Meeting, when he said: "Eleven Dimensional Supergravity (Ugh!)". 5

Ten to eleven: it is not too late

After the initial euphoria, however, nagging doubts about string theory began to creep in. Theorists love uniqueness; they like to think that the ultimate Theory of Everything 1 will one day be singled out, not merely because all rival theories are in disagreement with experiment, but because they are mathematically inconsistent. In other words, that the universe is the way it is because it is the only possible universe. But string theories are far from unique. Already in ten dimensions there are five mathematically consistent theories: the Type I SO(32), the heterotic SO(32), the heterotic Es x Es, the Type IIA and

19

the Type I lB. (Type I is an open string in that its ends are allowed to move freely in spacetime; the remaining four are closed strings which form a closed loop.) Thus the first problem is the uniqueness problem. The situation becomes even worse when we consider compactifying the extra six dimensions. There seem to be billions of different ways of compactifying the string from ten dimensions to four (billions of different Calabi-Yau manifolds) and hence billions of competing predictions ofthe real world (which is like having no predictions at all). This aspect of the uniqueness problem is called the vacuum-degeneracy problem. One can associate with each different phase of a physical system a vacuum state, so called because it is the quantum state corresponding to no real elementary particles at all. However, according to quantum field theory, this vacuum is actually buzzing with virtual particleantiparticle pairs that are continually being created and destroyed and consequently such vacuum states carry energy. The more energetic vacua, however, should be unstable and eventually decay into a (possibly unique) stable vacuum with the least energy, and this should describe the world in which we live. Unfortunately, all these Calabi-Yau vacua have the same energy and the string seems to have no way of preferring one to the other. By focussing on the fact that strings are formulated in ten spacetime dimensions and that they unify the forces at the Planck scale, many critics of string theory fail to grasp this essential point. The problem is not so much that strings are unable to produce four-dimensional models like the standard model with quarks and leptons held together by gluons, W-bosons, Z bosons and photons and of the kind that can be tested experimentally in current or forseeable accelerators. On the contrary, string theorists can dream up literally billions of them! The problem is that they have no way of discriminating between them. What is lacking is some dynamical mechanism that would explain why the theory singles out one particular Calabi-Yau manifold and hence why we live in one particular vacuum; in other words, why the world is the way it is. Either this problem will not be solved, in which case string theory will fall by the wayside like a hundred other failed theories, or else it will be solved and string theory will be put to the test experimentally. Neither string theory nor M-theory is relying for its credibility on building thousand-light-year accelerators capable of reaching the Planck energy, as some End-oj-Science Jeremiahs have suggested. Part and parcel of the vacuum degeneracy problem is the supersymmetrybreaking problem. If superstrings are to describe our world then supersymmetry must be broken, but the way in which strings achieve this, and at what energy scales, is still a great mystery. A third aspect of vacuum degeneracy is the cosmological constant problem.

20

Shortly after writing down the equations of general relativity, Einstein realized that nothing prevented him from adding an extra term, called the cosmological term because it affects the rate at which the universe as a whole is expanding. Current astrophysical data indicates that the coefficient of this term, called the cosmological constant, is zero or at least very small. Whenever an a priori allowed term in an equation seems to be absent, however, theorists always want to know the reason why. At first sight supersymmetry seems to provide the answer. The cosmological constant measures the energy of the vacuum, and in supersymmetric vacua the energy coming from virtual bosons is exactly cancelled by the energy coming from virtual fermions! Unfortunately, as we have already seen, the vacuum in which our universe currently finds itself can at best have broken supersymmetry and so all bets are off. As with cake, we can't have our cosmological constant and eat it too! In common with all other theories one can think of, superstrings as yet provide no resolution of this paradox. On the subject of gravity, let us not forget black holes. According to Cambridge University's Stephen Hawking, they are not as black as they are painted: quantum black holes radiate energy and hence grow smaller. Moreover, they radiate energy in the same way irrespective of what kind of matter went to make up the black hole in the first place. The rate of radiation increases with diminishing size and the black hole eventualy explodes leaving nothing behind, not even the grin on the Cheshire Cat. All the information about the original constituents of the black hole has been lost and this leads to the information loss paradox because such a scenario flies in the face of traditional quantum mechanics. On a more pragmatic level, another unsolved problem was that the thermodynamic entropy formula of the black hole radiation, first written down by Jacob Bekenstein (Hebrew University), had never received a microscopic explanation. The entropy of a system is a measure of its disorder, and is related to the number of quantum states that the system is allowed to occupy. For a black hole, this number seems incredibly high but what microscopic forces are at work to explain this? Not even strings, with their infinite number of vibrational modes seemed to have this capability. Given all the good news about string theory, though, string enthusiasts were reluctant to abandon the theory notwithstanding all these problems. Might the faults lie not with the theory itself but rather with the way the calculations are carried out? In common with the standard model and grand unified theories, the equations of string theory are just too complicated to solve exactly. We have to resort to an approximation scheme and the timehonored way of doing this in physics is perturbation theory. Let us recall quantum electrodynamics, for example, and denote by e the electric charge

21

on an electron. The ratio a = 21l"e 2 / he is a dimensionless number called, for historical reasons, the fine structure constant. Fortunately for physicists, a is about 1/137: much less than 1. Consequently, if we can express processes (such as the probability of one electron scattering off another) in a power series in this coupling constant a, then we can be confident that keeping just the first few terms in the series will be a good approximation to the exact result. As a simple example of approximating a mathematical function I(x) by a power series, consider (5.1) Provided x is very much less than unity, the first few terms provide a good approximation. This is precisely what Richard Feynman was doing when he devised his Feynman diagram technique. The same perturbative techniques work well in the weak interactions where the corresponding dimensionless coupling constant is about 10- 5 . Indeed, this is how the weak interactions justify their name. When we come to the strong interactions, however, we are not so lucky. Now the strong fine structure constant which governs the strength of low-energy nuclear processes, for example, is of order unity and perturbation theory can no longer be trusted: each term in the power series expansion is just as big as the others. The whole industry of lattice gauge theory is devoted to an attempt to avoid perturbation theory in the strong interactions by doing numerical simulations on supercomputers. It has proved enormously difficult. The point to bear in mind, however, (and one that even string theorists sometimes forget) is that "God does not do perturbation theory"; it is merely a technique dreamed up by poor physicists because it is the best they can do. Furthermore, although theories such as quantum electrodynamics manage to avoid it, there is a possible fatal flaw with perturbation theory. What happens if the process we are interested in depends on the coupling constant in an intrinsically non-perturbative way which does not even admit a power series expansion? Such mathematical functions are not difficult to come by: the function (5.2)

for example, cannot be approximated by a power series in x no matter how small x happens to be. The equations of string theory are sufficiently complicated that such non-perturbative behaviour cannot be ruled out. If so, might our failure to answer the really difficult problems be more the fault of string theorists than string theory?

22

An apparently different reason for having mixed feelings about superstrings, of course, especially for those who had been pursuing Kaluza-Klein supergravity prior to the 1984 superstring revolution, was the dimensionality of spacetime. If supersymmetry permits eleven spacetime dimensions, why should the theory of everything stop at ten? This problem rose to the surface again in 1987 when Eric Bergshoeff of the University of Groningen, Ergin Sezgin, now at Texas A&M University, and Paul Townsend from the University of Cambridge discovered The elevendimensional supermembrane. This membrane is a bubble-like extended object with two spatial dimensions which moves in a spacetime dictated by our old friend: eleven-dimensional supergravity! Moreover, Paul Howe (King's College, London University), Takeo Inami (Kyoto University), Kellogg Stelle (Imperial College) and I were then able to show that if one of the eleven dimensions is a circle, then we can wrap one of the membrane dimensions around it so that, if the radius of the circle is sufficiently small, it looks like a string in ten dimensions. In fact, it yields precisely the Type II A superstring. This suggested to us that maybe the eleven-dimensional theory was the more fundamental after all. 6

Supermembranes

Membrane theory has a strange history which goes back even further than strings. The idea that the elementary particles might correspond to modes of a vibrating membrane was put forward originally in 1960 by the British Nobel Prize winning physicist Paul Dirac, a giant of twentieth century science who was also responsible for two other daring postulates: the existence of antimatter and the existence of magnetic monopoles. Anti-particles carry the same mass but opposite charge from particles and were discovered experimentally in the 1930s. Magnetic monopoles carry a single magnetic charge and to this day have not yet been observed. As we shall see, however, they do feature prominently in M -theory. When string theory came along in the 1970s, there were some attempts to revive Dirac's membrane idea but without much success. The breakthrough did not come until 1986 when James Hughes, James Liu and Joseph Polchinski of the University of Texas showed that, contrary to the expectations of certain string theorists, it was possible to combine the membrane idea with supersymmetry: the supermembrane was born. Consequently, while all the progress in superstring theory was being made a small but enthusiastic group of theorists were posing a seemingly very different question: Once you have given up O-dimensional particles in favor of

23

I-dimensional strings, why not 2-dimensional membranes or in general ~ dimensional objects (inevitably dubbed p-branes)? Just as a a-dimensional particle sweeps out a I-dimensional worldline as it evolves in time, so a 1dimensional string sweeps out a 2-dimensional worldsheet and a ~brane sweeps out ad-dimensional worldvolume, where d = p + 1. Of course, there must be enough room for the ~brane to move about in spacetime, so d must be less than the number of spacetime dimensions D. In fact supersymmetry places further severe restrictions both on the dimension of the extended object and the dimension of spacetime in which it lives. One can represent these as points on a graph where we plot spacetime dimension D vertically and the ~brane dimension d = p + 1 horizontally. This graph is called the brane-scan. See Table 2. Curiously enough, the maximum spacetime dimension permitted is eleven, where Bergshoeff, Sezgin and Townsend found their 2-brane. In the early 80s Green and Schwarz had showed that spacetime supersymmetry allows classical superstrings moving in spacetime dimensions 3,4,6 and 10. (Quantum considerations rule out all but the ten-dimensional case as being truly fundamental. Of course some of these ten dimensions could be curled up to a very tiny size in the way suggested by Kaluza and Klein. Ideally six would be compactified in this way so as to yield the four spacetime dimensions with which we are familiar.) It was now realized, however, that there were twelve points on the scan which fall into four sequences ending with the superstrings or I-branes in D = 3,4,6 and 10, which were now viewed as but special cases of this more general class of supersymmetric extended object. These twelve points are the ones with d ~ 2 and denoted by S in Table 2. For completeness, we have also included the superparticles with d = 1 in D = 2,3, 5 and 9. The letters S, V and T refer to scalar, vector and tensor respectively and describe the different kinds of particles that live on the worldvolume of the ~brane. Spin a bosons and their spin 1/2 fermionic partners are said to form a scalar supermultiplet. An example is provided by the eleven-dimensional supermembrane that occupies the (D = 11, d = 3) slot on the branescan. It has 8 spin a and 8 spin 1/2 particles living on the three-dimensional (one time, two space) worldvolume of the membrane. But as we shall see, it was subsequently realized that there also exist branes which have higher spin bosons on their worldvolume and belong to vector and tensor supermultiplets. A particularly interesting solution of eleven-dimensional supergravity, found by Bergshoeff, Sezgin and myself in collaboration with Chris Pope of Texas A&M University, was called "The membrane at the end of the universe." It described a four-dimensional spacetime with the extra seven dimensions curled up into a seven-dimensional sphere and in which the supermembrane occupied the three-dimensional boundary of the four-dimensional

24 Table 2. The brane scan, where S, V and T denote scalar, vector and tensor multiplets.

Dt 11 10 9 8 7 6

5 4

3 2 1 0

S V SIV V S

T V V SIV V V V V S S T S V SIV V SIVV V S S V SIV SIV V . SIV SIV V S

0

1

2

3

4

5

6

7 8 9 1011 d ~

spacetime (rather as the two-dimensional surface of a soap bubble encloses a three-dimensional volume). This spacetime is of the kind first discussed earlier this century by the Dutch physicist Willem de Sitter and has a non-zero cosmological constant. It fact it is called anti-de Sitter space recause the cosmological constant is negative. Now shortly after he first wrote down the equations of the membrane, Dirac pointed out in a (at the time unrelated) paper that anti-de Sitter space admits some strange kinds of fields that he called singletons which have no analogue in ordinary flat spacetime. These were much studied by Christian Fronsdal and collaborators at the University of California, Los Angeles, who pointed out that they reside not in the bulk of the anti-de Sitter space but on the three dimensional boundary. In 1988, the present author noted that, in the case of the seven-sphere compactification of eleven-dimensional supergravity, the singletons correspond to the same 8 spin o plus 8 spin 1/2 scalar supermultiplet that lives on the worldvolume of the supermembrane, and it was natural to suggest that the membrane and the singletons should be identified. In this way, via the the membrane at the end of the universe, the physics in the bulk of the four-dimensional spacetime was really being determined by the physics on the three-dimensional boundary. Notwithstanding these and subsequent results, the supermembrane enterprise ( Type II A&M Theory?) was ignored by most adherents of conventional superstring theory. Those who had worked on eleven-dimensional supergravity and then on super membranes spent the early eighties arguing for spacetime

25

dimensions greater than four, and the late eighties and early nineties arguing for worldvolume dimensions greater than two. The latter struggle was by far the more bitter e • 7

Solitons, topology and duality

Another curious twist in the history of supermembranes concerns their interpretation as solitons. In their broadest definition, solitons are classical solutions of a field theory corresponding to lumps of field energy which are prevented from dissipating by a topological conservation law, and hence display particle-like properties. The classic example of a such soliton is provided by magnetic monopole solutions of four-dimensional grand unified theories found by Gerard 't Hooft of the the University of Utrecht in the Netherlands and Alexander Polyakov, now at Princeton. Solitons playa ubiquitous role in theoretical physics appearing in such diverse phenomena as condensed matter physics and cosmology, where they are frequently known as topological defects. To understand the meaning of a topological conservation law, we begin by recalling that in 1917 the German mathematician Emmy Noether had shown that to every global symmetry, there corresponds a quantity that is conserved in time. For example, invariance under time translations, space translations and rotations give rise to the laws of conservation of energy, momentum and angular momentum, respectively. Similarly, conservation of electric charge corresponds to a change in the phase of the quantum mechanical wave functions that describe the elementary particles. One might naively expect that conservation of magnetic charge would admit a similar explanation but, in fact, it has a completely different topological origin, and is but one example of what are now termed topological conservation laws. Topology is that branch of mathematics which concerns itself just with the shape of things. Topologically speaking, therefore, a teacup is equivalent to a doughnut because each two-dimensional surface has just one hole: one can continuosly deform one into the other. The surface of an orange, on the other hand, is topologically distinct having no holes: you cannot turn an orange into a doughnut. So it is with the intricate field configurations describing magnetic monopoles: you cannot turn a particle carrying n units of magnetic charge into one with n' units of magnetic charge, if n :I n'. Hence the charge is conserved but for eO ne string theorist I know would literally cover up his ears whenever the word "membrane" was mentioned within his earshot! Indeed, I used to chide my more conservative string theory colleagues by accusing them of being unable to utter the M-word. That the current theory ended up being called M-theory rather than Membrane theory was thus something of a Pyrrhic victory.

26

topological reasons; not for any reasons of symmetry. In 1977, however, Claus Montonen of the University of Helsinki and David Olive, now at the University of Wales at Swansea, made a bold conjecture. Might there exist a dual formulation of fundamental physics in which the roles of Noether charges and topological charges are reversed? In such a dual picture, the magnetic monopoles would be the fundamental objects and the quarks, W-bosons and Higgs particles would be the solitons! They were inspired by the observation that in certain supersymmetric grand unified theories, the masses M of all the particles whether elementary (carrying purely electric charge Q), solitonic (carrying purely magnetic charge P) or dyonic (carrying both) are described by a universal formula (7.1)

where v is a constant. Note that the mass formula remains unchanged if we exchange the roles of P and Q! The Montonen-Olive conjecture was that this electric/magnetic symmetry is a symmetry not merely of the mass formula but is an exact symmetry of the entire quantum theory! The reason why this idea remained merely a conjecture rather than a proof has to do with the whole question of perturbative versus non-perturbative effects. According to Dirac, the electric charge Q is quantized in units of e, the charge on the electron, whereas the magnetic charge is quantized in units of l/e. In other words, Q = me and P = n/e, where m and n are integers. The symmetry suggested by Olive and Montonen thus demanded that in the dual world, we not only exchange the integers m and n but we also replace e by l/e and go from a regime of weak coupling to a regime of strong coupling! This was very exciting because it promised a whole new window on non-perturbative effects. On the other hand, it also made a proof very difficult and the idea was largely forgotten for the next few years. Although the original paper by Hughes, Liu and Polchinski made use of the soliton idea, the subsequent impetus in supermembrane theory was to mimic superstrings and treat the p-branes as fundamental objects in their own right (analagous to particles carrying an electric Noether charge). Even within this framework, however, it was possible to postulate a certain kind of duality between one p-brane and another by relating them to the geometrical concept of p-forms. (Indeed, this is how p-branes originally got their name.) In their classic text on general relativity, Gravitation, Misner, Thorne and Wheeler 7 provide a way to visualize p-forms as describing the way in which surfaces are stacked. Open a cardbord carton containing a dozen bottles, and observe the honeycomb structure of intersecting north-south and east-west cardboard separators between the bottles. That honeycomb structure of tubes

27

is an example of a 2-form in the context of ordinary 3-dimensional space. It yields a number (number oftubes cut) for each choice of 2-dimensional surface slicing through the 3-dimensional space. Thus a 2-form is a device to produce a number out of a surface. All of electromagnetism can be summarized in the language of 2-forms, honeycomb-like structures filling all of 4-dimensional spacetime. There are two such structures, Faraday= F and Maxwell= *F each dual, or perpendicular, to the other. The amount of electric charge or magnetic charge in an elementary volume is equal respectively to the number of tubes of the Maxwell 2-form *F or Faraday 2-form F that end in that volume. (In a world with no magnetic monopoles, no tubes of F would ever end.) To summarize, in 4-dimensional spacetime, an electric O-brane is dual to a magnetic O-brane. An equivalent way to understand why O-dimensional point particles produce electric fields which are described by 2-forms is to note that in 4dimensional spacetime a pointlike electric charge can be surrounded by a twodimensional sphere. Similarly, a string (l-brane) in 4-dimensional spacetime can be "surrounded" by a I-dimensional circle, and so the electric charge per unit length of a string is described by a I-form Maxwell field but its magnetic dual perpendicular to it is described by a 3-form Faraday field. By contrast, in 5 spacetime dimensions, although the Faraday field of a O-brane is still a 2-form, the dual Maxwell field is a now a 3-form, consistent with the fact that you now need a 3-dimensional sphere to surround the point like electric charge. But a 3-form is just the Faraday field produced by a string. Consequently, in 5 spacetime dimensions, the magnetic dual of an electric O-brane is a string. Though in practice it is harder to visualize, it is straightforward in principle to generalize this duality idea to any p-brane in any spacetime dimension D. The rule is that the Faraday field is a (p + 2) form and the dual Maxwell field perpendicular to it is (D - p - 2)-form. Consequently, the magnetic dual of an electric p-brane is a fr brane where p = D - p - 4. In particular, in the critical D = 10 spacetime dimensions of superstring theory, a string (p = 1) is dual to a fivebrane (P = 5). (If you have trouble imagining that in 10 dimensions you need a 3-dimensional sphere to surround a 5-brane, don't worry, you are not alonel) Now the low energy limit of 10-dimensional string theory is a 10dimensional supergravity theory with a 3-form Faraday field and dual 7-form Maxwell field, just as one would expect if the fundamental object is a string. However, 10-dimensional supergravity had one puzzling feature that had long been an enigma from the point of view of string theory. In addition to the above version there existed a dual version in which the roles of the Faraday and Maxwell fields were interchanged: the Faraday field was a 7-form and the

28

Maxwell field was a 3-form! This suggested to the present author in 1987, in analogy with the Olive-Montonen conjecture, that perhaps this was indicative of a dual version of string theory in which the fundamental objects are fivebranes! This became known as the string/fivebrane duality conjecture. The analogy was still a bit incomplete, however, because at that time the fivebrane was not regarded as a soliton. The next development came in 1988 when Paul Townsend of Cambridge University revived the Hughes-Liu-Polchinski idea and showed that many of the super p-branes also admit an interpretation as topological defects (analogous to particles carrying a magnetic topological charge). Of course, this involved generalizing the usual notion of a soliton: it need not be restricted just to a O-brane in four dimensions but might be an extended object such as a p-brane in D-dimensions. Just like the monopoles studied by Montonen and Olive, these solitons preserve half of the spacetime supersymmetry and hence obey a relation which states that their mass per unit p-volume is given by their topological charge. Then in 1990, a major breakthrough for the string/fivebrane duality conjecture came along when Strominger found that the equations of the 10-dimensional heterotic string admit a fivebrane as a soliton solution which also preserves half the spacetime supersymmetry and whose mass per unit 5-volume is given by the topological charge associated with the Faraday 3-form ofthe string. Moreover, this mass became larger, the smaller the strength of the string coupling, exactly as one would expect for a soliton. He went on to suggest a complete strong/weak coupling duality with the strongly coupled string corresponding to the weakly coupled fivebrane. By generalizing some earlier work of Rafael Nepomechie (University of Florida, Gainesville) and Claudio Teitelboim (University of Santiago), moreover, it was possible to to show that the electric charge of the fundamental string and the magnetic charge of the soli tonic fivebrane obeyed a Dirac quantization rule. In this form, string/fivebrane duality was now much more closely mimicking the electric/magnetic duality of Montonen and Olive. Then Curtis Callan (Princeton University), Harvey and Strominger showed that similar results also appear in both Type II A and Type II B string theories; they also admit fivebrane solitons. However, since most physicists were already sceptical of electric/magnetic duality in four dimensions, they did not immediately embrace string/fivebrane duality in ten dimensions! Furthermore, there was one major problem with treating the fivebrane as a fundamental object in its own right; a problem that has bedevilled supermembrane theory right from the beginning: no-one knows how to quantize fundamental p-branes with p > 2. All the techniques that worked so well for fundamental strings and which allow us, for example, to calculate how

29

one string scatters off another, simply do not go through. Problems arise both at the level of the worldvolume equations where our old bete noir of non-renormalizability comes back to haunt us and also at the level of the spacetime equations. Each term in string perturbation theory corresponds to a two-dimensional worldsheet with more and more holes: we must sum over all topologies of the worldsheet. But for surfaces with more than two dimensions we do not know how to do this. Indeed, there are powerful theorems in pure mathematics which tell you that it is not merely hard but impossible. Of course, one could always invoke the dictum that God does not do perturbation theory, but that does not cut much ice unless you can say what He does do! So there were two major impediments to string/fivebrane duality in 10 dimensions. First, the electric/magnetic duality analogy was ineffective so long as most physicists were sceptical of this duality. Secondly, treating fivebranes as fundamental raised all the unresolved issues of non-perturbative quantization. The first of these impediments was removed, however, when Ashoke Sen (Tata Institute) revitalized the Olive-Montonen conjecture by establishing that certain dyonic states, which their conjecture demanded, were indeed present in the theory. Many duality sceptics were thus converted. Indeed this inspired Nathan Seiberg (Rutger's University) and Witten to look for duality in more realistic (though still supersymmetric) approximations to the standard model. The subsequent industry, known as Seiberg-Witten theory, provided a wealth of new information on non-perturbative effects in fourdimensional quantum field theories, such as quark-confinement and symmetrybreaking, which would have been unthinkable just a few years ago. The Olive-Montonen conjecture was originally intended to apply to fourdimensional grand unified field theories. In 1990, however, Anamarie Font, Luis Ibanez, Dieter Lust and Fernando Quevedo at CERN and, independently, Soo Yong Rey (University of Seoul) generalized the idea to four-dimensional superstrings, where in fact the idea becomes even more natural and goes by the name of S-duality. In fact, superstring theorists had already become used to a totally different kind of duality called T-duality. Unlike, S-duality which was a nonperturbative symmetry and hence still speculative, T-duality was a perturbative symmetry and rigorously established. If we compactify a string theory on a circle then, in addition to the Kaluza-Klein particles we would expect in an ordinary field theory, there are also extra winding particles that arise because a string can wind around the circle. T -duality states that nothing changes if we exchange the roles of the Kaluza-Klein and winding particles provided we also exchange the radius of the circle R by its inverse 1/ R. In short, a string

30

cannnot tell the difference between a big circle and a small one!

8

String/string duality in six dimensions

Recall that when wrapped around a circle, an ll-dimensional membrane behaves as if it were a lO-dimensional string. In a series of papers between 1991 and 1995, a team at Texas A&M University involving Ramzi Khuri, James T. Liu, Jianxin Lu, Ruben Minasian, Joachim Rahmfeld, and myself argued that this may also be the way out of the problems of 10dimensional string/fivebrane duality. If we allow four of the ten dimensions to be curled up and allow the soli tonic fivebrane to wrap around them, it will behave as if it were a 6-dimensional solitonic string! The fundamental string will remain a fundamental string but now also in 6-dimensions. So the lO-dimensional string/fivebrane duality conjecture gets replaced by a 6dimensional string/string duality conjecture. The obvious advantage is that, in contrast to the fivebrane, we do know how to quantize the string and hence we can put the predictions of string/string duality to the test. For example, one can show that the coupling constant of the solitonic string is indeed given by the inverse of the fundamental string's coupling constant, in complete agreement with the conjecture. When we spoke of string/string duality, we originally had in mind a duality between one heterotic string and another, but the next major development in the subject came in 1994 when Christopher Hull (Queen Mary and Westfield College, London University) and Townsend suggested that, if the four-dimensional compact space is chosen suitably, a six-dimensional heterotic string can also be dual to a six-dimensional Type I I A string! These authors also added futher support to the idea that the Type I I A string originates in eleven dimensions. It occurred to the present author that string/string duality has another unexpected pay-off. If we compactify the six-dimensional spacetime on two circles down to four dimensions, the fundamental string and the solitonic string will each acquire a T-duality. But here is the miracle: the T-duality of the soli tonic string is just the S-duality of the fundamental string, and viceversa! This phenomenon, in which the non-perturbative replacement of e by 1/e in one picture is just the perturbative replacement of R by 1/ R in the dual picture, goes by the name of Duality of Dualities. See Table 3. Thus fourdimensional electric/magnetic duality, which was previously only a conjecture, now emerges automatically if we make the more primitive conjecture of sixdimensional string/string duality.

31

Table 3. Duality of dualities

Fundamental string T - duality: Radius t+ l/(Radius) Kaluza - Klein t+ Winding S - duality: charge t+ l/(charge) Electric t+ Magnetic

9

Dual string

charge t+ l/(charge) Electric t+ Magnetic Radius t+ l/(Radius) Kaluza - Klein t+ Winding

M-theory

All this previous work on T-duality, S-duality, and string/string duality was suddenly pulled together under the umbrella of M -theory by Witten in his, by now famous, talk at the University of Southern California in February 1995. Curiously enough, however, Witten still played down the importance of supermembranes. But it was only a matter of time before he too succumbed to the conclusion that we weren't doing just string theory any more! In the coming months, literally hundreds of papers appeared in the internet confirming that, whatever M-theory may be, it certainly involves supermembranes in an important way. For example, in 1992 R. Giiven (Bosporus University) had shown that eleven-dimensional supergravity admits a solitonic fivebrane solution dual to the fundamental membrane solution found the year before by Stelle and myself. See the (D = 11, d = 6) point marked by a T in Table 2. It did not take long to realize that 6-dimensional string/string duality (and hence 4-dimensional electric/magnetic duality) follows from 11-dimensional membrane/fivebrane duality. The fundamental string is obtained by wrapping the membrane around a one-dimensional space and then compactifying on a four-dimensional space; whereas the solitonic string is obtained by wrapping the fivebrane around the four-dimensional space and then compactifying on the one-dimensional space. Nor did it take long before the more realistic kinds of electric/magnetic duality envisioned by Seiberg and Witten were also given an explanation in terms of string/string duality and hence M-theory. Even the chiral Es x Es string, which according to Witten's earlier theorem could never come from eleven-dimensions, was given an elevendimensional explanation by Petr Horava (Princeton University) and Witten. The no-go theorem is evaded by compactifying not on a circle (which has no ends), but on a line-segment (which has two ends). It is ironic that having driven the nail into the coffin of eleven-dimensions (and having driven GellMann to utter "Ugh!"), Witten was the one to pull the nail out again! He went

32

on to argue that if the size of this one-dimensional space is large compared to the six-dimensional Calabi-Yau manifold, then our world is approximately five-dimensional. This may have important consequences for confronting Mtheory with experiment. For example, it is known that the strengths of the four forces change with energy. In supersymmetric extensions of the standard model, one finds that the fine structure constants 0!3, 0!2, 0!1 associated with the SU(3) x SU(2) x U(I) all meet at about 10 16 GeV, entirely consistent with the idea of grand unification. The strength of the dimensionless number 2 O!G = GE , where G is Newton's contant and E is the energy, also almost meets the other three, but not quite. This near miss has been a source of great interest, but also frustration. However, in a universe of the kind envisioned by Witten, spacetime is approximately a narrow five dimensional layer bounded by four-dimensional walls. The particles of the standard model live on the walls but gravity lives in the five-dimensional bulk. As a result, it is possible to choose the size of this fifth dimension so that all four forces meet at this common scale. Note that this is much less than the Planck scale of 10 19 Ge V, so gravitational effects may be much closer in energy than we previously thought; a result that would have all kinds of cosmological consequences. Thus this eleven-dimensional framework now provides the starting point for understanding a wealth of new non-perturbative phenomena, including string/string duality, Seiberg-Witten theory, quark confinement, particle physics phenomenology and cosmology. 10

Black holes and D-branes

Type [[ string theories differ from heterotic theories in one important respect: in addition to the usual Faraday 3-form charge, called the Neveu-Schwarz charge after Andre Neveu (University of Montpelier) and Schwarz, they also carry so-called Ramond charges, named after Pierre Ramond of the University of Florida, Gainesville. These are associated with Faraday 2-forms and 4-forms in the case of Type [[A and Faraday 3-forms and 5-forms in the case of Type [[B. Accordingly in 1993, Jiaxin Lu and I were able to find new solutions of the Type [[A string equations describing super Jr branes with p = 0, 2 and their duals with p = 6,4 and new solutions of Type [[B string equations with p = 1,3 and their duals with p = 5,3. Interestingly enough, the Type II B superthreebrane is self-dual, carrying a magnetic charge equal to its electric charge. This meant that there were more points on the brane-scan than had previously been appreciated. These occupy the V slots in Table 2. For all these solutions, the mass per unit Jrvolume was given by the charge, as a consequence of the preservation half of the spacetime supersymmetry. However,

33

we recognized that they were in fact just the extremal mass=charge limit of more general non-supersymmetric solutions found previously by Horowitz and Strominger. These solutions, whose mass was greater than their charge, exhibit event horizons: surfaces from which nothing, not even light, can escape. They were black branes! Thus another by-product of these membrane breakthroughs has been an appreciation of the role played by black holes in particle physics and string theory. In fact they can be regarded as black branes wrapped around the compactified dimensions. These black holes are tiny (10- 35 meters) objects; not the multi-million solar mass objects that are gobbling up galaxies. However, the same physics applies to both and there are strong hints by Lenny Susskind (Stanford University) and others that M-theory may even clear up many of the apparent paradoxes of quantum black holes raised by Hawking. As we have already discussed, one of the biggest unsolved mysteries in string theory is why there seem to be billions of different ways of compactifying the string from ten dimensions to four and hence billions of competing predictions of the real world. Remarkably, Brian Greene of Cornell University, David Morrison of Duke University and Strominger have shown that these wrappped around black branes actually connect one Calabi-Yau vacuum to another. This holds promise of a dynamical mechanism that would explain why the world is as it is, in other words, why we live in one particular vacuum. A fuller discussion may be found in Greene's book 4. Another interconnection was recently uncovered by Polchinski who realized that the Type I I super p-branes carrying Ramond charges may be identified with the so-called Dirichlet-branes (or D-branes, for short) that he had studied some years ago by looking at strings with unusual boundary conditions. Dirichlet was a French mathematician who first introduced such boundary conditions. These D-branes are just the surfaces on which open strings can end. In the process, he discovered an 8-brane in Type IIA theory and a 7-brane and 9-brane in Type I I B which had previously been overlooked. See Table 2. This D-brane technology has opened up a whole new chapter in the history of supermembranes. In particular, it has enabled Strominger and Cumrun Vafa from Harvard to make a comparison of the black hole entropy calculated from the degeneracy of wrapped-around black brane states with the Bekenstein-Hawking entropy of an extreme black hole. Their agreement provided the first microscopic explanation of black hole entropy. Moreover, as Townsend had shown earlier, the extreme black hole solutions of the tendimensional Type I I A string (in other words, the Dirichlet O-branes) were just the Kaluza-Klein particles associated with wrapping the eleven-dimensional membrane around a circle. Moreover, four-dimensional black holes also ad-

34

mit the interpretation of intersecting membranes and fivebranes in elevendimensions. All this holds promise of a deeper understanding of black hole physics via supermembranes. 11

Eleven to twelve: is it still too early?

We have remarked that eleven spacetime dimensions are the maximum allowed by super p-branes. This is certainly true if we believe that the Universe has only one time dimension. Worlds with more than one time present all kind of headaches for theoretical physicists and they prefer not to think about them. For example, there would be no "before" and "after" in the conventional sense. Just for fun, however, in 1987 Miles Blencowe (Imperial College, University of London) and I imagined what would happen if one relaxed this one-time requirement. We found that we could not rule out the possibility of a supersymmetric extended object with a (2 space, 2 time) worldvolume living in a (10 space, 2 time) spacetime. We even suggested that the Type lIB string with its (1 space, 1 time) worldsheet living in a (9 space, 1 time) spacetime might be descended from this object in much the same way that the Type lIA string with its (1 space, 1 time) worldsheet living in a (9 space, 1 time) spacetime is descended from the (2 space, 1 time) worldvolume of the supermembrane living in a (10 space, 1 time) spacetime. This idea lay dormant for almost a decade but has recently been revived by Vafa and others in the context of F -theory. The utility of F -theory is certainly beyond dispute: it has yielded a wealth of new information on string/string duality. But should the twelve dimensions of F-theory be taken seriously? And if so, should Ftheory be regarded as more fundamental than M -theory? (If M stands for Mother, maybe F stands for Father.) To make sense of F-theory, however, it seems necessary to somehow freeze out the twelfth timelike dimension where there appears to be no dynamics. Moreover, Einstein's requirement that the laws of physics be invariant under changes in the spacetime coordinates seems to apply only to ten or eleven of the dimensions and not to twelve. So the symmetry of the theory, as far as we can tell, is only that of ten or eleven dimensions. The more conservative interpretation of F-theory, therefore, is that the twelfth dimension is just a mathematical artifact with no profound significance. Time (or perhaps I should say "Both times") will tell. 12

So what is M-theory?

Is M-theory to be regarded literally as membrane theory? In other words should we attempt to "quantize" the eleven dimensional membrane in some,

35

as yet unknown, non-perturbative way? Personally, I think the jury is still out on whether this is the right thing to do. Witten, for example, strongly believes that this is not the correct approach. He would say, in physicist's jargon, that we do not even know what the right degrees of freedom are. So although M-theory admits 2-branes and 5-branes, it is probably much more besides. Recently, Tom Banks and Stephen Shenker at Rutgers together with Willy Fischler from the Univerity of Texas and Susskind have even proposed a rigorous definiton of M-theory known as M(atrix) theory which is based on an infinite number of Dirichlet O-branes. In this picture spacetime is a fuzzy concept in which the spacetime coordinates x, y, Z, ... are matrices that do not commute e.g. xy =f. yx. This approch has generated great excitement but does yet seem to be the last word. It works well in high dimensions but as we descend in dimension it seems to break down before we reach the real four-dimensional world. Another interesting development has recently been provided by Juan Maldacena at Harvard, who has suggested that M-theory on anti-de Sitter space, including all its gravitational interactions, may be completely described by a non-gravitational theory on the boundary of anti-de Sitter space. This holds promise not only of a deeper understanding of M-theory, but may also throw light on non-perturbative aspects of the theories that live on the boundary, which in some circumstances can include the kinds of quark theories that govern the strong nuclear interactions. Models of this kind, where a bulk theory with gravity is equivalent to a boundary theory without gravity, have also been advocated by 't Hooft and independently by Susskind who call them holographic theories. The reader may notice a striking similarity to the earlier idea of "The membrane at the end of the universe" 6 and interconnections between the two are currently being explored. M -theory has sometimes been called the Second Superstring Revolution, but we feel this is really a misnomer. It certainly involves new ideas every bit as significant as those of the 1984 string revolution, but its reliance upon supermembranes and eleven dimensions makes it is sufficiently different from traditional string theory to warrant its own name. One cannot deny the tremendous historical influence of the last decade of superstrings on our current perspectives. Indeed, it is the pillar upon which our belief in a quantum consistent M-theory rests. In my opinion, however, the focus on the perturbative aspects of one-dimensional objects moving in a ten-dimensional spacetime that prevailed during this period will ultimately be seen to be a small (and perhaps physically insignificant) corner of M-theory. The overriding problem in superunification in the coming years will be to take the Mystery out of

36

M -theory, while keeping the Magic and the Membranes. Acknowledgements

I am grateful for correspondence with Freeman Dyson, Robert Low, Ergin Sezgin and Edward Witten. References 1. Steven Weinberg, Dreams of a Final Theory, Pantheon, 1992. 2. Michael Green, Superstrings, Scientific American 255 (1986) 44-56 (September). 3. Michael Duff, The theory formerly known as strings, Scientific American 278 (1998) 54-59 (February). 4. Brian Greene, The Elegant Universe, W. W. Norton, 1998. 5. Daniel Z. Freedman and Peter van Nieuwenhuizen, The Hidden Dimensions 0/ Spacetime, Scientific American 252 (1985) 62. 6. Michael Duff and Christine Sutton, The Membrane at the End of the Universe, New Scientist, June 30, 1988. 7. Charles W. Misner, Kip S. Thorne and John Archibald Wheeler, Gravitation, W. H. Freeman and Company, 1973.

MATHEMATICAL MODELS IN ECONOMICS IVAR EKELAND Institut de Finance Universit Paris-Dauphine 75775 Paris CEDEX 16, France E-mail: [email protected] It is claimed that economics, management and finance will have a deep influence on the future of mathematics. These fields are already a source of mathematical problems which are unlike anything we have seen up to now, either in pure mathematics or in the natural sciences

1

Introduction

At the present time, in developed countries, the majority of graduates in mathematics who find jobs outside the sector of education are employed in the tertiary sector (trade and services), not the secondary one (industrial production). The needs of the secondary sector have led to the inception and development of two mathematical subdisciplines: statistics and econometrics on the one hand, financial risk management on the other. I will begin by saying a few words about each of them. • Statistics and econometrics.

The days of large-scale macroeconomic modelling are over. On the other hand, individual entrepreneurs and businesses have to handle huge amounts of data (about customers, stocks and demands, etc ... ) which have suddenly become available and manageable, through computers and the internet. Organized markets have blossomed around the world for all kinds of assets and products, producing unprecedented streams of information. We now have as much data as we want, and even much more than we are able to handle. Statistics and econometrics were developed at a time when data was scarce. Now we have practically unbounded amounts of high-frequency, loosely structured, data. This calls for new techniques of analysis, and one can safely bet that such methods will be developed in the next century. • Financial risk management Since the seminal work of Fischer Black and Myron Scholes in 1973, the financial industry has created an enormous number of innovative 37

38

products: options, futures, derivatives of all kinds. Sophisticated mathematical techniques are required, both for pricing them correctly and for hedging them. Portfolio management and investment decisions have become very technical as well, relying the methods of stochastic control. All the models in use have to be calibrated and tested, calling into play the methods of statistical estimation. All these question require an immediate answer: there is no time to wait, or the customer will be lost. So the financial operator needs fast and reliable methods, which apply to a variety of situation. There is a true problem there, because one runs quickly into a dimension curse: the financial operator would like to solve his problems with three sources of risk or more, while the mathematician cannot solve reliably a parabolic equation in dimension three with the prescribed boundary conditions. All this points to progress in stochastic control and the numerical analysis of partial differential equations. Both statistics and econometrics are here to stay, and much more could be said about their previsible development. But it is another question which interests me: how will economic theory influence the development of mathematics in the coming century ? Before trying to answer it, I must state three principles, which are to be borne in mind if one is to do justice to economic theory: • Economics cannot be understood in isolation from other social phE.."nmena. Modern economic theory has laid a heavy stress on the influence of human capital (education, learning-by-doing) and political institutions on economic growth • Economics must be constructed from the bottom up. This is in contrast to an approach where society determines individual behaviour. On the contrary, modern economic theory tries to explain collective behaviour from individual behaviour, as a physicist would try to understand the properties of a cristal from those of its atoms. In other words, there will be no macroeconomics from first principles. • In economics, unlike physics, the past does not determine the future. The situation tomorrow depends, not only on the situation today (the so-called "fundamentals"), but also on peoples' expectations. What I believe today will happen tomorrow, event though it may have no factual

39

foundation, will play an important role in shaping tomorrow (think of self-fulfilling prophecies) I shall now describe three situations arising from basic considerations of economic theory, and which have led to unexpected mathematical developments. I believe this is a portent of what is to come in the next century. 2

A model for human behaviour

The standard model in use in economic theory is due to von Neumann and Morgenstern (see 3 for instance). It runs as follows. Each individual faces a set of possible actions, a E A, and a set of possible consequences of those actions, x EX. Each action a determines a probability law 7r a on X. Each consequence x is rated U(x), where U --+ R is the utility function of the individual under consideration. The individual then makes his choice by maximizing the expected utility of his action, that is, by solving the optimization problem:

(1) A very nice story, to be sure, but is it true? In other words, can the expected-utility model be tested ? We shall see that the answer ot this very basic economic question is by no means obvious. Consider the simple case of a consumer, having at his disposal the amount w to spend on K different goods, and facing market prices p = (PI, ... ,pK ), where Pk is the price of the k-th good. There is no uncertainty in this case, so that the consumer's problem is simply:

Max U(x) piX = LPk Xk ~ xk

(2)

W

(3)

2: 0 Vk

(4)

We are going to assume that the utilily function is smooth and strictly concave, so that this optimization problem has a unique solution x(P), depending smoothly on p. This is called the demand function of the individual, and it is observable, whereas the utility function is not. We are going to show that this function satisfies a certain system of partial differential equations, thereby providing a way to test the von Neumann Morgnestern model. If the model is true, then the demand function is characterized by the optimality conditions: UI(x(p)) = )..(p)p

(5)

40

where x(P) (the demand function) and P (the prices) are observed, while U(x) (the utility function) and A(P) (the Lagrange multiplier) are unknown

functions, the existence of which is in doubt. After suitable inversions and transformations, this equation becomes:

(6) so x(P) is collinear to a gradient. Setting:

(7) we rewrite the equation as: 1

(8)

w = - A(p)dV

By a theorem of Frobenius, a necessary and sufficient condition on the I-form w for such a relation to hold, with suitable A and V,is that:

(9)

w l\llw = 0

This condition is well-known in economic theory as the Slutsky or Antonelli condition. It can be restated directly in terms of the (observed) demand function x(P). Define the matrix S = (Sk.i) by: k.

S ,1

Bxk

= -

Bpi

'"

Bxk

i

BPi

.

- ~Pi _ X l

(10)

The Frobenius condition amounts to stating that the matrix S is symmetric:

(11) This relation was first obtained by Antonelli in 1886, and rediscovered by Slutsky in 1915. All ecometric tests were negative until, in 1996, Browing and Chiappori ? tested it positively by the simple device of separating individuals (singles) from households (couples and families). The demand functions of individuals do indeed satisfy the Slutsky relations. What about households? Two-person households are characterized by the fact that there are two, not one, utility functions, and only one budget constraint. A mathematical analysis of the problem leads to the fact that the demand function x(P) should satisfy the equation: w=!dV+gdW

(12)

41

where w = L xkdpk, as before, and /, g, V, W are unknown functions. By a well-known theorem of Darboux, this translates into the condition:

(13) In other words, the Slutsky matrix is no longer symmetric, as in the case of one individual, but is the sum of a symmetric matrix and a matrix of rank one. This condition has been econometrically tested as well by Browning and Chiappori, and the date has been found to be consistent wiht it. What about other aggregates? Economists have been studying market demand for al long time. It is defined as follows. There are N individuals, each of whom solves his own optimization problem:

(14)

Max Un(x) piX = LPkXk ~ xk

Wn

(15)

2': 0 Vk

(16)

the solution of which is xn(P), the individual demand of the n-th individual. It is not observed; only the aggregate demand X(P) is observed:

(17) n

We must have:

(18) n

n

which is the so-called Walras law. Further investigation leads to the following system of first-order partial differential equations: " 1 8Vn = Xk(P) ~ ".p. 8Vn 8Pk n

L..,.

(19)

• 8Pi

for which we seek concave solutions Vn . It has been proved by Chiappori and Ekeland 2 that, if N 2': K (there are more agents than goods) and if the function X (P) is real analytic, the preceding system has concave solutions Vn satisfying prescribed initial data. The proof relies on the Cartan-Khler theorem, which itself relies on the CauchyKowalewska theorem. Economically speaking, this means that we cannot check the von Neumann Morgenstern model by using aggregate data. However, it can still be done, as we saw, by using individual data. We are not going to dwell any longer on this line of research, and switch to another, by pointing out that markets are not the only types of organizations, and that most individuals function in more tightly run organizations, such as firms or institutions.

42 3

Types of organizations

Let us start with markets. They are characterized by a complet set of prices, at which all goods are traded, by complete information (everyone knows everything about the ggods being traded), and market clearing: prices are set by equating consumption with production. More common are non-market organizations. They are characterized by an incomplete set of prices (certain goods cannot be traded), and by assymetric information (some individuals know more than others about the available goods). The simplest non-market information is the contract, which is modelled as follows. Two people can enter into a contract together. The first one, the principal, has all the bargaining power: he offers the second one, the agent, a contract. The agent takes it or leaves it; that's all he can do. He cannot renegociate it, but he can turn it down. The contract itself consists in two parts: a quantity, x E R K , and a sum of money t E R. If the agent accepts the contract, he will have to produce x and deliver it to the principal, and for this he will be paid t. There are several types of agents, parametrized by 8 E RD. An agent of type 8 produces x at cost u(8, x), and he will not enter any contract which leaves him with less than v (8) (his reservation price). Each agent knows hes type, but the principal does not know the agent's type. In this way, the model gives all the bargaining power to the principal, and all the information to the agent. All the principal knows is the overall distribution of types, but not any individual's type. He will, however, try to take advantage of the latter by proposing a contract contingent on the agent type. So he will post a menu of contracts (x(8), t(8), in the hope that agents of type 8 will take the contract intended for them, and if this is the case, he will make the overall profit: P=

J

p(x(8)d8 -

J

(20)

r(8)d8

where p(x) is the profit the principal derives from getting the quantity x. Of course, this is meaningful only if the menu is devised in such a way that agents of type 8 take the contract (x(8), t(8) instead of any other contract (x(8'), t(8'). This translates into the condition: u(8, x(8))

+ t(8)

~

u(8, x(8'))

+ t(8')

V8,8'

(21)

In the simplest case, where the type comes in linearly into the agent's utility, u(x, 8) = L 8k x k , we end up with problems in the calculus of variations

43

with global convexity constraints: Max

In

L(B,x(B), \lx(B))d8

(22)

x(B) = xo(B) on en x(B) ~ Xl (B) on n u concaveon n

(23) (24) (25)

Very little is known about such problems (see 4 for the best available results). Existence is easy, but we do not have workable optimality conditions, nor reasonable algorithms, and we do not know how regular the solutions are. Economic considerations provide us with a nice qualitative picture, but do not know how to justify it. The scope of this kind of investigation is huge. It leads to understanding, and even optimizing, business organization. Procurement, regulation, tarification, privatization, institutions, all fall within its scope, and we are entering a period of social engineering, where future organizations will be designed to meet theoretical criteria. 4

How to predict the future

Let us go back to something we mentioned in the beginning: in economic theory, the past is not enough to determine the future. Consider an economic process, symbolized by an infinite sequence Xt, for t E Z. If we were dealing with physics, we would write Xt+l = cf>(Xt). But we are in economics, and we write Xt+1 = cf>(Xt, Yt), where Yt is the expected value of Xt+l at time t, knowing Xt. The consequences of this kind of modelling cannot be overestimated. It stresses the importance of commitment on the part of government: the economic effects of any policy will be extremely different according to whether one believes it is here in the long run or not. From the mathematical point of view, one must ask how to close the model, that is, how to derive the Yt endogeneously. One way to do so is to ask that expectations be rational, that is, that people will adopt prediction rules only if they are confirmed by experience. This can be done by a deterministic rule, for instance:

Yt = Xt+l

(26)

which, substituted into Xt+l = cf>(xt, Yt), yields:

Xt+l = cf>(Xt,XtH)

(27)

44

The dynamics of this equation are very interesting, and may give rise to nondeterminacies (several possibilities for Xt+l given Xt). One may also generate expectations by stochastic rules: P(Yt = Y

I

Xt = x) =

Pxy

(28)

which yields: (29) This can be solved for Pxy. Individuals will then coordinate on an (otherwies irrelevant) random variable, yielding what is known as a sunspot equilibrium. 5

Conclusion

Stakes are huge. I am convinced that scientific problems in the next century will come more form social organizations and biology than from engineering or physics. I have tried to show in this talk that, even now • we need sophisticated tools, like the exterior differential calculus, to answer very basic questions in economic theory • we get from economics well-posed, difficult problems, startingly different from anything we saw in physics References

1. M. Browning and P.A. Chiappori, "Efficient intra-household allocations", Econometrica 66 (1998), p. 1241-1278 2. P.A.Chiappori and I. Ekeland, "Aggregation and market deamnd: an exterior differential calculus viewpoint", Econometrica 67 (1999), p. 14351458 3. A. Mas-Colell, M.D. Whinston, J.R. Green, Microeconomic Theory, Oxford University Press, 1995 . 4. P. Chone and J.C. Rochet, "Sweeping, ironing and multidimensional screening" , Econometrica 66 (1998), p. 783-826

DUALITY AND STRINGS, SPACE AND TIME C.M. HULL Department of Physics Queen Mary and Westfield College Mile End Rd, London El 4NS, UK E-mail: [email protected]. uk Duality symmetries in M-theory and string theory are reviewed, with particular emphasis on the way in which string winding modes and brane wrapping modes can lead to new spatial dimensions. Brane world-volumes wrapping around Lorentzian tori can give rise to extra time dimensions and in this way dualities can change the number of time dimensions as well as the number of space dimensions. This suggests that brane wrapping modes and spacetime momenta should be on an equal footing and M-theory should not be formulated in a spacetime of definite dimension or signature.

1

String Theory, M-Theory and Duality

String theory is defined as a perturbation theory in the string coupling constant 9s, which is valid when 9s is small. The fundamental quanta are the p.xcitations of relativistic strings moving in spacetime and comprise of a finite set of massless particles plus an infinite tower of massive particles with the scale of the mass set by the string tension T = I/l~, expressed in terms of a string length scale Is. If the spacetime has some circular dimensions, or more generally has some non-contractible loops, the spectrum will also include winding modes in which a closed string winds around a non-contractible loop in spacetime. These have no analogue in local field theories and are responsible for some of the key differences between string theories and field theories. Physical quantities are calculated through a path integral over string histories, which can be calculated perturbatively in 9s using stringy Feynman rules, with string world-sheets of genus n contributing terms proportional to 9~. In the supersymmetric string theories, these contributions are believed to be finite at each order in 9s, giving a perturbatively finite quantum theory of gravity unified with other forces. There are five distinct perturbative supersymmetric finite string theories, all in 9 + I dimensions (Le. nine space and one time), the type I, type IIA and type lIB string theories, and the two heterotic string theories with gauge groups 80(32) and Es x Es. The massless degrees of freedom of each of these theories are governed by a IO-dimensional supergravity theory, which is the low-energy effective field theory. It has been a long-standing puzzle as to why 45

46

there should be five such theories of quantum gravity rather than one, and this has now been resolved. It is now understood that these are all equivalent nonperturbatively and that these distinct perturbation theories arise as different perturbative limits of a single underlying theory 1,2. We do not have an intrinsic formulation of this underlying non-perturbative theory yet, but the relationships between the string theories has been understood through the discovery of dualities linking them. A central role in the non-perturbative theory is played by the p-branes. These are p-dimensional extended objects, so that a O-brane is a particle, a 1-brane is a string, a 2-brane is a membrane and so on. In the perturbative superstring theories there is a 1-brane which is the fundamental string providing the perturbative states of the theory, while the other branes arise as solitons or as D-branes 3, which are branes on which fundamental strings can end. The type II string theories have a fundamental string and a solitonic 5-brane and a set of Dp-branes, where p = 0,2,4,6,8 for the IIA string theory and p = 1,3,5,7,9 for the lIB string theory. There are duality symmetries of string theories that relate brane degrees of freedom to fundamental quanta, so that all the branes are on the same footing. Ifsome ofthe spacetime dimensions are wrapped into some compact space K, so that the spacetime is M x K for some M, then branes can wrap around homology cycles of K and these give extra massive states in the compactified theory on M. For example, a p-brane wrapping around an n-cycle with n ~ p gives a p - n brane in the compactified theory. These brane wrapping modes generalise the string winding modes and are related to the perturbative states by V-dualities 1, and play an important role in the duality symmetries, as we shall see. One of the best-understood dualities is T-duality 4, which relates string theory on a spacetime S1 x M with a circular dimension of radius R to a string theory on S1 X M where the circular dimension is now of radius

R= l;

R

(1)

so that the radii R, R are inversely proportional. For bosonic and heterotic string theories, T-duality is a self-duality, so that heterotic (bosonic) string theory on a large circle is equivalent to heterotic (bosonic) string theory on a small circle, while it maps the type IIA string theory to the type lIB theory, with the result that type IIA string theory on a large circle is equivalent to type lIB string theory on a small circle 5,6. T-duality relates perturbative states to perturbative states, as does mirror symmetry which relates a superstring theory com pactified on a Calabi-Yau manifold K to a superstring theory compactified on a topologically distinct Calabi-Yau manifold, the mirror k

47 ofK. There are also non-perturbative dualities. For example the type IIA string theory compactified on K3 is equivalent to the heterotic string theory compactified on the 4-torus T4 1, while the type I theory with string coupling 9s is equivalent to the SO(32) heterotic string theory with string coupling 9s = 1/9s 2,7,8,9. This is an example of a strong-weak coupling duality relating the strong-coupling regime of one theory to the weak-coupling regime of another. Such dualities are important as they allow the description of strong-coupling physics in terms of a weakly-coupled dual theory. M-theory arises as the strong-coupling limit of the IIA string theory 2. The IIA string is interpreted as an 11-dimensional theory compactified on a circle of radius R = ls9s. Then at strong coupling, the extra dimension decompactifies to give a theory in 11 dimensions which has 11 dimensional supergravity as a low-energy limit We will refer to this 10+1 dimensional theory as M-theory. Duality transformations relate this to each of the five string theories, and the string theories and M-theory can all be thought of as arising as different limits of a single underlying theory. The IIA string theory is obtained by compactifying M-theory on a circle, the lIB string is obtained from the IIA by T-duality or directly from compactifying M-theory on a 2-torus and taking the limit in which it shrinks of zero size 10, the E8 x E8 heterotic string is obtained by modding out M-theory on a circle by a Z2 symmetry or equivalently from compactifying M-theory on a line interval 11 , the type I theory is obtained from the lIB string by orientifolding (modding out by world-sheet parity) 5,12, and the SO(32) heterotic string is the strong coupling limit ofthis 2. The type I theory and the SO(32) heterotic string (as well as the type I' string) can be obtained directly from M-theory compactified on a cylinder as on 13, while the massive IIA string theory is obtained from a limit of M-theory compactified on a T2 bundle over a circle 14

In D-dimensional general relativity or supergravity, a spacetime with a large circle SI is physically distinct from one with a small circle 81 , and a spacetime M x K is physically distinct from the mirror spacetime M x k, but in string theory these dual pairs of spacetimes define the same string theory and so define the same physics. The heterotic string on M x T4 is equivalent to the type IIA string on M x K3, even though T4 and K3 are very different spaces with different properties (e.g. they have different topologies and different curvatures) and there is no invariant answer to the question: what is the spacetime manifold? In the same way that spacetimes related by diffeomorphisms are regarded as equivalent, so too must spacetimes related by dualities, and the concept of spacetime manifold should be replaced by dual-

48

ity equivalence classes of spacetimes (or, more generally, duality equivalence classes of string or M-theory solutions). In the usual picture, the five superstring theories and the ll-dimensional theory arising as the strong coupling limit of the IIA string (referred to as M-theory here) are depicted as being different corners of the moduli space of the mysterious fundamental theory underpinning all of these theories (sometimes also referred to as M-theory, although we shall resist this usage here). More precisely, compactifying string theory or M-theory gives a theory depending on the moduli of metrics and antisymmetric tensor gauge fields on the compactification space. Each modulus gives rise to a scalar field in the compactified theory and the expectation value of any of the scalar fields can be used to define a coupling constant. One can then examine the perturbation theory in that constant. For some choices it will give a field theory, for others it will give a perturbative string theory and different perturbative string theories will correspond to different choices of coupling 15. The string theories and M-theory are each linked to each other by chains of dualities and so there is only one basic theory. More recently, other 'corners' corresponding to particular limits of the theory have been understood to correspond to field theories without gravity. For example the lIB string theory in the background given by the product of 5-dimensional anti-de Sitter space and a 5-sphere is equivalent to N = 4 supersymmetric Yang-Mills theory in four dimensions, with similar results for theories in other anti-de Sitter backgrounds 16, and certain null compactifications are equivalent to matrix models 17. Many dualities have now been found which can relate theories with different gauge groups, different spacetime dimensions, different spacetime geometries and topologies, different amounts of supersymmetry, and even relate theories of gravity to gauge theories. Thus many of the concepts that had been thought absolute are now understood as relative: they depend on the 'frame of reference' used, where the concept of frame of reference is generalised to include the values of the various coupling constants. For example, the description of a given system when a certain coupling is weak can be very different from the description at strong coupling, and the two regimes can have different spacetime dimension, for example. However, in all this, one thing that has remained unchanged is the number of time dimensions; all the theories considered are formulated in a Lorentzian signature with one time coordinate, although the number of spatial dimensions can change. Remarkably, it turns out that dualities can change the number of time dimensions as well, giving rise to exotic spacetime signatures 18. The resulting picture is that there should be some underlying fundamental theory and that different

49

spacetime signatures as well as different dimensions can arise in various limits. The new theories are different real forms of the complexification of the original M-theory and type II string theories, perhaps suggesting an underlying complex nature of spacetime. We will now proceed to examine some of these dualities in more detail, and in particular to focus on the way in which extra spacetime dimensions can emerge from brane wrapping modes. 2 2.1

Branes and Extra Dimensions Compactification on Sl

For a field theory compactified from D dimensions on a circle S1 of radius R, the momentum p in the circular dimension will be quantised with p = nj R for some integer n. In the limit R -t 0 this becomes divergent, so that finite-momentum states must move in the remaining D - 1 dimensions and are described by the dimensionally reduced theory in D - 1 dimensions. For finite R, the states carrying internal momentum can be interpreted as states in D - 1 dimensions with mass (taking the D-dimensional field theory to be massless for simplicity) M= InljR

(2)

The set of all such states for all n gives the 'Kaluza-Klein tower' of massive states arising from the compactification. If the original field theory includes gravity, there will be an infinite tower of massive gravitons, and if the theory is supersymmetric, then the tower fits into supersymmetry representations. In the limit R -t 0, the masses of all the states in these towers become infinite and they decouple, leaving the massless dimensionally reduced theory in D-1 dimensions. On the other hand, taking the decompactification limit R -t 00, all the states in the tower become massless and combine with the massless D - 1 dimensional fields to form the massless fields in D dimensions. Such a tower becoming massless is often a signal of the decompactification of an extra dimension. For a string theory the situation is very different, due to the presence of string winding modes which become light as the circle shrinks. A string can wind m times around the circular dimension, and the corresponding state in the D - 1 dimensional theory will have mass mRT where T is the string tension (into which a factor of 271" has been absorbed). The set of all such states for all m forms a tower of massive states and in the limit R -t 0 these become massless, so that there is an infinite tower of states becoming massless

50

(and fitting into supergravity multiplets, in the case of the superstring). This signals the opening up of a new circular dimension of radius

R= I/TR

(3)

with the string winding mode around the original circle of mass

M=mRT=m/R

(4)

reinterpreted as a momentum mode in the dual circle of radius R. Similarly, the momentum modes on the original circle (M = n/ R) can now be interpreted as string winding modes around the dual circle (M = nTR), and the new theory in D dimensions is again a string theory. A state with momentum n/ R and winding number m will have mass

n n m M = - + mRT = - + -;:;R R R

(5)

and this is clearly invariant under the T-duality transformation m f-+ n, R f-+ R interchanging the momentum and winding numbers and inverting the radius. Then the original string theory on M D - 1 x S1 (with M D - 1 some D - I-dimensional spacetime) is equivalent to a string theory on M D - 1 x [p where §1 has radius R, with the momentum modes of one theory corresponding to the winding modes of the other, and this equivalence is known as a T-duality. In the limit R -+ O,R -+ 00, the decompactified T-dual theory has full D dimensional Lorentz invariance. If the first string theory is a bosonic (heterotic) string, so is the second, while if one is a type IIA string theory, the other is a type IIB string theory. Then the type IIA string theory compactified on a circle of radius R is equivalent to the type IIB string theory compactified on a circle of radius R. For IO-dimensional supergravity compactified on a circle of radius R, taking R -+ 0 will give a 9-dimensional supergravity theory while for a string theory a new dimension opens up to replace the one of radius R that has shrunk to zero size. If R is much larger than Is, the desciption in terms of string theory on S1 is useful, while for R « Is, the T-dual description in terms of string theory on §1 is more appropriate, with the light states having a conventional description in terms of momentum modes on [Jl instead of winding modes on S1. 2.2

IIA String Theory and M- Theory

The type IIA string theory in 9+ I dimensions has DO-brane states, which are particle-like non-perturbative BPS states, with quantized charge n (for

51

integers n) and mass

(6) The state of charge n can be thought of as composed of n elementary DObranes. In the strong coupling limit g8 -+ 00, these states all become massless. Moreover, the DO-brane states for a given n fit into a short massive supergravity multiplet with spins ranging from zero to two and so at strong coupling there is an infinite number of gravitons becoming massless. It was proposed in 2 that this tower of massless states should be interpreted as a Kaluza-Klein tower for an extra circular dimension of radius

(7) Then the strong coupling limit of the IIA string theory is interpreted as the limit in which RM -+ 00 so that the extra dimension decompactifies to give a theory in 10+1 dimensions, and this is M-theory. Moreover, for the IIA string theory in D = 10 Minkowski space, the strong coupling limit is invariant under the full ll-dimensional Lorentz group and the effective field theory describing the massless degrees of freedom of M-theory is ll-dimensional supergravity. The radius can be rewritten in terms of the ll-dimensional Planck length tp as

(8) The IIA string theory is really only defined perturbatively for very small coupling g8' It can now be 'defined' at finite coupling g8 as M-theory compactified on a circle of radius RM, so that the problem is transferred to the one of defining M-theory. However, at low energies we see that the non-perturbative IIA theory is described by ll-dimensional supergravity compactified on a circle, and this leads to important non-perturbative predictions, so that this viewpoint can be useful even though we still know rather little about Mtheory. The IIA string has Dp-branes for all even p, while M-theory has a 2brane or membrane and a 5-brane. All the branes of the IIA string theory have an M-theory origin. For example, an M-theory membrane will give the fundamental string of the IIA theory if it wraps around the circular dimension and the D2-brane if it does not.

52

2.3

Compactijication on T2

For a D dimensional field theory compactified on a 2-torus there will be momentum modes with masses M",

(9)

where R I , R2 are the radii of the circular dimensions, p, q are integers and for simplicity we take the torus to be rectangular. These will decouple in the limit R I , R2 -t 0 leaving a theory in D - 2 dimensions. For example, for 11-dimensional supergravity, this limit will give the dimensionally reduced 9-dimensional maximal supergravity theory. We now compare this with M-theory compactified on T2. Consider first the circle of radius R 2 , say. M-theory compactified on this circle is equivalent to the IIA string theory with coupling constant 9s = (R 2/1 p )3/2, and so the limit R2 -t 0 is the weak coupling limit of this IIA string theory. We now have the IIA string theory compactified on a circle of radius R I , and by T-duality this is equivalent to the lIB string theory compactified on a circle of radius RI = l/T R I . Taking the limit Rl -t 0 is then the limit in which Rl -t 00 and an extra circle opens up to give the lIB string theory in 9+ 1 dimensions. The IIA string winding modes provide the tower of states that become massless in the limit and which are re-interpreted as momentum modes on the circle of radius Rl . Moreover, these IIA string winding modes are M-theory membranes wrapped around the 2-torus. These membrane wrapping modes have mass

(10) where the membrane tension is T2 = 1/1;. Then 10 M-theory compactified on a general 2-torus of area A and modulus T is equivalent to the lIB string theory compactified on a circle of radius 13 RB = ..E.

(11) A with string coupling 9s and axionic coupling () (defined as the expectation value of the scalar field in the Ramond-Ramond sector) given by .1 (12) T= () +Z9s The states of the lIB string carrying momentum in the circular dimension arise from membranes wrapping the 2-torus while the (p, q) string of the lIB theory winding round the circular dimension (with fundamental string charge

53

p and D-string charge q) arises from M-theory states carrying momentum pi R1 and ql R2 in the compact dimensions. Then in the limit A -+ 0, we lose two of the dimensions, as in the field theory, leaving a theory in 8+ 1 dimensions, but a new spatial dimension opens up to give a theory in 9+ 1 dimensions. 2.4

Compactijication on T3

For 11-dimensional supergravity compactified on a 3-torus, the limit in which the radii R 1,R2 ,R3 all tend to zero gives the dimensional reduction to the maximal supergravity in 8 dimensions. For M-theory on T 3 , membranes can wrap any of the three 2-cycles of T3. From the last section, we know that if R1 and R2 both shrink to zero while R3 stays fixed, an extra dimension opens up with radius R3 = l~1 R1R2 and the tower of membrane wrapping states is reinterpreted as a Kaluza-Klein tower for this extra dimension. The same picture applies to each of the three 2-cycles, and so if all 3 radii shrink, there are three extra dimensions opening up, with radii ~ given by _

lp3

~ = RjRk'

i

i- j i- k

(13)

Then when the original three torus shrinks to zero size, three dimensions are lost but three new ones emerge, so we are again back in 11 dimensions and the 11-dimensional theory is again M-theory 19. Thus M-theory compactified on a dual T3 with radii R 1, R 2, R3 is equivalent to M-theory compactified on the dual T3 with radii R 1,R2,R3 given by (13). 2.5

Compactijication on T4

It is tempting to apply these arguments to higher tori. For example, T4 has six 2-cycles, and membranes can wrap any of them. Compactifying D = 11 supergravity on a T4 and taking all four radii ~ -+ 0 gives a D = 7 field theory, but for M-theory there are six towers of states becoming massless in the limit arising from membranes wrapping each of the six shrinking 2-cycles. If each of these towers is interpreted as a Kaluza-Klein tower, this would give 6 extra dimensions in addition to the 7 original dimensions remaining, giving a total of 13 dimensions. However, there is no conventional supersymmetric theory in 13 dimensions, so it is difficult to see how such a theory could emerge. In fact the situation here is more complicated, and the 6 towers have a different interpretation here. The difference here is that there is also a string in the compactified theory arising from the M-theory 5-brane wrapped around the T4 which becomes 'light' at the same time as the 6 towers of

54

membrane wrapping modes 15 It turns out that M-theory on T4 is dual to lIB string theory on T3. The M-theory 5-brane wrapped around the T4 gives the fundamental string of the lIB theory moving in 7 dimensions 15. Compactifying the lIB string on T3 gives in addition three momentum modes and three winding modes, fitting into a 6 of the T-duality group SO(3,3), and these correspond to the 6 towers of membrane wrapping modes, which themselves transform as a 6 of the torus group SL(4) '" SO(3,3). Thus only 3 of the 6 towers can be interpreted as momentum modes for an extra dimension, the other three being interpreted as string winding modes, and the spacetime dimension of the dual theory is 10, not 13. Note that there is no invariant way of choosing which three of the six correspond to spacetime dimensions, as T-duality transformations will relate momentum and winding modes and change one subset ofthree to another. In this case, taking the limit in which the T4 on which the M-theory is compactified shrinks to zero size does not correspond to a decompactification limit of the dual theory, but to the weak coupling limit in which the coupling constant 98 of the compactified lIB string theory tends to zero 15. 3

Branes and Space and Time

We have seen that wrapped branes are associated with towers of massi ve states and that in some cases these can be interpreted as Kaluza-Klein towers for extra dimensions. In a limit in which such a tower becomes massless (e.g. Ri -t 0 for toroidal compactifications, or 98 -t 00 for the lIA DO-branes), the corresponding dimension decompactifies and new dimensions unfold. The presence of an enlarged Lorentz symmetry puts the new braney dimensions on an equal footing with the other dimensions, and the full theory includes gravity in the enlarged space. The number of dimensions lost in the limit is not always the same as the number of extra dimensions, so that the total number of spacetime dimensions can change (as in the relations between 11dimensional M-theory and 10-dimensional type II string theories considered in sections 2.2 and 2.3). We have also seen that, as in the case ofM-theory on T 4 , the towers of wrapped brane states cannot always be interpreted in terms of extra dimensions, and it is necessary to perform a more complete analysis to see what is going on. In all of the above cases, branes were wrapped around spacelike cycles and the extra dimensions that arose were all spacelike. A brane world-volume can also wrap around timelike cycles, and we will see that in such cases the extra dimensions can be timelike, so that the signature of spacetime can change. It is natural to ask whether it makes sense to consider compact time.

55

There are many classical solutions of gravity, supergravity, string and Mtheories with compact time and it is of interest to investigate their properties. Compact time does not appear to be a feature of our universe, but almost all spacetimes that are studied are also unrealistic. The presence of closed timelike loops means that the physics in such spaces is unusual, but it has often been fruitful in the past to study solutions that have little in common with the real world. An important issue with these solutions (as with many others) is whether a consistent quantum theory can be formulated in such backgrounds. If time were compact but with a huge period, it is not clear how that would manifest itself. With a compact time, it is straightforward to solve classical field equations, imposing periodic boundary conditions in time instead of developing Cauchy data. Can quantum theory make sense with compact time? There is no problem in solving Schrodinger or wave equations with periodic boundary conditions, but it is difficult to formulate any concept of measurement or collapse of a wave-function, as these would be inconsistent with periodic time: if a superposition of states collapsed to an eigenstate of an observable in some measurement process, it must already have been in that eigenstate from the last time it was measured. In string theory, it is straightforward to study the solutions of the physical state conditions, but there are new issues that arise from string world-sheets (and brane world-volumes) wrapping around the compact time. It has proved very fruitful to consider such compactifications in string theory. For example, the compactification of all 25+1 dimensions in bosonic string theory on a special Lorentzian torus played a central role in the work of Borcherds on the construction of vertex algebras and their application to the monster group 20. 4

4·1

Compactification on Lorentzian Tori and Signature Change

Compactijication on a Timelike Circle

Consider spacetimes of the form M D - 1 x Sl where Sl is a timelike circle of radius Rand M D - 1 is a Riemannian space. The time component of momentum is quantized o

n

p =-

R

(14)

and in the limit R --t 0, only the states with pO = 0 survive. For a field theory, the result is a dimensional reduction to a Euclidean field theory in D - 1 dimensions, on M D - 1 . For example, dimensionally reducing D = 11 supergravity on a timelike circle gives a supergravity theory in 10 Euclidean

56

dimensions, denoted the I I AE supergravity theory in 18. Timelike reductions of supergravity theories have been considered in 21,22,23. The field theory resulting from such a timelike reduction will in general have fields whose kinetic terms have the wrong sign. For example, the D dimensional graviton will give a graviton, a scalar and a vector field in D -1 dimensions on reducing on a circle, and if the circle is timelike, then the vector field will have a kinetic term of the wrong sign. Then the action for the physical matter fields of the reduced theory in D - 1 Euclidean dimensions will not be positive. This apparent problem is the result of the truncation to pO = 0 states. If this truncation is not made and if the full Kaluza-Klein towers of states with pO = nl R for all n are kept, then the theory is the full unitary D dimensional theory on a particular background, and the D dimensional gauge invariance can be used to choose a physical gauge locally with a positive action and states with positive norm. In general such a gauge choice cannot be made globally and there will be zero-mode states for which the action will not be positive. For example, the states with pO = 0 are governed by the non-positive dimensionally reduced action in D - 1 dimensions. For a Yang-Mills theory reduced on a timelike circle, the time component of the vector potential Ao gives a scalar field in D - 1 dimensions with a kinetic term of the wrong sign. For the full D dimensional theory compactified on the timelike circle, the negative-norm A o can be brought to a constant by D-dimensional gauge transformations, but one cannot gauge away the degrees of freedom associated with Wilson lines winding around the compact time dimension. (The fields with kinetic terms of the wrong sign can be handled in the path integral in the same way as the negative-action gravitational conformal mode is sometimes dealt with, namely by analytic continuation so that the offending field becomes imaginary 25 .) In a string theory, however, there will be winding modes in which the 1+ 1 dimensional string world-sheet winds around the compact time dimension, giving a spacelike 'world-line' in the compactified theory in D - 1 dimensions. As in the spacelike case, as R ~ 0, a dual circle opens up with radius R = liT R, and the new circle is again timelike. The winding number becomes the pO of the dual theory, and in this way a superstring theory in 9+ 1 dimensions compactified on a timelike circle of radius R is T-dual to a superstring theory in 9+ 1 dimensions compactified on a timelike circle of radius R. Such timelike T-dualities were considered for the bosonic and heterotic strings in e.g. 24, and they take the bosonic string theory to the bosonic string theory and the heterotic string theory to the heterotic string theory. However, for type II theories there is a surprise. It is straightforward to see that timelike T-duality cannot take the IIA string theory to either the lIB string or the IIA string

57

but must take it to a 'new' theory, denoted the II B* string theory in 25. Similarly, timelike T -duality takes the lIB string to a II A * string theory 25. The IIA* and IIB* strings are taken into each other by T-duality on a spacelike circle, and the IIA* (IIB*) theory is obtained from the IIA (lIB) string theory by acting with (i)FL 25 (where FL is the left-handed fermion number). The supergravity limits of the IIA* and IIB* have non-positive actions for the matter fields (the kinetic terms for the fields in the R-NS and R-R sectors have the wrong sign) so that the low-energy field theories are non-unitary, but the IIA* and IIB* string theories compactified on a timelike circle are equivalent to the IIA and lIB string theories on the dual timelike circle. Then, at least when on a timelike circle, the I I A * and II B* string theories are precisely the timelike compactifications of the usual IIA and lIB string theories, albeit written in dual variables. The supergravity limit for the IIA or lIB variables is the conventional one, while the supergravity limit for the dual variables is non-unitary. A physical gauge can then be chosen locally for the II A * and II B* string theories on a timelike circle, and any lack of unitarity or positivity is due to zero-modes. If time is compact and the physics is periodic in time, the requirements for a sensible theory are not the same as in Minkowski space. A theory that is unstable in Minkowski space (perhaps due to negative energy configurations) need not be pathological if time is compact: the periodic boundary conditions forbid any runaway solutions and the system will always return to its starting point after a period. A nonunitary theory in Minkowski space will not conserve probability, but with periodic time, any probability that is lost will always come back, as the solutions of the wave equations are required to be periodic. This suggests that the timelike compactifications of the IIA* and IIB* string theories should be consistent, although the question remains a.'l to the status of the decompactification limit in which the radius of the timelike circle becomes infinite. Similar considerations will apply to the other new theories of 18 described in this section. See 25 for further discussion of the type 11* theories. 25,

4.2

Compactijication on T 1 ,1

Consider now compactification on the Lorentzian torus T 1 ,1 with one spacelike circle and one timelike one. (We will use the notation Ts,t for a torus with s spacelike circles and t timelike ones.) For ll-dimensional supergravity, the limit R s , R t ~ 0 gives a 9-dimensional Euclidean supergravity theory. For Mtheory on a Euclidean torus T2, we saw in section 2.3 ·that in the limit in which the torus shrank to zero size, one new spacelike dimension opened up to give the lIB string theory in 9+ 1 dimensions. Here we expect something similar

58

to happen. Considering first the compactification on the spacelike circle of radius Rs, when Rs is small we obtain the IIA string theory with coupling constant gs = (R/l p )3/2. The compactification of this on a timelike circle of radius R t is T -dual to the II B* string theory compactified on a timelike circle of radius -

1

R t = TR t

(15)

Then taking the limit R t -+ 0, we obtain a theory in the expected 9 spacelike dimensions together with a new time dimension which opens up, the T-dual of the original timelike dimension. The membranes wrapping around T 1 ,1 have become the modes carrying the time component of momentum pO of the dual IIB* theory, and M-theory compactified on T 1 ,1 with radii Rs,Rt is dual to the II B* string theory compactified on a timelike circle of radius l!/ RsRt, as was shown in 18.

4.3

Compactijication on T 2 ,1

We have seen in section 2.3 that M-theory compactified on a Euclidean 2-torus T2 gains a new spatial dimension in the limit in which the 2-torus shrinks to zero size, replacing the two which have disappeared, so that the original theory in (10,1) dimensions becomes a theory in (9,1) dimensions: (9,1) = (10,1) - (2,0) + (1,0). Similarly, we have seen in section 4.1 that M-theory compactified on a Lorentzian 2-torus T 1 ,1 gains a new time dimension in the limit in which the 2-torus shrinks to zero size, replacing the (1,1) dimensions which have disappeared so that the original theory in (10,1) dimensions again becomes a theory in (9,1) dimensions: (9,1) = (10,1) - (1,1) + (0,1). Thus a shrinking T2 is associated with an extra space dimension while a shrinking T 1 ,1 is associated with an extra time dimension. For M-theory on a shrinking Euclidean T3, an extra space dimension emerges for each of the three shrinking 2-cycles, so that the three toroidal dimensions which are lost are replaced by three new spatial dimensions, and we end up back in M-theory in (10,1) dimensions: (10,1) = (10,1) - 3 x (1,0) + 3 x (1,0). Consider now the compactification on a Lorentzian 3-torus T 2 ,1 with two spacelike and one timelike circles. In the limit in which the torus shrinks to zero size, 2+1 dimensions are lost leaving 8 Euclidean dimensions and reducing ll-dimensional supergravity on T 2 ,1 indeed gives a supergravity in (8,0) dimensions. In M-theory, if the discussion above applies here, we expect an extra space dimension for every shrinking T2 and an extra time dimension for every shrinking T 1 ,1. The torus T 2 ,1 has two Lorentzian 2-cycles and

59

one Euclidean one, so that this suggests there should be an extra two time dimensions and one space dimension that open up in this limit, giving a theory in 11 dimensions with two-timing signature (9,2) = (8,0)+ (1, 0)+2 x (0, 1). If all the towers of wrapped membranes give extra dimensions, this must be the result, but we have seen that in some cases towers of wrapped brane states can have other meanings. A more careful analysis shows that this interpretation is indeed correct and taking M-theory on a shrinking T 2 ,1 gives a new theory in 9+2 dimensions 18. Then dualities can change the number of time dimensions as well as the number of space dimensions. This new theory in 9+2 dimensions was referred to as the M* theory in 18, and it has an effective field theory which is a new supergravity theory in 9+2 dimensions. M-theory compactified on T 2 ,1 is equivalent to M* theory compactified on a two-time torus T 1 ,2, with the sizes of the circles related by a formula similar to (13).

4·4

Compactijications of M* Theory

We can now investigate the compactifications of M* theory on various tori 18. Compactifying the M* theory on a timelike circle gives the I I A * string theory in 9+1 dimensions, while compactifying on a spacelike circle gives a new IIAlike string theory in 8+2 dimensions. Next consider the compactification on 2-tori in the limit in which they shrink to zero size. For TO,2 this gives the lIB string (compactification on the first circle gives the lIA* theory and the second then gives its T-dual on a timelike circle), for T 1 ,1 it gives the lIB* theory and for T 2 ,o it gives a new lIB-like theory in 7+3 dimensions. Thus a shrinking TO,2 gives an extra time dimension, a shrinking T 1 ,1 gives an extra space dimension and a shrinking T 2 ,o gives an extra time dimension. This can now be used to find the results of compactification on a shrinking three-torus. For T 1 ,2 there are two T 1 ,1 cycles and one Euclidean T2 cycle giving a theory in (9,2) - (1,2) + 2 x (1,0) + (0,1) = (10,1) dimensions and we are back in M-theory, for T 2,1 there are two T2 cycles and one T 1 ,1 cycle giving a theory in (9,2) - (1,2) + 2 x (0,1) + (1,0) = (9,2) dimensions and we are back in M* theory, while for T3,O there are three Euclidean T2 cycles giving a theory in (9,2) - (3,0) + 3 x (0,1) = (6,5) dimensions, giving a new theory in 6+5 dimensions. This theory was denoted the M' theory in 18, and M* theory compactified on T3,O is equivalent to the M' theory compactified on a dual TO,3. The above analysis can then be repeated for this new M' theory, and it turns out that only ll-dimensional theories that arise are the M, M* and M' theories, with signatures (10,1), (9,2) and (6,5), together with the mirror

60

theories in signatures (1,10), (2,9) and (5,6). Reduction on circles gives IIAlike theories in signatures 10+0,9+1, 8+2, 6+4 and 5+5 while reducing on 2-tori gives lIB-like theories in signatures 9+1, 7+3, and 5+5. (There are of course also mirror string theories in the signatures 1+9, 2+8 etc with space and time interchanged.) In each of these 10 and 11 dimensional cases there is a corresponding supergravity limit and it is a non-trivial result that these supergravities exist, and it is unlikely that there are maximal supergravities in signatures outside this list. These theories are linked to each other by an intricate web of dualities 18, some of which have been outlined above, and in particular all are linked by dualities to M-theory. Each of these theories has a set of branes of various world-volume signatures 26,27. For the M-type theories, M-theory has branes of world-volume signature 2+1 and 5+1 (the usual M2 and M5 branes), M* theory has branes of world-volume signature 3+0,1+2 and 5+1 while M' theory has branes of world-volume signature 2+1,0+3, 5+1, 3+3 and 1+5. 5

Discussion

In a field theory, compactification and then shrinking the internal space K to zero size gives a dimensionally reduced field theory in lower dimensions. In compactified string theory or M-theory, however, new dimensions can emerge when the internal space shrinks, with the Kaluza-Klein towers for the new dimensions corresponding to the brane wrapping modes in which branes wrap around cycles of K. In some cases (e.g. toroidal compactifications of string theory or M-theory on T3) the number of new dimensions equals the number that are lost and one regains the original spacetime dimension, while in others (such as M-theory compactified on T2) the number of new dimensions is different from the number that are lost and so the dimension of spacetime changes (for M-theory on T2 it changes from 11 to 10). Clearly, the notion of what is a spacetime dimension is not an invariant concept, but depends on the 'frame of reference', in the sense that it will depend on the values of various moduli. A given tower of BPS states could have a natural interpretation as a Kaluza-Klein tower associated with momentum in a particular compact spacetime dimension for one set of parameters, but could have an interpretation as a tower of brane wrapping modes for other values, and we have seen many examples of this in the preceding sections. We are used to considering field theories in spacetimes of given dimension and signature, but any attempt to formulate M-theory or string theory as a theory in a given spacetime dimension or signature will be misleading. In particular,

61

the theory underpinning all these theories has a limit which behaves like a theory in 10+1 dimensions with a supergravity limit and systematic corrections, but cannot at the fundamental level be a theory in 10+1 dimensions, as it has some limits which live 9+1 dimensions and others that live in 9+2 or 6+5 dimensions. The super symmetry algebra in 10+ 1 dimensions is 1 1 {Q, Q} = c(rM PM - 2!r M1M2 ZM1M 2 - 5!r M1 ... Ms ZM1 ... M s ) , (16) where C is the charge conjugation matrix, PM is the energy-momentum 11vector and ZM1M 2 and ZM1 ... Ms are 2-form and 5-form charges, associated with brane charges 28. There are 11+55+462=528 charges on the right-handside, which can be assembled into a symmetric bi-spinor XO/{3' Compactifying and then dualising, one finds that some of the brane charges become momenta of the dual theory and some of the momenta become brane charges of the dual theory, so that the split of the bi-spinor X into an 11-momentum and brane charges changes under duality. This suggests that rather than trying to formulate the theory in 10+1 dimensions, all 528 charges should be treated in the same way. There seem to be at least two ways in which this might be done. The first would be a geometrical one in which all 528 charges were treated as momenta and there is an underlying spacetime of perhaps 528 dimensions. The duality symmetries could then act geometrically, and there would be perhaps some dynamical way of choosing 11 of the dimensions as the preferred ones, e.g. through the 'world' being an 11-dimensional surface in this space. For example, in considering Tduality between a string theory on a space M x 3 1 and one in the dual space M x fP, it is sometimes useful to consider models on M X 3 1 x fJl in which both the circle of radius R and the dual circle of radius R are present, with different projections or gaugings giving the two T-dual models; see 4 and references therein. We have seen that different spacetimes related by dualities can define the same physics, so that the notion of spacetime geometry cannot be fundamental. This suggests that different degrees of freedom should be used, with spacetime emerging as a derived concept. An alternative 'anti-geometrical' formulation would be one in which none of the charges were geometrical, but instead an algebraic approach similar to that of matrix theory was used. For example, M-theory could be compactified to 0,1 or 2 dimensions to give a theory that would be expected to have duality symmetry 30 Ell, ElO or Eg where Eg is an affine E 8 , ElO is a hyperbolic algebra discussed, for example, in 31 and Ell might be some huge algebraic structure associated with the Ell Dynkin diagram. In one dimension the theory might be some matrix

62

quantum mechanics associated with ElO while in zero dimensions it would be some form of non-dynamical matrix theory. At special points in the moduli space, some of the charges would be associated with extra dimensions that are decompactifying. At different points, different numbers of space and time dimensions could emerge. Such formulations might be related to the reformulations of 11dimensional supergravity of 32,33,34 in which the tangent space group is enlarged so that some of the duality symmetries are manifest. For example, in the context of compactifications to 2+1 dimensions, the usual tangent space group SO(10, 1) is broken to SO(2, 1) x SO(8) and then anti-symmetric tensor degrees of freedom were used in 33 to reformulate the theory with tangent space group SO(2,1) x SO(16), with the SO(16) associated with the usual local SO(16) invariance of 3-dimensional supergravity. These formulations show that there are alternatives to the usual formulation in 11 spacetime dimensions and it would be interesting to consider others. The five superstring theories and M-theory are different corners of the moduli space of some as yet unknown fundamental theory and the dualities linking them all involve compactification on Riemannian spaces. If this is extended to include compactification on spaces with Lorentzian signature a richer structure emerges. The strong coupling limit of the type IIA superstring is M-theory in 10+1 dimensions whose low energy limit is 11-dimensional supergravity theory. The type I, type II and heterotic superstring theories and certain supersymmetric gauge theories emerge as different limits of M-theory. The M-theory in 10+1 dimensions is linked via dualities to M* theory in 9+2 dimensions and M'-theory in 6+5 dimensions. Various limits ofthese give rise to IIA-like string theories in 10+0, 9+1,8+2,6+4 and 5+5 dimensions, and to lIB-like string theories in 9+1,7+3, and 5+5 dimensions. The field theory limits are supergravity theories with 32 supersymmetries in 10 and 11 dimensions with these signatures, many of which are new. Further dualities similar to those of 16 relate these to supersymmetric gauge theories in various signatures and dimensions, such as 2+ 2, 3+ 1 and 4+0. These new string theories and M-type theories in various spacetime signatures can all be thought of as providing extra corners of the moduli space. Some corners are stranger than others, but in any case we can only live in one corner (perhaps M-theory compactified on the product of a line interval and a Calabi-Yau 3-fold) and there is no reason why other corners might not have quite unfamiliar properties. Theories in non-Lorentzian signatures usually have many problems, such as lack of unitarity and instability. However, the theories considered here are related to M-theory via dualities and so are just the usual theory expressed in terms of unusual variables. For example, the M* theory in 9+2 dimensions

63

compactified on T 1 ,2 is equivalent to M-theory compactified on T 2 ,1, and so the compactified M* theory will make sense provided M-theory compactified on a Lorentzian torus is a consistent theory. Then the problems with formulating a theory in 9+2 dimensions are in this case only apparent, as the theory can be rewritten as a theory in 10+1 dimensions using different variables, so that the extra time dimension is replaced by the degrees of freedom associated with branes wrapped around time. There are several possible generalisations of the notion of a particle to general signatures. A physical particle or an observer in Lorentzian spacetime with signature (8,1) follows a timelike (or nUll) world-line while a tachyon would follow a spacelike one. In a space of signature (8, T), one can again consider worldlines of signature (0,1), but other generalisations of particle might include branes with worldvolumes ('time-sheets') of signature (0, t) with t ~ T, sweeping out some or all of the times. In a general signature (8, T), it is natural to consider branes of arbitrary signature (s, t) with s ~ 8 and t ~ T, and the conditions on (s,t) for these to be supersymmetric were given in 26. In conclusion, we have reviewed part of the intricate web of duality symmetries linking many apparently different theories, but since the theories are all related in this way, they should all be regarded as corners of a single underlying theory. In particular, two dual theories can be formulated in spacetimes of different geometry, topology and even signature and dimension, and so all of these concepts must be relative rather than absolute, depending on the values of certain parameters or couplings, and such a relativity principle should be a feature of the fundamental theory that underlies all this. References References

1. C.M. Hull and P.K. Townsend, Nucl. Phys. B438 (1995) 109 hepth/9410167. 2. E. Witten, Nucl. Phys. B443 (1995) 85, hep-th/9503124 3. J. Polchinski, hep-th/9611050 4. A. Giveon, M. Porrati and E. Rabinovici, Phys. Rep. 244 (1994) 77. 5. J. Dai, R.G. Leigh and J. Polchinski, Mod. Phys. Lett. A4 (1989) 2073; 6. M. Dine, P. Huet and N. Seiberg, Nucl. Phys. B322 (1989) 301. 7. C.M. Hull, Phys. Lett. B357 (1995) 545, hep-th/9506194. 8. A. Dabholkar, Phys. Lett. B357 (1995) 307, hep-th/9506160. 9. J. Polchinski and E. Witten, Nucl.Phys. B460 (1996) 525, hep-

64

thj9510169. 10. P. Aspinwall, Nucl. Phys. Proc. Suppl. 46 (1996) 30, hep-thj9508154 ; J.H. Schwarz, Phys.Lett. B360 (1995) 13; Erratum-ibid. B364 (1995) 252, hep-thj9508143 . 11. P. Horava and E. Witten, Nucl. Phys. B460 (1996) 506, hep-thj9510209. 12. G. Pradisi and A. Sagnotti,Phys. Lett. B216 (1989) 59; M. Bianchi and A. Sagnotti, Phys. Lett. B247 (1990) 517; Nucl. Phys. B361 (1990) 519; P. Hoi'ava, Nucl. Phys. B327 (1989) 461; Phys. Lett. B231 (1989) 251. 13. E. Bergshoeff, E. Eyras, R. Halbersma, J.P. van der Schaar, C.M. Hull and Y. Lozano, hep-thj9812224. 14. C.M. Hull, JHEP 9811 (1998)27, hep-thj9811021 . 15. C.M. Hull, Nucl. Phys. B468 (1996) 113 hep-th/9512181. 16. J. Maldacena, hep-thj9711200. 17. T. Banks, hep-th/9710231; L. Bigatti and L. Susskind, hep-thj9712072 18. C.M. Hull, JHEP 11 (1998) 017 hep-thj9807127. 19. W. Fischler, E. Halyo, A. Rajaraman and L. Susskind, Nucl.Phys. B501 (1997) 409, hep-thj9703102. 20. P. Goddard, 'The Work of R.E. Borcherds',·to appear in Proceedings of the International Congress of Mathematicians, and references therein. 21. C. M. Hull and B. Julia, Nucl. Phys. B534 (1998) 250, hep-thj9803239. 22. E. Cremmer, I.V. Lavrinenko, H. Lu, C.N. Pope, K.S. Stelle and T.A. Tran, Nucl.Phys. B534 (1998) 250, hep-thj9803259. 23. K. S. Stelle, hep-thj9803116. 24. G. Moore, hep-th/9305139,9308052. 25. C.M. Hull, hep-th/9806146. 26. C.M. Hull and R.R. Khuri, Nucl. Phys. B536 (1998) 219 hep-thj9808069. 27. C.M. Hull and R.R. Khuri, hep-thj9911082. 28. C.M. Hull, Nucl. Phys. B509 (1998) 216, hep-thj9705162. 29. P.K. Townsend, hep-thj9712004 . 30. B. Julia, in Superspace and Supergravity, eds. S.W. Hawking and M. Rocek (Cambridge University Press, 1981) 31. B. Julia, in Lectures in Applied Mathematics, Vol. 21 (1985) 355; H. Nicolai, Phys. Lett. B276 (1992) 333 32. B. de Wit, H. Nicolai, Phys. Lett. 155B (1985) 47; Nucl. Phys. B274 (1986) 363 33. H. Nicolai, Phys.Lett. 187B (1987) 363. 34. H. Nicolai, hep-th/9801090.

FIELD THEORY - WHY HAVE SOME PHYSICISTS ABANDONED IT?*

ROMANJACKIW Massachusetts Institute of Technology, 77 Massachusetts Avenue, 6-320 Cambridge, MA 02139-4307, USA E-mail: [email protected] The present-day crisis in quantum field theory is described.

Our present-day theory for fundamental processes in Nature - and by this I mean our descriptions of elementary particles and forces - is phenomenally successful. Experimental data confirms theoretical prediction; where accurate calculations and experiments are attainable, agreement is achieved to many - six or seven - significant figures. Table ?? shows two examples. Of course mostly such precision cannot be achieved, neither theoretically nor experimentally. Yet no experiment has thus far contradicted our understanding of the gravitational interactions as described by Einstein's general relativity, nor of the strong nuclear interactions, nor of the electromagnetic and radioactivity-producing weak interactions that are now collected into the GlashowWeinberg-Salam "standard model". (A hint of physical phenomena beyond the standard model has recently been provided by experimentalists announcing the discovery of a neutrino mass, which is not predicted by the standard model. But if this much-anticipated result is independently confirmed, it can then be fitted very easily into a straightforward extension of the present-day model.) The strong and electro-weak theories make use of a quantum mechanical description, while classical physics suffices to account for all known gravitational phenomena. The theoretical structure within which this success has been achieved is local field theory, which offers physicists a tremendously wide variety of applications: it is both a language with which physical processes are discussed and also it provides a model for fundamental physical reality, as described by our theories of strong, electro-weak, and gravitational processes. No other framework exists in which one can calculate "so many phenomena with such ease and accuracy" (L.P. Williams, personal communication). Arising from a mathematical account of the propagation of fluids (both "ponderable" and "imponderable"), field theory emerged over 100 years ago in ·Proc. Nat!. Acad. Sci. USA 95, 12776 (1998); ©1998 National Academy of Sciences, USA

65

66 Table 1. Comparison between particle physics theory and experiment in two very favorable cases. Helium atom ground state energy (Rydbergs) -5.8071394 (14) -5.8071380 (5)

Experiment Theory

Muon magnetic dipole moment 2.00233184600 (1680) 2.00233183478 (308)

Experiment Theory

the discussion within classical physics of Faraday-Maxwell electromagnetism and soon thereafter of Einstein's gravity theory. Schrodinger's wave mechanics became a bridge between classical and quantum field theory: the quantum mechanical wave function is also a local field, which when "second" quantized gives rise to a true quantum field theory, albeit a nonrelativistic one. Quantization of electromagnetic waves produced the first relativistic quantum field theory, which when supplemented by the quantized Dirac field gave us quantum electrodynamics, whose further generalization to matrices of fields - the Yang-Mills construction - is the present-day standard model of elementary particles. This development carries with it an extrapolation over enormous scales: initial applications were at microscopic distances or at energies of a few electron volts, while contemporary studies of elementary particles involve 1011 electron volts or short distances of 10- 16 cm. The "quantization" procedure, which extended classical field theory's range of validity, consists of expanding a classical field in normal modes, and taking each mode to be a quantal oscillator. Field theoretic ideas also reach for the cosmos through the development of the "inflationary scenario" - a speculative, but completely physical analysis of the early universe, which appears to be consistent with available observations. Additionally, quantum field theories provide effective descriptions of many-body, condensed matter physics. Here the excitations are not elementary particles and fundamental interactions are not probed, but the collective phenomena that are described by many-body field theory exhibit many interesting effects, which in turn have been recognized as important for elementary particle theory. Such exchanges of ideas between different subfields of physics demonstrate vividly the vitality and flexibility of field theory. But in spite of these successes, today there is little confidence that field theory will advance our understanding of Nature at its fundamental workings, beyond what has been achieved. While in principle all observed phenomena

67

can be explained by present-day field theory (in terms of the quantal standard model for particle physics, perhaps slightly extended to incorporate massive neutrinos, and the classical Newton-Einstein model for gravity), these accounts are still imperfect. The particle physics model requires a list of ad hoc inputs that give rise to conceptual, general questions such as: Why is the dimensionality of space-time four? Why are there two types of elementary particles (bosons and fermions)? What determines the number of species of these particles? The standard model also leaves us with specific technical questions: What fixes the matrix structure, various mass parameters, mixing angles, and coupling strengths that must be specified for concrete prediction? Moreover, classical gravity theory has not been integrated into the quantum field description of nongravitational forces, again because of conceptual and technical obstacles: quantum theory makes use of a fixed space-time, so it is unclear how to quantize classical gravity, which allows space-time to fluctuate; even if this is ignored, quantizing the metric tensor of Einstein's theory produces a quantum field theory beset by infinities that cannot be controlled. But these shortcomings are actually symptoms of a deeper lack of understanding that has to do with symmetry and symmetry breaking. Physicists mostly agree that ultimate laws of Nature enjoy a high degree of symmetry, that is, the formulation of these laws is unchanged when various transformations are performed. Presence of symmetry implies absence of complicated and irrelevant structure, and our conviction that this is fundamentally true reflects an ancient aesthetic prejudice: physicists are happy in the belief that Nature in its fundamental workings is essentially simple. However, we must also recognize that actual, observed physical phenomena rarely exhibit overwhelming regularity. Therefore, at the very same time that we construct a physical theory with intrinsic symmetry, we must find a way to break the symmetry in physical consequences of the model. Progress in physics can frequently be seen as the resolution of this tension. In classical physics, the principal mechanism for symmetry breaking is through boundary and initial conditions on dynamical equations of motion. For example, Newton's rotationally symmetric gravitational equations admit the rotationally nonsymmetric solutions that describe actual orbits in the solar system, when appropriate, rotationally nonsymmetric, initial conditions are posited. The construction of physically successful quantum field theories makes use of symmetry for yet another reason. Quantum field theory models are notoriously difficult to solve and also explicit calculations are beset by infinities. Thus far we have been able to overcome these two obstacles only when the models possess a high degree of symmetry, which allows unravel-

68

ing the complicated dynamics and taming the infinities by renormalization. Our present-day model for quarks, leptons, and their interactions exemplifies this by enjoying a variety of chiral, scale/conformal, and gauge symmetries. But to agree with experiment, most of these symmetries must be absent in the solutions. At present we have available two mechanisms for achieving this necessary result. One is spontaneous symmetry breaking, which relies on energy differences between symmetric and nonsymmetric solutions: the dynamics may be such that the nonsymmetric solution has lower energy than the symmetric one, and the nonsymmetric one is realized in Nature while the symmetric solution is unstable. The second mechanism is anomalous or quantum mechanical symmetry breaking, which uses the infinities of quantum theory to effect a violation of the correspondence principle: the symmetries that appear in the model before quantization disappear after quantization, because the renormalization procedure - needed to tame the infinities and well define the theory - cannot be carried out in a fashion that preserves the symmetries. While these two methods of symmetry breaking successfully reduce the symmetries of the standard model to a phenomenologically acceptable level, this reduction is achieved in an ad hoc manner, and much of the previously mentioned arbitrariness, which must be fixed for physical prediction, arises precisely because of the uncertainties in the symmetry-breaking mechanisms. Spontaneous symmetry breaking is adopted from many-body, condensed matter physics, where it is well understood: the dynamical basis for the instability of symmetric configurations can be derived from first principles. In the particle physics application, we have not found the dynamical reason for the instability. Rather, we have postulated that additional fields exist, which are destabilizing and accomplish the symmetry breaking. But this ad hoc extension introduces additional, a priori unknown parameters and yet-unseen particles, the Higgs mesons. Anomalous symmetry breaking also carries with it arbitrariness: the "renormaiization scale" of quantum chromo dynamics (QCD), the theory that governs interactions between quarks; also, we invoke yet-unseen particles, the axions, which remove from QCD a time-reversal-violating angle that arises from anomalous breaking of a relevant symmetry, and which would otherwise have an undetermined magnitude. Moreover, the field theoretic infinities, which give rise to anomalous symmetry breaking, prevent the construction of an acceptable quantum gravity field theory, so it is peculiar to rely on them so critically for the viability of the standard model. Advancing our understanding of the above has been at an impasse for over two decades. In the absence of new experiments to channel theoretical speculation, some physicists have concluded that it will not be possible to

69

make progress on these questions within field theory, and have turned to a new structure, string theory. In field theory the quantized excitations are point particles with point interactions and this gives rise to the infinities. In string theory, the excitations are extended objects - strings - with interactions that extend over a finite space-time interval. There are no infinities, and this enormous defect of field theory is absent. Not only does quantum gravity exist in the new context, but it appears that some puzzles having to do with black holes can be answered. Moreover, string theory addresses precisely some of the questions that remained unanswered in field theory: dimensionality of space-time cannot be arbitrary because string theory cannot be formulated in arbitrary dimensions; fermions must coexist with bosons because of supersymmetry - a necessary ingredient of string theory, which requires bosons and fermions to be paired in a symmetry transformation; and so on. Yet in spite of these positive features, up to now string theory provides only a framework, rather than a definite structure. While present-day physics should be found in the low-energy limit of string theory, a precise derivation of the standard model has yet to be given. One thinks again about symmetry and symmetry breaking. The symmetries of quantum field theory surpass those of classical physics and require elaborate symmetry breaking mechanisms. The symmetries of string theory again vastly outpace those of field theory, and must be broken by yet-to-be-developed procedures, in order to explain the world around us. Are there any experimental facts - as opposed to theoretical ideas - that support string theory? One can identify only a few. Black holes are accepted as physical entities, and black hole radiance - Hawking radiation although not yet observed, appears to be a physical concomitant. The process is essentially quantum mechanical, and thus far only string theory gives us a consistent quantum theory of gravity, within which quantal properties of black holes can be calculated. (However, in dimensions lower than the physical3+1-dimensional space-time, gravity theory can be successfully quantized, and black hole physics can be described, without strings or supersymmetry.) Support for a different ingredient of string theory - that of supersymmetry - comes from our desire to unify all forces. Unification of the electro-weak and the strong forces, which at present are described by the three separate group theoretical entities SU(3) x SU(2) x U(l), may occur at sufficiently high energies. However, extrapolating present-day data to these high energies shows that the three do not merge into a single entity unless all the particles seen today possess supersymmetric partners (Fig. ??). Evidently, experimental support is tenuous. It requires extrapolation of 1015 to 1020 orders of magnitude from present knowledge. Nevertheless, string

70

Bupersymmetric

Bupersymmetric

partners

partners

Mw

10 15 Mw - - - = electro-weak

_ _ _ =strong

Figure 1. Schematic plot of(interaction strength) -1 in arbitrary units (vertical axis) versus energy in units of Mw = 0(100 GeV) (horizontal axis).

theory and supersymmetry now consume theorists' work. A recent compilation by the Institute of Scientific Information of the 1120 physicists with the most cited papers in the last 15 years a is dominated by experimentalists in highly populated fields like condensed matter and materials science. But the list is headed by the mathematicians' Fields medalist Edward Witten, with over 20,000 citations to his writings on supersymmetric string theory that fuel purely theoretical/mathematical speculation. One hopes that in the next millenium experimental data will become available with which we can assess this body of work, so that we can decide whether Nature, and not only the mathematics community, validates these ideas. On previous occasions when it appeared that quantum field theory was incapable of advancing our understanding of fundamental physics, new ideas and new approaches to the subject dispelled the pessimism. Today we do not know whether the impasse within field theory is due to a failure of imagination or whether indeed we have to present fundamental physical laws in a new framework, thereby replacing the field theoretic one, which has served us well for over 100 years.

ahttp://fluo.univ-lemans.fr:8001/1120physiciens.html

BLACK HOLES, THERMODYNAMICS AND POLYMERS RAMZI R. KHURI Baruch College, City University of New York, 17 Lexington Avenue, New York, NY, 10010 USA The Graduate School and University Center, City University of New York, 33 West 42nd Street, New York, NY 10036 USA Center for Advanced Mathematical Sciences, American University of Beirut, Beirut, Lebanon E-mail: [email protected] Quantum aspects of black holes represent an important testing ground for a theory of quantum gravity. The recent success of string theory in reproducing the Bekenstein-Hawking black hole entropy formula provides a link between general relativity and quantum mechanics via thermodynamics and statistical mechanics. Here we speculate on the existence of new and unexpected links between black holes and polymers and other soft-matter systems.

The standard model of elementary particle physics has been successful in describing three of the four fundamental forces of nature. In the most optimistic scenario, the standard model can be generalized to take the form of a grand unified theory, in which quantum chromodynamics, describing the strong force, and the electroweak theory, unifying the weak interaction with electromagnetism, are synthesized into a single theory in which all three forces have a common origin. The underlying framework of particle physics is quantum mechanics, in which the natural length scale associated with a particle of mass m (such as an elementary particle) is given by the Compton wavelength A = n/mc, where n is Planck's constant divided by 27r and c is the speed of light. Scales less than A are therefore unobservable within the context of the quantum mechanics of this particle. Quantum mechanics, however, has so far proven unsuccessful in describing the fourth fundamental force, gravitation. The successful theory in this case is that of general relativity, which, however, does not lend itself to a straightforward attempt at quantization. The main problem in such an endeavour is that the divergences associated with trying to quantize gravity cannot be circumvented (or "renormalized") as they are for the strong, weak and electromagnetic forces. Among the most interesting objects predicted by general relativity are black holes, which represent the endpoint of gravitational collapse. According 71

72

to relativity, an object of mass m under the influence of only the gravitational force (ie neutral with respect to the other three forces) will collapse into a region of spacetime bounded by a surface, the event horizon, beyond which signals cannot be transmitted to an outside observer. The event horizon for the simplest case of a static, spherically symmetric black hole of mass m is located at a radius R = 2Gm/2, the Schwarzschild radius, from the collapsed matter at the center of the sphere, where G is Newton's constant. In trying to reconcile general relativity and quantum mechanics, a natural question to ask is whether they have a common domain. This would arise when an elementary particle exhibits features associated with gravitation, R, which implies such as an event horizon. This may occur provided A

:s

that, even within the framework of quantum mechanics, an event horizon for an elementary particle may be observable. Such a condition is equivalent to m ~ mp = Vnc/G"" 10 19 GeV, the Planck mass, or A lp = VnG/c3, the

:s

Planck scale. It is in this domain that one may study a theory that combines quantum mechanics and gravity, the so-called quantum gravity (henceforth we use units in which n = c == 1). A problem, however, arises in this comparison, because most black holes are thermal objects, and hence cannot reasonably be identified with pure quantum states such as elementary particles. In fact, in accordance with the laws of black hole thermodynamics 1, black holes radiate with a (Hawking) temperature constant over the event horizon and proportional to the surface gravity: TH "" K. Furthermore, black holes possess an entropy S = A/4G, where A is the area of the horizon (the area law), and 8A ~ 0 in black hole processes. So only a black hole with zero area can correspond to a pure state with S = 0 such as an elementary particle, while a black hole with nonzero area, and therefore nonzero entropy, corresponds to an ensemble of states. A question, then, that can be posed of a theory of quantum gravity is the following: since the basis of ordinary thermodynamics is (quantum) statistical mechanics, can one recover the laws of black hole thermodynamics by the counting of microscopic states? In particular, can one recover the area law from a quantum mechanical entropy arising as the logarithm of the degeneracy of quantum states? At the present time, string theory, the theory of one-dimensional extended objects, is the only known reasonable candidate theory of quantum gravity. The divergences inherent in trying to quantize point-like gravity seem not to arise in string theory. Furthermore, string theory has the potential to unify all four fundamental forces within a common framework. At an intuitive level, one can see how point-like divergences may be avoided in string theory by con-

73

sidering scattering amplitudes in string theory 2. Unlike those of field theory, the four-point amplitudes in string theory do not have well-defined vertices at which the interaction can be said to take place, hence no corresponding divergences associated with the zero size of a particle. A simpler way of saying this is that the finite size of the string smooths out the divergence of the point particle. For the purpose of understanding black hole thermodynamics, an important feature of string theory is that classical solutions 3 may be easily constructed as composites of single-charged fundamental constituents. Identifying these constituents with states in string theory, one can compare the Bekenstein-Hawking entropy obtained from the area of the classical solution to the quantum-mechanical microcanonical counting of ensembles of states 4. For example, the extremal Reissner-Nordstr6m charged black hole solution of Einstein-Maxwell theory arises in string theory as the composite of four charges, N 1 , N 2, N3 and N 4, normalized to correspond to number operators in string theory. The area law then yields a Bekenstein-Hawking entropy SBH = 27rVNIN2N3N4. The counting of the degeneracy of the states forming this black hole leads to the same quantity SQM = Ind(Ni ) = SBH. Even in the black hole picture, this result can be seen to arise from the number of ways in which the various constituents combine. It is straightforward to show 5 that one can write four-centered solutions each with charge Ni of a given species. A black hole with nonzero area is formed when all charges are brought together to the same point. The precise partition function 6 yielding the correct degeneracy d(Ni ) = exp(SBH) is obtained provided both bosonic and fermionic excitations of a supersymmetric string-like object along various dimensions are taken into account. The recovery of the area law in a wide variety of contexts in string theory suggests that we have accounted for the microscopic degrees of freedom of the black hole. However, the ensemble of string states on the one hand and the black hole on the other represent two very different objects, so we must try to understand the correspondence between them 7. For simplicity, let us consider the case of a long, self-gravitating string in D = 4 dimensions 8. At level N, a free string has mass M '" .../NIls, size L '" N 1 / 4 1s and entropy S", .../N, where Is is the string scale. This picture is valid provided the string coupling 9 « 1, where 9 is related to Newton's constant G via G '" g21;. This picture represents a random walk 9 with n = .../N steps, each a single string "bit" of length Is 10. Let us now slowly increase the coupling g. As shown by Horowitz and Polchinski 8, gravitational effects start becoming strong at go '" N-3/8 = n- 3 /4, after which the string collapses until it reachers the size of the string

74

scale lB. At the critical coupling gc "" N- 1 / 4 = n- 1 / 2 , the Schwarzschild radius R = 2G M of a black hole with the same mass becomes of the order of the string scale, and one can sensibly start thinking of the string as a black hole. At this point, too, the entropies match: SBH "" R2/G = 1/ = VN = n. For 9 > gc, the black hole picture prevails. In the intermediate range go < 9 < gc, the size of the string state was shown using a thermal scalar field theory to be 8

g;

IB L "" g2 N1/2

(1)

IB g2n'

which smoothly interpolates between the random walk size and the string scale. Note that for n large, the coupling is small throughout the ranges we are considering. This is an interesting result with a specific prediction for the coupling dependence of the size of the string as it collapses into a black hole. A natural question to ask is whether this sort of result also arises in analogous physical systems already considered. Since random walks with interactions arise in polymer physics 11,10, the relation (1) should also hold for a self-attracting polymer chain. We start with a random walk with n steps each of size a, so that the size of the polymer is initially given by Lo = y'na. Suppose we place the polymer in a medium of scatterers of number density p and (dimensionless) potential strength u. Then the size of the polymer was shown to be 12 L2 = x- 2 (1 - exp( _nx 2a2)) , (2) where x = Upa2 can be thought of as an effective scattering cross section. To compare with a self-gravitating string with a = IB' the scatterers are taken to coincide with the positions of the string bits themselves. For large n and in a mean-field approximation, the number density of n bits in a volume L~ is given by

_ n _ -1/21-3 B· P - (n3/21~) - n

(3)

For 9 small, the leading order interaction potential is given by u g2 g2n 2 g2 n 3/2 I"" IT· - T·I "" = - 1- , 8

L i,j

•

J

La

(4)

8

where fi is the position of the ith link. It follows that x "" ng2 /1 B , so that L2 = 1;n- 2g- 4 (1- exp( _n 3g4 )) • (5) For 9 < go = n- 3 /4, L2 == nl; which is the random walk, corresponding to the free string. As in the string case, a transition occurs at 9 "" go. As 9

75

is increased past go, the size quickly shrinks to L2 ~ 1;/n2g4 = 1;/g2N, as in (1). This kind of relation holds a until 9 '" gc '" n- 1 / 2 , when L ""' R, the Schwarzschild radius of the polymer, and the black hole picture dominates. This connection between black holes, strings and polymers is very interesting and merits further investigation. Similar links with other soft-matter systems have also been noted by Callaway 13, where the area law was recovered for the case of a liquid field theory and where it was argued that the area law contributions to the free energy are primarily responsible for liquid surface tension. The speculation was also made that the area law arises in the context of protein folding. Connections between physical and biological systems are always exciting. The cases discussed above are especially so since quantum gravity is generally considered too remote to have relevance to other areas of physics, much less other fields of science. In particular, the fascinating possibility arises that mathematical techniques used to study black holes can be useful in understanding biological questions, such as protein dynamics, while methods of polymers physics can potentially shed light on quantum gravity. Acknowledgments

Work supported by NSF Grant 9900773, PSC-CUNY Award 669663 and by a Travel Grant from WSAS, Baruch College. References References 1. J. Bekenstein, Lett. Nuov. Cimento 4 (1972) 737; Phys. Rev. D7 (1973) 2333; Phys. Rev. D9 (1974) 3292; S. W. Hawking, Nature 248 (1974) 30; Comm. Math. Phys. 43 (1975) 199. 2. M. B. Green, J. H. Schwarz and E. Witten, Superstring Theory, Cambridge University Press, Cambridge (1987). 3. See M. J. Duff, R. R. Khuri and J. X. Lu, Phys. Rep. B259 (1995) 213, M. Cvetic and D. Youm, Phys.Rev. D54 (1996) 2612, M. Cvetic and A. A. Tseytlin, Nucl. Phys. B477 (1996) 499 and references therein. aO nce the self-interaction of the polymer becomes strong, the simple result (5) is no longer exact and a more precise computation is required. Nevertheless, it is clear that one obtains a smooth transition from the random walk to the Schwarzschild radius via a nonperturbative coupling dependence, so that even if (1) is not exactly recovered, it remains a good approximation for the collapse of the polymer.

76

4. A. Strominger and C. Vafa, Phys. Lett. B379 (1996) 99; J. Maldacena, hep-th/9607235 and references therein; K. Sfetsos and K. Skenderis, hepth/9711138; R. Arguiro. F. Englert and L. Houart, hep-th/9801053. 5. J. Rahmfeld, Phys. Lett. B372 (1996) 198. 6. T. M. Apostol, Introduction to Analytic Number Theory, Springer Verlag (1976). 7. L. Susskind, hep-th/9309145; G. T. Horowitz and J. Polchinski, Phys. Rev. D55 (1997) 6189. 8. G. T. Horowitz and J. Polchinski, Phys. Rev. D57 (1998) 2557. See also S. Kalyana RaIna, Phys. Lett. B424 (1998) 39. 9. P. Salomonson and B. S. Skagerstam, Nuc!. Phys. B268 (1986) 349; Physica A158 (1989) 499; D. Mitchell and N. Turok, Phys. Rev. Lett. 58 (1987) 1577; Nuc!. Phys. B294 (1987) 1138. 10. See C. B. Thorn, hep-th/9607204 and references therein; see also O. Bergman and C. B. Thorn, Nuc!. Phys. B502 (1997) 309. 11. See M. Doi and S. F. Edwards, The Theory of Polymer Dynamics, Clarendon Press, Oxford (1986) and references therein. 12. S. F. Edwards and M. Muthukumar, J. Chern. Phys. 89 (1988) 2435; S. F. Edwards and Y. Chen, J. Phys. A21 (1988) 2963. 13. D. J. E. Callaway, Phys. Rev. E53 (1996) 3738.

SPYING ON NATURE: HOW GENERAL PRINCIPLES LEAD TO PHYSICAL RESULTS A. MARTIN Theoretical Physics Division, CERN CH - 1211 Geneva 23 and LAPP *- F 74941 Annecy Le Vieux Cedex E-mail: [email protected] We show how the combination of analyticity properties derived from local field theory and the unitarity condition (in particular positivity) leads to non-trivial physical results, including the proof from first principles of the "Froissart bound", which might be qualitatively satured by nature, and the existence of absolute bounds on the pion-pion scattering amplitude. Dedication: To the memory of Harry Lehmann who contributed so much to this field.

At the present time, almost every physicist believes that the theory of strong interactions is QCD. However, nobody so far has been able to calculate directly from the QCD Lagrangian and from the quark masses the scattering amplitude for - say - proton-proton scattering. So far QCD can only handle "hard processes" or perhaps spectroscopy from the lattice. For soft processes there are of course models but these are only models, even if they are sometimes "QCD inspired". for this reason, use of general principles to obtain information on scattering amplitudes, which was done already back in 1954, is not yet out offashion as we shall try to show. In 1954, Gell-Mann, Goldberger and Thirring 1 proved that dispersion relations, previously developed in optics could be established for Compton Scattering: 'Y P -t 'Y P, from the existence of local fields satisfying the causality property

[A(x), A(y)] = 0 for (x - y)2 < 0 , Le., spacelike. This made it possible to express the real part of the forward scattering amplitude as an integral over the imaginary part of the forward scattering amplitude, Le., by the "optical theorem", an integral over the total cross-section for 'YP collisions. At the same time a general formulation of quantum field theory incorporating causality giving in particular general ·URA 1436. Assode

a I'Universite de Savoie. 77

78

expressions for scattering amplitudes was developed by Lehmann, Zimmermann and Symanzik (LSZ) in their pioneering paper (in german!) in Nuovo Cimento 2. On this basis, dispersion relations were "proposed" for massive particles in the work of Goldberger on the pion-nucleon scattering amplitude 3. Soon, his "heuristic proof" was turned into a real proof by various authors 4. One of these proofs is due to Harry Lehmann! Before going on, I would like to explain that if these results, even after the discovery that protons and pions are not elementary but made of quarks, are still valid, it is thanks to a fundamental contribution of Zimmermann entitled "On the bound state problem in quantum field theory" 5, in which it is proved that to a bound state we can associate a local operator. This constitutes an excellent answer to sceptics like Volodia Gribov 6 or Klaus Hepp 7 (qui brUle ce qu'il a adore!). Now I believe that it is necessary to give some technical details, even if many of you know about it. In 3+1 dimensions (3 space, 1 time) the scattering amplitude depends on two variables energy and angle. For a reaction A + B --t A + B

E c.m . =

JM1

+ k2 +

JM'tJ

+ k2

,

(1)

k being the centre-of-mass momentum. The angle is designated by (). There are alternative variables: 8

= (ECM)2, t = 2k2(COS() -1)

(2)

(Notice that physical t is NEGATIVE). We shall need later an auxiliary variable u, defined by , 8

+ t + u = 2M1 + 2M~

(3)

The Scattering amplitude (scalar case) can be written as a partial wave expansion, the convergence of which will be justified in a moment:

F(8, cos()) =

v;: ~)2£ + 1)!l(8)Pl(COS())

(4)

1£(8) is a partial wave amplitude. The Absorptive part, which coincides for cos() real (Le., physical) with the imaginary part of F, is defined as As(8,COS()) =

v;: 2:(2£ +

1) 1m 1£(8)(cos())

(5)

79

The Unitarity condition, implies, with the normalization we have chosen 1m h(s) 2: Ih(sW

(6)

which has, as a consequence

> 0,

1m h(s)

11£1 < 1 .

(7)

The differential cross-section is given by

du =! IFI2 dO. s ' and the total cross-section is given by the "optical theorem" 411"

Utotal

= k...jS As(s,cosO = 1) .

(8)

With these definitions, a dispersion relation can be written as:

-.!.f As(s',t)ds' + .!.f Au(u',t)du'

F( s,t,u ) -

11"

S' -

S

11"

U' - U

(9)

with possible subractions, i.e., for instance the replacement of l/(s' - s) by SN/S'N(S' - s) and the addition of a polynomial in s, with coefficients depending on t. The scattering amplitude in the s channel A + B -+ A + B is the boundary value of F for s + if, € > 0 -+ 0, s > (MA + MB)2. In the same way the amplitUde for A + B -+ A + B, B being the antiparticle of B is given by the boundary value of F for u+i€, € -+ 0 u> (MA +MB)2. Here we understand the need for the auxiliary variable u. The dispersion relation implies that, for fixed t the scattering amplitude can be continued in the s complex plane with two cuts. The scattering amplitude possesses the reality property, i.e., for t real it is real between the cuts and takes complex conjugate values above and below the cuts. In the most favourable cases, dispersion relations have been established for -T < t ~ 0, T > O. A list of these cases has been given in 1958 by Goldberger 8 and has not been enlarged since then. It is given in the Table, copied from the Goldberger review. In the general case, even if dispersion relations are not proved, the crossing property of Bros, Epstein and Glaser states that the scattering amplitude is analytic in a twice cut plane, minus a finite region, for any negative t 9. So it is possible to continue the amplitude directly from A + B -+ A + B to the complex conjugate of A + B -+ A + B. By a more subtle argument, using a path with fixed u and fixed s it is possible to continue directly from A+B-+A+BtoA+B-+A+B

80

DISPERSION RELATIONS a) Proved relations Process k+p~k'+p'

Limitation in invariant momentum transfer 32m2

2m p+m"

7r+N~7r+N

Tmax= ~

7r+7r~7r+7r

Tmax

= 28m;

'Y+N~'Y+N

Tmax

= 4m;

{{2m~+m,,)2

4(mp+m,,)2

2mp-m1r

+

T max

e+N~

Tmax = 4F("()j 'Y:=k5- k2

O~T 0 for s -t (MA + MB)2, T2(S) -t 0 for s -t 00. lt was thought by Mandelstam that these two analyticity properties, dispersion relations and Lehmann ellipses, were insufficient to carry very far the analyticity-unitarity program. he proposed the Mandelstam representation 12 which can be written schematically as

82

F= ~! p(sl,nds1dtl 7T 2 (Sl - S) (ti - t) +circular permutations in s, t, U

+one dimensional dispersion integrals +su btractions

(12)

This representation is nice. It gives back the ordinary dispersion relations and the Lehmann ellipse when one variable is fixed, but it was never proved nor disproved for all mass cases, even in perturbation theory,. One contributor, Jean Lascoux, refused to co-sign a "proof", which, in the end, turned out to be imperfect. One very impressive consequence of Mandelstam representation was the proof, by Marcel Froissart, that the total cross-section cannot increase faster than (log S)2, the so-called "Froissart Bound" 13. My own way to obtain the Froissart bound 14 was to use the fact that the Mandelstam representation implies the existence of an ellipse of analyticity in cosO qualitatively larger than the Lehmann ellipse, i.e., such that it contains a circle ItI < R, R fixed, independent of the energy. This has a consequence that 1m Ie (s) decreases with l at a certain exponential rate because of the convergence of the Legendre polynomial expansion and of the polynomial boundedness, but on the other hand the 1m Ie(s)'s are bounded by unity because of unitarity [Eq. (7)]. taking the best bound for each l gives the Froissart bound. To prove the Froissart bound without using the Mandelstam representation one must find a way to enlarge the "small" and the "large" Lehmann ellipses. In the autumn of 1965, I had very stimulating discussions with Harry Lehmann at the "Institut des Hautes Etudes Scientifiques" about an attempt made in this direction by Nakanishi in which he combined in a not very consistent way positivity and some analyticity properties derived from perturbation theory. He was using a domain shrinking to zero when the energy became physical and this lead nowhere. Finally, in December 1965 15, I found the way out. The positivity of 1m Ie implies, by using expansion (5),

d)n As(s,t)1

I(d t

-4k2:St~O

To calculate 1

F(s,t) = -

7T

:S I(dd)n As(S,t)1 t t=o

1 80

As(Slt)ds l sl - S

(13)

83

(forget the left-hand cut and subtractions!), for s real < So one can expand F(s, t) around t = o. From the property (13) one can prove that the successive derivatives can be obtained by differentiating under the integral. When one resums the series one discovers that this can be done not only for s real < So, but for any s and that the expansion has a domain of convergence in t independent of s. This means that the large Lehmann ellipse must contain a circle It I < R. This is exactly what is needed to get the Froissart bound. In fact, in favourable cases, R = 4m;, m7r being the pion mass. A recipe to get a lower bound for R was found by Sommer 16

(14) It was already known that for It I < 4m; the number of subtractions in the dispersion relations was at most two 17, and it lead to the more accurate bound 18 7r

< -2 (logS)2

(15) m7r Notice that this is only a bound, not an asymptotic estimate. In spite of many efforts the Froissart bound was never qualitatively improved, and it was shown by Kupsch 19 that if one uses only 1m it 2:: litl2 and full crossing symmetry one cannot do better than Froissart. Before 1972, rising cross-sections were a pure curiosity. Almost everybody believed that the proton-proton cross-section was approaching 40 millibarns at infinite energy. Only Cheng and Wu 20 had a QED inspired model in which cross-sections were rising and behaving like (log S)2 at extremely high energy. Yet, Khuri and Kinoshita 21 took seriously very early the possibility that cross-sections rise and proved, in particular, that if the scattering amplitude is dominantly crossing even, and if at rv (log S)2 then aT

ReF 7r p---rv-- ImF logs' where ReF and ImF are the real and imaginary part of the forward scattering amplitude. In 1972, it was discovered at the ISR, at CERN, that the p - p crosssection was rising by 3 millibarns from 30 GeV c.m. energy to 60 GeV c.m. energy 22 (see Fig. 1). I suggested to the experimentalists that they should measure p and test the Khuri-Kinoshita predictions. They did it 23 (see Fig. 2) , and, at Fermilab and at CERN it was found very early that p, negative at low energies, was becoming positive (If I say that it is because I was the first theoretician speaking after the talk of Ugo Amaldi at CERN, announcing the results of the CERN-Rome group at the ISR!). This kind of combined

84

I I I I I I

180 160 140

>.

!!'

Ii! en"

"-

Bestm from accelerators upto 550 GeV c::

e

iii >

~

I I

~I

120

Ii! I

:0 E

~I

li 100

,

,,

,,

I~

!ill

D

,,

, ,,

I I I I

80 60

i

40

0

::I: ...J

20 0 10

102

103

10 5

-Is (GeV) Figure 1. pp and pp total cross-sections.

measurements of aT and ReF are still going on. In aT we have now more than a 50 % increase with respect to low energy values at the SppS, and at least 75 % increase at the Tevatron (Fig. 1). The best fit of Augier et al. 25 with

gives "( = 2.2 ± 0.3, compatible with a qualitative saturation of the Froissart bound ("( = 2) and predicts atot = 107 mb at 14 TeV (hopefully the LHC energy). For an up to date review I refer to the article of Matthiae 24. it is my strong conviction that this activity should be continued with the future LHC. A breakdown of dispersion relation might be a sign of new physics due to the presence of extra com pact dimensions of space according to N.N. Khuri 26

The breakdown of dispersion relations can manifest itself either by the appearance of extra singularities in the complex plane or by the breakdown of polynomial boundedness. In the latter case one must multiply the forward scattering amplitude by a convergence factor, for instance

85

0.20

- - Best fit

-- --------- ---

0.15

--- ---

0.1 0.

0.05

10

103

rs(GeV)

Figure 2. Measurements of the p parameter for fip and pp scattering are shown together with the result of the dispersion relation fit of Augier et al. 25

.J

exp ( M R 2 E6 - Efab)' where R is the "size" of the extra compact dimensions, E lab the laboratory energy and M the mass of the target. At high energies, 2M E lab is essentially (ECM)2. If 1/ R is of the order of the supersymmetry breaking scale, which is believed to be at most the energy of the LHC one expects big effects at LHC energy. But, as pointed out by N.N. Khuri, the real part, because of its smallness, is a much more sensitive indicator than the total cross-section and will be seriously affected at appreciably lower energies. For that reason accurate measurements of p at lower energies (for instance re-doing the Tevatron measurement which is not sufficiently accurate to give any constraint) is extremely desirable. Future experiments, especially for p, will be difficult because of the necessity to go to very small angles, but not impossible 27. Before leaving the domain of high-energy scattering I would like to indicate the new version of the Pomeranchuk theorem. When it was believed that cross-sections were approaching finite limits, the Pomeranchuk theorem 28 stated that, under a certain assumption on the real part

ar(AB) - ar(AB) ~ 0 If cross-sections are rising to infinity, one can actually prove, according to

86 Upper bound

(1)

25 20 15 10

(2)

(3)

5

(4 (5)r 6 )

r

1965

1970

19{51.

-20 -40 -60 -80 -100 Lower bound

Figure 3. Bounds on the scattering amplitude at the symmetry point 8 = t as a function of time. Normalization: F(s = 4m;,O,O) = scattering length.

Eden

29

and Kinoshita

30

= u = 4/3m;

that aT(AB)/aT(AB) -* 1 .

Now I would like to turn to another aspect of analyticity-unitarity. A consequence of the enlargment of the Lehmann ellipse is that, in the special case of 11"11" -* 11"11" scattering, one can, by using crossing symmetry and analytic completion techniques, obtain a very large analyticity domain 31, but one can prove that the domain is smaller than the Mandelstam domain 32. By playing with crossing symmetry and unitarity in a clever way, one gets a bound on the scattering amplitude at the "symmetry point" which is 33

= t = u = 4m;/31 < 4 , normalized in such a way that F(s = u, t = 0, u = 0) W(s

where F is is the scattering length, aoo (the results, as a function of time, i.e., of the progress of theoreticians, are presented on Fig. 3). One can also obtain a lower bound on the scattering length, the best value being 34

11"011"0

aoo > -1.75 (m 1r )-1,

87

a number which is off the model predictions only by a factor 10. Though these latter results may seem "useless", they are remarkable, since they prove that the combination of analyticity and unitarity have a dynamical content. That kind of game, i.e., the obtention of absolute bounds, can be done equally well in 2+1 dimensions, where the 2-space dimensions are globally euclidean. In that situation, where the threshold scattering amplitude behaves like 7r / In k or k 2 35, the forward scattering amplitude has no zero in the complex plane and this might help to obtain much more impressive results, and allow to eliminate the unknown constants appearing in the old GlimmJaffe work 36.

References

1. M. Gell-Mann, M.L. Goldberger and W. Thirring, Phys.Rev. 95 (1954) 1612. 2. H. Lehmann, K. Symanzik and W. Zimmermann, Nuovo Cimento (Serie 10) 1 (1955) 205. 3. M.L. Goldberger, Phys.Rev. 99 (1975) 979. 4. N.N. Bogoliubov, B.V. Medvedev and M.K. Polivanov, Voprossy Teorii Dispersionnyk Sootnoshenii, V. Shirkov et al. Eds., Moscow 1958; K. Symanzik, Phys.Rev. 105 (1957) 743; H. Lehmann, Suppl. Nuovo Cimento 14 (1959) 153. 5. W. Zimmermann, Nuovo Cimento 10 (1958) 597. 6. V. Gribov, private communication (1997). 7. K. Hepp, private communication, Zurich, 1996. 8. M.L. Goldberger, Proceedings of the International Conference on High Energy Physics, CERN, Geneva, 1958, B. Ferretti ed., CERN Scientific Information Service, 1958, p. 208. 9. J. Bros, H. Epstein and V. Glaser, Commun. Math. Phys. 1 (1965) 240. 10. M. Gell-Mann, Proceedings of the 6th Annual Rochester Conference, J. Ballam, V.L.Fitch, T. Fulton, K. Huang, R.R. Rau and S.B. Treiman eds., Interscience Publishers, New York 1956, p. 30. 11. H. Lehmann, Nuovo Cimento 10 (1958) 579. 12. S. Mandelstam, Phys.Rev. 112 (1958) 1344. 13. M. Froissart, Phys.Rev. 123 (1961) 1053. 14. A. Martin, Phys.Rev. 129 (1963) 1432, and Proceedings of the 1962 Conference on High energy Physics at CERN, J. Prentki ed., CERN

88

Scientific Information Service, 1962, p. 567. 15. A. Martin, Nuovo Cimento 42 (1966) 901. 16. G. Sommer, Nuovo Cimento A48 (1967) 92. In the special case of pion-nucleon scattering a special argument gives R = 4m;. See D. Bessis and V. Glaser, Nuovo Cimento (Serie X) 50 (1967) 568. 17. Y.S. Jin and A. Martin, Phys.Rev. B135 (1964) 1375. 18. L. Lukaszuk and A. Martin, Nuovo Cimento 52 (1967) 122. 19. J. Kupsch, Nuovo Cimento B70 (1982) 85. 20. H. Cheng and T.T. Wu, Phys.Rev.Lett. 24 (1970) 1456. 21. N.N. Khuri and T. Kinoshita, Phys.Rev. B137 (1965) 720. 22. U. Amaldi et al., Phys.Lett. B44 (1973) 112; S.R. Amendolia et al., Phys.Lett. B44 (1973) 119. 23. V. Bartenev et al., Phys.Rev.Lett. 31 (1973) 1367; U. Amaldi et al., Phys.Lett. 66B (1977) 390. 24. G. Matthiae, Rep.Progr.Phys. 57 (1994) 743. 25. C. Augier et al., Phys.Lett. 316B (1993) 448. 26. N.N. Khuri, Rencontres de Physique de la vallee d' Aoste, 1994, M. Greco ed., Editions Frontieres 1994, p. 771; see also: N.N. Khuri and T.T. Wu, Phys.Rev. D56 (1997) 6779 and 6785. 27. Angela Faus-Golfe, private communication. 28. Y.Ya. Pomeranchuk, Soviet Phys. JETP 7 (1958) 499. 29. R.J. Eden, Phys.Rev.Lett. 16 (1966) 39. 30. T. Kinoshita, in Perspectives in Modern Physics, R.E. Marshak ed., New York 1966), p. 211; see also G. Grunberg and T.N. Truong, Phys.Rev.Lett. B31 (1973) 63. 31. A. Martin, Nuovo Cimento 44 (1966) 1219. 32. A. Martin, Proceedings of the 1967 International Conference on Particles and Fields, C. Hagen, G. Guralnik and V.A. Mathur, eds., John Wiley and Sons, New York 1967, p. 255. 33. A. Martin, Preprint, Institute of Theoretical Physics, Stanford University ITP-134 (1964), unpublished; A. Martin in "High Energy Physics and Elementary particles", ICTP Trieste 1965, International Atomic Energy Agency Vienna (1965), p. 155; L. Lukaszuk and A. Martin, Nuovo Cimento A47 (1967) 265; J.B. Healy, Phys.Rev. D8 (1973) 1907; G. Auberson, L. Epele, g. Mahoux and R.F.A. Sima6, Nucl.Phys. B94 (1975) 311; C. Lopez and G. Mennessier, Phys.Lett. B58 (1975) 437; B. Bonnier, C. Lopez and G. Mennessier, Phys.Lett. B60 (1975) 63;

89

C. Lopez and G. Mennessier, Nucl.Phys. BU8 (1977) 426. 34. I. Caprini and P. Dita, Preprint, Institute of Physics and Engineering, P.O. Box 5206, Bucharest (1978), unpublished. The initial work on this lower bound was: B. Bonnier and R. Vinh Mau, Phys.Rev. 165 (1968) 1923. 35. K. Chadan, N.N. Khuri, A. Martin and T.T. Wu, Phys.Rev. D58 (1998) 025814. 36. J. Glimm and A. Jaffe, Ann. Inst. Henri Poincare A22 (1975) 97.

PROBABILITY & ITS APPLICATIONS HENRY MCKEAN Courant Institute, New York University, 251 Mercer Street, New York, NY, 10012, USA E-mail: [email protected] In classical probability, there are two principal themes: the law of large numbers and the central limit theorem. They appear in their simplest form for independent Bernouilli trials: if the random variables Xl, X2, etc. = 0 or 1 are independent, with common law P(xo = 1) = p E (0,1), then you have the law of large numbers: P [lim

Xl

+ ... + Xn n

ntoo

_ _ _ _ more briefly, the sum = Sn

Xl

+ ... + Xn

is of the form

= np + y'np(1 - p) x a unit Gaussian variable,

in which you see some kind of "mean motion" np on the scale n, corrected by "Gaussian fluctuations" on the scale..;n. It is explained that in this regime of "small deviations" no further corrections are available. The question of "large deviations" from this type of typical behaviour is touched upon. Now it is of prime importance that, e.g., Bernoulli trials do not occur in nature. They are only the mathematician's model of what is going on. Contrariwise, the statistician, or more generally, any student of nature is confronted with raw observations in bulk, of which it is required to make (statistical) sense, and it is a piece of undeserved good luck that nature imitates the mathematics. Poincare made this very point. He says: Suppose you play roulette and you know precisely the mechanics of the wheel and the impulse the croupier gives it. Then, if you are very quick, you can compute the outcome. But how much simpler to say: "1/36". In this way, tossing an honest coin is fruitfully modelled by Bernoulli trials with p = 1/2, and what is remarkable is not only how well this works in these simple circumstances, but how the same pattern is found over and over in situations very far removed from games of chance. The rest of my talk is intended to illustrate this fact by a series of examples of (I hope) striking diversity.

Information & Coding (Shannon). The problem is to devise codes to permit the most accurate possible transmission of messages through a noisy channel (e.g. a poor telephone line). The model involves simple Markov chains analyzed by means of suitable law of large numbers (with Gaussian corrections). These are easily derived from the Benioulli case by a device 90

91

of the remarkable Roumanian mathematician Dablin. The outcome is the identification of a number,

C = the channel capacity = the "size of the pipe", with this property: that if you want to send information through the channel at a rate H < C, then you can encode it so as to have almost perfect success. Contrariwise, for simple channels, any attempt to send information at a rate H > C will almost surely produce garbage. The novelty of Shannon's ideas and their great success is, to my way of thinking, one of the high points of 20th century mathematics. Geometry in High Dimensions (Mehler, Gromov). Two innocent remarks: Mehler noticed that if the n-dimensional sphere of radius Vii is equipped with its customary round measure, normalized to volume 1, then, for fixed m and for n t 00, lim P

[n~l

(ak

~ Xk < bk))

ntoo

m

=

II

k=l

lh ak

_:z:2/2 e rn= dx, y

27f

i.e., the individual coordinates become independent Gaussian. This fact may be compared to the law of large numbers for the squares of independent Gaussian variables: P lim Xl2 + X 22 + ... + Xn2 =] 1 [ntoo

= 1,

n

which may be written (a little fancifully) as Jxi + X~ + etc. = J+oo and interpreted by saying that the oo-dimensional Gaussian product measure is the (unit) round measure on Soo(.JOO)-whatever that is! Gromov has taken this picture as the basic illustration of a whole new study of the statistical character of manifolds of very high dimension. Partial Differential Equations (Ito, Malliavin). Malliavin has developed this circle of ideas in a wholly different way: Bernoulli trials Xl, X2, etc. = ± 1 with common law P [xo = ±1) = 1/2 model the tossing of an honest coin. Putting the partial sums (winnings of player no. 1) on the scale Vii, as in

-

~n (t) -

Xl

+ ... + . r,;;

yn

Xm

£ m reCti) from him at the moment ti. Consequently, if the universe was homogeneous and isotropic in some domain with the scales r > re (ti), then near the center of this domain the universe can expand as a homogeneous, isotropic one forever, even if it was very inhomogeneous outside. Moreover, if during inflationary stage c+p« 10 then, as it follows from (4), the energy density 10 changes insignificantly compared to the relative change of scale factor, Le. ci/cCt) «a(t)Jai. Rigorously speaking this consideration refers only the eternal inflationary stage. If inflation finally ends then the above arguments should be a little modified. However the main idea remains basically the same also for inflation with finite duration. Thus we see an obvious indication that homogeneity and horizon problems can be solved in inflationary model. Actually one can start with a small causally connected homogeneous region in an inhomogeneous universe and increase tremendously its size during the inflation without significant change of the energy density in it. The particle horizon will also increase tremendously. It is important that if the size of this region is originally bigger than the size of the event horizon then the inhomogeneities which present outside this region will not disturb the evolution near the center of it. If the matter finally converts, e.g., into a radiation or dust as a result of the transition synchronized by the cosmic time t, one gets a big piece of an isotropic Friedmann universe. If inflation occurred very early (ti rv 10- 35 ..;- 10-43 sec) and after that the

100

universe evolved according to the "standard" scenario, the velocity at the end of the inflation 0,1 should be rv 1028 0,0. To avoid the problem with causality and have no time tuning for (ni -1) one should have ai ;S 0,0 at the beginning of inflation (see eqs. (6),(7)). So only if aJI ai is not less than 10 28 , inflation solves the horizon and flatness problems. When the "velocity" a increases significantly more than in 1028 time (no fine tuning for the duration of inflation) then, ai «0,0. Inspecting the formula (6) we see that in this case the size of a region where we could expect homogeneity at t = to, is much bigger than the size of the observed universe. Thus one can say that inflation predicts that the universe is isotropic in the scales much bigger than the present horizon. (Unfortunately this can only be checked in many billion years.) The other (verifiable) prediction directly follows from eq. (7). If Ini - 11 rv 0(1) and ai « 0,0, then Ino - 11 « 1, i.e. the energy density at present should be very close to the critical density. The problem with an overproduction of monopoles and other topological defects is also easily solved by inflation. Actually if monopoles were produced in a big amount before the inflation, their density drops to a negligible value after inflation.

3

Inflationary scenarios

In the previous consideration two questions were left completely open. One of them is how to realize the equation of state for which the energy-dominance condition is violated. The other one concerns the transition from inflation to the Friedmann stage. To answer these questions we should consider the concrete inflationary scenarios. At present there exist probably too many different scenarios, so it is hard to believe that we know the particular one which was actually realized in the very early universe. We just quote some of them: new, chaotic, extended, hybrid, natural, etc .. However, in spite of the fact that, at a first glance, they look different, all these scenarios have the same main physical features. Hence one can hope that we got, at least, the right underlying physical idea of what kind of processes were happening in the very early universe. The simplest way to break down the energy dominance condition is to consider the matter with the effective equation of state p ~ -c:, corresponding to the cosmological term in the Einstein equations. Of course, p and c: in such a case, can still be slowly varying functions of time. There are two natural ways to get such an equation of state: either using classical scalar field or considering vacuum polarization of quantum fields in the external gravi-

101

tational field. Vacuum polarization effects induce the corrections in Einstein equations. Such a theory is conformally equivalent to Einstein theory with the extra scalar fields (B. Whitt, 1984). Therefore in both cases the inflationary models look very similar. From the very beginning it was clear that the concrete model suggested by A. Guth (1981) ("old" inflation) does not work, since in this model the transition from inflation to the Friedmann universe is not smooth enough to avoid big inhomogeneities. However, in those days, there already existed a working model based on vacuum polarization effects. This model was proposed by A. Starobinsky (1980), who hoped to solve the problem of singularity in this way. The present author and G. Chibisov (1981) investigated the vacuum metric fluctuations in the Starobinsky model. They found that the phase transition is due to the longwave vacuum fluctuations and can lead to a smooth Friedmann universe. A somewhat similar model relying on the scalar field with Coleman-Weinberg type potential ("new" inflation) was invented by A. Linde (1982) and A. Albrecht and P. Steinhardt (1982). In both of these models the initial conditions necessary for inflation were of very delicate nature. In particular, to start inflation it was necessary to put the initial field in a particular state (maximum of the potential). In addition, the models required very unnatural values for parameters of the potential. The situation changed when A. Linde (1983) realized that the inflationary expansion is not special, but rather a general property of models with scalar fields and proposed chaotic inflationary scenario. This kind of scenarios can also be easily realized in higher derivative gravity without scalar fields. They are different from the previous ones mainly in two aspects. First, they do not require the very special shape of the potential and the initial values for scalar field. Second, the transition from inflation to the Friedmann era occurs as a result of the classical evolution of the field. After this realization, the building of various new models for inflation became to some extent a purely technical matter. It became clear that the inflationary stage arises under rather natural initial conditions. The main features which distinguish the existing inflationary scenarios are the following. First of all, it is the mechanism responsible for the emergence of the state in which the energy-dominance condition is violated. Such a state can be realized either via the classical scalar fields (or scalar condensates), or due to the vacuum polarization effects. The existence of the fundamental scalar fields is not a necessary condition for inflation. Second, the particular models are characterized by the type of transitions from inflationary to Friedmann epoch. The transition can be very violent accompanied by the formation of bubbles or very smooth. The reason for transition could be the

102

following: under barrier tunneling, instability due to the quantum fluctuations or just usual classical evolution of the scalar field. According to the most successful scenarios the transition is very smooth. There exist more complicated models which combine several features. Among them it is worth mentioning the extended inflation (D. La and P. Steinhardt, 1989). Most of the scenarios of inflation are based on the idea of slow roll of the scalar field, although the inflation can also be realised in the models where the scalar field is moving very fast (see, T. Damour, V. Mukhanov, 1998; Armendariz-Picon, et. al., 1999). The answers to the important questions about the energy scale of inflation and reheating after the inflation directly depend on the concrete model for inflation. Among all scenarios the chaotic inflation is the simplest and most representative one. It has features that have a lot in common with many inflationary models. Therefore, concluding this section, we would like briefly remind this scenario. The purpose of this consideration is just to give an idea about dynamical evolution of scale factor and scalar field during inflation. Therefore for simplicity we consider only the case of flat expanding universe filled by homogeneous scalar field tp(t) with the potential V(tp). The evolution of the scale factor a(t) and the field tp(t) can be described by the 0-0 Einstein equation and the equation for scalar field

(~r = 8; (~~2 + V)

(9)

+ 3~a ~ +

(10)

0,

3zEM

F(z) 0 small, F(z) 2: Co for some Co > O. Thus the origin lies in a valley: Izl < r, and the conditions of MPL above are satisfied. We may take the point Uo to be 21fi. But it is easily seen that the only critical value of F is zero. Here is the general MPL of Ambrosetti, Rabinowitz MPL 1. MPL. Let F be a C 1 real function in a Banach space X and suppose there is an open set U in X with 0 E U, and a point Uo rt U such that

F(O), F(uo)

< inf F au

= Co.

Consider the number c defined by (3); here p represents continuous paths joining Uo to 0, clearly c 2: Co. Then there exists a sequence of points {Uj} in X such that

F(Uj) ~ c and

1IF'(uj)11 ~ O.

(Here F', the Frechet derivative of F, is an element of the dual space.) To conclude then that c is in fact a critical value, one usually assumes the Palais-Smale condition: PS Condition. A C 1 function F in X is said to satisfy the PS condition if any sequence Uj E X with F(uj) converging and

1IF'(uj)11 ~ 0

119

has a strongly convergent subsequence. If one adds PS to the conditions above in MPL, it follows that c is a critical value. The PS Condition is a strong one, one has to get used to it. Often when trying to apply MPL, PS is the hardest thing to verify. In finite dimensions, if F is bounded below and satisfies PS then, necessarily, F(x) -+

00

as

Ixl -+ 00.

The way MPL is proved uses a method going back to Morse. If there is no critical value in an interval [a, bj, then one can deform the space by flowing, following the negative gradient of F. Here are some higher dimensional variants of MPL-higher dimensional in the sense that we do min max, as in (1), not with respect to paths, one dimensional things, but with respect to higher dimensional objects. Let F be a C 1 function in R 3 satisfying, for convenience, PS. Suppose r 1 and r 2 are two simple disjoint closed curves in R 3 which "link" each other

Figure 1.

Suppose F ::; C1 on r 1, F ;::: C2 on r 2 with C1 < C2. Then the following number c ;::: C2 is a critical value of F: Consider 2-dimensional surfaces spanning r 1 Le. think of r 1 as given by a one-one map ho of S1 onto r 1. Let h be any continuous extension of ho to the closed unit disk D 1 • The image of h is a surface spanning r 1. The number c is defined again by min max c = inf max F(h(z)). h zED,

The infinimum is taken with respect to all continuous maps h extending h o. Since the curves link, the image of every h intersects r 2 and thus c ;::: C2. Clearly there is also a critical value ::; C1.

120

Here is another "linking result". Theorem: Let F be a C l function in a Banach space X satisfying PS. Assume that X admits a direct sum decomposition

X

= Xl EBX2

with dimX2

= d < 00.

Suppose that on a (d - I)-sphere:

IIxll = R > 0,

x E X2, F satisfies F :$ a, while on Xl, F ~

~

b, with b > a. Then has a critical value

b.

I conclude with an attempt to use MPL to solve a long-standing problem in algebra. Jacobian Conjecture. Let 'IjJ be a holomorphic polynomial map of a: a: n with

det 'IjJ' (z)

==

n

into

1.

Then 'IjJ is 1 - 1 onto and the inverse is a polynomial. For real polynomial maps ofR N into RN the same conjecture was made by H. Keller in 1939 9. Those interested in the problem should look at 2. In the case of a polynomial map 'IjJ from a: n to a: n, to prove the conjecture it suffices to show that 'IjJ is 1 - 1. Some years ago N. Kerzman and I tried to prove the 1 - 1 property using MPL. Others have also tried using MPL see 5, where other results and conjectures are presented. Here is our attempt. We argue by contradiction. Suppose for some Zo :j:. 0 in a: n, 'IjJ(0) = 'IjJ(Zo) = 0,

say.

Consider the real function

F(z) = J'IjJ(Z)J2.

o and Zo are clearly minimum points of R and near the origin, 2 F(z) '" J'IjJ' (0)zJ ~ t

E

R, we add to the

splitting a subbundle in the direction of the vector field that generates the flow

and require for some Riemmanian metric and some constants C, 0 < A < 1, that

IIdX tiE s II, IldX -t lEu I

:-: ;

Ce A.t, t

E

R. See [PT], especially chapter

seven, for details and many of the notions presented here. A main conjecture in the 60's was precisely to show that hyperbolic systems were C dense in the set of all systems, r ~ 1. By the end of the 60's such a dream was lost due to many examples. The Global Conjecture above in some sense revive such a dream but in probabilistic terms both in the space of events by restricting our attention to attractors as in part I and III and in parameter space as in part II and III. Back to the above "dream" of the 60's, concerning the possibility that "generally" systems would be hyperbolic, we would have, in particular, that every system could be approximated by a hyperbolic one (denseness of hyperbolicity). By the end of the decade, the dream was lost as mentioned before and there was no meanful suggestion of how a global scenario should be for dynamics. It's interesting to notice that already in 1963, Lorenz in a remarkable numerical work, was pointing to the nongenericity of hyperbolic systems. He exhibited a relatively simple flow in three-dimensions, with quadratic equations, and so very deterministic, whose positive limit set was (numerically) nonhyperbolic, since it mixed a singularity of the flow (rest point) with periodic motions:

128

Lorenz Equations

x = -ax + oy y=rx-y-xz

i: = xy -bz

= 10

(J

r

=28

b=

7:

Lorenz Attractor

Figure 3

Moreover, Lorenz's system was sensitive with respect to initial conditions: when we consider two very nearby events (points), their orbits after a long time in the future may be very distant apart, as much as the diameter of the limit set, that is the "butterfly"above. This uncertainty occurs with total probability for the pair of initial events and, notice, this is happening with the long time behavior of a deterministic system! In brief, a system is chaotic if one of the attractors has exponential sensitivity to initial conditions in their basin of attraction. They do not have to be uniformly hyperbolic, as explained above. Important non-uniform examples are the Lorenz and Henon attractors; see figures. It is important to notice that Lorenz's example is robust: when we change very little the coefficients in the equations of the flow, we obtain a new one with similar properties, as mentioned above. On the other hand, Henon's numerical example is more elaborate, because is not robust but it only exists with positive probability in the coefficients of the system: in this case, just a quadratic transformation of the plane with inverse (a

129

diffeomorfism of the plane) and thus, again, deterministic. It is remarkable that only very recently it has been shown that Lorenz equations yields a Lorenz attractor [T]. In this direction, an important result is the characterization of the robust transitive singular attractors, like the Lorenz one [MPP] Henon Transformations

(1- ax

2

+ y, bX),

for a == 1.4

and

b ==.3

Henon Attractors

Figure 4

The concept became so much in evidence in the last two decades or so, that a controversy arose about to whom we should attribute the original idea of sensitivity or, equivalently, chaos. In fact, uncertainty and randomness are more than a century old concepts in Science and can be traced back to Maxwell, Boltzmann, Poincare ... So, the idea of chaoticity should be attributed to one of these great mathematicians/physicists or to Smale or to Lorenz or others? To avoid such a controversy, perhaps we can honor Edgar Alan Poe, who prior to all of them, wrote the following beautiful sentence: "For, in respect to the latter branch of the supposition, it should be considered that the most trifling variation in the facts of the two cases might give to the two important miscalculations, by diverting througly the two courses of events; very much as, in arithmetic, an error which, in its own individuality, may be

130

inappreciable, produces, at length, by dint of multiplication at all points, a result enormously at variance with truth." The mystery of Marie Roget Edgar Alan Poe. In the past, we were generally baffled by cycles in dynamics, such as in their fundamental role in the nondensity of hyperbolic systems: all the examples in the 60's exhibited cycles. First, we observe that cycles are common in dynamics. In fact, in many applications we deal with a "family" of systems and not a unique one, due to external parameters like, for instance, solar energy in the case of weather prediction, or external sensorial stimuli in the case of the brain. Then, the creation of cycles become unavoidable when we vary external parameters and the systems exhibiting such cycles may be "transient" ones: They are expected to be on the borderline of the chaotic systems and of the very simple ones, when attractors are just point attractors or periodic ones. Also, we believe that generally cycles should occupy a small part of the space of events: starting in most initial events, in the long run we will end up in an attractor. (See part I of the Global Conjecture before)

No cycle

Creating an unstable cycle

A stable cycle

Figure 5

In the set of figures above, Sj, S2 and S3 are hyperbolic sets or simply hyperbolic fixed points and the lines with arrows represent their stable and unstable manifolds (see [PTJ). According to Poincare's notation, they depicted the creation of an heteroclinic tangency. The most important case among heteroclinic tangencies is that of a homoclinic one for once we understand the dynamics of its unfolding, it is rather easy to obtain the general case from it.

131

Figure 6

The following set of results show how rich it becomes the dynamics when we create a homoclinic tangency and make it a transversal one, say through the variation of a parameter which in general we may assume to be along the line I in Figure 6. Theorem. When we generically unfold a homoclinic tangency for a one-parameter family of surface diffeomorphisms, we always obtain for arbitrary small variation of the parameter (assume that the tangency occurs at value zero of the parameter):

a) b) c) d)

Residual subsets of intervals in the parameter line whose corresponding diffeomorphisms display infinitely many coexisting sinks. 2ascade of period-doubling bifurcations of periodic points (sinks). Positive Lebesgue measure sets in the parameter line whose corresponding diffeomorphism display Henon-like attractors. Values of the parameter for which there are infinitely many coexisting Henonlike attractors

These facts were proved in the late 70's and the 80's by Newhouse, Yorke-Alligood and Mora-Viana extending the fundamental work of Benedicks-Carleson. See [PT] for a full discussion of these facts, except the last one, item d), which is due to Colli [C] and it is more recent. The above results are also valid in higher dimensions, when the dimension of the unstable manifold of the periodic point is one (codimension-one case) and the product of any two eigenvalues of the derivative of the map at this point has norm less than one (sectionally dissipative). See [PV] for the existence of infinitely many coexisting sinks in such a case. Much in line with Poincare sentence of the beginning of the paper, I have proposed sometime ago the following conjecture:

132

Conjecture. In any dimension, the diffeomorphisms exhibiting either a homoclinic tangency or a (finite) cycle of hyperbolic periodic orbits with different stable dimensions (heterodimensional cycles) are Cr dense in the complement of the closure of the hyperbolic ones, reI. Notice that for surface diffeomorphisms, it is not possible to have an heterodimensional cycle. So, if the above conjecture is true, then surface diffeomorphisms exhibiting Henon-like attractors or repellers (or infinitely many sinks or sources) are dense in the complement of the hyperbolic ones. The same question may be posed for non-invertible maps. Also for flows, but in this case one has to add, to homoclinic tangencies and heterodimensional cycles, flows exhibiting singular attractors, like the Lorenz-like ones, and singular cycles, i.e. cycles involving periodic orbits and singularities, studied in [LP]; see also [PT]. In a recent remarkable paper, Pujals-Sambarino have just shown that the conjecture is true for d surface diffeomorphisms [PS]. Moreover, for a parameterized family of such diffeomorphisms, if the topological entropy varies near certain value of the parameter then also nearby we have a parameter value such that the corresponding diffeomorphism exhibits a homoclinic tangency. Let us now consider the case where the homoclinic tangency for a surface diffeomorphism is associated to a horseshoe A like in Figure 7. Again, we denote by I the line of tangencies and also the Il-parameter line, when unfolding the homoclinic tangency. The set K2 in I represents the Cantor set obtained by the intersections of the stable manifolds of points in A with I and similarly K j represents the Cantor set obtained by intersections of the unstable manifolds with l. In first approximation, the dynamics of the unfolding of such a homoclinic tangency is translated to the understanding of the arithmetic difference of the Cantor sets K2 and K j that we denote by K2 - K j • In this setting, we have the following results: Theorem. a)

b) c) d)

If the Hausdorff dimension of A, HD(A), is smaller than one, then hyperbolicity is fully prevalent at the bifurcating parameter value Il = O. This is due to Newhouse-Palis and Palis-Takens, see [PT]. If HD(A) is bigger than one, then hyperbolicity is not fully prevalent at Il = O. This is due to Palis-Yoccoz, see [PY]. . If HD(A) is bigger than one, then K2 - K j contains nonempty intervals. This result is due to Moreira-Yoccoz, see [MY], and it implies b) in a strong sense. Again if HD(A) is bigger than one, then the parameter values such that the corresponding diffeomorphism display attractors have density zero at Il = O. This is due to Palis-Yoccoz in a work in development.

133

A

Figure 7 The results above concerning homoclinic bifurcations are now being extended to higher dimensions. For flows, the situation is even richer. The discussion of these topics, however, will be presented in a forthcoming paper. References [C]

[CLi]

E. Colli, Infinitely many coexisting strange attractors, to appear, Annales de i'Inst. Henri Poincare, Analyse Nonlineaire. M. L. Cartwright and J. E. Littlewood, On nonlinear differential equations of the second order: I. The equation

k(l-

[Lil] [Li2] [LP] [Lv] [Lvs]

yy2)y + Y = bAA cos (At + u), k large, 1. London Math. Soc. 20 (1945),180-189. J. E. Littlewood, On non-linear differential equations of second order: III, Acta Math. 97 (1957),267-308. J. E. Littlewood, On non-linear differential equations of second order: IV, Acta Math. 98 (1957), 1-110. R. Labarca and M. J. Pacifico, Stability of singular horseshoes, Topology 25 (1986), 337-352. M. Levi, Qualitative analysis of the periodically forced relaxation oscillations, Memoirs of the A.M.S. 32, n. 244 (1981). N. Levinson, A second order differential equation with singular solutions, Annals of Math. 50 (1949),127-153.

134

[MPP]

[MY] [P] [PT] [PV] [PY] [PS]

[T]

C. Morales, M. J. Pacffico and E. Pujals, On the C I robust singular transitive sets for three-dimensional flows, C. R. Acad. Sci. Paris, 326, Serie I: 81-86, 1998. c. Moreira and J-c. Yoccoz, Stable intersections of regular Cantor sets with large Hausdorff dimensions, preprint 1998. J. Palis, A Global View of Dynamics and a Conjecture on the Denseness of Finitude of Attractors, Asterisque 261, (1999),339-351) J. Palis and F. Takens, Hyperbolicity and sensitive-chaotic dynamics at homoclinic bifurcations, Cambridge University Press, 1993. J. Palis and M. Viana, High dimension diffeomorphisms displaying infinitely many periodic attractors, Annals of Math. 140 (1994), 207-250. J. Palis and J-C. Yoccoz, Homoclinic tangencies for hyperbolic sets of large Hausdorffdimension, Acta Mathematica, vol. 172 (1994), 91-136. E. Pujals and M. Sambarino, Homoclinic tangencies and hyperbolicity for surface diffeomorphisms : A conjecture of Palis, to appear in the Annals of Mathematics. W. Tucker, The Lorenz attractor exists, C. R. Acad. Sci. Paris, 328, Serie I: 1197-1202,1999.

BRANE CHARGE COHOMOLOGY AND TENSION LATTICES K.S. STELLE The Blackett Laboratory, Imperial College, London SW7 2BZ UK The anti symmetric tensor charges of supergravity theories form a representation of the cohomology ring of spacetime. This algebraic structure is related to the gauge symmetry algebra for cohomologically nontrivial gauge parameters in a doubled supergravity formalism, in which each original gauge potential is accompanied by its magnetic dual. These extended gauge symmetry algebras lead directly to product rules relating the units of the corresponding charge lattices for the form charges.

1

Introduction

Supergravity theories characteristically contain antisymmetric tensor gauge fields. In the specific case of D = 4 dimensions, however, their presence is camouflaged by the possibility of dualizing these fields to massless axionic scalar fields. In general, however, the presence of such gauge fields and of their corresponding form charges appear to be essential features of supergravity theories, both for their symmetry properties and also for their dynamics. It is these form charges that give rise to the BPS states, upon which the nonperturbative duality symmetries act in the simplest fashion. These charges also play a central role in determining the rules for brane intersections and brane interactions. In this article, based on 1 and following some of the ideas introduced in 2, we shall explore the nature of these form charges and shall show how the Diracquantized lattices of allowed charge values relate to the underlying cohomology of spacetime and to the local symmetries that arise when one doubles the formalism of the theory by including a dual magnetic gauge potential of rank ij = D - q - 2 for each rank q gauge potential in the original dimension D theory.

2

Brane charges and cohomology

Let us start from the bosonic sector of D = 11 super gravity, written in form notation: (1) 135

136

where F[41 = dA[sl is the field strength for the 3-form gauge field A[si field equation d * F{41

+ ~F{41

A F{41 = 0

A[si.

The (2)

is manifestly gauge invariant under the gauge transformation a where dA[31 = 0 .

(3)

Using the Bianchi identify dF[41 = 0, the A isl field equation can be written in terms of an overall exterior derivative, d( *F{41 + ~ A[31 A F(41) = O. This form of the field equation leads directly to the existence of a conserved "electric" charge

u=

r

JaM8

(*FI41

+ ~A[31 A F[41)

(4)

where Ms is some 8-dimensional spatial submanifold of the Mll spacetime. The Bianchi identity dF{41 == 0 also leads to the existence of a conserved "magnetic," charge,

v = Jr

F[41'

(5)

aMa

where Ms is some 5-dimensional spatial submanifold of Mu. The freedom of choice for the submanifold over which one integrates in the above conserved charges makes one suspect that these charges have an orientational sensitivity. The relation between the charge values and the choice of integration manifold is essentially topological, however 4: the charge values are invariant under continuous deformations of the charge surface, providing these deformations do not cause the integration-volume boundary 8M s ,s to pass through a zone where p-brane-current exists. In order for a given charge integral to have a non-zero value, the integration boundary 8M must "capture" the locus of brane-current, i. e. it must not be possible to continuously slide 8M away to a region of zero field strength without encountering current. Thus, the basic supergravity solutions displaying integrated electric or magnetic charges are either infinite or compactified p-branes, whose horizons may be taken to be the locus of brane-current, representing a delta-function source that should properly stand on the right-hand side of the field equation. For the case of the electric 2-brane of D = 11 supergravity, the source action is UThis version of the gauge transformation, which allows for cohomologically nontrivial parameters, has been discussed in particular in Ref. 3.

137

[source

=

T3

(6)

/W3 tfe [J-det(8I'x88tM8I1XN9MN) + ~€I'IIP8I'XM8I1XN8pXRAMNR(Z)]

where the locus x M (e) is the surface that the charge-integral boundary 8M cannot cross, for charges within the same topological class. For infinite brane solutions that differ by finite energy amounts from maximally symmetric (and maximally supersymmetric) "flat" solutions, the asymptotic orientation of the worldsheet within the surrounding D = 11 spacetime will be well-defined. The spatial volume element corresponding to this asymptotic orientation is the natural p-form that one can associate with the directional properties of the brane's charge; the magnitude of this charge is taken to be U or V according to the electric or magnetic case. In this way, one finds that p-branes are the natural carriers of p-form charge. These p-form charges also appear in the D = 11 supersymmetry algebra,

(7) where the dimensions of the Lorentz representations for the bosonic operators on the right-hand side are 11 + 55 + 462, adding up to the 528 independent components of a general symmetric product of a pair of 32-component D = 11 Majorana spinors. Now the question arises as to whether such charges could in some sense be Noether charges. Despite certain folklore about local symmetries not being associated with Noether charges, one can just try anyway to make an elementary construction. For a set offield transformations 8Acpi with parameter A, under which the Lagrangian transforms at most by a total D-dimensional spacetime divergence, 8AC = 8I'n~, the conserved Noether current is I' _ 8C i AI' J A - 8(8I'cpi) 8ACP - HA

.

(8)

Conservation of this current may be checked by straightforward application of the field equations. Given the current (8), one can directly construct the total integrated conserved charge, QA = IVD-I dD - 1 xJX, where the integral is taken over the full volume of a D - 1 dimensional spacelike hypersurface. Consider the example of Maxwell theory to illustrate this construction. Starting from the gauge transformation expressed in the generalised form 8AIII = AlII' with dAIII = 0, one derives the charge QA = IV3 AlII A *n = A FiiO, were h ni -- 8(88eoA;) -- FiO'18 the canomc . al momenturn. Nt 0 e I d3 X that for cohomologically trivial AlII = di.JJ, one may use the equation of motion

138

IV

0 to obtain the charge as a surface integral, Qw = 3 cf2E i (wFiO). This vanishes if w ~ 0 at the infinite boundary oV 3 , so one has non-vanishing charges only for a subset of the gauge transformations, e.g. those for which ow ~ const. at OV3. In this way, the usual expectation that there is no one-to-one relationship between charges and local symmetry transformations is borne out: the gauge symmetries with w ~ 0 at OV3, corresponding to the constraints in the Hamiltonian formalism and which imply the identification of gauge-related field configurations, have a vanishing Noether charge. The gauge transformations that have nonvanishing Noether charges, on the other hand, fall outside this "constraint" class, and are not generally interpreted as implying identifications. These include both those with parameters tending to constants at oV 3 infinity and also the cohomologically nontrivial transformations allowed by (3), in cases where the space V3 has nontrivial homology group HI. Now let us return to D = 11 supergravity. The above approach to deriving a Noether current now requires an n~ correction, since the third "ChernSimons" term in (1) transforms by a total derivative, 8AC l1 = d(~AI3] t\ A 13 ] t\ .F[4]), giving an integrated Noether charge OiFiO =

QA =

rA

JVlO

13 ]

t\

(*11 -

~AI3] t\ F[4])

,

(9)

where l1 ijk = 8(e:iijk) = Fi j kO+!*(F[4]t\A I3 ])Oi j k is the canonical momentum. The Chern-Simons term in the action has two effects here: it gives rise to the second term in the momentum and also requires the n~ second term in the current (9). Once again, for A13] cohomologically trivial, the charge QA can be written as a surface integral. Since we know that the basic carriers of such charges are brane solutions, one can try to evaluate the resulting charge directly for a p-brane. For infinite branes, however, this integral diverges with a factor proportional to the p-dimensional spatial volume of the brane. Accordingly, it is convenient to divide out this volume factor, or equivalently to take a specific form A13] = d(WI2] (y)8(x)), where the coordinates ym belong to the transverse space and the xlJ. correspond to the world volume of the brane (M = /l-,m). In order to have a non-vanishing result in the electric case, w12 ] should be taken to be oriented in alignment with the asymptotic spatial brane volume element, and also to tend to a nonvanishing value at transverse infinity, w12 ] y~ w12 ] 00, giving then Qw = Iworldvol w1 2 ] 008(x) I8Ms transverse (*11 - ~AI3] t\ .F[4]) = ~w~ UIJ.II' In this way, the original electric charge (4) emerges from the Noether charge (9) after factoring out the p-brane volume. As in the Maxwell example, there are also Noether charges (9) corresponding to cohomologically nontrivial gauge parameters AI3 ]. Unlike the Maxwell

139

case, which remains Abelian, one now finds for the supergravity charges a richer algebra. Imposing the standard Poisson bracket relation for canonically conjugate variables,

.. (x) , IIili2j3(x')]PB =68[{I~~2~{3]~(lO)(X_X') [A-~qq . . ~ q q]

,

(10)

one finds a nontrivial charge algebra [Q

(e) Q(e)] _ Q(m) AI3J' AI3J Al3J "A I3J '

(11)

Iv;

= AI6J /I. FI4J is the magnetic charge, related to the earlier where QA(m) 16J 10 charge V (5) after factorising out a 5-brane volume, by analogy to the above discussion for the electric case. Thus, the generalisation of gauge transformations to cohomologically nontrivial cases gives rise to a non-Abelian structure for the supergravity charge algebra, as a consequence of the Chern-Simons term in the action (1). This is actually a representation of the cohomology ring for the spacetime. (It is a ring because the combination operation (A~31' A~3J) -t A~3J /I. A~3J has no defined inverse.)

3

Brane tension relations

The cohomology ring structure of the integrated p-form charges may be interpreted another way, directly from local gauge symmetries of the theory, providing that one generalises 2 the supergravity formalism so as to include "doubled" gauge potentials for each q form gauge potential in the original formalism. To see how this can work, consider once again the D = 11 AI3J equation of motion (2), noting once more that it may be written d( *F[4J + AI3J /I. F I4J ) = O. If one now introduces a dual potential AI6J and its corresponding field strength

!

(1) then the equation of motion for the original gauge potential the duality constraint

AI3J

is implied by

(2) This doubled formalism has an extension of the original gauge symmetry: there are now independent transformations for AI3J and A I6J , under which both F[4J and .Fi7J are invariant: (3)

140

These extended gauge transformations have the algebra [8A l ,8A 2 [3] [3]

1= 8A[6[

[8A[31' 8 A[6]l

A[6]

= 0

= A~3]

/\

[8A[61' 8 A[6]l

A~3]

= 0.

(4)

This extended gauge algebra may be seen as the local origin of the cohomology ring for the integrated charges. It also has a direct implication for the charge units on the lattices of p-form charges allowed by the Dirac quantisation condition. Returning to the source coupling (7) and focussing only on the coupling of the brane to the background A[3] gauge field, the coupling term may be written in terms of a pull-back to the brane worldvolume M3: e[ectric I coupling

= To[3]

1

A [3]

•

(5)

M3

Consider once more the non-trivial gauge transformations with parameter A[3] in 3. Under 8A[3] = A[3]' the coupling term transforms by 8Icoupling = i M3 A[3]' and if M3 E H3(M l1 ) is a homology 3-cycle, with A[3] cohomologically nontrivial, this does not vanish. Thus, the coupling (5) fails to be fully gauge invariant under (3). Not all is lost, however. In quantum mechanics, it is not the action Icoupling that needs to be invariant, but the path-integral phase eilcoupHng. Thus, what we actually obtain is a condition for the parameters A[3] that selects a discrete allowable subset:

T3

r

1M3

A[3]

= 27rk ,

k E Z .

(6)

In the doubled formalism, one introduces also a coupling to the worldvolume M6 of the 5-brane, which carries the magnetic charge (5): Imagnetic coupling -

7:1 5

A[6]

•

(7)

M6

This coupling also fails to be invariant under cohomologically non-trivial transformations A6 in (3). Requiring quantum-level invariance just of the path-integral phase e iI , however, leads now to the discretisation condition

T6

r

A6 = 27ri ,

i EZ .

(8)

1M6

Now reconsider the algebra (4) in the light of (6,8). For A[6] the result of commuting two transformations with parameters A[3] and A;3P the transformation of the action is

(9)

141

For a toroidal spacetime, this integral decomposes into a product of integrals over 3-cycles. Recalling that fM AI31 is already quantised according to (6), one has consistency only if the 2-brane and 5-brane tensions are related by 1 2 T6 = -T3 . 211"

(10)

Relations such as (10) are not new, but what is interesting about the present derivation is that it is obtained purely within the context of a given theory, i.e. D = 11 supergravity, taken together with the supposition that there is a consistent means of quantising this theory. Previous derivations 5,4 have relied heavily upon conjectural symmetries of string theory. Specifically, the previous derivations of relations such as (10) have started from Dirac electric-magnetic charge quantisation constraints taken together with the implications of SL(2, Z) duality symmetry for type lIB string theory (which in D = 4k + 2 dimensions relates electric and magnetic branes charged under the same rank 2k + 1 field strength 4, and in particular in the case of the D = 10 lIB theory this relates the electric and magnetic 3-brane charge lattices supported by the self-dual 5-form field strength). A subsequent use of T dualities then relates the tensions of branes of differing worldvolume dimension, and in this way one is able to link the 2-brane and 5-brane charge lattice units. While this previous derivation is quite plausible in the context of a hoped-for duality-based M-theory synthesis, it obviously relies upon a number of unproven hypotheses at the present time. The present derivation of charge-lattice relations such as (10) of course also involves important assumptions, the most important one of these being that there is in the end some way to quantise M-theory that is consistent with its classical cohomologically nontrivial gauge symmetries. Thus, one now has two rather different ways of looking at the consistency requirements for M-theory quantisation. It is from such intersecting requirements that one may hope for a precise fix on just what quantised M-theory needs to be. 4

Abstract superalgebras and lower dimensions

One may extract the parameters of the extended local gauge algebra (4) underlying the brane-tension relations (10) and use this to streamline the analysis of these relations. Put the form parameters into a standard position in much the same way as one does for extracting the algebraic relations from commutators of supersymmetry transformations with stated parameters. In this way, one finds that the abstract generators corresponding to the odd-rank form parameters need to satisfy anticommutation relations: letting V t+ A13]!

142

VB

A[6['

one has the abstract algebra

2

{V, V} =-V

(V, V]

=0

[ii, V] = 0 .

(1)

From this graded abstract algebra, one may read off directly the brane tension relations (10) upon making the correspondence generator B tension/(27l'). To illustrate how this helps in the establishment of the brane tension relations in more complicated lower-dimensional cases, consider the type IIA theory 6, whose bosonic Lagrangian may be written

£10 = R * 1- ~ * d¢ /I. d¢ - ~e-~I/> * .1'[2) -

~el/>

* F(S) /I. F[s) -

~e- ~I/> * F(4J

/I. .1'[2)

/I. F(4J

- ~dA[sJ /I. dA[sJ /I. A[2) ,

(2)

where ¢ is the dilaton, F(4) = dA[s[ - dA[2J /I. A[l)' F[sJ = dA[2) and .1'[2) = dA[l)' with the "calligraphic" symbols denoting the Kaluza-Klein gauge field and the corresponding field strength, which arise upon deriving (2) by the dimensional reduction of the D = 11 theory (1). Following the doubling philosophy, one now introduces high-rank potentials {?,b[8J' A[7)' A[6)' A[5)} dual to the original gauge potentials (and dilaton) {¢, A[l),A[2)' A[3J}' The equations of motion for the original fields are then implied by the duality constraints 2, analogous to (1), _11/>

* F[4J = * F[3J = _31/> e ~ * F[2J = * d¢ = e ~ I/> e

-

F[6J

-

(3)

= dA[5) - A[2J /I. dA[3J

-

-

-

-

1

F(7) = dA[6J - 2" A[s) /I. dA[sJ F[8J

-

A[lJ /I.

-

(dA[5) - A[2) /I. dA[3))

1

-

1

= dA[7J - A[2) /I. (dA[5J - 2" A[2) /I. (dA[5J - 2" A[2) /I. dA[3J)

-

-

P[9) = d'lj;[8J

1 + 2"A[2J

3 +4A[lJ /I.

-

/I. dA[6)

1 + 4A[3J

-

-

/I. (dA[5) - A[2) /I. dA[3)) -

1

(dA[7) - A[2J /I. (dA[5J - 2"A[2) /I. dA[3J)) .

Analysing the extended gauge algebra for the constraint system (4), one find the abstract graded Lie algebra [H,WI] = ~W1 -

3 -

[H, VI]

= _VI

[H, WI] = -2"W1

[H, VI] = VI

[WI, VI] =-V [V1,V] = _WI

{WI, V} = -VI {V, V} = -VI

[VI, VI] =

-tiI

{V, V} = -~iI

[H,v] = ~ [H, V] = -~V [VI, V] =-V {WI, WI} =

,

-iiI (4)

143

where the abstract generators {H, WI, VIm V, V, VI, WI, if} correspond to the field strengths {dt/>, .r(2) , F{3J> F{4J> F(6) , F{T) , P8, .p{g)}. Upon making the correspondence generator t+ tension/(2rr), one obtains directly I the brane-tension relations for the IIA theory 1

T7 = 2rrT2TS 1 2 (5) 2rr T 3 Thus, the notions of space-time cohomology, extended gauge transformation algebras and the tensions of BPS branes are intimately interlinked. As our understanding of quantum M-theory develops, it may be hoped that these underlying mathematical structures can provide helpful clues in discovering the most appropriate formalism for the unified theory of quantised gravity and matter.

n=

References 1. I. Lavrinenko, H. Lii, C.N. Pope and K.S. Stelle, "Superdualities, brane

tensions and massive IIA/IIB duality," hep-th/9903057. 2. E. Cremmer, B. Julia, H. Lii and C.N. Pope, "Dualisation of dualities II: twisted self-duality of doubled fields and superdualities," Nucl. Phys. B535 (1998) 242, hep-th/9806106. 3. B. Julia and S. Silva, "Currents and superpotentials in classical gaugeinvariant theories: I. Local results with applications to perfect fluids and general relativity," Class. Quantum Grav. 15 (1998) 2215. 4. M. Bremer, H. Lii, C.N. Pope and K.S. Stelle, "Dirac quantisation conditions and Kaluza-Klein reduction," Nucl. Phys. B529 (1998) 259, hep-th/9708109. 5. J.H. Schwarz, "The power of M theory," Phys. Lett. B367 (1996) 97, hep-th/9510086. 6. I.C. Campbell and P.C. West, Nucl. Phys. 243 (1984) 112; F. Giani and M. Pernici, Phys. Rev. D30 (1984) 325; M. Huq and M.A. Namazie, Class. Quantum Gmv. 2 (1985) 293; ibid. 2 (1985) 597.

SCALING LIMITS OF LARGE INTERACTING SYSTEMS S.R.S. VARADHAN Courant Institute, New York University, 251 Mercer Street, New York, NY, 10012, USA E-mail: varadhan@cims. nyu. edu

1

From Classical Mechanics to Euler Equations

The basic example of a scaling limit is derivation of the 'Euler Equations 'from the equations of classical mechanics. Let us start with a collection of N ~ pf3 particles in a large periodic cube Ai of side f in R3. The motion of the particles is governed by the equations of motion of a classical Hamiltonian dynamical system with energy given by (1)

Here, qi E Ai is the position of the i-th particle and Pi E R3 is its velocity. We will use indices k = 1,2,3 to refer to the three components of position or velocity. The repulsive potential V ~ 0 is an even function and has compact support in Rd. The interaction in particular is short range. The equations of motion are ~ _ 8H(~.q) _ dt -

8P.

-

pk i

dp; _ _ 8H(~.q) _ _ " N

dt

-

8q.

-

L.Jj=l

Vik(q. _ q.) t

(2)

J

where Vk(q) = 8~)y) for k = 1,2,3 are the three components of the gradient of V. The dynamfcal system has five conserved quantities. The total number N of particles, the total momenta L~l p~ for k = 1,2,3 and the total energy H (p , q). The hydrodynamic scaling in this context consists of rescaling space and time by a factor of f. The rescaled space is the unit torus T3 in 3dimensions. The macroscopic quantities to be studied correspond to the five conserved quantities. The first one of these is the density, and is measured by a function p(t,x) of t and x. For each f < 00 it is approximated by Pi(t,X), 144

145

defined by

J'l[

1" qi(lt) J(X)Pl(t, x)dx = i3 ~ J(-i-) N

d

T

(3)

.=1

Differentiation with respect to t yields

!t fTd J(X)Pl(t,x)dx = !tis L~1 J(q;~lt») = -bL~I(V'J)(q;~lt»)'Pi(it) ~

(4)

fTd(V'J)(X)' Pl(t,X)Ul(t,x)dx

where Ul(t,X) = u~(t,x), k = 1,2,3 are the components of the 'average' velocity of the fluid at the rescaled space time point x, t. This introduces three other macroscopic variables corresponding to the three components of the momenta that are conserved. We can now write down the first of our five equations

ap - + V' . (pu)

at

(5)

= 0

To derive the next three equations, using a test functions J, we differentiate for k = 1,2,3

!t is L~1 J( q;~lt) )pf(lt) = Is L~1 pf (it) (V' J)( q;~lt») . Pi (it) -b L~1 Lf=1 J(q;~lt»)Vk(qi(it) If we now use the skew-symmetry of Vk = term of the right hand side of equation 6 as

88V, qk

(6)

qj(it))

we can rewrite the second

-~ L~1 Lf=1 (J(q;~lt») - J(qj~lt»)Vk(qi(it) - qj(it))

~ -~ L~1 Lf=1 Jr(q;~lt»)(qHlt) - qj (it))Vk (qi(it) - qj(it)) =

is L~1 Lf=1 Jr(q;~lt»)'Ij;k(qi(it) -

(7)

qj(it))

with

'lj;k(q)

= -~qrVk(q)

The next step is rather mysterious and requires considerable explanation. The quantities N

LPfpi, i=1

N

L 'lj;k(qi(t) - qj(t)) i,j=1

146

are not conserved. They depend on combinations of individual velocities and on spacings between particles both of which change in the microscopic time scale and therefore do so rapidly in the macroscopic scale. They should therefore be replaced by their space-time averages. By appealing to an 'Ergodic Theorem' they can be replaced instead by their averages with repect to their equilibrium distributions. The equilibrium 'ensemble' consists of an infinite collection of points {Pc., qoJ, in the phase space R3 x R3. There is a natural five parameter family of measures /Lp,u,T that are invariant under spatial translations as well as the Hamiltonian dynamics. The points {Pc.} are distributed according to a Gibbs Distribution with density p and formal interaction energy 1

2T

L V(qc. - q(3} c.,(3

In other words {qc.} is a point process obtained by taking infinite volume limit of N = fa p particles distributed in the cube of side '- in R3 according to the joint density 1 1 Z exp [ - 2T V(qi - Qj}]

L

lS.if.jS.N

where Z is the normalization constant. The velocities {Pc.} are distributed independently of each other as well as of {qc.}, having a common three dimensional Gaussian distribution with mean u and covariance T I. Assuming that the infinite volume limit exists in a reasonable sense, it will be a point process defined as an infinite volume Gibbs measure /Lp,T' The velocities {Pc.} will be an independent Gaussian ensemble Vu,T' In the first term the quantities p~pi are replaced by their expectations

uk(t, x}u r (t, x) + c5k,rT(t, x} and in the second term tPk,r are replaced by their expectations that involve the 'pressure' per unit volume in the Gibbs ensemble

P k(p , T) =

i~~ EJLp,T {'-~ L tPHqc. -

q(3} }

Iq",l$l Iql'll$l

This leads to the equation

It fTd J(x}uk(t,x}dx =

fTd L~=l

t;r (x}.(uk(t,x}ur(t,x) + c5k,rT (t,x))dx

+ lTd L~=l t~ (x}Pk(p(t,x},T(t,x}}dx

(8)

147

We now integrate by parts, remove the test function J and obtain from equation 8 d

dt (pu) + V . (p u 0 u + PT I + P (p ,T)) = 0

(9)

There is an equation of state that expresses the total energy per unit volume eas 1

e(p, u, T} = 2P(lu12 + 3T} + f(p, T}

(10)

where f(p, T}, the potential energy per unit volume, is given by

f(p,T} = lim

l-too

EIJP'T{ 2.[.~3

L

V(qa - q,B}}

Iq"l0

(29)

with the initial condition p(O,x)

= Po(x).

Tagging is only an exercise of the mind. The particles are totally oblivious to it. Since the empirical density is the same for the tagged or untagged system the solutions of equations 18 and 29 must coincide. This yields 1 1 "2VS(p(t,x))Vp(t,x) - V· (b(t,x)p(t,x)) = "2VDVp(t,x).

While it does not follow, it is quite likely that

~S(p(t,x»Vp(t,x) -

b(t,x}p(t,x) =

~DVp(t,x)

which can be solved to yield 1

b(t,x} = 2 ( ) [S(p(t,x)) -D]Vp(t,x) p t,x

155

If we now denote by Q a Markov process (total mass p) having initial density Po (x) and time dependent (backward) generator

L = !V'S( ( ))V' ! [S(p(t,x)) - D]V'p(t,x) . V' t 2 P t, x +2 p(t, x)

(30)

then one expects the empirical process defined in 16 to have the limit lim RN = Q

N-too

(31)

in probability. For simple exclusion processes 31 was indeed proved by Rezakhanlou (R]. Once the distribution of the tagged particle is shown to converge to a diffusion with generator L t given by 30 is not very hard to prove 31. One needs to show that the motion of two different tagged particles are asymptotically independent. However if we start with an initial configuration that is far away from equilibrium, like an arbitrary deterministic initial condition, to prove the limit theorm when an arbitrary particle in the initial configuration is tagged is a very hard result to prove. On the other hand proving 31 requires only that the limit theorem be valid for nearly all particles in the initial configuration. The proof of 31 in [R] uses earlier results in [Q] that study a situation that mediates between tagged and untagged versions. Let us devide the particles into a finite number of types that we will think of as colors. Colors do not affect the motion which is as before. We simply keep track of the colors. At any time t, we have the empirical distribution of each one of the k colors. In the scaled limit we will have k densities p(t, x) = {Pj(t, x) : 1 ~ j ~ k}. They should evolve as a nonlinear system

ap~;x)

= V'B(p(t,x))V'p(t,x)

(32)

with suitable initial conditions. The system 32 is equivalent to solving first the linear equation 18 for the total density k

p(t,x) = LPj(t,x) j=l

and then at the next step solving a set of (identical) equations that are again linear with the correct initial conditions for the densities of individual colors apj(t,x) _ L* .(t ) at tP3 ,x

which is just the forward equation for the probability density of the position of the tagged particle.

156

In [Q] the multicolor system was viewed in its own right and the law of large numbers under diffusive scaling was established for it. For the simple exclusion model, this turns out to be a nongradient system and the analysis becomes considerably harder. The transition from the multicolor system to 31 is essentially a question of compactness. Let P = {AI'···' Aq} be an arbitrary but fixed partition of the Td into nice sets. If E is a subset of the form E = {x: x(t 1 ) E A;p···,x(tn ) E A;n}

and the convergence in probability of RN(E) to Q(E) can be proved by induction on n. For n = 1, this just involves the empirical density and a single color. If assume the result for n time points we have qn possible past histories and we code them by qn distict colors. The emprical density of the diffrent colors at the next time tn+l is sufficient to yield information on subsets E at stage n + 1. Given the law of large numbers for the multicolor system the induction proceeds smoothly. Since the partition is arbitrary, if we have compactness, we can pass from finite dimensional distributions to weak convergence of the processes RN. This was carried out in [R]. 4

Large Deviations.

We now turn to the question of Large Deviation rates. We examine first the empirical densities vi'. The probability distribution of the empirical density satisfies a large deviation principle with the folowing rate functional l(p(·,·)) on the space of weakly continuous maps p from [0, T] into L1 (Td) that satisfy 0 ~ p( t, x) ~ 1.

l(p(·, .)) =

10 (p(O,

.)

liT

+"2

0

8p

118t -

1 2 "2V DV pll_1,p(t,.)(1_p(t,-»dt

(33)

The first term 10 is the rate function for deviations of the initial profile from its appropriate limit. For deterministic initial conditions it is 0 for the true value and 00 for all others. Otherwise it depends on the statistics of the initial configuration. The second term which is the dynamical part measures in some way by how much p fails to satisfy the heat equation 18. For any p(.) the norm II II-I,p(t,.)(l-p(t,.» refers to the dual of the weighted Sobolev norm

IIllIi ,p(.)

=

r < DV I(x), V/(x) > p(x)(1 lTd

p(x))dx

This was proved in [KOV] and it is necessary for us to understand how these large deviations arise. One can introduce a small bias in the system by per-

157

turbing the symmetric jump rates by a weak asymmeric term i.e. the probabilities of transition from x to x + z at time t, are changed from p(z), by a small perturbation, to p(z) + trq(t, N,z). This has an enrtopy cost as well as a macroscopic effect. To calculate these quantities we need to know that the system is locally close to an equlibrium (some Bernoulli measure) and the empirical density dictates which equlibrium we have to pick. Quntities not explicitly dependent on the empirical densities are replaced by their expectations in equlibrium which means by suitable functions of the empirical density. Macroscopically, with weak asymmetry, we saw that the equation 18 gets replaced by 21 and the entropy cost (after optimizing over q(.,.,.) for a given b(·, .) and normalization by a factor of N-d) is given by £(b(·, .)) =

~ 2

[T [ < Db(t, x), b(t, x) > p(t, x)(1 - p(t, x))dxdt

10 lTd

For a given p(.,.) we consider Bp (.,.)

= {b(.,·) : (5.2) holds}

The dynamical term in 33 is the result of optimizing £(b(·, .)) over b(·,·) E B p (.,.). The weak asymmetry has an efect on the behavior of a tagged particle which can be described through b(·, .). The generator L t gets replaced by Lt,b(t,.) = L t

+ (1 - p(t, x))b(t, x) . \7

and the tagged particle process with the new generator will be denoted by Qb. As long as b(·,·) E Bp (.,.), Qb will have p(t,·) for its marginal density at time t. Suppose Q is a general stochastic process on C[[O, T]; Td] with total mass p. Its marginal densities can be denoted by q(t, .). The 'current' is the collection of expected values of the stochastic integrals EQ

[loT < f(t, x(t)), dx(t) > ]

and we can try to find bQ E B(q(·, .)) such that the marginals as well as the 'current' of Q match those of the tagged particle distribution QbQ that we will obtain under our perturbation. The measure Q will then determine bQ and QbQ. The large deviation rate function for RN is the functional

I(Q) = Io(q(O, .)) + £(bQ(·, .)) + H(Q; QbQ)

(34)

158

The important point here is the interpretation of the terms. The first term is clearly the contribution of the statistics of the initial condition. The second term depends on the hydrodynamical scaling behavior of the model and depends on the objects that describe them. The last term is pure noise, and says that once the correct macroscopic behavior of density and current are assured the tagged particle motions are nearly independent. The details can be found in [QRVj References 1. L. Jensen and H.T. Yau, Hydrodynamical Scaling Limits of Simple Exclusion Models lAS/Park City Mathematics Series 167-221 1998 2. C. Kipnis and S.RS Varadhan, Central Limit Theorem for Additive Functionals of Reversible Markov Processes and Applications to Simple Exclusions. Commun. Math. Phys. 104 1-19. 1986 3. C. Kipnis, S. OHa and S.RS. Varadhan, Hydrodynamics and large deviations for simple exclusion processes Comm. Pure Appl. Math. 42 115-137 1989 4. J. Quastel, Diffusion of color in the simple exclusion process Comm. Pure Appl. Math. 45623-679 1992 5. J. Quastel, F. Rezakhanlou and S.RS. Varadhan, Large Deviations for the Symmetric Simple Exclusion Process Prob. Th. and Rel. Fields 113 1-841999 6. F. Rezakhanlou, Propagation of chaos for symmetric simple exclusion Comm. Pure Appl. Math. 117943-957 1994 7. H. Spohn, Large Scale Dynamics of Interacting Particles Text and Monographs in Physics Springer-Verlag 1991

THE MICROSCOPIC ORIGIN OF BLACK HOLE THERMODYNAMICS IN STRING THEORY SPENTA R. WADIA Department of Theoretical Physics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400 005, INDIA and Jawaharlal Nehru Centre for Advanced Scientific Research, Jakkur Campus, Jakkur PO, Bangalore 560064, INDIA E-mail:[email protected] The treatment of black holes in the framework of local quantum field theory predicts that Hawking radiation is exactly thermal. This violation of unitary evolution of standard quantum mechanics is popularly called the information puzzle. String theory, which is a new paradigm for the formulation of the laws of physics, enables a standard statistical mechanics explanation of black hole thermodynamics. We review some aspects of this development.

1

The String Paradigm

String theory is a new paradigm to formulate the fundamental laws of physics.

It is a radical departure from the Newtonian paradigm where the laws of nature are formulated in terms of point particles and used to describe more complex aggregates. This new paradigm involves in it's formulation higher dimensional structures called branes: a O-branes is a point particle, a I-brane is a string, a 2-brane is a membrane etc. It also comes comes with a new fundamental constant called the string length. In the following we explain the difficulties inherent in the application of local quantum field theory to the black holes of general relativity (GR). We then briefly summarize how string theory provides a setting to describe black holes and resolves some of these problems. 2

Quantum Mechanics and General Relativity

The application of local quantum field theory to GR leads to 2 basic problems: 1. The problem of ultra-violet divergences renders GR an ill-defined quantum theory. This specifically means that if we perform a perturbation expansion around flat Minkowski space-time (our world!) then to subtract infinities from the divergent diagrams we have to add an infinite number of terms to the 159

160

Einstein-Hilbert action with coefficients that are proportional to appropriate powers of the ultraviolet cutoff. There is good reason to believe that string theory solves this ultra-violet problem because the extended nature of string interactions have an inherent ultra-violet cutoff given by the fundamental string length H. One also knows that in string theory the Einstein-Hilbert action emerges as a low energy effective action for energy scales much larger than the string length and Newton's constant (in 10-dim.) is given by,

(1) where g8 is the string coupling. 2. While the above problem is related to the high energy behavior of GR, there exists another problem when we quantize matter fields in the presence of a black hole which does not involve high energy processes. This problem is called the information puzzle and in the following we shall explain the issue and also summarize the attempts within string theory to resolve the puzzle in a certain class of black holes. String theory has been proposed as a theory that describes all elementary particles and their interactions. Presently the theory is not in the stage of development where it can provide quantitative predictions in particle physics. However in case this framework resolves some logical problems that arise in the applications of quantum field theory to general relativity, then it is a step forward for string theory. 3

Classical Black Holes

Classically a black hole is a solution of the GR equations, and it is characterized by an event horizon, which is a null surface. The horizon is a one way gate, in the sense that once we are inside it we cannot get out because of the causal structure of the black hole space time. Physically one can imagine the formation of an event horizon due to the bending property of light by the matter that makes up the black hole. Let us list a few properties of classical black holes: Firstly the event horizon has an area and there is a area law which states that in any process involving black holes the final area of the event horizons is never less than the initial area(s):

(2)

161

The 'no hair theorems', tell us that the state of a classical black hole is completely characterized by its mass, angular momentum and global gauge charges. In particular the area of the event horizon depends only on these quantities. If we perturb a black hole then the perturbation decays in Planck time, and the new state of the black hole is again characterized by a event horizon whose area has increased and is characterized by the changed the mass, angular momentum or charge of the final state. 4

Quantum Mechanics and Black Holes

Application of quantum field theory to matter propagating in a black hole back-ground leads to the following results: 1. The black holes behave like black bodies. They emit thermal radiation and they are characterized by a temperature which again depends only on the mass, angular momentum and the global charges of the black hole. The fundamental formula for the temperature, due to Hawking, is given by, T

=

hI), 271'"

(3)

I), is surface gravity (acceleration due to gravity felt by a static observer) at the horizon of the black hole. For a Schwarzschild black hole:

I),

1 = 4G N M

(4)

The thermal black hole is characterized by an entropy (BekensteinHawking) which is proportional to the horizon area. The constant of proportionality is determined using the first law of thermodynamics and the temperature formula.

2 a = 4G N h

(5)

Using this we can now interpret the area law (2) as the second law of black hole thermodynamics. It is a striking fact that the constant 'a' in the entropy formula involves all the 3 fundamental constants and it is in fact an unit of Planck area. The fact that black holes emit radiation that is EXACTLY thermal leads to the information puzzle:

162

Initially the matter that formed the black hole is in a pure quantum mechanical state. Here in principle we know all the quantum mechanical correlations between the degrees of freedom of the system. In case the black hole evaporates completely then, according to Hawking, the final state of the system is purely thermal and hence it is a mixed state. This evolution of a pure state to a mixed state is in conflict with the standard laws of quantum mechanics which involve unitary time evolution of pure states into pure states. Hence we either have to modify quantum mechanics, as advocated by Hawking, or as we shall argue, replace the paradigm of quantum field theory by that of string theory. In string theory we retain quantum mechanics and resolve the information puzzle (for a certain class of black holes) by discovering the microscopic degrees of freedom of the black hole. In string theory the Hawking radiation is NOT thermal and in principle we can reconstruct the initial state of the system from the final state. However, just like in standard statistical mechanics we can introduce a density matrix to derive the thermodynamic description, the same thing can be done for black holes in string theory. In this way the thermodynamic formulas for black hole entropy and decay rates for Hawking radiation are derived. In particular the Bekenstein-Hawking formula is derived from Boltzmann's law: S = In O. It is well worth pointing out that the existence of black holes in nature (for which there is mounting evidence) compels us to resolve the conundrums that black holes present. One may take recourse to the fact that for black holes whose mass is a few solar masses the Hawking temperature is very tiny ('"" 10- 8 degs. Kelvin), and not of any observable consequence. However the logical problem that we have described above cannot be wished away and its resolution makes a definitive case for the string paradigm as a correct framework for fundamental physics as opposed to quantum field theory. 5

String Theory and Black Holes

In string theory one attempts to understand the geometric entropy formula of Hawking and Bekenstein in terms of Boltzmann's law: SBH

= logO

(6)

String theory provides a microscopic constituent model for a large class of black hole solutions. This provides the possibility of a calculation of 0 and hence SBH which can be matched with the GR formula. The basic point about the string theory description is that a black hole is described by a density matrix:

163

P=

nI Lli)(il

(7)

i

where Ii) is a micro-state. Once we make this "assumption" then we can calculate formulas of black hole thermodynamics just like we calculate the thermodynamic properties of macroscopic objects using standard methods of statistical mechanics. Here the quantum correlations that existed in the initial state of the system are erased by our procedure of defining the black hole state in terms of density matrix. Recall radiation coming out of a star, or a lump of hot coal. The 'thermal' description of the radiation coming out is merely the result of averaging over a large number of quantum states of the coal. In principle by making detailed measurements on the wave function of the emitted radiation we can infer the precise quantum state. For black holes the reasoning is similar. Clearly in this formulation the black hole can exist as a pure state: one among the highly degenerate set of states which are characterized by a small number of parameters. Let us also note that in Hawking's semi-classical analysis, which uses quantum field theory in a given black-hole space-time, there is no possibility of a microscopic construction of the black hole wave functions. What is not very clear at this stage is whether under all circumstances we can perform experiments to observe the hair (micro-states) of the black hole. 6

D-Branes and Micro-states

The black hole solution of Type lIB string theory is a well studied model system. This black hole is like an extremal (near extremal) Reissner-Nordstrom black hole with its charge equal to (nearly equal to) its mass. This extremal solution also has the BPS property. To describe the constituent model we need to first consider IO-dim. space as a product of R 5 ,1 x S1 X T4. So we have 5 non-compact dimensions, time, a circle and a 4-torus. Now consider 2 distinct types of solitons of this string theory. 5-branes and l-branes. These are simply domain walls of 5 and I dim. which carry a unit of charge (called Ramond charge, which is a generalized gauge charge). Now we consider wrapping a large number (Q5) of 5-branes on S1 X T4 and a large number (Qd of l-branes on S1. This is the so called DI- D5 system.

164

This assembly of solitons interact by means of open strings that end on them. Since we will presently restrict ourselves to low energy processes involving the black hole we need to address the question of the low lying excitations of the bound state formed by the system of branes. This is a complicated problem to which the answer is not known in complete detail. These studies bring out a deep and intimate connection between non-Abelian gauge theories and gravity theory. The low lying excitations (long wavelengths) are described by a 2-dim. super conformal field theory (SCFT) defined on the moduli space of instantons of a non-Abelian gauge theory on the torus T4. The gauge group is U(Q5) and the instanton number is Q1. The central charge of the conformal field theory is c = 4Q1Q5' The ground state of the black hole (T=O) is characterized by another charge viz. the value La = N in the SCFT. For large values of Q1 and Q5 this is a highly degenerate system and in fact the calculation of the entropy of this system can be done and it turns out to be the well known formula: (8)

This exactly coincides with the Bekenstein-Hawking result for the corresponding extremal black hole! The sigma model on the moduli space of instantons is singular in the limit of vanishing instanton size. This can be rectified by turning on certain marginal operators of the sigma model. This modified or deformed conformal field theory is non-singular.

7

Hawking Radiation

The study of Hawking radiation at low temperatures requires the DI - D5 system to be excited slightly above it's ground state. The black hole then decays back to its ground state by emission of Hawking radiation. The excited state of the black hole away from extremality corresponds to the level conditions: La = N + n and La = n, where n « N These conditions immediately enable us to calculate the entropy of the excited black hole in the micro-canonical ensemble,

(9) The small Hawking temperature (n «Q1Q5) of this near extremal black hole is given by,

165

2 JQ1Q5

1 _

TH - 7fR

-n-

(10)

These thermodynamic formulae match with the calculations done in general relativity. The next issue of importance is the calculation of the Hawking rates of emission of the various supergravity (closed string) modes by the near extremal black hole as it decays towards it's ground state. The formula for the Hawking rate, as calculated in general relativity, for a given channel is given by,

(11) where p(k) = (exp kiTH _1)-1 and a(k)abs is the absorption cross section of the wave incident on the black hole. In order to perform these calculations in the string theory one needs to know how these various modes couple to the bound state of the branes. Some progress in calculating these couplings has been made using the fact that near the horizon of the black hole there is a doubling of the number of super-symmetries from 4 to 8. This exactly matches with the eight supersymmetries of the SCFT. The resulting calculations are in precise agreement with general relativity. It is well worth outlining the principle of the string theory calculation. The picture of absorption is as follows. Consider throwing in a closed string mode, say a minimal scalar lPn, towards the DI-D5 configuration. A micro-state Ii) will couple to such a fluctuation in a certain way through some local interaction

Sint =

!

2

d zIPnIBOn(z, z)

and will get excited to a different state

If)

(12)

with the amplitude

(13) where l1Pc} represents the closed string mode. IPnlB means here the value of the supergravity mode on the brane.

166

Since we started from a density matrix description for the initial state rather than individual micro-states, the probability of the process would be given by the "unpolarized" expression:

(14)

Note that the "unpolarized" transition probability corresponds to averaging over initial states and summing over final states. n is the total number of initial micro-states representing the macroscopic charges of the black hole. The above expression for the absorption probability can then be used to calculate the Hawking decay rate in a standard fashion. The string calculation agrees at long wave lengths with the GR calculation.

8

Unsolved Problems

We conclude with a set of unsolved problems. Progress on any of them would be of interest. 1. Why is the black hole entropy proportional to the area of the horizon, rather than a volume as one would have expected in a standard statistical physics system? This fact has been elevated to the level of a principle : the Holographic Principle, which states that in a theory of gravity dynamics in d-dimensions is coded in dynamics in d-1 dimensions. Furthermore, the number of independent degrees of freedom needed to describe the system is at most one per Planck area. (In 3+1 dims. the Planck area is = 10- 66 cm 2 ) 2. The model which we discussed has a high degree of super-symmetry. This enables us to extrapolate certain weak coupling calculations, like the entropy and the Hawking rates of massless particles, to the strong coupling region, gQl « 1, Q5 « 1, relevant to the macroscopic black hole. In the absence of the high degree of super-symmetry, which includes the physical situation of the Schwarzschild or Kerr-Newmann black holes, what is the microscopic theory. 3. Even for the super-symmetric black hole the string theory calculations were restricted to low energies, with wavelengths much less than QIQ5' How do we go beyond this restriction? Below I have collected a few references as a guide to the issues we have discussed.

167

Acknowledglllents

It is a pleasure to acknowledge discussions with Justin David, Avinash Dhar and Gautam MandaI on the subject of black holes. I would also like to thank Ali Chamseddine and Nicola Khuri for a wonderful and exciting conference. References 1. S.W. Hawking and R. Penrose, 'The Nature of Space Time', Scientific American, July 1996. 2. L. Susskind, 'Black Holes and the Information Paradox', Scientific American, April 1997. 3. G. 't Hooft, 'The Scattering Matrix Approach for the Quantum Black Hole', gr-qc/9607022. 4. A. Strominger and C. Vafa, "Microscopic Origin of the BekensteinHawking Entropy," Phys. Lett. B379 (1996) 99, hep-th/9601029. 5. C. G. Callan and J. Maldacena, "D-brane Approach to Black Hole Quantum Mechanics, "Nucl. Phys. B472 (1996) 591, hep-th/9602043. 6. A. Dhar, G. MandaI and S. R. Wadia, "Absorption vs Decay of Black holes in string theory and T-symmetry," Phys. Lett. B388 (1996) 51, hep-th/9605234. 7. S. R. Das, and S. D. Mathur, "Interactions involving D-branes," Nucl. Phys. B482 (1996) 153, hep-th/9607149. 8. J. Maldacena and A. Strominger, "Black Hole Greybody Factors and D-Brane spectroscopy," Phys. Rev. D55 (1997) 861, hep-th/9609026. 9. C. G. Callan, S. S. Gubser, I. R. Klebanov and A. A. Tseytlin, "Absorption of Fixed scalars and the D-brane approach to Black Holes," Nucl. Phys. B489 (1997) 65, hep-th/9610172. 10. I. R. Klebanov and M. Krasnitz, "Fixed scalar greybody factors in five and four dimensions," Phys. Rev. D55 (1997) 3250, hep-th/9612051; "Testing Effective String Models of Black Holes with Fixed Scalars," Phys. Rev. D56 (1997) 2173, hep-th/9703216. 11. J. Maldacena, "Black Holes in String Theory," Ph.D. Thesis, hepth/9607235. 12. C. Vafa, "Instantons on D-branes," Nucl. Phys. B463 (1996) 435, hepth/9512078. 13. S. F. Hassan and S. R. Wadia, "Gauge Theory Description of D-brane Black Holes: Emergence of the Effective SCFT and Hawking Radiation," Nucl. Phys. B526 (1998) 311, hep-th/9712213. 14. M. R. Douglas, "Branes within Branes," hep-th/9512077.

168

15. E. Witten, "On the Conformal Field Theory Of The Higgs Branch," J. High Energy Phys. 07 (1997) 003, hep-th/9707093. 16. J. Maldacena, "The large N limit of superconformal field theories and supergravity," hep-th/9712200. 17. J. R. David, G. MandaI and S.R. Wadia, "Absorption and Hawking Radiation of Minimal and Fixed Scalars, and Ads/CFT Correspondence," Nucl. Phys. B544 (1999) 590, hep-th/9808168. 18. N. Seiberg and E. Witten, "The D1/D5 System And Singular CFT", hep-th/9903224. 19. J. R. David, G. MandaI and S.R. Wadia, "D1/D5 Moduli in SCFT and Gauge Theory, and Hawking Radiation" , hep-th/9907075.

ORGANIZING COMMITTEE

Khalil Bitar (American University of Beirut, Lebanon) Ali Chamseddine (American University of Beirut, Lebanon) Ahmad Nasri (American University of Beirut, Lebanon) Wafic Sabra (American University of Beirut, Lebanon) lalal Shatah (Courant Institute, NY, USA)

169

LIST OF SPEAKERS

Sir Michael Atiyah (University of Edinburgh, UK) John Ball (Oxford University, UK) Sir Michael Berry (University of Bristol, UK) Jean-Pierre Bourguignon (Institut des Hautes Etudes Scientifiques, France) Alain Connes (College de France, France) Michael Duff (Texas A&M, USA) Ivar Ekeland (University of Paris IX, France) Murray Gell-Mann (CaItech, USA) Ioannis Giannakis (New York University, USA) Chris Hull (Queen Mary College, London University, UK) Roman Jackiw (MIT, USA) Ramzi Khuri (City University of New York, USA) Andre Martin (CERN-Geneva, Switzerland) Henry McKean (Courant Institute, NY, USA) V. Mukhanov (Maxmilians University Munich, Germany) Hermann Nicolai (Einstein Institute, Germany) Louis Nirenberg (Courant Institute, NY, USA) Jacob Palis (IMPA, Brazil) George Papanicolaou (Stanford University, USA) Kelly Stelle, (Imperial College, London University, UK) S.R. Varadhan (Courant Institute NY, USA) Spenta Wadia (Tata Institute of Fundamental Research, India)

170

LIST OF PARTICIPANTS

F. Abi-Khuzam (American University of Beirut, Lebanon)

Michel Abou Ghantous (American University of Beirut, Lebanon) Ibrahim Abou Hamad (American University of Beirut, Lebanon) Ziad Abou Hatab (American University of Beirut, Lebanon) Manal Abou-Chakra (American University of Beirut, Lebanon) Rami Abou-Hamde (American University of Beirut, Lebanon) Hazar Abu-Khuzam (American University of Beirut, Lebanon) Mazen AI-Ghoul (American University of Beirut, Lebanon) Lina AI-Jader (American University of Beirut, Lebanon) Jawad AI-Khal (University of Maryland at College Park, USA) Safak Alpay (Middle East Technical University, Turkey) Aref Alsoufi (Lebanese University, Lebanon) Khalid Amin (University of Bahrain, Bahrain) Nabil Ayoub (Yarmouk University, Jordan) Monique Azar (American University of Beirut, Lebanon) George Batrouni (University of Nice, France) Verjouhi Bodakian (American University of Beirut, Lebanon) Diala Bouhadir (American University of Beirut, Lebanon) Mazen Bu-Khuzam (Cambridge University, UK) K. Chadan (Universite de Paris- Sud, France) Kevin Butcher (American University of Beirut, Lebanon) Caroline Chalouhi (American University of Beirut, Lebanon) Jean Chatila (Lebanese American University, Lebanon) Saleh M. Chehade (Beirut Arab University, Lebanon) Wissam Chemaissany (Lebanese University, Lebanon) Ajai Choudhry (Ambassador of India to Lebanon, India) Theodore Christidis (American University of Beirut, Lebanon) Firas Daboul (American University of Beirut, Lebanon) Roger Dagher (The Open University of London, UK) Tarek Dassouki (Lebanese University, Lebanon) Mohamed Debs (Lebanese University, Lebanon) Mary Deeb (American University of Beirut, Lebanon) George Dekermendjian (Eastwood College, Lebanon) Tekin Dereli (Middle East Technical University, Turkey) Ghassan Dibeh (Lebanese American University, Lebanon) Abir EI Rabih (American University of Beirut, Lebanon) Mirna EI-Bustani (Board of Trustees, American University of Beirut, Lebanon) Samir EI-Daher (Association of Lebanon for the United Nations, Lebanon) 171

172

Mounib El-Eid (American University of Beirut, Lebanon) Fadi Elias (American University of Beirut, Lebanon) Raymond El-Khoury (Eastwood College, Lebanon) Dina El-labban (American University of Beirut, Lebanon) Fouad El-Zein (University of Nantes, France) Mona El-Zein (American University of Beirut, Lebanon) Susanne Espenlaub (The University of Manchester, UK) Layla Ezzedeen (American University of Beirut, Lebanon) Jean Fares (Notre Dame University, Lebanon) Rami Farhat (American University of Beirut, Lebanon) Bassam Fayad (Ecole Poly technique, France) Tarek Gergaoui (American University of Beirut, Lebanon) Catherine Ghaddar (American University of Beirut, Lebanon) Ali Ghandour (Board of Trustees, American University of Beirut, USA) Nadine Ghandour (American University of Beirut, Lebanon) Sawsan Ghandour (American University of Beirut, Lebanon) Hassan Ghaziri (American University of Beirut, Lebanon) Diane Ghorayeb (American Uni versity of Beirut, Lebanon) Nassif Ghoussoub (University of Britsh Colombia, Canada) Ioannis Giannakis (New York University, USA) George Haddad (American University of Beirut-AUH, Lebanon) Rana Haddad (American University of Beirut, Lebanon) M. J. Haddadin (V.P., American University of Beirut, Lebanon) Jad Hage (American University of Beirut, Lebanon) Safouan Hage (American University of Beirut, Lebanon) Mehdi Hage-Hassan (Lebanese University, Lebanon) Ghassan Haidar (American University of Beirut, Lebanon) Nassar H. S. Haidar (American University of Beirut, Lebanon) Raja Hajjar (Lebanese American University, Lebanon) Roger Hajjar (Notre Dame University, Lebanon) Lara Halaoui (American University of Beirut, Lebanon) G. Hanna (University of Balamend, Lebanon) Michel Hebert (The American University in Cairo, Egypt) Oussama Hijazi (lnstiut Elie Cartan , University of Nantes, France) Samih Isber (American University of Beirut, Lebanon) Rim Jaber (American University of Beirut, Lebanon) Ghinwa Jalloul (American University of Beirut, Lebanon) Mona Jurdak (American University of Beirut, Lebanon) Wadi' Jureidini (American University of Beirut, Lebanon) Mohammad Amin Kayali (Texas A&M University, USA) Malhab Kayrouz (Notre Dame University, Lebanon) Kamal Khairallah (Eastwood College, Lebanon) Saad Khairallah (American University of Beirut, Lebanon)

173

Talal Khawaja (Lebanese University, Lebanon) Philip Khoury (MIT & Board of Trustees, AUB, USA) Elizabeth Khuri (Cornell University, USA) Nicola Khuri (Rockefeller University, USA) Soumaya Khuri (East Carolina University, USA) Kamal Khuri-Makdisi (McGill University, Canada) Leonid Klushin (American University of Beirut, Lebanon) Badrie Kojok (ESIB, Lebanon) Ulrich Kortz (American University of Beirut, Lebanon) Fouad Lahlou (Sidi Mohamed Ben Abdellah University, Morocco) James T. Liu (Rockefeller University, USA) Abdallah Lyzzaik (American University of Beirut, Lebanon) Hani Maalouf (Lebanese University, Lebanon) Wahib Mahjoub (Lebanese University, Lebanon) Samir Makdisi (American University of Beirut, Lebanon) Hadi Maktabi (American University of Beirut, Lebanon) Joseph Malkoun (American University of Beirut, Lebanon) Mahdi Mattar (Massachusetts Institute of Technology, USA) Issam Moghrabi (Lebanese American University, Lebanon) Nelly Mouawad (Lebanese University, Lebanon) Omar Mustafa (Eastern Mediterranean University, North Cyprus) Amin Muwafi (Alzaytoonah Jordanian University, Jordan) Nasrallah Nasrallah (Lebanese University, Lebanon) Diane Nawffal (University of Balamend, Lebanon) Abdul-Majid Nusayr (Jordan University of Science and Technology, Jordan) Nazim Nuwayhid (American University of Beirut, Lebanon) Rida Nuwayhid (American University of Beirut, Lebanon) Osmo Pekonen (University of Jyvaskyla, Finland) Ghassan Rizk (American University of Beirut, Lebanon) Nasser Saad (American University of Beirut, Lebanon) Nasser Saad (Notre Dame University, Lebanon) Adib Saad (American University of Beirut, Lebanon) Jawad Saade (American University of Beirut, Lebanon) Cihan Saclioglu (Bogazici University, Turkey) Nisrine Salam (American University of Beirut, Lebanon) Salim Salem (Universite Saint-Josesph, Lebanon) Marc Saliba (Eastwood College, Lebanon) Hisham Sati (Texas A & M University, USA) Toni Sayah (Ecole Polytechnique, France) Frederik Seitz (Rockefeller University, USA) Ghazi Serhan (American University of Beirut, Lebanon) Riad Shamseddine (Lebanese University, Lebanon) Malek Tabbal (American University of Beirut, Lebanon)

174

Khaled Tabbara (American University of Beirut, Lebanon) Jihad Touma (American University of Beirut, Lebanon) Fadi Twainy (Dar AI-Nahar, Lebanon) Kameshwar C. Wah (Syracuse University, USA) Emil K. Wehbeh (American University of Beirut, Lebanon) Rania Zaatari (American University of Beirut, Lebanon) Maya Zayour (American University of Beirut, Lebanon)