Table Of ContentCRM Series in Mathematical Physics
Springer Science+ Business Media, LLC
CRM Series in Mathematical Physics
Conte, The Painleve Property: One Century Later
MacKenzie, Paranjape, and Zakrzewski, Solitons: Properties,
Dynamics, Interactions, Applications
Saint-Aubin and Vinet, Algebraic Methods in Physics:
A Symposium for the 60th Birthdays of Jirf Patera and
Pavel Winternitz
Saint-Aubin and Vinet, Theoretical Physics at the End of the
Twentieth Century: Lecture Notes of the CRM Summer
School, Banff, Alberta
Semenoff and Vinet, Particles and Fields
van Diejen and Vinet, Calogero-Moser Sutherland Models
Yvan Saint-Aubin
Luc Vinet
Editors
Theoretical Physics
at the End of
the Twentieth Century
Lecture Notes of
the CRM Summer School,
Banff, Alberta
With 74 Illustrations
, Springer
Yvan Saint-Aubin Luc Vinet
Departement de Mathematiques Department of Physics
et Statistique and
Universite de Montreal Department of Mathematics and Statistics
c.P. 6128, Succursale Centre-viile McGiII University
Montreal, Quebec H3C 3J7 James Administration Building, Room 504
Canada Montreal, Quebec H3A 2T5
[email protected] Canada
[email protected]
Editorial Board
Joel S. Feldman Duong H. Phong
Department of Mathematics Department of Mathematics
University of British Columbia Columbia University
Vancouver, British Columbia V6T IZ2 New York, NY 10027-0029
Canada USA
[email protected] [email protected]
Yvan Saint-Aubin Luc Vinet
Departement de Mathematiques Department of Physics
et Statistique ami
Universite de Montreal Department of Mathematics and Statistics
C.P. 6128, Succursale Centre-viile McGi\I University
Montreal, Quebec H3C 3J7 James Administration Building, Room 504
Canada Montreal, Quebec H3A 2T5
[email protected] Canada
[email protected]
Library of Congress Cataloging-in-Publication Data
Theoretical physics at the end ofthe twentieth century : lecture notes ofthe CRM
summer school, Banff, Alberta I editors Yvan Saint-Aubin, Luc Vinet.
p. cm. - (The CRM series in mathematical physics)
Includes bibliographical references.
ISBN 978-1-4419-2948-8 ISBN 978-1-4757-3671-7 (eBook)
DOI 10.1007/978-1-4757-3671-7
1. Mathematical physics - Congresses. 2. Physics - Congresses. 1. Saint-Aubin, Yvan.
II. Vinet, Luc. III. CRM series on mathematical physics.
QA19.2 .T44 2001
530.15-dc21 2001032817
Printed on acid-free paper.
© 2002 Springer Science+Business Media New York
Originally published by Sprlnger-Verlag New York, Ine. in 2002
Softcover reprint of the hardcover 1 st edition 2002
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ten permission of the publisher Springer Scienee+ Business Media, LLC,
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The use of general descriptive names, trade names, trademarks, etc., in this publication, even if the
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Production managed by Yong-Soon Hwang; manufacturing supervised by Jerome Basma.
Photocomposed copy prepared from the CRM's LaTeX files.
987654321
ISBN 978-1-4419-2948-8 SPIN 10841204
Series Preface
The Centre de recherches matMmatiques (CRM) was created in 1968 by the
Universite de Montreal to promote research in the mathematical sciences.
It is now a national institute that hosts several groups and holds special
theme years, summer schools, workshops, and a postdoctoral program. The
focus of its scientific activities ranges from pure to applied mathematics and
includes statistics, theoretical computer science, mathematical methods in
biology and life sciences, and mathematical and theoretical physics. The
CRM also promotes collaboration between mathematicians and industry.
It is subsidized by the Natural Sciences and Engineering Research Council
of Canada, the Fonds FCAR of the Province de Quebec, and the Canadian
Institute for Advanced Research and has private endowments. Current ac
tivities, fellowships, and annual reports can be found on the CRM Web
page at www.CRM.UMontreal.CA.
The CRM Series in Mathematical Physics includes monographs, lecture
notes, and proceedings based on research pursued and events held at the
Centre de recherches matMmatiques.
Yvan Saint-Aubin
Montreal
Preface
When organizing a summer school at the end of the most fruitful century in
the history of physics one must carefully consider the question of topics. We
decided to avoid a retrospective of problems solved during the last century
and we shied away from a prediction of which ones might be settled during
the next. We opted for a snapshot of what theoretical physics is at the
end of this century (or, should we say, the last century), namely problems
actively researched in the last few years of the nineteen nineties. This choice
led to twelve courses that were timely, diverse and exciting.
This book contains seven of the courses given in the summer school.
They constitute up-to-date accounts of the following topics: supersymmet
ric Yang-Mills theory and integrable systems, branes, black holes and anti
de Sitter space, turbulence, Bose-Einstein condensation, integrability, de
formed Virasoro and elliptic algebras, mesoscopic physics, QeD in extreme
conditions. Young physicists will find in these texts pedagogical introduc
tions to subjects currently active in theoretical physics, and more seasoned
ones a chance to share the excitement of fields outside their immediate
research interests.
Yvan Saint-Aubin
Luc Vinet
Montreal
Contents
Series Preface v
Preface vii
Contributors xvii
1 Lectures on Supersymmetric Yang-Mills Theory and
Integrable Systems 1
Eric D'Hoker and D.H. Phong
1 Introduction.................... 1
1.1 Supersymmetry and the Standard Model . 2
1.2 Supersymmetry and Unification of Forces 4
1.3 Supersymmetric Yang-Mills Dynamics . 7
2 Supersymmetric Yang-Mills in 4 Dimensions 8
2.1 Supersymmetry Algebra . . . . . . 8
2.2 Massless Particle Representations . . . . 9
2.3 Massive Particle Representations . . . . 9
2.4 Field Contents of Supersymmetric Field Theories 11
2.5 N = 1 Supersymmetric Lagrangians 12
2.6 N = 1 Superfield Methods. . . . . . . . . . . . . 13
2.7 Irreducible Superfields of N = 1 . . . . . . . . . . 15
2.8 General N = 1 Susy Lagrangians via Superfields 18
2.9 Renormalizable N = 2,4 Susy Lagrangians 20
2.10 N = 2 Superfield Methods:
Unconstrained Superspace . . . . 21
2.11 N = 2 Superfield Methods:
Harmonic/ Analytic Superspaces . 24
3 Seiberg-Witten Theory ........ 28
3.1 Wilson Effective Couplings and Actions 28
3.2 Holomorphicity and Nonrenormalization 30
3.3 Low Energy Dynamics of N = 2 Super-Yang-Mills 33
3.4 Particle and Field Contents . . . . . . . . . . . . . 34
3.5 Form of the N = 2 Low Energy Effective Lagrangian 36
3.6 Physical Properties of the Prepotential . . 39
3.7 Electric-Magnetic Duality . . . . . . . . . 40
3.8 Monodromy via Elliptic Curves for SU(2)
Gauge Group . . . . . . . . . . . . . . 42
3.9 Physical Interpretation of Singularities . . 44
x Contents
3.10 Hypergeometric Function Representation. 45
4 More General Gauge Groups, Hypermultiplets . 47
4.1 Model of Riemann Surfaces . . . . . . . . 48
4.2 Identifying Seiberg-Witten and Riemann Surface Data. 49
4.3 SU(N) Gauge Algebras, Fundamental Hypermultiplets. 50
4.4 Classical Gauge Algebras, Fundamental Hypermultiplets 56
5 Mechanical Integrable Systems . . . . . . . . . . . . . . . 57
5.1 Lax Pairs with Spectral Parameter-Spectral Curves 58
5.2 The Toda Systems . . . . . . . . . . . . . . . . . . . 59
5.3 The Calogero-Moser Systems for SU(N) . . . . . . . 62
5.4 Relation between Calogero-Moser and Toda for SU(N). 64
5.5 Relations with KdV and KP Systems . . . . . . . . 66
5.6 Calogero-Moser Systems for General Lie Algebras 67
5.7 Scaling of Calogero-Moser to Toda for General Lie
Algebras . . . . . . . . . . . . . . . . . . . . . . . . 68
6 Calogero-Moser Lax Pairs for General Lie Algebras. . . 70
6.1 Lax Pairs with Spectral Parameter for Classical Lie
Algebras . . . . . . . . . . . . . . . . . . . . . . . 71
6.2 The General Ansatz ................ 73
6.3 Lax Pairs for Untwisted Calogero-Moser Systems 76
6.4 Lax Pairs for Twisted Calogero-Moser Systems 82
6.5 Scaling Limits of Lax Pairs ............. 85
7 Super-Yang-Mills and Calogero-Moser Systems . . . . . 87
7.1 Correspondence of Seiberg-Witten and Integrable
Systems . . . . . . . . . . . . . . . . . . . . . . . . 87
7.2 Calogero-Moser and Seiberg-Witten Theory for SU(N) 89
7.3 Four Fundamental Theorems .............. 90
7.4 Partial Decoupling of Hypermultiplet, Product Groups 95
8 Calogero-Moser and Seiberg-Witten for General 9 97
8.1 The General Case. . . . . . . . . . . 97
8.2 Spectral Curves for Low Rank. . . . 99
8.3 Perturbative Prepotential for SO(2n) 101
9 References.......... 103
A Notations and Conventions ..... 113
A.1 Spinors. . . . . . . . . . . . . . 113
A.2 Dirac Matrices in a Weyl Basis 114
A.3 Dirac Matrices in a Majorana Basis. 114
A.4 Two-Component Spinors . 114
B Lie Algebra Theory. . . . . . . . . . . 115
C Elliptic Functions. . . . . . . . . . . . 119
C.1 Basic Definitions and Properties 119
C.2 Half and Third Period Functions 122
C.3 The Function <P . . . . . . 123
C.4 The Functions A, <PI, <P2 . . . . . 124
Contents xi
2 Lectures on Branes, Black Holes, and
Anti-de Sitter Space 127
M.J. Duff
1 Introduction....................... 127
1.1 Supergravity, Supermembranes and M-Theory . 127
1.2 The Kaluza-Klein Idea. . . . . 129
1.3 The Field Content .. . . . . . 133
1.4 The AdS/CFT Correspondence 137
1.5 Plan of the Lectures . . . . 138
1.6 Problems 1 ......... 140
2 Eleven-Dimensional Supergravity 140
2.1 Bosonic Field Equations . . 140
2.2 AdS4 xS7 . . . . . . . . . . 140
2.3 Consistent Truncation to the Massless Modes 144
2.4 The Supermembrane Solution 146
2.5 AdS x S4 . . . . . . . . . . . 149
7
2.6 The Superfivebrane Solution. 150
2.7 Problems 2 ....... 152
3 Type IIB Supergravity . . . . 152
3.1 Bosonic Field Equations 152
3.2 AdS5 x S5 . . . . . . . . 153
3.3 The Self-Dual Superthreebrane Solution 154
3.4 Problems 3 .............. 155
4 The M2-Brane, D3-Brane and M5-Brane . 155
4.1 The M2-Brane 155
4.2 The M5-Brane 157
4.3 The D3-Brane . 159
4.4 Problems 4 .. 159
5 ADS/CFT: The Membrane at the End of the Universe 160
5.1 Singletons Live on the Boundary . . . . . . . . 160
5.2 The Membrane as a Singleton: The Membrane/
Supergravity Bootstrap ........... 162
5.3 Doubletons and Tripletons Revisited . . . . . . 165
5.4 The Membrane at the End of the Universe. . . 166
5.5 Near Horizon Geometry and p-Brane Aristocracy 169
5.6 Supermembranes with Fewer Supersymmetries.
Skew-Whiffing. . . . . . . . 171
5.7 The Maldacena Conjecture 173
5.8 Problems 5 ..... 174
6 Anti-de Sitter Black Holes . . . . 174
6.1 Introduction......... 174
6.2 S5 Reduction of Type IIB Supergravity 177
6.3 D = 5 AdS Black Holes ........ 180
6.4 Rotating D3-Brane . . . . . . . . . . . 180
6.5 S7 Reduction of D = 11 Supergravity 183
xii Contents
6.6 D = 4 AdS Black Holes ....... 186
6.7 Rotating M2-Brane . . . . . . . . . . 187
6.8 84 reduction of D = 11 Supergravity 188
6.9 D = 7 AdS Black Holes .... 190
6.10 Rotating M5-Brane . . . . . . . 191
6.11 Charge as Angular Momentum 192
6.12 Magnetic Black Holes ..... 193
6.13 Kaluza-Klein States as Black Holes. 194
6.14 Recent Developments. 195
6.15 Problems 6 ... 196
7 Solutions to Problems 196
7.1 Solutions 1 196
7.2 Solutions 2 198
7.3 Solutions 3 201
7.4 Solutions 4 204
7.5 Solutions 5 206
7.6 Solutions 6 207
8 References.... 212
A The Lagrangian, Symmetries and Transformation Rules of
D = 11 Supergravity . . . . . . . . . . . . . . . . . . . . .. 228
B The Field Equations, Symmetries and Transformation Rules
of Type lIB Supergravity .................... 230
C The Lagrangian, Symmetries and Transformation Rules of the
M2-Brane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
D The Field Equations, Symmetries and Transformation Rules
of the M5-Brane ......................... 234
E The Lagrangian, Symmetries and Transformation Rules of the
D3-Brane .............. . 238
F D = 4, N = 2 Gauged Supergravity . 240
3 Easy Turbulence 245
K rzysztoJ Gawf;dzki
Lecture 1 The Navier-Stokes Equations ....... . 245
Lecture 2 The Kolmogorov and Kraichnan-Batchelor
Theories of Turbulence. . . . . . 252
Lecture 3 The Richardson Dispersion Law. 259
3.1 Weakly Compressible Regime . 265
3.2 Strongly Compressible Regime 265
Lecture 4 Cascades and Intermittency . 267
4 BEC and the New World of Coherent Matter Waves 277
Allan Griffin
1 An Overview of Past and Recent Work . . . . 278
1.1 Some History before 1980 ....... . 278
1.2 More Recent Developments (1980-1995) 281
