Table Of ContentM-Theory and Quantum Geometry
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Series C: Mathematical and Physical Sciences - Vol. 556
M-Theory and Quantum Geometry
Edited by
Larus Thorlacius
Seienee Institute,
University of lee land
and
Thordur Jonsson
Seienee Institute,
University of leeland
....
"
Springer-Science+Business Media, BV.
Proceedings of the NATO Advanced Study Institute on
Quantum Geometry
Akureyri, Iceland
August 9-20, 1999
A C.I.P. Catalogue record for this book is available from the Library of Congress.
ISBN 978-0-7923-6475-7 ISBN 978-94-011-4303-5 (eBook)
DOI 10.1007/978-94-011-4303-5
Printed on acid-free paper
All Rights Reserved
© 2000 Springer Science+Business Media Dordrecht
Originally published by Kluwer Academic Publishers in 2000
Softcover reprint of the hardcover 1s t edition 2000
No part of the material protected by this copyright notice may be reproduced or utilized in
any form or by any means, electronic or mechanical, including photocopying, recording
or by any information storage and retrieval system, without written permission from the
copyright owner.
Table of Contents
Preface Xl
1 D BRANES IN STRING THEORY, I
P. Di Vecchia and A. Liccardo
1 Introduction............... 1
2 Perturbative String Theory . . . . . . 2
3 Conformal Field Theory Formulation . 11
4 T-Duality................ 20
5 Classical Solutions Of The Low-Energy String Effective Action 28
6 Bosonic Boundary State . . . . . . . . . . 30
7 Fermionic Boundary State . . . . . . . . . 39
8 Classical Solutions From Boundary State 46
9 Interaction Between a p and a p' Brane 48
2 MODULI SPACES OF CALABI-YAU
COMPACTIFICATIONS
J. Louis
1 Introduction. . . . . . . . 61
2 A short story about string theory, F-theory and M-theory 61
2.1 String Theory ........ . 61
2.2 Calabi-Yau compactifications 63
2.3 String Dualities. 64
2.4 F -Theory . . . . . . . . . . 66
2.5 M-Theory ......... . 67
2.6 Three Triplets of Dualities. 68
3 The q = 16 triplet .. 68
4 The q = 8 triplets 72
A Calabi-Yau manifolds 83
3 THE M(ATRIX) MODEL OF M-THEORY
w.
Taylor
1 Introduction................... 91
2 Matrix theory from the quantized supermembrane 92
2.1 Review of light-front string ........ . 95
vi
2.2 The bosonic membrane theory . . 96
2.3 The light-front bosonic membrane 98
2.4 Matrix regularization .... . . . 100
2.5 The bosonic membrane in a general background 103
2.6 The supermembrane . . . . . . .. 104
2.7 Covariant membrane quantization 110
3 The BFSS conjecture . . . . . . 111
3.1 Membrane "instability" 112
3.2 M-theory........ 114
3.3 The BFSS conjecture 115
3.4 Matrix theory as a second quantized theory 116
3.5 Matrix theory and DLCQ M-theory 118
4 M-theory objects from matrix theory. 123
4.1 Supergravitons 123
4.2 Membranes........... 125
4.3 5-branes . . . . . . . . . . . . . 133
4.4 Extended objects from matrices. 137
5 Interactions in matrix theory 139
5.1 Two-body interactions ..... . 140
5.2 The N-body problem ...... . 156
5.3 Longitudinal momentum transfer 160
6 Matrix theory in a general background . 160
6.1 T-duality .... . ....... . 161
6.2 Matrix theory on tori ..... . 163
6.3 Matrix theory in curved backgrounds. 165
7 Outlook . . .. . .. ... ...... . 168
4 THE HOLOGRAPHIC PRINCIPLE
D. Bigatti and L. Susskind
1 Black Hole Complementarity .... . 179
1.1 The Schwarz schild Black Hole . 180
1.2 Penrose Diagrams ..... . 184
1.3 Black Hole Thermodynamics 185
1.4 The Thermal Atmosphere ... 189
1.5 The Quantum Xerox Principle 190
1.6 Information Retention Time .. 192
1. 7 Quantum Xerox Censorship . . 194
1.8 Baryon Violation and Black Hole Horizons 195
1.9 String Theory at High Frequency ... 197
1.10 The Space Time Uncertainty Relation 199
2 Entropy Bounds ..... 201
2.1 Maximum Entropy .......... . 201
vii
2.2 Entropy on Light-Like Surfaces 203
2.3 Robertson Walker Geometry 205
2.4 Bousso's Generalization . . . 206
3 The AdS / CFT Correspondence and
the Holographic Principle . . . . 210
3.1 AdS Space .. .. ..... . 210
3.2 Holography in AdS Space . . 211
3.3 The AdS/CFT Correspondence. 212
3.4 The Infrared Ultraviolet Connection 214
3.5 Counting Degrees of Freedom 215
3.6 AdS Black Holes 216
3.7 The Horizon .... . 217
4 The Flat Space Limit ... . 218
4.1 The Flat Space Limit 219
4.2 High Energy Gravitons Deep in the Bulk 220
4.3 Kaluza Klein Modes . . . . . . . . . . . . 222
5 BORN-INFELD ACTIONS AND D-BRANE PHYSICS
C.G. Callan
1 D-Brane Solitons and the Born-Infeld Action . . . . . ... 227
2 Born-Infeld Dynamics of Branes in Flat Space. . . . . . .. 231
3 Branes in Curved Space and the Gauge Theory Connection 234
4 Born-Infeld Analysis of the Baryon Vertex. . . 240
5 Applications of the AdS/CFT Correspondence 245
6 Summary . . . . . . . . . .. . . . ... .. . . 252
6 LECTURESONSUPERCONFORMALQUANTUM
MECHANICS AND MULTI-BLACK HOLE MODULI
SPACES
R. Britto-Pacumio, J. Michelson, A. Strominger, and
A. Volovich
1 Introduction....... . .............. . . 255
2 A Simple Example of Conformal Quantum Mechanics 257
3 Conformally Invariant N-Particle Quantum Mechanics 259
4 Superconformal Quantum Mechanics . . . . . 261
4.1 A Brief Diversion on Supergroups ... 261
4.2 Quantum Mechanical Supermultiplets . 263
4.3 Osp(112)-Invariant Quantum Mechanics 264
4.4 D(2, 1; a)-Invariant Quantum Mechanics 267
5 The Quantum Mechanics of a Test Particle in a R eissner-
Nordstrom Background . ................... 271
V11l
6 Quantum Mechanics on the Black Hole Moduli Space 274
6.1 The black hole moduli space metric 274
6.2 The Near-Horizon Limit 276
6.3 Conformal Symmetry .. . . 277
7 Discussion . . . . . . . . . . . . . . . 278
A Differential Geometry with Torsion 279
7 LARGE-N GAUGE THEORIES
Y. Makeenko
1 . Introduction . . . ... .. .. . 285
2 O(N) Vector Models . .. .. . 286
2.1 Four-Fermi Interaction 287
2.2 Bubble graphs as zeroth order in l/N 290
2.3 Scale and Conformal Invariance of Four-Fermi Theory 292
2.4 Nonlinear sigma model .. .. . .... . 297
2.5 Large-N factorization in vector models . 300
3 Large-N QCD . . ... .. . .. .. .. 300
3.1 Index or ribbon graphs . . . . . . . . . . 301
3.2 Planar and non-planar graphs . . . . . . 305
3.3 Topological expansion and quark loops . 312
3.4 Large-Nc factorization . . . . . . 315
3.5 The master field . . .. . .... 317
3.6 1/N c as semiclassical expansion . 319
4 QCD in Loop Space . .... . .. ... 321
4.1 Observables in terms of Wilson loops . 322
4.2 Schwinger-Dyson equations for Wilson loop 324
4.3 P ath and area derivatives .. 326
4.4 Loop equations . . . . . . . . . . . . . . . 330
4.5 Relation to planar diagrams . . . . . . . . 332
4.6 Loop-space Laplacian and regularization . 333
4.7 Survey of non-perturbative solutions 337
4.8 Wilson loops in QCD2 . 338
5 Large-N Reduction . . .... . ... 342
5.1 Reduction of scalar field . . . 342
5.2 Reduction of Yang-Mills field 347
5·.3 Rd-symmetry in perturbation theory 349
5.4 Twisted reduced model .. ..... 350
8 INTRODUCTION TO RANDOM SURFACES
T. Jonsson
1 Introd uction. . . . . . . . . . . . . . . . . . ... . .. .. . 355
ix
2 Random paths ............. . 356
2.1 Lattice paths . . . . . . . . . . . 357
2.2 Dynamically triangulated paths. 359
3 Branched polymers . . . . . 362
3.1 Extrinsic properties .. . 362
3.2 Intrinsic properties ... . 366
4 Dynamicaly triangulated surfaces 368
4.1 Definitions . ... 368
4.2 Basic properties 369
4.3 The string tension 370
4.4 Further results 372
5 Lattice surfaces . . . . . . 373
5.1 Definitions .... 373
5.2 Critical behaviour 375
6 Conclusion ....... . 378
9 LORENTZIAN AND EUCLIDEAN QUANTUM GRAVITY
- ANALYTICAL AND NUMERICAL RESULTS
J. Ambj0rn, J. Jurkiewicz, and R. Loll
1 Introduction.. .. ..... 382
2 Lorentzian gravity in 2d . . 385
2.1 The discrete model . 385
2.2 The continuum limit 389
3 Topology changes and Euclidean quantum gravity 394
3.1 Baby universe creation .... . . . . . .. . 394
3.2 The fractal dimension of Euclidean 2d gravity. 401
4 Euclidean quantum gravity . . . 403
4.1 Some generalities .... . 403
4.2 Dynamical triangulations 404
4.3 The functional integral .. 407
4.4 Inclusion of matter fields 408
5 Numerical setup ... . .... . 410
5.1 Monte Carlo method and ergodic moves 410
5.2 Observables in 2d Euclidean gravity .. 417
5.3 Comments on the 2d results. . . . . . . 428
6 Dynamically triangulated quantum gravity in d > 2 . 433
6.1 Generalization to higher dimensions . . 433
6.2 Numerical results in higher dimensions . 438
7 Outlook 442
INDEX . 451
PREFACE
The fundamental structure of matter and spacetime at the shortest length
scales remains an exciting frontier of basic research in theoretical physics.
A unifying theme in this area is the quantization of geometrical objects.
The majority of lectures at the Advanced Study Institute on Quantum Ge
ometry in Akureyri was on recent advances in superstring theory, which is
the leading candidate for a unified description of all known elementary par
ticles and interactions. The geometric concept of one-dimensional extended
objects, or strings, has always been at the core of superstring theory but
in recent years the focus has shifted to include also higher-dimensional ob
jects, so called D-branes, which play a key role in the non-perturbative
dynamics of the theory.
A related development has seen the strong coupling regime of a given
string theory identified with the weak coupling regime of what was previ
ously believed to be a different theory, and a web of such" dualities" that
interrelates all known superstring theories has emerged. The resulting uni
fied theoretical framework, termed M-theory, has evolved at a rapid pace
in recent years.
D-branes have also advanced our understanding of quantum effects
in black hole physics, in particular the nature of black hole entropy and
thermodynamics. An unexpected correspondence between the near-horizon
physics of certain black holes and conformal quantum field theories has also
been uncovered. The conformal theories are related to the gauge theory of
the strong interaction so this line of development is of phenomenological in
terest besides offering a promising new approach to some quantum gravity
problems.
Some alternative approaches to quantization of gravity were also dis
cussed at the Advanced Study Institute. One of the most successful is "dy
namical triangulations" which endeavors to construct the quantum geome
try of spacetime using simple geometrical building blocks. This approach in
cludes extensive numerical simulations of systems in various dimensions and
has also provided techniques for quantizing geometric objects in a mathe
matically rigorous fashion.
Many individuals and organizations contributed to the success of the
meeting in Akureyri. First of all I would like to thank the ASI students for
their participation and enthusiasm for the program, and the lecturers for
Xl
