M-Theory and Quantum Geometry NATO Science Series

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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 1st 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 , 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 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/Nc 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 - 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 xii providing excellent reviews and up to date information on a wide range of rapidly developing subjects. The organizing of the Advanced Study Insti• tute has from beginning to end been shared with Thordur Jonsson, with valuable input from our fellow organizers Paolo DiVecchia and Andy Stro• minger. I especially wish to thank Gerlinde Xander of the Science Institute at the University of Iceland, for her efforts in preparing for and ensuring the smooth running of the meeting, and also at the later stages of finalizing reports and preparing the proceedings for publication. I also want to thank Ellen Pedersen of Nordita in for her valuable assistance with the accounts. The manager and staff at Hotel Edda in Akureyri provided an excellent environment for the participants, both inside and outside the lecture hall. The Advanced Study Institute was made possible by generous funding from a number of agencies. Our principal sponsor was the NATO Scientific and Environmental Affairs Division, with additional funding coming from NorFa and Nordita. The Icelandic Ministry of Education and the Town Council of Akureyri sponsored social events for the ASI participants. Finally, I thank the staff of Kluwer Academic Publishers for their pa• tience and cooperation.

Larus Thorlacius LECTURERS

Jan Ambjf2irn , Blegdamsvej 17, 2100 Copenhagen 0 , DANMARK [email protected] Curtis Callan Dept. of Physics, Princeton University, Princeton, NJ 08544, USA [email protected] Thordur Jonsson Science Institute, Univ. of Iceland, Dunhaga 3, IS-107 Reykjavik, ICELAND [email protected] Jerzy Jurkiewicz Institute ofPhyscis, Jagellonian University, Reymonta 4,30-059 Krakow, POLAND [email protected] Jan Louis MLU Halle, FB Physik, Friedemann-Bach-Platz 6, D-06108 Halle, GERMANY [email protected] Yuri Makeenko lust. of Theoretical and Experimental Physics, B. Chermushkinskaya 25 , 117218 Moscow, RUSSIA [email protected] Andrew Strominger Dept. of Physics, Harvard University, Cambridge, MA 02138, USA [email protected] Leonard Susskind Department of Physics, Stanford University, Stanford, CA 94305 , USA [email protected] Washington Taylor Center for Theoretical Physics, MIT, Cambridge, MA 02139, USA [email protected] Larus Thorlacius Science Institute, Univ. of Iceland, Dunhaga 3, IS-107 Reykjavik, ICELAND [email protected] Paolo Di Vecchia NORDITA, Blegdamsvej 17, DK-2100 Copenhagen, DENMARK [email protected] Herman Verlinde Princeton University, Dept. of Physics, Princeton University, Princeton, NJ 08544, USA ver [email protected]

Xlll xiv ASI Students

Carlo Angelantonj Centre de Physique Theorique, Ecole Poly technique, FR-91128 Palaiseau Cedex, FRANCE [email protected] Sergei Antropov Physical Faculty, Dniepropetrovsk State University, Naucnyj Str. 13, Dniepropetrovsk-lO, UKRAINE [email protected] Peter Austing Theoretical Physics, 1 Keble Road, Oxford OX1 3NP, UK [email protected] Maximo Banados Dept. de Fisica Teorica, Univ. de Zaragoza, Cuidad Universitaria, Zaragoza 50009, SPAIN [email protected] Daniela Bigatti Phys. Dept., Katholieke Univ. Leuven, Celestijnenleen 200 D, B-3001 Heverlee, BELGIUM daniela. [email protected] Bjorn Brinne FYSIKUM, Stockholm University, Box 6730, S-113 85 Stockholm, SWEDEN brinne@ physto.se Ruth Britto-Pacumio Dept. of Physics, Harvard University, Cambridge, MA 02138, USA [email protected] Zdzislaw Burda Institute of Physcis, Jagellonian University, Reymonta 4, 30-059 Krakow, POLAND [email protected] Gaspare Carbone S.I.S.S.A. - I.S.A.S. via Beirut No. 2-4, 1-34013 Trieste, ITALY [email protected] Daniel Cartin Dept. of Physics, Penn State University, State College, PA 16802, USA [email protected] Andrew H. Chamblin Pembroke College, Cambridge CB2 1RF, UK H.A. [email protected] Chang Shih Chan Graduate College, Princeton, NJ 08544, USA [email protected] xv

J oao Correia Niels Bohr Institute, Blegdamsvej 17, DK-2100 Copenhagen, DENMARK correia@nbLdk Vladimir Damgov Bulg. Academy of Sciences, Space Research Institute, JK"Mladost 1A" Bl. 522, Vh. 5, Ap. 98, 1784 Sofia, BULGARIA [email protected] Liana David The University of Edinburgh, Dept. of Mathematics and Statistics, Mayfield Road, Edinburgh EH9 3JZ, UK [email protected] Jaques Distler Phys. Dept., University of Texas, RLM 9-308, MS C1600, Austin, TX 78712 USA [email protected] Olivier Dore LAP T H, BP 110, F-74940 Annecy-Ie-Vieux, FRANCE [email protected] Brynjolfur Eyjolfsson Menntaskolinn a Akureyri, Eyrarlandsvegur 28, IS-600 Akureyri, ICELAND [email protected] Stephen Fairhurst Penn State University, University Park, PA 16802, USA [email protected] Daniel Friedan , 126 Frelinghuysen Road, Piscataway, NJ 08854 , USA [email protected] S. James Gates, Jr. Dept. of Physics, Univ. of Maryland at College Park, College Park, MD 20742-4111, USA [email protected] Florian Girelli Village, Montagagne 09240, FRANCE fiorian. [email protected] Vlf Gran Dept. of Theortical Physics and, Mechanics, Chalmers University of Technology, S-41296 G6teborg, SWEDEN [email protected] Marc Grisaru Dept. of Physics, Brandeis University, Waltham, MA 02454, USA [email protected] Vidar Gudmundsson Science Institute, University of Iceland, Dunhaga 3, IS-I07 Reykjavik, ICELAND [email protected] xvi Holger Gunther MLU Halle FB Physik, Friedemann-Bach-Platz 6, D-06108 Halle, GERMANY [email protected] Parvis Haggi Institute of Theoretical Physics, Box 6730, S-11385 Stockholm, SWEDEN [email protected] Troels Harmark Niels Bohr Institute, Blegdamsvej 17, DK-2100 Copenhagen, DENMARK [email protected] Carl Herrmann Centre de Phys. Theorique, CNRS - Luminy Case 907, F-13288 Marseille Cedex 9, FRANCE [email protected] Sigbjorn Hervik Dept. of Physics, UiO, Postbox 1048, Blindern, N-0316 Oslo, NORWAY sigb [email protected] Laur Jiirv Centre for Particle Theory, Dept. of Math. Sc., Science Labs., South Road, Durham, DH1 3LE, UK [email protected] Niels Karlsson Univ. of Akureyri, Akureyri, ICELAND [email protected] Justin Khoury Dept. of Physics, Princeton University, Princeton, NJ 08544, USA [email protected] Marcia E. Knutt Physics Dept., McGill University, 3600 University St., Montreal, QC H3A2T8, CANADA [email protected] Kristjan R. Kristjansson Science Institute, University of Iceland, Dunhaga 3, IS-107 Reykjavik, ICELAND kristk@hLis Morten Krogh Dept. of Physics, Princeton University, Princeton, NJ 08544, USA [email protected] Luis Martin Kruczenski Lindsbergsgatan 7B, S-75 240 Uppsala, SWEDEN [email protected] Teresia Mansson FYSIKUM, Box 6730, S-113 85 Stockholm, SWEDEN [email protected] xvii

Dennis Gerard McKeon Dept. of Applied Mathematics, University of Western Ontario, Ont., CANADA N6A 5B7 [email protected] Jeremy Michelson Jefferson Physical Labs., Harvard University, Cambridge, MA 02138, USA [email protected] Stefano Monni Dept. of Applied Mathematics and, Theoretical Physics, University of Cambridge, Silver Street, Cambridge CB3 9EW, UK [email protected] Eric Novac Dept. of Physics, Penn State University, University Park, PA 16802, USA [email protected] Sergei Odintsov Tomsk State Pedagogical University, 634041 Tomsk, RUSSIA [email protected] Aksel Hiorth Orsland Dept. of Physics, Postbox 1048, Blindern, N-0316 Oslo, NORWAY [email protected] Ossi Pasanen Helsinki , P.O. Box 9, FIN-00014 University of Helsinki, FINLAND [email protected] Konstantin Savvidy Dept. of Physics, Princeton University, Princeton, NJ 08544, USA [email protected] Jan Pieter van der Schaar Inst. for Theoretical Physics, Nyenbrogh 4,9747 AG Groningen, THE NETHERLANDS [email protected] Koenraad Schalm Inst. for Theoretical Physics, State University of New York, Stony Brook, NY 11794-3840, USA [email protected] Michael Schulz Dept. of Physics, LeConte Hall, University of California, Berkeley, CA 94720, USA [email protected] Khalil Shadiev 2-a Abdullo Kahhor Street, Samarkand, REPUBLIC OF UZBEKISTAN, 703024 [email protected] xviii Paul Shocklee Dept. of Physics, Princeton University, Princeton, NJ 08544, USA [email protected] Sergey Solodukhin Spinoza Institute, University of Utrecht, Leuvenlaan 4, , 3584 CE Utrecht, THE NETHERLANDS [email protected] Harald G . Svendsen 10 B 655 Kringsja, Sognsveien 218, N-0864 Oslo, NORWAY [email protected] Oyvind Tafjord 507 West Drive, Princeton, NJ 08540, USA [email protected] Gudmar Thorleifsson University of Bielefeld, FB Physik, P.O. Box lO 01 31, D-33501 Bielefeld, GERMANY [email protected] Graziano Vernizzi Physics Department, University of Parma, 1-43lO0 Parma, ITALY [email protected] Anastasia Volovich Dept. of Physics, Harvard University, Oxford Street # 17, Cambridge, MA 02138, USA [email protected] John Wheater Theoretical Physcis, 1 Keble Road, Oxford OX1 3NP, UK [email protected] Maxim Zahzine FYSIKUM, Stockholm University, Box 6730 S-11385 Stockholm, SWEDEN [email protected] Leopoldo Pando Zayas Dept. of Physics, University of Michigan, Ann Arbor, MI 48lO5, USA [email protected]