The se ries is published by an international board of publishers in conjunction with the NATO Scientific Affairs Division

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1 NATO ASI Series Advanced Science Institutes Series Aseries presenting the results of activities sponsored by the NA Ta Science Committee, which aims at the dissemination of advanced scientific and techn%gica/ know/edge, with a view to strengthening links between scientific communities. The se ries is published by an international board of publishers in conjunction with the NATO Scientific Affairs Division A B Life Sciences Physics Plenum Publishing Corporation New York and London C Mathematical and Physical Sciences o Behavioral and Social Sciences E Applied Sciences Kluwer Academic Publishers Dordrecht, Boston, and London F G H I Computer and Systems Sciences Ecological Seien ces Ce" Biology Global Environmental Change Springer-Verlag Berlin, Heidelberg, New York, London, Paris, TOkyo, Hong Kong, and Barcelona Recent Volumes in this Series Vo/ume 261-Z o Physics: Cargese 1990 edited by Maurice LElVY, Jean-Louis Basdevant, Maurice Jacob, David Speiser, Jacques Weyers, and Raymond Gastmans Vo/ume 262-Random Surfaces and Quantum Gravity edited by Orlando Alvarez, Enzo Marinari, and Paul Windey Vo/ume 263-Biologically Inspired Physics edited by L. Peliti Volume 264-Microscopic Aspects of Nonlinearity in Condensed Matter edited by A. R. Bishop, V. L. Pokrovsky, and V. Tognetti Volume 265-Fundamental Aspects of Heterogeneous Catalysis Studied by Particle Beams edited by H. H. Brongersma and R. A. van Santen Vo/ume 266-Diamond and Diamond-Like Films and Coatings edited by Robert E. Clausing, Linda L. Horton, John C. Angus, and Peter Koidl Volume 267-Phase Transitions in Surface Films 2 edited by H. Taub, G. Torzo, H. J. Lauter, and S. C. Fain, Jr. Series B: Physics

2 Random Surfaces and Quantum Gravity

3 Random Surfaces and Quantum Gravity Edited by Orlando Alvarez University of California Berkeley, California Enzo Marinari Universitâ di Roma II "Tor Vergata" Rome,ltaly and Paul Windey Universite Pierre et Marie Curie (Paris VI) Paris, France Springer Science+Business Media, LLC

4 Proceedings of a NATO Advanced Research Workshop on Random Surfaces and Quantum Gravity, held May 27-June 2,1990, in Cargese, France Llbrary of Congress Cataloglng-In-Publlcatlon Data NATO Advanced Research ~orkshop on Random Surfaces and Quantum Gravity (1990 : Cargese, France) Random surfaces and quantum gravity / edlted by Orlando Alvarez, Enzo Marinari, and Paul ~indey. p. cm. -- (NATD ASI series. Series B. Physics ; vol. 262) 'Pub 1 ished in cooperat ion with NATD Sclentlflc Affalrs Divislon." "Proceedings of a NATO Advanced Research ~orkshop on Random Surfaces and Quantum Gravity, held May 27-June 2, In Cargese, France"--T.p. verso. Includes bibllographical references and index. ISBN ISBN (ebook) DOI / Quantum gravity--congresses. 2. String models--congresses. 3. Surfaces (Physics)--Congresses. 1. Alvarez, Drlando II. Marinari. Enzo. III. ~indey, Paul. IV. North Atlantic Treaty Organizatlon. Scientific AffalrsDivision. V. Title. VI. Ser ies, NATO ASI series. Series B, Physics ; v QC178.N '. 14--dc CIP 1991 Springer Science+Business Media New York Originally published by Plenum Press, New York in 1991 Softcover reprint of the hardcover 1 st edition 1991 AII rights reserved No part of this book may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording, or otherwise, without written permission from the Publisher

5 PREFACE The Cargese Workshop Random Surfaces and Quantum Gravity was held from May 27 to June 2, Little was known about string theory in the non-perturbative regime before Oetober 1989 when non-perturbative equations for the string partition functions were found by using methods based on the random triangulations of surfaees. This set of methods provides a deseription of non-eritical string theory or equivalently of the coupling of matter fields to quantum gravity in two dimensions. The Cargese meeting was very successful in that it provided the first opportunity to gather most of the active workers in the field for a fuh week of lectures and extensive informal discussions about these exeiting new developments. The main results were reviewed, recent advances were explained, new results and conjectures (which appear for the first time in these proceedings) were presented and discussed. Among the most important topics discussed at the workshop were: The relation of KdV theory to loop equations and the Virasoro algebra, new results in Liouville field theory, effective (1 + 1) dimensional theory for 2 - D quantum gravity coupled to c = 1 matter and its fermionization, proposal for a new geometrical interpretation of the string equation and possible definition of quantum Riemann surfaces, discussion of the string equation for the multi-matrix models, links with topological field theories of gravity, issues in using target space supersymmetry to define good theories, definition of the partition function via analytic continuation, new models of random surfaces including non-orientable surfaces, proposal of how to distinguish purely stringy effects from ordinary (field theoretie) non-perturbative effects in string theory. The proceedings have been organized in three parts reflecting the main themes which were treated extensively during the meeting. We would like to point out that some of the contributions contain detailed reviews of the field whieh we hope will make this volume a useful tool even for the non specialist. The success of the meeting was made possible thanks to the crueial help of many people to all of whom we wish to express our gratitude. Firstly we are greatly indebted to the Nato Division for Scientific Affairs and particularly to Prof. Venturi for providing us a generous grant on very short notice and for constant help in organizing the meeting. v

6 We are grateful to Marie-France Hanseler for her unique organizational talent and for her smiling efforts at ironing out all last minute difficulties. Without her help the meeting would have been impossible to put together. We received the precious help of Liu Catena in editing the proceedings: without her constant attention this volume would be very different (and it would have been significantly delayed). Last but not least, enthusiastic speakers and participants were the key ingredient for the success of the meeting: we warmly thank all of them and especially those who did not hesitate to destroy a solid reputation for writer's block by submitting their contribution ahead of the most optimistic deadline. Orlando Alvarez Enzo Marinari Paul Windey VI

7 CONTENTS 2D GRAVITY AND NON PERTURBATIVE EFFECTS Diffusion Equation, Continuum Limit and Universality in 2D Quantum Gravity Orlando Alvarez and Paul Windey On Triangles and Squares C. Bachas Non-Perturbative Effects in 2D Gravity and Matrix Models Fran<;ois David Integrable Models of 2D Quantum Gravity P. Di Francesco and D. Kutasov Topological Strings and Loop Equations Robbert Dijkgraaf and Herman Verlinde The Two-Matrix Model Michael R. Douglas Action Principle and Large Order Behavior of Non-Perturbative Gravity P. Ginsparg and J. Zinn-Justin W-Geometry C. Itzykson Strings Embedded in Dynkin Diagrams I. K. Kostov Phase Transitions in One-Matrix Models Gautam Mandal Matrix Models of 2D Gravity and Isomonodromic Deformation Gregory Moore The Strength of Nonperturbative Effects in String Theory Stephen H. Shenker Approximating the Critical String Measure Using Dynamically Triangulated Surfaces Dirk-Jan Smit 201 VII

8 D = 1 STRINGS AND STRING FIELD THEORY Matrix Models, String Field Theory and Topology Tom Banks Multipoint Correlation Functions in One-Dimensional String Theory Dmitri Boulatov The Penner Model and D = 1 String Theory Jacques Distier and Cumrun Vafa Non-Perturbative String Theory David J. Gross Bosonic Strings and String Field Theories in One-Dimensional Target Space 269 Vladimir Kazakov A Few Remarks on Low Dimensional String Theory 307 Giorgio Parisi Excitations and Interactions in a One-Dimensional String Theory Spenta R. Wadia 317 LIOUVILLE THEORY AND D > 1 Random Surfaces in Dimensions Larger Than One Jan Ambj!2Srn On Gauge Invariances in Stochastic Quantization Laurent Baulieu The Quantum Group Structure of Quantum Gravity in Two Dimensions 347 Jean-Loup Gervais Notes on Quantum Liouville Theory and Quantum Gravity N athan Seiberg 363 INDEX 397 viii