Mathematical Sciences Research Institute Publications 3

Editors

S.S. Chern I. Kaplansky C.C. Moore I.M. Singer

Mathematical Sciences Research Institute Publications

Volume 1

Volume 2

Volume 3

Forthcoming

D. Freed and K. Uhlenbeck: Instantons and Four­Manifolds

S.S. Chern (ed.): Seminar on Nonlinear Partial Dif­ferential Equations

J. Lepowsky, S. Mandelstam, and I.M. Singer (eds.): Vertex Operators in Mathematics and Physics

S.S. Chern and P. Griffiths: Essays on Exterior Dif­ferential Systems

V. Kac (ed.): Infinite Dimensional Lie Groups

C.C. Moore (ed.): Group Representations, Ergodic Theory, Operator Algebras, and Mathematical Physics: Proceedings of a Conference in Honor of G.W. Mackey

Vertex Operators in Mathematics and Physics

Proceedings of a Conference November 10-17, 1983

Edited by J. Lepowsky . S. Mandelstam . I.M. Singer

With 37 Illustrations

Springer-Verlag N ew York Berlin Heidelberg Tokyo

J. Lepowsky Department of Mathematics Rutgers University New Brunswick, NJ 08903 U.S.A.

I.M. Singer Department of Mathematics University of California Berkeley, CA 94720 U.S.A.

Mathematical Sciences Research Institute 2223 Fulton Street, Room 603 Berkeley, CA 94720 U.S.A.

S. Mandelstam Department of Physics University of California Berkeley, CA 94720 U.S.A.

AMS Classification: 17-02,20-02,81-02, 05A19, 17B65, 20008, 35Q15, 35Q20, 58F07, 81E99, 81G99, 82A15, 83E50

Library of Congress Cataloging in Publication Data Main entry under title: Vertex operators in mathematics and physics.

(Mathematical Sciences Research Institute publications; 3) Includes proceedings from the Conference on Vertex Operators in Mathematics and Physics, held

at the Mathematical Sciences Research Institute, Nov. 10-17, 1983. Bibliography: p.

1. Nonassociative algebras-Congresses. 2. Groups, Theory of-Congresses. 3. Lie algebras­Congresses. 4. Quantum field theory-Congresses. I. Lepowsky, J. II. Mandelstarn, Stanley. III. Singer, I. M. (Isadore Manuel). IV. Conference on Vertex Operators in Mathematics and Physics (1983: Mathematical Sciences Research Institute) V. Mathematical Sciences Research Institute (Berkeley, Calif.) VI. Series. QA252.V47 1985 512'.24 84-26828

The camera-ready manuscript of this book was received by Springer-Verlag in October, 1984.

The Mathematical Sciences Research Institute wishes to acknowledge support from the National Science Foundation.

©1985 by Springer-Verlag New York Inc. Softcover reprint of the hardcover 1 st edition 1985

All rights reserved. No part of this book may be translated or reproduced in any form without written pennission from Springer-Verlag, 175 Fifth Avenue, New York, New York 10010, U.S.A.

Pennission to photocopy for internal or personal use, or the internal or personal use of specific clients, is granted by Springer-Verlag, New York Inc. for libraries and other users registered with the Copyright Clearance Center (CCC), provided that the base fee of $0.00 per copy, plus $0.20 per page is paid directly to CCC, 21 Congress Street, Salem, MA 01970, USA. Special requests should be addressed directly to Springer-Verlag, New York, 175 Fifth Avenue, New York, NY 10010, USA.

ISBN-13: 978-1-4613-9552-2 e-ISBN-13: 978-1-4613-9550-8 DOl: 10.1 007/978-1-4613-9550-8

9 8 7 654 3 2 1

PRBFACB

In January. 1972. F. Dyson delivered the J. W. Gibbs Lecture

at the Annual Meeting of the American Mathematical Society. 1 Called

"Missed Opportunities. " it was an inspiring encouragement to

mathematicians and physicists to communicate with one another. In

the talk. Dyson described several examples of mathematical discoveries

which were delayed because mathematicians were not paying sufficient

attention when "two disparate or incompatible mathematical concepts

were juxtaposed in the description of a single situation." In such a

case. he said. mathematicians should try to "create a wider conceptual

framework within which the pair of disparate elements would find a

harmonious coexistence."

As an example, Dyson mentioned his and I. G. Macdonald's

independent work on identities for certain powers of Dedekind's

eta-function. He regarded this episode as a personal "missed

opportunity" for two reasons -- his failure to notice the connection

with finite-dimensional simple Lie algebras "just because the number

theorist Dyson and the physicist Dyson were not speaking to each

other" and his failure to discuss his eta-function work with Macdonald,

because it never occurred to him that he might be studying the same

problem as a mathematician. He called this "a trivial episode from my

own experience, which illustrates vividly how the habit of

specialization can cause us to miss opportunities." asserting that even

in his undergraduate days at Cambridge "it was clear ... that number

theory in the style of Hardy and Ramanujan was old-fashioned and did

not have a great and glorious future ahead of it." He concluded by

suggesting that Macdonald had explained the juxtaposition of the "two

disparate concepts" of Lie algebras and modular functions and that the

subject was all but finished.

1Bulletin of the American Mathematical Society 78 (1972), 635-652.

Vertex Operators in Mathematics and Physics - Proceedings of a Conference November 10-17. 1983. Publications of the Mathematical Sciences Research Institute 113. Springer-Verlag. 1984.

v

But as we now know, the subject had hardly started. In fact,

the eta-function work of Dyson and Macdonald triggered an explosion

of ideas which now blend the "disparate concepts" of

infinite-dimensional Lie algebras, modular functions, the "dual-string"

theory in particle physics, two-dimensional quantum field theory, the

Rogers-Ramanwan identities, soliton theory and Monstrous Moonshine

into a rapidly developing theory.

In the last few years, interesting connections have been

discovered between the affine Kac-Moody Lie algebras and the

dual-string theory, through the use of vertex operators. In an effort

to explore these connections, a conference entitled "Vertex Operators

in Mathematics and Physics" was held November 10-17, 1983 as part

of the 1983-84 program on Kac-Moody algebras at the Mathematical

Sciences Research Institute.

The present time is especially appropriate for such a

conference, since the great current interest in Kac-Moody algebras

coincides with a renewed interest in string models and their relation

with supergravity. The interaction between the physical and

mathematical theories has already had fruitful applications in

mathematics: one may hope for further applications in both

mathematics and physics. Some recently discovered symmetries of

certain supergravity models appear to bear a striking resemblance to

the affine Kac-Moody algebra related to the vertex function of the

associated string model. This may indicate the presence of a hitherto

undiscovered symmetry or broken symmetry of the string model; such a

symmetry could be of crucial importance for the physicist.

Many two-dimensional models in quantum field theory or

statistical mechanics are exactly soluble. It has been found that most

if not all such models possess Backlund transformations and, more

recently, it has been shown that at least some of these models have

an affine Kac-Moody algebra, possibly without central extension, as a

symmetry algebra. Possible extensions to four dimensions have been

proposed. This is obviously another area where the interplay between

work in mathematics and physics may well be fruitful.

The present volume contains the proceedings of the conference.

including some papers by authors who were invited but were unable to

vi

attend. We have provided an introduction to help the reader with

terminology, notation and historical perspective.

There remain many mysteries to be explained. We hope this

volume will introduce the novice to the subject of thp, conference and

stimulate the expert toward deeper investigations.

We thank the Mathematical Sciences Research Institute for

inviting us to organize this conference and for providing us with a

cheerful and expert staff to run the conference and prepare this

volume.

Berkeley, July, 1984

vii

J. Lepowsky

S. Mandelstam

I. M. Singer

CONFERENCE ON VERTEX OPERATORS IN MATHEMATICS AND PHYSICS

NOVEMBER 10-17. 1983 MATHEMATICAL SCIENCES RESEARCH INSTITUTE

SCHEDULE OF' TALKS

Thursday. November 10

Vertex operators. 2'-algebras and the Fischer-Griess Monster James Lepowsky

Introduction to string models and vertex functions Stanley Mandelstam

Algebras. lattices and strings Peter Goddard

Sporadic groups and nonassociative algebras Robert L. Griess. Jr.

friday. November 11

Structure of the standard Ail) -modules (principal picture) Robert Lee Wilson

Structure of the standard Ail) -modules (homogeneous picture) Mirko Primc

Algebras. lattices and strings. II David Olive

A natural module for the Fischer-Griess Monster with the modular function J as character. I

Igor B. Frenkel

A natural module for the Fischer-Griess Monster with the modular function J as character. II

Arne Meurman

Saturday. November 12

Vertex operators and standard modules for some affine Lie algebras Kailash C. Misra

Solitons and infinite-dimensional Lie algebras Michio Jimbo

ix

On a duality of branching rules Tetsuji Miwa

Generalization of the Riemann-Hilbert problem and the KP, TL hierarchy

Kimio Ueno

Mondav. November 14

Vacuum vector representations of the Virasoro algebra Alvany Rocha-Caridi

Another viewpoint for studying instantons Howard Garland

Local charge algebra of quantum gauge field theory Itzhak Bars

Matrix coefficients of the wedge representation Dale H. Peterson

Supergeometry and Kac-Moody algebras Bernard Julia

Tuesday. November 15

Some applications of vertex operators to Kac-Moody algebras Alex J. Feingold

Massive Kaluza-Klein theories and bound states in Yang-Mills Louise Dolan

Towards a catalog of two-dimensional conformal field theories using the representation theory of the Virasoro algebra

Daniel Friedan

Integrability and hidden symmetries Bernard Julia

Infinite dimensional formal Lie groups - A bridge connecting the linear world and the non-linear world

Motohico Mulase

Wednesday. November 16

Lax pairs, the Riemann-Hilbert transform and Kac-Moody algebras Yong-Shi Wu

x

Bound state spectra in extended supergravity theories and infinite-dimensional super algebras

Mary K. Gaillard

An introduction to Polyakov's string model Orlando Alvarez

Conformally invariant field theories in two dimensions Thomas Curtright

Super Yang-Mills fields as integrable systems, and connections with other systems

Ling-Lie Chau

xi

TABLE OF CONTENTS

Preface

Schedule of talks

Introduction James Lepowsky

S t ri n g mo del s

Introduction to string models and vertex operators Stanley Mandelstam

An introduction to Polyakov's string model Orlando Alvarez

Conformally invariant field theories in two dimensions Thomas Curtright

Lie algebra representations

v

ix

1

15

37

49

Algebras, lattices and strings 51 Peter Goddard and David Olive

X-algebras and the Rogers-Ramanujan identities 97 James Lepowsky and Robert Lee Wilson

S'fu)ture of the standard modules for the affine Lie algebra 143 All in the homogeneous picture

James Lepowsky and Mirko Primc

Standard representations of some affine Lie algebras 163 Kailash C. Misra

Some applications of vertex operators to Kac-Moody algebras 185 Alex J. Feingold

On a duality of branching coefficients 207 Michio Jimbo and Tetsuji Miwa

The Monster

A brief introduction to the finite simple groups Robert L. Griess. Jr.

A Moonshine Module for the Monster Igor B. Frenkel. James Lepowsky and Arne Meurman

xiii

217

231

TABLE OF CONTENTS (CONT'D)

Integrable svstems

Monodromy, solitons and infinite dimensional Lie algebras 275 Michio Jimbo and Tetsuji Miwa

The Riemann-Hilbert decomposition and the KP hierarchy 291 Kimio Ueno

Supersymmetric Yang-Mills fields as an integrable system and 303 connections with other non-linear systems

Ling-Lie Chau

Lax pairs. Riemann-Hilbert transforms and affine algebras for 329 hidden symmetries in certain nonlinear field theories

Yong-Shi Wu and Mo-Lin Ge

Massive Kaluza-Klein theories and bound states in Yang-Mills 353 Louise Dolan

Local charge algebras in quantum chiral models and gauge 373 theories

Itzhak Bars

Supergeometry and Kac-Moody algebras 393 Bernard Julia

The Virasoro algebra

A proof of the no-ghost theorem using the Kac determinant 411 Charles B. Thorn

Conformal invariance. unitarity and two dimensional critical 419 exponents

Daniel Friedan. Zongan Qiu and Stephen Shenker

Vacuum vector representations of the Virasoro algebra Alvany Rocha-Caridi

Classical invariant theory and the Virasoro algebra Nolan R. Wallach

xiv

451

475