Selected Titles in This Series

58 Pavel I. Etingof, Igor B. Frenkel, and Alexander A. Kirillov, Jr., Lectures on representation theory and Knizhnik-Zamolodchikov equations, 1998

57 Marc Levine, Mixed motives, 1998 56 Leonid I. Korogodski and Yan S. Soibelman, Algebras of functions on quantum

groups: Part I, 1998 55 J. Scott Carter and Masahico Saito, Knotted surfaces and their diagrams, 1998 54 Casper Goffman, Togo Nishiura, and Daniel Waterman, Homeomorphisms in

analysis, 1997 53 Andreas Kriegl and Peter W . Michor, The convenient setting of global analysis, 1997 52 V . A. Kozlov, V. G. Maz'ya, and J. Rossmann, Elliptic boundary value problems in

domains with point singularities, 1997 51 Jan Maly and Wil l iam P. Ziemer, Fine regularity of solutions of elliptic partial

differential equations, 1997 50 Jon Aaronson, An introduction to infinite ergodic theory, 1997 49 R. E. Showalter, Monotone operators in Banach space and nonlinear partial differential

equations, 1997 48 Paul-Jean Cahen and Jean-Luc Chabert , Integer-valued polynomials, 1997 47 A. D . Elmendorf, I. Kriz, M. A. Mandell , and J. P. May (with an appendix by

M. Cole) , Rings, modules, and algebras in stable homotopy theory, 1997 46 S tephen Lipscomb, Symmetric inverse semigroups, 1996 45 George M. Bergman and A d a m O. Hausknecht , Cogroups and co-rings in

categories of associative rings, 1996 44 J. Amoros , M. Burger, K. Corlette , D . Kotschick, and D . Toledo, Fundamental

groups of compact Kahler manifolds, 1996 43 James E. Humphreys , Conjugacy classes in semisimple algebraic groups, 1995 42 Ralph Freese, Jaroslav Jezek, and J. B. Nat ion , Free lattices, 1995 41 Hal L. Smith , Monotone dynamical systems: an introduction to the theory of

competitive and cooperative systems, 1995 40.3 Daniel Gorenste in , Richard Lyons, and Ronald Solomon, The classification of the

finite simple groups, number 3, 1998 40.2 Daniel Gorenstein, Richard Lyons, and Ronald Solomon, The classification of the

finite simple groups, number 2, 1995 40.1 Daniel Gorenstein, Richard Lyons, and Ronald Solomon, The classification of the

finite simple groups, number 1, 1994 39 Sigurdur Helgason, Geometric analysis on symmetric spaces, 1994 38 Guy David and Stephen Semmes , Analysis of and on uniformly rectifiable sets, 1993 37 Leonard Lewin, Editor, Structural properties of polylogarithms, 1991 36 John B. Conway, The theory of subnormal operators, 1991 35 Shreeram S. Abhyankar, Algebraic geometry for scientists and engineers, 1990 34 Victor Isakov, Inverse source problems, 1990 33 Vladimir G. Berkovich, Spectral theory and analytic geometry over non-Archimedean

fields, 1990 32 Howard Jacobowitz , An introduction to CR structures, 1990 31 Paul J. Sally, Jr. and David A. Vogan, Jr., Editors, Representation theory and

harmonic analysis on semisimple Lie groups, 1989 30 T h o m a s W . Cusick and Mary E. Flahive, The Markoff and Lagrange spectra, 1989 29 Alan L. T. Paterson, Amenability, 1988 28 Richard Beals , Percy Deift , and Carlos Tomei, Direct and inverse scattering on the

line, 1988 (Continued in the back of this publication)

http://dx.doi.org/10.1090/surv/058

Lectures on Representation Theory and Knizhnik-Zamolodchikov Equations

Mathematical Surveys

and Monographs

Volume 58

Lectures on Representation Theory and Knizhnik-Zamolodchikov Equations

Pavel I. Etingof Igor B. Frenkel Alexander A. Kirillov, Jr.

American Mathematical Society

Editorial Board Georgia M. Benkart Tudor Stefan Ratiu, Chair

Michael Renardy

The authors were supported in part by the following NSF grants: RE.: DMS#9700477, I.F.: DMS#9700765, A.K.: DMS#9610201.

P.E. was also supported in part by an NSF postdoctoral fellowship.

1991 Mathematics Subject Classification. Primary 81R40, 81R50; Secondary 17B67, 17B69.

ABSTRACT. This book is devoted to the study of some of the mathematical structures arising in conformal field theory and their q-deformations. This field, though relatively young, is an area of intensive study by both mathematicians and physicists, and has already produced many beautiful results in mathematics and physics. In the book, we have tried to give a self-contained exposition of the theory of Knizhnik-Zamolodchikov equations and related topics that requires no previous knowledge of physics. The book would be useful to everyone interested in mathematical physics, from graduate students to experts. It can be used as a basis for a one-semester graduate course.

Library of Congress Cataloging-in-Publicat ion D a t a Etingof, P. I. (Pavel I.), 1969-

Lectures on representation theory and Knizhnik-Zamolodchikov equations / Pavel I. Etingof, Igor B. Frenkel, Alexander A. Kirillov, Jr.

p. cm. — (Mathematical surveys and monographs, ISSN 0076-5376 ; v. 58) Includes bibliographical references and index. ISBN 0-8218-0496-0 (hardcover : alk. paper) 1. Broken symmetry (Physics) 2. Quantum groups. 3. Kac-Moody algebras. 4. Mathe­

matical physics. I. Frenkel, Igor. II. Kirillov, Alexander A., 1967- . III. Title. IV. Series: Mathematical surveys and monographs ; no. 58. QC174.17.S9E88 1998 530.14/2—dc21 98-2948

CIP

Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given.

Republication, systematic copying, or multiple reproduction of any material in this publication (including abstracts) is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Assistant to the Publisher, American Mathematical Society, P. O. Box 6248, Providence, Rhode Island 02940-6248. Requests can also be made by e-mail to reprint-permission@ams.org.

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10 9 8 7 6 5 4 3 2 1 03 02 01 00 99 98

To our wives Tanya, Marina, and Varya

Contents

Preface xiii

Lecture 1. Introduction 1 1.1. Simple Lie algebras and Lie groups and their generalizations 1 1.2. Affine Lie algebras 1 1.3. Quantum groups 3 1.4. Knizhnik-Zamolodchikov equations 6 1.5. Quantum affine algebras and quantum Knizhnik-Zamolodchikov

equations 8 1.6. Further generalizations of affine Lie algebras and quantum groups. 11 1.7. Contents of the book 12

Lecture 2. Representations of finite-dimensional and affine Lie algebras 15 2.1. Simple Lie algebras 15 2.2. Cartan matrices of simple Lie algebras 16 2.3. Highest-weight modules over simple Lie algebras and contravariant

forms 17 2.4. Finite-dimensional representations and irreducibility of Verma mod­

ules 18 2.5. The maximal root, the Coxeter numbers, and the Casimir operator 19 2.6. Affine Lie algebras 20 2.7. Verma modules and Weyl modules for affine Lie algebras 22 2.8. Integrable representations of affine Lie algebras 24 2.9. The Virasoro algebra and its action on ^-modules 25

2.10. Generating functions and currents 26

Lecture 3. Knizhnik-Zamolodchikov equations 29 3.1. Classification of intertwining operators 29 3.2. Operator KZ equation 30 3.3. Gauge invariance of the intertwining operators 33 3.4. KZ equations for correlation functions 33 3.5. Consistency and g-invariance of the KZ equations 36 3.6. Analyticity of the correlation functions 37 3.7. Correlation functions span the space of solutions of the KZ equa­

tions 39 3.8. Trigonometric form of the KZ equations 42 3.9. Consistent systems of differential equations and the classical Yang-

Baxter equation 44

x CONTENTS

Lecture 4. Solutions of the Knizhnik-Zamolodchikov equations 49 4.1. The simplest solution of the KZ equations for Q = s^ 49 4.2. Simplest level one solution and Gauss hypergeometric function 51 4.3. Integral formulas for level one solutions 53 4.4. Solutions of the KZ equations for $[2: arbitrary level 56 4.5. Solutions of the KZ equations for a general simple Lie algebra 60

Lecture 5. Free field realization 63 5.1. Fock modules and vertex operators 63 5.2. Matrix elements of products of vertex operators 66 5.3. Interpretation of the rational part of solutions of the KZ equations

in terms of creation and annihilation operators. 67 5.4. Factorization of solutions of the KZ equations 69 5.5. Free field realization of Verma modules over SI2 69 5.6. Intertwining operators in the free field realization: level zero 73 5.7. Intertwining operators in the free field realization: positive level 75 5.8. Calculation of the correlation functions 77

Lecture 6. Quantum groups 79 6.1. Hopf algebras and their representations 79 6.2. Definition of quantum groups 81 6.3. Quasitriangular structure and braided tensor categories 84 6.4. Quantum Yang-Baxter equation and representations of braid

groups 87 6.5. Quantum double construction 88 6.6. Quantum double construction for Uq(ge) 89 6.7. Quantum Casimir element 92 6.8. Intertwining operators and their commutation relations 93

Lecture 7. Local systems and configuration spaces 97 7.1. Local systems 97 7.2. Cohomology and homology with coefficients in local systems 99 7.3. Configuration spaces and Orlik-Solomon algebra 101 7.4. Cohomology of configuration spaces with coefficients in local sys­

tems associated with the KZ equations for 5X2 103 7.5. Gauss-Manin connection 105 7.6. Relative homology 106 7.7. The case of arbitrary 9 110

Lecture 8. Monodromy of Knizhnik-Zamolodchikov equations 113 8.1. Monodromy of KZ equations and the braid group 113 8.2. Asymptotics of solutions of the KZ equations 115 8.3. Asymptotics of the correlation functions 118 8.4. Monodromy with respect to an infinite base point 119 8.5. Commutation relations for intertwining operators 122 8.6. Equivalence of categories and Drinfeld-Kohno theorem 124 8.7. Geometric approach to equivalence of categories 127

Lecture 9. Quantum affine algebras 131 9.1. Definition of quantum affine algebras 131 9.2. Evaluation representations of quantum affine algebras 132

CONTENTS xi

9.3. Intertwining operators 135 9.4. Quasitriangular structure in quantum affine algebras 136 9.5. Factorization of the i^-matrix 137 9.6. Evaluation representations and i?-matrix for Uq(sl2) 139 9.7. Quantum currents 144 9.8. Quantum Sugawara construction in degree zero 145

Lecture 10. Quantum Knizhnik-Zamolodchikov equations 147 10.1. Operator quantum KZ equation 147 10.2. Quantum correlation functions 150 10.3. Quantum KZ equations for correlation functions 151 10.4. A fundamental set of solutions of the quantum KZ equations 151 10.5. Holonomic systems of difference equations 152 10.6. Analyticity of the fundamental solution of the quantum KZ equa­

tions 153 10.7. The noncommutative product formula for the fundamental solution 155 10.8. Classical limit of the quantum KZ equations 155 10.9. Modified quantum KZ equations 156 10.10 Another proof of the quantum KZ equations 157

Lecture 11. Solutions of the quantum Knizhnik-Zamolodchikov equations for sl2 161

11.1. g-analogues of classical special functions 161 11.2. Jackson integral 163 11.3. The q- hyper geometric function 164 11.4. Some second order difference equations 165 11.5. The simplest solutions of the quantum KZ equations and the q-

hypergeometric function 166 11.6. Integral formulas for solutions 168

Lecture 12. Connection matrices for the quantum Knizhnik-Zamolodchikov equations and elliptic functions 171

12.1. Linear difference equations for functions of one complex variable 171 12.2. Connection relation for the g-hypergeometric equation 174 12.3. The connection matrix for the quantum KZ equations in the sim­

plest case 175 12.4. The connection matrix and the exchange matrix for intertwining

operators 176

Lecture 13. Current developments and future perspectives 179 13.1. KZ equations: quantum versus classical 179 13.2. Monodromy of the KZ equations, tensor categories, and quantum

groups 180 13.3. Vertex operator algebras, conformal field theory and their g-defor-

mations 182 13.4. Elliptic KZ equations and special values of the central charge 185 13.5. Double loop algebras and quantum affine algebras 186 13.6. Quantum KZ equations and physical models 187

References 189

Index 197

Preface

The last twenty years have seen an especially active interaction between mathe­matics and physics. This interaction has given birth to a number of remarkable new areas of mathematics and has provided powerful new tools in various fields of the­oretical physics. This book is devoted to one of those new areas, which deals with mathematical structures of conformal field theory and their g-deformations. It arose from the course of lectures on the classical and quantum Knizhnik-Zamolodchikov equations, given by one of the authors of this book (I. B. F.) in the Spring of 1992 at Yale University, and inspired by his recent (at that time) joint paper with Nicolai Reshetikhin. The instructor was lucky to find two enthusiastic graduate students and future co-authors, who improved and extended the exposition and later gave related courses at Harvard (P. I. E.) and MIT (A. A. K.).

By that time, all three of us had already been severely afflicted with the "q-disease", a dangerous mathematical illness whose earliest victim was Euler, but which was first diagnosed by Richard Askey. Mathematicians working in practically every field, be it algebra, geometry, analysis, differential equations - you name it - are vulnerable to its addictive charm. The first symptom of the g-disease is that one day you realize that most of the results obtained or acquired during your mathematical life admit a g-deformation. The second stage is indicated by the idea that the g-case is much more interesting than the classical one. It was at that stage that we started writing the second part of the book, with the intention to g-deform all structures of conformal field theory. This turned out to be a difficult task, which has taken five years, and is still not completed. Luckily, during these years the area grew up quite significantly, and we were able to use some of the more recent results and refer the reader to active new developments and promising research problems.

When writing this book, we followed one of the rules we learned from Israel Gelfand: for every new theory, choose the simplest non-trivial example and write down everything explicitly for this example. Therefore, all general constructions and theorems are accompanied by explicit calculations for the Lie algebra 0(2-

We also wanted to put this new area of mathematics into the general perspec­tive of development of representation theory. With this intention we wrote the first and the last lecture, trying to help the reader navigate in the mist of modern repre­sentation theory and mathematical physics. The bare scheme of the development of the theory is captured by the diagram at the end of Section 1.6 in the Introduction. Familiar to many mathematicians working in the area, it is closer to alchemical formulas than to mathematics. Nevertheless, it was instrumental in the discovery of some of the structures studied in this book, and it might still be useful again.

This book is written for people who are familiar with the representation theory of simple Lie algebras, but requires no knowledge of physics. It can be used for teaching an advanced one-semester graduate course, though the instructor and the

xiii

xiv P R E F A C E

students should work really hard, as it was in the Spring of 1992 at Yale. The number of lectures exactly corresponds to the number of weeks in one semester at Yale University. Each lecture is assumed to take two and a half hours and can be split into two or three weekly classes depending on the strength and enthusiasm of the participants. Our experience shows that the golden mean is always the best solution. Also the first and the last lectures should not be taken as seriously (for some people as lightly) as the rest of the book, giving the students and the instructor an opportunity to relax.

This theory was created by the efforts of many people, and some results were circulating as "folklore" for a number of years. We tried to give the references in the first lecture and especially in the introduction to each of the consecutive lectures, but we would like to apologize in advance for unintentional omissions.

One of the main purposes of this book was to simplify, extend and provide all the necessary details of the results in the original article [FR]. We hope that in our present exposition we have corrected more misprints and inaccuracies in that paper than we have added new ones.

The credit for the eventual existence of this book rightfully belongs to our editor, Sergei Gelfand, who achieved a seemingly impossible goal of persuading the authors - after a five year delay - to bring this book to a conclusion.

We are grateful for fruitful discussions and useful comments to many people, including M. Finkelberg, D. Kazhdan, A. Matsuo, N. Reshetikhin, O. Schiffman, K. Styrkas, A. Varchenko. We thank the National Science Foundation for par­tial support of this project, and the American Mathematical Society for its final materialization.

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Index

6.7-symbols, 95 /37-system, 67

quantum, 180 g-hypergeometric function, 164

Asymptotic solution for a difference equation, 173 of KZ equations, 117

Asymptotic zone, 117

Braid group, 87 Braided tensor category, 86

Cartan matrix, 16 generalized, 21 symmetrizable, 22, 82

Casimir element, 19 quantum, 92

Central charge, 23 Classical r-matrix, 45 Cohomology with coefficients in a local sys­

tem, 99 Conformal blocks, 183 Connection, 97

flat, 98 Connection matrices

for difference equations, 173 for KZ equations, 119

Correlation function, 34 for quantum affine algebras, 150

Coxeter number, 19 dual, 19

Drinfeld associator, 125 Drinfeld category, 124 Drinfeld-Kohno theorem, 126

Elliptic quantum group, 181

Evaluation homomorphisms, 132 Evaluation representation

of affine Lie algebras, 25 of quantum affine algebras, 134

Exchange matrix for affine Lie algebras, 122 for quantum affine algebras, 177 for quantum groups, 94

Fock module, 63, 68 Free field realization

of intertwining operators, 73-77 of Verma modules, 70

Gauss-Manin connection, 105

Heisenberg algebra, 63, 68 Hexagon axiom, 86 Highest-weight vector, 17 Holonomic system of difference equations,

153 Holonomy, 98 Homology

relative, 107 singular, 99

Hopf algebra, 79 cocommutative, 81 quasitriangular, 85

Hypergeometric function, 53 Hyper plane arrangement, 101

Jackson integral, 163

Knizhnik-Zamolodchikov equations, 35 elliptic, 185 operator, 32 quantum, 151

modified, 156

198 INDEX

operator, 149 trigonometric, 44

Knizhnik-Zamolodchikov-Bernard equation, 185

Level, 23 of a solution of KZ equations, 49 critical, 23 generic, 24

Lie algebras affine, 20 Kac-Moody, 21

extended, 22 simple finite-dimensional, 15 simply-laced, 16

Local system, 98

Monodromy, 114

Normal ordering, 64

Orlik-Solomon algebra, 101

Pentagon axiom, 86 Pochhammer loop, 53

Quantum affine algebra, 131 Quantum double, 89 Quantum groups, 83 Quasimeromorphic function, 172

Screening operators, 75 Shapovalov form, 17 Singular vectors, 19 Star-triangle relation, 94 Sugawara construcion, 26

Universal /^-matrix, 85 for Wq(0c), 90 for quantum affine algebras, 136

Vacuum vector, 64 Verma module, 17

contragredient, 18 for affine Lie algebras, 22

contragredient, 24 for quantum groups, 83

Vertex operator, 64 Virasoro algebra, 25

Wakimoto module, 70 Weight decomposition, 17

for quantum groups, 83 Weight subspace, 17 Weyl module, 23

Yang-Baxter equation classical, 45 dynamical, 96 quantum, 87

with a spectral parameter, 137

Selected Titles in This Series (Continued from the front of this publication)

27 N a t h a n J. Fine, Basic hypergeometric series and applications, 1988 26 Hari Bercovici , Operator theory and arithmetic in H°°, 1988 25 Jack K. Hale, Asymptotic behavior of dissipative systems, 1988 24 Lance W . Small, Editor, Noetherian rings and their applications, 1987 23 E. H. Rothe , Introduction to various aspects of degree theory in Banach spaces, 1986 22 Michael E. Taylor, Noncommutative harmonic analysis, 1986 21 Albert Baernste in , David Drasin, Peter Duren , and Albert Marden, Editors,

The Bieberbach conjecture: Proceedings of the symposium on the occasion of the proof, 1986

20 K e n n e t h R. Goodearl , Partially ordered abelian groups with interpolation, 1986 19 Gregory V. Chudnovsky, Contributions to the theory of transcendental numbers, 1984 18 Frank B. Knight , Essentials of Brownian motion and diffusion, 1981 17 Le Baron O. Ferguson, Approximation by polynomials with integral coefficients, 1980 16 O. T imothy O'Meara, Symplectic groups, 1978 15 J. Dieste l and J. J. Uhl , Jr. , Vector measures, 1977 14 V. Guil lemin and S. Sternberg, Geometric asymptotics, 1977 13 C. Pearcy, Editor, Topics in operator theory, 1974 12 J. R. Isbell, Uniform spaces, 1964 11 J. Cronin, Fixed points and topological degree in nonlinear analysis, 1964 10 R. Ayoub , An introduction to the analytic theory of numbers, 1963 9 Arthur Sard, Linear approximation, 1963 8 J. Lehner, Discontinuous groups and automorphic functions, 1964

7.2 A. H. Clifford and G. B . Pres ton , The algebraic theory of semigroups, Volume II, 1961 7.1 A. H. Clifford and G. B . Pres ton , The algebraic theory of semigroups, Volume I, 1961

6 C. C. Chevalley, Introduction to the theory of algebraic functions of one variable, 1951 5 S. Bergman, The kernel function and conformal mapping, 1950 4 O. F. G. Schilling, The theory of valuations, 1950 3 M. Marden, Geometry of polynomials, 1949 2 N . Jacobson, The theory of rings, 1943 1 J. A. Shohat and J. D . Tamarkin, The problem of moments, 1943

(See the AMS catalog for earlier titles)