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Home , Lie superalgebra, Operator algebra, Victor Kac

Introduction to Vertex and Their Modules and Its Applications

Managing Editor:

M. HAZEWINKEL Centre for Mathematics andComputer Science, Amsterdam, The Netherlands

Volume 456 Introduction to Vertex Operator Superalgebras and Their Modules

by

XiaopingXu Departement 0/Mathematics, The Hong Kong University 0/Science and Technology, Clear Water Bay, Kowloon Hong Kong , P.R. China

Springer-Science+Business Media, B.V. A C.I.P. Catalogue record for this book is available from the Library of Congress.

ISBN 978-90-481-5096-0 ISBN 978-94-015-9097-6 (eBook) DOI 10.1007/978-94-015-9097-6

Printedon acid-free paper

All Rights Reserved @1998 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 1998. Softcover reprint ofthe hardcover 1st edition 1998 No part ofthe 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 To My Wife Fan Wu Contents

Preface ix

Introduction xi Notational Conventions ...... xvi

I Self-Dual Lattices and Codes 1

1 Self-Dual Codes 3 1.1 Basic Definitions and Properties . 3 1.2 Gluing Technique . . . . . 10 1.2.1 Code Gluing Shells . . . . 10 1.2.2 Gluing Procedure . . . . . 13 1.2.3 Structure of Glue Codes . 16 1.3 Some Constructional Theorems 19 1.4 Four Constructions of Binary Self-Dual Codes 24 1.5 Constructions of Ternary Self-Dual Codes 28 1.6 Self-Dual Codes over Z4 . 32

2 Self-Dual Lattices 37 2.1 Definitions and Background . 37 2.2 Vntwisted Gluing Technique of Self-Dual Lattices 43 2.3 Twisted Gluing Technique of Self-Dual Lattices 47 2.4 V-Shellsof Type A ...... 50 2.5 T-Shells of Type A . 57 2.6 V-Shells and T-Shells of Type D . 72

11 Vertex Operator Superalgebras and Their Modules 83

3 Definitions and General Properties 85 3.1 Calculus of Formal Variables . . . 86

vii viii CONTENTS

3.2 Vertex Operator Superalgebras . 91 3.3 Modules . 100 3.4 Adjoint Operators and Intertwining Operat ors 114 3.4.1 Contragredient Modules . 114 3.4.2 Definition and Properties of Intertwining Operators 119 3.4.3 Intertwining Operators Induced by Imposing the Skew · 122 3.4.4 Adjoint Intertwining Operators . . . . . · 123 3.5 Invariant Bilinear Forms ...... · 126 3.6 Definitions and Properties of Lie Superalgebras . · 134 3.7 Simple Lie and Their Irreducible Representations · 138 3.8 Subalgebras of Invariants and the Constru ction . · 143

4 Conformal Superalgebras, Affine Kac-Moody Algebras and KZ Equa- tions 149 4.1 Conformal Superalgebras and Vertex Algebras . . . 149 4.2 VOAs Associated to Affine Kac-Moody Algebras . 169 4.3 Intertwining Operators and KZ Equations . . . . 179 4.3.1 Determination of the Intertwining Operators . 180 4.3.2 KZ Equations ...... 192

5 Analogue of the Highest-Weight Theory 203 5.1 Truncated Modules . · 203 5.2 Intertwining Operators and Bimodules · 217 5.3 Virasoro Vertex Operator .. . · 228

6 Lattice Vertex Operator Superalgebras 235 6.1 Construction of the Algebras . · 235 6.2 Construction of the Modules . · 244 6.3 Characterization of the Algebras . · 266 6.4 Characterization of the Modules . · 275 6.5 Intertwining Operators ...... · 283

7 VOSAs Generated by Their Subspaces of Small Weights 289 7.1 VOSAs Generated by Subspaces of Weights 1/2 and 1 . · 289 7.2 Super Extensions of the Virasoro VOA . . . 308 7.3 VOSAs from Graded Associative algebras . · 316

Bibliography 337

Index 351 Preface

Vertex algebra was introduced by Boreherds, and the slightly revised notion "vertex oper­ ator algebra" was formulated by Frenkel, Lepowsky and Meurman, in order to solve the problem of the moonshine representation of the Monster group - the largest sporadie group . On the one hand, vertex operator algebras ean be viewed as extensions of eertain infinite-dimensional Lie algebras such as affine Lie algebras and the . On the other hand, they are natural one-variable generalizations of commutative associative algebras with an identity element . In a certain sense, Lie algebras and commutative asso­ ciative algebras are reconciled in vertex operator algebras. Moreover, some other algebraie structures, such as integral linear lattiees, Jordan algebras and noncommutative associa­ tive algebras, also appear as subalgebraic structures of vertex operator algebras. The axioms of have geometrie interpretations in terms of Riemman spheres with punctures. The trace functions of a certain component of vertex operators enjoy the modular invariant properties. Vertex operator algebras appeared in physies as the fundamental algebraic structures of eonformal theory, whieh plays an important role in theory and statistieal meehanies .Moreover, eonformal fieldtheoryreveals an importantmathematieal property, the so called "mirror symmetry" among Calabi-Yau manifolds. The general correspondence between vertex operator algebras and Calabi-Yau manifolds still remains mysterious. Ever since the first book on vertex operator algebras by Frenkel, Lepowsky and Meur­ man was published in 1988, there has been a rapid development in vertex operator su­ peralgebras, which are slight generalizations of vertex operator algebras. It is estimated that there are about 400 related pap ers published. It seems to us that there is a need of giving a systematic update approach to the basic concepts, techniques and examples in vertex operator superalgebras. It was Prof. Miehiel Hazewinkel who suggested I write a book on this topie. Writing a book on the structure theory of vertex operator superalgebras and their irredueible modules is diffieult. First of all, such a theory has not reaehed a relativelycorn­ plete stage. Secondly, the diversity in vertex operator superalgebras and their applications implies that we cannot include all the achievements in a book of relatively short length. Nevertheless, we have tried to incorporate as many things as possible that are basic and

ix x PREFACE simple enough to be presented in a relatively short length. The selection of the materials in this book certainly reflects my personal bias. I regret that I am not able to include in this book the moonshine , an extraordinary exmaple of vertex operator algebra. Moreover, I apologize to those people whose important contributions to vertex operator superalgebras have not been included in the book, due to my intention of limiting the length of the book. Lattice vertex operator superalgebras are the largest dass of simple vertex operator superalgebras that have been relatively well studied. In particular, the vertex operator associated with a positive definite self-duallattice has a unique locaHy-finite irreducible module, that is, itself. In order to give the reader a better picture of this type of algebras, I include my work on self-dual codes and lattices as apart of this book. It is my intention that this book could become a one-semester graduate text book after a proper selection of the materials in the book based on the instructor's preference. A substantial number of exercises are given in order to help reader to better understand the content. The people to whom I owe most are my former Ph.D. thesis advisors Profs. James Lepowsky and Robert Wilson. They had patiently guided me into this fascinating field of vertex operator algebras. Their encouragement to my exploring new mathematical structures has prolonged impact on my mathematical research life after I completed my graduate study with them. I am very grateful to Prof. MichielHazewinkel for his sugges­ tion of writing this book and recommendation of publication. I thank Prof. for his interesting lectures on conformal superalgebras, the notions that have been inten­ sively used in the book. I would also like to thank Drs. Chongying Dong and Haisheng Li for their sending me their preprints and YongchangZhu for his explanations on certain things in his works. Last, but not least, I wish to express my gratitude to the publishers for their tremendous work on publishing this book. During my writing of this book, Hong Kong was returned to China. I would also like to dedicate the book to this great event of Chinese people.

Xiaoping Xu Hong Kong, P. R. China March 1998 Introduction

The commutator formulae offree fields appeared in the 1920s when was born. These may be counted as the earliestphenomena ofvertexoperator algebras.The mathematical turning point in this field came in the late 1960s when Kac-Moodyalgebras were introduced by Kac [KaI] and byMoody[Moo] ,independently, and theVirasoro algebra was studied by Gel'fand Fuck [GF]as the only one-dimensional central extension ofthe ofvector fields on a circle, although this algebra was actually discovered by Block [BI) towards theend of the 1950s. A subfamily of Kac-Moody algebras can be realized as one-dimensional central extensions ofthe loopalgebras, which are called affine Kac-Moody algebras . In the 1970s, Lepowsky and Wilson [LW1) introduced vertex operators for explicitly constructing the integral irreducible modules ofaffine Kac-Moodyalgebras. The exponential of a free bosonic field. We observed a few years aga that the linear Hamilto­ nian operators in the theory of Hamiltonian operators developed by Gel'fand , Dikii and Dorfman in middle 1970's are also quite closely related to vertex operator algebras . We believe that the not ion of "conformal superalgebra" formulated by Kac [Ka4] is equivalent to that of linear Hamitonian operator in Gel'fand-Dikii-Dorfman 's theory, where in the supercase, proper settings need to be added. In the1980s, vertex algebra was introduced by Boreherds [B02], and a slightly re­ vised notion "vertex operator algebra" was formulated by Frenkel, Lepowsky and Meur­ man [FLMl-3], in order to solve the problem of the moonshine representation of the Monster group - the largest sporadic group (cf. [ON]) . An axiomatic approach to ver­ tex operator algebras was given by Frenkel, Huang and Leposky [FHL]. Vertex operator superalgebras have been intensively developed in 1990s. Our intention of the book is to give a systematic update elucidation on the structures of vertex operator superalgebras and their modules. Since integral lattices play important roles in vertex operator super­ algebras, we shall include in this book part of our works on self-dual codes and lattices (cf. [Xl-4]). Below we shall give a chapter-by-chapter introduction. The book is divided into two parts. The first part is about self-dual codes and lat­ tices. In 1989, when I first understood the structures of self-duallattices from Conway­ Sloane's "gluing theory," I found it necessary to study self-dual codes over

xi xii INTRODUCTION

Zn relative to certain symmetrie bilinear forms. It turned out that this was one of the key points of my refining Conway-Sloane's gluing philosophy into more practical formats. In Chapter 1, we present a fundamental study on relatively self-dual codes over a finite commutative . A gluing technique for constructing relatively self-dual codes is given. Several exotic general constructions of self-dual codes over Zn are presented. Moreover, we give four interesting constructions of binary self-dual codes by our gluing technique and some constructional examples of ternary self-dual codes. In particular, we con­ struct new doubly-even self-dual binary codes with minimal weight 8 from a given one. Furthermore, we present some families of self-dual codes over Z4. The main results in this chapter were published in [X2]. Integral lattices are the major ingredients in lattice vertex operator superalgebras. The vertex operator superalgebra associated with a positive definite self-dual lattice en­ joys the property of having a unique locally-finite irreducible module, that is, itself. Such algebras can be used to construct more sophisticated and interesting simple vertex opera­ tor superalgebras such as the "Monster moonshine module" (cf. [B2], [FLM3], [DGMI]) . Automorphism groups of self-dual lattices sometimes possess important properties. For instance, Conway's three finite simple sporadic groups - denoted .1, .2 and .3 - are re­ lated to the automorphism group of the Leech lattice, which is the unique 24-dimensional even self-duallattice without elements of square 2 (cf. [C2]). Griess's construction [G] of the Monster group - the largest finite simple sporadic group - as the automor­ phism group of a new algebra, "the Griess algebra," is directly related to the Leech lattice. Another important example of self-duallattice is the Es root lattice, which plays substan­ tial roles in many branches of mathematics such as algebra, and . Classically, self-duallattices are fundamental algebraic structures in sphere packings and . In Chapter 2, we present two gluing techniques for constructing self-dual lattices and some concrete constructional examples. These techniques were developed in [Xl] and published in [X3]. The notion of "generalized complex lattice" is introduced. The first technique is called "untwisted gluing technique," where the objects to be glued are called "U-shells." We introduce classes of U-shells related to root lattices of the simple Lie algebra of types: A, D and E. Moreover, we construct a family of self-duallattices based on the U-shells of type A, which we call "untwisted type-A self-duallattices." By cutting this type of lattices, we can produce new U-shells. The second technique is called "twisted gluing technique," where the objects to be glued are called "T-shells." We also introduce classes of T-shells related to the root lattices and construct a family of self-duallattices based on the T-shells of type A, which we call "twisted type-A self-dual lattices." By cutting this type of lattices, we can produce new T-shells. Among these two families of type-A self-duallattice, there are two subfamilies that possess some features of complex INTRODUCTION xiii lattices. The lattices in these two subfamilies have a fixed-point-free automorphism that induced by the Coxeter automorphism of the type-A root lattices. The second part of this book is about vertex operator superalgebras and their mod­ ules. In Chapter 3, we present the basic concepts in vertex operator superalgebras, Lie superalgebras and general properties. First we introduce the calculus of formal variables, which mainly contains formal binomial expansions, the delta function , an analogue of Taylor's expansion, identities on the delta function and residue properties of formal func­ tions. Then we reformulate the definition of a vertex operator superalgebra and prove its equivalence to the definition given by Frenkel, Lepowsky and Meurman [FML3]. More­ over, we introduce the twisted modules of a vertex operator superalgebra with respect to a finite-erder automorphism, where the regular modules are treated as the special case when the automorphism is the identity map. Our definition of module is formulated in terms of the mathematical version of the "duality" in physics, which was written down in [X9] and [XlI] . We prove that the duality is equivalent to the Jacobi identity. Li's Theorem [Lifi] on the change of twisted vertex operators by the vertex operator of a certain "weight-l" element is given and his proof of the theorem is simplified. Furthermore, we present a the­ orem, which we found in [XlI], on the relation between the generators of vertex operator superalgebras and the duality of the twisted modules. The adjoint operators and inter­ twining operators of the twisted modules of a vertex operator superalgebra are discussed. The untwisted version of the theory is due to Frenkel, Huang and Lepowsky [FHL], and we generalized their theory to twisted modules in [X8-9]. In particular, the contragredient module of a given twisted module is constructed. In addition, we give a reformulation of Li's work on invariant bilinear forms of a vertex operator algebra [Li2] . Finally we present slight generalizations of the main results in [DM5] and [DLiM5] on subalgebras of invariants and the idea of orbifold construction of vertex operator superalgebras, which was originally introduced by Frenkel, Lepowsky and Meurman [FLMl-3]. The two-dimensional was initiated by Belavin, Polyakov and Zamolodchikov [BPZ], where the significance of the primary fields was pointed out . Knizh­ nik and Zamalodchikov [KZ] developed the theory with affine Kac-Moody algebras as the symmetry and gave the fundamental differential equations of the multipoint correlation functions, which are now called "KZ equations." Tsuchiya and Kanie [TK] presented a mathematically rigorous foundations to the work in [KZ] with the AP)-symmetry. The re­ sults in [TK] were generalized by Tsuchiya, Ueno and Yamada [TUY] and by Tsuchya and Yamada [TY]. Frenkel and Zhu [FZ] gave a direct construction of simple vertex operator algebras with arbitrary ranks associated with affine Kac-Moody algebras and the repre­ sentation theory of those algebras whose corresponding representations of the affine Lie algebras are integrable (cf. [Ka3]). Dong and Lepowsky [DL2] constructed the algebras of the integrable representations from the of level-l irreducible representations of xiv INTRODUCTION the affine Lie algebras. Kac [Ka4] formulated the notion of "conformal superalgebra," which we believe to be equivalent to the notion of linear Hamiltonian superoperator un­ der a proper setting. The later was introduced by Gel'fand, Dikii and Dorfman [GDil-2], [GDol-3]. Conformal superalgebras with "Virasoro element" are the local structures of vertex operator superalgebras. The main purpose of Chapter 4 is to present the precise connection between conformal superalgebras and vertex operator superalgebras and to give a reformulation of Frenkel-Zhu's work [FZ] by means of conformal superalgebras and a reformulation of the basic theory of KZ equations. In his 1990 thesis (which was published in [Z2]) , Zhu introduced an associated with a vertex operator algebra and established a one-to-one correspondence between the truncated irreducible modules of a vertex operator algebra and the irreducible modules of its associated associative algebra. Dong, Li and Mason [DLiM2] generalized Zhu's results to twisted modules of a vertex operator algebra. Frenkel and Zhu [FZ]gave a connection between intertwining operators among the truncated irreducible modules of a vertex operator algebra and the bimodules of its associated associative algebra. Wang [W] used Zhu's theory and Feigin-Fuch's results in [FFI-2]to prove that there 18 only a finite number of locally-finite irreducible modules of the vertex operator algebra associated with the Virasoro algebra with certain rational ranks and found the fusion rule of the algebra. We observed that Zhu's results and their generalizations are essentially analogues of highest-weight representation in Lie algebras. In fact, twisted modules of a vertex operator superalgebra are determined by modules of a certain "twisted " associated with it. From this point of view, the theory of vertex operator superalgebras is a subtheory of that of infinite-dimensional Lie superalgebras. In Chapter 5, we use the notion of conformal superalgebra to generalize Zhu's theory to twisted modules of a vertex operator superalgebra. We believe that our approach also gives a simplification to that in [Z2] and that in [DLiM2] . Moreover, we generalize Frenkel-Zhu's result on intertwining operators of a vertex operator algebra to those of a vertex operator superalgebra. The main feature of our approach is using of the conformal superalgebra associated with the semi-product of a vertex operator superalgebra with a module. A reformulation of Wang's work [W] is presented. The phenomena of "lattice vertex operator algebras" mathematically first appeared in Lepowsky and Wilson's work [LWI]. Vertex operator algebras associated with even lat­ tices were introduced in order to study the "moonshine module" of the Monster group, the largest finite simple sporadic group (cf. [B02], [FLMI-3]). Twisted irreducible modules of the vertex operator algebra associated with an even lattice with respect to the automor­ phism induced by a finite-erder automorphism of the lattice were essentially constructed by Lepowsky [Lepl] and a revised version was.given in [FLM2]. Dong and Lepowsky [DLl] generalized the structure of these modules to so-called "relative twisted vertex op- INTRODUCTION xv erators," whieh are related to a eertain projeetion of the even lattiee. This author [X5] generalized Lepowsky's construction to a more general one of irreducible twisted modules of the vertex operator superalgebra associated with an integrallattiee that is not neeessarily even. Our eonstruetion in [X5] is related to eertian cosets of the lattiee in the Z- of C with the lattice. In [Dcl], Dong classified all the untwisted irreducible modules of the vertex operator algebra assoeiated with an even lattiee (the nilpotent eondition of the positive Heisenberg operators should be added) . He also classified in [De2] all the irredueible twisted modules of the algebra with respect to the automorphism indueed by the negative identity map of the lattiee (the same eondition should be added) . Li and this author [LX] proved that a simple vertex operator algebra eontaining a certain Heisenberg algebra substrueture must be isomorphie to the vertex operator algebra associated with a finite-dimensional vector spaee with a nondegenerate symmetrie and an even additive subgroup, whieh is a natural generalization of the vertex operator algebra assoeiated with an even lattice. In [X9], this author classified all the irreducible twisted modules of the vertex operator algebra associated with an integrallattiee that is not neeessarily even with respeet to any finite-erder lattice-graded automorphism of the algebra , under eertain loeal conditions. Dong and Lepowsky [DL2] found the fusion rule of the untwisted irreducible modules of the vertex operator algebra associated with an even lattiee. This author introdueed in [X8] the notion of intertwining operator for twisted modules of a vertex operator superalgebra. In [X8] , we generalized the result on the fusion rule in [DL2] to eertain twisted modules of the vertex operator superalgebra associated with an integrallattiee. In Chapter 6, we present an updated approach to the theory of vertex operator su­ peralgebra eonstrueted from pairs of a finite-dimensional spaee with a nondegenerat e symmetrie bilinear form and an integral additive subgroup. We call these algebras "lat­ tiee vertex operator superalgebras" because any integral subgroup is a direct sum of a nondegenerate integrallattiee and an isotropie subgroup (cf. Lemma 6.1.2). In fact, the results mentioned in the above two paragraphs are improved and the arguments of the proofs are simplified. In Chapter 7, we are primarily concerned with simple vertex operator superalgebras that are N/2-graded and generated by their subspaees of small weights. First we give a classifieation of such algebras generated by their subspaces of weights 1 and 1/2. Lian's result in [Lia] on the classifieation of the algebras generated by their subspaces of weight 1 is improved. Moreover, we improve and simplify our early classifieation of the algebras generated their subspaces of weight 1/2 and their irredueible twisted modules in [XlI]. We remark that these algebras were first introdueed by Tsukada [Thl] and by Feingold, Frenkel and Ries [FFR], independently. A Z-graded twisted irredueible module of sueh an algebra was also constructed in [FFR]. A "Splitting Theorem" on the algebras generated XVI INTRODUCTION by both the subspaces of weights 1 and 1/2 was discovered by us during the writing of this book. Super extensions oftheVirasoro algebra were introduced and studied since the early sev­ enties (cf. [NS], [R], [AB-], [AB-S), [Ka2J). Axiomatic study on the structures of these extensions was carried out by Kac, et al. (cf. [KT], [KL], [CK), [Ka5J). These extensions can be viewed as the "extensions through finite-dimensional exterior algebras." Among these extensions, only the following algebras indeed have a Virasoro algebra substructure: the Neveu-Schwarz algebra, the N = 2, 3 and 4 superconformal algebras and the non­ trivial one-dimensiona~ central extension of the Lie superalgebra of all vector fields on the N = 2 supercircle (cf. [Ka5J) . These algebras are generated by certain conformal superalgebras. Therefore, they do "generate" vertex operator superalgebras by a theo­ rem in Chapter 4. We give a description of the conformal superalgebras generating these algebras based on the works in [KL] and [Ka5). Finally, we introduce a new family of infinite-dimensional Lie superalgebras, which we call "double affinizations" of Z2-graded associative algebras with respect a trace map. Frorn these Lie superalgebras, weconstruct new families of conformal superalgebras with a Virasoro element, which could generate new families of N/2-graded simple vertex operator superalgebras and generated by their subspaces of small weights. Examples related to matrix algebras and the Hecke algebras are given. Our part of this work was supported by Hong Kong RGC Competitive Earmarked Research Grant HKUST585/94P.

N otational Conventions

C: the field of complex numbers. CX : the multiplication group of nonzero complex numbers. i, i + j : {i, i + 1,i + 2, ..., i + j}, an index set.

8i ,j = 1 if i = j, 0 if i =1= j . N: {O, 1,2,3, ...}, the set of natural numbers iQ: the field of rational numbers. WT : exp 2rrH/T, a T-th root of unity. w: the Virasoro element. Z: the ring of . Z+: the set of positive integers. Zn: Z/Zn, a (or group). VOA: vertex operator algebra. VOSA: vertex operator superalgebra.