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Introduction to Vertex Operator Superalgebras and Their Modules Mathematics and Its Applications Introduction to Vertex Operator Superalgebras and Their Modules Mathematics 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 Symmetry · 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 Algebras and Their Irreducible Representations · 138 3.8 Subalgebras of Invariants and the Orbifold 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 Algebra .. · 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 Virasoro algebra. 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 vertex operator algebra 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 field theory, whieh plays an important role in string 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 module, 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 superalgebra 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. Victor Kac 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 quantum field theory 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 Lie algebra 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.
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