Introduction to Vertex Operator Algebras and Their Representations

329 Pages·2004·9.949 MB·English

by  James Lepowsky, Haisheng Li (auth.)

Progress in Mathematics
Volume 227
Series Editors
Hyman Bass
Joseph Oesterle
Alan Weinstein
James Lepowsky
Haisheng Li
Introduction to
Vertex Operator Algebras
and Their Representations
Springer Science+Business Media, LLC
James Lepowsky Haisheng Li
Department of Mathematics Department of Mathematics
Rutgers University Rutgers University
Hili Center, Busch Campus Armitage Hall
Piscataway, NJ 08854 Camden, NJ 08102
U.S.A. U.S.A.
Library of Congress Cataloging-in-Publication Data
Lepowsky, J. (James)
Introduction to vertex operator algebras and their representations / J ames Lepowsky,
Haisheng Li.
p. cm. - (Progress in mathematics ; v. 227)
Includes bibliographical references and index.
ISBN 978-1-4612-6480-4 ISBN 978-0-8176-8186-9 (eBook)
DOI 10.1007/978-0-8176-8186-9
I. Vertex operator algebras. 2. Representations of algebras. I. Li, Haisheng, 1962-11.
Title. III. Progress in mathematics (Boston, Mass.); v. 227.
QA326.L482004
512'.55-dc22 2003063839
CIP
AMS Subject Classifications: Primary: 17B69, 8IT40; Secondary: 17B67, 17B68, 81RlO
ISBN 978-1-4612-6480-4 Printed on acid-free paper.
©2004 Springer Science+Business Media New York
Originally published by Birkhäuser Boston in 2004
Softcover reprint ofthe hardcover 1st edition 2004
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ToLaelandMei
ToDongyuan, Joyce andJimmy
Contents
Preface " .... ... .... ix
1 Introduction ................................................... 1
1.1 Motivation .. .............. ......................... .......... 1
1.2 Example ofavertexoperator. ................................... 5
1.3 The notionofvertexoperatoralgebra............................. 8
1.4 Simplificationofthedefinition 12
1.5 Representations andmodules 13
1.6 Construction offamiliesofexamples ............................. 15
1.7 Somefurtherdevelopments ..................................... 17
2 Formal Calculus................................................ 21
2.1 Formalseriesandtheformaldeltafunction........................ 21
2.2 DerivationsandtheformalTaylorTheorem. ....................... 29
2.3 Expansions ofzeroandapplications.............................. 33
3 Vertex OperatorAlgebras:TheAxiomaticBasics .................... 49
3.1 Definitionsandsomefundamentalproperties. .... 49
3.2 Commutativityproperties....................................... 65
3.3 Associativityproperties 72
3.4 TheJacobiidentityfromcommutativityandassociativity. ........... 81
3.5 TheJacobiidentityfromcommutativity........................... 84
3.6 TheJacobiidentityfromskewsymmetryandassociativity. .......... 86
3.7 5 -symmetryoftheJacobiidentity 92
3
3.8 Theiterateformulaandnormal-orderedproducts ................... 94
3.9 Furtherelementarynotions 98
3.10Weaknilpotenceandnilpotence 101
3.11Centralizers andthecenter...................................... 105
3.12Directproductandtensorproductvertexalgebras 111
viii Contents
4 Modules 117
4.1 Definitionandsomeconsequences ............................... 118
4.2 Commutativity properties 121
4.3 Associativityproperties 124
4.4 TheJacobi identityasaconsequence ofassociativity and
commutativity properties 127
4.5 Further elementary notions 128
4.6 Tensorproduct modulesfortensorproduct vertexalgebras ........... 137
4.7 Vacuum-likevectors 138
4.8 Adjoining amoduletoavertexalgebra 141
5 Representations ofVertexAlgebras and the Construction ofVertex
Algebrasand Modules. .......................................... 145
5.1 Weakvertexoperators 148
5.2 The action of weak vertexoperators on the space of weak vertex
operators 151
5.3 Thecanonicalweakvertexalgebra£(W) andtheequivalencebetween
modules andrepresentations .................................... 156
5.4 Subalgebras of£(W) 163
5.5 Local subalgebrasandvertexsubalgebras of£(W) 165
5.6 Vertexsubalgebras of£(W) associated withtheVirasoroalgebra 173
5.7 Generalconstruction theoremsforvertexalgebras andmodules 179
6 ConstructionofFamiliesofVertexOperatorAlgebras andModules .... 191
6.1 Vertexoperatoralgebrasandmodules associated to
theVirasoroalgebra ..... ...................................... 193
6.2 Vertexoperatoralgebras andmodules associated to
affineLiealgebras. ............................................ 201
6.3 Vertexoperator algebrasandmodules associated to
Heisenbergalgebras 217
6.4 Vertexoperator algebrasandmodules associated to
evenlattices-thesetting ....................................... 226
6.5 Vertexoperator algebrasandmodules associated to
evenlattices-themainresults 239
6.6 Classificationoftheirreducible Lg(l, O)-modulesfor 9
finite-dimensionalsimpleandl apositiveinteger. .................. 264
References 289
Index ; 315
Preface
Vertexoperatoralgebra theoryisanewareaofmathematics.Ithasbeenanexciting and
ever-growing subject from thebeginning,starting evenbefore R.Borcherdsintroduced
theprecisemathematicalnotionof"vertexalgebra"inthe1980s[BI].Havingdeveloped
in conjunction with string theory in theoretical physics and with the theory of "mon
strous moonshine" and infinite-dimensionalLie algebratheory in mathematics,vertex
(operator) algebra theory is qualitatively different from traditional algebraic theories,
reflecting the "nonclassical" nature ofstring theory and of monstrous moonshine.The
theory hasrevealed newperspectives thatwereunavailable withoutit,andcontinuesto
doso.
"Monstrousmoonshine"beganasanastonishingsetofconjecturesrelatingtheMon
sterfinitesimplegroup tothetheory ofmodular functions innumbertheory.Asisnow
known,vertexoperatoralgebratheoryisafoundationalpillarofmonstrousmoonshine.
With the theory available, one can formulate and try to solve new problems that have
far-reaching implicationsinawiderangeoffieldsthathad notpreviously been thought
ofasbeing related.
This book systematically introduces the theory of vertex (operator) algebras from
the beginning, using "formalcalculus," and takes the reader through the fundamental
theory to the detailed construction of examples.The axiomatic foundations of vertex
operatoralgebras and modules arestudied indetail, general constructiontheorems for
vertex operator algebras and modules are presented, and the most basic families of
vertex operatoralgebras are constructedand their irreduciblemodules are constructed
and are also classified. The constructiontheorems for algebras and modules are based
onastudyofrepresentationsofavertexoperatoralgebra, asopposedtomodulesforthe
algebra, asdeveloped in[Li3].Asignificantfeature ofthe theory isthat ingeneral, the
constructionofmodules for(orrepresentationsof)avertex operatoralgebra isinsome
sense more subtle than the construction of the algebra itself. With the body of theory
presented in this book as background, the reader will be well prepared to embark on
anyofavastrange ofdirections inthetheory and itsapplications.
In the introduction, we shall sketch the theory and the contents of this book and
provide motivation. Wehavewritten thisbook tobeself-contained.The only prerequi-
x Preface
sitesaresomefamiliarity withLiealgebras andadesire tolearnvertexoperatoralgebra
theory.Ifthereadercan follow the mathematical material in the introduction, then he
orshe willhave notrouble following the whole book.
Thisbookissuitableforafirstgraduatecourseonvertexoperatoralgebratheory,and
infact, thatishowwehavebeen usingdrafts ofthis book while writing andexpanding
itovertheyears.
As we have mentioned, our treatment is axiomatic, leading to the construction of
examples. By coritrast, the original book [FLM6] developed examples and the the
ory of vertex operators before presenting the concept of vertex operatoralgebra as a
"conceptualsummary"ofthefundamental properties thathadbeen builtupthrough an
ever-deepeningstudyofcertain basicstructures (structuresthatwereneeded inthecon
struction ofthe "moonshinemodule" for the Monsterfinitesimple group in [FLM6]).
Thatbookisnotaprerequisiteforthisbook,butthereadermaywishtoconsult[FLM6]
to get a feeling for how vertex operator algebra theory originally developed through
examples andthrough thesolution ofproblems.
Wecontinue tousethehistorical terms"vertex algebra,"whichwastheoriginal term
of Borcherds, and "vertex operatoralgebra," which was used in [FLM6] for a certain
variantnotion.(Bothnotionsarediscussedatgreatlengthinthisbook.)Avertexoperator
algebra satisfies afew axioms that are not part of the definition of vertex algebra, but
thisisarelativelyminorissue;aswepointoutintheintroduction,therearemanyuseful
variants of the definition, depending on the context that one is interested in. It is the
conceptual features of the notion of vertex (operator) algebra that are emphasized in
thedefinitions in[B1]and [FLM6] that haveledustocontinuetouse these two pieces
of terminology as they have(usually) been used upto now.Theseconceptual features
are discussed in the introduction and in the main text of the book. For instance, the
twodifferent terms reflect theissue of whether thealgebras are viewed as"algebrasof
operators"orasalgebraswitha"multiplicationoperation"(orrather,aninfinitesequence
ofmultiplicationoperations); bothviewpointsare"correct,"andthebestthing istokeep
both inmind.
Primarily expository,this book does present anumberof new results developed by
theauthorsovertheyearsinwhichdraftsofthisbookwereusedincoursesandseminars.
These results are mentionedattheend of theintroduction.In addition,in many places
the proofs andtreatments gobeyond theoriginals,sometimesquite a bit.
Thenotionofvertex(operator)superalgebraisaslightgeneralizationofthenotionof
vertex(operator)algebra,andthebasicgeneraltheoryofvertex(operator)superalgebras
isalmostidenticaltothetheoryofvertex(operator)algebras;oneneedstosystematically
keep track of ± signs at every stage. For simplicity of exposition, in this book we
treat onlyvertex (operator) algebras. The original paper [Li3] was infact written inthe
generalityofvertex(operator)superalgebras,andtheinterestedreadercanconsult[Li3]
for the necessary (minor) changes.
Theworkonthisbookbeganwhenoneauthor(J.L.)usedtheJapanlecturenotes[Lil]
bytheotherauthor (H.L.) inlectures presentingthework [Li3]foragraduatecourse at
Rutgers University. H.L.'s work developedatheory of "representations" (in aspecific
Preface xi
sense) of a vertex operator algebra, and used it to construct families of examples of
vertexoperatoralgebrasandmodules; before [Li3],therehadbeennogeneralapproach
availableforconstructingfamilies ofexamplesofvertexoperatoralgebras andmodules.
Over a period of years, through many graduate courses and lectures by both authors,
this book grew from informal lecture notes toadetailed self-containedintroductionto
the theory of vertex operatoralgebras, starting from the elementary fundamentals and
leading to[Li3], and infact, asitturned out, toagreatly expandedtreatmentof[Li3].
Actually, [Li3] was the publishedform of half ofa 1993 preprintby H.L.,and the
otherhalfofthis1993preprint waspublishedas[Li4].H.L.isverygrateful toWeiqiang
Wang,thenagraduatestudentatM.1.T.,forhisearlyinterest inthispreprint. H.L.would
also like to thankMasahiko Miyamoto for organizing the wonderful workshop in the
summerof1994attheResearchInstituteforMathematicalSciences, KyotoUniversity,
inwhichChongyingDong,Yi-ZhiHuangandH.L.wereinvitedtogivelecturesonvertex
operatoralgebras.Weareverygrateful tomany students andcolleaguesinlectures and
seminarsovertheyears,especiallyChengmingBai,CorinaCalinescu,BenjaminDoyon,
Liang Kong, Machiel van Frankenhuysen and Lin Zhang.Their insightful comments
andquestionscontinuallyledustoseekincreasinglybetterwaysofilluminatingcritical
andsubtlepoints, andalsoofenhancing thereadabilitybymeansoffurtherexplanatory
remarks and cross-referencing.Wethank Martin Karelforreadingthe manuscript,and
ChongyingDong,Arne Meurman,Mirko Prime, and allthe reviewers for their helpful
comments. For the past two years, H.L. has used the text of this book in lectures at
HarbinNormal University,Harbin, China.Hewouldliketothank ShutaoChen,Yuwen
Wang,WendeLiu,ShuqingWangandtheMathematicsDepartmentfortheirhospitality,
and to thank all the participants,Wende Liu, Shuqing Wang,Yinhua He,Li Li,Liqin
Liu,Xuemei Liu and XiulingWang,fortheir interest andenthusiasm.
Theformal calculus formulationofthetheoryofvertexoperatoralgebras,including
theJacobi identity axiom, comes from [FLM6] and [FHL].The "weakcommutativity"
and "weak associativity" properties come from [DL3].J.L. would like to express his
thanks tohiscollaboratorsonthese works:IgorFrenkel, ArneMeurman,Yi-ZhiHuang
andChongyingDong.
The treatment in [DL3] actually included the vertex (operator) superalgebra case,
and in fact, this superalgebracase was itselfa very special case of the full generality
of [DL3]-that of "generalized vertex algebras" and "abelian intertwining algebras."
Structures similar to generalized vertex algebras were also introduced and studied in
[FFR]and[Mos1].Thespecialcaseofvertex(operator)superalgebrashadalready been
studied in[GoI] and [Ts2].
Itis a pleasure to thank Ann Kostant for encouraging us to publish this book with
Birkhauserand for her and her Birkhausercolleagues' wonderful work in making the
publicationprocess sosmooth.
Both authors gratefully acknowledge the support of the National ScienceFounda
tion.Inaddition, H.L. thanks theNational Security Agency and the Rutgers University
Research Council.