Selected Title s i n Thi s Serie s

13 Yako v M . Eliashber g an d Wil l ia m P . Thurston , Confoliations , 199 8

12 I . G . Macdonald , Symmetri c function s an d orthogona l polynomials , 199 8

11 Lar s Garding , Som e point s o f analysi s an d thei r history , 199 7

10 Victo r Kac , Verte x algebra s fo r beginners , Secon d Edition , 199 8

9 S tephe n Gelbart , Lecture s o n th e Arthur-Selber g trac e formula , 199 6

8 Bern d Sturmfels , Grobne r base s an d conve x polytopes , 199 6

7 A n d y R . Magid , Lecture s o n differentia l Galoi s theory , 199 4

6 Dus a McDuf F an d Die tma r Salamon , J-holomorphi c curve s an d quantu m cohomology ,

1994

5 V . I . Arnold , Topologica l invariant s o f plan e curve s an d caustics , 199 4

4 Davi d M . Goldschmidt , Grou p characters , symmetri c functions , an d th e Heck e algebra ,

1993

3 A . N . Varchenk o an d P . I . Etingof , Wh y th e boundar y o f a roun d dro p become s a

curve o f orde r four , 199 2

2 Frit z John , Nonlinea r wav e equations , formatio n o f singularities , 199 0

1 Michae l H . Freedma n an d Fen g Luo , Selecte d application s o f geometr y t o

low-dimensional topology , 198 9

http://dx.doi.org/10.1090/ulect/010

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University

LECTURE Series

Volume 1 0

Vertex Algebra s fo r Beginner s Second Editio n

Victor Ka c

American Mathematica l Societ y Providence, Rhod e Islan d

E D I T O R I A L C O M M I T T E E

Jerry L . Bon a Nicolai Reshetikhi n

Leonard L . Scot t (Chair )

2000 Mathematics Subject Classification. P r i m a r y 17B69 ; Secondary 17B65 , 81T05 , 81T40 .

ABSTRACT. Thi s boo k i s a n introductio n t o verte x algebras , a ne w mathematica l structur e tha t has appeare d recentl y i n quantu m physics . I t ca n b e use d b y researcher s an d graduat e student s working o n representatio n theor y an d mathematica l physics .

L i b r a r y o f C o n g r e s s Ca ta log ing - in -Pub l i ca t io n D a t a

Kac, Victo r G. , 1943 -Vertex algebra s fo r beginner s / Victo r Kac . — 2n d ed .

p. cm . — (Universit y lectur e series , ISS N 1047-399 8 ; v. 10 ) Includes bibliographica l reference s an d index .

ISBN 0-8218-1396- X (alk . paper ) 1. Verte x operato r algebras . 2 . Quantu m field theory . 3 . Mathematica l physics . I . Title .

II. Series : Universit y lectur e serie s (Providence , R.I. ) ; 10. QC174.52.06K33 199 8 512'.55—dc21 98-4127 6

CIP

C o p y i n g a n d r e p r i n t i n g . Individua l reader s o f thi s publication , an d nonprofi t librarie s actin g for them , ar e permitte d t o mak e fai r us e o f th e material , suc h a s t o cop y a chapte r fo r us e in teachin g o r research . Permissio n i s grante d t o quot e brie f passage s fro m thi s publicatio n i n reviews, provide d th e customar y acknowledgmen t o f th e sourc e i s given .

Republication, systemati c copying , o r multipl e reproductio n o f any materia l i n thi s publicatio n (including abstracts ) i s permitte d onl y unde r licens e fro m th e America n Mathematica l Society . Requests fo r suc h permissio n shoul d b e addresse d t o th e Assistan t t o th e Publisher , America n Mathematical Society , P . O. Bo x 6248 , Providence , Rhod e Islan d 02940-6248 . Request s ca n als o be mad e b y e-mai l t o reprint-permissionOams . org .

© Copyrigh t 1997 , 199 8 b y th e America n Mathematica l Society . First editio n publishe d 199 6

Second editio n publishe d 199 8 Second editio n reprinte d wit h correction s 200 1

Printed i n th e Unite d State s o f America . The America n Mathematica l Societ y retain s al l right s

except thos e grante d t o th e Unite d State s Government . @ Th e pape r use d i n thi s boo k i s acid-free an d fall s withi n th e guideline s

established t o ensur e permanenc e an d durability . Visit th e AM S hom e pag e a t URL : ht tp: / /www.ams.org /

10 9 8 7 6 5 4 3 2 0 6 0 5 0 4 0 3 0 2 0 1

Contents

Preface 1

Preface t o th e secon d editio n 3

Chapter 1 . Wightma n axiom s an d verte x algebra s 5

1.1. Wightma n axiom s o f a QF T 5

1.2. d = 2 QFT an d chira l algebra s 8

1.3. Definitio n o f a verte x algebr a 1 3

1.4. Holomorphi c verte x algebra s 1 5

Chapter 2 . Calculu s o f forma l distribution s 1 7

2.1. Forma l delta-functio n 1 7

2.2. A n expansio n o f a forma l distributio n a(z,w) an d forma l Fourie r

transform 1 9

2.3. Localit y o f two formal distribution s 2 4

2.4. Taylor' s formul a 2 9

2.5. Curren t algebra s 3 1

2.6. Conforma l weigh t an d th e Virasor o algebr a 3 4

2.7. Forma l distributio n Li e superalgebras an d conforma l superalgebra s 3 9

2.8. Conforma l module s an d module s ove r conforma l superalgebra s 5 0

2.9. Representatio n theor y o f finite conforma l algebra s 5 6

2.10. Associativ e conforma l algebra s an d th e genera l conforma l algebr a 6 1

2.11. Cohomolog y o f conforma l algebra s 6 7

Chapter 3 . Loca l fields 8 1

3.1. Normall y ordere d produc t 8 1

3.2. Dong' s lemm a 8 4

3.3. Wick' s theore m an d a "non-commutative " generalizatio n 8 7

vi C O N T E N T S

3.4. Bounde d an d fiel d representation s o f formal distributio n Li e

superalgebras 9 1

3.5. Fre e (super)boson s 9 3

3.6. Fre e (super)fermion s 9 8

Chapter 4 . Structur e theor y o f vertex algebra s 10 3

4.1. Consequence s o f translation covarianc e an d vacuu m axiom s 10 3

4.2. Skewsymmetr y 10 5

4.3. Subalgebras , ideals , an d tenso r product s 10 6

4.4. Uniquenes s theore m 10 8

4.5. Existenc e theore m 11 0

4.6. Borcherd s OP E formul a 11 1

4.7. Verte x algebra s associate d t o forma l distributio n Li e superalgebras 11 3

4.8. Borcherd s identit y 11 6

4.9. Grade d an d Mobiu s conforma l verte x algebra s 11 9

4.10. Conforma l verte x algebra s 12 5

4.11. Fiel d algebra s 12 9

Chapter 5 . Example s o f vertex algebra s an d thei r application s 13 3

5.1. Charge d fre e fermion s an d tripl e produc t identit y 13 3

5.2. Boson-fermio n correspondenc e an d K P hierarch y 13 7

5.3. gl^ an d W l+QO 14 3

5.4. Lattic e verte x algebra s 14 8

5.5. Simpl e lattic e verte x algebra s 15 2

5.6. Roo t lattic e verte x algebra s an d affln e verte x algebra s 15 8

5.7. Conforma l structur e fo r affln e verte x algebra s 16 1

5.8. Supe r boson-fermio n correspondenc e an d sum s o f square s 16 8

5.9. Supe r conformal verte x algebra s 17 8

5.10. O n classificatio n o f conforma l superalgebra s 18 5

Bibliography 19 3

Index 19 9

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Index

A-action, 5 3

A-bracket, 4 0

A-product, 2 7

2-cocycle, 46 , 144 , 15 2

afHne centra l charge , 9 5

affine Kac-Mood y algebra , 3 2

affine verte x algebra , 12 2

affinization o f a conforma l algebra , 4 1

affinization o f a loo p algebra , 3 2

affinization o f a verte x algebra , 10 8

annihilation algebra , 5 6

associative conforma l algebra , 6 1

associative superalgebra , 2 4

automorphism o f a verte x algebra , 10 7

basic cohomology , 7 0

Borcherds commutato r formula , 11 2

Borcherds identity , 11 7

Borcherds OP E formula , 11 1

boson-fermion correspondence , 13 9

bosonization, 13 4

bounded representation , 9 2

bracket, 2 4

Casimir operator , 16 3

causal order , 5

central charge , 37 , 12 5

character, 136 , 17 0

charge, 13 5

charge decomposition , 13 5

charge operator , 13 5

charged free fermions , 34 , 133 , 15 5

chiral algebra , 1 1

Clifford affinization , 3 3

conformal group , 7

conformal linea r map , 6 5

conformal module , 50 , 52 , 5 6

conformal quantu m field theory , 7

conformal superalgebra , 3 9

conformal vector , 12 5

conformal verte x algebra , 12 5

conformal weight , 34 , 127 , 18 6

coset model , 11 3

current algebra , 3 1

current conforma l superalgebra , 4 7

currents, 3 2

Dedekind 77-function , 13 7

199

200 INDEX

denominator identity , 136 , 17 6

derivation o f a verte x algebra , 107

Dong's lemma , 8 4

dual bases , 9 4

dual Coxete r number , 16 4

eigendistribution, 3 4

energy, 13 5

energy operator , 13 5

energy-momentum field, 12 5

Existence theorem , 11 0

extended annihilatio n algebra , 5 6

field, 14 , 81 , 82

field algebra , 12 9

field representation , 9 1

finite conforma l algebra , 4 0

finite forma l distributio n algebra , 4 5

finite module , 54 , 5 9

formal delta-function , 1 8

formal distribution , 1 7

formal distributio n Li e superalgebra , 37 , 9 1

formal Fourie r transform , 2 1

forward cone , 5

free boson , 31 , 94

free bosoni c verte x algebra , 11 6

free fermioni c verte x algebra , 11 6

free fermions , 9 9

free field theory , 8 8

free neutra l fermion , 3 4

Prenkel-Kac construction , 16 0

general conforma l algebra , 6 6

general linea r field algebra , 8 6

generating se t o f fields, 11 1

Goddard uniquenes s theorem , 10 8

Goddard-Kent-Olive construction , 16 7

graded conforma l superalgebra , 18 6

graded verte x algebra , 11 9

group GLQO , 14 2

Hamiltonian, 34 , 11 9

holomorphic verte x algebra , 1 5

homomorphism o f verte x algebras , 10 6

ideal o f a verte x algebra , 107

induced module , 9 1

infinitesimal translatio n operator , 14 , 10 3

inner automorphis m o f a vertex algebra , 12 1

integral lattice , 14 8

integration b y parts , 1 7

invariant bilinea r form , 3 1

irregular ideal , 4 4

Jacobi tripl e produc t identity , 13 6

Kac-Todorov model , 18 0

KP hierarchy , 141 , 14 2

lattice verte x algebra , 148 , 15 4

Lie algebr a V o f regula r differentia l opera -

tors o n C \ {0} , 14 5

Lie algebr a gl^i 14 4

Lie algebr a gl^, 14 3

Lie algebr a gloo, 14 0

Lie superalgebra , 2 4

light con e coordinates , 8

linear field algebra , 8 6

local forma l distribution , 2 1

local linea r field algebra , 8 6

locality, 2 4

locality axiom , 14 , 10 3

loop algebra , 3 2

Minkowski space-time , 5

Mobius-conformal verte x algebra , 12 3

module ove r a conforma l algebra , 5 2

mutually loca l forma l distributions , 2 4

n-th produc t o f fields , 82 , 8 4

n-th produc t o f forma l distributions , 2 6

N — 1 superconformal vector , 17 9

INDEX 201

N — 1 superconformal verte x algebra , 18 0

N — 1 vertex superalgebra , 18 3

N = 2 conforma l superalgebra , 18 6

N = 2 superconforma l Li e algebra , 18 2

N — 2 superconforma l verte x algebra , 18 2

N = 4 conforma l superalgebra , 187

Neveu-Schwarz algebra , 17 9

Neveu-Schwarz conforma l superalgebra , 18 5

"non-commutative" Wic k formula , 90 , 11 6

normally ordere d product , 81 , 87

OPE coefficients , 2 1

operator produc t expansio n (OPE) , 21 , 26

orbifold model , 11 3

oscillator algebra , 3 1

parity, 1 3

Pauli matrices , 187

Poincare group , 5

positive an d negativ e part s o f a forma l dis -

tribution, 2 5

primary field, 127

quantum field theory , 5

quasiassociativity, 119 , 13 0

quasiprimary field, 12 3

reduced cohomology , 7 0

regular forma l distribution s Li e superalge -

bra, 11 4

residue, 1 7

root lattice , 15 9

root system , 15 9

singular vector , 9 3

skew-supersymmetric bilinea r form , 3 3

skewsymmetry o f a verte x algebra , 10 5

space o f states , 1 4

space-like separate d subsets , 5

state-field correspondence , 1 4

strongly generatin g se t o f fields, 11 1

subalgebra o f a verte x algebra , 10 6

Sugawara construction , 16 5

super boson-fermio n correspondence , 17 2

superaffine verte x algebra , 12 2

superaffinization, 3 3

superconformal Li e algebra , 18 1

supercurrent, 3 3

supercurrent algebra , 11 5

superdimension, 1 4

superfields, 18 3

superspace, 1 3

supersymmetric bilinea r form , 3 2

Taylor's formula , 29 , 30 , 8 2

tensor produc t o f verte x algebras , 107

translation covarianc e axiom , 14 , 10 3

twisted grou p algebra , 15 2

universal affin e verte x algebra , 11 5

universal verte x algebr a associate d t o a for -

mal distributio n Li e superalgebra , 11 5

vacuum axiom , 14 , 10 3

vacuum subalgebra , 11 0

vacuum vector , 1 4

Veneziano field, 13 9

Verma module , 9 3

vertex algebra , 14 , 10 3

vertex algebr a Wi+oo,c » 14 7

vertex operator , 12 , 139 , 15 5

Virasoro algebra , 3 6

Virasoro conforma l algebra , 4 8

Virasoro field, 12 5

Virasoro forma l distribution , 3 7

Virasoro verte x algebra , 12 8

W-algebra, 11 3

Weyl affinization , 9 4

Wick theorem , 8 7