数理解析研究所講究録904 巻 1995 年 1-25

1Introduction to vertex operator algebras I
Chongying Dong1
Department of Mathematics, University of California, Santa Cruz, CA 95064

1

Introduction

The theory of vertex (operator) algebras has developed rapidly in the last few years.

These rich algebraic structures provide the proper formulation for the moonshine module

construction for the Monster group ([BI-B2], [FLMI], [FLM3]) and also give a lot of
new insight into the representation theory of the Virasoro algebra and affine Kac-Moody

algebras (see for instance [DL3], [DMZ], [FZ], [W]). The modern notion of chiral algebrain conformal field theory [BPZ] in physics essentially corresponds to the mathematical
notion of vertex operator algebra; see e.g. [MS].
This is the first part of three consecutive lectures by Huang, Li and myself. In this part
we are mainly concerned with the definitions of vertex operator algebras, twisted modules

and examples. The second part by Li is about the duality and local systems and the thirdpart by Huang is devoted to the contragradient modules and geometric interpretations ofvertex operator algebras. (We refer the reader to Li and Huang’s lecture notes for the

related topics.) So many exciting topics are not covered in these three lectures. The book
[FHL] is an excellent introduction to the subject. There are also existing papers [H1],[Ge] and [P] which review the axiomatic definition of vertex operator algebras, geometricinterpretation of vertex operator algebras, the connection with conformal field theory,
Borcherds algebras and the monster Lie algebra.

Most work on vertex operator algebras has been concentrated on the concrete examples of vertex operator algebras and the representation theory. In particular, the repre1Supported by NSF grant DMS-9303374 and a research grant from the Committee on Research, UC
Santa Cruz.

2

sentation theory for the concrete vertex operator algebras, which include the moonshine
vertex operator algebra$V^{\mathfrak{h}}([\mathrm{F}\mathrm{L}\mathrm{M}3],[\mathrm{D}3])$, the vertex operator algebras based on evenpositive definite lattices [D1], the vertex operator algebras associated with the integrable

representations of affine Lie algebras, Virasoro algebra and $W$-algebras$([\mathrm{D}\mathrm{M}\mathrm{Z}],$$[\mathrm{D}\mathrm{L}3]$,[FKRW], [FZ], [KW] and [W] , have been studied extensively. There are also abstract$)$

approaches such as one to one correspondence between the set of inequivalent irreducible(twisted) modules for a given vertex operator algebra and the set of inequivalent irre-

ducible modules for an associative algebra associated with the vertex operator algebra
and an automorphism of the algebra (see [DLM] and [Z]), the theory of local system[LI-L2], induced modules [DLi], the tensor products of modules ([HLI-HL4], [H3] and[L3] ; See also [FHL] for the results concerning intertwining operators and contragredient$)$

modules. Many of these results are analogues of the corresponding results in the classicalLie algebra theory.

This paper is organized as follows: In Section 2, we present the definitions of vertex
operator algebras and twisted modules. We also make some remarks. The Section 3 isabout the examples of vertex operator algebras. In particular, we discuss the vertex operator algebras associated with Heisenberg algebras, positive definite even lattices, affine
Lie algebras and Virasoro algebra. We emphasize how a vertex operator algebras withfinitely many generators can be constructed by using the Jacobi identity. In Section 5
we finally mention the orbifold theory which is not treated in the introductory text andvarious generalizations of the notion of vertex operator algebras.

Acknowledgment. This paper is an expanded version of my lecture in the workshop of “Moonshine and vertex operator algebra” at Research Institute of Mathematical

Science at Kyoto in the Fall of 1994. I thank the organizer Professor Miyamoto for theopportunity to present talk at this stimulating workshop. I also would like to thankProfessors Miyamoto and Yamada for their hospitality during my visit in Japan.

32Vertex operator algebras and modules
The notion of vertex (operator) algebra arose naturally from the problem of realizing themonster simple group as a symmetry group of certain algebraic structure. In this section
we shall review the definition of vertex operator algebras and their (twisted) modules (see[B1], [D2], [FFR], [FHL] and [FLM3] . We shall also discuss some consequences of the$)$definitions and basic properties.
First we introduce some notation. We shall use commuting formal variables$z_{2}$$z,$$z_{0},$$z_{1}$,, etc., and the basic generating function$\delta(z)=\sum_{n\in \mathrm{Z}}Z^{n}$

,

(2.1)

formally the expansion of the -function at $z=1$ . The fundamental (and elementary)$\delta$properties of the -function can be found in [FLM3], [FHL] and [DL3].
$\delta$

For a vector space$V$(we work overspace of formal Laurent series in$z^{1/r}$$\mathbb{C}$) and a positive integer

with coefficients in

$V$

$r$

we denote the vectorby.

$V[[z, Z-1/\gamma]1/t]=\{_{n\in\frac{\sum_{1}}{r}\mathrm{Z}}v_{n}Z|nv_{n}\in V\}$

(2.2)

We also define a formal residue notation:${\rm Res}_{z}v_{n^{Z}}=n \in\frac{\sum_{1}}{r}\mathrm{Z}nv-1$.

A vertex operator algebra is a -graded vector space:$\mathbb{Z}$$V= \prod_{n\in \mathrm{z}}V_{n}$such that $\dim V_{n}<\infty$ for all$n\in \mathbb{Z}$;for$v\in V_{n}$and $V_{n}=0$ if$n$,$n=\mathrm{w}\mathrm{t}v$

;

(2.3)

is sufficiently small; equipped with alinear map$\langle Varrow(\mathrm{E}\mathrm{n}\mathrm{d}V)[[z, Z^{-1}]]$.
$\langle$$v \vdasharrow \mathrm{Y}(v, Z)=\sum v_{n}zn\in \mathrm{z}-n-1$($v_{n}\in$

End )
$V$

(2.4)

4

and with two distinguished vectors $1\in V_{0},$satisfying the following conditions for$\omega\in V_{2}$$u,v\in V$ :for$u_{n}v=0$

suffi.ciently large;

$n$

(2.5)

$Y(1, z)=1$ ;

(2.6)

and

$\mathrm{Y}(v, z)1\in V[[z]]$$\lim_{zarrow 0}\mathrm{Y}(v, z)1=v$

;

(2.7)

$\langle_{Z_{0}^{-1}}\delta(\frac{z_{1}-z_{2}}{z_{0}})\mathrm{Y}(u, z_{1})\mathrm{Y}(v, z2)-z_{0}-1\delta(\frac{z_{2}-z_{1}}{-z_{0}})\mathrm{Y}(v, z_{2})\mathrm{Y}(u, Z_{1})$(2.8)$\langle=z_{2}^{-1}\delta(\frac{z_{1}-z0}{z_{2}})\mathrm{Y}(\mathrm{Y}(u, Z_{0})v,Z_{2})$(Jacobi identity) where all binomial expressions, for instance, $(z_{1}-z_{2})^{n}(n\in \mathbb{Z})$ areto be expanded in nonnegative integral powers of second variable$z_{2}$: This identity isinterpreted algebraically as follows: if this identity is applied to a single vector of

the coefficient of each monomial in$z_{0},$$z_{1},$ $z_{2}$is a finite sum in$V$$n\in \mathbb{Z}$

then

;

$[L(m), L(n)]=(m-n)L(m+n)+ \frac{1}{12}(m^{3}-m)\delta m+n,\mathrm{o}(\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}V)$

for $m,$

$V$

(2.9)

, where$L(n)=\omega_{n+1}$

for$n\in \mathbb{Z}$

,

i.e.,

$Y( \omega, z)=\sum_{n\in \mathrm{Z}}L(n)_{Z}-n-2$(2.10)

andrank$V\in \mathbb{Q}$;(2.11)$L(\mathrm{O})v=nv=(\mathrm{w}\mathrm{t}v)v$

rank$V\in \mathbb{Q}$for(2.12)$v\in V_{n}$;(2.13)$L(\mathrm{O})v=nv=(\mathrm{w}\mathrm{t}v)v$

for$\frac{d}{dz}\mathrm{Y}(v, z)=\mathrm{Y}(L(-1)v, Z)$$v\in V_{n}(n\in \mathbb{Z})$.

;(2.14)(2.15)

This completes the definition. We denote the vertex operator algebra just defined by$(V, Y, 1,\omega)$(or briefly, by$V$). The series$\mathrm{Y}(v, z)$

are called vertex operators.

(2.16)

5

The following are consequences of the definitions: For

Let

$S$

,

$u,$ $v\in

V$$[L(-1), \mathrm{Y}(v, Z)]=Y(L(-1)v, z)$(2.17)$[L(0), Y(v,z)]=Y(L(0)v, z)+z\mathrm{Y}(L(-1)v, z)$(2.18)$Y(e^{z\mathrm{o}L\mathrm{t}}-1)v,z)=\mathrm{Y}(v, Z+z_{0})$(2.19)$e^{z_{\mathrm{O}}L\mathrm{t}-}1)Y(v,z)e^{-zL\mathrm{t}}-1)=Y(0z0L\mathrm{t}^{-}1)zev,)$(2.20)$\mathrm{Y}(v, z)1=e^{zL(1}-)v$(2.21)$Y(u, z)v=e^{zL(-1)}Y(v, -Z)u$ .

(2.22)

be a subset of . The subalgebra$V$$\langle S\rangle$

generated by$S$, defined as the smallest

subalgebra containing , is given by$S$$\{v_{n_{2}}^{1}\cdots v_{n_{k}}k.

1|vi\in S\cup\{1,\omega\}, n_{i}\in \mathbb{Z}\}$is generated by

A vertex operator algebra$V$a finite set we say thatis finitely generated.

An ideal$v\in I$

$I$

of

$V$

$V$

is a subspace of

$V$

$S$

if

$V=\langle S\rangle$

such that$1\not\in I$.

. If we further assume that

$S$

is

and $u_{n}v\in I$ for any $u\in V$ and. Using the skew symmetry (2.22) one can verify that the quotient space $V/I$ has astructure of vertex operator algebra.
An automorphism
1 and$\omega$$g$of the vertex algebrasuch that the actions of

$g$

$V$

is a linear automorphism ofand $Y(v, z)$ on

$V$

$V$

preserving

are compatible in the sense that
(2.23)$g\mathrm{Y}(v, Z)g^{-1}=\mathrm{Y}(gv, z)$

for $v\in V$. Then$gV_{n}\subset V_{n}$for$n\in \mathbb{Z}$

and

$V$

is a direct sum of the eigenspaces of$r$is the order of$g,$$\eta=e^{2\pi i/\gamma}$:(2.24)$V=i \in \mathrm{z}/\mathrm{Z}\prod_{r}V^{j}$

where

$g$

and $V^{j}=\{v\in V|gv=\eta v\}j$ .

We also have the notion of -twisted module (see [D2], [FFR], [Le] and [FLM2]). Let
$g$

(V,

$\mathrm{Y},$$1,\omega$) be a vertex algebra and let$g$be an automorphism of$V$of order . A g-twisted$r$

module $M$ for $(Y, V, 1,\omega)$ is a -graded vector space:$\mathbb{C}$$M= \prod_{n\in \mathrm{c}}M_{n}$;for$w\in M_{n},$$n=\mathrm{w}\mathrm{t}w$;(2.25)

6such that for any fixed
for all$c\in \mathbb{C}$$\lambda\in \mathbb{C},$$M_{\lambda+n}=0$if$n \in\frac{1}{f}\mathbb{Z}$is sufficiently small and $\dim M_{c}<\infty$; equipped with a linear map$\langle Varrow(\mathrm{E}\mathrm{n}\mathrm{d}M)[[z^{1/}, z-1/’]\gamma]$$\langle$$v rightarrow \mathrm{Y}(v, z)=n\in\frac{\sum_{1}}{r}\mathrm{Z}v_{n}Z^{-}n-1$satisfying the following conditions for$u,$ $v\in

V$

for

$v_{n}\in$(2.26)

End $M$ )and $w\in M$ and$Y(v,z)=n \in l/\sum_{r+\mathrm{z}}v_{n}z-n-1$$u_{n}w=0$($l\in \mathbb{Z}$:for $v\in V^{l}$ ;(2.27)

sufficiently large;$n$(2.28)$Y(1, z)=1$ ;(2.29)$\langle z_{\overline{0}^{1}}\delta(\frac{z_{1}-z_{2}}{z_{0}}\mathrm{I}^{\mathrm{Y}(u,z_{1})\mathrm{Y}(v,z_{2})}-z\delta 0^{-1}(\frac{z_{2}-z_{1}}{-z_{0}})\mathrm{Y}(v, Z_{2})Y(u, z_{1})$(2.30)$\langle=z_{2}^{-1}(\frac{z_{1}-Z_{0}}{z_{2}})^{-i/r_{\delta}}(\frac{z_{1}-z_{0}}{z_{2}})\mathrm{Y}(\mathrm{Y}(u, z_{0})v,$ $z_{2})$where $u\in V^{i}$ and $Y(u, z_{0})$ is an operator on$V$;$[L(m), L(n)]=(m-n)L(m+n)+ \frac{1}{12}(m^{3}-m)\delta m+n,\mathrm{o}(\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}V)$

for $m,$$n\in \mathbb{Z}$(2.31), where$L(n)=\omega_{n+1}$

for$n\in \mathbb{Z}$,$L(0)w=nw=(\mathrm{w}\mathrm{t}w)w$

i.e.,
for

$\frac{d}{dz}\mathrm{Y}(v,z)=\mathrm{Y}(L(-1)v,Z)$$\mathrm{Y}(\omega, z)=\sum_{n\in \mathrm{Z}}L(n)Z^{-}n-2$$w\in M_{n}(n\in \mathbb{Q})$

.

;

;

(2.32)(2.33)(2.34)

This completes the definition. We denote this module by $(M, Y)$ (or briefly by $M$ ).

Remark 2.1 If $g=1$ , then $M$ is an ordinary module in the precise sense of [FLM3].
Note that any -twisted
$g$

$V$

-module is a$V^{0}$-module.

Remark 2.2 In the definition of twisted module if removing the finiteness condition$\dim M_{c}<\infty$

we get$a$weak -twisted module.
$g$7Remark 2.3 Taking${\rm Res}_{z_{0}}$in$(\mathit{2}.\mathit{3}\theta)$, we get the commutator formula:$[ \mathrm{Y}(u, z1), \mathrm{Y}(v, z2)]={\rm Res}_{z_{0}}\{z_{2}^{-1}(\frac{z_{1}-z_{0}}{z_{2}})^{-i/r}\delta(\frac{z_{1}-Z_{0}}{z_{2}})\mathrm{Y}(\mathrm{Y}(u, z0)v,$ $z2)\}$

Note that the factor$(_{z_{2}}^{z-z}\wedge \mathrm{A})^{-}i/r_{\delta}$.

only involves the nonnegative powers of(2.35)$z_{0}$. Thusin the computation of commutator $[Y(u, z_{1}), \mathrm{Y}(v, z2)]$ we only use the “singular terms”
in $Y(u, z_{0})v$ , namely,
Moreover, for$s,$ $t\in \mathbb{Q}$$\Sigma_{n\geq 0}\mathrm{o}^{u_{n}v}z^{-n-}1$. This fact was well-known in the physics literature.

, we can compare the coefficients of(2.35) to get the commutator$[u_{S}, v_{n}]$$z_{1}^{-S-}Z_{2}1-t-1$

on the both sides of:$[u_{S}, v_{t}]=u_{S}vt-vtu= \sum_{0}Sm\geq(u_{m}v)_{S}+t-m$

Remark 2.4 Acting on $M$ the operator$\mathrm{Y}(\mathrm{Y}(u, z_{0})v,$

$z2)$

.

(2.36)

is determined uniquely by theoperators $Y(u, Z_{1})$ and $Y(v, Z_{2})$ . In order to see this, we first recall a relation on

from$\delta$-functions$[FHL].\cdot$$z_{1}^{-1}( \frac{z_{2}+z_{0}}{z_{1}})^{i/\mathrm{r}_{\delta}}(\frac{z_{2}+z_{0}}{z_{1}})=z_{2}^{-1}(\frac{z_{1}-Z_{0}}{z_{2}})^{-i/\mathrm{r}_{\delta}}(\frac{z_{1}-Z_{0}}{z_{2}})$

Now multiplying (2.8) by$z_{1}^{i/\gamma}$$\mathrm{Y}(\mathrm{Y}(u, z\mathrm{o})v,$

and taking${\rm Res}_{z_{1}}$.

we see that

$Z2)={\rm Res}_{z}z_{1}-i/rz_{0^{1}}^{-} \delta 1(\frac{z_{1}-z_{2}}{z_{0}})\mathrm{Y}(u, z1)\mathrm{Y}(v, Z_{2})$.

$-{\rm Res}_{z_{1}}z_{1^{/_{z}1}}^{i} \delta f-0(\frac{z_{2}-z_{1}}{-Z_{0}})\mathrm{Y}(v, z2)\mathrm{Y}(u, Z_{1})$If $g=1$ noting thatand taking${\rm Res}_{z_{0}}$$u_{n}v$

is the coefficient of$z_{0^{n-1}}^{-}$(2.37)

in $Y(u, z_{0})v$ , multiplying (2.37) by$z_{0}^{n}$

we find that$\mathrm{Y}(u_{n}v, Z_{2})$$={\rm Res}_{z_{0}}\{z_{0}^{n-1}{\rm Res}_{z_{1}}\{$ $\delta(\frac{z_{1}-z_{2}}{z_{0}})\mathrm{Y}(u, z_{1})Y(v, Z_{2})$$- \delta(\frac{z_{2}-z_{1}}{-z_{0}})\mathrm{Y}(v, z2)\mathrm{Y}(u, z_{1})\}\}$$={\rm Res}_{z_{1}}\{(z_{1^{-z_{2}}})^{n}\mathrm{Y}(u, z1)\mathrm{Y}(v, z2)-(-z_{2}+z_{1})^{n}\mathrm{Y}(v, z2)\mathrm{Y}(u,z1)\}$. (2.38)

In fact, this suggests a way to construct vertex operator dgebras and their modules forfinitely generated vertex operator algebras. The vertex operator algebras associated with

8

highest weight representations for affine Lie algebras and Virasoro algebra were constructed in this spirit $[FZ]$. For a different construction by using so called “the local
systems” see [Ll-L2f.

Here we give an easy consequence of (2.38). In (2.38), if $n=-1$ , we have$\mathrm{Y}(u_{-1}v, z_{2})=\sum_{0m<}u_{m}Z-m-1\mathrm{Y}2(v, z_{2})+Y(v, z2)\sum u_{m2}m\geq 0Z^{-}m-1$

This suggests defining a “normal ordering” operation by: For$\langle$$u_{m}v_{n}$$\langle$$u,$ $v\in V,$$m,$ $n\in \mathbb{Z}$

if $m<0$$v_{n}u_{m}$$\langle$,(2.39)$\cross \mathrm{x}\mathrm{x}^{u_{m}v_{n}}\cross=\{$$\langle$.

if $m\geq 0$ .

Then (since $v=v_{-1}1$ )$Z_{2})Y(v, Z2)^{\mathrm{x}}$$\mathrm{Y}(u_{-1}v_{-1}1, z_{2})=Y(u_{-1}v, z_{2})=\mathrm{x}^{Y(}\cross\crossu,$.

(2.40)

However, this normal ordering is not in general a commutative operation, since if $m$ and$n$are both negative or both nonnegative,

and$\mathrm{x}^{u_{m}v_{n_{\cross}}}\mathrm{X}\cross$$\mathrm{x}_{v_{n}u_{m_{\cross}^{\mathrm{X}}}}\mathrm{x}$differ$\mathrm{b}\mathrm{y}\pm[u_{m}, v_{n}]$. Thefollowing proposition explains that the nilpotent property of vertex operators which holds

in the algebra will also hold for modules under a mild assumption [DL3]. In particular,
this is true for the vertex operator algebras associated with the integrable highest weight

representations of affine Lie algebras.

Proposition 2.5 Let$V$

be any vertex operator algebra and let $(W, Y_{W})$ be any V-module.

Let $v\in V$ be such that the component operators
one another, so that$\mathrm{Y}_{W}(v, z)N$

$v_{n}(n\in \mathbb{Z})$is well defined on $W$ forof$\mathrm{Y}_{W}(v, z)$$N\in \mathrm{N}$$\mathrm{Y}_{W}((v_{-1})^{N}1, Z)=Y_{W}(v, z)N$

all commute with. Then.

(2.41)

In particular, if $(v_{-1})^{N}1=0$ for a fixed $N$ , then
$Y_{W}(v, z)^{N}=0$

Let

$V$

be a vertex operator algebra andis -rational if
$g$

$V$

$V$

$V$

$g$

.

(2.42)$\square$a finite order automorphism. We say that$V$

has only finitely many irreducible -twisted modules and if any g-twisted$g$-module is completely reducible.
is rational and

$V$

$V$

is rational if$V$

is $id$-rational.

is the only irreducible module for itself.

$V$is holomorphic if

9Examples3

In this section we will present some well-known examples of vertex operator algebras
and their modules. We refer the reader to [D2], [DL4], [FFR], [FLMI-FLM3], [Le], [L2],[DM2-DM3] for various examples of twisted modules.

Vertex operator algebras associated with Heisenberg algebras. Let
vector space equipped with a symmetric nondegenerate bilinear form

identify$\mathrm{h}$with its dual

Viewing$\mathrm{h}$$\mathrm{h}^{*}$$\langle\cdot, \cdot\rangle$$\mathrm{h}$be a. So we cannaturally.

as an abelian Lie algebra, we consider the corresponding affine Lie algebra(3.1)$\hat{\mathrm{h}}=\mathrm{h}\otimes \mathbb{C}[t, t^{-1}]\oplus \mathbb{C}c$

with structure defined by$[x\otimes t^{m}, y\otimes t^{n}]=\langle x, y\rangle m\delta_{m}+n,0c$$[c,\hat{\mathrm{h}}]=0$

for$x,$ $y\in \mathrm{h},$$m,$ $n\in \mathbb{Z}$

,

(3.2)

.

(3.3)

Set
$\hat{\mathrm{h}}^{+}=\mathrm{h}\otimes t\mathbb{C}[t],\hat{\mathrm{h}}^{-}=\mathrm{h}\otimes t^{-1}\mathbb{C}[t-1]$

.

(3.4)

The subalgebra(3.5)$\hat{\mathrm{h}}_{\mathrm{Z}}=\hat{\mathrm{h}^{+}.}\oplus\hat{\mathrm{h}}^{-}\oplus \mathbb{C}c$of$\hat{\mathrm{h}}$is a Heisenberg algebra, in the sense that its commutator subalgebra coincides with

its center, which is one-dimensional. Let

and consider the induced -module$\hat{\mathrm{h}}$$\lambda\in \mathrm{h}$$M(1, \lambda)=U(\hat{\mathrm{h}})\otimes_{U(\mathrm{h}t}\otimes \mathrm{c}[]\oplus \mathrm{c}_{C})^{\mathbb{C}_{\lambda}}\simeq S(\hat{\mathrm{h}}^{-})$$\mathrm{h}\otimes t\mathbb{C}[t]$acting trivially on

write $M(1)$ for $M(1,0)$ . For$\mathrm{h}$$\mathbb{C},$acting as$\alpha\in \mathrm{h}$and$\langle h, \lambda\rangle$$n\in \mathbb{Z}$for$h\in \mathrm{h}$we writeset$\alpha(z)=\sum\alpha(n)_{Z}n\in \mathrm{z}-n-1$$\alpha(n)$(linearly),

and

$c$

(3.6)

acting as 1. We shallfor the operator$\alpha\otimes t^{n}$

and

10

and define$Y(v, z)= \mathrm{o}\mathrm{o}(\frac{1}{(n_{1}-1)!}(\frac{d}{dz})n1-1\mathrm{I}\alpha_{1}(z)\cdots(\frac{1}{(n_{k}-1)!}(\frac{d}{dz})^{n_{k}}-1\alpha_{k}(Z)\mathrm{I}\mathrm{o}$

’

where
$v=\alpha_{1}(-n_{1})\cdots\alpha_{k}(-n_{k})\in M(1)$for$\alpha_{1},$$\alpha_{k}\in \mathrm{h},$$n_{1},$$\ldots,$$\ldots,$$n_{k}\in \mathbb{Z}(n_{i}>0)$

and where we use a normal ordering procedure,

indicated by open colons, which signify that the expression is to be reordered if necessary
so that all the operators $\alpha(n)(\alpha\in \mathrm{h}, n<0)$ , are to be placed to the left of all theoperators$\alpha(n)(\alpha\in \mathrm{h}, n\geq 0)$

before the expression is evaluated. Extend this definition

to all $v\in M(1)$ by linearity. Set $1=1,$$\omega=\frac{1}{2}\sum_{i=1}^{d}\alpha i(-1)^{2}\in M(1)$where$\{\alpha_{i}\}$is anorthonomal basis of . The following result can be found in [Gu].
$\mathrm{h}$Proposition 3.1 The space $M(1)=(M(1), Y, 1,\omega)$ is a vertex operator algebra and$M(1, \lambda)=(M(1, \lambda),$$\lambda\in \mathrm{h}$$\mathrm{Y})$is a complete list of inequivalent irreducible module for $M(1)$ for.

Remark 3.2 It is easy to see that the vertex operator algebra $M(1)$ is generated by $S=$$\{\alpha_{i}(-1)\}$. If we identify $M(1)$ with a symmet$r\cdot ic$

algebra $\mathbb{C}[x_{i,n}|i=1, \ldots, d, n=1,2, \ldots]$ .

Then it is easy to see that all the vertex operators are built from the operators
is a multiplication operator on $M(1)$ and$\frac{\partial}{\partial x:,n}$$x_{i,n}$

which

.

Vertex operator algebras associated with even positive definite lattices. Let$L$

be an even lattice, i.e., a finite-rank free abelian group equipped with a symmetricnondegenerate$\langle\alpha, \alpha\rangle\in 2\mathbb{Z}$to$\mathrm{h}$$\mathbb{Q}$-valued -bilinear form

for all$\mathbb{Z}-$$\alpha\in L$$\langle\cdot, \cdot\rangle$. Set vector space, not necessarily positive definite such that$\mathrm{h}=\mathbb{C}\otimes_{\mathrm{Z}}L$and extend the form$\langle\cdot, \cdot\rangle$

from

$L$

by -linearity.
$\mathbb{C}$The dual lattice$L^{\mathrm{o}}$

of

$L$

is defined to be$L^{\mathrm{o}}=\{\beta\in \mathrm{h}|\langle\beta, L\rangle\subset \mathbb{Z}\}$

Then$L^{\mathrm{o}}$

.

(3.7)

is a rational lattice whose rank is equal to the rank ofextension of$L^{\mathrm{o}}$

$L$

. Let

$\hat{L}^{\mathrm{o}}$be a central:

1$arrow\langle\omega_{q}\ranglearrow\hat{L}^{\mathrm{O}}arrow-L^{\mathrm{O}}arrow 1$

,

(3.8)

11

with the commutator map
where$\langle\omega_{q}\rangle$for$c(\alpha,\beta)$$\alpha,\beta\in L^{\mathrm{O}}$such thatis the cyclic group generated by a primitive$c(\alpha,\beta)=(-1)^{(}\alpha,\beta\rangle$$q^{th}$root of unity

if$\alpha,\beta\in L$$\omega_{q}\in \mathbb{C}^{\cross}$

and

$q$

,

is

a positive even integer.
Form the induced$\hat{L}^{\mathrm{o}}$-module and -algebra$\mathbb{C}$$\mathbb{C}\{L^{\mathrm{o}}\}=\mathbb{C}[\hat{L}^{\mathrm{O}}]\otimes_{\mathbb{C}[\mathrm{t}^{\omega_{q}}})]\mathbb{C}\simeq \mathbb{C}[L^{\mathrm{O}}]$$\langle$where$\mathbb{C}[\cdot]$

denotes group algebra,

for the image of$\iota(a)$

$a$

in

$\mathbb{C}\{L^{\mathrm{O}}\}$$\omega_{q}$acts on$\mathbb{C}$(3.9)(linearly),

as multiplication by. Then the action of$a\cdot\iota(b)=\iota(ab)$$\hat{L}^{\mathrm{o}}$on$\mathbb{C}\{L^{\mathrm{o}}\}$$\omega_{q}$. For$a\in\hat{L}^{\mathrm{O}}$is given by:,(3.10)(3.11)$\omega_{q}\cdot\iota(b)=\omega_{q}\iota(b)$for$b\in\hat{L}^{\mathrm{o}}$$a,$. We givethe -gradation determined by:$\mathbb{C}$$\mathbb{C}\{L^{\mathrm{o}}\}$

wt

Also define an action of

for

for

$h\in \mathrm{h}$$h\in \mathrm{h}$$\mathrm{h}$on$\iota(a)=\frac{1}{2}\langle\overline{a},\overline{a}\rangle$$\mathbb{C}\{L^{\mathrm{O}}\}$

for$a\in\hat{L}^{\mathrm{O}}$.

(3.12)

by:$h\cdot\iota(a)=\langle h,\overline{a}\rangle\iota(a)$(3.13)$z^{h}\cdot\iota(a)=z^{\langle h,\overline{a})}\iota(a)$(3.14). Define. Set$V_{L^{0=}}M(1)\otimes_{\mathbb{C}}\mathbb{C}\{L\mathrm{O}\}\simeq S(\hat{\mathrm{h}}^{-})\otimes \mathbb{C}[L\circ]$

Then, write$\hat{L}^{\mathrm{o}},\hat{\mathrm{h}}_{\mathrm{Z}},$$h,$$z^{h}(h\in \mathrm{h})$act naturally on

indicated above. In particular,
For a subset $M$ of$L^{\mathrm{o}}$$c$$V_{L^{\mathrm{o}}}$(linearly)

by acting on either $M(1)$ or(3.15)$\mathbb{C}\{L^{\mathrm{O}}\}$

asacts as 1.

(not necessarily a sublattice), we write(3.16)$\hat{M}=\{b\in\hat{L}^{\mathrm{O}}|\overline{b}\in M\}$and$\mathbb{C}\{M\}=\mathrm{s}_{\mathrm{P}}\mathrm{a}\mathrm{n}\{\iota(b)|b\in\hat{M}\}\subset \mathrm{C}\{L^{\mathrm{o}}\}$,(3.17)

12$V_{M}=M(1)\otimes \mathbb{C}\{M\}\subset VL^{\circ}$.

(3.18)(Later we shall take $M$ to be a coset of a lattice.) We may apply the considerations of

the present section to$\hat{L}$in place of$\hat{L}^{\mathrm{O}}$. In particular, we have
(3.19)$\mathbb{C}\{L\}=\mathbb{C}[\hat{L}]\otimes_{\mathbb{C}[(v_{\mathrm{p}}}1^{\mathbb{C}}$and$V_{L}=M(1)\otimes \mathbb{C}\{L\}$.

(3.20)

Let(3.21)$L^{\mathrm{o}}= \in\bigcup_{iL\mathrm{Q}/L}(L+\lambda_{i})$be the coset decomposition such that$\lambda_{0}=0$. Let$\Lambda_{i}\in\hat{L}^{\mathrm{O}}$so that$\overline{\Lambda}_{i}=\lambda_{i}$. Then(3.22)$\hat{L}^{\mathrm{o}}--\bigcup_{Li\in L\circ/}\hat{L}\Lambda i$is the coset decomposition of$\hat{L}^{\mathrm{o}}$. For$i\in L^{\mathrm{O}}/L$, set$V(i)=VL+\lambda.\cdot=M(1)\otimes \mathbb{C}\{L+\lambda_{i}\}\simeq S(\hat{\mathrm{h}}-)\otimes \mathbb{C}[L+\lambda_{i}]$(linearly).

(3.23)

Then we have
$V_{L^{\mathrm{o}}}= \prod_{Li\in L^{\mathrm{o}}/}V(i)$.

(3.24)We shall next define the untwisted vertex operators $Y(v, z)$ forand$n\in \mathbb{Z}$we write$\alpha(n)$for the operator$\alpha\otimes t^{n}$$\mathrm{o}\mathrm{o}\mathrm{o}.0$.

evaluated. For$a\in\hat{L}^{\mathrm{o}}$$\alpha(n),$$\alpha\in \mathrm{h}$(3.25)

to signify that the enclosed expression is

to be reordered if necessary so that all the operators $\alpha(n)(\alpha\in \mathrm{h}, n<0),$
be placed to the left of all the operators. For

and set$\alpha(z)=\sum\alpha n\in \mathrm{z}(n)_{Z}-n-1$

As before we use a normal order notation$v\in V_{L^{\mathrm{o}}}$$z^{\alpha}(\alpha\in \mathrm{h}, n\geq 0)$$a\in\hat{L}^{\mathrm{o}}$are tobefore the expression is, set$\mathrm{Y}(\iota(a), z)=\mathrm{Y}(a, z)=00\circ\int e(z)-\overline{a}(0(\overline{a})z-1)aZ^{\overline{a}\mathrm{O}}$,(3.26)

13

using an obvious formal integration notation. Let$a\in\hat{L},$$\alpha_{1},$$\alpha_{k}\in \mathrm{h},$$\ldots,$$n_{1},$$\ldots,$$n_{k}\in \mathbb{Z}(n_{i}>0)$

and set$\langle v=\alpha_{1}(-n_{1})\cdots\alpha_{k}(-n_{k})\otimes\iota(a)$$=\alpha_{1}(-n_{1})\cdots\alpha_{k}(-n_{k})\cdot\iota(a)\in VL^{\circ}$$\langle$(3.27).

We define$\mathrm{Y}(v, z)=\mathrm{o}\mathrm{o}(\frac{1}{(n_{1}-1)!}(\frac{d}{dz})^{n-1}1\mathrm{I}\alpha_{1}(Z)\cdots(\frac{1}{(n_{k}-1)!}\alpha_{k}(z))Y(a, z)^{\circ}0^{\cdot}$$.\cdot\iota$:(3.28)

This gives us a well-defined linear map$\langle V_{L^{\circ}}arrow(\mathrm{E}\mathrm{n}\mathrm{d}V_{L^{\mathrm{o}}})\{z\}$$\langle vrightarrow Y(v, z)=n\sum_{\mathbb{C}\in}v_{n}z^{-}n-1$,$(v_{n}\in \mathrm{E}\mathrm{n}\mathrm{d}VL\mathrm{o})$(3.29),

where for any vector space $W$ , we define $W\{z\}$ to be the vector space of $W$-valued formal
series in , with arbitrary complex powers of
$z$

$z$

allowed:$W \{z\}=\{\sum_{n\in \mathbb{C}}w_{n^{Z}}n|w_{n}\in W\}$

We call$\mathrm{Y}(v, z)$(3.30)

the untwisted vertex operator associated with .
$v$

Theorem 3.3 1.

2. $V(i)$ for.

$(V(\mathrm{O}), \mathrm{Y}, 1,\omega)$$\dot{i}\in L^{\mathrm{O}}/L$

is a simple

vertex’ operator algebra (see).

$[Bl\mathit{1},[FLM\mathit{3}]$is a complete list of inequivalent irreducible modules for$V(\mathrm{O})$(see $fDlf$, [FLM3]).

3. Any$V(\mathrm{O})$-module is completely reducible, that is,

Note that from this theorem,$V_{L}$$V(\mathrm{O})$is rational (see $[Gu]$).

is holomorphic if and only if$L$

is self dual:$L=L^{\mathrm{O}}$.

So a holomorphic vertex operator algebra is also called a self dual vertex operator algebra[Go].

Vertex operator algebras associated with the highest weight representationsof affine Lie algebras. Let$\triangle$$\mathrm{g}$be a simple Lie algebra over$\mathrm{h}$$\mathbb{C},$the corresponding root system. We fix a set of positive root

its Cartan subalgebra and$\triangle+\mathrm{a}\mathrm{n}\mathrm{d}$a nondegenerate

14

symmetric invariant bilinear form
2. The affine Lie algebraof$(\cdot, \cdot)$$\mathrm{g}$such that the square length of a long root is

is defined as$\hat{\mathrm{g}}$$\hat{\mathrm{g}}=g\otimes \mathbb{C}[t,t^{-}1]\oplus \mathbb{C}c$

with Lie algebra structure$[x\otimes t^{m}, y\otimes t^{n}]=[x,y]\otimes t^{m+n}+(x,y)m\delta_{m}+n,0c$

for$x,$ $y\in \mathrm{h}$for$x(n)$

Let

and

$c$

and $m,$$x\mathrm{x}t^{n}$$n\in \mathbb{Z}$. Then$\lambda\in \mathrm{h}^{*}$

is an center element. For convenience we shall write
and$\mathrm{g}$(identified with withbe a -module which is extended to a

$V$

$k$

$c$

$\hat{\mathrm{g}}\pm=\hat{\mathrm{g}}\otimes t^{\pm}\mathbb{C}[t\pm]$$\mathrm{g}$act as a fixed scalarof level

where

. If

$V=V(\lambda)$$k\in \mathbb{C}$. Let

is linearly isomorphic to$\hat{V}_{k}$on$\hat{V}_{k,\lambda}$by$U(\hat{\mathrm{g}}-)$

vertex operator algebra structure on
also define an action of-module by letting$\hat{V}_{k}$if$\hat{V}_{k,\lambda}$. Then$\hat{V}_{k,\lambda}$$\hat{\mathrm{g}}_{+}$

act as$0$

be the induced -module$\hat{V}_{k}=U(\hat{\mathrm{g}})\otimes_{U\langle\oplus\oplus \mathrm{c}c}\hat{\mathrm{g}}+\mathrm{g})V$$\hat{V}_{k}$) are subalgebras.

$\hat{\mathrm{g}}_{+}\oplus \mathrm{g}\oplus \mathbb{C}c$$\hat{\mathrm{g}}$is an irreducible highest weight module forwe denote the corresponding$\hat{V}_{k}=\hat{V}_{k,0}$$\mathrm{g}\otimes 1$$\mathrm{g}$with highest weightis a Verma module. Note thatas vector spaces. Our goal next is to define a$k$is not equal to the dual Coxeter number and.

First for $u(-1)\in\hat{V}_{k}$ we define$\mathrm{Y}(u(-1), z)=u(z)\sum_{n\in \mathrm{Z}}u(n)_{Z}-n-1$.

(3.31)

Then using the $L(-1)$ -derivation property (2.15) to define$\mathrm{Y}(u(-n-1), z)=\frac{1}{n!}\frac{d^{n}}{dz^{n}}\mathrm{Y}(u(-1), z)$In general if$0<n\in \mathbb{Z}$$\mathrm{Y}(v, z)$.

(3.32)

has been defined, we use (2.38) define $\mathrm{Y}(u(-n)v, Z)$ forand$u\in \mathrm{g}$as$\mathrm{Y}(u(-n)v, z2)={\rm Res}_{z_{1}}\{(z_{1}-z2)^{-}n\mathrm{Y}(u, z1)\mathrm{Y}(v, Z_{2})-(-z_{2}+z_{1})^{-n}\mathrm{Y}(v, z2)\mathrm{Y}(u, z1)\}$.

(3.33)

Then we get a linear map$\frac{1}{2(k+h^{\vee})}\Sigma_{i}v_{i}(-1)2$of$\mathrm{g}$$h^{\vee}$with respect the form

[L1] :$)$

where$\mathrm{Y}$from$\hat{V}_{k}$to$(\mathrm{E}\mathrm{n}\mathrm{d}\hat{V}_{k})[[z, Z-1]]$is the dual Coxeter number and$(\cdot, \cdot)$. Set 1 $=1\cross 1$ and$\{v_{i}\}$$\omega=$

is an orthonomal basis. The following theorem can be found in [FZ] (also see

15

Theorem 3.4 (1) If$k\neq-h^{\mathrm{v}}$(2) Define an action ofmodule for$\hat{V}_{k}$then$\hat{V}_{k}$on$(\hat{V}_{k}, \mathrm{Y}, 1,\omega)$$\hat{V}_{k,\lambda}$

is a vertex operator algebra.

by the the same formulas. Then.

Now we turn our attention to the irreducible quotients. It is well-known thatunique maximal submodule
with$1\otimes V(\lambda))$$k\neq 0$

is a$(\hat{V}_{k,\lambda}, \mathrm{Y})$is . Let$0$$I(k, \lambda)$$L(k, \lambda)$

whose intersection with

$V(\lambda)$$\hat{V}_{k,\lambda}$

has a(which can be identified

be the corresponding irreducible quotient. Note that if

and $k\neq h^{\vee}I(k, 0)$ is an ideal and $L(k, 0)$ is a quotient vertex operator algebra.

The following theorem can be also found in [FZ] (see [DL3] for a different approach):
Theorem 3.5 If$k$is a positive integer, $L(k, 0)$ is rational vertex operator algebra whose

irreducible modules are$\{L(k, \lambda)|(\lambda, \theta)\leq k\}$where$\theta$

is the longest positive root in$\Delta$.

Vertex operator algebras associated with Virasoro algebras. Recall that theVirasoro algebra Vir has a basis$\{L_{n}|n\in \mathbb{Z}\}\cup\{c\}$with relation:$[L_{m}, L_{n}]=(m-n)Lm+n+ \frac{m-m^{3}}{12}\delta_{m}+n,0C$

and

$c$

is in the center. Define two subalgebras

Vir$\geq 0=\oplus_{n=0n}^{\infty}\mathbb{C}L$,

Vir$<0=\oplus_{n=1}^{\infty}\mathbb{C}L_{-n}$.

Given a pair of complex numbers $(k, h)$ , the Verma module $V(k, h)$ is an induced module$V(k, h)=U(\mathrm{V}\mathrm{i}\mathrm{r})\otimes_{U(^{}\mathrm{r})^{\mathbb{C}}}\mathrm{i}\geq 0k,h\simeq U(\mathrm{V}\mathrm{i}\mathrm{r}<0)$

where$\mathbb{C}_{k,h}=\mathbb{C}$is a module for Vir$\geq 0$such that $L_{n}1=0$ for $n>0$ and $L_{0}1=h,$ $c1=k$ .

As in the case of affine algebra, we expect $V(k, 0)$ is a vertex operator algebra. But
this is not quite true. The relations $Y(L_{-1}1, Z)= \frac{d}{dz}\mathrm{Y}(1, z)=0$ andforce $L_{-1}1=0$ . So it is natural to consider the quotient module

instead of $V(k, 0)$ where $1=1\otimes 1$ . Set$1=1+U(\mathrm{V}\mathrm{i}\mathrm{r})L_{-1}1$$\lim_{z\mapsto 0}\mathrm{Y}(v, \mathcal{Z})1=v$$V_{k}=V(k, 0)/U(\mathrm{V}\mathrm{i}\mathrm{r})L_{-1}1$. Then$V_{k}$

has a basis$\{L-n_{1}\cdots L_{-n_{s}}1|ni\in \mathrm{N}, n_{1}\geq n_{2}\geq\cdots\geq n_{2}\geq 2,s\geq 0\}$.

16

Define operator$Y(L_{-2}1, Z)=L(z)= \sum_{n\in \mathrm{z}}L_{n}z-n-2$.

(3.34)

Again using the $L(-1)$ -derivation property (2.15) to define$\mathrm{Y}(L_{-n-2}1, z)=\frac{1}{n!}\frac{d^{n}}{dz^{n}}L(z)$.

(3.35)

and use (2.38)$\mathrm{Y}(L(-n)v, Z2)={\rm Res}_{z}1\{(z_{1}-Z_{2})^{-}n+1L(z1)Y(v, z_{2})-(-z2+Z1)-n+1\mathrm{Y}(v, Z_{2})L(_{Z}1)\}$(3.36)

if $Y(v, z)$ has been defined. This yields a linear map$\omega=L_{-2}1$$\mathrm{Y}$from$V_{k}$

to$(\mathrm{E}\mathrm{n}\mathrm{d}V_{k}))[[z, z-1]]$

. Set

.

Theorem 3.6 (1)$(V_{k}, \mathrm{Y}, 1,\omega)$(2) Define an action ofa module for$V_{k}$$V_{k}$

is vertex operator algebra.

on $V(k, h)$ by the the same formulas. Then $(V(k, h),$$\mathrm{Y})$is.

This theorem was proved in [FZ]. Also see [H1]. Now it is easy corollary of the general
theory on local systems [L1].

Now we denote by $W(k, h)$ the irreducible quotient of $V(k, h)$ module the uniquemaximal submodule whose intersection with

$C_{k,h}$is . Then as before $W(k, 0)$ is a vertex$0$operator algebra. Next we shall discuss the rationality of $W(k, 0)$ . This is related to thediscrete series of unitary representations of the Virasoro algebra ([FQS] and [GKO]).
The work in [FQS] and [GKO] gives a complete classification of unitary highest weight
representations of the Virasoro algebra. The highest weight representation $W(k, h)$ is

unitary if and only if either $(k, h)$ satisfies $k\geq 1$ and $h\geq 0$ , or else $(k, h)$ is among thefollowing list:$k=c_{m}=1- \frac{6}{(m+2)(m+3)}$$h=h_{r,s}^{m}= \frac{[(m+3)r-(m+2)s]^{2}-1}{4(m+2)(m+3)}$

The unitary representations$L(c_{m}, h_{t,S}^{m})$for$(m=0,1,2\cdots)$ ,$(r, s\in \mathrm{N}, 1\leq s\leq r\leq m+1)$$(c_{m}h_{\Gamma}^{m})$.

in the discrete series as above arecalled the discrete series of the Virasoro algebra. If $m=0$ , the only unitary representation

17

is the trivial representation$L( \frac{1}{2},0),$and$L( \frac{1}{2}, \frac{1}{2})$$L(\mathrm{O},0)$$L( \frac{1}{2}, \frac{1}{16})$. If $m=1$ , there are three unitary representations.

Theorem 3.7 $W(k, 0)$ is a rational vertex operator algebra if and only if $k=c_{m}$ . In thiscase all irreducible modules are given by$L(c_{m}, h_{fs}m,)$.

The rationality of $L(c_{m},0)$ was proved and all its irreducible modules were found in[DMZ]. This theorem was proved completely later in [W].

4

Twisted modules and associative algebras

In [Z], Zhu introduced an associative algebra $A(V)$ associated to a VOA$V$

which isextremely useful in study the representation theory of . These are analogues for g-twisted$V$

modules in [DLMI]. We shall review these results form [DLMI].

Fix a VOA$V$

and an automorphism$g$of finite . For$r$$u,$ $v\in

V$

with

$u$

homogeneous,define a product $u*v$ as follows:$u*v={\rm Res}_{z}( \mathrm{Y}(u, Z)\frac{(z+1)^{\mathrm{w}\mathrm{t}u}}{z}v)=\sum i=0\infty u_{i-1}v$

Then extends (4.1) to a linear

Recall from (2.24) that$V^{j}$

$u$

(4.1)

.

$\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{d}\mathrm{u}\mathrm{c}\mathrm{t}*\mathrm{o}\mathrm{n}V$is the eigenspace of$j<r$ . Define a subspace $O_{g}(V)$ offollowing type if

.

$V$

$g$

with eigenvalue$e^{2\pi ij/r}$

to be the linear span of all elements

where $0\leq$$u\mathrm{o}_{g}v$

is homogeneous and $u\in V^{j}$ and $v\in V$,$\langle$${\rm Res}_{z}( \mathrm{Y}(u,Z)\frac{(z+1)^{\mathrm{W}\mathrm{t}u}}{z^{2}}v)$$\langle$${\rm Res}_{z}(Y(u,Z) \frac{(z+1)^{\mathrm{W}\mathrm{t}u+}\mathrm{j}/r-1}{z}v)$$\langle$

if $j=0$(4.2)$u\mathrm{o}_{g}v=\{$

Note that if $0<j<r,$Lemma 4.1 If
Let$(M, \mathrm{Y}_{g})$of the$V^{(g\rangle}$$u\mathrm{o}_{g}1=u$

if $j>0$ .

. Thus we haveis the sub $VOA$ of -invariants of$g$

be a weak -twisted$g$$\langle$

$V$

$V$

then $V=V^{(g\}}+O_{g}(V)$ .

-module. Consider for each

. which, in fact, is a -twisted submodule of$M(\lambda)=\Sigma_{i\in \mathrm{Z}}M_{\lambda+\overline{K}}$

$g$

$M$

$\lambda\in \mathbb{C}$the subspaceby (2.27). Then we

18

have

$M=\oplus_{\lambda\in \mathrm{c}/\frac{\mathrm{z}}{K}(\lambda)}M$. Thus it is enough to study the twisted module of type:$M= \prod_{n\in\frac{1}{K}\mathrm{z},n\geq 0}Mc+n$

where

is a fixed number and $M_{c}\neq 0$ . We will call$c\in \mathbb{C}$For homogeneous $u\in V$, the component operator$u_{\mathrm{w}\mathrm{t}u-1}$

subspace of $M$ and in particular acts on the top levelof$u_{\mathrm{w}\mathrm{t}u-1}$to$M_{c}$$M_{c}$$M_{c}$

the “top level” of $M$ .

preserves each homogeneousof $M$ . Let$o_{g}(u)$be the restriction, so that we have a linear map$\langle$V$\langle u$$\langle$$arrow$$\langlerightarrow$$\langle$End$(M_{\mathrm{C}})$$\langle_{\mathit{0}_{g}}(u)$

(4.3)

.

Note that if $u\in V^{j/r}$ with $0<j<r,$ $o_{g}(u)=0$ from (2.27). Set $A_{g}(V)=V/O_{g}(V)$ . Then
we have [DLMI]:

Theorem 4.2 (i) $A_{g}(V)$ is an associative algebra with $multiplication*and$ the centralizer$C(g)$

of

(ii)

in

$g$

$\mathrm{A}\mathrm{u}\mathrm{t}(V)$$urightarrow o_{g}(u)$induces a group of algebra automorphisms of $A_{g}(V)$ .

gives a representation of $A_{g}(V)$ on$M_{c}$. Moreover, if any weak g-twisted

module is completely reducible, $A_{g}(V)$ is semisimple.
(iii) $Marrow M_{c}$ gives a bijection between the set of equivalence classes of simple weak$g$-twisted$V$-modules and the set of equivalence classes of simple $A_{g}(V)$ -modules.

The associative algebra $A(V)=A_{1}(V)$ was discovered in [Z] and the theorem above
was also established in this case. It was proved in [FZ] that for the vertex operator

algebra$\hat{V}_{k}$, the associative algebra

is isomorphic toby$e_{\theta}^{k+1}$. Moreover,isomorphic to

5$U(\mathrm{g})/\langle e_{\theta}^{k}\rangle$

is canonically isomorphic to$0\neq e_{\theta}\in \mathrm{g}_{\theta}$

is isomorphic to$A(V_{k})$$\mathbb{C}[x_{1,\ldots,d}X]$

where$A(\hat{V}_{k})$and$\mathbb{C}[x]$$\langle e_{\theta}\rangle$$U(\mathrm{g})$and $A(L(k,0))$is the two-sided ideal generated. The reader can verify that $A(M(1))$ is.

Some remarks

Finally we want to comment briefly on orbifold theory and on the generalizations of vertex
operator algebras.

19

It is believed that the module category for a rational VOA is equivalent to the module

category of a Hopf algebra associated with the vertex operator algebra. In the case of

VOAs associated with the level$k$highest weight unitary representations for affine Liealgebras , the Hopf algebras are essentially the quantum group$U_{q}(\mathrm{g})$$\hat{\mathrm{g}}$where$\mathbb{C}$$h$is the dual Coxeter number. If$V$is holomorphic the corresponding Hopf algebra is. It is natural next to determine the Hopf algebra for$G$

is a finite automorphism group andwhere $q=e^{2\pi i/k}h+$$V^{G}$

is the$G$$V^{G}$

where

$V$

is a holomorphic VOA,

-invariants which is a vertex operator

subalgebra of . This is the so-called orbifold theory in the physical literature. In fact, the$V$moonshine module is an$\mathbb{Z}_{2}$-orbifold theory constructed from the vertex operator algebra

associated with Leech lattice and an order 2 automorphism induced from-l isometry of
the Leech lattice (a-orbifold construction of the moonshine module have been studied$\mathbb{Z}_{p}$in [DM1] and [Mon] .
$)$The main new feature of the orbifold theory is the introduction of twisted modules
because any -twisted module is an ordinary module for$g$that for holomorphic vertex operator algebra$V$$V^{G}$

for $g\in G$ . It is conjectured

and finite automorphism groupcorresponding Hopf algebra is the quasi quantum double $D^{c}(G)$ whereuniquely determined by

$V$

$G$

, the

$c\in H^{3}(G,\mathbb{C})$is(see [DVVV], [DPR], [DM2-DM5]). There have been a lot

progress in this direction in [DM4-DM5] for nilpotent group$G$. The results concerning

the modular invariance of trace functions (or correlation functions on the torus) developedin [Z] and [DM6] play important roles in the orbifold theory. (These results also explain
mysterious relations among affine Lie algebras, Virasoro algebra, monster group and themodular group.)
In [DLI-DL3], the theory of vertex operator algebras has been generalized in a system-

atic way at three successively more general levels, all of which incorporate one.dimensional

braid group representations intrinsically into the algebraic structure: First, the notion of“generalized vertex operator algebra” incorporates such structure as -algebras, parafer$Z$

minon algebras, and the vertex operator superalgebras. Next, what they term “general-ized vertex algebras” further encompass the algebras of vertex operators associated withrational lattices. Finally, the most general of three notions, that of “abelian intertwin-

20

ing algebras,” also intertwining operators for certain classes of vertex operator algebras.
See [L4] and [DLM2] for more examples of abelian intertwining algebras related to thesimple currents. The notion of generalized vertex algebra has been also independently

introduced and studied in [FFR] with different motivations, examples (involving spinor
constructions) and axiomatic approach from ours, and also see [Mos]. See also [LZ] and[H2] for the notion of topological vertex algebra.

References[B1]

R. E. Borcherds, Vertex algebras, Kac-Moody algebras, and the Monster, Proc.
Natl. Acad. Sci. USA 83 (1986), 3068-3071.

[B2]

R. E. Borcherds, Monstrous moonshine and monstrous Lie superalgebras, Invent.
Math. 109 (1992), 405-444.

[BPZ]

A. A. Belavin, A. M. Polyakov and A. B. Zamolodchikov, Infinite conformal
symmetries in two-dimensional quantum field theory, Nucl. Phys. B241 (1984),

333-380.
[DPR]

R. Dijkgraaf, V. Pasquier and P. Roche, Quasi-quantum groups related to orbifold models, Ed. by M. Carfora, M. Martellini, A Marguolis, World Scientific,
1992, 75-89.

[DVVV] R. Dijkgraaf, C. Vafa, E. Verlinde and H. Verlinde, The operator algebra oforbifold models, Comm. Math. Phys. 123 (1989), 485-526.
[D1]

C. Dong, Vertex algebras associated with even lattices, J. Algebra 160 (1993),

245-265.
[D2]

C. Dong, Twisted modules for vertex algebras associated with even lattices, J.
Algebra 165 (1994), 91-1.1.2.

21

[D3]

C. Dong, Representations of the moonshine module vertex operator algebra,
Contemporary Math. 175 (1994).

[DL1]

C. Dong and J. Lepowsky, A Jacobi identity for relative vertex operators andthe equivalence of -algebras and parafermion algebras, in: Proc. XVIIth Intl.
$Z$

Colloq. on Group Theoretical Methods in Physics, Ste-Ad\‘ele, June, 1988, ed. Y.

Saint-Aubin and L. Vinet,[DL2]$\mathrm{W}\dot{\mathrm{o}}$rld Scientific, Singapore, 1989, 235-238.

C. Dong and J. Lepowsky, Abelian intertwining algebras –a generalization ofvertex operator algebras, Proc. Symp. Pure Math., American Math. Soc. 56(1994), 261-293.

[DL3]

C. Dong, J. Lepowsky, Generalized Vertex Algebras and Relative Vertex Operators, Progress in Math. Vol. 112, Birkh\"auser, Boston 1993.

[DL4]

C. Dong, J. Lepowsky, The algebraic structure of relative twisted vertex operators, preprint.

[DLMI] C. Dong, H. Li and G. Mason, Twisted representations of vertex operator algebras, preprint.

[DLM2] C. Dong, H. Li and G. Mason, Simple currents and abelian intertwining algebras,

preprint.
[DLi]

C. Dong and Zongzhu Lin, Induced modules for vertex operator algebras,
preprint.

[DM1]

C. Dong and G. Mason, The construction of the moonshine module as a$\mathbb{Z}_{p^{-}}$

orbifold, Contemporary Math. 175 (1994).
[DM2]

C. Dong and G. Mason, Nonabelian orbifolds and boson-fermion correspondence,
Commu. Math. Phys. 163 (1994), 523-559.

[DM3]

C. Dong and G. Mason, Conformal field theory of genus zero associated with thesporadic simple group$M_{24}$, Commu. Math. Phys. 164 (1994), 87-104.

22

[DM4]

C. Dong and G. Mason, On quantum Galois theory, preprint.

[DM5]

C. Dong and G. Mason, On the operator content of orbifold models, preprint.

[DM6]

C. Dong and G. Mason, Modular invariance of trace functions in orbifold theory,
preprint.

[DMZ]

C. Dong, G. Mason and Z. Zhu, Discrete series of the Virasoro algebra and themoonshine module, Proc. Symp. Pure. Math.,$Amer\cdot i_{C}an$

Math. Soc. 56 II (1994),

295-316.
[FFR]

A. J. Feingold, I. B. Frenkel and J. F. X. Ries, Spinor construction of vertex
operator algebras, triality and$E_{8}^{(1)}$, Contemporary Math. 121, 1991.

[FKRW] E. Frenkel, V. Kac, A Radul and W. Wang,$W_{1+\infty}$

and $W(gl_{N})$ with central

charge N, preprint.
[F]

I. B. Frenkel, Orbital theory for affine Lie algebras, Invent. Math. 77 (1984),

301-352.
[FHL]

I. B. Frenkel, Y.-Z. Huang and J. Lepowsky, On axiomatic approaches to vertex

operator algebras and modules, Memoirs Amer. Math. Soc. 104, 1993.
[FLMI] I. Frenkel, J. Lepowsky and A. Meurman, A natural representation of the Fischer-

Griess Monster with the modular function J as character, Proc. Acad. Sci. USA
81 (1984), 3256-3260.
[FLM2] I. B. Frenkel, J. Lepowsky and A. Meurman, Vertex operator calculus, in: Mathematical Aspects of String Theory, Proc. 1986 Conference, San Diego. Ed. by

S.-T. Yau, World Scientific, Singapore, 1987, 150-188.
[FLM3] I. B. Frenkel, J. Lepowsky and A. Meurman, Vertex Operator Algebras and theMonster, Pure and Applied Math., Vol. 134, Academic Press, 1988.
[FZ]

I. B. Frenkel and Y. Zhu, Vertex operator algebras associated to representationsof affine and Virasoro algebras, Duke Math. J. 66 (1992), 123-168.

23

[FQS]

D. Friedan, Z. Qiu and S. Shenker, Details of the non-unitarity proof for highest

weight representations of Virasoro Algebra, Commun. Math. Phys. 107 (1986),

535-542.
[Ge]

R. W. Gebert, Introduction to vertex algebras, Borcherds algebras, and themonster algebra, Intl. J. Modern Phys. A8 (1993), 5441-5503.
[Go]

P. Goddard, Meromorphic conformal field theory, in: Proc. Conference on In-

finite Dimensional Lie Algebras and Groups, Luming, 1988, ed. V. Kac, World
Scientific, Singapore, 1989, Advanced Series in Math. Phys. Vol7, 556-587.
[GKO]

P. Goddard, A. Kent and D. Olive, Unitary representations of the Virasoro

Algebra and super-Virasoro algebras Commun. Math. Phys. 103 (1986), 105-

119.
[Gu]

H. Guo, On abelian intertwining algebras and modules, Ph.D. Dissertation, Rut-gers University, 1994.
[H1]

Y.-Z. Huang, Vertex operator algebras and conformal field theory, Intl. J. Modern Phys. A7 (1992), 2109-2151.

[H2]

Y.-Z. Huang, Operadic formulation of topological vertex algebras and Gerstenhaber or Batalin-Vilkovisky algebras, Comm. Math. Phys., 164 (1994), 105-144.

[H3]

Y.-Z. Huang, A theory of tensor product for module categories for a vertex

operator algebra, VI, to appear.
[HL1]

Y.-Z. Huang and J. Lepowsky, Toward a theory of tensor products for representa-

tions of a vertex operator algebra, in: Proc. 20th Intl. Conference on Differential

Geometric Methods in Theoretical Physics, New York, 1991, ed. S. Catto and A.
Rocha, World Scientific, Singapore, 1992, Vol. 1, 344-354.
[HL2]

Y.-Z. Huang and J. Lepowsky, A theory of tensor product for module categories

for a vertex operator algebra, I, to appear.

24

[HL3]

Y.-Z. Huang and J. Lepowsky, A theory of tensor product for module categories

for a vertex operator algebra, II, to appear.
[HL4]

Y.-Z. Huang and J. Lepowsky, Tensor products of modules for a vertex operator
algebras and vertex tensor categories, to appear.

[HL5]

Y.-Z. Huang and J. Lepowsky, A theory of tensor product for module categories

for a vertex operator algebra, III, to appear.
[Le]

J. Lepowsky, Calculus of twisted vertex operators, Proc. Natl. Acad Sci. USA
82 (1985), 8295-8299.

[L1]

H. Li, Local systems of vertex operators, vertex superalgebras and modules, J.
Pure Appl. Alg., to appear.

[L2]

H. Li, Local systems of twisted vertex operators, vertex superalgebras andtwisted modules, J. Pure Appl. Alg., to appear.

[L3]

H. Li, Representation theory and tensor product theory for vertex operator al-

gebras, Ph.D. Dissertation, Rutgers University, 1994.
[L4]

H. Li, The theory of physical super selection sectors in terms of vertex operator

algebra language, preprint.
[MS]

G. Moore and N. Seiberg, Classical and quantum conformal field theory, Comm.
Math. Phys. 123 (1989), 177-254.

[LZ]

B. Lian and G. Zuckerman, New perspectives on the BRST-algebraic structureof string theory, Comm. Math. Phys. 154 (1993), 613-646.

[KP]

V. Kac and D. Petersdn, Infinite-dimensional Lie algebras, theta functions andmodular forms, Adv. Math. 53 (1984), 124-264.

[KW]

V. Kac and W. Wang, Vertex operator superalgebras and their representations,Contemporary Math. 175 (1994).

25

[Mon]

P. Montague, Codes, lattices and conformal field theories, Ph.

$\mathrm{D}$dissertation,

Cambridge University, 1991.
[Mos]

G. Mossberg, Axiomatic

ver.tex algebras and the Jacobi identity, Ph. disserta$\mathrm{D}$tion, University of Lund, Sweden, 1993.
[P]

M. Primc, Introduction to vertex operator algebra, preprint.

[TK]

A. Tsuchiya and Y. Kanie, Vertex operators in conformal field theory on
monodromy representations of braid group, in:$\mathrm{P}^{1}$

andConformal Field Theory andSolvable Lattice Models, Advanced Studies in Pure Math., Vol. 16, Kinokuniya
Company Ltd., Tokyo, 1988, 297-372.
[V]

E. Verlinde, Fusion rules and modular transformations in$2\mathrm{D}$conformal field

theory, Nucl. Phys. B300 (1988), 360-376.
[W]

W. Wang, Rationality of Virasoro vertex operator algebras, Duke Math.

$J$.

IMRN, Vol. 71, No. 1 (1993), 197-211.
[Z]

Y. Zhu, Vertex operator algebras, elliptic functions and modular forms, Ph.D.
dissertation, Yale University (1990).

Address: Department of Mathematics, University of California, Santa Cruz, CA 95064

dong@dong.ucsc.edu