VERTEX OPERATOR ALGEBRAS AND MODULES

11

2.2. Definition of vertex operator algebras

DEFINITION

2.2.1. A vertex operator algebraic a Z-graded vector space (graded

by weights)

V

= I I

v(n)l f o r v e V(n),

n = wtv; (2.2.1)

such that

dim V(n) oo for n G Z, (2.2.2)

V(n) = 0 for n sufficiently small, (2.2.3)

equipped with a linear map V ® V — • V[[2,

2-1]],

or equivalently,

V-(End V)^,*" 1 ]]

v -+ y(v, z) = ] P vnz-n-1 (where vn G End V), (2.2.4)

Y(v, z) denoting the vertex operator associated with v, and equipped also with

two distinguished homogeneous vectors 1 (the vacuum) and w E V. The following

conditions are assumed for u,v £V :

i£nu = 0 for n sufficiently large; (2.2.5)

Y(l,z) = 1 (1 on the right being the identity operator); (2.2.6)

the creation property holds:

Y(v, z)l E V[[z}} and lim Y(y, z)l = v (2.2.7)

z—0

(that is, Y(v, z)\ involves only nonnegative integral powers of z and the constant

term is v)\ the Jacobi identity holds:

Zo'S

(^^)Y(U,Z1)Y(V,Z2)

- z^S { ^ ^ ) Y(v,z2)Y{u,z1)

=

Z-^S

(^Y^1)

Y(Y(u, zo)v, z2) (2.2.8)

(note that when each expression in (2.2.8) is applied to any element of V, the

coefficient of each monomial in the formal variables is a finite sum; on the right-

hand side, the notation Y(•, z2) is understood to be extended in the obvious way

to V[[^o, ^o"1]]); the Virasoro algebra relations hold:

[L(m), L(n)] = (ra — n)L(m + n) + —(m3 — ™)$m+n,o(rank V) for m,n E Z,

(2.2.9)