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)
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