2
CRISTIANO HUSU
The first section of this paper is devoted to a study of the Jacobi identity
for a vertex operator algebra V over a field of characteristic zero. The first
subsection of Section 1 recalls the definition of a vertex operator algebra from
[6], Section 8.10 and [5]. We present the Jacobi identity in the following form:
Z°H (~^r) r ( u ' 2 i ) F ( u ' ^ ) - tf6 ( £ 3 ^ L ) Y(v,z2)Y(u,Zl) =
=
z-i6 ( ^ ^ ) Y(Y(u,z0)v,z2),
where
ZQ,Z1,Z2
are independent commuting formal variables,
nGZ
each expression of the form
(zi -
Zj)n,
neZ
is to be understood as binomial expansion in nonnegative integral powers of
the second term, and the vertex operator K(u, z) is a formal Laurent series
in the variable z, linear with respect to the vertex operator algebra vector
u, with operators on the vertex operator algebra V as its coefficients (cf.
Subsection 1.1). The left-hand side of the Jacobi identity can be viewed as the
generating function of a sequence of generalized commutators of Y(u, zx) and
F(t, z2), and the right-hand side is an iterated vertex operator multiplied by
the appropriate ^-function. Each expression appearing in the Jacobi identity
"converges" in a precise algebraic sense: when applied to a vector in V,
the coefficient of any monomial in z0,
zij a
^ d z2 is a finite sum of vectors
in V. The analogy between the Jacobi identity for vertex operator algebras
and the Jacobi identity for Lie algebras, and a result in the Appendix of
[6] (Proposition A.3.3) suggest and motivate the constructions of the multi-
operator identities presented in Subsection 1. Here we systematically relate
products (cf. the left-hand side of the identity above) and iterates of several
vertex operators, all multiplied by suitable products of 6-functions. The
existence in the algebraic sense of such products and/or iterates of
operators and (^-functions corresponds, in the case of the field C and the
complex realization of vertex operator algebras, to the fact that each vertex
operator expression converges in a certain domain defined by inequalities
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