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