1 A multi-operator extension of the Jacobi

identity

In this section, we work over an arbitrary field of characteristic zero. (In

Sections 2 and 3, it will be more convenient to work over C.) The symbols

z ^o? zi, etc., shall designate independent commuting formal variables.

1.1 Vertex operator algebras

In this subsection we recall a basic definition: A vertex operator algebra (cf.

[6], Section 8.10) is a Z-graded vector space

V = U

y

(n)5 n = wtv for v G V(n), (1.1)

such that

dimV(n) oo for n G Z, (1.2)

V(n) = 0 for n sufficiently small, (1-3)

equipped with a linear map

V

—*(EndV)[[z,*-1]]

v*-*Y(v,z) =

J^n*-"-1

(1.4)

(vn G EndV), and with two distinguished homogeneous vectors l,u G V,

satisfying the following conditions for u , u £ V :

uni; = 0 for n sufficiently large, (1-5)

or, equivalently,

Y(l,* ) = l; (1.6)

y(w,z) l € V[[z]] and l i m

r

^

0

* , z ) l = v; (1.7)

^ ( ^ ) y ( y ( u , ^ , : 2 ) (1.8)

Z2