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
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