EXTENSIONS OF THE JACOBI IDENTITY 3
among the variables ^1,^2, etc.. See [6] Chapter 8 and the Appendix
and [5] for background on vertex operator algebras and for the relation
between the algebraic and the analytic viewpoints.
The second subsection of Section 1 presents identities involving the cases
of three and four operators Y{y^Zi)^ and the third (and last) subsection
of Section 1 treats the general case of any (finite) number of Y{y^Zi). We
choose the name "n-operator (or multi-operator) Jacobi identity" for a cer-
tain formula which expresses a single iterate of vertex operators multiplied
by suitable ^-functions in terms of products of Y(vi, Zi) and ^-functions. The
multi-operator Jacobi identity is part of the statement of Theorem 1.11.
In Section 2 we give the detailed construction of relative Z2-twisted vertex
operators over the field of complex numbers, combining the constructions
in [2] and in Chapter 9 of [6]. The starting point is a lattice L with a
symmetric Z-bilinear form , . We fix a subspace of the complexification
of L where the form, induced by , , is nondegenerate. Then relative
Z2-twisted vertex operators Y+(u,z) are defined (relative, that is, to this
subspace). As in the case of the (non-relative) twisted vertex operators of
[6], Chapter 9, K(u , z) depends linearly on the vector u of the vertex operator
algebra VL and is a generalized Laurent series involving fractional powers of
the formal variable z, whose coefficients are operators on the (Z2-twisted)
vertex operator module V^f. The underlying space of the vertex operator
algebra VL is, roughly speaking, the symmetric algebra on the span of the
vectors a ®
tn
(where a is a lattice element, t a formal variable and n a
negative integer) tensored with the group algebra of the lattice. On the other
hand, the underlying space of the vertex operator module is, roughly
speaking, the symmetric algebra on the span of the vectors a ® W
2
(where
n is a negative odd integer) tensored with a finite-dimensional space T. The
relative vertex operators generalize the vertex operators of [6] in two ways:
first, all integrality assumptions are removed and second, L is not assumed
nondegenerate and the role of h = L ®z C in [6] is played by the fixed
nondegenerate subspace of h.
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