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 V£ 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.