As we mentioned, the identity obtained by equating (1.51) and (1.52)
is called the four-operator (or four-vertex operator) Jacobi identity. This
achieves the goal of expressing a combination of products of four vertex
operators Y(v{,Zi) and suitable 5-functions in terms of the product of an
"iterated" vertex operator and the resulting 6-iunctions. Moreover, we recall
that both (1.51) and (1.52) equal the expression (1.50).
1.3 A multi-operator extension of the Jacobi identity
The following is a doubly recursive (on the indices r and k j) definition of
a linear combination of products of vertex operators Y{y^ z-), i 1,2,..., n,
and £-functions.
Let n 1 and let vi, v2j..., vn be a family of vectors in a vertex operator
algebra V and let (in?*n-i»--*i) be a permutation of (n,n 1,...,1). For
1 j k n
Y(vin,vin_1,...,Vi1\Zin,zin_1,...,Zi1-,{zilij, le {j + l j + 2,...,fc}};0;[j,£]) =
= [Y(vik,zik) xZikij [ K K . , , ^ ) x ^ . (1.60)
This is the recursive step r = 0 of the definition. Note that if j = fc, (1.60)
equals y(v^ , z
). Then for
1 r j k n
and for
ii r, j + 1 lk,
Previous Page Next Page