Extensions of the Jacobi Identity for Vertex Operators, and Standard A1(1)-Modules

This book extends the Jacobi identity, the main axiom for a vertex operator algebra, to multi-operator identities. Based on constructions of Dong and Lepowsky, relative ${\mathbf Z}_2$-twisted vertex operators are then introduced, and a Jacobi identity for these operators is established. Husu uses these ideas to interpret and recover the twisted Z -operators and corresponding generating function identities developed by Lepowsky and Wilson for the construction of the standard $A^{(1)}_1$-modules. The point of view of the Jacobi identity also shows the equivalence between these twisted Z-operator algebras and the (twisted) parafermion algebras constructed by Zamolodchikov and Fadeev. The Lepowsky-Wilson generating function identities correspond to the identities involved in the construction of a basis for the space of C-disorder fields of such parafermion algebras.