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.

Readership

Mathematicians and physicists interested in vertex operators, Lie theory, conformal field theory, and string theory.

Table of Contents

• Introduction
• A multi-operator extension of the Jacobi identity
• A relative twisted Jacobi identity
• Standard representations of the twisted affine Lie algebra \(A^{(1)}_1\)
• References