$N$-point locality for vertex operators: normal ordered products, operator product expansions, twisted vertex algebras

TL;DR

This paper explores $N$-point local fields, deriving residue formulas, introducing descendant spaces, and demonstrating their role in twisted vertex algebras and Lie algebra representations.

Contribution

It develops the theory of $N$-point local fields, including residue formulas, descendant spaces, and their applications to vertex operator algebras and Lie algebra representations.

Findings

Residue formulas for $N$-point local fields in terms of delta functions and Bell polynomials.

Examples include vertex operators for boson-fermion correspondences of types B, C, D-A.

The generated field theory forms a twisted vertex algebra.

Abstract

In this paper we study fields satisfying $N$-point locality and their properties. We obtain residue formulae for $N$-point local fields in terms of derivatives of delta functions and Bell polynomials. We introduce the notion of the space of descendants of $N$-point local fields which includes normal ordered products and coefficients of operator product expansions. We show that examples of $N$-point local fields include the vertex operators generating the boson-fermion correspondences of type B, C and D-A. We apply the normal ordered products of these vertex operators to the setting of the representation theory of the double-infinite rank Lie algebras $b_{\infty}, c_{\infty}, d_{\infty}$. Finally, we show that the field theory generated by $N$-point local fields and their descendants has a structure of a twisted vertex algebra.