Twisted vertex algebras, bicharacter construction and boson-fermion correspondences

TL;DR

This paper introduces a new boson-fermion correspondence of type D-A, defines twisted vertex algebras of order N, and develops a bicharacter construction for these algebras, unifying several known correspondences.

Contribution

It presents a novel boson-fermion correspondence of type D-A and introduces twisted vertex algebras of order N, expanding the framework of vertex algebra theory.

Findings

New boson-fermion correspondence of type D-A

Definition of twisted vertex algebra of order N

Bicharacter formulas for vacuum expectation values

Abstract

The boson-fermion correspondences are an important phenomena on the intersection of several areas in mathematical physics: representation theory, vertex algebras and conformal field theory, integrable systems, number theory, cohomology. Two such correspondences are well known: the types A and B (and their super extensions). As a main result of this paper we present a new boson-fermion correspondence, of type D-A. Further, we define a new concept of twisted vertex algebra of order $N$, which generalizes super vertex algebra. We develop the bicharacter construction which we use for constructing classes of examples of twisted vertex algebras, as well as for deriving formulas for the operator product expansions (OPEs), analytic continuations and normal ordered products. By using the underlying Hopf algebra structure we prove general bicharacter formulas for the vacuum expectation values for three important groups of examples. We show that the correspondences of type B, C and D-A are isomorphisms of twisted vertex algebras.