Boson-fermion correspondence of type B and twisted vertex algebras

TL;DR

This paper introduces twisted vertex algebras to generalize super vertex algebras, enabling the boson-fermion correspondence of type B, which involves singularities at both z=w and z=-w, to be understood as an isomorphism between such algebras.

Contribution

The paper develops the concept of twisted vertex algebras to accommodate singularities at multiple points, extending the framework of super vertex algebras for type B correspondence.

Findings

Defined twisted vertex algebra as a generalization of super vertex algebra.

Demonstrated that the type B boson-fermion correspondence is an isomorphism between two twisted vertex algebras.

Provided examples of twisted vertex algebras illustrating the correspondence.

Abstract

The boson-fermion correspondence of type A is an isomorphism between two super vertex algebras (and so has singularities in the operator product expansions only at $z = w$). The boson-fermion correspondence of type B plays similarly important role in many areas, including representation theory, integrable systems, random matrix theory and random processes. But the vertex operators describing it have singularities in their operator product expansions at both $z = w$ and $z = -w$, and thus need a more general notion than that of a super vertex algebra. In this paper we present such a notion: the concept of a twisted vertex algebra, which generalizes the concept of super vertex algebra. The two sides of the correspondence of type B constitute two examples of twisted vertex algebras. The boson-fermion correspondence of type B is thus an isomorphism between two twisted vertex algebras.