Vertex Operator Superalgebras and Their Representations

TL;DR

This paper develops a framework for vertex operator superalgebras, constructs an associative algebra for their representations, and applies this to classify modules, determine fusion rules, and analyze specific superalgebra cases.

Contribution

It generalizes Zhu's construction to superalgebras, providing new tools for understanding their modules, fusion rules, and rationality conditions.

Findings

Constructed associative algebra for vertex operator superalgebras

Classified representations of superalgebras like Neveu-Schwarz and super affine Kac-Moody

Derived explicit formulas for singular vectors and relations in super affine algebras

Abstract

After giving some definitions for vertex operator SUPERalgebras and their modules, we construct an associative algebra corresponding to any vertex operator superalgebra, such that the representations of the vertex operator algebra are in one-to-one correspondence with those of the corresponding associative algebra. A way is presented to decribe the fusion rules for the vertex operator superalgebras via modules of the associative algebra. The above are generalizations of Zhu's constructions for vertex operator algebras. Then we deal in detail with vertex operator superalgebras corresponding to Neveu-Schwarz algebras, super affine Kac-Moody algebras, and free fermions. We use the machinery established above to find the rationality conditions, classify the representations and compute the fusion rules. In the appendix, we present explicit formulas for singular vectors and defining relations for the integrable representations of super affine algebras. These formulas are not only crucial for the theory of the corresponding vertex operator superalgebras and their representations, but also of independent interest.