Vertex Operator Superalgebras and Odd Trace Functions

TL;DR

This paper reviews Zhu's theorem on modular invariance for vertex operator superalgebras, introduces odd trace functions, and provides a new representation-theoretic interpretation of Jacobi's identity related to the Dedekind eta function.

Contribution

It extends Zhu's modular invariance results to superalgebras via odd trace functions and offers a novel interpretation of a classical identity.

Findings

Computed an odd trace function for the N=1 superconformal algebra

Provided a new representation-theoretic perspective on Jacobi's identity

Extended modular invariance to superalgebras using odd trace functions

Abstract

We begin by reviewing Zhu's theorem on modular invariance of trace functions associated to a vertex operator algebra, as well as a generalisation by the author to vertex operator superalgebras. This generalisation involves objects that we call `odd trace functions'. We examine the case of the N=1 superconformal algebra. In particular we compute an odd trace function in two different ways, and thereby obtain a new representation theoretic interpretation of a well known classical identity due to Jacobi concerning the Dedekind eta function.