Classification of the irreducible ordinary modules for affine vertex operator superalgebras

TL;DR

This paper classifies irreducible ordinary modules for affine vertex operator superalgebras associated with basic classical Lie superalgebras at boundary admissible levels, revealing the number of such modules depends on the algebra type.

Contribution

It provides a complete classification of irreducible ordinary modules for these superalgebras, distinguishing cases based on algebra type and boundary levels.

Findings

Type I superalgebras have exactly u inequivalent modules.

Type II superalgebras have only the algebra itself as the module.

The classification depends on the boundary admissible level u.

Abstract

Let $\mathfrak{g}$ be a basic classical Lie superalgebra, $\mathcal{k}=\frac{h^{\vee}}{u}-h^{\vee}$ a boundary admissible level of $\widehat{\mathfrak{g}}$, where $u$ is a positive integer and $h^{\vee}$ is the dual Coxeter number of $\mathfrak{g}$. In this paper, we classify the irreducible ordinary modules for the affine vertex operator superalgebra $L_{\widehat{\mathfrak{g}}}(\mathcal{k},0)$ associated to any basic classical Lie superalgebra $\mathfrak{g}$. More specifically, if $\mathfrak{g}$ is a basic classical Lie superalgebra of type I, we prove that $L_{\widehat{\mathfrak{g}}}(\mathcal{k},0)$ has exactly $u$ inequivalent irreducible ordinary modules. If $\mathfrak{g}$ is a finite dimensional simple Lie algebra or a basic classical Lie superalgebra of type II, we prove that $L_{\widehat{\mathfrak{g}}}(\mathcal{k},0)$ itself is the only irreducible ordinary $L_{\widehat{\mathfrak{g}}}(\mathcal{k},0)$-module.