Classification of the irreducible ordinary modules for affine vertex operator superalgebras

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.