Semisimplicity of module categories of certain affine vertex operator superalgebras

TL;DR

This paper proves that certain categories of modules for affine vertex operator superalgebras at non-admissible levels are semisimple, leading to braided tensor structures for their finite-length modules.

Contribution

It establishes the semisimplicity of Kazhdan-Lusztig categories for specific affine vertex operator superalgebras at conformal non-admissible levels, a novel result.

Findings

Kazhdan-Lusztig categories are semisimple for these superalgebras

Categories of finite-length modules have braided tensor structures

Results apply to non-admissible levels at conformal points

Abstract

In this paper, we show Kazhdan-Lusztig categories, that is, the categories of lower bounded generalized weight modules for certain affine vertex operator superalgebras that are locally finite modules of the underlying finite dimensional Lie superalgebra, are semisimple. Those are all representation categories of affine vertex operator superalgebras at conformal but non admissible levels. As a consequence, the categories of finite length generalized modules for these affine vertex operator superalgebras have braided tensor category structures.