Simplicity of vacuum modules over affine Lie superalgebras

TL;DR

This paper establishes explicit conditions on the level for the irreducibility of vacuum modules over affine Lie superalgebras, impacting the understanding of their associated W-algebras and confirming a conjecture by Gorelik and Kac.

Contribution

It provides a precise criterion for the irreducibility of vacuum modules over affine Lie superalgebras, confirming a conjecture and extending understanding of W-algebra simplicity.

Findings

Explicit irreducibility condition for vacuum modules

Simplicity criteria for minimal W-algebras

Confirmation of Gorelik and Kac's conjecture

Abstract

We prove an explicit condition on the level $k$ for the irreducibility of a vacuum module $V^{k}$ over a (non-twisted) affine Lie superalgebra, which was conjectured by M. Gorelik and V.G. Kac. An immediate consequence of this work is the simplicity conditions for the corresponding minimal W-algebras obtained via quantum reduction, in all cases except when the level $k$ is a non-negative integer.