Rationality of vertex operator superalgebras with rational conformal weights

TL;DR

This paper proves the rationality of certain affine vertex algebras, superalgebras, and W-algebras with respect to specific Virasoro elements, advancing understanding of their module categories and semisimplicity.

Contribution

It establishes the rationality of affine vertex superalgebras and W-algebras with rational conformal weights relative to particular Virasoro structures.

Findings

Subcategories of weak modules are semisimple.

Affine vertex algebras are rational with respect to certain Virasoro elements.

Affine vertex superalgebras and W-algebras are also proven rational.

Abstract

For the affine vertex algebra $V_k(\mathfrak{g})$ at an admissible level $k$ of $\hat{\mathfrak{g}}$, we prove that certain subcategory of weak $V_k(\mathfrak{g})$-module category is semisimple. As a consequence, we show that $V_k(\mathfrak{g})$ is rational with respect to a family of Virasoro elements. We also prove that certain affine vertex operator superalgebras and minimal $W$-algebras are rational with respect to a family of Virasoro elements.