Fusion categories for affine vertex algebras at admissible levels

TL;DR

This paper proves that the module categories of affine vertex algebras at admissible levels form braided fusion categories, with some being modular, and extends the Verlinde formula beyond rational vertex operator algebras.

Contribution

It establishes the rigidity and modularity of module categories for affine vertex algebras at admissible levels, and provides a fusion rule formula for principal W-algebras.

Findings

Categories are rigid and braided at admissible levels

Certain levels yield modular tensor categories

Open Hopf links match normalized S-matrix entries

Abstract

The main result is that the category of ordinary modules of an affine vertex operator algebra of a simply laced Lie algebra at admissible level is rigid and thus a braided fusion category. If the level satisfies a certain coprime property then it is even a modular tensor category. In all cases open Hopf links coincide with the corresponding normalized S-matrix entries of torus one-point functions. This is interpreted as a Verlinde formula beyond rational vertex operator algebras. A preparatory Theorem is a convenient formula for the fusion rules of rational principal W-algebras of any type.