Rationality of admissible affine vertex algebras in the category O

TL;DR

This paper proves the conjecture that admissible affine vertex algebras at fractional levels are rational in the category O, by classifying their modules via Joseph's characteristic varieties and integral root systems.

Contribution

It establishes the rationality of admissible affine vertex algebras in category O, confirming a conjecture by Adamovic and Milas, and characterizes modules over these algebras.

Findings

Admissible affine vertex algebras are rational in category O.

Modules are classified by Joseph's characteristic varieties.

Irreducible modules correspond to admissible representations with matching root systems.

Abstract

We study the vertex algebras associated with modular invariant representations of affine Kac-Moody algebras at fractional levels, whose simple highest weight modules are classified by Joseph's characteristic varieties. We show that an irreducible highest weight representation of a non-twisted affine Kac-Moody algebra at an admissible level k is a module over the associated simple affine vertex algebra if and only if it is an admissible representation whose integral root system is isomorphic to that of the vertex algebra itself. This in particular proves the conjecture of Adamovic and Milas on the rationality of admissible affine vertex algebras in the category O.