Vertex operator algebras associated to modular invariant representations for $A_1 ^{(1)}$

TL;DR

This paper studies vertex operator algebras linked to modular-invariant representations of the affine Lie algebra A_1^{(1)}, demonstrating their rationality and classifying irreducible modules for admissible levels.

Contribution

It establishes the rationality of these VOAs in category O and classifies all irreducible weight modules for admissible levels.

Findings

VOA L(k,0) is rational in category O

All irreducible weight modules are classified

Results apply to admissible rational levels

Abstract

We investigate vertex operator algebras $L(k,0)$ associated with modular-invariant representations for an affine Lie algebra $A_1 ^{(1)}$ , where k is 'admissible' rational number. We show that VOA $L(k,0)$ is rational in the category $\cal O$ and find all irreducible representations in the category of weight modules.