Vertex operator algebras associated to admissible representations of $\hat{sl}_2$

Abstract

The admissible modules for $\hat{sl}_2$ are studied from the point of view of vertex operator algebra. If $l$ is rational such that $l+2={p\over q}$ for some coprime positive integers $p\ge 2$ and $q$, Kac and Wakimoto found finitely many distinguished irreducible representations for $\hat{sl}_2$, called admissible representations. In this paper we prove that the vertex operator algebra $L(l,0)$ associated to irreducible highest weight representation of $l$ is not rational if $l$ is not a positive integer. However if we change the Virasoro algebra in certain way, $L(l,0)$ becomes a rational vertex operator algebra whose irreducible representations are exactly those admissible representations. We show that the $q$-dimensions with respect to the new Virasoro algebra are modular functions. We aslo calculate the fusions rules.