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

TL;DR

This paper explores the structure of vertex operator algebras linked to admissible representations of _2, showing how modifications can make them rational and analyzing their modular properties and fusion rules.

Contribution

It demonstrates how changing the Virasoro algebra makes certain vertex operator algebras rational and characterizes their irreducible modules and fusion rules.

Findings

L(l,0) is not rational if l is not a positive integer

Modified Virasoro algebra yields rational vertex operator algebra

q-dimensions are modular functions and fusion rules are computed

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.