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

Abstract

The admissible modules for ${\widehat{sl}}_2$ are studied from the point of view of vertex operator algebra. If $l$ is rational such that $l+2=\frac{p}{q}$ for some coprime positive integers $p\ge 2$ and $q$, Kac and Wakimoto found finitely many distinguished irreducible representations for ${\widehat{sl}}_2$, 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.

Latex 42 pages

References in corpus (1)

• Some Rational Vertex Algebras

Cited by in corpus (4)

• Logarithmic Conformal Field Theory: Beyond an Introduction
• Modular Data and Verlinde Formulae for Fractional Level WZW Models I
• Relaxed singular vectors, Jack symmetric functions and fractional level $\widehat{\mathfrak{sl}}(2)$ models
• Modularity of logarithmic parafermion vertex algebras