Representations and fusion rules for the orbifold vertex operator algebras $L_{\widehat{\frak{sl}_2}}(k,0)^{\mathbb{Z}_3}$

TL;DR

This paper classifies and constructs all irreducible modules of the orbifold vertex operator algebra formed by the $ ext{sl}_2$ affine algebra at level $k$ under a $ ext{Z}_3$ symmetry, and determines its fusion rules and quantum dimensions.

Contribution

It provides a complete classification, explicit construction, and determination of fusion rules and quantum dimensions for the orbifold VOA $L_{\widehat{\mathfrak{sl}_2}}(k,0)^{\mathbb{Z}_3}$, advancing understanding of orbifold VOAs.

Findings

All irreducible modules classified and explicitly constructed.

Quantum dimensions of modules determined.

Fusion rules for the orbifold VOA established.

Abstract

For the cyclic group $\mathbb{Z}_3$ and positive integer $k$, we study the representations of the orbifold vertex operator algebra $L_{\widehat{\mathfrak{sl}_2}}(k,0)^{\mathbb{Z}_3}$. All the irreducible modules for $L_{\widehat{\mathfrak{sl}_2}}(k,0)^{\mathbb{Z}_3}$ are classified and constructed explicitly. Quantum dimensions and fusion rules for the orbifold vertex operator algebra $L_{\widehat{\mathfrak{sl}_2}}(k,0)^{\mathbb{Z}_3}$ are completely determined.