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

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.