Representations of the orbifold VOAS $L_{\hat{\frak{sl}_2}}(k,0)^K$ and the commutant VOAS $C_{{L_{\hat{\mathfrak{so}_m}}(1,0)}^{\otimes 3}}({L_{\hat{\mathfrak{so}_m}}(3,0)})$

Abstract: For the Klein group $K$, $k\in\mathbb{Z}{\geqslant 1}$ and $m\in\mathbb{Z}{\geqslant 4}$, we study the representations of the orbifold vertex operator algebra $L_{\hat{\mathfrak{sl}2}}(k,0)^{K}$ and the commutant vertex operator algebra of $L{\hat{\mathfrak{so}m}}(3,0)$ in $L{\hat{\mathfrak{so}m}}(1,0)^{\otimes 3}$ which can be realized as the orbifold vertex operator subalgebra $L{\hat{\mathfrak{sl}2}}(2m,0)^{K}$ or its extension. All the irreducible modules for $L{\hat{\mathfrak{sl}2}}(k,0)^{K}$ and $C{{L_{\hat{\mathfrak{so}m}}(1,0)}^{\otimes 3}}({L{\hat{\mathfrak{so}_m}}(3,0)})$ are classified and constructed explicitly.