Commutant of $\mathcal{L}_{\widehat{\mathfrak{sl}}_2}(4,0)$ in the cyclic permutation orbifold of $\mathcal{L}_{\widehat{\mathfrak{sl}}_2}(1,0)^{\otimes 4}$

Abstract: We study the commutant of the vertex operator algebra $\mathcal{L}{\widehat{\mathfrak{sl}}_2}(4,0)$ in the cyclic permutation orbifold model $(\mathcal{L}{\widehat{\mathfrak{sl}}_2}(1,0)^{\otimes 4})^\tau$ with $\tau=(1\,2\,3\,4)$. It is shown that the commutant is isomorphic to a ${\mathbb Z}_2\times{\mathbb Z}_2$-orbifold model of a tensor product of two lattice type vertex operator algebras of rank one.