Permutation orbifolds of vertex operator superalgebra and associative algebras

Let $V$ be a vertex operator superalgebra and $g=\left(1\ 2\ \cdots k\right)$ be a $k$-cycle which is viewed as an automorphism of the tensor product vertex operator superalgebra $V^{\otimes k}$. In this paper, we construct an explicit isomorphism from $A_{g}\left(V^{\otimes k}\right)$ to $A\left(V\right)$ if $k$ is odd and to $A_{\sigma}\left(V\right)$ if $k$ is even where $\sigma$ is the canonical automorphism of $V$ of order 2 determined by the superspace structure of $V.$ These recover previous results by Barron and Barron-Werf that there is a one-to-one correspondence between irreducible $g$-twisted $V^{\otimes k}$-modules and irreducible $V$-modules (resp. irreducible $\sigma$-twisted $V$-modules) when $k$ is odd (resp. even). This explicit isomorphism is expected to be useful in our further study on the Zhu algebra of fixed point subalgebra.