Permutation orbifolds of vertex operator superalgebra and associative algebras

Abstract

Let $V$ be a vertex operator superalgebra and $g=\left(12\cdots k\right)$ 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.

arXiv comment21 pages. The article was accepted for publication in SCIENCE CHINA Mathematics