Permutation orbifolds of vertex operator superalgebra and associative algebras

TL;DR

This paper constructs explicit isomorphisms between associative algebras related to permutation orbifolds of vertex operator superalgebras, clarifying the correspondence between twisted modules and modules of the original algebra.

Contribution

It provides explicit isomorphisms between certain associative algebras associated with permutation orbifolds and the original algebra, extending previous results.

Findings

Explicit isomorphism for odd k: $A_g(V^{ ensor k}) o A(V)$

Explicit isomorphism for even k: $A_g(V^{ ensor k}) o A_\sigma(V)$

Clarifies correspondence between twisted modules and original modules

Abstract

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.