Permutation orbifolds and associative algebras

Let $V$ be a vertex operator algebra and $g=(12\cdots k)$ be a $k$-cycle which is viewed as an automorphism of the vertex operator algebra $V^{\otimes k}$. It is proved that Dong-Li-Mason’s associated associative algebra $A_g\left(V^{\otimes k}\right)$ is isomorphic to Zhu’s algebra $A\left(V\right)$ explicitly. This result recovers a previous result that there is a one-to-one correspondence between irreducible $g$-twisted $V^{\otimes k}$-modulesandirreducible$V$-modules.