Bimodules over twisted Zhu algebras and twisted fusion rules theorem for vertex operator algebras

Let $V$ be a strongly rational vertex operator algebra, and let $g_1,g_2,g_3$ be three commuting finitely ordered automorphisms of $V$ such that $g_1{g}_2=g_3$ and $g_i^T=1$ for $i=1,2,3$ and $T\in \mathbb{N}$. Suppose $M^1$ is a $g_1$-twisted module. For any $n,m\in \frac{1}{T}\mathbb{N}$, we construct an $A_{g_3,n}(V)$-$A_{g_2,m}(V)$-bimodule ${\mathcal{A}}_{g_3,g_2,n,m}(M^1)$ associated to the quadruple $(M^1,g_1,g_2,g_3)$. Given an $A_{g_2,m}(V)$-module $U$, an admissible $g_3$-twisted module $\mathcal{M}(M^1,U)$ is constructed. For the quadruple $(V,1,g,g)$ with some finitely ordered $g\in \text{Aut}(V)$, ${\mathcal{A}}_{g,g,n,m}(V)$ coincides with the $A_{g,n}(V)$-$A_{g,m}(V)$-bimodules $A_{g,n,m}(V)$ constructed by Dong-Jiang, and $\mathcal{M}(V,U)$ is the generalized Verma type admissible $g$-twisted module generated by $U$. When $U=M^2(m)$ is the $m$-th component of a $g_2$-twisted module $M^2$ for some $m\in \frac{1}{T}\mathbb{N}$, we show that the submodule of $\text{M}(M^1,M^2(m))$ generated by the $m$-th component satisfies the universal property of the tensor product of $M^1$ and $M^2$. Using this result, we obtain a twisted version of Frenkel-Zhu-Li’s fusion rules theorem.