Bimodules over twisted Zhu algebras and a construction of tensor product of twisted modules for vertex operator algebras

Let $V$ be a simple, non-negatively-graded, rational, $C_2$-cofinite, and self dual vertex operator algebra, $g_1, g_2, g_3$ be three commuting finitely ordered automorphisms of $V$ such that $g_1g_2=g_3$ and $g_i^T=1$ for $i=1, 2, 3$ and $T\in \N$. Suppose $M^1$ is a $g_1$-twisted module. For any $n, m\in \frac{1}{T}\N$, 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 $(V, 1, g, g)$ for some $g\in \text{Aut}(V)$, $\mathcal{A}_{g, g, n, m}(V)$ 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 $U$. For an irreducible $g_1$-twisted module $M^1$ and an irreducible $g_2$-twisted module $M^2$, we give a construction of tensor product of $M^1$ and $M^2$ using the bimodule theory developed in this paper. As an application, a twisted version of the fusion rules theorem is established.