Twisted $\phi$-coordinated modules for vertex algebras and Zhu's correspondence theorem

Let $V$ be a vertex algebra and $g$ be an automorphism of $V$ of order $T$. For any $n, m \in (1/T)\mathbb{N}$, we construct an $\tilde{A}_{g,n}(V)\!-\!\tilde{A}_{g,m}(V)$-bimodule $\tilde{A}_{g,n,m}(V)$, where $\tilde{A}_{g,n}(V)$ denotes the associative algebra constructed by the authors in \cite{Shun1}. We introduce the notion of $(1/T)\mathbb{N}$-graded $g$-twisted $\phi$-coordinated $V$-modules and prove that there exists a bijection between the simple $\tilde{A}_{g}(V)$-modules and the irreducible $(1/T)\mathbb{N}$-graded $g$-twisted $\phi$-coordinated $V$-modules, where $\tilde{A}_{g}(V)=\tilde{A}_{g,0}(V)$. We construct the universal enveloping algebra $U(V[g])$, showing that $\tilde{A}_{g}(V)$ is subquotient of $U(V[g])$. When $V$ is vertex operator algebra, we show that each $\tilde{A}_{g,n,m}(V)$ is isomorphic to the $A_{g,n}(V)-A_{g,m}(V)$-bimodule $A_{g,n,m}(V)$ constructed by Dong and Jiang~\cite{DJ2}. Also we prove that there exists a bijection between the irreducible admissible $g$-twisted $V$-modules and the irreducible $(1/T)\mathbb{N}$-graded $g$-twisted $\phi$-coordinated $V$-modules.