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

Abstract

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 $\varphi$-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 $\varphi$-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 $\varphi$-coordinated $V$-modules.

44 pages