Twisted $φ$-coordinated modules for vertex algebras and Zhu's correspondence theorem
Shun Xu
TL;DR
The paper develops a twisted Zhu-type framework for vertex algebras with a finite-order automorphism g by introducing tilde A_{g,n}(V) and tilde A_{g,n,m}(V) and establishing a bimodule theory around them. It builds a universal enveloping algebra U(V[g]), constructs functors tildeOmega_n and tildeL to connect module categories, and proves a bijection between simple tilde A_g(V)-modules and irreducible (1/T)-graded g-twisted phi-coordinated V-modules, extending to irreducible admissible g-twisted V-modules. In the VOA case, the new algebras recover the established A_{g,n}(V) and A_{g,n,m}(V) structures of Dong–Li–Mason and Dong–Jiang, respectively, tying Zhu-like correspondences to the phi-coordinated module formalism. The results also support the independence of g-rationality and g-regularity from the conformal vector and unify several strands of twisted representation theory within a single algebraic framework.
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 $φ$-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 $φ$-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 $φ$-coordinated $V$-modules.