Refining twisted bimodules associated to VOAs

TL;DR

This work resolves a conjecture on the structure of twisted Zhu bimodules by showing $O_{g,n,m}(V)=O'_{g,n,m}(V)$ for all $m,n\in(1/T)\mathbb{N}$. The authors leverage the twisted regular representation $(\mathfrak D(V),Y^R)$ and its dual connections to $\Omega_n(M,Y_M)$ to obtain a concise proof, linking the subspace $O_{g,n,m}(V)$ directly to $O'_{g,n,m}(V)$. The result simplifies the construction of $A_{g,n,m}(V)$ as an $A_{g,n}(V)$-$A_{g,m}(V)$-bimodule and strengthens the relation to representations of the universal enveloping algebra $U(V[g])$. Overall, the paper clarifies the algebraic underpinnings of twisted module theory and enhances the toolkit for classifying admissible $g$-twisted $V$-modules.

Abstract

Let $V$ be a vertex operator algebra and $g$ an automorphism of $V$ of finite order $T$. For any $m, n \in(1/T) \mathbb N$, an $A_{g,n}(V)\!-\!A_{g,m}(V)$ bimodule $A_{g,n, m}(V)=V/O_{g,n,m}(V)$ was defined by Dong and Jiang, where $O_{g,n,m}(V)$ is the sum of three certain subspaces $O_{g,n, m}^{\prime}(V), O_{g,n, m}^{\prime \prime}(V)$ and $O_{g,n, m}^{\prime \prime \prime}(V)$. In this paper, we show that $O_{g,n, m}(V)=O_{g,n, m}^{\prime}(V)$.

Theorems & Definitions (12)

• Theorem 1.1
• Definition 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• Definition 3.1
• Theorem 3.2
• Remark 3.3
• Proposition 3.4
• proof