Regular representations and $A_{n}(V)$-$A_{m}(V)$ bimodules

TL;DR

This paper establishes a connection between regular representations of vertex operator algebras and bimodules over associative algebras, providing new structures and confirming a conjecture by Dong and Jiang.

Contribution

It introduces a natural link between regular representations and bimodules, constructing new bimodule structures and confirming a conjecture in the theory of vertex operator algebras.

Findings

Identifies the dual space of certain quotient spaces with vacuum subspaces of regular modules.

Constructs bimodule structures on these quotient spaces.

Recovers known results and confirms a conjecture of Dong and Jiang.

Abstract

This paper is to establish a natural connection between regular representations for a vertex operator algebra $V$ and $A_{n}(V)$-$A_{m}(V)$ bimodules of Dong and Jiang. Let $W$ be a weak $V$-module and let $(m,n)$ be a pair of nonnegative integers. We study two quotient spaces $A_{n,m}^{\dagger}(W)$ and $A^{\diamond}_{n,m}(W)$ of $W$. It is proved that the dual space $A^{\dagger}_{n,m}(W)^{*}$ viewed as a subspace of $W^*$ coincides with the level-$(m,n)$ vacuum subspace of the regular representation module $\mathfrak{D}_{(-1)}(W)$. By making use of this connection, we obtain an $A_{n}(V)$-$A_m(V)$ bimodule structure on both $A_{n,m}^{\dagger}(W)$ and $A^{\diamond}_{n,m}(W)$. Furthermore, we obtain an $\N$-graded weak $V$-module structure together with a commuting right $A_m(V)$-module structure on $A^{\diamond}_{\Box,m}(W):=\oplus_{n\in \N}A^{\diamond}_{n,m}(W)$. Consequently, we recover the corresponding results and roughly confirm a conjecture of Dong and Jiang.