Regular representations and $A_n(V)$-$A_m(V)$ bimodules

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 $A_{n,m}^{†}(W)$ and $A_{n,m}^{\diamond }(W)$ of $W$. It is proved that the dual space $A_{n,m}^{†}(W{)}^{\ast }$ viewed as a subspace of $W^{\ast }$ 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 $A_{n,m}^{†}(W)$ and $A_{n,m}^{\diamond }(W)$. Furthermore, we obtain an $\mathbb{N}$-graded weak $V$-module structure together with a commuting right $A_m(V)$-module structure on $A_{\square ,m}^{\diamond }(W):={\oplus }_{n\in \mathbb{N}}{A}_{n,m}^{\diamond }(W)$. Consequently, we recover the corresponding results and roughly confirm a conjecture of Dong and Jiang.

28 pages