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

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.