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}^{d a g g e r} (W)

and

A_{n, m}^{d i a m o n d} (W)

of

W

. It is proved that the dual space

A_{n, m}^{d a g g e r} (W)^{*}

viewed as a subspace of

W^{*}

coincides with the level-

(m, n)

vacuum subspace of the regular representation module

m a t h f r a k 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}^{d a g g e r} (W)

and

A_{n, m}^{d i a m o n d} (W)

. Furthermore, we obtain an

N

-graded weak

V

-module structure together with a commuting right

A_{m} (V)

-module structure on

A_{B o x, m}^{d i a m o n d} (W) : = o p l u s_{n i n N} A_{n, m}^{d i a m o n d} (W)

. Consequently, we recover the corresponding results and roughly confirm a conjecture of Dong and Jiang.

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