Bimodule and twisted representation of vertex operator algebras

TL;DR

This paper constructs bimodules associated with vertex operator algebras and automorphisms, exploring their properties and connections to intertwining operators, especially under the assumption of $g$-rationality.

Contribution

It introduces a new $A_{g,n}(V)$-bimodule construction for vertex operator algebras with automorphisms and studies its properties and relations to intertwining operators.

Findings

Bimodule $ ext{AA}_{g,n}(M)$ is constructed and analyzed.

Connections between bimodules and intertwining operators are established.

Isomorphism between intertwining operator spaces and homomorphism spaces is proved for $g$-rational VOAs.

Abstract

In this paper, for a vertex operator algebra $V$ with an automorphism $g$ of order $T,$ an admissible $V$-module $M$ and a fixed nonnegative rational number $n\in\frac{1}{T}\Bbb{Z}_{+},$ we construct an $A_{g,n}(V)$-bimodule $\AA_{g,n}(M)$ and study its some properties, discuss the connections between bimodule $\AA_{g,n}(M)$ and intertwining operators. Especially, bimodule $\AA_{g,n-\frac{1}{T}}(M)$ is a natural quotient of $\AA_{g,n}(M)$ and there is a linear isomorphism between the space ${\cal I}_{M\,M^j}^{M^k}$ of intertwining operators and the space of homomorphisms $\rm{Hom}_{A_{g,n}(V)}(\AA_{g,n}(M)\otimes_{A_{g,n}(V)}M^j(s), M^k(t))$ for $s,t\leq n, M^j, M^k$ are $g$-twisted $V$ modules, if $V$ is $g$-rational.