Bimodules and $g$-rationality of vertex operator algebras

This paper studies the twisted representations of vertex operator algebras. Let $V$ be a vertex operator algebra and $g$ an automorphism of $V$ of finite order $T.$ For any $m,n\in \frac {1}{T}\mathbb {Z}_+$, an $A_{g,n}(V)$-$A_{g,m}(V)$-bimodule $A_{g,n,m}(V)$ is constructed. The collection of these bimodules determines any admissible $g$-twisted $V$-module completely. A Verma type admissible $g$-twisted $V$-module is constructed naturally from any $A_{g,m}(V)$-module. Furthermore, it is shown with the help of bimodule theory that a simple vertex operator algebra $V$ is $g$-rational if and only if its twisted associative algebra $A_g(V)$ is semisimple and each irreducible admissible $g$-twisted $V$-module is ordinary.