Bimodule and twisted representation of vertex operator algebras

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 \(n \in \tfrac{1}{T}\mathbb{Z}_ +\), we construct an A_g,n(V)-bimodule A_g,n(M) and study its properties, discuss the connections between bimodule A_g,n(M) and intertwining operators. Especially, bimodule \(A_{g,n - \tfrac{1}{T}} (M)\) (M) is a natural quotient of A_g,n(M) and there is a linear isomorphism between the space \(\mathcal{I}_{M M^j }^{M^k }\) of intertwining operators and the space of homomorphisms \(Hom_{A_{g,n} (V)} \left( {A_{g,n} \left( M \right) \otimes _{A_{g,n} (V)} M^j \left( s \right),M^k \left( t \right)} \right)\) for s, t ? n, M^j, M^k are g-twisted V modules, if V is g-rational.