Twisted Intertwining Operators and Tensor Products of (Generalized) Twisted Modules

We study the general twisted intertwining operators (intertwining operators among twisted modules) for a vertex operator algebra V. We give the skew-symmetry and contragredient isomorphisms between spaces of twisted intertwining operators and also prove some other properties of twisted intertwining operators. Using twisted intertwining operators,we introduce a notion of $P(z)$-tensor product of two objects for z∈ C× in a category of suitable g-twisted V-modules for g in a group of automorphisms of V and give a construction of such a $P(z)$-tensor product under suitable assumptions. We also construct G-crossed commutativity isomorphisms and G-crossed braiding isomorphisms. We formulate a $P(z)$-compatibility condition and a $P(z)$-grading-restriction condition and use these conditions to give another construction of the $P(z)$-tensor product.