A Theory of Tensor Products for Vertex Operator Algebra Satisfying C 2 -cofiniteness

The recent researchs show that C2-cofiniteness is a natural conditition to consider a vertex operator algebra with finitely many simple modules. Therefore, we extended the tensor product theory of vertex operator algebras developed by Huang and Lepowsky without assuming the compatibility condition nor the semisimplicity of grading operator so that we could apply it to all vertex operator algebras satisfying only C2-cofiniteness. We also showed that the tensor product theory develops naturally if we include not only ordinary modules, but also weak modules with a composition series of finite length (we call it an Artin module). Actually, a C2-cofiniteness on V is enough to show that a tensor product of two Artin modules is again an Arting module and we have natural commutativity and associativity of tensor products. Namely, the category of Artin modules becomes a braided tensor category. Our main purpose is an application of the tensor product theory under C2cofiniteness. We determined the representation theory of orbifold models. For example, if a vertex operator algebra V has a finite automorphism group G and the fixed point vertex operator subalgebra V G is C2-cofinite, then for g ∈ G and any irreducible V 〈g〉-module W , there is an element h ∈ 〈g〉 such that W is contained in some h-twisted V -module. Furthermore, if V G is rational, then V 〈g〉 is also rational for any g ∈ G.