A theory of tensor products for vertex operator algebra satisfying C

We reformed the tensor product theory of vertex operator algebras developed by Huang and Lepowsky so that we could apply it to all vertex operator algebras satisfying 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). In particular, we don’t assume the semisimplicity of the weight operator L(0). Actually, without the assumption of rationality, a C2-cofiniteness on V is enough to obtain the existence of a tensor product of two Artin modules and natural associativity of tensor products. Namely, the category of Artin modules becomes a braided tensor category. As an application of the tensor product theory under C2-cofiniteness, we proved the rationality of some orbifold models. For example, if a vertex operator algebra V has a finite automorphism group and the fixed point vertex operator subalgebra V G is C2-cofinite, then for any irreducible V -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 is also rational for any g ∈ G.