A theory of tensor products for vertex operator algebra satsifying C_2-cofiniteness

Details A theory of tensor products for vertex operator algebra satsifying C_2-cofiniteness A theory of tensor products for vertex operator algebra satsifying C_2-cofiniteness

2003-09-22

arxiv.org/abs/math/0309350v3

Mathematics

Quantum Algebra

27 pages. Latex

Scientific paper

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 C_2-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 C_2-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 C_2-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 C_2-cofinite, then for any irreducible V^{}-module W, there is an element h\in 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\in G.

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