Modular Invariance of Vertex Operator Algebras Satisfying C 2 -cofiniteness

We investigate trace functions of modules for vertex operator algebras satisfying C2-cofiniteness. For the modular invariance property, Zhu assumed two conditions in [Zh]: A(V ) is semisimple and C2-cofiniteness. We show that C2-cofiniteness is enough to prove a modular invariance property. For example, if a VOA V is C2cofinite, then the space spanned by generalized characters (pseudo-trace functions of the vacuum element) of V -modules is a finite dimensional SL2(Z)-invariant space and the central charge and conformal weights are all rational numbers. Namely we show that C2-cofiniteness implies ”rational conformal field theory” in a sense as expected in [GN]. Viewing a trace map as one of symmetric linear maps and using a result of symmetric algebras, we introduce ”pseudo-traces” and pseudo-trace functions and then show that the space spanned by such pseudo-trace functions has a modular invariance property. We also show that C2-cofiniteness is equivalent to the condition that every weak module is an admissible module which is a direct sum of generalized eigenspaces of L(0).