Modular invariance of vertex operator algebras satisfying C2-cofiniteness

We investigate trace functions of modules for vertex operator algebras (VOA) satisfying C-2-cofiniteness. For the modular invariance property, Zhu assumed two conditions in [Z]: (1) A(V) is semisimple and (2) C-2-cofiniteness. We show that C-2-cofiniteness is enough to prove a modular invariance property. For example, if a VOA V = +(infinity)(m=0) V-m is C-2-cofinite, then the space spanned by generalized characters (pseudotrace 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 C-2-cofiniteness implies "rational conformal field theory" in a sense as expected in Gaberdiel and Neitzke [GN]. Viewing a trace map as a symmetric linear map and using a result of symmetric algebras, we introduce "pseudotraces" and pseudotrace functions and then show that the space spanned by such pseudotrace functions has a modular invariance property. We also show that C-2-cofiniteness is equivalent to the condition that every weak module is an N-graded weak module that is a direct sum of generalized eigenspaces of L (0).