Modular invariance of vertex operator algebras satisfying C 2 -cofiniteness

Abstract

We investigate trace functions of modules for vertex operator algebras (VOA) satisfying C2 -cofiniteness. For the modular invariance property, Zhu assumed two conditions in [Z]: (1) A(V) is semisimple and (2) C2 -cofiniteness. We show that C2 -cofiniteness is enough to prove a modular invariance property. For example, if a VOA V= ⊕ m=0 ∞ V m is C2 -cofinite, then the space spanned by generalized characters (pseudotrace functions of the vacuum element) of V -modules is a finite-dimensional $\SL_2(\mathbb{Z})$ SL 2 ( 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 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 C2 -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) .