Modular invariance of vertex operator algebras satisfying C 2 -cofiniteness

Abstract

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 $V={\oplus }_{m=0}^{\infty }{V}_m$ is $C_2$-cofinite, then the space spanned by generalized characters (pseudotrace functions of the vacuum element) of $V$-modules is a finite-dimensional $\SL_2(\mathbb{Z})$${\text{SL}}_{\text{2}}(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 $L(0)$.