Abstract
We investigate trace functions of modules for vertex operator algebras (VOA) satisfying -cofiniteness. For the modular invariance property, Zhu assumed two conditions in [Z]: (1) is semisimple and (2) -cofiniteness. We show that -cofiniteness is enough to prove a modular invariance property. For example, if a VOA is -cofinite, then the space spanned by generalized characters (pseudotrace functions of the vacuum element) of -modules is a finite-dimensional $\SL_2(\mathbb{Z})$-invariant space and the central charge and conformal weights are all rational numbers. Namely, we show that -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 -cofiniteness is equivalent to the condition that every weak module is an -graded weak module that is a direct sum of generalized eigenspaces of .
Citation
Masahiko Miyamoto. "Modular invariance of vertex operator algebras satisfying -cofiniteness." Duke Math. J. 122 (1) 51 - 91, 15 March 2004. https://doi.org/10.1215/S0012-7094-04-12212-2
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