Duke Mathematical Journal

Modular invariance of vertex operator algebras satisfying C 2 -cofiniteness

Masahiko Miyamoto

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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) .

Article information

Duke Math. J., Volume 122, Number 1 (2004), 51-91.

First available in Project Euclid: 24 March 2004

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 17B69: Vertex operators; vertex operator algebras and related structures
Secondary: 11F22: Relationship to Lie algebras and finite simple groups


Miyamoto, Masahiko. Modular invariance of vertex operator algebras satisfying $C_{2}$ -cofiniteness. Duke Math. J. 122 (2004), no. 1, 51--91. doi:10.1215/S0012-7094-04-12212-2. https://projecteuclid.org/euclid.dmj/1080137202

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