## Duke Mathematical Journal

### Modular invariance of vertex operator algebras satisfying $C_{2}$-cofiniteness

Masahiko Miyamoto

#### 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 $V=\bigoplus_{m=0}^{\infty}V_m$ 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})$$\SL_2(\mathbb{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 $\mathbb{N}$-graded weak module that is a direct sum of generalized eigenspaces of $L(0)$.

#### Article information

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

Dates
First available in Project Euclid: 24 March 2004

http://projecteuclid.org/euclid.dmj/1080137202

Digital Object Identifier
doi:10.1215/S0012-7094-04-12212-2

Mathematical Reviews number (MathSciNet)
MR2046807

Zentralblatt MATH identifier
02133133

#### Citation

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. http://projecteuclid.org/euclid.dmj/1080137202.

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