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

By Masahiko Miyamoto. 2002

220 citations recorded by the source corpus.

We show that C_2-cofiniteness is enough to prove a modular invariance property of vertex operator algebras without assuming the semisimplicity of Zhu algebra. For example, if a VOA V=\oplus_{m=0}^{\infty}V_m is C_2-cofinite, then the space spanned by generalized characters of V-modules is invariant under the action of SL_2(\Z). In this case, the central charge and conformal weights are all rational numbers. Namely, a VOA satisfying C_2-cofiniteness is a rational conformal field theory in a sense. We also show that C_2-cofiniteness is equivalent to the condition that every weak module is an \N-graded weak module which is a direct sum of generalized eigenspaces of L(0).

Subfields: Algebra and Number Theory, Geometry and Topology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/ad7e7e0269a6c0f0583032dc36bcc780e411239a

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In Algebra and Number Theory

1. A. Gainutdinov
2. I. Tipunin
3. A. M. Semikhatov
4. B. Feigin
5. T. Arakawa
6. M. Gaberdiel

In Geometry and Topology

1. M. Gaberdiel
2. Rajesh Gopakumar
3. A. Gainutdinov
4. Lorenz Eberhardt
5. B. Feigin
6. I. Tipunin
7. A. M. Semikhatov
8. T. Arakawa

In Statistical and Nonlinear Physics

1. Rajesh Gopakumar
2. M. Gaberdiel
3. Lorenz Eberhardt
4. I. Tipunin
5. A. M. Semikhatov
6. B. Feigin
7. A. Gainutdinov
8. T. Arakawa

Citations in the corpus

• Logarithmic extensions of minimal models: characters and modular transformations. (2006, background): “The second is the general category-theory aspects of conformal field theo ry involving vertex-operator algebras with nonsemisimple representation categories [10, 11 ] (also see [12] and the references therein).”
• The worldsheet dual of the symmetric product CFT (2018, background): “The fact that the S-matrix depends on τ is typical of logarithmic conformal field theories [38, 39], and it will cancel out once left- and right-movers are correctly combined.”
• A remark on the C2-cofiniteness condition on vertex algebras (2010, background): “This condition has been assumed in many significant theories of vertex operator algebras, such as [6,16,17,25,26,29].”