Vol. 9, No. 9, 2015

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Congruence property in conformal field theory

Chongying Dong, Xingjun Lin and Siu-Hung Ng

Vol. 9 (2015), No. 9, 2121–2166

The congruence subgroup property is established for the modular representations associated to any modular tensor category. This result is used to prove that the kernel of the representation of the modular group on the conformal blocks of any rational, C2-cofinite vertex operator algebra is a congruence subgroup. In particular, the q-character of each irreducible module is a modular function on the same congruence subgroup. The Galois symmetry of the modular representations is obtained and the order of the anomaly for those modular categories satisfying some integrality conditions is determined.

Frobenius–Schur indicator, modular tensor category, modular group, vertex operator algebra
Mathematical Subject Classification 2010
Primary: 17B69
Secondary: 18D10, 20H05, 81R05
Received: 5 March 2015
Revised: 20 July 2015
Accepted: 19 August 2015
Published: 4 November 2015
Chongying Dong
Department of Mathematics
UC Santa Cruz
194 Baskin Engineering
Santa Cruz, CA 95064
United States
Xingjun Lin
Department of Mathematics
Sichuan University
Chengdu, 610064
Siu-Hung Ng
Department of Mathematics
Louisiana State University
Baton Rouge, LA 70803
United States