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
Abstract

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, ${C}_{2}$-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.

Keywords
Frobenius–Schur indicator, modular tensor category, modular group, vertex operator algebra
Mathematical Subject Classification 2010
Primary: 17B69
Secondary: 18D10, 20H05, 81R05