#### Vol. 9, No. 9, 2015

 Recent Issues
 The Journal About the Journal Editorial Board Subscriptions Editors' Interests Submission Guidelines Submission Form Editorial Login Ethics Statement ISSN: 1944-7833 (e-only) ISSN: 1937-0652 (print) Author Index To Appear Other MSP Journals
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
##### Milestones
Received: 5 March 2015
Revised: 20 July 2015
Accepted: 19 August 2015
Published: 4 November 2015
##### Authors
 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 China Siu-Hung Ng Department of Mathematics Louisiana State University Baton Rouge, LA 70803 United States