Congruence Subgroups and Generalized Frobenius-Schur Indicators

Authors

Siu Hung Ng, Iowa State University
Peter Schauenburg, Ludwig-Maximilians-Universität München

Document Type

Article

Publication Date

9-2-2010

Abstract

We introduce generalized Frobenius-Schur indicators for pivotal categories. In a spherical fusion category C, an equivariant indicator of an object in C is defined as a functional on the Grothendieck algebra of the quantum double C via generalized Frobenius-Schur indicators. The set of all equivariant indicators admits a natural action of the modular group. Using the properties of equivariant indicators, we prove a congruence subgroup theorem for modular categories. As a consequence, all modular representations of a modular category have finite images, and they satisfy a conjecture of Eholzer. In addition, we obtain two formulae for the generalized indicators, one of them a generalization of Bantay's second indicator formula for a rational conformal field theory. This formula implies a conjecture of Pradisi-Sagnotti-Stanev, as well as a conjecture of Borisov-Halpern-Schweigert. © 2010 Springer-Verlag.

Publication Source (Journal or Book title)

Communications in Mathematical Physics

First Page

1

Last Page

46

Recommended Citation

Ng, S., & Schauenburg, P. (2010). Congruence Subgroups and Generalized Frobenius-Schur Indicators. Communications in Mathematical Physics, 300 (1), 1-46. https://doi.org/10.1007/s00220-010-1096-6