Mathematics – Quantum Algebra
Scientific paper
2008-06-16
Communications in Mathematical Physics, 300 (2010), no. 1, 1--46
Mathematics
Quantum Algebra
42 pages Latex, corrected typos, added some references, slightly rewritten abstract of the previous version
Scientific paper
10.1007/s00220-010-1096-6
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 Z(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.
Ng Siu-Hung
Schauenburg Peter
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