Congruence Subgroups and Generalized Frobenius-Schur Indicators

TL;DR

This paper develops generalized Frobenius-Schur indicators for pivotal categories, proving a congruence subgroup theorem for modular categories and deriving formulas that confirm several longstanding conjectures in the field.

Contribution

It introduces a new framework for equivariant indicators in spherical fusion categories and proves a key congruence subgroup theorem with broad implications.

Findings

All modular representations have finite images.

Derived formulas generalizing Bantay's second indicator formula.

Confirmed conjectures of Eholzer, Pradisi-Sagnotti-Stanev, and Borisov-Halpern-Schweigert.

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 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.