Communications in Mathematical Physics

, Volume 300, Issue 1, pp 1–46 | Cite as

Congruence Subgroups and Generalized Frobenius-Schur Indicators

  • Siu-Hung NgEmail author
  • Peter Schauenburg


We introduce generalized Frobenius-Schur indicators for pivotal categories. In a spherical fusion category \({\mathcal {C}}\) , an equivariant indicator of an object in \({\mathcal {C}}\) is defined as a functional on the Grothendieck algebra of the quantum double \({Z(\mathcal {C})}\) 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.


Monoidal Category Vertex Operator Algebra Congruence Subgroup Fusion Category Forgetful Functor 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  1. 1.Department of MathematicsIowa State UniversityAmesUSA
  2. 2.Mathematisches Institut der Universität MünchenMünchenGermany

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