Rank-finiteness for modular categories

We prove a rank-finiteness conjecture for modular categories: up to equivalence, there are only finitely many modular categories of any fixed rank. Our technical advance is a generalization of the Cauchy theorem in group theory to the context of spherical fusion categories. For a modular category \m...

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Bibliographic Details
Published in:Journal of the American Mathematical Society 2016-07, Vol.29 (3), p.857-881
Main Authors: Bruillard, Paul, Ng, Siu-Hung, Rowell, Eric C., Wang, Zhenghan
Format: Article
Language:English
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Summary:We prove a rank-finiteness conjecture for modular categories: up to equivalence, there are only finitely many modular categories of any fixed rank. Our technical advance is a generalization of the Cauchy theorem in group theory to the context of spherical fusion categories. For a modular category \mathcal {C} with N= {\rm ord}(T), the order of the modular T-matrix, the Cauchy theorem says that the set of primes dividing the global quantum dimension D^2 in the Dedekind domain \mathbb{Z}[e^{\frac {2\pi i}{N}}] is identical to that of N.
ISSN:0894-0347
1088-6834
DOI:10.1090/jams/842