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Rank-finiteness for modular categories


Authors: Paul Bruillard, Siu-Hung Ng, Eric C. Rowell and Zhenghan Wang
Journal: J. Amer. Math. Soc. 29 (2016), 857-881
MSC (2010): Primary 18D10; Secondary 57R56, 16T05, 81R50, 17B37
Published electronically: July 21, 2015
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Abstract: 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$.


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Additional Information

Paul Bruillard
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843
Address at time of publication: Pacific Northwest National Laboratory, 902 Battelle Boulevard, Richland, Washington 99354
Email: pjb2357@gmail.com

Siu-Hung Ng
Affiliation: Department of Mathematics, Louisiana State University, Baton Rouge, Louisiana 70803
Email: rng@math.lsu.edu

Eric C. Rowell
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843
Email: rowell@math.tamu.edu

Zhenghan Wang
Affiliation: Microsoft Research Station Q and Department of Mathematics, University of California, Santa Barbara, California 93106
Email: zhenghwa@microsoft.com

DOI: https://doi.org/10.1090/jams/842
Keywords: Modular categories, Cauchy theorem, Frobenius-Schur indicator
Received by editor(s): August 29, 2014
Received by editor(s) in revised form: March 13, 2015, and May 26, 2015
Published electronically: July 21, 2015
Additional Notes: The first, third, and fourth authors were partially supported by NSF grant DMS1108725.
The second author was partially supported by NSF grants DMS1001566, DMS1303253, and DMS1501179.
Article copyright: © Copyright 2015 American Mathematical Society