Rank-finiteness for modular categories

Bruillard, Paul; Ng, Siu-Hung; Rowell, Eric; Wang, Zhenghan

doi:10.1090/jams/842

Rank-finiteness for modular categories
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by Paul Bruillard, Siu-Hung Ng, Eric C. Rowell and Zhenghan Wang

J. Amer. Math. Soc. 29 (2016), 857-881

DOI: https://doi.org/10.1090/jams/842

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 $N= \textrm {ord}(T)$, 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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