Rank-finiteness for modular categories

Bruillard, Paul^1, 2 ; Ng, Siu-Hung³ ; Rowell, Eric¹ ; Wang, Zhenghan⁴

¹ Department of Mathematics, Texas A&M University, College Station, Texas 77843
² Pacific Northwest National Laboratory, 902 Battelle Boulevard, Richland, Washington 99354
³ Department of Mathematics, Louisiana State University, Baton Rouge, Louisiana 70803
⁴ Microsoft Research Station Q and Department of Mathematics, University of California, Santa Barbara, California 93106

Journal of the American Mathematical Society, Tome 29 (2016) no. 3, pp. 857-881

Résumé

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 $N$.

Détail

DOI : 10.1090/jams/842

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