Rank-finiteness for modular categories

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 C \mathcal {C} with N = ord ( T ) N= \textrm {ord}(T) , the order of the modular T T -matrix, the Cauchy theorem says that the set of primes dividing the global quantum dimension D 2 D^2 in the Dedekind domain Z [ e 2 π i N ] \mathbb {Z}[e^{\frac {2\pi i}{N}}] is identical to that of N N .