RANK-FINITENESS FOR MODULAR CATEGORIES
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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 with , the order of the modular -matrix, the Cauchy theorem says that the set of primes dividing the global quantum dimension in the Dedekind domain is identical to that of .