Rank-finiteness for modular categories

摘要: We prove a rank-finiteness conjecture for modular categories: up to equivalence, there are only finitely many categories of any fixed rank. Our technical advance is generalization the Cauchy theorem in group theory context spherical fusion categories. For category $\mathcal{C}$ with $N=ord(T)$, order $T$-matrix, says that set primes dividing global quantum dimension $D^2$ Dedekind domain $\mathbb{Z}[e^{\frac{2\pi i}{N}}]$ identical $N$.