Rank-finiteness for modular categories
Institutional Repository Document
Overview
Research
Identity
Other
View All
Overview
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 $N=ord(T)$, 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 $mathbb{Z}[e^{frac{2pi i}{N}}]$ is identical to that of $N$.
