Rank-finiteness for modular categories (Q2802073)

Fix \(r\in\mathbb{N}\). The authors show that there are, up to equivalence, finitely many modular categories over \(\mathbb{C}\) with exactly \(r\) isomorphism classes of simple objects. Recall that a modular category over \(\mathbb{C}\) is a non-degenerate braided spherical fusion category over \(\mathbb{C}\). That is, a semisimple rigid monoidal fusion category with finitely many simple objects. This problem has been known for some time as the rank-finiteness conjecture, introduced by the fourth named author. We remark that for each choice of \(r\in\mathbb{N}\), the bound of the number of possible modular categories found by the authors grows exponentially fast. Nevertheless, it triggers a classification program for low rank cases. The authors announce such a classification for rank 5, cases up to 4 already being known by the work of \textit{J. de Boer} and \textit{J. Goeree} [Commun. Math. Phys. 139, No. 2, 267--304 (1991; Zbl 0760.57002)].NEWLINENEWLINEThe idea of the proof is inspired on the work of \textit{E. Landau} [Math. Ann. 56, 671--676 (1903; JFM 34.0241.09)], showing that for each choice \(r\in\mathbb{N}\) of the number of irreducible complex representations, there are only finitely many finite groups \(G\) with exactly \(r\) such representations. However, due to technical matters, a new approach is needed and they obtain a generalization of the Cauchy theorem in group theory to the context of spherical fusion categories. As a byproduct, the authors deduce new arithmetic properties of the family of quantum dimensions of the simple objects, namely that these are \(S\)-units with respect to a certain set \(S\) of prime ideals in a Dedekind domain. A result of \textit{J.-H. Evertse} [Compos. Math. 53, 225--244 (1984; Zbl 0547.10008)] allows them to show that there are only finitely many possible fusion rules for any given rank, which proves the rank-finiteness conjecture, by the so-called Ocneanu rigidity, a result of \textit{P. Etingof} et al. [Ann. Math. (2) 162, No. 2, 581--642 (2005; Zbl 1125.16025)], stating that there are only finitely many equivalence classes of modular categories for any fixed set of fusion rules.