On classification of modular categories by rank

TL;DR

This work develops a rank-based program for classifying modular categories by exploiting the Rank-Finiteness Theorem, combining arithmetic, representation theory, and number-theoretic methods. By carefully analyzing modular data $(S,T)$, their SL$(2,\mathbb{Z})$ representations, and Galois symmetries, the authors derive strong constraints on admissible modular data and field extensions, enabling a complete rank-5 classification. They show rank-5 modular categories are Grothendieck-equivalent to one of four families $SU(2)_4$, $SU(2)_9/\mathbb{Z}_2$, $SU(5)_1$, or $SU(3)_4/\mathbb{Z}_3$, and give a route to monoidal classification via Kazhdan–Wenzl theory. The results illuminate the structure of low-rank modular categories, with implications for topological quantum field theories and topological quantum computation, and lay groundwork for extending the program to higher ranks.

Abstract

The feasibility of a classification-by-rank program for modular categories follows from the Rank-Finiteness Theorem. We develop arithmetic, representation theoretic and algebraic methods for classifying modular categories by rank. As an application, we determine all possible fusion rules for all rank=$5$ modular categories and describe the corresponding monoidal equivalence classes.

Theorems & Definitions (70)

• Definition 2.1
• Definition 2.2
• Theorem 2.3
• Definition 2.4
• Theorem 2.5
• Lemma 2.6
• Definition 2.7
• Remark 3.1
• Lemma 3.2
• proof