On classification of modular categories by rank

TL;DR

This paper advances the classification of modular categories by rank, developing new algebraic methods and explicitly determining fusion rules for rank-5 categories, thus contributing to the broader understanding of their structure.

Contribution

It introduces novel arithmetic and algebraic techniques for classifying modular categories by rank and explicitly classifies all rank-5 cases.

Findings

All fusion rules for rank-5 modular categories are determined.

The monoidal equivalence classes for these categories are described.

The methods facilitate classification based on rank, supporting the Rank-Finiteness Theorem.

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.