A Computational Approach to Classifying Low Rank Modular Categories

TL;DR

This paper presents a computational method for classifying low rank modular categories by analyzing their modular data, using algebraic and group-theoretic techniques to systematically identify valid categories.

Contribution

It introduces a novel computational framework that combines group theory and algebraic geometry to classify low rank modular categories based on their modular data.

Findings

Successfully enumerated all possible modular data for low rank categories

Identified constraints on the Galois group acting on the S matrix

Provided a systematic approach to classify categories up to their modular data

Abstract

This paper introduces a computational approach to classifying low rank modular categories up to their modular data. The modular data of a modular category is a pair of matrices, $(S,T)$. Virtually all the numerical information of the category is contained within or derived from the modular data. The modular data satisfy a variety of criteria that Bruillard, Ng, Rowell, and Wang call the admissibility criteria. Of note is the Galois group of the $S$ matrix is an abelian group that acts faithfully on the columns of the eigenvalue matrix, $s = (\frac{S_{ij}}{S_{0j}})$. This gives an injection from Gal$(\mathbb{Q}(S),\mathbb{Q}) \to $ Sym$_r$, where $r$ is the rank of the category. Our approach begins by listing all the possible abelian subgroups of Sym$_6$ and building all the possible modular data for each group. We run each set of modular data through a series of Gr\"obner basis calculations until we either find a contradiction or solve for the modular data.