A Computational Approach to Classifying Low Rank Modular Categories

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.