[Paper Review] A Computational Approach to Classifying Low Rank Modular Categories

Daniel Creamer|arXiv (Cornell University)|Dec 4, 2019

Algebraic structures and combinatorial models10 references44 citations

TL;DR

The paper presents a computational workflow to classify low-rank modular categories via their modular data (S,T), leveraging Galois actions and Gröbner basis computations to test admissibility criteria.

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}) o $ 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 Gröbner basis calculations until we either find a contradiction or solve for the modular data.

Motivation & Objective

• Motivate the classification of low-rank modular categories through modular data (S,T).
• Exploit admissibility criteria to constrain possible categories.
• Develop a computational pipeline combining group-theoretic enumeration with Gröbner basis solving.

Proposed method

• Represent a modular category by its modular data (S,T).
• Note that the Galois group of S acts faithfully on the eigenvalue-derived matrix s, giving an injection Gal(Q(S),Q) → Sym_r.
• List all abelian subgroups of Sym_6 and construct modular data compatible with each group.
• Apply Gröbner basis calculations to derive constraints and solve for modular data, detecting contradictions when present.

Experimental results

Research questions

• RQ1 What modular data (S,T) are consistent with the admissibility criteria for low-rank modular categories?
• RQ2 How does the abelian Galois action constrain possible modular data for rank up to 6?
• RQ3 Can a complete computational workflow enumerate and certify all low-rank modular categories up to a given rank?
• RQ4 What contradictions arise in the Gröbner basis computations that eliminate impossible data sets?

Key findings

• A computational framework is proposed to enumerate abelian subgroups of Sym_6 and generate corresponding modular data.
• Gröbner basis calculations are used to test consistency and prune infeasible data, stopping when contradictions are found or solutions are obtained.
• The approach integrates the Galois action on the S-matrix to constrain the eigenvalue structure and drive the search through modular data space.