Computing Modular Data for Pointed Fusion Categories

TL;DR

This paper derives an explicit formula for the modular data of the Drinfeld center of pointed fusion categories using quasi-Hopf algebra representations, and provides computational tools and a database for small finite groups and 3-cocycles.

Contribution

It offers a rigorous derivation of the modular data formula for arbitrary 3-cocycles and develops optimized code to compute and catalog this data for small groups.

Findings

Derived explicit modular data formula for pointed fusion categories.

Created optimized computational code for modular data calculation.

Compiled a comprehensive database for categories of dimension less than 64.

Abstract

A formula for the modular data of $\mathcal{Z}(Vec^{\omega}G)$ was given by Coste, Gannon and Ruelle in arXiv:hep-th/0001158, but without an explicit proof for arbitrary 3-cocycles. This paper presents a derivation using the representation category of the quasi Hopf algebra $D^{\omega}G$. Further, we have written code to compute this modular data for many pairs of small finite groups and 3-cocycles. This code is optimised using Galois symmetries of the S and T matrices. We have posted a database of modular data for the Drinfeld center of every Morita equivalence class of pointed fusion categories of dimension less than 64.