F-symbols and R-symbols for the Drinfeld center of the Haagerup subfactor
Fabian Mäurer, Ulrich Thiel, Gert Vercleyen
TL;DR
This work develops and applies an algorithm to compute the $F$-symbols, $R$-symbols, and $P$-symbols of the Drinfeld center $\mathcal{Z}(\mathcal{H}_3)$ by expressing them over a minimal field, enabling explicit data access and efficient computation. The authors implement these methods in TensorCategories.jl and demonstrate that the center data for the Haagerup category $\mathcal{H}_3$ can be realized with greatly reduced field degree, yielding over $1.6$ million $F$-symbols and $1400$ $R$-symbols, alongside modular data. They introduce the minimal-field gauge approach for multiplicity-free pivotal fusion categories, with explicit minimal fields $M(\mathcal{H}_2)=\mathbb{Q}(\alpha)$ and $M(\mathcal{H}_3)=\mathbb{Q}(\beta)$, and provide constructive procedures (GSB) to place the center data in these fields. The results facilitate reliable, scalable computations of center data and modular invariants, with data and code publicly available, advancing the computational study of exotic centers and RCFT connections. Overall, the paper delivers a practical toolkit for accessing and manipulating center data and a theoretical framework for minimizing the arithmetic field needed for symbolic fusion-category data.
Abstract
We have recently devised and implemented an algorithm for computing the Drinfeld center of a fusion category in our software package \textsc{TensorCategories.jl}. By simplifying the field of definition of the F-symbols, R-symbols, and pivotal coefficients of the Haagerup H3 category, we were now able to compute explicit F-symbols, R-symbols, and pivotal coefficients for its Drinfeld center Z(H3). In this paper, we discuss how to access this data. We also present our new general algorithm for finding a gauge of a multiplicity-free fusion category in which the symbols belong to a minimal field.