[Paper Review] Drinfeld centers of fusion categories arising from generalized Haagerup subfactors

Pinhas Grossman, Masaki Izumi|arXiv (Cornell University)|Jan 30, 2015

Algebraic structures and combinatorial models25 references4 citations

TL;DR

This paper computes the modular data of the Drinfeld center for fusion categories arising from generalized Haagerup subfactors, particularly focusing on the $4442$ subfactor and related categories. Using a Cuntz algebra framework and tube algebra techniques, it derives explicit formulas for the $S$ and $T$ matrices, identifies the full set of simple objects in the Drinfeld center, and provides evidence for new infinite families of quadratic modular categories.

ABSTRACT

We consider generalized Haagerup categories such that $1 \oplus X$ admits a $Q$-system for every non-invertible simple object $X$. We show that in such a category, the group of order two invertible objects has size at most four. We describe the simple objects of the Drinfeld center and give partial formulas for the modular data. We compute the remaining corner of the modular data for several examples and make conjectures about the general case. We also consider several types of equivariantizations and de-equivariantizations of generalized Haagerup categories and describe their Drinfeld centers. In particular, we compute the modular data for the Drinfeld centers of a number of examples of fusion categories arising in the classification of small-index subfactors: the Asaeda-Haagerup subfactor; the $3^{\Z_4} $ and $3^{\Z_2 imes \Z_2} $ subfactors; the $2D2$ subfactor; and the $4442$ subfactor. The results suggest the possibility of several new infinite families of quadratic categories. A description and generalization of the modular data associated to these families in terms of pairs of metric groups is taken up in the accompanying paper \cite{GI19_2}.

Motivation & Objective

• To determine the structure of the Drinfeld center for fusion categories derived from generalized Haagerup subfactors.
• To compute the modular data (S and T matrices) for the Drinfeld center of the $4442$ subfactor and related categories.
• To investigate the existence and classification of invertible objects in generalized Haagerup categories with $1\oplus X$ admitting a $Q$-system.
• To explore equivariantizations and de-equivariantizations of generalized Haagerup categories and their Drinfeld centers.
• To provide evidence for new infinite families of quadratic modular tensor categories through explicit modular data computation.

Proposed method

• Utilizes the Cuntz $C^*$-algebra framework to realize generalized Haagerup categories via endomorphisms of $\mathcal{O}_{|G|+1}$.
• Applies the tube algebra formalism to compute half-braidings and identify minimal central projections in the Drinfeld center.
• Employs eigenvalue analysis of the $\mathbf{t}$-operator to classify irreducible half-braidings and determine $T$-eigenvalues.
• Derives explicit formulas for the $S$-matrix using the Verlinde formula and block-wise indexing of simple objects.
• Uses the $Q$-system condition on $1\oplus X$ to define generalized Haagerup subfactors and ensure unitarity.
• Applies equivariantization and de-equivariantization techniques to relate different fusion categories and their centers.

Experimental results

Research questions

• RQ1 What is the structure of the Drinfeld center for fusion categories arising from generalized Haagerup subfactors with $1\oplus X$ admitting a $Q$-system?
• RQ2 What are the modular data (S and T matrices) for the Drinfeld center of the $4442$ subfactor and related fusion categories?
• RQ3 How many invertible objects of order two can exist in a generalized Haagerup category where $1\oplus X$ admits a $Q$-system?
• RQ4 Can the modular data of these centers be described in terms of pairs of metric groups, as suggested in the companion paper [GI19]?
• RQ5 What are the implications of the computed modular data for the existence of new infinite families of quadratic modular tensor categories?

Key findings

• The group of order two invertible objects in a generalized Haagerup category with $1\oplus X$ admitting a $Q$-system has size at most four.
• The Drinfeld center of the $4442$ fusion category has 48 simple objects, grouped into eight blocks of sizes 6, 2, 6, 2, 6, 2, 12, and 12.
• The $S$-matrix for the Drinfeld center of the $4442$ subfactor is explicitly computed and displayed in Figure 7, with entries involving $\omega$, $\zeta_5$, and trigonometric functions of $\pi/5$.
• The $T$-matrix eigenvalues are determined as $1, -1, \zeta_5^\varepsilon, \omega, \omega\zeta_5^{\varepsilon_1}$, with $\omega$ a cube root of unity.
• The modular data computation provides strong numerical evidence for the existence of new infinite families of quadratic modular tensor categories.
• The results for the Asaeda-Haagerup, $3^{\mathbb{Z}_4}$, $3^{\mathbb{Z}_2\times\mathbb{Z}_2}$, and $2D2$ subfactors are consistent with a unified framework for generalized Haagerup categories and their centers.