Infinite families of potential modular data related to quadratic categories

TL;DR

This paper constructs infinite families of potential modular data inspired by quadratic categories, generalizing previous conjectures and including many known and new examples, with verified modular relations and non-negative Verlinde coefficients.

Contribution

It introduces new infinite families of potential modular data derived from involutive metric groups, extending known conjectures and including most known quadratic categories.

Findings

Constructed $S$ and $T$ matrices satisfying modular relations.

Verlinde coefficients are non-negative integers.

Includes many potential modular data not corresponding to known categories.

Abstract

We present several infinite families of potential modular data motivated by examples of Drinfeld centers of quadratic categories. In each case, the input is a pair of involutive metric groups with Gauss sums differing by a sign, along with some conditions on the fixed points of the involutions and the relative sizes of the groups. From this input we construct $S$ and $T$ matrices which satisfy the modular relations and whose Verlinde coefficients are non-negative integers. We also check certain restrictions coming from Frobenius-Schur indicators. These families generalize Evans and Gannon's conjectures for the modular data associated to generalized Haagerup and near-group categories for odd groups, and include the modular data of the Drinfeld centers of almost all known quadratic categories. In addition to the subfamilies which are conjecturally realized by centers of quadratic categories, these families include many examples of potential modular data which do not correspond to known types of modular tensor categories.