Realizing modular data from centers of near-group categories

TL;DR

This paper constructs and analyzes the modular data of certain near-group categories and their Drinfeld centers, revealing new connections and realizations of modular data through condensation and fusion categories.

Contribution

It demonstrates the existence of specific near-group categories and computes their Drinfeld center modular data, linking them to known quantum group categories.

Findings

Existence of a near-group category of type Z/4Z x Z/4Z+16.

Rank 10 modular data obtained via condensation of the Drinfeld center.

Modular data of the Drinfeld center of Z/8Z+8 matches that of quantum group category C(g2,4).

Abstract

In this paper, we show the existence of a near-group category of type $\mathbb{Z} / 4\mathbb{Z} \times \mathbb{Z} / 4\mathbb{Z}+16$ and compute the modular data of its Drinfeld center. We prove that a modular data of rank $10$ can be obtained through condensation of the Drinfeld center of the near-group category $\mathbb{Z} / 4\mathbb{Z} \times \mathbb{Z} / 4\mathbb{Z}+16$, and it can also be realized as the Drinfeld center of a fusion category of rank $4$. Moreover, we compute the modular data for the Drinfeld center of a near-group category $\mathbb{Z} / 8\mathbb{Z}+8$ and show that the non-pointed factor of its condensation has the same modular data as the quantum group category $C(\mathfrak{g}_2, 4)$.