Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Realizing modular data from centers of near-group categories

Zhiqiang Yu, Qing Zhang

TL;DR

This work constructs and analyzes modular data arising from centers of near-group fusion categories, focusing on a category of type $\mathbb{Z}/4\mathbb{Z}\times\mathbb{Z}/4\mathbb{Z}+16$ and its Drinfeld center, which has rank $304$, and shows that condensing a Tannakian subcategory $\operatorname{Rep}(\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/4\mathbb{Z})$ yields a rank-10 modular category whose data matches a known rank-10 modular category from ng2023classification; it also demonstrates that this condensation is the center of a rank-4 self-dual fusion category. Additionally, the center of a near-group category $\mathbb{Z}/8\mathbb{Z}+8$ is computed and shown to possess a $\mathbb{Z}/2\mathbb{Z}$-Tannakian subcategory whose condensation produces a braided product $\mathcal{C}(\mathbb{Z}/4\mathbb{Z},q)\boxtimes \mathcal{D}$ with $\mathcal{D}$ sharing modular data with the quantum group category $\mathcal{C}(\mathfrak{g}_2,4)$, up to Galois conjugation. These results illustrate how condensation and equivariantization produce new modular data and connect near-group categories to quantum groups and VOAs, advancing realizability questions for modular data.

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 $C(\mathfrak{g}_2, 4)$.

Realizing modular data from centers of near-group categories

TL;DR

This work constructs and analyzes modular data arising from centers of near-group fusion categories, focusing on a category of type and its Drinfeld center, which has rank , and shows that condensing a Tannakian subcategory yields a rank-10 modular category whose data matches a known rank-10 modular category from ng2023classification; it also demonstrates that this condensation is the center of a rank-4 self-dual fusion category. Additionally, the center of a near-group category is computed and shown to possess a -Tannakian subcategory whose condensation produces a braided product with sharing modular data with the quantum group category , up to Galois conjugation. These results illustrate how condensation and equivariantization produce new modular data and connect near-group categories to quantum groups and VOAs, advancing realizability questions for modular data.

Abstract

In this paper, we show the existence of a near-group category of type and compute the modular data of its Drinfeld center. We prove that a modular data of rank can be obtained through condensation of the Drinfeld center of the near-group category , and it can also be realized as the Drinfeld center of a fusion category of rank . Moreover, we compute the modular data for the Drinfeld center of a near-group category and show that the non-pointed factor of its condensation has the same modular data as the quantum group category .
Paper Structure (13 sections, 16 theorems, 72 equations, 5 tables)

This paper contains 13 sections, 16 theorems, 72 equations, 5 tables.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Modular data
  4. Near-group categories
  5. Condensation of modular categories
  6. Realization of modular data
  7. Near-group categories $\mathbb Z/4\mathbb Z\times \mathbb Z/4\mathbb Z+16$
  8. Existence of $\mathbb{Z}/4 \mathbb{Z}\times\mathbb{Z}/4 \mathbb{Z} + 16$
  9. Modular data of the center
  10. Condensation of the Drinfeld center of $\mathbb{Z}/4\mathbb Z\times\mathbb{Z}/4\mathbb Z+16$
  11. Near-group category $\mathbb Z/ 8\mathbb Z+8$
  12. The center of $\mathbb Z/8\mathbb Z+8$
  13. Solutions for $(\omega_j, \tau_j, \xi_j)$

Key Result

Theorem 1

Let $\mathcal{C}$ be a near-group category of type $\mathbb Z/4\mathbb Z\times \mathbb Z/4\mathbb Z +16$ associated with the symmetric bi-character $\langle(g_1, g_2),(h_1, h_2)\rangle=\left(\zeta_4\right)^{g_1 h_1- g_2h_2}$, $g_1,g_2,h_1,h_2\in \mathbb Z/4\mathbb Z\times \mathbb Z/4\mathbb Z$.Then

Theorems & Definitions (32)

  • Theorem 1
  • Theorem 2
  • Theorem 2.1
  • Proposition 2.2
  • proof
  • Example 2.3.1
  • Proposition 3.1
  • proof
  • Remark 3.2
  • Lemma 3.3
  • ...and 22 more