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The $F$-Symbols for Transparent Haagerup-Izumi Categories with $G = \mathbb{Z}_{2n+1}$

Tzu-Chen Huang, Ying-Hsuan Lin

TL;DR

<3-5 sentence high-level summary> This work introduces and exploits the notion of transparency for Haagerup-Izumi fusion categories with G = Z_{2n+1}, showing that demanding transparency dramatically reduces the independent F-symbol data and renders the pentagon equations tractable. By combining transparency with A4 or S4 tetrahedral invariance, the authors construct and classify unitary and non-unitary F-symbols up to G = Z_{15}, explicitly populating unitary cases including Haagerup H_2 and H_3, and providing extensive polynomial root data that encode the non-invertible sector. The results connect to prior Cuntz algebra approaches and gauging relations, offering a practical route to new fusion categories for new fusion rings and providing concrete data for applications in topological phases, string-net models, and defect crossing symmetry. The work also suggests broader applicability to nearby quadratic and near-group fusion rings and highlights the potential for partial transparency to yield solvable pentagon identities in broader families.

Abstract

A fusion category is called transparent if the associator involving any invertible object is the identity map. For the Haagerup-Izumi fusion rings with $G = \mathbb{Z}_{2n+1}$ (the $\mathbb{Z}_3$ case is the Haagerup fusion ring with six simple objects), the transparent ansatz reduces the number of independent $F$-symbols from order $\mathcal{O}(n^6)$ to $\mathcal{O}(n^2)$, rendering the pentagon identity practically solvable. Transparent Haagerup-Izumi fusion categories are thereby constructively classified up to $G = \mathbb{Z}_9$, recovering all known Haagerup-Izumi fusion categories to this order, and producing new ones. Transparent Haagerup-Izumi fusion categories additionally satisfying $S_4$ tetrahedral invariance are further classified up to $G = \mathbb{Z}_{15}$, and the explicit $F$-symbols for the unitary ones, including the Haagerup $\mathcal{H}_3$ fusion category, are compactly presented. The $F$-symbols for the Haagerup $\mathcal{H}_2$ fusion category are also presented. Going beyond, the transparent ansatz offers a viable course towards constructing novel fusion categories for new fusion rings.

The $F$-Symbols for Transparent Haagerup-Izumi Categories with $G = \mathbb{Z}_{2n+1}$

TL;DR

<3-5 sentence high-level summary> This work introduces and exploits the notion of transparency for Haagerup-Izumi fusion categories with G = Z_{2n+1}, showing that demanding transparency dramatically reduces the independent F-symbol data and renders the pentagon equations tractable. By combining transparency with A4 or S4 tetrahedral invariance, the authors construct and classify unitary and non-unitary F-symbols up to G = Z_{15}, explicitly populating unitary cases including Haagerup H_2 and H_3, and providing extensive polynomial root data that encode the non-invertible sector. The results connect to prior Cuntz algebra approaches and gauging relations, offering a practical route to new fusion categories for new fusion rings and providing concrete data for applications in topological phases, string-net models, and defect crossing symmetry. The work also suggests broader applicability to nearby quadratic and near-group fusion rings and highlights the potential for partial transparency to yield solvable pentagon identities in broader families.

Abstract

A fusion category is called transparent if the associator involving any invertible object is the identity map. For the Haagerup-Izumi fusion rings with (the case is the Haagerup fusion ring with six simple objects), the transparent ansatz reduces the number of independent -symbols from order to , rendering the pentagon identity practically solvable. Transparent Haagerup-Izumi fusion categories are thereby constructively classified up to , recovering all known Haagerup-Izumi fusion categories to this order, and producing new ones. Transparent Haagerup-Izumi fusion categories additionally satisfying tetrahedral invariance are further classified up to , and the explicit -symbols for the unitary ones, including the Haagerup fusion category, are compactly presented. The -symbols for the Haagerup fusion category are also presented. Going beyond, the transparent ansatz offers a viable course towards constructing novel fusion categories for new fusion rings.

Paper Structure

This paper contains 31 sections, 2 theorems, 93 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Transparent fusion categories
  4. Transparent Haagerup-Izumi fusion categories
  5. Classification of $F$-symbols
  6. Main theorems
  7. Explicit $F$-symbols for $G = \mathbb{Z}_{2n+1}$ with $1 \le n \le 7$
  8. Haagerup ${\mathcal{H}}_2$ ($G = \mathbb{Z}_3$)
  9. Theorem \ref{['Main']}(b).
  10. Haagerup ${\mathcal{H}}_3$ ($G = \mathbb{Z}_3$)
  11. Theorem \ref{['Main2']}(b).
  12. $G = \mathbb{Z}_5$
  13. Theorem \ref{['Main2']}(c).
  14. $G = \mathbb{Z}_7$
  15. Theorem \ref{['Main2']}(d).
  16. ...and 16 more sections

Key Result

Theorem 5.1

For the Haagerup-Izumi fusion rings with $G = \mathbb{Z}_{2n+1}$, let Under the transparent constraints T and imposing $A_4$ tetrahedral invariance (necessary by transparency), the pentagon identity has the following solutions:

Theorems & Definitions (5)

  • Definition 2.1: Cyclic-permutation invariance
  • Definition 3.1: Transparency
  • Definition 4.1: Transparent constraints
  • Theorem 5.1
  • Theorem 5.2