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Topological Field Theory with Haagerup Symmetry

Tzu-Chen Huang, Ying-Hsuan Lin

TL;DR

This work constructs a (1+1)d topological field theory extended by defects that realizes the Haagerup ${\mathcal{H}}_3$ fusion category through six vacua and six defect lines. By solving the full set of crossing and modular constraints and exploiting the transparent $F$-symbol gauge, the authors explicitly determine the local operator algebra, defect operator data, and their intertwinings, culminating in a concrete TFT with fully specified axiomatic data. They verify that boundary data form a NIM-rep and show how gauging algebra objects connects ${\mathcal{H}}_3$ to ${\mathcal{H}}_1$ and ${\mathcal{H}}_2$, situating Haagerup exotica within a gauging framework. The paper offers a constructive bootstrap approach to exotica, providing a concrete template for realizing nontrivial fusion categories in (1+1)d TFTs and clarifying how such theories can relate to potential CFTs via RG flows and gauging.

Abstract

We construct a (1+1)$d$ topological field theory (TFT) whose topological defect lines (TDLs) realize the transparent Haagerup $\mathcal{H}_3$ fusion category. This TFT has six vacua, and each of the three non-invertible simple TDLs hosts three defect operators, giving rise to a total of 15 point-like operators. The TFT data, including the three-point coefficients and lasso diagrams, are determined by solving all the sphere four-point crossing equations and torus one-point modular invariance equations. We further verify that the Cardy states furnish a non-negative integer matrix representation under TDL fusion. While many of the constraints we derive are not limited to the this particular TFT with six vacua, we leave open the construction of TFTs with two or four vacua. Finally, TFTs realizing the Haagerup $\mathcal{H}_1$ and $\mathcal{H}_2$ fusion categories can be obtained by gauging algebra objects. This note makes a modest offering in our pursuit of exotica and the quest for their eventual conformity.

Topological Field Theory with Haagerup Symmetry

TL;DR

This work constructs a (1+1)d topological field theory extended by defects that realizes the Haagerup fusion category through six vacua and six defect lines. By solving the full set of crossing and modular constraints and exploiting the transparent -symbol gauge, the authors explicitly determine the local operator algebra, defect operator data, and their intertwinings, culminating in a concrete TFT with fully specified axiomatic data. They verify that boundary data form a NIM-rep and show how gauging algebra objects connects to and , situating Haagerup exotica within a gauging framework. The paper offers a constructive bootstrap approach to exotica, providing a concrete template for realizing nontrivial fusion categories in (1+1)d TFTs and clarifying how such theories can relate to potential CFTs via RG flows and gauging.

Abstract

We construct a (1+1) topological field theory (TFT) whose topological defect lines (TDLs) realize the transparent Haagerup fusion category. This TFT has six vacua, and each of the three non-invertible simple TDLs hosts three defect operators, giving rise to a total of 15 point-like operators. The TFT data, including the three-point coefficients and lasso diagrams, are determined by solving all the sphere four-point crossing equations and torus one-point modular invariance equations. We further verify that the Cardy states furnish a non-negative integer matrix representation under TDL fusion. While many of the constraints we derive are not limited to the this particular TFT with six vacua, we leave open the construction of TFTs with two or four vacua. Finally, TFTs realizing the Haagerup and fusion categories can be obtained by gauging algebra objects. This note makes a modest offering in our pursuit of exotica and the quest for their eventual conformity.

Paper Structure

This paper contains 26 sections, 111 equations, 1 figure, 3 tables.

Table of Contents

  1. Introduction
  2. Topological field theory extended by defects
  3. Fusion category of topological defect lines
  4. Local operators and commutative Frobenius algebra
  5. Defect operators, defect operator algebra, and lassos
  6. General observables, crossing symmetry and modular invariance
  7. Spectral constraints by Haagerup symmetry
  8. The Haagerup fusion ring with six simple objects
  9. Action on local operators and representation theory
  10. Modular invariance and vacuum degeneracy
  11. Transparency and $\mathbb{Z}_3$ symmetry
  12. $\mathbb{Z}_3$ relations for lassos and dumbbells
  13. $\mathbb{Z}_3$ action on defect operators
  14. Bootstrap constraints
  15. Local operator algebra and associativity
  16. ...and 11 more sections

Figures (1)

  • Figure 1: Gauging relations among theories realizing the three Haagerup fusion categories.