Topological Field Theory with Haagerup Symmetry

TL;DR

This paper constructs a 1+1 dimensional topological field theory with defect lines realizing the Haagerup fusion category, determining its data through crossing and modular invariance equations, and exploring its vacua and related categories.

Contribution

It introduces a novel TFT with Haagerup symmetry, explicitly solving for its data and analyzing its defect operators and vacua, expanding the understanding of exotic fusion categories in TFTs.

Findings

Constructed a TFT with six vacua and Haagerup symmetry.

Determined the TFT data via crossing and modular invariance equations.

Verified the non-negative integer matrix representation of Cardy states.

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.