Topological Field Theory with Haagerup Symmetry

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.

41+11 pages, 1 figure, 3 tables; v2: corrected statements about the literature, revised Appendix A

References in corpus (13)

• Generalized Global Symmetries
• Interacting anyons in topological quantum liquids: The golden chain
• Symmetries and Strings of Adjoint QCD${}_2$
• Fusion Category Symmetry I: Anomaly In-Flow and Gapped Phases
• Topological Defects on the Lattice: Dualities and Degeneracies
• D-branes and K-theory in 2D topological field theory
• ${\mathbb{Z}}_N$ Symmetries, Anomalies, and the Modular Bootstrap
• Lorentzian Dynamics and Factorization Beyond Rationality
• Perfect Metal Phases of One-Dimensional and Anisotropic Higher-Dimensional Systems
• Perturbative search for dead-end CFTs
• The F-Symbols for the H3 Fusion Category
• Open Verlinde line operators
• Microscopic Models for Fusion Categories

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• Fusion Surface Models: 2+1d Lattice Models from Fusion 2-Categories
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