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We construct a (1 + 1)d topological field theory (TFT) whose topological defect lines (TDLs) realize the transparent Haagerup H3 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 this particular TFT with six vacua, we leave open the construction of TFTs with two or four vacua. Finally, TFTs realizing the Haagerup H1 and H2 fusion categories can be obtained by gauging algebra objects. This article makes a modest offering in our pursuit of exotica and the quest for their eventual conformity.
Huang et al. (Fri,) studied this question.