On triality defects in 2d CFT

We consider the triality fusion category discovered in the c = 1 Kosterlitz-Thouless theory [1]. We analyze this fusion category using the tools from the group theoretical fusion category and compute the simple lines, fusion rules and F-symbols. We then studied the physical implication of this fusion category including deriving the spin selection rule, computing the asymptotic density of states of irreducible representations of the fusion category symmetries, and analyzing its anomaly and constraints under the renormalization group flow. There is another set of F-symbols for the fusion categories with the same fusion rule known in the literature [2]. We find these two solutions are different as they lead to different spin selection rules. This gives a complete list of the fusion categories with the same fusion rule by the classification result in [3].