On triality defects in 2d CFT

We consider the triality fusion category discovered in $c = 1$ KT theory \cite{Thorngren:2021yso}. analyze this using tools from group theoretical and describe how to compute simple lines, rules $F$-symbols. then study physical implication of including deriving spin selection rule, computing asymptotic density states irreps symmetries, analyzing its anomaly constraints on renormalization flow. There is another set $F$-symbols for categories with same rule known literature \cite{teo2015theory} which we compare with, find two are different as they lead rules. This gives a complete list by classification result \cite{jordan2009classification}.