On Triality Defects in 2d CFT

TL;DR

This paper analyzes the triality fusion category in 2d conformal field theory, exploring its mathematical structure and physical implications, including spin rules, state density, anomalies, and RG constraints.

Contribution

It provides a detailed group-theoretical analysis of the triality fusion category, computes its simple lines, fusion rules, and F-symbols, and compares different sets of F-symbols with physical consequences.

Findings

Different F-symbol sets lead to distinct spin selection rules.

The fusion category's asymptotic state density is computed.

Constraints on RG flow from the fusion category are analyzed.

Abstract

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