On Triality Defects in 2d CFT

TL;DR

This work characterizes the triality fusion category arising in the c=1 KT theory as a group-theoretical fusion category C(A4,1,Z2,1), and computes its simple lines, fusion rules, and F-symbols via A–A bimodule techniques. It then translates these algebraic data into physical consequences, deriving spin selection rules, defect-state densities, and anomaly/RG-flow constraints, and tests the framework in the explicit c=1 KT example by computing twisted partition functions and spectra. The authors also compare two distinct F-symbol frameworks, showing they yield different spin rules and thus inequivalent fusion categories, and they classify all triality categories consistent with the given fusion rules, including both group-theoretical and intrinsic (non-intrinsic) classes. The results establish that finite non-intrinsic non-invertible 0-form symmetries in 2d bosonic theories are completely captured by group-theoretical fusion categories, while intrinsic variants exhibit distinct spin spectra and FS structures. Overall, the paper provides a concrete, bimodule-based construction of triality defects, clarifies when such symmetries admit symmetric gapped phases, and connects categorical data to modular bootstrap and density-of-states analyses, with explicit KT-model verification. The findings offer a robust framework for classifying and utilizing non-invertible 2d symmetries in CFTs and related quantum field theories.

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}.

Figures (9)

• Figure 1: The fusion vertex of TDLs $\mathcal{L}_1,\mathcal{L}_2$ fuse into $\mathcal{L}_3$ and split vertex of TDLs $\mathcal{L}_3$ splits into $\mathcal{L}_1,\mathcal{L}_2$. The red cross marks the last leg, which determines the ordering of each vertex.
• Figure 2: The fusion of three TDLs has two different ways, but they are related by the $F$-move and characterized by the $F$-symbol defined in this diagram.
• Figure 3: The upper 2 $F$-moves and the lower 3 $F$-moves yield the same diagram, which gives the pentagon equation in \ref{['eq:pentaequ']}.
• Figure 4: Convention on the twisted partition function $Z_{\mathcal{L}_1 \mathcal{L}_2}^{\mathcal{L}_3}(\tau,\overline{\tau})$.
• Figure 5: Generalized modular bootstrap equations from $S$ modular transformation of the twisted partition function $Z^{\mathcal{L}_3,\mu,\nu}_{\mathcal{L}_1,\mathcal{L}_2}(\tau,\overline{\tau})$.