On Triality Defects in 2d CFT

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}.

v4, a mistake on the anomaly of triality fusion category is corrected with section 4.3 rewritten thanks to the referee

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• SymTFTs and Duality Defects from 6d SCFTs on 4-manifolds
• Symmetry-resolved Entanglement Entropy, Spectra & Boundary Conformal Field Theory
• Realizing triality and $p$-ality by lattice twisted gauging in (1+1)d quantum spin systems
• Fermionic Non-Invertible Symmetries in (1+1)d: Gapped and Gapless Phases, Transitions, and Symmetry TFTs
• Noninvertible Symmetry-Resolved Affleck-Ludwig-Cardy Formula and Entanglement Entropy from the Boundary Tube Algebra
• Self-$G$-ality in 1+1 dimensions
• von Neumann Subfactors and Non-invertible Symmetries
• Non-invertible translation from Lieb-Schultz-Mattis anomaly