Exploring G-ality defects in 2-dim QFTs

The Tambara-Yamagami (TY) fusion category symmetry TY(A,χ,ε) describes the enhanced non-invertible self-duality symmetry of a $2$-dim QFT under gauging a finite Abelian group A. We generalize the enhanced non-invertible symmetries by considering twisted gauging which allows stacking A-SPT before and after the gauging. Such non-invertible symmetries correspond to invertible anyon permutation symmetries of the $3$-dim SymTFT. Consider a finite group G formed by (un)twisted gaugings of A, a $2$-dim QFT invariant under topological manipulations in G admits non-invertible G-ality defects. We study the classification and the physical implication of the G-ality defects using SymTFT and the group-theoretical fusion categories, with three concrete examples. 1) Triality with A = ZN × ZN where N is coprime with $3$. The classification is acquired previously by Jordan and Larson where the data is similar to the TY fusion categories, and we determine the anomaly of these fusion categories. 2) p-ality with A = Zₚ × Zₚ where p is an odd prime. We consider two such categories P±,m which are distinguished by different choices of the symmetry fractionalization, a new data that does not appear in the TY classification, and show that they have distinct anomaly structures and spin selection rules. 3) S₃-ality with A = ZN × ZN. We study their classification explicitly for $N < 20$ via SymTFT, and provide a group-theoretical construction for certain N. We find $N=5$ is the minimal N to admit an S₃-ality and $N=11$ is the minimal N to admit a group-theoretical S₃-ality.