N-ality symmetry and SPT phases in 1+1d

Duality symmetries have been extensively investigated in various contexts, playing a crucial role in understanding quantum field theory and condensed matter theory. In this work, we extend this framework by studying N-ality symmetries, which are a generalization of duality symmetries and are mathematically described by Z_N-graded fusion categories. In particular, we focus on an N-ality symmetry obtained by gauging a non-anomalous subgroup of Z_N³ symmetry with a type III anomaly. We determine the corresponding fusion rules via two complementary approaches: a direct calculation and a representation-theoretic method. As an application, we study the symmetry-protected topological (SPT) phases associated with the N-ality symmetry. We classify all such SPT phases using the SymTFT framework and explicitly construct lattice Hamiltonians for some of them.