This analysis shows fusion rules in quantum field theory and identifies symmetry-protected topological phases, indicating novel characteristics of N-ality symmetry.
A bstract Duality symmetries have been extensively investigated in various contexts, playing a crucial role in understanding quantum field theory and condensed matter theory. In this paper, 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 N × ℤ N × ℤ N 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.
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Maeda et al. (2025) studied this question.