When are duality defects group-theoretical?

A bstract A quantum field theory with a finite abelian symmetry G may be equipped with a non-invertible duality defect associated with gauging G . For certain G , duality defects admit an alternative construction where one starts with invertible symmetries with certain ’t Hooft anomaly, and gauging a non-anomalous subgroup. This special type of duality defects are termed group theoretical. In this work, we determine when duality defects are group theoretical, among $$G={\mathbb{Z}}_{N}^{\left(0\right)}$$ and $${\mathbb{Z}}_{N}^{\left(1\right)}$$ in 2d and 4d quantum field theories, respectively. A duality defect is group theoretical if and only if its Symmetry TFT is a Dijkgraaf-Witten theory, and we argue that this is equivalent to a certain stability condition of the topological boundary conditions of the G gauge theory. By solving the stability condition, we find that a $${\mathbb{Z}}_{N}^{\left(0\right)}$$ duality defect in 2d is group theoretical if and only if N is a perfect square, and under certain assumptions a $${\mathbb{Z}}_{N}^{\left(1\right)}$$ duality defect in 4d is group theoretical if and only if N = L 2 M where −1 is a quadratic residue of M . For these subset of N , we construct explicit topological manipulations that map the non-invertible duality defects to invertible defects. We also comment on the connection between our results and the recent discussion of obstruction to duality-preserving gapped phases.