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 = L2M 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.
