When are Duality Defects Group-Theoretical?

Abstract

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 $G={\mathbb{Z}}_N^{(0)}$ and ${\mathbb{Z}}_N^{(1)}$ in $2$d 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 ${\mathbb{Z}}_N^{(0)}$ duality defect in 2d is group theoretical if and only if $N$ is a perfect square, and under certain assumptions a ${\mathbb{Z}}_N^{(1)}$ 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.

35 pages

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