When are Duality Defects Group-Theoretical?

TL;DR

This paper characterizes when duality defects in certain quantum field theories are group theoretical, linking this property to the structure of the symmetry TFT and providing explicit constructions for specific cases.

Contribution

It determines criteria for duality defects to be group theoretical in 2d and 4d theories, relating this to Dijkgraaf-Witten theories and stability conditions, with explicit examples.

Findings

A duality defect is group theoretical iff its Symmetry TFT is a Dijkgraaf-Witten theory.

In 2d, a $ ext{Z}_N$ duality defect is group theoretical iff N is a perfect square.

In 4d, a $ ext{Z}_N$ duality defect is group theoretical iff N=L^2 M with -1 a quadratic residue of M.

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.