When are Duality Defects Group-Theoretical?

TL;DR

The paper investigates when non-invertible duality defects arising from gauging a finite abelian symmetry G are group theoretical, focusing on G = Z_N^{(0)} in 2d and G = Z_N^{(1)} in 4d. Using the Symmetry Topological Field Theory (SymTFT) formalism, the authors relate group-theoreticality to the existence of a Dijkgraaf-Witten theory as the bulk, which in turn requires the presence of a stable Lagrangian boundary condition for the underlying gauge theory. They prove that in 2d, a Z_N^{(0)} duality defect is group theoretical if and only if N is a perfect square; in 4d, a Z_N^{(1)} duality defect on spin manifolds is group theoretical if and only if N = L^2 M with -1 a quadratic residue of M, and they construct explicit topological manipulations mapping the non-invertible defects to invertible ones for those N. The results illuminate the obstructions to duality-preserving gapped phases and connect to broader DW/SymTFT perspectives, with some higher-dimensional aspects remaining conjectural under certain assumptions. Overall, the work provides a systematic criterion and constructive protocol for identifying and transforming group-theoretical duality defects across dimensions.

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.

Figures (4)

• Figure 1: A $d$ dimensional QFT ${\mathcal{X}}$ with a non-anomalous $\mathbb{Z}_N^{(d/2-1)}$ symmetry can be expanded into a $d+1$ dimensional slab, where the bulk of the slab is the SymTFT of the $\mathbb{Z}_N^{(d/2-1)}$ symmetry---$\mathbb{Z}_N$$d/2$-form gauge theory, the left boundary is a topological boundary condition setting the electric 2-form gauge field to classical background value $B^{(2)}$, and the right boundary is a non-topological dynamical boundary condition.
• Figure 2: Idea of determining when a $\mathbb{Z}_N^{(d/2-1)}$ duality defect is group theoretical. We will argue (at the physics level of rigor) for the solid arrows, while for the dashed arrow, our argument is based on an assumption that is only proved in $d=2$.
• Figure 3: For 2d theory: space of $N$ where group theoretical $\mathbb{Z}_N^{(0)}$ duality defects exist (bounded by orange circle), and where $\mathbb{Z}_N^{(0)}$ duality defect is anomaly free (bounded by blue circle). The orange circle bounds the regime where $N$ is a perfect square, and the blue circle bounds only a point $N=1$. As a consequence, the cases satisfying (b) also only contains a trivial point $N=1$. This further implies that the condition for (b) to be satisfied is also $N=1$ since (b) is more restricted than (a).
• Figure 4: For 4d theory: space of $N$ where group theoretical $\mathbb{Z}_N^{(1)}$ duality defects exist (bounded by orange circle) which coincides with the situation where there exists a $\mathbb{Z}_N^{(1)}$-TQFT invariant under gauging $\mathbb{Z}_N^{(1)}$ (hence also bounded by orange circle), where there exists a $\mathbb{Z}_N^{(1)}$ SPT invariant under gauging $\mathbb{Z}_N^{(1)}$ (bounded by red circle), and where the $\mathbb{Z}_N^{(1)}$ duality defect is anomaly free (bounded by blue circle). The orange circle bounds the regime where $N$ takes the form $L^2 M$ and $-1$ is a quadratic residue of $M$. The red circle bounds the regime where $-1$ is a quadratic residue of $N$. The condition for the blue circle depends on the choice of the (higher dimensional generalization of) FS indicator, which is not determined by this current work, but is later determined in Antinucci:2023ezlCordova:2023bja.