Non-Invertible Duality Defects in 3+1 Dimensions

TL;DR

This work constructs and analyzes non-invertible topological defects arising from gauging a higher-form symmetry in half of spacetime, generalizing the 1+1d Kramers-Wannier duality to 3+1d. Focusing on a Z_N^(1) one-form symmetry in 3+1d, the authors derive the fusion rules with one-form symmetry surfaces, and show such duality defects are incompatible with trivially gapped IR phases for broad classes of N through SPT analyses. They provide explicit continuum realizations in free U(1) and SO(8) gauge theories and lattice realizations via modified Villain models, where the duality defect is implemented by a Chern-Simons coupling across the interface. The results imply strong dynamical constraints on RG flows and offer a unified framework for constructing and studying non-invertible defects across dimensions, with potential extensions to more general half-space couplings to TQFTs. Overall, the paper develops a robust, gauge-theoretic mechanism for duality defects and elucidates their implications for IR phases and symmetry-protected topological data.

Abstract

For any quantum system invariant under gauging a higher-form global symmetry, we construct a non-invertible topological defect by gauging in only half of spacetime. This generalizes the Kramers-Wannier duality line in 1+1 dimensions to higher spacetime dimensions. We focus on the case of a one-form symmetry in 3+1 dimensions, and determine the fusion rule. From a direct analysis of one-form symmetry protected topological phases, we show that the existence of certain kinds of duality defects is intrinsically incompatible with a trivially gapped phase. We give an explicit realization of this duality defect in the free Maxwell theory, both in the continuum and in a modified Villain lattice model. The duality defect is realized by a Chern-Simons coupling between the gauge fields from the two sides. We further construct the duality defect in non-abelian gauge theories and the $\mathbb{Z}_N$ lattice gauge theory.

Figures (8)

• Figure 1: The spacetime manifold is divided into two regions along the interface which is depicted as a red line. The arrow on the interface indicates that the interface is generally oriented.
• Figure 2: In the shaded region the system is coupled to a $\mathbb{Z}_N^{(q)}$ gauge theory, with Dirichlet boundary conditions imposed on the boundaries.
• Figure 3: In the 1+1d critical Ising CFT, when the duality line crosses the order operator $\sigma$, it becomes a disorder operator $\mu$ attached to the $\mathbb{Z}_2^{(0)}$ line. Here the red line denotes the duality line $\cal D$ and the dotted line denotes the $\mathbb{Z}_2^{(0)}$ line.
• Figure 4: As the three-dimensional duality defect $\cal D$ (shown in red) sweeps past a $\mathbb{Z}_N^{(1)}$-charged line operators (shown in black), the latter is attached to the $\mathbb{Z}_N^{(1)}$ topological two-surface $\eta$ (shown in blue) on the other side.
• Figure 5: By moving the duality defect $\cal D$ around the charged line operator, the $\mathbb{Z}_N^{(1)}$ topological surfaces $\eta$ can be freely generated and be absorbed by the duality defect, i.e. $\eta\times {\cal D}={\cal D}\times \eta={\cal D}$.