On anomalies and gauging of U(1) non-invertible symmetries in 4d QED

TL;DR

This work promotes the anomalous axial U(1) in 4d QED to an exact non-invertible symmetry by coupling to a 3d scalar on a manifold and defining a gauge-invariant, topological operator \tilde{U}_\alpha. Gauging this non-invertible symmetry introduces a dynamical 4d field $\tilde{b}$ and enforces the crucial constraint $d\tilde{b}\wedge da=0$, yielding constrained, non-invertible gauge theories that cancel would-be gauge anomalies. By coupling to background fields, the authors study 't-Hooft anomalies of non-invertible symmetries and show that anomaly matching can be realized in the IR via solitons or vortices, with the non-commuting nature of flow and background gauging playing a key role. Extensions to 2d and non-abelian 4d cases reveal obstructions, where the non-invertible construction yields trivial results unless $\alpha$ is a multiple of $2\pi$. The discussion contrasts this approach with alternative schemes (e.g., Karasik’s) and highlights how non-invertible symmetries, anomalies, and soliton dynamics intertwine to constrain low-energy physics in abelian gauge theories.

Abstract

In this work we propose a way to promote the anomalous axial U(1) transformations to exact non-invertible U(1) symmetries. We discuss the procedure of coupling the non-invertible symmetry to a (dynamical or background) gauge field. We show that as part of the gauging procedure, certain constraints are imposed to make the gauging consistent. The constraints emerge naturally from the form of the non-invertible U(1) conserved current. In the case of dynamical gauging, this results in new type of gauge theories we call non-invertible gauge theories: These are gauge theories with additional constraints that cancel the would-be gauge anomalies. By coupling to background gauge fields, we can discuss 't-Hooft anomalies of non-invertible symmetries. We show in an example that the matching conditions hold but they are realized in an unconventional way. Turning on non-trivial background for the non-invertible gauge field changes the vacuum even when the symmetry is not broken and the background is very weak. The anomalies are then matched by the appearance of solitons in the new vacuum.

Figures (2)

• Figure 1: The set of insertions of topological operators $\tilde{U}$ can be described by a vector field $\tilde{b}_\mu$ orthogonal to the manifolds on which $\tilde{U}$ are defined. Effectively, inserting into the path integral ${\left< {\tilde{U}_\alpha(\mathcal{M}_3)\tilde{U}_{\alpha'}(\mathcal{M}_3')...} \right>}$ is equivalent to inserting ${\left< {\int D\phi e^{i\int \tilde{b}_\mu \tilde{J}^\mu}} \right>}$ where $\phi$ now lives in the entire 4d spacetime.
• Figure 2: Formally, anomalies are matched along the green line. Only after we couple to gauge fields we can probe the anomaly and derive rigorous consistency conditions. If $[flow,background]=0$ as for ordinary symmetries, we can pull back the matching to the blue line that connects the uv and the IR without coupling to gauge fields. When $[flow,background]\neq 0$ as was the case for the $\tilde{U}(1)_b^3$ anomaly, the matching occurs only along the green line and cannot be pulled back to the blue line.