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

TL;DR

This paper introduces a framework for promoting anomalous axial U(1) symmetries in 4d QED to exact non-invertible symmetries, leading to new gauge theories with unique anomaly cancellation mechanisms and vacuum structures.

Contribution

It develops a method to gauge non-invertible U(1) symmetries, resulting in non-invertible gauge theories with novel anomaly cancellation and vacuum phenomena.

Findings

Non-invertible gauge theories with anomaly-canceling constraints

Unconventional 't Hooft anomaly matching in non-invertible symmetries

Vacuum changes induced by background gauge fields without symmetry breaking

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.