Action of the axial $U(1)$ non-invertible symmetry on the 't~Hooft line operator: A simple argument

TL;DR

This paper provides a simplified derivation showing that the axial $U(1)$ non-invertible symmetry operator has no effect on certain 't~Hooft line operators in axion quantum electrodynamics, using a modified Villain lattice formulation.

Contribution

It offers a more straightforward derivation of the invariance of 't~Hooft lines under axial symmetry in axion QED, building on prior results.

Findings

The axial $U(1)$ symmetry operator does not affect 't~Hooft lines with magnetic charge.

A boundary of a non-topological defect represents the 't~Hooft line invariant under axial transformation.

The modified Villain lattice formulation simplifies the analysis of symmetry actions.

Abstract

Employing the modified Villain lattice formulation of the axion quantum electrodynamics, we present an alternative and much simpler derivation of the conclusion of~Ref.~\cite{Honda:2024sdz} that the sweep of the axial $U(1)$ non-invertible symmetry operator over the (non-genuine) gauge invariant 't~Hooft line operator with an integer magnetic charge does not leave any effect. The point is that such a 't~Hooft line can be represented by a boundary of a (non-topological) defect that is invariant under the axial transformation on the axion field.