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

Employing the modified Villain lattice formulation of the axion QED, we present an alternative and much simpler derivation of the conclusion of~Ref.~{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.