Action of the Axial <i>U</i>(1) Noninvertible Symmetry on the ’t Hooft Line Operator: A Simple Argument

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. [1] that the sweep of the axial $U(1)$ noninvertible symmetry operator over the (nongenuine) 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 (nontopological) defect that is invariant under the axial transformation on the axion field.