Lattice realization of the axial $U(1)$ noninvertible symmetry

Understand

In $U(1)$ lattice gauge theory with compact $U(1)$ variables, we construct the symmetry operator, i.e.\ the topological defect, for the axial $U(1)$ noninvertible symmetry.

• This requires a lattice formulation of chiral gauge theory with an anomalous matter content and we employ the lattice formulation on the basis of the Ginsparg--Wilson relation.
• The invariance of the symmetry operator under the gauge transformation of the gauge field on the defect is realized, imitating the prescription by Karasik in continuum theory, by integrating the lattice Chern--Simons term on the defect over \emph{smooth\/} lattice gauge transformations.
• The projection operator for allowed magnetic fluxes on the defect then emerges with lattice regularization.