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

TL;DR

This paper constructs a lattice realization of the axial $U(1)$ noninvertible symmetry in $U(1)$ gauge theory, using Ginsparg--Wilson fermions and topological quantum field theory techniques, ensuring gauge invariance and proper flux projections.

Contribution

It introduces a lattice formulation of the axial $U(1)$ noninvertible symmetry using Ginsparg--Wilson fermions and topological quantum field theory methods, bridging continuum and lattice approaches.

Findings

Constructed the symmetry operator on the lattice with gauge invariance.

Implemented the lattice Chern--Simons term over smooth gauge transformations.

Provided an alternative construction using 3D $bZ_N$ BF theory.

Abstract

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. The resulting symmetry operator is manifestly invariant under lattice gauge transformations. In an appendix, we give another way of constructing the symmetry operator on the basis of a 3D $\mathbb{Z}_N$ topological quantum field theory, the level-$N$ BF theory on the lattice.