Action of the axial $U(1)$ non-invertible symmetry on the 't~Hooft line operator: A lattice gauge theory study

We study how the symmetry operator of the axial $U(1)$ non-invertible symmetry acts on the 't~Hooft line operator in the $U(1)$ gauge theory by employing the modified Villain-type lattice formulation. We model the axial anomaly by a compact scalar boson, the ``QED axion''. For the gauge invariance, the simple 't~Hooft line operator, which is defined by a line integral of the dual $U(1)$ gauge potential, must be ``dressed'' by the scalar and $U(1)$ gauge fields. The anomalous Ward--Takahashi identity with the dressing factor shows that, when the symmetry operator sweep out a 't~Hooft loop, the loop acquires a surface operator of the $U(1)$ field strength with a correction of a higher cup product. This result appears consistent with the phenomenon observed in the continuum theory. A similar analysis for the axion string operator shows that it is not affected by the symmetry operator.