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

TL;DR

This study investigates how the axial $U(1)$ non-invertible symmetry acts on 't~Hooft line operators in lattice gauge theory, revealing that the symmetry operator has no effect on these operators, contrasting with continuum theory expectations.

Contribution

The paper provides a lattice formulation analysis showing the axial $U(1)$ symmetry operator leaves 't~Hooft line operators unaffected, challenging previous continuum theory results.

Findings

The symmetry operator does not affect 't~Hooft line operators on the lattice.

The result contrasts with continuum theory predictions.

Similar conclusions hold for the axion string operator.

Abstract

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. A careful consideration on the basis of the anomalous Ward--Takahashi identity containing the 't~Hooft operator with the dressing factor and a precise definition of the symmetry operator on the lattice shows that the symmetry operator leaves no effect when it sweeps out a 't~Hooft loop operator. This result appears inequivalent with the phenomenon concluded in the continuum theory. In an appendix, we demonstrate that the half-space gauging of the magnetic $\mathbb{Z}_N$ 1-form symmetry, when formulated in an appropriate lattice framework, leads to the same conclusion as above. A similar result is obtained for the axion string operator.