Axion QED as a Lattice Gauge Theory and Non-Invertible Symmetry

TL;DR

This paper explores the lattice formulation of axion QED, revealing how non-invertible symmetries affect gauge invariance and magnetic objects, and introduces new gauge-invariant operators on the lattice.

Contribution

It constructs a lattice action for axion QED, demonstrates gauge invariance breaking, and develops gauge-invariant magnetic objects, providing new insights into non-invertible symmetries in lattice gauge theories.

Findings

Gauge invariance is broken in naive lattice axion QED.

New gauge-invariant magnetic objects are constructed.

Explicit response of magnetic objects under non-invertible symmetry is computed.

Abstract

We investigate the non-invertible symmetry associated with chiral symmetry in axion quantum electrodynamics (QED) using the modified Villain formulation. In axion QED, it is known that naive magnetic objects such as 't Hooft loops and axion strings lose their gauge invariance due to the violation of the Bianchi identity for the field strength of the photon or "field strength" of the axion. First, we construct the action of axion QED on the square lattice, which is more intricate than its counterpart in the continuum theory. We then observe the breaking of gauge invariance. Subsequently, we construct gauge-invariant magnetic objects by introducing new degrees of freedom localized at the positions of the magnetic objects. Furthermore, we explicitly compute the response of the magnetic objects under the action of the non-invertible symmetry operator constructed in Ref. [1]. In this analysis, we employ a method different from the so-called half-space gauging, which is the standard method to study non-invertible symmetries.