Fractional topological charge in lattice Abelian gauge theory

TL;DR

This paper constructs a lattice model for fractional topological charge in Abelian gauge theory, revealing a mixed anomaly between one-form symmetry and time reversal, extending continuum results to the lattice setting.

Contribution

It introduces a lattice construction of fractional topological charge incorporating 't Hooft flux, connecting lattice gauge theory with continuum anomaly analysis.

Findings

Constructed a $U(1)/Z_q$ principal bundle on $T^4$ from lattice gauge fields.

Defined a fractional topological charge invariant under one-form gauge transformations.

Provided a lattice realization of the mixed 't Hooft anomaly at $ heta=\pi$.

Abstract

We construct a non-trivial $U(1)/\mathbb{Z}_q$ principal bundle on~$T^4$ from the compact $U(1)$ lattice gauge field by generalizing L\"uscher's constriction so that the cocycle condition contains $\mathbb{Z}_q$ elements (the 't~Hooft flux). The construction requires an admissibility condition on lattice gauge field configurations. From the transition function so constructed, we have the fractional topological charge that is $\mathbb{Z}_q$ one-form gauge invariant and odd under the lattice time reversal transformation. Assuming a rescaling of the vacuum angle $\theta\to q\theta$ suggested from the Witten effect, our construction provides a lattice implementation of the mixed 't~Hooft anomaly between the $\mathbb{Z}_q$ one-form symmetry and the time reversal symmetry in the $U(1)$ gauge theory with matter fields of charge~$q\in2\mathbb{Z}$ when $\theta=\pi$, which was studied by Honda and Tanizaki [J. High Energy Phys. \textbf{12}, 154 (2020)] in the continuum framework.