Fractional topological charge in lattice Abelian gauge theory

We construct a non-trivial $U(1)\mathrm{/}{\mathbb{Z}}_q$ principal bundle on~$T^4$ from the compact $U(1)$ lattice gauge field by generalizing Lüscher’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~$q\in 2\mathbb{Z}$ when $\theta =\pi$, which was studied by Honda and Tanizaki [J. High Energy Phys. $\mathbf{\text{12}}$, 154 (2020)] in the continuum framework.