Fractional topological charge in lattice Abelian gauge theory

Abstract

AbstractWe construct a non-trivial $U(1)/\mathbb {Z}_q$ principal bundle on T4 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 θ → qθ 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 θ = π, which was studied by Honda and Tanizaki [J. High Energy Phys. 12, 154 (2020)] in the continuum framework.