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Abelian Topological Order on Lattice Enriched with Electromagnetic Background

In topological phases of matter, the interplay between intrinsic topological order and global symmetry is an interesting task. In the study of topological orders with discrete global symmetry, an important systematic approach is the construction of exactly soluble lattice models. However, for continuous global symmetry, in particular the electromagnetic $U(1)$, the lattice approach has been less systematically developed. In this paper, we introduce a systematic construction of effective theories for a large class of abelian topological orders on three-dimensional spacetime lattice with electromagnetic background. We discuss the associated topological properties, including the Hall conductivity and the spin-c nature of the electromagnetic background. Some of these effective spacetime lattice theories can be readily mapped to microscopic Hamiltonians on spatial lattice; others may also shed light on their possible microscopic Hamiltonian realizations. Our approach is based on the gauging of $1$-form $\mathbb{Z}$ symmetries. Our construction is naturally related to the continuum path integral of (doubled) $U(1)$ Chern-Simons theory, through the latter's formal description in terms of Deligne-Beilinson cohomology; when the global symmetry is dropped, our construction can be reduced to the Dijkgraaf-Witten model of associated abelian topological orders, as expected.