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.

在物质的拓扑阶段，固有拓扑顺序和全局对称性之间的相互作用是一项有趣的任务。在研究具有离散全局对称性的拓扑阶时，一种重要的系统方法是构建完全可溶的晶格模型。但是，对于连续的全局对称性，尤其是电磁U（1），格子方法没有得到系统的发展。在本文中，我们介绍了在具有电磁背景的三维时空晶格上针对一大类阿贝尔拓扑阶的有效理论的系统构建。我们讨论了相关的拓扑属性，包括霍尔电导率和电磁背景的自旋c性质。这些有效的时空晶格理论中的一些可以很容易地映射到空间晶格上的微观哈密顿量。其他人也可能会阐明他们可能的微观哈密顿量实现。我们的方法基于1形式\（\ mathbb {Z} \）对称性的度量。我们的构造自然与（加倍）U的连续路径积分有关（1）Chern-Simons理论，通过后者对Deligne-Beilinson同调的形式描述；当全局对称性下降时，可以按预期将我们的构造简化为关联的阿贝尔拓扑阶数的Dijkgraaf–Witten模型。