Abelian topological order on lattice enriched with electromagnetic background (Q2223727)

The author introduces a construction of effective theories for a class of abelian topological orders on a 3D spacetime lattice with electromagnetic background. The author does this by gauging 1-form \(\mathbb{Z}\) symmetries, and in doing so, further insights are obtained, including the spin-c nature of the background field when the topological order is fermionic. This gauging of 1-form symmetries is similar to the Villain model.The Hall conductivity is also discussed. Some of the effective lattice theories can be mapped to microscopic Hamiltonians on a spatial lattice. The author also makes an explicit connection between the lattice construction and the doubled \(U(1)\) Chern-Simons path integral in the continuum. When the assumption of global symmetry is removed, the construction reduces to the Dijkgraaf-Witten model of associated abelian topological orders.