Abelian Topological Phases with Background Electromagnetic Field on Lattice, and Deligne-Beilinson Double Cohomology in Continuum

Constructing exactly soluble lattice model is an important approach towards understanding topological phases of matter. The coupling of exactly soluble lattice topological models to continuous background gauge field is less well-studied compared to discrete background gauge fields. The former is however responsible for important topological phenomena such as the Hall conductivity and the spin-charge relation. In this talk, I introduce a systematic approach to this problem for abelian topological phases, which is to exactly retrieve the spacetime lattice model from the corresponding Chern-Simons theory in the continuum, via the latter's formal structure known as Deligne-Beilinson double cohomology.