Polarization measured by magnetic monopole and boundary Luttinger theorem

We develop a topological approach to bulk polarization of electric insulators in d>1 dimensions. We define polarization through topological Theta terms involving background gauge fields of the U(1) charge conservation and lattice translation symmetries. In this approach, the bulk polarization is related to properties of magnetic monopoles under translation symmetries. Specifically, in 2d the monopole is a 2pi-flux insertion in space, and the polarization is determined by the crystal momentum of the flux. In 3d the polarization is determined by the projective representation of translation symmetries on Dirac monopoles. Our approach also leads to a practical scheme to calculate polarization in 2d, which in principle can be applied even to strongly interacting systems. For open boundary condition, the bulk polarization leads to a violation of  Luttinger theorem on the boundary, and is more generally related to Lieb-Schultz-Mattis type of quantum anomalies for the boundary low-energy theory.