  We develop a nonperturbative approach to the bulk polarization of crystalline
electric insulators in $d\geq1$ dimensions. Formally, we define polarization
via the response to background fluxes of both charge 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
source of $2\pi$-flux, and the polarization is determined by the crystal
momentum of the $2\pi$-flux. In $3d$ the polarization is determined by the
projective representation of translation symmetries on Dirac monopoles. Our
approach also leads to a concrete 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 an altered `boundary'
Luttinger theorem (constraining the Fermi surface of surface states) and also
to modified Lieb-Schultz-Mattis theorems on the boundary, which we derive.
