  We propose and analyze a physical system that naturally admits
two-dimensional topological nearly flat bands. Our approach utilizes an array
of three-level dipoles (effective S = 1 spins) driven by inhomogeneous
electromagnetic fields. The dipolar interactions produce arbitrary uniform
background gauge fields for an effective collection of conserved hardcore
bosons, namely, the dressed spin-flips. These gauge fields result in
topological band structures, whose bandgap can be larger than the corresponding
bandwidth. Exact diagonalization of the full interacting Hamiltonian at
half-filling reveals the existence of superfluid, crystalline, and supersolid
phases. An experimental realization using either ultra-cold polar molecules or
spins in the solid state is considered.
