  We show that polar molecules driven by microwave fields give naturally rise
to strong three-body interactions, while the two-particle interaction can be
independently controlled and even switched off. The derivation of these
effective interaction potentials is based on a microscopic understanding of the
underlying molecular physics, and follows from a well controlled and systematic
expansion into many-body interaction terms. For molecules trapped in an optical
lattice, we show that these interaction potentials give rise to Hubbard models
with strong nearest-neighbor two-body and three-body interaction. As an
illustration, we study the one-dimensional Bose-Hubbard model with dominant
three-body interaction and derive its phase diagram.
