  We study the superfluid and insulating phases of interacting bosons on the
triangular lattice with an inverted dispersion, corresponding to frustrated
hopping between sites. The resulting single-particle dispersion has multiple
minima at nonzero wavevectors in momentum space, in contrast to the unique
zero-wavevector minimum of the unfrustrated problem. As a consequence, the
superfluid phase is unstable against developing additonal chiral order that
breaks time reversal (T) and parity (P) symmetries by forming a condensate at
nonzero wavevector. We demonstrate that the loss of superfluidity can lead to
an even more exotic phase, the chiral Mott insulator, with nontrivial current
order that breaks T, P. These results are obtained via variational estimates,
as well as a combination of bosonization and DMRG of triangular ladders, which
taken together permit a fairly complete characterization of the phase diagram.
We discuss the relevance of these phases to optical lattice experiments, as
well as signatures of chiral symmetry breaking in time-of-flight images.
