  We report on a phenomenological study of superfluid to Mott insulator
transitions of bosons on the triangular lattice, focusing primarily on the
interplay between Mott localization and geometrical charge frustration at
1/2-filling. A general dual vortex field theory is developed for arbitrary
rational filling factors f, based on the appropriate projective symmetry group.
At the simple non-frustrated density f=1/3, we uncover an example of a
deconfined quantum critical point very similar to that found on the half-filled
square lattice. Turning to f=1/2, the behavior is quite different. Here, we
find that the low-energy action describing the Mott transition has an emergent
nonabelian SU(2)\times U(1) symmetry, not present at the microscopic level.
This large nonabelian symmetry is directly related to the frustration-induced
quasi-degeneracy between many charge-ordered states not related by microscopic
symmetries. Through this ``pseudospin'' SU(2)symmetry, the charged excitations
in the insulator close to the Mott transition develop a skyrmion-like
character. This leads to an understanding of the recently discovered supersolid
phase of the triangular lattice XXZ model (cond-mat/0505258, cond-mat/0505257,
cond-mat/0505298) as a ``partially melted'' Mott insulator. The latter picture
naturally explains a number of puzzling numerical observations of the
properties of this supersolid. Moreover, we predict that the nearby quantum
phase transition from this supersolid to the Mott insulator is in the
recently-discovered non-compact CP^1 critical universality class (PRB 70,
075104 (2004)). A description of a broad range of other Mott and supersolid
states, and a diverse set of quantum critical points between them, is also
provided.
