  Several simple models of strongly correlated bosons on three-dimensional
lattices have been shown to possess exotic fractionalized Mott insulating
phases with a gapless `photon' excitation. In this paper we show how to view
the physics of this `Coulomb' state in terms of the excitations of proximate
superfluid. We argue for the presence of ordered vortex cores with a broken
discrete symmetry in the nearby superfluid phase, and that proliferating these
degenerate but distinct vortices with equal amplitudes produces the Coulomb
phase. This provides a simple physical description of the origin of the exotic
excitations of the Coulomb state. The physical picture is formalized by means
of a dual description of three-dimensional bosonic systems in terms of
fluctuating quantum mechanical vortex loops. Such a dual formulation is
extensively developed. It is shown how the Coulomb phase (as well as various
other familiar phases) of three-dimensional bosonic systems may be described in
this vortex loop theory. For bosons at half-filling and the closely related
system of spin-1/2 quantum magnets on a cubic lattice, fractionalized phases as
well as bond or `box' ordered states are possible. The latter are analyzed by
an extension of techniques previously developed in two spatial dimensions. The
relation between these `confining' phases with broken translational symmetry
and the fractionalized Coulomb phase is exposed.
