  We present a general framework for describing the quantum phases obtained by
doping paramagnetic Mott insulators on the square lattice. The undoped
insulators are efficiently characterized by the projective transformations of
various fields under the square lattice space group (the PSG). We show that the
PSG also imposes powerful constraints on the doped system, and on the effective
action for the vortex and Bogoliubov quasiparticle excitations of
superconducting states. This action can also be extended across transitions to
supersolid or insulating states at nonzero doping. For the case of a valence
bond solid (VBS) insulator, we show that the doped system has the same PSG as
that of elementary bosons with density equal to the density of electron Cooper
pairs. We also discuss aspects of the action for a d-wave superconductor
obtained by doping a ``staggered-flux'' spin liquid state.
