  We consider a SU(2) lattice gauge theory on the square lattice, with a single
fundamental complex fermion and a single fundamental complex boson on each
lattice site. Projective symmetries of the gauge-charged fermions are chosen so
that they match with those of the spinons of the $\pi$-flux spin liquid. Global
symmetries of all gauge-invariant observables are chosen to match with those of
the particle-hole symmetric electronic Hubbard model at half-filling.
Consequently, both the fundamental fermion and fundamental boson move in an
average background $\pi$-flux, their gauge-invariant composite is the physical
electron, and eliminating gauge fields in a strong gauge-coupling expansion
yields an effective extended Hubbard model for the electrons. The SU(2) gauge
theory displays several confining/Higgs phases: a nodal $d$-wave
superconductor, and states with N\'eel, valence-bond solid, charge, or
staggered current orders. There are also a number of quantum phase transitions
between these phases which are very likely described by 2+1 dimensional
deconfined conformal gauge theories, and we present large flavor expansions for
such theories. These include the phenomenologically attractive case of a
transition between a conventional insulator with a charge gap and N\'eel order,
and a conventional $d$-wave superconductor with gapless Bogoliubov
quasiparticles at 4 nodal points in the Brillouin zone. We also apply our
approach to the honeycomb lattice, where we find a bicritical point at the
junction of N\'eel, valence bond solid (Kekul\'e), and Dirac semi-metal phases.
