The ground state of a spin-1/2 nearest-neighbor quantum Heisenberg antiferromagnet on the pyrochlore lattice is investigated using a large N SU(N) fermionic mean-field theory. We find several mean-field states, of which the state of lowest energy upon Gutzwiller projection is a parity and time-reversal breaking chiral phase with a unit monopole flux exiting each tetrahedron. This "monopole flux" state has a Fermi surface consisting of four lines intersecting at a point. At mean field the low-energy excitations about the Fermi surface are gapless spinons. An analysis using the projective symmetry group of this state suggests that the state is stable to small fluctuations which neither induce a gap nor alter the unusual Fermi surface.