  Recent experiments indicate that Na$_4$Ir$_3$O$_8$, a material in which s=1/2
iridium local moments form a three dimensional network of corner-sharing
triangles, may have a quantum spin liquid ground state with gapless spin
excitations. Using a combination of exact diagonalization, symmetry analysis of
fermionic mean field ground states and Gutzwiller projected variational
wavefunction studies, we propose a quantum spin liquid with spinon Fermi
surfaces as a favorable candidate for the ground state of the Heisenberg model
on the hyper-kagome lattice of Na$_4$Ir$_3$O$_8$. We present a renormalized
mean field theory of the specific heat of this spin liquid and also discuss
possible low temperature instabilities of the spinon Fermi surfaces.
