  We study a mixture of ultracold spin-half fermionic and spin-one bosonic
atoms in a shallow optical lattice where the bosons are coupled to the fermions
via both density-density and spin-spin interactions. We consider the parameter
regime where the bosons are in a superfluid ground state, integrate them out,
and obtain an effective action for the fermions. We carry out a renormalization
group analysis of this effective fermionic action at low temperatures, show
that the presence of the spinor bosons may lead to a separation of Fermi
surfaces of the spin-up and spin-down fermions, and investigate the parameter
range where this phenomenon occurs. We also calculate the susceptibilities
corresponding to the possible superfluid instabilities of the fermions and
obtain their possible broken-symmetry ground states at low temperatures and
weak interactions.
