  We describe how color superfluidity is modified in the presence of color-flip
and color-orbit fields in the context of ultra-cold atoms, and discuss
connections between this problem and that of color superconductivity in quantum
chromodynamics. We consider s-wave contact interactions between different
colors, and we identify superfluid phases, with five being nodal and one being
fully gapped. When our system is described in a mixed color basis, the
superfluid order parameter tensor is characterized by six independent
components with explicit momentum dependence induced by color-orbit coupling.
The nodal superfluid phases are topological in nature, and the low temperature
phase diagram of color-flip field versus interaction parameter exhibits a
pentacritical point, where all five nodal color superfluid phases converge.
These results are in sharp contrast to the case of zero color-flip and
color-orbit fields, where the system has perfect U(3) symmetry and possess a
superfluid phase that is characterized by fully gapped quasiparticle
excitations with a single complex order parameter with no momentum dependence
and by inert unpaired fermions representing a non-superfluid component.
Furthermore, we analyse the order parameter tensor in a total pseudo-spin
basis, investigate its momentum dependence in the singlet, triplet and quintet
sectors, and compare the results with the simpler case of spin-1/2 fermions in
the presence of spin-flip and spin-orbit fields. Finally, we analyse in detail
spectroscopic properties of color superfluids in the presence of color-flip and
color-orbit fields, such as the quasiparticle excitation spectrum, momentum
distribution, and density of states to help characterize all the encountered
topological quantum phases, which can be realized in fermionic isotopes of
Lithium, Potassium and Ytterbium atoms with three internal states trapped.
