  A fully gapped state of matter, whether insulator or superconductor, can be
asked if it is topologically trivial or nontrivial. Here we investigate
topological properties of superconducting Dirac fermions in 3D having a color
superconductor as an application. In the chiral limit, when the pairing gap is
parity even, the right-handed and left-handed sectors of the free space
Hamiltonian have nontrivial topological charges with opposite signs.
Accordingly, a vortex line in the superconductor supports localized gapless
right-handed and left-handed fermions with the dispersion relations E=+/-vp_z
(v is a parameter dependent velocity) and thus propagating in opposite
directions along the vortex line. However, the presence of the fermion mass
immediately opens up a mass gap for such localized fermions and the dispersion
relations become E=+/-v(m^2+p_z^2)^(1/2). When the pairing gap is parity odd,
the situation is qualitatively different. The right-handed and left-handed
sectors of the free space Hamiltonian in the chiral limit have nontrivial
topological charges with the same sign and therefore the presence of the small
fermion mass does not open up a mass gap for the fermions localized around the
vortex line. When the fermion mass is increased further, there is a topological
phase transition at m=(\mu^2+\Delta^2)^(1/2) and the localized gapless fermions
disappear. We also elucidate the existence of gapless surface fermions
localized at a boundary when two phases with different topological charges are
connected. A part of our results is relevant to the color superconductivity of
quarks.
