  The physical properties of arbitrary half-integer spins F = N - 1/2 fermionic
cold atoms loaded into a one-dimensional optical lattice are investigated by
means of a conformal field theory approach. We show that for attractive
interactions two different superfluid phases emerge for F \ge 3/2: A BCS
pairing phase, and a molecular superfluid phase which is formed from
bound-states made of 2N fermions. In the low-energy approach, the competition
between these instabilities and charge-density waves is described in terms of
Z_N parafermionic degrees of freedom. The quantum phase transition for
F=3/2,5/2 is universal and shown to belong to the Ising and three-state Potts
universality classes respectively. For a filling of one atom per site, a Mott
transition occurs and the nature of the possible Mott-insulating phases are
determined.
