  We present a theory for a lattice array of weakly coupled one-dimensional
ultracold attractive Fermi gases (1D `tubes') with spin imbalance, where strong
intratube quantum fluctuations invalidate mean field theory. We first construct
an effective field theory, which treats spin-charge mixing exactly, based on
the Bethe ansatz solution of the 1D single tube problem. We show that the 1D
Fulde-Ferrel-Larkin-Ovchinnikov (FFLO) state is a two-component Luttinger
liquid, and its elementary excitations are fractional states carrying both
charge and spin. We analyze the instability of the 1D FFLO state against
inter-tube tunneling by renormalization group analysis, and find that it flows
into either a polarized Fermi liquid or a FFLO superfluid, depending on the
magnitude of interaction strength and spin imbalance. We obtain the phase
diagram of the quasi-1D system and further determine the scaling of the
superfluid transition temperature with intertube coupling.
