  Coherent coupling generated by laser light between the hyperfine states of
atoms, loaded in a 1D optical lattice, gives rise to the "synthetic dimension"
system which is equivalent to a Hofstadter model in a finite strip of square
lattice. An SU(M) symmetric attractive interaction in conjunction with the
synthetic gauge field present in this system gives rise to unusual effects. We
study the two- body problem of the system using the T-matrix formalism. We show
that the two-body ground states pick up a finite momentum and can transform
into two-body resonance like features in the scattering continuum with a large
change in the phase shift. As a result, even for this 1D system, a critical
amount of attraction is needed to form bound states. These phenomena have
spectacular effects on the many body physics of the system analyzed using the
numerical density matrix renormalization group technique. We show that the
Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) states form in the system even for a
"balanced" gas and the FFLO momentum of the pairs scales linearly with flux.
Considering suitable measures, we investigate interesting properties of these
states. We also discuss a possibility of realization of a generalized
interesting topological model, called the Creutz ladder.
