  We apply the method of coadjoint orbits of \winf-algebra to the problem of
non-relativistic fermions in one dimension. This leads to a geometric
formulation of the quantum theory in terms of the quantum phase space
distribution of the fermi fluid. The action has an infinite series expansion in
the string coupling, which to leading order reduces to the previously discussed
geometric action for the classical fermi fluid based on the group $w_\infty$ of
area-preserving diffeomorphisms. We briefly discuss the strong coupling limit
of the string theory which, unlike the weak coupling regime, does not seem to
admit of a two dimensional space-time picture. Our methods are equally
applicable to interacting fermions in one dimension.
