We study the Cauchy problem for the cubic fractional nonlinear Schr\"odinger
equation (fNLS) on the real line and on the circle. In particular, we prove
global well-posedness of the cubic fNLS with all orders of dispersion higher
than the usual Schr\"odinger equation in negative Sobolev spaces. On the real
line, our well-posedness result is sharp in the sense that a contraction
argument does not work below the threshold regularity. On the circle, due to
ill-posedness of the cubic fNLS in negative Sobolev spaces, we study the
renormalized cubic fNLS. In order to overcome the failure of local uniform
continuity of the solution map in negative Sobolev spaces, by applying a gauge
transform and partially iterating the Duhamel formulation, we study the
resulting equation with a cubic-quintic nonlinearity. In proving uniqueness, we
present full details justifying the use of the normal form reduction for rough
solutions, which seem to be missing from the existing literature. Our
well-posedness result on the circle extends those in Miyaji-Tsutsumi (2018) and
Oh-Wang (2018) to the endpoint regularity.Comment: 49 page