Schrödinger maps are maps from the space-time into a Kähler manifold with a metric and a complex structure satisfying
where denotes the covariant derivative on . By using a pullback frame on , a gauge invariant nonlinear Schrödinger equation is associated to (SM); in the Coulomb gauge, this equation is given schematically by
The authors are interested in studying the correspondence between solutions of (SM) and solutions of (GNLS) for low-regularity data. They establish the equivalence when the target is or . They also prove the existence of global weak solutions in for two space dimensions. The ideas are extended to the maps into compact Hermitian symmetric manifolds with trivial first cohomology.