An unconstrained Lagrangian formulation and conservation laws for the Schr\"odinger map system

We consider energy-critical Schr"odinger maps from R^2 into the sphere and hyperbolic plane. Viewing such maps with respect to orthonormal frames on the pullback bundle provides a gauge field formulation of the evolution. We show that this gauge field system is the set of Euler-Lagrange equations corresponding to an action that includes a Chern-Simons term. We also introduce the stress-energy tensor and derive conservation laws. In conclusion we offer comparisons between Schr"odinger maps and the closely related Chern-Simons-Schr"odinger system.