  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.
