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We consider energy-critical Schr\"odinger maps from R^2 into the sphere andhyperbolic plane. Viewing such maps with respect to orthonormal frames on thepullback bundle provides a gauge field formulation of the evolution. We showthat this gauge field system is the set of Euler-Lagrange equationscorresponding to an action that includes a Chern-Simons term. We also introducethe stress-energy tensor and derive conservation laws. In conclusion we offercomparisons between Schr\"odinger maps and the closely relatedChern-Simons-Schr\"odinger system.

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