1301.7483 (Paul Smith)

Schrödinger maps as a gauge theory    [PDF]

Paul Smith

We consider energy-critical Schr\"odinger maps from the Euclidean plane into the 2-sphere and the 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, which is seen to include a Chern-Simons term. We also consider harmonic map heat flow and harmonic maps from a related point of view. For the Schr\"odinger evolution, we derive several conservation laws. These conservation laws give rise to virial identities and suggest a framework for proving Morawetz estimates. We conclude by comparing the gauged Schr\"odinger map system with the Chern-Simons-Schr\"odinger system.

View original: http://arxiv.org/abs/1301.7483