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

TL;DR

This paper develops a gauge field formulation for energy-critical Schr"odinger maps into spheres and hyperbolic planes, deriving conservation laws and connecting to Chern-Simons theories.

Contribution

It introduces an unconstrained Lagrangian framework with a Chern-Simons term for Schr"odinger maps, providing new insights into their geometric and physical properties.

Findings

Derived conservation laws for the gauge field system.

Established the Lagrangian formulation including a Chern-Simons term.

Compared Schr"odinger maps with Chern-Simons-Schr"odinger systems.

Abstract

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.