Gauge theory of Faddeev-Skyrme functionals

TL;DR

This paper investigates the existence of energy-minimizing maps, called Hopfions, in nonlinear sigma-models using gauge theory techniques to handle the variational problem on 3-manifolds with various target spaces.

Contribution

It introduces a gauge-theoretic framework to prove the existence of Hopfions for symmetric space targets and extends results to more general spaces with weaker conditions.

Findings

Existence of Hopfions as finite energy Sobolev maps in symmetric spaces.

Weaker existence results for general target spaces.

Development of a gauge calculus to manage non-compactness in minimization.

Abstract

We study geometric variational problems for a class of nonlinear sigma-models in quantum field theory. Mathematically, one needs to minimize an energy functional on homotopy classes of maps from closed 3-manifolds into compact homogeneous spaces G/H. The minimizers are known as Hopfions and exhibit localized knot-like structure. Our main results include proving existence of Hopfions as finite energy Sobolev maps in each (generalized) homotopy class when the target space is a symmetric space. For more general spaces we obtain a weaker result on existence of minimizers in each 2-homotopy class. Our approach is based on representing maps into G/H by equivalence classes of flat connections. The equivalence is given by gauge symmetry on pullbacks of G-->G/H bundles. We work out a gauge calculus for connections under this symmetry, and use it to eliminate non-compactness from the minimization problem by fixing the gauge.