Hopf solitons and area preserving diffeomorphisms of the sphere

TL;DR

This paper studies a 3+1D field theory on the sphere with Hopf solitons, revealing an algebraic structure linked to area-preserving diffeomorphisms and identifying conserved currents as Noether currents for this symmetry.

Contribution

It demonstrates the isomorphism between the Poisson algebra of topological charges and the algebra of area-preserving diffeomorphisms, and links conserved currents to this symmetry.

Findings

Exact soliton solutions with Hopf charges found.

Poisson algebra matches area-preserving diffeomorphisms.

Conserved currents are Noether currents for the symmetry.

Abstract

We consider a (3+1)-dimensional local field theory defined on the sphere. The model possesses exact soliton solutions with non trivial Hopf topological charges, and infinite number of local conserved currents. We show that the Poisson bracket algebra of the corresponding charges is isomorphic to that of the area preserving diffeomorphisms of the sphere. We also show that the conserved currents under consideration are the Noether currents associated to the invariance of the Lagrangian under that infinite group of diffeomorphisms. We indicate possible generalizations of the model.