Hopf solitons and area preserving diffeomorphisms of the sphere

We consider a (3+1)-dimensional local field theory defined on the sphere S. 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 S. 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. We consider a Lagrangian field theory model defined on four-dimensional Minkowski space-time with coordinates x. The fields of the model form a three-dimensional vectorvalued function n(x) satisfying the relation n (x) = 1. (1) and so, the target space is the two-dimensional sphere S. The Lagrangian density is defined as [1, 2] L = − 2 3 ( HμνH μν 8 3/4 , (2) where the tensor Hμν is defined in terms of the three component unit vector field n ∈ S 2 as Hμν ≡ n (∂μn × ∂νn) , (3) where ∂μn(x) = ∂n(x)/∂x . The power 3/4 of HμνH μν in (2) is such that the theory circumvents the usual obstacle of the Derrick’s scaling argument against existence of stable solitons. We are interested in the boundary condition n = (0, 0, 1) at spatial infinity. This condition compactifies effectively the Euclidean space R to the three-sphere S. Accordingly, n becomes a map: S → S. Due to the fact that π3(S ) = Z, the field configurations fall into disjoint classes characterized by the value of the Hopf invariant QH . Using the stereographic projection of S, we introduce instead of the vector-valued function n(x) satisfying (1) the scalar complex field u(x) so that