Abstract
We extend the definition of the topological charge pertaining to the CP1 (i.e. O(3)) Skyrme-Fadde'ev Hopfion on ℝ3 to candidates for topological charges of ℂPn sigma models on ℝ2n+1 for all n. For this, the Abelian composite connections of the ℂPn sigma models are employed. In higher dimensions (n ≥ 1) it turns out that such charges, described by the nonAbelian composite connections of suitable Grassmannian sigma models, can also be constructed. A concrete discussion of the non-Abelian case for n = 2 is presented.
Highlights
Hopfions are static “soliton like” solutions to the field equations of nonlinear sigma models
Requiring the field configurations in question have the asymptotic values (18) as in the Abelian case, one finds the topological charge. The scope of this talk is restricted to the consideration of the topological charges of possible Hopfions in 2n + 1 dimensions
Since the topological charges are the volume integrals of Chern-Simons densities in the appropriate dimension. The latter are not total divergence, and to qualify as topological charge densities they must be subjected to the appropriate symmetries that render them total divergences
Summary
Hopfions are static “soliton like” solutions to the field equations of nonlinear sigma models. It should be noted that this aim of rendering the residual Chern-Simons density a total divergence can be achieved only when the sigma model constraint is satisfied This is automatic if the Ansatz in question is parametrised in terms of functions that satisfy this constraint. The above account of topological charges of Hopfions relies entirely on the existence of suitable composite connections and curvatures of the nonlinear sigma model in question In this sense, both Abelian and non-Abelian systems can be considered. The talk consists of two main sections, 2 and 3, describing our prescriptions for Abelian and non-Abelian Hopfions respectively Each of these sections is subdivided in a subsection defining the appropriate nonlinear sigma model, followed by the imposition of symmetries yielding the desired toplogical charge densities. Eqn (9) trivialises only under the appropriate symmetries, i.e. (8) becomes a density that is “essentially total divergence”, as required of a topological charge density
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