Abelian and non-Abelian Hopfions in all odd dimensions
Seminar Room 1, Newton Institute
AbstractHopfions are field configurations of scalar matter systems characterised prominently by the fact that they describe knots in configuration space. Like the 'usual solitons', e.g. Skyrmions, monopoles, vortices and instantons, Hopfions are static and finite energy solutions that are stabilised by a topological charge, which supplies the energy lower bound. In contrast to the 'usual solitons' however, the topological charge of Hopfions is not the volume integral of a total divergence. While the topological charge densities of the 'usual solitons', namely the Chern-Pontryagin (CP) densities or their descendants, are total divergence, the corresponding quantities for Hopfions are the Chern-Simons (CS) densities which are not total divergence. Subject to the appropriate symmetries however, these CS densities do reduce to total divergence and become candidates for topological charges. Thus, Hopfion field are necessarily subject to the appropriate symmetry to decsribe knots, excluding spherically symmetry, in contrast to the 'usual solitons'. The construction of these CS densities is enabled by employing complex nonlinear sigma models, which feature composite connections. The CS densities are defined in terms of these connections and their curvatures. (In some dimensions the complex sigma model can be equivalent to a real sigma model, e.g. in D=3 Skyrme-Fadde'ev O(3) model and the corresponding CP^1 model.) It is natural to propose Hopfion fields in all odd space dimensions where a CS density can be defined. This covers both Abelian and non-Abelian theories, namely empolying projective-complex and Grassmannian models, respectively. It is in this sense that we have used the terminology of Abelian and non-Abelian Hopfions. Explicit field configurations displaying the appropriate symmetries and specific asymptotic behaviours in several (higher) dimensions are proposed, and it is verified that for these configurations the CS densities do indeed become total divergence.
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