Topological Solitons

Skyrmions Skyrmions are candidates for a solitonic description of nuclei, where the number of solitons is identified with baryon number. The Skyrme model is a modified nonlinear sigma model and its classical solutions are known only through numerical computations. However, it is possible to use an approximation in which Skyrmions can be constructed from rational maps between Riemann spheres and this has proved useful in understanding the structure of Skyrmions. The figure displays baryon density isosurfaces for various charges. It can be seen that some Skyrmions are very symmetric, and this can be understood in terms of the existence of particularly symmetric rational maps.

Monopoles Monopoles arise in Yang-Mills-Higgs gauge theories, and are solitons which carry magnetic charge. They lie at the heart of recent developments using electric-magnetic duality, where fundamental particles carrying electric charge are dual to the magnetically charge monopoles. The equations describing static BPS monopoles are integrable and this allows various sophisticated twistor methods to be applied in their study. Monopole dynamics is not an integrable system but for slowly moving monopoles their dynamics can be approximated by geodesic motion on the hyperkaler moduli space of static solutions. The figure shows a particular scattering of three monopoles at increasing times. Monopoles often resemble Skyrmions and although this is not yet well understood, monopoles can also be described by rational maps, so there are some hints at a connection.

Knot Solitons Solitons stabilized by the Hopf invariant (which is a linking number between field lines) arise in a modified sigma model known as the Skyrme-Faddeev model. The figure shows a certain field line for the solitons with Hopf charge one to eight. It can be seen that the first few solitons consist of a single loop, which in some cases is twisted, but for solitons above charge five, links and knots appear. The charge seven configuration is a trefoil knot. Is is expected that many other links and knots will appear at higher charges.