Integrability and Twistor Theory
David M.J. Calderbank
Two dimensional reductions of self-dual conformal geometry.
I will review the twistorial framework of integrable background geometries for studying reductions of dispersionless integrable systems. The general principles will be illustrated by some reductions of four-dimensional self-dual conformal geometry to two dimensions, which include projective structures (studied by N. Hitchin and M. Dunajski et al.), spinor-vortex geometries (first introduced by S. Burtsev et al.) and closed 1-forms (introduced in this context by J. Tafel). These geometries provide backgrounds for gauge field equations, and when the gauge group is a subgroup of the diffeomorphism group of an auxiliary surface, solutions provide constructions of self-dual conformal manifolds.
Anti-self-dual Einstein metrics and projective structures.
Any projective structure on a surface gives rise to an anti--self--dual Einstein metric with neutral signature. I shall discuss this construction, and show how it gives explicit solutions of the SU(\infinity) Toda equation. This is based on joint work with Thomas Mettler, Rod Gover, and Alice Waterhouse.
Magnetic Skyrmions at Critical Coupling.
We introduce a family of models for magnetic skyrmions in the plane for which infinitely many solutions can be given explicitly. The energy defining the models is bounded below by a topological invariant, and the configurations attaining the bound satisfy first order Bogomol'nyi equations. We give explicit solutions which depend on an arbitrary holomorphic function. The simplest solutions are the basic Bloch and Néel skyrmions, but we also exhibit distorted and rotated single skyrmions as well as line defects, and configurations consisting of skyrmions and anti-skyrmions.