Nora Gavrea (School of Mathematics, University of Leeds) – Integrable vortices in the Abelian Higgs model

Vortices are 2-dimensional topological solitons defined on a Riemann surface in the context of the Abelian Higgs model. Physically, they model magnetic flux tubes in superconductors. At critical coupling, they satisfy a 1st order system of PDEs called the Bogomolny equations. I will first review the derivation of these equations using a Bogomolny argument, and then derive the Taubes equation. Next, I will introduce a generalised Abelian Higgs energy functional, which gives rise to 5 different vortex equations (this has been carefully investigated by Nicholas Manton). For a constant curvature base surface, these equations turn out to be integrable, reducing to a Liouville equation. One can obtain further integrable vortex equations by choosing suitable conformal factors, and in this case, the Taubes equation becomes the sinh-Gordon or Tzitzeica equation. If we assume radial symmetry, these are equivalent to a Painlevé III ODE. I will discuss the construction of these vortices for the rest of the talk, which is a joint work with Maciej Dunajski.