The geometry of integrable vortices

Abstract

This thesis is concerned with geometric interpretations of vortices. We demonstrate that all five of the integrable Abelian vortex equations can be encoded in terms of the flatness and holomorphic trivialisation of a non-Abelian connection. There is a natural lift of this story to three dimensional group manifolds where the flat connection is related to the Maurer-Cartan one-form. In particular we present a detailed study of vortices on the two-sphere and on two dimensional hyperbolic space. For these cases the lifted vortices give rise to solutions of coupled equations including a massless gauged Dirac equation on flat three dimensional space. Squaring the Dirac equation arising from vortices on the 2-sphere gives rise to a Schrodinger like equation for a spinor wave function in the background of a magnetic field with non-trivial linking. Both the wave function and the magnetic field pick up non-trivial topology related to the vortex. In the final Section a potential realisation scheme for magnetic fields with non-trivial linking as synthetic fields in a Bose-Einstein condensate is discussed.