  Starting from the geometrical interpretation of integrable vortices on
two-dimensional hyperbolic space as conical singularities, we explain how this
picture can be expressed in the language of Cartan connections, and how it can
be lifted to the double cover of three-dimensional Anti-de Sitter space viewed
as a trivial circle bundle over hyperbolic space. We show that vortex
configurations on the double cover of AdS space give rise to solutions of the
Dirac equation minimally coupled to the magnetic field of the vortex. After
stereographic projection to (2+1)-dimensional Minkowski space we obtain, from
each lifted hyperbolic vortex, a Dirac field and an abelian gauge field which
solve a Lorentzian, (2+1)-dimensional version of the Seiberg-Witten equations.
