Cartan connections for an infinite family of integrable vortices

An infinite family of integrable vortex equations is studied and related to the Cartan geometry of the underlying Riemann surfaces. This Cartan picture gives an interpretation of the vortex equations as the flatness of a non-Abelian connection. Solutions of the vortex equations also give rise to magnetic zero-modes for a certain Dirac operator on the lifted geometry. The family of integrable vortex equations is parametrised by a positive number $n$, that is equal to unity in the standard case and an integer in the case of polynomial vortex equations; finally, it may be extended to any positive real number.