Nineteen vortex equations and integrability

Abstract: The class of five integrable vortex equations discussed recently by Manton is extended so it includes the relativistic BPS Chern-Simons vortices, yielding a total of nineteen vortex equations. Not all the nineteen vortex equations are integrable, but four new integrable equations are discovered and we generalize them to infinitely many sets of four integrable vortex equations, with each set denoted by its integer order $n$. Their integrability is similar to the known cases, but give rise to different (generalized) Baptista geometries, where the Baptista metric is a conformal rescaling of the background metric by the Higgs field. In particular, the Baptista manifolds have conical singularities. Where the Jackiw-Pi, Taubes, Popov and Ambj{\o}rn-Olesen vortices have conical deficits of $2\pi$ at each vortex zero in their Baptista manifolds, the higher-order generalizations of these equations are also integrable with larger constant curvatures and a $2\pi n$ conical deficit at each vortex zero. We then generalize a superposition law, known for Taubes vortices of how to add vortices to a known solution, to all the integrable vortex equations. We find that although the Taubes and the Popov equations relate to themselves, the Ambj{\o}rn-Olesen and Jackiw-Pi vortices are added by using the Baptista metric and the Popov equation. Finally, we find many further relations between vortex equations, e.g. we find that the Chern-Simons vortices can be interpreted as Taubes vortices on the Baptista manifold of their own solution.