Five Vortex Equations

The Taubes equation for Abelian Higgs vortices is generalised to five distinct U(1) vortex equations. These include the Popov and Jackiw--Pi vortex equations, and two new equations. The Baptista metric, a conformal rescaling of the background metric by the squared Higgs field, gives insight into these vortices, and shows that vortices can be interpreted as conical singularities superposed on the background geometry. When the background has a constant curvature adapted to the vortex type, then the vortex equation is integrable by a reduction to Liouville's equation, and the Baptista metric has a constant curvature too, apart from its conical singularities. The conical geometry is fairly easy to visualise in some cases.