Magnetic Monopoles over Topologically Nontrivial Riemann Surfaces

An explicit canonical construction of monopole connections on nontrivial U(1) bundles over Riemann surfaces of any genus is given. The class of monopole solutions depends on the conformal class of the given Riemann surface and a set of integer weights. The reduction of Seiberg--Witten 4-monopole equations to Riemann surfaces is performed. It is then shown that the monopole connections constructed are solutions to these equations.