Gravitating vortices, cosmic strings, and the Kähler--Yang--Mills equations

Abstract

In this paper we construct new solutions of the K ahler{Yang{Mills equations, by applying dimensional reduction methods to the product of the complex projective line with a compact Riemann surface. The resulting equations, that we call gravitating vortex equations, describe abelian vortices on the Riemann surface with back reaction of the metric. As a particular case of these gravitating vortices on the Riemann sphere we nd solutions of the Einstein{Bogomol'nyi equations, which physically correspond to Nielsen{ Olesen cosmic strings in the Bogomol'nyi phase. We use this to provide a Geometric Invariant Theory interpretation of an existence result by Y. Yang for the Einstein{Bogomol'nyi equations, applying a criterion due to G. Sz ekelyhidi.