Gravitating Vortices, Cosmic Strings, and the Kähler–Yang–Mills Equations

Abstract

In this paper we construct new solutions of the Kahler–Yang–Mills equations, by applying dimensional reduction methods to the product of the complex projective line with a compact Riemann surface. The resulting equations, which 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 find 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. Szekelyhidi.