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, 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 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.

arXiv commentv1: 42 pages. v2: 24 pages. The first version of this paper was split up into two parts. This is the first part. To appear in Comm. Math. Phys. The paper presentation has been significantly improved