Gravitating vortices, cosmic strings, and the K\"ahler--Yang--Mills equations

TL;DR

This paper introduces gravitating vortex equations derived from Kähler-Yang-Mills equations, providing new solutions that describe cosmic strings with back reaction effects, and offers a geometric invariant theory perspective on their existence.

Contribution

It constructs new solutions to the gravitating vortex equations via dimensional reduction, linking them to cosmic strings and geometric invariant theory.

Findings

Solutions of the gravitating vortex equations on Riemann surfaces.

Existence of Einstein-Bogomol'nyi solutions as cosmic strings.

GIT interpretation of existence results for these 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.