Gravitating vortices and the Einstein--Bogomol'nyi equations

TL;DR

This paper studies gravitating vortex equations on Riemann surfaces, establishing a link with geometric invariant theory, proving a conjecture about cosmic strings, and demonstrating existence and uniqueness results for higher genus surfaces.

Contribution

It provides a converse to Yang's existence theorem for Einstein--Bogomol'nyi equations, linking them to GIT, and proves existence and uniqueness of solutions for gravitating vortices on higher genus surfaces.

Findings

Established a correspondence with Geometric Invariant Theory for Einstein--Bogomol'nyi equations.

Proved a conjecture on the non-existence of cosmic strings on f1f1 f1f1 superimposed at a point.

Proved existence and uniqueness of gravitating vortex solutions for genus > 1.

Abstract

In this work we consider the gravitating vortex equations. These equations couple a metric over a compact Riemann surface with a hermitian metric over a holomorphic line bundle equipped with a fixed global section --- the Higgs field ---, and have a symplectic interpretation as moment-map equations. As a particular case of the gravitating vortex equations on $\mathbb{P}^1$, we find the Einstein--Bogomol'nyi equations, previously studied in the theory of cosmic strings in physics. We prove two main results in this paper. Our first main result gives a converse to an existence theorem of Y. Yang for the Einstein--Bogomol'nyi equations, establishing in this way a correspondence with Geometric Invariant Theory for these equations. In particular, we prove a conjecture by Y. Yang about the non-existence of cosmic strings on $\mathbb{P}^1$ superimposed at a single point. Our second main result is an existence and uniqueness result for the gravitating vortex equations in genus greater than one.