Gravitating vortices and the Einstein–Bogomol’nyi equations

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 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 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.