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 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 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. 1

Cahen–Gutt moment map, closed Fedosov star product and structure of the automorphism group

“…As can be seen in other similar problems (see e.g. [14], [16], [21], [1]), the two obstructions of the Lie algebra character and the reductiveness come always in pair. The purpose of this paper is to show that this is the case, namely the following is the main result of this paper.…”

Gravitating vortices with positive curvature

“…Nonetheless, as the recent result of Han-Sohn [15] showed, Yang's existence result does not exhaust all the possible solutions (see a detailed discussion in Section 2.2). For c 0, the first two authors jointly with Álvarez-Cónsul and García-Prada found a new obstruction to the existence of solutions of (2.2), thus establishing a relation with Geometric Invariant Theory (GIT) for these equations [2,4]. For c < 0, the existence and uniqueness of solutions has been established in [4] in genus greater than one for a suitable range of the coupling constant α, depending only on the topology of the surface and the line bundle.…”

“…For c 0, the first two authors jointly with Álvarez-Cónsul and García-Prada found a new obstruction to the existence of solutions of (2.2), thus establishing a relation with Geometric Invariant Theory (GIT) for these equations [2,4]. For c < 0, the existence and uniqueness of solutions has been established in [4] in genus greater than one for a suitable range of the coupling constant α, depending only on the topology of the surface and the line bundle. For c > 0, the analytical techniques in [4,15,29] do not apply, and the existence problem has hitherto remained open.…”

“…The gravitating vortex equations (1.3) are fundamental, in the following sense: firstly, they admit a Hamiltonian interpretation [1,4] akin to the existence problem for Kähler-Einstein metrics, where algebro-geometric stability obstructions appear on compact Kähler manifolds with c 1 > 0. Secondly, being a particular case of the Kähler-Yang-Mills equations [1], the coupled system (1.3) is motivated by the question of understanding moduli spaces for smooth polarised varieties equipped with vector bundles.…”

“…For c < 0, the existence and uniqueness of solutions has been established in [4] in genus greater than one for a suitable range of the coupling constant α, depending only on the topology of the surface and the line bundle. For c > 0, the analytical techniques in [4,15,29] do not apply, and the existence problem has hitherto remained open.…”