Gravitating vortices and Symplectic Reduction by Stages

TL;DR

This paper introduces a new symplectic reduction approach to gravitating vortices on Riemann surfaces, establishing existence, uniqueness, and stability results using advanced pluripotential theory techniques.

Contribution

It applies symplectic reduction by stages to PDE and gauge theory, proving existence, polystability, and uniqueness of solutions for gravitating vortex equations.

Findings

Existence of solutions on the sphere implies polystability of the divisor.

Uniqueness of vortices in any K"ahler class without automorphisms.

Solutions exist for genus g ≥ 1 under certain parameter ranges.

Abstract

We undertake a novel approach to the existence problem for gravitating vortices on a Riemann surface based on symplectic reduction by stages, which seems to be new in the PDE as well as the gauge theory literature. The main technical tool for our study is the reduced $\alpha$-K-energy, for which we establish convexity properties by means of finite-energy pluripotential theory, as recently applied to the study of constant scalar curvature K\"ahler metrics. Using these methods, we prove that the existence of solutions to the gravitating vortex equations on the sphere implies the polystability of the effective divisor defined by the zeroes of the Higgs field. This approach also enables us to establish the uniqueness of gravitating vortices in any admissible K\"ahler class, in the absence of automorphisms. Lastly, we also prove the existence of solutions for the gravitating vortex equations for genus $g\geq 1$ for certain ranges of the coupling constant $\alpha$ and the volume.