Vortices on Hyperbolic Surfaces

TL;DR

This paper demonstrates a geometric construction of abelian Higgs vortices on hyperbolic surfaces using holomorphic maps, revealing new solutions and their relation to Yang--Mills fields.

Contribution

It introduces a novel geometric method to construct vortices on hyperbolic surfaces from holomorphic maps, including new examples on compact and revolution surfaces.

Findings

Vortices are linked to ramification points of holomorphic maps.

The Higgs field magnitude indicates local isometry of the map.

New vortex solutions on compact and revolution hyperbolic surfaces are presented.

Abstract

It is shown that abelian Higgs vortices on a hyperbolic surface $M$ can be constructed geometrically from holomorphic maps $f:M \to N$, where $N$ is also a hyperbolic surface. The fields depend on $f$ and on the metrics of $M$ and $N$. The vortex centres are the ramification points, where the derivative of $f$ vanishes. The magnitude of the Higgs field measures the extent to which $f$ is locally an isometry. Witten's construction of vortices on the hyperbolic plane is rederived, and new examples of vortices on compact surfaces and on hyperbolic surfaces of revolution are obtained. The interpretation of these solutions as SO(3)-invariant, self-dual SU(2) Yang--Mills fields on $\R^4$ is also given.