Vortices on Hyperbolic Surfaces

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 ${\mathbb{R}}^4$ is also given.

Revised version: new section on four-dimensional interpretation of hyperbolic vortices added.

References in corpus (3)

• Integrability of Vortex Equations on Riemann Surfaces
• The Kähler Potential of Abelian Higgs Vortices
• Hyperbolic calorons, monopoles, and instantons

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• Ginzburg-Landau equations on Riemann surfaces of higher genus
• Higgs and Coulomb Branch Descriptions of the Volume of the Vortex Moduli Space
• Hyperbolic vortices and Dirac fields in 2+1 dimensions
• Cartan Connections and Integrable Vortex Equations
• BPS Boojums in N=2 supersymmetric gauge theories I
• Non-abelian vortices on CP^1 and Grassmannians