Analytic vortex solutions on compact hyperbolic surfaces

TL;DR

This paper constructs Abelian-Higgs vortex solutions on compact hyperbolic surfaces with constant negative curvature, using Schwarz triangle functions for explicit solutions in symmetric cases.

Contribution

It is the first to explicitly construct Abelian-Higgs vortices on compact hyperbolic surfaces, expanding the understanding of vortex solutions in curved geometries.

Findings

Explicit vortex solutions on hyperbolic surfaces are obtained.

Solutions are expressed using Schwarz triangle functions.

Analytic solutions are available for symmetric configurations.

Abstract

We construct, for the first time, Abelian-Higgs vortices on certain compact surfaces of constant negative curvature. Such surfaces are represented by a tessellation of the hyperbolic plane by regular polygons. The Higgs field is given implicitly in terms of Schwarz triangle functions and analytic solutions are available for certain highly symmetric configurations.