Orbifold Riemann Surfaces and the Yang-Mills-Higgs Equations

TL;DR

This paper extends Hitchin's work to orbifold Riemann surfaces, proving existence of solutions to Yang-Mills-Higgs equations and constructing their moduli spaces, which are rich geometric structures with various interpretations.

Contribution

It introduces the first analysis of Yang-Mills-Higgs solutions on orbifold surfaces and constructs their moduli spaces, revealing new geometric and algebraic insights.

Findings

Existence of orbifold solutions to Yang-Mills-Higgs equations

Construction of moduli spaces as hyper-Kahler manifolds

Identification of moduli spaces with orbifold Higgs bundles and Fuchsian group representations

Abstract

We extend Hitchin's results on "The self-duality equations on a Riemann surface" (Proc. LMS (3), vol. 55, 1987) to orbifold Riemann surfaces. We prove existence results for orbifold solutions of the Yang-Mills-Higgs equations and construct the moduli space of solutions. These moduli spaces provide interesting examples of non-compact hyper-Kahler manifolds in all dimensions divisible by 4 and of completely integrable Hamiltonian systems. We also reinterpret these moduli spaces as spaces of orbifold Higgs bundles and as representation varieties of Fuchsian groups.