The SO(3) Vortex Equations over Orbifold Riemann Surfaces

2021/03/22 by Mariano Echeverria, Echeverria, Mariano
Mathematics · Physics and Astronomy · #57R57 #57R58 #Black Holes and Theoretical Physics #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric Topology (math.GT) #Geometry and complex manifolds

paper · pdf · doi:10.48550/arxiv.2103.11957

openalex publication_date 2021/03/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study the general properties of the moduli spaces of SO(3) vortices over orbifold Riemann surfaces and use these to characterize the solutions of the SO(3) monopole equations on Seifert manifolds following in the footsteps of Mrowka, Ozsváth and Yu. We also study the solutions to the SO(3) monopole equations on S1 x (Riemann Surface) in order to motivate the construction of a version of monopole Floer homology, which we call framed monopole Floer homology, in analogy with the construction given by Kronheimer and Mrowka for the case of instanton Floer homology. Finally, the SO(3) vortex moduli spaces provide a nice toy model for recent work due to Feehan and Leness regarding the study of a natural Morse-Bott function on the moduli space of SO(3) monopoles over Kahler manifolds. In particular, we compute the Morse-Bott indices of this function.

Citations

• The Self-Duality Equations on a Riemann Surface
• Vector Bundles Over an Elliptic Curve
• Orbifold Riemann Surfaces and the Yang-Mills-Higgs Equations
• A Generalization of the Tristram-Levine Knot Signatures as a Singular Furuta-Ohta Invariant for Tori
• On the essential (p)-dimension of parabolic bundles on a curve

Related

• A monopole homology of integral homology 3-spheres
• A Morse-Bott approach to monopole Floer homology and the Triangulation conjecture
• Symplectic fibrations and the abelian vortex equations
• Monopole Floer homology for codimension-3 Riemannian foliation
• The moduli space of multi-monopoles on a Riemann surface