The SO(3) Vortex Equations over Orbifold Riemann Surfaces

TL;DR

This paper investigates the moduli spaces of SO(3) vortices on orbifold Riemann surfaces and their applications to monopole equations, Floer homology, and Morse-Bott theory, providing new insights and models in gauge theory.

Contribution

It characterizes SO(3) vortex moduli spaces on orbifold surfaces and introduces framed monopole Floer homology, connecting vortex solutions to monopole equations and Floer theories.

Findings

Characterized moduli spaces of SO(3) vortices on orbifold Riemann surfaces.

Constructed a version of framed monopole Floer homology.

Computed Morse-Bott indices for a natural Morse-Bott function.

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, Ozsv\'ath 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.