The SO(3) Vortex Equations over Orbifold Riemann Surfaces

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 S 1 × Σ 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.