Abelian Vortices with Singularities

TL;DR

This paper studies abelian vortex equations on line bundles over Riemann surfaces with singular metrics and parabolic structures, showing the moduli space remains unchanged and computing geometric invariants with explicit solutions.

Contribution

It demonstrates that the moduli space of singular vortex solutions is identical to the regular case and provides formulas for its volume and scalar curvature, including explicit solutions.

Findings

Moduli space of singular vortices matches the regular case

Computed total volume and scalar curvature of the moduli space

Constructed explicit vortex solutions on the thrice punctured hyperbolic sphere

Abstract

Let L --> X be a complex line bundle over a compact connected Riemann surface. We consider the abelian vortex equations on L when the metric on the surface has finitely many point degeneracies or conical singularities and the line bundle has parabolic structure. These conditions appear naturally in the study of vortex configurations with constraints, or configurations invariant under the action of a finite group. We first show that the moduli space of singular vortex solutions is the same as in the regular case. Then we compute the total volume and total scalar curvature of the moduli space singular vortex solutions. These numbers differ from the case of regular vortices by a very natural term. Finally we exhibit explicit non-trivial vortex solutions over the thrice punctured hyperbolic sphere.