Removal of singularities and Gromov compactness for symplectic vortices

TL;DR

This paper establishes a Gromov compactification for the moduli space of symplectic vortices with bounded energy, extending previous results to arbitrary compact Lie groups and introducing a removable singularity theorem.

Contribution

It generalizes Gromov compactness results for symplectic vortices from circle actions to all compact Lie groups, including a new removable singularity theorem.

Findings

Gromov compactification of vortex moduli space established

Removable singularity theorem for vortices proved

Extension from circle actions to arbitrary compact Lie groups

Abstract

We prove that the moduli space of gauge equivalence classes of symplectic vortices with uniformly bounded energy in a compact Hamiltonian manifold admits a Gromov compactification by polystable vortices. This extends results of Mundet i Riera and Tian for circle actions to the case of arbitrary compact Lie groups. Our argument relies on an a priori estimate for vortices that allows us to apply techniques used by McDuff and Salamon in their proof of Gromov compactness for pseudoholomorphic curves. As an intermediate result we prove a removable singularity theorem for vortices.