Local model for the moduli space of affine vortices

TL;DR

This paper proves that the moduli space of regular affine vortices over the complex plane and upper half plane is a smooth manifold, using Fredholm theory, which is essential for defining the open quantum Kirwan map.

Contribution

It establishes the smooth manifold structure of the moduli space of affine vortices, extending previous results to new settings and supporting the development of quantum invariants.

Findings

Moduli space of affine vortices is a smooth manifold.

Extension of results to affine vortices over the upper half plane.

Foundation for defining the open quantum Kirwan map.

Abstract

We show that the moduli space of regular affine vortices, which are solutions of the symplectic vortex equation over the complex plane, has the structure of a smooth manifold. The construction uses Ziltener's Fredholm theory results [31]. We also extend the result to the case of affine vortices over the upper half plane. These results are necessary ingredients in defining the "open quantum Kirwan map" proposed by Woodward [24].