Deformation theory of the blown-up Seiberg-Witten equation in dimension three

TL;DR

This paper studies the deformation theory of three-dimensional Seiberg-Witten equations linked to quaternionic representations, constructing models around boundary points and exploring conditions for deforming Fueter sections into solutions.

Contribution

It develops Kuranishi models for boundary points of moduli spaces and analyzes the deformation of Fueter sections into Seiberg-Witten solutions in dimension three.

Findings

Construction of Kuranishi models around boundary points.

Identification of boundary points with Fueter sections via Haydys correspondence.

Conditions under which Fueter sections deform into Seiberg-Witten solutions.

Abstract

Associated with every quaternionic representation of a compact, connected Lie group there is a Seiberg-Witten equation in dimension three. The moduli spaces of solutions to these equations are typically non-compact. We construct Kuranishi models around boundary points of a partially compactified moduli space. The Haydys correspondence identifies such boundary points with Fueter sections - solutions of a non-linear Dirac equation - of the bundle of hyperk\"ahler quotients associated with the quaternionic representation. We discuss when such a Fueter section can be deformed to a solution of the Seiberg-Witten equation.