# 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.

## Key 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.

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Source: https://tomesphere.com/paper/1704.02954