An introduction to Seiberg-Witten theory on closed 3-manifolds

TL;DR

This paper introduces the construction and analysis of Seiberg-Witten invariants for closed 3-manifolds, exploring the moduli space structure and invariants' behavior under metric deformation, serving as an educational resource.

Contribution

It provides a detailed exposition of existing results on Seiberg-Witten invariants for 3-manifolds, including background and foundational aspects, aimed at newcomers to gauge field theory.

Findings

Definition of Seiberg-Witten invariants for 3-manifolds

Analysis of moduli space structure

Behavior of invariants under metric deformation

Abstract

This is a version of the author's diploma thesis written at the University of Cologne in 2002/03. The topic is the construction of Seiberg-Witten invariants of closed 3-manifolds. In analogy to the four dimensional case, the structure of the moduli space is investigated. The Seiberg-Witten invariants are defined and their behaviour under deformation of the Riemannian metric is analyzed. Since it is essentially an exposition of results which were already known during the time of writing, the thesis has not been published. In particular, the author does not claim any originality concerning the results. Moreover, new developments of the theory are not included. However, the detailed account--together with the appendices on the required functional analytic and geometric background--might be of interest for people starting to work in the area of gauge field theory.