Seiberg-Witten theory on 4-manifolds

Abstract

This thesis is an exposition on Seiberg-Witten theory for 4-manifolds. We begin by describing basic notions of gauge theory like bundles, connections, curvature, gauge group, configuration space etc. in the first chapter. This is followed by brief account of Spin geometry for 4-manifolds with main objective of defining Dirac operators and describing Wietzenböck formula and the Atiyah-Singer index theorem. In the third chapter, we set up the monopole moduli space based on the Seiberg-Witten equations and prove compactness and transversality. We also define the Seiberg-Witten invariants. In the final chapter, we compute the Seiberg-Witten invariants for Kahler surfaces and find exotic smooth structures on 4-manifolds.