On the Seiberg-witten invariants of smooth 4-Manifolds

Abstract

This thesis reviews Seiberg-Witten Gauge theory and the Seiberg-Witten invariants of smooth 4D manifolds. After reviewing some preliminaries on Clifford Algebras, Spinbundles, Dirac Operators, We go into discussing a system of mildly nonlinear partial differential equations on a U(1) bundle which are commonly known as Seiberg-Witten equations. We discuss its properties, consider their solution space and then quotient it by the equivalence due to gauge transformations. The moduli space that we get after moding on the space of solutions has some nicer properties as compared to Donaldson’s. In the last chapter, we briefly talk about the Witten conjecture which makes a connection between the Seiberg-Witten Invariants and the Donaldson invariants. Many physicists argue that using S-duality, SW theory and Donaldson theory can be viewed as the two extreme cases (one N → ∞, and the other N → 0) of a common theory, but S-duality is not yet mathematically understood fully rigorously. Even with seminal progresses regarding proving this conjecture which is widely believed to be true by many professional physicists- it still remains to be proven true in the general sense. This thesis acts as a review of these ideas as an introduction to Seiberg-Witten theory.