Seiberg-Witten theory for symplectic manifolds

Abstract

In this thesis we give an introduction to Seiberg-Witten gauge theory used to study compact oriented four-dimensional manifolds X. Seiberg-Witten theory uses a Spin c structure to create two vector bundles over X called the spinor bundle and determinant line bundle. One then considers the set of solutions to the Seiberg-Witten equations, which are expressed in terms of a section of the spinor bundle and a Dirac operator formed out of a connection on the determinant line bundle. After taking the quotient by an action of a U(1)-gauge group, one constructs an invariant by integrating cohomology classes over the resulting moduli space. In this thesis we show these Seiberg-Witten invariants can be used to find obstructions to the existence of a symplectic structure on X.