Instantons and Electric-Magnetic Duality on Del Pezzo surfaces

Abstract

Instantons and electric-magnetic duality are two phenomena that arise in the interplay between gauge theory and four-dimensional geometry. An important class of instantons in four-dimensional Yang--Mills theory is formed by anti-self-dual (ASD) connections. The first part of this thesis covers the moduli space of ASD connections with a focus on SU(2) bundles. Del Pezzo surfaces are four-manifolds which have a Kähler structure and this allows for a description of ASD connections on SU(2) bundles in terms of stable holomorphic vector bundles. This description is worked out in more detail for S^2xS^2. The second part of the thesis discusses electric-magnetic duality in Euclidean Maxwell theory on closed, connected, oriented, Riemannian four-manifolds. Results by Verlinde, Witten, Olive and Alvarez describe this duality in terms of the modular behaviour of the associated partition functions. The theory will be applied to compute and study the partition functions for del Pezzo surfaces. Necessary background information on four-dimensional geometry and del Pezzo surfaces will be given as well.