Electromagnetic duality, del Pezzo surfaces and Instantons

Abstract

Electromagnetism can be described as an Abelian gauge theory. A mathematical overview of gauge theory and the concept of instantons is given. Electromagnetic duality is the phenomenon that Maxwell's equations in vacuum are invariant under interchanging of the electric and magnetic field. We can generalize this idea by describing Maxwell theory on an arbitrary four-dimensional curved space-time and enhance the duality to an action of the group SL(2,R). One can ask to what extent this duality remains in the quantized theory. We will see that the partition function of this Maxwell theory transforms under an action of a discrete subgroup of SL(2,R). The transformations properties depend on the topology of the space-time manifold under consideration. Del Pezzo surfaces are a type of four-dimensional surfaces that are interesting for string theory. The del Pezzo surfaces can be described by blowing up points of the projective plane. Using topological data of these manifolds, the Maxwell partition function for the del Pezzo surfaces is calculated. The partition function is related to the moduli space of Abelian 'instantons'. A short overview is given of the theory of the moduli space of instantons of non-abelian gauge theories. This moduli spaces is used to produce Donaldson invariants of four-manifolds.