K-Theory, D-Branes and Ramond-Ramond Fields

This thesis is dedicated to the study of K-theoretical properties of D-branes and Ramond-Ramond fields. We construct abelian groups which define a homology theory on the category of CW-complexes, and prove that this homology theory is equivalent to the bordism representation of KO-homology, the dual theory to KO-theory. We construct an isomorphism between our geometric representation and the analytic representation of KO-homology, which induces a natural equivalence of homology functors. We apply this framework to describe mathematical properties of D-branes in type I String theory. We investigate the gauge theory of Ramond-Ramond fields arising from type II String theory defined on global orbifolds. We use the machinery of Bredon cohomology and the equivariant Chern character to construct abelian groups which generalize the properties of differential K-theory defined by Hopkins and Singer to the equivariant setting, and can be considered as a differential extension of equivariant K-theory for finite groups. We show that the Dirac quantization condition for Ramond-Ramond fieldstrengths on a good orbifold is dictated by equivariant K-theory and the equivariant Chern character, and study the group of flat Ramond-Ramond fields in the particular case of linear orbifolds in terms of our orbifold differential K-theory.