On applications of manifest, maximal supersymmetry & U-duality

Abstract: Maximal supersymmetry is made manifest through the introduction of the pure spinor formalism. In many ways, this formalism represents a powerful tool in examinations of characteristics of different theories in high energy physics, though it is far from extensively exploited. In this thesis, we review the ideas behind the manifestly, maximally supersymmetric formulations of supersymmetric Yang--Mills theory and supergravity, as well as the concept of pure spinors. We then proceed to the two research areas of Born--Infeld theory in the context of D-branes and loop amplitude descriptions of supersymmetric Yang--Mills theory and supergravity. Both are examples of theories in which interesting applications for the pure spinor formulations can be found. For example, the question of finiteness of the amplitude description in perturbative maximal supergravity in four dimensions is not yet quite settled. The theory is not believed to be finite, but an actual divergence still remains to be found. In addition, we briefly review the concept of U-duality and its description in terms of generalised, exceptional geometry. Apart from being an interesting research area on its own, e.g. with respect to its tensor formalism, it also connects to restrictions of loop amplitude descriptions of maximal supergravity and ongoing research.