Off-shell, maximal supersymmetry & exceptional geometry

Sammanfattning: Two lines of research are presented in this thesis, both with a focus on fundamental properties within the supersymmetric theories of high energy physics. The first features analyses based on the actions for maximal supersymmetric Yang–Mills theory and supergravity, provided by the pure spinor formalism. The second is the development of the theory of exceptional geometry. The analyses within maximally supersymmetric Yang–Mills theory and supergravity, benefitting from the pure spinor formalism, centres around the investigations of the UV divergences of the amplitude diagrams, where the case of maximal supergravity is subject to ongoing research. There is currently an interesting development in connection to the four-dimensional theory, regarding a possible finiteness, indicated in the pure spinor formalism setting of the research articles appended to this thesis. The results are contrary to the expected divergence, currently of amplitude diagrams with more than six loops. Exceptional geometry is an extension of supergravity that is constructed to incorporate U-duality. It is a geometric formulation with an extended space, in a way similar to in doubled geometry, where T-duality is the symmetry accommodated for. Constituting a rather recent area of research, the theory is under development with respect to the inherent symmetries, the tensor formalism, etc., regarding the properties affected by the extended space. Its recognised features, construction and concepts to be investigated are the objects of interest in this thesis.