# Maximally Supersymmetric Models in Four and Six Dimensions

Sammanfattning: We consider two examples of maximally supersymmetric models; the N=4 Yang-Mills theory in four dimensions and the (2,0) theory in six dimensions. The first part of the thesis serves as an introduction to the topics covered in the appended research papers, and begins with a self-contained discussion of principal fibre bundles and symplectic geometry. These two topics in differential geometry find applications throughout the thesis in terms of gauge theory and canonical quantization. Subsequently, we consider the N=4 supersymmetry algebra and the massless Yang-Mills multiplet representation. In particular, we discuss the vacuum structure of the N=4 theory in a space-time with the geometry of a torus, and the computation of its weak coupling spectrum. We investigate the case with a gauge group of adjoint form and discuss the implications of non-trivial bundle topology for the moduli space of flat connections. Furthermore, we consider gauging the R-symmetry of the theory by introducing a corresponding background connection. We identify a special class of terms in the partition function, which are BPS and can (in principle) be computed at weak coupling, and discuss the action of S-duality on this sector. Finally, we consider the (2,0) theory in six dimensions, provide a general introduction and derive the free tensor multiplet representation of the supersymmetry algebra. We then proceed to study (2,0) theory defined on a manifold which can be described as a circle fibred over some five-dimensional manifold. We discuss the dimensional reduction of the free (2,0) tensor multiplet on the circle and derive the (maximally supersymmetric) abelian Yang-Mills theory obtained in five dimensions for the most general metric of such a fibration. We discuss the properties of the Yang-Mills theory corresponding to the superconformal symmetry of the (2,0) theory and propose a generalization to the interacting (non-abelian) case, where a field theory description in six dimensions is problematic. The case when the circle fibration description becomes singular is also considered and we give a concrete example of such a manifold and discuss the degrees of freedom located at the singularity.

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