LOST IN LOCALISATION searching for exact results in supersymmetric gauge theories

Abstract: This thesis deals with one of the very basics of theoretical physics: computing observable quantities. In the language commonly used to describe the subatomic world, gauge theories, this problem is far from trivial as the observables are expressed in terms of infinite-dimensional integrals. This holds true even in supersymmetric gauge theories, but in some cases, this additional symmetry may be used to reduce the infinite-dimensional integrals to finite-dimensional ones - which naturally simplifies the expressions significantly. This thesis revolves around one of these techniques: Localisation. In general, this poses strict requirements on the theory as well as the manifold on which the theory is placed. However, by first twisting the theory so as to obtain a topological field theory, localisation can be carried out on any background, whereas one otherwise is confined to manifolds with a large amount of symmetry such as for example d-dimensional spheres. The explicit calculation of the path integral is nonetheless in general still complicated even after localisation, and it is only in certain limits that it may be computed exactly. For example, simplifications often occur in the limit of infinitely many colours (the large N limit). Of the five papers appended to this thesis, the first three deal with topological twists of maximally supersymmetric Yang-Mills theory and (2,0) theory, whereas the last two revolve around the behaviour of the free energy of massive ABJM theory in the large N limit.