Exact Results in Supersymmetric Quantum Field Theories

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In recent years new techniques have allowed exact calculations in quantum field theories to be performed. This allows quantum field theories to be investigated beyond the weak coupling regime where they may be defined as an asymptotic series in the coupling constant generated by Feynman diagrams. The aim of this thesis is to apply these new techniques in novel areas to help build theoretical understanding.Supersymmetric localisation is an example of one of these new techniques. It allows infinite dimensional Feynman integrals to be reduced to finite dimensional integrals over a compact group. Many models with supersymmetry are amenable to this technique. In particular, many three dimensional partition functions related to the ABJM model, a conjectured fundamental worldvolume theory on the M branes of M theory, can be localised. It has also been argued that three dimensional partition functions can be obtained as a suitable limit of the "superconformal index" of appropriate four dimensional superconformal field theories, which are expressed as similar finite dimensional integrals. In order to investigate the ABJM model we will investigate this link in order to perform exact calculations for quantum field theories related to the ABJM model, but defined on novel manifolds.