Exact Results in Supersymmetric Quantum Field Theories

近年来，新技术使量子场论中的精确计算成为可能。这使得量子场论可以在弱耦合状态之外进行研究，在弱耦合状态下，它们可以被定义为费曼图生成的耦合常数中的渐近级数。本文的目的是将这些新技术应用于新的领域，以帮助建立理论认识。超对称定位就是这些新技术的一个例子。它允许将无限维费曼积分化为紧群上的有限维积分。许多具有超对称的模型都适用于这种技术。特别是，与ABJM模型相关的许多三维配分函数是可以局部化的，ABJM模型是基于M理论的M膜的一个推测的基本世界体积理论。本文还论证了三维配分函数可以作为适当的四维超共形场论的“超共形指数”的适当极限而得到，这些超共形场论可以表示为类似的有限维积分。为了研究ABJM模型，我们将研究这一联系，以便对与ABJM模型相关的量子场论进行精确计算，但定义在新的流形上。

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.