Exact Results in Supersymmetric Quantum Field Theories

近年来，新技术已经允许量子场论中的精确计算被执行。这使得量子场论的研究超越了弱耦合制度，在那里它们可以被定义为费曼图产生的耦合常数的渐近级数。本论文的目的是将这些新技术应用于新的领域，以帮助建立理论上的理解。超对称定位是这些新技术的一个例子。它允许无限维的费曼积分减少到有限维积分在一个紧凑的群体。许多具有超对称性的模型都适用于这种技术。特别是，许多与ABJM模型相关的三维配分函数，M理论的M膜上的一个简化的基本世界体积理论，可以被局部化。也有人认为，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，它们被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.