The Geometry and Physics of M-theory on $G_2$ -Holonomy Manifolds

目的：刻画$G_2$流形上的M-理论紧化，并利用这些刻画得到现象学上有趣的4维规范理论。研究方法的创新：利用最近在数学文献中得到的$G_2$完整流形的新构造。探索它们在弦紧化中的含义以及与规范理论的模空间的联系。弦理论和M理论是在完全一致的量子理论中研究规范理论和引力的独特框架。然而，激励这些理论的许多初始问题仍然没有得到回答：4d物理与什么是相关的？近年来，出现了一种系统刻画弦理论真空的方法。在一般的基础上，4D物理的性质被编码在所谓的紧凑几何中：弦/M理论被定义在更高的维度(10或11)中，并且为了获得4D理论，剩余的维度被扩展到紧致几何上。它的性质编码了决定4D规范理论的大部分数据：规范群、物质场、耦合，最重要的是对称性。确保这个框架对微小波动具有健壮性的一个关键性质是，在由此产生的4D理论中存在超对称性。超对称性，一个理论的玻色场和费米场之间的对称性，有着深刻的含义，因为它对紧凑几何的完整约束施加了约束。在M理论的情况下，它是一个11D理论，紧致几何必须是7维的，并且为了保持超对称性，必须具有简化的完整系统$G_2$(而不是$SO(7)$)。该项目将纯数学的结果，特别是微分几何的结果，与数学物理中的应用，如弦理论和场论联系起来。这个项目是及时的，因为最近数学家们已经构造了一大类$G_2$流形，即所谓的“扭曲连通和构造”，而这些几何的含义以及这些构造的推广还没有被发现。博士项目的一个中心目标将是探索如何构造这种$G_2$流形的奇异极限，因为这些在M-理论的应用中将是关键的。我们的目标是修改扭曲的连通和结构，以包括在四维压缩中产生手性物质的奇点。一个将被广泛使用的工具是杂交化和F理论紧凑化的对偶，就像最近由Braun和Schafer-Namek获得的那样。这个项目属于EPSRC数学物理研究领域。

Aims: Characterizing M-theory compactifications on $G_2$ manifolds, and using these to obtain phenomenologically interesting 4-dimensional gauge theories.Novelty of research methodology: Utilizing new constructions of $G_2$ holonomy manifolds obtained recently in the mathematics literature. Exploring their implications in String Compactifications and connection to moduli spaces of gauge theories. String theory and M-theory are unique frameworks to study gauge theory and gravity in a fully consistent quantum theory. Nevertheless, much of the initial questions, motivating these theories have remained unanswered: what is the relevance for 4d physics? In recent years an approach to systematically characterize string theory vacua has emerged. On general grounds properties of the 4d physics are encoded in so-called compactification geometries: string/M-theory are defined in higher dimensions (10 or 11) and to obtain a 4d theory, the remaining dimensions are extended on a compact geometry. The properties of this encode most of the data that determine a 4d gauge theory: the gauge group, matter fields, couplings, and most importantly, symmetries. A key property, which ensures that this framework is robust against small fluctuations is the existence of supersymmetry in the resulting 4d theory. Supersymmetry, a symmetry between the bosonic and fermionic fields of a theory, has profound implications, in that it imposes constraints on the holonomies of the compactification geometries. In the case of M-theory, which is an 11d theory, the compactification geometry has to be 7-dimensional, and to preserve supersymmetry, has to have reduced holonomy $G_2$ (instead of $SO(7)$). The project gives a connection between results in pure mathematics, specifically differential geometry, and applications in mathematical physics, such as string theory and field theory. The project is timely, in that recently a large class of $G_2$ manifolds have been constructed by mathematicians, the so-called "twisted connected sum construction", and the implications of these geometries, as well as generalizations of these constructions are yet to be uncovered. One central goal of the PhD project will be to explore how singular limits of such $G_2$ manifolds can be constructed, as these will be of key importance in applications to M-theory. The goal is to modify the twisted connected sum constructions to include singularities that give rise to chiral matter in the four-dimensional compactification. One tool that will be used heavily is the duality to heterotic and F-theory compactifications, as obtained recently by Braun and Schafer-Nameki.This project falls within the EPSRC Mathematical Physics research area.