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

$G_2$ 上的 M 理论的几何和物理 -完整流形

• 批准号: 1942180
• 金额: --
• 依托单位: University of Oxford
• 依托单位国家: 英国
• 项目类别: Studentship
• 财政年份: 2017
• 资助国家: 英国
• 起止时间: 2017 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=studentship-1942180
• 关键词: Geometry; Physics; theory; G_2; Holonomy

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.

目的：刻画了G_2流形上的M理论紧化，并利用它们得到唯象有趣的四维规范理论.研究方法的新奇：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意义以及与规范理论模空间的联系。弦论和M理论是在完全相容的量子理论中研究规范理论和引力的独特框架。然而，激发这些理论的许多最初的问题仍然没有答案：4d物理学的相关性是什么？近年来，出现了一种系统地描述弦理论真空的方法。一般来说，4d物理学的性质被编码在所谓的紧化几何中：弦/M理论被定义在更高的维度（10或11）中，为了获得4d理论，其余的维度被扩展到紧几何中。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，以及最重要的对称性。一个关键的属性，这确保了这个框架是强大的对小波动是超对称性的存在，在所产生的4d理论。超对称性，一个理论的玻色子场和费米子场之间的对称性，具有深刻的含义，因为它对紧化几何的完整性施加了约束。在M理论的情况下，这是一个11维的理论，紧化几何必须是7维的，并且为了保持超对称性，必须有约化的完整性G_2（而不是SO（7））。该项目给出了纯数学，特别是微分几何的结果之间的联系，以及数学物理中的应用，如弦理论和场论。这个项目是及时的，因为最近数学家们构造了一大类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有待于揭示。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，因为这些极限在M理论的应用中至关重要。我们的目标是修改扭曲连接和结构，包括奇异性，从而产生手征物质的四维紧化。一个工具，将大量使用的是对偶杂种优势和F-理论紧化，最近获得的布劳恩和Schafer-Nameki.This项目福尔斯属于EPSRC数学物理研究领域。