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New geometry from string dualities

来自弦对偶性的新几何

基本信息

批准号：
EP/V049089/1
负责人：
Daniel Waldram
金额：
$ 25.8万
依托单位：
Imperial College London
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2021
资助国家：
英国
起止时间：
2021 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FV049089%2F1
关键词：
New geometry string dualities

项目摘要

The aim of our proposed research is to develop new ideas in geometry using structures that naturally arise in string theory, which in turn feeds back to advance our understanding of the nature of gravity and particle physics. String theory is a putative quantum theory of gravity, that defines particular, natural extensions of General Relativity, Einstein's description of gravity in terms of curved geometry. These extensions include analogues of the electromagnetic field, collectively known as fluxes. Certain special spacetimes have additional symmetries (known as supersymmetries) that give additional structure to the geometry. Such structures, such as Kahler, Calabi-Yau, Sasaki-Einstein and Joyce manifolds have long been studied by mathematicians. In some cases powerful theorems exist, where the existence of solutions to the differential equations the structures have to satisfy can be translated into a more algebraic condition known as stability. In addition, there can be remarkable duality symmetries between spaces with such structures (notably mirror symmetry of Calabi-Yau manifolds, first discovered in the context of string theory). The theme of this proposal is that both ideas of stability and mirror symmetry have physical interpretations using string theory and furthermore have extensions to a larger class of natural string structures. A key ingredient is the remarkable duality between gravitational theories on certain spacetimes and certain conventional (non-gravitational) quantum field theories, known as the AdS-cft correspondence. We hope to develop these ideas in multiple ways: to ask if one can propose new existence conjectures which ultimately will be important for building string models of particle physics; to understand the relation between stability and the notion of quantum corrections in the dual quantum field theory and the connection to the algebraic structure of field theory defining so-called Calabi-Yau algebras; and to understand extensions of the topological string theories that underlie mirror symmetry.

我们提出的研究目的是利用弦理论中自然产生的结构来发展几何学的新思想，这反过来又促进了我们对引力和粒子物理本质的理解。弦理论是一种假定的量子引力理论，它定义了广义相对论的特殊的、自然的延伸，广义相对论是爱因斯坦用弯曲几何来描述引力的。这些扩展包括电磁场的类似物，统称为通量。某些特殊时空具有额外的对称性（称为超对称性），赋予几何结构额外的结构。数学家们长期以来一直在研究Kahler、Calabi-Yau、Sasaki-Einstein和Joyce流形等结构。在某些情况下，存在强大的定理，其中结构必须满足的微分方程解的存在性可以转化为更代数的条件，称为稳定性。此外，具有这种结构的空间之间可能存在显著的对偶对称性（特别是在弦理论背景下首次发现的Calabi-Yau流形的镜像对称）。这一提议的主题是，稳定性和镜像对称的概念都可以用弦理论进行物理解释，并且可以扩展到更大的自然弦结构类别。一个关键因素是某些时空上的引力理论和某些传统（非引力）量子场论之间显著的对偶性，即AdS-cft对应关系。我们希望以多种方式发展这些想法：询问是否可以提出最终对建立粒子物理的弦模型很重要的新的存在猜想；理解对偶量子场论中稳定性与量子修正概念之间的关系，以及与定义所谓Calabi-Yau代数的场论代数结构的联系；并理解作为镜像对称基础的拓扑弦理论的扩展。