Geometry, dualities and the string landscape

弦理论是一种假定的量子引力理论，它也有可能对构成我们宇宙中物质和力的基本组成部分的粒子和相互作用给出统一的描述。此外，弦理论有一组显著的对偶对称性。因为根据广义相对论，引力本质上是一种几何理论，这些对偶性对数学有重要的影响。这个项目的目的是把弦理论的两个重要的最新发展结合起来。广义地说，第一个是在黎曼几何的弦论推广中出现的一组新的几何结构概念，它统一了弦论的自由度。第二个是对“弦景观”的研究，即弦理论中出现的粒子物理低能理论的一般问题。由于超对称弦理论（以及它们的对偶，被称为“m理论”）描述的是十维或十一维的几何形状，为了实现我们自己的四维宇宙，我们必须假设时空的一部分是一个小的六维或七维紧致空间。这个“紧化空间”的几何形状强烈地影响着低能理论的性质。因此，管柱景观规划的一个核心问题是了解可能的紧致几何形状的范围及其性质。弦几何的新发展意味着我们现在有了更完整的背景，而且，至关重要的是，我们有了新的工具来研究弦的“模量”（只需要很少或不需要能量的几何变形）等特性。发展我们对这些几何形状的理解，与二元性的联系以及对串状景观方案的影响和见解是本提案的中心目标。本研究的中长期目标是对量子引力的本质、弦理论的基本对称性有新的认识，并有可能找到现实的引力和量子力学统一理论。在数学中也有潜在的新的和有趣的含义，特别是弦驱动的特殊完整和代数几何概念的扩展。

String theory is a putative theory of quantum gravity that also has the potential to give a unified description of the particles and interactions that form the fundamental building blocks of the matter and forces in our universe. In addition, string theory has a remarkable set of duality symmetries.Since according to general relativity gravity is intrinsically a theory of geometry, these dualities have had important implications for mathematics. This project aims to bring together two important recent developments in string theory. Broadly, the first is a set of new notions of geometrical structure that appears in the string theory generalisation of Riemannian geometry and unifies the string theory degrees of freedom. The second is the study of the "string landscape", that is, the general question of which low-energy theories of particle physics can arise in string theory. Since supersymmetric string theories (and their duals, known as "M-theory") describe ten- or eleven-dimensional geometries, one must assume that part of spacetime is a small six- or seven-dimensional compact space in order to realise our own four-dimensional universe. The geometry of this "compactification space" strongly influences the nature of the low-energy theory. Thus, a central question in the string landscape programme is to understand the range of possible compact geometries, and their properties. The new developments in string geometry mean that we now have access to a much more complete set of backgrounds, and, crucially, have new tools for studying properties such as their "moduli" (geometrical deformations that cost little or no energy). Developing our understanding of these geometries, the connections to duality and the implications for and insights from the string landscape programme are the central goals of this proposal. The medium- to long-term goals of this research are gain new understanding of the nature of quantum gravity, the fundamental symmetries of string theory, and the possibility of finding realistic unified theories of gravity and quantum mechanics. There are also potentially new and intriguing implications for mathematics, notably string-motivated extension of notions of special holonomy and algebraic geometry.