New geometry from string dualities

我们提出的研究的目的是利用弦论中自然产生的结构来发展几何学的新思想，这反过来又会促进我们对引力和粒子物理学本质的理解。弦论是一种假定的引力量子理论，它定义了广义相对论的特殊自然扩展，爱因斯坦用弯曲几何描述引力。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），为几何提供了额外的结构。这类结构，如Kahler流形、Calabi-Yau流形、Sasaki-Einstein流形和Joyce流形，早已被数学家研究。在某些情况下，强有力的定理存在，其中结构必须满足的微分方程的解的存在可以转化为称为稳定性的更代数条件。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（特别是在弦理论中首次发现的卡-丘流形的镜像对称性）。这个提议的主题是，稳定性和镜像对称的概念都可以用弦论来进行物理解释，而且还可以扩展到更大类的自然弦结构。其中一个关键因素是某些时空上的引力理论与某些常规（非引力）量子场论之间的显著对偶性，即AdS-cft对应。我们希望以多种方式发展这些想法：询问是否可以提出新的存在假设，这些假设最终将对建立粒子物理学的弦模型很重要;理解对偶量子场论中稳定性和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之间的联系;并理解镜像对称基础上的拓扑弦理论的扩展。

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.