Geometry in String Theory , Geometry in General Relativity

几何在任何量子引力理论中都扮演着重要的角色，比如弦理论。弦理论的最新进展已经证明了以前没有得到太多研究的新型几何空间的重要性。对于通过紧化额外维度的基本粒子的现象学现实模型的弦理论构造，用于固定模的通量的存在，从而确定耦合常数和质量，现在被认为是必不可少的。有了通量，一般的内部几何不再是Calabi-Yau，而是涉及到具有扭转的非kahler流形。然而，非卡勒几何的大部分结构尚不为人所知。在一个相当不同的方向上，最近在理解重力/规范理论（AdS/CFT）在相当丰富的几何背景中的对应关系方面取得了相当大的进展。特别是，我们对sasaki - einstein几何的理解有了很大的提高，现在有许多sasaki - einstein AdS/CFT对偶的无限族，其中对偶的两边都是明确已知的。总的来说，AdS/CFT对应关系预测了共形场论的几何和性质之间的迷人关系。这个项目有两个主要组成部分。PI将研究异弦理论中通量紧化的几何。目标是了解非kahler几何的潜在数学结构，以便回答诸如无质量粒子的数量及其相互作用等重要的物理问题。除了几何分析外，还将应用弦对偶性和弦世界表方法进行研究。此外，PI将进一步发展佐佐木-爱因斯坦几何，应用于AdS/CFT对应和共形场理论。目标包括制定，作为一个精确的数学陈述，从几何到共形场论的映射，并解决Sasaki-Einstein度量的存在性和唯一性的重要问题。另一个目标是描述具有通量的更一般的AdS/CFT背景的几何结构，并研究重整化群流的几何描述。该项目处于弦理论研究的前沿，预计将对标准模型、共形场论和弦理论对偶性之外的物理理解做出重大贡献。这将通过解决最近弦理论发展中出现的一些基础数学问题来实现。预计该计划的结果也将对数学的其他领域产生深远的影响，如微分几何和代数几何。更广泛的影响：该项目的一个核心方面是培养研究生和博士后将现代数学工具应用于理论物理和发展新的数学来解决物理问题。PI在组织研讨会和会议方面有着良好的记录，包括在北京举办的2006年弦和数学物理界年度会议。他还将继续与媒体接触，宣传和教育公众对数学界和物理界的研究。

Geometry plays an important role in any theory of quantum gravity, such as stringtheory. Recent advances in string theory have demonstrated the importance of newtypes of geometrical spaces that previously have not been much studied. For stringtheory constructions of phenomenologically realistic models of elementary particles viacompactifying extra dimensions,the presence of fluxes for fixing moduli, and hencedetermining coupling constants and masses, has now been deemed essential. Withfluxes, the generic internal geometry is no longer Calabi-Yau, but involves non-Kahlermanifolds with torsion. Yet much of the structure of non-Kahler geometries is not known.In a rather different direction, there has been considerable progress recently inunderstanding the gravity/gauge theory (AdS/CFT) correspondence in quite a rich classof geometrical backgrounds. In particular, our understanding of Sasaki-Einsteingeometry has improved greatly, and there are now a number of infinite families ofSasaki-Einstein AdS/CFT duals where both sides of the duality are known explicitly. Ingeneral, the AdS/CFT correspondence predicts a fascinating relationship betweengeometry and properties of conformal field theory. This project has two maincomponents. The PI will investigate the geometry of flux compactifications in heteroticstring theory. The objective is to understand the underlying mathematical structure ofnon-Kahler geometries of interest so that important physical questions like the numberof massless particles and their interactions can be answered. Besides geometricalanalysis, the application of string dualities and string worldsheet methods will be utilizedin the investigations. In addition, the PI will further develop Sasaki-Einstein geometry,with application to the AdS/CFT correspondence and conformal field theory. Goalsinclude formulating, as a precise mathematical statement, the map from geometry toconformal field theory, and addressing the important issue of existence and uniquenessof Sasaki-Einstein metrics. Another goal is to describe the geometrical structuresunderlying more general AdS/CFT backgrounds with fluxes, and also to study thegeometric description of renormlization group flows. The proposed project is at thefrontier of string theory research and is expected to make a substantial contribution tothe understanding of physics beyond the Standard Model, conformal field theory, andstring theory dualities. This will be achieved by addressing some of the foundationalmathematical problems that have arisen in the recent development of string theory. It isexpected that the results of the proposed program will also have a profound impact indifferent areas of mathematics, such as differential geometry and algebraic geometry.Broader Impact: A central aspect of this project is the training of graduate students andpostdoctoral fellows in applying the modern tools of mathematics to theoretical physicsand developing new mathematics to solve physical problems. The PI has a strongrecord in organizing workshops and conferences, including the annual Strings 2006conference in Beijing, for the string and mathematical physics community. He will alsocontinue to reach out to the media to publicize and educate the general public on theresearch of the math and physics communities.