Geometry in String Theory , Geometry in General Relativity

几何学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太多研究的新型几何空间的重要性。对于弦理论通过压缩额外维度来构造基本粒子的唯象现实模型，固定模的通量的存在，以及由此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，而是包含了带挠的非Kahler流形。然而，非Kahler几何的许多结构是未知的，在一个相当不同的方向上，最近在理解相当丰富的几何背景下的引力/规范理论（AdS/CFT）对应方面取得了相当大的进展。特别是，我们对Sasaki-Einsteingeometry的理解有了很大的提高，现在有许多Sasaki-Einstein AdS/CFT的无限族，其中对偶性的两边都是明确的。总的来说，AdS/CFT对应预言了几何学与共形场论性质之间的迷人关系。该项目有两个主要组成部分。PI将研究heteroticstring理论中通量紧致化的几何。目的是了解潜在的数学结构ofnon-Kahler几何的利益，使重要的物理问题，如数量的无质量粒子和它们的相互作用可以回答。除了几何分析，弦对偶性和弦世界面方法的应用也将在研究中使用。此外，PI将进一步发展Sasaki-Einstein几何，应用于AdS/CFT对应和共形场论。目标包括制定，作为一个精确的数学陈述，从几何映射到共形场论，并解决重要问题的存在性和唯一性的佐佐木-爱因斯坦度量。另一个目标是描述更一般的AdS/CFT背景下的几何结构，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，有望对理解标准模型、共形场论和弦理论对偶之外的物理学做出实质性贡献。这将通过解决弦理论最近发展中出现的一些基础数学问题来实现。预计该项目的成果也将对数学的各个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：该项目的一个中心方面是培养研究生和博士后研究员，将现代数学工具应用于理论物理，并开发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，包括在北京为弦和数学物理社区举办的2006年弦会议。他还将继续接触媒体，宣传和教育公众对数学和物理社区的研究。

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.