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。弦理论的最新进展证明了以前没有太多研究的新型几何空间的重要性。对于通过压缩额外维度建立现象学上现实的基本粒子模型的弦理论而言，确定模数的通量的存在，并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，而是涉及到具有扭转的非Kahler流形。然而，非Kahler几何的大部分结构尚不清楚。在一个相当不同的方向上，最近在相当丰富的几何背景下理解引力/规范理论(ADS/CFT)的对应关系方面取得了相当大的进展。特别是，我们对佐佐木-爱因斯坦计量学的理解有了很大的提高，现在有许多佐佐木-爱因斯坦ADS/CFT对偶的无穷族，其中对偶的两边都是明确的。总体而言，ADS/CFT对应预言了几何和共形场论性质之间的有趣关系。这个项目有两个主要组成部分。PI将研究异质弦理论中的磁通紧凑化的几何。其目的是了解感兴趣的非卡勒几何的基本数学结构，以便回答重要的物理问题，如无质量粒子的数量及其相互作用。除了几何分析外，还将利用弦对偶和弦世界表方法进行调查。此外，PI还将进一步发展佐佐木-爱因斯坦几何学，并将其应用于ADS/CFT对应和保形场理论。目标包括，作为一种精确的数学表述，建立从几何到共形场论的映射，并解决佐佐木-爱因斯坦度规的存在和唯一性这一重要问题。另一个目标是用通量描述更一般的ADS/CFT背景下的几何结构，并研究重整化群流的几何描述。该项目是弦理论研究的前沿，有望为理解超越标准模型、共形场理论和弦理论对偶的物理学做出重大贡献。这将通过解决最近弦理论发展中出现的一些基本数学问题来实现。预计该计划的结果还将对不同的数学领域产生深远的影响，如微分几何和代数几何。广泛的影响：该计划的一个核心方面是培训研究生和博士后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。国际弦协会在为弦和数学物理界组织研讨会和会议方面有着良好的记录，包括在北京举行的年度Strings 2006会议。他还将继续接触媒体，宣传和教育公众关于数学和物理界的研究。

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.