Positive Mass, Singularities, and Supersymmetry

正质量、奇点和超对称性

• 批准号: 0504890
• 负责人: Mark Stern
• 金额: $ 10.8万
• 依托单位: Duke University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-08-01 至 2010-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0504890&HistoricalAwards=false
• 关键词: Positive; Mass; Singularities; Supersymmetry

AbstractAward: DMS-0504890Principal Investigator: Mark A. SternIn collaboration with Anda Degeratu, the principal investigatorwill continue their study of harmonic spinors on manifolds withdegenerating spin structures. Their primary goal is to use thesedegenerating spin structures to extend Witten's technique forproving the positive mass theorem to new classes of higherdimensional nonspin manifolds. They will also investigate theapplication of index theory to these degenerate spin structuresin order to find new relations for the A hat genus on nonspinmanifolds. The principal investigator will also continue hisstudy of a geometric model of B fields and the higher p-formpotentials arising in string theory, which is suitable forunderstanding how these fields affect the application ofgeometric analysis to string theory. He will then apply thisimproved understanding to various questions in string theory,including joint work with Savdeep Sethi on the analysis ofgravitational anomalies of field theories arising from stringtheories on singular spaces.At low speeds and masses, general relativity should reduce toNewton's theory of gravity. This correspondence is confused bythe fact that there is no definition of mass in generalrelativity which allows one to define the mass density at asingle point in space. One can only define the mass of largeregions. Physicists have introduced several notions of mass toobviate this difficulty, the best known of which is the ADMmass. The positive mass conjecture posits that this complicatedADM mass is never negative. This is important because a negativemass universe seems physically unreasonable. Moreover, the truthof the positive mass conjecture has strong implications for thevery stability of space. In lower dimensions (including 4) thisconjecture was proved by Schoen and Yau. Recent work in stringtheory has led to the study of gravity in more than 4 spacetimedimensions and thus interest in higher dimensional positive masstheorems under investigation in this project. Many relatedquestions arising in the gravity theories produced by stringtheory are not currently amenable to careful mathematicalanalysis because the correct mathematical translation of some ofthe important string theory concepts, such as higher p formpotentials, remains unclear. One of the goals of this project isto provide and apply such a translation.

摘要奖项：DMS-0504890 首席研究员：Mark A. Stern 与 Anda Degeratu 合作，首席研究员将继续研究具有简并自旋结构的流形上的谐波旋量。他们的主要目标是利用这些简并自旋结构来扩展维滕的技术，以证明正质量定理到新类别的高维非自旋流形。他们还将研究指数理论在这些简并自旋结构中的应用，以便找到非自旋流形上 A hat 属的新关系。首席研究员还将继续研究 B 场的几何模型和弦理论中出现的更高的 p 型势，这适合于理解这些场如何影响几何分析在弦理论中的应用。然后，他将把这种改进的理解应用于弦理论中的各种问题，包括与 Savdeep Sethi 合作分析由奇异空间弦理论引起的场论引力异常。在低速和质量下，广义相对论应该简化为牛顿的引力理论。这种对应关系因广义相对论中没有质量定义而无法定义空间中单个点的质量密度这一事实而令人困惑。人们只能定义大区域的质量。物理学家引入了几种质量概念来消除这一困难，其中最著名的是 ADM 质量。正质量猜想假设这个复杂的 ADM 质量永远不会是负的。这很重要，因为负质量宇宙在物理上似乎是不合理的。此外，正质量猜想的真实性对于空间的稳定性具有重要意义。在低维（包括4维）这个猜想被Schoen和Yau证明了。 最近弦理论的工作导致了对超过 4 个时空维度的引力的研究，因此对本项目中正在研究的高维正质量定理产生了兴趣。 由弦理论产生的引力理论中出现的许多相关问题目前无法进行仔细的数学分析，因为一些重要的弦理论概念（例如更高的p形式势）的正确数学翻译仍然不清楚。该项目的目标之一是提供并应用这样的翻译。