Positive Mass, Singularities, and Supersymmetry

摘要奖：DMS-0504890首席研究员：马克·A·斯特恩与Anda Degeratu合作，主要研究员将继续他们对具有简并自旋结构的流形上的简谐自旋的研究。他们的主要目标是利用这些简并的自旋结构将Witten证明正质量定理的技术扩展到新的高维非自旋流形。他们还将研究指数理论在这些简并自旋结构中的应用，以便为非自旋流形上的A-Hat亏格找到新的关系。主要研究人员还将继续研究B场的几何模型和弦理论中出现的更高的p形势，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，他将把改进后的理解应用到弦理论的各种问题上，包括与萨夫迪普·塞西合作分析奇异空间上的弦理论产生的场论的引力反常。在低速和低质量时，广义相对论应该简化为牛顿的引力理论。广义相对论中没有质量的定义，这一事实混淆了这种对应关系，广义相对论允许人们定义空间中某一点的质量密度。人们只能定义大区域的质量。物理学家已经提出了几个质量的概念来避免这一困难，其中最著名的是ADM质量。正质量猜想假定这个复杂的ADM质量永远不会是负的。这一点很重要，因为负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空间的稳定性有很强的影响。在低维(包括4维)上，Schoen和Yau证明了这一猜想。最近弦理论的工作已经导致了在四个以上的空间介质中研究引力，因此对这个项目中正在研究的高维正质量定理感兴趣。由弦理论产生的引力理论中出现的许多相关问题目前还不能进行仔细的数学分析，因为一些重要的弦理论概念的正确数学翻译仍然不清楚，例如更高的p型势。这个项目的目标之一就是提供和应用这样的翻译。

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.