Positive Mass, Singularities, and Supersymmetry

摘要奖：DMS-0504890主要研究者：Mark A. Stern在与Anda Degeratu的合作中，主要的研究者将继续他们对具有退化自旋结构的流形上的调和旋量的研究。他们的主要目标是利用这些生成自旋的结构，将维滕的正质量定理的证明方法推广到新的高维非自旋流形上。他们还将研究指数理论在这些简并自旋结构中的应用，以找到非自旋流形上A帽亏格的新关系。首席研究员也将继续他的研究几何模型的B领域和更高的p-形式的潜力出现在弦理论，这是适合于了解这些领域如何影响应用ofgeometric分析弦理论。然后，他将把这种改进的理解应用到弦理论的各种问题中，包括与萨夫迪普·塞西（Savdeep Sethi）合作分析奇异空间上的弦理论所产生的场论的引力异常。在低速和低质量下，广义相对论应该简化为牛顿的引力理论。这种对应关系被这样一个事实所混淆，即广义相对论中没有质量的定义，它允许人们定义空间中一个点的质量密度。人们只能定义大区域的质量。物理学家已经引入了几个质量的概念来克服这个困难，其中最著名的是ADMmass。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，因为一个负质量的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）中，Schoen和Yau证明了这个猜想。 弦理论的最新工作导致了对4个以上时空中引力的研究，因此对本项目正在研究的高维正质量定理产生了兴趣。 弦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理论分析，因为一些重要的弦论概念的正确数学翻译，如更高的p型势，仍然不清楚。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.