NonFredholm Index Theory, Matrix Models, and Hodge Theory

非Fredholm指数理论、矩阵模型和霍奇理论

• 批准号: 9870161
• 负责人: Mark Stern
• 金额: $ 6.61万
• 依托单位: Duke University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1998
• 资助国家: 美国
• 起止时间: 1998-07-15 至 2003-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9870161&HistoricalAwards=false
• 关键词: NonFredholm; Index; Theory; Matrix; Models

AbstractProposal: DMS 9870161Principal Investigator: Mark Stern In collaboration with S. Sethi, the PI will analyze the supersymmetricmatrix model quantum mechanics which has been conjectured to provide adefinition of M theory. Crucial to the conjecture is the existence ofnormalizable ground states in quantum mechanical gauge theories with16 supersymmetries. These ground states correspond to gravitons inthe model. We seek to show existence and uniqueness of a groundstatewavefunction, by computing an L2 index for a large family ofassociated non Fredholm, (generalized) Dirac operators. We willanalyze the structure of the wavefunction in order to determine, forexample, the "size" of the graviton. In further work related to thematrix model, in collaboration with S. Paban and Savdeep Sethi, the PIwill study constraints on the effective action in Yang Mills theorieswith 16 supersymmetries. Mathematically, this may be viewed as amoduli problem for deformations of systems with 16 supersymmetries.In collaboration with W. Pardon, the PI proposes to establish aharmonic theory for the L2 cohomology of singular projective varietieswith the metric induced by a projective embedding, extending theirharmonic theory for varieties with isolated singularities. The goalof such an investigation is to bring to the study of the L2 cohomology(and conjecturally the intersection cohomology) of these varieties thesame array of tools available for studying complete Kahler manifolds.This includes Hodge and Lefschetz decompositions and potentiallyBochner type vanishing theorems.The primary goal of this project is to explore a recently proposedmodel for a theory of quantum gravity known as the matrix model.Mathematically this model has many advantages, but it is not yetcomplete and is not yet clear that it gives a physically reasonabletheory. We will perform various (mathematical) tests to determine ifthe model does have the conjectured desired physical properties. Forexample, we seek to determine whether it predicts the existence of astable particle, "the graviton". Then we will try to determine thestructure of this particle and see if the model gives a framework forcomputing predicted outcomes to simple processes such as scattering ofgravitons.

在与S. Sethi的合作下，PI将分析量子力学的超对称矩阵模型，该模型已经被推测提供M理论的定义。这一猜想的关键在于具有16个超对称性的量子力学规范理论中可归一化基态的存在。这些基态对应于模型中的重子。通过计算一大族相关的非Fredholm（广义）Dirac算子的L2指标，我们试图证明基态波函数的存在性和唯一性。我们将分析波函数的结构，以确定，例如，引力子的“大小”。在进一步与矩阵模型相关的工作中，与S. Paban和Savdeep Sethi合作，pi将研究具有16个超对称的杨米尔斯理论中有效作用的约束。在数学上，这可以看作是16个超对称系统变形的模量问题。在与W. Pardon的合作中，PI提出了奇异射影变体与由射影嵌入诱导的度量的L2上同调的非调和理论，扩展了它们对于具有孤立奇异点的变体的调和理论。这样一项研究的目的是为研究这些变体的L2上同调（以及推测的交上同调）带来与研究完全Kahler流形相同的工具阵列。这包括Hodge和Lefschetz分解和潜在的bochner型消失定理。该项目的主要目标是探索最近提出的量子引力理论模型，即矩阵模型。这个模型在数学上有许多优点，但它还不完整，也不清楚它是否给出了一个物理上合理的理论。我们将进行各种（数学）测试，以确定模型是否确实具有推测所需的物理性质。例如，我们试图确定它是否预测了不稳定粒子“引力子”的存在。然后，我们将尝试确定这种粒子的结构，并看看该模型是否为计算简单过程（如引力子散射）的预测结果提供了一个框架。