NonFredholm Index Theory, Matrix Models, and Hodge Theory

摘要提案：DMS 9870161主要研究者：Mark Stern与S.塞西，PI将分析超矩阵模型量子力学，它已被证明提供M理论的定义。 这个猜想的关键是在具有16个超对称的量子力学规范理论中存在可归一化的基态。 这些基态对应于模型中的引力子。 我们试图证明基态波函数的存在性和唯一性，通过计算一个大家庭ofassociated非Fredholm，（广义）狄拉克运营商的L2指数。 我们将分析波函数的结构，以便确定，例如，引力子的“大小”。 在与矩阵模型相关的进一步工作中，与S. Paban和Savdeep Sethi，PI将研究具有16个超对称的杨米尔斯理论中有效作用量的限制。 在数学上，这可以看作是具有16个超对称的系统变形的无模问题。P.I.提出了用射影嵌入引入的度量建立奇异射影簇的L2上同调的调和理论，将他们的调和理论推广到孤立奇点簇。 这样的研究的目的是将L2上同调的研究带入到（和交叉上同调）在这些种类中，有一系列相同的工具可用于研究完整的Kahler流形。这包括Hodge和Lefschetz分解以及潜在的Bochner型消失定理。这个项目的主要目标是探索最近提出的量子引力理论的模型，称为矩阵模型。数学上，这个模型它有许多优点，但它还不完全，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，以确定该模型是否具有所需的物理特性。 例如，我们试图确定它是否预言了非稳定粒子“引力子”的存在。 然后，我们将尝试确定这种粒子的结构，并看看该模型是否给出了一个框架，用于计算诸如引力子散射等简单过程的预测结果。

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.