Bound States, Singularities, and Supersymmetry

摘要DMS-0204188主要研究人员将寻求具有平凡正则丛的奇异变种的“弦”上同调的解析实现。他将研究这种解析实现(及其推广)的含义，以便为弦理论中的规范对称性增强机制提供新的解释。在与S.Sethi的合作下，PI将继续他的程序来分析超对称矩阵模型量子力学，该模型已经被猜想提供了M理论的定义。研究生G·费尔斯通将研究非FredholmDirac算子的L2指数理论，这些算子的L2核给出了这一理论的基态。Sethi和PI将研究由奇异空间上的弦理论引起的场论引力反常的计算方法。S.Paban，S.Sethi和PI将研究超对称性对与孤立的五膜相关联的超对称代数的形变施加的限制。PI将研究Witten证明正质量猜想的旋量技术的扩展，将光滑自旋结构的假设替换为余维2子流形上自旋结构简并的假设。M理论给出了一个潜在的框架，用于提供一个结合了引力和描述高能粒子的量子物理的物理理论。这一理论的确切形式还没有完全发展起来。可衡量的后果甚至还远未完全计算出来。这个项目试图为这个理论的分支确定基本的结果，例如当这些粒子结合成更复杂的物质时，不同类型的粒子的数量，以及这些粒子及其组合的大小和形状。除了发展M和弦理论的物理结果外，这样的计算还可以用来检查M理论基本假设中的潜在错误。这个项目还寻求更好地理解一个数学模型，该模型描述了被称为5膜的某些“粒子”之间的物理相互作用，它的存在是由M理论预测的。

ABSTRACT DMS - 0204188.The principal investigator will seek analytic realizations of "stringy"cohomology of singular varieties with trivial canonical bundle. He will studythe implications of such analytic realizations (and their generalizations)towards providing new explanations of gauge symmetry enhancement mechanisms instring theory. In collaboration with S. Sethi, the PI will continue his programto analyze the supersymmetric matrix model quantum mechanics which has been conjectured to provide a definition of M theory. Graduate student G. Firestone, will study the L2 index theory for the nonFredholm Dirac operators whose L2kernels give the ground states for this theory. Sethi and the PI will examinemethods for computing gravitational anomalies of field theories arising fromstring theories on singular spaces. S. Paban, S. Sethi, and the PI will examinethe constraints imposed by supersymmetry on deformations of the supersymmetryalgebra associated to an isolated five brane. The PI will investigate theextension of Witten's spinor technique of proving the positive mass conjecture by replacing the hypothesis of a smooth spin structure by the assumption of a spin structure degenerating on a codimension 2 submanifold.M theory gives a potential framework for providing a physical theory whichincorporates both gravitation and the quantum physics describing high energy particles. The precise form of this theory is not yet fully developed. The measurable consequences are even further from being fully worked out. This project attempts to determine, for branches of this theory, fundamental consequences, such as the number of particles of different types, when these particles combine to make more complex matter, and the size and shape of these particles and their combinations.In addition to developing the physical consequences of M and string theory,such computations can be used to check for potential errors in the underlying hypotheses of M theory. This project also seeks a better understanding of amathematical model for the physical interactions between certain "particles" known as 5 branes, whose existence is predicted by M theory.