Bound States, Singularities, and Supersymmetry

摘要DMS - 0204188。主要研究者将寻求具有平凡正则束的奇异变异的“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，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的合作中，PI将继续他的项目来分析超对称矩阵模型量子力学，该模型已经被推测为M理论的定义。研究生G. Firestone，将研究非fredholm狄拉克算子的L2指标理论，该算子的l2核给出了该理论的基态。Sethi和PI将研究计算奇异空间上由弦理论产生的场论引力异常的方法。S. Paban， S. Sethi和PI将研究超对称对与孤立五膜相关的超对称代数变形的约束。PI将研究Witten的旋量技术的扩展，该技术通过用在余维2子流形上退化的自旋结构的假设取代光滑自旋结构的假设来证明正质量猜想。M理论提供了一个潜在的框架来提供一个结合了引力和描述高能粒子的量子物理学的物理理论。这一理论的确切形式尚未得到充分发展。可衡量的后果甚至还远未得到充分考虑。这个项目试图为这个理论的分支确定基本的结果，比如不同类型的粒子的数量，当这些粒子组合成更复杂的物质时，这些粒子的大小和形状以及它们的组合。除了发展M理论和弦理论的物理结果，这样的计算可以用来检查M理论的潜在假设中的潜在错误。该项目还寻求更好地理解某些被称为5膜的“粒子”之间物理相互作用的数学模型，这些粒子的存在是由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.