Quantum Cohomology, Representation Theory, and Feynman Amplitudes

摘要：量子上同调、表示理论和费曼振幅主要研究方向为：基本群和量子上同调的表示，以及费曼振幅的结构研究（第二个方向是与加州大学洛杉矶分校的Patrick Brosnan合作）。在第一个领域，PI想把他最近对霍恩猜想和饱和度猜想的几何证明推广到这些猜想的量子类比。这个项目的灵感来自于射影n空间的基本群p1的幺正表示问题，其中n个点被规定的局部单点移除。首席研究员计划调查其他群的Horn型递归的存在性，并研究饱和猜想的类似物。这项工作的最终目标是确定一个最优不等式集的存在问题的酉表示与规定的单。在第二个领域（与Brosnan合作），首席研究员将继续研究费曼振幅与代数几何之间的关系。这项工作的第一步是用一般的代数几何术语来理解物理学家的积分计算。表征理论（“对称”）与代数几何之间关系的研究是一个非常重要的研究领域。这项工作的部分动机来自特征值问题，这在数值计算和波动力学中很重要。关于费曼振幅几何的研究在数学和物理学中都很有趣。其目的是更好地从数学上理解费曼振幅，这是量子理论的基础。

Principal Investigator: Prakash BelkaleProposal Number: 0300356Institution: University of North Carolina at Chapel HillAbstract: Quantum Cohomology, Representation theory and Feynman AmplitudesThe principal investigator wants to pursue research in two areas: Representations of the fundamental group and Quantum Cohomology, and the study of the structure of Feynman amplitudes (the second area is in collaboration with Patrick Brosnan of UCLA). In the first field, the PI wants to generalise his recent geometric proof of Horn and Saturation conjectures to the Quantum analogues of these conjectures. This project is inspired by the the problem of unitary representations of the fundamental group of p1 of projective n-space with n points removed with prescribed local monodromies. The principal investigator plans to investigate the existence of Horn type recursion for other groups and study analogues of the Saturation conjecture. A final goal of this work is to determine an optimal set of inequalities for the problem of existence of unitary representations with prescribed monodromies. In the second field (in collaboration with Brosnan), the principal investigator will continue the study of relations between Feynman amplitudes and algebraic geometry. The first step of this work is to understand in general Algebro-geometric terms the integral computations of the physicists.The study of relations between Representation theory (`symmetries') and Algebraic Geometry is a very important area of research. Part of the motivation for this work comes from eigenvalue problems, which are important in numerical computing and in wave mechanics. The work on the geometry of Feynman amplitudes is of interest both in mathematics and physics. The aim is a better mathematical understanding of Feynman amplitudes, which are fundamental to the quantum theory.