Quantum Cohomology, Representation Theory, and Feynman Amplitudes

量子上同调、表示论和费曼振幅

• 批准号: 0300356
• 负责人: Prakash Belkale
• 金额: $ 10万
• 依托单位: University of North Carolina at Chapel Hill
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2003
• 资助国家: 美国
• 起止时间: 2003-05-01 至 2007-04-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0300356&HistoricalAwards=false
• 关键词: Quantum; Cohomology; Representation; Theory; Feynman

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.

主要研究者：Prakash Belkale提案编号：0300356机构：北卡罗来纳州教堂山大学摘要：量子上同调，表示论和费曼振幅主要研究者希望在两个领域进行研究：基本群和量子上同调的表示，以及费曼振幅结构的研究（第二个领域是与加州大学洛杉矶分校的帕特里克布鲁斯南合作）。在第一个领域中，PI希望将他最近对Horn和Saturation的几何证明推广到这些几何的量子类似物。这个项目的灵感来自于射影n-空间的p1的基本群的酉表示问题，其中n个点被指定的局部单值性去除。主要研究人员计划调查其他群体的霍恩型递归的存在性，并研究类似的饱和猜想。 这项工作的最终目标是确定一个最佳的一组不等式的问题存在的酉表示与规定的monodromies。在第二个领域（与布鲁斯南合作），主要研究员将继续研究费曼振幅和代数几何之间的关系。这项工作的第一步是理解一般代数几何术语的积分计算的物理学家。研究之间的关系表示论（“对称性”）和代数几何是一个非常重要的研究领域。这项工作的部分动机来自本征值问题，这是重要的数值计算和波动力学。 费曼振幅的几何学研究在数学和物理学中都很有意义。目的是更好地数学理解费曼振幅，这是量子理论的基础。