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.

首席研究员：Prakash Belkale 提案编号：0300356 机构：北卡罗来纳大学教堂山分校 摘要：量子上同调、表示论和费曼振幅 首席研究员希望在两个领域进行研究：基本群的表示和量子上同调，以及结构的研究费曼振幅（第二个领域是与加州大学洛杉矶分校的帕特里克·布罗斯南合作）。在第一个领域，PI 希望将他最近的霍恩猜想和饱和猜想的几何证明推广到这些猜想的量子类似物。该项目的灵感来自于射影 n 空间 p1 的基本群的酉表示问题，其中 n 个点通过规定的局部单调被移除。首席研究员计划调查其他群体的 Horn 型递归的存在性，并研究饱和猜想的类似情况。 这项工作的最终目标是确定一组最优的不等式，以解决具有规定的单一性的单一表示的存在问题。在第二个领域（与布罗斯南合作），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。表示论（“对称性”）和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，这在数值计算和波力学中很重要。 关于费曼振幅几何的工作在数学和物理学中都很有趣。目的是更好地从数学角度理解费曼振幅，这是量子理论的基础。