Infinite Dimensional Lie algebras, Quantum Groups and their Applications

无限维李代数、量子群及其应用

基本信息

批准号：
1902226
负责人：
Nicolai Reshetikhin
金额：
$ 18万
依托单位：
University of California-Berkeley
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2019
资助国家：
美国
起止时间：
2019-08-01 至 2023-07-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1902226&HistoricalAwards=false
关键词：
Infinite Dimensional Lie algebras Quantum

项目摘要

The concept behind proposed research can be characterized as the study of mathematical aspects of symmetry properties in quantum theory and statistical mechanics. The first part of the project is focused on the objects which appear in the analysis of models of quantum field theory in two-dimensional space time with special conditions on the boundary. In the second part the PI will study the reasons why some mechanical systems can be solved exactly analytically and the symmetry arguments that go with this. The third and the fifth parts are related to questions involving statistical aspects of symmetry. The fourth part of the project is to connect quantum mechanics and a special two dimensional quantum field theory. This part is expected to bring new insight into now relatively the old subject of quantum mechanics.In the first part of the project the PI proposes to investigate Harish-Chandra series in a broad setting of finite dimensional simple Lie groups, affine Lie groups and their quantum counterparts. The second part can be viewed as a classical counterpart of the first part. It appears that a classical Hamiltonian counterpart of Harish-Chandra theory is a superintegrable system. The third and the fifth parts outline projects at the interface of representation theory and statistical mechanics. One of the specific directions here is the study of of limit shapes in integrable models of statistical mechanics. The fourth part of the project is built of prior results of the PI and the main goal is to give a description of a full semiclassical asymptotic of eigenfunctions of integrable systems in terms of Feynman diagrams for the corresponding Poisson sigma model.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

所提出的研究背后的概念可以描述为量子理论和统计力学中对称性的数学方面的研究。该项目的第一部分重点关注在边界上具有特殊条件的二维时空量子场论模型分析中出现的物体。在第二部分中，PI 将研究某些机械系统可以精确解析求解的原因以及与之相关的对称性论证。第三和第五部分涉及涉及对称性统计方面的问题。该项目的第四部分是将量子力学和特殊的二维量子场论联系起来。这部分预计将为现在相对古老的量子力学学科带来新的见解。在该项目的第一部分中，PI 建议在有限维简单李群、仿射李群及其的广泛设置中研究 Harish-Chandra 级数。量子对应物。第二部分可以被视为第一部分的经典对应部分。 Harish-Chandra 理论的经典哈密顿对应物似乎是一个超可积系统。第三和第五部分概述了表示论和统计力学接口的项目。这里的具体方向之一是统计力学可积模型中极限形状的研究。该项目的第四部分是根据 PI 的先前结果构建的，主要目标是根据相应的泊松西格玛模型的费曼图来描述可积系统本征函数的完整半经典渐近。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。