Infinite Dimensional Lie algebras, Quantum Groups and their Applications

拟议研究背后的概念可以被描述为量子理论和统计力学中对称性的数学方面的研究。该项目的第一部分是集中在对象出现在二维时空与边界上的特殊条件的量子场论模型的分析。在第二部分中，PI将研究为什么某些力学系统可以精确地解析求解的原因以及与此相关的对称性参数。第三和第五部分涉及对称性的统计方面的问题。第四部分是将量子力学与一个特殊的二维量子场论联系起来。这一部分有望为量子力学这个相对古老的学科带来新的见解。在项目的第一部分中，PI提出在有限维简单李群、仿射李群及其量子对应物的广泛背景下研究Harish-Chandra级数。第二部分可以被看作是第一部分的经典对应部分。看来，一个经典的哈密尔顿对应的哈里什-钱德拉理论是一个超可积系统。第三和第五部分概述了表征理论和统计力学的接口项目。这里的一个具体方向是研究统计力学可积模型中的极限形状。该项目的第四部分是建立在PI的先前结果和主要目标是给出一个完整的半经典渐近的可积系统的特征函数的Feynman图为相应的泊松sigma模型的描述。这个奖项反映了NSF的法定使命，并已被认为是值得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的支持。

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.