Infinite Dimensional Lie Algebras, Quantum Groups and their Applications

无限维李代数、量子群及其应用

• 批准号: 1601947
• 负责人: Nicolai Reshetikhin
• 金额: $ 16.2万
• 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: 2016
• 资助国家: 美国
• 起止时间: 2016-09-01 至 2019-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1601947&HistoricalAwards=false
• 关键词: Infinite; Dimensional; Lie; Algebras; Quantum

This research project lies at the interface of algebra, geometry, and mathematical physics. One of the central problems in modern theoretical physics is to construct a mathematically consistent theory of fundamental interactions. The paradigm for this construction is known as quantum field theory. An underlying theme of the research is the role of the boundary of "space-time." One way to think about this is the usual evolution of time over a finite interval. In this case the boundary of space-time is the union of the space at the initial time and at the target time. Quantum field theory on a space-time with boundary is important not only for high energy physics, but also for low energy physics, for example insulators and other states of matter where the surface (the boundary) has distinguishing physical properties. This project will develop algebraic tools to study two and three dimensional quantum field theories. It will also construct quantum field theories with infinite dimensional symmetries.The proposal consists of three parts. The common theme is the effect of the boundary on local quantum field theory and on local spacial models in statistical mechanics. The first part is focused on representation theory. The main goals are to connect coideal subalgebras to boundary q-Knizhnik-Zamolodchikov equation by studying vertex operators for principal series representations of the corresponding quantum affine algebras; to study integrable systems on symmetric spaces; and to construct invariants of knots with flat connections in the complement using R-matrix intertwining representations of quantum sl(2) at roots of unity and construct the corresponding invariants of 3-manifolds via surgery. In the second part the goals are to construct perturbative topological quantum field theories, which involves constructing perturbative topoplogical quantum field theory for manifolds with boundary; and to construct semiclassical asymptotic for q-6j symbols for simple Lie algebras of rank grater then one. The goal of the third part is the study of limit shapes in statistical mechanics. This part is closely related to the second part and can be regarded as the study of the semiclassical limit of spin models in statistical mechanics. The main concept is that the thermodynamical limit is similar to the semiclassical limit. The models in question involve the stochastic 6-vertex model, ASEP models and others.

本研究项目是代数、几何和数学物理的交叉领域。现代理论物理学的中心问题之一是建立一个数学上一致的基本相互作用理论。这种构造的范式被称为量子场论。这项研究的一个潜在主题是“时空”边界的作用。考虑这个问题的一种方法是时间在有限区间内的演化。在这种情况下，时空的边界是空间在初始时间和目标时间的并集。具有边界的时空的量子场论不仅对高能物理很重要，而且对低能物理也很重要，例如绝缘体和其他表面（边界）具有不同物理性质的物质状态。这个项目将开发代数工具来研究二维和三维量子场论。它还将构建具有无限维对称性的量子场论。该提案由三部分组成。共同的主题是边界对局部量子场论和统计力学中局部空间模型的影响。第一部分主要介绍表征理论。主要目的是通过研究量子仿射代数的主级数表示的顶点算子，将共理想子代数与边界q-Knizhnik-Zamolodchikov方程连接起来；研究对称空间上的可积系统；利用量子sl(2)在单位根处的r -矩阵交织表示构造补中具有平连接的结的不变量，并通过手术构造3流形的相应不变量。第二部分的目标是构造微扰拓扑量子场理论，包括构造具有边界的流形的微扰拓扑量子场理论；构造秩大于1的简单李代数的q-6j符号的半经典渐近。第三部分的目的是研究统计力学中的极限形状。这一部分与第二部分密切相关，可以看作是统计力学中自旋模型的半经典极限的研究。主要的概念是，热力学极限类似于半经典极限。所讨论的模型包括随机六顶点模型、ASEP模型和其他模型。