Infinite Dimensional Lie Algebras, Quantum Groups and their Applications

这项研究项目位于代数、几何和数学物理的交界处。现代理论物理学的核心问题之一是构建一个数学上一致的基本相互作用理论。这种构建的范例被称为量子场论。这项研究的一个基本主题是“时空边界”的作用。考虑这一点的一种方法是，通常情况下，时间会在有限的时间间隔内演化。在这种情况下，时空的边界是初始时间和目标时间的空间的并集。有边界时空的量子场论不仅对于高能物理，而且对于低能物理，例如绝缘体和其他表面(边界)具有独特物理性质的物质状态，都具有重要的意义。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。共同的主题是边界对局域量子场理论和统计力学中的局域空间模型的影响。第一部分是关于表征理论的研究。主要目的是通过研究相应量子仿射代数主级数表示的顶点算子，将上理想子代数连接到边界Q-Knizhnik-Zamolodchikov方程；研究对称空间上的可积系统；利用单位根量子sl(2)的R-矩阵缠绕表示，构造补中具有平坦连接的纽结的不变量，并通过手术构造相应的三维流形的不变量。在第二部分中，我们的目标是构造微扰拓扑量子场论，这涉及到构造带边界流形的微扰拓扑量子场论，以及对于秩比1的单李代数，构造q-6J符号的半经典渐近。第三部分的目标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，可以看作是对统计力学中自旋模型半经典极限的研究。其主要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模型等。

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.