Infinite Dimensional Lie Algebras, Quantum Groups and their Applications

该提案由三部分组成。第一部分讨论了量子化泛李代数表示的泊松李群和聚类代数方面的问题。本研究的结果将用于构造和研究结点的不变量和3流形的不变量。第二部分的目标是发展经典场论的半经典量子化。它主要有两个方向：一是有边界时空流形配分函数的定义，二是通过配分函数的胶合运算验证量子场论的局部性。项目的这一部分主要集中在拓扑规范理论上，特别是chen - simons理论。提案的第三部分集中讨论平衡统计力学中的问题。PI计划继续研究具有固定边界条件的状态模型的相关函数和配分函数的行为，特别是半经典类型的缩放限制。这种半经典极限出现在超对称杨-米尔斯理论中对“长算子”相关函数的研究中。提案第一部分的重要部分旨在提供构建量子拓扑规范理论的代数工具。粒子物理学的一个基本问题是建立一个理论来解释实验检测到的粒子的动力学和解释它们的变化。标准模型是这种理论的候选。这个理论的一个关键方面是，它的经典对应物具有无限维的内部对称性，称为规范对称。这使得量子理论的构建异常复杂。严格地说，这个理论在数学上还处于起步阶段。该建议的第二部分旨在解决拓扑规范理论的一个更简单的情况下的这些问题，其中动力学要简单得多，但对称性是完全相同的。提案的最后一部分侧重于研究与概率论中的大偏差非常相似的现象（在N次大N试验中，估计硬币在一侧落下xN次，在另一侧落下（1-x）N次的概率）和流体力学的随机起源（我们知道水的运动在大尺度上是确定的，但在分子尺度上是随机的）。PI将继续在一些二维模型中研究类似的现象。

The proposal is composed of three parts. The first part is aimed at problems in Poisson Lie groups and cluster algebra aspects of representations of quantized universal Lie algebras. The results of this investigation will be used to construct and study invariants of knots and invariants of 3-manifolds. The goal of the second part is the development of the semiclassical quantization of classical field theories. It has two major directions: the first one is the definition of the partition function for space time manifolds with boundaries, the second one is the verification of locality of quantum field theory through the gluing operation for partition functions. This portion of the project is focused mainly on topological gauge theories, in particular on the Chern-Simons theory. The third part of the proposal is focused on problems in equilibrium statistical mechanics. The PI plans to continue to investigate the behavior of correlation functions and of the partition functions for models with fixed boundary conditions on states and, in particular, scaling limits of semiclassical type. Such semiclassical limits appear in the study of correlation functions of `long operators" in the supersymmetric Yang-Mills theory.Significant portion of the first part of the proposal is aimed at providing algebraic tools for constructing quantum topological gauge theories. One of the fundamental problems in particle physics is to construct a theory which explains the dynamics of experimentally detected particles and would explain their variety. The Standard Model is a candidate for such theory. One of the key aspects of this theory is that its classical counterpart has infinite dimensional internal symmetry known as gauge symmetry. This makes the construction of the quantum theory incredibly complicated. Strictly speaking, mathematically the theory is still in its childhood. The second part of the proposal is aimed at resolving such problems in a simpler case of topological gauge theory, where the dynamics is much simpler but the symmetry is exactly the same. The last part of the proposal focuses on the study of phenomena very similar to large deviations in probability theory (estimate the probability that a coin will fall xN times on one side and (1-x)N times on the other side in N trials for large N) and to the stochastic origins of hydrodynamics (we know that the motion of water is deterministic at large scale, but is random at the molecular scale). The PI will continue to study similar phenomena in a number of two dimensional models.