Infinite Dimensional Lie Algebras, Quantum Groups and their Applications

该提案由三部分组成。第一部分是针对问题的泊松李群和集群代数方面的表示量化的普遍李代数。研究结果将用于构造和研究纽结不变量和三维流形不变量。第二部分的目标是发展经典场论的半经典量子化。它有两个主要方向：第一个是有边界的时空流形的配分函数的定义，第二个是通过配分函数的胶合操作验证量子场论的定域性。这一部分的项目主要集中在拓扑规范理论，特别是陈-西蒙斯理论。第三部分的建议是集中在平衡统计力学的问题。PI计划继续研究相关函数和配分函数的行为，特别是半经典类型的标度极限。这种半经典极限出现在对超对称杨-米尔斯理论中“长算子”的关联函数的研究中。建议的第一部分的重要部分旨在为构造量子拓扑规范理论提供代数工具。粒子物理学的基本问题之一是建立一个理论来解释实验检测到的粒子的动力学，并解释它们的多样性。标准模型就是这种理论的候选者。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，称为规范对称性。这使得量子理论的构建变得极其复杂。严格地说，从数学上讲，这个理论还处于幼年期。该计划的第二部分旨在在拓扑规范理论的一个更简单的情况下解决这些问题，其中动力学要简单得多，但对称性完全相同。该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（估计硬币在N次试验中一侧落下xN次而另一侧落下（1-x）N次的概率）以及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性的，但在分子尺度上是随机的）。PI将继续研究一些二维模型中的类似现象。

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.