The Quantization of the Moduli Space of Flat Connections on a Surface

曲面上平面连接模空间的量化

• 批准号: 0604694
• 负责人: Razvan Gelca
• 金额: $ 10.56万
• 依托单位: Texas Tech University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-07-01 至 2012-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0604694&HistoricalAwards=false
• 关键词: Quantization; Moduli; Space; Flat; Connections

The present research lies on the boundary between low dimensional topology and quantum physics. It aims to construct a geometric model for the quantization of the moduli space of flat SU(2)-connections on a surface, with the Lagrangian given by the Chern-Simons functional. This would yield a new method for studying the Jones polynomial of a knot, a fundamental topological invariant, with the tools of quantum mechanics. The main problem is to understand the observables of the quantum mechanical system associated with the Jones polynomial. The quantum physical model in discussion is currently applied to the study of the behaviour of electrons in a strong magnetic field at low temperatures. Physicists use the Chern-Simons functional to couple a magnetic charge to the electric charge of the electron and transform it formally into a particle whose physics is easier to describe. It is believed that the behaviour of electrons is sophisticated enough so that it can be used for writing the code of a quantum computer. We are concerned with the mathematical aspects of this theory.

目前的研究处于低维拓扑和量子物理之间的边界上。利用Chern-Simons泛函给出的拉格朗日函数，构造了曲面上平坦SU(2)-连通模空间量子化的几何模型。这将产生一种新的方法，用量子力学的工具来研究纽结的琼斯多项式，这是一个基本的拓扑不变量。主要的问题是理解与琼斯多项式相关的量子力学系统的可观性。所讨论的量子物理模型目前被应用于研究低温强磁场中电子的行为。物理学家使用Chern-Simons泛函将磁电荷耦合到电子的电荷上，并将其正式转化为物理更容易描述的粒子。人们相信，电子的行为足够复杂，可以用来编写量子计算机的代码。我们关心这一理论的数学方面。