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

目前的研究主要集中在低维拓扑与量子物理的边界上。它的目的是建立一个几何模型的量子化的平坦SU（2）-连接的模空间的表面上，与拉格朗日给出的Chern-Simons功能。这将产生一个新的方法来研究琼斯多项式的一个结，一个基本的拓扑不变量，与量子力学的工具。主要问题是理解与琼斯多项式相关的量子力学系统的可观测量。 所讨论的量子物理模型目前被应用于研究低温下强磁场中电子的行为。物理学家使用陈-西蒙斯泛函将磁荷耦合到电子的电荷，并将其形式上转化为物理学更容易描述的粒子。人们认为电子的行为足够复杂，可以用来编写量子计算机的代码。我们关心的是这个理论的数学方面。

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.