Teichmuller Theory and Quantum Topology

本课题旨在研究结点和三流形的量子不变量与Teichmuller理论的关系。Teichmuller空间，定义为一个表面上的双曲度量空间，直到同位素，是理解低维流形几何和最终拓扑的关键对象。它的量子化，产生了所谓的量子Teichmuller空间，在数学物理学中找到了它的起源，更具体地说，是在2+1量子引力的研究中。为了更深入地理解这种联系，将发展两种方法。第一种方法将包括研究构造与量子Teichmuller空间表示理论相关的有限维模函子的可能性。如果这种方法是成功的，它将导致构建穿孔表面的映射类群的一系列表示，并最终构建拓扑量子场论。第二种方法将包括研究量子Teichmuller空间与由圆弧和连杆组成的绞结代数之间的关系，这是PI和一位合作者最近开发的。这两种观点之间的可能联系预计类似于SU(2)-TQFT的几何方法和组合方法之间的关系。对三流形的研究本质上是对宇宙可能形状的研究，因此自然处于数学和理论物理的交叉点。在过去的三十年里，这个学科经历了几次重大的突破，其中有两种先验的不相关的方法：一种来自对三流形所允许的可能几何的研究，另一种来自数学物理和量子不变量的概念。近年来，许多结果和猜想旨在调和这两种观点。PI打算进一步研究这些联系，以实现从一种方法到另一种方法的翻译。

This project aims at studying the relationship between Teichmuller theory and quantum invariants of knots and three-manifolds. The Teichmuller space, defined as the space of hyperbolic metrics on a surface up to isotopy, is a key object in understanding the geometry and ultimately the topology of low-dimensional manifolds. Its quantization, giving rise to the so-called quantum Teichmuller space, finds its origin in mathematical physics, more specifically in the study of 2+1 quantum gravity. Two approaches will be developed in understanding this connection more deeply. The first approach will consist in studying the possibility of constructing a finite dimensional modular functor associated to the representation theory of the quantum Teichmuller space. If this approach is successful, it would lead to the construction of a family of representations of the mapping class groups of punctured surfaces and ultimately to the construction of a topological quantum field theory. The second approach will consist in studying the relationship between the quantum Teichmuller space and the skein algebra of arcs and links developed recently by the PI and a collaborator. The possible connections between the two points of view is expected to be similar to the relationship between the geometrical and the combinatorial approach to SU(2)-TQFT.The study of three-manifolds is in essence the study of the possible shapes of the universe and as such sits naturally at the intersection of mathematics and theoretical physics. The subject experienced several major breakthrough in the past thirty years following two a priori unrelated approaches: one coming from the study of the possible geometries a three-manifold admits and the other coming from mathematical physics and the notion of quantum invariants. In recent years, numerous results and conjectures have aimed at reconciling the two points of view. The PI intends on studying these connections further with the goal of translating methods from one to the other.