Classical and quantum homomorphisms from discrete groups to Lie groups

从离散群到李群的经典和量子同态

• 批准号: 1406559
• 负责人: Francis Bonahon
• 金额: $ 31.4万
• 依托单位: University of Southern California
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-07-01 至 2018-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1406559&HistoricalAwards=false
• 关键词: Classical; quantum; homomorphisms; discrete; groups

The modern formulation of various physical problems involves families of matrices (that is tables of numbers) called Lie groups. This is true of the foundations of classical mechanics, and even more so for general relativity and quantum physics. For this reason much mathematics has been developed in this framework for the past 150 years. In the 1980s, several breakthroughs have emphasized the role of specific Lie groups in various geometric problems. This includes the surprisingly effective use of non-euclidean hyperbolic geometry to analyze the knotting of curves in space. Other developments, drawing their inspiration from quantum physics, have provided tools to attack the same knotting problems based on certain deformations of Lie groups called quantum groups. The Project investigates several problems involving classical Lie groups and their quantum group deformations, and their applications to the study of knots and 3-dimensional spaces. It draws its motivation and technical tools from several different branches of mathematics, including geometry, topology, algebra and dynamical systems. The Project is articulated along two themes that are very different in nature, but united by the fact that hyperbolic geometry can be used as an intellectual guide in each of them. The Project also has a strong educational component, as its research themes are designed so that they can nurture the doctoral work of several graduate students, and provide a broad postdoctoral training to junior faculty in the research group of the Principal Investigator (PI). The first theme of the Project is focused on the classical geometry of homomorphisms from the fundamental group of a surface to a Lie group. When the Lie group is split real, for instance for the special linear group SL(n,R), the so-called Hitchin homomorphisms satisfy many important geometric and dynamical properties. A first goal of the proposal is to develop a differential calculus for the spectrum of the images under Hitchin homomorphisms of simple closed curves on the surface. This includes the development of a parametrization of the space of Hitchin homomorphisms that is well-adapted to such a calculus. The Project will also use these methods to investigate the boundary at infinity of the space of Hitchin homomorphisms. Moving to the complex set-up, the PI will investigate the geometry of homomorphisms valued in complex Lie groups but close to real Hitchin homomorphisms. The second theme of the Project involves quantum group invariants of knots in 3-dimensional manifolds. The PI will continue his investigation of representations of the Kauffman skein algebra on a surface, considered as points of a quantization of the space of homomorphism from the fundamental group of the surface to the Lie group SL(2,C). He will then build on the results and tools developed in this investigation, and on earlier work of Kashaev-Baseilhac-Benedetti, to build a (2+1)-dimensional topological quantum field theory that mixes quantum topology and hyperbolic geometry. The long term goals of this work is to provide conceptual and technical tools to attack the Volume Conjecture, which predicts a precise relationship between the asymptotic behavior of certain quantum invariants of a knot in 3-space and the hyperbolic volume of its complement. The technology provided by the Kauffman skein algebra is more intrinsic than earlier approaches, and should be particularly useful.

各种物理问题的现代表述涉及矩阵家族（即数字表）称为谎言组。对于经典力学的基础，对于一般相对论和量子物理学而言，这是正确的。因此，在过去的150年中，在此框架中已经开发了许多数学。在1980年代，一些突破强调了特定谎言群体在各种几何问题中的作用。这包括令人惊讶的有效使用非欧国双曲线几何形状来分析太空中曲线的打结。其他发展是从量子物理学中汲取灵感的，它提供了基于称为量子组的谎言组的某些变形，以攻击相同的打结问题。该项目研究了涉及经典谎言组及其量子组变形的几个问题，以及它们在结和3维空间的研究中的应用。它从几个不同的数学分支中汲取了动力和技术工具，包括几何，拓扑，代数和动态系统。 该项目沿两个本质上截然不同的主题阐明，但是通过双曲线几何形状可以用作每个主题的智力指南。该项目还具有强大的教育组成部分，因为其研究主题的设计是为了培养几位研究生的博士学位，并向首席研究员（PI）研究小组的初级教师提供广泛的博士后培训。 该项目的第一个主题侧重于从表面的基本组到谎言组的同构的经典几何形状。当Lie组实时拆分时，例如，对于特殊的线性组SL（N，R），所谓的Hitchin同构同态满足许多重要的几何和动力学特性。该提案的第一个目标是为在表面上简单封闭曲线的Hitchin同态下形成图像谱的差分微积分。这包括开发Hitchin同态空间的参数化，该空间适应了这种演算。该项目还将使用这些方法来研究Hitchin同态空间无穷大的边界。转向复杂的设置，PI将研究在复杂的谎言基团中有价值的同构的几何形状，但接近实际的Hitchin同态。 该项目的第二个主题涉及三维流形中的量子组不变。 PI将继续他对表面上的Kauffman绞线代数的表示，被认为是从表面基本组到Lie off sl（2，c）的同态空间量化的点。然后，他将建立在这项研究中开发的结果和工具的基础上，并基于Kashaev-Baseilhac-Benedetti的早期工作，以建立（2+1）维拓扑量子场理论，该理论将量子拓扑结构和双曲线几何形状混合在一起。这项工作的长期目标是提供概念和技术工具来攻击音量猜想，这预测了在三个空间中某些量子不变的渐近行为与补充的双曲线体积之间的精确关系。 Kauffman Skein代数提供的技术比以前的方法更为内在，应该特别有用。