A Quantum Field Theory Approach to the Study of Low-dimensional Topology Invaraints and their Categorification

研究低维拓扑不变量及其分类的量子场论方法

• 批准号: 0509793
• 负责人: Lev Rozansky
• 金额: $ 11.65万
• 依托单位: University of North Carolina at Chapel Hill
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-06-15 至 2008-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0509793&HistoricalAwards=false
• 关键词: Quantum; Field; Theory; Approach; Study

The Jones polynomial invariant of links presents two puzzles. The first one is its interpretation within the framework of classical topology and its relation to the fundamental group of the link complement. This relation is well-known for the Alexander polynomial, but its Jones analog is missing. I search for it by using the quantum field theory approach. I decompose the colored Jones polynomial in the semi-classical limit into a sum of many simpler invariants, which turn out to be multi-variable polynomial invariants of links. These invariants are arranged naturally into a `tower' and the Alexander polynomial lies at its foundation. My goal is to study the properties of the other `higher level' polynomials and establish their relation to the topology of the knot complement. The second puzzle of the Jones polynomial is its polynomial structure, which does not follow easily from its Chern-Simons-Witten path integral presentation. An amazing explanation for the polynomial nature of the Jones polynomial comes from Khovanov's categorification program, which interprets the Jones polynomial as a graded Euler characteristic of a chain complex of graded modules, associated to a link up to a homotopy. I will study the possible quantum field theory interpretations of the Khovanov homology. Namely, the homology of the categorification complex should be a space of states for a 4-dimensional topological quantum field theory, which has to be constructed. This construction should help us to extend Khovanov's results to quantum polynomial invariants of 3-manifolds and to interpret their relation to 4-dimensional topology.The problem of knot classification is very old and although its formulation is simple and transparent, it has not been solved yet despite concerted efforts of many mathematicians. One of the most successful approaches to knot classification is the construction of knot invariants, that is, the numbers, which can be easily computed by looking at a picture of a knot and which would be the same for all pictures representing the same knot. This area has undergone significant developments in the last 30 years starting with the invention of the Jones polynomial. This polynomial and the other invariants which followed it, are intimately related to quantum field theory and can be expressed as so-called path integrals. Thus many of the new ideas in the theory of knot invariants are inspired by methods and approaches of quantum physics. The goal of my research is to use the methods of quantum field theory in order to interpret the new `quantum' topological invariants within the framework of classical topology and to apply them to the solution of the knot classification problem.

链环的琼斯多项式不变量有两个难题。第一个是在经典拓扑学框架内的解释及其与环补的基本群的关系。这个关系在亚历山大多项式中很有名，但它的琼斯类比却不见了。我用量子场论的方法来寻找它。将半经典极限下的有色琼斯多项式分解为若干简单不变量之和，得到了链环的多元多项式不变量。这些不变量被安排成一个自然的“塔”和亚历山大多项式在于其基础。 我的目标是研究其他“高级别”多项式的性质，并建立它们与结补拓扑的关系。琼斯多项式的第二个难题是它的多项式结构，这并不容易从它的陈-西蒙斯-维滕路径积分表示。对琼斯多项式的多项式性质的一个令人惊讶的解释来自霍瓦诺夫的分类程序，该程序将琼斯多项式解释为分次模的链复形的分次欧拉特征，与同伦的链接相关联。我将研究Khovanov同调的可能的量子场论解释。也就是说，分类复合体的同调应该是一个四维拓扑量子场论的态空间，这是必须构造的。这个结构应该有助于我们将Khovanov的结果扩展到3-流形的量子多项式不变量，并解释它们与4维拓扑的关系。结的分类问题是非常古老的，虽然它的公式简单而透明，但尽管许多数学家的共同努力，它仍然没有得到解决。结分类的最成功的方法之一是构造结不变量，即数字，可以通过查看结的图片轻松计算，并且对于表示相同结的所有图片都是相同的。在过去的30年里，这一领域经历了重大的发展，从琼斯多项式的发明开始。这个多项式和它后面的其他不变量与量子场论密切相关，可以表示为所谓的路径积分。因此，纽结不变量理论中的许多新思想都受到量子物理方法和途径的启发。我的研究的目标是使用量子场论的方法，以解释新的“量子”拓扑不变量的经典拓扑结构的框架内，并将其应用到解决的结分类问题。