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

连杆的琼斯多项式不变量提出了两个难题。第一部分是在经典拓扑框架下的解释及其与链补基群的关系。这个关系因亚历山大多项式而闻名，但它的琼斯类比却缺失了。我用量子场论的方法来寻找它。我将半经典极限中的有色琼斯多项式分解为许多简单不变量的和，这些不变量是连杆的多变量多项式不变量。这些不变量自然地排列成一个“塔”，亚历山大多项式是它的基础。我的目标是研究其他“高级”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，从它的chen - simons - witten路径积分表示中很难推导出来。一个关于琼斯多项式多项式性质的惊人解释来自Khovanov的分类程序，该程序将琼斯多项式解释为一个与同伦相连的链式复链的梯度欧拉特征。我将研究对霍万诺夫同调可能的量子场论解释。也就是说，对一个四维拓扑量子场论来说，范畴复合体的同调应该是一个状态空间，它必须被构造。这种构造有助于我们将Khovanov的结果推广到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。结的分类问题由来已久，虽然它的公式简单明了，但在许多数学家的共同努力下仍未得到解决。结分类最成功的方法之一是结不变量的构造，也就是说，这些数字可以很容易地通过查看结的图片来计算，并且对于表示同一结的所有图片都是相同的。从琼斯多项式的发明开始，这个领域在过去的30年里经历了重大的发展。这个多项式和紧随其后的其他不变量，与量子场论密切相关，可以用所谓的路径积分来表示。因此，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。我的研究目标是利用量子场论的方法，在经典拓扑的框架内解释新的“量子”拓扑不变量，并将其应用于结分类问题的解决。

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.