Categorification and Topological Quantum Field Theories

分类和拓扑量子场论

• 批准号: 0808974
• 负责人: Lev Rozansky
• 金额: $ 14.46万
• 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: 2008
• 资助国家: 美国
• 起止时间: 2008-06-15 至 2013-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0808974&HistoricalAwards=false
• 关键词: Categorification; Topological; Quantum; Field; Theories

One of the most important developments in 3-dimensional topology over the last 25 years was the discovery of the so-called quantum invariants of links and 3-manifolds such as the Jones, HOMFLYPT and Kauffman polynomials and the Reshetikhin-Turaev invariants. These invariants were described combinatorially, their relation to classical 3-dimensional topology was mysterious, and the only conceptual explanation for their existence came from physics: Witten constructed a family of topological quantum field theories, whose path integral partition functions were conjectured to coincide with quantum invariants. Recently, Khovanov, Ozsvath ans Szabo discovered the categorification of polynomial link invariants: to a link one associates a chain complex of graded vector spaces, whose graded Euler characteristic coincides with a given polynomial invariant. Also, to a cobordism between two links one associates a chain map between the link categorification complexes. Most of the known categorification constructions are either combinatorial or `semi-combinatorial' and their intrinsic 3-dimensional origin remains mysterious. The goal of this proposal is to extend the categorification to a wider class of link polynomial invariants and, more importantly, to the Reshetikhin-Turaev invariants of 3-manifolds. Rozansky suggests two approaches to this problem. First, he intends to use combinatorial methods based upon commutative algebra (matrix factorizations) and upon the topology of virtual links. These methods have shown promise in the categorification of the HOMFLYPT and SO(2N) Kauffman polynomials. The second approach is based on the methods of quantum field theory. Categorification implies that Witten's Chern-Simons theory is a dimensional reduction of a yet unknown 4-dimensional theory. The candidate theories were considered in the papers of Gukov, Kapustin and Witten, but the correct theory has not been found yet. A construction of this topological quantum field theory may lead to combinatorial categorification constructions for link and manifold invariants. It should also provide a conceptual explanation for the existence of cobordism invariants. 3-dimensional topology deals with classification of knots, links and 3-dimensional surfaces. The main approach to this problem is to construct topological invariants, that is, the numbers or polynomials that can be assigned to a topological object, computed readily from its presentation (say, from the picture of a knot) and used to distinguish these objects. A major breakthrough it 3-dimensional topology came about 25 years ago with the discovery of a wide variety of the so-called quantum invariants.Surprisingly, these invariants were rooted in physics: they come from a special kind of a 3-dimensional quantum field theory. Thus quantum invariants provide an important bridge between topology and advanced physical theories. A more recent development came in the form of categorification: it turned out that quantum invariants were just dimensions of special vector spaces associated with topological objects. From the physics point of view, this means that the 3-dimensional quantum field theory describing quantum invariants is the dimensional reduction of the yet unknown 4-dimensional theory (the idea of dimensional reduction is familiar in string theory which requires a 9-dimensional space, 6 extra dimensions being wrapped up tightly in order to make them invisible for a general observer). Rozansky proposes to search for the 4-dimensional physical theory related to categorification and to use the topology-physics relation in order to get a better understanding of both fields of science.

在过去的25年里，三维拓扑学最重要的发展之一是发现了所谓的链路和3流形的量子不变量，如琼斯多项式、HOMFLYPT多项式和考夫曼多项式以及Reshetikhin-Turaev不变量。这些不变量是组合描述的，它们与经典三维拓扑的关系是神秘的，它们存在的唯一概念解释来自物理学：Witten构建了一组拓扑量子场论，其路径积分配分函数被推测与量子不变量相一致。最近，Khovanov， Ozsvath和Szabo发现了多项式连杆不变量的范畴性：一个连杆对应一个由梯度向量空间组成的链复合体，其梯度欧拉特征与给定的多项式不变量相一致。此外，对于两个链接之间的协同关系，人们在链接分类复合物之间关联链映射。大多数已知的分类结构要么是组合的，要么是“半组合的”，它们内在的三维起源仍然是神秘的。本提案的目标是将分类扩展到更广泛的连杆多项式不变量类，更重要的是，扩展到3流形的Reshetikhin-Turaev不变量。罗赞斯基提出了两种解决这个问题的方法。首先，他打算使用基于交换代数（矩阵分解）和虚拟链路拓扑结构的组合方法。这些方法在HOMFLYPT和SO(2N) Kauffman多项式的分类中显示出了希望。第二种方法是基于量子场论的方法。分类意味着Witten的chen - simons理论是一个未知的四维理论的降维。候选理论在Gukov， Kapustin和Witten的论文中被考虑过，但正确的理论尚未找到。这种拓扑量子场论的构造可能导致链路不变量和流形不变量的组合分类构造。它还应该为协共不变量的存在提供一个概念性的解释。三维拓扑处理结点、链路和三维曲面的分类。这个问题的主要方法是构造拓扑不变量，也就是说，可以分配给拓扑对象的数或多项式，可以从其表示（例如，从一个结的图片）中轻松计算，并用于区分这些对象。三维拓扑学的一个重大突破出现在大约25年前，当时发现了各种各样的所谓量子不变量。令人惊讶的是，这些不变量根植于物理学：它们来自一种特殊的三维量子场论。因此，量子不变量在拓扑学和高级物理理论之间提供了一个重要的桥梁。最近的一个发展是以分类的形式出现的：它证明了量子不变量只是与拓扑对象相关的特殊向量空间的维度。从物理学的角度来看，这意味着描述量子不变量的三维量子场论是未知的四维理论的降维（降维的概念在弦理论中很熟悉，弦理论需要一个9维空间，6个额外的维度被紧密地包裹起来，以使一般观察者看不见它们）。Rozansky建议寻找与分类相关的四维物理理论，并使用拓扑物理关系，以便更好地理解这两个科学领域。