Categorification and Topological Quantum Field Theories

在过去的25年中，三维拓扑学最重要的发展之一是发现了所谓的量子不变量，如Jones、HOMFLYPT和Kauffman多项式以及Reshetikhin-Turaev不变量。这些不变量被组合地描述，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性解释来自物理学：维滕构建了拓扑量子场论家族，其路径积分配分函数被证明与量子不变量一致。最近，Khovanov，Ozsvath ans Szabo发现了多项式链不变量的分类：一个链与一个分次向量空间的链复形相关联，其分次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配边，人们将链接分类复合体之间的链映射联系起来。大多数已知的分类结构要么是组合的或“半组合的”，其内在的三维起源仍然是神秘的。这个建议的目标是将分类扩展到更广泛的一类链接多项式不变量，更重要的是，3流形的Reshetikhin-Turaev不变量。Rozansky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。范畴化意味着维滕的陈-西蒙斯理论是一个未知的四维理论的降维。Gukov，Kapustin和维滕的论文中考虑了候选理论，但至今还没有找到正确的理论。这种拓扑量子场论的构造可能导致链接和流形不变量的组合分类构造。它还应该为协边不变量的存在提供概念上的解释。3-三维拓扑学处理结、链接和三维表面的分类。解决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字或多项式，很容易从它的表示（比如说，从一个结的图片）计算出来，并用于区分这些对象。三维拓扑学的一个重大突破是在大约25年前发现了各种各样的所谓量子不变量，令人惊讶的是，这些不变量植根于物理学：它们来自一种特殊的三维量子场论。因此，量子不变量提供了一个重要的桥梁之间的拓扑结构和先进的物理理论。最近的一个发展是以分类的形式出现的：量子不变量只是与拓扑对象相关的特殊向量空间的维度。从物理学的观点来看，这意味着描述量子不变量的3维量子场论是未知的4维理论的降维（降维的想法在弦理论中很熟悉，弦理论需要9维空间，6个额外的维度被紧紧包裹起来，以便使它们对一般观察者不可见）。Rozansky建议寻找与范畴化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。

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.