Categorification and Topological Quantum Field Theories

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环和三维流形的量子不变量，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性解释来自物理学：Witten构建了一族拓扑量子场理论，其路径积分配分函数被猜想为与量子不变量重合。最近，Khovanov，Ozsvath ans Szabo发现了多项式链环不变量的分类：一个链环上有一个由分次向量空间组成的链复合体，它的分次欧拉特征与给定的多项式不变量重合。此外，对于两个环之间的余边图，人们将环分类复合体之间的链映射关联起来。大多数已知的范畴化结构要么是组合式的，要么是‘半组合式的’，其内在的三维起源仍然是个谜。这一建议的目的是将分类扩展到更广泛的链接多项式不变量，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)和虚链的拓扑的组合方法。这些方法在HOMFLYPT和SO(2N)Kauffman多项式的分类中显示了良好的应用前景。第二种方法是基于量子场论的方法。范畴化意味着Witten的Chern-Simons理论是一个尚未知晓的四维理论的降维。古科夫、卡普斯汀和维腾的论文都考虑过候选理论，但还没有找到正确的理论。这种拓扑型量子场论的构造可能导致链式和流形不变量的组合范畴化构造。它还应该为协边不变量的存在提供一个概念性的解释。三维拓扑学涉及结点、链节和三维曲面的分类。解决这个问题的主要方法是构造拓扑不变量，即可以赋给拓扑对象的数字或多项式，从其表示形式(例如，从纽结的图像)很容易地计算出来，并用于区分这些对象。大约25年前，随着各种所谓的量子不变量的发现，三维拓扑学取得了重大突破。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，量子不变量在拓扑学和高级物理理论之间架起了一座重要的桥梁。更新的发展是以分类的形式出现的：原来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(降维的想法在弦理论中很常见，它需要一个9维空间，额外的6个维度被紧紧地包裹起来，以便让普通观察者看不见)。为了更好地理解这两个科学领域，罗赞斯基建议寻找与范畴化相关的四维物理理论，并利用拓扑-物理关系。

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.