标题： Reshetikhin-Turaev形式主义的算子提升到Khovanov-Rozansky TQFTs 显示英文标题

标题： Operator lift of Reshetikhin-Turaev formalism to Khovanov-Rozansky TQFTs

摘要： 拓扑量子场论（TQFT）是一种强大的工具，用于描述同调，这通常涉及复形和各种映射/态射，使得一种仅对单一类型的映射求和的泛函积分方法似乎存在问题。 在TQFT中，这个问题是通过利用BRST算子丰富的零模来解决的，这些零模足以描述复形。 我们解释了这种方法对于重要的Khovanov-Rozansky（KR）上同调类看起来是什么样子，这些上同调对3d Chern-Simons理论中的可观测量（Wilson线或纽结多项式）进行分类。 我们开发了一种与所有链图相关的奇微分算子的构造，包括带有开放端的辫子。 这些算子仅在没有外部腿的图中才是幂零的，但对于开放的辫子，也可以发展出一种分解形式，该形式保持Reidemeister/拓扑不变性——问题的对称性。 这种方法似乎比传统的同调代数语言更“物理”，并且应该在超越Chern-Simons理论的各种问题中有许多应用。 我们也希望这种语言将提供高效的算法，并最终允许计算机化KR上同调的计算——对于闭合图和开放辫子。 显示英文摘要

摘要： Topological quantum field theory (TQFT) is a powerful tool to describe homologies, which normally involve complexes and a variety of maps/morphisms, what makes a functional integration approach with a sum over a single kind of maps seemingly problematic. In TQFT this problem is overcame by exploiting the rich set of zero modes of BRST operators, which appear sufficient to describe complexes. We explain what this approach looks like for the important class of Khovanov-Rozansky (KR) cohomologies, which categorify the observables (Wilson lines or knot polynomials) in 3d Chern-Simons theory. We develop a construction of odd differential operators, associated with all link diagrams, including tangles with open ends. These operators become nilpotent only for diagram with no external legs, but even for open tangles one can develop a factorization formalism, which preserve Reidemeister/topological invariance -- the symmetry of the problem. This technique seems much more ``physical'' than conventional language of homological algebra and should have many applications to various problems beyond Chern-Simons theory. We also hope that this language will provide efficient algorithms, and finally allow to computerize the calculation of KR cohomologies -- for closed diagrams and for open tangles.