Categorification and Double Categorification of Quantum Topological Invariants of Links and 3-Manifolds

在经典拓扑框架下解释3流形中链路的量子不变量一直是一个挑战。对它们的性质最令人信服的解释来自威滕的工作，他将这些不变量与量子陈-西蒙斯理论联系起来。然而，为什么Jones和HOMFLY-PT多项式是q中的多项式，而不是（q-1）中的形式幂级数（正如量子场论所暗示的那样），以及这些多项式系数的含义仍然是一个谜。Khovanov的分类构造带来了一个突破：它证明了量子多项式是z级同调的欧拉特征，该特征通过组合构造与链路相关联。将这个结果推广到3流形中连杆的Witten-Reshetikhin-Turaev （WRT）不变量是一个具有挑战性的问题，部分原因是WRT不变量不是q的多项式，并且仅当q是1的根时才定义。最近，Khovanov和Rozansky对2球与圆乘积中连杆的WRT不变量的“稳定”多项式部分进行了分类。它们的构造使用了Khovanov代数H_n上模的派生范畴。Rozansky将把这个结果进一步推广到一般的3流形。他推测，这可能是通过将代数H_n变形为a -∞代数来实现的，从而将它们的z分级降低为周期性的Z_r分级，这将对应于相关参数q是1的根。一个类似的技巧可以从SU(N) HOMFLY-PT多项式的2变量版本构建一个分类。除了对WRT不变量进行组合分类外，Rozansky还将尝试对Khovanov的Jones多项式的分类构造进行分类。这个想法是基于Kamnitzer-Cautis版本的Khovanov分类中使用的对象与Kapustin， Rozansky和Saulina联合工作中与全纯辛流形相关的2范畴之间的相似性。三维球面连杆的Jones多项式和HOMFLY-PT多项式以及三维流形中有色连杆的Witten-Reshetikhin-Turaev不变量的发现，打开了三维拓扑学的新篇章。与Alexander多项式在建立链路拓扑性质方面非常有效不同，这些新的不变量与经典拓扑之间的关系是间接的。量子不变量的目的似乎是建立三维拓扑与数学和量子场论（QFT）的其他分支之间的深层联系。Khovanov对Jones多项式的分类是朝这个方向迈出的重要一步：它表明代数几何可以与三维拓扑“联姻”。Witten提出一个超弦启发的6维QFT将Khovanov同调和Langlands对偶联系在一起。通过使用同调代数的方法，Rozansky试图将Khovanov的分类规划从3球中的连杆推广到一般3流形中的连杆。他还将尝试通过关联一个类别而不是同源性来将分类提高一个层次。如果这是真的，这可能表明底层的QFT是7维的而不是6维的。

An interpretation of quantum invariants of links in 3-manifolds within the framework of classical topology presented a challenge for a long time. The most convincing explanation of their nature came from Witten's work which related these invariants to quantum Chern-Simons theory. However the reason why Jones and HOMFLY-PT polynomials were polynomials in q rather than formal power series in (q-1), as a Quantum Field Theory would suggest, and what is the meaning of the coefficients of these polynomials remained a mystery. A breakthrough came from Khovanov's categorification construction: it turned out that a quantum polynomial is an Euler characteristic of a Z-graded homology associated by a combinatorial construction to a link. An extension of this result to the Witten-Reshetikhin-Turaev (WRT) invariant of links in 3-manifolds is a challenging problem, in part because the WRT invariant is not exactly a polynomial of q and is defined only for q being a root of 1. Recently Khovanov and Rozansky categorified the "stable" polynomial part of the WRT invariant of links in the product of a 2-sphere with a circle. Their construction uses the derived categories of modules over Khovanov's algebras H_n. Rozansky will try to extend this result further to general 3-manifolds. He conjectures that this might be done by deforming the algebras H_n into A-infinity algebras, thus reducing their Z-grading to a periodic Z_r grading which would correspond to the associated parameter q being the root of 1. A similar trick worked to construct a categorification of the SU(N) HOMFLY-PT polynomial from its 2-variable version. In addition to categorifying combinatorially the WRT invariant, Rozansky will try to categorify Khovanov's categorificaiton construction of the Jones polynomial. This idea is based on a similarity between the objects used in the Kamnitzer-Cautis version of Khovanov's categorification and the 2-category associated to a holomorphic symplectic manifold in the joint work of Kapustin, Rozansky and Saulina.The discovery of quantum invariants such as the Jones and HOMFLY-PT polynomials of links in a 3-sphere and the Witten-Reshetikhin-Turaev invariant of colored links in a 3-manifold opened a new chapter in 3-dimensional topology. In contrast to the Alexander polynomial which was very efficient in establishing the topological properties of links, the relation between these new invariants and classical topology is indirect. It seems that the purpose of quantum invariants is to establish deep links between 3-dimensional topology and other branches of Mathematics and Quantum Field Theory (QFT). Khovanov's categorification of the Jones polynomial was an important step in this direction: it showed that algebraic geometry could be "married" to 3-dimensional topology. Witten suggests that a superstring-inspired 6-dimensional QFT links together Khovanov homology and Langlands duality. By using the methods of homological algebra, Rozansky will try to extend Khovanov's categorification program from links in a 3-sphere to links in general 3-manifolds. He will also try to raise categorification by one level through associating a category rather than a homology to a link. If true, this might suggest that the underlying QFT is 7-dimensional rather than 6-dimensional.