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

在经典拓扑学的框架内，对三维流形中链接的量子不变量的解释在很长一段时间内都是一个挑战。最令人信服的解释，他们的性质来自维滕的工作有关这些不变量的量子陈-西蒙斯理论。然而，为什么琼斯和HOMFLY-PT多项式是多项式在q，而不是正式的幂级数在（q-1），作为一个量子场论将建议，什么是意义的系数，这些多项式仍然是一个谜。一个突破来自于霍瓦诺夫的分类构造：原来量子多项式是一个Z-阶同调的欧拉特征，由一个组合构造与一个链接相关联。将这个结果推广到3-流形中链环的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，并且只定义为q是1的根。最近Khovanov和Rozansky归类的“稳定”多项式部分的WRT不变量的产品中的链接的2球的一个圆圈。它们的构造使用Khovanov代数H_n上模的导范畴。Rozansky将尝试将这个结果进一步推广到一般的3-流形。他认为这可以通过将代数H_n变形为A-无穷代数来实现，从而将它们的Z-分次化为周期Z_r分次，这对应于相关参数q是1的根。一个类似的技巧可以从它的2变量版本构造SU（N）HOMFLY-PT多项式的分类。除了分类组合WRT不变量，Rozansky将尝试分类Khovanov的非线性构造的琼斯多项式。这个想法是基于在Khovanov的分类的Kamnitzer-Cautis版本中使用的对象与Kapustin的联合工作中与全纯辛流形相关联的2-范畴之间的相似性，Rozansky和Saulina.量子不变量的发现，例如3-球面中链接的Jones和HOMFLY-PT多项式以及3-球面中有色链接的Witten-Reshetikhin-Turaev不变量。流形揭开了三维拓扑学的新篇章。相对于亚历山大多项式，这是非常有效的建立链路的拓扑性质，这些新的不变量和经典拓扑之间的关系是间接的。量子不变量的目的似乎是在三维拓扑与数学和量子场论（QFT）的其他分支之间建立深层联系。霍瓦诺夫对琼斯多项式的分类是朝着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”。维滕认为，一个受超弦启发的6维QFT将霍瓦诺夫同调和朗兰兹对偶联系在一起。通过使用同调代数的方法，Rozansky将试图将Khovanov的分类程序从3-球面中的链接扩展到一般3-流形中的链接。他还试图通过将一个范畴而不是同源性与一个联系联系起来，将范畴化提高一个层次。如果这是真的，这可能意味着潜在的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.