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

连杆和3-流形的量子拓扑不变量的分类和双分类

• 批准号: 1108727
• 负责人: Lev Rozansky
• 金额: $ 15.66万
• 依托单位: University of North Carolina at Chapel Hill
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-06-01 至 2015-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1108727&HistoricalAwards=false
• 关键词: Categorification; Double; Quantum; Topological; Invariants

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.

在经典拓扑框架内解释三流形中的量子不变量长期以来一直是一个挑战。对它们本质最令人信服的解释来自威滕的工作，该工作将这些不变量与量子陈-西蒙斯理论联系起来。然而，正如量子场论所暗示的那样，琼斯和 HOMFLY-PT 多项式为何是 q 中的多项式而不是 (q-1) 中的形式幂级数，以及这些多项式的系数的含义仍然是个谜。霍瓦诺夫的分类构造取得了突破：事实证明，量子多项式是通过与链接的组合构造相关联的 Z 分级同调性的欧拉特征。将该结果扩展到 3 流形中的 Witten-Reshetikhin-Turaev (WRT) 链接不变量是一个具有挑战性的问题，部分原因是 WRT 不变量并不完全是 q 的多项式，并且仅在 q 为 1 的根时定义。最近 Khovanov 和 Rozansky 将 WRT 链接不变量的“稳定”多项式部分分类为 a 的乘积 2-带圆的球体。它们的构造使用了 Khovanov 代数 H_n 上模的派生类别。 Rozansky 将尝试将此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-无穷代数来完成的，从而将它们的 Z 分级减少为周期性 Z_r 分级，这将对应于相关参数 q 作为 1 的根。类似的技巧可以从其 2 变量版本构造 SU(N) HOMFLY-PT 多项式的分类。除了对 WRT 不变量进行组合分类之外，Rozansky 还将尝试对 Jones 多项式的 Khovanov 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 的联合工作中与全纯辛流形相关的 2-范畴之间的相似性。量子不变量的发现，例如 3-球体中的 Jones 和 HOMFLY-PT 多项式以及 3 流形中彩色链接的 Witten-Reshetikhin-Turaev 不变量开启了 3 维拓扑的新篇章。与在建立链接的拓扑性质方面非常有效的亚历山大多项式相反，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 3 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”。威滕认为，受超弦启发的 6 维 QFT 将霍瓦诺夫同调性和朗兰兹对偶性联系在一起。通过使用同调代数的方法，Rozansky 将尝试将 Khovanov 的分类程序从 3 球体中的链接扩展到一般 3 流形中的链接。他还将尝试通过将类别而不是同源性与链接相关联来将分类提高一个级别。如果属实，这可能表明底层 QFT 是 7 维而不是 6 维。