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.

在古典拓扑框架内，对3个manifolds中链接的量子不变性的解释提出了很长时间的挑战。对它们的性质的最令人信服的解释是Witten的作品，这些作品与量子Chern-Simons理论有关。然而，琼斯和霍姆菲普多项式是Q中的多项式，而不是（Q-1）中的正式功率序列，正如量子场理论所暗示的那样，这些多项式的系数的含义是什么，这是一个谜。霍瓦诺夫（Khovanov）的分类结构取得了突破：事实证明，量子多项式是组合结构与链接相关的Z级同源性的欧拉（Euler）特征。该结果扩展到3个manifolds中链接的Witten-Reshetikhin-turaev（WRT）不变性的问题是一个充满挑战的问题，部分是因为WRT不变并不完全是Q的多项式，并且仅定义为Q是1的根。 2个圆圈。他们的构造使用Khovanov的代数H_N上的模块类别。罗赞斯基（Rozansky）将尝试将此结果进一步扩展到一般的3型曼尼弗。他猜想这可以通过将代数H_N变形为一个内代代数来完成，从而将它们的Z分级降低到定期的Z_R分级，该分级与1的词根q是1的根。一种类似的技巧可用于构建SU（N）homfly-pty-ptynomial从其2-Varial-varial-varial-varial-varial-varial semply构建分类。除了对WRT不变性进行分类外，Rozansky还将尝试对Khovanov的分类为琼斯多项式的构建。这个想法是基于Khovanov的Kamnitzer-Cautis版本中使用的对象之间的相似性，以及Kapustin，Rozansky和Saulina联合工作中与Holomormorphic Symphic sempledeclic歧管相关的2类别。 Witten-Reshetikhin-turaev在3个曼尼佛中的彩色链接不变，开了一个三维拓扑的新章节。与亚历山大多项式（Alexander）多项式相反，该多项式在建立链接的拓扑特性非常有效，这些新不变性和经典拓扑之间的关系是间接的。量子不变的目的似乎是在数学和量子场理论（QFT）之间建立3维拓扑与其他分支之间的深层联系。 Khovanov对琼斯多项式的分类是朝这个方向迈出的重要一步：它表明代数几何形状可以“已婚”到3维拓扑。维滕（Witten）表明，以超弦为灵感的6维QFT将Khovanov同源性和Langlands二重性联系在一起。通过使用同源代数的方法，Rozansky将尝试将Khovanov的分类程序从3个球体的链接扩展到一般3个manifolds中的链接。他还将尝试通过将类别而不是与链接的同源性关联来提高一个级别的分类。如果是真的，这可能表明基础QFT是7维而不是6维。