Representation theoretic methods in geometry and topology

几何和拓扑中的表示理论方法

• 批准号: RGPIN-2014-04841
• 负责人: Cautis, Sabin
• 金额: $ 2.04万
• 依托单位: University of British Columbia
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2018
• 资助国家: 加拿大
• 起止时间: 2018-01-01 至 2019-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=646992
• 关键词: Representation; theoretic; methods; geometry; topology

The common theme of this proposal is the use of representation theoretic methods (often inspired by ideas in categorification) to study*a) 3-manifold invariants (e.g. Reshetikhin-Turaev invariants),*b) categories arising in algebraic geometry (e.g. categories of coherent sheaves or D-modules on moduli spaces),*c) (categorified) quantum groups and vertex operator algebras.**One of the main tools in low dimensional topology are invariants which can distinguish between different 3-dimensional (or 4-dimensional) manifolds. A series of such invariant were introduced by Reshetikhin and Turaev. These Reshetikhin-Turaev (RT) invariants have deep connections to representation theory and, in particular, to quantum groups. In some ways, these connections are even more fundamental than the original relation to topology.**The simplest RT invariant for knots in the 3-sphere is called the Jones polynomial (since it was discovered earlier by Vaughan Jones). In 2001 Mikhail Khovanov showed that the Jones polynomial can be lifted to a more powerful homological invariant (now called Khovanov homology).**In subsequent work, jointly with Joel Kamnitzer, I showed that Khovanov homology can also be defined using categories of coherent sheaves on certain iterated Grassmannian bundles. These varieties can be defined using the affine Grassmannian which in turn is related (via geometric Satake) to the representation theory of semisimple Lie algebras. This is part of a sequence of ideas and results highlighting certain deep relations between algebraic geometry, representation theory and topology. **The main part of this proposal involves extending this story further. On the topology side one would like to lift the RT invariants to homological invariants of arbitrary 3-manifolds (not just knots in the 3-sphere). On the algebro-geometric side one would like to understand certain categories of coherent sheaves (and D-modules) on more complicated varieties appearing in geometric representation theory. This involves developing new techniques in representation theory of quantum groups as well as categorification (the study of their higher, homological analogues).**On the representation theoretic side one has the rich theory of vertex operators. These operators form one way to define the RT 3-manifold invariants. In earlier work with Anthony Licata, we showed how these operators can be categorified by relating them to categories of coherent sheaves on Hilbert schemes of points on surfaces (generalizing work of Nakajima and Grojnowski). I hope to continue this study of "categorified" vertex operators and their relation to geometry. One of the main aims is to use this to lift the RT invariants to homological invariants of all 3-manifolds.**The original introduction of vertex operator algebras by Borcherds and Frenkel-Lepowski-Meurman was for the purposes of proving the moonshine conjecture (which is a remarkable relationship between the largest sporadic finite group and the j-function from number theory). Subsequently, understanding the "higher" theory of vertex operators could also lead to a categorical version of moonshine theory.**To conclude, the aim of this proposal is to relate certain aspects from several fields: algebraic geometry, representation theory, 3-manifold invariants and categorification. There is a promising, rich and largely unexplored interplay between these areas along the lines sketched out above.

这个提议的共同主题是使用表示论方法（通常受到范畴化思想的启发）来研究 *a）3-流形不变量（例如Reshetikhin-Turaev不变量），*B）代数几何中出现的范畴（例如模空间上的相干层或D-模的范畴），*c）（范畴化的）量子群和顶点算子代数。低维拓扑学的主要工具之一是不变量，它可以区分不同的3维（或4维）流形。Reshetikhin和Turaev引入了一系列这样的不变量。这些Reshetikhin-Turaev（RT）不变量与表示论，特别是量子群有着深刻的联系。在某些方面，这些联系甚至比拓扑学的原始关系更基本。最简单的RT不变量称为琼斯多项式（因为它是由沃恩·琼斯发现的）。2001年，米哈伊尔·霍瓦诺夫证明了琼斯多项式可以提升为更强大的同调不变量（现在称为霍瓦诺夫同调）。在随后的工作中，我与乔尔·卡姆尼策（Joel Kamnitzer）一起证明了霍瓦诺夫同调也可以用某些迭代格拉斯曼丛上的相干层范畴来定义。这些变种可以使用仿射格拉斯曼来定义，而格拉斯曼又与半单李代数的表示理论有关（通过几何佐竹）。这是一系列的想法和结果的一部分，突出了代数几何，表示论和拓扑之间的某些深刻关系。 ** 本书的主要部分是进一步扩展这个故事。在拓扑学方面，人们希望将RT不变量提升为任意3-流形的同调不变量（而不仅仅是3-球面中的结点）。在代数几何方面，人们希望理解某些范畴的相干层（和D-模）在更复杂的品种出现在几何表示论。这包括发展量子群表示论和分类（研究它们的高级同调类似物）的新技术。在表示论方面，人们有丰富的顶点算子理论。这些运算符形成了定义RT 3-流形不变量的一种方法。在与Anthony Licata的早期工作中，我们展示了如何通过将这些算子与曲面上的点的希尔伯特方案上的相干层的范畴相关联来将它们归类（中岛和格罗伊诺夫斯基的推广工作）。我希望继续研究“范畴化”顶点算子及其与几何的关系。其中一个主要目的是使用它来提升RT不变量到所有3-流形的同调不变量。Borcherds和Frenkel-Lepowski-Meurman最初引入顶点算子代数是为了证明月光猜想（这是数论中最大的零星有限群和j函数之间的一个显着关系）。随后，理解顶点算子的“高级”理论也可能导致月光理论的范畴化版本。最后，这个建议的目的是从几个领域的某些方面：代数几何，表示论，3-流形不变量和分类。沿着上面概述的路线，这些领域之间存在着有希望的、丰富的和基本上未经探索的相互作用。