Representation theoretic methods in geometry and topology

这一建议的共同主题是使用表征理论方法(通常是受分类中的想法启发)来研究 A)3-流形不变量(例如Reshetikhin-Turaev不变量)， B)出现在代数几何中的范畴(例如，模空间上的凝聚层或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，特别是与量子群有很深的联系。在某些方面，这些联系甚至比原始的拓扑学关系更基本。 3-球面中纽结最简单的RT不变量称为琼斯多项式(因为它是由沃恩·琼斯较早发现的)。2001年，Mikhail Khovanov证明了Jones多项式可以提升到一个更强大的同调不变量(现在称为Khovanov同调)。 在随后的工作中，我与Joel Kamnitzer一起证明了Khovanov同调也可以用某些迭代Grassmanian丛上的凝聚层范畴来定义。这些簇可以用仿射Grassman来定义，而仿射Grassman又与半单李代数的表示理论有关(通过几何Satake)。这是一系列思想和结果的一部分，突出了代数几何、表示论和拓扑学之间的某些深层关系。 这项提议的主要部分涉及进一步扩展这个故事。在拓扑学方面，人们希望将RT不变量提升到任意3-流形(不仅仅是3-球面中的纽结)的同调不变量。在代数几何方面，人们想要了解几何表示理论中出现的更复杂簇上的某些凝聚层(和D-模)范畴。这涉及到在量子群的表示理论以及分类(研究它们更高的同源类似物)方面开发新的技术。 在表示理论方面，我们拥有丰富的顶点算子理论。这些算子形成了定义RT3-流形不变量的一种方法。在安东尼·利卡塔的早期工作中，我们展示了如何通过将这些算子与曲面上点的Hilbert方案上的相干层范畴相联系来分类(推广了Nakajima和Grojnowski的工作)。我希望继续研究“范畴化”的顶点运算符及其与几何的关系。其中一个主要目的是利用这一点将RT不变量提升为所有3-流形的同调不变量。 Borcherds和Frenkel-Lepowski-Meurman最初引入顶点算子代数的目的是为了证明Moonlight猜想(这是最大的零星有限群与数论中的j-函数之间的显著关系)。随后，理解顶点运算符的“更高”理论也可能导致月光理论的一个范畴版本。 总之，这个建议的目的是从几个领域联系某些方面：代数几何、表示论、3-流形不变量和范畴化。沿着上面勾勒出的路线，这些地区之间存在着一种充满希望、丰富且基本上未被探索的相互作用。

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.