Moduli Spaces and Integrable Systems

模空间和可积系统

• 批准号: RGPIN-2015-04393
• 负责人: Hurtubise, Jacques
• 金额: $ 2.26万
• 依托单位: McGill University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2016
• 资助国家: 加拿大
• 起止时间: 2016-01-01 至 2017-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=612938
• 关键词: Moduli; Spaces; Integrable; Systems

My proposed research centres on the application of techniques of algebraic geometry, symplectic geometry, and differential geometry to the study of the intertwined topics of moduli spaces and integrable systems. These are subjects of both intrinsic and extrinsic interest. Moduli spaces of various sorts are central not only in geometry, but occur in physics (typically as critical points of an action; as such they have been the focal point for the interaction of geometry and physics), and in areas as wide afield as number theory and combinatorics. One recurrent theme is a strong intertwining of algebraic geometry and symplectic geometry, with the two providing complementary insight into the structure of the spaces. The geometry of moduli spaces is tied intimately to the geometry of integrable systems, and even more tightly in recent years with the development of cohomological invariants for moduli, typically bundled together as generating functions, in which guise they appear as tau-functions. These functions, more properly sections of a determinant bundle on an infinite dimensional Grassmannian, have been the mainstay of the theory of integrable systems in the last twenty years. Specific problems: A) Moduli 1) Monopoles with Dirac type singularities, as considered Witten and Kapustin in their study of the geometric Langlands theory; they mediate Hecke transforms. 2) Linking various gauge theoretical moduli spaces via Nahm transforms: for specific manifolds (R^n, ALE or ALF manifolds), the Nahm transform allows a description of moduli. 3) G-bundles and compactification. This is linked to understanding of how these objects behave in a limit; one wants a description that admits a good deformation theory. 4) Real moduli. I have recently looked at some examples of moduli space of real (i.e. complex, but conjugation invariant) geometric objects; one area of interest is character varieties. B) Integrable systems. 1) Singular connections and isomonodromy. Following on a recent success in describing deformations of irregular singularities, I would now like to work on their Poisson geometry. 2) Poisson geometry of networks. Poisson spaces can be associated to various graphs, and there are interesting links to cluster algebras. 3) The general theory of tau functions: These functions, associated to determinant bundles over Grassmannians, admit several interesting generalisations which should clarify their nature and their role. 4) Tau functions and counting problems: Since the proof by Kontsevich of the Witten conjecture, one abiding mystery is the role of the tau function (and so integrable systems) in various enumerative problems. 5) Tau functions and the Eynard Orantin invariants: The understanding of the previous problem seems to go through a better understanding of these quite remarkable invariants.

我建议的研究集中在应用代数几何、辛几何和微分几何的技巧来研究模空间和可积系统的相互交织的主题。这些都是既有内在利益又有外在利益的主题。 各种各样的模空间不仅在几何学中是中心，而且在物理学中也存在(通常作为作用的临界点，因此它们一直是几何和物理相互作用的焦点)，并且在数论和组合学等广泛的领域中都存在。一个反复出现的主题是代数几何和辛几何的紧密交织，两者提供了对空间结构的补充洞察。 模空间的几何与可积系统的几何密切相关，近年来随着模的上同调不变量的发展更加紧密，模的上同调不变量通常作为生成函数捆绑在一起，伪装成tau-函数。这些函数，更确切地说，是无穷维格拉斯曼上行列式丛的部分，在过去的二十年里一直是可积系统理论的支柱。 具体问题： A)模数 1)具有狄拉克奇点的单极子，如Witten和Kapustin在几何朗兰兹理论研究中所认为的那样；它们调解Hecke变换。 2)通过Nahm变换连接各种规范理论模空间：对于特定的流形(R^n、ALE或ALF流形)，Nahm变换允许模的描述。 3)G丛与紧致化。这与理解这些物体在极限中的行为有关；人们需要一种允许良好的变形理论的描述。 4)实模。我最近看了一些实(即复数，但共轭不变)几何对象的模空间的例子；一个感兴趣的领域是特征变量。 B)可积系统。 1)单联结和等向单向度。在最近成功地描述了不规则奇点的变形之后，我现在想研究它们的泊松几何。 2)网络的泊松几何。Poisson空间可以与各种图联系在一起，并且有到簇代数的有趣的链接。 3)tau函数的一般理论：这些函数与Grassmannians上的行列式丛有关，有几个有趣的推广，应该能澄清它们的性质和作用。 4)Tau函数和计数问题：自从Kontsevich证明了Witten猜想以来，一个永恒的谜团是Tau函数(以及可积系统)在各种计数问题中的作用。 5)Tau函数和Eynard Orantin不变量：对前面问题的理解似乎要经过对这些相当显著的不变量的更好的理解。