Quantum topology of lagrangian submanifolds

拉格朗日子流形的量子拓扑

• 批准号: 92913-2011
• 负责人: Lalonde, Francois
• 金额: $ 3.64万
• 依托单位: Université de Montréal
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2015
• 资助国家: 加拿大
• 起止时间: 2015-01-01 至 2016-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=569019
• 关键词: Quantum; topology; lagrangian; submanifolds

The goal of this proposal is to investigate the topology of Lagrangian submanifolds. We are going to establish a new research program that develops a Floer theory for Lagrangian intersections without obstructions. The main new object is the cluster moduli space associated to any Lagrange submanifold L in M and a generic Morse function f from L to M. It is obtained by attaching to each homotopy class lambda and set x, x_1,..., x_k in Crit(f) the space of all tree-like configurations starting at x and ending at x_1,..., x_k where each edge is sent to a negative flowline of f and each interior vertex is replaced by a holomorphic disc D mapped to M with boundary mapped to L so that the sum of the homotopy classes of all discs be equal to lambda. This is made from more elementary moduli spaces by gluing them along their common boundary strata (the point being that the dimension loss in the bubbling off of a disc A from a disc B is compensated by the additional real parameter that specifies the time of the flowline between two discs). We consider coefficients which reflect the loop structure of the boundary of such configurations. Some applications are presented - for instance the proof in some instances of the well-known conjecture that for any almost complex structure J and any Lagrangian submanifold L that can be disjoined from itself, there is a J-holomorphic disc with boundary passing through any point of L, or the proof of inequalities on the Maslov indices of Lagrangian submanifolds, and applications to the existence of periodic orbits of the ambient space obtained by stretching the neck of specific almost complex structures near the Lagrangian submanifold. We will also study the Homological Lagrangian Monodromy using the tools of absolute and relative Seidel morphisms. This monodromy, for Hamiltonian loops taking a given Lagrangian submanifold L back to itself, has been computed in some special cases by Mei-Lin Yau. Recently, we established the fundamental result on this monodromy stating that the rational homological monodromy of a weakly exact Lagrangian submanifold is always trivial. The next goal is to compute this monodromy in many cases, starting from the Clifford and Chekanov tori.

这个提议的目的是研究拉格朗日子流形的拓扑。 我们将建立一个新的研究计划，发展拉格朗日交点的Floer理论 没有障碍物。主要的新对象是与任何拉格朗日子流形相关联的簇模空间 L在M中和一般的莫尔斯函数f从L到M。它是通过附加到每个同伦类lambda和集合x，x_1，.，Crit（f）中的x_k是从x开始到x_1，...，x_k，其中每个边被发送到f的负流线，每个内部顶点被映射到M的全纯圆盘D替换，边界映射到L，使得所有圆盘的同伦类之和等于λ。这是由更多的基本模量空间通过沿沿着它们的公共边界层将它们粘合而形成的（要点是，在从圆盘B鼓泡出圆盘A时的尺寸损失由指定两个圆盘之间流线的时间的附加真实的参数补偿）。我们考虑的系数，反映了这种配置的边界的环路结构。给出了一些应用，例如证明了著名的猜想：对于任意几乎复结构J和任意可自分离的Lagrange子流形L，存在一个J-全纯圆盘，其边界通过L的任意点，或者证明了Lagrange子流形的Maslov指数的不等式，并应用于通过拉伸拉格朗日子流形附近的特定几乎复结构的颈部而获得的环境空间的周期轨道的存在性。我们还将使用绝对和相对Seidel态射的工具来研究同调拉格朗日单值性。这monodromy，为哈密顿回路采取一个给定的拉格朗日子流形L回本身，已计算在某些特殊情况下，由Mei-Lin Yau。最近，我们建立了关于这种单值性的基本结果，即弱正合拉格朗日子流形的有理同调单值性总是平凡的。下一个目标是在许多情况下计算这种单值性，从Clifford和Chekanov环面开始。