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.

该提案的目的是调查拉格朗日亚曼菲尔德的拓扑。 我们将建立一个新的研究计划，该计划为拉格朗日交叉口开发浮动理论 没有障碍。主要的新对象是与任何Lagrange Submanifold关联的群集模量空间 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 tr​​ee-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因此，所有圆盘的同质类别的总和都等于lambda。这是由更基本的模量空间沿其共同边界层粘合的（要点是，从圆盘B中冒出圆盘A的尺寸损失是由指定两个光盘之间流动线的附加真实参数所补偿的。我们考虑反映此类配置边界的循环结构的系数。 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拉格朗日submanifold附近的特定几乎复杂结构的颈部。我们还将使用绝对和相对Seidel形态的工具来研究同源性Lagrangian单片。对于在某些特殊情况下，Mei-lin Yau计算了这种单型，用于将给定的Lagrangian submanifold l回到本身的哈密顿循环。最近，我们确定了这种单曲子的基本结果，表明弱精确的Lagrangian Submanifold的合理同源性单曲始终是微不足道的。下一个目标是在许多情况下，从克利福德（Clifford）和雪卡诺夫·托里（Chekanov Tori）开始计算这种单片。