Adiabatic Limits of Quantum Symplectic Invariants

量子辛不变量的绝热极限

• 批准号: 2105417
• 负责人: Christopher Woodward
• 金额: $ 48.42万
• 依托单位: Rutgers University New Brunswick
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2021
• 资助国家: 美国
• 起止时间: 2021-07-01 至 2025-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2105417&HistoricalAwards=false
• 关键词: Adiabatic; Limits; Quantum; Symplectic; Invariants

Symplectic geometry is the mathematical foundation for particle motion in classical physics. Recently, research in this area has centered on the study of quantum invariants defined using geometric objects known as holomorphic curves. These quantum invariants have appeared not only in fields of mathematics known as geometric analysis and low-dimensional topology but also in certain models in high-energy physics. The investigator will study the behavior of these quantum invariants under adiabatic limits in which directions in space or time are re-scaled. Applications will be of interest in topology and physics. The investigator will also continue his mentoring and outreach activities in mathematics education.The investigator will study three types of adiabatic limits of quantum invariants in symplectic geometry. First, he will study the limit of the Fukaya category of a symplectic manifold under the multi-directional symplectic field theory limit, which is equivalent in many cases to shrinking the fibers of a Lagrangian torus fibration. The investigator will extend previous results to Lagrangians compatible with this tropical limit and apply the results to examples arising in mirror symmetry, such as the computation of disk potentials. Secondly, the investigator will study the behavior of the Fukaya category and quantum cohomology under flips with non-trivial centers, or equivalently, mean curvature flow in which a subset of the symplectic manifold collapses over some non-trivial base, with the aim of constructing generators for the Fukaya category. A related project in gauge theory will study the behavior of higher rank monopole Floer homology under variation of monopole parameter, and relate these invariants with instanton homology and abelian monopole Floer homology. In a third project, the investigator will study the limits of flow spaces arising from Fukaya-isomorphic Lagrangians in cotangent bundles as the Lagrangians approach the zero section via rescaling, and in particular higher-dimensional moduli spaces of Morse flow trees that appear in the limit.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

辛几何是经典物理中粒子运动的数学基础。最近，这一领域的研究主要集中在研究使用被称为全纯曲线的几何对象定义的量子不变量。这些量子不变量不仅出现在几何分析和低维拓扑等数学领域，还出现在高能物理的某些模型中。研究人员将研究这些量子不变量在绝热极限下的行为，在绝热极限中，空间或时间的方向会被重新调整。应用程序将在拓扑学和物理学方面引起人们的兴趣。研究人员还将继续他在数学教育方面的指导和外展活动。研究人员将研究辛几何中量子不变量的三种绝热极限。首先，他将研究在多向辛场理论极限下辛流形的Fukaya范畴的极限，这在许多情况下等价于收缩拉格朗日环面纤维。研究人员将把以前的结果推广到与热带极限相容的拉格朗日，并将结果应用于镜像对称的例子，如盘势的计算。其次，研究者将研究Fukaya范畴和量子上同调在具有非平凡中心的翻转下的行为，或者等价地，在辛流形的子集在一些非平凡基上坍塌的平均曲率流下的行为，目的是构造Fukaya范畴的生成元。规范理论中的一个相关项目将研究高阶单极Floer同调在单极参数变化时的行为，并将这些不变量与瞬子同调和交换单极Floer同调联系起来。在第三个项目中，研究人员将通过重新缩放来研究余切丛中Fukaya同构拉格朗日所产生的流空间的极限，特别是出现在极限中的莫尔斯流树的高维模空间。这一裁决反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

项目成果

Invariance of immersed Floer cohomology under Maslov flows

Maslov流下浸没Floer上同调的不变性

• DOI: 10.2140/agt.2021.21.2313
• 发表时间: 2021
• 期刊: Algebraic & Geometric Topology
• 影响因子: 0.7
• 作者: Palmer, Joseph;Woodward, Chris
• 通讯作者: Woodward, Chris