New Avenues in Symplectic Geometry and its Applications

辛几何及其应用的新途径

• 批准号: 1211819
• 负责人: Jonathan Weitsman
• 金额: $ 25.68万
• 依托单位: Northeastern University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-07-01 至 2018-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1211819&HistoricalAwards=false
• 关键词: New; Avenues; Symplectic; Geometry; its

We describe several problems in symplectic geometry which new results and methods, appearing in both Mathematics and Physics, may provide new avenues for. The geometry of the Lagrangian subvarieties of a symplectic manifold has been the subject of a great deal of current research. One version of this is a construction by Weinstein of the symplectic ``category'' whose objects are symplectic manifolds and whose morphisms are Lagrangian subvarieties. A recent monograph by Guillemin and Sternberg shows that Weinstein's category and variants of it are powerful tools in semi-classical analysis. We show hints that this category reveals symmetries in semiclassical systems which are not apparent in the underlying symplectic manifold. This may be a semiclassical version of a principle advocated by Witten, that a quantum system may be have much more symmetry than the underlying symplectic manifold. Another area we propose investigating is the relation between symplectic geometry and quantum manifold invariants. Many topological quantum field theories may be interpreted mathematically as counts of points in moduli spaces. One major exception seemed to be Chern-Simons gauge theory; in Physics language, this theory is not supersymmetric. Recent work in Physics (due to Beasley-Witten, Kapustin-Willett-Yaakov, Kallen, and others) has shows that Chern Simons Gauge theory has supersymmetric avatars. We use this insight to conjecture formulas for quantum manifold invariants which may give more insight into their topological nature. We also speculate on other possibilities raised by these constructions.The interplay of ideas from Mathematics and Theoretical Physics has been a productive one for both fields. This project develops new perspectives on the relations between these two areas. The planned project would give both concrete mathematical results and, hopefully, new avenues of interaction between the two fields. More broadly, problems arising from Physics, and the Mathematics arising from grappling with these problems, have been at the core of Mathematics for centuries. Quantum Field Theory and String Theory show every promise of playing a similar role in the future, and stimulating new discoveries in Mathematics, which invariably lead to applications far removed from their area of origin.

我们描述了辛几何中的几个问题，这些问题在数学和物理学中出现的新结果和新方法可能为解决这些问题提供新的途径。辛流形的拉格朗日子变种的几何性质一直是当前大量研究的课题。其中一个版本是Weinstein对辛“范畴”的构造，它的对象是辛流形，其态射是拉格朗日子变种。吉列明和斯滕伯格最近的一篇专著表明，韦恩斯坦的分类及其变体是半经典分析的有力工具。我们展示了这个范畴揭示了在潜在的辛流形中不明显的半经典系统中的对称性的暗示。这可能是威腾所提倡的一个原理的半经典版本，即量子系统可能比底层的辛流形具有更多的对称性。我们提出研究的另一个领域是辛几何和量子流形不变量之间的关系。许多拓扑量子场论在数学上可以解释为模空间中点的计数。一个主要的例外似乎是陈-西蒙斯规范理论；用物理学的语言来说，这个理论不是超对称的。最近在物理学上的工作（由于Beasley-Witten， Kapustin-Willett-Yaakov， Kallen和其他人）表明，chen Simons规范理论具有超对称的虚拟体。我们利用这一见解来推测量子流形不变量的公式，这可能会让我们更深入地了解它们的拓扑性质。我们还推测了这些构造所带来的其他可能性。数学和理论物理思想的相互作用对两个领域都是有益的。这个项目为这两个领域之间的关系提供了新的视角。计划中的项目将给出具体的数学结果，并有望为两个领域之间的相互作用提供新的途径。更广泛地说，几个世纪以来，物理学产生的问题以及解决这些问题所产生的数学一直是数学的核心。量子场论和弦理论显示出在未来扮演类似角色的希望，并刺激数学的新发现，这些发现总是导致远离其起源领域的应用。