New Avenues in Symplectic Geometry and its Applications

我们描述了辛几何中的几个问题，数学和物理中出现的新结果和新方法可能会为这些问题提供新的途径。辛流形的拉格朗日子簇的几何一直是当前大量研究的主题。其中一个版本是温斯坦对辛“范畴”的构造，其对象是辛流形，其态射是拉格朗日子簇。吉列曼和斯特恩伯格最近发表的一本专著表明，温斯坦的范畴及其变体是半经典分析中的强大工具。我们表明，这一范畴揭示了半经典系统中的对称性，而这些对称性在基本辛流形中并不明显。这可能是Witten所倡导的一个原理的半经典版本，即量子系统可能比底层辛流形具有更多的对称性。我们建议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。许多拓扑量子场论可以在数学上解释为模空间中的点的计数。一个主要的例外似乎是Chern-Simons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(归功于Beasley-Witten，Kapustin-Willett-Yaakov，Kallen和其他人)表明，Chern Simons规范理论具有超对称化身。我们利用这一洞察力来猜想量子流形不变量的公式，这可能会使我们更深入地了解它们的拓扑性质。我们还推测了这些结构带来的其他可能性。来自数学和理论物理的思想的相互作用对这两个领域都是富有成效的。这个项目为这两个领域之间的关系开辟了新的视角。计划中的项目将给出具体的数学结果，并有望给出这两个领域之间互动的新途径。更广泛地说，从物理学产生的问题，以及在解决这些问题时产生的数学，几个世纪以来一直是数学的核心。量子场论和弦理论展示了在未来扮演类似角色的每一个希望，并刺激了数学的新发现，这些发现总是导致应用远离它们的起源领域。

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.