Knots and contact topology through holomorphic curves

通过全纯曲线的结和接触拓扑

基本信息

批准号：
1406371
负责人：
Lenhard Ng
金额：
$ 43.67万
依托单位：
Duke University
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2014
资助国家：
美国
起止时间：
2014-09-01 至 2018-08-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1406371&HistoricalAwards=false
关键词：
Knots contact topology through holomorphic

项目摘要

The subjects of mathematics and physics have always been closely intertwined, with each one motivating and informing progress in the other. This project focuses on one current area of interplay, between topology (the study of shapes) on the mathematical side and string theory on the physics side. The jumping-off point for the project is an intriguing and unexpected connection, recently discovered by the Principal Investigator and collaborators on both sides, between two separate algebraic structures associated to knots in space: one in topology developed by the Principal Investigator, and one in string theory that has been the focus of much research in the past few years. In the course of this project, the Principal Investigator will establish this connection, which is currently only supported circumstantially; it is hoped that this work will create and strengthen new lines of communication between mathematics and physics, introducing techniques from each discipline into the other. The Principal Investigator will also use this project to train future mathematicians at all levels, from contributing to the annual USA Mathematical Olympiad for high school students, to supervising the research of undergraduates, graduate students, and postdoctoral fellows, to organizing conferences and seminars for established researchers.In the past decade, the Principal Investigator has introduced and studied a package of knot invariants called knot contact homology, which arises by counting holomorphic curves in certain symplectic manifolds, using a method in symplectic geometry pioneered by Gromov and Floer and more recently culminating in the Symplectic Field Theory of Eliashberg, Givental, and Hofer. Previous work has shown that knot contact homology is a robust invariant that is effective at distinguishing knots and contains classical topological information about the knot. In 2012, it was discovered by Aganagic, Ekholm, Vafa, and the Principal Investigator that knot contact homology has an unexpected and potentially powerful relation to string theory and mirror symmetry: the augmentation polynomial, a knot invariant derived from knot contact homology, is conjectured to be equal to Aganagic and Vafa's Q-deformed A-polynomial, which arises in the context of topological strings. The Principal Investigator will approach this conjecture using Lagrangian fillings and Gromov-Witten potentials. This could have significant ramifications in different directions: in knot theory, it would establish a variant of the AJ conjecture; in mirror symmetry, it would produce a new approach via Symplectic Field Theory to constructing mirror Calabi-Yau 3-folds; and in topological string theory, it would provide a mathematical foundation for recent results. In related work, the Principal Investigator will develop and strengthen the algebraic framework underneath certain aspects of Symplectic Field Theory, including knot contact homology and symplectic homology. New algebraic tools in this context, such as representation theory for differential graded algebras, would enable one to more effectively attack problems in symplectic geometry, in particular by analyzing Weinstein structures on symplectic manifolds and Legendrian and transverse knots in contact manifolds.

数学和物理学科始终紧密地交织在一起，每一学科都激励和促进另一学科的进步。该项目重点关注当前相互作用的一个领域，即数学方面的拓扑（形状研究）和物理方面的弦理论之间的相互作用。该项目的起点是首席研究员和双方合作者最近发现的一种有趣且意想不到的联系，它位于与空间结相关的两个独立的代数结构之间：一个是首席研究员开发的拓扑结构，另一个是由首席研究员开发的拓扑结构。弦理论是近几年研究的热点。在该项目的过程中，首席研究员将建立这种连接，目前仅在情况下提供支持；希望这项工作能够创建并加强数学和物理学之间的新交流途径，将每个学科的技术引入到另一个学科中。首席研究员还将利用该项目来培训各个级别的未来数学家，从为每年一度的美国高中生数学奥林匹克竞赛做出贡献，到监督本科生、研究生和博士后研究员的研究，再到为已建立的数学家组织会议和研讨会。在过去的十年中，首席研究员引入并研究了一系列称为结接触同源性的结不变量，它是通过计算某些辛中的全纯曲线而产生的流形，使用由 Gromov 和 Floer 首创的辛几何方法，最近在 Eliashberg、Givental 和 Hofer 的辛场论中达到顶峰。先前的工作表明，结接触同源性是一个稳健的不变量，可以有效区分结，并且包含有关结的经典拓扑信息。 2012 年，Aganagic、Ekholm、Vafa 和首席研究员发现结接触同源性与弦理论和镜像对称性具有意想不到的潜在强大关系：推测增广多项式，即从结接触同源性导出的结不变量等于 Aganagic 和 Vafa 的 Q 变形 A 多项式，它出现在拓扑弦的上下文中。首席研究员将使用拉格朗日填充和格罗莫夫-维滕势来解决这个猜想。这可能会在不同的方向上产生重大影响：在纽结理论中，它将建立 AJ 猜想的一个变体；在镜像对称方面，它将产生一种通过辛场论构建镜像 Calabi-Yau 3 倍的新方法；在拓扑弦理论中，它将为最近的结果提供数学基础。在相关工作中，首席研究员将开发和加强辛场论某些方面的代数框架，包括结接触同调和辛同调。在这种情况下，新的代数工具，例如微分分级代数的表示论，将使人们能够更有效地解决辛几何中的问题，特别是通过分析辛流形上的韦恩斯坦结构以及接触流形中的勒让结和横结。