Symplectic Field Theory, its interactions and applications

辛场论、其相互作用和应用

• 批准号: 0707103
• 负责人: Yakov Eliashberg
• 金额: $ 56.33万
• 依托单位: Stanford University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-07-01 至 2014-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0707103&HistoricalAwards=false
• 关键词: Symplectic; Field; Theory; its; interactions

Symplectic Field Theory (SFT) project, initiated a few years ago by A. Givental, H. Hofer and the PI of this proposal, had two main initial goals. First, to develop a formalism for the theory of holomorphic curves in symplectic manifolds (Gromov-Witten theory) in a spirit of a Topological Field Theory, and second, to find new invariants of contact manifolds and their Legendrian submanifolds. However, later on there were discovered many new connections of SFT with other areas of Mathematics, and in particular with the theory of integrable systems and String Topology. The main directions of research of the proposed phase of the project are: completion of analytic foundations SFT; further development and expansion of the algebraic formalism of the SFT to include TCFT and Givental's loop groop formalism in Gromov-Witten theory; development of the full relative version of SFT, which would adequatelly describe the structure of the compactified moduli spaces of holomorphic curves with mixed asymptotic and Lagrangian boundary conditions; study of the relation between quantum integrable systems of SFT and classical integrable systems of Gromov-Witten theory; development of computational techniques for invariants arising in SFT; applications of SFT to low-dimensional topology. Symplectic Field Theory is at the crossroads of several mathematical disciplines and its development already had, and as a result of the current research should have even a bigger impact on a number of different areas of Mathematics, such as Symplectic Geometry, Hamiltonian Dynamics, String Topology, Low-dimensional Topology, Topology of Moduli Spaces and Theory of Integrable Systems. It is also of interest for Mathematical Physics. The current project should further clarify the relations between SFT and these disciplines. One of the goals of the project is to write a book intended as a basic reference, as well as a ``user guide" for mathematicians interested in applications of SFT.

辛场论（SFT）项目，几年前由A。Givental，H.霍费尔和本提案的PI， 有两个主要的初始目标。第一，在拓扑场论的精神下，发展辛流形中全纯曲线理论的形式（Gromov-Witten理论），第二，找到接触流形及其Legendrian子流形的新不变量。然而，后来发现了许多新的连接SFT与其他领域的数学，特别是与理论的可积系统和弦拓扑。研究的主要方向 拟议阶段 该项目的主要内容是： 完成分析基础SFT;进一步发展和扩展SFT的代数形式主义，包括TCFT和Givental的循环群形式主义， Gromov-Witten理论; SFT的完整相对版本的发展，这将充分描述与混合渐近和拉格朗日边界条件的全纯曲线的紧化模空间的结构; SFT的量子可积系统和Gromov-Witten理论的经典可积系统之间的关系的研究; SFT中产生的不变量的计算技术的发展; SFT低维拓扑的应用。 辛场论是几个数学学科的交叉点，它的发展已经有了，并且由于目前的研究应该对数学的一些不同领域产生更大的影响，例如辛几何，哈密顿动力学，弦拓扑，低维拓扑，模空间拓扑和可积系统理论。 它也是数学物理的兴趣。 本项目应进一步明确SFT与这些学科之间的关系。这个项目的目标之一是写一本书 预期 作为一个基本的参考，以及“用户指南”的数学家感兴趣的应用程序的SFT。