Symplectic Field Theory, its interactions and applications

辛场论（SFT）项目是几年前由a . Givental， H. Hofer和该提案的PI发起的，最初有两个主要目标。首先，以拓扑场论的精神发展辛流形中全纯曲线理论的形式主义（Gromov-Witten理论）；其次，寻找接触流形及其Legendrian子流形的新不变量。然而，后来人们发现了SFT与其他数学领域的许多新的联系，特别是与可积系统理论和弦拓扑的联系。项目拟建阶段的主要研究方向是：完成分析基础SFT；进一步发展和扩展了SFT的代数形式，使其在Gromov-Witten理论中包括TCFT和Givental的环群形式；发展了完整的相对版本的SFT，它将充分描述具有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量子可积系统与Gromov-Witten理论经典可积系统关系的研究SFT中不变量计算技术的发展；SFT在低维拓扑中的应用。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许多不同的数学领域产生更大的影响，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。数学物理也对它感兴趣。本课题应进一步明确科学理论与这些学科之间的关系。该项目的目标之一是写一本书作为基础参考，以及对SFT应用感兴趣的数学家的“用户指南”。

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.