Symplectic Field Theory, its interactions and applications

辛场理论(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，在拓扑场论的精神下，建立了辛流形上的全纯曲线理论(Gromov-Witten理论)的形式；其次，找到了接触流形及其Legendrian子流形的新的不变量。然而，后来发现了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学的联系。该项目拟议阶段的主要研究方向是：完成解析基础SFT；进一步发展和扩展SFT的代数形式，将TCFT和Givental环槽形式包括在Gromov-Witten理论中；开发完整的相对版本的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致模空间的结构；研究SFT的量子可积系统与Gromov-Witten理论的经典可积系统之间的关系；SFT中不变量的计算技术的发展；SFT在低维拓扑中的应用。辛场论是几个数学学科的交叉学科，它的发展已经有了，作为目前研究的结果，应该会对数学的一些不同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。这也是数学物理的兴趣所在。目前的项目应该进一步澄清SFT和这些学科之间的关系。该项目的目标之一是编写一本书作为基本参考，并为对SFT应用感兴趣的数学家编写一本“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.