Symplectic Field Theory, its interactions and applications

Stanford University, Stanford CA

Investigators

Abstract

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.