Final Report Summary - OPENGWTRIANGLE (Three ideas in open Gromov-Witten theory)
Open Gromov-Witten theory is a branch of symplectic geometry that concerns the enumeration of J-holomorphic curves with boundary in a Lagrangian submanifold. It features rich interactions with physics as well as real algebraic geometry and classical closed Gromov-Witten theory. The project has contributed to the foundations of open Gromov-Witten theory, establishing a systematic and general method for defining and computing invariant counts of J-holomorphic disks based on the obstruction theory of Fukaya A-infinity algebras. The project has also formulated and proved an open analog of Witten’s conjecture regarding descendent integrals on the moduli space of curves. Other results include the proof that Floer cohomology is well defined for the Lagrangian tori of the SYZ conjecture, and that the Fukaya category of a hyperkahler manifold is local. In another direction, the project has established fundamental results on geodesics in the space of positive Lagrangian submanifolds, including examples of smooth solutions to the geodesic equation in any dimension, a theory of weak solutions, and a correspondence between geodesics and families of special Lagrangians with boundary.