Open Gromov–Witten theory on symplectic manifolds and symplectic cutting

Let (X,ω)(X,ω) be a symplectic manifold and LL be a Lagrangian submanifold diffeomorphic to SnSn, RPnRPn, or a Lens space of a certain type. Using the symplectic cut and symplectic sum constructions, we express the open Gromov–Witten invariants of (X,L)(X,L) in terms of open Gromov–Witten invariants of a pair (X−,L)(X−,L) determined by LL and the standard Gromov–Witten invariants of a symplectic manifold X+X+ determined by (X,L)(X,L). We also describe other applications of this approach.