Open GW theory on symplectic manifolds and symplectic cutting

TL;DR

This paper develops a method to compute open Gromov-Witten invariants of symplectic manifolds with Lagrangian submanifolds using symplectic cutting and summing techniques, linking them to known invariants.

Contribution

It introduces a novel approach to express open Gromov-Witten invariants via symplectic cut and sum constructions, connecting them to standard invariants.

Findings

Expressed open Gromov-Witten invariants in terms of simpler invariants.

Applied symplectic cut and sum techniques to compute invariants.

Provided new applications of the symplectic cutting approach.

Abstract

Let $(X,\om)$ be a symplectic manifold and $L$ be a Lagrangian submanifold diffeomorphic to $S^n$, $\R\P^n$, or a Lens space of a certain type. Using the symplectic cut and symplectic sum constructions, we express the open Gromov-Witten invariants of $(X,L)$ in terms of open Gromov-Witten invariants of a pair $(X_-,L)$ determined by $L$ and the standard Gromov-Witten invariants of a symplectic manifold $X_+$ determined by $(X,L)$. We also describe other applications of this approach.