Open/closed correspondence for the projective line

TL;DR

This paper establishes a correspondence linking disk invariants of the complex projective line with boundary conditions to genus-zero closed Gromov-Witten invariants of toric surfaces, providing a new bridge between open and closed string theories.

Contribution

It introduces a novel correspondence between open disk invariants and closed Gromov-Witten invariants for the projective line and toric surfaces.

Findings

Disk invariants of $P^1$ relate to closed Gromov-Witten invariants of toric surfaces.

The correspondence applies to boundary conditions specified by $S^1$-invariant Lagrangian submanifolds.

This bridges open and closed enumerative invariants in symplectic geometry.

Abstract

We establish a correspondence between the disk invariants of the complex projective line $\bP^1$ with boundary condition specified by an $S^1$-invariant Lagrangian sub-manifold $L$ and the genus-zero closed Gromov-Witten invariants of a toric surface $X$.