Gross-Siebert's slab functions and open GW invariants for toric Calabi-Yau manifolds

TL;DR

This paper establishes a connection between slab functions in the Gross-Siebert program and open Gromov-Witten invariants for toric Calabi-Yau manifolds, confirming a conjecture and illustrating tropical-symplectic correspondence.

Contribution

It proves the conjecture that slab functions equal generating functions of open GW invariants, linking tropical and symplectic geometry in the open sector.

Findings

Confirmed the Gross-Siebert conjecture on slab functions and open GW invariants.

Established a correspondence between tropical and symplectic geometry.

Utilized the open mirror theorem to prove the equality.

Abstract

This paper derives an equality between the slab functions in Gross-Siebert program and generating functions of open Gromov-Witten invariants for toric Calabi-Yau manifolds, and thereby confirms a conjecture of Gross-Siebert on symplectic enumerative meaning of slab functions. The proof is based on the open mirror theorem of Chan-Cho-Lau-Tseng \cite{CCLT13}. It shows an instance of correspondence between tropical and symplectic geometry in the open sector.