The log-open correspondence for two-component Looijenga pairs

TL;DR

This paper establishes a deep correspondence between logarithmic Gromov-Witten invariants of two-component Looijenga pairs and open Gromov-Witten invariants of certain toric Calabi-Yau threefolds, confirming a conjecture and enabling new computational methods.

Contribution

It proves an all-genus correspondence linking surface and threefold Gromov-Witten theories, settling a conjecture and introducing a new computational approach.

Findings

Confirmed the conjecture for two boundary components.

Derived BPS integrality for the invariants.

Enabled computations via the topological vertex method.

Abstract

A two-component Looijenga pair is a rational smooth projective surface with an anticanonical divisor consisting of two transversally intersecting curves. We establish an all-genus correspondence between the logarithmic Gromov-Witten theory of a two-component Looijenga pair and open Gromov-Witten theory of a toric Calabi-Yau threefold geometrically engineered from the surface geometry. This settles a conjecture of Bousseau, Brini and van Garrel in the case of two boundary components. We also explain how the correspondence implies BPS integrality for the logarithmic invariants and provides a new means for computing them via the topological vertex method.