Multi-Component Open/Relative/Local Correspondence

TL;DR

This paper establishes a comprehensive correspondence between genus-zero open Gromov-Witten invariants of toric Calabi-Yau 3-orbifolds with multiple components and various closed and relative invariants of higher-dimensional and intermediate geometries, advancing the understanding of open/closed Gromov-Witten theory.

Contribution

It introduces a multi-component correspondence linking open invariants with closed and relative invariants across different geometries, extending existing conjectures and principles in Gromov-Witten theory.

Findings

Proves a correspondence among open and closed invariants in multi-component settings.

Provides examples supporting the log/local principle and refined conjectures.

Establishes the multi-component case of the open/closed Gromov-Witten correspondence.

Abstract

For a toric Calabi-Yau 3-orbifold relative to s Aganagic-Vafa outer branes, we prove a correspondence among the genus-zero open Gromov-Witten invariants with maximal winding at each brane and: (i) closed invariants of a toric Calabi-Yau (3+s)-orbifold; (ii) formal relative invariants of a formal toric Calabi-Yau (FTCY) 3-orbifold with maximal tangency to s divisors; (iii) formal relative invariants of a sequence of FTCY intermediate geometries interpolating dimensions 3 and 3+s. The correspondence provides examples of the log/local principle of van Garrel-Graber-Ruddat in the multi-component setting and the refined conjecture of Brini-Bousseau-van Garrel via intermediate geometries. It also establishes the multi-component case of the open/closed correspondence proposed by Lerche-Mayr and studied by Liu-Yu. As an application, we obtain examples of the conjecture of Klemm-Pandharipande on the integrality of BPS invariants of higher-dimensional toric Calabi-Yau manifolds. Along the way, we set the basic stages of the relative Gromov-Witten theory of higher-dimensional FTCY orbifolds, generalizing the case of smooth 3-folds by Li-Liu-Liu-Zhou.