Multi-Component Open/Relative/Local Correspondence

TL;DR

This work constructs and analyzes a comprehensive multi-component open/relative/local Gromov-Witten framework for toric Calabi-Yau orbifolds with multiple Aganagic-Vafa branes. By introducing relative formal toric Calabi-Yau (FTCY) orbifolds and a hierarchy of intermediate geometries, the authors establish a sequence of precise correspondences that connect genus-zero open invariants to formal relative and then to closed invariants, via localization on FTCY graphs. They prove open/relative/local correspondences and their iteration across intermediate steps, thereby realizing a multi-component log/local principle and refining BBvG20 in this setting; they also derive BPS integrality results for higher-dimensional toric CY manifolds. The framework unifies and extends open/closed GW theory, relative GW theory, and the log/local/open-closed web, and it is illustrated with a detailed C^3 example. The results provide new evidence for integrality conjectures (Klemm-Pandharipande) in higher dimensions and offer a robust toolkit for computing and comparing GW invariants across open, relative, and closed theories in multi-brane configurations.

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.

Figures (6)

• Figure 2: Partitions of $V(\Gamma)$ and $E(\Gamma)$.
• Figure 3: Subgraphs $\Gamma_i$ and $\Gamma_i^c$. In the situation of Remark \ref{['rem:U3+iVanish']}, the unique vertex $\widetilde{v}_i$ in $\Gamma_i$ is connected to each component of $\Gamma^c_i$ by an edge in $E_{0i}$.
• Figure 4: New vertex and edge near marking $n+i$ in the construction of the decorated graph $\epsilon(\vec{\Gamma})$ from $\vec{\Gamma}$.
• Figure 5: FTCY graph for the 3-dimensional formal relative geometry $(\hat{\mathcal{Y}}, \hat{\mathcal{D}})$.
• Figure 6: FTCY graph for the 4-dimensional formal relative geometry $(\hat{\mathcal{Y}}^{(1)}, \hat{\mathcal{D}}^{(1)})$.

Theorems & Definitions (30)

• Theorem 1.1: See Theorem \ref{['thm:OpenClosedStatement']}
• Theorem 1.2: See Theorem \ref{['thm:RelativeStatement']}
• Remark 2.1
• Definition 2.2
• Definition 2.3
• Example 2.4
• Definition 3.1
• Lemma 4.3
• Definition 5.1
• Theorem 5.2: FLT12