Open/closed Correspondence via Relative/local Correspondence

TL;DR

This paper establishes a novel correspondence linking open disk invariants of a toric Calabi-Yau 3-fold with boundary conditions to closed Gromov-Witten invariants of a related Calabi-Yau 4-fold, advancing understanding of open/closed string dualities.

Contribution

It provides a new proof of the open/closed correspondence without formal geometry and clarifies the relationship through intermediate relative Gromov-Witten invariants.

Findings

Established a correspondence between disk invariants and genus-zero closed Gromov-Witten invariants.

Connected open invariants to relative Gromov-Witten invariants of a partial compactification.

Linked relative invariants to closed invariants of a Calabi-Yau 4-fold.

Abstract

We establish a correspondence between the disk invariants of a smooth toric Calabi-Yau 3-fold $X$ with boundary condition specified by a framed Aganagic-Vafa outer brane $(L, f)$ and the genus-zero closed Gromov-Witten invariants of a smooth toric Calabi-Yau 4-fold $\widetilde{X}$, proving the open/closed correspondence proposed by Mayr and developed by Lerche-Mayr. Our correspondence is the composition of two intermediate steps: $\circ$ First, a correspondence between the disk invariants of $(X,L,f)$ and the genus-zero maximally-tangent relative Gromov-Witten invariants of a relative Calabi-Yau 3-fold $(Y,D)$, where $Y$ is a toric partial compactification of $X$ by adding a smooth toric divisor $D$. This correspondence can be obtained as a consequence of the topological vertex (Li-Liu-Liu-Zhou) and Fang-Liu where the all-genus open Gromov-Witten invariants of $(X,L,f)$ are identified with the formal relative Gromov-Witten invariants of the formal completion of $(Y,D)$ along the toric 1-skeleton. Here, we present a proof without resorting to formal geometry. $\circ$ Second, a correspondence in genus zero between the maximally-tangent relative Gromov-Witten invariants of $(Y,D)$ and the closed Gromov-Witten invariants of the toric Calabi-Yau 4-fold $\widetilde{X} = \mathcal{O}_Y(-D)$. This can be viewed as an instantiation of the log-local principle of van Garrel-Graber-Ruddat in the non-compact setting.