Open/closed Correspondence via Relative/local Correspondence

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.