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 X˜, proving the open/closed correspondence proposed by Mayr and developed by Lerche-Mayr. Our correspondence is the composition of two intermediate steps:•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.•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 X˜=OY(−D). This can be viewed as an instantiation of the log-local principle of van Garrel-Graber-Ruddat in the non-compact setting.