Hodge-theoretic Open/Closed Correspondence

TL;DR

This paper develops a Hodge-theoretic framework for the open/closed string correspondence in mirror symmetry, extending Picard-Fuchs systems and relating periods of dual geometries.

Contribution

It introduces a Hodge-theoretic perspective on the open/closed correspondence, extending Picard-Fuchs systems and establishing a cycle-period correspondence.

Findings

Extended Picard-Fuchs system for the combined geometry.

Constructed a correspondence between cycles in dual geometries.

Identified variations of mixed Hodge structures with relative cohomology.

Abstract

We continue the B-model development of the open/closed correspondence proposed by Mayr and Lerche-Mayr, complementing the A-model study in the preceding joint works with Liu and providing a Hodge-theoretic perspective. Given a corresponding pair of open geometry on a toric Calabi-Yau 3-orbifold $\mathcal{X}$ relative to a framed Aganagic-Vafa brane $\mathcal{L}$ and closed geometry on a toric Calabi-Yau 4-orbifold $\widetilde{\mathcal{X}}$, we consider the Hori-Vafa mirrors $\mathcal{X}^\vee$ and $\widetilde{\mathcal{X}}^\vee$, where the mirror of $\mathcal{L}$ can be given by a family of hypersurfaces $\mathcal{Y} \subset \mathcal{X}^\vee$. We show that the Picard-Fuchs system associated to $\widetilde{\mathcal{X}}$ extends that associated to $\mathcal{X}$ and characterize the full solution space in terms of the open string data. Furthermore, we construct a correspondence between integral 4-cycles in $\widetilde{\mathcal{X}}^\vee$ and relative 3-cycles in $(\mathcal{X}^\vee, \mathcal{Y})$ under which the periods of the former match the relative periods of the latter. On the dual side, we identify the variations of mixed Hodge structures on the middle-dimensional cohomology of $\widetilde{\mathcal{X}}^\vee$ with that on the middle-dimensional relative cohomology of $(\mathcal{X}^\vee, \mathcal{Y})$ up to a Tate twist.