Hodge-theoretic Open/Closed Correspondence

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 $\tilde{\mathcal{X}}$, we consider the Hori-Vafa mirrors ${\mathcal{X}}^{\vee }$ and ${\tilde{\mathcal{X}}}^{\vee }$, 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 $\tilde{\mathcal{X}}$ 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 ${\tilde{\mathcal{X}}}^{\vee }$ 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 ${\tilde{\mathcal{X}}}^{\vee }$ with that on the middle-dimensional relative cohomology of $({\mathcal{X}}^{\vee },\mathcal{Y})$ up to a Tate twist.

48 pages, 5 figures