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 X relative to a framed Aganagic-Vafa brane L and closed geometry on a toric Calabi-Yau 4-orbifold X, we consider the Hori-Vafa mirrors X^ and X^, where the mirror of L can be given by a family of hypersurfaces Y X^. We show that the Picard-Fuchs system associated to X extends that associated to X and characterize the full solution space in terms of the open string data. Furthermore, we construct a correspondence between integral 4-cycles in X^ and relative 3-cycles in (X^, 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 X^ with that on the middle-dimensional relative cohomology of (X^, Y) up to a Tate twist.
Song Yu (Mon,) studied this question.