  Associativity of the quantum product ensures flatness of the Dubrovin
connection and is the basis for Hodge-theoretic mirror symmetry of Calabi-Yau
threefolds. We use ring and module structure on cohomology pertaining to a
Lagrangian submanifold to define an extension of the A-model Variation of mixed
Hodge structure that matches the predictions from extended mirror symmetry. Our
construction makes contact with axioms for open Gromov-Witten theory recently
proposed by Solomon-Tukachinsky.
