Hints for off-shell mirror symmetry in type II/F-theory compactifications

We perform a Hodge theoretic study of parameter dependent families of D-branes on compact Calabi-Yau manifolds in type II and F-theory compactifications. Starting from a geometric Gauss-Manin connection for B-type branes we study the integrability and flatness conditions. The B-model geometry defines an interesting ring structure of operators. For the mirror A-model this indicates the existence of an open-string extension of the so-called A-model connection, whereas the discovered ring structure should be part of the open-string A-model quantum cohomology. We obtain predictions for genuine Ooguri-Vafa invariants for Lagrangian branes on the quintic in P4 that pass some non-trivial consistency checks. We discuss the lift of the brane compactifications to F-theory on Calabi-Yau four-folds and the effective couplings in the effective supergravity action as determined by the N=1 special geometry of the open-closed deformation space.