**Off Shell Mirror Symmetry**

In these projects we performed 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 studied 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 obtained predictions for genuine Ooguri-Vafa invariants for Lagrangian branes on the quintic in P4 that pass some non-trivial consistency checks. We discussed the lift of the brane compactifications to F-theory on Calabi-Yau 4-folds and the effective couplings in the effective supergravity action as determined by the N = 1 special geometry of the open-closed deformation space.

#### Further Literature

- Flat Connections in Open String Mirror Symmetry.

Murad Alim, Michael Hecht, Hans Jockers, Peter Mayr, Adrian Mertens, Masoud Soroush

arXiv:1110.6522. JHEP 1206 (2012) 138. - Type II/F-theory Superpotentials with Several Deformations and N=1 Mirror Symmetry.

Murad Alim, Michael Hecht, Hans Jockers, Peter Mayr, Adrian Mertens, Masoud Soroush

arXiv:1010.0977. JHEP 1106 (2011) 103. - Hints for Off-Shell Mirror Symmetry in type II/F-theory Compactifications.

Murad Alim, Michael Hecht, Hans Jockers, Peter Mayr, Adrian Mertens, Masoud Soroush

arXiv:0909.1842. Nucl.Phys. B841 (2010) 303-338. - Mirror Symmetry for Toric Branes on Compact Hypersurfaces.

Murad Alim, Michael Hecht, Peter Mayr, Adrian Mertens

arXiv:0901.2937. JHEP 0909 (2009) 126. - Talk at KITP