Type II/F-theory Superpotentials with Several Deformations and N=1 Mirror Symmetry

We present a detailed study of D-brane superpotentials depending on several open and closed-string deformations. The relative cohomology group associated with the brane defines a generalized hypergeometric GKZ system which determines the off-shell superpotential and its analytic properties under deformation. Explicit expressions for the N=1 superpotential for families of type II/F-theory compactifications are obtained for a list of multi-parameter examples. Using the Hodge theoretic approach to open-string mirror symmetry, we obtain new predictions for integral disc invariants in the A model instanton expansion. We study the behavior of the brane vacua under extremal transitions between different Calabi-Yau spaces and observe that the web of Calabi-Yau vacua remains connected for a particular class of branes.