  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.
