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We present a detailed study of D-brane superpotentials depending on severalopen and closed-string deformations. The relative cohomology group associatedwith the brane defines a generalized hypergeometric GKZ system which determinesthe off-shell superpotential and its analytic properties under deformation.Explicit expressions for the N=1 superpotential for families of typeII/F-theory compactifications are obtained for a list of multi-parameterexamples. Using the Hodge theoretic approach to open-string mirror symmetry, weobtain new predictions for integral disc invariants in the A model instantonexpansion. We study the behavior of the brane vacua under extremal transitionsbetween different Calabi-Yau spaces and observe that the web of Calabi-Yauvacua remains connected for a particular class of branes.

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