  We study a flat connection defined on the open-closed deformation space of
open string mirror symmetry for type II compactifications on Calabi-Yau
threefolds with D-branes. We use flatness and integrability conditions to
define distinguished flat coordinates and the superpotential function at an
arbitrary point in the open-closed deformation space. Integrability conditions
are given for concrete deformation spaces with several closed and open string
deformations. We study explicit examples for expansions around different limit
points, including orbifold Gromov-Witten invariants, and brane configurations
with several brane moduli. In particular, the latter case covers stacks of
parallel branes with non-Abelian symmetry.
