  In this paper we study several issues related to the generation of
superpotential induced by background Ramond-Ramond fluxes in compactification
of Type IIA string theory on Calabi-Yau four-folds. Identifying BPS solitons
with D-branes wrapped over calibrated submanifolds in a Calabi-Yau space, we
propose a general formula for the superpotential and justify it comparing the
supersymmetry conditions in D=2 and D=10 supergravity theories. We also suggest
a geometric interpretation to the supersymmetric index in the two-dimensional
effective theory in terms of topological invariants of the Calabi-Yau
four-fold, and estimate the asymptotic growth of these invariants from BTZ
black hole entropy. Finally, we explicitly construct new supersymmetric vacua
for Type IIA string theory compactification on a Calabi-Yau four-fold with
Ramond-Ramond fluxes.
