  From the worldsheet perspective, the superpotential on a D-brane wrapping
internal cycles of a Calabi-Yau manifold is given as a generating functional
for disk correlation functions. On the other hand, from the geometric point of
view, D-brane superpotentials are captured by certain chain integrals. In this
work, we explicitly show for branes wrapping internal 2-cycles how these two
different approaches are related. More specifically, from the worldsheet point
of view, D-branes at the Landau-Ginzburg point have a convenient description in
terms of matrix factorizations. We use a formula derived by Kapustin and Li to
explicitly evaluate disk correlators for families of D2-branes. On the geometry
side, we then construct a three-chain whose period gives rise to the effective
superpotential and show that the two expressions coincide. Finally, as an
explicit example, we choose a particular compact Calabi-Yau hypersurface and
compute the effective D2-brane superpotential in different branches of the open
moduli space, in both geometric and worldsheet approaches.
