On Matrix Factorizations, Residue Pairings and Homological Mirror Symmetry

We argue how boundary $B$-type Landau-Ginzburg models based on matrix factorizations can be used to compute exact superpotentials for intersecting $D$-brane configurations on compact Calabi-Yau spaces. In this paper, we consider the dependence of open-string, boundary changing correlators on bulk moduli. This determines, via mirror symmetry, non-trivial disk instanton corrections in the $A$-model. As crucial ingredient we propose a differential equation that involves matrix analogs of Saito's higher residue pairings. As example, we compute from this for the elliptic curve certain quantum products $m_2$ and $m_3$, which reproduce genuine boundary changing, open Gromov-Witten invariants.