Effective superpotentials of Type II D-brane/F-theory on compact complete intersection Calabi–Yau threefolds

In this paper, by generalizing the GKZ system for Picard–Fuchs equations under the toric description of compact complete intersection Calabi–Yau (CICY) construction, we extend the calculation of effective superpotentials to the case of compact CICY. As seen in several one and two bulk deformation moduli models, this generalized method is effective to calculate both on-shell and off-shell superpotentials. Furthermore, through the mirror symmetry technique, the Ooguri–Vafa invariants are extracted from the A-model instanton expansion. It is also noteable that the chosen discrete symmetrical groups, ℤ2 and ℤ3, between the holomorphic curves wrapped by D-branes, play important roles in superpotential calculations.