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In this paper, we extend the GKZ-system method to the more general case:compact Complete Intersection Calabi-Yau manifolds (CICY). For severalone-deformation modulus compact CICYs with D-branes, the on-shellsuperpotentials in this paper from the extended GKZ-system method are exactlyconsistent with published results obtained from other methods. We furthercompute the off-shell superpotentials of these models. Then we obtain both theon-shell and off-shell superpotentials for sev- eral two-deformation modulicompact CICYs with D-branes by using the extended GKZ-system method. Thediscrete symmetrical groups, Z2, Z3 and Z4, of the holo- morphic curves wrappedby D-branes play the important roles in computing the superpotentials, in somesense, they are the quantum symmetries of these models. Furthermore, throughthe mirror symmetry, the Ooguri-Vafa invariants are extracted from the A-modelinstanton expansion.

Effective Superpotentials of Type II D-brane/F-theory on Compact Complete Intersection Calabi-Yau Threefolds