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For D-brane system with three D-branes on compact Calabi-Yau threefolds, thedual F-theory fourfolds for parallel/complete coincident/part coincidentD-brane system is constructed by the type II/F-theory duality. Completecoincident means that the three D-branes coincide and part coincidentrepresents the coincident of two of the three D branes. The low energyeffective superpotentials are calculated by mirror symmetry, GKZ-system methodand the type II/F-theory duality on the B-model side, respectively. Using themirror symmetry, A-model superpotentials and the Ooguri-Vafa invariants areobtained from the B-model side. These results indicate that the superpotentialcontributed by one of three parallel D-branes is identical with the D-branesystem with only one D-brane, which is a signal of decoupling of the paralleltopological D-branes. However, the superpotential and Oogrui-Vafa invariantsare different among parallel, complete coincident and part coincident D-branesystem, which show the evidence of the phase transition due to the enhancedgauge symmetry in the low energy theories and the geometrical singularity.

Superpotentials and Geometric Invariants of Parallel/Complete Coincident/Part Coincident D-brane System on Compact Calabi-Yau Manifold