The superpotential and geometric invariants of D-brane system on compact Calabi-Yau manifold

The effective D-brane superpotential determines the vacuum structure and is the generating function of the Ooguri-Vafa invariants. It is of great significance in both physics and mathematics. In this work, the superpotential of the system with three D-branes is calculated by mirror symmetry, GKZ-system (hypergeometric system of Gel'fand, Kapranov, and Zelevinski) method, and the type Ⅱ/F-theory duality. On compact Calabi-Yau manifold, the superpotenial and geometric invariants of D-brane system in the three phases are studied. The calculations show that the superpotentials in the three phases are different. This implies that the enhanced gauge symmetry in low energy theory and geometric singularity may lead to the phase transition.