D-brane superpotentials, SU(2) Ooguri-Vafa invariants and TypeII/F -theory duality

The phase transitions are studied for the D-brane systems with multiple open-string moduli in terms of toric geometry: between the parallel D-brane phase corresponding to the Coulomb branch and the coincident phase corresponding to the Higgs branch. The two separated D-branes on compact Calabi-Yau 3-fold coincide developing the geometric singularity in the corresponding F-theory Calabi-Yau 4-fold in terms of TypeII/F theory duality, and the enhancement of gauge group $U(1)\times U(1)\rightarrow SU(2)$ in terms of gauge theory. For several D-brane system with various closed-string moduli, using the mirror symmetry and the typeII/F theory duality the A-model superpotentials are obtained from the B-model side for the two phases, and the Ooguri-Vafa invariants are extracted from the A-model superpotential. We find the discrete $Z_2$ symmetry of superpotentials in the two parallel D-branes phase of all the models which is a signal of decoupling of the parallel topological D-branes. Furthermore, the \emph{$U(1)$ Ooguri-Vafa invariants} for one of the two parallel D-branes are the same as the invariants for the D-brane system with only one D-brane. However they are different from the \emph{$SU(2)$ Ooguri-Vafa invariants} corresponding to the two D-branes coincident phase. We present these Ooguri-Vafa invariants for two phases with figures to observe the difference, and find that in two phases the points for invariants form a wave-packet respectively. The wave-packets for coincident phase are higher and wider than ones for parallel phase which means more complicate spectrum structure. This is an evidence of the phase transition between the Coulomb branch and the Higgs branch.