利用类型Ⅱ弦理论/F理论对偶，第一次在平行和重合相区计算一个具体的双D膜系统的超势，并提取Ooguri-Vafa不变量。平行D膜相与重合D膜相间的相变也对应着D膜世界叶上规范理论规范对称性的提升U（1）×…×U（1）→U（n）。计算显示这两个相区的超势截然不同，并给出不同的Ooguri-Vafa不变量。这意味着相变的发生导致两个相区能谱结构的差异。
The superpotential is very important physical observation in the topological string theory, and it determines the F-term of the low-energy effective theory and the string vacuum structure. Ooguri-Vafa invariants, as the counting number of BPS states, can be extracted from the superpotential, giving rise to the energy spectrum of the D-brane system. We compute the superpotentials and Ooguri-Vafa invariants for complicated D-brane system, namely double D-branes system, in the parallel and coincident phases using the Type Ⅱ/F theory duality for the first time. The coincidence of parallel D-branes leads to the enhancement of gauge symmetry U(1)×…×U(1)→U(n) in terms of gauge theory on the worldvolume of the D-branes being known as the phase transition between the parallel D-brane phase and the coincident D-brane phase. We find the difference between the Ooguri-Vafa invariants for the two phases giving rise to the distinct spectra. It can be viewed as an evidence of the phase transition.
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