有效的D膜超势决定真空结构，是Ooguri-Vafa不变量的生成函数。它对物理和数学都具有重要意义。利用镜像对称性、GKZ系统和Type Ⅱ/F-理论对偶的方法，计算带有3个D膜的系统的超势；研究在紧致Calabi-Yau流形中，处于3个相中的D膜系统的超势和几何不变量。计算结果显示，这3个相区中得到的超势是不同的，这意味着增加的规范对称性和几何奇异性导致了相变。
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.
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