We examine the vacuum structure of 4D effective theories of moduli fields in spacetime compactifications with quantized background fluxes. Imposing the no-scale structure for the volume deformations, we numerically investigate the distributions of flux vacua of the effective potential in complex structure moduli and axio-dilaton directions for two explicit examples in Type IIB string theory and F-theory compactifications. It turns out that distributions of non-supersymmetric flux vacua exhibit a non-increasing functional behavior of several on-shell quantities with respect to the string coupling. We point out that this phenomena can be deeply connected with a previously-reported possible correspondence between the flux vacua in moduli stabilization problem and the attractor mechanism in supergravity, and our explicit demonstration implies that such a correspondence generically exist even in the framework of F-theory. In particular, we confirm that the solutions of the effective potential we explicitly evaluated in Type IIB and F-theory flux compactifications indeed satisfy the generalized form of the attractor equations simultaneously.

中文翻译：

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F 理论通量真空和吸引子方程

我们研究了具有量化背景通量的时空压缩模场的 4D 有效理论的真空结构。对体积变形施加无尺度结构，我们对 IIB 型弦理论和 F 理论紧化中的两个明确例子的复杂结构模量和轴-膨胀方向的有效势的通量真空分布进行了数值研究。事实证明，非超对称通量真空的分布表现出几个关于串耦合的壳上量的非增加函数行为。我们指出，这种现象可以与先前报道的模稳定问题中的通量真空与超重力中的吸引器机制之间的可能对应关系密切相关，并且我们的显式证明暗示即使在 F 理论的框架中也普遍存在这种对应关系。特别是，我们确认我们在 IIB 型和 F 理论通量压缩中明确评估的有效势的解确实同时满足吸引子方程的广义形式。