In this note, we apply mathematical results for the volume of certain symmetric spaces to the problem of counting flux vacua in simple IIB Calabi–Yau compactifications. In particular, we obtain estimates for the number of flux vacua including the geometric factor related to the Calabi–Yau moduli space, in the large flux limit, for the FHSV model and some closely related models. We see that these geometric factors give rise to contributions to the counting formula that are typically not of order one and might potentially affect the counting qualitatively in some cases. We also note, for simple families of Calabi–Yau moduli spaces, an interesting dependence of the moduli space volumes on the dimension of the flux space, which in turn is governed by the Betti numbers of the Calabi–Yaus.

中文翻译：

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Flux vacua：大量的叙述

在这篇笔记中，我们将某些对称空间体积的数学结果应用于计算简单 IIB Calabi-Yau 紧化中的通量真空的问题。特别是，我们获得了对 FHSV 模型和一些密切相关模型在大通量限制中的通量真空数量的估计，包括与 Calabi-Yau 模量空间相关的几何因子。我们看到这些几何因素对计数公式的贡献通常不是一阶的，并且在某些情况下可能会定性地影响计数。我们还注意到，对于简单的 Calabi-Yau 模空间族，模空间体积对通量空间维度的有趣依赖性，而通量空间维度又受 Calabi-Yaus 的 Betti 数控制。