We present a collection of explicit formulas for the minimum volume of Sasaki-Einstein 5-manifolds. The cone over these 5-manifolds is a toric Calabi-Yau 3-fold. These toric Calabi-Yau 3-folds are associated with an infinite class of 4d N=1 supersymmetric gauge theories, which are realized as worldvolume theories of D3-branes probing the toric Calabi-Yau 3-folds. Under the AdS/CFT correspondence, the minimum volume of the Sasaki-Einstein base is inversely proportional to the central charge of the corresponding 4d N=1 superconformal field theories. The presented formulas for the minimum volume are in terms of geometric invariants of the toric Calabi-Yau 3-folds. These explicit results are derived by implementing machine learning regularization techniques that advance beyond previous applications of machine learning for determining the minimum volume. Moreover, the use of machine learning regularization allows us to present interpretable and explainable formulas for the minimum volume. Our work confirms that, even for extensive sets of toric Calabi-Yau 3-folds, the proposed formulas approximate the minimum volume with remarkable accuracy.

通过实施机器学习规范化技术，我们提出了最小维度规卷的一系列明确公式，这些公式以几何不变量的形式给出了拟物理概述。

机器学习正则化对于拓扑的Calabi-Yau 3折面最小体积公式的应用