Machine learning regularization for the minimum volume formula of toric Calabi-Yau 3-folds

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 world volume theories of D3-branes probing the toric Calabi-Yau 3-folds. Under the $\mathrm{AdS}/\mathrm{CFT}$ 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.