Machine Learning Regularization for the Minimum Volume Formula of Toric Calabi-Yau 3-folds

TL;DR

The paper addresses the challenge of obtaining explicit, interpretable formulas for the minimum volume $V_{min}$ of Sasaki-Einstein bases $Y_5$ over toric Calabi-Yau 3-folds, which in AdS/CFT are tied to the central charge of 4d $\mathcal{N}=1$ SCFTs. Building on prior work, it employs Lasso regularization on polynomial and logarithmic regression using features derived from toric diagrams, including $n$-enlarged diagrams, to yield sparse, explainable formulas with high predictive accuracy ($R^2$ around 0.98–0.993) across multiple datasets. The results demonstrate that a small set of toric-diagram features suffices to approximate $1/V_{min}$, enhancing interpretability and potential applicability to large classes of toric CY3-folds. This approach offers a practical, transparent tool for estimating central charges via geometric data, and suggests broader use of regularization to extract concise mathematical relations in string theory contexts.

Abstract

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.

Figures (9)

• Figure 1: (a) The brane tiling for the second phase of the zeroth Hirzebruch surface $F_0$, and (b) its corresponding toric diagram hirzebruch1968singularitiesbrieskorn1966beispieleMorrison:1998csFeng:2000mi.
• Figure 2: (a) The triangulated toric diagram for the zeroth Hirzebruch surface $F_0$, and (b) the corresponding normal vectors $\mathbf{u}_{i,j}$ for each unit triangle $\Delta_i$ in the triangulation.
• Figure 3: (a) The toric diagram $\Delta_1$ for the cone over $\text{dP}_1$, and (b) the corresponding $2$-enlarged toric diagram $\Delta_2$ with $n=2$.
• Figure 4: (a) Toric diagrams in datasets $S_\text{1a}$ and $S_\text{2a}$ are constrained by a $n_x\times n_y$ lattice box in $\mathbb{Z}^2$, whereas (b) toric diagrams in datasets $S_\text{1b}$ and $S_\text{2b}$ are constrained by a circle of radius $r$ with the center at $(0,0)\in\mathbb{Z}^2$.
• Figure 5: The distribution of expected minimum volumes $y=1/V_{min}$ for the datasets (a) $S_\text{1a}$, (b) $S_\text{1b}$, (c) $S_\text{2a}$ and (c) $S_\text{2b}$. The mean expected value $\overline{y}$ is indicated by a white line. The histograms for values of $y=1/V_{min}$ are obtained for bin sizes $\Delta y$ with the number of toric diagrams in $\text{bin}_h$ given by $N(\text{bin}_h)$.