Symbolic Approximations to Ricci-flat Metrics Via Extrinsic Symmetries of Calabi-Yau Hypersurfaces

TL;DR

The paper addresses the long-standing challenge of constructing explicit Ricci-flat metrics on Calabi–Yau manifolds by leveraging machine learning and novel symmetry ideas. It introduces extrinsic ambient-space symmetries that constrain the flat metric and enable compact representations, then demonstrates how neural nets (notably ModNet) can efficiently learn the correction potential $ ext{phi}$ for Fermat CYs and related families. By distilling ML outputs into symbolic expressions, the work yields closed-form approximations with far fewer parameters while preserving essential geometric properties, and it derives exact metric form on particular loci. Overall, the results offer a pathway toward interpretable, near-zero scalar curvature Kähler metrics with potential applications in string theory and algebraic geometry.

Abstract

Ever since Yau's non-constructive existence proof of Ricci-flat metrics on Calabi-Yau manifolds, finding their explicit construction remains a major obstacle to development of both string theory and algebraic geometry. Recent computational approaches employ machine learning to create novel neural representations for approximating these metrics, offering high accuracy but limited interpretability. In this paper, we analyse machine learning approximations to flat metrics of Fermat Calabi-Yau n-folds and some of their one-parameter deformations in three dimensions in order to discover their new properties. We formalise cases in which the flat metric has more symmetries than the underlying manifold, and prove that these symmetries imply that the flat metric admits a surprisingly compact representation for certain choices of complex structure moduli. We show that such symmetries uniquely determine the flat metric on certain loci, for which we present an analytic form. We also incorporate our theoretical results into neural networks to reduce Ricci curvature for multiple Calabi--Yau manifolds compared to previous machine learning approaches. We conclude by distilling the ML models to obtain for the first time closed form expressions for Kahler metrics with near-zero scalar curvature.

Figures (14)

• Figure 1: The symmetry group of a K3 defining polynomial (right), $\mathbb{S}_4$, is also the automorphism group of the complete graph (left).
• Figure 2: The symmetry of a Tian-Yau manifold (right), and a matching graph (left).
• Figure 3: A complete pipeline that incorporates $\mathbb{C}^*$, $\mathbb{Z}_2$, $\mathbb{S}_n$, and $\mathbb{Z}_{n+1}^n$ symmetries into a NN model. Note that spectral features $Z_i\overline{Z_j}$ must be encoded into edge features of GNN layers in order to align with the permutation symmetries.
• Figure 4: Computation graph of attribution weights, with arrows denoting dependencies. Left: differentiation with respect to features is impossible as it overlaps with an earlier one. Right: correct implementation with both differentiations in feature space.
• Figure 5: Relative salience $a$ on K3 (log scale) on the test set. Larger score means higher attribution relative to other features so that total salience sums to $1$. Here $\mathbb{C}^*$ represents projective scaling symmetry, $\mathbb{Z}_2$ is conjugation invariance, $\mathbb{S}_n$ is the permutation group, $\mathbb{Z}_{n+1}^n$ is toric symmetry $(Z_j \mapsto e^{\frac{2i\pi}{n+1} k_j} Z_j)$, and $\operatorname{U}(1)^n$ is phase invariance $(Z_j \mapsto e^{i\phi_j} Z_j)$.

Theorems & Definitions (31)

• Proposition 3.1
• proof
• Conjecture 3.2: Extrinsic Symmetries
• Proposition 3.3: Existence of Individual $U(1)$ Symmetry
• proof
• Theorem 3.4: Integration Weights Identity
• proof
• Proposition 4.1
• Proposition 4.2
• proof