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

TL;DR

This paper combines theoretical symmetry analysis with machine learning to construct explicit Ricci-flat metrics on Calabi-Yau hypersurfaces, providing new analytic forms and reducing curvature with neural networks.

Contribution

It formalizes symmetry conditions that lead to compact metric representations and integrates these insights into neural networks to improve metric approximation accuracy.

Findings

Symmetries imply compact Ricci-flat metric representations.

Analytic forms derived for specific Calabi-Yau loci.

Neural networks reduced Ricci curvature and produced near-zero scalar curvature metrics.

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.