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

Ever since Yau’s non-constructive existence proof of Ricci-flat metrics on Calabi–Yau (CY) 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 CY 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 on the Fermat manifold, for which we present an analytic form. We also incorporate our theoretical results into neural networks to reduce Ricci curvature for multiple CY manifolds compared to previous machine learning approaches. We conclude with distilling the ML models to obtain closed form expressions for Kähler metrics with near-zero scalar curvature, discovering them directly from numerical data.