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

2024-12-27

AI Summary

Explicit construction of Ricci-flat metrics on Calabi–Yau manifolds remains a long-standing challenge, as conventional geometric and analytic methods rarely yield closed-form solutions. Method: We propose a novel paradigm integrating differential geometry, algebraic geometry, and physics-informed neural networks (PINNs), leveraging newly discovered extrinsic symmetries—beyond the manifold’s intrinsic structure—that Ricci-flat metrics can possess. Contribution/Results: We prove the existence of such symmetries and derive the unique compact analytic solution on the complex structure moduli space. Using model distillation and symbolic regression, we enhance interpretability and obtain the first closed-form Kähler metrics with near-zero scalar curvature. Our approach achieves state-of-the-art Ricci curvature error across multiple benchmark Calabi–Yau manifolds, providing new computational tools for string theory compactifications and geometric machine learning.

Technology Category

Machine Learning: Learning with ManifoldsNatural Language Processing: Interpretability, Analysis, and Evaluation of NLP ModelsHumans and AI: Explainable AI (XAI) for Human Understanding

Application Category

Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphsEconomics, Online Markets and Human Computation: Economic aspects and information design of Web, GenAI, and cloud computingUser Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization

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 achieve state-of-the-art reductions in Ricci curvature for multiple Calabi-Yau manifolds. We conclude by distilling the ML models to obtain for the first time closed form expressions for Kahler metrics with near-zero scalar curvature.

Problem

• Calabi-Yau Manifolds
• Smooth Spatial Structures
• Machine Learning Estimation

Innovation

• Machine Learning
• Mathematical Symmetry
• Kahler Metric Estimation