利用对称性发现卡拉比-丘流形度量的简洁表达，提升机器学习精度并实现闭式解。
Symbolic Approximations to Ricci-flat Metrics Via Extrinsic Symmetries of Calabi-Yau Hypersurfaces
- 基于外在对称性分析，揭示度量对称性超越流形本身的规律。
- 在特定复结构模空间中，唯一确定平坦度量并给出解析表达式。
- 结合理论结果改进神经网络，首次获得近零标量曲率的闭式凯勒度量。
自丘成桐证明卡拉比-丘流形上存在平坦度量以来，其显式构造始终是弦论与代数几何的核心难题。本文研究三维费马型卡拉比-丘n-fold及其单参数变形的机器学习逼近度量，发现某些情况下度量对称性高于底流形本身。我们证明此类对称性导致平坦度量在特定复结构模空间中具有惊人紧凑的表示形式，并在这些点上唯一确定度量，给出解析表达式。将理论结果融入神经网络后，显著降低多个卡拉比-丘流形的里奇曲率。最终，通过提炼机器学习模型，首次得到标量曲率接近零的闭式凯勒度量。
原文摘要 · Abstract (English)
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.
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