Calabi-Yau four-, five-, sixfolds as Pwn hypersurfaces: Machine learning, approximation, and generation

Abstract

Calabi-Yau fourfolds may be constructed as hypersurfaces in weighted projective spaces of complex dimension five defined via weight systems of six weights. In this work, neural networks were implemented to learn the Calabi-Yau Hodge numbers from the weight systems, where gradient saliency and symbolic regression then inspired a truncation of the Landau-Ginzburg model formula for the Hodge numbers of any dimensional Calabi-Yau constructed in this way. The approximation always provides a tight lower bound, is shown to be dramatically quicker to compute (with computation times reduced by up to 4 orders of magnitude), and gives remarkably accurate results for systems with large weights. Additionally, complementary datasets of weight systems satisfying the necessary but insufficient conditions for transversality were constructed, including considerations of the interior point, reflexivity, and intradivisibility properties, overall producing a classification of this weight system landscape, further confirmed with machine learning methods. Using the knowledge of this classification and the properties of the presented approximation, a novel dataset of transverse weight systems consisting of seven weights was generated for a sum of weights ≤200, producing a new database of Calabi-Yau fivefolds, with their respective topological properties computed. Furthermore, an equivalent database of candidate Calabi-Yau sixfolds was generated with approximated Hodge numbers. Published by the American Physical Society 2024