Machine Learning Sasakian and G2 topology on contact Calabi-Yau 7-manifolds

Aug 5, 2026

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A machine learning framework studies topological quantities tied to the Sasakian and G2 geometries of contact Calabi-Yau 7-manifolds. Datasets are computed for Sasakian Hodge numbers and for the Crowley-Nrdström invariant of the natural G2 structure on the 7-dimensional link of weighted projective Calabi-Yau 3-fold hypersurface singularities. Learning from P4(w) weights alone with neural networks and symbolic regression reaches R2 scores 0.969 and 0.993, respectively.

LanguageEnglish

CategoryResearch & Report

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Pages12

Reading Time30 min

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Introduction

Motivation

Topological invariants of CY links

Machine Learning Sasakian and G2 topology on contact Calabi-Yau 7-manifolds Daattavya Aggarwala , Yang-Hui Heb,c,d,e , Elli Heyesb,c , Edward Hirstf, Henrique N. S Earpg , Toms S. R. Silvag a Department of Computer Science and Technology, University of Cambridge, CB3 0FD, UK b Department of Mathematics, City, University of London, EC1V 0HB, UK c London Institute for Mathematical Sciences, Royal Institution, London, W1S 4BS, UK d Merton College, University of Oxford, OX1 4JD, UK e School of Physics, NanKai University, Tianjin, 300071, P. R. China f Centre for Theoretical Physics, Queen Mary, University of London, E1 4NS, UK g Institute of Mathematics, Statistics and Scientific Computing, University of Campinas (Unicamp), 13083 -859, Brazil Abstract We propose a machine learning approach to study topological quantities related to the Sasakian and G2-geometries of contact Calabi-Yau 7-manifolds. Specifically, we compute datasets for certain Sasakian Hodge numbers and for the Crowley-Nrdstrom invariant of the natural G2-structure of the 7-dimensional link of a weighted projective Calabi-Yau 3-fold hypersurface singularity, for 7549 of the 7555 possible P4 (w) projective spaces. These topological quantities are then machine learnt with high performance scores, where learning the Sasakian Hodge numbers from the P4 (w) weights alone, using both neural networks and a symbolic regressor which achieve R2 scores of 0 .969 and 0 .993 respectively. Additionally, properties of the respective Grbner bases are well-learnt, leading to a vast improvement in computation speeds which may be of independent interest. The data generation and analysis further induced novel conjectures to be raised. Keywords: G2-manifolds, machine learning, Hodge numbers, Crowley-Nrdstrom invariant, contact Calabi-Yau manifolds Report Number: QMUL-PH-23-14 1. Introduction Motivation. Since its inception in 2017 [1–4], the study of CalabiYau (CY) manifolds in the context of string theory compactifications, using machine learning techniques has flourished, encompassing a wide array of investigations. Notably, these methods have been employed to predict Hodge numbers [5– 9], learn Ricci-flat Calabi-Yau metrics [10–14], forecast line bundle cohomologies [15], generate new Calabi-Yau manifolds [16], and uncover volume bounds on Sasaki-Einstein manifolds [17] . Furthermore, machine learning techniques have found various other applications in geometry and physics [18–28] . For an extensive review, see [29, 30] . As important as CalabiYau compactification is to string theory, 7-manifolds of holonomy G2 are crucial to M-theory compactification [31, 32] . Nonetheless, there have yet been no successful applications of machine learning in the context of G2-geometry, let alone on compact both closed and coclosed. We propose therefore to work with coclosed G2-structures on certain contact Calabi-Yau (cCY) 7-manifolds, which are closely related to the weighted projective Calabi-Yau 3-folds famously studied in [33] . Despite their unsuitability to M-theory, torsionful G2 structures retain relevance in the context of (3+7)-dimensional heterotic supergravity with flux, as demonstrated by [34–36] . Indeed, as shown by [37], one can explicitly solve the corresponding Strominger system on cCY 7-manifolds, by way of coclosed G2-structures together yielding non-trivial scalar and G2-instanton gauge fields, with constant dilaton, as well as an H-flux with prescribed Chern-Simons defect, in accordance to the ‘anomaly-free’ condition referred to as the heterotic Bianchi identity. Topological invariants of CY links. Contact Calabi-Yau manifolds were introduced by Tomassini and Vezzoni in [38], and arXiv :2310 .03064v2 [math .DG] 23 Feb 2024 manifolds with such special holonomy. This is no doubt due to the scarcity of dedicated databases for G2-manifolds, which in turn reflects the difficulty in describing (torsion-free) G2- manifolds systematically in

FAQs

What topological quantities are computed for contact Calabi-Yau 7-manifolds?

The work computes datasets for Sasakian Hodge numbers and for the Crowley-Nrdström invariant of the natural G2 structure on the corresponding 7-dimensional links.

How are the machine learning models trained in the study?

Models learn the target topological quantities from the P4(w) weight data alone, using both neural networks and a symbolic regressor.

What performance do the learning methods achieve?

Learning the Sasakian Hodge numbers from P4(w) weights attains R2 scores of 0.969 (neural networks) and 0.993 (symbolic regression), while related properties of Gröbner bases improve computation speeds.

Aug 5, 2026

Upload by Aurora