Machine learning Sasakian and $G_2$ topology on contact Calabi-Yau 7-manifolds

Physics Letters B, Vol. 850 – arXiv:2310.03064

We propose a machine learning approach to study topological quantities related to the Sasakian and \(G_2\)-geometries of contact Calabi–Yau \(7\)-manifolds. Specifically, we compute datasets for certain Sasakian Hodge numbers and for the Crowley–Nordström invariant of the natural \(G_2\)-structure of the \(7\)-dimensional link of a weighted projective Calabi–Yau \(3\)-fold hypersurface singularity, for \(7549\) of the \(7555\) possible \(\mathbb{P}^4(w)\) projective spaces.

These topological quantities are then learned with high performance scores, where the Sasakian Hodge numbers are predicted from the \(\mathbb{P}^4(w)\) weights alone using both neural networks and a symbolic regressor, achieving \(R^2\) scores of \(0.969\) and \(0.993\), respectively.

Additionally, properties of the respective Gröbner bases are well learned, leading to a vast improvement in computation speeds which may be of independent interest. The data generation and analysis further led to several novel conjectures.