Machine Learned Calabi--Yau Metrics and Curvature

Finding Ricci-flat (Calabi--Yau) metrics is a long standing problem in geometry with deep implications for string theory and phenomenology. A new attack on this problem uses neural networks to engineer approximations to the Calabi--Yau metric within a given K\"ahler class. In this paper we investigate numerical Ricci-flat metrics over smooth and singular K3 surfaces and Calabi--Yau threefolds. Using these Ricci-flat metric approximations for the Cefal\'u and Dwork family of quartic twofolds and the Dwork family of quintic threefolds, we study characteristic forms on these geometries. Using persistent homology, we show that high curvature regions of the manifolds form clusters near the singular points, but also elsewhere. For our neural network approximations, we observe a Bogomolov--Yau type inequality $3c_2 \geq c_1^2$ and observe an identity when our geometries have isolated $A_1$ type singularities. We sketch a proof that $\chi(X~\smallsetminus~\mathrm{Sing}\,{X}) + 2~|\mathrm{Sing}\,{X}| = 24$ also holds for our numerical approximations.

Machine Learning Kreuzer--Skarke Calabi--Yau Threefolds

Using a fully connected feedforward neural network we study topological invariants of a class of Calabi--Yau manifolds constructed as hypersurfaces in toric varieties associated with reflexive polytopes from the Kreuzer--Skarke database. In particular, we find the existence of a simple expression for the Euler number that can be learned in terms of limited data extracted from the polytope and its dual.