Machine Learned Calabi-Yau Metrics and Curvature

TL;DR

This paper employs neural networks to approximate Ricci-flat Calabi-Yau metrics on complex manifolds, analyzing their topological and geometric properties, including stability, curvature clustering, and characteristic inequalities, with implications for geometry and string theory.

Contribution

It introduces neural network methods for approximating Calabi-Yau metrics on smooth and singular varieties, and investigates their topological and curvature properties.

Findings

Neural network models influence the numerical stability of topological characteristic computations.

High curvature regions cluster near singular points, as shown by persistent homology.

Numerical approximations satisfy a Bogomolov--Yau type inequality and a related topological identity.

Abstract

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 family of quartic twofolds and the Dwork family of quintic threefolds, we study characteristic forms on these geometries. We observe that the numerical stability of the numerically computed topological characteristic is heavily influenced by the choice of the neural network model, in particular, we briefly discuss a different neural network model, namely Spectral networks, which correctly approximate the topological characteristic of a Calabi-Yau. Using persistent homology, we show that high curvature regions of the manifolds form clusters near the singular points. 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.