Numerical Metrics for Complete Intersection and Kreuzer-Skarke Calabi-Yau Manifolds

TL;DR

This paper introduces neural network techniques and a software package to compute numerical Ricci-flat Calabi-Yau metrics on complete intersection and Kreuzer-Skarke manifolds, enabling precise geometric and topological calculations.

Contribution

It develops and implements neural network methods for Ricci-flat metric computation on complex Calabi-Yau manifolds, including a new software package cymetric.

Findings

Reliable volume and line bundle slope computations from Ricci-flat metrics

Successful application to various manifolds including quintic, bi-cubic, and Kreuzer-Skarke

Approximate Hermitian-Yang-Mills connection computation on a line bundle

Abstract

We introduce neural networks to compute numerical Ricci-flat CY metrics for complete intersection and Kreuzer-Skarke Calabi-Yau manifolds at any point in K\"ahler and complex structure moduli space, and introduce the package cymetric which provides computation realizations of these techniques. In particular, we develop and computationally realize methods for point-sampling on these manifolds. The training for the neural networks is carried out subject to a custom loss function. The K\"ahler class is fixed by adding to the loss a component which enforces the slopes of certain line bundles to match with topological computations. Our methods are applied to various manifolds, including the quintic manifold, the bi-cubic manifold and a Kreuzer-Skarke manifold with Picard number two. We show that volumes and line bundle slopes can be reliably computed from the resulting Ricci-flat metrics. We also apply our results to compute an approximate Hermitian-Yang-Mills connection on a specific line bundle on the bi-cubic.