Numerical Ricci-flat metrics on K3

TL;DR

This paper introduces numerical algorithms for computing Ricci-flat metrics on Calabi-Yau manifolds, demonstrating efficiency and applicability to K3 surfaces with potential extension to three-folds.

Contribution

The authors develop and validate numerical methods exploiting Kahler geometry for efficient Ricci-flat metric computation on Calabi-Yau manifolds, specifically applied to K3 surfaces.

Findings

High-resolution metrics obtained in days on a desktop

Methods successfully compute geometric and spectral quantities

Potential extension to Calabi-Yau three-folds with symmetries

Abstract

We develop numerical algorithms for solving the Einstein equation on Calabi-Yau manifolds at arbitrary values of their complex structure and Kahler parameters. We show that Kahler geometry can be exploited for significant gains in computational efficiency. As a proof of principle, we apply our methods to a one-parameter family of K3 surfaces constructed as blow-ups of the T^4/Z_2 orbifold with many discrete symmetries. High-resolution metrics may be obtained on a time scale of days using a desktop computer. We compute various geometric and spectral quantities from our numerical metrics. Using similar resources we expect our methods to practically extend to Calabi-Yau three-folds with a high degree of discrete symmetry, although we expect the general three-fold to remain a challenge due to memory requirements.