Numerical Ricci-flat metrics on K3

TL;DR

This work demonstrates that numerical Ricci-flat metrics on Calabi–Yau manifolds can be computed by exploiting their Kähler structure, encoding the geometry in a Kähler potential and reducing the Einstein equations to a Monge–Ampère problem. The authors develop a Gauss–Seidel–type relaxation on a patchwise lattice, fix complex and Kähler moduli a priori, and dynamically adjust the volume parameter to obtain high-resolution Ricci-flat metrics. As a proof of principle, they apply the method to highly symmetric Kummer K3 surfaces, obtaining metrics across a one-parameter moduli family and extracting curvature, induced geometry on exceptional divisors, and Laplacian spectra. The results indicate that the approach is practical for highly symmetric Calabi–Yau spaces and may extend to more general K3s and, with greater memory, to Calabi–Yau threefolds, with broad implications for mathematical geometry and string-theoretic applications.

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.