Approximate Ricci-flat Metrics for Calabi-Yau Manifolds

Abstract

We outline a method to determine analytic K\"ahler potentials with associated approximately Ricci-flat K\"ahler metrics on Calabi-Yau manifolds. Key ingredients are numerically calculating Ricci-flat K\"ahler potentials via machine learning techniques and fitting the numerical results to Donaldson's Ansatz. We apply this method to the Dwork family of quintic hypersurfaces in ${\mathbb{P}}^4$ and an analogous one-parameter family of bi-cubic CY hypersurfaces in ${\mathbb{P}}^2\times {\mathbb{P}}^2$. In each case, a relatively simple analytic expression is obtained for the approximately Ricci-flat K\"ahler potentials, including the explicit dependence on the complex structure parameter. We find that these K\"ahler potentials only depend on the modulus of the complex structure parameter.