Approximate Ricci-flat Metrics for Calabi-Yau Manifolds

Abstract

We outline a method to determine analytic Kähler potentials with associated approximately Ricci-flat Kähler metrics on Calabi-Yau manifolds. Key ingredients are numerically calculating Ricci-flat Kähler 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 Kähler potentials, including the explicit dependence on the complex structure parameter. We find that these Kähler potentials only depend on the modulus of the complex structure parameter.