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 and an analogous one‐parameter family of bi‐cubic CY hypersurfaces in . 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.