Approximate Ricci-flat Metrics for Calabi-Yau Manifolds

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 mathbbP4 and an analogous one-parameter family of bi-cubic CY hypersurfaces in mathbbP2×mathbbP2. 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.