Approximate Ricci-flat Metrics for Calabi-Yau Manifolds

TL;DR

This paper introduces a machine learning-based method to approximate Ricci-flat metrics on Calabi-Yau manifolds, providing explicit analytic expressions that depend on complex structure parameters, aiding in geometric and string theory research.

Contribution

It presents a novel approach combining numerical techniques and Donaldson's Ansatz to derive explicit approximate Ricci-flat Kähler potentials for Calabi-Yau manifolds.

Findings

Successfully applied to Dwork quintic hypersurfaces and bi-cubic CY hypersurfaces.

Obtained simple analytic expressions for Kähler potentials with explicit complex structure dependence.

Found that Kähler potentials depend only on the modulus of the complex structure parameter.

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.