Calabi-Yau Metrics with K\"ahler Moduli Dependence

Abstract

We present a method to construct approximate analytic expressions for Ricci-flat K\"ahler metrics on Calabi-Yau threefolds with explicit dependence on the K\"ahler moduli. Our strategy combines numerical data obtained from machine learning with an explicit analytic Ansatz for the K\"ahler potential and symbolic regression methods. Specifically, we use neural networks to learn the K\"ahler potential at selected points in K\"ahler moduli space, fit this data to analytic expressions with K\"ahler moduli-dependent parameters, and determine an analytic form of these coefficients as functions of the K\"ahler moduli using symbolic regression. In this way, we reconstruct closed-form approximations to the Ricci-flat metric that retain explicit K\"ahler-moduli dependence. We apply this method to two Calabi-Yau threefolds with $h^{1,1}=2$, namely a bicubic hypersurface in $\mathbb{P}^2 \times \mathbb{P}^2$ and a bi-degree $(2,4)$ hypersurface in $\mathbb{P}^1 \times \mathbb{P}^3$, both of which admit nontrivial discrete symmetry groups that simplify the structure of the metric. In both cases, the resulting analytic expressions reproduce the numerically learned K\"ahler potentials with percent-level accuracy and yield a Ricci-flatness measure that remains sufficiently small across the sampled region. Our results represent a concrete bridge between purely numerical results for Calabi-Yau metrics and analytic constructions, opening the door to a systematic study of their dependence on K\"ahler moduli.