[Paper Review] Calabi-Yau Metrics with Kähler Moduli Dependence

Andrei Constantin, Andre Lukas+1 more|arXiv (Cornell University)|Mar 12, 2026

Geometry and complex manifolds0 citations

TL;DR

The paper presents a hybrid method combining neural networks and analytic reconstruction to produce closed-form Ricci-flat Kähler potentials with explicit dependence on Kähler moduli for Calabi–Yau threefolds, demonstrated on two h^{1,1}=2 examples.

ABSTRACT

We present a method to construct approximate analytic expressions for Ricci-flat Kähler metrics on Calabi-Yau threefolds with explicit dependence on the Kähler moduli. Our strategy combines numerical data obtained from machine learning with an explicit analytic Ansatz for the Kähler potential and symbolic regression methods. Specifically, we use neural networks to learn the Kähler potential at selected points in Kähler moduli space, fit this data to analytic expressions with Kähler moduli-dependent parameters, and determine an analytic form of these coefficients as functions of the Kähler moduli using symbolic regression. In this way, we reconstruct closed-form approximations to the Ricci-flat metric that retain explicit Kähler-moduli dependence. We apply this method to two Calabi-Yau threefolds with $h^{1,1}=2$, namely a bicubic hypersurface in $\mathbb{P}^2 imes \mathbb{P}^2$ and a bi-degree $(2,4)$ hypersurface in $\mathbb{P}^1 imes \mathbb{P}^3$, 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 Kähler potentials with percent-level accuracy and respect the discrete symmetry of the underlying manifold. 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 Kähler moduli.

Motivation & Objective

• Motivate the need for analytic, moduli-dependent Ricci-flat metrics in string phenomenology.
• Develop a hybrid strategy that combines numerical learning with analytic reconstruction to obtain explicit Kähler moduli dependence.
• Apply the method to Calabi–Yau threefolds with h^{1,1}=2 to demonstrate accuracy and symmetry properties.
• Provide closed-form approximations that reproduce numerically learned Kähler potentials within percent-level accuracy.

Proposed method

• Use neural networks to learn the Ricci-flat Kähler potential at selected points in Kähler moduli space.
• Propose an explicit analytic Ansatz for the Kähler potential with coefficients that depend on Kähler moduli.
• Impose a symmetry-adapted basis to reduce coefficients to N-invariant singlets in the presence of a discrete symmetry group.
• Apply symbolic regression to fit the moduli-dependent coefficients as analytic functions.
• Insert the moduli-dependent coefficients into the Kähler potential Ansatz to obtain a closed-form approximate metric.
• Demonstrate accuracy by comparing to numerically learned Kähler potentials with percent-level deviations.

Figure 1 : Left: final $\sigma$ -measure achieved by the neural network as a function of the moduli ratio $t_{1}/t_{2}$ . Right: average percentage deviation $\langle|K_{N}-K|/K_{N}\rangle$ as a function of $t_{1}/t_{2}$ , comparing different ansätze for the Kähler potential. The blue points corresp

Experimental results

Research questions

• RQ1 How can one construct analytic, moduli-dependent approximations to Ricci-flat Calabi–Yau metrics starting from numerical data?
• RQ2 Can a finite-dimensional, symmetry-respecting Ansatz capture explicit dependence on multiple Kähler moduli while remaining computationally tractable?
• RQ3 What is the accuracy of hybrid neural-network plus analytic reconstruction in reproducing the Kähler potential across moduli space?
• RQ4 Do discrete symmetries simplify the form of the Kähler potential without compromising validity or symmetry properties?
• RQ5 How does the method perform on different Calabi–Yau geometries with two Kähler moduli (h^{1,1}=2)?

Key findings

• The method yields closed-form analytic expressions for the Kähler potential with explicit Kähler-moduli dependence.
• For the bicubic X in P^2×P^2, percent-level agreement is achieved between analytic and numerically learned Kähler potentials.
• For the (2,4) hypersurface in P^1×P^3, a symmetry-adapted invariant basis yields accurate moduli-dependent coefficients.
• Symbolic regression provides analytic expressions for the moduli-dependent coefficients that reproduce the numerical data well.
• Off-diagonal coefficients are found to be suppressed, and the resulting expressions respect the discrete symmetries of the manifolds.
• The approach creates a bridge between numeric metric data and analytic constructions, enabling systematic moduli-space studies.

Figure 2 : Best-fit coefficients as functions of $t_{1}/t_{2}$ for the $(k_{1},k_{2})=(2,2)$ truncation. Out of the $36\times 36$ entries of the Hermitian matrix $\alpha_{IJ}$ , only $36$ , the diagonal entries, are non-vanishing within numerical accuracy; these naturally organise into eight groups,