Machine Learned Calabi-Yau Metrics and Curvature

TL;DR

The paper investigates neural-network-based approaches to approximate Ricci-flat Calabi–Yau metrics on both smooth and singular Calabi–Yau spaces, focusing on the Cefalú and Dwork deformation families. It introduces PhiModel and spectral networks within cymetric to generate CY metrics and uses persistent homology and Chern-class-based diagnostics to assess curvature distributions and topological invariants. Key findings show that spectral networks yield numerically stable topological quantities near singularities, reproduce BY-type inequalities, and achieve Euler-characteristic checks close to the expected 24, even in near-singular regimes. The work demonstrates the importance of global, well-defined φ-models for reliable global invariants and highlights practical implications for string phenomenology, including Yukawa couplings and Kähler moduli stabilization. Overall, the results advocate spectral-network–based approaches as a robust path toward scalable, topologically faithful Calabi–Yau metric approximations across broad CY datasets.

Abstract

Finding Ricci-flat (Calabi-Yau) metrics is a long standing problem in geometry with deep implications for string theory and phenomenology. A new attack on this problem uses neural networks to engineer approximations to the Calabi-Yau metric within a given Kähler class. In this paper we investigate numerical Ricci-flat metrics over smooth and singular K3 surfaces and Calabi-Yau threefolds. Using these Ricci-flat metric approximations for the Cefalú family of quartic twofolds and the Dwork family of quintic threefolds, we study characteristic forms on these geometries. We observe that the numerical stability of the numerically computed topological characteristic is heavily influenced by the choice of the neural network model, in particular, we briefly discuss a different neural network model, namely Spectral networks, which correctly approximate the topological characteristic of a Calabi-Yau. Using persistent homology, we show that high curvature regions of the manifolds form clusters near the singular points. For our neural network approximations, we observe a Bogomolov--Yau type inequality $3c_2 \geq c_1^2$ and observe an identity when our geometries have isolated $A_1$ type singularities. We sketch a proof that $χ(X~\smallsetminus~\mathrm{Sing}\,{X}) + 2~|\mathrm{Sing}\,{X}| = 24$ also holds for our numerical approximations.

Figures (32)

• Figure 1: The Cefalú (complex structure) deformation family of quartics \ref{['e:Cefalu']} (lower $\lambda$-plane), and their $\mathbb{Z}_2$ quotients (upper plane), together with the identifications $X_1\,{\approx}\,\{T^4/\mathbb{Z}_2\}$ and $X_{\frac{3}{4}}\,{\approx}\,(X_0/\mathbb{Z}_2)$, the latter identification labeled by "$\spadesuit$".
• Figure 2: Neural-network architecture for building the PhiModel using cymetric.
• Figure 3: Euler number for the Fermat quintic $Y_{\psi=0}$ computed using different numbers of sample points.
• Figure 4: Numerical values of \ref{['e:chiK3']} along the Cefalú pencil. Black points and error bars showing a $95\%$ confidence interval are associated to Fubini--Study results, while the red dots correspond to the machine learned metric approximation using fully-connected networks. See the Appendix for details on integration.
• Figure 5: Visualization of the real subset of $X_{\lambda^\sharp + \epsilon}$ in a patch $\{z_0 \neq 0\}$. The coloring is defined by the values of the trained spectral network $\phi$. (See Section \ref{['sec:spectralNNs']}.)

Theorems & Definitions (6)

• Proposition 1
• proof
• Conjecture 1
• Conjecture 2
• Proposition 2
• proof