Moduli-dependent Calabi-Yau and SU(3)-structure metrics from Machine Learning

TL;DR

This work tackles the challenge of obtaining moduli-dependent Calabi–Yau and SU(3)-structure metrics for string compactifications. It introduces neural-network–based frameworks that either learn the Kähler potential (via an Hermitian matrix H) or directly learn the metric, and it extends to SU(3) structures with torsion by employing an ansatz or direct learning approaches. The methods are demonstrated on a one-parameter quintic hypersurface family, showing competitive accuracy and significant speed-ups over traditional Donaldson-based computations, while naturally incorporating complex structure moduli. The results open avenues for computing moduli-dependent kinetic terms, Yukawa couplings, and swampland-relevant spectra, and lay groundwork for extending to non-Kähler geometries, more general CYs, and other special holonomy manifolds.

Abstract

We use machine learning to approximate Calabi-Yau and SU(3)-structure metrics, including for the first time complex structure moduli dependence. Our new methods furthermore improve existing numerical approximations in terms of accuracy and speed. Knowing these metrics has numerous applications, ranging from computations of crucial aspects of the effective field theory of string compactifications such as the canonical normalizations for Yukawa couplings, and the massive string spectrum which plays a crucial role in swampland conjectures, to mirror symmetry and the SYZ conjecture. In the case of SU(3) structure, our machine learning approach allows us to engineer metrics with certain torsion properties. Our methods are demonstrated for Calabi-Yau and SU(3)-structure manifolds based on a one-parameter family of quintic hypersurfaces in $\mathbb{P}^4.$

Figures (12)

• Figure 1: Schematic overview of how models predicting the Hermitian matrix $H$ ( left) and the metric $g_{a\bar{b}}$ ( right) are designed. The respective models are neural networks of different complexity.
• Figure 2: (Left): Errors $\sigma$ for Donaldson's algorithm as a function of $\psi$ for random sampling. (Middle): Values of $\psi$ used in the training and evaluation set. (Right): Comparison of $\sigma$ for different approximation approaches.
• Figure 3: (Left): Errors $\sigma$ for Donaldson's algorithm as a function of $\psi$ for sparse sampling. (Middle): Values of $\psi$ used in the training and evaluation set. (Right): Comparison of $\sigma$ for different approximation approaches.
• Figure 4: $\sigma$ accuracies at $k=6$ achieved by the dense network with one and two hidden layers. The shaded area indicates the range of $|\psi|$ that was not used during training, and thus shows the extrapolation behavior of the networks. For reference, the $\sigma$ accuracy achieved by Donaldson's algorithm for the same range of $|\psi|$ is shown. The dashed line corresponds to the extrapolation of using Donaldson's balanced metric at $\psi=100$ over real values of $\psi$. The error band in each case corresponds to the maximal and minimal value obtained respectively when evaluating the $\sigma$ accuracy at different angles.
• Figure 5: Evolution of the training loss during training. Left: Optimizing the NN with all three losses. Middle: Optimizing the NN without Kähler loss (i.e. $\lambda_2=0$). Right: Optimizing the NN without overlap loss (i.e. $\lambda_3=0$).