Constructing and Machine Learning Calabi-Yau Five-folds

TL;DR

This work catalogs complete intersection Calabi–Yau five-folds realized as zeros of up to four polynomials in ambient products of projective spaces, generating 27,068 inequivalent configuration matrices and computing Euler numbers for all. Using adjunction and a novel symmetrised adjunction sequence, the authors determine a large portion of the Hodge data, obtaining 2,375 distinct Hodge diamonds across 12,433 non-product cases (53.7% of the non-product subset). They train classifier and regressor neural networks (including a convolutional variant) on the cohomological data to predict $h^{1,1}, h^{1,4}, h^{2,3}$, and $\eta$, finding highly successful learning for $h^{1,1}$ (about 96% exact accuracy; $R^2\approx0.91$) and strong but less exact performance for invariants with broader ranges. The results demonstrate promising avenues for leveraging machine learning to explore vast Calabi–Yau landscapes in higher dimensions and motivate future expansion to the full dataset and additional topological quantities relevant to string compactifications.

Abstract

We construct all possible complete intersection Calabi-Yau five-folds in a product of four or less complex projective spaces, with up to four constraints. We obtain $27068$ spaces, which are not related by permutations of rows and columns of the configuration matrix, and determine the Euler number for all of them. Excluding the $3909$ product manifolds among those, we calculate the cohomological data for $12433$ cases, i.e. $53.7 \%$ of the non-product spaces, obtaining $2375$ different Hodge diamonds. The dataset containing all the above information is available at https://www.dropbox.com/scl/fo/z7ii5idt6qxu36e0b8azq/h?rlkey=0qfhx3tykytduobpld510gsfy&dl=0 . The distributions of the invariants are presented, and a comparison with the lower-dimensional analogues is discussed. Supervised machine learning is performed on the cohomological data, via classifier and regressor (both fully connected and convolutional) neural networks. We find that $h^{1,1}$ can be learnt very efficiently, with very high $R^2$ score and an accuracy of $96\%$, i.e. $96 \%$ of the predictions exactly match the correct values. For $h^{1,4},h^{2,3}, η$, we also find very high $R^2$ scores, but the accuracy is lower, due to the large ranges of possible values.

Figures (2)

• Figure 1: These plots show the distribution of the invariants that we computed. Since $h^{1,1}$ has the smallest range, it is the most suitable piece of data in the machine learning context. Conversely, we can anticipate that the wide ranges of $h^{1,4}$, $h^{2,2}$ and $h^{2,3}$ make it very hard for the neural network to predict those numbers exactly. Finally, it is interesting to note how $h^{1,2}$ and $h^{1,3}$ vanish in the vast majority of cases, resulting in a very skewed sampling.
• Figure 2: The plots represent Hodge data of Calabi-Yau spaces of different dimensions, obtained via different constructions. CICY three-folds and CICY four-folds data were taken from 3F_data and 4F_data, respectively. The toric hypersurfaces data can be found at KSdata.