Constructing and Machine Learning Calabi-Yau Five-folds
This work provides a comprehensive dataset and machine learning benchmarks for Calabi-Yau manifolds in theoretical physics and mathematics, but it is incremental as it extends similar analyses to five-folds.
The authors constructed all complete intersection Calabi-Yau five-folds in specific projective spaces, identifying 27,068 spaces and computing Euler numbers, with cohomological data for 12,433 cases yielding 2,375 distinct Hodge diamonds. They applied supervised machine learning to this dataset, achieving a 96% accuracy in predicting the Hodge number h^{1,1} and high R² scores for other invariants.
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.