Constructing and Machine Learning Calabi-Yau Five-Folds

Motivated by their role in M-theory, F-theory, and S-theory compactifications, all possible complete intersections Calabi-Yau five-folds in a product of four or less complex projective spaces are constructed, with up to four constraints. A total of 27 068 spaces are obtained, 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, the cohomological data for 12 433 cases are calculated, i.e., 53.7% of the non-product spaces, obtaining 2375 different Hodge diamonds. The dataset containing all the above information is available here. 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. h^{1, 1} can be learnt very efficiently, with very high R² score and an accuracy of 96% is found, i.e., 96% of the predictions exactly match the correct values. For $h^{1,4},h^{2,3},\eta$, very high R² scores are also found, but the accuracy is lower, due to the large ranges of possible values.