Machine Learned Calabi-Yau Metrics and Curvature

We present new invariant machine learning models that approximate the Ricci-flat metric on Calabi-Yau (CY) manifolds with discrete symmetries. We accomplish this by combining the $\phi$-model of the cymetric package with non-trainable, $G$-invariant, canonicalization layers that project the $\phi$-model's input data (i.e. points sampled from the CY geometry) to the fundamental domain of a given symmetry group $G$. These $G$-invariant layers are easy to concatenate, provided o...

We review briefly the characteristic topological data of Calabi--Yau threefolds and focus on the question of when two threefolds are equivalent through related topological data. This provides an interesting test case for machine learning methodology in discrete mathematics problems motivated by physics.

Using a fully connected feedforward neural network we study topological invariants of a class of Calabi--Yau manifolds constructed as hypersurfaces in toric varieties associated with reflexive polytopes from the Kreuzer--Skarke database. In particular, we find the existence of a simple expression for the Euler number that can be learned in terms of limited data extracted from the polytope and its dual.

In these lecture notes, we survey the landscape of Calabi-Yau threefolds, and the use of machine learning to explore it. We begin with the compact portion of the landscape, focusing in particular on complete intersection Calabi-Yau varieties (CICYs) and elliptic fibrations. Non-compact Calabi-Yau manifolds are manifest in Type II superstring theories, they arise as representation varieties of quivers, used to describe gauge theories in the bulk familiar four dimensions. Final...

The latest techniques from Neural Networks and Support Vector Machines (SVM) are used to investigate geometric properties of Complete Intersection Calabi-Yau (CICY) threefolds, a class of manifolds that facilitate string model building. An advanced neural network classifier and SVM are employed to (1) learn Hodge numbers and report a remarkable improvement over previous efforts, (2) query for favourability, and (3) predict discrete symmetries, a highly imbalanced problem to w...

In this work, we report the results of applying deep learning based on hybrid convolutional-recurrent and purely recurrent neural network architectures to the dataset of almost one million complete intersection Calabi-Yau four-folds (CICY4) to machine-learn their four Hodge numbers $h^{1,1}, h^{2,1}, h^{3,1}, h^{2,2}$. In particular, we explored and experimented with twelve different neural network models, nine of which are convolutional-recurrent (CNN-RNN) hybrids with the R...

Calabi-Yau spaces, or Kahler spaces admitting zero Ricci curvature, have played a pivotal role in theoretical physics and pure mathematics for the last half-century. In physics, they constituted the first and natural solution to compactification of superstring theory to our 4-dimensional universe, primarily due to one of their equivalent definitions being the admittance of covariantly constant spinors. Since the mid-1980s, physicists and mathematicians have joined forces in c...

We present a numerical computation, based on neural network techniques, of the physical Yukawa couplings in a heterotic string theory compactification on a smooth Calabi-Yau threefold with non-standard embedding. The model belongs to a large class of heterotic line bundle models that have previously been identified and whose low-energy spectrum precisely matches that of the MSSM plus fields uncharged under the Standard Model group. The relevant quantities for the calculation,...

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...

We develop numerical algorithms for solving the Einstein equation on Calabi-Yau manifolds at arbitrary values of their complex structure and Kahler parameters. We show that Kahler geometry can be exploited for significant gains in computational efficiency. As a proof of principle, we apply our methods to a one-parameter family of K3 surfaces constructed as blow-ups of the T^4/Z_2 orbifold with many discrete symmetries. High-resolution metrics may be obtained on a time scale o...