Numerical Metrics for Complete Intersection and Kreuzer-Skarke Calabi-Yau Manifolds

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...

Generalized Complete Intersection Calabi-Yau Manifold (gCICY) is a new construction of Calabi-Yau manifolds established recently. However, the generation of new gCICYs using standard algebraic method is very laborious. Due to this complexity, the number of gCICYs and their classification still remain unknown. In this paper, we try to make some progress in this direction using neural network. The results showed that our trained models can have a high precision on the existing ...

We revisit the question of predicting both Hodge numbers $h^{1,1}$ and $h^{2,1}$ of complete intersection Calabi-Yau (CICY) 3-folds using machine learning (ML), considering both the old and new datasets built respectively by Candelas-Dale-Lutken-Schimmrigk / Green-H\"ubsch-Lutken and by Anderson-Gao-Gray-Lee. In real world applications, implementing a ML system rarely reduces to feed the brute data to the algorithm. Instead, the typical workflow starts with an exploratory dat...

We continue earlier efforts in computing the dimensions of tangent space cohomologies of Calabi-Yau manifolds using deep learning. In this paper, we consider the dataset of all Calabi-Yau four-folds constructed as complete intersections in products of projective spaces. Employing neural networks inspired by state-of-the-art computer vision architectures, we improve earlier benchmarks and demonstrate that all four non-trivial Hodge numbers can be learned at the same time using...

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.

We develop a theory of flows in the space of Riemannian metrics induced by neural network gradient descent. This is motivated in part by recent advances in approximating Calabi-Yau metrics with neural networks and is enabled by recent advances in understanding flows in the space of neural networks. We derive the corresponding metric flow equations, which are governed by a metric neural tangent kernel, a complicated, non-local object that evolves in time. However, many archite...

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 the first version of CYJAX, a package for machine learning Calabi-Yau metrics using JAX. It is meant to be accessible both as a top-level tool and as a library of modular functions. CYJAX is currently centered around the algebraic ansatz for the K\"ahler potential which automatically satisfies K\"ahlerity and compatibility on patch overlaps. As of now, this implementation is limited to varieties defined by a single defining equation on one complex projective space....

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...

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...