Calabi-Yau Links and Machine Learning

Calabi-Yau four-folds may be constructed as hypersurfaces in weighted projective spaces of complex dimension 5 defined via weight systems of 6 weights. In this work, neural networks were implemented to learn the Calabi-Yau Hodge numbers from the weight systems, where gradient saliency and symbolic regression then inspired a truncation of the Landau-Ginzburg model formula for the Hodge numbers of any dimensional Calabi-Yau constructed in this way. The approximation always prov...

We review recent efforts to machine learn relations between knot invariants. Because these knot invariants have meaning in physics, we explore aspects of Chern-Simons theory and higher dimensional gauge theories. The goal of this work is to translate numerical experiments with Big Data to new analytic results.

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 survey some recent applications of machine learning to problems in geometry and theoretical physics. Pure mathematical data has been compiled over the last few decades by the community and experiments in supervised, semi-supervised and unsupervised machine learning have found surprising success. We thus advocate the programme of machine learning mathematical structures, and formulating conjectures via pattern recognition, in other words using artificial intelligence to hel...

We use the latest techniques in machine-learning to study whether from the landscape of Calabi-Yau manifolds one can distinguish elliptically fibred ones. Using the dataset of complete intersections in products of projective spaces (CICY3 and CICY4, totalling about a million manifolds) as a concrete playground, we find that a relatively simple neural network with forward-feeding multi-layers can very efficiently distinguish the elliptic fibrations, much more so than using the...

M-theory compactification leads one to consider 7-manifolds obtained by rolling Calabi-Yau threefolds in the web of Calabi-Yau moduli spaces. The resulting 7-space in general has singularities governed by the extremal transition undergone. After providing some background in Sec. 1, the simplest case of conifold transitions is studied in Sec. 2. In Sec. 3 we employ topological methods, Smale's classification theorem of smooth simply-connected spin closed 5-manifolds, and a com...

We present a collection of explicit formulas for the minimum volume of Sasaki-Einstein 5-manifolds. The cone over these 5-manifolds is a toric Calabi-Yau 3-fold. These toric Calabi-Yau 3-folds are associated with an infinite class of 4d N=1 supersymmetric gauge theories, which are realized as worldvolume theories of D3-branes probing the toric Calabi-Yau 3-folds. Under the AdS/CFT correspondence, the minimum volume of the Sasaki-Einstein base is inversely proportional to the ...

We investigate the mathematical properties of the class of Calabi-Yau four-folds recently found in [arXiv:1303.1832]. This class consists of 921,497 configuration matrices which correspond to manifolds that are described as complete intersections in products of projective spaces. For each manifold in the list, we compute the full Hodge diamond as well as additional topological invariants such as Chern classes and intersection numbers. Using this data, we conclude that there a...

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...

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.