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

We discuss the extent to which numerical techniques for computing approximations to Ricci-flat metrics can be used to investigate hierarchies of curvature scales on Calabi-Yau manifolds. Control of such hierarchies is integral to the validity of curvature expansions in string effective theories. Nevertheless, for seemingly generic points in moduli space it can be difficult to analytically determine if there might be a highly curved region localized somewhere on the Calabi-Yau...

We introduce a neural network inspired by Google's Inception model to compute the Hodge number $h^{1,1}$ of complete intersection Calabi-Yau (CICY) 3-folds. This architecture improves largely the accuracy of the predictions over existing results, giving already 97% of accuracy with just 30% of the data for training. Moreover, accuracy climbs to 99% when using 80% of the data for training. This proves that neural networks are a valuable resource to study geometric aspects in b...

We present a pedagogical introduction to the recent advances in the computational geometry, physical implications, and data science of Calabi-Yau manifolds. Aimed at the beginning research student and using Calabi-Yau spaces as an exciting play-ground, we intend to teach some mathematics to the budding physicist, some physics to the budding mathematician, and some machine-learning to both. Based on various lecture series, colloquia and seminars given by the author in the past...

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...

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...

Hodge numbers of Calabi-Yau manifolds depend non-trivially on the underlying manifold data and they present an interesting challenge for machine learning. In this letter we consider the data set of complete intersection Calabi-Yau four-folds, a set of about 900,000 topological types, and study supervised learning of the Hodge numbers h^1,1 and h^3,1 for these manifolds. We find that h^1,1 can be successfully learned (to 96% precision) by fully connected classifier and regress...

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...

We identify a set of "energy" functionals on the space of metrics in a given Kaehler class on a Calabi-Yau manifold, which are bounded below and minimized uniquely on the Ricci-flat metric in that class. Using these functionals, we recast the problem of numerically solving the Einstein equation as an optimization problem. We apply this strategy, using the "algebraic" metrics (metrics for which the Kaehler potential is given in terms of a polynomial in the projective coordinat...

We numerically study whether there exist nowhere vanishing harmonic $1$-forms on the real locus of some carefully constructed examples of Calabi-Yau manifolds, which would then give rise to potentially new examples of $G_2$-manifolds and an explicit description of their metrics. We do this in two steps: first, we use a neural network to compute an approximate Calabi-Yau metric on each manifold. Second, we use another neural network to compute an approximately harmonic $1$-for...

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...