Complete Intersection Calabi-Yau Manifolds Research Articles

Machine learning on generalized complete intersection Calabi-Yau manifolds

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 type $(1,1)$ and type $(2,1)$ gCICYs in the literature. Moreover, They can achieve a $97\%$ precision in predicting new gCICY which is generated differently from those used for training and testing. This shows that machine learning could be an effective method to classify and generate new gCICY.

Machine learning for complete intersection Calabi-Yau manifolds: A methodological study

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 data analysis (EDA) which aims at understanding better the input data and finding an optimal representation. It is followed by the design of a validation procedure and a baseline model. Finally, several ML models are compared and combined, often involving neural networks with a topology more complicated than the sequential models typically used in physics. By following this procedure, we improve the accuracy of ML computations for Hodge numbers with respect to the existing literature. First, we obtain 97% (resp. 99%) accuracy for $h^{1,1}$ using a neural network inspired by the Inception model for the old dataset, using only 30% (resp. 70%) of the data for training. For the new one, a simple linear regression leads to almost 100% accuracy with 30% of the data for training. The computation of $h^{2,1}$ is less successful as we manage to reach only 50% accuracy for both datasets, but this is still better than the 16% obtained with a simple neural network (SVM with Gaussian kernel and feature engineering and sequential convolutional network reach at best 36%). This serves as a proof of concept that neural networks can be valuable to study the properties of geometries appearing in string theory.

Heterotic line bundle models on generalized complete intersection Calabi Yau manifolds

The systematic program of heterotic line bundle model building has resulted in a wealth of standard-like models (SLM) for particle physics. In this paper, we continue this work in the setting of generalised Complete Intersection Calabi Yau (gCICY) manifolds. Using the gCICYs constructed in ref. [1], we identify two geometries that, when combined with line bundle sums, are directly suitable for heterotic GUT models. We then show that these gCICYs admit freely acting ℤ2 symmetry groups, and are thus amenable to Wilson line breaking of the GUT gauge group to that of the standard model. We proceed to a systematic scan over line bundle sums over these geometries, that result in 99 and 33 SLMs, respectively. For the first class of models, our results may be compared to line bundle models on homotopically equivalent Complete Intersection Calabi Yau manifolds. This shows that the number of realistic configurations is of the same order of magnitude.

Analytic structure of all loop banana integrals

Using the Gelfand-Kapranov-Zelevinskĭ system for the primitive cohomology of an infinite series of complete intersection Calabi-Yau manifolds, whose dimension is the loop order minus one, we completely clarify the analytic structure of all banana integrals with arbitrary masses. In particular, we find that the leading logarithmic structure in the high energy regime, which corresponds to the point of maximal unipotent monodromy, is determined by a novel hat{Gamma}hbox{-} mathrm{class} evaluation in the ambient spaces of the mirror, while the imaginary part of the integral in this regime is determined by the hat{Gamma}hbox{-} mathrm{class} of the mirror Calabi-Yau manifold itself. We provide simple closed all loop formulas for the former as well as for the Frobenius κ-constants, which determine the behaviour of the integrals when the momentum square equals the sum of the masses squared, in terms of zeta values. We extend our previous work from three to four loops by providing for the latter case a complete set of (inhomogeneous) Picard-Fuchs differential equations for arbitrary masses. This allows to evaluate the banana integral in very short time to very high numerical precision for all values of the physical parameters. Using modular properties of the periods we determine the value of the maximal cut equal mass four-loop integral at the attractor points in terms of periods of modular weight two and four Hecke eigenforms and the quasiperiods of their meromorphic cousins.

Detecting symmetries with neural networks

Identifying symmetries in data sets is generally difficult, but knowledge about them is crucial for efficient data handling. Here we present a method how neural networks can be used to identify symmetries. We make extensive use of the structure in the embedding layer of the neural network which allows us to identify whether a symmetry is present and to identify orbits of the symmetry in the input. To determine which continuous or discrete symmetry group is present we analyse the invariant orbits in the input. We present examples based on rotation groups SO(n) and the unitary group SU(2). Further we find that this method is useful for the classification of complete intersection Calabi-Yau manifolds where it is crucial to identify discrete symmetries on the input space. For this example we present a novel data representation in terms of graphs.

Calabi–Yau generalized complete intersections and aspects of cohomology of sheaves

We consider generalized complete intersection manifolds in the product space of projective spaces and work out useful aspects pertaining to the cohomology of sheaves over them. First, we present and prove a vanishing theorem on the cohomology groups of sheaves for subvarieties of the ambient product space of projective spaces. We then prove a birational equivalence between configuration matrices of complete intersection Calabi–Yau manifolds. We also present a formula of the genus of curves in generalized complete intersection manifolds. Some of these curves arise as the fixed point locus of certain symmetry group action on the generalized complete intersection Calabi–Yau manifolds. We also make a blowing-up along curves by which one can generate new Calabi–Yau manifolds. Moreover, an approach on spectral sequences is used to compute Hodge numbers of generalized complete intersection Calabi–Yau manifolds and the genus of curves therein.

Explore and Exploit with Heterotic Line Bundle Models

AbstractWe use deep reinforcement learning to explore a class of heterotic GUT models constructed from line bundle sums over Complete Intersection Calabi Yau (CICY) manifolds. We perform several experiments where A3C agents are trained to search for such models. These agents significantly outperform random exploration, in the most favourable settings by a factor of 1700 when it comes to finding unique models. Furthermore, we find evidence that the trained agents also outperform random walkers on new manifolds. We conclude that the agents detect hidden structures in the compactification data, which is partly of general nature. The experiments scale well with h(1, 1), and may thus provide the key to model building on CICYs with large h(1, 1).

Line Bundle Cohomologies on CICYs with Picard Number Two

AbstractWe analyse line bundle cohomologies on all favourable co‐dimension two Complete Intersection Calabi Yau (CICY) manifolds of Picard number two. Our results provide further evidence that the cohomology dimensions of such line bundles are given by analytic expressions, which change between regions in the line bundle charge space. This agrees with recent observations of CY line bundles presented in Refs. [1, 2]. In many cases, the expressions for bundle cohomology dimensions are polynomial functions of the line bundle charges (of degree at most 3), and the regions are cones. A more novel observation is that for some CICY manifolds, the cohomologies are more succinctly determined by recursive relationships. There can also be boundaries between regions where a polynomial fit fails, and we link these exceptional cases to irregular behaviour of the index of the line bundle. Finally, our observations provide evidence for similarities in the line bundle cohomologies for CICY manifolds that share rows in the configuration matrix. Among such related CICY manifolds, we find both that the line bundle charge space is partitioned in the same manner, and that the same, or closely related, analytical descriptions apply for the cohomology dimensions in these regions.

Calabi-Yau manifolds and SU(3) structure

We show that non-trivial SU(3) structures can be constructed on large classes of Calabi-Yau three-folds. Specifically, we focus on Calabi-Yau three-folds constructed as complete intersections in products of projective spaces, although we expect similar methods to apply to other constructions and also to Calabi-Yau four-folds. Among the wide range of possible SU(3) structures we find Strominger-Hull systems, suitable for heterotic or type II string compactifications, on all complete intersection Calabi-Yau manifolds. These SU(3) structures of Strominger-Hull type have a non-vanishing and non-closed three-form flux which needs to be supported by source terms in the associated Bianchi identity. We discuss the possibility of finding such source terms and present first steps towards their explicit construction. Provided suitable sources exist, our methods lead to Calabi-Yau compactifications of string theory with a non Ricci-flat, physical metric which can be written down explicitly and in analytic form.

Complete intersection Calabi–Yau manifolds with respect to homogeneous vector bundles on Grassmannians

Based on the method by Kuchle (Math Z 218(4), 563–575, 1995), we give a procedure to list up all complete intersection Calabi–Yau manifolds with respect to direct sums of irreducible homogeneous vector bundles on Grassmannians for each dimension. In particular, we give a classification of such Calabi–Yau 3-folds and determine their topological invariants. We also give alternative descriptions for some of them.

Movable vs Monodromy Nilpotent Cones of Calabi-Yau Manifolds

We study mirror symmetry of complete intersection Calabi-Yau manifolds which have birational automorphisms of infinite order. We observe that movable cones in birational geometry are transformed, under mirror symmetry, to the monodromy nilpotent cones which are naturally glued together.

Heterotic instanton superpotentials from complete intersection Calabi-Yau manifolds

We study Pfaffians that appear in non-perturbative superpotential terms arising from worldsheet instantons in heterotic theories. A result by Beasley and Witten shows that these instanton contributions cancel among curves within a given homology class for Calabi-Yau manifolds that can be described as hypersurfaces or complete intersections in projective or toric ambient spaces. We provide a prescription that identifies all ℙ1 curves in certain homology classes of complete intersection Calabi-Yau manifolds in products of projective spaces (CICYs) and cross-check our results by a comparison with the genus zero Gromov-Witten invariants. We then use this construction to study instanton superpotentials on those manifolds and their quotients. We identify a non-toric quotient of a non-favorable CICY with a single genus zero curve in a certain homology class, so that a cancellation à la Beasley-Witten is not possible. In another example, we study a non-toric quotient of a favorable CICY and check that the superpotential still vanishes. From this and related examples, we conjecture that the Beasley-Witten cancellation result can be extended to toric and non-toric quotients of CICYs, but can be avoided if the CICY is non-favorable.

Holomorphic Yukawa couplings for complete intersection Calabi-Yau manifolds

We develop methods to compute holomorphic Yukawa couplings for heterotic compactifications on complete intersection Calabi-Yau manifolds, generalising results of an earlier paper for Calabi-Yau hypersurfaces. Our methods are based on constructing the required bundle-valued forms explicitly and evaluating the relevant integrals over the projective ambient space. We also show how our approach relates to an earlier, algebraic one to calculate the holomorphic Yukawa couplings. A vanishing theorem, which we prove, implies that certain Yukawa couplings allowed by low-energy symmetries are zero due to topological reasons. To illustrate our methods, we calculate Yukawa couplings for SU(5)-based standard models on a co-dimension two complete intersection manifold.

Special Lagrangian torus fibrations of complete intersection Calabi–Yau manifolds: A geometric conjecture

For complete intersection Calabi–Yau manifolds in toric varieties, Gross and Haase–Zharkov have given a conjectural combinatorial description of the special Lagrangian torus fibrations whose existence was predicted by Strominger, Yau and Zaslow. We present a geometric version of this construction, generalizing an earlier conjecture of the first author.

A Comprehensive Scan for Heterotic SU(5) GUT models

Compactifications of heterotic theories on smooth Calabi-Yau manifolds remains one of the most promising approaches to string phenomenology. In two previous papers, http://arXiv.org/abs/arXiv:1106.4804 and http://arXiv.org/abs/arXiv:1202.1757, large classes of such vacua were constructed, using sums of line bundles over complete intersection Calabi-Yau manifolds in products of projective spaces that admit smooth quotients by finite groups. A total of 10^12 different vector bundles were investigated which led to 202 SU(5) Grand Unified Theory (GUT) models. With the addition of Wilson lines, these in turn led, by a conservative counting, to 2122 heterotic standard models. In the present paper, we extend the scope of this programme and perform an exhaustive scan over the same class of models. A total of 10^40 vector bundles are analysed leading to 35,000 SU(5) GUT models. All of these compactifications have the right field content to induce low-energy models with the matter spectrum of the supersymmetric standard model, with no exotics of any kind. The detailed analysis of the resulting vast number of heterotic standard models is a substantial and ongoing task in computational algebraic geometry.

Monad constructions of omalous bundles

We consider a particular class of holomorphic vector bundles relevant for supersymmetric string theory, called omalous, over nonsingular projective varieties. We use monads to construct examples of such bundles over 3-fold hypersurfaces in P4, complete intersection Calabi–Yau manifolds in Pk, blow-ups of P2 at n distinct points, and products Pm×Pn.

Completing the web of ℤ3‐quotients of complete intersection Calabi‐Yau manifolds

AbstractWe complete the study [1] of smooth ℤ3‐quotients of complete intersection Calabi‐Yau threefolds by discussing the six new manifolds that admit free ℤ3 actions that were discovered in [2]. These manifolds were missed in [1] and complete the web of smooth ℤ3‐quotients in a nice way. We discuss the transitions between these manifolds and include also the other manifolds of the web. This leads to the conclusion that the web of ℤ3‐free quotients of complete intersection Calabi‐Yau threefolds is connected by conifold transitions.

Two hundred heterotic standard models on smooth Calabi-Yau threefolds

We construct heterotic standard models by compactifying on smooth Calabi-Yau three-folds in the presence of purely Abelian internal gauge fields. A systematic search over complete intersection Calabi-Yau manifolds with less than six K\"ahler parameters leads to over 200 such models which we present. Each of these models has precisely the matter spectrum of the minimal supersymmetric standard model, at least one pair of Higgs doublets, the standard model gauge group, and no exotics. For about 100 of these models there are four additional $U(1)$ symmetries which are Green-Schwarz anomalous and, hence, massive. In the remaining cases, three $U(1)$ symmetries are anomalous, while the fourth, massless one can be spontaneously broken by singlet vacuum expectation values. The presence of additional global $U(1)$ symmetries, together with the possibility of switching on singlet vacuum expectation values, leads to a rich phenomenology which is illustrated for a particular example. Our database of standard models, which can be further enlarged by simply extending the computer-based search, allows for a detailed and systematic phenomenological analysis of string standard models, covering issues such as the structure of Yukawa couplings, $R$-parity violation, proton stability, and neutrino masses.

On free quotients of complete intersection Calabi-Yau manifolds

In order to find novel examples of non-simply connected Calabi-Yau threefolds, free quotients of complete intersections in products of projective spaces are classified by means of a computer search. More precisely, all automorphisms of the product of projective spaces that descend to a free action on the Calabi-Yau manifold are identified.

Calabi-Yau manifolds from pairs of non-compact Calabi-Yau manifolds

Most of Calabi-Yau manifolds that have been considered by physicists are complete intersection Calabi-Yau manifolds of toric varieties or some quotients of product types. Purpose of this paper is to introduce a different and rather new kind of construction method of Calabi-Yau manifolds by pasting two non-compact Calabi-Yau manifolds. We will also in some details explain a curious and mysterious similarity with construction of some $G_2$-manifolds (also called Joyce manifolds), which are base spaces for M-theory.