Calabi‐Yau Threefolds with Small Hodge Numbers

Candelas, Philip; Constantin, Andrei; Mishra, Challenger

doi:10.1002/prop.201800029

Cited by 35 publications

(39 citation statements)

References 39 publications

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“…Second, many properties of the CICYs have already been computed over the years, like their Hodge numbers [9,10] and discrete isometries [11][12][13][14]. The Hodge numbers of their quotients by freely acting discrete isometries have also been computed [11,[15][16][17][18].…”

Section: Introductionmentioning

confidence: 99%

Machine learning CICY threefolds

Bull

Jejjala

et al. 2018

Physics Letters B

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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 which both Synthetic Minority Oversampling Technique (SMOTE) and permutations of the CICY matrix are used to decrease the class imbalance and improve performance. In each case study, we employ a genetic algorithm to optimise the hyperparameters of the neural network. We demonstrate that our approach provides quick diagnostic tools capable of shortlisting quasi-realistic string models based on compactification over smooth CICYs and further supports the paradigm that classes of problems in algebraic geometry can be machine learned. * kieran.bull@seh.ox.ac.uk † hey@maths.ox.ac.uk ‡ vishnu@neo.phys.wits.ac.za

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Section: Introductionmentioning

confidence: 99%

Machine learning CICY threefolds

Bull

Jejjala

et al. 2018

Physics Letters B

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“…Therefore, generically, it is expected that Calabi-Yau manifolds with a small fundamental group π 1 (X), should far exceed in number those with a large π 1 (X) (this should be contrasted with the relative paucity of Calabi-Yau manifolds of small Hodge numbers [35,36,38]). The smallest possible π 1 (X) that breaks the SU (5) GUT group to the Standard Model gauge group is Γ = Z 2 and this setup is expected to dominate.…”

Section: The Kahler Cone;mentioning

confidence: 99%

Counting string theory standard models

Constantin

Lukas

2019

Physics Letters B

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We derive an approximate analytic relation between the number of consistent heterotic Calabi-Yau compactifications of string theory with the exact charged matter content of the standard model of particle physics and the topological data of the internal manifold: the former scaling exponentially with the number of Kähler parameters. This is done by an estimate of the number of solutions to a set of Diophantine equations representing constraints satisfied by any consistent heterotic string vacuum with three chiral massless families, and has been computationally checked to hold for complete intersection Calabi-Yau threefolds (CICYs) with up to seven Kähler parameters. When extrapolated to the entire CICY list, the relation gives ∼10 23 string theory standard models; for the class of Calabi-Yau hypersurfaces in toric varieties, it gives ∼10 723 standard models.

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“…These are the common roots of the above polynomial with the relation 6 = ζ . The polynomial above is 15 ( ), the fifteenth cyclotomic polynomial, and the roots are k for k coprime to 15, that is for the eight values The two common roots of (3.2) and the equation 6 = ζ are and 11 . This quantity is important to us because it is the quantity that appears in Table 4.…”

Section: Galois Actionsmentioning

confidence: 99%

“…Further work in [2][3][4][5] calculated the Hodge numbers for each of these quotients, and was summarised in [6]. Not all of the quotient manifolds seem interesting, but some are.…”

Section: Introductionmentioning

confidence: 99%

Highly Symmetric Quintic Quotients

Candelas

Mishra

2018

Fortschritte der Physik

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The quintic family must be the most studied family of Calabi‐Yau threefolds. Particularly symmetric members of this family are known to admit quotients by freely acting symmetries isomorphic to double-struckZ5×double-struckZ5. The corresponding quotient manifolds may themselves be symmetric. That is, they may admit symmetries that descend from the symmetries that the manifold enjoys before the quotient is taken. The formalism for identifying these symmetries was given a long time ago by Witten and instances of these symmetric quotients were given also, for the family double-struckP7false[2,2,2,2false], by Goodman and Witten. We rework this calculation here, with the benefit of computer assistance, and provide a complete classification. Our motivation is largely to develop methods that apply also to the analysis of quotients of other CICY manifolds, whose symmetries have been classified recently. For the double-struckZ5×double-struckZ5 quotients of the quintic family, our list contains families of smooth manifolds with symmetry Z4, Dic3 and Dic5, families of singular manifolds with four conifold points, with symmetry Z6 and Q8, and rigid manifolds, each with at least a curve of singularities, and symmetry Z10. We intend to return to the computation of the symmetries of the quotients of other CICYs elsewhere.

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Calabi‐Yau Threefolds with Small Hodge Numbers

Cited by 35 publications

References 39 publications

Machine learning CICY threefolds

Machine learning CICY threefolds

Counting string theory standard models

Highly Symmetric Quintic Quotients

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