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Signed-off-by: Riccardo Finotello <riccardo.finotello@gmail.com>
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@@ -2,24 +2,24 @@ In the previous parts we presented mathematical tools for the theoretical interp
The ultimate goal of the analysis is to provide some insights on the predictive capabilities of the string theory framework applied to phenomenological data.
As already argued in~\Cref{sec:CYmanifolds} the procedure is however quite challenging as there are different ways to match string theory with the experimental reality, that is there are several different vacuum configurations arising from the compactification of the extra-dimensions.
The investigation of feasible phenomenological models in a string framework has therefore to deal also with computational aspects related to the exploration of the \emph{landscape}~\cite{Douglas:2003:StatisticsStringTheory} of possible vacua.
Unfortunately the number of possibilities is huge (numbers as high as $\num{e272000}$ have been suggested for some models)~\cite{Lerche:1987:ChiralFourdimensionalHeterotic, Douglas:2003:StatisticsStringTheory, Ashok:2004:CountingFluxVacua, Douglas:2004:BasicResultsVacuum, Douglas:2007:FluxCompactification, Taylor:2015:FtheoryGeometryMost, Schellekens:2017:BigNumbersString, Halverson:2017:AlgorithmicUniversalityFtheory, Taylor:2018:ScanningSkeleton4D, Constantin:2019:CountingStringTheory}, the mathematical objects entering the compactifications are complex and typical problems are often NP-complete, NP-hard, or even undecidable~\cite{Denef:2007:ComputationalComplexityLandscape, Halverson:2019:ComputationalComplexityVacua, Ruehle:2020:DataScienceApplications}, making an exhaustive classification impossible.
Unfortunately the number of possibilities is huge (numbers as high as $\num{e272000}$ have been suggested for some models)~\cite{Douglas:2003:StatisticsStringTheory, Ashok:2004:CountingFluxVacua, Taylor:2015:FtheoryGeometryMost, Taylor:2018:ScanningSkeleton4D, Constantin:2019:CountingStringTheory}, the mathematical objects entering the compactifications are complex and typical problems are often NP-complete, NP-hard, or even undecidable~\cite{Denef:2007:ComputationalComplexityLandscape, Halverson:2019:ComputationalComplexityVacua}, making an exhaustive classification impossible.
Additionally there is no single framework to describe all the possible (flux) compactifications.
As a consequence each class of models must be studied with different methods.
This has in general discouraged, or at least rendered challenging, precise connections to the existing and tested theories (in particular, the \sm of particle physics).
Until recently the string landscape has been studied using different methods such as analytic computations for simple examples, general statistics, random scans or algorithmic enumerations of possibilities.
This has been a large endeavor of the string community~\cite{Grana:2006:FluxCompactificationsString, Lust:2009:SeeingStringLandscape, Ibanez:2012:StringTheoryParticle, Brennan:2018:StringLandscapeSwampland, Halverson:2018:TASILecturesRemnants, Ruehle:2020:DataScienceApplications}.
This has been a large endeavor of the string community~\cite{Grana:2006:FluxCompactificationsString, Brennan:2018:StringLandscapeSwampland}.
The main objective of such studies is to understand what are the generic predictions of string theory.
The first conclusion of these studies is that compactifications giving an effective theory close to the Standard Model are scarce~\cite{Dijkstra:2005:ChiralSupersymmetricStandard, Dijkstra:2005:SupersymmetricStandardModel, Blumenhagen:2005:StatisticsSupersymmetricDbrane, Gmeiner:2006:OneBillionMSSMlike, Douglas:2007:LandscapeIntersectingBrane, Anderson:2014:ComprehensiveScanHeterotic}.
The first conclusion of these studies is that compactifications giving an effective theory close to the Standard Model are scarce~\cite{Dijkstra:2005:ChiralSupersymmetricStandard, Blumenhagen:2005:StatisticsSupersymmetricDbrane, Douglas:2007:LandscapeIntersectingBrane, Anderson:2014:ComprehensiveScanHeterotic}.
The approach however has limitations mainly given by lack of a general understanding or high computational power required to run the algorithms.
In reaction to these difficulties and starting with the seminal paper~\cite{Abel:2014:GeneticAlgorithmsSearch} new investigations based on Machine Learning (\ml) appeared in the recent years, focusing on different aspects of the string landscape and of the geometries used in compactifications~\cite{Krefl:2017:MachineLearningCalabiYau, Ruehle:2017:EvolvingNeuralNetworks, He:2017:MachinelearningStringLandscape, Carifio:2017:MachineLearningString, Altman:2019:EstimatingCalabiYauHypersurface, Bull:2018:MachineLearningCICY, Cole:2019:TopologicalDataAnalysis, Klaewer:2019:MachineLearningLine, Mutter:2019:DeepLearningHeterotic, Wang:2018:LearningNonHiggsableGauge, Ashmore:2019:MachineLearningCalabiYau, Brodie:2020:MachineLearningLine, Bull:2019:GettingCICYHigh, Cole:2019:SearchingLandscapeFlux, Faraggi:2020:MachineLearningClassification, Halverson:2019:BranesBrainsExploring, He:2019:DistinguishingEllipticFibrations, Bies:2020:MachineLearningAlgebraic, Bizet:2020:TestingSwamplandConjectures, Halverson:2020:StatisticalPredictionsString, Krippendorf:2020:DetectingSymmetriesNeural, Otsuka:2020:DeepLearningKmeans, Parr:2020:ContrastDataMining, Parr:2020:PredictingOrbifoldOrigin} (see also~\cite{Erbin:2018:GANsGeneratingEFT, Betzler:2020:ConnectingDualitiesMachine, Chen:2020:MachineLearningEtudes, Gan:2017:HolographyDeepLearning, Hashimoto:2018:DeepLearningAdS, Hashimoto:2018:DeepLearningHolographic, Hashimoto:2019:AdSCFTCorrespondence, Tan:2019:DeepLearningHolographic, Akutagawa:2020:DeepLearningAdS, Yan:2020:DeepLearningBlack, Comsa:2019:SupergravityMagicMachine, Krishnan:2020:MachineLearningGauged} for related works and~\cite{Ruehle:2020:DataScienceApplications} for a comprehensive summary of the state of the art).
In reaction to these difficulties and starting with the seminal paper~\cite{Abel:2014:GeneticAlgorithmsSearch} new investigations based on Machine Learning (\ml) appeared in the recent years, focusing on different aspects of the string landscape and of the geometries used in compactifications~\cite{Krefl:2017:MachineLearningCalabiYau, Ruehle:2017:EvolvingNeuralNetworks, He:2017:MachinelearningStringLandscape, Carifio:2017:MachineLearningString, Altman:2019:EstimatingCalabiYauHypersurface, Bull:2018:MachineLearningCICY, Mutter:2019:DeepLearningHeterotic, Ashmore:2020:MachineLearningCalabiYau, Brodie:2020:MachineLearningLine, Bull:2019:GettingCICYHigh, Cole:2019:SearchingLandscapeFlux, Faraggi:2020:MachineLearningClassification, Halverson:2019:BranesBrainsExploring, Bizet:2020:TestingSwamplandConjectures, Halverson:2020:StatisticalPredictionsString, Krippendorf:2020:DetectingSymmetriesNeural, Otsuka:2020:DeepLearningKmeans, Parr:2020:ContrastDataMining, Parr:2020:PredictingOrbifoldOrigin} (see~\cite{Ruehle:2020:DataScienceApplications} for a comprehensive summary of the state of the art).
In fact \ml is definitely adequate when it comes to pattern search or statistical inference starting from large amount of data.
This motivates two main applications to string theory: the systematic exploration of the space of possibilities (if they are not random then \ml should be able to find a pattern) and the deduction of mathematical formulas from the \ml approximation.
The last few years have seen a major uprising of \ml, and more particularly of neural networks (\nn)~\cite{Bengio:2017:DeepLearning, Chollet:2018:DeepLearningPython, Geron:2019:HandsOnMachineLearning}.
The last few years have seen a major uprising of \ml, and more particularly of neural networks (\nn)~\cite{Goodfellow:2017:DeepLearning, Chollet:2018:DeepLearningPython, Geron:2019:HandsOnMachineLearning}.
This technology is efficient at discovering and predicting patterns and now pervades most fields of applied sciences and of the industry.
One of the most critical places where progress can be expected is in understanding the geometries used to describe string compactifications and this will be the object of study in the following analysis.
We mainly refer to~\cite{Geron:2019:HandsOnMachineLearning, Chollet:2018:DeepLearningPython, Bengio:2017:DeepLearning} for reviews in \ml and deep learning techniques, and to~\cite{Ruehle:2020:DataScienceApplications, Skiena:2017:DataScienceDesign, Zheng:2018:FeatureEngineeringMachine} for applications of data science techniques.
We mainly refer to~\cite{Geron:2019:HandsOnMachineLearning, Chollet:2018:DeepLearningPython, Goodfellow:2017:DeepLearning} for reviews in \ml and deep learning techniques, and to~\cite{Ruehle:2020:DataScienceApplications, Skiena:2017:DataScienceDesign, Zheng:2018:FeatureEngineeringMachine} for applications of data science techniques.
We address the question of computing the Hodge numbers $\hodge{1}{1} \in \N$ and $\hodge{2}{1} \in \N$ for \emph{complete intersection Calabi--Yau} (\cicy) $3$-folds~\cite{Green:1987:CalabiYauManifoldsComplete} using different \ml algorithms.
A \cicy is completely specified by its \emph{configuration matrix} (whose entries are positive integers) which is the basic input of the algorithms.
@@ -68,7 +68,7 @@ Code is available on \href{https://thesfinox.github.io/ml-cicy/}{Github}.
As presented in~\Cref{sec:CYmanifolds}, a \cy $n$-fold is a $n$-dimensional complex manifold $X$ with \SU{n} holonomy (dimension in \R is $2n$).
An equivalent definition is the vanishing of its first Chern class.
A standard reference for the physicist is~\cite{Hubsch:1992:CalabiyauManifoldsBestiary} (see also~\cite{Anderson:2018:TASILecturesGeometric, He:2020:CalabiYauSpacesString} for useful references).
A standard reference for the physicist is~\cite{Hubsch:1992:CalabiyauManifoldsBestiary} (see also~\cite{Anderson:2018:TASILecturesGeometric} for useful references).
The compactification on a \cy leads to the breaking of large part of the supersymmetry which is phenomenologically more realistic than the very high energy description with intact supersymmetry.
\cy manifolds are characterised by a certain number of topological properties (see~\Cref{sec:cohomology_hodge}), the most salient being the Hodge numbers \hodge{1}{1} and \hodge{2}{1}, counting respectively the Kähler and complex structure deformations, and the Euler characteristics:\footnotemark{}
@@ -79,14 +79,14 @@ The compactification on a \cy leads to the breaking of large part of the supersy
\chi = 2 \qty(\hodge{1}{1} - \hodge{2}{1}).
\label{eq:cy:euler}
\end{equation}
Interestingly topological properties of the manifold directly translate into features of the $4$-dimensional effective action (in particular the number of fields, the representations and the gauge symmetry)~\cite{Hubsch:1992:CalabiyauManifoldsBestiary, Becker:2006:StringTheoryMTheory}.\footnotemark{}
Interestingly topological properties of the manifold directly translate into features of the $4$-dimensional effective action (in particular the number of fields, the representations and the gauge symmetry)~\cite{Hubsch:1992:CalabiyauManifoldsBestiary}.\footnotemark{}
\footnotetext{%
Another reason for sticking to topological properties is that there is no \cy manifold for which the metric is known.
Hence it is not possible to perform explicitly the Kaluza--Klein reduction in order to derive the $4$-dimensional theory.
}%
}
In particular the Hodge numbers count the number of chiral multiplets (in heterotic compactifications) and the number of hyper- and vector multiplets (in type II compactifications): these are related to the number of fermion generations ($3$ in the Standard Model) and is thus an important measure of the distance to the Standard Model.
The simplest \cy manifolds are constructed by considering the complete intersection of hypersurfaces in a product $\cA$ of projective spaces $\mathds{P}^{n_i}$ (called the ambient space)~\cite{Green:1987:CalabiYauManifoldsComplete, Green:1987:PolynomialDeformationsCohomology, Candelas:1988:CompleteIntersectionCalabiYau, Green:1989:AllHodgeNumbers, Anderson:2017:FibrationsCICYThreefolds, Anderson:2018:TASILecturesGeometric}:
The simplest \cy manifolds are constructed by considering the complete intersection of hypersurfaces in a product $\cA$ of projective spaces $\mathds{P}^{n_i}$ (called the ambient space)~\cite{Green:1987:CalabiYauManifoldsComplete, Green:1987:PolynomialDeformationsCohomology, Candelas:1988:CompleteIntersectionCalabiYau, Green:1989:AllHodgeNumbers, Anderson:2017:FibrationsCICYThreefolds}:
\begin{equation}
\cA = \mathds{P}^{n_1} \times \cdots \times \mathds{P}^{n_m}.
\end{equation}
@@ -173,7 +173,7 @@ Below we show a list of the \cicy properties and of their configuration matrices
\item unique Hodge number combinations: $266$
\end{itemize}
\item ``original dataset''~\cite{Candelas:1988:CompleteIntersectionCalabiYau, Green:1989:AllHodgeNumbers}
\item ``original dataset''~\cite{Candelas:1988:CompleteIntersectionCalabiYau, Green:1989:AllHodgeNumbers}:
\begin{itemize}
\item maximal size of the configuration matrices: $12 \times 15$
\item number of favourable matrices (excluding product spaces): $4874$ ($\num{61.8}\%$)
@@ -181,7 +181,7 @@ Below we show a list of the \cicy properties and of their configuration matrices
\item number of different ambient spaces: $235$
\end{itemize}
\item ``favourable dataset''~\cite{Anderson:2017:FibrationsCICYThreefolds}
\item ``favourable dataset''~\cite{Anderson:2017:FibrationsCICYThreefolds}:
\begin{itemize}
\item maximal size of the configuration matrices: $15 \times 18$
\item number of favourable matrices (excluding product spaces): $7820$ ($\num{99.1}\%$)