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Signed-off-by: Riccardo Finotello <riccardo.finotello@gmail.com>
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Riccardo
2020-10-21 15:24:18 +02:00
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sec/part1/conclusion.tex
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We thus showed that the specific geometry of the intersecting D-branes leads to different results when computing the value of the classical action, that is the leading contribution to the Yukawa couplings in string theory.
In particular in the Abelian case the value of the action is exactly the area formed by the intersecting D-branes in the $\R^2$ plane, i.e.\ the string worldsheet is completely contained in the polygon on the plane.
In particular in the Abelian case the value of the action is exactly the area formed by the intersecting D-branes in the $\R^2$ plane, that is the string worldsheet is completely contained in the polygon on the plane.
The difference between the \SO{4} case and \SU{2} is more subtle as in the latter there are complex coordinates in $\R^4$ for which the classical string solution is holomorphic in the upper half plane.
In the generic case presented so far this is in general no longer true.
The reason can probably be traced back to supersymmetry, even though we only dealt with the bosonic string.

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sec/part2/conclusion.tex
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@@ -5,7 +5,7 @@ This would explain in an intuitive way why the eikonal approach to gravitational
In fact it appears that the classical and quantum scattering on an electromagnetic wave~\cite{Jackiw:1992:ElectromagneticFieldsMassless} or gravitational wave~\cite{tHooft:1987:GravitonDominanceUltrahighenergy} in \bo and \nbo are well behaved.
From this point of view the ACV approach~\cite{Soldate:1987:PartialwaveUnitarityClosedstring, Amati:1987:SuperstringCollisionsPlanckian} may be more sensible, especially when considering massive external states~\cite{Black:2012:HighEnergyString}.
Finally it seems that all issues are related with the Laplacian associated with the space-like subspace with vanishing volume at the singularity.
As a matter of fact if there is a discrete zero eigenvalue the theory develops divergences.
As a matter of fact if there is a discrete zero eigenvalue the theory automatically develops divergences.
% vim: ft=tex

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sec/part3/conclusion.tex
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@@ -15,7 +15,6 @@ Techniques such as (variational) autoencoders~\cite{Kingma:2014:AutoEncodingVari
Finally our techniques apply directly to \cicy $4$-folds~\cite{Gray:2013:AllCompleteIntersection, Gray:2014:TopologicalInvariantsFibration}.
However there are many more manifolds in this case (around \num{e6}) and more Hodge numbers, such that one can expect to reach a better accuracy for the different Hodge numbers (the different learning curves for the $3$-folds indicate that the model training would benefit from more data).
Another interesting class of manifolds to explore with our techniques are generalized \cicy $3$-folds~\cite{Anderson:2016:NewConstructionCalabiYau}.
These and others will indeed be ground for future investigations.

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sec/part3/introduction.tex
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In the previous parts we presented mathematical tools for the theoretical interpretation of amplitudes in field theory and string theory.
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 ultimate goal of the analysis is to provide some insights on the predictive capabilities of the string theory framework applied to (semi-)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.
There are in fact 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{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).
This has in general discouraged, or at least rendered challenging or unfeasible, precise connections to the existing and tested theories (for instance 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, Brennan:2018:StringLandscapeSwampland}.
This has been a large endeavor of the string community (see for instance~\cite{Grana:2006:FluxCompactificationsString, Brennan:2018:StringLandscapeSwampland} and references therein).
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, Blumenhagen:2005:StatisticsSupersymmetricDbrane, Douglas:2007:LandscapeIntersectingBrane, Anderson:2014:ComprehensiveScanHeterotic}.
The immediate conclusions are 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, 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 reaction to these difficulties, and starting with the seminal paper by Abel and Rizos in 2014~\cite{Abel:2014:GeneticAlgorithmsSearch}, new investigations based on Machine Learning (\ml) appeared in the recent years.
The have been 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{Goodfellow:2017:DeepLearning, Chollet:2018:DeepLearningPython, Geron:2019:HandsOnMachineLearning}.
@@ -84,7 +86,7 @@ Interestingly topological properties of the manifold directly translate into fea
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.
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 (three 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}:
\begin{equation}