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Signed-off-by: Riccardo Finotello <riccardo.finotello@gmail.com>
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In this thesis we present topics in phenomenology of string theory ranging from particle physics amplitudes and Big Bang-like singularities to the study of state-of-the-art deep learning techniques based on recent advancements in artificial intelligence for string compactifications.
We present topics in phenomenology of string theory ranging from particle physics amplitudes and Big Bang-like singularities to the study of state-of-the-art deep learning techniques for string compactifications based on recent advancements in artificial intelligence.
In particular we show the computation of the leading contribution to amplitudes in the presence of non Abelian twist fields in intersecting D-branes scenarios in non factorised tori.
We show the computation of the leading contribution to amplitudes in the presence of non Abelian twist fields in intersecting D-branes scenarios in non factorised tori.
This is a generalisation to the current literature which mainly covers factorised internal spaces.
We also study a new method to compute amplitudes in the presence of an arbitrary number of spin fields introducing point-like defects on the string worldsheet.
This method can then be treated as an alternative computation with respect to bosonization and older approaches based on the Reggeon vertex.
This method can then be treated as an alternative computation with respect to bosonization and approaches based on the Reggeon vertex.
We then present an analysis of Big Bang-like cosmological divergences in string theory on time-dependent orbifolds.
We show that the nature of the divergences are not due to gravitational feedback but to the lack of an underlying effective field theory.
We also introduce a new orbifold structure capable of fixing the issue and reinstate a distributional interpretation to field theory amplitudes.
@@ -14,7 +14,7 @@ We also include a methodological study of machine learning applied to data in st
We thus show how such an approach can help in improving results by processing the data before using it.
We then show how deep learning can reach the highest accuracy in the task with smaller networks with less parameters.
This is a novel approach to the task: differently from previous attempts we focus on using convolutional neural networks capable of reaching higher accuracy on the predictions and ensuring phenomenological relevance to results.
The approach is inspired by recent advancements in computer science and inspired by Google's research in the field.
In fact parameter sharing and concurrent scans of the configuration matrix retain better generalisation properties and adapt better to the task than fully connected networks.
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The thesis follows my research work as Ph.D.\ student and candidate at the \emph{Università degli Studi di Torino}, Italy.
During my programme I mainly dealt with the topic of string theory and its relation with phenomenology: I tried to cover mathematical aspects related to amplitudes in intersecting D-branes scenarios and in the presence of defects on the worldsheet, but I also worked on computational issues of the application of recent deep learning and machine learning techniques to the compactification of the extra-dimensions of string theory.
This thesis follows my research work as a Ph.D.\ student and candidate at the \emph{Università degli Studi di Torino}, Italy.
During my programme I mainly dealt with the topic of string theory and its relation with a viable formulation of phenomenology in this framework.
I tried to cover mathematical aspects related to amplitudes in intersecting D-branes scenarios and in the presence of defects on the worldsheet, but I also worked on computational issues such as the application of recent deep learning and machine learning techniques to the compactification of the extra-dimensions of superstrings.
In this manuscript I present the original results obtained over the course of my Ph.D.\ programme.
They are mainly based on published work~\cite{Finotello:2019:ClassicalSolutionBosonic, Arduino:2020:OriginDivergencesTimeDependent} and preprints~\cite{Finotello:2019:2DFermionStrip, Erbin:2020:InceptionNeuralNetwork, Erbin:2020:MachineLearningComplete}.
However I also include some hints to future directions to cover which might expand the work shown here.
The thesis is organised in three main parts plus a fourth with appendices and notes.
In this manuscript we present the original results obtained over the course of my Ph.D.\ programme.
They are mainly based on published~\cite{Finotello:2019:ClassicalSolutionBosonic, Arduino:2020:OriginDivergencesTimeDependent} and preprint~\cite{Finotello:2019:2DFermionStrip, Erbin:2020:InceptionNeuralNetwork, Erbin:2020:MachineLearningComplete} works.
We however also include some hints to future directions to cover which might expand the work shown here.
The thesis is organised in three main parts plus a fourth with appendices and useful notes.
We dedicate~\Cref{part:cft} to set the stage for the entire thesis and to present mathematical tools used to compute amplitudes with phenomenological relevance in string theory.
Namely we begin with and introduction on conformal symmetry (clearly we focus only on aspects relevant to the discussion as many reviews on the subject have already been written) and the role of compactification and D-branes in replicating results obtained in particle physics.
We then move to analyse a specific setup involving angled D-branes in non factorised internal space: this is a generalisation of the usual scenario with D6-branes embedded as lines in factorised tori.\footnotemark{}
\Cref{part:cft} of the manuscript is dedicated to set the stage for the entire discussion and to present mathematical tools used to compute amplitudes with phenomenological relevance in string theory.
Namely it starts with an introduction on conformal symmetry (clearly focusing only on aspects relevant to the discussion as many reviews on the subject have already been written) and the role of compactification and D-branes in replicating results obtained in particle physics.
Then the analysis of a specific setup involving angled D6-branes intersecting in non factorised internal space is presented.\footnotemark{}
\footnotetext{%
For instance this is a generalisation of the typical setup involving D6-branes filling entirely the $4$-dimensional spacetime and embedded as lines in $T^2 \times T^2 \times T^2$, where the possible rotations performed by the branes are Abelian $\SO{2} \sim \U{1}$ rotations.
For instance this is a generalisation of the typical setup involving D6-branes filling entirely the $4$-dimensional spacetime and embedded as lines in $T^2 \times T^2 \times T^2$, where the possible rotations performed by the D-branes are parametrised by Abelian $\SO{2} \simeq \U{1}$ rotations.
}
Here we present a general framework to deal with \SO{4} rotated D-branes.
We then compute the leading term of amplitudes involving an arbitrary number of non Abelian twist fields located at their intersection, that is we calculate the exponential contribution of the classical bosonic string in this geometry.
We finally consider point-like defects along the time direction of the (super)string worldsheet and study the propagation of fermions.
We discover that the stress-energy tensor presents a time dependence but still respects the usual operator product expansion.
Here a general framework to deal with \SO{4} rotated D-branes is presented alongside the computation of the leading term of amplitudes involving an arbitrary number of non Abelian twist fields located at their intersection, that is the exponential contribution of the classical bosonic string in this geometry.
Finally point-like defects along the time direction of the (super)string worldsheet are introduced and the propagation of fermions on such surface studied in detail.
In this setup the stress-energy tensor presents a time dependence but it still respects the usual operator product expansion.
Thus the theory is still conformal though time dependence is due to the defects where spin fields are located.
Through the study of the operator algebra we find a way to compute amplitudes in the presence of spin fields and matter fields alternative to the usual bosonization, but which may be expanded also to twist fields and to more general configurations.
Through the study of the operator algebra the computation of amplitudes in the presence of spin fields and matter fields are computed with a method alternative to the usual bosonization, but which may be expanded also to twist fields and to more general configurations.
In~\Cref{part:cosmo} we deal with cosmological singularities in string and field theory.
We specifically focus on time-dependent orbifolds as simple models of Big Bang-like singularities in string theory: we first briefly introduce the concept of orbifold from the mathematical and the physical point of views and then immediately move to define the Null Boost Orbifold.
Differently from what usually referred, we find that the divergences appearing in the amplitudes are not a consequence of gravitational feedback, but they appear also at the tree level of open string amplitudes.
We therefore first show the source of the divergences in string and field theory amplitudes due to the presence of the compact dimension and its conjugated momentum which prevents the interpretation of the amplitudes even as a distribution.
We then show that the introduction of a non compact direction of motion on the orbifold restores the physical interpretation of the amplitude, hence the origin of the divergences comes from geometrical aspects of the orbifold models.
\Cref{part:cosmo} deals with cosmological singularities in string and field theory.
The main focus is on time-dependent orbifolds as simple models of Big Bang-like singularities in string theory: after a brief introduction on the concept of orbifold from the mathematical and the physical point of views, the Null Boost Orbifold is introduced as first example.
Differently from what usually referred, the divergences appearing in the amplitudes are not a consequence of gravitational feedback, but they appear also at the tree level of open string amplitudes.
The source of the divergences are shown in string and field theory amplitudes due to the presence of the compact dimension and its conjugated momentum which prevents the interpretation of the amplitudes even as a distribution.
In fact the introduction of a non compact direction of motion on the orbifold restores the physical interpretation of the amplitude, hence the origin of the divergences comes from geometrical aspects of the orbifold models.
Namely it is hidden in contact terms and interaction with massive string states which are no longer spectators, thus invalidating the usual approach with the inheritance principle.
\Cref{part:deeplearning} is dedicated to state-of-the-art application of deep learning techniques to the field of string theory compactifications.
We focus on predicting the Hodge numbers of Complete Intersection Calabi--Yau $3$-folds through a rigorous machine learning analysis.
In fact we show that the blind application of neural networks to the configuration matrix of the manifolds can be improved by exploratory data analysis and feature engineering, which allow us to infer behaviour and relations in topological quantities invisibly hidden in the configuration matrix.
We then concentrate on deep learning techniques applied to the manifolds to obtain the Hodge numbers as a regression task.\footnotemark{}
The Hodge numbers of Complete Intersection Calabi--Yau $3$-folds are computed through a rigorous data science and machine learning analysis.
In fact the blind application of neural networks to the configuration matrix of the manifolds can be improved by exploratory data analysis and feature engineering, from which to infer behaviour and relations of topological quantities invisibly hidden in the configuration matrix.
Deep learning techniques are then applied to the manifolds to obtain the Hodge numbers as a regression task.\footnotemark{}
\footnotetext{%
Many previous approaches have proposed classification tasks to get the best performance out of machine learning models.
This however implies specific knowledge of the definition interval of the Hodge numbers and does not generalise well to unknown examples of Complete Intersection Calabi--Yau manifolds.
}
We introduce a new neural network architecture based on recent computer vision advancements in the field of computer science: we use parallel convolutional kernels to extract the Hodge numbers from the configuration matrix and reach near perfect accuracy on the prediction of \hodge{1}{1}.
We also include good preliminary results for \hodge{2}{1}.
We also include details and reviews in~\Cref{part:appendix} for completeness.
A new neural network architecture based on recent computer vision advancements in the field of computer science is eventually introduced: it utilises parallel convolutional kernels to extract the Hodge numbers from the configuration matrix and it reaches near perfect accuracy on the prediction of \hodge{1}{1}.
Such model also leads to good preliminary results for \hodge{2}{1} which has been mostly ignored by previous attempts.

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@@ -8,17 +8,9 @@ In fact when considering \SU{2} rotated D-branes part of the spacetime supersymm
In the general case there does not seem to be any possible way of computing the action~\eqref{eq:action_with_imaginary_part} in term of the global data.
Most probably the value of the action is larger than in the holomorphic case since the string is no longer confined to a plane.
Given the nature of the rotation its worldsheet has to bend in order to be attached to the D-brane as pictorially shown in~\Cref{fig:brane3d} in the case of a $3$-dimensional space.
The general case we considered then differs from the known factorized case by an additional contribution in the on-shell action which can be intuitively understood as a small ``bump'' of the string worldsheet in proximity of the boundary.
The general case we considered then differs from the known factorised case by an additional contribution in the on-shell action which can be intuitively understood as a small ``bump'' of the string worldsheet in proximity of the boundary.
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.
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.
In fact when considering \SU{2} rotated D-branes part of the spacetime supersymmetry is preserved, while this is not the case for \SO{4} rotations.
In a technical and direct way we showed the computation of amplitudes involving an arbitrary number of Abelian spin and matter fields.
In a technical and direct way we also showed the computation of amplitudes involving an arbitrary number of Abelian spin and matter fields.
The approach we introduced does not generally rely on \cft techniques and can be seen as an alternative to bosonization and old methods based on the Reggeon vertex.
Starting from this work the future direction may involve the generalisation to non Abelian spin fields and the application to twist fields.
In this sense this approach might be the only way to compute the amplitudes involving these complicated scenarios.

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@@ -3,25 +3,25 @@
As seen in the previous sections, the study of viable phenomenological models in string theory involves the analysis of the properties of systems of D-branes.
The inclusion of the physical requirements deeply constrains the possible scenarios.
In particular the chiral spectrum of the \sm acts as a strong restriction on the possible setup.
In this section we study models based on \emph{intersecting branes}, which represent a relevant class of such models with interacting chiral matter.
In this section we study \emph{intersecting D-branes}, which represent a relevant class of models with interacting chiral matter.
We focus on the development of technical tools for the computation of Yukawa interactions for D-branes at angles~\cite{Chamoun:2004:FermionMassesMixing,Cremades:2003:YukawaCouplingsIntersecting,Cvetic:2010:BranesInstantonsIntersecting,Abel:2007:RealisticYukawaCouplings,Chen:2008:RealisticWorldIntersecting,Chen:2008:RealisticYukawaTextures,Abel:2005:OneloopYukawasIntersecting}.
The fermion--boson couplings and the study of flavour changing neutral currents~\cite{Abel:2003:FlavourChangingNeutral} are keys to the validity of the models.
These and many other computations require the ability to calculate correlation functions of (excited) twist and (excited) spin fields.
Furthermore these and many other computations require the ability to calculate correlation functions of (excited) twist and (excited) spin fields.
The goal of the section is therefore to address such challenges in specific scenarios.
The computation of the correlation functions of Abelian twist fields can be found in literature and plays a role in many scenarios, such as magnetic branes with commuting magnetic fluxes~\cite{Angelantonj:2000:TypeIStringsMagnetised,Bertolini:2006:BraneWorldEffective,Bianchi:2005:OpenStoryMagnetic,Pesando:2010:OpenClosedString,Forste:2018:YukawaCouplingsMagnetized}, strings in gravitational wave background~\cite{Kiritsis:1994:StringPropagationGravitational,DAppollonio:2003:StringInteractionsGravitational,Berkooz:2004:ClosedStringsMisner,DAppollonio:2005:DbranesBCFTHppwave}, bound states of D-branes~\cite{Gava:1997:BoundStatesBranes,Duo:2007:NewTwistField,David:2000:TachyonCondensationD0} and tachyon condensation in Superstring Field Theory~\cite{David:2000:TachyonCondensationD0,David:2001:TachyonCondensationUsing,David:2002:ClosedStringTachyon,Hashimoto:2003:RecombinationIntersectingDbranes}.
The computation of the correlation functions of Abelian twist fields can be found in literature and plays a role in many scenarios such as magnetic branes with commuting magnetic fluxes~\cite{Angelantonj:2000:TypeIStringsMagnetised,Bertolini:2006:BraneWorldEffective,Bianchi:2005:OpenStoryMagnetic,Pesando:2010:OpenClosedString,Forste:2018:YukawaCouplingsMagnetized}, strings in gravitational wave background~\cite{Kiritsis:1994:StringPropagationGravitational,DAppollonio:2003:StringInteractionsGravitational,Berkooz:2004:ClosedStringsMisner,DAppollonio:2005:DbranesBCFTHppwave}, bound states of D-branes~\cite{Gava:1997:BoundStatesBranes,Duo:2007:NewTwistField,David:2000:TachyonCondensationD0} and tachyon condensation in Superstring Field Theory~\cite{David:2000:TachyonCondensationD0,David:2001:TachyonCondensationUsing,David:2002:ClosedStringTachyon,Hashimoto:2003:RecombinationIntersectingDbranes}.
A similar analysis can be extended to excited twist fields even though they are more subtle to treat and hide more delicate aspects~\cite{Burwick:1991:GeneralYukawaCouplings,Stieberger:1992:YukawaCouplingsBosonic,Erler:1993:HigherTwistedSector,Anastasopoulos:2011:ClosedstringTwistfieldCorrelators,Anastasopoulos:2012:LightStringyStates,Anastasopoulos:2013:ThreeFourpointCorrelators}.
Results were however found starting from dual models up to modern interpretations of string theory~\cite{Sciuto:1969:GeneralVertexFunction,DellaSelva:1970:SimpleExpressionSciuto}.
Results were however found starting from dual models~\cite{Sciuto:1969:GeneralVertexFunction,DellaSelva:1970:SimpleExpressionSciuto} up to modern interpretations of string theory.
Correlation functions involving arbitrary numbers of plain and excited twist fields were more recently studied~\cite{Pesando:2014:CorrelatorsArbitraryUntwisted,Pesando:2012:GreenFunctionsTwist,Pesando:2011:GeneratingFunctionAmplitudes} blending the CFT techniques with the path integral approach and the canonical quantization~\cite{Pesando:2008:MultibranesBoundaryStates,DiVecchia:2007:WrappedMagnetizedBranes,Pesando:2011:StringsArbitraryConstant,DiVecchia:2011:OpenStringsSystem,Pesando:2013:LightConeQuantization}.
We consider D6-branes intersecting at angles in the case of non Abelian relative rotations presenting non Abelian twist fields at the intersections.
We try to understand subtleties and technical issues arising from a scenario which has been studied only in the formulation of non Abelian orbifolds \cite{Inoue:1987:NonAbelianOrbifolds,Inoue:1990:StringInteractionsNonAbelian,Gato:1990:VertexOperatorsNonabelian,Frampton:2001:ClassificationConformalityModels} and for D-branes relative \SU{2} rotations~\cite{Pesando:2016:FullyStringyComputation}.
We try to understand subtleties and technical issues arising from a scenario which has been studied only in the formulation of non Abelian orbifolds \cite{Inoue:1987:NonAbelianOrbifolds,Inoue:1990:StringInteractionsNonAbelian,Gato:1990:VertexOperatorsNonabelian,Frampton:2001:ClassificationConformalityModels} and for relative \SU{2} rotations of the D-branes ~\cite{Pesando:2016:FullyStringyComputation}.
In this configuration we study three D6-branes in $10$-dimensional Minkowski space $\ccM^{1,9}$ with an internal space of the form $\R^4 \times \R^2$ before the compactification.
The D-branes are embedded as lines in $\R^2$ and as two-dimensional surfaces inside $\R^4$.
We focus on the relative rotations which characterise each D-brane in $\R^4$ with respect to the others.
In total generality, they are non commuting \SO{4} matrices.
We study the classical solution of the bosonic string which dominates the behaviour of the correlator of twist fields.
Using the path integral approach we can in fact separate the classical contribution from the quantum fluctuations and write the correlators of $N_B$ twist fields $\sigma_{\rM_{(t)}}( x_{(t)} )$ as:\footnotemark{}
Using the path integral approach we can in fact separate the classical contribution from the quantum fluctuations and write the correlators of $N_B$ twist fields $\sigma_{\rM_{(t)}}\qty( x_{(t)} )$ as:\footnotemark{}
\footnotetext{%
Ultimately $N_B = 3$ in our case.
}
@@ -46,7 +46,7 @@ Their calculations requires the correlator of four twist fields which in turn re
We therefore study the boundary conditions for the open string describing the D-branes embedded in $\R^4$.
In particular we first address the issue connected to the global description of the embedding of the D-branes.
In conformal coordinates we rephrase such problem into the study of the monodromies acquired by the string coordinates.
These additional phase factors can then be specialised to \SO{4}, which can be studied in spinor representation as a double copy of \SU{2}.
These additional phase factors can then be specialised to \SO{4} and be studied in spinor representation as a tensor product of \SU{2} elements.
We thus recast the issue of finding the solution as $4$-dimensional real vector to a tensor product of two solutions in the fundamental representation of \SU{2}.
We then see that these solutions are well represented by hypergeometric functions, up to integer factors.
Physical requirements finally restrict the number of possible solutions.
@@ -2568,7 +2568,7 @@ Each term of the action can be interpreted again as an area of a triangle where
In the general case there does not seem to be any possible way of computing the action~\eqref{eq:action_with_imaginary_part} in term of the global data.
Most probably the value of the action is larger than in the holomorphic case since the string is no longer confined to a plane.
Given the nature of the rotation its worldsheet has to bend in order to be attached to the D-brane as pictorially shown in~\Cref{fig:brane3d} in the case of a $3$-dimensional space.
The general case we considered then differs from the known factorized case by an additional contribution in the on-shell action which can be intuitively understood as a small ``bump'' of the string worldsheet in proximity of the boundary.
The general case we considered then differs from the known factorised case by an additional contribution in the on-shell action which can be intuitively understood as a small ``bump'' of the string worldsheet in proximity of the boundary.
% vim: ft=tex

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As previously pointed out, the computation of quantities such as Yukawa couplings involves correlators of excited spin and twist fields.
After the analysis of the main contribution to amplitudes involving twist fields at the intersection of D-branes, we focus on the computation of correlators of (excited) spin fields.
This has been a research subject for many years until the formulation found in the seminal paper by Friedan, Martinec and Shenker~\cite{Friedan:1986:ConformalInvarianceSupersymmetry} based on bosonization.
In general the available techniques allow to compute only correlators involving ``Abelian'' configurations, that is configurations which can be factorized in sub-configurations having \U{1} symmetry.
Non Abelian cases have been considered~\cite{Inoue:1987:NonAbelianOrbifolds,Inoue:1990:StringInteractionsNonAbelian,Gato:1990:VertexOperatorsNonabelian,Frampton:2001:ClassificationConformalityModels,Pesando:2016:FullyStringyComputation} which is mathematically by far more complicated.
In general the available techniques allow to compute only correlators involving Abelian configurations, that is configurations which can be factorized in sub-configurations having \U{1} symmetry.
Non Abelian cases have also been considered~\cite{Inoue:1987:NonAbelianOrbifolds,Inoue:1990:StringInteractionsNonAbelian,Gato:1990:VertexOperatorsNonabelian,Frampton:2001:ClassificationConformalityModels,Pesando:2016:FullyStringyComputation}, though their mathematical formulation is by far more complicated.
Despite the existence of an efficient method based on bosonization~\cite{Friedan:1986:ConformalInvarianceSupersymmetry} and old methods based on the Reggeon vertex~\cite{Sciuto:1969:GeneralVertexFunction,DellaSelva:1970:SimpleExpressionSciuto,Schwarz:1973:EvaluationDualFermion,DiVecchia:1990:VertexIncludingEmission,Nilsson:1990:GeneralNSRString,DiBartolomeo:1990:GeneralPropertiesVertices,Engberg:1993:AlgorithmComputingFourRamond,Petersen:1989:CovariantSuperreggeonCalculus}, we take into examination the computation of spin field correlators and propose a new method to compute them.
We hope to be able to extend this approach to correlators involving twist fields and non Abelian spin and twist fields.
We would also like to investigate the reason of the non existence of an approach equivalent to bosonization for twist fields.
At the same time we are interested to explore what happens to a \cft in presence of defects.
It turns out that, despite the defects, it is still possible to define a radial time dependent stress-energy tensor which satisfies the canonical \ope
It turns out that despite the defects it is still possible to define a radial time dependent stress-energy tensor which satisfies the canonical \ope
Moreover the boundary changing defects in the construction can be associated with excited spin fields enabling the computation of correlators involving excited spin fields without resorting to bosonization.

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@@ -1,48 +1,48 @@
In this first part we focus on aspects of string theory directly connected with its worldsheet description and symmetries.
The underlying idea is to build technical tools to address the study of viable phenomenological models in this framework.
The construction of realistic string models of particle physics is the key to better understanding the nature of a theory of everything such as string theory.
The construction of realistic string models of particle physics is key to better understanding the nature of a theory of everything such as string theory.
As a first test of validity, the string theory should properly extend the known Standard Model (\sm) of particle physics, which is arguably one of the most experimentally backed theoretical frameworks in modern physics.
As a first test of validity, the string theory should properly extend the known Standard Model (\sm) of particle physics which is arguably one of the most experimentally backed theoretical frameworks in modern physics.
In particular its description in terms of fundamental strings should be able to include a gauge algebra isomorphic to the algebra of the group
\begin{equation}
\SU{3}_{\rC} \otimes \SU{2}_{\rL} \otimes \U{1}_{\rY}
\label{eq:intro:smgroup}
\end{equation}
in order to reproduce known results.
For instance, a good string theory could provide a unified framework by predicting the existence of a larger gauge group containing the \sm as a subset.
For instance a good string theory could provide a unified framework by predicting the existence of a larger gauge group containing the \sm as a subset.
In what follows we deal with the definition of mathematical tools to compute amplitudes to be used in phenomenological calculations related to the study of particles.
In this introduction we present instruments and preparatory frameworks used throughout the manuscript as many other aspects are strongly connected and their definitions are interdependent.
In particular we recall some results on the symmetries of string theory and how to recover a realistic description of physics.
In this introduction we present instruments and frameworks used throughout the manuscript as many other aspects are strongly connected and their definitions are interdependent.
In particular we recall some results on the symmetries of string theory and how to recover a realistic description of $4$-dimensional physics.
\subsection{Properties of String Theory and Conformal Symmetry}
Strings are extended one-dimensional objects.
They are curves in spacetime parametrized by a coordinate $\sigma \in \qty[0, \ell ]$.
When propagating they span a two-dimensional surface, the \emph{worldsheet}, described by the position of the string at given values of $\sigma$ at a time $\tau$, i.e.\ $X^{\mu}(\tau, \sigma)$ where $\mu = 0, 1, \dots, D - 1$ indexes the coordinates.
Such surface can have different topologies according to the nature of the object propagating in spacetime: strings can be \emph{closed} if $X^{\mu}(\tau, 0) = X^{\mu}(\tau, \ell)$ or \emph{open} if the endpoints in $\sigma = 0$ and $\sigma = \ell$ do not coincide.
They are curves in spacetime parametrized by a coordinate $\sigma \in \qty[ 0, \ell ]$.
When propagating they span a two-dimensional surface, the \emph{worldsheet}, described by the position of the string at given values of $\sigma$ at a time $\tau$, i.e.\ $X^{\mu}\qty(\tau, \sigma)$ where $\mu = 0, 1, \dots, D - 1$ indexes the coordinates.
Such surface can have different topologies according to the nature of the object propagating in spacetime: strings can be \emph{closed} if $X^{\mu}\qty(\tau, 0) = X^{\mu}\qty(\tau, \ell)$ or \emph{open} if the endpoints in $\sigma = 0$ and $\sigma = \ell$ do not coincide.
\subsubsection{Action Principle}
As the action of a point particle is proportional to the length of its trajectory (its \emph{worldline}), the same object for a string is proportional to the area of the worldsheet in the original formulation by Nambu and Goto.
The solutions of the classical equations of motion (\eom) are therefore strings spanning a worldsheet of extremal area.
While Nambu and Goto's formulation is fairly direct in its definition, it usually best to work with Polyakov's action~\cite{Polyakov:1981:QuantumGeometryBosonic}:
While Nambu and Goto's formulation is fairly direct in its definition, it si usually best to work with Polyakov's action~\cite{Polyakov:1981:QuantumGeometryBosonic}:
\begin{equation}
S_P[ \gamma, X ]
S_P\qty[ \gamma, X ]
=
-\frac{1}{4 \pi \ap}
\infinfint{\tau}
\finiteint{\sigma}{0}{\ell}
\sqrt{- \det \gamma(\tau, \sigma)}\,
\gamma^{\alpha\beta}(\tau, \sigma)\,
\ipd{\alpha} X^{\mu}(\tau, \sigma)\,
\ipd{\beta} X^{\nu}(\tau, \sigma)\,
\sqrt{- \det \gamma\qty(\tau, \sigma)}\,
\gamma^{\alpha\beta}\qty(\tau, \sigma)\,
\ipd{\alpha} X^{\mu}\qty(\tau, \sigma)\,
\ipd{\beta} X^{\nu}\qty(\tau, \sigma)\,
\eta_{\mu\nu}.
\label{eq:conf:polyakov}
\end{equation}
The \eom for the string $X^{\mu}(\tau, \sigma)$ is therefore:
The \eom for the string $X^{\mu}\qty(\tau, \sigma)$ is therefore:
\begin{equation}
\frac{1}{\sqrt{- \det \gamma}}\,
\ipd{\alpha}
@@ -58,18 +58,18 @@ The \eom for the string $X^{\mu}(\tau, \sigma)$ is therefore:
\qquad
\alpha,\, \beta = 0, 1.
\end{equation}
In this formulation $\gamma_{\alpha\beta}$ is the worldsheet metric with Lorentzian signature $(-, +)$.
In this formulation $\gamma_{\alpha\beta}$ is the worldsheet metric with Lorentzian signature $\qty(-,\, +)$.
As there are no derivatives of $\gamma_{\alpha\beta}$, its \eom is a constraint ensuring the equivalence of Polyakov's and Nambu and Goto's formulations.
In fact
\begin{equation}
\fdv{S_P[\gamma, X]}{\gamma^{\alpha\beta}}
\fdv{S_P\qty[\gamma,\, X]}{\gamma^{\alpha\beta}}
=
- \frac{1}{4 \pi \ap}
\sqrt{- \det \gamma}\,
\qty(
\ipd{\alpha} X \cdot \ipd{\beta} X
-
\frac{1}{2}
\frac{1}{2}\,
\gamma_{\alpha\beta}\,
\gamma^{\lambda\rho}\,
\ipd{\lambda} X \cdot \ipd{\rho} X
@@ -80,7 +80,7 @@ In fact
\end{equation}
implies
\begin{equation}
\eval{S_P[\gamma, X]}_{\fdv{S_P[\gamma, X]}{\gamma^{\alpha\beta}} = 0}
\eval{S_P\qty[\gamma,\, X]}_{\fdv{S_P\qty[\gamma,\, X]}{\gamma^{\alpha\beta}} = 0}
=
- \frac{1}{2 \pi \ap}
\infinfint{\tau}
@@ -91,35 +91,35 @@ implies
\end{equation}
where $S_{NG}[X]$ is the Nambu--Goto action for the classical string, $\dotX = \ipd{\tau} X$ and $\pX = \ipd{\sigma} X$.
The symmetries of $S_P[\gamma, X]$ are the keys to the success of the string theory framework~\cite{Polchinski:1998:StringTheoryIntroduction}.
The symmetries of $S_P\qty[\gamma,\, X]$ are keys to the success of the string theory framework~\cite{Polchinski:1998:StringTheoryIntroduction}.
Specifically~\eqref{eq:conf:polyakov} displays symmetries under:
\begin{itemize}
\item $D$-dimensional Poincaré transformations
\begin{equation}
\begin{split}
X'^{\mu}(\tau, \sigma)
X'^{\mu}\qty(\tau, \sigma)
& =
\tensor{\Lambda}{^{\mu}_{\nu}}\, X^{\mu}(\tau, \sigma) + c^{\nu},
\tensor{\Lambda}{^{\mu}_{\nu}}\, X^{\nu}\qty(\tau, \sigma) + c^{\mu},
\\
\gamma'_{\alpha\beta}(\tau, \sigma)
\gamma'_{\alpha\beta}\qty(\tau, \sigma)
& =
\gamma_{\alpha\beta}(\tau, \sigma)
\gamma_{\alpha\beta}\qty(\tau, \sigma)
\end{split}
\end{equation}
where $\Lambda \in \SO{1, D-1}$ and $c \in \R^D$,
where $\Lambda \in \SO{1,\, D-1}$ and $c \in \R^D$,
\item 2-dimensional diffeomorphisms
\begin{equation}
\begin{split}
X'^{\mu}(\tau', \sigma')
X'^{\mu}\qty(\tau', \sigma')
& =
X^{\mu}(\tau, \sigma)
X^{\mu}\qty(\tau, \sigma)
\\
\gamma'_{\alpha\beta}(\tau', \sigma')
\gamma'_{\alpha\beta}\qty(\tau', \sigma')
& =
\pdv{\sigma'^{\lambda}}{\sigma^{\alpha}}\,
\pdv{\sigma'^{\rho}}{\sigma^{\beta}}\,
\gamma_{\lambda\rho}(\tau, \sigma)
\gamma_{\lambda\rho}\qty(\tau, \sigma)
\end{split}
\end{equation}
where $\sigma^0 = \tau$ and $\sigma^1 = \sigma$,
@@ -127,29 +127,29 @@ Specifically~\eqref{eq:conf:polyakov} displays symmetries under:
\item Weyl transformations
\begin{equation}
\begin{split}
X'^{\mu}(\tau', \sigma')
X'^{\mu}\qty(\tau', \sigma')
& =
X^{\mu}(\tau, \sigma)
X^{\mu}\qty(\tau, \sigma)
\\
\gamma'_{\alpha\beta}(\tau, \sigma)
\gamma'_{\alpha\beta}\qty(\tau, \sigma)
& =
e^{2 \omega(\tau, \sigma)}\, \gamma_{\alpha\beta}(\tau, \sigma)
e^{2 \omega\qty(\tau, \sigma)}\, \gamma_{\alpha\beta}\qty(\tau, \sigma)
\end{split}
\end{equation}
for arbitrary $\omega(\tau, \sigma)$.
for an arbitrary function $\omega\qty(\tau, \sigma)$.
\end{itemize}
Notice that the last is not a symmetry of the Nambu--Goto action and it only appears in Polyakov's formulation of the action.
Notice that the last is not a symmetry of the Nambu--Goto action and it only appears in Polyakov's formulation.
\subsubsection{Conformal Invariance}
The definition of the 2-dimensional stress-energy tensor is a direct consequence of~\eqref{eq:conf:worldsheetmetric} \cite{Green:1988:SuperstringTheoryIntroduction}.
The definition of the 2-dimensional stress-energy tensor is a direct consequence of~\eqref{eq:conf:worldsheetmetric}~\cite{Green:1988:SuperstringTheoryIntroduction}.
In fact the classical constraint on the tensor is simply
\begin{equation}
T_{\alpha\beta}
=
\frac{4 \pi}{\sqrt{- \det \gamma}}
\fdv{S_P[\gamma, X]}{\gamma^{\alpha\beta}}
\fdv{S_P\qty[\gamma,\, X]}{\gamma^{\alpha\beta}}
=
-\frac{1}{\ap}
\qty(
@@ -172,10 +172,10 @@ In other words, the $(1 + 1)$-dimensional theory of massless scalars $X^{\mu}$ i
Using the invariances of the actions we can set $\gamma_{\alpha\beta}(\tau, \sigma) = e^{\phi(\tau, \sigma)}\, \eta_{\alpha\beta}$, known as \emph{conformal gauge} where $\eta_{\alpha\beta} = \mathrm{diag}(-1, 1)$.
This gauge choice is however preserved by the residual \emph{pseudoconformal} transformations
\begin{equation}
\tau \pm \sigma = \sigma_{\pm} \mapsto f_{\pm}(\sigma_{\pm}),
\tau \pm \sigma = \sigma_{\pm} \quad \mapsto \quad f_{\pm}\qty(\sigma_{\pm}),
\label{eq:conf:residualgauge}
\end{equation}
where $f_{\pm}(\xi)$ are arbitrary functions.
where $f_{\pm}$ is an arbitrary function of its argument.
It is natural to introduce a Wick rotation $\tau_E = i \tau$ and the complex coordinates $\xi = \tau_E + i \sigma$ and $\bxi = \xi^*$.
The transformation maps the Lorentzian worldsheet to a new surface: an infinite Euclidean strip for open strings or a cylinder for closed strings.
@@ -185,36 +185,48 @@ In these terms, the tracelessness of the stress-energy tensor translates to
\end{equation}
while its conservation $\partial^{\alpha} T_{\alpha\beta} = 0$ becomes:\footnotemark{}
\footnotetext{%
Since we fix $\gamma_{\alpha\beta}(\tau, \sigma) \propto \eta_{\alpha\beta}$ we do not need to account for the components of the connection and we can replace the covariant derivative $\nabla^{\alpha}$ with a standard derivative $\partial^{\alpha}$.
Since we fix $\gamma_{\alpha\beta}\qty(\tau, \sigma) \propto \eta_{\alpha\beta}$ we do not need to account for the components of the connection and we can replace the covariant derivative $\nabla^{\alpha}$ with a standard derivative $\partial^{\alpha}$.
}
\begin{equation}
\bpd T_{\xi\xi}( \xi,\, \bxi ) = \pd \barT_{\bxi\bxi}( \xi,\, \bxi ) = 0.
\bpd T_{\xi\xi}\qty( \xi,\, \bxi )
=
\pd \barT_{\bxi\bxi}\qty( \xi,\, \bxi )
=
0.
\end{equation}
The last equation finally implies
\begin{equation}
T_{\xi\xi}( \xi,\, \bxi ) = T_{\xi\xi}( \xi ) = T( \xi ),
T_{\xi\xi}\qty( \xi,\, \bxi )
=
T_{\xi\xi}\qty( \xi )
=
T\qty( \xi ),
\qquad
\barT_{\bxi\bxi}( \xi,\, \bxi ) = \barT_{\bxi\bxi}( \bxi ) = \barT( \bxi ),
\barT_{\bxi\bxi}\qty( \xi,\, \bxi )
=
\barT_{\bxi\bxi}\qty( \bxi )
=
\barT\qty( \bxi ),
\end{equation}
which are respectively the holomorphic and the anti-holomorphic components of the bidimensional stress energy tensor.
The previous properties define what is known as a bidimensional \emph{conformal field theory} (\cft).
Ordinary tensor fields
\begin{equation}
\phi_{\omega, \bomega}( \xi,\, \bxi )
\phi_{\omega, \bomega}\qty( \xi, \bxi )
=
\phi_{\underbrace{\xi \dots \xi}\limits_{\omega~\text{times}}\, \underbrace{\bxi \dots \bxi}\limits_{\bomega~\text{times}}}( \xi,\, \bxi )
\phi_{\underbrace{\xi \dots \xi}\limits_{\omega~\text{times}}\, \underbrace{\bxi \dots \bxi}\limits_{\bomega~\text{times}}}\qty( \xi, \bxi )
\qty( \dd{\xi} )^{\omega}
\qty( \dd{\bxi} )^{\bomega}
\end{equation}
are classified according to their weight $\qty( \omega,\, \bomega )$ referring to the holomorphic and anti-holomorphic parts respectively.
In fact a transformation $\xi \mapsto \chi(\xi)$ and $\bxi \mapsto \bchi(\bxi)$ maps the conformal fields to
\begin{equation}
\phi_{\omega, \bomega}( \chi, \bchi )
\phi_{\omega, \bomega}\qty( \chi, \bchi )
=
\qty( \dv{\chi}{\xi} )^{\omega}\,
\qty( \dv{\bchi}{\bxi} )^{\bomega}\,
\phi_{\omega, \bomega}( \xi,\, \bxi ).
\phi_{\omega, \bomega}\qty( \xi, \bxi ).
\end{equation}
\begin{figure}[tbp]
@@ -262,17 +274,17 @@ The transformation on a field $\phi_{\omega, \bomega}$ of weight $(\omega, \bome
\begin{split}
\delta_{\epsilon, \bepsilon} \phi_{\omega, \bomega}
& =
\liebraket{Q_{\epsilon, \bepsilon}}{\phi_{\omega, \bomega}( w, \barw )}
\liebraket{Q_{\epsilon, \bepsilon}}{\phi_{\omega, \bomega}\qty( w, \barw )}
\\
& =
\cint{0} \ddz \epsilon(z) \qty[ T(z), \phi_{\omega, \bomega}( w, \barw ) ]
\cint{0} \ddz \epsilon(z) \qty[ T(z), \phi_{\omega, \bomega}\qty( w, \barw ) ]
+
\cint{0} \ddbz \bepsilon(\barz) \qty[ \barT(\barz), \phi_{\omega, \bomega}( w, \barw ) ]
\cint{0} \ddbz \bepsilon(\barz) \qty[ \barT(\barz), \phi_{\omega, \bomega}\qty( w, \barw ) ]
\\
& =
\cint{w} \ddz \epsilon(z)\, \rR\!\qty( T(z)\, \phi_{\omega, \bomega}( w, \barw ) )
\cint{w} \ddz \epsilon(z)\, \rR\qty( T(z)\, \phi_{\omega, \bomega}( w, \barw ) )
+
\cint{\barw} \ddbz \bepsilon(\barz)\, \rR\!\qty( \barT(\barz)\, \phi_{\omega, \bomega}( w, \barw ) ),
\cint{\barw} \ddbz \bepsilon(\barz)\, \rR\qty( \barT(\barz)\, \phi_{\omega, \bomega}( w, \barw ) ),
\end{split}
\end{equation}
where in the last passage we used the fact that the difference of ordered integrals becomes the contour integral of the radially ordered product computed surrounding $w$.
@@ -281,32 +293,32 @@ Equating the result with the expected variation
\begin{split}
\delta_{\epsilon, \bepsilon} \phi_{\omega, \bomega}
& =
\omega\, \ipd{w} \epsilon( w )\, \phi_{\omega, \bomega}( w, \barw )
\omega\, \ipd{w} \epsilon( w )\, \phi_{\omega, \bomega}\qty( w, \barw )
+
\epsilon( w )\, \ipd{w} \phi_{\omega, \bomega}( w, \barw )
\epsilon( w )\, \ipd{w} \phi_{\omega, \bomega}\qty( w, \barw )
\\
& +
\bomega\, \ipd{\barw} \bepsilon( \barw )\, \phi_{\omega, \bomega}( w, \barw )
\bomega\, \ipd{\barw} \bepsilon( \barw )\, \phi_{\omega, \bomega}\qty( w, \barw )
+
\epsilon( \barw )\, \ipd{\barw} \phi_{\omega, \bomega}( w, \barw )
\epsilon( \barw )\, \ipd{\barw} \phi_{\omega, \bomega}\qty( w, \barw )
\end{split}
\end{equation}
we find the short distance singularities of the components of the stress-energy tensor with the field $\phi_{\omega, \bomega}( w, \barw )$:
\begin{equation}
\begin{split}
T( z )\, \phi_{\omega, \bomega}( w, \barw )
T( z )\, \phi_{\omega, \bomega}\qty( w, \barw )
& =
\frac{\omega}{(z - w)^2}\, \phi_{\omega, \bomega}( w, \barw )
\frac{\omega}{(z - w)^2}\, \phi_{\omega, \bomega}\qty( w, \barw )
+
\frac{1}{z - w}\, \ipd{w} \phi_{\omega, \bomega}( w, \barw )
\frac{1}{z - w}\, \ipd{w} \phi_{\omega, \bomega}\qty( w, \barw )
+
\order{1},
\\
\barT( \barz )\, \phi_{\omega, \bomega}( w, \barw )
\barT( \barz )\, \phi_{\omega, \bomega}\qty( w, \barw )
& =
\frac{\bomega}{(\barz - \barw)^2}\, \phi_{\omega, \bomega}( w, \barw )
\frac{\bomega}{(\barz - \barw)^2}\, \phi_{\omega, \bomega}\qty( w, \barw )
+
\frac{1}{\barz - \barw}\, \ipd{\barw} \phi_{\omega, \bomega}( w, \barw )
\frac{1}{\barz - \barw}\, \ipd{\barw} \phi_{\omega, \bomega}\qty( w, \barw )
+
\order{1},
\end{split}
@@ -335,10 +347,9 @@ which is an asymptotic expansion containing the full information on the singular
\frac{\delta_{ij}}{(z - w)^{\omega_i + \omega_j} (\barz - \barw)^{\bomega_i + \bomega_j}}.
\end{equation*}
}
The constant coefficients $\cC_{ijk}$ are subject to restrictive constraints given by the properties of the conformal theories to the point that a \cft is completely specified by the spectrum of the weights $(\omega_i, \bomega_i)$ and the coefficients $\cC_{ijk}$ \cite{Friedan:1986:ConformalInvarianceSupersymmetry}.
The constant coefficients $\cC_{ijk}$ are subject to restrictive constraints given by the properties of the conformal theories to the point that a \cft is completely specified by the spectrum of the weights $(\omega_i, \bomega_i)$ and the coefficients $\cC_{ijk}$ \cite{Friedan:1986:ConformalInvarianceSupersymmetry, Ginsparg:1988:AppliedConformalField}.
The \ope can also be computed on the stress-energy tensor itself.
Focusing on the holomorphic component we find
The \ope can also be computed on the stress-energy tensor itself:
\begin{equation}
\begin{split}
T( z )\, T( w )
@@ -382,7 +393,7 @@ This ultimately leads to the quantum algebra
\\
\liebraket{\barL_n}{\barL_m}
& =
(n - m)\, \barL_{n + m} + \frac{\barc}{12}\, n\, (n^2 - 1)\, \delta_{n, -m},
(n - m)\, \barL_{n + m} + \frac{c}{12}\, n\, (n^2 - 1)\, \delta_{n, -m},
\\
\liebraket{L_n}{\barL_m}
& =
@@ -393,10 +404,10 @@ This ultimately leads to the quantum algebra
known as Virasoro algebra, unique central extension of the classical de Witt algebra, with central charge $c$.
Operators $L_n$ and $\barL_n$ are called Virasoro operators.\footnotemark{}
\footnotetext{%
Notice that the subset of Virasoro operators $\qty{ L_{-1}, L_0, L_1 }$ forms a closed subalgebra generating the group $\SL{2}{\R}$.
Notice that the subset of Virasoro operators $\qty{ L_{-1},\, L_0,\, L_1 }$ forms a closed subalgebra generating the group $\SL{2}{\R}$.
}
Notice that $L_0 + \barL_0$ is the generator of the dilations on the complex plane.
In terms of radial quantization this translates to time translations and $L_0 + \barL_0$ can be considered to be the Hamiltonian of the theory.
In terms of radial quantization this maps to time translations and $L_0 + \barL_0$ can be considered to be the Hamiltonian of the theory.
In the same fashion as~\eqref{eq:conf:Texpansion}, fields can be expanded in modes:
\begin{equation}
@@ -424,7 +435,7 @@ The regularity of \eqref{eq:conf:expansion} requires
0,
\qquad
n > \omega,
\quad
\qquad
m > \bomega.
\end{equation}
As a consequence also
@@ -453,7 +464,7 @@ Finally the definitions of the primary operators~\eqref{eq:conf:primary} define
From this definition we can build an entire representation of \emph{descendant} states applying any operator $L_{-n}$ (or $\barL_{-n}$) with $n \ge 1$ to $\ket{\phi_{\omega, \bomega}}$.
Let $\phi_{\omega}( w )$ be a holomorphic field in the \cft for simplicity, and let $\ket{\phi_{\omega}}$ be its corresponding state.
The generic state at level $m$ build from such state is
The generic state at level $m$ built from such state is
\begin{equation}
\ket{\phi_{\omega}^{\lbrace n_1, n_2, \dots, n_m \rbrace}}
=
@@ -462,7 +473,7 @@ The generic state at level $m$ build from such state is
\qquad
\finitesum{i}{1}{m} n_i = m \ge 0.
\end{equation}
From the commutation relations~\eqref{eq:conf:virasoro} we finally compute its conformal weight as eigenvalue of $L_0$:
From the commutation relations~\eqref{eq:conf:virasoro} we finally compute its conformal weight as eigenvalue of the (holomorphic) Hamiltonian $L_0$:
\begin{equation}
L_0 \ket{\phi_{\omega}^{\lbrace n_1, n_2, \dots, n_m \rbrace}}
=
@@ -489,13 +500,13 @@ and the components of the stress-energy tensor~\eqref{eq:conf:stringT} are
\end{split}
\label{eq:conf:bosonicstringT}
\end{equation}
Using the normalisation of the 2-points function $\left\langle X^{\mu}( z, \barz ) X^{\nu}( w, \barw ) \right\rangle = - \frac{1}{2} \eta^{\mu\nu} \ln\left| z - w \right|$ we can prove that $c = D$ in~\eqref{eq:conf:TTexpansion}, where $D$ is the dimensions of spacetime (or equivalently the number of scalar fields $X^{\mu}$ in the action).
Using the normalisation of the 2-points function $\left\langle X^{\mu}( z, \barz )\, X^{\nu}( w, \barw ) \right\rangle = - \frac{1}{2} \eta^{\mu\nu} \ln\left| z - w \right|$ and the Wick theorem, we can prove that $c = D$ in~\eqref{eq:conf:TTexpansion}, where $D$ is the dimensions of spacetime (or equivalently the number of scalar fields $X^{\mu}$ in the action).
It can be shown that in order to cancel the central charge in bosonic string theory we need to introduce a pair of conformal ghosts $b(z)$ and $c(z)$ with conformal weights $(2, 0)$ and $(-1, 0)$ respectively, together with their anti-holomorphic counterparts $\barb(z)$ and $\barc(z)$.
The non vanishing components of their stress-energy tensor can be computed as:\footnotemark{}
\footnotetext{%
In general a system of ghosts $b( z )$ and $c( z )$ with weight $(\lambda, 0)$ and $(1 - \lambda, 0)$ can be introduced as a standalone \cft with action~\cite{Friedan:1986:ConformalInvarianceSupersymmetry,Polchinski:1998:StringTheorySuperstring}
In general a system of ghosts $b( z )$ and $c( z )$ with weight $(\lambda,\, 0)$ and $(1 - \lambda,\, 0)$ can be introduced as a standalone \cft with action~\cite{Friedan:1986:ConformalInvarianceSupersymmetry, Polchinski:1998:StringTheorySuperstring}
\begin{equation*}
S = \frac{1}{2 \pi} \iint \dd{z} \dd{\barz} b( z )\, \ipd{\barz} c( z ).
S = \frac{1}{2 \pi} \iint \dd{z} \dd{\barz}\, b( z )\, \ipd{\barz} c( z ).
\end{equation*}
The equations of motion are $\ipd{\barz} c( z ) = \ipd{\barz} b( z ) = 0$.
The \ope is
@@ -509,7 +520,7 @@ The non vanishing components of their stress-energy tensor can be computed as:\f
\end{equation*}
Their central charge is therefore $c_{\text{ghost}} = \varepsilon\, ( 1 - 3 \cQ^2)$, where $\cQ = \varepsilon\,( 1 - 2 \lambda )$.
Notice finally that this ghost \cft has in general an additional \emph{ghost number} \U{1} symmetry generated by the current
The ghost \cft has in general an additional \emph{ghost number} \U{1} symmetry generated by the current
\begin{equation*}
j( z ) = - b( z )\, c( z ).
\end{equation*}
@@ -560,7 +571,7 @@ we get the \ope of the components of their stress-energy tensor:
\end{split}
\end{equation}
which show that $c_{\text{ghost}} = - 26$.
The central charge is therefore cancelled in the full theory (bosonic string and ghosts) when the spacetime dimensions are $D = 26$.
The central charge is therefore cancelled in the full theory (bosonic string and reparametrisation ghosts) when the spacetime dimensions are $D = 26$.
In fact let $T_{\text{full}} = T + T_{\text{ghost}}$ and $\barT_{\text{full}} = \barT + \barT_{\text{ghost}}$, then:
\begin{equation}
\eval{T_{\text{full}}( z )}_{\order{(z - w)^{-4}}}
@@ -581,14 +592,14 @@ In fact let $T_{\text{full}} = T + T_{\text{ghost}}$ and $\barT_{\text{full}} =
\subsection{Superstrings}
As bosonic string theory deals with commuting fields $X^{\mu}$, it is impossible to build spacetime fermions and a consistent phenomenology.
As bosonic string theory deals with commuting fields $X^{\mu}$, it is impossible to build spacetime fermions and consequently a consistent phenomenology.
It is in fact necessary to introduce worldsheet fermions (i.e.\ anti-commuting variables on the string worldsheet) as an extension to the bosonic coordinates \cite{Friedan:1986:ConformalInvarianceSupersymmetry,Polchinski:1998:StringTheorySuperstring}.
We schematically and briefly recall some results due to the extension of bosonic string theory to the superstring as they will be used in what follows and mainly follow from the previous discussion.
We schematically and briefly recall some results due to the extension of bosonic string theory to the superstring as they will be used in what follows and mainly descend from the previous discussion.
The superstring action is built as an addition to the bosonic equivalent~\eqref{eq:conf:polyakov}.
In complex coordinates on the plane it is:
In complex coordinates on the plane it is~\cite{Polchinski:1998:StringTheorySuperstring}:
\begin{equation}
S[ X, \psi ]
S\qty[ X,\, \psi ]
=
- \frac{1}{4 \pi}
\iint \dd{z} \dd{\barz}
@@ -602,7 +613,7 @@ In complex coordinates on the plane it is:
\eta_{\mu\nu}.
\label{eq:super:action}
\end{equation}
In the last expression $\psi^{\mu}$ are $D$ two-dimensional holomorphic fermion fields with conformal weight $(\frac{1}{2}, 0)$ and $\bpsi^{\mu}$ are their anti-holomorphic counterparts with weight $(0, \frac{1}{2})$. Their short-distance behaviour is
In the last expression, $\psi^{\mu}$ are $D$ two-dimensional holomorphic fermion fields with conformal weight $\qty(\frac{1}{2}, 0)$ and $\bpsi^{\mu}$ are their anti-holomorphic counterparts with weight $\qty(0, \frac{1}{2})$. Their short-distance behaviour is
\begin{equation}
\psi^{\mu}( z )\, \psi^{\nu}( w ) = \frac{\eta^{\mu\nu}}{z - w},
\qquad
@@ -680,8 +691,8 @@ The central charge associated to the Virasoro algebra is in this case given by b
\end{equation}
The central charge is therefore $c = \frac{3}{2} D$ for the \cft defined in~\eqref{eq:super:action}.
As in the case of the bosonic string, in order to cancel the central charge we introduce the reparametrisation anti-commuting ghosts $b( z )$ and $c( z )$ and their anti-holomorphic components as well as the commuting \emph{superghosts} $\beta( z )$ and $\gamma( z )$ and their anti-holomorphic counterparts.
These are conformal fields with conformal weights $\qty( \frac{3}{2}, 0 )$ and $\qty( -\frac{1}{2}, 0 )$.
As in the case of the bosonic string, in order to cancel the central charge of superstring theory we introduce the reparametrisation anti-commuting ghosts $b( z )$ and $c( z )$ and their anti-holomorphic components as well as the commuting \emph{superghosts} $\beta( z )$ and $\gamma( z )$ and their anti-holomorphic counterparts.
These are conformal fields with conformal weights $\qty( \frac{3}{2},\, 0 )$ and $\qty( -\frac{1}{2},\, 0 )$.
Their central charge becomes $c_{\text{ghost}} = c_{bc} + c_{\beta\gamma} = -26 + 11 = -15$ (see \cref{note:conf:ghosts} for the general computation).
When considering the full theory $T_{\text{full}} = T + T_{\text{ghost}}$ and $\barT_{\text{full}} = \barT + \barT_{\text{ghost}}$ the central charge vanishes only in 10-dimensional spacetime:
\begin{equation}
@@ -706,8 +717,8 @@ When considering the full theory $T_{\text{full}} = T + T_{\text{ghost}}$ and $\
\label{sec:CYmanifolds}
We are ultimately interested in building a consistent phenomenology in the framework of string theory.
Any theoretical infrastructure has then to be able to support matter states made of fermions.
In what follows we thus consider the superstring formulation in $D = 10$ dimensions even when we deal with bosonic string theory only.
Any theoretical infrastructure has to be able to support matter states made of fermions.
In what follows we thus consider the superstring formulation in $D = 10$ dimensions even when we focus only on its bosonic components.
It is however clear that low energy phenomena need to be explained by a $4$-dimensional theory in order to be comparable with other theoretical frameworks and experimental evidence.
In this section we briefly review for completeness the necessary tools to be able to reproduce consistent models capable of describing particle physics and beyond.
@@ -725,10 +736,10 @@ where $\ccX_6$ is a generic $6$-dimensional manifold at this stage.
This \emph{internal} manifold $\ccX_6$ is however subject to very stringent restrictions due to mathemtical consistency conditions and physical requests.
In particular $\ccX_6$ should be a compact manifold to ``hide'' the 6 extra-dimensions computed in~\eqref{eq:super:dimensions}.
Moreover the geometry of $\ccM^{1,3}$ should be a maximally symmetric space and there should be a $N = 1$ unbroken supersymmetry in $4$ dimensions.
Finally the gauge group of and the spectrum of fermions should be realistic (e.g.\ it should be possible to define chiral fermion states) \cite{Candelas:1985:VacuumConfigurationsSuperstrings}.
These manifolds were first conjectured to exist by Eugenio Calabi~\cite{Calabi:1957:KahlerManifoldsVanishing}.
Their existence was later proved by Shing-Tung Yau~\cite{Yau:1977:CalabiConjectureNew}, hence the name Calabi-Yau (\cy) manifolds.
They are defined as complex Ricci-flat Kähler manifolds $M$ of dimensions $2m$ and with holonomy \SU{3} (see for instance \cite{Joyce:2000:CompactManifoldsSpecial,Joyce:2002:LecturesCalabiYauSpecial,Greene:1997:StringTheoryCalabiYau}).
Finally the arising gauge group and the spectrum of fermions should be realistic (e.g.\ it should be possible to define chiral fermion states)~\cite{Candelas:1985:VacuumConfigurationsSuperstrings}.
These manifolds were first conjectured to exist by Eugenio Calabi~\cite{Calabi:1957:KahlerManifoldsVanishing} and their existence was later proved by Shing-Tung Yau~\cite{Yau:1977:CalabiConjectureNew}, hence the name Calabi-Yau (\cy) manifolds.
They are defined as complex Ricci-flat Kähler manifolds $M$ of dimensions $2m$ and with holonomy \SU{m} (see for instance~\cite{Joyce:2000:CompactManifoldsSpecial,Joyce:2002:LecturesCalabiYauSpecial,Greene:1997:StringTheoryCalabiYau}).
More on this topic is also presented in~\Cref{part:deeplearning} of this thesis where we compute topological properties of a subset of \cy manifolds.
\subsubsection{Complex and Kähler Manifolds}
@@ -753,28 +764,34 @@ The tensor $J$ is called \emph{complex structure} if there exist a tensor $N$ su
for any $v_p,\, w_p \in \rT_p M$, where $\liebraket{\cdot}{\cdot}\colon\, \rT_p M \times \rT_p M \to \rT_p M$ is the Lie braket of vector fields.
A manifold $M$ is a \emph{complex} manifold if it is possible to define a complex structure $J$ on it.\footnotemark{}
\footnotetext{%
Notice that a smooth function $f\colon\, M \to \C$ whose pushforward of $v_p \in \rT_p M$ is $f_{*\, p}\colon\, \rT_p M \to \rT_{f(p)} \C$ is called holomorphic if $(J\, f_{*\, p}( v_p ))^a = i ( f_{*\, p}( v_p ) )^a$ as such expression encodes the Cauchy-Riemann equations.
Notice that a smooth function $f\colon\, M \to \C$ whose pushforward of $v_p \in \rT_p M$ is $f_{*\, p}\colon\, \rT_p M \to \rT_{f(p)} \C \simeq \C$ is called holomorphic if $(J\, f_{*\, p}( v_p ))^a = i ( f_{*\, p}( v_p ) )^a$ as such expression encodes the Cauchy-Riemann equations.
Let in fact $f( x, y ) = f_1( x, y ) + i\, f_2( x, y )$, then the expression implies
\begin{equation*}
\begin{cases}
\ipd{x} f_1( x, y )
\ipd{x} f_1\qty( x, y )
& =
\ipd{y} f_2( x, y )
\ipd{y} f_2\qty( x, y )
\\
\ipd{x} f_2( x, y )
\ipd{x} f_2\qty( x, y )
& =
-\ipd{y} f_1( x, y )
-\ipd{y} f_1\qty( x, y )
\end{cases}
\quad
\Rightarrow
\ipd{x} f( x, y ) = -i \ipd{y} f( x, y )
\quad
\ipd{x} f\qty( x, y ) = -i \ipd{y} f\qty( x, y )
\quad
\Rightarrow
\ipd{\barz} f( z, \barz ) = 0
\quad
\ipd{\barz} f\qty( z, \barz ) = 0
\quad
\Rightarrow
f( z, \barz ) = f( z ).
\quad
f\qty( z, \barz ) = f( z ).
\end{equation*}
}
Let then $(M, J, g)$ be a complex manifold with a Riemannian metric $g$.
Let then $\qty(M,\, J,\, g)$ be a complex manifold with a Riemannian metric $g$.
The metric is \emph{Hermitian} if
\begin{equation}
g( v_p, w_p ) = g( J\, v_p, J\, w_p )
@@ -815,7 +832,7 @@ $(M, J, g)$ is a \emph{Kähler} manifold if:
\label{eq:cy:kaehler}
\end{equation}
or equivalently $\nabla J = 0$ or $\nabla \omega = 0$, where $\nabla$ is the connection of $g$.
Notice that the operators $\pd$ and $\bpd$ are operators such that $\pd^2 = \bpd^2 = 0$: they replace the \emph{de Rham cohomology} operator $\mathrm{d}^2 = 0$ in complex space with the holomorphic and antiholomorphic \emph{Dolbeault cohomology} operators.
Notice that the operators $\pd$ and $\bpd$ are such that $\pd^2 = \bpd^2 = 0$: they replace the \emph{de Rham cohomology} operator $\mathrm{d}^2 = 0$ in complex space with the holomorphic and antiholomorphic \emph{Dolbeault cohomology} operators.
The covariant conservation of $J$ and $\omega$ implies that the holonomy group must preserve these objects in $\R^{2m}$.
Thus we have $\mathrm{Hol}(g) \subseteq \U{m} \subset \OO{2m}$.
@@ -823,7 +840,7 @@ Thus we have $\mathrm{Hol}(g) \subseteq \U{m} \subset \OO{2m}$.
\subsubsection{Calabi-Yau Manifolds}
With the general definitions of the Kähler geometry we can now explicitly compute the conditions needed for a \cy manifold.
In local coordinates a Hermitian metric is such that
In local complex coordinates a Hermitian metric is such that
\begin{equation}
g
=
@@ -834,7 +851,9 @@ In local coordinates a Hermitian metric is such that
thus the Kähler form becomes $\omega = i g_{a\barb}\, \dd{z}^a \wedge \dd{\barz}^{\barb}$.
The relation~\eqref{eq:cy:kaehler} then translates into:
\begin{equation}
\dd{\omega} = i\, \qty( \pd + \bpd )\, g_{a\barb}\, \dd{z}^a \wedge \dd{\barz}^{\barb}
\dd{\omega}
=
i\, \qty( \pd + \bpd )\, g_{a\barb}\, \dd{z}^a \wedge \dd{\barz}^{\barb}
=
0
\quad
@@ -853,20 +872,20 @@ This ultimately leads to
=
\pdv{\phi( z, \barz )}{z^a}{\barz^{\barb}}
=
\ipd{a} \ipd{\barb}\, \phi( z, \barz ),
\ipd{z^a} \ipd{\barz^b}\, \phi( z, \barz ),
\end{equation}
Since $\omega$ is the Kähler form then the Levi-Civita connection has only fully holomorphic and anti-holomorphic components:
\begin{equation}
\tensor{\Gamma}{^a_{bc}}
=
\tensor{g}{^{a\bard}}\,
\ipd{b}\,
\ipd{z^b}\,
\tensor{g}{_{\bard c}},
\qquad
\tensor{\Gamma}{^{\bara}_{\barb\barc}}
\tensor{\Gamma}{^{\bara}_{\barb\barc}}
=
\tensor{g}{^{\bara d}}\,
\ipd{\barb}\,
\ipd{\barz^b}\,
\tensor{g}{_{d\barc}}.
\end{equation}
As a consequence the Ricci tensor becomes
@@ -877,8 +896,8 @@ As a consequence the Ricci tensor becomes
\pdv{\tensor{\Gamma}{^{\barc}_{\bara\barc}}}{z^b}.
\end{equation}
Since \cy manifolds have \SU{m} holonomy, the trace part of the coefficients of the connection vanishes.
\cy manifolds thus have $\tensor{R}{_{\bara b}} = 0$, that is they are complex Ricci-flat Kähler manifolds.
Since \cy manifolds present $\mathrm{Hol}(g) \subseteq \SU{m}$, the trace part of the coefficients of the connection vanishes.
\cy manifolds thus have $\tensor{R}{_{\bara b}} = 0$, that is they are complex Ricci-flat Kähler manifolds with \SU{m} holonomy.
\subsubsection{Cohomology and Hodge Numbers}
@@ -888,18 +907,18 @@ Since \cy manifolds have \SU{m} holonomy, the trace part of the coefficients of
They can be characterised in different ways.
For instance the study of the cohomology groups of the manifold has a direct connection with the analysis of topological invariants.
For real manifolds $M$ of dimension $2m$, closed $p$-forms $\omega$ are always defined up to an \emph{exact} term.
For real manifolds $\tildeM$ of dimension $2m$, closed $p$-forms $\omega$ are always defined up to an \emph{exact} term.
In fact:
\begin{equation}
\dd{\omega'_{(p)}} = \dd{(\omega_{(p)} + \dd{\eta_{(p-1)}})} = 0
\label{eq:cy:closedform}
\end{equation}
implies an equivalence relation $\omega'_{(p)} \sim \omega_{(p)} + \dd{\eta_{(p-1)}}$.
This translates in the fact that elements of the de Rham cohomology group $H^{(p)}_{\mathrm{d}}(M, \R)$ are equivalence classes $[ \omega ]$ computed through the operator $\mathrm{d}$.
The term $b^{p} = \dim{H^{(p)}_{\mathrm{d}}( M, \R )}$ counts the total number of possible $p$-forms we can build on $X$, up to \emph{gauge transformations}.
This translates to the fact that elements of the de Rham cohomology group $H^{(p)}_{\mathrm{d}}\qty(\tildeM, \R)$ are equivalence classes $[ \omega ]$ computed through the operator $\mathrm{d}$.
The term $b^{p} = \dim{H^{(p)}_{\mathrm{d}}( \tildeM, \R )}$ counts the total number of possible $p$-forms we can build on $\tildeM$, up to \emph{gauge transformations}.
These are known as \emph{Betti numbers}.
The extension to the Dolbeault cohomology in complex space is possible through the operators $\pd$ and $\bpd$ over $(r, s)$-forms on manifolds $M$ of real dimension $2m$.
The extension to the Dolbeault cohomology in complex space is possible through the operators $\pd$ and $\bpd$ over $(r, s)$-forms on manifolds $M$ of complex dimension $m$.
The equivalence relation~\eqref{eq:cy:closedform} has a similar expression in complex space as
\begin{equation}
\omega'_{(r,s)} \sim \omega_{(r,s)} + \bpd \omega_{(r,s-1)},
@@ -972,14 +991,14 @@ We are ultimately interested in their study to construct Yukawa couplings in str
As a first approach to the definition of D-branes, consider the action~\eqref{eq:conf:polyakov}.
The variation of such action with respect to $\delta X$ leads to the equation of motion
\begin{equation}
\partial_{\alpha} \partial^{\alpha}\, X^{\mu}( \tau, \sigma ) = 0
\partial_{\alpha} \partial^{\alpha} X^{\mu}( \tau, \sigma ) = 0
\qquad
\mu = 0, 1, \dots, D - 1,
\label{eq:tduality:eom}
\end{equation}
and naturally to the \emph{Neumann} boundary conditions:\footnotemark{}
\footnotetext{%
As~\cite{Polchinski:1996:TASILecturesDBranes} shows, \emph{Dirichlet} conditions can be shown to descend from T-duality.
As~\cite{Polchinski:1996:TASILecturesDBranes} shows, \emph{Dirichlet} conditions can descend from T-duality which is introduced later.
}
\begin{equation}
\eval{\ipd{\sigma} X^{\mu}( \tau, \sigma )}_{\sigma = 0}^{\sigma = \ell} = 0,
@@ -987,7 +1006,6 @@ and naturally to the \emph{Neumann} boundary conditions:\footnotemark{}
\mu = 0, 1, \dots, D - 1.
\label{eq:tduality:bc}
\end{equation}
Closed strings are such that $X^{\mu}( \tau, \sigma + \ell ) = X^{\mu}( \tau, \sigma )$.
The usual mode expansion in conformal coordinates $X^{\mu}( z, \barz ) = X( z ) + \barX( \barz )$ leads to
\begin{equation}
@@ -1054,7 +1072,7 @@ We finally have
\end{equation}
An interesting phenomenon involving these quantities appears when computing the mass spectrum of the theory.
From~\eqref{eq:conf:Texpansion} and \eqref{eq:conf:bosonicstringT} we find
From~\eqref{eq:conf:Texpansion} and~\eqref{eq:conf:bosonicstringT} we find
\begin{equation}
\begin{split}
L_0
@@ -1063,9 +1081,9 @@ From~\eqref{eq:conf:Texpansion} and \eqref{eq:conf:bosonicstringT} we find
\qty(
\qty( \alpha_0^{D-1} )^2
+
\sum\limits_{i = 0}^{D-2}\, \qty( \alpha_0^i )^2
\finitesum{i}{0}{D-2}\, \qty( \alpha_0^i )^2
+
\sum\limits_{n = 1}^{+\infty}\, \qty( 2 \alpha_{-n}^{\mu} \alpha_n^{\nu}\, \eta_{\mu\nu} + a )
\finitesum{n}{1}{+\infty}\, \qty( 2\, \alpha_{-n}^{\mu} \alpha_n^{\nu}\, \eta_{\mu\nu} + a )
),
\\
\barL_0
@@ -1074,9 +1092,9 @@ From~\eqref{eq:conf:Texpansion} and \eqref{eq:conf:bosonicstringT} we find
\qty(
\qty( \balpha_0^{D-1} )^2
+
\sum\limits_{i = 0}^{D-2}\, \qty( \balpha_0^i )^2
\finitesum{i}{0}{D-2}\, \qty( \balpha_0^i )^2
+
\sum\limits_{n = 1}^{+\infty}\, \qty( 2 \balpha_{-n}^{\mu} \balpha_n^{\nu}\, \eta_{\mu\nu} + a )
\finitesum{n}{1}{+\infty}\, \qty( 2\, \balpha_{-n}^{\mu} \balpha_n^{\nu}\, \eta_{\mu\nu} + a )
),
\end{split}
\end{equation}
@@ -1086,18 +1104,18 @@ Imposing physical conditions~\eqref{eq:conf:physical} and the \emph{level matchi
\begin{split}
M^2
& =
\frac{1}{(\ap)^2}\, \qty( n\, \frac{\ap}{R} + m\, R )^2
\frac{1}{\qty(\ap)^2}\, \qty( n\, \frac{\ap}{R} + m\, R )^2
+
\frac{4}{\ap}\, \qty( \rN + a )
\\
& =
\frac{1}{(\ap)^2}\, \qty( n\, \frac{\ap}{R} - m\, R )^2
\frac{1}{\qty(\ap)^2}\, \qty( n\, \frac{\ap}{R} - m\, R )^2
+
\frac{4}{\ap}\, \qty( \brN + a ),
\end{split}
\label{eq:dbranes:closedspectrum}
\end{equation}
where $\rN = \sum\limits_{n = 1}^{+\infty}\, \alpha_{-n} \cdot \alpha_n$ and $\brN = \sum\limits_{n = 1}^{+\infty}\, \balpha_{-n} \cdot \balpha_n$.
where $\rN = \finitesum{n}{1}{+\infty}\, \alpha_{-n} \cdot \alpha_n$ and $\brN = \finitesum{n}{1}{+\infty}\, \balpha_{-n} \cdot \balpha_n$.
We then notice that as $R \to \infty$ all states with $m \neq 0$ become infinitely massive while the states for $m = 0$ and all values of $n$ become a continuum.
Conversely, as $R \to 0$ all states with $n \neq 0$ become infinitely heavy.
In field theory this would translate into a reduction of the number of dimensions since the remaining fields would be independent of the compact coordinate~\cite{Polchinski:1996:TASILecturesDBranes,Zwiebach::FirstCourseString}.
@@ -1121,7 +1139,7 @@ defining the dual coordinate
\subsubsection{D-branes from T-duality}
Consider the case of open strings satisfying the \eom~\eqref{eq:tduality:eom} and the coundary conditions~\eqref{eq:tduality:bc}.
The usual mode expansion~\eqref{eq:tduality:modes} here leads to
The usual mode expansion~\eqref{eq:tduality:modes} here leads to:
\begin{equation}
X^{\mu}( z, \barz )
=
@@ -1187,7 +1205,7 @@ The coordinate of the endpoint in the compact direction is therefore fixed and c
\end{equation}
The only difference in the position of the endpoints can only be a multiple of the radius of the compactified dimension.
Otherwise they lie on the same hypersurface.
The procedure can be generalises to $p$ coordinates and constraining the string to live on a $(D - p - 1)$-brane.
The procedure can be generalised to $p$ coordinates, constraining the string to live on a $(D - p - 1)$-brane.
This geometric interpretation of the Dirichlet branes and boundary conditions is the basis for the definition of more complex scenarios in which multiple D-branes are inserted in spacetime.
D-branes are however much more than mathematical entities.
@@ -1203,14 +1221,14 @@ Reproducing the \sm or beyond \sm spectra are however strong constraints on the
Many of the physical requests usually involve the introduction of D-branes and the study of open strings in order to be able to define chiral fermions and realist gauge groups.
As seen in the previous section, D-branes introduce preferred directions of motion by restricting the hypersurface on which the open string endpoints live.
Specifically a Dp-branes breaks the original \SO{1, D-1} symmetry to $\SO{1, p} \otimes \SO{D - 1 - p}$.\footnotemark{}
Specifically a Dp-brane breaks the original \SO{1,\, D-1} symmetry to $\SO{1,\, p} \otimes \SO{D - 1 - p}$.\footnotemark{}
\footnotetext{%
Notice that usually $D = 10$, but we keep a generic indication of the spacetime dimensions when possible.
Notice that usually $D = 10$ in the superstring formulation ($D = 26$ for purely bosonic strings), but we keep a generic indication of the spacetime dimensions when possible.
}
The massless spectrum of the theory on the D-brane is easily computed in lightcone gauge~\cite{Goddard:1973:QuantumDynamicsMassless,Polchinski:1998:StringTheoryIntroduction,Green:1988:SuperstringTheoryIntroduction,Angelantonj:2002:OpenStrings}.
Using the residual symmetries~\eqref{eq:conf:residualgauge} of the two-dimensional diffeomorphism (i.e.\ armonic functions of $\tau$ and $\sigma$) we can set
Using the residual symmetries~\eqref{eq:conf:residualgauge} of the two-dimensional diffeomorphism (i.e.\ harmonic functions of $\tau$ and $\sigma$) we can set
\begin{equation}
X^+( \tau, \sigma ) = x_0^+ + 2 \ap\, p^+\, \tau,
X^+\qty( \tau, \sigma ) = x_0^+ + 2 \ap\, p^+\, \tau,
\end{equation}
where $X^{\pm} = \frac{1}{\sqrt{2}} (X^0 \pm X^{D-1})$.
The vanishing of the stress-energy tensor fixes the oscillators in $X^-$ in terms of the physical transverse modes.
@@ -1238,8 +1256,7 @@ we find that at the massless level we have a single \U{1} gauge field in the rep
\qquad
\alpha_{-1}^i \regvacuum.
\end{equation}
The introduction of a Dp-brane however breaks the Lorentz invariance down to $\SO{1, p} \otimes \SO{D - 1 - p}$.
The introduction of a Dp-brane however breaks the Lorentz invariance down to $\SO{1,\, p} \otimes \SO{D - 1 - p}$.
Thus the gauge field in the original theory is split into
\begin{equation}
\begin{split}
@@ -1288,14 +1305,14 @@ They have no dynamics and do not spoil Poincaré or conformal invariance in the
Each state can then be labelled by $i$ and $j$ running from $1$ to $N$.
Matrices $\tensor{\lambda}{^a_{ij}}$ thus form a basis for expanding wave functions and states:
\begin{equation}
\ket{n;\, a} = \sum\limits_{i,\, j = 1}^N \ket{n;\, i, j}\, \lambda^a_{ij}.
\ket{n;\, a} = \sum\limits_{i,\, j = 1}^N \ket{n;\, i, j}\, \tensor{\lambda}{^a_i_j}.
\end{equation}
In general Chan-Paton factors label the D-brane on which the endpoint of the string lives as in the left of~\Cref{fig:dbranes:chanpaton}.
Notice that strings stretching across different D-branes present an additional term in the mass shell condition proportional to the distance between the hypersurfaces: fields built using strings with Chan-Paton factors $\lambda^a_{ij}$ for which $i \neq j$ will therefore be massive.
However when $N$ D-branes coincide in space and form a stack their mass vanishes again: it then possible to organise the $N^2$ resulting massless fields in a representation of the gauge group \U{N}, thus promoting the symmetry $\bigotimes\limits_{a = 1}^N \rU_a( 1 )$ of $N$ separate D-branes.
Notice that strings stretching across different D-branes present an additional term in the mass shell condition proportional to the distance between the hypersurfaces: fields built using strings with Chan-Paton factors $\tensor{\lambda}{^a_i_j}$ for which $i \neq j$ will therefore be massive.
However when $N$ D-branes coincide in space and form a stack their mass vanishes again: it then possible to organise the $N^2$ resulting massless fields in a representation of the gauge group \U{N}, thus promoting the symmetry $\bigotimes\limits_{a = 1}^N \rU_a( 1 )$ of $N$ separate D-branes to a larger gauge group.
It is also possible to show that in the field theory limit the resulting gauge theory is a Super Yang-Mills gauge theory.
Eventually the massless spectrum of $N$ coincident $Dp-branes$ is formed by \U{N} gauge bosons in the adjoint representation, $N^2 \times (D - 1 - p)$ scalars and $N^2$ sets of $(p+1)$-dimensional fermions~\cite{Uranga:2005:TASILecturesString}.
Eventually the massless spectrum of $N$ coincident Dp-branes is formed by \U{N} gauge bosons in the adjoint representation, $N^2 \times (D - 1 - p)$ scalars and $N^2$ sets of $(p+1)$-dimensional fermions~\cite{Uranga:2005:TASILecturesString}.
These are the basic building blocks for a consistent string phenomenology involving both gauge bosons and matter.
@@ -1307,25 +1324,24 @@ For instance there is no way to describe chirality by simply using parallel D-br
For instance, in the low energy limit it is possible to build a gauge theory of the strong force using a stack of $3$ coincident D-branes and an electroweak sector using $2$ D-branes.
These stacks would separately lead to a $\U{3} \times \U{2}$ gauge theory.
It would however be theory of pure force, without matter content.
Moreover we should also worry about the extra \U{1} groups appearing: these need careful consideration but go beyond the necessary analysis for what follows.
It would however be a theory of pure force, without matter content.
Moreover we should also worry about the extra \U{1} groups appearing: these need careful consideration but go beyond the necessary analysis presented in what follows.
Matter fields are notoriously fermions transforming in the bi-fundamental representation $(\vec{N}, \vec{M})$ of the \sm gauge group~\eqref{eq:intro:smgroup}.
For example left handed quarks in the \sm transform under the $(\vec{3}, \vec{2})$ representation of the group $\SU{3}_C \otimes \SU{2}_L$.
Matter fields are fermions transforming in the bi-fundamental representation $\qty(\vb{N}, \vb{M})$ of the \sm gauge group~\eqref{eq:intro:smgroup}.
For example left handed quarks in the \sm transform under the $\qty(\vb{3}, \vb{2})$ representation of the group $\SU{3}_C \otimes \SU{2}_L$.
This is realised in string theory by a string stretched across two stacks of $3$ and $2$ D-branes as in the right of~\Cref{fig:dbranes:chanpaton}.
The fermion would then be characterised by the charge under the gauge bosons living on the D-branes.
The corresponding anti-particle would then simply be a string oriented in the opposite direction.
Things get complicated when introducing also left handed leptons transforming in the $(\vec{1}, \vec{2})$ representation: they cannot have endpoints on the same stack of D-branes as quarks since they do not have colour charge.
The corresponding anti-particle would then be modelled as a string oriented in the opposite direction.
Things get complicated when introducing also left handed leptons transforming in the $\qty(\vb{1}, \vb{2})$ representation: they cannot have endpoints on the same stack of D-branes as quarks since they do not have colour charge.
We therefore need to introduce more D-branes to account for all the possible combinations.
An additional issue comes from the requirement of chirality.
Strings stretched across D-branes are naturally massive but, in the field theory limit, a mass term would mix different chiralities.
We thus need to include a symmetry preserving mechanism for generating the mass of fermions.
In string theory there are ways to deal with the requirement~\cite{Uranga:2003:ChiralFourdimensionalString,Uranga:2005:TASILecturesString,Zwiebach::FirstCourseString,Aldazabal:2000:DBranesSingularitiesBottomUp}.
These range from D-branes located at singular points of orbifolds to D-branes intersecting at angles~\cite{Finotello:2019:ClassicalSolutionBosonic}.
We focus in particular on the latter.
Specifically we focus on intersecting D6-branes filling the $4$-dimensional spacetime and whose additional $3$ dimensions are embedded in a \cy 3-fold (e.g.\ as lines in a factorised torus $T^6 = T^2 \times T^2 \times T^2$).
This D-brane geometry supports chiral fermion at their intersection: while some of the modes of the stretched string become indeed massive, the spectrum of the fields is proportional to combinations of the angles and some of the modes can stay massless.
These range from D-branes located at singular points of orbifolds to D-branes intersecting at angles.
In this manuscript we focus on intersecting D6-branes filling the $4$-dimensional spacetime and whose additional $3$ dimensions are embedded in a \cy 3-fold (e.g.\ as lines in a factorised torus $T^6 = T^2 \times T^2 \times T^2$).
This D-brane geometry supports chiral fermion states at their intersection: while some of the modes of the stretched string become indeed massive, the spectrum of the fields is proportional to combinations of the angles and some of the modes can remain massless.
The light spectrum is thus composed of the desired matter content alongside with other particles arising from the string compactification.
\begin{figure}[tbp]
@@ -1340,20 +1356,21 @@ The light spectrum is thus composed of the desired matter content alongside with
\end{figure}
It is therefore possible to recover a \sm-like construction using multiple D-branes at angles as in~\Cref{fig:dbranes:smbranes}, where the angles have been drawn perpendicular but can in principle be arbitrary~\cite{Ibanez:2001:GettingJustStandard,Grimm:2005:EffectiveActionType,Sheikh-Jabbari:1998:ClassificationDifferentBranes,Berkooz:1996:BranesIntersectingAngles}.
For instance quarks are localised at the intersection of the \emph{baryonic} stack of D-branes, yielding the colour symmetry generators, with the \emph{left} and \emph{right} stacks, leading to the $( \vec{3}, \vec{2} )$ and $( \vec{3}, \vec{1})$ representations.
For instance quarks are localised at the intersection of the \emph{baryonic} stack of D-branes, yielding the colour symmetry generators, with the \emph{left} and \emph{right} stacks, leading to the $\qty( \vb{3}, \vb{2} )$ and $\qty( \vb{3}, \vb{1})$ representations.
The same applies to leptons created by strings attached to the \emph{leptonic} stack.
Combinations of the additional \U{1} factors in the resulting gauge group finally lead to the definition of the hypercharge $Y$.
Physics in $4$ dimensions is eventually recovered by compactifying the extra-dimensions of the superstring.\footnotemark{}
\footnotetext{%
In general we reviewed particle physics interactions.
We specifically reviewed particle physics interactions.
Gravitational interactions in general remain untouched by these constructions and still propagate in $10$-dimensional spacetime.
}
Fermions localised at the intersection of the D-branes are however naturally $4$-dimensional as they only propagate in the non compact Minkowski space $\ccM^{1,3}$.
The presence of compactified dimensions however leads to phenomena such as \emph{family replications} of the fermions.
With accurate calibration it is in fact possible to recover the quark and lepton families in the \sm.
Consider for example the simple \cy factorised manifold $T^6 = T^2 \times T^2 \times T^2$ and introduce stacks of D6-branes as lines in each of the bi-tori.
Even though the lines might never intersect on a plane, they can come across on a torus due to the identifications~\cite{Zwiebach::FirstCourseString}.
Even though the lines might never intersect on a plane, they can have points in common on a torus due to the identifications~\cite{Zwiebach::FirstCourseString}.
Since each intersections supports a different set of fermions with different spectrum, the angles of the intersecting branes can be calibrated to reproduce the separation in mass of the families of quarks and leptons in the \sm.
% vim: ft=tex

10
sec/part2/conclusion.tex
View File

@@ -1,11 +1,11 @@
In the previous analysis it seems that string theory cannot do better than field theory when the latter does not exist, at least at the perturbative level where one deals with particles.
Moreover when spacetime becomes singular, the string massive modes are not anymore spectators.
From the previous analysis it seems that string theory cannot do better than field theory when the latter does not exist, at least at the perturbative level where one deals with particles.
Moreover when spacetime becomes singular the string massive modes are not spectators anymore.
Everything seems to suggest that issues with spacetime singularities are hidden into contact terms and interactions with massive states.
This would explain in an intuitive way why the eikonal approach to gravitational scattering works well: the eikonal is indeed concerned with three point massless interactions.
This would explain in an intuitive way why the eikonal approach to gravitational scattering works well: it is indeed concerned with three point massless interactions.
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}.
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.
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 develops divergences.
% vim: ft=tex

2
sec/part2/divergences.tex
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@@ -9,7 +9,7 @@ In what follows we show a direct computation showing that the presence of the di
Unnoticed in the Null Boost Orbifold (\nbo) \cite{Liu:2002:StringsTimeDependentOrbifold}, even the four \emph{open string} tachyons amplitude is divergent.
Since we are working at tree level gravity is not an issue.
In fact in Equation (6.16) of \cite{Liu:2002:StringsTimeDependentOrbifold} the four tachyons amplitude in the divergent region reads
In fact in~\cite{Liu:2002:StringsTimeDependentOrbifold} the four tachyons amplitude in the divergent region reads
\begin{equation}
A_4 \sim \int\limits_{q \sim \infty} \frac{\dd{q}}{\abs{q}} \ccA( q )
\end{equation}

23
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@@ -1,14 +1,14 @@
In the previous part we mainly focused on the mathematical tools needed to compute amplitudes in a phenomenologically valid string theory framework of particle physics.
This ultimately led to the introduction of intersecting D-branes and point-like defects to perform the calculation of correlation functions involving twist and spin fields, inevitably necessary fields when considering chiral matter fields.
In the previous part we mainly focused on the mathematical tools needed to compute amplitudes in a (semi-)phenomenologically viable string theory framework of particle physics.
This ultimately led to the introduction of intersecting D-branes and point-like defects to perform the calculation of correlation functions involving twist and spin fields, inevitably necessary when considering chiral matter fields.
While this is indeed a good starting point to build an entire string phenomenology, the theory cannot be limited to the study of particle physics models.
String Theory is in fact considered to be one of the candidate theories for the description of quantum gravity alongside the nuclear interactions.
String theory is in fact considered to be one of the candidate theories for the description of quantum gravity alongside the nuclear interactions.
As a \emph{theory of everything} it is therefore fascinating to analyse cosmological implications as seen from its description.
In this part of the thesis we focus on the implications of the string theory when considering for instance the Big Bang singularity, or, broadly speaking, singularities which exist in one point in time.\footnotemark{}
In this part of the thesis we focus on the implications of the string theory when considering for instance the Big Bang singularity, or, broadly speaking, singularities which exist in one point in time (i.e.\ space-like).\footnotemark{}
\footnotetext{%
They are intended as distinct from time-like singularities such as black holes which are present for extended periods of time in one spatial point.
The space-like singularities we consider are the opposite: they exist in a given instant.
The space-like singularities we consider are the opposite: they exist in a given instant in time but could in principle cover an extended hypersurface in space.
}
Among the different possible descriptions of such space-like singularities~\cite{Berkooz:2007:ShortReviewTime} we concentrate on string theory solutions on time-dependent orbifolds.
Among the different possible descriptions of such space-like singularities~\cite{Berkooz:2007:ShortReviewTime} we concentrate on string theory solutions on time-dependent orbifolds as they represent the simplest models describing such phenomena.
Before delving into the subject we briefly present their definition and the reason behind their relevance in what follows~\cite{CaramelloJr:2019:IntroductionOrbifolds,Bachas:2002:NullBraneIntersections,Bachas:2003:RelativisticStringPulse}.
@@ -16,9 +16,9 @@ Before delving into the subject we briefly present their definition and the reas
First of all we recall the formal definition of orbifold to better introduce the idea of a manifold locally isomorphic to a quotient space.
Let therefore $M$ be a topological space and $G$ a group with an action $\ccG: G \times M \to M$ defined by $\ccG(g,\, p) = g p$ for $g \in G$ and $p \in M$.
Then the \emph{isotropy subgroup} (or \emph{stabiliser}) of $p \in M$ is $G_p = \qty{ g \in G \mid g p = p }$ such that $G_{gp} = g\, G_p\, g^{-1}$.
Then the \emph{isotropy subgroup} (or \emph{stabiliser}) of $p \in M$ is $G_p = \qty{ g \in G \mid g p = p }$ such that $G_{gp} = g^{-1}\, G_p\, g$.
Given an element $p \in M$ its \emph{orbit} is $Gp = \qty{ gp \in M \mid g \in G }$.
The action of the group is said \emph{transitive} if $Gp = M$ and \emph{effective} if its $\ker{\ccG} = \qty{ \1 }$.
The action of the group is said \emph{transitive} if $Gp = M$ and \emph{effective} if its kernel is trivial, i.e.\ $\ker{\ccG} = \qty{ \1 }$.
The \emph{orbit space} $M / G$ is the set of equivalence classes given by the orbital partitions and inherits the quotient topology from $M$.
Let now $M$ be a manifold and $G$ a Lie group acting continuously and transitively on $M$.
@@ -29,8 +29,8 @@ For every point $p \in M$ we can define a continuous bijection $\lambda_p \colon
}
Such map is a diffeomorphism if $M$ and $G$ are locally compact spaces and $M / G$ is in turn a manifold itself.
If $G$ is a discrete or finite group the action is called \emph{properly discontinuous}, that is for every $U \subset M$ then $\qty{ g \in G \mid U \cap g U \neq \emptyset }$ is finite.
The definition of orbifold intuitively includes quotient manifolds such as $M / G$: analogously to manifold which are locally Euclidean, in the broad sense orbifolds are locally modelled by quotients with actions given by finite groups.
The definition of orbifold intuitively includes quotient manifolds such as $M / G$: analogously to manifold which are locally Euclidean, in the broad sense orbifolds are locally modelled by quotients with actions given by finite groups.
An \emph{orbifold chart} $\qty( \tildeU,\, G,\, \phi )$ of dimension $n \in \N$ for an open subset $U \in M$ is made of:
\begin{itemize}
\item a connected open subset $\tildeU \subset \R^n$,
@@ -51,12 +51,13 @@ The $n$-dimensional \emph{orbifold} $\ccO$ is finally defined as a paracompact H
}
\subsection{Orbifolds and Strings}
\subsubsection{Orbifolds and Strings}
In string theory the notion of orbifold has a more stringent characterisation with respect to pure mathematics.
Differently from the general definition, orbifolds in physics usually appear as a global orbit space $M / G$ where $M$ is a manifold and $G$ the group of its isometries, often leading to the presence of \emph{fixed points} (i.e.\ points in the manifold which are left invariant by the action of $G$) where singularities emerge due to the presence of additional degrees of freedom given by \emph{twisted states} of the string~\cite{Dixon:1985:StringsOrbifolds,Dixon:1986:StringsOrbifoldsII}.
They are commonly introduced as singular limits of \cy manifolds~\cite{Candelas:1985:VacuumConfigurationsSuperstrings}, which in turn can be recovered using algebraic geometry to smoothen the singular points.
However they can also be used to model peculiar time-dependent backgrounds~\cite{Horowitz:1991:SingularStringSolutions,Khoury:2002:BigCrunchBig,Figueroa-OFarrill:2001:GeneralisedSupersymmetricFluxbranes,Cornalba:2002:NewCosmologicalScenario,Cornalba:2004:TimedependentOrbifoldsString,Craps:2006:BigBangModels,Bachas:2002:NullBraneIntersections,Bachas:2003:RelativisticStringPulse}.
They are in fact good toy models to study Big Bang scenarios in string theory and we focus specifically on the study of such cosmological singularity in the framework of string theory.
They are in fact good toy models to study Big Bang scenarios in string theory.
We focus specifically on the study of such cosmological singularities in the framework of string theory defined on time-dependent orbifolds.
% vim: ft=tex

8
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@@ -1,5 +1,5 @@
We have proved how a proper data analysis can lead to improvements in predictions of Hodge numbers \hodge{1}{1} and \hodge{2}{1} for \cicy $3$-folds.
Moreover considering more complex neural networks inspired by the computer vision applications~\cite{Szegedy:2015:GoingDeeperConvolutions, Szegedy:2016:RethinkingInceptionArchitecture, Szegedy:2016:Inceptionv4InceptionresnetImpact} allowed us to reach close to \SI{100}{\percent} accuracy for \hodge{1}{1} with much less data and less parameters than in previous works.
We have proved that a proper data analysis can lead to improvements in predictions of Hodge numbers \hodge{1}{1} and \hodge{2}{1} for \cicy $3$-folds.
Moreover more complex neural networks inspired by computer vision applications~\cite{Szegedy:2015:GoingDeeperConvolutions, Szegedy:2016:RethinkingInceptionArchitecture, Szegedy:2016:Inceptionv4InceptionresnetImpact} allowed us to reach close to \SI{100}{\percent} accuracy for \hodge{1}{1} with much less data and less parameters than in previous works.
While our analysis improved the accuracy for \hodge{2}{1} over what can be expected from a simple sequential neural network, we barely reached \SI{50}{\percent}.
Hence it would be interesting to push further our study to improve the accuracy.
Possible solutions would be to use a deeper Inception network, find a better architecture including engineered features, and refine the ensembling.
@@ -12,8 +12,8 @@ Or on the contrary one could generate more matrices for the same manifold in ord
Another possibility is to use the graph representation of the configuration matrix to which is automatically invariant under permutations~\cite{Hubsch:1992:CalabiyauManifoldsBestiary} (another graph representation has been decisive in~\cite{Krippendorf:2020:DetectingSymmetriesNeural} to get a good accuracy).
Techniques such as (variational) autoencoders~\cite{Kingma:2014:AutoEncodingVariationalBayes, Rezende:2014:StochasticBackpropagationApproximate, Salimans:2015:MarkovChainMonte}, cycle GAN~\cite{Zhu:2017:UnpairedImagetoimageTranslation}, invertible neural networks~\cite{Ardizzone:2019:AnalyzingInverseProblems}, graph neural networks~\cite{Gori:2005:NewModelLearning, Scarselli:2004:GraphicalbasedLearningEnvironments} or more generally techniques from geometric deep learning~\cite{Monti:2017:GeometricDeepLearning} could be helpful.
Finally, our techniques apply directly to \cicy $4$-folds~\cite{Gray:2013:AllCompleteIntersection, Gray:2014:TopologicalInvariantsFibration}.
However there are much more manifolds in this case, 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).
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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@@ -3,9 +3,9 @@ The ultimate goal of the analysis is to provide some insights on the predictive
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.
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 prevented any precise connection to the existing and tested theories (in particular, the \sm of particle physics).
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}.
@@ -19,6 +19,7 @@ This motivates two main applications to string theory: the systematic exploratio
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}.
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 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.
@@ -48,8 +49,8 @@ A good validation strategy is also needed to ensure that the predictions appropr
For instance, we find that a simple linear regression using the configuration matrix as input gives \SIrange{43.6}{48.8}{\percent} for \hodge{1}{1} and \SIrange{9.6}{10.4}{\percent} for \hodge{2}{1} using from $20\%$ to $80\%$ of data for training.
Hence any algorithm \emph{must} do better than this to be worth considering.
In the dataset we use for accomplishing the task there is a finite number of $7890$ \cicy $3$-folds.
Due to the freedom in representing the configuration matrix, two datasets have been constructed: the \emph{original dataset}~\cite{Candelas:1988:CompleteIntersectionCalabiYau, Green:1989:AllHodgeNumbers} and the \emph{favourable dataset}~\cite{Anderson:2017:FibrationsCICYThreefolds}.
The datasets we use for task contains $7890$ \cicy $3$-folds.
Due to the freedom in representing the configuration matrix, we need to consider two datasets which have been constructed: the \emph{original dataset}~\cite{Candelas:1988:CompleteIntersectionCalabiYau, Green:1989:AllHodgeNumbers} and the \emph{favourable dataset}~\cite{Anderson:2017:FibrationsCICYThreefolds}.
Our analysis continues and generalises~\cite{He:2017:MachinelearningStringLandscape, Bull:2018:MachineLearningCICY} at different levels.
For example we compute \hodge{2}{1} which has been ignored in~\cite{He:2017:MachinelearningStringLandscape, Bull:2018:MachineLearningCICY}, where the authors argue that it can be computed from \hodge{1}{1} and from the Euler characteristics (a simple formula exists for the latter).
In our case, we want to push the idea of using \ml to learn about the physics (or the mathematics) of \cy to its very end: we assume that we do not know anything about the mathematics of the \cicy, except that the configuration matrix is sufficient to derive all quantities.
@@ -58,8 +59,8 @@ Thus getting also \hodge{2}{1} from \ml techniques is an important first step to
Finally regression is also more useful for extrapolating results: a classification approach assumes that we already know all the possible values of the Hodge numbers and has difficulties to predict labels which do not appear in the training set.
This is necessary when we move to a dataset for which not all topological quantities have been computed, for instance CY constructed from the Kreuzer--Skarke list of polytopes~\cite{Kreuzer:2000:CompleteClassificationReflexive}.
The data analysis and \ml are programmed in Python using standard open-source packages: \texttt{pandas}~\cite{WesMcKinney:2010:DataStructuresStatistical}, \texttt{matplotlib}~\cite{Hunter:2007:Matplotlib2DGraphics}, \texttt{seaborn}~\cite{Waskom:2020:MwaskomSeabornV0}, \texttt{scikit-learn}~\cite{Pedregosa:2011:ScikitlearnMachineLearning}, \texttt{scikit-optimize}~\cite{Head:2020:ScikitoptimizeScikitoptimize}, \texttt{tensorflow}~\cite{Abadi:2015:TensorFlowLargescaleMachine} (and its high level API \emph{Keras}).
The code and its description are available on \href{https://thesfinox.github.io/ml-cicy/}{Github}.
The data analysis and \ml are programmed in Python using open-source packages: \texttt{pandas}~\cite{WesMcKinney:2010:DataStructuresStatistical}, \texttt{matplotlib}~\cite{Hunter:2007:Matplotlib2DGraphics}, \texttt{seaborn}~\cite{Waskom:2020:MwaskomSeabornV0}, \texttt{scikit-learn}~\cite{Pedregosa:2011:ScikitlearnMachineLearning}, \texttt{scikit-optimize}~\cite{Head:2020:ScikitoptimizeScikitoptimize}, \texttt{tensorflow}~\cite{Abadi:2015:TensorFlowLargescaleMachine} (and its high level API \emph{Keras}).
Code is available on \href{https://thesfinox.github.io/ml-cicy/}{Github}.
\subsection{Complete Intersection Calabi--Yau Manifolds}
@@ -70,7 +71,7 @@ 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).
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.
Calabi--Yau 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{}
\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{}
\footnotetext{%
In full generality, the Hodge numbers \hodge{p}{q} count the numbers of harmonic $\qty(p,\, q)$-forms.
}%
@@ -78,21 +79,21 @@ Calabi--Yau manifolds are characterised by a certain number of topological prope
\chi = 2 \qty(\hodge{1}{1} - \hodge{2}{1}).
\label{eq:cy:euler}
\end{equation}
Interestingly, topological properties of the manifold directly translates 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, Becker:2006:StringTheoryMTheory}.\footnotemark{}
\footnotetext{%
Another reason for sticking to topological properties is that there is no CY 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.
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 ($3$ in the Standard Model) and is thus an important measure of the distance to the Standard Model.
The simplest CYs 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, Anderson:2018:TASILecturesGeometric}:
\begin{equation}
\cA = \mathds{P}^{n_1} \times \cdots \times \mathds{P}^{n_m}.
\end{equation}
Such hypersurfaces are defined by homogeneous polynomial equations: a Calabi--Yau $X$ is described by the solution to the system of equations, i.e.\ by the intersection of all these surfaces.
Such hypersurfaces are defined by homogeneous polynomial equations: a \cicy manifold $X$ is described by the solution to the system of equations, i.e.\ by the intersection of all these surfaces.
The intersection is ``complete'' in the sense that the hypersurface is non-degenerate.
To gain some intuition, consider the case of a single projective space $\mathds{P}^n$ with (homogeneous) coordinates $Z^I$, $I = 0, \ldots, n$.
To gain some intuition, consider the case of a single projective space $\mathds{P}^n$ with (homogeneous) coordinates $Z^I$, where $I = 0,\, 1,\, \dots,\, n$.
A codimension $1$ subspace is obtained by imposing a single homogeneous polynomial equation of degree $a$ on the coordinates:
\begin{equation}
\begin{split}
@@ -109,7 +110,7 @@ A codimension $1$ subspace is obtained by imposing a single homogeneous polynomi
Each choice of the polynomial coefficients $P_{I_1 \dots I_a}$ leads to a different manifold.
However it can be shown that the manifolds are in general topologically equivalent.
Since we are interested only in classifying the \cy as topological manifolds and not as complex manifolds, the information on $P_{I_1 \dots I_a}$ can be discarded and it is sufficient to keep track only of the dimension $n$ of the projective space and of the degree $a$ of the equation.
The resulting hypersurface is denoted equivalently as $\qty[\mathds{P}^n \mid a] = \qty[n \mid a]$.
The resulting hypersurface is denoted as $\qty[\mathds{P}^n \mid a] = \qty[n \mid a]$.
Notice that $\qty[\mathds{P}^n \mid a]$ is $3$-dimensional if $n = 4$ (the equation reduces the dimension by one), and it is a \cy (the ``quintic'') if $a = n + 1 = 5$ (this is required for the vanishing of its first Chern class).
The simplest representative of this class if Fermat's quintic defined by the equation:
\begin{equation}
@@ -208,12 +209,9 @@ Below we show a list of the \cicy properties and of their configuration matrices
\label{fig:data:hist-hodge}
\end{figure}
The configuration matrix completely encodes the information of the \cicy and all topological quantities can be derived from it.
However the computations are involved and there is often no closed-form expression.
This situation is typical in algebraic geometry and it can be even worse for some problems, in the sense that it is not even known how to compute the desired quantity (e.g. the metric of \cy manifolds).
For these reasons it is interesting to study how to retrieve these properties using \ml algorithms.
In what follows we focus on the prediction of the Hodge numbers.
We then move to the data science analysis of the data.
To provide a good test case for the use of \ml in context where the mathematical theory is not completely understood, we make no use of known formulas.
In fact we try to push as far as possible the capabilities of \ml algorithms to play a role in discovering patterns which can be used in phenomenology and algebraic geometry.
% vim: ft=tex

20
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@@ -1398,6 +1398,7 @@
pages = {651--686},
issn = {05503213},
doi = {10.1016/0550-3213(90)90379-R},
file = {/home/riccardo/.local/share/zotero/files/di_bartolomeo_et_al_1990_general_properties_of_vertices_with_two_ramond_or_twisted_states4.pdf},
keywords = {archived},
langid = {english},
number = {3}
@@ -1464,6 +1465,7 @@
pages = {63--70},
issn = {03702693},
doi = {10.1016/0370-2693(90)90098-Q},
file = {/home/riccardo/.local/share/zotero/files/di_vecchia_et_al_1990_a_vertex_including_emission_of_spin_fields4.pdf},
keywords = {archived},
langid = {english},
number = {1-2}
@@ -2093,7 +2095,8 @@
date = {2014},
pages = {2672--2680},
publisher = {{Curran Associates, Inc.}},
url = {http://papers.nips.cc/paper/5423-generative-adversarial-nets.pdf}
url = {http://papers.nips.cc/paper/5423-generative-adversarial-nets.pdf},
file = {/home/riccardo/.local/share/zotero/files/goodfellow_et_al_2014_generative_adversarial_nets.pdf}
}
@inproceedings{Gori:2005:NewModelLearning,
@@ -2104,6 +2107,7 @@
volume = {2},
pages = {729--734},
doi = {10.1109/IJCNN.2005.1555942},
file = {/home/riccardo/.local/share/zotero/files/gori_et_al_2005_a_new_model_for_learning_in_graph_domains2.pdf},
organization = {{IEEE}}
}
@@ -2142,7 +2146,7 @@
@article{Gray:2013:AllCompleteIntersection,
title = {All {{Complete Intersection Calabi}}-{{Yau Four}}-{{Folds}}},
author = {Gray, James and Haupt, Alexander S. and Lukas, Andre},
date = {2013-07},
date = {2013},
journaltitle = {Journal of High Energy Physics},
shortjournal = {J. High Energ. Phys.},
volume = {2013},
@@ -2160,7 +2164,7 @@
@article{Gray:2014:TopologicalInvariantsFibration,
title = {Topological {{Invariants}} and {{Fibration Structure}} of {{Complete Intersection Calabi}}-{{Yau Four}}-{{Folds}}},
author = {Gray, James and Haupt, Alexander S. and Lukas, Andre},
date = {2014-09},
date = {2014},
journaltitle = {Journal of High Energy Physics},
shortjournal = {J. High Energ. Phys.},
volume = {2014},
@@ -2747,7 +2751,7 @@
@online{Kingma:2014:AutoEncodingVariationalBayes,
title = {Auto-{{Encoding Variational Bayes}}},
author = {Kingma, Diederik P. and Welling, Max},
date = {2014-05-01},
date = {2014},
url = {http://arxiv.org/abs/1312.6114},
urldate = {2020-10-10},
abstract = {How can we perform efficient inference and learning in directed probabilistic models, in the presence of continuous latent variables with intractable posterior distributions, and large datasets? We introduce a stochastic variational inference and learning algorithm that scales to large datasets and, under some mild differentiability conditions, even works in the intractable case. Our contributions is two-fold. First, we show that a reparameterization of the variational lower bound yields a lower bound estimator that can be straightforwardly optimized using standard stochastic gradient methods. Second, we show that for i.i.d. datasets with continuous latent variables per datapoint, posterior inference can be made especially efficient by fitting an approximate inference model (also called a recognition model) to the intractable posterior using the proposed lower bound estimator. Theoretical advantages are reflected in experimental results.},
@@ -3491,7 +3495,7 @@
@online{Rezende:2014:StochasticBackpropagationApproximate,
title = {Stochastic {{Backpropagation}} and {{Approximate Inference}} in {{Deep Generative Models}}},
author = {Rezende, Danilo Jimenez and Mohamed, Shakir and Wierstra, Daan},
date = {2014-05-30},
date = {2014},
url = {http://arxiv.org/abs/1401.4082},
urldate = {2020-10-10},
abstract = {We marry ideas from deep neural networks and approximate Bayesian inference to derive a generalised class of deep, directed generative models, endowed with a new algorithm for scalable inference and learning. Our algorithm introduces a recognition model to represent approximate posterior distributions, and that acts as a stochastic encoder of the data. We develop stochastic back-propagation -- rules for back-propagation through stochastic variables -- and use this to develop an algorithm that allows for joint optimisation of the parameters of both the generative and recognition model. We demonstrate on several real-world data sets that the model generates realistic samples, provides accurate imputations of missing data and is a useful tool for high-dimensional data visualisation.},
@@ -3566,6 +3570,7 @@
author = {Salimans, Tim and Kingma, Diederik and Welling, Max},
date = {2015},
pages = {1218--1226},
file = {/home/riccardo/.local/share/zotero/files/salimans_et_al_2015_markov_chain_monte_carlo_and_variational_inference.pdf},
keywords = {⛔ No DOI found}
}
@@ -3575,8 +3580,9 @@
author = {Scarselli, Franco and Tsoi, Ah Chung and Gori, Marco and Hagenbuchner, Markus},
date = {2004},
pages = {42--56},
keywords = {⛔ No DOI found},
organization = {{Springer}}
doi = {10.1007/978-3-540-27868-9_4},
file = {/home/riccardo/.local/share/zotero/files/scarselli_et_al_2004_graphical-based_learning_environments_for_pattern_recognition3.pdf},
isbn = {978-3-540-22570-6 978-3-540-27868-9}
}
@online{Schellekens:2017:BigNumbersString,