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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.
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.
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.
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.
We finally present a new artificial intelligence approach to algebraic geometry and string compactifications.
We compute the Hodge numbers of Complete Intersection Calabi--Yau $3$-folds using deep learning techniques based on computer vision and object recognition techniques.
We also include a methodological study of machine learning applied to data in string theory: as in most applications machine learning almost never relies on the blind application of algorithms to the data but it requires a careful exploratory analysis and feature engineering.
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.
% vim: ft=tex

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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.
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{}
\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.
}
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.
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.
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.
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{}
\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.

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sec/part1/introduction.tex
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@@ -28,7 +28,7 @@ Such surface can have different topologies according to the nature of the object
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 usually best to work with Polyakov's action~\cite{Polyakov:1981:QuantumGeometryBosonic}:
\begin{equation}
S_P[ \gamma, X ]
=
@@ -42,7 +42,7 @@ While Nambu and Goto's formulation is fairly direct in its definition, it usuall
\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 $X^{\mu}(\tau, \sigma)$ is therefore:
\begin{equation}
\frac{1}{\sqrt{- \det \gamma}}\,
\ipd{\alpha}

31
thesis.bib
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@@ -324,7 +324,7 @@
@online{Ardizzone:2019:AnalyzingInverseProblems,
title = {Analyzing {{Inverse Problems}} with {{Invertible Neural Networks}}},
author = {Ardizzone, Lynton and Kruse, Jakob and Wirkert, Sebastian and Rahner, Daniel and Pellegrini, Eric W. and Klessen, Ralf S. and Maier-Hein, Lena and Rother, Carsten and Köthe, Ullrich},
date = {2019-02-06},
date = {2019},
url = {http://arxiv.org/abs/1808.04730},
urldate = {2020-10-10},
abstract = {In many tasks, in particular in natural science, the goal is to determine hidden system parameters from a set of measurements. Often, the forward process from parameter- to measurement-space is a well-defined function, whereas the inverse problem is ambiguous: one measurement may map to multiple different sets of parameters. In this setting, the posterior parameter distribution, conditioned on an input measurement, has to be determined. We argue that a particular class of neural networks is well suited for this task -- so-called Invertible Neural Networks (INNs). Although INNs are not new, they have, so far, received little attention in literature. While classical neural networks attempt to solve the ambiguous inverse problem directly, INNs are able to learn it jointly with the well-defined forward process, using additional latent output variables to capture the information otherwise lost. Given a specific measurement and sampled latent variables, the inverse pass of the INN provides a full distribution over parameter space. We verify experimentally, on artificial data and real-world problems from astrophysics and medicine, that INNs are a powerful analysis tool to find multi-modalities in parameter space, to uncover parameter correlations, and to identify unrecoverable parameters.},
@@ -336,17 +336,22 @@
primaryClass = {cs, stat}
}
@online{Arduino:2020:OriginDivergencesTimeDependent,
@article{Arduino:2020:OriginDivergencesTimeDependent,
title = {On the {{Origin}} of {{Divergences}} in {{Time}}-{{Dependent Orbifolds}}},
author = {Arduino, Andrea and Finotello, Riccardo and Pesando, Igor},
date = {2020},
journaltitle = {The European Physical Journal C},
shortjournal = {Eur. Phys. J. C},
volume = {80},
pages = {476},
issn = {1434-6044, 1434-6052},
doi = {10.1140/epjc/s10052-020-8010-y},
abstract = {We consider time-dependent orbifolds in String Theory and we show that divergences are not associated with a gravitational backreaction since they appear in the open string sector too. They are related to the non existence of the underlying effective field theory as in several cases fourth and higher order contact terms do not exist. Since contact terms may arise from the exchange of string massive states, we investigate and show that some three points amplitudes with one massive state in the open string sector are divergent on the time-dependent orbifolds. To check that divergences are associated with the existence of a discrete zero eigenvalue of the Laplacian of the subspace with vanishing volume, we construct the Generalized Null Boost Orbifold where this phenomenon can be turned on and off.},
archivePrefix = {arXiv},
eprint = {2002.11306},
eprinttype = {arxiv},
file = {/home/riccardo/.local/share/zotero/files/arduino_et_al_2020_on_the_origin_of_divergences_in_time-dependent_orbifolds.pdf},
primaryClass = {gr-qc, physics:hep-th}
file = {/home/riccardo/.local/share/zotero/files/arduino_et_al_2020_on_the_origin_of_divergences_in_time-dependent_orbifolds2.pdf;/home/riccardo/.local/share/zotero/storage/QNJXWD2H/2002.html},
number = {5}
}
@online{Ashmore:2020:MachineLearningCalabiYau,
@@ -1722,19 +1727,15 @@
}
@online{Erbin:2020:InceptionNeuralNetwork,
ids = {Erbin:2020:InceptionNeuralNetworka},
title = {Inception {{Neural Network}} for {{Complete Intersection Calabi}}-{{Yau}} 3-Folds},
author = {Erbin, Harold and Finotello, Riccardo},
date = {2020},
url = {http://arxiv.org/abs/2007.13379},
urldate = {2020-08-06},
abstract = {We introduce a neural network inspired by Google's Inception model to compute the Hodge number \$h\^\{1,1\}\$ of complete intersection Calabi-Yau (CICY) 3-folds. This architecture improves largely the accuracy of the predictions over existing results, giving already 97\% of accuracy with just 30\% of the data for training. Moreover, accuracy climbs to 99\% when using 80\% of the data for training. This proves that neural networks are a valuable resource to study geometric aspects in both pure mathematics and string theory.},
archivePrefix = {arXiv},
eprint = {2007.13379},
eprinttype = {arxiv},
file = {/home/riccardo/.local/share/zotero/files/erbin_finotello_2020_inception_neural_network_for_complete_intersection_calabi-yau_3-folds.pdf},
keywords = {⛔ No DOI found},
primaryClass = {hep-th}
keywords = {⛔ No DOI found}
}
@online{Erbin:2020:MachineLearningComplete,
@@ -1742,15 +1743,12 @@
shorttitle = {Machine Learning for Complete Intersection {{Calabi}}-{{Yau}} Manifolds},
author = {Erbin, Harold and Finotello, Riccardo},
date = {2020},
url = {http://arxiv.org/abs/2007.15706},
urldate = {2020-08-06},
abstract = {We revisit the question of predicting both Hodge numbers \$h\^\{1,1\}\$ and \$h\^\{2,1\}\$ of complete intersection Calabi-Yau (CICY) 3-folds using machine learning (ML), considering both the old and new datasets built respectively by Candelas-Dale-Lutken-Schimmrigk / Green-H\textbackslash "ubsch-Lutken and by Anderson-Gao-Gray-Lee. In real world applications, implementing a ML system rarely reduces to feed the brute data to the algorithm. Instead, the typical workflow starts with an exploratory data analysis (EDA) which aims at understanding better the input data and finding an optimal representation. It is followed by the design of a validation procedure and a baseline model. Finally, several ML models are compared and combined, often involving neural networks with a topology more complicated than the sequential models typically used in physics. By following this procedure, we improve the accuracy of ML computations for Hodge numbers with respect to the existing literature. First, we obtain 97\% (resp. 99\%) accuracy for \$h\^\{1,1\}\$ using a neural network inspired by the Inception model for the old dataset, using only 30\% (resp. 70\%) of the data for training. For the new one, a simple linear regression leads to almost 100\% accuracy with 30\% of the data for training. The computation of \$h\^\{2,1\}\$ is less successful as we manage to reach only 50\% accuracy for both datasets, but this is still better than the 16\% obtained with a simple neural network (SVM with Gaussian kernel and feature engineering and sequential convolutional network reach at best 36\%). This serves as a proof of concept that neural networks can be valuable to study the properties of geometries appearing in string theory.},
archivePrefix = {arXiv},
eprint = {2007.15706},
eprinttype = {arxiv},
file = {/home/riccardo/.local/share/zotero/files/erbin_finotello_2020_machine_learning_for_complete_intersection_calabi-yau_manifolds.pdf},
keywords = {⛔ No DOI found},
primaryClass = {hep-th}
keywords = {⛔ No DOI found}
}
@article{Erler:1993:HigherTwistedSector,
@@ -1854,19 +1852,15 @@
title = {{{2D Fermion}} on the {{Strip}} with {{Boundary Defects}} as a {{CFT}} with {{Excited Spin Fields}}},
author = {Finotello, Riccardo and Pesando, Igor},
date = {2019},
url = {http://arxiv.org/abs/1912.07617},
urldate = {2020-02-27},
abstract = {We consider a two-dimensional fermion on the strip in the presence of an arbitrary number of zero-dimensional boundary changing defects. We show that the theory is still conformal with time dependent stress-energy tensor and that the allowed defects can be understood as excited spin fields. Finally we compute correlation functions involving these excited spin fields without using bosonization.},
archivePrefix = {arXiv},
eprint = {1912.07617},
eprinttype = {arxiv},
file = {/home/riccardo/.local/share/zotero/files/finotello_pesando_2019_2d_fermion_on_the_strip_with_boundary_defects_as_a_cft_with_excited_spin_fields.pdf},
keywords = {⛔ No DOI found},
primaryClass = {hep-th}
keywords = {⛔ No DOI found}
}
@article{Finotello:2019:ClassicalSolutionBosonic,
ids = {Finotello:2019:ClassicalSolutionBosonica},
title = {The {{Classical Solution}} for the {{Bosonic String}} in the {{Presence}} of {{Three D}}-Branes {{Rotated}} by {{Arbitrary SO}}(4) {{Elements}}},
author = {Finotello, Riccardo and Pesando, Igor},
date = {2019},
@@ -1877,7 +1871,6 @@
issn = {05503213},
doi = {10.1016/j.nuclphysb.2019.02.010},
abstract = {We consider the classical instantonic contribution to the open string configuration associated with three D-branes with relative rotation matrices in SO(4) which corresponds to the computation of the classical part of the correlator of three non Abelian twist fields. We write the classical solution as a sum of a product of two hypergeometric functions. Differently from all the previous cases with three D-branes, the solution is not holomorphic and suggests that the classical bosonic string knows when the configuration may be supersymmetric. We show how this configuration reduces to the standard Abelian twist field computation. From the phenomenological point of view, the Yukawa couplings between chiral matter at the intersection in this configuration are more suppressed with respect to the factorized case in the literature.},
annotation = {ZSCC: 0000000},
archivePrefix = {arXiv},
eprint = {1812.04643},
eprinttype = {arxiv},

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\thispagestyle{plain}
\noindent {\Large \sc Abstract} \\
\rule{0.99\linewidth}{\sepwidth}
\rule{0.99\linewidth}{\sepwidth} \\[2ex]
}
{%
\vfill
@@ -214,7 +214,7 @@
\thispagestyle{plain}
\noindent {\Large \sc Acknowledgements} \\
\rule{0.99\linewidth}{\sepwidth}
\rule{0.99\linewidth}{\sepwidth} \\[2ex]
}
{%
\vspace{\parskip}

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@@ -7,7 +7,7 @@
\usepackage{import}
\author{Riccardo Finotello}
\title{String Theory and Phenomenology: \\ Theoretical and Computational Aspects From D-branes to Artificial Intelligence}
\title{D-branes and Deep Learning: \\ Theoretical and Computational Aspects In String Theory}
\advisor{Igor Pesando}
\institution{Università degli Studi di Torino}
\school{Scuola di Dottorato}
@@ -20,7 +20,7 @@
\fancyhead[R]{\rightmark}
\hypersetup{%
pdftitle={String Theory and Phenomenology: Theoretical and Computational Aspects From D-branes to Artificial Intelligence},
pdftitle={D-branes and Deep Learning: Theoretical and Computational Aspects In String Theory},
pdfauthor={Riccardo Finotello}
}
@@ -126,6 +126,7 @@
%---- PARTICLE PHYSICS
\thesispart{Conformal Symmetry and Geometry of the Worldsheet}
\label{part:cft}
\section{Introduction}
\input{sec/part1/introduction.tex}
\section{D-branes Intersecting at Angles}
@@ -137,6 +138,7 @@
%---- COSMOLOGY
\thesispart{Cosmological Backgrounds and Divergences}
\label{part:cosmo}
\section{Introduction}
\input{sec/part2/introduction.tex}
\section{Time Dependent Orbifolds}
@@ -158,6 +160,7 @@
%---- APPENDIX
\thesispart{Appendix}
\label{part:appendix}
\appendix
\section{The Isomorphism in Details}
\label{sec:isomorphism}