Corrections and adjustments to the introduction

Signed-off-by: Riccardo Finotello <riccardo.finotello@gmail.com>

sec/abstract.tex

@@ -1,4 +1,4 @@
-We present topics of phenomenological relevance in 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.
+We present topics of (semi-)phenomenological relevance in 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.
 
 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.
@@ -11,8 +11,8 @@ We also introduce a new orbifold structure capable of fixing the issue and reins
 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 that the deep learning approach can reach the highest accuracy in the task with smaller networks, less parameters and less data.
+We thus show how such an approach can help in improving results by processing the data before utilising them.
+We then show that deep learning the configuration matrix of the manifolds reaches the highest accuracy in the task with smaller networks, less parameters and less data.
 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.
 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.
 

sec/outline.tex

@@ -1,6 +1,6 @@
 This thesis follows my research work as a Ph.D.\ student and candidate at the \emph{Università degli Studi di Torino} in 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.
+I covered 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.
@@ -13,14 +13,14 @@ Then the analysis of a specific setup involving angled D6-branes intersecting in
   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 a general framework to deal with \SO{4} rotated D-branes is introduced 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.
+Finally point-like defects in 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 has 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 the computation of amplitudes in the presence of spin fields and matter fields is eventually displayed by means of a method alternative to the usual bosonization, but which might be expanded also to twist fields and to more general configurations (e.g.\ non Abelian spin fields).
 
 \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 a first example.
-Differently from what usually referred in the literature, 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 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 points of view, the Null Boost Orbifold is introduced as a first example.
+Differently from what usually referred in the literature, the divergences appearing in the amplitudes are not a consequence of gravitational feedback, but they are present 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 significance 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.

sec/part1/introduction.tex

@@ -12,7 +12,7 @@ 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.
 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 frameworks used throughout the manuscript as many aspects are strongly connected and their definitions are interdependent.
+In this introduction we present instruments used throughout the manuscript as many 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.
 
 
@@ -20,7 +20,7 @@ In particular we recall some results on the symmetries of string theory and how
 
 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.\ $X^{\mu}\qty(\tau, \sigma)$ where $\mu = 0, 1, \dots, D - 1$ indexes the coordinates.
+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.
 
 
@@ -91,7 +91,7 @@ implies
 \end{equation}
 where $S_{NG}[X]$ is the Nambu--Goto action of the classical string, and $\dotX = \ipd{\tau} X$ and $\pX = \ipd{\sigma} X$.
 
-The symmetries of $S_P\qty[\gamma,\, X]$ are keys to the success of the string theory framework~\cite{Polchinski:1998:StringTheoryIntroduction}.
+The symmetries of $S_P\qty[\gamma,\, X]$ are key 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
@@ -1159,7 +1159,7 @@ In fact the original Neumann boundary condition~\eqref{eq:tduality:bc} becomes a
     \eval{i\, \ipd{\tau_E} Y^{D-1}( e^{\tau_E + i \sigma} ) + i\, \ipd{\tau_E} \barY^{D-1}( e^{\tau_E - i \sigma} )}^{\sigma = \pi}_{\sigma = 0}
     \\
     & =
-    \eval{i\, \ipd{\tau} Y^{D-1}( \tau, \sigma )}^{\sigma = \pi}_{\sigma = 0}
+    \eval{\ipd{\tau} Y^{D-1}( \tau, \sigma )}^{\sigma = \pi}_{\sigma = 0}
     \\
     & =
     0.
@@ -1248,7 +1248,7 @@ Thus the gauge field in the original theory is split into
     \qquad
     \alpha_{-1}^A \regvacuum,
     \qquad
-    A = 1, \dots, p - 2,
+    A = 0, 1, \dots, p,
     \\
     \cA^a
     \qquad
@@ -1347,7 +1347,7 @@ Physics in four dimensions is eventually recovered by compactifying the extra-di
 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.
+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-torus.
 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:2009: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 mass separation of the families of quarks and leptons in the \sm.
 

sec/part3/ml.tex

@@ -985,7 +985,8 @@ Training and evaluation were performed on a \texttt{NVidia GeForce 940MX} laptop
 
 \subsubsection{Fully Connected Network}
 
-First we reproduce the analysis in~\cite{Bull:2018:MachineLearningCICY} for the prediction of \hodge{1}{1}.
+Before we move to the complete deep learning study using convolutional kernels, first we reproduce the analysis in~\cite{Bull:2018:MachineLearningCICY} for the prediction of \hodge{1}{1}.
+We will then use this as a baseline for our investigation.
 
 
 \paragraph{Model}

thesis.tex

@@ -152,7 +152,7 @@
 \label{part:deeplearning}
 \section{Introduction}
 \input{sec/part3/introduction.tex}
-\section{Machine and Deep Learning for CICY Manifolds}
+\section{Machine Learning and Deep Learning for CICY Manifolds}
 \input{sec/part3/ml.tex}
 \section{Summary and Conclusion}
 \label{sec:conclusion}