Update images and references

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

sciencestuff.sty

@@ -50,10 +50,10 @@
 
 \providecommand{\sm}{\textsc{sm}\xspace}
 \providecommand{\eom}{\textsc{e.o.m.}\xspace}
-\providecommand{\cft}{\textsc{CFT}\xspace}
-\providecommand{\qft}{\textsc{QFT}\xspace}
-\providecommand{\qed}{\textsc{QED}\xspace}
-\providecommand{\qcd}{\textsc{QCD}\xspace}
+\providecommand{\cft}{\textsc{cft}\xspace}
+\providecommand{\qft}{\textsc{qft}\xspace}
+\providecommand{\qed}{\textsc{qed}\xspace}
+\providecommand{\qcd}{\textsc{qcd}\xspace}
 \providecommand{\ope}{\textsc{o.p.e.}\xspace}
 \providecommand{\ode}{\textsc{o.d.e.}\xspace}
 \providecommand{\dof}{\textsc{d.o.f.}\xspace}

sec/abstract.tex

@@ -12,7 +12,7 @@ We finally present a new artificial intelligence approach to algebraic geometry
 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.
+We then show that the deep learning approach can reach the highest accuracy in the task with smaller networks and 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.
 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/app/ml.tex

@@ -11,7 +11,7 @@ A linear model learns a function
 \end{equation}
 where $w$ and $b$ are the \emph{weights} and \emph{intercept} of the fit.
 
-One of the key assumptions behind a linear fit is the independence of the residual error between the predicted point and the value of the model, which can therefore be assumed to be sampled from a normal distribution peaked at the average value~\cite{Lista:2017:StatisticalMethodsData, Caffo::DataScienceSpecialization}.
+One of the key assumptions behind a linear fit is the independence of the residual error between the predicted point and the value of the model, which can therefore be assumed to be sampled from a normal distribution peaked at the average value~\cite{Skiena:2017:DataScienceDesign, Caffo::DataScienceSpecialization}.
 The parameters of the fit are then chosen to maximise their \emph{likelihood} function, or conversely to minimise its logarithm with a reversed sign (the $\chi^2$ function).
 A related task is to minimise the mean squared error without assuming a statistical distribution of the residual error: \ml for regression usually implements this as loss function of the estimators.
 In this sense loss functions for regression are more general than a likelihood approach but they are nonetheless related.
@@ -147,9 +147,9 @@ They are called \textit{support vectors} (accessible using the attribute \texttt
 As a consequence any sum involving $\alpha^{(i)}$ or $\beta^{(i)}$ can be restricted to the subset of support vectors.
 Using the kernel notation, the predictions will therefore be
 \begin{equation}
-  y_{pred}^{(i)}
+  y_{\text{pred}}^{(i)}
   =
-  y_{pred}\qty(x^{(i)})
+  y_{\text{pred}}\qty(x^{(i)})
   =
   \finitesum{n}{1}{F'} w_n \phi_n\qty(x^{(i)}) + b
   =
@@ -215,7 +215,7 @@ In regression tasks it is usually given by the $l_1$ and $l_2$ norms of the devi
   \begin{equation}
     H^{[l]}_n\qty(x;\, t_{j,\, n})
     =
-    \frac{1}{\abs{\cM^{[l]}_n( t_{j,\, n} )}} \sum\limits_{i \in A^{[l]}_n} \abs{y^{(i)} - \tilde{y}^{[l]}_{pred,\, n}( x )},
+    \frac{1}{\abs{\cM^{[l]}_n( t_{j,\, n} )}} \sum\limits_{i \in A^{[l]}_n} \abs{y^{(i)} - \tilde{y}^{[l]}_{\text{pred},\, n}( x )},
     \quad
     \qty( x^{(i)},\, y^{(i)} ) \in \cM_n\qty( t_{j,\, n} ),
   \end{equation}
@@ -224,20 +224,20 @@ In regression tasks it is usually given by the $l_1$ and $l_2$ norms of the devi
   \begin{equation}
   H^{[l]}_n\qty(x;\, t_{j,\, n})
   =
-  \frac{1}{\abs{\cM^{[l]}_n( t_{j,\, n} )}} \sum\limits_{i \in A^{[l]}_n} \qty( y^{(i)} - \bar{y}^{[l]}_{pred,\, n}( x ) )^2,
+  \frac{1}{\abs{\cM^{[l]}_n( t_{j,\, n} )}} \sum\limits_{i \in A^{[l]}_n} \qty( y^{(i)} - \bar{y}^{[l]}_{\text{pred},\, n}( x ) )^2,
   \quad
   \qty( x^{(i)}, y^{(i)} ) \in \cM_n( t_{j,\, n} ),
   \end{equation}
 \end{itemize}
 where $\abs{\cM^{[l]}_n\qty( t_{j,\, n} )}$ is the cardinality of the set $\cM^{[l]}_n\qty( t_{j,\, n} )$ for $l = 1, 2$ and
 \begin{equation}
-  \tilde{y}^{[l]}_{pred,\, n}( x )
+  \tilde{y}^{[l]}_{\text{pred},\, n}( x )
   =
-  \underset{i \in A^{[l]}_n}{\mathrm{median}}~ y_{pred}\qty(x^{(i)}),
+  \underset{i \in A^{[l]}_n}{\mathrm{median}}~ y_{\text{pred}}\qty(x^{(i)}),
   \qquad
-  \bar{y}^{[l]}_{pred,\, n}( x )
+  \bar{y}^{[l]}_{\text{pred},\, n}( x )
   =
-  \frac{1}{\abs{A^{[l]}_n}} \sum\limits_{i \in A^{[l]}_n} y_{pred}\qty(x^{(i)}),
+  \frac{1}{\abs{A^{[l]}_n}} \sum\limits_{i \in A^{[l]}_n} y_{\text{pred}}\qty(x^{(i)}),
 \end{equation}
 where $A_n^{[l]} \subset A_n$ are the subset of labels in the left and right splits ($l = 1$ and $l = 2$, that is) of the node $n$.
 
@@ -280,7 +280,7 @@ Also random forests of trees provide a variable ranking system by averaging the
 
 As a reference, \textit{random forests} of decision trees (as in \texttt{ensemble.RandomForestRegressor} in \texttt{scikit-learn}) are ensemble learning algorithms based on fully grown (deep) decision trees.
 They were created to overcome the issues related to overfitting and variability of the input data and are based on random sampling of the training data~\cite{Ho:1995:RandomDecisionForests}.
-The idea is to take $K$ random partitions of the training data and train a different decision tree for each of them and combine the results: for a classification task this would resort to averaging the \textit{a posteriori} (or conditional) probability of predicting the class $c$ given an input $x$ (i.e.\ the Bayesan probability $P\qty(c \mid x)$) over the $K$ trees, while for regression this amount to averaging the predictions of the trees $y_{pred,\, \hatn}^{(i)\, \lbrace k \rbrace}$ where $k = 1, 2, \dots, K$ and $\hatn$ is the final node (i.e. the node containing the final predictions).
+The idea is to take $K$ random partitions of the training data and train a different decision tree for each of them and combine the results: for a classification task this would resort to averaging the \textit{a posteriori} (or conditional) probability of predicting the class $c$ given an input $x$ (i.e.\ the Bayesan probability $P\qty(c \mid x)$) over the $K$ trees, while for regression this amount to averaging the predictions of the trees $y_{\text{pred},\, \hatn}^{(i)\, \lbrace k \rbrace}$ where $k = 1, 2, \dots, K$ and $\hatn$ is the final node (i.e. the node containing the final predictions).
 This defines what has been called a \textit{random forest} of trees which can usually help in improving the predictions by reducing the variance due to trees adapting too much to training sets.
 
 \textit{Boosting} methods are another implementation of ensemble learning algorithms in which more \textit{weak learners}, in this case shallow decision trees, are trained over the training dataset~\cite{Friedman:2001:GreedyFunctionApproximation, Friedman:2002:StochasticGradientBoosting}.
@@ -343,11 +343,13 @@ In \fc networks the input of layer $l$ is a feature vector $a^{(i)\, \qty{l}} \i
 }
 In other words, each entry of the vectors $a^{(i)\, \qty{l}}_j$ (for $j = 1, 2, \dots, n_l$) is mapped through a function $\psi$ to all the components of the following layer $a^{\qty{l+1}} \in \R^{n_{l+1}}$:
 \begin{equation}
-  \begin{split}
-  \psi\colon & \R^{n_l} \quad \longrightarrow \quad \R^{n_{l+1}}
+  \centering
+  \begin{tabular}{@{}rlll@{}}
+    $\psi\colon$ & $\R^{n_l}$ & $\longrightarrow$ & $\R^{n_{l+1}}$
     \\
-  & a^{(i)\, \qty{l}} \quad \longmapsto \quad a^{(i)\, \qty{l+1}} = \psi_j( a^{(i)\, \qty{l}} ),
-  \end{split}
+               & $a^{\qty(i)\, \qty{l}}$ & $\longmapsto$ & $a^{\qty(i)\, \qty{l+1}} = \psi_j\qty( a^{\qty(i)\, \qty{l}} )$,
+    \\
+  \end{tabular}
 \end{equation}
 such that
 \begin{equation}
@@ -367,7 +369,7 @@ A common choice is the \textit{rectified linear unit} ($\mathrm{ReLU}$) function
 \end{equation}
 which has been proven to be better at training deep learning architectures~\cite{Glorot:2011:DeepSparseRectifier}, or its modified version $\mathrm{LeakyReLU}( z ) = \max( \alpha z, z )$ which introduces a slope $\alpha > 0$ to improve the computational performance near the non differentiable point in the origin.
 
-\cnn architectures were born in the context of computer vision and object localisation~\cite{Tompson:2015:EfficientObjectLocalization}.
+\cnn architectures rose to fame in the context of computer vision and object localisation~\cite{Tompson:2015:EfficientObjectLocalization}.
 As one can suspect looking at~\Cref{fig:nn:lenet} for instance, the fundamental difference with \fc networks is that they use a convolution operation $K^{\qty{l}} * a^{(i)\, \qty{l}}$ instead of a linear map to transform the output of the layers, before applying the activation function.\footnotemark{}
 \footnotetext{%
   In general the input of each layer can be a generic tensor with an arbitrary number of axis.

sec/outline.tex

@@ -6,7 +6,7 @@ They are mainly based on published work~\cite{Finotello:2019:ClassicalSolutionBo
 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.
 
-\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.
+\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 (semi-)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{%
@@ -28,7 +28,7 @@ Namely it is hidden in contact terms and interaction with massive string states
 \Cref{part:deeplearning} is dedicated to state-of-the-art application of deep learning techniques to the field of string theory compactifications.
 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{}
+Deep learning techniques are then applied to the configuration matrix of 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.

sec/part1/dbranes.tex

@@ -9,13 +9,13 @@ The fermion--boson couplings and the study of flavour changing neutral currents~
 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}.
-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~\cite{Sciuto:1969:GeneralVertexFunction,DellaSelva:1970:SimpleExpressionSciuto} up to modern interpretations of string theory.
+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,Bianchi:2005:OpenStoryMagnetic,Pesando:2010:OpenClosedString,Forste:2018:YukawaCouplingsMagnetized}, strings in gravitational wave background~\cite{Kiritsis:1994:StringPropagationGravitational,DAppollonio:2003:StringInteractionsGravitational}, bound states of D-branes~\cite{Gava:1997:BoundStatesBranes,Duo:2007:NewTwistField} and tachyon condensation in Superstring Field Theory~\cite{David:2000:TachyonCondensationD0,David:2001:TachyonCondensationUsing,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,Anastasopoulos:2012:LightStringyStates,Anastasopoulos:2013:ThreeFourpointCorrelators}.
+Results were however found starting from dual models~\cite{Sciuto:1969:GeneralVertexFunction} 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 relative \SU{2} rotations of the D-branes ~\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} 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.
@@ -1128,7 +1128,7 @@ Using the \rP symbol the solutions can be symbolically written as
 The normalisation parameters $K$ cannot however be guessed from the \rP symbol.
 
 As we are interested in finding the truly independent solutions to the original problem, we can use properties of the hypergeometric functions to reduce the number of possible choices of the integer factors in the definition of the parameters.
-It is possible to show that any hypergeometric function $\hyp{a + \ffa}{b + \ffb}{c + \ffc}{z}$ can be written as a combination of \hyp{a}{b}{c}{z} and any of its contiguous functions~\cite{::NISTDigitalLibrary}.
+It is possible to show that any hypergeometric function $\hyp{a + \ffa}{b + \ffb}{c + \ffc}{z}$ can be written as a combination of \hyp{a}{b}{c}{z} and any of its contiguous functions~\cite{Olver:2020:NISTDigitalLibrary}.
 For instance we can choose:
 \begin{equation}
   \hyp{a + \ffa}{b + \ffb}{c + \ffc}{z}

sec/part1/fermions.tex

@@ -4,9 +4,9 @@ As previously pointed out, the computation of quantities such as Yukawa coupling
 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 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.
+Non Abelian cases have also been considered~\cite{Inoue:1987:NonAbelianOrbifolds,Inoue:1990:StringInteractionsNonAbelian,Gato:1990:VertexOperatorsNonAbelian,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.
+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,DiVecchia:1990:VertexIncludingEmission,Nilsson:1990:GeneralNSRString,DiBartolomeo:1990:GeneralPropertiesVertices,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.

sec/part1/introduction.tex

@@ -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 si 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 is usually best to work with Polyakov's action~\cite{Polyakov:1981:QuantumGeometryBosonic}:
 \begin{equation}
   S_P\qty[ \gamma, X ]
   =
@@ -58,8 +58,8 @@ The \eom for the string $X^{\mu}\qty(\tau, \sigma)$ is therefore:
   \qquad
   \alpha,\, \beta = 0, 1.
 \end{equation}
-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 this formulation $\gamma_{\alpha\beta}$ are the components of the worldsheet metric with Lorentzian signature $\qty(-,\, +)$.
+As there are no derivatives of $\gamma_{\alpha\beta}$, the \eom of the metric is a constraint ensuring the equivalence of Polyakov's and Nambu and Goto's formulations.
 In fact
 \begin{equation}
   \fdv{S_P\qty[\gamma,\, X]}{\gamma^{\alpha\beta}}
@@ -89,7 +89,7 @@ implies
   =
   S_{NG}[X],
 \end{equation}
-where $S_{NG}[X]$ is the Nambu--Goto action for the classical string, $\dotX = \ipd{\tau} X$ and $\pX = \ipd{\sigma} X$.
+where $S_{NG}[X]$ is the Nambu--Goto action of the classical string, $\dotX = \ipd{\tau} X$ 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}.
 Specifically~\eqref{eq:conf:polyakov} displays symmetries under:
@@ -146,7 +146,7 @@ Notice that the last is not a symmetry of the Nambu--Goto action and it only app
 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}
+  \cT_{\alpha\beta}
   =
   \frac{4 \pi}{\sqrt{- \det \gamma}}
   \fdv{S_P\qty[\gamma,\, X]}{\gamma^{\alpha\beta}}
@@ -163,54 +163,54 @@ In fact the classical constraint on the tensor is simply
   0.
   \label{eq:conf:stringT}
 \end{equation}
-While its conservation $\nabla^{\alpha} T_{\alpha\beta} = 0$ is somewhat trivial, Weyl invariance also ensures the tracelessness of the tensor
+While its conservation $\nabla^{\alpha} T_{\alpha\beta} = 0$ is somewhat trivial, Weyl invariance also ensures the vanishing trace of the tensor
 \begin{equation}
-  \trace{T} = \tensor{T}{^{\alpha}_{\alpha}} = 0.
+  \trace{\cT} = \tensor{\cT}{^{\alpha}_{\alpha}} = 0.
 \end{equation}
-In other words, the $(1 + 1)$-dimensional theory of massless scalars $X^{\mu}$ in~\eqref{eq:conf:polyakov} is \emph{conformally invariant} (for review and details see \cite{Friedan:1986:ConformalInvarianceSupersymmetry,DiFrancesco:1997:ConformalFieldTheory,Ginsparg:1988:AppliedConformalField,Blumenhagen:2009:IntroductionConformalField}).
+In other words, the $(1 + 1)$-dimensional theory of massless scalars $X^{\mu}$ in~\eqref{eq:conf:polyakov} is \emph{conformally invariant} (for review and details see \cite{Friedan:1986:ConformalInvarianceSupersymmetry,DiFrancesco:1997:ConformalFieldTheory,Blumenhagen:2009:IntroductionConformalField}).
 
-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
+Using the invariances of the actions we 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{pseudo-conformal} transformations
 \begin{equation}
   \tau \pm \sigma = \sigma_{\pm} \quad \mapsto \quad f_{\pm}\qty(\sigma_{\pm}),
   \label{eq:conf:residualgauge}
 \end{equation}
-where $f_{\pm}$ is an arbitrary function of its argument.
+where $f_{\pm}$ is an arbitrary function of its argument (the subscript $\pm$ distinguishes the combination of the variables $\tau$ and $\sigma$ in it).
 
 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.
-In these terms, the tracelessness of the stress-energy tensor translates to
+In these terms, the vanishing trace of the stress-energy tensor translates to
 \begin{equation}
-  T_{\xi \bxi} = 0,
+  \cT_{\xi \bxi} = 0,
 \end{equation}
-while its conservation $\partial^{\alpha} T_{\alpha\beta} = 0$ becomes:\footnotemark{}
+while its conservation $\partial^{\alpha} \cT_{\alpha\beta} = 0$ becomes:\footnotemark{}
 \footnotetext{%
   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}\qty( \xi,\, \bxi )
+  \ipd{\bxi} \cT_{\xi\xi}\qty( \xi,\, \bxi )
   =
-  \pd \barT_{\bxi\bxi}\qty( \xi,\, \bxi )
+  \ipd{\xi} \overline{\cT}_{\bxi\bxi}\qty( \xi,\, \bxi )
   =
   0.
 \end{equation}
 The last equation finally implies
 \begin{equation}
-  T_{\xi\xi}\qty( \xi,\, \bxi )
+  \cT_{\xi\xi}\qty( \xi,\, \bxi )
   =
-  T_{\xi\xi}\qty( \xi )
+  \cT_{\xi\xi}\qty( \xi )
   =
-  T\qty( \xi ),
+  \cT\qty( \xi ),
   \qquad
-  \barT_{\bxi\bxi}\qty( \xi,\, \bxi )
+  \overline{\cT}_{\bxi\bxi}\qty( \xi,\, \bxi )
   =
-  \barT_{\bxi\bxi}\qty( \bxi )
+  \overline{\cT}_{\bxi\bxi}\qty( \bxi )
   =
-  \barT\qty( \bxi ),
+  \overline{\cT}\qty( \bxi ),
 \end{equation}
-which are respectively the holomorphic and the anti-holomorphic components of the bidimensional stress energy tensor.
+which are respectively the holomorphic and the anti-holomorphic components of the stress energy tensor.
 
-The previous properties define what is known as a bidimensional \emph{conformal field theory} (\cft).
+The previous properties define what is known as a two-dimensional \emph{conformal field theory} (\cft).
 Ordinary tensor fields
 \begin{equation}
   \phi_{\omega, \bomega}\qty( \xi, \bxi )
@@ -254,17 +254,17 @@ An additional conformal transformation
 \end{equation}
 maps the worldsheet of the string to the complex plane.
 On this Riemann surface the usual time ordering becomes a \emph{radial ordering} as constant time surfaces are circles around the origin (see the contours $\ccC_{(0)}$ and $\ccC_{(1)}$ in \Cref{fig:conf:complex_plane}). 
-In these coordinates the conserved charge associated to the transformation $z \mapsto z + \epsilon(z)$ in radial quantization is
+In these coordinates the conserved charge associated to the transformation $z \mapsto z + \epsilon(z)$ in radial quantization is:
 \begin{equation}
   Q_{\epsilon, \bepsilon}
   =
   \cint{0}
   \ddz
-  \epsilon(z)\, T(z)
+  \epsilon(z)\, \cT(z)
   +
   \cint{0}
   \ddbz
-  \bepsilon(\barz)\, \barT(\barz),
+  \bepsilon(\barz)\, \overline{\cT}(\barz),
 \end{equation}
 where $\ccC_0$ is an anti-clockwise constant radial time path around the origin.
 The transformation on a field $\phi_{\omega, \bomega}$ of weight $(\omega, \bomega)$ is thus given by the commutator with $Q_{\epsilon, \bepsilon}$:
@@ -275,17 +275,17 @@ The transformation on a field $\phi_{\omega, \bomega}$ of weight $(\omega, \bome
     \liebraket{Q_{\epsilon, \bepsilon}}{\phi_{\omega, \bomega}\qty( w, \barw )}
     \\
     & =
-    \cint{0} \ddz \epsilon(z) \qty[ T(z), \phi_{\omega, \bomega}\qty( w, \barw ) ]
+    \cint{0} \ddz \epsilon(z) \qty[ \cT(z), \phi_{\omega, \bomega}\qty( w, \barw ) ]
     +
-    \cint{0} \ddbz \bepsilon(\barz) \qty[ \barT(\barz), \phi_{\omega, \bomega}\qty( w, \barw ) ]
+    \cint{0} \ddbz \bepsilon(\barz) \qty[ \overline{\cT}(\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( \cT(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( \overline{\cT}(\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 $w$.
+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 as a infinitesimally small anti-clockwise loop around $w$.
 Equating the result with the expected variation
 \begin{equation}
   \begin{split}
@@ -304,7 +304,7 @@ Equating the result with the expected variation
 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}\qty( w, \barw )
+    \cT( z )\, \phi_{\omega, \bomega}\qty( w, \barw )
     & =
     \frac{\omega}{(z - w)^2}\, \phi_{\omega, \bomega}\qty( w, \barw )
     +
@@ -312,7 +312,7 @@ we find the short distance singularities of the components of the stress-energy
     +
     \order{1},
     \\
-    \barT( \barz )\, \phi_{\omega, \bomega}\qty( w, \barw )
+    \overline{\cT}( \barz )\, \phi_{\omega, \bomega}\qty( w, \barw )
     & =
     \frac{\bomega}{(\barz - \barw)^2}\, \phi_{\omega, \bomega}\qty( w, \barw )
     +
@@ -345,26 +345,26 @@ 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, Ginsparg:1988:AppliedConformalField}.
+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 \ope can also be computed on the stress-energy tensor itself:
 \begin{equation}
   \begin{split}
-    T( z )\, T( w )
+    \cT( z )\, \cT( w )
     & =
     \frac{\frac{c}{2}}{(z - w)^4}
     +
-    \frac{2}{(z - w)^2}\, T(w)
+    \frac{2}{(z - w)^2}\, \cT(w)
     +
-    \frac{1}{z - w}\, \ipd{w} T(w),
+    \frac{1}{z - w}\, \ipd{w} \cT(w),
     \\
-    \barT( \barz )\, \barT( \barw )
+    \overline{\cT}( \barz )\, \overline{\cT}( \barw )
     & =
     \frac{\frac{\barc}{2}}{(\barz - \barw)^4}
     +
-    \frac{2}{(\barz - \barw)^2}\, \barT(\barw)
+    \frac{2}{(\barz - \barw)^2}\, \overline{\cT}(\barw)
     +
-    \frac{1}{\barz - \barw}\, \ipd{\barw} \barT(\barw).
+    \frac{1}{\barz - \barw}\, \ipd{\barw} \overline{\cT}(\barw).
   \end{split}
   \label{eq:conf:TTexpansion}
 \end{equation}
@@ -372,13 +372,13 @@ The components of the stress-energy tensor are therefore not primary fields and
 This is a reflection of the anomalous algebra of the operator modes $L_n$ and $\barL_n$ computed from the Laurent expansion
 \begin{equation}
   \begin{split}
-    T( z ) = \infinfsum{n} L_n\, z^{-n -2}
+    \cT( z ) = \infinfsum{n} L_n\, z^{-n -2}
     & \Rightarrow
-    L_n = \cint{0} \ddz z^{n + 1} T(z),
+    L_n = \cint{0} \ddz z^{n + 1} \cT(z),
     \\
-    \barT( \barz ) = \infinfsum{n} \barL_n\, \barz^{-n -2}
+    \overline{\cT}( \barz ) = \infinfsum{n} \barL_n\, \barz^{-n -2}
     & \Rightarrow
-    \barL_n = \cint{0} \ddbz \barz^{n + 1} \barT(\barz).
+    \barL_n = \cint{0} \ddbz \barz^{n + 1} \overline{\cT}(\barz).
   \end{split}
   \label{eq:conf:Texpansion}
 \end{equation}
@@ -402,7 +402,7 @@ 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 sub-algebra 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 maps to time translations and $L_0 + \barL_0$ can be considered to be the Hamiltonian of the theory.
@@ -425,7 +425,7 @@ From the previous relations we can finally define the ``asymptotic'' in-states a
   \phi_{\omega, \bomega}
   \regvacuum.
 \end{equation}
-The regularity of \eqref{eq:conf:expansion} requires
+Regularity of \eqref{eq:conf:expansion} requires
 \begin{equation}
   \phi_{\omega, \bomega}^{(n, m)}
   \regvacuum
@@ -492,9 +492,9 @@ In particular the solutions to the \eom factorise into a holomorphic and an anti
 and the components of the stress-energy tensor~\eqref{eq:conf:stringT} are
 \begin{equation}
   \begin{split}
-    T( z ) & = \ipd{z} X( z ) \cdot \ipd{z} X( z ),
+    \cT( z ) & = \ipd{z} X( z ) \cdot \ipd{z} X( z ),
     \\
-    \barT( \barz ) & = \ipd{\barz} \barX( \barz ) \cdot \ipd{\barz} \barX( \barz ).
+    \overline{\cT}( \barz ) & = \ipd{\barz} \barX( \barz ) \cdot \ipd{\barz} \barX( \barz ).
   \end{split}
   \label{eq:conf:bosonicstringT}
 \end{equation}
@@ -502,7 +502,7 @@ Using the normalisation of the 2-points function $\left\langle X^{\mu}( z, \barz
 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}
   \begin{equation*}
     S = \frac{1}{2 \pi} \iint \dd{z} \dd{\barz}\, b( z )\, \ipd{\barz} c( z ).
   \end{equation*}
@@ -514,17 +514,17 @@ The non vanishing components of their stress-energy tensor can be computed as:\f
   where $\varepsilon = +1$ for anti-commuting fields and $\varepsilon = -1$ for Bose statistic.
   Their stress-energy tensor is
   \begin{equation*}
-    T_{\text{ghost}}( z ) = - \lambda\, b( z )\, \ipd{z} c( z ) - \varepsilon\, (1 - \lambda)\, c( z )\, \ipd{z} b( z ).
+    \cT_{\text{ghost}}( z ) = - \lambda\, b( z )\, \ipd{z} c( z ) - \varepsilon\, (1 - \lambda)\, c( z )\, \ipd{z} b( z ).
   \end{equation*}
   Their central charge is therefore $c_{\text{ghost}} = \varepsilon\, ( 1 - 3 \cQ^2)$, where $\cQ = \varepsilon\,( 1 - 2 \lambda )$.
 
-  The ghost \cft has in general an additional \emph{ghost number} \U{1} symmetry generated by the current
+  The ghost \cft has an additional \emph{ghost number} \U{1} symmetry generated by the current
   \begin{equation*}
     j( z ) = - b( z )\, c( z ).
   \end{equation*}
-  In general this current is a primary field (i.e.\ it is not anomalous) when $\cQ = 0$ since
+  The current is a primary field (i.e.\ it is not anomalous) when $\cQ = 0$ since
   \begin{equation*}
-    T_{\text{ghost}}( z )\, j( w ) = \frac{Q}{( z - w )^3} + \order{(z - w)^{-2}}.
+    \cT_{\text{ghost}}( z )\, j( w ) = \frac{Q}{( z - w )^3} + \order{(z - w)^{-2}}.
   \end{equation*}
   This is the case of the worldsheet fermions in~\eqref{eq:super:action} for which $\lambda = \frac{1}{2}$.
   For instance the reparametrisation ghosts with $\lambda = 2$ have $Q = -3$, while the superghosts with $\lambda = \frac{3}{2}$ present $Q = 2$.
@@ -532,11 +532,11 @@ The non vanishing components of their stress-energy tensor can be computed as:\f
 }
 \begin{equation}
   \begin{split}
-    T_{\text{ghost}}( z )
+    \cT_{\text{ghost}}( z )
     & =
     c( z )\, \ipd{z} b( z ) - 2\, b( z )\, \ipd{z} c( z ),
     \\
-    \barT_{\text{ghost}}( \barz )
+    \overline{\cT}_{\text{ghost}}( \barz )
     & =
     \barc( \barz )\, \ipd{\barz} \barb( \barz ) - 2\, \barb( \barz )\, \ipd{\barz} \barc( \barz ).
   \end{split}
@@ -551,30 +551,30 @@ From their 2-points functions
 we get the \ope of the components of their stress-energy tensor:
 \begin{equation}
   \begin{split}
-    T_{\text{ghost}}(z)\, T_{\text{ghost}}(w)
+    \cT_{\text{ghost}}(z)\, \cT_{\text{ghost}}(w)
     & =
     \frac{-13}{(z - w)^4}
     +
-    \frac{2}{(z - w)^2}\, T_{\text{ghost}}(z)
+    \frac{2}{(z - w)^2}\, \cT_{\text{ghost}}(z)
     +
-    \frac{1}{z - w}\, \ipd{z} T_{\text{ghost}}(z),
+    \frac{1}{z - w}\, \ipd{z} \cT_{\text{ghost}}(z),
     \\
-    \barT_{\text{ghost}}(\barz)\, \barT_{\text{ghost}}(\barw)
+    \overline{\cT}_{\text{ghost}}(\barz)\, \overline{\cT}_{\text{ghost}}(\barw)
     & =
     \frac{-13}{(\barz - \barw)^4}
     +
-    \frac{2}{(\barz - \barw)^2}\, \barT_{\text{ghost}}(\barz)
+    \frac{2}{(\barz - \barw)^2}\, \overline{\cT}_{\text{ghost}}(\barz)
     +
-    \frac{1}{\barz - \barw}\, \ipd{\barz} \barT_{\text{ghost}}(\barz),
+    \frac{1}{\barz - \barw}\, \ipd{\barz} \overline{\cT}_{\text{ghost}}(\barz),
   \end{split}
 \end{equation}
 which show that $c_{\text{ghost}} = - 26$.
 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:
+In fact let $\cT_{\text{full}} = \cT + \cT_{\text{ghost}}$ and $\overline{\cT}_{\text{full}} = \overline{\cT} + \overline{\cT}_{\text{ghost}}$, then:
 \begin{equation}
-  \eval{T_{\text{full}}( z )}_{\order{(z - w)^{-4}}}
+  \eval{\cT_{\text{full}}( z )}_{\order{(z - w)^{-4}}}
   =
-  \eval{\barT_{\text{full}}( \barz )}_{\order{(\barz - \barw)^{-4}}}
+  \eval{\overline{\cT}_{\text{full}}( \barz )}_{\order{(\barz - \barw)^{-4}}}
   =
   c + c_{\text{ghost}}
   =
@@ -586,12 +586,13 @@ In fact let $T_{\text{full}} = T + T_{\text{ghost}}$ and $\barT_{\text{full}} =
   \quad
   D = 26.
 \end{equation}
+$\cT_{\text{full}}$ and $\overline{\cT}_{\text{full}}$ are then primary fields with conformal weight $-2$.
 
 
 \subsection{Superstrings}
 
 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}.
+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.
 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}.
@@ -611,7 +612,7 @@ In complex coordinates on the plane it is~\cite{Polchinski:1998:StringTheorySupe
   \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 $\qty(\frac{1}{2}, 0)$ and $\bpsi^{\mu}$ are their anti-holomorphic counterparts with weight $\qty(0, \frac{1}{2})$. Their short-distance behaviour is
+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
@@ -620,11 +621,11 @@ In the last expression, $\psi^{\mu}$ are $D$ two-dimensional holomorphic fermion
 In this case the components of the stress-energy tensor of the theory are:
 \begin{equation}
   \begin{split}
-    T( z )
+    \cT( z )
     & =
     -\frac{1}{\ap}\, \ipd{z} X( z ) \cdot \ipd{z} X( z ) - \frac{1}{2}\, \psi( z ) \cdot \ipd{z} \psi( z ),
     \\
-    \barT( \barz )
+    \overline{\cT}( \barz )
     & =
     -\frac{1}{\ap}\, \ipd{\barz} \barX( \barz ) \cdot \ipd{\barz} \barX( \barz ) - \frac{1}{2}\, \bpsi( \barz ) \cdot \ipd{\barz} \bpsi( \barz ).
   \end{split}
@@ -650,14 +651,14 @@ The action~\eqref{eq:super:action} is also invariant under the \emph{supersymmet
     - \bepsilon( \barz )\, \ipd{\barz} \barX^{\mu}( \barz )
   \end{split}
 \end{equation}
-generated by the currents $J( z ) = \epsilon( z )\, T_F( z )$ and $\barJ( \barz ) = \bepsilon( \barz )\, \barT_F( \barz )$, where $\epsilon( z )$ and $\bepsilon( \barz ) = \qty( \epsilon( z ) )^*$ are anti-commuting fermions and
+generated by the currents $J( z ) = \epsilon( z )\, \cT_F( z )$ and $\barJ( \barz ) = \bepsilon( \barz )\, \overline{\cT}_F( \barz )$, where $\epsilon( z )$ and $\bepsilon( \barz ) = \qty( \epsilon( z ) )^*$ are anti-commuting fermions and
 \begin{equation}
   \begin{split}
-    T_F( z )
+    \cT_F( z )
     & =
     i\, \sqrt{\frac{2}{\ap}}\, \psi( z ) \cdot \ipd{z} X( z ),
     \\
-    \barT_F( \barz )
+    \overline{\cT}_F( \barz )
     & =
     i\, \sqrt{\frac{2}{\ap}}\, \bpsi( \barz ) \cdot \ipd{\barz} \barX( \barz )
   \end{split}
@@ -666,23 +667,23 @@ are the \emph{supercurrents}.
 The central charge associated to the Virasoro algebra is in this case given by both bosonic and fermionic contributions:
 \begin{equation}
   \begin{split}
-    T( z )\, T( w )
+    \cT( z )\, \cT( w )
     & =
     \frac{\frac{3 D}{4}}{( z - w )^4}
     +
-    \frac{2}{( z - w )^2} T( w )
+    \frac{2}{( z - w )^2} \cT( w )
     +
-    \frac{1}{z - w} \ipd{w} T( w )
+    \frac{1}{z - w} \ipd{w} \cT( w )
     +
     \order{1},
     \\
-    \barT( \barz )\, \barT( \barw )
+    \overline{\cT}( \barz )\, \overline{\cT}( \barw )
     & =
     \frac{\frac{3 D}{4}}{( \barz - \barw )^4}
     +
-    \frac{2}{( \barz - \barw )^2} \barT( \barw )
+    \frac{2}{( \barz - \barw )^2} \overline{\cT}( \barw )
     +
-    \frac{1}{\barz - \barw} \ipd{\barw} \barT( \barw )
+    \frac{1}{\barz - \barw} \ipd{\barw} \overline{\cT}( \barw )
     +
     \order{1}.
   \end{split}
@@ -692,11 +693,11 @@ The central charge is therefore $c = \frac{3}{2} D$ for the \cft defined in~\eqr
 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:
+When considering the full theory $\cT_{\text{full}} = \cT + \cT_{\text{ghost}}$ and $\overline{\cT}_{\text{full}} = \overline{\cT} + \overline{\cT}_{\text{ghost}}$ the central charge vanishes only in 10-dimensional spacetime:
 \begin{equation}
-  \eval{T_{\text{full}}( z )}_{\order{(z - w)^{-4}}}
+  \eval{\cT_{\text{full}}( z )}_{\order{(z - w)^{-4}}}
   =
-  \eval{\barT_{\text{full}}( \barz )}_{\order{(\barz - \barw)^{-4}}}
+  \eval{\overline{\cT}_{\text{full}}( \barz )}_{\order{(\barz - \barw)^{-4}}}
   =
   c + c_{\text{ghost}}
   =
@@ -721,18 +722,20 @@ In what follows we thus consider the superstring formulation in $D = 10$ dimensi
 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.
 These results represent the background knowledge necessary to better understand more complicated scenarios involving strings.
-As we will never deal directly with $4$-dimensional physics this is not a complete review and we refer to \cite{Anderson:2018:TASILecturesGeometric,Blumenhagen:2007:FourdimensionalStringCompactifications,Blumenhagen:2013:BasicConceptsString,Grana:2005:FluxCompactificationsString,Grana:2017:StringTheoryCompactifications,Krippendorf:2010:CambridgeLecturesSupersymmetry,Uranga:2005:TASILecturesString} for more in-depth explanations.
+As we will never deal directly with $4$-dimensional physics this is not a complete review and we refer to \cite{Anderson:2018:TASILecturesGeometric,Blumenhagen:2007:FourdimensionalStringCompactifications,Grana:2006:FluxCompactificationsString,Grana:2017:StringTheoryCompactifications,Uranga:2005:TASILecturesString} for more in-depth explanations.
 
 In general we consider Minkowski space in $10$ dimensions $\ccM^{1,9}$.
 To recover $4$-dimensional spacetime we let it be defined as a product
 \begin{equation}
-  \ccM^{1,9}
-  =
-  \ccM^{1,3} \otimes \ccX_6,
+  \ccM^{1,9} = \ccM^{1,3} \otimes \ccX_6,
 \end{equation}
 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}.
+This \emph{internal} manifold $\ccX_6$ is however subject to very stringent restrictions due to mathematical consistency conditions and physical requests.
+In particular $\ccX_6$ should be a \emph{compact} manifold to ``hide'' the 6 extra-dimensions computed in~\eqref{eq:super:dimensions}.\footnotemark{}
+\footnotetext{%
+  A compact manifold \ccX is defined as a Hausdorff topological space whose open covers all have a finite subcover.
+  In other words \ccX is compact if for each covering atlas $\ccA = \qty{ U_{\alpha} }_{\alpha \in A}$ such that $\ccX = \bigcup\limits_{\alpha \in A} U_{\alpha}$, then $\exists \ccB = \qty{ V_{\beta} }_{\beta \in B} \subset \ccA$ finite such that $\ccX = \bigcup\limits_{\beta \in B} V_{\beta}$.
+}
 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 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.
@@ -795,14 +798,6 @@ The metric is \emph{Hermitian} if
   g( v_p, w_p ) = g( J\, v_p, J\, w_p )
   \quad
   \forall v_p,\, w_p \in \rT_p M
-  % \quad
-  % \Leftrightarrow
-  % \quad
-  % \tensor{g}{_{ab}}
-  % =
-  % \tensor{J}{_a^c}\,
-  % \tensor{J}{_b^d}\,
-  % \tensor{g}{_{cd}}.
 \end{equation}
 In this case we can define a $(1, 1)$-form $\omega$ as
 \begin{equation}
@@ -811,13 +806,6 @@ In this case we can define a $(1, 1)$-form $\omega$ as
   g( J\, v_p, w_p )
   \quad
   \forall v_p,\, w_p \in \rT_p M.
-  % \quad
-  % \Leftrightarrow
-  % \quad
-  % \tensor{\omega}{_{ab}}
-  % =
-  % \tensor{J}{_a^c}\,
-  % \tensor{g}{_{cb}}.
 \end{equation}
 $(M, J, g)$ is a \emph{Kähler} manifold if:
 \begin{equation}
@@ -830,7 +818,7 @@ $(M, J, g)$ 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 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 anti-holomorphic \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 $\mathrm{Hol}(g) \subseteq \U{m} \subset \OO{2m}$.
 
@@ -847,7 +835,7 @@ In local complex coordinates a Hermitian metric is such that
   g_{\bara b}\, \dd{\barz}^{\bara} \otimes \dd{z}^b,
 \end{equation}
 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:
+Relation~\eqref{eq:cy:kaehler} translates into:
 \begin{equation}
   \dd{\omega}
   =
@@ -894,7 +882,7 @@ As a consequence the Ricci tensor becomes
   \pdv{\tensor{\Gamma}{^{\barc}_{\bara\barc}}}{z^b}.
 \end{equation}
 
-Since \cy manifolds present $\mathrm{Hol}(g) \subseteq \SU{m}$, the trace part of the coefficients of the connection vanishes.
+Since for \cy manifolds $\mathrm{Hol}(g) \subseteq \SU{m}$, 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.
 
 
@@ -905,21 +893,21 @@ Since \cy manifolds present $\mathrm{Hol}(g) \subseteq \SU{m}$, the trace part o
 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 $\tildeM$ 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 $\tomega$ are always defined up to an \emph{exact} term.
 In fact:
 \begin{equation}
-  \dd{\omega'_{(p)}} = \dd{(\omega_{(p)} + \dd{\eta_{(p-1)}})} = 0
+  \dd{\tomega'_{(p)}} = \dd{\qty(\tomega_{(p)} + \dd{\teta_{(p-1)}})} = 0
   \label{eq:cy:closedform}
 \end{equation}
-implies an equivalence relation $\omega'_{(p)} \sim \omega_{(p)} + \dd{\eta_{(p-1)}}$.
-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 $\mathrm{d}$.
+implies an equivalence relation $\tomega'_{(p)} \sim \tomega_{(p)} + \dd{\teta_{(p-1)}}$.
+This translates to the fact that elements of the de Rham cohomology group $H^{(p)}_{\mathrm{d}}\qty(\tildeM, \R)$ are equivalence classes $[ \tomega ]$ 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 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)},
+  \omega'_{(r,s)} \sim \omega_{(r,s)} + \bpd \eta_{(r,s-1)},
 \end{equation}
 or an equivalent formulation using $\pd$.
 The cohomology group in this case is $H^{(r,s)}_{\bpd}( M, \C )$ and the relation with the real counterpart is
@@ -979,8 +967,8 @@ These results will also be the starting point of~\Cref{part:deeplearning} in whi
 \subsection{D-branes and Open Strings}
 
 Dirichlet branes, or \emph{D-branes}, are another key mathematical object in string theory.
-They are naturally included as extended hypersurfaces supporting strings with open topology and as physical objects with charge and tension~\cite{Polchinski:1995:DirichletBranesRamondRamond,Polchinski:1996:TASILecturesDBranes,DiVecchia:1999:DbranesStringTheory,DiVecchia:2000:BranesStringTheory,DiVecchia:1997:ClassicalPbranesBoundary,Taylor:2003:LecturesDbranesTachyon,Taylor:2004:DBranesTachyonsString,Johnson:2000:DBranePrimer}.
-They are relevant in the definition of phenomenological models in string theory as they can be arranged to support chiral fermions and bosons in \sm-like scenarios as well as beyond~\cite{Honecker:2012:FieldTheoryStandard,Lust:2009:LHCStringHunter,Zwiebach::FirstCourseString}.
+They are naturally included as extended hypersurfaces supporting strings with open topology and as physical objects with charge and tension~\cite{Polchinski:1995:DirichletBranesRamondRamond,Polchinski:1996:TASILecturesDBranes,DiVecchia:1999:DbranesStringTheory,DiVecchia:2000:BranesStringTheory,DiVecchia:1997:ClassicalPbranesBoundary}.
+They are relevant in the definition of phenomenological models in string theory as they can be arranged to support chiral fermions and bosons in \sm-like scenarios as well as beyond~\cite{Honecker:2012:FieldTheoryStandard,Lust:2009:LHCStringHunter}.
 We are ultimately interested in their study to construct Yukawa couplings in string theory.
 
 
@@ -1116,7 +1104,7 @@ Imposing physical conditions~\eqref{eq:conf:physical} and the \emph{level matchi
 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}.
+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.
 However in closed string theory as $R \to 0$ the compactified dimension is again present.
 
 As seen in~\eqref{eq:dbranes:closedspectrum} the mass spectra of the theories compactified at radius $R$ or $\ap\, R^{-1}$ are the same under the exchange of $n$ and $m$.
@@ -1136,7 +1124,7 @@ 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}.
+Consider the case of open strings satisfying the \eom~\eqref{eq:tduality:eom} and the boundary conditions~\eqref{eq:tduality:bc}.
 The usual mode expansion~\eqref{eq:tduality:modes} here leads to:
 \begin{equation}
   X^{\mu}( z, \barz )
@@ -1207,14 +1195,14 @@ The procedure can be generalised to $p$ coordinates, constraining the string to
 
 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.
-They also present physical properties such as tension and charge~\cite{DiVecchia:1997:ClassicalPbranesBoundary,DiVecchia:2006:BoundaryStateMagnetized,Polchinski:1995:DirichletBranesRamondRamond}.
+They also present physical properties such as tension and charge~\cite{Polchinski:1995:DirichletBranesRamondRamond,DiVecchia:1997:ClassicalPbranesBoundary,DiVecchia:2006:BoundaryStateMagnetized}.
 However these aspects will not be discussed here as the following analysis will mainly focus on geometrical aspects of D-branes in spacetime.
 
 
 \subsubsection{Gauge Groups from D-branes}
 
 As previously stated, in order to recover $4$-dimensional physics we need to compactify the $6$ extra-dimensions of the superstring.
-There are in general multiple ways to do such operation consistently~\cite{Brown:1988:NeutralizationCosmologicalConstant,Bousso:2000:QuantizationFourformFluxes,Susskind:2003:AnthropicLandscapeString,tHooft:2009:DimensionalReductionQuantum,Kachru:2003:SitterVacuaString}.
+There are in general multiple ways to do such operation consistently~\cite{Bousso:2000:QuantizationFourformFluxes,Susskind:2003:AnthropicLandscapeString,Kachru:2003:SitterVacuaString}.
 Reproducing the \sm or beyond \sm spectra are however strong constraints on the possible compactification procedures~\cite{Cleaver:2007:SearchMinimalSupersymmetric,Lust:2009:LHCStringHunter}.
 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.
 
@@ -1223,7 +1211,7 @@ Specifically a Dp-brane breaks the original \SO{1,\, D-1} symmetry to $\SO{1,\,
 \footnotetext{%
   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}.
+The massless spectrum of the theory on the D-brane is easily computed in lightcone gauge~\cite{Goddard:1973:QuantumDynamicsMassless,Angelantonj:2002:OpenStrings}.
 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^+\qty( \tau, \sigma ) = x_0^+ + 2 \ap\, p^+\, \tau,
@@ -1316,7 +1304,7 @@ These are the basic building blocks for a consistent string phenomenology involv
 
 Being able to describe gauge bosons and fermions is not enough.
 Physics as we test it in experiments poses stringent constraints on what kind of string models we can use.
-For instance there is no way to describe chirality by simply using parallel D-branes and strings stretching among them, while requiring the existence of fermions transforming in different representations of the gauge group is necessary to reproduce \sm results~\cite{Aldazabal:2000:DBranesSingularitiesBottomUp}.
+For instance there is no way to describe chirality by simply using parallel D-branes and strings stretching among them, while requiring the existence of fermions transforming in different representations of the gauge group is necessary to reproduce \sm results~\cite{Aldazabal:2000:DBranesSingularitiesBottomUp, Ibanez:2012:StringTheoryParticle}.
 
 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.
@@ -1334,7 +1322,7 @@ We therefore need to introduce more D-branes to account for all the possible com
 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}.
+In string theory there are ways to deal with the requirement~\cite{Uranga:2003:ChiralFourdimensionalString,Aldazabal:2000:DBranesSingularitiesBottomUp,Zwiebach:2009:FirstCourseString}.
 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.
@@ -1350,7 +1338,7 @@ The light spectrum is thus composed of the desired matter content alongside with
   \label{fig:dbranes:smbranes}
 \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}.
+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,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 $\qty( \vb{3}, \vb{2} )$ 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 $Y$.
@@ -1364,7 +1352,7 @@ Fermions localised at the intersection of the D-branes are however naturally $4$
 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 have points in common 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: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 separation in mass of the families of quarks and leptons in the \sm.
 
 

sec/part2/divergences.tex

@@ -49,14 +49,14 @@ We then go back to string theory and we verify that in the \nbo the open string
 
 We then introduce the generalised Null Boost Orbifold (\gnbo) as a generalisation of the \nbo which still has a light-like singularity and is generated by one Killing vector.
 However in this model there are two directions associated with $\cA$, one compact and one non compact.
-We can then construct the scalar \qed and the effective field theory which extends it with the inclusion of higher order terms since all terms have a distributional interpretation~\cite{Estrada:2012:GeneralIntegral}.
+We can then construct the scalar \qed and the effective field theory which extends it with the inclusion of higher order terms since all terms have a distributional interpretation.
 However if a second Killing vector is used to compactify the formerly non compact direction, the theory has again the same problems as in the \nbo.
 In the literature there are however also other attempts at regularizing the \nbo such as the Null Brane.
 This kind of orbifold was originally defined in \cite{Figueroa-OFarrill:2001:GeneralisedSupersymmetricFluxbranes,Cornalba:2004:TimedependentOrbifoldsString} and studied in perturbation theory in \cite{Liu:2002:StringsTimeDependentOrbifolds}.
 The Null Brane shares with the \gnbo the existence of a non compact direction on the orbifold.
 In this case it is indeed possible to build single particle wave functions which leads to the convergence of the smeared amplitudes.
 
-We finally present also a brief examination of the Boost Orbifold (\bo) where the divergences are generally milder~\cite{Horowitz:1991:SingularStringSolutions,Khoury:2002:BigCrunchBig}.
+We finally present also a brief examination of the Boost Orbifold (\bo) where the divergences are generally milder~\cite{Horowitz:1991:SingularStringSolutions}.
 The scalar eigenfunctions behave in time $t$ as $\abs{t}^{\pm i\, \frac{l}{\Delta}}$ near the singularity but there is one eigenfunction which behaves as $\log(\abs{t})$ and again it is the constant eigenfunction along the compact direction which is the origin of all divergences.
 In particular the scalar \qed on the \bo can be defined and the first term which gives a divergent contribution is of the form $\abs{\phi~\dphi}^2$, i.e.\ divergences are hidden into the derivative expansion of the effective field theory.
 Again three points open string amplitudes with one massive state diverge.

sec/part2/introduction.tex

@@ -56,7 +56,7 @@ The $n$-dimensional \emph{orbifold} $\ccO$ is finally defined as a paracompact H
 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}.
+However they can also be used to model peculiar time-dependent backgrounds~\cite{Horowitz:1991:SingularStringSolutions,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.
 We focus specifically on the study of such cosmological singularities in the framework of string theory defined on time-dependent orbifolds.
 

sec/part3/conclusion.tex

@@ -10,7 +10,7 @@ For instance we could try to set up a map from any matrix to its favourable repr
 This could be the basis for the use of adversarial networks~\cite{Goodfellow:2014:GenerativeAdversarialNets} capable of generating the favourable embedding from the first.
 Or on the contrary one could generate more matrices for the same manifold in order to increase the size of the training set.
 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.
+Techniques such as (variational) autoencoders~\cite{Kingma:2014:AutoEncodingVariationalBayes, Rezende:2014:StochasticBackpropagationApproximate}, cycle GAN~\cite{Zhu:2017:UnpairedImagetoimageTranslation}, invertible neural networks~\cite{Ardizzone:2019:AnalyzingInverseProblems}, graph neural networks~\cite{Gori:2005:NewModelLearning, Scarselli:2004:GraphicalbasedLearningEnvironments} or 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 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).

sec/part3/introduction.tex

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

sec/part3/ml.tex

@@ -302,7 +302,7 @@ Obviously the very small percentage of outliers makes the effect of removing the
 We compare the performances of different \ml algorithms: linear regression, support vector machines (\svm), random forests, gradient boosted trees and (deep) neural networks.
 We obtain the best results using deep \emph{convolutional} neural networks.
 In fact we present a new neural network architecture, inspired by the Inception model~\cite{Szegedy:2015:GoingDeeperConvolutions, Szegedy:2016:RethinkingInceptionArchitecture, Szegedy:2016:Inceptionv4InceptionresnetImpact} which has been developed in the field of computer vision.
-We provide some details on the different algorithms in~\Cref{app:ml-algo} and refer the reader to the literature~\cite{Bengio:2017:DeepLearning, Chollet:2018:DeepLearningPython, Geron:2019:HandsOnMachineLearning, Skiena:2017:DataScienceDesign, Mehta:2019:HighbiasLowvarianceIntroduction, Carleo:2019:MachineLearningPhysical, Ruehle:2020:DataScienceApplications} for more details.
+We provide some details on the different algorithms in~\Cref{app:ml-algo} and refer the reader to the literature~\cite{Goodfellow:2017:DeepLearning, Chollet:2018:DeepLearningPython, Geron:2019:HandsOnMachineLearning, Skiena:2017:DataScienceDesign, Ruehle:2020:DataScienceApplications} for more details.
 
 
 \subsubsection{Feature Extraction}
@@ -394,7 +394,7 @@ For the same reason, the latter are not displayed for the favourable dataset.
 \paragraph{Visualisation of the performance}
 
 Complementary to the predictions and the accuracy results, we also provide different visualisations of the performance of the models in the form of univariate plots (histograms) and multivariate distributions (scatter plots).
-In fact the usual assumption behind the statistical inference of a distribution is that the difference between the observed data and the predicted values can be modelled by a random variable called \textit{residual}~\cite{Lista:2017:StatisticalMethodsData,Caffo::DataScienceSpecialization}.\footnotemark{}
+In fact the usual assumption behind the statistical inference of a distribution is that the difference between the observed data and the predicted values can be modelled by a random variable called \textit{residual}~\cite{Skiena:2017:DataScienceDesign,Caffo::DataScienceSpecialization}.\footnotemark{}
 \footnotetext{%
   The difference between the non observable \textit{true} value of the model and the observed data is known as \textit{statistical error}.
   The difference between residuals and errors is subtle but the two definitions have different interpretations in the context of the regression analysis: in a sense, residuals are an estimate of the errors.
@@ -1232,7 +1232,7 @@ In fact this neural network is much more powerful than the previous networks we
 When predicting only \hodge{1}{1} it surpasses \SI{97}{\percent} accuracy using only \SI{30}{\percent} of the data for training.
 While it seems that the predictions suffer when using a single network for both Hodge numbers this remains much better than any other algorithm.
 It may seem counter-intuitive that convolutions work well on this data since they are not translation or rotation invariant but only permutation invariant.
-However convolution alone is not sufficient to ensure invariances under these transformations but it must be supplemented with pooling operations~\cite{Bengio:2017:DeepLearning} which we do not use.
+However convolution alone is not sufficient to ensure invariances under these transformations but it must be supplemented with pooling operations~\cite{Goodfellow:2017:DeepLearning} which we do not use.
 Moreover convolution layers do more than just taking translation properties into account: they allow to make highly complicated combinations of the inputs and to share weights among components to find subtler patterns than standard fully connected layers.
 This network is more studied in more details in~\cite{Erbin:2020:InceptionNeuralNetwork}.
 

thesis.cls

@@ -201,7 +201,8 @@
 \newenvironment{abstractpage}
 {%
   \thispagestyle{plain}
-
+  \phantomsection
+  \addcontentsline{toc}{section}{Abstract}
   \noindent {\Large \sc Abstract} \\
   \rule{0.99\linewidth}{\sepwidth} \\[2ex]
 }
@@ -214,7 +215,8 @@
 \newenvironment{acknowledgmentspage}
 {%
   \thispagestyle{plain}
-
+  \phantomsection
+  \addcontentsline{toc}{section}{Acknowledgements}
   \noindent{\Large \sc Acknowledgements} \\
   \rule{0.99\linewidth}{\sepwidth} \\[2ex]
 }
@@ -238,6 +240,7 @@
 \newcommand{\outline}[1]
 {%
   \thispagestyle{plain}
+  \phantomsection
   \section*{#1}
   \addcontentsline{toc}{section}{#1}
 }

thesis.tex

@@ -14,6 +14,7 @@
 \specialisation{Dottorato in Fisica ed Astrofisica}
 \logo{img/unito}
 
+\renewcommand*{\bibfont}{\small}
 \addbibresource{thesis.bib}
 
 \fancyhead[L]{}

tikz/complex_plane.pgf

@@ -10,7 +10,7 @@
 
 % draw arrows
 \draw[-latex] (0,0) -- (1.75cm, 1.75cm) node[anchor=south] (z1) {$\abs{z_{(1)}} = e^{\tau_{E\, (1)}}$};
-\draw[-latex] (0,0) -- (0.45cm, -0.9cm) node[anchor=north] (z0) {$\abs{z_{(0)}} = e^{\tau_{E\, (0)}}$};
+\draw[-latex] (0,0) -- (0.45cm, -0.9cm) node[anchor=north west] (z0) {$\abs{z_{(0)}} = e^{\tau_{E\, (0)}}$};
 
 % draw isolated point
 \draw[fill] (-1.5cm, 1.1cm) circle (2pt) node[anchor=south west] (w) {$w$};