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Begin of the introduction

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Signed-off-by: Riccardo Finotello <riccardo.finotello@gmail.com>
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*.bbl
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In this first part we focus on aspects of string theory directly connected with its worldsheet description and symmetries.
The underlying idea is to build technical tools to address the study of viable phenomenological models in this framework.
In fact the construction of realistic string models of particle physics is the key to better understanding the nature of a theory of everything such as string theory.
As a first test of validity, the string theory should properly extend the known Standard Model (\sm) of particle physics, which is arguably one of the most experimentally backed theoretical frameworks in modern physics.
In particular its description in terms of fundamental strings should be able to include a gauge algebra locally isomorphic to that of
\begin{equation}
\SU{3}_{\rC} \otimes \SU{2}_{\rL} \otimes \U{1}_{\rY}
\end{equation}
in order to reproduce known results.
For instance, 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 string theory.
\subsection{Properties of String Theory and Conformal Symmetry}
Strings are extended one-dimensional objects.
They are curves in space parametrized by a coordinate $\sigma \in \left[0, \ell \right]$.
Propagating in $D$-dimensional spacetime they span a two-dimensional surface, the \emph{worldsheet}, described by the position of the string at given values of $\sigma$ at a time $\tau$, i.e.\ $X^{\mu}(\tau, \sigma)$ where $\mu = 0, 1, \dots, D - 1$ indexes the coordinates.
\subsubsection{Action Principle}
As the action of a point particle is proportional to the length of its trajectory (its \emph{worldline}), the same object for string is proportional to the area of the worldsheet in the original formulation by Nambu and Goto.
The solutions of the classical equations of motion (\eom) are therefore strings spanning a worldsheet of extremal area.
While Nambu and Goto's formulation is fairly direct in its definition, it usually best to work with Polyakov's action~\cite{Polyakov:1981:QuantumGeometryBosonic}
\begin{equation}
S_P[ \gamma, X ]
=
-\frac{T}{2}
\infinfint{\tau}
\finiteint{\sigma}{0}{\ell}
\sqrt{- \det \gamma}\,
\gamma^{\alpha\beta}\,
\ipd{\alpha} X^{\mu}(\tau, \sigma)\,
\ipd{\beta} X^{\nu}(\tau, \sigma)\,
\eta_{\mu\nu}.
\label{eq:conf:polyakov}
\end{equation}
In this formulation $\gamma_{\alpha\beta}$ is the worldsheet metric with Lorentzian signature $(-, +)$.
As there are no derivatives of $\gamma_{\alpha\beta}$, its \eom is a constraint ensuring the equivalence of Polyakov's and Nambu and Goto's formulations.
In fact
\begin{equation}
\fdv{S_P[\gamma, X]}{\gamma^{\alpha\beta}}
=
\ipd{\alpha} X \cdot \ipd{\beta} X
-
\frac{1}{2}
\gamma_{\alpha\beta}\,
\gamma^{\lambda\rho}\,
\ipd{\lambda} X \cdot \ipd{\rho} X
=
0
\label{eq:conf:worldsheetmetric}
\end{equation}
implies
\begin{equation}
\eval{S_P[\gamma, X]}_{\fdv{S_P[\gamma, X]}{\gamma^{\alpha\beta}} = 0}
=
- T
\infinfint{\tau}
\finiteint{\sigma}{0}{\sigma}
\sqrt{\dX \cdot \dX - \pX \cdot \pX}
=
S_{NG}[X],
\end{equation}
where $S_{NG}[X]$ is the Nambu--Goto action for the classical string.
The symmetries of $S_P[\gamma, X]$ are the keys to the success of the string theory framework~\cite{Polchinski:1998:StringTheoryIntroduction}.
Specifically~\eqref{eq:conf:polyakov} displays symmetries under:
\begin{itemize}
\item $D$-dimensional Poincaré invariance
\begin{equation}
\begin{split}
X'^{\mu}(\tau, \sigma)
& =
\tensor{\Lambda}{^{\mu}_{\nu}}\, X^{\mu}(\tau, \sigma) + c^{\nu},
\\
\gamma'_{\alpha\beta}(\tau, \sigma)
& =
\gamma_{\alpha\beta}(\tau, \sigma)
\end{split}
\end{equation}
where $\Lambda \in \SO{1, D-1}$ and $c \in \R^D$,
\item 2-dimensional diffeomorphism invariance
\begin{equation}
\begin{split}
X'^{\mu}(\tau', \sigma')
& =
X^{\mu}(\tau, \sigma)
\\
\gamma'_{\alpha\beta}(\tau', \sigma')
& =
\pdv{\sigma'^{\lambda}}{\sigma^{\alpha}}\,
\pdv{\sigma'^{\rho}}{\sigma^{\beta}}\,
\gamma_{\lambda\rho}(\tau, \sigma)
\end{split}
\end{equation}
where $\sigma^0 = \tau$ and $\sigma^1 = \sigma$,
\item Weyl invariance
\begin{equation}
\begin{split}
X'^{\mu}(\tau', \sigma')
& =
X^{\mu}(\tau, \sigma)
\\
\gamma'_{\alpha\beta}(\tau, \sigma)
& =
e^{2 \omega(\tau, \sigma)}\, \gamma_{\alpha\beta}(\tau, \sigma)
\end{split}
\end{equation}
for arbitrary $\omega(\tau, \sigma)$.
\end{itemize}
Notice that the last is not a symmetry of the Nambu--Goto action and it only appears in Polyakov's formulation of the action.
\subsubsection{Conformal Invariance}
The definition of the 2-dimensional stress-energy tensor is a direct consequence of~\eqref{eq:conf:worldsheetmetric} \cite{Green:1988:SuperstringTheoryIntroduction}.
In fact the classical constraint on the tensor is simply
\begin{equation}
T_{\alpha\beta}
=
-\frac{2 \pi}{\sqrt{- \det \gamma}}
\fdv{S_P[\gamma, X]}{\gamma^{\alpha\beta}}
=
0.
\end{equation}
While its conservation $\nabla^{\alpha} T_{\alpha\beta} = 0$ is somewhat trivial, Weyl invariance also ensures the tracelessness of the tensor
\begin{equation}
\trace{T} = \tensor{T}{^{\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:ConformalInvarianceSupersymmetrya,DiFrancesco:1997:ConformalFieldTheorya}).
Finally we can set $\gamma_{\alpha\beta}(\tau, \sigma) = e^{\phi(\tau, \sigma)}\, \eta_{\alpha\beta}$, known as \emph{conformal gauge} where $\eta_{\alpha\beta} = \mathrm{diag}(-1, 1)$, using the invariances of the action.
This gauge choice is however preserved by the residual \emph{pseudoconformal} transformations
\begin{equation}
\tau \pm \sigma = \sigma_{\pm} \mapsto f_{\pm}(\sigma_{\pm}),
\end{equation}
where $f_{\pm}(\xi)$ are arbitrary functions.
It is natural to introduce a Wick rotation $\tau_E = i \tau$ and the complex coordinates $\xi = \tau_E + i \sigma$ and $\bxi = \xi^*$.
In these terms, the tracelessness of the stress-energy tensor translates to
\begin{equation}
T_{z \bz} = 0,
\end{equation}
while its conservation $\partial^{\alpha} T_{\alpha\beta} = 0$ becomes:\footnotemark{}
\footnotetext{%
Since we fix $\gamma_{\alpha\beta}(\tau, \sigma) \propto \eta_{\alpha\beta}$ we do not need to account for the components of the connection and we can replace the covariant derivative $\nabla^{\alpha}$ with a standard derivative $\partial^{\alpha}$.
}
\begin{equation}
\bpd T_{\xi\xi}( \xi, \bxi ) = \pd \bT_{\bxi\bxi}( \xi, \bxi ) = 0.
\end{equation}
The last equation finally implies
\begin{equation}
T_{\xi\xi}( \xi, \bxi ) = T_{\xi\xi}( \xi ) = T( \xi ),
\qquad
\bT_{\bxi\bxi}( \xi, \bxi ) = \bT_{\bxi\bxi}( \bxi ) = \bT( \bxi ),
\end{equation}
which are respectively the holomorphic and the anti-holomorphic components of the 2-dimensional stress energy tensor.
The previous properties define what is known as a 2-dimensional \emph{conformal field theory} (\cft).
Ordinary tensor fields
\begin{equation}
\phi_{\omega, \bomega}( \xi, \bxi )
=
\phi_{\underbrace{\xi \dots \xi}\limits_{\omega~\text{times}}\, \underbrace{\bxi \dots \bxi}\limits_{\bomega~\text{times}}}( \xi, \bxi )
\left( \dd{\xi} \right)^{\omega}
\left( \dd{\bxi} \right)^{\bomega}
\end{equation}
can be classified according to their weights $\omega$ and $\bomega$.
In fact a transformation $\xi \mapsto \chi(\xi)$ and $\bxi \mapsto \bchi(\bxi)$ maps the conformal fields to
\begin{equation}
\phi_{\omega, \bomega}( \chi, \bchi )
=
\left( \dv{\chi}{\xi} \right)^{\omega}\,
\left( \dv{\bchi}{\bxi} \right)^{\bomega}\,
\phi_{\omega, \bomega}( \xi, \bxi ).
\end{equation}
\subsection{Extra Dimensions and Compactification}
\subsection{D-branes and Open Strings}
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@article{Calabi:1957:KahlerManifoldsVanishing,
title = {On {{Kähler}} Manifolds with Vanishing Canonical Class, {{Algebraic}} Geometry and Topology. {{A}} Symposium in Honor of {{S}}. {{Lefschetz}}},
author = {Calabi, Eugenio},
date = {1957},
journaltitle = {Princeton Math. Ser., Princeton University Press, Princeton},
volume = {78},
pages = {89},
keywords = {⛔ No DOI found}
}
@article{Candelas:1985:VacuumConfigurationsSuperstringsa,
title = {Vacuum Configurations for Superstrings},
author = {Candelas, P. and Horowitz, Gary T. and Strominger, Andrew and Witten, Edward},
date = {1985-01},
journaltitle = {Nuclear Physics B},
shortjournal = {Nuclear Physics B},
volume = {258},
pages = {46--74},
issn = {05503213},
doi = {10.1016/0550-3213(85)90602-9},
url = {10.1016/0550-3213(85)90602-9},
urldate = {2020-08-31},
file = {/home/riccardo/.local/share/zotero/files/candelas_et_al_1985_vacuum_configurations_for_superstrings2.pdf},
keywords = {archived},
langid = {english}
}
@book{DiFrancesco:1997:ConformalFieldTheorya,
title = {Conformal Field Theory},
author = {Di Francesco, P. and Mathieu, P. and Senechal, D.},
date = {1997},
publisher = {{Springer-Verlag}},
location = {{New York}},
doi = {10.1007/978-1-4612-2256-9},
isbn = {978-0-387-94785-3 978-1-4612-7475-9},
series = {Graduate Texts in Contemporary Physics}
}
@article{Friedan:1986:ConformalInvarianceSupersymmetrya,
title = {Conformal Invariance, Supersymmetry and String Theory},
author = {Friedan, Daniel and Martinec, Emil J. and Shenker, Stephen H.},
date = {1986},
journaltitle = {Nuclear Physics B},
shortjournal = {Nucl. Phys. B},
volume = {271},
pages = {93--165},
doi = {10.1016/0550-3213(86)90356-1},
url = {10.1016/0550-3213(86)90356-1},
file = {/home/riccardo/.local/share/zotero/files/friedan_et_al_1986_conformal_invariance,_supersymmetry_and_string_theory4.pdf},
number = {PRINT-86-0024 (CHICAGO), EFI-85-89-CHICAGO}
}
@mvbook{Green:1988:SuperstringTheoryIntroduction,
title = {Superstring {{Theory}}. {{Introduction}}.},
author = {Green, Michael B. and Schwarz, J.H. and Witten, Edward},
date = {1988-07},
volume = {1},
file = {/home/riccardo/.local/share/zotero/storage/G7BXB7V4/GreenSchwarzWitten_SuperstringTheory_vol1.pdf},
isbn = {978-0-521-35752-4},
series = {Cambridge Monographs on Mathematical Physics},
volumes = {2}
}
@mvbook{Green:1988:SuperstringTheoryLoop,
title = {Superstring {{Theory}}. {{Loop Amplitudes}}, {{Anomalies}} and {{Phenomenology}}.},
author = {Green, Michael B. and Schwarz, J.H. and Witten, Edward},
date = {1988-07},
volume = {2},
file = {/home/riccardo/.local/share/zotero/storage/QG8G8I3V/GreenSchwarzWitten_SuperstringTheory_vol2.pdf},
isbn = {978-0-521-35753-1},
volumes = {2}
}
@article{Joyce:2002:LecturesCalabiYauSpecial,
title = {Lectures on {{Calabi}}-{{Yau}} and Special {{Lagrangian}} Geometry},
author = {Joyce, Dominic},
date = {2002-06-25},
url = {http://arxiv.org/abs/math/0108088},
urldate = {2020-08-31},
abstract = {This paper gives a leisurely introduction to Calabi-Yau manifolds and special Lagrangian submanifolds from the differential geometric point of view, followed by a survey of recent results on singularities of special Lagrangian submanifolds, and their application to the SYZ Conjecture. It is aimed at graduate students in Geometry, String Theorists, and others wishing to learn the subject, and is designed to be fairly self-contained. It is based on lecture courses given at Nordfjordeid, Norway and MSRI, Berkeley in June and July 2001. We introduce Calabi-Yau m-folds via holonomy groups, Kahler geometry and the Calabi Conjecture, and special Lagrangian m-folds via calibrated geometry. `Almost Calabi-Yau m-folds' (a generalization of Calabi-Yau m-folds useful in special Lagrangian geometry) are explained and the deformation theory and moduli spaces of compact special Lagrangian submanifolds in (almost) Calabi-Yau m-folds is described. In the final part we consider isolated singularities of special Lagrangian m-folds, focussing mainly on singularities locally modelled on cones, and the expected behaviour of singularities of compact special Lagrangian m-folds in generic (almost) Calabi-Yau m-folds. String Theory, Mirror Symmetry and the SYZ Conjecture are briefly discussed, and some results of the author on singularities of special Lagrangian fibrations of Calabi-Yau 3-folds are described.},
archivePrefix = {arXiv},
eprint = {math/0108088},
eprinttype = {arxiv},
file = {/home/riccardo/.local/share/zotero/files/joyce_2002_lectures_on_calabi-yau_and_special_lagrangian_geometry.pdf},
keywords = {⛔ No DOI found,High Energy Physics - Theory,Mathematics - Algebraic Geometry,Mathematics - Differential Geometry}
}
@mvbook{Polchinski:1998:StringTheoryIntroduction,
title = {String {{Theory}}. {{An}} Introduction to the Bosonic String.},
author = {Polchinski, Joseph},
date = {1998-10-13},
edition = {1},
volume = {1},
publisher = {{Cambridge University Press}},
doi = {10.1017/CBO9780511816079; http://web.archive.org/web/20200831133025/https://www.cambridge.org/core/books/string-theory/30409AF2BDE27D53E275FDA395AB667A},
url = {https://doi.org/10.1017/CBO9780511816079},
urldate = {2020-08-31},
file = {/home/riccardo/.local/share/zotero/storage/IBTZMGEY/Polchinski_StringTheory_vol1.pdf},
isbn = {978-0-521-67227-6 978-0-521-63303-1 978-0-511-81607-9},
keywords = {archived},
volumes = {2}
}
@mvbook{Polchinski:1998:StringTheorySuperstring,
title = {String {{Theory}}. {{Superstring}} Theory and Beyond.},
author = {Polchinski, Joseph},
date = {1998-10-13},
edition = {1},
volume = {2},
publisher = {{Cambridge University Press}},
doi = {10.1017/CBO9780511618123; http://web.archive.org/web/20200831133053/https://www.cambridge.org/core/books/string-theory/2D456468D20AA8A9CE10CEB08B95B9DC},
url = {https://doi.org/10.1017/CBO9780511618123},
urldate = {2020-08-31},
file = {/home/riccardo/.local/share/zotero/storage/HYIV7348/Polchinski_StringTheory_vol2.pdf},
isbn = {978-0-521-63304-8 978-0-521-67228-3 978-0-511-61812-3},
keywords = {archived},
volumes = {2}
}
@article{Polyakov:1981:QuantumGeometryBosonic,
title = {Quantum Geometry of Bosonic Strings},
author = {Polyakov, A.M.},
date = {1981-07},
journaltitle = {Physics Letters B},
shortjournal = {Physics Letters B},
volume = {103},
pages = {207--210},
issn = {03702693},
doi = {10.1016/0370-2693(81)90743-7},
url = {10.1016/0370-2693(81)90743-7},
urldate = {2020-08-31},
file = {/home/riccardo/.local/share/zotero/files/polyakov_1981_quantum_geometry_of_bosonic_strings.pdf},
keywords = {archived},
langid = {english},
number = {3}
}
@article{Yau:1977:CalabiConjectureNew,
title = {Calabi's Conjecture and Some New Results in Algebraic Geometry},
author = {Yau, S.-T.},
date = {1977-05-01},
journaltitle = {Proceedings of the National Academy of Sciences},
shortjournal = {Proceedings of the National Academy of Sciences},
volume = {74},
pages = {1798--1799},
issn = {0027-8424, 1091-6490},
doi = {10.1073/pnas.74.5.1798},
url = {10.1073/pnas.74.5.1798},
urldate = {2020-08-31},
file = {/home/riccardo/.local/share/zotero/files/yau_1977_calabi's_conjecture_and_some_new_results_in_algebraic_geometry.pdf},
keywords = {archived},
langid = {english},
number = {5}
}

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%---- packages
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@@ -100,32 +99,22 @@
\providecommand{\theschool}{\@school}
\providecommand{\thelogo}{\@logo}
\makeatother
\hypersetup
{%
pdftitle={\thetitle},
pdftitle={\@title},
pdfsubject={Thesis},
pdfauthor={\theauthor},
pdfauthor={\@author}
pdfkeywords={thesis, graduation, doctoral, phd},
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\makeatother
%---- titlepage
\newcommand{\maketitlepage}

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\usepackage[british]{babel}
\usepackage{csquotes}
\usepackage{debug}
\usepackage{sciencestuff}
\usepackage{lipsum}
\author{Riccardo Finotello}
\title{Title of the Thesis}
\title{Topics on Theoretical and Computational Aspects of String Theory and Their Phenomenological Implications}
\advisor{Igor Pesando}
\institution{Universit\`{a} degli Studi di Torino}
\school{Scuola di Dottorato}
\specialisation{Dottorato in Fisica ed Astrofisica}
\logo{img/unito}
\addbibresource{thesis.bib}
%---- additional commands
\newcommand{\sm}{\textsc{sm}\xspace}
\newcommand{\eom}{\textsc{e.o.m.}\xspace}
\newcommand{\cft}{\textsc{cft}\xspace}
%---- derivatives
\newcommand{\pd}{\ensuremath{\partial}}
\newcommand{\bpd}{\ensuremath{\overline{\partial}}}
%---- operators
\newcommand{\bT}{\ensuremath{\overline{T}}}
%---- coordinates
\newcommand{\dX}{\ensuremath{\dot{X}}}
\newcommand{\pX}{\ensuremath{X'}}
\newcommand{\bxi}{\ensuremath{\overline{\xi}}}
\newcommand{\bchi}{\ensuremath{\overline{\chi}}}
\newcommand{\bz}{\ensuremath{\overline{z}}}
\newcommand{\bw}{\ensuremath{\overline{w}}}
\newcommand{\bomega}{\ensuremath{\overline{\omega}}}
%---- BEGIN DOCUMENT
\begin{document}
\maketitlepage{}
%---- TITLE
\maketitlepage{}
\cleardoubleplainpage{}
\cleardoubleplainpage{}
%---- FRONTESPICE
\makefrontespice{}
\cleardoubleplainpage{}
\makefrontespice{}
%---- ABSTRACT
\begin{abstractpage}
\input{sec/abstract.tex}
\end{abstractpage}
\cleardoubleplainpage{}
\cleardoubleplainpage{}
%---- ACKNOWLEDGENTS
\begin{acknowledgmentspage}
\input{sec/acknowledgments.tex}
\end{acknowledgmentspage}
\cleardoubleplainpage{}
\begin{abstractpage}
%---- TOC AND OUTLINE
\plaintoc{}
\outline{Outline}
\input{sec/outline.tex}
\lipsum[1-2]
%---- PARTICLE PHYSICS
\thesispart{Conformal Symmetry and Geometry of the Wordlsheet}
\section{Introduction}
\input{sec/part1/introduction.tex}
\section{D-branes Intersecting at Angles}
\input{sec/part1/dbranes.tex}
\section{Fermions With Boundary Defects}
\input{sec/part1/fermions.tex}
\end{abstractpage}
%---- COSMOLOGY
\thesispart{Cosmology and Time Dependent Divergences}
\section{Introduction}
\input{sec/part2/introduction.tex}
\cleardoubleplainpage{}
%---- DEEP LEARNING
\thesispart{Deep Learning the Geometry of String Theory}
\section{Introduction}
\input{sec/part3/introduction.tex}
\begin{acknowledgmentspage}
%---- APPENDIX
\appendix
\lipsum[3-4]
\end{acknowledgmentspage}
\cleardoubleplainpage{}
\plaintoc{}
\outline{Outline}
\lipsum[1-3]
\thesispart{This Is the First Part}
\section{AAA}
\lipsum[1]
\subsection{AAA1}
\lipsum[2-5]
\subsection{AAA2}
\lipsum[6-11]
\thesispart{This Is the Second Part 2}
\section{BBB}
\lipsum[1]
\subsection{BBB1}
\lipsum[2-5]
\subsection{BBB2}
\lipsum[6-10]
\appendix
\section{CCC}
\lipsum[1-3]
\section{DDD}
\lipsum[4-6]
%---- BIBLIOGRAPHY
\printbibliography[heading=bibintoc]
\end{document}