In this first part we focus on aspects of string theory directly connected with its worldsheet description and symmetries.
The underlying idea is to build technical tools to address the study of viable phenomenological models in this framework.
The construction of realistic string models of particle physics is 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 isomorphic to the algebra of the group
\begin{equation}
  \SU{3}_{\rC} \otimes \SU{2}_{\rL} \otimes \U{1}_{\rY}
  \label{eq:intro:smgroup}
\end{equation}
in order to reproduce known results.
For instance a good string theory could provide a unified framework by predicting the existence of a larger gauge group containing the \sm as a subset.
In what follows we deal with the definition of mathematical tools to compute amplitudes to be used in phenomenological calculations related to the study of particles.

In this introduction we present instruments used throughout the manuscript as many aspects are strongly connected and their definitions are interdependent.
In particular we recall some results on the symmetries of string theory and how to recover a realistic description of $4$-dimensional physics.


\subsection{Properties of String Theory and Conformal Symmetry}

Strings are extended one-dimensional objects.
They are curves in spacetime parametrized by a coordinate $\sigma \in \qty[ 0, \ell ]$.
When propagating they span a two-dimensional surface, the \emph{worldsheet}, described by the position of the string at given values of $\sigma$ at a time $\tau$, i.e.\ $X^{\mu}\qty(\tau, \sigma)$ where $\mu = 0,\, 1,\, \dots,\, D - 1$ indexes the coordinates.
Such surface can have different topologies according to the nature of the object propagating in spacetime: strings can be \emph{closed} if $X^{\mu}\qty(\tau, 0) = X^{\mu}\qty(\tau, \ell)$ or \emph{open} if the endpoints in $\sigma = 0$ and $\sigma = \ell$ do not coincide.


\subsubsection{Action Principle}

As the action of a point particle is proportional to the length of its trajectory (its \emph{worldline}), the same object for a string is proportional to the area of the worldsheet in the original formulation by Nambu and Goto.
The solutions of the classical equations of motion (\eom) are therefore strings spanning a worldsheet of extremal area.
While Nambu and Goto's formulation is fairly direct in its definition, it is usually best to work with Polyakov's action~\cite{Polyakov:1981:QuantumGeometryBosonic}:
\begin{equation}
  S_P\qty[ \gamma, X ]
  =
  -\frac{1}{4 \pi \ap}
  \infinfint{\tau}
  \finiteint{\sigma}{0}{\ell}
  \sqrt{- \det \gamma\qty(\tau, \sigma)}\,
  \gamma^{\alpha\beta}\qty(\tau, \sigma)\,
  \ipd{\alpha} X^{\mu}\qty(\tau, \sigma)\,
  \ipd{\beta} X^{\nu}\qty(\tau, \sigma)\,
  \eta_{\mu\nu}.
  \label{eq:conf:polyakov}
\end{equation}
The \eom for the string $X^{\mu}\qty(\tau, \sigma)$ is therefore:
\begin{equation}
  \frac{1}{\sqrt{- \det \gamma}}\,
  \ipd{\alpha}
  \qty(
    \sqrt{- \det \gamma}\,
    \gamma^{\alpha\beta}\,
    \ipd{\beta} X^{\mu}
  )
  =
  0,
  \qquad
  \mu = 0, 1, \dots, D - 1,
  \qquad
  \alpha,\, \beta = 0, 1.
\end{equation}
In this formulation $\gamma_{\alpha\beta}$ are the components of the worldsheet metric.
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}}
  =
  - \frac{1}{4 \pi \ap}
  \sqrt{- \det \gamma}\,
  \qty(
    \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\qty[\gamma,\, X]}_{\fdv{S_P\qty[\gamma,\, X]}{\gamma^{\alpha\beta}} = 0}
  =
  - \frac{1}{2 \pi \ap}
  \infinfint{\tau}
  \finiteint{\sigma}{0}{\sigma}
  \sqrt{\dotX \cdot \dotX - \pX \cdot \pX}
  =
  S_{NG}[X],
\end{equation}
where $S_{NG}[X]$ is the Nambu--Goto action of the classical string, and $\dotX = \ipd{\tau} X$ and $\pX = \ipd{\sigma} X$.

The symmetries of $S_P\qty[\gamma,\, X]$ are key to the success of the string theory framework~\cite{Polchinski:1998:StringTheoryIntroduction}.
Specifically~\eqref{eq:conf:polyakov} displays symmetries under:
\begin{itemize}
  \item $D$-dimensional Poincaré transformations
    \begin{equation}
      \begin{split}
        X'^{\mu}\qty(\tau, \sigma)
        & =
        \tensor{\Lambda}{^{\mu}_{\nu}}\, X^{\nu}\qty(\tau, \sigma) + c^{\mu},
        \\
        \gamma'_{\alpha\beta}\qty(\tau, \sigma)
        & =
        \gamma_{\alpha\beta}\qty(\tau, \sigma)
      \end{split}
    \end{equation}
    where $\Lambda \in \SO{1,\, D-1}$ and $c \in \R^D$,

  \item 2-dimensional diffeomorphisms
    \begin{equation}
      \begin{split}
        X'^{\mu}\qty(\tau', \sigma')
        & =
        X^{\mu}\qty(\tau, \sigma)
        \\
        \gamma'_{\alpha\beta}\qty(\tau', \sigma')
        & =
        \pdv{\sigma'^{\lambda}}{\sigma^{\alpha}}\,
        \pdv{\sigma'^{\rho}}{\sigma^{\beta}}\,
        \gamma_{\lambda\rho}\qty(\tau, \sigma)
      \end{split}
    \end{equation}
    where $\sigma^0 = \tau$ and $\sigma^1 = \sigma$,

  \item Weyl transformations
    \begin{equation}
      \begin{split}
        X'^{\mu}\qty(\tau', \sigma')
        & =
        X^{\mu}\qty(\tau, \sigma)
        \\
        \gamma'_{\alpha\beta}\qty(\tau, \sigma)
        & =
        e^{2 \omega\qty(\tau, \sigma)}\, \gamma_{\alpha\beta}\qty(\tau, \sigma)
      \end{split}
    \end{equation}
    for an arbitrary function $\omega\qty(\tau, \sigma)$.
\end{itemize}
Notice that the last is not a symmetry of the Nambu--Goto action and it only appears in Polyakov's formulation.


\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}
  \cT_{\alpha\beta}
  =
  \frac{4 \pi}{\sqrt{- \det \gamma}}
  \fdv{S_P\qty[\gamma,\, X]}{\gamma^{\alpha\beta}}
  =
  -\frac{1}{\ap}
  \qty(
    \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:stringT}
\end{equation}
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{\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,Blumenhagen:2009:IntroductionConformalField}).

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 (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 vanishing trace of the stress-energy tensor translates to
\begin{equation}
  \cT_{\xi \bxi} = 0,
\end{equation}
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}
  \ipd{\bxi} \cT_{\xi\xi}\qty( \xi,\, \bxi )
  =
  \ipd{\xi} \overline{\cT}_{\bxi\bxi}\qty( \xi,\, \bxi )
  =
  0.
\end{equation}
The last equation finally implies
\begin{equation}
  \cT_{\xi\xi}\qty( \xi,\, \bxi )
  =
  \cT_{\xi\xi}\qty( \xi )
  =
  \cT\qty( \xi ),
  \qquad
  \overline{\cT}_{\bxi\bxi}\qty( \xi,\, \bxi )
  =
  \overline{\cT}_{\bxi\bxi}\qty( \bxi )
  =
  \overline{\cT}\qty( \bxi ),
\end{equation}
which are respectively the holomorphic and the anti-holomorphic components of the stress energy tensor.

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 )
  =
  \phi_{\underbrace{\xi \dots \xi}\limits_{\omega~\text{times}}\, \underbrace{\bxi \dots \bxi}\limits_{\bomega~\text{times}}}\qty( \xi, \bxi )
  \qty( \dd{\xi} )^{\omega}
  \qty( \dd{\bxi} )^{\bomega}
\end{equation}
are classified according to their weight $\qty( \omega,\, \bomega )$ referring to the holomorphic and anti-holomorphic parts respectively.
In fact a transformation $\xi \mapsto \chi(\xi)$ and $\bxi \mapsto \bchi(\bxi)$ maps the conformal fields to
\begin{equation}
  \phi_{\omega, \bomega}\qty( \chi, \bchi )
  =
  \qty( \dv{\chi}{\xi} )^{\omega}\,
  \qty( \dv{\bchi}{\bxi} )^{\bomega}\,
  \phi_{\omega, \bomega}\qty( \xi, \bxi ).
\end{equation}

\begin{figure}[tbp]
  \centering
  \begin{subfigure}[b]{0.45\linewidth}
    \centering
    \import{tikz}{complex_plane.pgf}
    \caption{Radial ordering.}
  \end{subfigure}
  \hfill
  \begin{subfigure}[b]{0.45\linewidth}
    \centering
    \import{tikz}{radial_ordering.pgf}
    \caption{Difference of time ordered contours.}
  \end{subfigure}
  \caption{Map to the complex plane.}
  \label{fig:conf:complex_plane}
\end{figure}

An additional conformal transformation
\begin{equation}
  z = e^{\xi} = e^{\tau_e + i \sigma} \in \qty{ z \in \C | \Im z \ge 0 },
  \qquad
  \barz = e^{\bxi} = e^{\tau_e - i \sigma} \in \qty{ z \in \C | \Im z \le 0 }
\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:
\begin{equation}
  Q_{\epsilon, \bepsilon}
  =
  \cint{0}
  \ddz
  \epsilon(z)\, \cT(z)
  +
  \cint{0}
  \ddbz
  \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 $Q_{\epsilon, \bepsilon}$:
\begin{equation}
  \begin{split}
    \delta_{\epsilon, \bepsilon} \phi_{\omega, \bomega}
    & =
    \liebraket{Q_{\epsilon, \bepsilon}}{\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[ \overline{\cT}(\barz), \phi_{\omega, \bomega}\qty( w, \barw ) ]
    \\
    & =
    \cint{w} \ddz \epsilon(z)\, \rR\qty( \cT(z)\, \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 as a infinitesimally small anti-clockwise loop around $w$ and $\barw$.
Equating the result with the expected variation
\begin{equation}
  \begin{split}
    \delta_{\epsilon, \bepsilon} \phi_{\omega, \bomega}
    & =
    \omega\, \ipd{w} \epsilon( w )\, \phi_{\omega, \bomega}\qty( w, \barw )
    +
    \epsilon( w )\, \ipd{w} \phi_{\omega, \bomega}\qty( w, \barw )
    \\
    & +
    \bomega\, \ipd{\barw} \bepsilon( \barw )\, \phi_{\omega, \bomega}\qty( w, \barw )
    +
    \epsilon( \barw )\, \ipd{\barw} \phi_{\omega, \bomega}\qty( w, \barw )
  \end{split}
\end{equation}
we find the short distance singularities of the components of the stress-energy tensor with the field $\phi_{\omega, \bomega}( w, \barw )$:
\begin{equation}
  \begin{split}
    \cT( z )\, \phi_{\omega, \bomega}\qty( w, \barw )
    & =
    \frac{\omega}{(z - w)^2}\, \phi_{\omega, \bomega}\qty( w, \barw )
    +
    \frac{1}{z - w}\, \ipd{w} \phi_{\omega, \bomega}\qty( w, \barw )
    +
    \order{1},
    \\
    \overline{\cT}( \barz )\, \phi_{\omega, \bomega}\qty( w, \barw )
    & =
    \frac{\bomega}{(\barz - \barw)^2}\, \phi_{\omega, \bomega}\qty( w, \barw )
    +
    \frac{1}{\barz - \barw}\, \ipd{\barw} \phi_{\omega, \bomega}\qty( w, \barw )
    +
    \order{1},
  \end{split}
  \label{eq:conf:primary}
\end{equation}
where we drop the radial ordering symbol $\rR$ for simplicity.
Since the contour $\ccC_{w}$ is infinitely small around $w$, the conformal properties of $\phi_{\omega, \bomega}( w, \barw )$ are entirely defined by these relations.
In fact $\phi_{\omega, \bomega}( w, \barw )$ is a \emph{primary field} if its short distance behaviour with the stress-energy tensor is as in~\eqref{eq:conf:primary}.
This is an example of an \emph{operator product expansion} (\ope)
\begin{equation}
  \phi^{(i)}_{\omega_i, \bomega_i}( z, \barz )\, \phi^{(j)}_{\omega_j, \bomega_j}( w, \barw )
  =
  \sum\limits_{k}
  \cC_{ijk}
  (z - w)^{\omega_k - \omega_i - \omega_j}\,
  (\barz - \barw)^{\bomega_k - \bomega_i - \bomega_j}\,
  \phi^{(k)}_{\omega_k, \bomega_k}( w, \barw )
  \label{eq:conf:ope}
\end{equation}
which is an asymptotic expansion containing full information on the singularities.\footnotemark{}
\footnotetext{%
  Expression~\eqref{eq:conf:ope} is valid assuming the normalisation of the 2-points function
  \begin{equation*}
    \left\langle \phi^{(i)}_{\omega_i, \bomega_i}( z, \barz )\, \phi^{(j)}_{\omega_j, \bomega_j}( w, \barw ) \right\rangle
    =
    \frac{\delta_{ij}}{(z - w)^{\omega_i + \omega_j} (\barz - \barw)^{\bomega_i + \bomega_j}}.
  \end{equation*}
}
The constant coefficients $\cC_{ijk}$ are subject to restrictive constraints given by the properties of the conformal theories to the point that a \cft is completely specified by the spectrum of the weights $(\omega_i, \bomega_i)$ and the coefficients $\cC_{ijk}$ \cite{Friedan:1986:ConformalInvarianceSupersymmetry}.

The \ope can also be computed on the stress-energy tensor itself:
\begin{equation}
  \begin{split}
    \cT( z )\, \cT( w )
    & =
    \frac{\frac{c}{2}}{(z - w)^4}
    +
    \frac{2}{(z - w)^2}\, \cT(w)
    +
    \frac{1}{z - w}\, \ipd{w} \cT(w),
    \\
    \overline{\cT}( \barz )\, \overline{\cT}( \barw )
    & =
    \frac{\frac{\barc}{2}}{(\barz - \barw)^4}
    +
    \frac{2}{(\barz - \barw)^2}\, \overline{\cT}(\barw)
    +
    \frac{1}{\barz - \barw}\, \ipd{\barw} \overline{\cT}(\barw).
  \end{split}
  \label{eq:conf:TTexpansion}
\end{equation}
The components of the stress-energy tensor are therefore not primary fields and show an anomaly in the behaviour encoded by the constant $c \in \R$.
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}
    \cT( z ) = \infinfsum{n} L_n\, z^{-n -2}
    & \Rightarrow
    L_n = \cint{0} \ddz z^{n + 1} \cT(z),
    \\
    \overline{\cT}( \barz ) = \infinfsum{n} \barL_n\, \barz^{-n -2}
    & \Rightarrow
    \barL_n = \cint{0} \ddbz \barz^{n + 1} \overline{\cT}(\barz).
  \end{split}
  \label{eq:conf:Texpansion}
\end{equation}
This ultimately leads to the quantum algebra
\begin{equation}
  \begin{split}
    \liebraket{L_n}{L_m}
    & =
    (n - m)\, L_{n + m} + \frac{c}{12}\, n\, (n^2 - 1)\, \delta_{n, -m},
    \\
    \liebraket{\barL_n}{\barL_m}
    & =
    (n - m)\, \barL_{n + m} + \frac{c}{12}\, n\, (n^2 - 1)\, \delta_{n, -m},
    \\
    \liebraket{L_n}{\barL_m}
    & =
    0,
  \end{split}
  \label{eq:conf:virasoro}
\end{equation}
known as Virasoro algebra, unique central extension of the classical de Witt algebra, with central charge $c$.
Coefficients $L_n$ 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 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.

In the same fashion as~\eqref{eq:conf:Texpansion}, fields can be expanded in modes:
\begin{equation}
  \phi_{\omega, \bomega}( w, \barw )
  =
  \sum\limits_{n,\, m = -\infty}^{+\infty}
  \phi_{\omega, \bomega}^{(n, m)}\,
  z^{-n -\omega}\,
  \barz^{-m -\bomega}.
  \label{eq:conf:expansion}
\end{equation}
From the previous relations we can finally define the ``asymptotic'' in-states as one-to-one correspondence with conformal operators:
\begin{equation}
  \ket{\phi_{\omega, \bomega}}
  =
  \lim\limits_{z,\, \barz \to 0}
  \phi_{\omega, \bomega}
  \regvacuum.
\end{equation}
Regularity of \eqref{eq:conf:expansion} requires
\begin{equation}
  \phi_{\omega, \bomega}^{(n, m)}
  \regvacuum
  =
  0,
  \qquad
  n > \omega,
  \qquad
  m > \bomega.
\end{equation}
As a consequence also
\begin{equation}
  L_n \regvacuum
  =
  \barL_n \regvacuum
  =
  0,
  \qquad
  n > -2.
\end{equation}
Finally the definitions of the primary operators~\eqref{eq:conf:primary} define the \emph{physical} states as
\begin{equation}
  \begin{split}
    L_0 \ket{\phi_{\omega, \bomega}} & = \omega \ket{\phi_{\omega, \bomega}},
    \\
    \barL_0 \ket{\phi_{\omega, \bomega}} & = \bomega \ket{\phi_{\omega, \bomega}},
    \\
    L_n \ket{\phi_{\omega, \bomega}} & = \barL_n \ket{\phi_{\omega, \bomega}} = 0,
    \quad
    n \ge 1.
  \end{split}
  \label{eq:conf:physical}
\end{equation}

From this definition we can build an entire representation of \emph{descendant} states applying any operator $L_{-n}$ (or $\barL_{-n}$) with $n \ge 1$ to $\ket{\phi_{\omega, \bomega}}$.
Let $\phi_{\omega}( w )$ be a holomorphic field in the \cft for simplicity, and let $\ket{\phi_{\omega}}$ be its corresponding state.
The generic state at level $m$ built from such state is
\begin{equation}
  \ket{\phi_{\omega}^{\lbrace n_1, n_2, \dots, n_m \rbrace}}
  =
  L_{-n_1}\, L_{-n_2}\, \dots L_{-n_m}
  \ket{\phi_{\omega}},
  \qquad
  \finitesum{i}{1}{m} n_i = m \ge 0.
\end{equation}
From the commutation relations~\eqref{eq:conf:virasoro} we finally compute its conformal weight as eigenvalue of the (holomorphic) Hamiltonian $L_0$:
\begin{equation}
  L_0 \ket{\phi_{\omega}^{\lbrace n_1, n_2, \dots, n_m \rbrace}}
  =
  (\omega + m) \ket{\phi_{\omega}^{\lbrace n_1, n_2, \dots, n_m \rbrace}}.
\end{equation}
States corresponding to primary operators have therefore the lowest energy (intended as eigenvalue of the Hamiltonian) in the entire representation.
They are however called \emph{highest weight} states from the mathematical literature which uses the opposite sign for the Hamiltonian operator.

The particular case of the \cft in \eqref{eq:conf:polyakov} can be cast in this language.
In particular the solutions to the \eom factorise into a holomorphic and an anti-holomorphic contributions:
\begin{equation}
  \ipd{z} \ipd{\barz} X( z, \barz ) = 0
  \qquad
  \Rightarrow
  \qquad
  X( z, \barz ) = X( z ) + \barX( \barz ),
\end{equation}
and the components of the stress-energy tensor~\eqref{eq:conf:stringT} are
\begin{equation}
  \begin{split}
    \cT( z ) & = \ipd{z} X( z ) \cdot \ipd{z} X( z ),
    \\
    \overline{\cT}( \barz ) & = \ipd{\barz} \barX( \barz ) \cdot \ipd{\barz} \barX( \barz ).
  \end{split}
  \label{eq:conf:bosonicstringT}
\end{equation}
Using the normalisation of the 2-points function $\left\langle X^{\mu}( z, \barz )\, X^{\nu}( w, \barw ) \right\rangle = - \frac{1}{2} \eta^{\mu\nu} \ln\left| z - w \right|$ and the Wick theorem, we can prove that $c = D$ in~\eqref{eq:conf:TTexpansion}, where $D$ is the dimensions of spacetime (or equivalently the number of scalar fields $X^{\mu}$ in the action).
It can be shown that in order to cancel the central charge in bosonic string theory we need to introduce a pair of conformal ghosts $b(z)$ and $c(z)$ with conformal weights $(2, 0)$ and $(-1, 0)$ respectively, together with their anti-holomorphic counterparts $\barb(z)$ and $\barc(z)$, as a first order Lagrangian theory.
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}
  \begin{equation*}
    S = \frac{1}{2 \pi} \iint \dd{z} \dd{\barz}\, b( z )\, \ipd{\barz} c( z ).
  \end{equation*}
  The \eom are $\ipd{\barz} c( z ) = \ipd{\barz} b( z ) = 0$.
  The \ope is
  \begin{equation*}
    b( z )\, c( z ) = \frac{\varepsilon}{z - w} + \order{1},
  \end{equation*}
  where $\varepsilon = +1$ for anti-commuting fields and $\varepsilon = -1$ for Bose statistic.
  Their stress-energy tensor is
  \begin{equation*}
    \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 an additional \emph{ghost number} \U{1} symmetry generated by the current
  \begin{equation*}
    j( z ) = - b( z )\, c( z ).
  \end{equation*}
  The current is a primary field (i.e.\ it is not anomalous) when $\cQ = 0$ since
  \begin{equation*}
    \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 $Q = 2$.
  \label{note:conf:ghosts}
}
\begin{equation}
  \begin{split}
    \cT_{\text{ghost}}( z )
    & =
    c( z )\, \ipd{z} b( z ) - 2\, b( z )\, \ipd{z} c( z ),
    \\
    \overline{\cT}_{\text{ghost}}( \barz )
    & =
    \barc( \barz )\, \ipd{\barz} \barb( \barz ) - 2\, \barb( \barz )\, \ipd{\barz} \barc( \barz ).
  \end{split}
\end{equation}

From their 2-points functions
\begin{equation}
  \left\langle b(z)\, c(w) \right\rangle = \frac{1}{z - w},
  \qquad
  \left\langle \barb(\barz)\, \barc(\barw) \right\rangle = \frac{1}{\barz - \barw},
\end{equation}
we get the \ope of the components of their stress-energy tensor:
\begin{equation}
  \begin{split}
    \cT_{\text{ghost}}(z)\, \cT_{\text{ghost}}(w)
    & =
    \frac{-13}{(z - w)^4}
    +
    \frac{2}{(z - w)^2}\, \cT_{\text{ghost}}(z)
    +
    \frac{1}{z - w}\, \ipd{z} \cT_{\text{ghost}}(z),
    \\
    \overline{\cT}_{\text{ghost}}(\barz)\, \overline{\cT}_{\text{ghost}}(\barw)
    & =
    \frac{-13}{(\barz - \barw)^4}
    +
    \frac{2}{(\barz - \barw)^2}\, \overline{\cT}_{\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 $\cT_{\text{full}} = \cT + \cT_{\text{ghost}}$ and $\overline{\cT}_{\text{full}} = \overline{\cT} + \overline{\cT}_{\text{ghost}}$, then:
\begin{equation}
  \eval{\cT_{\text{full}}( z )}_{\order{(z - w)^{-4}}}
  =
  \eval{\overline{\cT}_{\text{full}}( \barz )}_{\order{(\barz - \barw)^{-4}}}
  =
  c + c_{\text{ghost}}
  =
  \frac{D}{2} - 13
  =
  0
  \quad
  \Leftrightarrow
  \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 \emph{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}.
In complex coordinates on the plane it is~\cite{Polchinski:1998:StringTheorySuperstring}:
\begin{equation}
  S\qty[ X,\, \psi ]
  =
  - \frac{1}{4 \pi}
  \iint \dd{z} \dd{\barz}
  \qty(
    \frac{2}{\ap}\, \ipd{\barz} X^{\mu}\, \ipd{z} X^{\nu}
    +
    \psi^{\mu}\, \ipd{\barz} \psi^{\nu}
    +
    \bpsi^{\mu}\, \ipd{z} \bpsi^{\nu}
  )
  \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
\begin{equation}
  \psi^{\mu}( z )\, \psi^{\nu}( w ) = \frac{\eta^{\mu\nu}}{z - w},
  \qquad
  \bpsi^{\mu}( \barz )\, \bpsi^{\nu}( \barw ) = \frac{\eta^{\mu\nu}}{\barz - \barw}.
\end{equation}
In this case the components of the stress-energy tensor of the theory are:
\begin{equation}
  \begin{split}
    \cT( z )
    & =
    -\frac{1}{\ap}\, \ipd{z} X( z ) \cdot \ipd{z} X( z ) - \frac{1}{2}\, \psi( z ) \cdot \ipd{z} \psi( z ),
    \\
    \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}
\end{equation}

The action~\eqref{eq:super:action} is also invariant under the \emph{supersymmetric} transformations
\begin{equation}
  \begin{split}
    \sqrt{\frac{2}{\ap}}\,
    \delta_{\epsilon, \bepsilon}
    X^{\mu}( z, \barz )
    & =
    \epsilon( z )\, \psi^{\mu}( z ) + \bepsilon( \barz )\, \bpsi^{\mu}( \barz ),
    \\
    \sqrt{\frac{2}{\ap}}\,
    \delta_{\epsilon} \psi^{\mu}( z )
    & =
    - \epsilon( z )\, \ipd{z} X^{\mu}( z ),
    \\
    \sqrt{\frac{2}{\ap}}\,
    \delta_{\bepsilon} \bpsi^{\mu}( \barz )
    & =
    - \bepsilon( \barz )\, \ipd{\barz} \barX^{\mu}( \barz )
  \end{split}
\end{equation}
generated by $J( z ) = \epsilon( z )\, \cT_F( z )$ 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}
    \cT_F( z )
    & =
    i\, \sqrt{\frac{2}{\ap}}\, \psi( z ) \cdot \ipd{z} X( z ),
    \\
    \overline{\cT}_F( \barz )
    & =
    i\, \sqrt{\frac{2}{\ap}}\, \bpsi( \barz ) \cdot \ipd{\barz} \barX( \barz )
  \end{split}
\end{equation}
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}
    \cT( z )\, \cT( w )
    & =
    \frac{\frac{3 D}{4}}{( z - w )^4}
    +
    \frac{2}{( z - w )^2} \cT( w )
    +
    \frac{1}{z - w} \ipd{w} \cT( w )
    +
    \order{1},
    \\
    \overline{\cT}( \barz )\, \overline{\cT}( \barw )
    & =
    \frac{\frac{3 D}{4}}{( \barz - \barw )^4}
    +
    \frac{2}{( \barz - \barw )^2} \overline{\cT}( \barw )
    +
    \frac{1}{\barz - \barw} \ipd{\barw} \overline{\cT}( \barw )
    +
    \order{1}.
  \end{split}
\end{equation}
The central charge is therefore $c = \frac{3}{2} D$ for the \cft defined in~\eqref{eq:super:action}.

As in the case of the bosonic string, in order to cancel the central charge 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 $\cT_{\text{full}} = \cT + \cT_{\text{ghost}}$ and $\overline{\cT}_{\text{full}} = \overline{\cT} + \overline{\cT}_{\text{ghost}}$ the central charge vanishes only in 10-dimensional spacetime:
\begin{equation}
  \eval{\cT_{\text{full}}( z )}_{\order{(z - w)^{-4}}}
  =
  \eval{\overline{\cT}_{\text{full}}( \barz )}_{\order{(\barz - \barw)^{-4}}}
  =
  c + c_{\text{ghost}}
  =
  \frac{3}{2}\, D - 15
  =
  0
  \quad
  \Leftrightarrow
  \quad
  D = 10.
  \label{eq:super:dimensions}
\end{equation}


\subsection{Extra Dimensions and Compactification}
\label{sec:CYmanifolds}

We are ultimately interested in building a consistent phenomenology in the framework of string theory.
Any theoretical infrastructure has to be able to support matter states made of fermions.
In what follows we thus consider the superstring formulation in $D = 10$ dimensions even when we focus only on its bosonic components.

It is however clear that low energy phenomena need to be explained by a $4$-dimensional theory in order to be comparable with other theoretical frameworks and experimental evidence.
In this section we briefly review for completeness the necessary tools to be able to reproduce consistent models capable of describing particle physics and beyond.
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,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,
\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 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.
They are defined as complex Ricci-flat Kähler manifolds $M$ of dimensions $2m$ and with holonomy \SU{m} (see for instance~\cite{Joyce:2000:CompactManifoldsSpecial,Joyce:2002:LecturesCalabiYauSpecial,Greene:1997:StringTheoryCalabiYau}).
More on this topic is also presented in~\Cref{part:deeplearning} of this thesis where we compute topological properties of a subset of \cy manifolds.


\subsubsection{Complex and Kähler Manifolds}

In general an \emph{almost complex structure} $J$ is a tensor such that $\tensor{J}{^a_b}\, \tensor{J}{^b_c} = - \tensor{\delta}{^a_c}$.
For any vector field $v_p \in \rT_p M$ defined in $p \in M$ we then define $(J v)^a = \tensor{J}{^a_b} v^b$, thus the tangent space $\rT_p M$ has the structure of a \emph{complex vector space}.
The tensor $J$ is called \emph{complex structure} if
\begin{equation}
  \liebraket{v_p}{w_p}
  +
  J
  \qty( \liebraket{J\, v_p}{w_p} + \liebraket{v_p}{J\, w_p} )
  -
  \liebraket{J\, v_p}{J\, w_p}
  =
  0
\end{equation}
for any $v_p,\, w_p \in \rT_p M$, where $\liebraket{\cdot}{\cdot}\colon\, \rT_p M \times \rT_p M \to \rT_p M$ is the usual Lie braket of vector fields.
A manifold $M$ is a \emph{complex} manifold if it is possible to define a complex structure $J$ on it.\footnotemark{}
\footnotetext{%
  Notice that a smooth function $f\colon\, M \to \C$ whose pushforward of $v_p \in \rT_p M$ is $f_{*\, p}\colon\, \rT_p M \to \rT_{f(p)} \C \simeq \C$ is called holomorphic if $(J\, f_{*\, p}( v_p ))^a = i ( f_{*\, p}( v_p ) )^a$ as such expression encodes the Cauchy-Riemann equations.
  Let in fact $f( x, y ) = f_1( x, y ) + i\, f_2( x, y )$, then the expression implies
  \begin{equation*}
    \begin{cases}
      \ipd{x} f_1\qty( x, y )
      & =
      \ipd{y} f_2\qty( x, y )
      \\
      \ipd{x} f_2\qty( x, y )
      & =
      -\ipd{y} f_1\qty( x, y )
    \end{cases}
    \quad
    \Rightarrow
    \quad
    \ipd{x} f\qty( x, y ) = -i \ipd{y} f\qty( x, y )
    \quad
    \Rightarrow
    \quad
    \ipd{\barz} f\qty( z, \barz ) = 0
    \quad
    \Rightarrow
    \quad
    f\qty( z, \barz ) \equiv f( z ).
  \end{equation*}
}

Let then $\qty(M,\, J,\, g)$ be a complex manifold with a Riemannian metric $g$.
The metric is \emph{Hermitian} if
\begin{equation}
  g( v_p, w_p ) = g( J\, v_p, J\, w_p )
  \quad
  \forall v_p,\, w_p \in \rT_p M
\end{equation}
In this case we can define a $(1, 1)$-form $\omega$ as
\begin{equation}
  \omega( v_p, w_p )
  =
  g( J\, v_p, w_p )
  \quad
  \forall v_p,\, w_p \in \rT_p M.
\end{equation}
$(M, J, g)$ is a \emph{Kähler} manifold if:
\begin{equation}
  \dd{\omega}
  =
  \qty( \pd + \bpd )
  \omega(z, \barz)
  =
  0,
  \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 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 $\mathrm{Hol}(g) \subseteq \U{m} \subset \OO{2m}$.


\subsubsection{Calabi-Yau Manifolds}

With the general definitions of the Kähler geometry we can now explicitly compute the conditions needed for a \cy manifold.
In local complex coordinates a Hermitian metric is such that
\begin{equation}
  g
  =
  g_{a \barb}\, \dd{z}^a \otimes \dd{\barz}^{\barb}
  +
  g_{\bara b}\, \dd{\barz}^{\bara} \otimes \dd{z}^b,
\end{equation}
thus the Kähler form becomes $\omega = i g_{a\barb}\, \dd{z}^a \wedge \dd{\barz}^{\barb}$.
Relation~\eqref{eq:cy:kaehler} translates into:
\begin{equation}
  \dd{\omega}
  =
  i\, \qty( \pd + \bpd )\, g_{a\barb}\, \dd{z}^a \wedge \dd{\barz}^{\barb}
  =
  0
  \quad
  \Leftrightarrow
  \quad
  \begin{cases}
    \ipd{z^c} g_{a\barb} & = \ipd{z^a} g_{c\barb}
    \\
    \ipd{\barz^c} g_{\bara b} & = \ipd{\barz^a} g_{\barc b}
  \end{cases}.
\end{equation}
The $(1,1)$-form $\omega$ can locally be written as $\omega = i\, \pd \bpd\, \phi( z, \barz )$ up to a constant.
This ultimately leads to
\begin{equation}
  g_{a\barb}
  =
  \pdv{\phi( z, \barz )}{z^a}{\barz^{\barb}}
  =
  \ipd{z^a} \ipd{\barz^b}\, \phi( z, \barz ),
\end{equation}
Since $\omega$ is the Kähler form then the Levi-Civita connection has only fully holomorphic and anti-holomorphic components:
\begin{equation}
  \tensor{\Gamma}{^a_{bc}}
  =
  \tensor{g}{^{a\bard}}\,
  \ipd{z^b}\,
  \tensor{g}{_{\bard c}},
  \qquad
  \tensor{\Gamma}{^{\bara}_{\barb\barc}}
  =
  \tensor{g}{^{\bara d}}\,
  \ipd{\barz^b}\,
  \tensor{g}{_{d\barc}}.
\end{equation}
As a consequence the Ricci tensor becomes
\begin{equation}
  \tensor{R}{_{\bara b}}
  =
  -
  \pdv{\tensor{\Gamma}{^{\barc}_{\bara\barc}}}{z^b}.
\end{equation}

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.


\subsubsection{Cohomology and Hodge Numbers}
\label{sec:cohomology_hodge}

\cy manifolds $M$ of complex dimension $m$ present geometric characteristics of general interest both in pure mathematics and string theory.
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 $\tomega$ are always defined up to an \emph{exact} term.
In fact:
\begin{equation}
  \dd{\tomega'_{(p)}} = \dd{\qty(\tomega_{(p)} + \dd{\teta_{(p-1)}})} = 0
  \label{eq:cy:closedform}
\end{equation}
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 \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
\begin{equation}
  H^{(p)}_{\mathrm{d}}( M, \R )
  =
  \bigoplus\limits_{p = r + s}\,
  H^{(r,s)}_{\bpd}( M, \C ).
\end{equation}
As in the case of Betti numbers, we can define the complex equivalents, the \emph{Hodge numbers}, $\hodge{r}{s} = \dim\limits_{\C} H^{(r,s)}_{\bpd}( M, \C )$ which count the number of harmonic $(r, s)$-forms on $M$.
Notice that in this case $\hodge{r}{s}$ is the complex dimension $\dim\limits_{\C}$ of the cohomology group.

For \cy manifolds it is possible to show that the \SU{m} holonomy of $g$ implies that the vector space of $(r, 0)$-forms is \C if $r = 0$ or $r = m$.
Therefore $\hodge{0}{0} = \hodge{m}{0} = 1$, while $\hodge{r}{0} = 0$ if $r \neq 0,\, m$.
Exploiting symmetries of the cohomology groups, Hodge numbers are usually collected in \emph{Hodge diamonds}.
In string theory we are ultimately interested in \cy manifolds of real dimensions $6$, thus we focus mainly on \cy $3$-folds (i.e.\ having $m = 3$).
The diamond in this case is
\begin{equation}
  \mqty{%
            &         &         & \hodge{0}{0} &         &         & 
    \\
            &         & \hodge{1}{0} &         & \hodge{0}{1} &         & 
    \\
            & \hodge{2}{0} &         & \hodge{1}{1} &         & \hodge{0}{2} & 
    \\
    \hodge{3}{0} &         & \hodge{2}{1} &         & \hodge{1}{2} &         & \hodge{0}{3}
    \\
            & \hodge{3}{1} &         & \hodge{2}{2} &         & \hodge{1}{3} & 
    \\
            &         & \hodge{3}{2} &         & \hodge{2}{3} &         & 
    \\
            &         &         & \hodge{3}{3} &         &         & 
  }
  \quad
  =
  \quad
  \mqty{%
            &         &         &    1    &         &         & 
    \\
            &         &    0    &         &    0    &         & 
    \\
            &    0    &         & \hodge{1}{1} &         &    0    & 
    \\
        1   &         & \hodge{2}{1} &         & \hodge{2}{1} &         &    1
    \\
            &    0    &         & \hodge{1}{1} &         &    0    & 
    \\
            &         &    0    &         &    0    &         & 
    \\
            &         &         &    1    &         &         & 
  },
\end{equation}
where we used $\hodge{r}{s} = h^{m-r, m-s}$ to stress the fact that the only independent Hodge numbers are $\hodge{1}{1}$ and $\hodge{2}{1}$ for $m = 3$.
These results will also be the starting point of~\Cref{part:deeplearning} in which the ability to predict the values of the Hodge numbers using \emph{artificial intelligence} is tested.


\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}.
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.


\subsubsection{Compactification of Closed Strings}

As a first approach to the definition of D-branes, consider the action~\eqref{eq:conf:polyakov}.
The variation of such action with respect to $\delta X$ leads to the equation of motion
\begin{equation}
  \partial_{\alpha} \partial^{\alpha} X^{\mu}( \tau, \sigma ) = 0
  \qquad
  \mu = 0, 1, \dots, D - 1,
  \label{eq:tduality:eom}
\end{equation}
and naturally to the \emph{Neumann} boundary conditions:\footnotemark{}
\footnotetext{%
  As~\cite{Polchinski:1996:TASILecturesDBranes} shows, \emph{Dirichlet} conditions can descend from T-duality which is introduced later.
}
\begin{equation}
  \eval{\ipd{\sigma} X^{\mu}( \tau, \sigma )}_{\sigma = 0}^{\sigma = \ell} = 0,
  \qquad
  \mu = 0, 1, \dots, D - 1.
  \label{eq:tduality:bc}
\end{equation}
Closed strings are such that $X^{\mu}( \tau, \sigma + \ell ) = X^{\mu}( \tau, \sigma )$.
The usual mode expansion in conformal coordinates $X^{\mu}( z, \barz ) = X( z ) + \barX( \barz )$ leads to
\begin{equation}
  \begin{split}
    X^{\mu}( z )
    & =
    x_0^{\mu}
    +
    i\, \sqrt{\frac{\ap}{2}}\,
    \qty(
      - \alpha_0^{\mu}\, \ln{z}
      + \sum\limits_{n \in \Z \setminus \{0\}} \frac{\alpha_n^{\mu}}{n} z^{-n}
    ),
    \\
    \barX^{\mu}( \barz )
    & =
    \barx_0^{\mu}
    +
    i\, \sqrt{\frac{\ap}{2}}\,
    \qty(
      - \balpha_0^{\mu}\, \ln{\barz}
      + \sum\limits_{n \in \Z \setminus \{0\}} \frac{\balpha_n^{\mu}}{n} \barz^{-n}
    ),
  \end{split}
  \label{eq:tduality:modes}
\end{equation}
where $\alpha_0^{\mu} = \balpha_0^{\mu}$ and $\ell = 2 \pi$.
When the string is free to move in the entire $D$-dimensional space, then the momentum of the center of mass is $p^{\mu} = \frac{1}{\sqrt{2 \ap}} ( \alpha_0^{\mu} + \balpha_0^{\mu} )$.

Now let
\begin{equation}
  \ccM^{1, D - 1} = \ccM^{1, D - 2} \otimes S^1( R ),
  \label{eq:tduality:compactification}
\end{equation}
where $S^1( R )$ is a compact $1$-dimensional circle of radius $R$ such that the boundary conditions for the compact coordinate are
\begin{equation}
  X^{D - 1}( z\, e^{2\pi i}, \barz\, e^{-2\pi i} )
  =
  X^{D - 1}( z, \barz ) + 2 \pi\, m\, R,
  \qquad
  m \in \Z.
  \label{eq:dbranes:winding}
\end{equation}
This is cast into
\begin{equation}
  \begin{split}
    \alpha_0^{D-1} + \balpha_0^{D-1} & = \sqrt{\frac{2}{\ap}}\, n\, \frac{\ap}{R},
    \qquad
    n \in \Z,
    \\
    \alpha_0^{D-1} - \balpha_0^{D-1} & = \sqrt{\frac{2}{\ap}}\, m\, R,
    \qquad
    m \in \Z,
  \end{split}
\end{equation}
respectively encoding the quantisation of the momentum for a compact coordinate and the \emph{winding} in the compact direction~\eqref{eq:dbranes:winding}.
We finally have
\begin{equation}
  \begin{split}
    \alpha_0^{D-1} &= \frac{1}{\sqrt{2 \ap}}\, \qty( n\, \frac{\ap}{R} + m\, R ),
    \\
    \balpha_0^{D-1} &= \frac{1}{\sqrt{2 \ap}}\, \qty( n\, \frac{\ap}{R} - m\, R ),
  \end{split}
\end{equation}

An interesting phenomenon involving these quantities appears when computing the mass spectrum of the theory.
From~\eqref{eq:conf:Texpansion} and~\eqref{eq:conf:bosonicstringT} we find
\begin{equation}
  \begin{split}
    L_0
    &=
    \frac{\ap}{2}\,
    \qty(
      \qty( \alpha_0^{D-1} )^2
      +
      \finitesum{i}{0}{D-2}\, \qty( \alpha_0^i )^2
      +
      \finitesum{n}{1}{+\infty}\, \qty( 2\, \alpha_{-n}^{\mu} \alpha_n^{\nu}\, \eta_{\mu\nu} + a )
    ),
    \\
    \barL_0
    &=
    \frac{\ap}{2}\,
    \qty(
      \qty( \balpha_0^{D-1} )^2
      +
      \finitesum{i}{0}{D-2}\, \qty( \balpha_0^i )^2
      +
      \finitesum{n}{1}{+\infty}\, \qty( 2\, \balpha_{-n}^{\mu} \balpha_n^{\nu}\, \eta_{\mu\nu} + a )
    ),
  \end{split}
\end{equation}
where $a$ is constant given by normal ordering, representing the zero point energy of the theory.
Imposing physical conditions~\eqref{eq:conf:physical} and the \emph{level matching} $(L_0 - \barL_0) \ket{\phi} = 0$ for closed strings, we find
\begin{equation}
  \begin{split}
    M^2
    & =
    \frac{1}{\qty(\ap)^2}\, \qty( n\, \frac{\ap}{R} + m\, R )^2
    +
    \frac{4}{\ap}\, \qty( \rN + a )
    \\
    & =
    \frac{1}{\qty(\ap)^2}\, \qty( n\, \frac{\ap}{R} - m\, R )^2
    +
    \frac{4}{\ap}\, \qty( \brN + a ),
  \end{split}
  \label{eq:dbranes:closedspectrum}
\end{equation}
where $\rN = \finitesum{n}{1}{+\infty}\, \alpha_{-n} \cdot \alpha_n$ 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.
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 $m$.
At the level of the modes this \emph{T-duality} acts by swapping the sign of the right zero-modes in the compact direction
\begin{equation}
  \alpha_0^{D-1} \stackrel{T}{\longmapsto} \alpha_0^{D-1},
  \qquad
  \balpha_0^{D-1} \stackrel{T}{\longmapsto} - \balpha_0^{D-1},
\end{equation}
defining the dual coordinate
\begin{equation}
  Y^{D-1}( z, \barz ) = Y^{D-1}( z ) + \barY^{D-1}( \barz ) =  X^{D-1}( z ) - \barX^{D-1}( \barz ).
  \label{eq:tduality:compactdirection}
\end{equation}



\subsubsection{D-branes from T-duality}

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 )
  =
  x_0^{\mu}
  -
  i\, \ap\, p^{\mu}\, \ln( z \barz )
  +
  i\, \sqrt{\frac{\ap}{2}}\,
  \sum\limits_{n \in \Z \setminus \{0\}}
  \frac{\alpha_n^{\mu}}{n} \qty( z^{-n} + \barz^{-n} )
\end{equation}
and $\ell = \pi$.

Under the compactification~\eqref{eq:tduality:compactification} open strings do not wind around the compact cycle.
Thus they do not present a quantum number $m$ as closed strings do.
When $R \to 0$ modes with non vanishing momentum (i.e.\ with $n \neq 0$) become infinitely massive:
\begin{equation}
  p^{D-1}
  =
  \frac{n}{R} \stackrel{R \to 0}{\longrightarrow} \infty.
\end{equation}
The behaviour is similar to the traditional field theory: the compactified dimension disappears and open string endpoints live in a $(D-1)$-dimensional hypersurface.
This is a consequence of the T-duality transformation applied on the compact direction.
In fact the original Neumann boundary condition~\eqref{eq:tduality:bc} becomes a \emph{Dirichlet condition} for $Y^{D-1}$ defined as in~\eqref{eq:tduality:compactdirection}:
\begin{equation}
  \begin{split}
    \eval{\ipd{\sigma} X^{D-1}( \tau, \sigma )}^{\sigma = \pi}_{\sigma = 0}
    & =
    \eval{\ipd{\sigma} X^{D-1}( e^{\tau_E + i \sigma} ) + \ipd{\sigma} \barX^{D-1}( e^{\tau_E - i \sigma} )}^{\sigma = \pi}_{\sigma = 0}
    \\
    & =
    \eval{i\, \ipd{\xi} X^{D-1}( e^{\xi} ) - i\, \ipd{\bxi} \barX^{D-1}( e^{\bxi} )}^{\Im \xi = \pi}_{\Im \xi = 0}
    \\
    & =
    \eval{i\, \ipd{\tau_E} Y^{D-1}( e^{\tau_E + i \sigma} ) + i\, \ipd{\tau_E} \barY^{D-1}( e^{\tau_E - i \sigma} )}^{\sigma = \pi}_{\sigma = 0}
    \\
    & =
    \eval{\ipd{\tau} Y^{D-1}( \tau, \sigma )}^{\sigma = \pi}_{\sigma = 0}
    \\
    & =
    0.
  \end{split}
\end{equation}
The coordinate of the endpoint in the compact direction is therefore fixed and constrained on a hypersurface called \emph{Dp-brane}, where $p+1$ is the dimension of the surface (in this case $p = D - 2$):
\begin{equation}
  \begin{split}
    Y^{D-1}( \tau, \pi ) - Y^{D-1}( \tau, 0 )
    & =
    \finiteint{\sigma}{0}{\pi} \ipd{\sigma} Y^{D-1}( \tau, \sigma )
    \\
    & =
    i\, \finiteint{\sigma}{0}{\pi} \ipd{\tau} X^{D-1}( \tau, \sigma)
    \\
    & =
    2 \pi \ap p^{D-1}
    \\
    & = 
    2 \pi n\, \frac{\ap}{R}
    \\
    & =
    2 \pi n\, R'.
  \end{split}
\end{equation}
The only difference in the position of the endpoints can only be a multiple of the radius of the compactified dimension.
Otherwise they lie on the same hypersurface.
The procedure can be generalised to $p$ coordinates, constraining the string to live on a $(D - p - 1)$-brane.

This geometric interpretation of the Dirichlet branes and boundary conditions is the basis for the definition of more complex scenarios in which multiple D-branes are inserted in spacetime.
D-branes are however much more than mathematical entities.
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{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.

As seen in the previous section, D-branes introduce preferred directions of motion by restricting the hypersurface on which the open string endpoints live.
Specifically a Dp-brane breaks the original \ISO{1,\, D-1} symmetry to $\ISO{1,\, p} \otimes \SO{D - 1 - p}$.\footnotemark{}
\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,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,
\end{equation}
where $X^{\pm} = \frac{1}{\sqrt{2}} (X^0 \pm X^{D-1})$.
The vanishing of the stress-energy tensor fixes the oscillators in $X^-$ in terms of the physical transverse modes.
The mass shell condition for open strings then becomes:\footnotemark{}
\footnotetext{%
  The constant $a$ in~\eqref{eq:dbranes:closedspectrum} takes here the value $-1$ from the imposition of the canonical commutation relations and a $\zeta$-regularisation.
}
\begin{equation}
  M^2 = \frac{1}{\ap} \qty( N - 1 ).
\end{equation}

Consider for a moment bosonic string theory and define the usual vacuum as
\begin{equation}
  \alpha_n^i \regvacuum = 0,
  \qquad
  n \ge 0,
  \qquad
  i = 1, 2, \dots, D - 2,
\end{equation}
we find that at the massless level we have a single \U{1} gauge field in the representation of the Little Group \SO{D-2}:
\begin{equation}
  \cA^i
  \qquad
  \rightarrow
  \qquad
  \alpha_{-1}^i \regvacuum.
\end{equation}
The introduction of a Dp-brane however breaks the Lorentz invariance.
Thus the gauge field in the original theory is split into
\begin{equation}
  \begin{split}
    \cA^A
    \qquad
    & \rightarrow
    \qquad
    \alpha_{-1}^A \regvacuum,
    \qquad
    A = 0, 1, \dots, p,
    \\
    \cA^a
    \qquad
    & \rightarrow
    \qquad
    \alpha_{-1}^a \regvacuum,
    \qquad
    a = 1, 2, \dots, D - 1 -p.
  \end{split}
\end{equation}
In the last expression $\cA^A$ forms a representation of the Little Group \SO{p-2} and as such it is a vector gauge field in $p$ dimensions.
The field $\cA^a$ forms a vector representation of the group \SO{D-1-p} and from the point of view of the Lorentz group they are $D - 1 - p$ scalars in the light spectrum.

\begin{figure}[tbp]
  \centering
  \begin{subfigure}[b]{0.45\linewidth}
    \centering
    \import{tikz}{chanpaton.pgf}
    \caption{Chan-Paton factors labelling strings.}
  \end{subfigure}
  \hfill
  \begin{subfigure}[b]{0.45\linewidth}
    \centering
    \import{tikz}{quark.pgf}
    \caption{Naive model of left handed massive quarks.}
  \end{subfigure}
  \caption{Strings attached to different D-branes.}
  \label{fig:dbranes:chanpaton}
\end{figure}

It is also possible to add non dynamical degrees of freedom (\dof) to the open string endpoints.
They are known as \emph{Chan-Paton factors}~\cite{Paton:1969:GeneralizedVenezianoModel}.
They have no dynamics and do not spoil Poincaré or conformal invariance in the action of the string.
Each state can then be labelled by $i$ and $j$ running from $1$ to $N$.
Matrices $\tensor{\lambda}{^a_{ij}}$ thus form a basis for expanding wave functions and states:
\begin{equation}
  \ket{n;\, a} = \sum\limits_{i,\, j = 1}^N \ket{n;\, i, j}\, \tensor{\lambda}{^a_i_j}.
\end{equation}
In general Chan-Paton factors label the D-brane on which the endpoint of the string lives as in the left of~\Cref{fig:dbranes:chanpaton}.
Notice that strings stretching across different D-branes present an additional term in the mass shell condition proportional to the distance between the hypersurfaces: fields built using strings with Chan-Paton factors $\tensor{\lambda}{^a_i_j}$ for which $i \neq j$ will therefore be massive.
However when $N$ D-branes coincide in space and form a stack their mass vanishes again: it is then possible to organise the $N^2$ resulting massless fields in a representation of the gauge group \U{N}, thus promoting the symmetry $\bigotimes\limits_{a = 1}^N \rU_a( 1 )$ of $N$ separate D-branes to a larger gauge group.
It is also possible to show that in the field theory limit the resulting gauge theory is a Super Yang-Mills gauge theory.

Eventually the massless spectrum of $N$ coincident Dp-branes is formed by \U{N} gauge bosons in the adjoint representation, $N^2 \times (D - 1 - p)$ scalars and $N^2$ sets of $(p+1)$-dimensional fermions~\cite{Uranga:2005:TASILecturesString}.
These are the basic building blocks for a consistent string phenomenology involving both gauge bosons and matter.


\subsubsection{Standard Model Scenarios}

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{Ibanez:2012:StringTheoryParticle,Aldazabal:2000:DBranesSingularitiesBottomUp}.

For instance, in the low energy limit it is possible to build a gauge theory of the strong force using a stack of $3$ coincident D-branes and an electroweak sector using $2$ D-branes.
These stacks would separately lead to a $\U{3} \times \U{2}$ gauge theory.
It would however be a theory of pure force, without matter content.
Moreover we should also worry about the extra \U{1} groups appearing: these need careful consideration but go beyond the necessary analysis presented in what follows.

Matter fields are fermions transforming in the bi-fundamental representation $\qty(\vb{N}, \vb{M})$ of the \sm gauge group~\eqref{eq:intro:smgroup}.
For example left handed quarks in the \sm transform under the $\qty(\vb{3}, \vb{2})$ representation of the group $\SU{3}_C \otimes \SU{2}_L$.
This is realised in string theory by a string stretched across two stacks of $3$ and $2$ D-branes as in the right of~\Cref{fig:dbranes:chanpaton}.
The fermion would then be characterised by the charge under the gauge bosons living on the D-branes.
The corresponding anti-particle would then be modelled as a string oriented in the opposite direction.
Things get complicated when introducing also left handed leptons transforming in the $\qty(\vb{1}, \vb{2})$ representation: they cannot have endpoints on the same stack of D-branes as quarks since they do not have colour charge.
We therefore need to introduce more D-branes to account for all the possible combinations.

An additional issue comes from the requirement of chirality.
Strings stretched across D-branes are naturally massive but, in the field theory limit, a mass term would mix different chiralities.
We thus need to include a symmetry preserving mechanism for generating the mass of fermions.
In string theory there are ways to deal with the requirement.
These range from D-branes located at singular points of orbifolds to D-branes intersecting at angles~\cite{Uranga:2003:ChiralFourdimensionalString,Aldazabal:2000:DBranesSingularitiesBottomUp,Zwiebach:2009:FirstCourseString}.
In this manuscript we focus on intersecting D6-branes filling the $4$-dimensional spacetime and whose additional $3$ dimensions are embedded in a \cy 3-fold (e.g.\ as lines in a factorised torus $T^6 = T^2 \times T^2 \times T^2$).
This D-brane geometry supports chiral fermion states at their intersection: while some of the modes of the stretched string become indeed massive, the spectrum of the fields is proportional to combinations of the angles and some of the modes can remain massless.
The light spectrum is thus composed of the desired matter content alongside with other particles arising from the string compactification.

\begin{figure}[tbp]
  \centering
  \import{tikz}{smbranes.pgf}
  \caption{%
    Example of \sm-like construction using intersecting D-branes with the indications of the hypercharge $Y$.
    Perpendicular angles are only a matter of convenience: they are in principal arbitrary.
  }
  \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,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 $Y$.

Physics in four dimensions is eventually recovered by compactifying the extra-dimensions of the superstring.\footnotemark{}
\footnotetext{%
  We specifically reviewed particle physics interactions.
  Gravitational interactions in general remain untouched by D-branes constructions and still propagate in $10$-dimensional spacetime unless the internal space is compact.
}
Fermions localised at the intersection of the D-branes are however naturally $4$-dimensional as they only propagate in the non compact Minkowski space $\ccM^{1,3}$.
The presence of compactified dimensions however leads to phenomena such as \emph{family replications} of the fermions.
With accurate calibration it is in fact possible to recover the quark and lepton families in the \sm.
Consider for example the simple \cy factorised manifold $T^6 = T^2 \times T^2 \times T^2$ and introduce stacks of D6-branes as lines in each of the bi-torus.
Even though the lines might never intersect on a plane, they can have points in common on a torus due to the identifications~\cite{Zwiebach:2009:FirstCourseString}.
Since each intersections supports a different set of fermions with different spectrum, the angles of the intersecting branes can be calibrated to reproduce the mass separation of the families of quarks and leptons in the \sm.


% vim: ft=tex
