Corrections and adjustments to the introduction

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

sec/part1/introduction.tex

@@ -12,7 +12,7 @@ in order to reproduce known results.
 For instance a good string theory could provide a unified framework by predicting the existence of a larger gauge group containing the \sm as a subset.
 In what follows we deal with the definition of mathematical tools to compute amplitudes to be used in phenomenological calculations related to the study of particles.
 
-In this introduction we present instruments and frameworks used throughout the manuscript as many aspects are strongly connected and their definitions are interdependent.
+In this introduction we present instruments used throughout the manuscript as many aspects are strongly connected and their definitions are interdependent.
 In particular we recall some results on the symmetries of string theory and how to recover a realistic description of $4$-dimensional physics.
 
 
@@ -20,7 +20,7 @@ In particular we recall some results on the symmetries of string theory and how
 
 Strings are extended one-dimensional objects.
 They are curves in spacetime parametrized by a coordinate $\sigma \in \qty[ 0, \ell ]$.
-When propagating they span a two-dimensional surface, the \emph{worldsheet}, described by the position of the string at given values of $\sigma$ at a time $\tau$, i.e.\ $X^{\mu}\qty(\tau, \sigma)$ where $\mu = 0, 1, \dots, D - 1$ indexes the coordinates.
+When propagating they span a two-dimensional surface, the \emph{worldsheet}, described by the position of the string at given values of $\sigma$ at a time $\tau$, i.e.\ $X^{\mu}\qty(\tau, \sigma)$ where $\mu = 0,\, 1,\, \dots,\, D - 1$ indexes the coordinates.
 Such surface can have different topologies according to the nature of the object propagating in spacetime: strings can be \emph{closed} if $X^{\mu}\qty(\tau, 0) = X^{\mu}\qty(\tau, \ell)$ or \emph{open} if the endpoints in $\sigma = 0$ and $\sigma = \ell$ do not coincide.
 
 
@@ -91,7 +91,7 @@ implies
 \end{equation}
 where $S_{NG}[X]$ is the Nambu--Goto action of the classical string, and $\dotX = \ipd{\tau} X$ and $\pX = \ipd{\sigma} X$.
 
-The symmetries of $S_P\qty[\gamma,\, X]$ are keys to the success of the string theory framework~\cite{Polchinski:1998:StringTheoryIntroduction}.
+The symmetries of $S_P\qty[\gamma,\, X]$ are key to the success of the string theory framework~\cite{Polchinski:1998:StringTheoryIntroduction}.
 Specifically~\eqref{eq:conf:polyakov} displays symmetries under:
 \begin{itemize}
   \item $D$-dimensional Poincaré transformations
@@ -1159,7 +1159,7 @@ In fact the original Neumann boundary condition~\eqref{eq:tduality:bc} becomes a
     \eval{i\, \ipd{\tau_E} Y^{D-1}( e^{\tau_E + i \sigma} ) + i\, \ipd{\tau_E} \barY^{D-1}( e^{\tau_E - i \sigma} )}^{\sigma = \pi}_{\sigma = 0}
     \\
     & =
-    \eval{i\, \ipd{\tau} Y^{D-1}( \tau, \sigma )}^{\sigma = \pi}_{\sigma = 0}
+    \eval{\ipd{\tau} Y^{D-1}( \tau, \sigma )}^{\sigma = \pi}_{\sigma = 0}
     \\
     & =
     0.
@@ -1248,7 +1248,7 @@ Thus the gauge field in the original theory is split into
     \qquad
     \alpha_{-1}^A \regvacuum,
     \qquad
-    A = 1, \dots, p - 2,
+    A = 0, 1, \dots, p,
     \\
     \cA^a
     \qquad
@@ -1347,7 +1347,7 @@ Physics in four dimensions is eventually recovered by compactifying the extra-di
 Fermions localised at the intersection of the D-branes are however naturally $4$-dimensional as they only propagate in the non compact Minkowski space $\ccM^{1,3}$.
 The presence of compactified dimensions however leads to phenomena such as \emph{family replications} of the fermions.
 With accurate calibration it is in fact possible to recover the quark and lepton families in the \sm.
-Consider for example the simple \cy factorised manifold $T^6 = T^2 \times T^2 \times T^2$ and introduce stacks of D6-branes as lines in each of the bi-tori.
+Consider for example the simple \cy factorised manifold $T^6 = T^2 \times T^2 \times T^2$ and introduce stacks of D6-branes as lines in each of the bi-torus.
 Even though the lines might never intersect on a plane, they can have points in common on a torus due to the identifications~\cite{Zwiebach:2009:FirstCourseString}.
 Since each intersections supports a different set of fermions with different spectrum, the angles of the intersecting branes can be calibrated to reproduce the mass separation of the families of quarks and leptons in the \sm.