Add explanations and fix typos

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

thesis.tex

@@ -41,7 +41,7 @@
   \\
   I.N.F.N.\ -- sezione di Torino
 }
-\date{15th December 2020}
+\date{18th December 2020}
 
 \usetikzlibrary{decorations.markings}
 \usetikzlibrary{decorations.pathmorphing}
@@ -378,21 +378,48 @@
     % \pause
 
     \begin{block}{T-duality}
-      \textbf{Dirichlet b.c.} consequence of \textbf{T-duality} on $p$ directions:
+      \only<2>{%
-      \begin{equation*}
+        Consider \textbf{closed strings} on $\mathscr{M}^{1,D-1} = \mathscr{M}^{1,D-2} \otimes \mathrm{S}^1( R )$:
-        \overline{X}( z ) \mapsto - \overline{X}( z )
+        \begin{equation*}
-        \quad
+          \begin{split}
-        \Rightarrow
+            \begin{cases}
-        \quad
+              \upalpha_0^{D-1} & = \frac{1}{\sqrt{2 \upalpha'}} \qty( n \frac{\upalpha'}{R} + m R )
-        \eval{\partial_{\upsigma} X^i\qty( \uptau, \upsigma )}_{\upsigma = 0}^{\upsigma = \ell} = 0
+              \\
-        \quad
+              \widetilde{\upalpha}_0^{D-1} & = \frac{1}{\sqrt{2 \upalpha'}} \qty( n \frac{\upalpha'}{R} - m R )
-        \stackrel{T-duality}{\longrightarrow}
+            \end{cases}
-        \quad
+            \quad
-        \eval{\partial_{\uptau} \widetilde{X}^i\qty( \uptau, \upsigma )}_{\upsigma = 0}^{\upsigma = \ell} = 0
+            \Rightarrow
-      \end{equation*}
+            \quad
-      thus \textbf{open strings} can be \textbf{constrained} to $D(D - p - 1)$-branes.
+            M^2
-      \hfill
+            =
-      \cite{Polchinski (1995, 1996)}
+            -p^{\upmu} p_{\upmu}
+            & =
+            \frac{2}{\upalpha'} \qty( \upalpha_0^{D-1} )^2 + \frac{4}{\upalpha'} \qty( \mathrm{N} + a )
+            \\
+            & =
+            \frac{2}{\upalpha'} \qty( \widetilde{\upalpha}_0^{D-1} )^2 + \frac{4}{\upalpha'} \qty( \widetilde{\mathrm{N}} + a )
+          \end{split}
+        \end{equation*}
+        \vfill
+      }
+      \only<3->{%
+        \textbf{Dirichlet b.c.} consequence of \textbf{T-duality} on $p$ directions:
+        \begin{equation*}
+          \overline{X}( z ) \mapsto - \overline{X}( z )
+          \quad
+          \Rightarrow
+          \quad
+          \eval{\partial_{\upsigma} X^i\qty( \uptau, \upsigma )}_{\upsigma = 0}^{\upsigma = \ell} = 0
+          \quad
+          \stackrel{T-duality}{\longrightarrow}
+          \quad
+          \eval{\partial_{\uptau} \widetilde{X}^i\qty( \uptau, \upsigma )}_{\upsigma = 0}^{\upsigma = \ell} = 0
+        \end{equation*}
+        thus \textbf{open strings} can be \textbf{constrained} to $D(D - p - 1)$-branes.
+        \hfill
+        \cite{Polchinski (1995, 1996)}
+        \vfill
+      }
     \end{block}
   \end{frame}
 
@@ -744,9 +771,9 @@
 
               \item classical action \textbf{larger} than factorised case
             \end{itemize}
+            \hspace{0.65\columnwidth}\cite{RF, Pesando (2019)}
           \end{column}
         \end{columns}
-        \vfill
       }
     \end{block}
   \end{frame}
@@ -975,6 +1002,7 @@
       +
       \frac{1}{2} \qty( \sum\limits_{t = 1}^N \frac{n_{(t)} + \frac{\upepsilon_{(t)}}{2}}{z - x_{(t)}} )^2
     \end{equation*}
+    \hfill\cite{RF, Pesando (2019)}
 
     \pause
 
@@ -1144,7 +1172,7 @@
     Use \textbf{time-dependent orbifolds} to model \textbf{space-like singularities}:
     
     \begin{center}
-      divergent \highlight{closed string} aplitudes
+      divergent \highlight{closed string} amplitudes
       $\Rightarrow$
       gravitational backreaction?
     \end{center}
@@ -1257,6 +1285,10 @@
     So far:
     \begin{itemize}
       \item field theory presents \textbf{divergences} (even sQED $\rightarrow$ eikonal?)
+
+        \pause
+
+      \item obvious ways to regularise (Wilson lines, higher derivative couplings, etc.) \textbf{do not work}
         
         \pause
 
@@ -1280,7 +1312,7 @@
         \colon
         \qty(%
           \frac{i}{\sqrt{2 \upalpha'}}\,
-          \upxi \cdot \partial^2_x X( x,\, x )
+          \upxi_{\upalpha} \partial^2_x X^{\upalpha}( x,\, x )
           +
           \qty( \frac{i}{\sqrt{2 \upalpha'}} )^2\,
           S_{\upalpha\upbeta}
@@ -1296,7 +1328,7 @@
 
     \begin{center}
       \it
-      string theory cannot do \textbf{better than field theory} (EFT) if the latter \textbf{does not exist} (even a Wilson line around $z$ does not prevent such behaviour)
+      string theory cannot do \textbf{better than field theory} (EFT) if the latter \textbf{does not exist}
     \end{center}
   \end{frame}
 
@@ -1361,12 +1393,14 @@
     \vspace{2em}
     \begin{center}
       \it
-      spacetime singularities are \textbf{hidden into contact terms} and interactions with \textbf{massive states} (the gravitational eikonal deals with massless interactions)
+      divergences are \textbf{hidden into contact terms} and interactions with \textbf{massive states} (the gravitational eikonal deals with massless interactions)
     \end{center}
 
     \begin{tikzpicture}[remember picture, overlay]
       \draw[line width=4pt, red] (0em, 4.5em) rectangle (40em, 1em);
     \end{tikzpicture}
+
+    \hfill\cite{Arduino, RF, Pesando (2020)}
   \end{frame}
 
 
@@ -1453,13 +1487,15 @@
     \pause
 
     \begin{block}{Machine Learning Approach}
-      What is $\mathscr{R}$?
+      What is $\mathscr{R}$ in \textbf{machine learning} approach?
       \begin{equation*}
         \mathscr{R}( M ) \longrightarrow \mathscr{R}_n( M;\, w )
         \qquad
         \text{s.t.}
         \qquad
-        \lim\limits_{n \to \infty} \abs{\mathscr{R}( M ) - \mathscr{R}_n( M;\, w )} = 0
+        \exists n > M > 0 \mid \mathcal{L}\qty(\mathscr{R}( M ),\, \mathscr{R}_n( M;\, w )) < \upepsilon
+        \quad
+        \forall \upepsilon > 0
       \end{equation*}
     \end{block}
   \end{frame}
@@ -1532,6 +1568,8 @@
       Hodge numbers
     \end{center}
 
+    \hfill\cite{Ruehle (2020); Erbin, RF (2020)}
+
     \pause
 
     \begin{columns}
@@ -1598,13 +1636,14 @@
       \begin{column}{0.5\linewidth}
         \centering
         \textbf{Configuration Matrix Only}
-        \includegraphics[width=0.8\columnwidth, trim={0 0 3.3in 0}, clip]{img/cicy_matrix_plots.pdf}
+        \includegraphics[width=0.75\columnwidth, trim={0 0 3.3in 0}, clip]{img/cicy_matrix_plots.pdf}
       \end{column}
       \hfill\pause
       \begin{column}{0.5\linewidth}
         \centering
         \textbf{Best Training Set}
-        \includegraphics[width=0.8\columnwidth, trim={0 0 3.3in 0}, clip]{img/cicy_best_plots.pdf}
+        \cite{Erbin, RF (2020)}
+        \includegraphics[width=0.75\columnwidth, trim={0 0 3.3in 0}, clip]{img/cicy_best_plots.pdf}
       \end{column}
     \end{columns}
   \end{frame}
@@ -1699,7 +1738,9 @@
       \begin{column}{0.5\linewidth}
         \centering
         \textbf{Best Training Set}
-        \includegraphics[width=\columnwidth]{img/cicy_best_plots.pdf}
+        \cite{Erbin, RF (2020)}
+        \only<1>{\includegraphics[width=0.8\columnwidth, trim={0 0 1.65in 0}, clip]{img/cicy_best_plots.pdf}}
+        \only<2->{\includegraphics[width=\columnwidth]{img/cicy_best_plots.pdf}}
       \end{column}
       \hfill
       \begin{column}{0.5\linewidth}