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Add explanations and fix typos

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Signed-off-by: Riccardo Finotello <riccardo.finotello@gmail.com>
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Riccardo
2020-12-03 16:49:07 +01:00
parent 798a7c5c3e
commit 86905c9434
1 changed files with 67 additions and 26 deletions
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93
thesis.tex
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@@ -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:
\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)}
\only<2>{%
Consider \textbf{closed strings} on $\mathscr{M}^{1,D-1} = \mathscr{M}^{1,D-2} \otimes \mathrm{S}^1( R )$:
\begin{equation*}
\begin{split}
\begin{cases}
\upalpha_0^{D-1} & = \frac{1}{\sqrt{2 \upalpha'}} \qty( n \frac{\upalpha'}{R} + m R )
\\
\widetilde{\upalpha}_0^{D-1} & = \frac{1}{\sqrt{2 \upalpha'}} \qty( n \frac{\upalpha'}{R} - m R )
\end{cases}
\quad
\Rightarrow
\quad
M^2
=
-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}
@@ -1260,6 +1288,10 @@
\pause
\item obvious ways to regularise (Wilson lines, higher derivative couplings, etc.) \textbf{do not work}
\pause
\item divergences are \textbf{not (only) gravitational}
\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}