From 86905c9434af9ef18100c68bc7a037837e5ade0a Mon Sep 17 00:00:00 2001 From: Riccardo Finotello Date: Thu, 3 Dec 2020 16:49:07 +0100 Subject: [PATCH] Add explanations and fix typos Signed-off-by: Riccardo Finotello --- thesis.tex | 93 +++++++++++++++++++++++++++++++++++++++--------------- 1 file changed, 67 insertions(+), 26 deletions(-) diff --git a/thesis.tex b/thesis.tex index a05b582..3e0c285 100644 --- a/thesis.tex +++ b/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: - \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} @@ -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}