From e2f4bae1560a06dd4a65618f6fa27653a7bd9df9 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.pdf | Bin 15335700 -> 15348791 bytes thesis.tex | 93 ++++++++++++++++++++++++++++++++++++++--------------- 2 files changed, 67 insertions(+), 26 deletions(-) diff --git a/thesis.pdf b/thesis.pdf index d2784705a44600fc1462d4ece607860610920124..cc1cac4782713a5db301a5d76c796237db9e00be 100644 GIT binary patch delta 358866 zcmaIdb95w8+bH^AV%v5yu`#i2O>EohI5Y7?9ox3;OzdQ0+xG4EJLjJJ-E-GHYwut8 zvr*OcN7s6KS5+qkm%_9hm%`LWK!Q6Z{({u|wrZh_2_!=`#|U|YjCg|t2Y~?jPy7u8 z3IrMi1_Txa4g?+q0R#~Q2?QAg1q2lY4Fnwo0|XNU3j`Yk2Lu-c4+I~C0E7^P2!t4f z1cVfX41^qn0)!HT3WOSj280%b4ul?r0pvRfBM1`+GYAU^D+n71I|v5|CkPh^HwX_1 zF9;t9KZpRx4-i2RArN5@5fD)jF%WSO2@pvTDG+H884y_zIS_de1rS9LB@ks06%bVr zH4t?W4G>KbEf8%G9S~iRpCEc5`XB}%h9E{D#vmpjrXXe@<{%azmLOIwZ;;lvG_baZ 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z6D8Z&x;$TD^wAYk2E(~EmD{TW!-)$%&0;vuk#_g$lPlbBB&pikA}?VSs*HqBvIkku z+Ark8SAnRoAArS^#kncYmkvwz8MI}o$l2-)+nJny3*1;<><@+x}r!o?fp70Mg6p1py2Ypl)=OE0O22=DZJ5TD(Se%OurxXa!2=$L(7#i)VP-E>^1f3~K1 zA5gsWzS|noyGD@{% + 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}