diff --git a/thesis.tex b/thesis.tex index 872b9f2..e8fa25f 100644 --- a/thesis.tex +++ b/thesis.tex @@ -56,6 +56,7 @@ } \newcommand{\highlight}[1]{\fcolorbox{yellow}{yellow!20}{#1}} +\renewcommand{\cite}[1]{{\tiny{\fcolorbox{red}{red!10}{[#1]}}}} \newcommand{\firstlogo}{img/unito} \newcommand{\thefirstlogo}{% @@ -304,6 +305,8 @@ ) \end{equation*} where $\uplambda_b = 2$ and $\uplambda_c = -1$, and $\uplambda_{\upbeta} = \frac{3}{2}$ and $\uplambda_{\upgamma} = -\frac{1}{2}$. + \hfill + \cite{Friedan, Martinec, Shenker (1986)} \end{block} \pause @@ -338,6 +341,7 @@ \end{column} \begin{tikzpicture}[remember picture, overlay] + \node[anchor=base] at (8em,-3.3em) {\cite{code in Hanson (1994)}}; \node[anchor=base] at (3em,-3.75em) {\includegraphics[width=0.3\linewidth]{img/cy}}; \end{tikzpicture} \begin{column}{0.3\linewidth} @@ -411,6 +415,8 @@ \forall i = 1, 2,\, \dots,\, p \end{equation*} thus \textbf{open strings} can be \textbf{constrained} to \highlight{$D(D - p - 1)$-branes.} + \hfill + \cite{Polchinski (1995, 1996)} \end{frame} \begin{frame}{D-branes and Open Strings} @@ -418,7 +424,7 @@ \pause - \begin{equationblock}{Massless Spectrum (\emph{irrep} of \textbf{little group} $\mathrm{SO}( D - 2 )$} + \begin{equationblock}{Massless Spectrum (\emph{irrep} of \textbf{little group} $\mathrm{SO}( D - 2 )$)} \begin{equation*} \mathcal{A}^{\upmu} \quad \leftrightarrow \quad @@ -456,6 +462,7 @@ \begin{column}{0.5\linewidth} Introducing \textbf{Chan--Paton factors} $\tensor{\uplambda}{^r_{ij}}$, when branes are \textbf{coincident}: + \hfill\cite{Chan, Paton (1969)} \begin{equation*} \bigoplus\limits_{r = 1}^N \mathrm{U}_r(1) \quad @@ -472,6 +479,7 @@ \begin{frame}{Standard Model-like Scenarios} \centering \resizebox{0.8\linewidth}{!}{\import{img}{smbranes.pgf}} + \hfill\cite{Zwiebach (2009)} \end{frame} @@ -498,12 +506,12 @@ \pause \begin{columns} - \begin{column}{0.5\linewidth} + \begin{column}{0.3\linewidth} \centering - \resizebox{0.5\columnwidth}{!}{\import{img}{branesangles.pgf}} + \resizebox{0.8\columnwidth}{!}{\import{img}{branesangles.pgf}} \end{column} - \begin{column}{0.5\linewidth} + \begin{column}{0.7\linewidth} D-branes in \textbf{factorised} internal space: \begin{itemize} \item \textbf{embedded as lines} in $\mathds{R}^2 \times \mathds{R}^2 \times \mathds{R}^2$ @@ -512,6 +520,8 @@ \item $S_{E\, (\text{cl})}\qty( \qty{ x_{(t)},\, \mathrm{M}_{(t)} }_{1 \le t \le N_B} ) \sim \text{Area}\qty( \qty{ f_{(t)},\, \mathrm{R}_{(t)} }_{1 \le t \le N_B} )$ \end{itemize} + + \hfill\cite{Cremades, Ibanez, Marchesano (2003); Pesando (2012)} \end{column} \end{columns} \end{frame} @@ -813,7 +823,9 @@ % \mathcal{T}_{\uptau\uptau}( \uptau, \upsigma ) % ) = - 0 \Leftrightarrow \uptau \in \qty( \uptau_{(t)},\, \uptau_{(t-1)} ) + 0 + \quad \Leftrightarrow \quad + \uptau \in \qty( \uptau_{(t)},\, \uptau_{(t-1)} ) \\ \dot{\mathrm{P}}( \uptau ) & @@ -1082,6 +1094,7 @@ \highlight{time-dependent orbifold models} \end{center} + \hfill\cite{Craps, Kutasov, Rajesh (2002); Liu, Moore, Seiberg (2002)} \end{column} \end{columns} \end{frame} @@ -1097,9 +1110,9 @@ \item (Lie) group $G$ - \item \emph{stabilizer} $G_p = \qty{g \in G \mid gp = p \in M}$ + \item \emph{stabilizer}: $G_p = \qty{g \in G \mid gp = p \in M}$ - \item \emph{orbit} $Gp = \qty{gp \in M \mid g \in G}$ + \item \emph{orbit}: $Gp = \qty{gp \in M \mid g \in G}$ \item charts $\upphi = \uppi \circ \mathscr{P}$ where: @@ -1136,12 +1149,13 @@ \pause + \vspace{2em} \begin{center} - time-dependent orbifolds + Use \textbf{time-dependent orbifolds} to model singularities in time \end{center} \begin{tikzpicture}[remember picture, overlay] - \draw[line width=4pt, red] (13em,3.5em) rectangle (27em, 1em); + \draw[line width=4pt, red] (5em,3.5em) rectangle (35em, 1em); \end{tikzpicture} \end{frame} @@ -1253,26 +1267,30 @@ \begin{center} \it - most terms \textbf{do not converge} and cannot be recovered even with a \textbf{distributional interpretions} due to the term $\propto u^{-1}$ in the exponentatial + most terms \textbf{do not converge} due to \textbf{isolated zeros} ($l_{(*)} \equiv 0$) and cannot be recovered even with a \textbf{distributional interpretions} due to the term $\propto u^{-1}$ in the exponentatial \end{center} \end{frame} \begin{frame}{String and Field Theory} So far: \begin{itemize} - \item field theory presents \textbf{divergences} + \item field theory presents \textbf{divergences} (even sQED $\rightarrow$ eikonal?) \pause - \item issues are \textbf{still present} in sQED (eikonal?) + \item divergences are \textbf{not (only) gravitational} \pause - \item divergences are \textbf{not (only) gravitational} + \item \textbf{vanishing volume} in phase space of the compact direction is responsible for the divergence \end{itemize} \pause + What about \highlight{string theory?} + + \pause + \begin{equationblock}{Massive String States} \begin{equation*} V_M\qty( x;\, k,\, S,\, \upxi ) @@ -1410,6 +1428,7 @@ p_a\qty( \uplambda Z^0,\, \dots,\, \uplambda Z^n ) & = \uplambda^a p_a\qty( Z^0,\, \dots,\, Z^n ) \end{cases} \end{equation*} + \hfill\cite{Green, Hübsch (1987); Hübsch (1992)} \end{block} \end{frame} @@ -1452,7 +1471,7 @@ \qquad \text{s.t.} \qquad - \lim\limits_{n \to \infty} f( M;\, w ) = \lim\limits_{n \to \infty} \abs{\mathscr{R}( M ) - \mathscr{R}_n( M;\, w )} = 0 + \lim\limits_{n \to \infty} \abs{\mathscr{R}( M ) - \mathscr{R}_n( M;\, w )} = 0 \end{equation*} \end{block} \end{frame} @@ -1541,18 +1560,22 @@ \begin{frame}{Machine Learning} \centering \includegraphics[width=0.85\linewidth]{img/ml_map} - - \pause - + \begin{tikzpicture}[remember picture, overlay] - \draw[line width=10pt, red, -latex] (-18em,1em) -- (-14.5em, 6em); - \draw[line width=10pt, red, -latex] (19em, 7em) -- (14em, 4em); + \node[anchor=base] at (16em,18em) {\cite{from scikit-learn.org}}; \end{tikzpicture} \pause \begin{tikzpicture}[remember picture, overlay] - \draw[line width=4pt, red] (12em,12em) ellipse (2cm and 1.5cm); + \draw[line width=10pt, red, -latex] (-18em,2em) -- (-14.5em,7.5em); + \draw[line width=10pt, red, -latex] (19em,9em) -- (14em,5em); + \end{tikzpicture} + + \pause + + \begin{tikzpicture}[remember picture, overlay] + \draw[line width=4pt, red] (12em,13em) ellipse (2cm and 1.5cm); \end{tikzpicture} \end{frame} @@ -1632,6 +1655,7 @@ \begin{column}{0.4\linewidth} \centering \resizebox{\columnwidth}{!}{\import{img}{fc.pgf}} + \hfill\cite{rendition of the neural network in Bull et al.\ (2018)} \end{column} \end{columns} \end{frame}