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Signed-off-by: Riccardo Finotello <riccardo.finotello@gmail.com>
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Riccardo
2020-11-13 12:14:47 +01:00
parent 77139f0386
commit b6f3da5b5b
1 changed files with 44 additions and 20 deletions
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thesis.tex
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@@ -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}
\pause
\item issues are \textbf{still present} in sQED (eikonal?)
\item field theory presents \textbf{divergences} (even sQED $\rightarrow$ eikonal?)
\pause
\item divergences are \textbf{not (only) gravitational}
\pause
\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}
@@ -1542,17 +1561,21 @@
\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}