From 6538ed4ebd0d30d1c83727792926c1f227aad3e1 Mon Sep 17 00:00:00 2001 From: Riccardo Finotello Date: Thu, 5 Nov 2020 23:46:30 +0100 Subject: [PATCH] Add branes at angles Signed-off-by: Riccardo Finotello --- img/branesangles.pgf | 35 ++++++++++++++++++++++++++ thesis.pdf | Bin 6594190 -> 6612734 bytes thesis.tex | 58 +++++++++++++++++++++++++++++++++---------- 3 files changed, 80 insertions(+), 13 deletions(-) create mode 100644 img/branesangles.pgf diff --git a/img/branesangles.pgf b/img/branesangles.pgf new file mode 100644 index 0000000..56cf643 --- /dev/null +++ b/img/branesangles.pgf @@ -0,0 +1,35 @@ +\begin{tikzpicture} + +% draw axis +\draw[thick, ->] (-3cm, 0cm) -- (3cm, 0cm) node[anchor=south] {$X^1$}; +\draw[thick, ->] (0cm, -3cm) -- (0cm, 3cm) node[anchor=east] {$X^2$}; + +% draw the D-branes +\draw[dash pattern=on 5pt off 2pt on 5pt off 2pt on 4cm off 2pt] (-2.5cm, -1cm) -- (2.75cm, -2.15cm); +\draw[dash pattern=on 5pt off 2pt on 5pt off 2pt on 5pt off 2pt on 5pt off 2pt on 4.5cm off 2pt] (-2.5cm, -2cm) -- (1cm, 2.75cm); +\draw[dash pattern=on 5pt off 2pt on 5pt off 2pt on 4.5cm off 2pt] (0.6cm, 2.9cm) -- (2cm, -2.75cm); + +% give the D-branes names +\node[anchor=base] (D1) at (-0.75cm, -2.25cm) {$D_{(1)}$}; +\node[anchor=base] (D2) at (-1.25cm, 1cm) {$D_{(2)}$}; +\node[anchor=base] (D3) at (1.7cm, 0.8cm) {$D_{(3)}$}; + +% draw the distance of each side from the center +\draw[thin] (0cm, 0cm) -- (-0.3cm, -1.47cm) node[anchor=north] {$g_{(1)}$}; +\draw[thin] (0cm, 0cm) -- (-0.7cm, 0.45cm) node[anchor=east] {$g_{(2)}$}; +\draw[thin] (0cm, 0cm) -- (1.27cm, 0.22cm) node[anchor=west] {$g_{(3)}$}; + +% prolong the end points +\draw[dotted] (1.3cm, -1.925cm) node[anchor=north] {$f_{(1)}$} -- (3cm, -1.925cm); +\draw[dotted] (0.3cm, 2.4cm) -- (2cm, 2.4cm) node[anchor=north east] {$f_{(2)}$}; +\draw[dotted] (-2.3cm, -1.14cm) node[anchor=north east] {$f_{(3)}$} -- (-0.6cm, -1.14cm); + +% draw the arcs for the angles + +\draw[thin, <-] (2.6cm, -2.12cm) arc (0:30:0.4cm) node[anchor=south] {$\pi \alpha_{(1)}$}; +\draw[thin, <-] (-2.1cm, -1.5cm) arc (250:360:0.4cm) node[anchor=south west] {$\pi \alpha_{(2)}$}; +\draw[thin, ->] (1.1cm, 2.4cm) arc (0:100:0.4cm) node[anchor=south] {$\pi \alpha_{(3)}$}; + +\end{tikzpicture} + +% vim: ft=tex \ No newline at end of file diff --git a/thesis.pdf b/thesis.pdf index 0b344d7bff33fe0c16d7be84b84b86c3c115a5fe..63cae2ec0daa0a575cb81233774d2a747a2e75e4 100644 GIT binary patch delta 179526 zcmcHA^LL)x!|(f`v2EM7ZQE&VwsCSd%^llzW7}?Q+qRR&J8P}y`)K2gv&T8V%z6C@ zukoIrE2<+7{Czwg{09&e2nGZPegQ%NA%RdpXdny_76=D~2OH7AOal z2Pyy+fl5GSpbAhGs0LIAY5+BXT0m`}4p0}USFf)W1j?428*WUO>>Z?!004LN2WRCgs1M;aYh1{YSCC^dzS@~u&Y z&>yUlgSs9jFT`{Lxd-^0dc2(v2~IYnylNgyRtCsX#z$_NQmpCrEzA0wsz>tG>cQ1? z2&Zg0M`RDDW`ZjyMUm?7V!)Sc@Vv6RLp`AgE_o~TRULEw9c9)E>AeVtM7z3BirBH% zWO2PqmiY)CJUt4&v0j70pVft+`M7e@%O+lFD#6EwajVwy;wI*|;na}@3R+gZ>pAD7 z*z=U+EE*v&ASmTJ@1&R8Dn~4q#Jjo@M_4!5n)=j1)bx?sIpvpMQvfKvzg1iEmO0gI z6yFErm%9;H3ez|=^Z;{Z 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\usebeamercolor{footnote} +% \hfill +% \insertframenumber{}~/~\inserttotalframenumber{} +% \hspace{1em} +% \vspace{1em} +% \par +% } % \AtBeginSection[] % {% @@ -157,7 +157,7 @@ } {% - \setbeamertemplate{footline}{} + % \setbeamertemplate{footline}{} \usebackgroundtemplate{% \transparent{0.1} \includegraphics[width=\paperwidth]{img/torino.png} @@ -331,13 +331,16 @@ \item $N = 1$ \textbf{supersymmetry} is preserved in 4D - \item algebra of $\mathrm{SU}(3) \otimes \mathrm{SU}(2) \otimes \mathrm{U}(1)$ contained in arising \textbf{gauge group} + \item algebra of $\mathrm{SU}(3) \otimes \mathrm{SU}(2) \otimes \mathrm{U}(1)$ in arising \textbf{gauge group} \end{itemize} \end{column} + \begin{tikzpicture}[remember picture, overlay] + \node[anchor=base] at (3em,-3.75em) {\includegraphics[width=0.3\linewidth]{img/cy}}; + \end{tikzpicture} \begin{column}{0.3\linewidth} - \centering - \includegraphics[width=0.9\columnwidth]{img/cy} + % \centering + % \includegraphics[width=0.9\columnwidth]{img/cy} \end{column} \end{columns} \end{block} @@ -478,8 +481,37 @@ \begin{equationblock}{Twist Fields Correlators} \begin{equation*} \left\langle \prod\limits_{t = 1}^N \upsigma_{\mathrm{M}_{(t)}}\qty( x_{(t)} ) \right\rangle + = + \mathcal{N}\qty( \qty{ x_{(t)},\, \mathrm{M}_{(t)} }_{1 \le t \le N_B} ) + e^{- S_{E\, (\text{cl})}\qty( \qty{ x_{(t)},\, \mathrm{M}_{(t)} }_{1 \le t \le N_B} )} \end{equation*} \end{equationblock} + + \pause + + \begin{tikzpicture}[remember picture, overlay] + \draw[line width=4pt, red] (29.5em,3.5em) ellipse (2cm and 1cm); + \end{tikzpicture} + + \pause + + \begin{columns} + \begin{column}{0.5\linewidth} + \centering + \resizebox{0.5\columnwidth}{!}{\import{img}{branesangles.pgf}} + \end{column} + + \begin{column}{0.5\linewidth} + D-branes in \textbf{factorised} internal space: + \begin{itemize} + \item \textbf{embedded as lines} in $\mathds{R}^2 \times \mathds{R}^2 \times \mathds{R}^2$ + + \item \textbf{relative rotations} are $\mathrm{SO}(2) \simeq \mathrm{U}(1)$ elements + + \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} + \end{column} + \end{columns} \end{frame} \section[Time Divergences]{Cosmological Backgrounds and Divergences}