Add branes at angles

Signed-off-by: Riccardo Finotello <riccardo.finotello@gmail.com>

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

thesis.tex

@@ -117,15 +117,15 @@
   \end{center}
 }
 
-\setbeamertemplate{footline}{%
+% \setbeamertemplate{footline}{%
-  \usebeamerfont{footnote}
+%   \usebeamerfont{footnote}
-  \usebeamercolor{footnote}
+%   \usebeamercolor{footnote}
-  \hfill
+%   \hfill
-  \insertframenumber{}~/~\inserttotalframenumber{}
+%   \insertframenumber{}~/~\inserttotalframenumber{}
-  \hspace{1em}
+%   \hspace{1em}
-  \vspace{1em}
+%   \vspace{1em}
-  \par
+%   \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
+        %   \centering
-          \includegraphics[width=0.9\columnwidth]{img/cy}
+        %   \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}