From 6dc21a9be27f312e2c500eb79105330e91871924 Mon Sep 17 00:00:00 2001
From: Riccardo Finotello <riccardo.finotello@gmail.com>
Date: Fri, 6 Nov 2020 19:40:16 +0100
Subject: [PATCH] Add branes in conformal plane

Signed-off-by: Riccardo Finotello <riccardo.finotello@gmail.com>
---
 img/threebranes_plane.pgf | 21 ++++++++++
 thesis.tex                | 80 +++++++++++++++++++++++++++++++++++++--
 2 files changed, 97 insertions(+), 4 deletions(-)
 create mode 100644 img/threebranes_plane.pgf

diff --git a/img/threebranes_plane.pgf b/img/threebranes_plane.pgf
new file mode 100644
index 0000000..d7e3f12
--- /dev/null
+++ b/img/threebranes_plane.pgf
@@ -0,0 +1,21 @@
+\usetikzlibrary{decorations.pathreplacing}
+\usetikzlibrary{decorations.pathmorphing}
+
+\begin{tikzpicture}
+
+% draw axis
+\draw[thick, ->] (-0.5cm, 0cm) -- (5cm, 0cm) node[anchor=south] {$\Re \omega$};
+\draw[thick, ->] (0cm, -1cm) -- (0cm, 3cm) node[anchor=east] {$\Im \omega$};
+
+% draw branching cuts
+\filldraw[fill=black!30, draw=black, dashed] (0cm, 2pt) rectangle (4.8cm, -2pt);
+\filldraw[fill=black!10, draw=black, dashed] (1cm, 1pt) rectangle (4.8cm, -1pt);
+
+% draw branching points
+\filldraw[fill=white, draw=black] (4.8cm, 0cm) circle (2pt) node[anchor=north] (x1) {$\infty$};
+\filldraw[fill=white, draw=black] (1cm, 0cm) circle (2pt) node[anchor=north] (x2) {$1$};
+\filldraw[fill=white, draw=black] (0cm, 0cm) circle (2pt) node[anchor=north east] (x3) {$0$};
+
+\end{tikzpicture}
+
+% vim: ft=tex
\ No newline at end of file
diff --git a/thesis.tex b/thesis.tex
index f8d66c5..1a7392b 100644
--- a/thesis.tex
+++ b/thesis.tex
@@ -521,7 +521,7 @@
     \pause
 
     \begin{columns}
-      \begin{column}{0.5\linewidth}
+      \begin{column}{0.4\linewidth}
         \centering
         \resizebox{0.9\columnwidth}{!}{\import{img}{welladapted.pgf}}
       \end{column}
@@ -531,9 +531,10 @@
           \qty( X_{(t)} )^I
           =
           \tensor{\qty( R_{(t)} )}{^I_J}\, X^J - g_{(t)}^I
-          \quad
-          \text{s.t.}
-          \quad
+        \end{equation*}
+        \pause
+        where
+        \begin{equation*}
           R_{(t)} \in \frac{\mathrm{SO}(4)}{\mathrm{S}\qty( \mathrm{O}(2) \times \mathrm{O}(2) )}
         \end{equation*}
         \pause
@@ -598,6 +599,77 @@
     \end{equationblock}
   \end{frame}
 
+  \begin{frame}{Doubling Trick and Spinor Representation}
+    \begin{block}{Doubling Trick}
+      \begin{equation*}
+        \partial_z \mathcal{X}( z )
+        =
+        \begin{cases}
+          \partial_u X( u )
+          & \text{if}~z \in \mathscr{H}_{>}^{(\overline{t})}
+          \\
+          U_{(\overline{t})}\, \partial_{\overline{u}} \overline{X}( \overline{u} )
+          & \text{if}~z \in \mathscr{H}_{<}^{(\overline{t})}
+        \end{cases}
+        \quad
+        \Rightarrow
+        \quad
+        \mqty{%
+          \partial_{z} \mathcal{X}( x_{(t)} + e^{2 \uppi i} \updelta_+ )
+          =
+          \mathcal{U}_{(t,\, t+1)}\,
+          \partial_{z} \mathcal{X}( x_{(t)} + \updelta_+ ),
+          \\
+          \partial_{z} \mathcal{X}( x_{(t)} + e^{2 \uppi i} \updelta_- )
+          =
+          \widetilde{\mathcal{U}}_{(t,\, t+1)}\,
+          \partial_{z} \mathcal{X}( x_{(t)} + \updelta_- ),
+        }
+      \end{equation*}
+      where $\mathscr{H}_{\gtrless}^{(t)} = \qty{z \in \mathds{C} \mid \Im z \gtrless 0~\text{or}~z \in D_{(t)} }$ and $\updelta_{\pm} = \upeta \pm i 0^+$.
+    \end{block}
+
+    \pause
+
+    \begin{tikzpicture}[remember picture, overlay]
+      \draw[line width=4pt, red] (31em,6em) ellipse (0.8cm and 1.2cm);
+    \end{tikzpicture}
+
+    \pause
+
+    Use \highlight{Pauli matrices} $\uptau = \qty( i\, \mathds{1}_2, \vec{\upsigma} )$:
+    \begin{equation*}
+      \partial_z \mathcal{X}_{(s)}( z )
+      =
+      \partial_z \mathcal{X}^I( z )\, \uptau_I
+      \quad
+      \Rightarrow
+      \quad
+      \partial_{z} \mathcal{X}( x_{(t)} + e^{2 \uppi i}\, \updelta_{\pm} )
+      =
+      \overset{\qty(\sim)}{\mathcal{L}}_{(t,\, t+1)}\,
+      \partial_{z} \mathcal{X}( x_{(t)} + \updelta_{\pm} )\,
+      \overset{\qty(\sim)}{\mathcal{R}}_{(t,\, t+1)}\,
+    \end{equation*}
+    where
+    \begin{equation*}
+      \overset{\qty(\sim)}{\mathcal{L}}_{(t,\, t+1)} \in \mathrm{SU}(2)_L
+      \quad
+      \text{and}
+      \quad
+      \overset{\qty(\sim)}{\mathcal{R}}_{(t,\, t+1)} \in \mathrm{SU}(2)_R
+    \end{equation*}
+  \end{frame}
+
+  \begin{frame}{Hypergeometric Basis}
+    \begin{columns}
+      \begin{column}{0.5\linewidth}
+        \centering
+        \resizebox{0.8\columnwidth}
+      \end{column}
+    \end{columns}
+  \end{frame}
+
   \section[Time Divergences]{Cosmological Backgrounds and Divergences}
 
   \begin{frame}{BBB}