Add branch cuts

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

img/branchcuts.pgf

@@ -0,0 +1,29 @@
+\begin{tikzpicture}
+
+% draw axis
+\draw[thick, ->] (-3cm, 0cm) -- (3cm, 0cm) node[anchor=south] {$x$};
+\draw[thick, ->] (-0.4cm, -1cm) -- (-0.4cm, 3cm) node[anchor=east] {$y$};
+
+% draw branching cuts
+\filldraw[fill=black!50, draw=black, dashed] (-2cm, 3pt) rectangle (2.5cm, -3pt);
+\filldraw[fill=black!30, draw=black, dashed] (-0.75cm, 2pt) rectangle (2.5cm, -2pt);
+\filldraw[fill=black!10, draw=black, dashed] (1cm, 1pt) rectangle (2.5cm, -1pt);
+
+% draw branching points
+\filldraw[fill=white, draw=black] (2.5cm, 0cm) circle (2pt) node[anchor=north, below=3pt] (x1) {$x_{(1)}$};
+\filldraw[fill=white, draw=black] (1cm, 0cm) circle (2pt) node[anchor=north, below=3pt] (x2) {$x_{(2)}$};
+\filldraw[fill=white, draw=black] (-0.75cm, 0cm) circle (2pt) node[anchor=north, below=3pt] (x3) {$x_{(3)}$};
+\filldraw[fill=white, draw=black] (-2cm, 0cm) circle (2pt) node[anchor=north, below=3pt] (x4) {$x_{(4)}$};
+
+% assign the D-branes
+\draw[thin, decorate, decoration={brace}] (1cm, 0.3cm) -- (2.5cm, 0.3cm) node[midway, anchor=south] {$D_{(2)}$};
+\draw[thin, decorate, decoration={brace}] (-0.75cm, 0.3cm) -- (1cm, 0.3cm) node[midway, anchor=south] {$D_{(3)}$};
+\draw[thin, decorate, decoration={brace}] (-2cm, 0.3cm) -- (-0.75cm, 0.3cm) node[midway, anchor=south] {$D_{(4)}$};
+
+% draw the remaining D-brane
+\draw[thin, ->, dash pattern=on 2pt off 2pt on 2pt off 2pt on 2pt off 2pt on 2pt off 2pt on 2pt off 2pt on 2pt off 2pt on 1cm] (-2cm, 2pt) -- (-2cm, 1cm)-- (-2.75cm, 1cm) node[midway, anchor=south] {$D_{(1)}$} ;
+\draw[thin, ->, dash pattern=on 2pt off 2pt on 2pt off 2pt on 2pt off 2pt on 2pt off 2pt on 2pt off 2pt on 2pt off 2pt on 1cm] (2.5cm, 2pt) -- (2.5cm, 1cm)-- (3cm, 1cm) node[midway, anchor=south] {$D_{(1)}$} ;
+
+\end{tikzpicture}
+
+% vim: ft=tex

img/welladapted.pgf

@@ -0,0 +1,13 @@
+\begin{tikzpicture}
+
+% draw the D-brane
+\draw (0cm, 0cm) -- (5cm, -1cm) -- (6cm, 3cm) -- (1.5cm, 4cm) node[anchor=south west] {$D_{(t)}$} -- cycle;
+
+% draw a triad of axis
+\draw[thick, ->] (3cm, 1cm) -- (2.7cm, 0.1cm) node[anchor=east] {$X^1_{(t)}$};
+\draw[thick, ->] (3cm, 1cm) -- (5cm, 0.6cm) node[anchor=south] {$X^2_{(t)}$};
+\draw[thick, ->] (3cm, 1cm) -- (3cm, 2.75cm) node[anchor=east] {$X^{3,\, 4}_{(t)}$};
+
+\end{tikzpicture}
+
+% vim: ft=tex

thesis.tex

@@ -44,6 +44,7 @@
 
 \usetikzlibrary{decorations.markings}
 \usetikzlibrary{decorations.pathmorphing}
+\usetikzlibrary{decorations.pathreplacing}
 \usetikzlibrary{arrows}
 
 \newenvironment{equationblock}[1]{%
@@ -480,7 +481,7 @@
 
     \begin{equationblock}{Twist Fields Correlators}
       \begin{equation*}
-        \left\langle \prod\limits_{t = 1}^N \upsigma_{\mathrm{M}_{(t)}}\qty( x_{(t)} ) \right\rangle
+        \left\langle \prod\limits_{t = 1}^{N_B} \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} )}
@@ -514,6 +515,89 @@
     \end{columns}
   \end{frame}
 
+  \begin{frame}{$\mathrm{SO}(4)$ Rotations}
+    Consider \highlight{$\mathds{R}^4 \times \mathds{R}^2$} (focus on $\mathds{R}^4$):
+
+    \pause
+
+    \begin{columns}
+      \begin{column}{0.5\linewidth}
+        \centering
+        \resizebox{0.9\columnwidth}{!}{\import{img}{welladapted.pgf}}
+      \end{column}
+
+      \begin{column}{0.6\linewidth}
+        \begin{equation*}
+          \qty( X_{(t)} )^I
+          =
+          \tensor{\qty( R_{(t)} )}{^I_J}\, X^J - g_{(t)}^I
+          \quad
+          \text{s.t.}
+          \quad
+          R_{(t)} \in \frac{\mathrm{SO}(4)}{\mathrm{S}\qty( \mathrm{O}(2) \times \mathrm{O}(2) )}
+        \end{equation*}
+        \pause
+        that is
+        \begin{equation*}
+          \qty[ R_{(t)} ]
+          =
+          \qty{ R_{(t)} \sim \mathcal{O}_{(t)} R_{(t)} }
+        \end{equation*}
+      \end{column}
+    \end{columns}
+  \end{frame}
+
+  \begin{frame}{Boundary Conditions}
+    What are the consequences for \highlight{open strings?}
+
+    \pause
+
+    \begin{columns}
+      \begin{column}{0.6\linewidth}
+        \begin{itemize}
+          \item consider $u = x + i y = e^{\uptau_e + i \upsigma}$ and $\overline{u} = u^*$
+
+          \item let $x_{(t)} < x_{(t-1)}$ be the \textbf{worldsheet intersection points} on \textbf{real axis}
+
+          \item $X_{(t)}^{1,\, 2}$ are \textbf{Neumann}, $X_{(t)}^{3,\, 4}$ are \textbf{Dirichlet}
+        \end{itemize}
+      \end{column}
+
+      \begin{column}{0.4\linewidth}
+        \centering
+        \resizebox{0.9\columnwidth}{!}{\import{img}{branchcuts.pgf}}
+      \end{column}
+    \end{columns}
+
+    \pause
+
+    \begin{equationblock}{Branch Cuts and Discontinuities for $x \in D_{(t)}$}
+      \begin{equation*}
+        \begin{cases}
+          \partial_u X( x + i 0^+ )
+          & =
+          U_{(t)}
+          \cdot
+          \partial_{\overline{u}} \overline{X}( x - i 0^+ )
+          =
+          \qty[%
+            R_{(t)}^{-1}
+            \cdot
+            \qty( \upsigma_3 \otimes \mathds{1}_2 )
+            \cdot
+            R_{(t)}
+          ]
+          \cdot
+          \partial_{\overline{u}} \overline{X}( x - i 0^+ )
+          \\
+          X( x_{(t)},\, x_{(t)} )
+          & =
+          f_{(t)}
+        \end{cases}
+      \end{equation*}
+    \end{equationblock}
+  \end{frame}
+
   \section[Time Divergences]{Cosmological Backgrounds and Divergences}
 
   \begin{frame}{BBB}