Begin part with fermions and defect CFT

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

img/brane3d.pgf

@@ -0,0 +1,25 @@
+\usetikzlibrary{decorations.pathmorphing}
+
+\begin{tikzpicture}
+
+% draw branes
+\draw[thick] (-0.35cm, 0cm) -- (-3cm, -2cm) -- (-3cm, 3cm) -- (1cm, 5cm) -- (1cm, 4cm);
+\draw[dashed] (1cm, 4cm) -- (1cm, 1cm) -- (-0.35cm, 0cm);
+
+\draw[thick] (-3cm, 2cm) -- (-4cm, 4cm) -- (1.5cm, 4cm) -- (3.5cm, 0cm) -- (-0.35cm, 0cm);
+\draw[dashed] (-3cm, 2cm) -- (-2cm, 0cm) -- (-0.35cm, 0cm);
+
+\draw[dotted] (-0.35cm, 0cm) -- (-1cm, 4.01cm);
+
+% draw names
+\node[anchor=base] at (0.15cm, 5cm) {$D_{(t)}$};
+\node[anchor=base] at (2.5cm, -0.5cm) {$D_{(t+1)}$};
+
+% draw string
+\draw[thick, decorate, decoration={snake, segment length=1cm}] (1.5cm, 2cm) .. controls (1.1cm, 1.1cm) and (-1cm, 0.1cm) .. (-1.8cm, 0.3cm);
+\filldraw[fill=black, draw=black] (1.5cm, 2cm) circle (2pt);
+\filldraw[fill=black, draw=black] (-1.8cm, 0.3cm) circle (2pt);
+
+\end{tikzpicture}
+
+% vim: ft=tex

img/defects.pgf

@@ -0,0 +1,27 @@
+\begin{tikzpicture}
+
+% draw axis
+\draw[thick, ->] (-3cm, 0cm) -- (3cm, 0cm) node[anchor=south] {$\uptau$};
+\draw[thick, ->] (-0.75cm, -1cm) -- (-0.75cm, 3cm) node[anchor=east] {$\upsigma$};
+
+% draw defects
+\filldraw[fill=white, draw=black] (-2cm, 0cm) circle (2pt) node[anchor=north] {$\hat{\uptau}_{(t+1)}$};
+\filldraw[fill=white, draw=black] (0.25cm, 0cm) circle (2pt) node[anchor=north] {$\hat{\uptau}_{(t)}$};
+\filldraw[fill=white, draw=black] (2.25cm, 0cm) circle (2pt) node[anchor=north] {$\hat{\uptau}_{(t-1)}$};
+
+% draw the endlines
+\draw[dotted] (-2cm, 2pt) -- (-2cm, 2cm);
+\draw[dotted] (0.25cm, 2pt) -- (0.25cm, 2cm);
+\draw[dotted] (2.25cm, 2pt) -- (2.25cm, 2cm);
+
+% draw the second D-brane
+\draw[dashed] (-3cm, 2cm) -- (3cm, 2cm);
+\node[anchor=south west] at (-0.75cm, 2cm) {$\uppi$};
+
+% draw the string
+\draw[decorate, decoration={snake, segment length=0.75cm}] (1cm, 0cm) -- (1cm, 2cm);
+\draw[->] (1.1cm, 1cm) -- (1.6cm, 1cm);
+
+\end{tikzpicture}
+
+% vim: ft=tex

img/threebranes_plane.pgf

@@ -1,11 +1,8 @@
-\usetikzlibrary{decorations.pathreplacing}
+\begin{tikzpicture}
-\usetikzlibrary{decorations.pathmorphing}
-
-\begin{tikzpicture}
 
 % draw axis
-\draw[thick, ->] (-0.5cm, 0cm) -- (5cm, 0cm) node[anchor=south] {$\Re \omega$};
+\draw[thick, ->] (-0.5cm, 0cm) -- (5cm, 0cm) node[anchor=south] {$\Re \upomega$};
-\draw[thick, ->] (0cm, -1cm) -- (0cm, 3cm) node[anchor=east] {$\Im \omega$};
+\draw[thick, ->] (0cm, -1cm) -- (0cm, 3cm) node[anchor=east] {$\Im \upomega$};
 
 % draw branching cuts
 \filldraw[fill=black!30, draw=black, dashed] (0cm, 2pt) rectangle (4.8cm, -2pt);

thesis.tex

@@ -663,13 +663,174 @@
 
   \begin{frame}{Hypergeometric Basis}
     \begin{columns}
-      \begin{column}{0.5\linewidth}
+      \begin{column}{0.3\linewidth}
         \centering
-        \resizebox{0.8\columnwidth}
+        \resizebox{0.9\columnwidth}{!}{\import{img}{threebranes_plane.pgf}}
+      \end{column}
+      \hfill
+      \begin{column}{0.7\linewidth}
+        Sum over \highlight{all contributions:}
+        \begin{equation*}
+          \begin{split}
+            \partial_z \mathcal{X}( z )
+            & =
+            \sum\limits_{l,\, r} c_{lr}\,
+            \qty( - \upomega_z )^{A_{lr}}\,
+            \qty( 1 - \upomega_z )^{B_{lr}}\,
+            B_{0,\, l}^{(L)}( \omega_z )\,
+            \qty( B_{0,\, r}^{(R)}( \omega_z ) )^T
+          \end{split}
+        \end{equation*}
       \end{column}
     \end{columns}
+
+    \pause
+
+    \begin{equationblock}{Basis of Solutions}
+      \begin{equation*}
+        B_{0,\, n}( \upomega_z )
+        =
+        \mqty(%
+          1 & 0
+          \\
+          0 & K_n
+        )
+        \mqty(%
+          \frac{1}{\Upgamma( c_n )}\,
+          \tensor[_2]{F}{_1}( a_n,\, b_n;\, c_n;\, \upomega_z )
+          \\
+          \qty( -\upomega_z )^{1 - c_n}\,
+          \frac{1}{\Upgamma( 2 - c_n )}\,
+          \tensor[_2]{F}{_1}( a_n + 1 - c_n,\, b_n + 1 - c_n;\, 2 - c_n;\, \upomega_z )
+        )
+      \end{equation*}
+    \end{equationblock}
   \end{frame}
 
+  \begin{frame}{The Solution}
+    \highlight{Operations sequence:}
+    \begin{enumerate}
+      \item rotation matrix $=$ monodromy matrix
+
+        \pause
+
+      \item contiguity relations $\Rightarrow$ independent hypergeometrics
+
+        \pause
+
+      \item finite action $\Rightarrow$ $2$ solutions (no.\ of d.o.f.\ is correctly saturated)
+
+        \pause
+
+      \item boundary conditions $\Rightarrow$ fix free constants $c_{lr}$
+    \end{enumerate}
+
+    \pause
+
+    \begin{block}{Physical Interpretation}
+      \only<5>{%
+        \begin{columns}
+          \begin{column}{0.4\linewidth}
+            \centering
+            \resizebox{0.607\columnwidth}{!}{\import{img}{branesangles.pgf}}
+          \end{column}
+          \hfill
+          \begin{column}{0.6\linewidth}
+            \begin{equation*}
+              \begin{split}
+                \eval{S_{\mathds{R}^4}}_{\text{on-shell}}
+                & =
+                \frac{1}{2\pi \alpha'}
+                \sum\limits_{t = 1}^3
+                \qty( \frac{1}{2} \abs{g_{(t)}^{\perp}} \abs{f_{(t-1)} - f_{(t)}} )
+                \\
+                & =
+                \text{Area}\qty( \qty{ f_{(t)} } )
+              \end{split}
+            \end{equation*}
+          \end{column}
+        \end{columns}
+        \vfill
+      }
+      \only<6->{%
+        \centering
+        \resizebox{0.25\columnwidth}{!}{\import{img}{brane3d.pgf}}
+      }
+    \end{block}
+  \end{frame}
+
+
+  \subsection[Fermions]{Fermions and Point-like Defect CFT}
+
+  \begin{frame}{Fermions on the Strip}
+    \begin{columns}
+      \begin{column}{0.4\linewidth}
+        \centering
+        \resizebox{0.9\columnwidth}{!}{\import{img}{defects.pgf}}
+      \end{column}
+      \hfill
+      \begin{column}{0.6\linewidth}
+        \begin{equationblock}{Action of Boundary Changing Operators}
+          \begin{equation*}
+            \begin{cases}
+              \uppsi_-^i( \uptau, 0 )
+              & =
+              \tensor{\qty( R_{(t)} )}{^I_J}\,
+              \uppsi_+^J( \uptau, 0 )
+              \quad \text{for}~
+              \uptau \in \qty( \hat{\uptau}_{(t)},\, \hat{\uptau}_{(t-1)} )
+              \\
+              \uppsi_-^I( \uptau, \uppi )
+              & =
+              - \uppsi_+^I( \uptau, \uppi )
+              \quad \text{for}~
+              \uptau \in \mathds{R}
+            \end{cases}
+          \end{equation*}
+        \end{equationblock}
+      \end{column}
+    \end{columns}
+
+    \pause
+
+    \begin{block}{Stress-energy Tensor}
+      \begin{equation*}
+        \mathcal{T}_{\pm\pm}( \upxi_{\pm} )
+        =
+        -i\, \frac{T}{4}\,
+        \uppsi^*_{\pm,\, I}( \upxi_{\pm} )\,
+        \overset{\leftrightarrow}{\partial} \uppsi^I_{\pm}( \upxi_{\pm} )
+        \quad
+        \Rightarrow
+        \quad
+        \begin{cases}
+          \dot{\mathrm{H}}( \uptau )
+          &
+          % =
+          % \partial_{\uptau}
+          % \qty(%
+          %   \int\limits_0^{\uppi} \dd{\upsigma}
+          %   \mathcal{T}_{\uptau\uptau}( \uptau, \upsigma )
+          % )
+          =
+          0 \Leftrightarrow \uptau \in \qty( \uptau_{(t)},\, \uptau_{(t-1)} )
+          \\
+          \dot{\mathrm{P}}( \uptau )
+          &
+          % =
+          % \partial_{\uptau}
+          % \qty(%
+          %   \int\limits_0^{\uppi} \dd{\upsigma}
+          %   \mathcal{T}_{\uptau\upsigma}( \uptau, \upsigma )
+          % )
+          \neq
+          0
+        \end{cases}
+      \end{equation*}
+    \end{block}
+  \end{frame}
+
+
   \section[Time Divergences]{Cosmological Backgrounds and Divergences}
 
   \begin{frame}{BBB}