Add fade animation and content up to d-branes

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

img/chanpaton.pgf

@@ -0,0 +1,23 @@
+\begin{tikzpicture}
+
+% draw the D-branes
+\draw[thick] (0cm, 0cm) -- (1cm, 0.5cm) -- (1cm, 4cm) -- (0cm, 3.5cm) -- cycle;
+\draw[thick] (2.5cm, 0cm) -- (3.5cm, 0.5cm) -- (3.5cm, 4cm) -- (2.5cm, 3.5cm) -- cycle;
+\draw[thick] (5cm, 0cm) -- (6cm, 0.5cm) -- (6cm, 4cm) -- (5cm, 3.5cm) -- cycle;
+
+% draw strings
+\draw[decorate, decoration={snake, segment length=1cm}, dash pattern=on 2.45cm off 1pt on 1pt off 1pt on 1pt off 1pt on 1pt off 1pt on 1pt off 1pt on 1pt off 1pt on 1pt off 1pt on 1pt off 1pt on 1pt off 1pt, postaction={decoration={markings, mark=at position 0.5 with {\arrow{>}}}, decorate}] (0.5cm, 3cm) -- (3cm, 1.5cm);
+\node[anchor=base] (l1) at (1.75cm, 2.75cm) {$\tensor{\lambda}{^1_1_2}$};
+
+\draw[decorate, decoration={snake, segment length=1cm}, dash pattern=on 2.45cm off 1pt on 1pt off 1pt on 1pt off 1pt on 1pt off 1pt on 1pt off 1pt on 1pt off 1pt on 1pt off 1pt on 1pt off 1pt on 1pt off 1pt, postaction={decoration={markings, mark=at position 0.5 with {\arrow{>}}}, decorate}] (3cm, 3cm) -- (5.5cm, 1.5cm);
+\node[anchor=base] (l2) at (4.25cm, 2.75cm) {$\tensor{\lambda}{^2_2_3}$};
+
+\draw[decorate, decoration={snake, segment length=1cm}, postaction={decoration={markings, mark=at position 0.765 with {\arrow{>}}}, decorate}] (3cm, 2.5cm) .. controls (3.5cm, 2cm) and (3.5cm, 1cm) .. (3cm, 0.5cm);
+\node[anchor=base] (l3) at (4cm, 1.5cm) {$\tensor{\lambda}{^3_2_2}$};
+
+\draw[decorate, decoration={snake, segment length=1cm}, dash pattern=on 5.15cm off 1pt on 1pt off 1pt on 1pt off 1pt on 1pt off 1pt on 1pt off 1pt on 1pt off 1pt on 1pt off 1pt on 1pt off 1pt on 1pt off 1pt on 1pt off 1pt, postaction={decoration={markings, mark=at position 0.5 with {\arrow{<}}}, decorate}] (0.5cm, 1cm) .. controls (1.5cm, 0cm) and (4cm, 0cm) .. (5.5cm, 1cm);
+\node[anchor=base] (l4) at (3cm, -0.5cm) {$\tensor{\lambda}{^4_3_1}$};
+
+\end{tikzpicture}
+
+% vim: ft=tex

thesis.tex

@@ -26,6 +26,7 @@
 \usecolortheme{crane}
 \usefonttheme{structurebold}
 \setbeamertemplate{navigation symbols}{}
+\addtobeamertemplate{background canvas}{\transfade[duration=0.25]}{}
 
 \author[Finotello]{Riccardo Finotello}
 \title[D-branes and Deep Learning]{D-branes and Deep Learning}
@@ -41,6 +42,10 @@
 }
 \date{15th December 2020}
 
+\usetikzlibrary{decorations.markings}
+\usetikzlibrary{decorations.pathmorphing}
+\usetikzlibrary{arrows}
+
 \newenvironment{equationblock}[1]{%
   \begin{block}{#1}
   \vspace*{-\baselineskip}\setlength\belowdisplayshortskip{0pt}
@@ -48,6 +53,8 @@
   \end{block}
 }
 
+\newcommand{\highlight}[1]{\fcolorbox{yellow}{yellow!20}{#1}}
+
 \newcommand{\firstlogo}{img/unito}
 \newcommand{\thefirstlogo}{%
   \begin{figure}
@@ -138,6 +145,7 @@
       \transparent{0.1}
       \includegraphics[width=\paperwidth]{img/torino.png}
     }
+    \addtobeamertemplate{background canvas}{\transfade[duration=0.25]}{}
     \begin{frame}[noframenumbering, plain]
       \titlepage{}
     \end{frame}
@@ -145,6 +153,11 @@
 
   {%
     \setbeamertemplate{footline}{}
+    \usebackgroundtemplate{%
+      \transparent{0.1}
+      \includegraphics[width=\paperwidth]{img/torino.png}
+    }
+    \addtobeamertemplate{background canvas}{\transfade[duration=0.25]}{}
     \begin{frame}[noframenumbering]{\contentsname}
       \tableofcontents{}
     \end{frame}
@@ -180,9 +193,11 @@
       \end{equation*}
     \end{equationblock}
 
+    \pause
+
     \begin{columns}
       \begin{column}[t]{0.5\linewidth}
-        \fcolorbox{yellow}{yellow!20}{Symmetries:}
+        \highlight{Symmetries:}
         \begin{itemize}
           \item \textbf{Poincaré transf.}\ $X'^{\upmu} = \tensor{\Uplambda}{^{\upmu}_{\upnu}} X^{\upnu} + c^{\upmu}$
 
@@ -192,8 +207,10 @@
         \end{itemize}
       \end{column}
 
+      \pause
+
       \begin{column}[t]{0.5\linewidth}
-        \fcolorbox{yellow}{yellow!20}{Conformal symmetry:}
+        \highlight{Conformal symmetry:}
         \begin{itemize}
           \item \textbf{vanishing} stress-energy tensor: $\mathcal{T}_{\upalpha \upbeta} = 0$
           
@@ -209,7 +226,7 @@
   \begin{frame}{Action Principle and Conformal Symmetry}
     \begin{columns}
       \begin{column}{0.6\linewidth}
-        \fcolorbox{yellow}{yellow!20}{%
+        \highlight{%
           Let $z = e^{\uptau_E + i \upsigma} \Rightarrow \overline{\partial} \mathcal{T}( z ) = \partial \overline{\mathcal{T}}( \overline{z} ) = 0$:
         }
         \begin{equation*}
@@ -229,9 +246,9 @@
 
         \begin{equationblock}{Virasoro algebra $\mathscr{V} \oplus \overline{\mathscr{V}}$}
           \begin{eqnarray*}
-            \qty[ L_n,\, L_m ]
+            \qty[ \overset{\scriptscriptstyle(-)}{L}_n,\, \overset{\scriptscriptstyle(-)}{L}_m ]
             & = &
-            (n - m) L_{n + m} + \frac{c}{12} n \qty(n^2 - 1) \updelta_{n + m,\, 0}
+            (n - m) \overset{\scriptscriptstyle(-)}{L}_{n + m} + \frac{c}{12} n \qty(n^2 - 1) \updelta_{n + m,\, 0}
             \\
             \qty[ L_n,\, \overline{L}_m ]
             & = &
@@ -243,14 +260,14 @@
       \begin{column}{0.4\linewidth}
         \begin{figure}[h]
           \centering
-          \resizebox{0.8\columnwidth}{!}{\import{img}{complex_plane.pgf}}
+          \resizebox{0.9\columnwidth}{!}{\import{img}{complex_plane.pgf}}
         \end{figure}
       \end{column}
     \end{columns}
   \end{frame}
 
   \begin{frame}{Action Principle and Conformal Symmetry}
-    \fcolorbox{yellow}{yellow!20}{Superstrings in $D$ dimensions:}
+    \highlight{Superstrings in $D$ dimensions:}
     \begin{equation*}
       \mathcal{T}( z )
       =
@@ -264,6 +281,8 @@
       c = \frac{3}{2} D
     \end{equation*}
 
+    \pause
+
     \begin{block}{$\qty( \uplambda, 0 )~/~\qty( 1 - \uplambda, 0 )$ Ghost System}
       Introduce anti-commuting $\qty( b,\, c )$ and commuting $\qty( \upbeta,\, \upgamma )$ conformal fields:
       \begin{equation*}
@@ -277,10 +296,12 @@
           \upbeta( z )\, \overline{\partial} \upgamma( z )
         )
       \end{equation*}
-      where $\uplambda_b = 2$ and $\uplambda_{\upbeta} = \frac{3}{2}$.
+      where $\uplambda_b = 2$ and $\uplambda_c = -1$, and $\uplambda_{\upbeta} = \frac{3}{2}$ and $\uplambda_{\upgamma} = -\frac{1}{2}$.
     \end{block}
 
-    \fcolorbox{yellow}{yellow!20}{Consequence:}
+    \pause
+
+    \highlight{Consequence:}
     \begin{equation*}
       c_{\text{full}} = c + c_{\text{ghost}} = 0
       \quad
@@ -292,8 +313,153 @@
 
 
   \begin{frame}{Extra Dimensions and Compactification}
+    \begin{block}{Compactification}
+      \begin{columns}
+        \begin{column}{0.6\linewidth}
+          \begin{equation*}
+            \mathscr{M}^{1,\, 9}
+            =
+            \mathscr{M}^{1,\, 3} \otimes \mathscr{X}_6
+          \end{equation*}
+          \begin{itemize}
+            \item $\mathscr{X}_6$ is a \textbf{compact} manifold
+
+            \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}
+          \end{itemize}
+        \end{column}
+
+        \begin{column}{0.4\linewidth}
+          \centering
+          \includegraphics[width=0.7\columnwidth]{img/cy}
+        \end{column}
+      \end{columns}
+    \end{block}
+
+    \pause
+
+    \begin{columns}
+      \begin{column}[t]{0.5\linewidth}
+        \highlight{Kähler manifolds} $\qty( M,\, g )$ such that
+        \begin{itemize}
+          \item $\dim\limits_{\mathds{C}} M = m$
+
+          \item $\mathrm{Hol}( g ) \subseteq \mathrm{SU}( m )$
+
+          \item $g$ is Ricci-flat (equiv.\ $c_1\qty( M )$ vanishes)
+        \end{itemize}
+      \end{column}
+
+      \pause
+
+      \begin{column}[t]{0.5\linewidth}
+        Characterised by \highlight{Hodge numbers}
+        \begin{equation*}
+          h^{r,\,s} = \dim\limits_{\mathds{C}} H_{\overline{\partial}}^{r,\,s}\qty( M,\, \mathds{C} )
+        \end{equation*}
+        counting the no.\ of harmonic $(r,\,s)$-forms.
+      \end{column}
+    \end{columns}
   \end{frame}
 
+  \begin{frame}{D-branes and Open Strings}
+    Polyakov's action naturally introduces \highlight{Neumann b.c.:}
+    \begin{equation*}
+      \eval{\partial_{\upsigma} X\qty( \uptau, \upsigma )}_{\upsigma = 0}^{\upsigma = \ell} = 0
+    \end{equation*}
+    satisfied by \textbf{open and closed} strings living in $D$ dimensions s.t.\ $\square X = 0$.
+
+    \pause
+
+    \begin{block}{T-duality}
+      \begin{equation*}
+        X( z, \overline{z} )
+        =
+        X( z ) + \overline{X}( \overline{z} )
+        \quad
+        \stackrel{T}{\Rightarrow}
+        \quad
+        X( z ) - \overline{X}( \overline{z} )
+        =
+        Y( z, \overline{z} )
+        =
+        Y( z ) + \overline{Y}( \overline{z} )
+      \end{equation*}
+    \end{block}
+
+    \pause
+
+    Resulting effect (repeated $p \le D - 1$ times) leads to \highlight{Dirichlet b.c.:}
+    \begin{equation*}
+      \eval{\partial_{\upsigma} X^i\qty( \uptau, \upsigma )}_{\upsigma = 0}^{\upsigma = \ell} = 0
+      \quad
+      \stackrel{T}{\Rightarrow}
+      \quad
+      \eval{\partial_{\uptau} Y^i\qty( \uptau, \upsigma )}_{\upsigma = 0}^{\upsigma = \ell} = 0
+      \quad
+      \forall i = 1, 2,\, \dots,\, p
+    \end{equation*}
+    thus \textbf{open strings} can be \textbf{constrained} to \highlight{$D(D - p - 1)$-branes.}
+  \end{frame}
+
+  \begin{frame}{D-branes and Open Strings}
+    Introducing $Dp$-branes breaks \highlight{$\mathrm{ISO}(1,\, D-1) \rightarrow \mathrm{ISO}( 1, p ) \otimes \mathrm{SO}( D - 1 - p )$.}
+
+    \pause
+
+    \begin{block}{Spectrum}
+      At massless level (\emph{irrep} of \textbf{little group} $\mathrm{SO}( D - 2 )$):
+      \begin{equation*}
+        \mathcal{A}^{\upmu}
+        \quad \leftrightarrow \quad
+        \alpha_{-1}^{\upmu} \ket{0}
+        \qquad
+        \longrightarrow
+        \qquad
+        \begin{tabular}{@{}llll@{}}
+          $\mathcal{A}^A$
+          &
+          $\leftrightarrow$
+          &
+          $\alpha_{-1}^A \ket{0},$
+          &
+          $A = 0,\, 1,\, \dots,\, p$
+          \\
+          $\mathcal{A}^a$
+          &
+          $\leftrightarrow$
+          &
+          $\alpha_{-1}^a \ket{0},$
+          &
+          $a = 1,\, 2,\, \dots,\, D - p - 1$
+        \end{tabular}
+      \end{equation*}
+    \end{block}
+
+    \pause
+
+    \begin{columns}
+      \begin{column}{0.5\linewidth}
+        \centering
+        \resizebox{0.55\columnwidth}{!}{\import{img}{chanpaton.pgf}}
+      \end{column}
+
+      \begin{column}{0.5\linewidth}
+        Introducing \textbf{Chan--Paton factors} $\tensor{\uplambda}{^r_{ij}}$, when branes are \textbf{coincident}:
+        \begin{equation*}
+          \bigoplus\limits_{r = 1}^N \mathrm{U}_r(1)
+          \quad
+          \longrightarrow
+          \quad
+          \mathrm{U}( N )
+        \end{equation*}
+        \pause
+        \highlight{Build gauge bosons, fermions and scalars.}
+      \end{column}
+
+    \end{columns}
+  \end{frame}
 
   \section[Time Divergences]{Cosmological Backgrounds and Divergences}