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Add fade animation and content up to d-branes

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Signed-off-by: Riccardo Finotello <riccardo.finotello@gmail.com>
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Riccardo
2020-11-04 13:31:03 +01:00
parent d563c8b47d
commit 814675e1e4
3 changed files with 198 additions and 9 deletions
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184
thesis.tex
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@@ -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}