From 814675e1e4d3c682ddb1bb3e78ca625c6af2d558 Mon Sep 17 00:00:00 2001 From: Riccardo Finotello Date: Wed, 4 Nov 2020 13:31:03 +0100 Subject: [PATCH] Add fade animation and content up to d-branes Signed-off-by: Riccardo Finotello --- img/chanpaton.pgf | 23 ++++++ img/cy.png | Bin 0 -> 253494 bytes thesis.tex | 184 +++++++++++++++++++++++++++++++++++++++++++--- 3 files changed, 198 insertions(+), 9 deletions(-) create mode 100644 img/chanpaton.pgf create mode 100644 img/cy.png diff --git a/img/chanpaton.pgf b/img/chanpaton.pgf new file mode 100644 index 0000000..811c788 --- /dev/null +++ b/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 diff --git a/img/cy.png b/img/cy.png new file mode 100644 index 0000000000000000000000000000000000000000..d3b2fce7625683cd76c538ba7898fe321a051cef GIT binary patch literal 253494 zcmeFYby(cZvM!3dyUU=#-QC^YZE$CBclV$P?vUUFcMYz=EqDm-!4Ao{_FCV%XWxD9 zex7^&+wd^Vue<84>bI-9x|t?gMM)YNfdByv3=COTMnVk?3}WEzM-vX_t!3G12M7j6 z7ww~=15z{fBy)Ckvb3=Wl7YOPfn-208%r=Sua%OIHt9sIj^VFv*hUaE&N?W=ZXJ-! zo14JW>$ZhC8-3e~%Bp$*h0GQT7_H~aL3hm~bHI&W!)$c-m(+ZeD@j2{#iF`moV-s zGw!c0)mh@BiQ7A|9p3^h;Z3-QM~%I!x@)b#JL^r)sG_E3DB;t?VH#im&AR7r#4v!r*cZB@_H8`10Uaen|c|4eRtuhu;)v|FrPT=(QVk1 z*S$jOd3<2i{qtM3Fv&^A`sl@D%Jmo^Baig;`j}}c?`%~6*Jg*CdXyX{mQ!Gv_@s!?*4ern?`l>W}Nm9_qI>o%-fv~ zcVnHNb{$tJ9UL%BI9Ld;x6O~u!p=1SZm7>x`lLdSghe_r{M3`dkvr!c zB7i_#Udr$NEbVGu582D$=*X5r7T#&2>NJr!e{Mv;cnKU{1ZVf?QU)GFWCj^Pw!u+ZWJBz^Y}M`YrT1JeTj`MO>$y9?ipwQZ{EgdX>@c7h)}sk9=*==2;N2>Q`+)Wv|>$ 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0 HcmV?d00001 diff --git a/thesis.tex b/thesis.tex index ff70be0..d8b895c 100644 --- a/thesis.tex +++ b/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}