312 lines
7.9 KiB
TeX
312 lines
7.9 KiB
TeX
\documentclass[10pt, aspectratio=169]{beamer}
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\usepackage[utf8]{inputenc}
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\usepackage[T1]{fontenc}
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\usepackage[british]{babel}
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\usepackage{csquotes}
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\usepackage{amsmath}
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\usepackage{amsfonts}
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\usepackage{amssymb}
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\usepackage{mathrsfs}
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\usepackage{dsfont}
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\usepackage{upgreek}
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\usepackage{physics}
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\usepackage{tensor}
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\usepackage{graphicx}
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\usepackage{transparent}
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\usepackage{tikz}
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\usepackage{import}
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\usepackage{booktabs}
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\usepackage{multicol}
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\usepackage{multirow}
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\usepackage{bookmark}
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\usepackage{xspace}
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\usetheme{Singapore}
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\usecolortheme{crane}
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\usefonttheme{structurebold}
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\setbeamertemplate{navigation symbols}{}
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\author[Finotello]{Riccardo Finotello}
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\title[D-branes and Deep Learning]{D-branes and Deep Learning}
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\subtitle{Theoretical and Computational Aspects in String Theory}
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\institute[UniTO]{%
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Scuola di Dottorato in Fisica e Astrofisica
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\\[0.5em]
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Università degli Studi di Torino
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\\
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and
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\\
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I.N.F.N.\ -- sezione di Torino
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}
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\date{15th December 2020}
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\newenvironment{equationblock}[1]{%
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\begin{block}{#1}
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\vspace*{-\baselineskip}\setlength\belowdisplayshortskip{0pt}
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}{%
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}
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\newcommand{\firstlogo}{img/unito}
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\newcommand{\thefirstlogo}{%
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\begin{figure}
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\centering
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\includegraphics[width=7em]{\firstlogo}
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\end{figure}
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}
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\newcommand{\secondlogo}{img/infn}
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\newcommand{\thesecondlogo}{%
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\begin{figure}
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\centering
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\includegraphics[width=7em]{\secondlogo}
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\end{figure}
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}
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\begin{center}
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{%
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{\Huge \inserttitle}\xspace
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}
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\vspace{0.5em}
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}\par
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{%
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\usebeamercolor{subtitle}
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{\large \it \insertsubtitle}\xspace
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\vspace{2em}
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}\par
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{%
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\usebeamercolor{author}
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{\Large \insertauthor}\xspace
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\vspace{1em}
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}\par
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{%
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\centering
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\begin{column}{0.3\linewidth}
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\centering
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\thefirstlogo
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\end{column}
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\begin{column}{0.4\linewidth}
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\centering
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\usebeamercolor{institute}
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\usebeamerfont{institute}
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\insertinstitute{}
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\\[1em]
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\insertdate{}
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\begin{column}{0.3\linewidth}
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\centering
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\thesecondlogo
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}\par
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\end{center}
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}
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\setbeamertemplate{footline}{%
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\usebeamerfont{footnote}
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\hfill
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\insertframenumber{}~/~\inserttotalframenumber{}
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\hspace{1em}
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\vspace{1em}
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\par
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}
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% \AtBeginSection[]
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% {%
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% {%
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% \setbeamertemplate{footline}{}
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% \begin{frame}[noframenumbering]{\contentsname}
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% \tableofcontents[currentsection]
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% \end{frame}
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% }
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% }
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\begin{document}
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{%
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\usebackgroundtemplate{%
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\transparent{0.1}
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\includegraphics[width=\paperwidth]{img/torino.png}
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}
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\begin{frame}[noframenumbering, plain]
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\titlepage{}
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\end{frame}
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}
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{%
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\setbeamertemplate{footline}{}
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\begin{frame}[noframenumbering]{\contentsname}
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\tableofcontents{}
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\end{frame}
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}
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\section[CFT]{Conformal Symmetry and Geometry of the Worldsheet}
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\subsection[Preliminary]{Preliminary Concepts and Tools}
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\begin{frame}{Action Principle and Conformal Symmetry}
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\begin{equationblock}{Polyakov's Action}
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\begin{equation*}
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S_P\qty[ \upgamma,\, X,\, \uppsi ]
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=
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-\frac{1}{4\pi}
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\int\limits_{-\infty}^{+\infty} \dd{\uptau}
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\int\limits_0^{\ell} \dd{\upsigma}
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\sqrt{-\det \upgamma}\,
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\upgamma^{\upalpha \upbeta}\,
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\qty(%
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\frac{2}{\upalpha'}\,
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\partial_{\upalpha} X^{\upmu}\,
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\partial_{\upbeta} X^{\upnu}
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+
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\uppsi^{\upmu}\,
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\uprho_{\upalpha}
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\partial_{\upbeta}
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\uppsi^{\upnu}
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)\,
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\upeta_{\upmu\upnu}
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\end{equation*}
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\end{equationblock}
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\begin{columns}
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\begin{column}[t]{0.5\linewidth}
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\fcolorbox{yellow}{yellow!20}{Symmetries:}
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\begin{itemize}
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\item \textbf{Poincaré transf.}\ $X'^{\upmu} = \tensor{\Uplambda}{^{\upmu}_{\upnu}} X^{\upnu} + c^{\upmu}$
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\item \textbf{2D diff.}\ $\upgamma'_{\upalpha \upbeta} = \tensor{\qty( \mathrm{J}^{-1} )}{_{\upalpha \upbeta}^{\uplambda \uprho}}\, \gamma_{\uplambda \uprho}$
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\item \textbf{Weyl transf.}\ $\upgamma'_{\upalpha \upbeta} = e^{2 \upomega}\, \gamma_{\upalpha \upbeta}$
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\end{itemize}
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\end{column}
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\begin{column}[t]{0.5\linewidth}
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\fcolorbox{yellow}{yellow!20}{Conformal symmetry:}
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\begin{itemize}
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\item \textbf{vanishing} stress-energy tensor: $\mathcal{T}_{\upalpha \upbeta} = 0$
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\item \textbf{traceless} stress-energy tensor: $\trace{\mathcal{T}} = 0$
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\item \textbf{conformal gauge} $\upgamma_{\upalpha \upbeta} = e^{\upphi}\, \upeta_{\upalpha \upbeta}$
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\end{itemize}
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\end{column}
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\end{columns}
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\end{frame}
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\begin{frame}{Action Principle and Conformal Symmetry}
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\begin{columns}
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\begin{column}{0.6\linewidth}
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\fcolorbox{yellow}{yellow!20}{%
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Let $z = e^{\uptau_E + i \upsigma} \Rightarrow \overline{\partial} \mathcal{T}( z ) = \partial \overline{\mathcal{T}}( \overline{z} ) = 0$:
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}
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\begin{equation*}
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\mathcal{T}( z )\, \Upphi_h( w )
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\stackrel{z \to w}{\sim}
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\frac{h}{(z - w)^2} \Upphi_h( w )
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+
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\frac{1}{z - w} \partial_w \Upphi_h( w )
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\end{equation*}
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\begin{equation*}
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\mathcal{T}( z )\, \mathcal{T}( w )
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\stackrel{z \to w}{\sim}
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\frac{\frac{c}{2}}{(z - w)^4}
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+
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\order{(z - w)^{-2}}
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\end{equation*}
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\begin{equationblock}{Virasoro algebra $\mathscr{V} \oplus \overline{\mathscr{V}}$}
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\begin{eqnarray*}
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\qty[ L_n,\, L_m ]
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& = &
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(n - m) L_{n + m} + \frac{c}{12} n \qty(n^2 - 1) \updelta_{n + m,\, 0}
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\\
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\qty[ L_n,\, \overline{L}_m ]
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& = &
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0
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\end{eqnarray*}
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\end{equationblock}
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\end{column}
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\begin{column}{0.4\linewidth}
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\begin{figure}[h]
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\centering
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\resizebox{0.8\columnwidth}{!}{\import{img}{complex_plane.pgf}}
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\end{figure}
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\end{column}
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\end{columns}
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\end{frame}
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\begin{frame}{Action Principle and Conformal Symmetry}
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\fcolorbox{yellow}{yellow!20}{Superstrings in $D$ dimensions:}
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\begin{equation*}
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\mathcal{T}( z )
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=
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-\frac{1}{\upalpha'}
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\partial X( z ) \cdot \partial X( z )
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-\frac{1}{2}
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\uppsi( z ) \cdot \partial \uppsi( z )
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\quad
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\Rightarrow
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\quad
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c = \frac{3}{2} D
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\end{equation*}
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\begin{block}{$\qty( \uplambda, 0 )~/~\qty( 1 - \uplambda, 0 )$ Ghost System}
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Introduce anti-commuting $\qty( b,\, c )$ and commuting $\qty( \upbeta,\, \upgamma )$ conformal fields:
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\begin{equation*}
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S_{\text{ghost}}\qty[ b,\, c,\, \upbeta,\, \upgamma ]
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=
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\frac{1}{2\uppi}
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\iint \dd{z} \dd{\overline{z}}
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\qty(%
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b( z )\, \overline{\partial} c( z )
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+
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\upbeta( z )\, \overline{\partial} \upgamma( z )
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)
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\end{equation*}
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where $\uplambda_b = 2$ and $\uplambda_{\upbeta} = \frac{3}{2}$.
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\end{block}
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\fcolorbox{yellow}{yellow!20}{Consequence:}
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\begin{equation*}
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c_{\text{full}} = c + c_{\text{ghost}} = 0
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\quad
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\Leftrightarrow
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\quad
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D = 10.
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\end{equation*}
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\end{frame}
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\begin{frame}{Extra Dimensions and Compactification}
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\end{frame}
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\section[Time Divergences]{Cosmological Backgrounds and Divergences}
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\begin{frame}{BBB}
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b
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\end{frame}
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\section[Deep Learning]{Deep Learning the Geometry of String Theory}
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\begin{frame}{CCC}
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c
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\end{frame}
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\end{document}
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