1762 lines
50 KiB
TeX
1762 lines
50 KiB
TeX
\documentclass[10pt, aspectratio=169, compress]{beamer}
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\usepackage{physics}
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\usepackage{tensor}
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\usepackage{graphicx}
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\usepackage{tikz}
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\usepackage{import}
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\setbeamertemplate{navigation symbols}{}
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\addtobeamertemplate{background canvas}{\transfade[duration=0.25]}{}
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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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\usetikzlibrary{decorations.markings}
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\newenvironment{equationblock}[1]{%
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\vspace*{-0.75\baselineskip}\setlength\belowdisplayshortskip{0.25\baselineskip}
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}{%
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}
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\newcommand{\highlight}[1]{\fcolorbox{yellow}{yellow!20}{#1}}
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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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\\[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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}
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\tableofcontents[currentsection]
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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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\addtobeamertemplate{background canvas}{\transfade[duration=0.25]}{}
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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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\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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\addtobeamertemplate{background canvas}{\transfade[duration=0.25]}{}
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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\uppi}
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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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\pause
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\begin{columns}
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\begin{column}[t]{0.5\linewidth}
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\highlight{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.}\ $\upgamma'_{\upalpha \upbeta} = \tensor{\qty( \mathrm{J}^{-1} )}{_{\upalpha \upbeta}^{\uplambda \uprho}}\, \upgamma_{\uplambda \uprho}$
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\item \textbf{Weyl transf.}\ $\upgamma'_{\upalpha \upbeta} = e^{2 \upomega}\, \upgamma_{\upalpha \upbeta}$
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\end{itemize}
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\end{column}
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\pause
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\begin{column}[t]{0.5\linewidth}
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\highlight{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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% \highlight{%
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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[ \overset{\scriptscriptstyle(-)}{L}_n,\, \overset{\scriptscriptstyle(-)}{L}_m ]
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% & = &
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% (n - m) \overset{\scriptscriptstyle(-)}{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{0.9\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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\highlight{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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\pause
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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 $\uplambda_c = -1$, and $\uplambda_{\upbeta} = \frac{3}{2}$ and $\uplambda_{\upgamma} = -\frac{1}{2}$.
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\end{block}
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\pause
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\highlight{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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\begin{block}{Compactification}
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\begin{columns}
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\begin{column}{0.7\linewidth}
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\begin{equation*}
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\mathscr{M}^{1,\, 9}
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=
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\mathscr{M}^{1,\, 3} \otimes \mathscr{X}_6
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\end{equation*}
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\begin{itemize}
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\item $\mathscr{X}_6$ is a \textbf{compact} manifold
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\item $N = 1$ \textbf{supersymmetry} is preserved in 4D
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\item algebra of $\mathrm{SU}(3) \otimes \mathrm{SU}(2) \otimes \mathrm{U}(1)$ in arising \textbf{gauge group}
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\end{itemize}
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\end{column}
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\begin{tikzpicture}[remember picture, overlay]
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\node[anchor=base] at (3em,-3.75em) {\includegraphics[width=0.3\linewidth]{img/cy}};
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\end{tikzpicture}
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\begin{column}{0.3\linewidth}
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% \centering
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% \includegraphics[width=0.9\columnwidth]{img/cy}
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\end{column}
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\end{columns}
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\end{block}
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\pause
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\begin{columns}
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\begin{column}[t]{0.5\linewidth}
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\highlight{Kähler manifolds} $\qty( M,\, g )$ such that
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\begin{itemize}
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\item $\dim\limits_{\mathds{C}} M = m$
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\item $\mathrm{Hol}( g ) \subseteq \mathrm{SU}( m )$
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\item $g$ is Ricci-flat (equiv.\ $c_1\qty( M )$ vanishes)
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\end{itemize}
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\end{column}
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\pause
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\begin{column}[t]{0.5\linewidth}
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Characterised by \highlight{Hodge numbers}
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\begin{equation*}
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h^{r,\,s} = \dim\limits_{\mathds{C}} H_{\overline{\partial}}^{r,\,s}\qty( M,\, \mathds{C} )
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\end{equation*}
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counting the no.\ of harmonic $(r,\,s)$-forms.
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\end{column}
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\end{columns}
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\end{frame}
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\begin{frame}{D-branes and Open Strings}
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Polyakov's action naturally introduces \highlight{Neumann b.c.:}
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\begin{equation*}
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\eval{\partial_{\upsigma} X\qty( \uptau, \upsigma )}_{\upsigma = 0}^{\upsigma = \ell} = 0
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\end{equation*}
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satisfied by \textbf{open and closed} strings living in $D$ dimensions s.t.\ $\square X = 0$.
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\pause
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\begin{equationblock}{T-duality}
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\begin{equation*}
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X( z, \overline{z} )
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=
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X( z ) + \overline{X}( \overline{z} )
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\quad
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\stackrel{T}{\Rightarrow}
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\quad
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X( z ) - \overline{X}( \overline{z} )
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=
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Y( z, \overline{z} )
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=
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Y( z ) + \overline{Y}( \overline{z} )
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\end{equation*}
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\end{equationblock}
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\pause
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Resulting effect (repeated $p \le D - 1$ times) leads to \highlight{Dirichlet b.c.:}
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\begin{equation*}
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\eval{\partial_{\upsigma} X^i\qty( \uptau, \upsigma )}_{\upsigma = 0}^{\upsigma = \ell} = 0
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\quad
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\stackrel{T}{\Rightarrow}
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\quad
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\eval{\partial_{\uptau} Y^i\qty( \uptau, \upsigma )}_{\upsigma = 0}^{\upsigma = \ell} = 0
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\quad
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\forall i = 1, 2,\, \dots,\, p
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\end{equation*}
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thus \textbf{open strings} can be \textbf{constrained} to \highlight{$D(D - p - 1)$-branes.}
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\end{frame}
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\begin{frame}{D-branes and Open Strings}
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Introducing $Dp$-branes breaks \highlight{$\mathrm{ISO}(1,\, D-1) \rightarrow \mathrm{ISO}( 1, p ) \otimes \mathrm{SO}( D - 1 - p )$.}
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\pause
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\begin{equationblock}{Massless Spectrum (\emph{irrep} of \textbf{little group} $\mathrm{SO}( D - 2 )$}
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\begin{equation*}
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\mathcal{A}^{\upmu}
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\quad \leftrightarrow \quad
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\upalpha_{-1}^{\upmu} \ket{0}
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\qquad
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\longrightarrow
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\qquad
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\begin{tabular}{@{}llll@{}}
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$\mathcal{A}^A$
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&
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$\leftrightarrow$
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&
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$\upalpha_{-1}^A \ket{0},$
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&
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$A = 0,\, 1,\, \dots,\, p$
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\\
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$\mathcal{A}^a$
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&
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$\leftrightarrow$
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&
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$\upalpha_{-1}^a \ket{0},$
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&
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$a = 1,\, 2,\, \dots,\, D - p - 1$
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\end{tabular}
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\end{equation*}
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\end{equationblock}
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\pause
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\begin{columns}
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\begin{column}{0.5\linewidth}
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\centering
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\resizebox{0.55\columnwidth}{!}{\import{img}{chanpaton.pgf}}
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\end{column}
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\begin{column}{0.5\linewidth}
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Introducing \textbf{Chan--Paton factors} $\tensor{\uplambda}{^r_{ij}}$, when branes are \textbf{coincident}:
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\begin{equation*}
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\bigoplus\limits_{r = 1}^N \mathrm{U}_r(1)
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\quad
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\longrightarrow
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\quad
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\mathrm{U}( N )
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\end{equation*}
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\pause
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\highlight{Build gauge bosons, fermions and scalars.}
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\end{column}
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\end{columns}
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\end{frame}
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\begin{frame}{Standard Model-like Scenarios}
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\centering
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\resizebox{0.8\linewidth}{!}{\import{img}{smbranes.pgf}}
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\end{frame}
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|
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\subsection[D-branes at Angles]{D-branes Intersecting at Angles}
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\begin{frame}{Intersecting D-branes}
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Consider \highlight{$N$ intersecting $D6$-branes} filling $\mathscr{M}^{1,3}$ and \textbf{embedded in} $\mathds{R}^6$
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\begin{equationblock}{Twist Fields Correlators}
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\begin{equation*}
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\left\langle \prod\limits_{t = 1}^{N_B} \upsigma_{\mathrm{M}_{(t)}}\qty( x_{(t)} ) \right\rangle
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=
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\mathcal{N}\qty( \qty{ x_{(t)},\, \mathrm{M}_{(t)} }_{1 \le t \le N_B} )
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e^{- S_{E\, (\text{cl})}\qty( \qty{ x_{(t)},\, \mathrm{M}_{(t)} }_{1 \le t \le N_B} )}
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\end{equation*}
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\end{equationblock}
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\pause
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\begin{tikzpicture}[remember picture, overlay]
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\draw[line width=4pt, red] (29.5em,3.5em) ellipse (2cm and 1cm);
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\end{tikzpicture}
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\pause
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\begin{columns}
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\begin{column}{0.5\linewidth}
|
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\centering
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\resizebox{0.5\columnwidth}{!}{\import{img}{branesangles.pgf}}
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\end{column}
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\begin{column}{0.5\linewidth}
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D-branes in \textbf{factorised} internal space:
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\begin{itemize}
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|
\item \textbf{embedded as lines} in $\mathds{R}^2 \times \mathds{R}^2 \times \mathds{R}^2$
|
|
|
|
\item \textbf{relative rotations} are $\mathrm{SO}(2) \simeq \mathrm{U}(1)$ elements
|
|
|
|
\item $S_{E\, (\text{cl})}\qty( \qty{ x_{(t)},\, \mathrm{M}_{(t)} }_{1 \le t \le N_B} ) \sim \text{Area}\qty( \qty{ f_{(t)},\, \mathrm{R}_{(t)} }_{1 \le t \le N_B} )$
|
|
\end{itemize}
|
|
\end{column}
|
|
\end{columns}
|
|
\end{frame}
|
|
|
|
\begin{frame}{$\mathrm{SO}(4)$ Rotations}
|
|
Consider \highlight{$\mathds{R}^4 \times \mathds{R}^2$} (focus on $\mathds{R}^4$):
|
|
|
|
\pause
|
|
|
|
\begin{columns}
|
|
\begin{column}{0.4\linewidth}
|
|
\centering
|
|
\resizebox{0.9\columnwidth}{!}{\import{img}{welladapted.pgf}}
|
|
\end{column}
|
|
|
|
\begin{column}{0.6\linewidth}
|
|
\begin{equation*}
|
|
\qty( X_{(t)} )^I
|
|
=
|
|
\tensor{\qty( R_{(t)} )}{^I_J}\, X^J - g_{(t)}^I
|
|
\end{equation*}
|
|
\pause
|
|
where
|
|
\begin{equation*}
|
|
R_{(t)} \in \frac{\mathrm{SO}(4)}{\mathrm{S}\qty( \mathrm{O}(2) \times \mathrm{O}(2) )}
|
|
\end{equation*}
|
|
\pause
|
|
that is
|
|
\begin{equation*}
|
|
\qty[ R_{(t)} ]
|
|
=
|
|
\qty{ R_{(t)} \sim \mathcal{O}_{(t)} R_{(t)} }
|
|
\end{equation*}
|
|
\end{column}
|
|
\end{columns}
|
|
\end{frame}
|
|
|
|
\begin{frame}{Boundary Conditions}
|
|
What are the consequences for \highlight{open strings?}
|
|
|
|
\pause
|
|
|
|
\begin{columns}
|
|
\begin{column}{0.6\linewidth}
|
|
\begin{itemize}
|
|
\item consider $u = x + i y = e^{\uptau_e + i \upsigma}$ and $\overline{u} = u^*$
|
|
|
|
\item let $x_{(t)} < x_{(t-1)}$ be the \textbf{worldsheet intersection points} on \textbf{real axis}
|
|
|
|
\item $X_{(t)}^{1,\, 2}$ are \textbf{Neumann}, $X_{(t)}^{3,\, 4}$ are \textbf{Dirichlet}
|
|
\end{itemize}
|
|
\end{column}
|
|
|
|
\begin{column}{0.4\linewidth}
|
|
\centering
|
|
\resizebox{0.9\columnwidth}{!}{\import{img}{branchcuts.pgf}}
|
|
\end{column}
|
|
\end{columns}
|
|
|
|
\pause
|
|
|
|
\begin{equationblock}{Branch Cuts and Discontinuities for $x \in D_{(t)}$}
|
|
\begin{equation*}
|
|
\begin{cases}
|
|
\partial_u X( x + i 0^+ )
|
|
& =
|
|
U_{(t)}
|
|
\cdot
|
|
\partial_{\overline{u}} \overline{X}( x - i 0^+ )
|
|
=
|
|
\qty[%
|
|
R_{(t)}^{-1}
|
|
\cdot
|
|
\qty( \upsigma_3 \otimes \mathds{1}_2 )
|
|
\cdot
|
|
R_{(t)}
|
|
]
|
|
\cdot
|
|
\partial_{\overline{u}} \overline{X}( x - i 0^+ )
|
|
\\
|
|
X( x_{(t)},\, x_{(t)} )
|
|
& =
|
|
f_{(t)}
|
|
\end{cases}
|
|
\end{equation*}
|
|
\end{equationblock}
|
|
\end{frame}
|
|
|
|
\begin{frame}{Doubling Trick and Spinor Representation}
|
|
\begin{equationblock}{Doubling Trick}
|
|
\begin{equation*}
|
|
\partial_z \mathcal{X}( z )
|
|
=
|
|
\begin{cases}
|
|
\partial_u X( u )
|
|
& \text{if}~z \in \mathscr{H}_{>}^{(\overline{t})}
|
|
\\
|
|
U_{(\overline{t})}\, \partial_{\overline{u}} \overline{X}( \overline{u} )
|
|
& \text{if}~z \in \mathscr{H}_{<}^{(\overline{t})}
|
|
\end{cases}
|
|
\quad
|
|
\Rightarrow
|
|
\quad
|
|
\mqty{%
|
|
\partial_{z} \mathcal{X}( x_{(t)} + e^{2 \uppi i} \updelta_+ )
|
|
=
|
|
\mathcal{U}_{(t,\, t+1)}\,
|
|
\partial_{z} \mathcal{X}( x_{(t)} + \updelta_+ ),
|
|
\\
|
|
\partial_{z} \mathcal{X}( x_{(t)} + e^{2 \uppi i} \updelta_- )
|
|
=
|
|
\widetilde{\mathcal{U}}_{(t,\, t+1)}\,
|
|
\partial_{z} \mathcal{X}( x_{(t)} + \updelta_- ),
|
|
}
|
|
\end{equation*}
|
|
where $\mathscr{H}_{\gtrless}^{(t)} = \qty{z \in \mathds{C} \mid \Im z \gtrless 0~\text{or}~z \in D_{(t)} }$ and $\updelta_{\pm} = \upeta \pm i 0^+$.
|
|
\end{equationblock}
|
|
|
|
\pause
|
|
|
|
\begin{tikzpicture}[remember picture, overlay]
|
|
\draw[line width=4pt, red] (31em,6em) ellipse (0.8cm and 1cm);
|
|
\end{tikzpicture}
|
|
|
|
\pause
|
|
|
|
Use \highlight{Pauli matrices} $\uptau = \qty( i\, \mathds{1}_2, \vec{\upsigma} )$:
|
|
\begin{equation*}
|
|
\partial_z \mathcal{X}_{(s)}( z )
|
|
=
|
|
\partial_z \mathcal{X}^I( z )\, \uptau_I
|
|
\quad
|
|
\Rightarrow
|
|
\quad
|
|
\partial_{z} \mathcal{X}( x_{(t)} + e^{2 \uppi i}\, \updelta_{\pm} )
|
|
=
|
|
\overset{\qty(\sim)}{\mathcal{L}}_{(t,\, t+1)}\,
|
|
\partial_{z} \mathcal{X}( x_{(t)} + \updelta_{\pm} )\,
|
|
\overset{\qty(\sim)}{\mathcal{R}}_{(t,\, t+1)}\,
|
|
\end{equation*}
|
|
where
|
|
\begin{equation*}
|
|
\overset{\qty(\sim)}{\mathcal{L}}_{(t,\, t+1)} \in \mathrm{SU}(2)_L
|
|
\quad
|
|
\text{and}
|
|
\quad
|
|
\overset{\qty(\sim)}{\mathcal{R}}_{(t,\, t+1)} \in \mathrm{SU}(2)_R
|
|
\end{equation*}
|
|
\end{frame}
|
|
|
|
\begin{frame}{Hypergeometric Basis}
|
|
\begin{columns}
|
|
\begin{column}{0.3\linewidth}
|
|
\centering
|
|
\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 )
|
|
\\
|
|
\frac{\qty( -\upomega_z )^{1 - c_n}}{\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\uppi \upalpha'}
|
|
\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}
|
|
|
|
\begin{frame}{Conserved Product and Operators}
|
|
Expand on a \highlight{basis of solutions}
|
|
\begin{equation*}
|
|
\uppsi_{\pm}( \upxi_{\pm} )
|
|
=
|
|
\sum\limits_{n = -\infty}^{+\infty} b_n\, \uppsi_n( \upxi_{\pm} )
|
|
\qquad
|
|
\Rightarrow
|
|
\qquad
|
|
\Uppsi( z )
|
|
=
|
|
\begin{cases}
|
|
\uppsi_{E,\, +}( u ) \quad \text{if}~z \in \mathscr{H}_{>}^{(\overline{t})}
|
|
\\
|
|
\uppsi_{E,\, -}( u ) \quad \text{if}~z \in \mathscr{H}_{<}^{(\overline{t})}
|
|
\end{cases}
|
|
\end{equation*}
|
|
|
|
\pause
|
|
|
|
\begin{equationblock}{Conserved Product and Dual Basis}
|
|
\begin{equation*}
|
|
\left\langle\!\left\langle
|
|
\tensor[^*]{\uppsi}{_n},\,
|
|
\uppsi_m
|
|
\right. \right\rangle
|
|
=
|
|
2\uppi \mathcal{N}\,
|
|
\oint
|
|
\frac{\dd{z}}{2\uppi i}\,
|
|
\tensor[^*]{\Uppsi}{_n^*}\,
|
|
\tensor{\Uppsi}{_m}
|
|
=
|
|
\updelta_{n,\, m}
|
|
\quad
|
|
\Rightarrow
|
|
\quad
|
|
\left\langle\!\left\langle
|
|
\tensor[^*]{\Uppsi}{_n^{(*)}},\,
|
|
\Uppsi^{(*)}
|
|
\right. \right\rangle
|
|
=
|
|
b_n^{(\dagger)}
|
|
\end{equation*}
|
|
\end{equationblock}
|
|
|
|
\pause
|
|
|
|
Derive the \highlight{algebra of operators:}
|
|
\begin{equation*}
|
|
\qty[ b_n,\, b_m^{\dagger} ]_+
|
|
=
|
|
\frac{2 \mathcal{N}}{T}\,
|
|
\left\langle\!\left\langle
|
|
\tensor[^*]{\Uppsi}{_n^*},\,
|
|
\Uppsi_m^*
|
|
\right. \right\rangle
|
|
\end{equation*}
|
|
\end{frame}
|
|
|
|
\begin{frame}{Twisted Complex Fermions}
|
|
Consider the case $R_{(t)} = e^{i \uppi \upalpha_{(t)}} \in \mathrm{U}( 1 )$:
|
|
\begin{equation*}
|
|
\Uppsi( x_{(t)} + e^{2\uppi i} \updelta )
|
|
=
|
|
e^{i \uppi \upepsilon_{(t)}}\,
|
|
\Uppsi( x_{(t)} + \updelta )
|
|
\end{equation*}
|
|
where
|
|
\begin{equation*}
|
|
\upepsilon_{(t)}
|
|
=
|
|
\upalpha_{(t+1)} - \upalpha_{(t)}
|
|
+
|
|
\uptheta\qty( \upalpha_{(t)} - \upalpha_{(t+1)} - 1 )
|
|
-
|
|
\uptheta\qty( \upalpha_{(t+1)} - \upalpha_{(t)} - 1 )
|
|
\end{equation*}
|
|
|
|
\pause
|
|
|
|
\begin{equationblock}{Basis of Solutions}
|
|
\begin{equation*}
|
|
\begin{split}
|
|
\Uppsi_n\qty( z;\, \qty{ x_{(t)} } )
|
|
& =
|
|
\mathcal{N}_{\Uppsi}\,
|
|
z^{-n}\,
|
|
\prod\limits_{t = 1}^N
|
|
\qty( 1 - \frac{z}{x_{(t)}} )^{n_{(t)} + \frac{\upepsilon_{(t)}}{2}}
|
|
\\
|
|
\tensor[^*]{\Uppsi}{_n}\qty( z;\, \qty{ x_{(t)} } )
|
|
& =
|
|
\frac{1}{2\uppi \mathcal{N} \mathcal{N}_{\Uppsi}}\,
|
|
z^{n - 1}\,
|
|
\prod\limits_{t = 1}^N
|
|
\qty( 1 - \frac{z}{x_{(t)}} )^{-\widetilde{n}_{(t)} + \frac{\upepsilon_{(t)}}{2}}
|
|
\end{split}
|
|
\end{equation*}
|
|
\end{equationblock}
|
|
\end{frame}
|
|
|
|
\begin{frame}{Vacua}
|
|
Define the \textbf{vacuum} with respect to $b_n$:
|
|
\begin{equation*}
|
|
\begin{split}
|
|
b_n \ket{\qty{ x_{(t)} }} = 0 &\quad \text{for} \quad n \ge 1
|
|
\\
|
|
b_n \ket{\widetilde{0}} = 0 &\quad \text{for} \quad n \ge n_{(t)} + \frac{\upepsilon_{(t)}}{2} + \frac{1}{2}
|
|
\end{split}
|
|
\end{equation*}
|
|
|
|
\pause
|
|
|
|
Theories are subject to \highlight{consistency conditions:}
|
|
\begin{columns}
|
|
\begin{column}{0.6\linewidth}
|
|
\begin{equation*}
|
|
\mathrm{L}
|
|
=
|
|
n_{(t)} + \widetilde{n}_{(t)}
|
|
\uncover<3->{%
|
|
\alert{= 0}
|
|
}
|
|
\end{equation*}
|
|
\end{column}
|
|
\hfill
|
|
\begin{column}{0.4\linewidth}
|
|
\centering
|
|
\resizebox{\columnwidth}{!}{\import{img}{inconsistent_theories.pgf}}
|
|
\end{column}
|
|
\end{columns}
|
|
\end{frame}
|
|
|
|
\begin{frame}{Stress-energy Tensor and CFT Approach}
|
|
Compute the OPEs leading to the \highlight{stress-energy tensor:}
|
|
\begin{equation*}
|
|
\mathcal{T}( z )
|
|
=
|
|
\frac{\uppi T}{2} \mathcal{N}_{\Uppsi}^2
|
|
\sum\limits_{n,\, m = -\infty}^{+\infty}
|
|
\colon b_n\, b_m^* \colon\,
|
|
z^{-n -m}\,
|
|
\qty[%
|
|
\frac{m - n}{2}
|
|
+
|
|
2 \sum\limits_{t = 1}^N \frac{n_{(t)} + \frac{\upepsilon_{(t)}}{2}}{z - x_{(t)}}
|
|
]
|
|
+
|
|
\frac{1}{2} \qty( \sum\limits_{t = 1}^N \frac{n_{(t)} + \frac{\upepsilon_{(t)}}{2}}{z - x_{(t)}} )^2
|
|
\end{equation*}
|
|
|
|
\pause
|
|
|
|
\begin{equationblock}{Invariant Vacuum and Spin Fields}
|
|
\begin{equation*}
|
|
\ket{\qty{ x_{(t)} }}
|
|
=
|
|
\mathcal{N}\qty( \qty{ x_{(t)} } )\,
|
|
\mathrm{R}\qty[ \prod\limits_{t = 1}^M S_{(t)}( x_{(t)} ) ]\,
|
|
\ket{0}_{\mathrm{SL}_2( \mathds{R} )}
|
|
\end{equation*}
|
|
\end{equationblock}
|
|
\end{frame}
|
|
|
|
\begin{frame}{Spin Fields Amplitudes}
|
|
\begin{equationblock}{Equivalence with Bosonization}
|
|
\begin{equation*}
|
|
\begin{split}
|
|
\partial_{x_{(t)}} \braket{\qty{x_{(t)}}}
|
|
& =
|
|
\oint\limits_{x_{(t)}} \frac{\dd{z}}{2\uppi i}
|
|
\frac{%
|
|
\bra{\qty{x_{(t)}}} \mathcal{T}( z ) \ket{\qty{x_{(t)}}}
|
|
}{%
|
|
\braket{\qty{x_{(t)}}}
|
|
}
|
|
\\
|
|
\Rightarrow
|
|
\quad
|
|
\braket{\qty{x_{(t)}}}
|
|
& =
|
|
\mathcal{N}\qty( \qty{ \upepsilon_{(t)} } )
|
|
\prod\limits_{\substack{t = 1 \\ t > u}}^N
|
|
\qty( x_{(u)} - x_{(t)} )^{\qty( n_{(u)} + \frac{\upepsilon_{(u)}}{2} )\qty( n_{(t)} + \frac{\upepsilon_{(t)}}{2} )}
|
|
\end{split}
|
|
\end{equation*}
|
|
\end{equationblock}
|
|
|
|
\pause
|
|
|
|
\begin{itemize}
|
|
\item (semi-)phenomenological models involve \textbf{twist and spin} fields and \textbf{open strings}
|
|
|
|
\pause
|
|
|
|
\item general framework for \textbf{bosonic} open strings with \textbf{intersecting D-branes}
|
|
|
|
\pause
|
|
|
|
\item leading contribution for \textbf{twist fields}
|
|
|
|
\pause
|
|
|
|
\item \textbf{spin fields} as \textbf{boundary changing operators} on \textbf{defects}
|
|
|
|
\pause
|
|
|
|
\item alternative framework for amplitudes (extension to (non) Abelian twist/spin fields?)
|
|
\end{itemize}
|
|
\end{frame}
|
|
|
|
|
|
\section[Time Divergences]{Cosmological Backgrounds and Divergences}
|
|
|
|
|
|
\subsection[Orbifold]{Orbifolds and Cosmological Toy Models}
|
|
|
|
\begin{frame}{A Few Words on a Theory of Everything}
|
|
\begin{center}
|
|
string theory = theory of everything = nuclear forces + gravity
|
|
\end{center}
|
|
|
|
\pause
|
|
|
|
\begin{columns}
|
|
\begin{column}{0.5\linewidth}
|
|
\centering
|
|
\includegraphics[width=0.9\columnwidth]{img/cone}
|
|
\end{column}
|
|
\hfill
|
|
\begin{column}{0.5\linewidth}
|
|
From the phenomenological point of view:
|
|
\begin{itemize}
|
|
\item cosmological implications
|
|
|
|
\pause
|
|
|
|
\item Big Bang(-like) singularities
|
|
|
|
\pause
|
|
|
|
\item toy models of \textbf{space-like singularities}
|
|
\end{itemize}
|
|
|
|
\pause
|
|
|
|
\begin{center}
|
|
$\Downarrow$
|
|
|
|
\highlight{time-dependent orbifold models}
|
|
\end{center}
|
|
\end{column}
|
|
\end{columns}
|
|
\end{frame}
|
|
|
|
\begin{frame}{Orbifolds}
|
|
\begin{columns}[c]
|
|
\begin{column}{0.475\linewidth}
|
|
\begin{center}
|
|
\textbf{Mathematics}
|
|
|
|
\begin{itemize}
|
|
\item manifold $M$
|
|
|
|
\item (Lie) group $G$
|
|
|
|
\item \emph{stabilizer} $G_p = \qty{g \in G \mid gp = p \in M}$
|
|
|
|
\item \emph{orbit} $Gp = \qty{gp \in M \mid g \in G}$
|
|
|
|
\item charts $\upphi = \uppi \circ \mathscr{P}$ where:
|
|
|
|
\begin{itemize}
|
|
\item $\mathscr{P}\colon U \subset \mathds{R}^n \to U / G$
|
|
|
|
\item $\uppi\colon U / G \to M$
|
|
\end{itemize}
|
|
\end{itemize}
|
|
\end{center}
|
|
\end{column}
|
|
\begin{column}{0.05\linewidth}
|
|
\centering
|
|
$\Rightarrow$
|
|
\end{column}
|
|
\begin{column}{0.475\linewidth}
|
|
\begin{center}
|
|
\textbf{Physics}
|
|
|
|
\begin{itemize}
|
|
\item global orbit space $M / G$
|
|
|
|
\item $G$ group of isometries
|
|
|
|
\item fixed points
|
|
|
|
\item additional d.o.f.\ (\emph{twisted states})
|
|
|
|
\item singular limits of CY manifolds
|
|
\end{itemize}
|
|
\end{center}
|
|
\end{column}
|
|
\end{columns}
|
|
|
|
\pause
|
|
|
|
\begin{center}
|
|
time-dependent orbifolds
|
|
\end{center}
|
|
|
|
\begin{tikzpicture}[remember picture, overlay]
|
|
\draw[line width=4pt, red] (13em,3.5em) rectangle (27em, 1em);
|
|
\end{tikzpicture}
|
|
\end{frame}
|
|
|
|
\begin{frame}{Cosmological Singularities}
|
|
Use \textbf{time-dependent orbifolds} to model \textbf{space-like singularities}:
|
|
|
|
\begin{center}
|
|
divergent \highlight{closed string} aplitudes
|
|
$\Rightarrow$
|
|
gravitational backreaction?
|
|
\end{center}
|
|
|
|
\pause
|
|
|
|
\begin{block}{Divergences}
|
|
Even in simple models (e.g.\ NBO, more on this later) the $4$ tachyons amplitude is divergent \textbf{at tree level}:
|
|
\begin{equation*}
|
|
A_4 \sim \int\limits_{q \sim \infty} \frac{\dd{q}}{\abs{q}} \mathscr{A}( q )
|
|
\end{equation*}
|
|
where
|
|
\begin{equation*}
|
|
\mathscr{A}_{\text{closed}}( q ) \sim q^{4 - \upalpha' \norm{\vec{p}_{\perp}}^2}
|
|
\qquad
|
|
\text{and}
|
|
\qquad
|
|
\mathscr{A}_{\text{open}}( q ) \sim q^{1 - \upalpha' \norm{\vec{p}_{\perp}}^2} \trace(\qty[T_1,\, T_2]_+\, \qty[T_3,\, T_4]_+)
|
|
\end{equation*}
|
|
\end{block}
|
|
\end{frame}
|
|
|
|
|
|
\subsection[NBO]{Null Boost Orbifold}
|
|
|
|
\begin{frame}{Null Boost Orbifold}
|
|
Start from $\qty( x^+,\, x^-,\, x^2,\, \vec{x} ) \in \mathscr{M}^{1,\, D-1}$:
|
|
\begin{equation*}
|
|
\begin{cases}
|
|
u & = x^-
|
|
\\
|
|
z & = \frac{x^2}{\Updelta\, x^-}
|
|
\\
|
|
v & = x^+ - \frac{1}{2} \frac{\qty( x^2 )^2}{x^-}
|
|
\end{cases}
|
|
\qquad
|
|
\Rightarrow
|
|
\qquad
|
|
\dd{s}^2 = -2 \dd{u} \dd{v} + \qty( \Updelta\, u )^2\, \dd{z}^2 + \updelta_{ij} \dd{x}^i \dd{x}^j
|
|
\end{equation*}
|
|
|
|
\pause
|
|
|
|
\begin{equationblock}{Killing Vector and Null Boost Oribfold}
|
|
\begin{equation*}
|
|
\upkappa = -i \qty( 2 \uppi \Updelta ) J_{+2} = 2 \uppi \partial_z
|
|
\Rightarrow
|
|
z \sim z + 2 \uppi n
|
|
\end{equation*}
|
|
\end{equationblock}
|
|
|
|
\pause
|
|
|
|
Consider \highlight{scalar QED:}
|
|
\begin{equation*}
|
|
\upphi_{\qty{ k_+,\, l,\, \vec{k},\, r}}\qty( u,\, v,\, z,\, \vec{x} )
|
|
=
|
|
e^{i \qty( k_+ v + l z + \vec{k} \cdot \vec{x} )}\,
|
|
\widetilde{\upphi}_{\qty{ k_+,\, l,\, \vec{k},\, r}}\qty( u )
|
|
=
|
|
\frac{e^{i \qty( k_+ v + l z + \vec{k} \cdot \vec{x} )}}{\sqrt{\qty( 2 \uppi )^D\, \abs{2 \Updelta k_+ u}}}\,
|
|
e^{-i \frac{l^2}{2 \Updelta^2 k_+} \frac{1}{u} + i \frac{\norm{\vec{k}}^2 + r}{2 k_+} u}
|
|
\end{equation*}
|
|
\end{frame}
|
|
|
|
\begin{frame}{Scalar QED Interactions}
|
|
Scalar--photon interactions:
|
|
\begin{equation*}
|
|
S_{\text{sQED}}^{\text{(int)}}
|
|
=
|
|
\int\limits_{\Upomega} \dd[D]{x} \sqrt{- g }\,
|
|
\qty(%
|
|
-i\, e\, g^{\upalpha\upbeta} a_{\upalpha} \qty( \upphi^*\, \partial_{\upbeta} \upphi - \partial_{\upbeta} \upphi^*\, \upphi )
|
|
+ e^2\, g^{\upalpha\upbeta} a_{\upalpha} a_{\upbeta} \abs{\upphi}^2
|
|
- \frac{g_4}{4}\, \abs{\upphi}^4
|
|
)
|
|
\end{equation*}
|
|
|
|
\pause
|
|
|
|
Terms involved:
|
|
\begin{equation*}
|
|
\begin{split}
|
|
\mathcal{I}^{\qty[\upnu]}_{\qty{N}}
|
|
& =
|
|
\int\limits_{-\infty}^{+\infty} \dd{u}
|
|
\abs{\Updelta\, u} u^{\upnu}
|
|
\prod\limits_{i = 1}^N
|
|
\widetilde{\upphi}_{\qty{ k_{+\, (i)},\, l_{(i)},\, \vec{k}_{(i)},\, r_{(i)}}}\qty( u )
|
|
\\
|
|
\mathcal{J}^{\qty[\upnu]}_{\qty{N}}
|
|
& =
|
|
\int\limits_{-\infty}^{+\infty} \dd{u}
|
|
\abs{\Updelta} \abs{u}^{1 + \upnu}
|
|
\prod\limits_{i = 1}^N
|
|
\widetilde{\upphi}_{\qty{ k_{+\, (i)},\, l_{(i)},\, \vec{k}_{(i)},\, r_{(i)}}}\qty( u )
|
|
\end{split}
|
|
\end{equation*}
|
|
|
|
\pause
|
|
|
|
\begin{center}
|
|
\it
|
|
most terms \textbf{do not converge} and cannot be recovered even with a \textbf{distributional interpretions} due to the term $\propto u^{-1}$ in the exponentatial
|
|
\end{center}
|
|
\end{frame}
|
|
|
|
\begin{frame}{String and Field Theory}
|
|
So far:
|
|
\begin{itemize}
|
|
\item field theory presents \textbf{divergences}
|
|
|
|
\pause
|
|
|
|
\item issues are \textbf{still present} in sQED (eikonal?)
|
|
|
|
\pause
|
|
|
|
\item divergences are \textbf{not (only) gravitational}
|
|
\end{itemize}
|
|
|
|
\pause
|
|
|
|
\begin{equationblock}{Massive String States}
|
|
\begin{equation*}
|
|
V_M\qty( x;\, k,\, S,\, \upxi )
|
|
=
|
|
\colon
|
|
\qty(%
|
|
\frac{i}{\sqrt{2 \upalpha'}}\,
|
|
\upxi \cdot \partial^2_x X( x,\, x )
|
|
+
|
|
\qty( \frac{i}{\sqrt{2 \upalpha'}} )^2\,
|
|
S_{\upalpha\upbeta}
|
|
\partial_x X^{\upalpha}( x,\, x )
|
|
\partial_x X^{\upbeta}( x,\, x )
|
|
)
|
|
e^{i k \cdot X( x,\, x )}
|
|
\colon
|
|
\end{equation*}
|
|
\end{equationblock}
|
|
|
|
\pause
|
|
|
|
\begin{center}
|
|
\it
|
|
string theory cannot do \textbf{better than field theory} (EFT) if the latter \textbf{does not exist} (even a Wilson line around $z$ does not prevent such behaviour)
|
|
\end{center}
|
|
\end{frame}
|
|
|
|
\begin{frame}{Resolution and Motivation}
|
|
Introduce the \highlight{generalised NBO:}
|
|
\begin{equation*}
|
|
\begin{cases}
|
|
u & = x^-
|
|
\\
|
|
z & = \frac{1}{2 x^-} \qty( \frac{x^2}{\Updelta_2} + \frac{x^3}{\Updelta_3} )
|
|
\\
|
|
w & = \frac{1}{2 x^-} \qty( \frac{x^2}{\Updelta_2} - \frac{x^3}{\Updelta_3} )
|
|
\\
|
|
v & = x^+ - \frac{1}{2 x^-} \qty( \qty( x^2 )^2 + \qty( x^3 )^2 )
|
|
\end{cases}
|
|
\qquad
|
|
\Rightarrow
|
|
\qquad
|
|
\upkappa
|
|
=
|
|
-2 \uppi i \qty( \Updelta_2 J_{+2} + \Updelta_3 J_{+3} )
|
|
=
|
|
2 \uppi \partial_z
|
|
\end{equation*}
|
|
|
|
\pause
|
|
|
|
\begin{equationblock}{Distributional Interpretation}
|
|
\begin{equation*}
|
|
\widetilde{\upphi}_{\qty{ k_+,\, p,\, l,\, \vec{k},\, r}}\qty( u )
|
|
=
|
|
\frac{1}{2 \sqrt{\qty(2 \uppi)^D \abs{\Updelta_2 \Updelta_3 k_+}}}
|
|
\frac{1}{\abs{u}}
|
|
e^{-i\, \qty( \frac{1}{8 k_+ u} \qty[ \frac{(l + p)^2}{\Updelta_2^2} + \frac{(l - p)^2}{\Updelta_3^2} ] - \frac{\norm{\vec{k}}^2 + r}{2 k_+} u )}
|
|
\end{equation*}
|
|
\end{equationblock}
|
|
\end{frame}
|
|
|
|
\begin{frame}{On the Divergences and Their Nature}
|
|
\begin{itemize}
|
|
\item divergences are present in sQED and \textbf{open string} sector
|
|
|
|
\pause
|
|
|
|
\item singularities $\Rightarrow$ \textbf{massive states} are no longer spectators
|
|
|
|
\pause
|
|
|
|
\item vanishing volume (\textbf{compact orbifold directions}) $\Rightarrow$ particles ``cannot escape''
|
|
|
|
\pause
|
|
|
|
\item \textbf{non compact} orbifold directions $\Rightarrow$ interpretation of \textbf{amplitudes as distributions}
|
|
|
|
\pause
|
|
|
|
\item issue not restricted to NBO/GNBO but also BO, null brane, etc. (it is a \textbf{general issues} connected to the geometry of the underlying space)
|
|
\end{itemize}
|
|
|
|
\pause
|
|
|
|
\vspace{2em}
|
|
\begin{center}
|
|
\it
|
|
spacetime singularities are \textbf{hidden into contact terms} and interactions with \textbf{massive states} (the gravitational eikonal deals with massless interactions)
|
|
\end{center}
|
|
|
|
\begin{tikzpicture}[remember picture, overlay]
|
|
\draw[line width=4pt, red] (0em, 4.5em) rectangle (40em, 1em);
|
|
\end{tikzpicture}
|
|
\end{frame}
|
|
|
|
|
|
\section[Deep Learning]{Deep Learning the Geometry of String Theory}
|
|
|
|
|
|
\subsection[Introduction]{Machine Learning and Deep Learning}
|
|
|
|
\begin{frame}{The Simplest Calabi--Yau}
|
|
Focus on Calabi--Yau \highlight{3-folds:}
|
|
\begin{equation*}
|
|
h^{r,\, s} = \dim\limits_{\mathds{C}} H_{\overline{\partial}}^{r,\, s}\qty( M,\, \mathds{C} )
|
|
\qquad
|
|
\Rightarrow
|
|
\qquad
|
|
\begin{cases}
|
|
h^{0,\, 0} & = h^{3,\, 0} = 1
|
|
\\
|
|
h^{r,\, 0} & = 0 \quad \text{if} \quad r \neq 3
|
|
\\
|
|
h^{r,\, s} & = h^{3 - r,\, 3 - s}
|
|
\\
|
|
h^{1,\, 1},\, h^{2,\, 1} \in \mathds{N}
|
|
\end{cases}
|
|
\end{equation*}
|
|
|
|
\pause
|
|
|
|
\begin{block}{Complete Intersection Calabi--Yau Manifolds}
|
|
Intersection of hypersurfaces in
|
|
\begin{equation*}
|
|
\mathcal{A} = \mathds{P}^{n_1} \times \dots \times \mathds{P}^{n_m}
|
|
\end{equation*}
|
|
where
|
|
\begin{equation*}
|
|
\mathds{P}^n\colon
|
|
\qquad
|
|
\begin{cases}
|
|
p_a\qty( Z^0,\, \dots,\, Z^n ) & = P_{I_1 \dots I_a} Z^{I_1} \dots Z^{I_a} = 0
|
|
\\
|
|
p_a\qty( \uplambda Z^0,\, \dots,\, \uplambda Z^n ) & = \uplambda^a p_a\qty( Z^0,\, \dots,\, Z^n )
|
|
\end{cases}
|
|
\end{equation*}
|
|
\end{block}
|
|
\end{frame}
|
|
|
|
\begin{frame}{Representation of the Output}
|
|
CICY can be generalised to \highlight{$m$ projective spaces and $k$ equations.}
|
|
The problem is thus mapped to:
|
|
\begin{equation*}
|
|
\begin{tabular}{@{}lccc@{}}
|
|
$\mathscr{R}\colon$
|
|
&
|
|
$\mathds{Z}^{m \times k}$
|
|
&
|
|
$\longrightarrow$
|
|
&
|
|
$\mathds{N}$
|
|
\\[1em]
|
|
&
|
|
$\qty[%
|
|
\begin{tabular}{@{}c|ccc@{}}
|
|
$\mathds{P}^{n_1}$ & $a_1^1$ & $\dots$ & $a_k^1$
|
|
\\
|
|
$\vdots$ & $\vdots$ & $\ddots$ & $\vdots$
|
|
\\
|
|
$\mathds{P}^{n_m}$ & $a_1^m$ & $\dots$ & $a_k^m$
|
|
\end{tabular}
|
|
]$
|
|
&
|
|
$\longrightarrow$
|
|
&
|
|
$h^{1,\, 1} \quad \text{or} \quad h^{2,\, 1}$
|
|
\end{tabular}
|
|
\end{equation*}
|
|
|
|
\pause
|
|
|
|
\begin{block}{Machine Learning Approach}
|
|
What is $\mathscr{R}$?
|
|
\begin{equation*}
|
|
\mathscr{R}( M ) \longrightarrow \mathscr{R}_n( M;\, w )
|
|
\qquad
|
|
\text{s.t.}
|
|
\qquad
|
|
\lim\limits_{n \to \infty} f( M;\, w ) = \lim\limits_{n \to \infty} \abs{\mathscr{R}( M ) - \mathscr{R}_n( M;\, w )} = 0
|
|
\end{equation*}
|
|
\end{block}
|
|
\end{frame}
|
|
|
|
\begin{frame}{Machine Learning}
|
|
\begin{itemize}
|
|
\item exchange \textbf{analytical solution} with \textbf{optimisation problem}
|
|
|
|
\pause
|
|
|
|
\item use \textbf{various algorithms} and exploit \textbf{large datasets}
|
|
|
|
\pause
|
|
|
|
\item learn a \textbf{representation} rather than a \textbf{solution}
|
|
|
|
\pause
|
|
|
|
\item effectively use knowledge from \textbf{computer science, mathematics and physics} to solve problems
|
|
\end{itemize}
|
|
|
|
\pause
|
|
|
|
\begin{center}
|
|
\includegraphics[width=0.7\linewidth]{img/label-distribution_orig}
|
|
\end{center}
|
|
\end{frame}
|
|
|
|
|
|
\subsection[Machine Learning]{Machine Learning for String Theory}
|
|
|
|
|
|
\begin{frame}{Exploratory Data Analysis}
|
|
Machine Learning \highlight{pipeline:}
|
|
\begin{center}
|
|
\textbf{exploratory} data analysis
|
|
$\rightarrow$
|
|
feature \textbf{selection}
|
|
$\rightarrow$
|
|
Hodge numbers
|
|
\end{center}
|
|
|
|
\pause
|
|
|
|
\begin{columns}
|
|
\begin{column}{0.5\linewidth}
|
|
\centering
|
|
\includegraphics[width=0.9\columnwidth]{img/corr-matrix_orig}
|
|
\end{column}
|
|
\hfill
|
|
\begin{column}{0.5\linewidth}
|
|
\centering
|
|
\includegraphics[width=0.9\columnwidth, trim={0 0 6in 0}, clip]{img/scalar-features_orig}
|
|
\end{column}
|
|
\end{columns}
|
|
\end{frame}
|
|
|
|
\begin{frame}{Dataset}
|
|
\begin{itemize}
|
|
\item $7890$ CICY manifolds (full dataset)
|
|
|
|
\pause
|
|
|
|
\item \textbf{dataset pruning}: no product spaces, no ``very far'' outliers (reduction of $0.49\%$)
|
|
|
|
\pause
|
|
|
|
\item $h^{1,\, 1} \in \qty[ 1,\, 16 ]$ and $h^{2,\, 1} \in \qty[ 15,\, 86 ]$
|
|
|
|
\pause
|
|
|
|
\item $80\%$ training, $10\%$ validation, $10\%$ test
|
|
|
|
\pause
|
|
|
|
\item choose \textbf{regression}, but evaluate using \textbf{accuracy} (round the result)
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\end{itemize}
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\pause
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|
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|
\begin{center}
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|
\includegraphics[width=0.7\linewidth]{img/label-distribution-compare_orig}
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\end{center}
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\end{frame}
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\begin{frame}{Machine Learning}
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\centering
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|
\includegraphics[width=0.85\linewidth]{img/ml_map}
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\pause
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|
|
|
\begin{tikzpicture}[remember picture, overlay]
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|
\draw[line width=10pt, red, -latex] (-18em,1em) -- (-14.5em, 6em);
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\draw[line width=10pt, red, -latex] (19em, 7em) -- (14em, 4em);
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\end{tikzpicture}
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\pause
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|
|
|
\begin{tikzpicture}[remember picture, overlay]
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\draw[line width=4pt, red] (12em,12em) ellipse (2cm and 1.5cm);
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\end{tikzpicture}
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\end{frame}
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|
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\begin{frame}{A Word on PCA}
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\begin{columns}
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|
\begin{column}{0.4\linewidth}
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|
What is PCA for a $X \in \mathds{R}^{n \times p}$?
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\begin{itemize}
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|
\item find new coordinates to \textbf{``put the variance in order''}
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|
|
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\item \highlight{equivalently} compute the \textbf{eigenvectors} of $X X^T$ or the \textbf{singular values} of $X$
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|
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\item isolate \textbf{the signal} from the \textbf{background}
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\item ease the machine learning job of finding a better representation of the input
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\end{itemize}
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\end{column}
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|
\hfill
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|
\begin{column}{0.6\linewidth}
|
|
\centering
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|
\includegraphics[width=0.5\columnwidth]{img/marchenko-pastur}
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|
\includegraphics[width=\columnwidth]{img/svd_orig}
|
|
\end{column}
|
|
\end{columns}
|
|
\end{frame}
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|
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\begin{frame}{Machine Learning Results}
|
|
\begin{columns}
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|
\begin{column}{0.5\linewidth}
|
|
\centering
|
|
\textbf{Configuration Matrix Only}
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|
\includegraphics[width=0.8\columnwidth, trim={0 0 3.3in 0}, clip]{img/cicy_matrix_plots}
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|
\end{column}
|
|
\hfill\pause
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|
\begin{column}{0.5\linewidth}
|
|
\centering
|
|
\textbf{Best Training Set}
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|
\includegraphics[width=0.8\columnwidth, trim={0 0 3.3in 0}, clip]{img/cicy_best_plots}
|
|
\end{column}
|
|
\end{columns}
|
|
\end{frame}
|
|
|
|
\subsection[Deep Learning]{AI Implementations for Geometry and Strings}
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|
|
|
\begin{frame}{Artificial Intelligence and Neural Networks}
|
|
\begin{columns}
|
|
\begin{column}{0.6\linewidth}
|
|
\begin{itemize}
|
|
\item use \textbf{gradient descent} to optimise \textbf{weights}
|
|
|
|
\item learn highly \textbf{non linear} representations of the input
|
|
|
|
\item can be \highlight{``large''} to have enough parameters
|
|
|
|
\item can be \highlight{``deep''} to to learn \textbf{complicated functions}
|
|
\end{itemize}
|
|
|
|
\begin{block}{Neural Networks}
|
|
\vspace{0.5em}
|
|
\begin{tabular}{@{}lc@{}}
|
|
fully connected:
|
|
&
|
|
$a^{\qty(i)\, \qty{l+1}} = \upphi\qty( a^{\qty(i)\, \qty{l}} \cdot W^{\qty{l}} + b^{\qty{l}} \mathds{1} )$
|
|
\\
|
|
convolutional:
|
|
&
|
|
$a^{\qty(i)\, \qty{l+1}} = \upphi\qty( a^{\qty(i)\, \qty{l}}\, *\, W^{\qty{l}} + b^{\qty{l}} \mathds{1} )$
|
|
\end{tabular}
|
|
\end{block}
|
|
|
|
Non linearity ensured by:
|
|
\begin{equation*}
|
|
\upphi( z ) = \mathrm{ReLU}\qty( z ) = \max\qty(0,\, z)
|
|
\end{equation*}
|
|
\end{column}
|
|
\hfill
|
|
\begin{column}{0.4\linewidth}
|
|
\centering
|
|
\resizebox{\columnwidth}{!}{\import{img}{fc.pgf}}
|
|
\end{column}
|
|
\end{columns}
|
|
\end{frame}
|
|
|
|
\begin{frame}{Convolutional Neural Networks}
|
|
Why convolutional?
|
|
\begin{columns}
|
|
\begin{column}{0.4\linewidth}
|
|
\begin{itemize}[<+->]
|
|
\item retain \textbf{spacial awareness}
|
|
|
|
\item smaller \textbf{no.\ of parameters} ($\approx 2 \times 10^5$ vs.\ $\approx 2 \times 10^6$)
|
|
|
|
\item weights are \textbf{shared}
|
|
|
|
\item CNN can isolate \textbf{``defining features''}
|
|
|
|
\item find patterns as in \textbf{computer vision}
|
|
\end{itemize}
|
|
\end{column}
|
|
\hfill
|
|
\begin{column}{0.6\linewidth}
|
|
\centering
|
|
\only<1>{\includegraphics[width=0.75\columnwidth, trim={12in 10in 0 0}, clip]{img/input_mat}}
|
|
\only<2>{\includegraphics[width=0.75\columnwidth, trim={12in 5in 0 5in}, clip]{img/input_mat}}
|
|
\only<3>{\includegraphics[width=0.75\columnwidth, trim={12in 0 0 10in}, clip]{img/input_mat}}
|
|
\only<4->{\resizebox{\columnwidth}{!}{\import{img}{ccnn.pgf}}}
|
|
\end{column}
|
|
\end{columns}
|
|
\end{frame}
|
|
|
|
\begin{frame}{Inception Neural Networks}
|
|
Recent development by Google's deep learning teams led to:
|
|
\begin{itemize}
|
|
\item neural networks with \textbf{better generalisation properties}
|
|
|
|
\item smaller networks (both parameters and depth)
|
|
|
|
\item different \textbf{concurrent kernels} (e.g.\ one over \textbf{equations} one over \textbf{coordinates})
|
|
\end{itemize}
|
|
|
|
\pause
|
|
|
|
\begin{center}
|
|
\resizebox{0.75\linewidth}{!}{\import{img}{icnn.pgf}}
|
|
\end{center}
|
|
\end{frame}
|
|
|
|
\begin{frame}{Deep Learning Results and Generalisation Properties}
|
|
\begin{columns}
|
|
\begin{column}{0.5\linewidth}
|
|
\centering
|
|
\textbf{Best Training Set}
|
|
\includegraphics[width=\columnwidth]{img/cicy_best_plots}
|
|
\end{column}
|
|
\hfill
|
|
\begin{column}{0.5\linewidth}
|
|
\centering
|
|
\includegraphics[width=0.75\columnwidth]{img/inc_nn_learning_curve_h11}
|
|
\includegraphics[width=0.75\columnwidth]{img/inc_nn_learning_curve}
|
|
\end{column}
|
|
\end{columns}
|
|
\end{frame}
|
|
|
|
\begin{frame}{A Few Comments and Future Directions}
|
|
Why \highlight{deep learning in physics?}
|
|
\begin{itemize}
|
|
\item reliable \textbf{predictive method} \pause (provided good data analysis)
|
|
|
|
\pause
|
|
|
|
\item reliable \textbf{source of inspiration} \pause (provided good data analysis)
|
|
|
|
\pause
|
|
|
|
\item reliable \textbf{generalisation method} \pause (provided good data analysis)
|
|
|
|
\pause
|
|
|
|
\item \textbf{CNNs are powerful tools} (this is the \emph{first time in physics!})
|
|
|
|
\pause
|
|
|
|
\item interdisciplinary approach $=$ win-win situation!
|
|
\end{itemize}
|
|
|
|
\pause
|
|
|
|
What now?
|
|
|
|
\begin{itemize}
|
|
\item representation learning $\Rightarrow$ what is the best way to represent CICYs?
|
|
|
|
\pause
|
|
|
|
\item study invariances $\Rightarrow$ invariances should not influence the result (graph representations?)
|
|
|
|
\pause
|
|
|
|
\item higher dimensions $\Rightarrow$ what about CICY 4-folds?
|
|
|
|
\pause
|
|
|
|
\item geometric deep learning $\Rightarrow$ explain the geometry of the ``AI'' behind deep learning!
|
|
|
|
\pause
|
|
|
|
\item reinforcement learning $\Rightarrow$ give the rules, not the result!
|
|
\end{itemize}
|
|
\end{frame}
|
|
|
|
{%
|
|
\setbeamertemplate{footline}{}
|
|
\usebackgroundtemplate{%
|
|
\transparent{0.1}
|
|
\includegraphics[width=\paperwidth]{img/torino.png}
|
|
}
|
|
\addtobeamertemplate{background canvas}{\transfade[duration=0.25]}{}
|
|
\begin{frame}[noframenumbering]{The End?}
|
|
\begin{center}
|
|
\Huge
|
|
THANK YOU!
|
|
\end{center}
|
|
\end{frame}
|
|
}
|
|
|
|
\end{document}
|