1190 lines
33 KiB
TeX
1190 lines
33 KiB
TeX
\documentclass[10pt, aspectratio=169]{beamer}
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\usepackage{physics}
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\usepackage{tensor}
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\usepackage{graphicx}
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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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\begin{figure}
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\centering
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\includegraphics[width=7em]{\firstlogo}
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% \AtBeginSection[]
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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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\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}
|
|
\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{block}{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{block}
|
|
|
|
\pause
|
|
|
|
\begin{tikzpicture}[remember picture, overlay]
|
|
\draw[line width=4pt, red] (31em,6em) ellipse (0.8cm and 1.2cm);
|
|
\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 )
|
|
\\
|
|
\qty( -\upomega_z )^{1 - c_n}\,
|
|
\frac{1}{\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)}}}
|
|
}
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\\
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\Rightarrow
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\quad
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\braket{\qty{x_{(t)}}}
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& =
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\mathcal{N}\qty( \qty{ \upepsilon_{(t)} } )
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\prod\limits_{\substack{t = 1 \\ t > u}}^N
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\qty( x_{(u)} - x_{(t)} )^{\qty( n_{(u)} + \frac{\upepsilon_{(u)}}{2} )\qty( n_{(t)} + \frac{\upepsilon_{(t)}}{2} )}
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\end{split}
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\end{equation*}
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\end{equationblock}
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\pause
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\begin{itemize}
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\item (semi-)phenomenological models involve \textbf{twist and spin} fields and \textbf{open strings}
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\pause
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|
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\item general framework for \textbf{bosonic} open strings with \textbf{intersecting D-branes}
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|
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|
\pause
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|
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\item leading contribution for \textbf{twist fields}
|
|
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|
\pause
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|
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|
\item \textbf{spin fields} as \textbf{boundary changing operators} on \textbf{defects}
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|
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|
\pause
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|
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|
\item alternative framework for amplitudes (extension to (non) Abelian twist/spin fields?)
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\end{itemize}
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\end{frame}
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|
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\section[Time Divergences]{Cosmological Backgrounds and Divergences}
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\subsection[Orbifold]{Orbifolds and Cosmological Toy Models}
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\begin{frame}{A Few Words on a Theory of Everything}
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\begin{center}
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string theory = theory of everything = nuclear forces + gravity
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\end{center}
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|
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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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|
\includegraphics[width=0.9\columnwidth]{img/cone}
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\end{column}
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|
\hfill
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|
\begin{column}{0.5\linewidth}
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|
From the phenomenological point of view:
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\begin{itemize}
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|
\item cosmological implications
|
|
|
|
\pause
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|
|
|
\item Big Bang(-like) singularities
|
|
|
|
\pause
|
|
|
|
\item toy models of \textbf{space-like singularities}
|
|
\end{itemize}
|
|
|
|
\pause
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|
|
|
\begin{center}
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|
$\Downarrow$
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|
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|
\highlight{time-dependent orbifold models}
|
|
\end{center}
|
|
\end{column}
|
|
\end{columns}
|
|
\end{frame}
|
|
|
|
\begin{frame}{Orbifolds}
|
|
\begin{columns}[c]
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|
\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}
|
|
\end{frame}
|
|
|
|
|
|
\section[Deep Learning]{Deep Learning the Geometry of String Theory}
|
|
|
|
\begin{frame}{CCC}
|
|
c
|
|
\end{frame}
|
|
|
|
\end{document}
|