615 lines
17 KiB
TeX
615 lines
17 KiB
TeX
\documentclass[10pt, aspectratio=169]{beamer}
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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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{%
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% \setbeamertemplate{footline}{}
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\begin{frame}[noframenumbering]{\contentsname}
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\tableofcontents{}
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\end{frame}
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}
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\section[CFT]{Conformal Symmetry and Geometry of the Worldsheet}
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\subsection[Preliminary]{Preliminary Concepts and Tools}
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\begin{frame}{Action Principle and Conformal Symmetry}
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\begin{equationblock}{Polyakov's Action}
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\begin{equation*}
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S_P\qty[ \upgamma,\, X,\, \uppsi ]
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=
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-\frac{1}{4\pi}
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\int\limits_{-\infty}^{+\infty} \dd{\uptau}
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\int\limits_0^{\ell} \dd{\upsigma}
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\sqrt{-\det \upgamma}\,
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\upgamma^{\upalpha \upbeta}\,
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\qty(%
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\frac{2}{\upalpha'}\,
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\partial_{\upalpha} X^{\upmu}\,
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\partial_{\upbeta} X^{\upnu}
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+
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\uppsi^{\upmu}\,
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\uprho_{\upalpha}
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\partial_{\upbeta}
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\uppsi^{\upnu}
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)\,
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\upeta_{\upmu\upnu}
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\end{equation*}
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\end{equationblock}
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\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.}\ $\upgamma'_{\upalpha \upbeta} = \tensor{\qty( \mathrm{J}^{-1} )}{_{\upalpha \upbeta}^{\uplambda \uprho}}\, \gamma_{\uplambda \uprho}$
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\item \textbf{Weyl transf.}\ $\upgamma'_{\upalpha \upbeta} = e^{2 \upomega}\, \gamma_{\upalpha \upbeta}$
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\end{itemize}
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\end{column}
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\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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\alpha_{-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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$\alpha_{-1}^A \ket{0},$
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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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$\alpha_{-1}^a \ket{0},$
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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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\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 $\mathds{R}^2 \times \mathds{R}^2 \times \mathds{R}^2$
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\item \textbf{relative rotations} are $\mathrm{SO}(2) \simeq \mathrm{U}(1)$ elements
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\item $S_{E\, (\text{cl})}\qty( \qty{ x_{(t)},\, \mathrm{M}_{(t)} }_{1 \le t \le N_B} ) \sim \text{Area}\qty( \qty{ f_{(t)},\, \mathrm{R}_{(t)} }_{1 \le t \le N_B} )$
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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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|
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|
\begin{frame}{$\mathrm{SO}(4)$ Rotations}
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|
Consider \highlight{$\mathds{R}^4 \times \mathds{R}^2$} (focus on $\mathds{R}^4$):
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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{0.9\columnwidth}{!}{\import{img}{welladapted.pgf}}
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\end{column}
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|
|
|
\begin{column}{0.6\linewidth}
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|
\begin{equation*}
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|
\qty( X_{(t)} )^I
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=
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\tensor{\qty( R_{(t)} )}{^I_J}\, X^J - g_{(t)}^I
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|
\quad
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|
\text{s.t.}
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\quad
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R_{(t)} \in \frac{\mathrm{SO}(4)}{\mathrm{S}\qty( \mathrm{O}(2) \times \mathrm{O}(2) )}
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|
\end{equation*}
|
|
\pause
|
|
that is
|
|
\begin{equation*}
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|
\qty[ R_{(t)} ]
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|
=
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|
\qty{ R_{(t)} \sim \mathcal{O}_{(t)} R_{(t)} }
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|
\end{equation*}
|
|
\end{column}
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|
\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}
|
|
|
|
\section[Time Divergences]{Cosmological Backgrounds and Divergences}
|
|
|
|
\begin{frame}{BBB}
|
|
b
|
|
\end{frame}
|
|
|
|
|
|
\section[Deep Learning]{Deep Learning the Geometry of String Theory}
|
|
|
|
\begin{frame}{CCC}
|
|
c
|
|
\end{frame}
|
|
|
|
\end{document}
|