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\author[Finotello]{Riccardo Finotello}
\title[D-branes and Deep Learning]{D-branes and Deep Learning}
\subtitle{Theoretical and Computational Aspects in String Theory}
\institute[UniTO]{%
  Scuola di Dottorato in Fisica e Astrofisica
  \\[0.5em]
  Università degli Studi di Torino
  \\
  and
  \\
  I.N.F.N.\ -- sezione di Torino
}
\date{15th December 2020}

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  \section[CFT]{Conformal Symmetry and Geometry of the Worldsheet}
  

  \subsection[Preliminary]{Preliminary Concepts and Tools}

  \begin{frame}{Action Principle and Conformal Symmetry}
    \begin{equationblock}{Polyakov's Action}
      \begin{equation*}
        S_P\qty[ \upgamma,\, X,\, \uppsi ]
        =
        -\frac{1}{4\pi}
        \int\limits_{-\infty}^{+\infty} \dd{\uptau}
        \int\limits_0^{\ell} \dd{\upsigma}
        \sqrt{-\det \upgamma}\,
        \upgamma^{\upalpha \upbeta}\,
        \qty(%
          \frac{2}{\upalpha'}\,
          \partial_{\upalpha} X^{\upmu}\,
          \partial_{\upbeta} X^{\upnu}
          +
          \uppsi^{\upmu}\,
          \uprho_{\upalpha}
          \partial_{\upbeta}
          \uppsi^{\upnu}
        )\,
        \upeta_{\upmu\upnu}
      \end{equation*}
    \end{equationblock}

    \begin{columns}
      \begin{column}[t]{0.5\linewidth}
        \fcolorbox{yellow}{yellow!20}{Symmetries:}
        \begin{itemize}
          \item \textbf{Poincaré transf.}\ $X'^{\upmu} = \tensor{\Uplambda}{^{\upmu}_{\upnu}} X^{\upnu} + c^{\upmu}$

          \item \textbf{2D diff.}\ $\upgamma'_{\upalpha \upbeta} = \tensor{\qty( \mathrm{J}^{-1} )}{_{\upalpha \upbeta}^{\uplambda \uprho}}\, \gamma_{\uplambda \uprho}$

          \item \textbf{Weyl transf.}\ $\upgamma'_{\upalpha \upbeta} = e^{2 \upomega}\, \gamma_{\upalpha \upbeta}$
        \end{itemize}
      \end{column}

      \begin{column}[t]{0.5\linewidth}
        \fcolorbox{yellow}{yellow!20}{Conformal symmetry:}
        \begin{itemize}
          \item \textbf{vanishing} stress-energy tensor: $\mathcal{T}_{\upalpha \upbeta} = 0$
          
          \item \textbf{traceless} stress-energy tensor: $\trace{\mathcal{T}} = 0$

          \item \textbf{conformal gauge} $\upgamma_{\upalpha \upbeta} = e^{\upphi}\, \upeta_{\upalpha \upbeta}$
        \end{itemize}
      \end{column}
    \end{columns}
  \end{frame}


  \begin{frame}{Action Principle and Conformal Symmetry}
    \begin{columns}
      \begin{column}{0.6\linewidth}
        \fcolorbox{yellow}{yellow!20}{%
          Let $z = e^{\uptau_E + i \upsigma} \Rightarrow \overline{\partial} \mathcal{T}( z ) = \partial \overline{\mathcal{T}}( \overline{z} ) = 0$:
        }
        \begin{equation*}
          \mathcal{T}( z )\, \Upphi_h( w )
          \stackrel{z \to w}{\sim}
          \frac{h}{(z - w)^2} \Upphi_h( w )
          +
          \frac{1}{z - w} \partial_w \Upphi_h( w )
        \end{equation*}
        \begin{equation*}
          \mathcal{T}( z )\, \mathcal{T}( w )
          \stackrel{z \to w}{\sim}
          \frac{\frac{c}{2}}{(z - w)^4}
          +
          \order{(z - w)^{-2}}
        \end{equation*}

        \begin{equationblock}{Virasoro algebra $\mathscr{V} \oplus \overline{\mathscr{V}}$}
          \begin{eqnarray*}
            \qty[ L_n,\, L_m ]
            & = &
            (n - m) L_{n + m} + \frac{c}{12} n \qty(n^2 - 1) \updelta_{n + m,\, 0}
            \\
            \qty[ L_n,\, \overline{L}_m ]
            & = &
            0
          \end{eqnarray*}
        \end{equationblock}
      \end{column}

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          \centering
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        \end{figure}
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    \end{columns}
  \end{frame}

  \begin{frame}{Action Principle and Conformal Symmetry}
    \fcolorbox{yellow}{yellow!20}{Superstrings in $D$ dimensions:}
    \begin{equation*}
      \mathcal{T}( z )
      =
      -\frac{1}{\upalpha'}
      \partial X( z ) \cdot \partial X( z )
      -\frac{1}{2}
      \uppsi( z ) \cdot \partial \uppsi( z )
      \quad
      \Rightarrow
      \quad
      c = \frac{3}{2} D
    \end{equation*}

    \begin{block}{$\qty( \uplambda, 0 )~/~\qty( 1 - \uplambda, 0 )$ Ghost System}
      Introduce anti-commuting $\qty( b,\, c )$ and commuting $\qty( \upbeta,\, \upgamma )$ conformal fields:
      \begin{equation*}
        S_{\text{ghost}}\qty[ b,\, c,\, \upbeta,\, \upgamma ]
        =
        \frac{1}{2\uppi}
        \iint \dd{z} \dd{\overline{z}}
        \qty(%
          b( z )\, \overline{\partial} c( z )
          +
          \upbeta( z )\, \overline{\partial} \upgamma( z )
        )
      \end{equation*}
      where $\uplambda_b = 2$ and $\uplambda_{\upbeta} = \frac{3}{2}$.
    \end{block}

    \fcolorbox{yellow}{yellow!20}{Consequence:}
    \begin{equation*}
      c_{\text{full}} = c + c_{\text{ghost}} = 0
      \quad
      \Leftrightarrow
      \quad
      D = 10.
    \end{equation*}
  \end{frame}


  \begin{frame}{Extra Dimensions and Compactification}
  \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}