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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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\begin{document}

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  \section[CFT]{Conformal Symmetry and Geometry of the Worldsheet}
  
  \begin{frame}{Action Principle and Conformal Symmetry}
    \begin{block}{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}{\alpha'}\,
          \partial_{\upalpha} X^{\upmu}\,
          \partial_{\upbeta} X^{\upnu}
          +
          \uppsi^{\upmu}\,
          \uprho_{\upalpha}
          \partial_{\upbeta}
          \uppsi^{\upnu}
        )\,
        \upeta_{\upmu\upnu}
      \end{equation*}
    \end{block}

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

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

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

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

          \item 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}
        Let $z = e^{\uptau_E + i \upsigma} \Rightarrow \overline{\partial} \mathcal{T}( z ) = \partial \overline{\mathcal{T}}( \overline{z} ) = 0$:
        \begin{equation*}
          T( z )\, \Upphi_{\upomega}( w )
          \stackrel{z \to w}{\sim}
          \frac{\upomega}{(z - w)^2} \Upphi_{\upomega}( w )
          +
          \frac{1}{z - w} \partial_w \Upphi_{\upomega}( w )
        \end{equation*}
      \end{column}

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        \begin{figure}[h]
          \centering
          \resizebox{0.8\columnwidth}{!}{\import{img}{complex_plane.pgf}}
        \end{figure}
      \end{column}
    \end{columns}
  \end{frame}


  \subsection[Tools]{Preliminary Tools and Definitions}

  \begin{frame}{AAA}
    a1
  \end{frame}


  \subsection[D-branes]{D-branes Intersecting at Angles}

  \begin{frame}{AAA}
    a2
  \end{frame}


  \subsection[Fermions]{Fermions With Boundary Defects}

  \begin{frame}{AAA}
    a3
  \end{frame}


  \section[Time Divergences]{Cosmological Backgrounds and Divergences}

  \begin{frame}{BBB}
    b
  \end{frame}


  \subsection[Orbifolds]{Orbifolds and Cosmological Models}

  \begin{frame}{BBB}
    b1
  \end{frame}


  \subsection[Time Dependency]{Time Dependent Orbifolds}

  \begin{frame}{BBB}
    b2
  \end{frame}


  \section[Deep Learning]{Deep Learning the Geometry of String Theory}

  \begin{frame}{CCC}
    c
  \end{frame}

  \subsection[CICY]{Complete Intersection Calabi--Yau Manifolds}

  \begin{frame}{CCC}
    c1
  \end{frame}


  \subsection[Machine Learning]{Machine Learning and Deep Learning for CICY Manifolds}

  \begin{frame}{CCC}
    c2
  \end{frame}


\end{document}