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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\uppi}
        \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}

    \pause

    \begin{columns}[T, totalwidth=0.935\linewidth]
      \begin{column}{0.45\linewidth}
        \begin{tabular}{@{}ll@{}}
          Symmetries: &
          \\
          \toprule
          \textbf{Poincaré transf.}: & 
          $X'^{\upmu} = \tensor{\Uplambda}{^{\upmu}_{\upnu}} X^{\upnu} + c^{\upmu}$
          \\
          \textbf{2D diff.}: &
          $\upgamma'_{\upalpha \upbeta} = \tensor{\qty( \mathrm{J}^{-1} )}{_{\upalpha \upbeta}^{\uplambda \uprho}}\, \upgamma_{\uplambda \uprho}$
          \\
          \textbf{Weyl transf.}: &
          $\upgamma'_{\upalpha \upbeta} = e^{2 \upomega}\, \upgamma_{\upalpha \upbeta}$
          \\
        \end{tabular}
      \end{column}
      \hfill
      \begin{column}{0.45\linewidth}
        \begin{tabular}{@{}ll@{}}
          Conformal symmetry: &
          \\
          \toprule
          \textbf{vanishing} stress-energy tensor: &
          $\mathcal{T}_{\upalpha \upbeta} = 0$
          \\
          \textbf{traceless} stress-energy tensor: &
          $\trace{\mathcal{T}} = 0$
          \\
          \textbf{conformal gauge}: &
          $\upgamma_{\upalpha \upbeta} = e^{\upphi}\, \upeta_{\upalpha \upbeta}$
          \\
        \end{tabular}

        \pause

        \begin{center}
          \highlight{Conformal properties fixed by \textbf{OPE}s.}
        \end{center}
      \end{column}
    \end{columns}
  \end{frame}


  \begin{frame}{Action Principle and Conformal Symmetry}
    Superstrings in $D$ dimensions $\longrightarrow$ \emph{Virasoro algebra} (central extension of de Witt's algebra):
    \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*}

    \pause

    \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 $\uplambda_c = -1$, and $\uplambda_{\upbeta} = \frac{3}{2}$ and $\uplambda_{\upgamma} = -\frac{1}{2}$.
      \hfill
      \cite{Friedan, Martinec, Shenker (1986)}
    \end{block}

    \pause

    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}
    \begin{block}{Compactification}
      \begin{columns}[T, totalwidth=0.95\linewidth]
        \begin{column}{0.8\linewidth}
          \begin{equation*}
            \mathscr{M}^{1,\, 9}
            =
            \mathscr{M}^{1,\, 3} \otimes \mathscr{X}_6
          \end{equation*}
          \vspace{-1em}
          \begin{itemize}
            \item $\mathscr{X}_6$ is a \textbf{compact} manifold

            \item $N = 1$ \textbf{supersymmetry} preserved in 4D

            \item algebra of $\mathrm{SU}(3) \otimes \mathrm{SU}(2) \otimes \mathrm{U}(1)$ in arising \textbf{gauge group}
          \end{itemize}
        \end{column}

        \begin{tikzpicture}[remember picture, overlay]
          \node[anchor=base] at (-2em,-6em) {\cite{code in Hanson (1994)}};
          \node[anchor=base] at (-7em,-6em) {\includegraphics[width=0.25\linewidth]{img/cy.png}};
        \end{tikzpicture}
      \end{columns}
    \end{block}

    \pause

    \vfill
    \begin{columns}[T, totalwidth=0.95\linewidth]
      \begin{column}{0.475\linewidth}
        \textbf{Kähler manifolds} $\qty( M,\, g )$ such that:
        \begin{itemize}
          \item $\dim\limits_{\mathds{C}} M = m$

          \item $\mathrm{Hol}( g ) \subseteq \mathrm{SU}( m )$

          \item $\mathrm{Ric}( g ) \equiv 0$ (equiv.\ $c_1\qty( M ) \equiv 0$)
        \end{itemize}
      \end{column}
      \hfill

      \pause

      \begin{column}{0.475\linewidth}
        Characterised by \textbf{Hodge numbers}
        \begin{equation*}
          h^{r,\,s} = \dim\limits_{\mathds{C}} H_{\overline{\partial}}^{r,\,s}\qty( M,\, \mathds{C} )
        \end{equation*}
        counting the no.\ of harmonic $(r,s)$-forms.
      \end{column}
    \end{columns}
  \end{frame}

  \begin{frame}{D-branes and Open Strings}
    Polyakov's action naturally introduces \textbf{Neumann b.c.} for \textbf{open strings}:
    \begin{equation*}
      \eval{\partial_{\upsigma} X\qty( \uptau, \upsigma )}_{\upsigma = 0}^{\upsigma = \ell} = 0
    \end{equation*}
    satisfied by \highlight{\textbf{open and closed strings} in $D$ dim.} s.t.\ $\square X = 0 \Rightarrow X( z, \overline{z} ) = X( z ) + \overline{X}( \overline{z} )$.

    \pause

    % \begin{equationblock}{Equivalent Theories of Closed String Compactification}
    %   \begin{equation*}
    %     X( z, \overline{z} )
    %     =
    %     X( z ) + \overline{X}( \overline{z} )
    %     \quad
    %     \stackrel{T-dual}{\Rightarrow}
    %     \quad
    %     X( z ) - \overline{X}( \overline{z} )
    %     =
    %     Y( z, \overline{z} )
    %     =
    %     Y( z ) + \overline{Y}( \overline{z} )
    %   \end{equation*}
    % \end{equationblock}

    % \pause

    \begin{block}{T-duality}
      \textbf{Dirichlet b.c.} consequence of \textbf{T-duality} on $p$ directions:
      \begin{equation*}
        \overline{X}( z ) \mapsto - \overline{X}( z )
        \quad
        \Rightarrow
        \quad
        \eval{\partial_{\upsigma} X^i\qty( \uptau, \upsigma )}_{\upsigma = 0}^{\upsigma = \ell} = 0
        \quad
        \stackrel{T-duality}{\longrightarrow}
        \quad
        \eval{\partial_{\uptau} \widetilde{X}^i\qty( \uptau, \upsigma )}_{\upsigma = 0}^{\upsigma = \ell} = 0
      \end{equation*}
      thus \textbf{open strings} can be \textbf{constrained} to $D(D - p - 1)$-branes.
      \hfill
      \cite{Polchinski (1995, 1996)}
    \end{block}
  \end{frame}

  \begin{frame}{D-branes and Open Strings}
    Introducing $Dp$-branes breaks \highlight{$\mathrm{ISO}(1,\, D-1) \rightarrow \mathrm{ISO}( 1, p ) \otimes \mathrm{SO}( D - 1 - p )$}:
    \begin{equation*}
      \mathcal{A}^{\upmu} \rightarrow \qty( \mathcal{A}^A,\, \mathcal{A}^a )
      \quad
      \Rightarrow
      \quad
      \mathrm{U}( 1 )~\text{theory~in}~p+1~\text{dimensions~(and~scalars)}
    \end{equation*}

    \pause

    \vspace{2em}
    \begin{columns}[T, totalwidth=0.95\linewidth]
      \begin{column}{0.475\linewidth}
        \centering
        \resizebox{0.5\columnwidth}{!}{\import{img}{chanpaton.pgf}}

        \cite{Chan, Paton (1969)}
        \begin{equation*}
          \ket{n;\, r}
          =
          \sum\limits_{i,\, j = 1}^N
          \ket{n;\, i,\, j}\,
          \tensor{\uplambda}{^r_{ij}}
        \end{equation*}
      \end{column}
      \hfill
      \pause

      \begin{column}{0.475\linewidth}
        \begin{block}{Chan--Paton Factors}
          When branes are \textbf{coincident}:
          \begin{equation*}
            \bigoplus\limits_{r = 1}^N \mathrm{U}_r(1)
            \quad
            \longrightarrow
            \quad
            \mathrm{U}( N )
          \end{equation*}
        \end{block}
        \pause
        \highlight{Build gauge bosons, fermions and scalars.}
      \end{column}
    \end{columns}
  \end{frame}

  \begin{frame}{Standard Model-like Scenarios}
    \centering
    \resizebox{0.8\linewidth}{!}{\import{img}{smbranes.pgf}}
    \hfill\cite{Zwiebach (2009)}
  \end{frame}


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

  \begin{frame}{Intersecting D-branes}
    Consider \highlight{$N$ intersecting $D6$-branes} filling $\mathscr{M}^{1,3}$ and \textbf{embedded in} $\mathds{R}^6$

    \begin{equationblock}{Twist Fields Correlators}
      \begin{equation*}
        \left\langle \prod\limits_{t = 1}^{N_B} \upsigma_{\mathrm{M}_{(t)}}\qty( x_{(t)} ) \right\rangle
        =
        \mathcal{N}\qty( \qty{ x_{(t)},\, \mathrm{M}_{(t)} }_{1 \le t \le N_B} )
        e^{- S_{E\, (\text{cl})}\qty( \qty{ x_{(t)},\, \mathrm{M}_{(t)} }_{1 \le t \le N_B} )}
      \end{equation*}
    \end{equationblock}

    \pause

    \begin{tikzpicture}[remember picture, overlay]
      \draw[line width=4pt, red] (29.5em,3.5em) ellipse (2cm and 1cm);
    \end{tikzpicture}

    \pause

    \begin{columns}
      \begin{column}{0.3\linewidth}
        \centering
        \resizebox{0.8\columnwidth}{!}{\import{img}{branesangles.pgf}}
      \end{column}

      \begin{column}{0.7\linewidth}
        D-branes in \textbf{factorised} internal space:
        \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}

        \hfill\cite{Cremades, Ibanez, Marchesano (2003); Pesando (2012)}
      \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
          \in \mathds{R}^4
        \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 $u = x + i y = e^{\uptau_e + i \upsigma}$ and $\overline{u} = u^*$

          \item $x_{(t)} < x_{(t-1)}$ \textbf{worldsheet intersection points}

          \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 = -\infty}^{+\infty} 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}
    Sequence of the operations:
    \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{0.6\columnwidth}{!}{\import{img}{branesangles.pgf}}
          \end{column}
          \hfill
          \begin{column}{0.6\linewidth}
            \begin{equation*}
              \begin{split}
                2 \uppi \upalpha' \eval{S_{\mathds{R}^4}}_{\text{on-shell}}
                & =
                \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)} }_{1 \le t \le N_B} )
              \end{split}
            \end{equation*}
          \end{column}
        \end{columns}
        \vfill
      }
      \only<6->{%
        \begin{columns}
          \begin{column}{0.35\linewidth}
            \centering
            \resizebox{0.8\columnwidth}{!}{\import{img}{brane3d.pgf}}
          \end{column}
          \hfill
          \begin{column}{0.6\linewidth}
            \begin{itemize}
              \item strings no longer confined to plane

              \item strings form a \emph{small bump} from the D-brane

              \item classical action \textbf{larger} than factorised case
            \end{itemize}
          \end{column}
        \end{columns}
        \vfill
      }
    \end{block}
  \end{frame}


  \subsection[Fermions]{Fermions and Point-like Defect CFT}

  \begin{frame}{Fermions on the Strip}
    \begin{columns}[totalwidth=0.95\linewidth]
      \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{equationblock}{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
          \quad \Leftrightarrow \quad
          \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{equationblock}
  \end{frame}

  \begin{frame}{Conserved Product and Operators}
    Expand on a \textbf{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 \textbf{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 \textbf{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 \textbf{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.pdf}
      \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}
        \hfill\cite{Craps, Kutasov, Rajesh (2002); Liu, Moore, Seiberg (2002)}
      \end{column}
    \end{columns}
  \end{frame}

  \begin{frame}{Orbifolds}
    \begin{columns}[T]
      \begin{column}{0.475\linewidth}
        \begin{tabular}{@{}l@{}}
          \textbf{Mathematics}
          \\
          \toprule
          manifold $M$
          \\
          (Lie) group $G$
          \\
          \emph{stabilizer}: $G_p = \qty{g \in G \mid gp = p \in M}$
          \\
          \emph{orbit}: $Gp = \qty{gp \in M \mid g \in G}$
          \\
          charts $\upphi = \uppi \circ \mathscr{P}$ where:
          \\
          \vspace{1em}$\mathscr{P}\colon U \subset \mathds{R}^n \to U / G$
          \\
          \vspace{1em}$\uppi\colon U / G \to M$
          \\
        \end{tabular}
      \end{column}
      \hfill
      \begin{column}{0.475\linewidth}
        \begin{tabular}{@{}l@{}}
          \textbf{Physics}
          \\
          \toprule
          global orbit space $M / G$
          \\
          $G$ group of isometries
          \\
          fixed points
          \\
          additional d.o.f.\ (\emph{twisted states})
          \\
          singular limits of CY manifolds
          \\
        \end{tabular}
      \end{column}
    \end{columns}

    \pause

    \vspace{2em}
    \begin{center}
      Use \textbf{time-dependent orbifolds} to model singularities in time
    \end{center}

    \begin{tikzpicture}[remember picture, overlay]
      \draw[line width=4pt, red] (5em,3.5em) rectangle (35em, 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

    Scalars on NBO:
    \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}
      \emph{
        most terms \textbf{do not converge} due to \textbf{isolated zeros} \emph{($l_{(*)} \equiv 0$)} 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} (even sQED $\rightarrow$ eikonal?)
        
        \pause

      \item divergences are \textbf{not (only) gravitational}

        \pause

      \item \textbf{vanishing volume} in phase space of the compact direction is responsible for the divergence
    \end{itemize}

    \pause

    What about \highlight{string theory?}

    \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*}
      \hfill\cite{Green, Hübsch (1987); Hübsch (1992)}
    \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{tikzpicture}[remember picture, overlay]
      \draw[line width=4pt, red] (13em, 5.5em) rectangle (22em, 0em);
    \end{tikzpicture}

    \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} \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 knowledge from \textbf{computer science, mathematics and physics} to solve problems

        \pause

      \item provide in-depth \textbf{data analysis} of the datasets
    \end{itemize}

    \begin{center}
      \includegraphics[width=0.7\linewidth]{img/label-distribution_orig.pdf}
    \end{center}
  \end{frame}

  
  \subsection[Machine Learning]{Machine Learning for String Theory}


  \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)
    \end{itemize}

    \pause

    \begin{center}
      \includegraphics[width=0.7\linewidth]{img/label-distribution-compare_orig.pdf}
    \end{center}
  \end{frame}

  \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.pdf}
      \end{column}
      \hfill
      \begin{column}{0.5\linewidth}
        \centering
        \includegraphics[width=0.9\columnwidth, trim={0 0 6in 0}, clip]{img/scalar-features_orig.pdf}
      \end{column}
    \end{columns}
  \end{frame}

  \begin{frame}{Machine Learning}
    \centering
    \includegraphics[width=0.85\linewidth]{img/ml_map.png}
    
    \begin{tikzpicture}[remember picture, overlay]
      \node[anchor=base] at (16em,18em) {\cite{from scikit-learn.org}};
    \end{tikzpicture}

    \pause

    \begin{tikzpicture}[remember picture, overlay]
      \draw[line width=10pt, red, -latex] (-18em,2em) -- (-14.5em,7.5em);
      \draw[line width=10pt, red, -latex] (19em,9em) -- (14em,5em);
    \end{tikzpicture}

    \pause

    \begin{tikzpicture}[remember picture, overlay]
      \draw[line width=4pt, red] (12em,13em) ellipse (2cm and 1.5cm);
    \end{tikzpicture}
  \end{frame}

  \begin{frame}{A Word on PCA}
    \begin{columns}
      \begin{column}{0.4\linewidth}
        What is PCA for a $X \in \mathds{R}^{n \times p}$?
        \begin{itemize}
          \item project data onto a \textbf{lower dimensional} space where variance is maximised

          \item equivalently compute the \textbf{eigenvectors} of $X X^T$ or the \textbf{singular values} of $X$

          \item isolate \textbf{the signal} from the \textbf{background}

          \item ease the machine learning job of finding a better representation of the input
        \end{itemize}
      \end{column}
      \hfill
      \pause
      \begin{column}{0.6\linewidth}
        \centering
        \includegraphics[width=0.5\columnwidth]{img/marchenko-pastur.pdf}
        \includegraphics[width=\columnwidth]{img/svd_orig.pdf}
      \end{column}
    \end{columns}
  \end{frame}

  \begin{frame}{Machine Learning Results}
    \begin{columns}
      \begin{column}{0.5\linewidth}
        \centering
        \textbf{Configuration Matrix Only}
        \includegraphics[width=0.8\columnwidth, trim={0 0 3.3in 0}, clip]{img/cicy_matrix_plots.pdf}
      \end{column}
      \hfill\pause
      \begin{column}{0.5\linewidth}
        \centering
        \textbf{Best Training Set}
        \includegraphics[width=0.8\columnwidth, trim={0 0 3.3in 0}, clip]{img/cicy_best_plots.pdf}
      \end{column}
    \end{columns}
  \end{frame}

  \subsection[Deep Learning]{AI Implementations for Geometry and Strings}

  \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}{Layers}
          \vspace{0.5em}
          \begin{tabular}{@{}ll@{}}
            \textbf{fully connected}:
            &
            $\upphi\qty( a^{\qty(i)\, \qty{l}} \cdot W^{\qty{l}} + b^{\qty{l}} \mathds{1}_l )$
            \\
            \textbf{convolutional}:
            &
            $\upphi\qty( a^{\qty(i)\, \qty{l}}\, *\, W^{\qty{l}} + b^{\qty{l}} \mathds{1}_l )$
          \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}}
        \hfill\cite{rendition of the neural network in Bull et al.\ (2018)}
      \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<1-3>{\includegraphics[width=0.75\columnwidth, trim={12in 5in 0 5in}, clip]{img/input_mat.png}}
        \only<4>{\animategraphics[autoplay,loop,controls={play,stop},width=\linewidth]{8}{img/animation/sequence/conv-}{0}{79}}
        \only<5>{\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.pdf}
      \end{column}
      \hfill
      \begin{column}{0.5\linewidth}
        \centering
        \includegraphics[width=0.75\columnwidth]{img/inc_nn_learning_curve_h11.pdf}
        \includegraphics[width=0.75\columnwidth]{img/inc_nn_learning_curve.pdf}
      \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}
