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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}
  

  \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}

    \pause

    \begin{columns}
      \begin{column}[t]{0.5\linewidth}
        \highlight{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}

      \pause

      \begin{column}[t]{0.5\linewidth}
        \highlight{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}
        \highlight{%
          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[ \overset{\scriptscriptstyle(-)}{L}_n,\, \overset{\scriptscriptstyle(-)}{L}_m ]
            & = &
            (n - m) \overset{\scriptscriptstyle(-)}{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}

      \begin{column}{0.4\linewidth}
        \begin{figure}[h]
          \centering
          \resizebox{0.9\columnwidth}{!}{\import{img}{complex_plane.pgf}}
        \end{figure}
      \end{column}
    \end{columns}
  \end{frame}

  \begin{frame}{Action Principle and Conformal Symmetry}
    \highlight{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*}

    \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 $\uplambda_{\upgamma} = -\frac{1}{2}$.
    \end{block}

    \pause

    \highlight{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}
        \begin{column}{0.7\linewidth}
          \begin{equation*}
            \mathscr{M}^{1,\, 9}
            =
            \mathscr{M}^{1,\, 3} \otimes \mathscr{X}_6
          \end{equation*}
          \begin{itemize}
            \item $\mathscr{X}_6$ is a \textbf{compact} manifold

            \item $N = 1$ \textbf{supersymmetry} is 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 (3em,-3.75em) {\includegraphics[width=0.3\linewidth]{img/cy}};
        \end{tikzpicture}
        \begin{column}{0.3\linewidth}
        %   \centering
        %   \includegraphics[width=0.9\columnwidth]{img/cy}
        \end{column}
      \end{columns}
    \end{block}

    \pause

    \begin{columns}
      \begin{column}[t]{0.5\linewidth}
        \highlight{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 $g$ is Ricci-flat (equiv.\ $c_1\qty( M )$ vanishes)
        \end{itemize}
      \end{column}

      \pause

      \begin{column}[t]{0.5\linewidth}
        Characterised by \highlight{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 \highlight{Neumann b.c.:}
    \begin{equation*}
      \eval{\partial_{\upsigma} X\qty( \uptau, \upsigma )}_{\upsigma = 0}^{\upsigma = \ell} = 0
    \end{equation*}
    satisfied by \textbf{open and closed} strings living in $D$ dimensions s.t.\ $\square X = 0$.

    \pause

    \begin{equationblock}{T-duality}
      \begin{equation*}
        X( z, \overline{z} )
        =
        X( z ) + \overline{X}( \overline{z} )
        \quad
        \stackrel{T}{\Rightarrow}
        \quad
        X( z ) - \overline{X}( \overline{z} )
        =
        Y( z, \overline{z} )
        =
        Y( z ) + \overline{Y}( \overline{z} )
      \end{equation*}
    \end{equationblock}

    \pause

    Resulting effect (repeated $p \le D - 1$ times) leads to \highlight{Dirichlet b.c.:}
    \begin{equation*}
      \eval{\partial_{\upsigma} X^i\qty( \uptau, \upsigma )}_{\upsigma = 0}^{\upsigma = \ell} = 0
      \quad
      \stackrel{T}{\Rightarrow}
      \quad
      \eval{\partial_{\uptau} Y^i\qty( \uptau, \upsigma )}_{\upsigma = 0}^{\upsigma = \ell} = 0
      \quad
      \forall i = 1, 2,\, \dots,\, p
    \end{equation*}
    thus \textbf{open strings} can be \textbf{constrained} to \highlight{$D(D - p - 1)$-branes.}
  \end{frame}

  \begin{frame}{D-branes and Open Strings}
    Introducing $Dp$-branes breaks \highlight{$\mathrm{ISO}(1,\, D-1) \rightarrow \mathrm{ISO}( 1, p ) \otimes \mathrm{SO}( D - 1 - p )$.}

    \pause

    \begin{equationblock}{Massless Spectrum (\emph{irrep} of \textbf{little group} $\mathrm{SO}( D - 2 )$}
      \begin{equation*}
        \mathcal{A}^{\upmu}
        \quad \leftrightarrow \quad
        \alpha_{-1}^{\upmu} \ket{0}
        \qquad
        \longrightarrow
        \qquad
        \begin{tabular}{@{}llll@{}}
          $\mathcal{A}^A$
          &
          $\leftrightarrow$
          &
          $\alpha_{-1}^A \ket{0},$
          &
          $A = 0,\, 1,\, \dots,\, p$
          \\
          $\mathcal{A}^a$
          &
          $\leftrightarrow$
          &
          $\alpha_{-1}^a \ket{0},$
          &
          $a = 1,\, 2,\, \dots,\, D - p - 1$
        \end{tabular}
      \end{equation*}
    \end{equationblock}

    \pause

    \begin{columns}
      \begin{column}{0.5\linewidth}
        \centering
        \resizebox{0.55\columnwidth}{!}{\import{img}{chanpaton.pgf}}
      \end{column}

      \begin{column}{0.5\linewidth}
        Introducing \textbf{Chan--Paton factors} $\tensor{\uplambda}{^r_{ij}}$, when branes are \textbf{coincident}:
        \begin{equation*}
          \bigoplus\limits_{r = 1}^N \mathrm{U}_r(1)
          \quad
          \longrightarrow
          \quad
          \mathrm{U}( N )
        \end{equation*}
        \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}}
  \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}{0.5\linewidth}
        \centering
        \resizebox{0.5\columnwidth}{!}{\import{img}{branesangles.pgf}}
      \end{column}

      \begin{column}{0.5\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}
      \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}{0.5\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
          \quad
          \text{s.t.}
          \quad
          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}

  \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}