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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\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}
      \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.}\ $\upgamma'_{\upalpha \upbeta} = \tensor{\qty( \mathrm{J}^{-1} )}{_{\upalpha \upbeta}^{\uplambda \uprho}}\, \upgamma_{\uplambda \uprho}$

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

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  %         \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
        \upalpha_{-1}^{\upmu} \ket{0}
        \qquad
        \longrightarrow
        \qquad
        \begin{tabular}{@{}llll@{}}
          $\mathcal{A}^A$
          &
          $\leftrightarrow$
          &
          $\upalpha_{-1}^A \ket{0},$
          &
          $A = 0,\, 1,\, \dots,\, p$
          \\
          $\mathcal{A}^a$
          &
          $\leftrightarrow$
          &
          $\upalpha_{-1}^a \ket{0},$
          &
          $a = 1,\, 2,\, \dots,\, D - p - 1$
        \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}{0.4\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
        \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 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}

  \begin{frame}{Doubling Trick and Spinor Representation}
    \begin{block}{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{block}

    \pause

    \begin{tikzpicture}[remember picture, overlay]
      \draw[line width=4pt, red] (31em,6em) ellipse (0.8cm and 1.2cm);
    \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} 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 )
          \\
          \qty( -\upomega_z )^{1 - c_n}\,
          \frac{1}{\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}
    \highlight{Operations sequence:}
    \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{0.607\columnwidth}{!}{\import{img}{branesangles.pgf}}
          \end{column}
          \hfill
          \begin{column}{0.6\linewidth}
            \begin{equation*}
              \begin{split}
                \eval{S_{\mathds{R}^4}}_{\text{on-shell}}
                & =
                \frac{1}{2\uppi \upalpha'}
                \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)} } )
              \end{split}
            \end{equation*}
          \end{column}
        \end{columns}
        \vfill
      }
      \only<6->{%
        \centering
        \resizebox{0.25\columnwidth}{!}{\import{img}{brane3d.pgf}}
      }
    \end{block}
  \end{frame}


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

  \begin{frame}{Fermions on the Strip}
    \begin{columns}
      \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{block}{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 \Leftrightarrow \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{block}
  \end{frame}

  \begin{frame}{Conserved Product and Operators}
    Expand on a \highlight{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 \highlight{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 \highlight{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 \highlight{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}
      \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}
      \end{column}
    \end{columns}
  \end{frame}

  \begin{frame}{Orbifolds}
    \begin{columns}[c]
      \begin{column}{0.475\linewidth}
        \begin{center}
          \textbf{Mathematics}

          \begin{itemize}
            \item manifold $M$

            \item (Lie) group $G$

            \item \emph{stabilizer} $G_p = \qty{g \in G \mid gp = p \in M}$

            \item \emph{orbit} $Gp = \qty{gp \in M \mid g \in G}$

            \item charts $\upphi = \uppi \circ \mathscr{P}$ where:

            \begin{itemize}
              \item $\mathscr{P}\colon U \subset \mathds{R}^n \to U / G$

              \item $\uppi\colon U / G \to M$
            \end{itemize}
          \end{itemize}
        \end{center}
      \end{column}
      \begin{column}{0.05\linewidth}
        \centering
        $\Rightarrow$
      \end{column}
      \begin{column}{0.475\linewidth}
        \begin{center}
          \textbf{Physics}

          \begin{itemize}
            \item global orbit space $M / G$

            \item $G$ group of isometries

            \item fixed points

            \item additional d.o.f.\ (\emph{twisted states})

            \item singular limits of CY manifolds
          \end{itemize}
        \end{center}
      \end{column}
    \end{columns}

    \pause

    \begin{center}
      time-dependent orbifolds
    \end{center}

    \begin{tikzpicture}[remember picture, overlay]
      \draw[line width=4pt, red] (13em,3.5em) rectangle (27em, 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}
  \end{frame}


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

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

\end{document}