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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} - \begin{frame}{BBB} - b + + \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}