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(0cm,0cm) circle (2pt) node[anchor=north] {$1$}; +\node[anchor=north] at (1cm, 0cm) {$\cdots$}; +\filldraw[fill=white, draw=black] (2cm,0cm) circle (2pt) node[anchor=north] {$\mathrm{L}$}; +\filldraw[fill=white, draw=black] (3cm,0cm) circle (2pt) node[anchor=north] {$\mathrm{L}+1$}; +\node[anchor=north] at (4cm, 0cm) {$\cdots$}; + +% draw limits +\draw[->] (0cm, 2pt) -- (0cm, 2cm) -- (4cm, 2cm) node[midway, anchor=south west] {in-annihilators} node[anchor=north east] {$b_{n}$}; +\draw[->] (2cm, 2pt) -- (2cm, 1.8cm) -- (-3cm, 1.8cm) node[midway, anchor=south east] {out-annihilators} node[anchor=north west] {$b^*_{\mathrm{L} + 1 - n}$}; + +\end{tikzpicture} + +% vim: ft=tex diff --git a/thesis.tex b/thesis.tex index 8e7d8b5..9b5232a 100644 --- a/thesis.tex +++ b/thesis.tex @@ -46,6 +46,7 @@ \usetikzlibrary{decorations.pathmorphing} \usetikzlibrary{decorations.pathreplacing} \usetikzlibrary{arrows} +\usetikzlibrary{patterns} \newenvironment{equationblock}[1]{% \begin{block}{#1} @@ -180,7 +181,7 @@ \begin{equation*} S_P\qty[ \upgamma,\, X,\, \uppsi ] = - -\frac{1}{4\pi} + -\frac{1}{4\uppi} \int\limits_{-\infty}^{+\infty} \dd{\uptau} \int\limits_0^{\ell} \dd{\upsigma} \sqrt{-\det \upgamma}\, @@ -207,9 +208,9 @@ \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{2D diff.}\ $\upgamma'_{\upalpha \upbeta} = \tensor{\qty( \mathrm{J}^{-1} )}{_{\upalpha \upbeta}^{\uplambda \uprho}}\, \upgamma_{\uplambda \uprho}$ - \item \textbf{Weyl transf.}\ $\upgamma'_{\upalpha \upbeta} = e^{2 \upomega}\, \gamma_{\upalpha \upbeta}$ + \item \textbf{Weyl transf.}\ $\upgamma'_{\upalpha \upbeta} = e^{2 \upomega}\, \upgamma_{\upalpha \upbeta}$ \end{itemize} \end{column} @@ -229,48 +230,48 @@ \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{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{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{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:} @@ -421,7 +422,7 @@ \begin{equation*} \mathcal{A}^{\upmu} \quad \leftrightarrow \quad - \alpha_{-1}^{\upmu} \ket{0} + \upalpha_{-1}^{\upmu} \ket{0} \qquad \longrightarrow \qquad @@ -430,7 +431,7 @@ & $\leftrightarrow$ & - $\alpha_{-1}^A \ket{0},$ + $\upalpha_{-1}^A \ket{0},$ & $A = 0,\, 1,\, \dots,\, p$ \\ @@ -438,7 +439,7 @@ & $\leftrightarrow$ & - $\alpha_{-1}^a \ket{0},$ + $\upalpha_{-1}^a \ket{0},$ & $a = 1,\, 2,\, \dots,\, D - p - 1$ \end{tabular} @@ -740,7 +741,7 @@ \begin{split} \eval{S_{\mathds{R}^4}}_{\text{on-shell}} & = - \frac{1}{2\pi \alpha'} + \frac{1}{2\uppi \upalpha'} \sum\limits_{t = 1}^3 \qty( \frac{1}{2} \abs{g_{(t)}^{\perp}} \abs{f_{(t-1)} - f_{(t)}} ) \\ @@ -830,11 +831,352 @@ \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} - \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}