Begin NBO part
Signed-off-by: Riccardo Finotello <riccardo.finotello@gmail.com>
This commit is contained in:
BIN
img/cone.pdf
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img/cone.pdf
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27
img/inconsistent_theories.pgf
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27
img/inconsistent_theories.pgf
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@@ -0,0 +1,27 @@
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\begin{tikzpicture}
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% fill the overlap area
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\draw[black!0, pattern=north east lines, pattern color=black!15] (0cm, 0cm) rectangle (2cm, 2cm);
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\node[anchor=base, text width=2cm, align=center] at (1cm, 1.4cm) {overlap region};
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\node[anchor=base, text width=3cm, align=center, scale=0.5] at (1cm, 0.5cm) {inconsistent theories};
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% draw the horizontal axis
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\draw[thick, ->] (-3cm, 0cm) -- (4cm, 0cm) node[anchor=south west] {$n$};
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% draw points
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\node[anchor=north] at (-3cm, 0cm) {$\cdots$};
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\filldraw[fill=white, draw=black] (-2cm,0cm) circle (2pt) node[anchor=north] {$-1$};
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\filldraw[fill=white, draw=black] (-1cm,0cm) circle (2pt) node[anchor=north] {$0$};
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\filldraw[fill=white, draw=black] (0cm,0cm) circle (2pt) node[anchor=north] {$1$};
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\node[anchor=north] at (1cm, 0cm) {$\cdots$};
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\filldraw[fill=white, draw=black] (2cm,0cm) circle (2pt) node[anchor=north] {$\mathrm{L}$};
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\filldraw[fill=white, draw=black] (3cm,0cm) circle (2pt) node[anchor=north] {$\mathrm{L}+1$};
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\node[anchor=north] at (4cm, 0cm) {$\cdots$};
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% draw limits
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\draw[->] (0cm, 2pt) -- (0cm, 2cm) -- (4cm, 2cm) node[midway, anchor=south west] {in-annihilators} node[anchor=north east] {$b_{n}$};
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\draw[->] (2cm, 2pt) -- (2cm, 1.8cm) -- (-3cm, 1.8cm) node[midway, anchor=south east] {out-annihilators} node[anchor=north west] {$b^*_{\mathrm{L} + 1 - n}$};
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\end{tikzpicture}
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% vim: ft=tex
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440
thesis.tex
440
thesis.tex
@@ -46,6 +46,7 @@
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\usetikzlibrary{decorations.pathmorphing}
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\usetikzlibrary{decorations.pathreplacing}
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\usetikzlibrary{arrows}
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\usetikzlibrary{patterns}
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\newenvironment{equationblock}[1]{%
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\begin{block}{#1}
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@@ -180,7 +181,7 @@
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\begin{equation*}
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S_P\qty[ \upgamma,\, X,\, \uppsi ]
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=
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-\frac{1}{4\pi}
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-\frac{1}{4\uppi}
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\int\limits_{-\infty}^{+\infty} \dd{\uptau}
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\int\limits_0^{\ell} \dd{\upsigma}
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\sqrt{-\det \upgamma}\,
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@@ -207,9 +208,9 @@
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\begin{itemize}
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\item \textbf{Poincaré transf.}\ $X'^{\upmu} = \tensor{\Uplambda}{^{\upmu}_{\upnu}} X^{\upnu} + c^{\upmu}$
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\item \textbf{2D diff.}\ $\upgamma'_{\upalpha \upbeta} = \tensor{\qty( \mathrm{J}^{-1} )}{_{\upalpha \upbeta}^{\uplambda \uprho}}\, \gamma_{\uplambda \uprho}$
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\item \textbf{2D diff.}\ $\upgamma'_{\upalpha \upbeta} = \tensor{\qty( \mathrm{J}^{-1} )}{_{\upalpha \upbeta}^{\uplambda \uprho}}\, \upgamma_{\uplambda \uprho}$
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\item \textbf{Weyl transf.}\ $\upgamma'_{\upalpha \upbeta} = e^{2 \upomega}\, \gamma_{\upalpha \upbeta}$
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\item \textbf{Weyl transf.}\ $\upgamma'_{\upalpha \upbeta} = e^{2 \upomega}\, \upgamma_{\upalpha \upbeta}$
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\end{itemize}
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\end{column}
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@@ -229,48 +230,48 @@
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\end{frame}
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\begin{frame}{Action Principle and Conformal Symmetry}
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\begin{columns}
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\begin{column}{0.6\linewidth}
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\highlight{%
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Let $z = e^{\uptau_E + i \upsigma} \Rightarrow \overline{\partial} \mathcal{T}( z ) = \partial \overline{\mathcal{T}}( \overline{z} ) = 0$:
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}
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\begin{equation*}
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\mathcal{T}( z )\, \Upphi_h( w )
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\stackrel{z \to w}{\sim}
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\frac{h}{(z - w)^2} \Upphi_h( w )
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+
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\frac{1}{z - w} \partial_w \Upphi_h( w )
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\end{equation*}
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\begin{equation*}
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\mathcal{T}( z )\, \mathcal{T}( w )
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\stackrel{z \to w}{\sim}
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\frac{\frac{c}{2}}{(z - w)^4}
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+
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\order{(z - w)^{-2}}
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\end{equation*}
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% \begin{frame}{Action Principle and Conformal Symmetry}
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% \begin{columns}
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% \begin{column}{0.6\linewidth}
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% \highlight{%
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% Let $z = e^{\uptau_E + i \upsigma} \Rightarrow \overline{\partial} \mathcal{T}( z ) = \partial \overline{\mathcal{T}}( \overline{z} ) = 0$:
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% }
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% \begin{equation*}
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% \mathcal{T}( z )\, \Upphi_h( w )
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% \stackrel{z \to w}{\sim}
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% \frac{h}{(z - w)^2} \Upphi_h( w )
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% +
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% \frac{1}{z - w} \partial_w \Upphi_h( w )
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% \end{equation*}
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% \begin{equation*}
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% \mathcal{T}( z )\, \mathcal{T}( w )
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% \stackrel{z \to w}{\sim}
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% \frac{\frac{c}{2}}{(z - w)^4}
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% +
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% \order{(z - w)^{-2}}
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% \end{equation*}
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\begin{equationblock}{Virasoro algebra $\mathscr{V} \oplus \overline{\mathscr{V}}$}
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\begin{eqnarray*}
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\qty[ \overset{\scriptscriptstyle(-)}{L}_n,\, \overset{\scriptscriptstyle(-)}{L}_m ]
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& = &
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(n - m) \overset{\scriptscriptstyle(-)}{L}_{n + m} + \frac{c}{12} n \qty(n^2 - 1) \updelta_{n + m,\, 0}
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\\
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\qty[ L_n,\, \overline{L}_m ]
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& = &
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0
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\end{eqnarray*}
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\end{equationblock}
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\end{column}
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% \begin{equationblock}{Virasoro algebra $\mathscr{V} \oplus \overline{\mathscr{V}}$}
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% \begin{eqnarray*}
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% \qty[ \overset{\scriptscriptstyle(-)}{L}_n,\, \overset{\scriptscriptstyle(-)}{L}_m ]
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% & = &
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% (n - m) \overset{\scriptscriptstyle(-)}{L}_{n + m} + \frac{c}{12} n \qty(n^2 - 1) \updelta_{n + m,\, 0}
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% \\
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% \qty[ L_n,\, \overline{L}_m ]
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% & = &
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% 0
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% \end{eqnarray*}
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% \end{equationblock}
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% \end{column}
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\begin{column}{0.4\linewidth}
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\begin{figure}[h]
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\centering
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\resizebox{0.9\columnwidth}{!}{\import{img}{complex_plane.pgf}}
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\end{figure}
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\end{column}
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\end{columns}
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\end{frame}
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% \begin{column}{0.4\linewidth}
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% \begin{figure}[h]
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% \centering
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% \resizebox{0.9\columnwidth}{!}{\import{img}{complex_plane.pgf}}
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% \end{figure}
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% \end{column}
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% \end{columns}
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% \end{frame}
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\begin{frame}{Action Principle and Conformal Symmetry}
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\highlight{Superstrings in $D$ dimensions:}
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@@ -421,7 +422,7 @@
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\begin{equation*}
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\mathcal{A}^{\upmu}
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\quad \leftrightarrow \quad
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\alpha_{-1}^{\upmu} \ket{0}
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\upalpha_{-1}^{\upmu} \ket{0}
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\qquad
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\longrightarrow
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\qquad
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@@ -430,7 +431,7 @@
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&
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$\leftrightarrow$
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&
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$\alpha_{-1}^A \ket{0},$
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$\upalpha_{-1}^A \ket{0},$
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&
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$A = 0,\, 1,\, \dots,\, p$
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\\
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@@ -438,7 +439,7 @@
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&
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$\leftrightarrow$
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&
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$\alpha_{-1}^a \ket{0},$
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$\upalpha_{-1}^a \ket{0},$
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&
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$a = 1,\, 2,\, \dots,\, D - p - 1$
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\end{tabular}
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@@ -740,7 +741,7 @@
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\begin{split}
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\eval{S_{\mathds{R}^4}}_{\text{on-shell}}
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& =
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\frac{1}{2\pi \alpha'}
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\frac{1}{2\uppi \upalpha'}
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\sum\limits_{t = 1}^3
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\qty( \frac{1}{2} \abs{g_{(t)}^{\perp}} \abs{f_{(t-1)} - f_{(t)}} )
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\\
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@@ -830,11 +831,352 @@
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\end{block}
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\end{frame}
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\begin{frame}{Conserved Product and Operators}
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Expand on a \highlight{basis of solutions}
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\begin{equation*}
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\uppsi_{\pm}( \upxi_{\pm} )
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=
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\sum\limits_{n = -\infty}^{+\infty} b_n\, \uppsi_n( \upxi_{\pm} )
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\qquad
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\Rightarrow
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\qquad
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\Uppsi( z )
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=
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\begin{cases}
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\uppsi_{E,\, +}( u ) \quad \text{if}~z \in \mathscr{H}_{>}^{(\overline{t})}
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\\
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\uppsi_{E,\, -}( u ) \quad \text{if}~z \in \mathscr{H}_{<}^{(\overline{t})}
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\end{cases}
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\end{equation*}
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\pause
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\begin{equationblock}{Conserved Product and Dual Basis}
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\begin{equation*}
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\left\langle\!\left\langle
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\tensor[^*]{\uppsi}{_n},\,
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\uppsi_m
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\right. \right\rangle
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=
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2\uppi \mathcal{N}\,
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\oint
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\frac{\dd{z}}{2\uppi i}\,
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\tensor[^*]{\Uppsi}{_n^*}\,
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\tensor{\Uppsi}{_m}
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=
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\updelta_{n,\, m}
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\quad
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\Rightarrow
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\quad
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\left\langle\!\left\langle
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\tensor[^*]{\Uppsi}{_n^{(*)}},\,
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\Uppsi^{(*)}
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\right. \right\rangle
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=
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b_n^{(\dagger)}
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\end{equation*}
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\end{equationblock}
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\pause
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Derive the \highlight{algebra of operators:}
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\begin{equation*}
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\qty[ b_n,\, b_m^{\dagger} ]_+
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=
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\frac{2 \mathcal{N}}{T}\,
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\left\langle\!\left\langle
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\tensor[^*]{\Uppsi}{_n^*},\,
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\Uppsi_m^*
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\right. \right\rangle
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\end{equation*}
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\end{frame}
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\begin{frame}{Twisted Complex Fermions}
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Consider the case $R_{(t)} = e^{i \uppi \upalpha_{(t)}} \in \mathrm{U}( 1 )$:
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\begin{equation*}
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\Uppsi( x_{(t)} + e^{2\uppi i} \updelta )
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=
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e^{i \uppi \upepsilon_{(t)}}\,
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\Uppsi( x_{(t)} + \updelta )
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\end{equation*}
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where
|
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\begin{equation*}
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\upepsilon_{(t)}
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=
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\upalpha_{(t+1)} - \upalpha_{(t)}
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+
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\uptheta\qty( \upalpha_{(t)} - \upalpha_{(t+1)} - 1 )
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-
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\uptheta\qty( \upalpha_{(t+1)} - \upalpha_{(t)} - 1 )
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\end{equation*}
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\pause
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\begin{equationblock}{Basis of Solutions}
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\begin{equation*}
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\begin{split}
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\Uppsi_n\qty( z;\, \qty{ x_{(t)} } )
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& =
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\mathcal{N}_{\Uppsi}\,
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z^{-n}\,
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\prod\limits_{t = 1}^N
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\qty( 1 - \frac{z}{x_{(t)}} )^{n_{(t)} + \frac{\upepsilon_{(t)}}{2}}
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\\
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\tensor[^*]{\Uppsi}{_n}\qty( z;\, \qty{ x_{(t)} } )
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& =
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\frac{1}{2\uppi \mathcal{N} \mathcal{N}_{\Uppsi}}\,
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z^{n - 1}\,
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\prod\limits_{t = 1}^N
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\qty( 1 - \frac{z}{x_{(t)}} )^{-\widetilde{n}_{(t)} + \frac{\upepsilon_{(t)}}{2}}
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\end{split}
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\end{equation*}
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\end{equationblock}
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\end{frame}
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\begin{frame}{Vacua}
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Define the \textbf{vacuum} with respect to $b_n$:
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\begin{equation*}
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\begin{split}
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b_n \ket{\qty{ x_{(t)} }} = 0 &\quad \text{for} \quad n \ge 1
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\\
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b_n \ket{\widetilde{0}} = 0 &\quad \text{for} \quad n \ge n_{(t)} + \frac{\upepsilon_{(t)}}{2} + \frac{1}{2}
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\end{split}
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\end{equation*}
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\pause
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Theories are subject to \highlight{consistency conditions:}
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\begin{columns}
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\begin{column}{0.6\linewidth}
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\begin{equation*}
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\mathrm{L}
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=
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n_{(t)} + \widetilde{n}_{(t)}
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\uncover<3->{%
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\alert{= 0}
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}
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\end{equation*}
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\end{column}
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\hfill
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\begin{column}{0.4\linewidth}
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\centering
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\resizebox{\columnwidth}{!}{\import{img}{inconsistent_theories.pgf}}
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\end{column}
|
||||
\end{columns}
|
||||
\end{frame}
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\begin{frame}{Stress-energy Tensor and CFT Approach}
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Compute the OPEs leading to the \highlight{stress-energy tensor:}
|
||||
\begin{equation*}
|
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\mathcal{T}( z )
|
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=
|
||||
\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}
|
||||
|
||||
|
||||
|
||||
Reference in New Issue
Block a user