From 3d1d20debb61619d00f94970cc94ee2450383c82 Mon Sep 17 00:00:00 2001 From: Riccardo Finotello Date: Mon, 9 Nov 2020 18:40:17 +0100 Subject: [PATCH] Begin part with fermions and defect CFT Signed-off-by: Riccardo Finotello --- img/brane3d.pgf | 25 ++++++ img/defects.pgf | 27 +++++++ img/threebranes_plane.pgf | 11 +-- thesis.tex | 165 +++++++++++++++++++++++++++++++++++++- 4 files changed, 219 insertions(+), 9 deletions(-) create mode 100644 img/brane3d.pgf create mode 100644 img/defects.pgf diff --git a/img/brane3d.pgf b/img/brane3d.pgf new file mode 100644 index 0000000..910701a --- /dev/null +++ b/img/brane3d.pgf @@ -0,0 +1,25 @@ +\usetikzlibrary{decorations.pathmorphing} + +\begin{tikzpicture} + +% draw branes +\draw[thick] (-0.35cm, 0cm) -- (-3cm, -2cm) -- (-3cm, 3cm) -- (1cm, 5cm) -- (1cm, 4cm); +\draw[dashed] (1cm, 4cm) -- (1cm, 1cm) -- (-0.35cm, 0cm); + +\draw[thick] (-3cm, 2cm) -- (-4cm, 4cm) -- (1.5cm, 4cm) -- (3.5cm, 0cm) -- (-0.35cm, 0cm); +\draw[dashed] (-3cm, 2cm) -- (-2cm, 0cm) -- (-0.35cm, 0cm); + +\draw[dotted] (-0.35cm, 0cm) -- (-1cm, 4.01cm); + +% draw names +\node[anchor=base] at (0.15cm, 5cm) {$D_{(t)}$}; +\node[anchor=base] at (2.5cm, -0.5cm) {$D_{(t+1)}$}; + +% draw string +\draw[thick, decorate, decoration={snake, segment length=1cm}] (1.5cm, 2cm) .. controls (1.1cm, 1.1cm) and (-1cm, 0.1cm) .. (-1.8cm, 0.3cm); +\filldraw[fill=black, draw=black] (1.5cm, 2cm) circle (2pt); +\filldraw[fill=black, draw=black] (-1.8cm, 0.3cm) circle (2pt); + +\end{tikzpicture} + +% vim: ft=tex \ No newline at end of file diff --git a/img/defects.pgf b/img/defects.pgf new file mode 100644 index 0000000..8851e9a --- /dev/null +++ b/img/defects.pgf @@ -0,0 +1,27 @@ +\begin{tikzpicture} + +% draw axis +\draw[thick, ->] (-3cm, 0cm) -- (3cm, 0cm) node[anchor=south] {$\uptau$}; +\draw[thick, ->] (-0.75cm, -1cm) -- (-0.75cm, 3cm) node[anchor=east] {$\upsigma$}; + +% draw defects +\filldraw[fill=white, draw=black] (-2cm, 0cm) circle (2pt) node[anchor=north] {$\hat{\uptau}_{(t+1)}$}; +\filldraw[fill=white, draw=black] (0.25cm, 0cm) circle (2pt) node[anchor=north] {$\hat{\uptau}_{(t)}$}; +\filldraw[fill=white, draw=black] (2.25cm, 0cm) circle (2pt) node[anchor=north] {$\hat{\uptau}_{(t-1)}$}; + +% draw the endlines +\draw[dotted] (-2cm, 2pt) -- (-2cm, 2cm); +\draw[dotted] (0.25cm, 2pt) -- (0.25cm, 2cm); +\draw[dotted] (2.25cm, 2pt) -- (2.25cm, 2cm); + +% draw the second D-brane +\draw[dashed] (-3cm, 2cm) -- (3cm, 2cm); +\node[anchor=south west] at (-0.75cm, 2cm) {$\uppi$}; + +% draw the string +\draw[decorate, decoration={snake, segment length=0.75cm}] (1cm, 0cm) -- (1cm, 2cm); +\draw[->] (1.1cm, 1cm) -- (1.6cm, 1cm); + +\end{tikzpicture} + +% vim: ft=tex diff --git a/img/threebranes_plane.pgf b/img/threebranes_plane.pgf index d7e3f12..d743482 100644 --- a/img/threebranes_plane.pgf +++ b/img/threebranes_plane.pgf @@ -1,11 +1,8 @@ -\usetikzlibrary{decorations.pathreplacing} -\usetikzlibrary{decorations.pathmorphing} - -\begin{tikzpicture} +\begin{tikzpicture} % draw axis -\draw[thick, ->] (-0.5cm, 0cm) -- (5cm, 0cm) node[anchor=south] {$\Re \omega$}; -\draw[thick, ->] (0cm, -1cm) -- (0cm, 3cm) node[anchor=east] {$\Im \omega$}; +\draw[thick, ->] (-0.5cm, 0cm) -- (5cm, 0cm) node[anchor=south] {$\Re \upomega$}; +\draw[thick, ->] (0cm, -1cm) -- (0cm, 3cm) node[anchor=east] {$\Im \upomega$}; % draw branching cuts \filldraw[fill=black!30, draw=black, dashed] (0cm, 2pt) rectangle (4.8cm, -2pt); @@ -18,4 +15,4 @@ \end{tikzpicture} -% vim: ft=tex \ No newline at end of file +% vim: ft=tex diff --git a/thesis.tex b/thesis.tex index 1a7392b..8e7d8b5 100644 --- a/thesis.tex +++ b/thesis.tex @@ -663,13 +663,174 @@ \begin{frame}{Hypergeometric Basis} \begin{columns} - \begin{column}{0.5\linewidth} + \begin{column}{0.3\linewidth} \centering - \resizebox{0.8\columnwidth} + \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\pi \alpha'} + \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} + + \section[Time Divergences]{Cosmological Backgrounds and Divergences} \begin{frame}{BBB}