Begin part with fermions and defect CFT
Signed-off-by: Riccardo Finotello <riccardo.finotello@gmail.com>
This commit is contained in:
25
img/brane3d.pgf
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25
img/brane3d.pgf
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\usetikzlibrary{decorations.pathmorphing}
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\begin{tikzpicture}
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% draw branes
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\draw[thick] (-0.35cm, 0cm) -- (-3cm, -2cm) -- (-3cm, 3cm) -- (1cm, 5cm) -- (1cm, 4cm);
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\draw[dashed] (1cm, 4cm) -- (1cm, 1cm) -- (-0.35cm, 0cm);
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\draw[thick] (-3cm, 2cm) -- (-4cm, 4cm) -- (1.5cm, 4cm) -- (3.5cm, 0cm) -- (-0.35cm, 0cm);
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\draw[dashed] (-3cm, 2cm) -- (-2cm, 0cm) -- (-0.35cm, 0cm);
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\draw[dotted] (-0.35cm, 0cm) -- (-1cm, 4.01cm);
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% draw names
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\node[anchor=base] at (0.15cm, 5cm) {$D_{(t)}$};
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\node[anchor=base] at (2.5cm, -0.5cm) {$D_{(t+1)}$};
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% draw string
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\draw[thick, decorate, decoration={snake, segment length=1cm}] (1.5cm, 2cm) .. controls (1.1cm, 1.1cm) and (-1cm, 0.1cm) .. (-1.8cm, 0.3cm);
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\filldraw[fill=black, draw=black] (1.5cm, 2cm) circle (2pt);
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\filldraw[fill=black, draw=black] (-1.8cm, 0.3cm) circle (2pt);
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\end{tikzpicture}
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% vim: ft=tex
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27
img/defects.pgf
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27
img/defects.pgf
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\begin{tikzpicture}
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% draw axis
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\draw[thick, ->] (-3cm, 0cm) -- (3cm, 0cm) node[anchor=south] {$\uptau$};
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\draw[thick, ->] (-0.75cm, -1cm) -- (-0.75cm, 3cm) node[anchor=east] {$\upsigma$};
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% draw defects
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\filldraw[fill=white, draw=black] (-2cm, 0cm) circle (2pt) node[anchor=north] {$\hat{\uptau}_{(t+1)}$};
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\filldraw[fill=white, draw=black] (0.25cm, 0cm) circle (2pt) node[anchor=north] {$\hat{\uptau}_{(t)}$};
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\filldraw[fill=white, draw=black] (2.25cm, 0cm) circle (2pt) node[anchor=north] {$\hat{\uptau}_{(t-1)}$};
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% draw the endlines
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\draw[dotted] (-2cm, 2pt) -- (-2cm, 2cm);
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\draw[dotted] (0.25cm, 2pt) -- (0.25cm, 2cm);
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\draw[dotted] (2.25cm, 2pt) -- (2.25cm, 2cm);
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% draw the second D-brane
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\draw[dashed] (-3cm, 2cm) -- (3cm, 2cm);
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\node[anchor=south west] at (-0.75cm, 2cm) {$\uppi$};
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% draw the string
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\draw[decorate, decoration={snake, segment length=0.75cm}] (1cm, 0cm) -- (1cm, 2cm);
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\draw[->] (1.1cm, 1cm) -- (1.6cm, 1cm);
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\end{tikzpicture}
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% vim: ft=tex
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@@ -1,11 +1,8 @@
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\usetikzlibrary{decorations.pathreplacing}
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\usetikzlibrary{decorations.pathmorphing}
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\begin{tikzpicture}
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\begin{tikzpicture}
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% draw axis
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\draw[thick, ->] (-0.5cm, 0cm) -- (5cm, 0cm) node[anchor=south] {$\Re \omega$};
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\draw[thick, ->] (0cm, -1cm) -- (0cm, 3cm) node[anchor=east] {$\Im \omega$};
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\draw[thick, ->] (-0.5cm, 0cm) -- (5cm, 0cm) node[anchor=south] {$\Re \upomega$};
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\draw[thick, ->] (0cm, -1cm) -- (0cm, 3cm) node[anchor=east] {$\Im \upomega$};
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% draw branching cuts
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\filldraw[fill=black!30, draw=black, dashed] (0cm, 2pt) rectangle (4.8cm, -2pt);
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165
thesis.tex
165
thesis.tex
@@ -663,13 +663,174 @@
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\begin{frame}{Hypergeometric Basis}
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\begin{columns}
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\begin{column}{0.5\linewidth}
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\begin{column}{0.3\linewidth}
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\centering
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\resizebox{0.8\columnwidth}
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\resizebox{0.9\columnwidth}{!}{\import{img}{threebranes_plane.pgf}}
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\end{column}
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\hfill
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\begin{column}{0.7\linewidth}
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Sum over \highlight{all contributions:}
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\begin{equation*}
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\begin{split}
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\partial_z \mathcal{X}( z )
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& =
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\sum\limits_{l,\, r} c_{lr}\,
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\qty( - \upomega_z )^{A_{lr}}\,
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\qty( 1 - \upomega_z )^{B_{lr}}\,
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B_{0,\, l}^{(L)}( \omega_z )\,
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\qty( B_{0,\, r}^{(R)}( \omega_z ) )^T
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\end{split}
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\end{equation*}
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\end{column}
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\end{columns}
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\pause
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\begin{equationblock}{Basis of Solutions}
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\begin{equation*}
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B_{0,\, n}( \upomega_z )
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=
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\mqty(%
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1 & 0
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\\
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0 & K_n
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)
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\mqty(%
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\frac{1}{\Upgamma( c_n )}\,
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\tensor[_2]{F}{_1}( a_n,\, b_n;\, c_n;\, \upomega_z )
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\\
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\qty( -\upomega_z )^{1 - c_n}\,
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\frac{1}{\Upgamma( 2 - c_n )}\,
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\tensor[_2]{F}{_1}( a_n + 1 - c_n,\, b_n + 1 - c_n;\, 2 - c_n;\, \upomega_z )
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)
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\end{equation*}
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\end{equationblock}
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\end{frame}
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\begin{frame}{The Solution}
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\highlight{Operations sequence:}
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\begin{enumerate}
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\item rotation matrix $=$ monodromy matrix
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\pause
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\item contiguity relations $\Rightarrow$ independent hypergeometrics
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\pause
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\item finite action $\Rightarrow$ $2$ solutions (no.\ of d.o.f.\ is correctly saturated)
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\pause
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\item boundary conditions $\Rightarrow$ fix free constants $c_{lr}$
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\end{enumerate}
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\pause
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\begin{block}{Physical Interpretation}
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\only<5>{%
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\begin{columns}
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\begin{column}{0.4\linewidth}
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\centering
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\resizebox{0.607\columnwidth}{!}{\import{img}{branesangles.pgf}}
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\end{column}
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\hfill
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\begin{column}{0.6\linewidth}
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\begin{equation*}
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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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\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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& =
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\text{Area}\qty( \qty{ f_{(t)} } )
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\end{split}
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\end{equation*}
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\end{column}
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\end{columns}
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\vfill
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}
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\only<6->{%
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\centering
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\resizebox{0.25\columnwidth}{!}{\import{img}{brane3d.pgf}}
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}
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\end{block}
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\end{frame}
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\subsection[Fermions]{Fermions and Point-like Defect CFT}
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\begin{frame}{Fermions on the Strip}
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\begin{columns}
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\begin{column}{0.4\linewidth}
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\centering
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\resizebox{0.9\columnwidth}{!}{\import{img}{defects.pgf}}
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\end{column}
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\hfill
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\begin{column}{0.6\linewidth}
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\begin{equationblock}{Action of Boundary Changing Operators}
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\begin{equation*}
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\begin{cases}
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\uppsi_-^i( \uptau, 0 )
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& =
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\tensor{\qty( R_{(t)} )}{^I_J}\,
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\uppsi_+^J( \uptau, 0 )
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\quad \text{for}~
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\uptau \in \qty( \hat{\uptau}_{(t)},\, \hat{\uptau}_{(t-1)} )
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\\
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\uppsi_-^I( \uptau, \uppi )
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& =
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- \uppsi_+^I( \uptau, \uppi )
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\quad \text{for}~
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\uptau \in \mathds{R}
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\end{cases}
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\end{equation*}
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\end{equationblock}
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\end{column}
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\end{columns}
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\pause
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\begin{block}{Stress-energy Tensor}
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\begin{equation*}
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\mathcal{T}_{\pm\pm}( \upxi_{\pm} )
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=
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-i\, \frac{T}{4}\,
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\uppsi^*_{\pm,\, I}( \upxi_{\pm} )\,
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\overset{\leftrightarrow}{\partial} \uppsi^I_{\pm}( \upxi_{\pm} )
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\quad
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\Rightarrow
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\quad
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\begin{cases}
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\dot{\mathrm{H}}( \uptau )
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&
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% =
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% \partial_{\uptau}
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% \qty(%
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% \int\limits_0^{\uppi} \dd{\upsigma}
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% \mathcal{T}_{\uptau\uptau}( \uptau, \upsigma )
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% )
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=
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0 \Leftrightarrow \uptau \in \qty( \uptau_{(t)},\, \uptau_{(t-1)} )
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\\
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\dot{\mathrm{P}}( \uptau )
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&
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% =
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% \partial_{\uptau}
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% \qty(%
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% \int\limits_0^{\uppi} \dd{\upsigma}
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% \mathcal{T}_{\uptau\upsigma}( \uptau, \upsigma )
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% )
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\neq
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0
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\end{cases}
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\end{equation*}
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\end{block}
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\end{frame}
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\section[Time Divergences]{Cosmological Backgrounds and Divergences}
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\begin{frame}{BBB}
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