From ae8149954411761b413c9d012fbe8f433232e694 Mon Sep 17 00:00:00 2001 From: Riccardo Finotello Date: Wed, 4 Nov 2020 18:47:16 +0100 Subject: [PATCH] Add up to intersecting branes Signed-off-by: Riccardo Finotello --- img/chanpaton.pgf | 8 ++++---- img/smbranes.pgf | 31 +++++++++++++++++++++++++++++++ thesis.tex | 41 +++++++++++++++++++++++++++++++---------- 3 files changed, 66 insertions(+), 14 deletions(-) create mode 100644 img/smbranes.pgf diff --git a/img/chanpaton.pgf b/img/chanpaton.pgf index 811c788..70be2bc 100644 --- a/img/chanpaton.pgf +++ b/img/chanpaton.pgf @@ -7,16 +7,16 @@ % draw strings \draw[decorate, decoration={snake, segment length=1cm}, dash pattern=on 2.45cm off 1pt on 1pt off 1pt on 1pt off 1pt on 1pt off 1pt on 1pt off 1pt on 1pt off 1pt on 1pt off 1pt on 1pt off 1pt on 1pt off 1pt, postaction={decoration={markings, mark=at position 0.5 with {\arrow{>}}}, decorate}] (0.5cm, 3cm) -- (3cm, 1.5cm); -\node[anchor=base] (l1) at (1.75cm, 2.75cm) {$\tensor{\lambda}{^1_1_2}$}; +\node[anchor=base] (l1) at (1.75cm, 2.75cm) {$\tensor{\uplambda}{^1_1_2}$}; \draw[decorate, decoration={snake, segment length=1cm}, dash pattern=on 2.45cm off 1pt on 1pt off 1pt on 1pt off 1pt on 1pt off 1pt on 1pt off 1pt on 1pt off 1pt on 1pt off 1pt on 1pt off 1pt on 1pt off 1pt, postaction={decoration={markings, mark=at position 0.5 with {\arrow{>}}}, decorate}] (3cm, 3cm) -- (5.5cm, 1.5cm); -\node[anchor=base] (l2) at (4.25cm, 2.75cm) {$\tensor{\lambda}{^2_2_3}$}; +\node[anchor=base] (l2) at (4.25cm, 2.75cm) {$\tensor{\uplambda}{^2_2_3}$}; \draw[decorate, decoration={snake, segment length=1cm}, postaction={decoration={markings, mark=at position 0.765 with {\arrow{>}}}, decorate}] (3cm, 2.5cm) .. controls (3.5cm, 2cm) and (3.5cm, 1cm) .. (3cm, 0.5cm); -\node[anchor=base] (l3) at (4cm, 1.5cm) {$\tensor{\lambda}{^3_2_2}$}; +\node[anchor=base] (l3) at (4cm, 1.5cm) {$\tensor{\uplambda}{^3_2_2}$}; \draw[decorate, decoration={snake, segment length=1cm}, dash pattern=on 5.15cm off 1pt on 1pt off 1pt on 1pt off 1pt on 1pt off 1pt on 1pt off 1pt on 1pt off 1pt on 1pt off 1pt on 1pt off 1pt on 1pt off 1pt on 1pt off 1pt, postaction={decoration={markings, mark=at position 0.5 with {\arrow{<}}}, decorate}] (0.5cm, 1cm) .. controls (1.5cm, 0cm) and (4cm, 0cm) .. (5.5cm, 1cm); -\node[anchor=base] (l4) at (3cm, -0.5cm) {$\tensor{\lambda}{^4_3_1}$}; +\node[anchor=base] (l4) at (3cm, -0.5cm) {$\tensor{\uplambda}{^4_3_1}$}; \end{tikzpicture} diff --git a/img/smbranes.pgf b/img/smbranes.pgf new file mode 100644 index 0000000..04a7565 --- /dev/null +++ b/img/smbranes.pgf @@ -0,0 +1,31 @@ +\begin{tikzpicture} + +% draw the baryonic stack +\draw[thick] (-2.5cm, 5cm) -- (2.5cm, 5cm); +\draw[thick] (-2.5cm, 4.9cm) -- (2.5cm, 4.9cm) node[anchor=west] (baryonic) {baryonic $\qty(Y = \frac{1}{3} )$}; +\draw[thick] (-2.5cm, 4.8cm) -- (2.5cm, 4.8cm); + +% draw the left stack +\draw[thick] (-2.25cm, 5.25cm) -- (-2.25cm, 0cm); +\draw[thick] (-2.15cm, 5.25cm) -- (-2.15cm, 0cm) node[anchor=north east] (left) {left $\qty( Y = - \frac{1}{2} )$}; + +% draw leptonic and right stacks +\draw[thick] (-2.5cm, 2.5cm) -- (2.5cm, 2.5cm) node[anchor=west] (l1) {leptonic $\qty( Y = 0 )$}; +\draw[thick] (-2.5cm, 0.5cm) -- (2.5cm, 0.5cm) node[anchor=west] (l2) {leptonic $\qty( Y = -1 )$}; +\draw[thick] (0cm, 5.25cm) -- (0cm, 0cm) node[anchor=north] (r1) {right $\qty( Y = -1 )$}; +\draw[thick] (2.25cm, 5.25cm) -- (2.25cm, 0cm) node[anchor=north west] (r2) {right $\qty( Y = 0 )$}; + +% draw left quarks +\draw[decorate, decoration={snake, segment length=0.5cm}, postaction={decoration={markings, mark=at position 0.5 with {\arrow{>}}}, decorate}] (-2.15cm, 3.5cm) node[anchor=south east] {$\mqty( u_L \\ d_L )$} -- (-1cm, 4.8cm); +% draw up right quark +\draw[decorate, decoration={snake, segment length=0.5cm}, postaction={decoration={markings, mark=at position 0.5 with {\arrow{<}}}, decorate}] (0cm, 3.5cm) node[anchor=east] {$\overline{u}_R$} -- (1cm, 4.8cm); +% draw down right quark +\draw[decorate, decoration={snake, segment length=0.5cm}, postaction={decoration={markings, mark=at position 0.5 with {\arrow{<}}}, decorate}] (2.25cm, 3.5cm) node[anchor=west] {$\overline{d}_R$} -- (1.15cm, 4.8cm); +% draw left leptons +\draw[decorate, decoration={snake, segment length=0.5cm}, postaction={decoration={markings, mark=at position 0.5 with {\arrow{>}}}, decorate}] (-2.15cm, 0.9cm) node[anchor=south east] {$\mqty( \upnu^e_L \\ e^-_L )$} -- (-1cm, 2.5cm); +% draw right leptons +\draw[decorate, decoration={snake, segment length=0.5cm}, postaction={decoration={markings, mark=at position 0.5 with {\arrow{<}}}, decorate}] (0cm, 0.9cm) node[anchor=east] {$\overline{e}^+_R$} -- (1cm, 2.5cm); + +\end{tikzpicture} + +% vim: ft=tex diff --git a/thesis.tex b/thesis.tex index d8b895c..05a9d64 100644 --- a/thesis.tex +++ b/thesis.tex @@ -48,7 +48,7 @@ \newenvironment{equationblock}[1]{% \begin{block}{#1} - \vspace*{-\baselineskip}\setlength\belowdisplayshortskip{0pt} + \vspace*{-0.75\baselineskip}\setlength\belowdisplayshortskip{0.25\baselineskip} }{% \end{block} } @@ -131,6 +131,11 @@ % {% % {% % \setbeamertemplate{footline}{} +% \usebackgroundtemplate{% +% \transparent{0.1} +% \includegraphics[width=\paperwidth]{img/torino.png} +% } +% \addtobeamertemplate{background canvas}{\transfade[duration=0.25]}{} % \begin{frame}[noframenumbering]{\contentsname} % \tableofcontents[currentsection] % \end{frame} @@ -315,7 +320,7 @@ \begin{frame}{Extra Dimensions and Compactification} \begin{block}{Compactification} \begin{columns} - \begin{column}{0.6\linewidth} + \begin{column}{0.7\linewidth} \begin{equation*} \mathscr{M}^{1,\, 9} = @@ -330,9 +335,9 @@ \end{itemize} \end{column} - \begin{column}{0.4\linewidth} + \begin{column}{0.3\linewidth} \centering - \includegraphics[width=0.7\columnwidth]{img/cy} + \includegraphics[width=0.9\columnwidth]{img/cy} \end{column} \end{columns} \end{block} @@ -372,7 +377,7 @@ \pause - \begin{block}{T-duality} + \begin{equationblock}{T-duality} \begin{equation*} X( z, \overline{z} ) = @@ -386,7 +391,7 @@ = Y( z ) + \overline{Y}( \overline{z} ) \end{equation*} - \end{block} + \end{equationblock} \pause @@ -408,8 +413,7 @@ \pause - \begin{block}{Spectrum} - At massless level (\emph{irrep} of \textbf{little group} $\mathrm{SO}( D - 2 )$): + \begin{equationblock}{Massless Spectrum (\emph{irrep} of \textbf{little group} $\mathrm{SO}( D - 2 )$} \begin{equation*} \mathcal{A}^{\upmu} \quad \leftrightarrow \quad @@ -435,7 +439,7 @@ $a = 1,\, 2,\, \dots,\, D - p - 1$ \end{tabular} \end{equation*} - \end{block} + \end{equationblock} \pause @@ -457,10 +461,27 @@ \pause \highlight{Build gauge bosons, fermions and scalars.} \end{column} - \end{columns} \end{frame} + \begin{frame}{Standard Model-like Scenarios} + \centering + \resizebox{0.8\linewidth}{!}{\import{img}{smbranes.pgf}} + \end{frame} + + + \subsection[D-branes at Angles]{D-branes Intersecting at Angles} + + \begin{frame}{Intersecting D-branes} + Consider \highlight{$N$ intersecting $D6$-branes} filling $\mathscr{M}^{1,3}$ and \textbf{embedded in} $\mathds{R}^6$ + + \begin{equationblock}{Twist Fields Correlators} + \begin{equation*} + \left\langle \prod\limits_{t = 1}^N \upsigma_{\mathrm{M}_{(t)}}\qty( x_{(t)} ) \right\rangle + \end{equation*} + \end{equationblock} + \end{frame} + \section[Time Divergences]{Cosmological Backgrounds and Divergences} \begin{frame}{BBB}