From d563c8b47de6571ede69616d106f983348359b14 Mon Sep 17 00:00:00 2001 From: Riccardo Finotello Date: Tue, 3 Nov 2020 22:05:04 +0100 Subject: [PATCH] Change the structure and adjustments Signed-off-by: Riccardo Finotello --- thesis.tex | 164 +++++++++++++++++++++++++++++++---------------------- 1 file changed, 96 insertions(+), 68 deletions(-) diff --git a/thesis.tex b/thesis.tex index 986e452..ff70be0 100644 --- a/thesis.tex +++ b/thesis.tex @@ -41,11 +41,18 @@ } \date{15th December 2020} +\newenvironment{equationblock}[1]{% + \begin{block}{#1} + \vspace*{-\baselineskip}\setlength\belowdisplayshortskip{0pt} +}{% + \end{block} +} + \newcommand{\firstlogo}{img/unito} \newcommand{\thefirstlogo}{% \begin{figure} \centering - \includegraphics[width=5em]{\firstlogo} + \includegraphics[width=7em]{\firstlogo} \end{figure} } @@ -53,7 +60,7 @@ \newcommand{\thesecondlogo}{% \begin{figure} \centering - \includegraphics[width=5em]{\secondlogo} + \includegraphics[width=7em]{\secondlogo} \end{figure} } @@ -113,15 +120,15 @@ \par } -\AtBeginSection[] -{% - {% - \setbeamertemplate{footline}{} - \begin{frame}[noframenumbering]{\contentsname} - \tableofcontents[currentsection] - \end{frame} - } -} +% \AtBeginSection[] +% {% +% {% +% \setbeamertemplate{footline}{} +% \begin{frame}[noframenumbering]{\contentsname} +% \tableofcontents[currentsection] +% \end{frame} +% } +% } \begin{document} @@ -146,8 +153,11 @@ \section[CFT]{Conformal Symmetry and Geometry of the Worldsheet} + + \subsection[Preliminary]{Preliminary Concepts and Tools} + \begin{frame}{Action Principle and Conformal Symmetry} - \begin{block}{Polyakov's Action} + \begin{equationblock}{Polyakov's Action} \begin{equation*} S_P\qty[ \upgamma,\, X,\, \uppsi ] = @@ -157,7 +167,7 @@ \sqrt{-\det \upgamma}\, \upgamma^{\upalpha \upbeta}\, \qty(% - \frac{2}{\alpha'}\, + \frac{2}{\upalpha'}\, \partial_{\upalpha} X^{\upmu}\, \partial_{\upbeta} X^{\upnu} + @@ -168,44 +178,66 @@ )\, \upeta_{\upmu\upnu} \end{equation*} - \end{block} + \end{equationblock} \begin{columns} \begin{column}[t]{0.5\linewidth} - Symmetries: + \fcolorbox{yellow}{yellow!20}{Symmetries:} \begin{itemize} - \item Poincaré transf.\ $X'^{\upmu} = \tensor{\Uplambda}{^{\upmu}_{\upnu}} X^{\upnu} + c^{\upmu}$ + \item \textbf{Poincaré transf.}\ $X'^{\upmu} = \tensor{\Uplambda}{^{\upmu}_{\upnu}} X^{\upnu} + c^{\upmu}$ - \item 2D diff.\ $\upgamma'_{\upalpha \upbeta} = \tensor{\qty( \mathrm{J}^{-1} )}{_{\upalpha \upbeta}^{\uplambda \uprho}}\, \gamma_{\uplambda \uprho}$ + \item \textbf{2D diff.}\ $\upgamma'_{\upalpha \upbeta} = \tensor{\qty( \mathrm{J}^{-1} )}{_{\upalpha \upbeta}^{\uplambda \uprho}}\, \gamma_{\uplambda \uprho}$ - \item Weyl transf.\ $\upgamma'_{\upalpha \upbeta} = e^{2 \upomega}\, \gamma_{\upalpha \upbeta}$ + \item \textbf{Weyl transf.}\ $\upgamma'_{\upalpha \upbeta} = e^{2 \upomega}\, \gamma_{\upalpha \upbeta}$ \end{itemize} \end{column} \begin{column}[t]{0.5\linewidth} - Conformal symmetry: + \fcolorbox{yellow}{yellow!20}{Conformal symmetry:} \begin{itemize} - \item vanishing stress-energy tensor: $\mathcal{T}_{\upalpha \upbeta} = 0$ + \item \textbf{vanishing} stress-energy tensor: $\mathcal{T}_{\upalpha \upbeta} = 0$ - \item traceless stress-energy tensor: $\trace{\mathcal{T}} = 0$ + \item \textbf{traceless} stress-energy tensor: $\trace{\mathcal{T}} = 0$ - \item conformal gauge $\upgamma_{\upalpha \upbeta} = e^{\upphi}\, \upeta_{\upalpha \upbeta}$ + \item \textbf{conformal gauge} $\upgamma_{\upalpha \upbeta} = e^{\upphi}\, \upeta_{\upalpha \upbeta}$ \end{itemize} \end{column} \end{columns} \end{frame} + \begin{frame}{Action Principle and Conformal Symmetry} \begin{columns} \begin{column}{0.6\linewidth} - Let $z = e^{\uptau_E + i \upsigma} \Rightarrow \overline{\partial} \mathcal{T}( z ) = \partial \overline{\mathcal{T}}( \overline{z} ) = 0$: + \fcolorbox{yellow}{yellow!20}{% + Let $z = e^{\uptau_E + i \upsigma} \Rightarrow \overline{\partial} \mathcal{T}( z ) = \partial \overline{\mathcal{T}}( \overline{z} ) = 0$: + } \begin{equation*} - T( z )\, \Upphi_{\upomega}( w ) + \mathcal{T}( z )\, \Upphi_h( w ) \stackrel{z \to w}{\sim} - \frac{\upomega}{(z - w)^2} \Upphi_{\upomega}( w ) + \frac{h}{(z - w)^2} \Upphi_h( w ) + - \frac{1}{z - w} \partial_w \Upphi_{\upomega}( 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[ L_n,\, L_m ] + & = & + (n - m) 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} @@ -217,25 +249,49 @@ \end{columns} \end{frame} + \begin{frame}{Action Principle and Conformal Symmetry} + \fcolorbox{yellow}{yellow!20}{Superstrings in $D$ dimensions:} + \begin{equation*} + \mathcal{T}( z ) + = + -\frac{1}{\upalpha'} + \partial X( z ) \cdot \partial X( z ) + -\frac{1}{2} + \uppsi( z ) \cdot \partial \uppsi( z ) + \quad + \Rightarrow + \quad + c = \frac{3}{2} D + \end{equation*} - \subsection[Tools]{Preliminary Tools and Definitions} + \begin{block}{$\qty( \uplambda, 0 )~/~\qty( 1 - \uplambda, 0 )$ Ghost System} + Introduce anti-commuting $\qty( b,\, c )$ and commuting $\qty( \upbeta,\, \upgamma )$ conformal fields: + \begin{equation*} + S_{\text{ghost}}\qty[ b,\, c,\, \upbeta,\, \upgamma ] + = + \frac{1}{2\uppi} + \iint \dd{z} \dd{\overline{z}} + \qty(% + b( z )\, \overline{\partial} c( z ) + + + \upbeta( z )\, \overline{\partial} \upgamma( z ) + ) + \end{equation*} + where $\uplambda_b = 2$ and $\uplambda_{\upbeta} = \frac{3}{2}$. + \end{block} - \begin{frame}{AAA} - a1 + \fcolorbox{yellow}{yellow!20}{Consequence:} + \begin{equation*} + c_{\text{full}} = c + c_{\text{ghost}} = 0 + \quad + \Leftrightarrow + \quad + D = 10. + \end{equation*} \end{frame} - \subsection[D-branes]{D-branes Intersecting at Angles} - - \begin{frame}{AAA} - a2 - \end{frame} - - - \subsection[Fermions]{Fermions With Boundary Defects} - - \begin{frame}{AAA} - a3 + \begin{frame}{Extra Dimensions and Compactification} \end{frame} @@ -246,38 +302,10 @@ \end{frame} - \subsection[Orbifolds]{Orbifolds and Cosmological Models} - - \begin{frame}{BBB} - b1 - \end{frame} - - - \subsection[Time Dependency]{Time Dependent Orbifolds} - - \begin{frame}{BBB} - b2 - \end{frame} - - \section[Deep Learning]{Deep Learning the Geometry of String Theory} \begin{frame}{CCC} c \end{frame} - \subsection[CICY]{Complete Intersection Calabi--Yau Manifolds} - - \begin{frame}{CCC} - c1 - \end{frame} - - - \subsection[Machine Learning]{Machine Learning and Deep Learning for CICY Manifolds} - - \begin{frame}{CCC} - c2 - \end{frame} - - \end{document}