Change the structure and adjustments

Signed-off-by: Riccardo Finotello <riccardo.finotello@gmail.com>

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[]
+% \AtBeginSection[]
-{%
+% {%
-  {%
+%   {%
-    \setbeamertemplate{footline}{}
+%     \setbeamertemplate{footline}{}
-    \begin{frame}[noframenumbering]{\contentsname}
+%     \begin{frame}[noframenumbering]{\contentsname}
-      \tableofcontents[currentsection]
+%       \tableofcontents[currentsection]
-    \end{frame}
+%     \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}
+    \fcolorbox{yellow}{yellow!20}{Consequence:}
-    a1
+    \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}{Extra Dimensions and Compactification}
-
-  \begin{frame}{AAA}
-    a2
-  \end{frame}
-
-
-  \subsection[Fermions]{Fermions With Boundary Defects}
-
-  \begin{frame}{AAA}
-    a3
   \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}