Begin NBO part

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

img/inconsistent_theories.pgf

@@ -0,0 +1,27 @@
+\begin{tikzpicture}
+
+% fill the overlap area
+\draw[black!0, pattern=north east lines, pattern color=black!15] (0cm, 0cm) rectangle (2cm, 2cm);
+\node[anchor=base, text width=2cm, align=center] at (1cm, 1.4cm) {overlap region};
+\node[anchor=base, text width=3cm, align=center, scale=0.5] at (1cm, 0.5cm) {inconsistent theories};
+
+% draw the horizontal axis
+\draw[thick, ->] (-3cm, 0cm) -- (4cm, 0cm) node[anchor=south west] {$n$};
+
+% draw points
+\node[anchor=north] at (-3cm, 0cm) {$\cdots$};
+\filldraw[fill=white, draw=black] (-2cm,0cm) circle (2pt) node[anchor=north] {$-1$};
+\filldraw[fill=white, draw=black] (-1cm,0cm) circle (2pt) node[anchor=north] {$0$};
+\filldraw[fill=white, draw=black] (0cm,0cm) circle (2pt) node[anchor=north] {$1$};
+\node[anchor=north] at (1cm, 0cm) {$\cdots$};
+\filldraw[fill=white, draw=black] (2cm,0cm) circle (2pt) node[anchor=north] {$\mathrm{L}$};
+\filldraw[fill=white, draw=black] (3cm,0cm) circle (2pt) node[anchor=north] {$\mathrm{L}+1$};
+\node[anchor=north] at (4cm, 0cm) {$\cdots$};
+
+% draw limits
+\draw[->] (0cm, 2pt) -- (0cm, 2cm) -- (4cm, 2cm) node[midway, anchor=south west] {in-annihilators} node[anchor=north east] {$b_{n}$};
+\draw[->] (2cm, 2pt) -- (2cm, 1.8cm) -- (-3cm, 1.8cm) node[midway, anchor=south east] {out-annihilators} node[anchor=north west] {$b^*_{\mathrm{L} + 1 - n}$};
+
+\end{tikzpicture}
+
+% vim: ft=tex

thesis.tex

@@ -46,6 +46,7 @@
 \usetikzlibrary{decorations.pathmorphing}
 \usetikzlibrary{decorations.pathreplacing}
 \usetikzlibrary{arrows}
+\usetikzlibrary{patterns}
 
 \newenvironment{equationblock}[1]{%
   \begin{block}{#1}
@@ -180,7 +181,7 @@
       \begin{equation*}
         S_P\qty[ \upgamma,\, X,\, \uppsi ]
         =
-        -\frac{1}{4\pi}
+        -\frac{1}{4\uppi}
         \int\limits_{-\infty}^{+\infty} \dd{\uptau}
         \int\limits_0^{\ell} \dd{\upsigma}
         \sqrt{-\det \upgamma}\,
@@ -207,9 +208,9 @@
         \begin{itemize}
           \item \textbf{Poincaré transf.}\ $X'^{\upmu} = \tensor{\Uplambda}{^{\upmu}_{\upnu}} X^{\upnu} + c^{\upmu}$
 
-          \item \textbf{2D diff.}\ $\upgamma'_{\upalpha \upbeta} = \tensor{\qty( \mathrm{J}^{-1} )}{_{\upalpha \upbeta}^{\uplambda \uprho}}\, \gamma_{\uplambda \uprho}$
+          \item \textbf{2D diff.}\ $\upgamma'_{\upalpha \upbeta} = \tensor{\qty( \mathrm{J}^{-1} )}{_{\upalpha \upbeta}^{\uplambda \uprho}}\, \upgamma_{\uplambda \uprho}$
 
-          \item \textbf{Weyl transf.}\ $\upgamma'_{\upalpha \upbeta} = e^{2 \upomega}\, \gamma_{\upalpha \upbeta}$
+          \item \textbf{Weyl transf.}\ $\upgamma'_{\upalpha \upbeta} = e^{2 \upomega}\, \upgamma_{\upalpha \upbeta}$
         \end{itemize}
       \end{column}
 
@@ -229,48 +230,48 @@
   \end{frame}
 
 
-  \begin{frame}{Action Principle and Conformal Symmetry}
+  % \begin{frame}{Action Principle and Conformal Symmetry}
-    \begin{columns}
+  %   \begin{columns}
-      \begin{column}{0.6\linewidth}
+  %     \begin{column}{0.6\linewidth}
-        \highlight{%
+  %       \highlight{%
-          Let $z = e^{\uptau_E + i \upsigma} \Rightarrow \overline{\partial} \mathcal{T}( z ) = \partial \overline{\mathcal{T}}( \overline{z} ) = 0$:
+  %         Let $z = e^{\uptau_E + i \upsigma} \Rightarrow \overline{\partial} \mathcal{T}( z ) = \partial \overline{\mathcal{T}}( \overline{z} ) = 0$:
-        }
+  %       }
-        \begin{equation*}
+  %       \begin{equation*}
-          \mathcal{T}( z )\, \Upphi_h( w )
+  %         \mathcal{T}( z )\, \Upphi_h( w )
-          \stackrel{z \to w}{\sim}
+  %         \stackrel{z \to w}{\sim}
-          \frac{h}{(z - w)^2} \Upphi_h( w )
+  %         \frac{h}{(z - w)^2} \Upphi_h( w )
-          +
+  %         +
-          \frac{1}{z - w} \partial_w \Upphi_h( w )
+  %         \frac{1}{z - w} \partial_w \Upphi_h( w )
-        \end{equation*}
+  %       \end{equation*}
-        \begin{equation*}
+  %       \begin{equation*}
-          \mathcal{T}( z )\, \mathcal{T}( w )
+  %         \mathcal{T}( z )\, \mathcal{T}( w )
-          \stackrel{z \to w}{\sim}
+  %         \stackrel{z \to w}{\sim}
-          \frac{\frac{c}{2}}{(z - w)^4}
+  %         \frac{\frac{c}{2}}{(z - w)^4}
-          +
+  %         +
-          \order{(z - w)^{-2}}
+  %         \order{(z - w)^{-2}}
-        \end{equation*}
+  %       \end{equation*}
 
-        \begin{equationblock}{Virasoro algebra $\mathscr{V} \oplus \overline{\mathscr{V}}$}
+  %       \begin{equationblock}{Virasoro algebra $\mathscr{V} \oplus \overline{\mathscr{V}}$}
-          \begin{eqnarray*}
+  %         \begin{eqnarray*}
-            \qty[ \overset{\scriptscriptstyle(-)}{L}_n,\, \overset{\scriptscriptstyle(-)}{L}_m ]
+  %           \qty[ \overset{\scriptscriptstyle(-)}{L}_n,\, \overset{\scriptscriptstyle(-)}{L}_m ]
-            & = &
+  %           & = &
-            (n - m) \overset{\scriptscriptstyle(-)}{L}_{n + m} + \frac{c}{12} n \qty(n^2 - 1) \updelta_{n + m,\, 0}
+  %           (n - m) \overset{\scriptscriptstyle(-)}{L}_{n + m} + \frac{c}{12} n \qty(n^2 - 1) \updelta_{n + m,\, 0}
-            \\
+  %           \\
-            \qty[ L_n,\, \overline{L}_m ]
+  %           \qty[ L_n,\, \overline{L}_m ]
-            & = &
+  %           & = &
-            0
+  %           0
-          \end{eqnarray*}
+  %         \end{eqnarray*}
-        \end{equationblock}
+  %       \end{equationblock}
-      \end{column}
+  %     \end{column}
 
-      \begin{column}{0.4\linewidth}
+  %     \begin{column}{0.4\linewidth}
-        \begin{figure}[h]
+  %       \begin{figure}[h]
-          \centering
+  %         \centering
-          \resizebox{0.9\columnwidth}{!}{\import{img}{complex_plane.pgf}}
+  %         \resizebox{0.9\columnwidth}{!}{\import{img}{complex_plane.pgf}}
-        \end{figure}
+  %       \end{figure}
-      \end{column}
+  %     \end{column}
-    \end{columns}
+  %   \end{columns}
-  \end{frame}
+  % \end{frame}
 
   \begin{frame}{Action Principle and Conformal Symmetry}
     \highlight{Superstrings in $D$ dimensions:}
@@ -421,7 +422,7 @@
       \begin{equation*}
         \mathcal{A}^{\upmu}
         \quad \leftrightarrow \quad
-        \alpha_{-1}^{\upmu} \ket{0}
+        \upalpha_{-1}^{\upmu} \ket{0}
         \qquad
         \longrightarrow
         \qquad
@@ -430,7 +431,7 @@
           &
           $\leftrightarrow$
           &
-          $\alpha_{-1}^A \ket{0},$
+          $\upalpha_{-1}^A \ket{0},$
           &
           $A = 0,\, 1,\, \dots,\, p$
           \\
@@ -438,7 +439,7 @@
           &
           $\leftrightarrow$
           &
-          $\alpha_{-1}^a \ket{0},$
+          $\upalpha_{-1}^a \ket{0},$
           &
           $a = 1,\, 2,\, \dots,\, D - p - 1$
         \end{tabular}
@@ -740,7 +741,7 @@
               \begin{split}
                 \eval{S_{\mathds{R}^4}}_{\text{on-shell}}
                 & =
-                \frac{1}{2\pi \alpha'}
+                \frac{1}{2\uppi \upalpha'}
                 \sum\limits_{t = 1}^3
                 \qty( \frac{1}{2} \abs{g_{(t)}^{\perp}} \abs{f_{(t-1)} - f_{(t)}} )
                 \\
@@ -830,11 +831,352 @@
     \end{block}
   \end{frame}
 
+  \begin{frame}{Conserved Product and Operators}
+    Expand on a \highlight{basis of solutions}
+    \begin{equation*}
+      \uppsi_{\pm}( \upxi_{\pm} )
+      =
+      \sum\limits_{n = -\infty}^{+\infty} b_n\, \uppsi_n( \upxi_{\pm} )
+      \qquad
+      \Rightarrow
+      \qquad
+      \Uppsi( z )
+      =
+      \begin{cases}
+        \uppsi_{E,\, +}( u ) \quad \text{if}~z \in \mathscr{H}_{>}^{(\overline{t})}
+        \\
+        \uppsi_{E,\, -}( u ) \quad \text{if}~z \in \mathscr{H}_{<}^{(\overline{t})}
+      \end{cases}
+    \end{equation*}
+
+    \pause
+
+    \begin{equationblock}{Conserved Product and Dual Basis}
+      \begin{equation*}
+        \left\langle\!\left\langle
+          \tensor[^*]{\uppsi}{_n},\,
+          \uppsi_m
+        \right. \right\rangle
+        =
+        2\uppi \mathcal{N}\,
+        \oint
+        \frac{\dd{z}}{2\uppi i}\,
+        \tensor[^*]{\Uppsi}{_n^*}\,
+        \tensor{\Uppsi}{_m}
+        =
+        \updelta_{n,\, m}
+        \quad
+        \Rightarrow
+        \quad
+        \left\langle\!\left\langle
+          \tensor[^*]{\Uppsi}{_n^{(*)}},\,
+          \Uppsi^{(*)}
+        \right. \right\rangle
+        =
+        b_n^{(\dagger)}
+      \end{equation*}
+    \end{equationblock}
+
+    \pause
+
+    Derive the \highlight{algebra of operators:}
+    \begin{equation*}
+      \qty[ b_n,\, b_m^{\dagger} ]_+
+      =
+      \frac{2 \mathcal{N}}{T}\,
+      \left\langle\!\left\langle
+        \tensor[^*]{\Uppsi}{_n^*},\,
+        \Uppsi_m^*
+      \right. \right\rangle
+    \end{equation*}
+  \end{frame}
+
+  \begin{frame}{Twisted Complex Fermions}
+    Consider the case $R_{(t)} = e^{i \uppi \upalpha_{(t)}} \in \mathrm{U}( 1 )$:
+    \begin{equation*}
+      \Uppsi( x_{(t)} + e^{2\uppi i} \updelta )
+      =
+      e^{i \uppi \upepsilon_{(t)}}\,
+      \Uppsi( x_{(t)} + \updelta )
+    \end{equation*}
+    where
+    \begin{equation*}
+      \upepsilon_{(t)}
+      =
+      \upalpha_{(t+1)} - \upalpha_{(t)}
+      +
+      \uptheta\qty( \upalpha_{(t)} - \upalpha_{(t+1)} - 1 )
+      -
+      \uptheta\qty( \upalpha_{(t+1)} - \upalpha_{(t)} - 1 )
+    \end{equation*}
+
+    \pause
+
+    \begin{equationblock}{Basis of Solutions}
+      \begin{equation*}
+        \begin{split}
+          \Uppsi_n\qty( z;\, \qty{ x_{(t)} } )
+          & =
+          \mathcal{N}_{\Uppsi}\,
+          z^{-n}\,
+          \prod\limits_{t = 1}^N
+          \qty( 1 - \frac{z}{x_{(t)}} )^{n_{(t)} + \frac{\upepsilon_{(t)}}{2}}
+          \\
+          \tensor[^*]{\Uppsi}{_n}\qty( z;\, \qty{ x_{(t)} } )
+          & =
+          \frac{1}{2\uppi \mathcal{N} \mathcal{N}_{\Uppsi}}\,
+          z^{n - 1}\,
+          \prod\limits_{t = 1}^N
+          \qty( 1 - \frac{z}{x_{(t)}} )^{-\widetilde{n}_{(t)} + \frac{\upepsilon_{(t)}}{2}}
+        \end{split}
+      \end{equation*}
+    \end{equationblock}
+  \end{frame}
+
+  \begin{frame}{Vacua}
+    Define the \textbf{vacuum} with respect to $b_n$:
+    \begin{equation*}
+      \begin{split}
+        b_n \ket{\qty{ x_{(t)} }} = 0 &\quad \text{for} \quad n \ge 1
+        \\
+        b_n \ket{\widetilde{0}} = 0 &\quad \text{for} \quad n \ge n_{(t)} + \frac{\upepsilon_{(t)}}{2} + \frac{1}{2}
+      \end{split}
+    \end{equation*}
+
+    \pause
+
+    Theories are subject to \highlight{consistency conditions:}
+    \begin{columns}
+      \begin{column}{0.6\linewidth}
+        \begin{equation*}
+          \mathrm{L}
+          =
+          n_{(t)} + \widetilde{n}_{(t)}
+          \uncover<3->{%
+            \alert{= 0}
+          }
+        \end{equation*}
+      \end{column}
+      \hfill
+      \begin{column}{0.4\linewidth}
+        \centering
+        \resizebox{\columnwidth}{!}{\import{img}{inconsistent_theories.pgf}}
+      \end{column}
+    \end{columns}
+  \end{frame}
+
+  \begin{frame}{Stress-energy Tensor and CFT Approach}
+    Compute the OPEs leading to the \highlight{stress-energy tensor:}
+    \begin{equation*}
+      \mathcal{T}( z )
+      =
+      \frac{\uppi T}{2} \mathcal{N}_{\Uppsi}^2
+      \sum\limits_{n,\, m = -\infty}^{+\infty}
+      \colon b_n\, b_m^* \colon\,
+      z^{-n -m}\,
+      \qty[%
+        \frac{m - n}{2}
+        +
+        2 \sum\limits_{t = 1}^N \frac{n_{(t)} + \frac{\upepsilon_{(t)}}{2}}{z - x_{(t)}}
+      ]
+      +
+      \frac{1}{2} \qty( \sum\limits_{t = 1}^N \frac{n_{(t)} + \frac{\upepsilon_{(t)}}{2}}{z - x_{(t)}} )^2
+    \end{equation*}
+
+    \pause
+
+    \begin{equationblock}{Invariant Vacuum and Spin Fields}
+      \begin{equation*}
+        \ket{\qty{ x_{(t)} }}
+        =
+        \mathcal{N}\qty( \qty{ x_{(t)} } )\,
+        \mathrm{R}\qty[ \prod\limits_{t = 1}^M S_{(t)}( x_{(t)} ) ]\,
+        \ket{0}_{\mathrm{SL}_2( \mathds{R} )}
+      \end{equation*}
+    \end{equationblock}
+  \end{frame}
+
+  \begin{frame}{Spin Fields Amplitudes}
+    \begin{equationblock}{Equivalence with Bosonization}
+      \begin{equation*}
+        \begin{split}
+          \partial_{x_{(t)}} \braket{\qty{x_{(t)}}}
+          & =
+          \oint\limits_{x_{(t)}} \frac{\dd{z}}{2\uppi i}
+          \frac{%
+            \bra{\qty{x_{(t)}}} \mathcal{T}( z ) \ket{\qty{x_{(t)}}}
+          }{%
+            \braket{\qty{x_{(t)}}}
+          }
+          \\
+          \Rightarrow
+          \quad
+          \braket{\qty{x_{(t)}}}
+          & =
+          \mathcal{N}\qty( \qty{ \upepsilon_{(t)} } )
+          \prod\limits_{\substack{t = 1 \\ t > u}}^N
+          \qty( x_{(u)} - x_{(t)} )^{\qty( n_{(u)} + \frac{\upepsilon_{(u)}}{2} )\qty( n_{(t)} + \frac{\upepsilon_{(t)}}{2} )}
+        \end{split}
+      \end{equation*}
+    \end{equationblock}
+
+    \pause
+
+    \begin{itemize}
+      \item (semi-)phenomenological models involve \textbf{twist and spin} fields and \textbf{open strings}
+
+        \pause
+
+      \item general framework for \textbf{bosonic} open strings with \textbf{intersecting D-branes}
+
+        \pause
+
+      \item leading contribution for \textbf{twist fields}
+
+        \pause
+
+      \item \textbf{spin fields} as \textbf{boundary changing operators} on \textbf{defects}
+
+        \pause
+
+      \item alternative framework for amplitudes (extension to (non) Abelian twist/spin fields?)
+    \end{itemize}
+  \end{frame}
+
 
   \section[Time Divergences]{Cosmological Backgrounds and Divergences}
 
-  \begin{frame}{BBB}
+
-    b
+  \subsection[Orbifold]{Orbifolds and Cosmological Toy Models}
+
+  \begin{frame}{A Few Words on a Theory of Everything}
+    \begin{center}
+      string theory = theory of everything = nuclear forces + gravity
+    \end{center}
+
+    \pause
+
+    \begin{columns}
+      \begin{column}{0.5\linewidth}
+        \centering
+        \includegraphics[width=0.9\columnwidth]{img/cone}
+      \end{column}
+      \hfill
+      \begin{column}{0.5\linewidth}
+        From the phenomenological point of view:
+        \begin{itemize}
+          \item cosmological implications
+            
+            \pause
+
+          \item Big Bang(-like) singularities
+
+            \pause
+
+          \item toy models of \textbf{space-like singularities}
+        \end{itemize}
+
+        \pause
+
+        \begin{center}
+          $\Downarrow$
+
+          \highlight{time-dependent orbifold models}
+        \end{center}
+      \end{column}
+    \end{columns}
+  \end{frame}
+
+  \begin{frame}{Orbifolds}
+    \begin{columns}[c]
+      \begin{column}{0.475\linewidth}
+        \begin{center}
+          \textbf{Mathematics}
+
+          \begin{itemize}
+            \item manifold $M$
+
+            \item (Lie) group $G$
+
+            \item \emph{stabilizer} $G_p = \qty{g \in G \mid gp = p \in M}$
+
+            \item \emph{orbit} $Gp = \qty{gp \in M \mid g \in G}$
+
+            \item charts $\upphi = \uppi \circ \mathscr{P}$ where:
+
+            \begin{itemize}
+              \item $\mathscr{P}\colon U \subset \mathds{R}^n \to U / G$
+
+              \item $\uppi\colon U / G \to M$
+            \end{itemize}
+          \end{itemize}
+        \end{center}
+      \end{column}
+      \begin{column}{0.05\linewidth}
+        \centering
+        $\Rightarrow$
+      \end{column}
+      \begin{column}{0.475\linewidth}
+        \begin{center}
+          \textbf{Physics}
+
+          \begin{itemize}
+            \item global orbit space $M / G$
+
+            \item $G$ group of isometries
+
+            \item fixed points
+
+            \item additional d.o.f.\ (\emph{twisted states})
+
+            \item singular limits of CY manifolds
+          \end{itemize}
+        \end{center}
+      \end{column}
+    \end{columns}
+
+    \pause
+
+    \begin{center}
+      time-dependent orbifolds
+    \end{center}
+
+    \begin{tikzpicture}[remember picture, overlay]
+      \draw[line width=4pt, red] (13em,3.5em) rectangle (27em, 1em);
+    \end{tikzpicture}
+  \end{frame}
+
+  \begin{frame}{Cosmological Singularities}
+    Use \textbf{time-dependent orbifolds} to model \textbf{space-like singularities}:
+    
+    \begin{center}
+      divergent \highlight{closed string} aplitudes
+      $\Rightarrow$
+      gravitational backreaction?
+    \end{center}
+
+    \pause
+
+    \begin{block}{Divergences}
+      Even in simple models (e.g.\ NBO, more on this later) the $4$ tachyons amplitude is divergent \textbf{at tree level}:
+      \begin{equation*}
+        A_4 \sim \int\limits_{q \sim \infty} \frac{\dd{q}}{\abs{q}} \mathscr{A}( q )
+      \end{equation*}
+      where
+      \begin{equation*}
+        \mathscr{A}_{\text{closed}}( q ) \sim q^{4 - \upalpha' \norm{\vec{p}_{\perp}}^2}
+        \qquad
+        \text{and}
+        \qquad
+        \mathscr{A}_{\text{open}}( q ) \sim q^{1 - \upalpha' \norm{\vec{p}_{\perp}}^2} \trace(\qty[T_1,\, T_2]_+\, \qty[T_3,\, T_4]_+)
+      \end{equation*}
+    \end{block}
+  \end{frame}
+
+  
+  \subsection[NBO]{Null Boost Orbifold}
+
+  \begin{frame}{Null Boost Orbifold}
   \end{frame}