riccardo/phd-thesis-beamer
1
0
Fork 0
Code Issues Pull Requests Actions Packages Projects Releases Wiki Activity

Shorter and contained version

Browse Source 
Signed-off-by: Riccardo Finotello <riccardo.finotello@gmail.com>
This commit is contained in:
Riccardo
2020-12-14 15:05:42 +01:00
parent 86905c9434
commit 8277f4503f
4 changed files with 246 additions and 274 deletions
Download Patch File Download Diff File Expand all files Collapse all files

BIN
img/unito.pdf
View File

Binary file not shown.

BIN
img/unito.png Normal file
View File

Binary file not shown.
Side by Side

After

Width:  |  Height:  |  Size: 199 KiB

BIN
img/vector-tensor-features_orig.pdf Normal file
View File

Binary file not shown.

520
thesis.tex
View File

@@ -59,7 +59,7 @@
\newcommand{\highlight}[1]{\fcolorbox{yellow}{yellow!20}{#1}} \newcommand{\highlight}[1]{\fcolorbox{yellow}{yellow!20}{#1}}
\renewcommand{\cite}[1]{{\tiny{\fcolorbox{red}{red!10}{[#1]}}}} \renewcommand{\cite}[1]{{\tiny{\fcolorbox{red}{red!10}{[#1]}}}}
\newcommand{\firstlogo}{img/unito.pdf} \newcommand{\firstlogo}{img/unito.png}
\newcommand{\thefirstlogo}{% \newcommand{\thefirstlogo}{%
\begin{figure} \begin{figure}
\centering \centering
@@ -237,12 +237,6 @@
$\upgamma_{\upalpha \upbeta} = e^{\upphi}\, \upeta_{\upalpha \upbeta}$ $\upgamma_{\upalpha \upbeta} = e^{\upphi}\, \upeta_{\upalpha \upbeta}$
\\ \\
\end{tabular} \end{tabular}
\pause
\begin{center}
\highlight{Conformal properties fixed by \textbf{OPE}s.}
\end{center}
\end{column} \end{column}
\end{columns} \end{columns}
\end{frame} \end{frame}
@@ -311,7 +305,7 @@
\item $N = 1$ \textbf{supersymmetry} preserved in 4D \item $N = 1$ \textbf{supersymmetry} preserved in 4D
\item algebra of $\mathrm{SU}(3) \otimes \mathrm{SU}(2) \otimes \mathrm{U}(1)$ in arising \textbf{gauge group} \item contains algebra of $\mathrm{SU}(3) \otimes \mathrm{SU}(2) \otimes \mathrm{U}(1)$
\end{itemize} \end{itemize}
\end{column} \end{column}
@@ -326,8 +320,8 @@
\vfill \vfill
\begin{columns}[T, totalwidth=0.95\linewidth] \begin{columns}[T, totalwidth=0.95\linewidth]
\begin{column}{0.475\linewidth} \begin{column}{0.55\linewidth}
\textbf{Kähler manifolds} $\qty( M,\, g )$ such that: \textbf{Calabi--Yau manifolds} $\qty( M,\, g )$ such that:
\begin{itemize} \begin{itemize}
\item $\dim\limits_{\mathds{C}} M = m$ \item $\dim\limits_{\mathds{C}} M = m$
@@ -335,23 +329,25 @@
\item $\mathrm{Ric}( g ) \equiv 0$ (equiv.\ $c_1\qty( M ) \equiv 0$) \item $\mathrm{Ric}( g ) \equiv 0$ (equiv.\ $c_1\qty( M ) \equiv 0$)
\end{itemize} \end{itemize}
\hspace{1em}\cite{Calabi (1957), Yau (1977), Candelas \emph{et al.} (1985)}
\end{column} \end{column}
\hfill \hfill
\pause \pause
\begin{column}{0.475\linewidth} \begin{column}{0.4\linewidth}
Characterised by \textbf{Hodge numbers} Characterised by \textbf{Hodge numbers}
\begin{equation*} \begin{equation*}
h^{r,\,s} = \dim\limits_{\mathds{C}} H_{\overline{\partial}}^{r,\,s}\qty( M,\, \mathds{C} ) h^{r,\,s} = \dim\limits_{\mathds{C}} H_{\overline{\partial}}^{r,\,s}\qty( M,\, \mathds{C} )
\end{equation*} \end{equation*}
counting the no.\ of harmonic $(r,s)$-forms. (no.\ of harmonic $(r,s)$-forms).
\end{column} \end{column}
\end{columns} \end{columns}
\end{frame} \end{frame}
\begin{frame}{D-branes and Open Strings} \begin{frame}{D-branes and Open Strings}
Polyakov's action naturally introduces \textbf{Neumann b.c.} for \textbf{open strings}: Polyakov's action naturally introduces \textbf{Neumann b.c.}:
\begin{equation*} \begin{equation*}
\eval{\partial_{\upsigma} X\qty( \uptau, \upsigma )}_{\upsigma = 0}^{\upsigma = \ell} = 0 \eval{\partial_{\upsigma} X\qty( \uptau, \upsigma )}_{\upsigma = 0}^{\upsigma = \ell} = 0
\end{equation*} \end{equation*}
@@ -441,46 +437,51 @@
\centering \centering
\resizebox{0.5\columnwidth}{!}{\import{img}{chanpaton.pgf}} \resizebox{0.5\columnwidth}{!}{\import{img}{chanpaton.pgf}}
\cite{Chan, Paton (1969)} \hfill\cite{Chan, Paton (1969)}
\begin{equation*} \begin{equation*}
\ket{n;\, r} \ket{n;\, r}
= =
\sum\limits_{i,\, j = 1}^N \sum\limits_{i,\, j = 1}^N
\ket{n;\, i,\, j}\, \ket{n;\, i,\, j}\,
\tensor{\uplambda}{^r_{ij}} \tensor{\uplambda}{^r_{ij}}
\quad
\Rightarrow
\quad
\highlight{\mathrm{U}(N)}
\end{equation*} \end{equation*}
\end{column} \end{column}
\hfill \hfill
\pause \pause
\begin{column}{0.475\linewidth} \begin{column}{0.5\linewidth}
\begin{block}{Chan--Paton Factors} \centering
When branes are \textbf{coincident}: \resizebox{\columnwidth}{!}{\import{img}{smbranes.pgf}}
\begin{equation*}
\bigoplus\limits_{r = 1}^N \mathrm{U}_r(1) % \begin{block}{Symmetry Enhancement}
\quad % When branes are \textbf{coincident}:
\longrightarrow % \begin{equation*}
\quad % \bigoplus\limits_{r = 1}^N \mathrm{U}_r(1)
\mathrm{U}( N ) % \quad
\end{equation*} % \longrightarrow
\end{block} % \quad
\pause % \mathrm{U}( N )
\highlight{Build gauge bosons, fermions and scalars.} % \end{equation*}
% \end{block}
\end{column} \end{column}
\end{columns} \end{columns}
\end{frame} \end{frame}
\begin{frame}{Standard Model-like Scenarios} % \begin{frame}{Standard Model-like Scenarios}
\centering % \centering
\resizebox{0.8\linewidth}{!}{\import{img}{smbranes.pgf}} % \resizebox{0.8\linewidth}{!}{\import{img}{smbranes.pgf}}
\hfill\cite{Zwiebach (2009)} % \hfill\cite{Zwiebach (2009)}
\end{frame} % \end{frame}
\subsection[D-branes at Angles]{D-branes Intersecting at Angles} \subsection[D-branes at Angles]{D-branes Intersecting at Angles}
\begin{frame}{Intersecting D-branes} \begin{frame}{Intersecting D-branes}
Consider \highlight{$N$ intersecting $D6$-branes} filling $\mathscr{M}^{1,3}$ and \textbf{embedded in} $\mathds{R}^6$ Consider \highlight{$3$ intersecting $D6$-branes} filling $\mathscr{M}^{1,3}$ and \textbf{embedded in} $\mathds{R}^6$
\begin{equationblock}{Twist Fields Correlators} \begin{equationblock}{Twist Fields Correlators}
\begin{equation*} \begin{equation*}
@@ -512,7 +513,7 @@
\item \textbf{relative rotations} are $\mathrm{SO}(2) \simeq \mathrm{U}(1)$ elements \item \textbf{relative rotations} are $\mathrm{SO}(2) \simeq \mathrm{U}(1)$ elements
\item $S_{E\, (\text{cl})}\qty( \qty{ x_{(t)},\, \mathrm{M}_{(t)} }_{1 \le t \le N_B} ) \sim \text{Area}\qty( \qty{ f_{(t)},\, \mathrm{R}_{(t)} }_{1 \le t \le N_B} )$ \item $S_{E}^{(\text{cl})}\qty( \qty{ x_{(t)},\, \mathrm{M}_{(t)} }_{1 \le t \le N_B} ) \sim \text{Area}\qty( \qty{ f_{(t)},\, \mathrm{R}_{(t)} }_{1 \le t \le N_B} )$
\end{itemize} \end{itemize}
\hfill\cite{Cremades, Ibanez, Marchesano (2003); Pesando (2012)} \hfill\cite{Cremades, Ibanez, Marchesano (2003); Pesando (2012)}
@@ -523,8 +524,6 @@
\begin{frame}{$\mathrm{SO}(4)$ Rotations} \begin{frame}{$\mathrm{SO}(4)$ Rotations}
Consider \highlight{$\mathds{R}^4 \times \mathds{R}^2$} (focus on $\mathds{R}^4$): Consider \highlight{$\mathds{R}^4 \times \mathds{R}^2$} (focus on $\mathds{R}^4$):
\pause
\begin{columns} \begin{columns}
\begin{column}{0.4\linewidth} \begin{column}{0.4\linewidth}
\centering \centering
@@ -538,12 +537,10 @@
\tensor{\qty( R_{(t)} )}{^I_J}\, X^J - g_{(t)}^I \tensor{\qty( R_{(t)} )}{^I_J}\, X^J - g_{(t)}^I
\in \mathds{R}^4 \in \mathds{R}^4
\end{equation*} \end{equation*}
\pause
where where
\begin{equation*} \begin{equation*}
R_{(t)} \in \frac{\mathrm{SO}(4)}{\mathrm{S}\qty( \mathrm{O}(2) \times \mathrm{O}(2) )} R_{(t)} \in \frac{\mathrm{SO}(4)}{\mathrm{S}\qty( \mathrm{O}(2) \times \mathrm{O}(2) )}
\end{equation*} \end{equation*}
\pause
that is that is
\begin{equation*} \begin{equation*}
\qty[ R_{(t)} ] \qty[ R_{(t)} ]
@@ -554,11 +551,7 @@
\end{columns} \end{columns}
\end{frame} \end{frame}
\begin{frame}{Boundary Conditions} \begin{frame}{Boundary Conditions and Open Strings}
What are the consequences for \highlight{open strings?}
\pause
\begin{columns} \begin{columns}
\begin{column}{0.6\linewidth} \begin{column}{0.6\linewidth}
\begin{itemize} \begin{itemize}
@@ -566,13 +559,13 @@
\item $x_{(t)} < x_{(t-1)}$ \textbf{worldsheet intersection points} \item $x_{(t)} < x_{(t-1)}$ \textbf{worldsheet intersection points}
\item $X_{(t)}^{1,\, 2}$ are \textbf{Neumann}, $X_{(t)}^{3,\, 4}$ are \textbf{Dirichlet} % \item $X_{(t)}^{1,\, 2}$ are \textbf{Neumann}, $X_{(t)}^{3,\, 4}$ are \textbf{Dirichlet}
\end{itemize} \end{itemize}
\end{column} \end{column}
\hfill
\begin{column}{0.4\linewidth} \begin{column}{0.4\linewidth}
\centering \centering
\resizebox{0.9\columnwidth}{!}{\import{img}{branchcuts.pgf}} \resizebox{\columnwidth}{!}{\import{img}{branchcuts.pgf}}
\end{column} \end{column}
\end{columns} \end{columns}
@@ -668,27 +661,31 @@
\end{frame} \end{frame}
\begin{frame}{Hypergeometric Basis} \begin{frame}{Hypergeometric Basis}
\begin{columns} \begin{columns}[totalwidth=0.95\linewidth]
\begin{column}{0.3\linewidth} \begin{column}{0.4\linewidth}
\centering \centering
\resizebox{0.9\columnwidth}{!}{\import{img}{threebranes_plane.pgf}} \resizebox{\columnwidth}{!}{\import{img}{threebranes_plane.pgf}}
\end{column} \end{column}
\hfill \hfill
\begin{column}{0.7\linewidth} \begin{column}{0.6\linewidth}
Sum over \highlight{all contributions:} Sum over \highlight{all contributions:}
\begin{equation*} \begin{equation*}
\begin{split} \begin{split}
\partial_z \mathcal{X}( z ) \partial_z \mathcal{X}( z )
& = & =
\pdv{\omega_z}{z}\,
\sum\limits_{l,\, r = -\infty}^{+\infty} c_{lr}\, \sum\limits_{l,\, r = -\infty}^{+\infty} c_{lr}\,
\qty( - \upomega_z )^{A_{lr}}\, \qty( - \upomega_z )^{A_{lr}}\,
\qty( 1 - \upomega_z )^{B_{lr}}\, \qty( 1 - \upomega_z )^{B_{lr}}\,
\\
& \times
B_{0,\, l}^{(L)}( \omega_z )\, B_{0,\, l}^{(L)}( \omega_z )\,
\qty( B_{0,\, r}^{(R)}( \omega_z ) )^T \qty( B_{0,\, r}^{(R)}( \omega_z ) )^T
\end{split} \end{split}
\end{equation*} \end{equation*}
\end{column} \end{column}
\end{columns} \end{columns}
\vfill
\pause \pause
@@ -717,23 +714,17 @@
\begin{enumerate} \begin{enumerate}
\item rotation matrix $=$ monodromy matrix \item rotation matrix $=$ monodromy matrix
\pause
\item contiguity relations $\Rightarrow$ independent hypergeometrics \item contiguity relations $\Rightarrow$ independent hypergeometrics
\pause
\item finite action $\Rightarrow$ $2$ solutions (no.\ of d.o.f.\ is correctly saturated) \item finite action $\Rightarrow$ $2$ solutions (no.\ of d.o.f.\ is correctly saturated)
\pause
\item boundary conditions $\Rightarrow$ fix free constants $c_{lr}$ \item boundary conditions $\Rightarrow$ fix free constants $c_{lr}$
\end{enumerate} \end{enumerate}
\pause \pause
\begin{block}{Physical Interpretation} \begin{block}{Physical Interpretation}
\only<5>{% \only<2>{%
\begin{columns} \begin{columns}
\begin{column}{0.4\linewidth} \begin{column}{0.4\linewidth}
\centering \centering
@@ -756,7 +747,7 @@
\end{columns} \end{columns}
\vfill \vfill
} }
\only<6->{% \only<3->{%
\begin{columns} \begin{columns}
\begin{column}{0.35\linewidth} \begin{column}{0.35\linewidth}
\centering \centering
@@ -771,7 +762,7 @@
\item classical action \textbf{larger} than factorised case \item classical action \textbf{larger} than factorised case
\end{itemize} \end{itemize}
\hspace{0.65\columnwidth}\cite{RF, Pesando (2019)} \hspace{0.65\columnwidth}\cite{\textbf{RF}, Pesando (2019)}
\end{column} \end{column}
\end{columns} \end{columns}
} }
@@ -809,46 +800,45 @@
\end{equationblock} \end{equationblock}
\end{column} \end{column}
\end{columns} \end{columns}
\vfill
\pause \pause
\begin{equationblock}{Stress-energy Tensor} \begin{equation*}
\begin{equation*} \mathcal{T}_{\pm\pm}( \upxi_{\pm} )
\mathcal{T}_{\pm\pm}( \upxi_{\pm} ) =
-i\, \frac{T}{4}\,
\uppsi^*_{\pm,\, I}( \upxi_{\pm} )\,
\overset{\leftrightarrow}{\partial} \uppsi^I_{\pm}( \upxi_{\pm} )
\quad
\Rightarrow
\quad
\begin{cases}
\dot{\mathrm{H}}( \uptau )
&
% =
% \partial_{\uptau}
% \qty(%
% \int\limits_0^{\uppi} \dd{\upsigma}
% \mathcal{T}_{\uptau\uptau}( \uptau, \upsigma )
% )
= =
-i\, \frac{T}{4}\, 0
\uppsi^*_{\pm,\, I}( \upxi_{\pm} )\, \quad \Leftrightarrow \quad
\overset{\leftrightarrow}{\partial} \uppsi^I_{\pm}( \upxi_{\pm} ) \uptau \in \qty( \uptau_{(t)},\, \uptau_{(t-1)} )
\quad \\
\Rightarrow \dot{\mathrm{P}}( \uptau )
\quad &
\begin{cases} % =
\dot{\mathrm{H}}( \uptau ) % \partial_{\uptau}
& % \qty(%
% = % \int\limits_0^{\uppi} \dd{\upsigma}
% \partial_{\uptau} % \mathcal{T}_{\uptau\upsigma}( \uptau, \upsigma )
% \qty(% % )
% \int\limits_0^{\uppi} \dd{\upsigma} \neq
% \mathcal{T}_{\uptau\uptau}( \uptau, \upsigma ) 0
% ) \end{cases}
= \end{equation*}
0
\quad \Leftrightarrow \quad
\uptau \in \qty( \uptau_{(t)},\, \uptau_{(t-1)} )
\\
\dot{\mathrm{P}}( \uptau )
&
% =
% \partial_{\uptau}
% \qty(%
% \int\limits_0^{\uppi} \dd{\upsigma}
% \mathcal{T}_{\uptau\upsigma}( \uptau, \upsigma )
% )
\neq
0
\end{cases}
\end{equation*}
\end{equationblock}
\end{frame} \end{frame}
\begin{frame}{Conserved Product and Operators} \begin{frame}{Conserved Product and Operators}
@@ -912,7 +902,7 @@
\end{frame} \end{frame}
\begin{frame}{Twisted Complex Fermions} \begin{frame}{Twisted Complex Fermions}
Consider the case $R_{(t)} = e^{i \uppi \upalpha_{(t)}} \in \mathrm{U}( 1 )$: Consider $R_{(t)} = e^{i \uppi \upalpha_{(t)}} \in \mathrm{U}( 1 )$:
\begin{equation*} \begin{equation*}
\Uppsi( x_{(t)} + e^{2\uppi i} \updelta ) \Uppsi( x_{(t)} + e^{2\uppi i} \updelta )
= =
@@ -969,6 +959,10 @@
\begin{columns} \begin{columns}
\begin{column}{0.6\linewidth} \begin{column}{0.6\linewidth}
\begin{equation*} \begin{equation*}
\braket{\qty{x_{(t)}}} = 1
\quad
\Rightarrow
\quad
\mathrm{L} \mathrm{L}
= =
n_{(t)} + \widetilde{n}_{(t)} n_{(t)} + \widetilde{n}_{(t)}
@@ -986,7 +980,7 @@
\end{frame} \end{frame}
\begin{frame}{Stress-energy Tensor and CFT Approach} \begin{frame}{Stress-energy Tensor and CFT Approach}
Compute the OPEs leading to the \textbf{stress-energy tensor:} Compute the OPEs leading to the \highlight{time dependent} \textbf{stress-energy tensor:}
\begin{equation*} \begin{equation*}
\mathcal{T}( z ) \mathcal{T}( z )
= =
@@ -1002,7 +996,7 @@
+ +
\frac{1}{2} \qty( \sum\limits_{t = 1}^N \frac{n_{(t)} + \frac{\upepsilon_{(t)}}{2}}{z - x_{(t)}} )^2 \frac{1}{2} \qty( \sum\limits_{t = 1}^N \frac{n_{(t)} + \frac{\upepsilon_{(t)}}{2}}{z - x_{(t)}} )^2
\end{equation*} \end{equation*}
\hfill\cite{RF, Pesando (2019)} \hfill\cite{\textbf{RF}, Pesando (2019)}
\pause \pause
@@ -1021,7 +1015,7 @@
\begin{equationblock}{Equivalence with Bosonization} \begin{equationblock}{Equivalence with Bosonization}
\begin{equation*} \begin{equation*}
\begin{split} \begin{split}
\partial_{x_{(t)}} \braket{\qty{x_{(t)}}} \partial_{x_{(t)}} \ln \braket{\qty{x_{(t)}}}
& = & =
\oint\limits_{x_{(t)}} \frac{\dd{z}}{2\uppi i} \oint\limits_{x_{(t)}} \frac{\dd{z}}{2\uppi i}
\frac{% \frac{%
@@ -1046,21 +1040,11 @@
\begin{itemize} \begin{itemize}
\item (semi-)phenomenological models involve \textbf{twist and spin} fields and \textbf{open strings} \item (semi-)phenomenological models involve \textbf{twist and spin} fields and \textbf{open strings}
\pause \item framework for \textbf{bosonic} open strings with \textbf{intersecting D-branes}
\item general framework for \textbf{bosonic} open strings with \textbf{intersecting D-branes} \item \textbf{spin fields} as \textbf{boundary changing operators} (hidden in \textbf{defects})
\pause \item framework for amplitudes (extension to (non) Abelian twist/spin fields?)
\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{itemize}
\end{frame} \end{frame}
@@ -1072,7 +1056,7 @@
\begin{frame}{A Few Words on a Theory of Everything} \begin{frame}{A Few Words on a Theory of Everything}
\begin{center} \begin{center}
string theory = theory of everything = nuclear forces + gravity \textbf{string theory} = theory of everything = \textbf{nuclear forces + gravity}
\end{center} \end{center}
\pause \pause
@@ -1109,64 +1093,64 @@
\end{columns} \end{columns}
\end{frame} \end{frame}
\begin{frame}{Orbifolds} % \begin{frame}{Orbifolds}
\begin{columns}[T] % \begin{columns}[T]
\begin{column}{0.475\linewidth} % \begin{column}{0.475\linewidth}
\begin{tabular}{@{}p{0.975\columnwidth}@{}} % \begin{tabular}{@{}p{0.975\columnwidth}@{}}
\textbf{Mathematics} % \textbf{Mathematics}
\\ % \\
\toprule % \toprule
\begin{itemize} % \begin{itemize}
\item manifold $M$ % \item manifold $M$
\item (Lie) group $G$ % \item (Lie) group $G$
\item \emph{stabilizer} $G_p = \qty{g \in G \mid gp = p \in M}$ % \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 \emph{orbit} $Gp = \qty{gp \in M \mid g \in G}$
\item charts $\upphi = \uppi \circ \mathscr{P}$ where: % \item charts $\upphi = \uppi \circ \mathscr{P}$ where:
\begin{itemize} % \begin{itemize}
\item $\mathscr{P}\colon U \subset \mathds{R}^n \to U / G$ % \item $\mathscr{P}\colon U \subset \mathds{R}^n \to U / G$
\item $\uppi\colon U / G \to M$ % \item $\uppi\colon U / G \to M$
\end{itemize} % \end{itemize}
\end{itemize} % \end{itemize}
\end{tabular} % \end{tabular}
\end{column} % \end{column}
\hfill % \hfill
\begin{column}{0.475\linewidth} % \begin{column}{0.475\linewidth}
\begin{tabular}{@{}p{0.975\columnwidth}@{}} % \begin{tabular}{@{}p{0.975\columnwidth}@{}}
\textbf{Physics} % \textbf{Physics}
\\ % \\
\toprule % \toprule
\begin{itemize} % \begin{itemize}
\item global orbit space $M / G$ % \item global orbit space $M / G$
\item $G$ group of isometries % \item $G$ group of isometries
\item fixed points % \item fixed points
\item additional d.o.f.\ (\emph{twisted states}) % \item additional d.o.f.\ (\emph{twisted states})
\item singular limits of CY manifolds % \item singular limits of CY manifolds
\end{itemize} % \end{itemize}
\end{tabular} % \end{tabular}
\end{column} % \end{column}
\end{columns} % \end{columns}
\pause % \pause
\vspace{2em} % \vspace{2em}
\begin{center} % \begin{center}
Use \textbf{time-dependent orbifolds} to model singularities in time % Use \textbf{time-dependent orbifolds} to model singularities in time
\end{center} % \end{center}
\begin{tikzpicture}[remember picture, overlay] % \begin{tikzpicture}[remember picture, overlay]
\draw[line width=4pt, red] (5em,3.5em) rectangle (35em, 1em); % \draw[line width=4pt, red] (5em,3.5em) rectangle (35em, 1em);
\end{tikzpicture} % \end{tikzpicture}
\end{frame} % \end{frame}
\begin{frame}{Cosmological Singularities} \begin{frame}{Cosmological Singularities}
Use \textbf{time-dependent orbifolds} to model \textbf{space-like singularities}: Use \textbf{time-dependent orbifolds} to model \textbf{space-like singularities}:
@@ -1216,10 +1200,12 @@
\pause \pause
\begin{equationblock}{Killing Vector and Null Boost Oribfold} \begin{equationblock}{Killing Vector and Null Boost Orbifold}
\begin{equation*} \begin{equation*}
\upkappa = -i \qty( 2 \uppi \Updelta ) J_{+2} = 2 \uppi \partial_z \upkappa = -i \qty( 2 \uppi \Updelta ) J_{+2} = 2 \uppi \partial_z
\quad
\Rightarrow \Rightarrow
\quad
z \sim z + 2 \uppi n z \sim z + 2 \uppi n
\end{equation*} \end{equation*}
\end{equationblock} \end{equationblock}
@@ -1276,7 +1262,7 @@
\begin{center} \begin{center}
\emph{ \emph{
most terms \textbf{do not converge} due to \textbf{isolated zeros} \emph{($l_{(*)} \equiv 0$)} and cannot be recovered even with a \textbf{distributional interpretions} due to the term $\propto u^{-1}$ in the exponentatial most terms \textbf{do not converge} due to \textbf{isolated zeros} \emph{($l_{(*)} \equiv 0$)} and cannot be recovered even with a \textbf{distributional interpretion} due to the term $\propto u^{-1}$ in the exponential
} }
\end{center} \end{center}
\end{frame} \end{frame}
@@ -1284,20 +1270,15 @@
\begin{frame}{String and Field Theory} \begin{frame}{String and Field Theory}
So far: So far:
\begin{itemize} \begin{itemize}
\item field theory presents \textbf{divergences} (even sQED $\rightarrow$ eikonal?) \item field theory presents \textbf{divergences} (see sQED)
\pause
\item obvious ways to regularise (Wilson lines, higher derivative couplings, etc.) \textbf{do not work} \item obvious ways to regularise (Wilson lines, higher derivative couplings, etc.) \textbf{do not work}
\pause
\item divergences are \textbf{not (only) gravitational} \item divergences are \textbf{not (only) gravitational}
\pause \item \textbf{vanishing volume} in phase space responsible for the divergence
\item \textbf{vanishing volume} in phase space of the compact direction is responsible for the divergence
\end{itemize} \end{itemize}
\vfill
\pause \pause
@@ -1356,7 +1337,7 @@
\pause \pause
\begin{equationblock}{Distributional Interpretation} \begin{equationblock}{No isolated zeros $\Rightarrow$ distributional Interpretation}
\begin{equation*} \begin{equation*}
\widetilde{\upphi}_{\qty{ k_+,\, p,\, l,\, \vec{k},\, r}}\qty( u ) \widetilde{\upphi}_{\qty{ k_+,\, p,\, l,\, \vec{k},\, r}}\qty( u )
= =
@@ -1371,36 +1352,28 @@
\begin{itemize} \begin{itemize}
\item divergences are present in sQED and \textbf{open string} sector \item divergences are present in sQED and \textbf{open string} sector
\pause
\item singularities $\Rightarrow$ \textbf{massive states} are no longer spectators \item singularities $\Rightarrow$ \textbf{massive states} are no longer spectators
\pause
\item vanishing volume (\textbf{compact orbifold directions}) $\Rightarrow$ particles ``cannot escape'' \item vanishing volume (\textbf{compact orbifold directions}) $\Rightarrow$ particles ``cannot escape''
\pause
\item \textbf{non compact} orbifold directions $\Rightarrow$ interpretation of \textbf{amplitudes as distributions} \item \textbf{non compact} orbifold directions $\Rightarrow$ interpretation of \textbf{amplitudes as distributions}
\pause
\item issue not restricted to NBO/GNBO but also BO, null brane, etc. (it is a \textbf{general issues} connected to the geometry of the underlying space) \item issue not restricted to NBO/GNBO but also BO, null brane, etc. (it is a \textbf{general issues} connected to the geometry of the underlying space)
\end{itemize} \end{itemize}
\vfill
\pause \pause
\vspace{2em}
\begin{center} \begin{center}
\it \it
divergences are \textbf{hidden into contact terms} and interactions with \textbf{massive states} (the gravitational eikonal deals with massless interactions) divergences are \textbf{hidden into EFT contact terms} and interactions with \textbf{string massive states}: gravity is not the only cause as the same problems are present also in gauge theories.
\end{center} \end{center}
\begin{tikzpicture}[remember picture, overlay] \begin{tikzpicture}[remember picture, overlay]
\draw[line width=4pt, red] (0em, 4.5em) rectangle (40em, 1em); \draw[line width=4pt, red] (0em, 4.5em) rectangle (40em, 1em);
\end{tikzpicture} \end{tikzpicture}
\hfill\cite{Arduino, RF, Pesando (2020)} \hfill\cite{Arduino, \textbf{RF}, Pesando (2020)}
\end{frame} \end{frame}
@@ -1439,9 +1412,9 @@
\mathds{P}^n\colon \mathds{P}^n\colon
\qquad \qquad
\begin{cases} \begin{cases}
p_a\qty( Z^0,\, \dots,\, Z^n ) & = P_{I_1 \dots I_a} Z^{I_1} \dots Z^{I_a} = 0 p_i\qty( Z^0,\, \dots,\, Z^n ) & = P_{a_1 \dots a_i} Z^{a_1} \dots Z^{a_i} = 0
\\ \\
p_a\qty( \uplambda Z^0,\, \dots,\, \uplambda Z^n ) & = \uplambda^a p_a\qty( Z^0,\, \dots,\, Z^n ) p_i\qty( \uplambda Z^0,\, \dots,\, \uplambda Z^n ) & = \uplambda^i p_i\qty( Z^0,\, \dots,\, Z^n )
\end{cases} \end{cases}
\end{equation*} \end{equation*}
\hfill\cite{Green, Hübsch (1987); Hübsch (1992)} \hfill\cite{Green, Hübsch (1987); Hübsch (1992)}
@@ -1489,11 +1462,11 @@
\begin{block}{Machine Learning Approach} \begin{block}{Machine Learning Approach}
What is $\mathscr{R}$ in \textbf{machine learning} approach? What is $\mathscr{R}$ in \textbf{machine learning} approach?
\begin{equation*} \begin{equation*}
\mathscr{R}( M ) \longrightarrow \mathscr{R}_n( M;\, w ) \mathscr{R}( M ) \rightarrow \mathscr{R}_n( M;\, w ) \rightarrow \widehat{h}^{p,\,q}
\qquad \qquad
\text{s.t.} \text{s.t.}
\qquad \qquad
\exists n > M > 0 \mid \mathcal{L}\qty(\mathscr{R}( M ),\, \mathscr{R}_n( M;\, w )) < \upepsilon \exists n > M > 0 \quad \mid \quad \mathcal{L}_n\qty(\widehat{h}^{p,\,q},\, h^{p,\,q}) < \upepsilon
\quad \quad
\forall \upepsilon > 0 \forall \upepsilon > 0
\end{equation*} \end{equation*}
@@ -1502,19 +1475,15 @@
\begin{frame}{Machine Learning} \begin{frame}{Machine Learning}
\begin{itemize} \begin{itemize}
\item exchange \textbf{analytical solution} with \textbf{optimisation problem} \item \textbf{optimisation problem} $\Rightarrow$ \highlight{gradient descent} (or similar)
\pause \pause
\item use \textbf{various algorithms} and exploit \textbf{large datasets} \item use \textbf{various algorithms} and exploit \textbf{large datasets} (more training)
\pause \pause
\item learn a \textbf{representation} rather than a \textbf{solution} \item intersection of \textbf{computer science, mathematics and physics}
\pause
\item knowledge from \textbf{computer science, mathematics and physics} to solve problems
\pause \pause
@@ -1523,6 +1492,8 @@
\begin{center} \begin{center}
\includegraphics[width=0.7\linewidth]{img/label-distribution_orig.pdf} \includegraphics[width=0.7\linewidth]{img/label-distribution_orig.pdf}
\hfill\cite{Green \emph{et al.} (1987)}
\end{center} \end{center}
\end{frame} \end{frame}
@@ -1533,21 +1504,13 @@
\begin{frame}{Dataset} \begin{frame}{Dataset}
\begin{itemize} \begin{itemize}
\item $7890$ CICY manifolds (full dataset) \item $7890$ CICY manifolds (full dataset)
\pause
\item \textbf{dataset pruning}: no product spaces, no ``very far'' outliers (reduction of $0.49\%$) \item \textbf{dataset pruning}: no product spaces, no ``very far'' outliers (reduction of $0.49\%$)
\pause
\item $h^{1,\, 1} \in \qty[ 1,\, 16 ]$ and $h^{2,\, 1} \in \qty[ 15,\, 86 ]$ \item $h^{1,\, 1} \in \qty[ 1,\, 16 ]$ and $h^{2,\, 1} \in \qty[ 15,\, 86 ]$
\pause
\item $80\%$ training, $10\%$ validation, $10\%$ test \item $80\%$ training, $10\%$ validation, $10\%$ test
\pause
\item choose \textbf{regression}, but evaluate using \textbf{accuracy} (round the result) \item choose \textbf{regression}, but evaluate using \textbf{accuracy} (round the result)
\end{itemize} \end{itemize}
@@ -1559,7 +1522,6 @@
\end{frame} \end{frame}
\begin{frame}{Exploratory Data Analysis} \begin{frame}{Exploratory Data Analysis}
Machine Learning \highlight{pipeline:}
\begin{center} \begin{center}
\textbf{exploratory} data analysis \textbf{exploratory} data analysis
$\rightarrow$ $\rightarrow$
@@ -1568,19 +1530,25 @@
Hodge numbers Hodge numbers
\end{center} \end{center}
\hfill\cite{Ruehle (2020); Erbin, RF (2020)} \hfill\cite{Ruehle (2020); Erbin, \textbf{RF} (2020)}
\vfill
\pause \pause
\begin{columns} \begin{columns}
\begin{column}{0.5\linewidth} \begin{column}{0.33\linewidth}
\centering \centering
\includegraphics[width=0.9\columnwidth]{img/corr-matrix_orig.pdf} \includegraphics[width=\columnwidth]{img/corr-matrix_orig.pdf}
\end{column} \end{column}
\hfill \hfill
\begin{column}{0.5\linewidth} \begin{column}{0.33\linewidth}
\centering \centering
\includegraphics[width=0.9\columnwidth, trim={0 0 6in 0}, clip]{img/scalar-features_orig.pdf} \includegraphics[width=\columnwidth, trim={0 0 6in 0}, clip]{img/scalar-features_orig.pdf}
\end{column}
\hfill
\begin{column}{0.33\linewidth}
\centering
\includegraphics[width=\columnwidth, trim={0 0.5in 6in 0}, clip]{img/vector-tensor-features_orig.pdf}
\end{column} \end{column}
\end{columns} \end{columns}
\end{frame} \end{frame}
@@ -1590,7 +1558,7 @@
\includegraphics[width=0.85\linewidth]{img/ml_map.png} \includegraphics[width=0.85\linewidth]{img/ml_map.png}
\begin{tikzpicture}[remember picture, overlay] \begin{tikzpicture}[remember picture, overlay]
\node[anchor=base] at (16em,18em) {\cite{from scikit-learn.org}}; \node[anchor=base] at (16em,18em) {\cite{from \href{https://scikit-learn.org/stable/tutorial/machine_learning_map/index.html}{scikit-learn.org}}};
\end{tikzpicture} \end{tikzpicture}
\pause \pause
@@ -1612,13 +1580,13 @@
\begin{column}{0.4\linewidth} \begin{column}{0.4\linewidth}
What is PCA for a $X \in \mathds{R}^{n \times p}$? What is PCA for a $X \in \mathds{R}^{n \times p}$?
\begin{itemize} \begin{itemize}
\item project data onto a \textbf{lower dimensional} space where variance is maximised \item project data such that \textbf{variance is maximised}
\item equivalently compute the \textbf{eigenvectors} of $X X^T$ or the \textbf{singular values} of $X$ \item \textbf{eigenvectors} of $X X^T$ or the \textbf{singular values} of $X$
\item isolate \textbf{the signal} from the \textbf{background} \item isolate \textbf{signal} from \textbf{background}
\item ease the machine learning job of finding a better representation of the input \item ease the ML job of finding a better representation of the input
\end{itemize} \end{itemize}
\end{column} \end{column}
\hfill \hfill
@@ -1642,7 +1610,7 @@
\begin{column}{0.5\linewidth} \begin{column}{0.5\linewidth}
\centering \centering
\textbf{Best Training Set} \textbf{Best Training Set}
\cite{Erbin, RF (2020)} \cite{Erbin, \textbf{RF} (2020)}
\includegraphics[width=0.75\columnwidth, trim={0 0 3.3in 0}, clip]{img/cicy_best_plots.pdf} \includegraphics[width=0.75\columnwidth, trim={0 0 3.3in 0}, clip]{img/cicy_best_plots.pdf}
\end{column} \end{column}
\end{columns} \end{columns}
@@ -1666,13 +1634,13 @@
\begin{block}{Layers} \begin{block}{Layers}
\vspace{0.5em} \vspace{0.5em}
\begin{tabular}{@{}ll@{}} \begin{tabular}{@{}ll@{}}
\textbf{fully connected}: fully connected:
& &
$\upphi\qty( a^{\qty(i)\, \qty{l}} \cdot W^{\qty{l}} + b^{\qty{l}} \mathds{1}_l )$ $\upphi\qty( a^{\qty(i)\, \qty{l}} \cdot W^{\qty{l}} + b^{\qty{l}} \mathds{1} )$
\\ \\
\textbf{convolutional}: convolutional:
& &
$\upphi\qty( a^{\qty(i)\, \qty{l}}\, *\, W^{\qty{l}} + b^{\qty{l}} \mathds{1}_l )$ $\upphi\qty( a^{\qty(i)\, \qty{l}}\, *\, W^{\qty{l}} + b^{\qty{l}} \mathds{1} )$
\end{tabular} \end{tabular}
\end{block} \end{block}
@@ -1694,14 +1662,14 @@
Why convolutional? Why convolutional?
\begin{columns} \begin{columns}
\begin{column}{0.4\linewidth} \begin{column}{0.4\linewidth}
\begin{itemize}[<+->] \begin{itemize}
\item retain \textbf{spacial awareness} \item retain \textbf{spacial awareness}
\item smaller \textbf{no.\ of parameters} ($\approx 2 \times 10^5$ vs.\ $\approx 2 \times 10^6$) \item smaller \textbf{no.\ of parameters} ($\approx 2 \times 10^5$ vs.\ $\approx 2 \times 10^6$)
\item weights are \textbf{shared} \item weights are \textbf{shared}
\item CNN can isolate \textbf{``defining features''} \item CNNs isolate \textbf{``defining features''}
\item find patterns as in \textbf{computer vision} \item find patterns as in \textbf{computer vision}
\end{itemize} \end{itemize}
@@ -1709,21 +1677,21 @@
\hfill \hfill
\begin{column}{0.6\linewidth} \begin{column}{0.6\linewidth}
\centering \centering
\only<1-3>{\includegraphics[width=0.75\columnwidth, trim={12in 5in 0 5in}, clip]{img/input_mat.png}} \only<1>{\includegraphics[width=0.75\columnwidth, trim={12in 5in 0 5in}, clip]{img/input_mat.png}}
\only<4>{\animategraphics[autoplay,loop,controls={play,stop},width=\linewidth]{8}{img/animation/sequence/conv-}{0}{79}} \only<2>{\animategraphics[autoplay,loop,controls={play,stop},width=\linewidth]{8}{img/animation/sequence/conv-}{0}{79}}
\only<5>{\resizebox{\columnwidth}{!}{\import{img}{ccnn.pgf}}} \only<3>{\resizebox{\columnwidth}{!}{\import{img}{ccnn.pgf}}}
\end{column} \end{column}
\end{columns} \end{columns}
\end{frame} \end{frame}
\begin{frame}{Inception Neural Networks} \begin{frame}{Inception Neural Networks}
Recent development by Google's deep learning teams led to: Recent development by deep learning research at \highlight{Google} led to:
\begin{itemize} \begin{itemize}
\item neural networks with \textbf{better generalisation properties} \item neural networks with better \textbf{generalisation properties}
\item smaller networks (both parameters and depth) \item \textbf{smaller} networks (both parameters and depth)
\item different \textbf{concurrent kernels} (e.g.\ one over \textbf{equations} one over \textbf{coordinates}) \item different \textbf{concurrent kernels}
\end{itemize} \end{itemize}
\pause \pause
@@ -1738,64 +1706,50 @@
\begin{column}{0.5\linewidth} \begin{column}{0.5\linewidth}
\centering \centering
\textbf{Best Training Set} \textbf{Best Training Set}
\cite{Erbin, RF (2020)} \cite{Erbin, \textbf{RF} (2020)}
\only<1>{\includegraphics[width=0.8\columnwidth, trim={0 0 1.65in 0}, clip]{img/cicy_best_plots.pdf}} \only<1>{\includegraphics[width=0.8\columnwidth, trim={0 0 1.65in 0}, clip]{img/cicy_best_plots.pdf}}
\only<2->{\includegraphics[width=\columnwidth]{img/cicy_best_plots.pdf}} \only<2->{\includegraphics[width=\columnwidth]{img/cicy_best_plots.pdf}}
\end{column} \end{column}
\hfill \hfill
\begin{column}{0.5\linewidth} \visible<2->{
\centering \begin{column}{0.5\linewidth}
\includegraphics[width=0.75\columnwidth]{img/inc_nn_learning_curve_h11.pdf} \centering
\includegraphics[width=0.75\columnwidth]{img/inc_nn_learning_curve.pdf} \includegraphics[width=0.75\columnwidth]{img/inc_nn_learning_curve_h11.pdf}
\end{column} \includegraphics[width=0.75\columnwidth]{img/inc_nn_learning_curve.pdf}
\end{column}
}
\end{columns} \end{columns}
\end{frame} \end{frame}
\begin{frame}{A Few Comments and Future Directions} \begin{frame}{A Few Comments and Future Directions}
Why \highlight{deep learning in physics?} \begin{tabular}{@{}l@{}}
\begin{itemize} Why \highlight{deep learning in physics?}
\item reliable \textbf{predictive method} \pause (provided good data analysis) \\
\toprule
\pause $\circ$ reliable \textbf{predictive method} \pause (provided good data analysis)
\\
\item reliable \textbf{source of inspiration} \pause (provided good data analysis) $\circ$ reliable \textbf{source of inspiration} \pause (provided good data analysis)
\\
\pause $\circ$ reliable \textbf{generalisation method} \pause (provided good data analysis)
\\
\item reliable \textbf{generalisation method} \pause (provided good data analysis) $\circ$ \textbf{CNNs are powerful tools} (this is the \emph{first time in physics!})
\\
\pause $\circ$ interdisciplinary approach $=$ win-win situation!
\\[1em]
\item \textbf{CNNs are powerful tools} (this is the \emph{first time in physics!}) \pause
What now?
\pause \\
\toprule
\item interdisciplinary approach $=$ win-win situation! $\circ$ representation learning $\Rightarrow$ what is the best way to represent CICYs?
\end{itemize} \\
$\circ$ study invariances $\Rightarrow$ invariances should not influence the result (graph representations?)
\pause \\
$\circ$ higher dimensions $\Rightarrow$ what about CICY 4-folds?
What now? \\
$\circ$ geometric deep learning $\Rightarrow$ explain the geometry of the ``AI'' behind deep learning!
\begin{itemize} \\
\item representation learning $\Rightarrow$ what is the best way to represent CICYs? $\circ$ reinforcement learning $\Rightarrow$ give the rules, not the result!
\end{tabular}
\pause
\item study invariances $\Rightarrow$ invariances should not influence the result (graph representations?)
\pause
\item higher dimensions $\Rightarrow$ what about CICY 4-folds?
\pause
\item geometric deep learning $\Rightarrow$ explain the geometry of the ``AI'' behind deep learning!
\pause
\item reinforcement learning $\Rightarrow$ give the rules, not the result!
\end{itemize}
\end{frame} \end{frame}
{% {%
@@ -1806,6 +1760,24 @@
} }
\addtobeamertemplate{background canvas}{\transfade[duration=0.25]}{} \addtobeamertemplate{background canvas}{\transfade[duration=0.25]}{}
\begin{frame}[noframenumbering]{The End?} \begin{frame}[noframenumbering]{The End?}
\begin{columns}[T, totalwidth=\linewidth]
\begin{column}{0.7\linewidth}
\begin{itemize}
\item general framework for \textbf{D-branes at angles}
\item alternative computations of \textbf{correlators of spin fields}
\item strings and divergences in \textbf{time dependent orbifolds}
\item string compactifications and \textbf{deep learning techniques}
\end{itemize}
\end{column}
\begin{column}{0.3\linewidth}
\centering
\includegraphics[width=0.5\columnwidth]{\firstlogo}
\end{column}
\end{columns}
\vfill
\begin{center} \begin{center}
\Huge \Huge
THANK YOU THANK YOU