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Add part of the deep learning paper

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Signed-off-by: Riccardo Finotello <riccardo.finotello@gmail.com>
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Riccardo
2020-10-07 23:53:50 +02:00
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sec/part1/introduction.tex
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@@ -703,6 +703,7 @@ When considering the full theory $T_{\text{full}} = T + T_{\text{ghost}}$ and $\
\subsection{Extra Dimensions and Compactification}
\label{sec:CYmanifolds}
We are ultimately interested in building a consistent phenomenology in the framework of string theory.
Any theoretical infrastructure has then to be able to support matter states made of fermions.
@@ -881,6 +882,7 @@ Since \cy manifolds have \SU{m} holonomy, the trace part of the coefficients of
\subsubsection{Cohomology and Hodge Numbers}
\label{sec:cohomology_hodge}
\cy manifolds $M$ of complex dimension $m$ present geometric characteristics of general interest both in pure mathematics and string theory.
They can be characterised in different ways.
@@ -910,29 +912,29 @@ The cohomology group in this case is $H^{(r,s)}_{\bpd}( M, \C )$ and the relatio
\bigoplus\limits_{p = r + s}\,
H^{(r,s)}_{\bpd}( M, \C ).
\end{equation}
As in the case of Betti numbers, we can define the complex equivalents, the \emph{Hodge numbers}, $h^{r,s} = \dim\limits_{\C} H^{(r,s)}_{\bpd}( M, \C )$ which count the number of harmonic $(r, s)$-forms on $M$.
Notice that in this case $h^{r,s}$ is the complex dimension $\dim\limits_{\C}$ of the cohomology group.
As in the case of Betti numbers, we can define the complex equivalents, the \emph{Hodge numbers}, $\hodge{r}{s} = \dim\limits_{\C} H^{(r,s)}_{\bpd}( M, \C )$ which count the number of harmonic $(r, s)$-forms on $M$.
Notice that in this case $\hodge{r}{s}$ is the complex dimension $\dim\limits_{\C}$ of the cohomology group.
For \cy manifolds it is possible to show that the \SU{m} holonomy of $g$ implies that the vector space of $(r, 0)$-forms is \C if $r = 0$ or $r = m$.
Therefore $h^{0,0} = h^{m,0} = 1$, while $h^{r,0} = 0$ if $r \neq 0,\, m$.
Therefore $\hodge{0}{0} = \hodge{m}{0} = 1$, while $\hodge{r}{0} = 0$ if $r \neq 0,\, m$.
Exploiting symmetries of the cohomology groups, Hodge numbers are usually collected in \emph{Hodge diamonds}.
In string theory we are ultimately interested in \cy manifolds of real dimensions $6$, thus we focus mainly on \cy $3$-folds (i.e.\ having $m = 3$).
The diamond in this case is
\begin{equation}
\mqty{%
& & & h^{0,0} & & &
& & & \hodge{0}{0} & & &
\\
& & h^{1,0} & & h^{0,1} & &
& & \hodge{1}{0} & & \hodge{0}{1} & &
\\
& h^{2,0} & & h^{1,1} & & h^{0,2} &
& \hodge{2}{0} & & \hodge{1}{1} & & \hodge{0}{2} &
\\
h^{3,0} & & h^{2,1} & & h^{1,2} & & h^{0,3}
\hodge{3}{0} & & \hodge{2}{1} & & \hodge{1}{2} & & \hodge{0}{3}
\\
& h^{3,1} & & h^{2,2} & & h^{1,3} &
& \hodge{3}{1} & & \hodge{2}{2} & & \hodge{1}{3} &
\\
& & h^{3,2} & & h^{2,3} & &
& & \hodge{3}{2} & & \hodge{2}{3} & &
\\
& & & h^{3,3} & & &
& & & \hodge{3}{3} & & &
}
\quad
=
@@ -942,18 +944,18 @@ The diamond in this case is
\\
& & 0 & & 0 & &
\\
& 0 & & h^{1,1} & & 0 &
& 0 & & \hodge{1}{1} & & 0 &
\\
1 & & h^{2,1} & & h^{2,1} & & 1
1 & & \hodge{2}{1} & & \hodge{2}{1} & & 1
\\
& 0 & & h^{1,1} & & 0 &
& 0 & & \hodge{1}{1} & & 0 &
\\
& & 0 & & 0 & &
\\
& & & 1 & & &
},
\end{equation}
where we used $h^{r,s} = h^{m-r, m-s}$ to stress the fact that the only independent Hodge numbers are $h^{1,1}$ and $h^{2,1}$ for $m = 3$.
where we used $\hodge{r}{s} = h^{m-r, m-s}$ to stress the fact that the only independent Hodge numbers are $\hodge{1}{1}$ and $\hodge{2}{1}$ for $m = 3$.
These results will also be the starting point of~\Cref{part:deeplearning} in which the ability to predict the values of the Hodge numbers using \emph{artificial intelligence} is tested.