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\section{Toric Elliptic Fibrations}

\subsection{Toric Sections}

A \emph{toric} section is a ambient toric divisor that intersects the
Calabi-Yau hypersurface in one point in each toric fiber. While not
every section has to be of this toric form,\footnote{In fact, there
  can be at most $6$ toric sections in a toric Calabi-Yau threefold
  hypersurface~\cite{2012arXiv1201.0930G} so there are necessarily
  non-toric sections if the Mordell-Weil group has more than $6$
  elements.} it turns out that at least one section usually is. For
example, among the single-block $SU(n)$ torus fibrations to be
discussed in \autoref{sec:blockSUn}, every section is toric.


\subsection{Matter Content}

The matter content at codimension-two fibers can be read off from the
degree of vanishing $(\deg a, \deg b, \deg \delta)$ along a generic
curve that intersects the discriminant at the given point. If the
degree of vanishing equals the one predicted by the Tate algorithm
(the generic case), then there is no matter concentrated at this
fiber. Other 

\begin{table}
  \centering
  \renewcommand{\arraystretch}{1.5}
  \begin{tabular}{cc|cc}
    Group & Tate & Enhanced & Matter
    \\ \hline\hline
    \multirow{2}{*}{$SU(2)$} & 
    \multirow{2}{*}{$(0,0,2)$}  
    & $(0,0,3)$ & $\Rep{2} = \tiny\yng(1)$\strut \\&
    & $(1,2,3)$ & unchanged = $\tiny\yng(1,1)$\strut \\
    \hline
    \multirow{2}{*}{$SU(3)$} & 
    \multirow{2}{*}{$(0,0,3)$}  
    & $(0,0,4)$ & $\Rep{3} = \tiny\yng(1)$\strut \\&
    & $(2,2,4)$ & $\barRep{3} = \tiny\yng(1,1)$\strut \\
    \hline
    \multirow{2}{*}{
      \begin{math}
        \begin{array}{c}
          SU(n),
          \\
          n=4,5 \text{ or } n\geq 6
        \end{array}
      \end{math}
    } & 
    \multirow{2}{*}{$(0,0,n)$}  
    & $(0,0,n+1)$ & $\tiny\yng(1)$\strut \\&
    & $(2,3,n+1)$ & $\tiny\yng(1,1)$\strut \\
    \hline
    \multirow{3}{*}{$SU(6)$} & 
    \multirow{3}{*}{$(0,0,6)$}  
    & $(0,0,7)$ & $\Rep{6} = \tiny\yng(1)$\strut \\&
    & $(2,3,8)$ & $\Rep{15} = \tiny\yng(1,1)$\strut \\&
    & $(3,4,8)$ & $\Rep{20} = \tiny\yng(1,1,1)$\strut \\
    \hline
  \end{tabular}
  \caption{Matter content of codimension-two fibers.}
\end{table}





\section{Single-Block SU(n)}
\label{sec:blockSUn}

Here we list the single-block $SU(n)$ models, that is the toric
elliptic fibrations with a single $SU(n)$ non-Abelian gauge group from
a discriminant component over a toric $\CP^1\subset \CP^2$ such that
$h^{11}=n+1$. This implies that there are no $U(1)$ gauge groups (the
Mordell-Weil group is torsion), no non-toric discriminant component
supporting a non-trivial gauge group, and that the Calabi-Yau
hypersurface is a flat fibration (there is no space for the cohomology
class of a vertical surface).


\subsection{The Toric Discriminant Case}

\begin{table}
  \centering
  \begin{tabular}{c|cccccc}
    $SU(n)$ & $h^{11}$ & $h^{21}$ & \# sections & 
    {\small\yng(1)} & {\tiny\yng(1,1)} & {\tiny\yng(1,1,1)} \\
    \hline
    $SU(2)$  & ~3 & 231 & 1 & 22 & 6 \\ 
    $SU(3)$  & ~4 & 208 & 1 & 24 & 3 \\
    $SU(4)$  & ~5 & 189 & 1 & 20 & 3 \\
    $SU(5)$  & ~6 & 171 & 1 & 19 & 3 \\
    \multirow{2}{*}{$SU(6)$}
             & ~7 & 151 & 1 & 21 & & 3 \\
             & ~7 & 154 & 1 & 18 & 3 \\
    $SU(7)$  & ~8 & 138 & 1 & 17 & 3 \\
    $SU(8)$  & ~9 & 123 & 1 & &  \\
    $SU(9)$  & 10 & 109 & 1 & &  \\
    $SU(10)$ & 11 & ~96 & 1 & &  \\
    $SU(11)$ & 12 & ~84 & 1 & &  \\
    $SU(12)$ & 13 & ~73 & 1 & 12 & 3 \\
    $SU(13)$ & 14 & ~63 & 1 & 11 & 3 \\
    $SU(14)$ & 15 & ~54 & 1 & &  \\
    $SU(15)$ & 16 & ~46 & 1 & &  \\
    $SU(16)$ & 17 & ~39 & 1 & &  \\
    $SU(17)$ & 18 & ~33 & 1 & &  \\
    $SU(24)$ & 25 & ~19 & 2 & &  \\
  \end{tabular}
  \caption{Hodge numbers, number of toric sections, and matter content 
    for the toric ($\Rightarrow b=1$) single-block $SU(n)$
    models.}
  \label{tab:toricSUn}
\end{table}
In \autoref{tab:toricSUn}, the single-block $SU(n)$ models with a
toric non-Abelian discriminant are listed. That is, the $SU(n)$ gauge
group arises from a discriminant component that is one of the three
coordinate hyperplanes $\{u=0\}$, $\{v=0\}$,
$\{w=0\}\subset\CP^2_{[u:v:w]}$. Note that the Kodaira fiber is $I_n$
in all of these cases. In principle, a $SU(2)=Sp(1)$ gauge group can
also arise from a $III$ and $IV$ Kodaira fiber, but these do not arise
in generic Calabi-Yau hypersurfaces in toric varieties.




\subsection{The Non-Toric Discriminant Case}

\begin{table}
  \centering
  \begin{tabular}{c|ccccccc}
    $SU(n)$ & $b$ & $h^{11}$ & $h^{21}$ & \# sections & 
    {\small\yng(1)} & {\tiny\yng(1,1)} & {\tiny\yng(1,1,1)} \\
    \hline
    $SU(2)$  & ~1 & ~3 & 231 & 1 &  &  \\ 
    $SU(2)$  & ~2 & ~3 & 195 & 1 &  &  \\ 
    $SU(2)$  & ~3 & ~3 & 165 & 1 &  &  \\ 
    $SU(2)$  & ~4 & ~3 & 141 & 1 &  &  \\ 
    $SU(2)$  & ~5 & ~3 & 123 & 1 &  &  \\ 
    $SU(2)$  & ~6 & ~3 & 111 & 1 &  &  \\ 
    $SU(2)$  & ~7 & ~3 & 105 & 1 &  &  \\ 
    $SU(2)$  & ~8 & ~3 & 105 & 1 &  &  \\ 
    $SU(2)$  & 12 & ~3 & 165 & 2 &  &  \\ 
  \end{tabular}
  \caption{Class $b$ of the non-Abelian discriminant
    component, number of toric sections, Hodge numbers, and matter content 
    for the toric single-block $SU(n)$
    models.}
  \label{tab:nonSU2}
\end{table}
The discriminant component supporting the non-Abelian gauge group need
not be a coordinate hyperplane. It can be a generic line, or a curve
of higher degree. In \autoref{tab:nonSU2}, we list these non-toric
models for gauge group $SU(2)$.




\section{Abelian Theories}

\subsection{Single U(1)}



\begin{table}
  \centering
  \begin{tabular}{c|cccc@{$~=~$}c@{$~+~$}cc}
    $G$ & $h^{11}$ & $h^{21}$ & \# sections & 
    $H_c$ &
    $\mathbf{1}_1$ & $\mathbf{1}_2$ \\
    \hline
    $U(1)$ & ~3 & 165 & $\{2^2,3\}$ & 108 & 108 & 0 &  \\ 
    $U(1)$ & ~3 & 141 & 2 & 132 & 128 & 4 &  \\ 
    $U(1)$ & ~3 & 123 & 2 & 150 & 140 & 10 &  \\ 
    $U(1)$ & ~3 & 111 & $\{1,2\}$ & 162 & 144 & 18 &  \\ 
    $U(1)$ & ~3 & 105 & $\{2^2,3^2\}$ & 168 & 140 & 28 &  two \\ 
    $U(1)$ & ~3 & 105 & $\{2^2,3^2\}$ & 168 & 128 & 40 &  possibilites \\ 
    $U(1)$ & ~3 & ~95 & 1 & 178 &  &  \\ 
    $U(1)$ & ~3 & ~93 & 1 & 180 &  &  \\ 
    $U(1)$ & ~3 & ~91 & 1 & 182 &  &  \\ 
    $U(1)$ & ~3 & ~85 & 1 & 188 &  &  \\ 
    $U(1)$ & ~3 & ~83 & 1 & 190 &  &  \\ 
    $U(1)$ & ~3 & ~81 & 1 & 192 &  &  \\ 
    $U(1)$ & ~3 & ~75 & $1^2$ & 198 &  &  \\ 
    $U(1)$ & ~3 & ~73 & 1 & 200 &  &  \\ 
    $U(1)$ & ~3 & ~71 & $1^2$ & 202 &  &  \\ 
  \end{tabular}
  \caption{Single $U(1)$ models. Exponents denote that the given value
    of $h^{21}$ is realized by multiple polytopes.}
  \label{tab:singleU1}
\end{table}
In \autoref{tab:singleU1}, the models with a single $U(1)$ are listed.








\section{Toric SU(n) Models with Abelian Factors}

% Now we relax the requirement that $h^{11}(X) = n+1$, that is, allow a
% non-vanishing Mordell-Weil group. This amounts to allowing $U(1)$
% factors in the gauge group next to the single non-Abelian factor
% $SU(n)$.








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\renewcommand{\refname}{Bibliography}
\addcontentsline{toc}{section}{Bibliography} 
\bibliography{Main}

\end{document}


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