# 6-d Anomaly Cancellation / Main.tex

  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 \documentclass[12pt]{article} \input{Preamble} \newcommand{\cone}[1]{\ensuremath{\left<#1\right>}} \begin{document} % \input{Titlepage.tex} \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 % $SU(n)$. \bibliographystyle{utcaps} \renewcommand{\refname}{Bibliography} \addcontentsline{toc}{section}{Bibliography} \bibliography{Main} \end{document} %%% Local Variables: %%% mode: TeX-pdf %%% End: