TFPT scientific source excerpt
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000175 | The lattice glue $(D_5{\oplus}A_3){+}\mu_4\to E_8$ has an exact \emph{conformal-field-theory} mirror. The
000176 | level-$1$ WZW central charges $c=\dim\mathfrak g/(1{+}h^\vee)$ are the compiler atoms:
000177 | \[
000178 |   c(E_8)_1=\tfrac{248}{31}=8=\operatorname{rank}E_8,\qquad c(D_5)_1=\tfrac{45}{9}=5=\gcar,\qquad c(A_3)_1=\tfrac{15}{5}=3=\Nfam,
000179 | \]
000180 | and the coset has $c=8-5-3=0$ --- the criterion for a \emph{conformal embedding}
000181 | $(D_5)_1{\times}(A_3)_1\subset(E_8)_1$. So the $(5,3)$ split is central-charge \emph{additivity}, and the
000182 | ``$8$'' in $\cthree=\tfrac1{8\pi}$ is $c(E_8)_1$. (\emph{Honest:} $c=248/31{=}8$ is \emph{not} sold as a
000183 | closure --- the content is the embedding $c_{\mathrm{coset}}{=}0$.)
000184 | 
000185 | \textbf{$SU(3)\leftrightarrow SU(4)$ reconciliation.} Why is the flavor monodromy $\rho$ valued in $SU(3)$
000186 | while the carrier glue is $A_3{=}SU(4)$? Both are sides of $\mathbb P^1\setminus\mu_4$: with $n{=}4$ punctures,
000187 | $H_1=\Z^{\,n-1}$, so $\Nfam=|\mu_4|-1=3$. The \emph{carrier} side is the puncture/center $Z_4{=}\mu_4$
000188 | ($A_3{=}SU(4)$); the \emph{flavor} side is the monodromy on $H_1{=}\Z^3$, $SU(3)$-valued, with center
000189 | $Z_3$ (triality) whose phases are the parabolic weights $\{0,\tfrac13,\tfrac23\}$ ($\neq$ the $SU(4)_1$
000190 | weights $\{0,\tfrac38,\tfrac12\}$), and $SU(3)\subset SU(4)$ via $4\to3{+}1$. So it is \emph{not} one WZW for
000191 | all gates, but one geometry $\mathbb P^1\setminus\mu_4$ with two dual sides. (\emph{Checked}: the Cardy route