  We count the number of independent solutions to crossing constraints of four
point functions involving charged scalars and charged fermions in a CFT with
large gap in the spectrum. To find the CFT data we employ recently developed
analytical functionals to charged fields. We compute the corresponding higher
dimensional flat space S matrices in an independent group theoretic manner and
obtain agreement with our CFT counting of ambiguities. We also write down the
local lagrangians explicitly. Our work lends further evidence to
\cite{Heemskerk:2009pn} that any CFT with a large charge expansion and a gap in
the spectrum has an AdS bulk dual.
