Bulk locality for scalars and fermions with global symmetry

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.