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

Bulk locality for scalars and fermions with global symmetry