AdS Bulk Locality from Sharp CFT Bounds

Abstract

It is a long-standing conjecture that any CFT with a large central charge and a large gap ${\Delta }_{\text{gap}}$ in the spectrum of higher-spin single-trace operators must be dual to a local effective field theory in AdS. We prove a sharp form of this conjecture by deriving numerical bounds on bulk Wilson coefficients in terms of ${\Delta }_{\text{gap}}$ using the conformal bootstrap. Our bounds exhibit the scaling in ${\Delta }_{\text{gap}}$ expected from dimensional analysis in the bulk. Our main tools are dispersive sum rules that provide a dictionary between CFT dispersion relations and S-matrix dispersion relations in appropriate limits. This dictionary allows us to apply recently-developed flat-space methods to construct positive CFT functionals. We show how AdS${}_4$ naturally resolves the infrared divergences present in 4D flat-space bounds. Our results imply the validity of twice-subtracted dispersion relations for any S-matrix arising from the flat-space limit of AdS/CFT.

70+21 pages, 18 figures

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