AdS Bulk Locality from Sharp CFT Bounds

TL;DR

This paper proves a conjecture linking large-gap conformal field theories to local effective field theories in AdS by deriving bounds on bulk interactions using conformal bootstrap and dispersion relations.

Contribution

It establishes a quantitative connection between CFT spectral gaps and bulk locality through numerical bounds on Wilson coefficients, employing dispersive sum rules and flat-space limits.

Findings

Bounds on bulk Wilson coefficients scale with $\Delta_{ ext{gap}}$ as expected.

AdS$_{4}$ resolves infrared divergences in flat-space bounds.

Supports twice-subtracted dispersion relations for S-matrices from AdS/CFT.

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.