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 ∆ 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 ∆ gap using the conformal bootstrap. Our bounds exhibit the scaling in ∆ 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.

Mixed Correlator Dispersive CFT Sum Rules

“…k ij < 0. In particular, for equal operators, the k = (0, −1, −1, 0) subtraction scheme, or equivalently (0, 0, −1, −1), applied to the B k;v|34 functional corresponds to the anti-subtracted functional B −2 introduced in [30]. We will show that anti-subtractions change the Mellin-space domain of convergence.…”

“…Moreover, the presence of the double discontinuity suppresses doubletwist operators thereby allowing us to probe non-perturbative features of CFTs. Finally, they enjoy positivity properties due to the double-zeros that make them desirable for the numerical 2 Our convention differs from that of [29,30]: for equal operators, the u-channel identity is always present. Therefore, they define their sum rules to be normalized as follows:…”

“…In recent years, with the introduction of analytic functionals [20][21][22][23][24][25][26][27] and dispersion relation methods [28][29][30][31][32][33][34], the gap between numerics and analytics has greatly narrowed revealing new horizons for the bootstrap program. This paper builds upon this bridge by introducing new analytic dispersive functionals for correlators with unequal external scalar operators.…”

“…When evaluated in the Regge limit, these dispersive functionals are further capable of probing bulk AdS physics. This insight led to the construction of holographic functionals in [30] where the collinear B k,v functional served as a seed in their construction. Such holographic functionals are capable of bounding couplings in AdS, and they become dispersion relations in the flat-space limit [35,36].…”

“…We see that subtractions along the τ 12 and τ 34 trajectories probe subleading double-twists trajectories. We further note that k mn < 0 is allowed leading to "anti-subtracted" functionals such as the B −2 functional introduced in [30] which lead to superconvergent sum rules [44].…”