AdS Bulk Locality from Sharp CFT Bounds

TL;DR

This work proves a sharp form of the bulk locality conjecture in AdS/CFT by deriving numerical bounds on bulk Wilson coefficients in terms of the higher-spin gap $ riangle_{ ext{gap}}$ using dispersive CFT sum rules. By constructing the holographic functionals $C_{k, u}$ and exploiting Regge moments and harmonic analysis on $H^{d-1}$, the authors map flat-space S-matrix positivity to AdS EFT constraints, showing how AdS$_4$ regularizes infrared divergences and enabling twice-subtracted dispersion relations in the flat-space limit. The results yield two-sided, parametricly correct bounds on higher-derivative bulk couplings $g_n$, demonstrate bulk locality down to scales $oxed{R_{ m AdS}/ riangle_{ ext{gap}}}$, and reproduce known flat-space bounds in the appropriate limit. The formalism also provides bounds on anomalous dimensions of light double-trace operators and clarifies the role of gravity via gravitons and eikonalization, with extensions to maximal supersymmetry and potential spinning correlator generalizations. Overall, the paper establishes a robust, rigorous mechanism by which bulk locality in AdS emerges from CFT bootstrap data through dispersive sum rules and holographic dictionary.

Abstract

It is a long-standing conjecture that any CFT with a large central charge and a large gap $Δ_{\text{gap}}$ 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 $Δ_{\text{gap}}$ using the conformal bootstrap. Our bounds exhibit the scaling in $Δ_{\text{gap}}$ 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.

Figures (18)

• Figure 1: A four-point function near the Regge limit. Bulk excitations localize along two null sheets, which intersect in a transverse hyperbolic space $H_{d-1}$ (thick blue line). The red and blue arrows show the boundary and bulk actions of $\mathrm{SO}(d-1,1)$ isometries, which we exploit to focus in impact parameter.
• Figure 2: Allowed region (shaded) for $g_2$ and $g_3$ from Caron-Huot:2021rmr, in a theory of a scalar coupled to gravity in flat space in dimensions $D=5$ and heavy mass scale $M$. Asymptotically to the right, the region is a cone which matches the non-gravitational bounds; note the offset to the left. The solid lines show the non-optimal bounds from the simple functionals \ref{['g2_g3_functionals']}.
• Figure 3: Action of the ${\cal C}^{(i)}$ functionals in \ref{['g2_g3_functionals']} for $i=1,2$ respectively, on various heavy states with $m\geq M$ in units where $M=1$. Individual points, representing different spins $J$, are lined up with constant impact parameter $b=\frac{2J}{m}$ in order to highlight the regularity as $m\to\infty$ and $J\to\infty$. This helps confirm positivity on all heavy states. The blue outliers (at $J=2$ and $J=0,4$ respectively) are caused by single zeros slightly below threshold $m=M$.
• Figure 4: Relation between impact parameter and angular momentum in the Regge limit. (a) In flat space, a pair of massless particles with center-of-mass energy $m$ and transverse separation $b$ carries total angular momentum $J=m\frac{b}{2}$. (b) In a holographic CFT${}_d$, the Regge limit of the four-point function localizes along two null sheets. They intersect in the transverse hyperbolic space $H_{d-1}$ depicted by the blue line. A pair of massless particles with center-of-mass energy $\Delta/R_{\rm AdS}$ separated in $H_{d-1}$ by geodesic distance $\beta R_{\rm AdS}$ carries total angular momentum $J=\Delta\tanh\frac{\beta}{2}$.
• Figure 5: In the CFT, the angle of approach $\eta$ to the Regge limit (in which operators 3 and 4 become null from 1 and 2, respectively) acts as a smeared version of the bulk impact parameter ${\eta_{\rm AdS}}=\cosh\beta$. Since both variables transform like hyperbolic cosines under $\mathrm{SO}(d-1,1)$ transverse isometries, shown as blue and red arrows, harmonic analysis can be used to "undo" the smearing and achieve bulk focusing. The operators always stay within the "Rindler wedge" in which $1$ is spacelike from $3$ (and $4$ is spacelike from $2$).