The Analytic Bootstrap and AdS Superhorizon Locality

Abstract: We take an analytic approach to the CFT bootstrap, studying the 4-pt correlators of d > 2 dimensional CFTs in an Eikonal-type limit, where the conformal cross ratios satisfy |u| << |v| < 1. We prove that every CFT with a scalar operator \phi must contain infinite sequences of operators O_{\tau,l} with twist approaching \tau -> 2\Delta_\phi + 2n for each integer n as l -> infinity. We show how the rate of approach is controlled by the twist and OPE coefficient of the leading twist operator in the \phi x \phi OPE, and we discuss SCFTs and the 3d Ising Model as examples. Additionally, we show that the OPE coefficients of other large spin operators appearing in the OPE are bounded as l -> infinity. We interpret these results as a statement about superhorizon locality in AdS for general CFTs.

• The paper demonstrates that CFTs with scalar operators exhibit infinite towers of operators with twist values converging to 2Δφ+2n at large spin.
• It employs an analytic exploration of four-point correlators in the eikonal limit to establish bounded OPE coefficients in higher-dimensional CFTs.
• The findings confirm that dual AdS interactions weaken at large separations, reinforcing superhorizon locality and the connection to local bulk physics.

Analyzing the Analytic Bootstrap and AdS Superhorizon Locality

The paper "The Analytic Bootstrap and AdS Superhorizon Locality" presents a theoretical analysis of the conformal field theory (CFT) bootstrap, particularly focusing on the implications for Anti-de Sitter (AdS) space locality at superhorizon scales. The authors adopt an analytic approach, contrasting with the more common numerical methods, to explore the structure of four-point correlators in CFTs with dimensions greater than two, specifically examining the eikonal limit where the conformal cross-ratios satisfy $|u| \ll |v| < 1$.

Key Results and Contributions

The authors establish that any CFT containing a scalar operator $\phi$ must possess infinite sequences of operators with twist values tending towards $\tau \to 2 \Delta_{\phi} + 2n$ for each integer $n$, as the spin $\ell$ of the operators approaches infinity. This result underlines a non-trivial structure in the operator product expansion (OPE) near large spin limits, indicating that the rate at which these twists approach is governed by the twist and OPE coefficients of the leading twist operator in the $\phi \times \phi$ OPE. Notably, the paper explores specific instances, such as superconformal field theories (SCFTs) and the three-dimensional Ising model, to exemplify these findings.

Additionally, the research demonstrates that the OPE coefficients of large spin operators appearing in the $\phi \times \phi$ OPE are bounded as the spin increases. These findings offer insights into the semblance of locality in AdS at superhorizon distances for general CFTs, as the CFT correlators reflect dual behaviors akin to interactions shutting off as bulk impact parameters exceed the AdS length scale.

Implications and Interpretation

This work's primary theoretical implication is its assertion that all CFTs in dimensions greater than two must feature correlators dual to local physics in AdS at superhorizon distances. This conclusion is rooted in the observation that the interactions between orbiting entities in the dual AdS space diminish at large separations, implying a form of "coarse locality."

From a practical perspective, these results extend the conventional understanding of locality in AdS/CFT correspondence, typically concerned with sub-AdS-scale phenomena. Moreover, they align with existing AdS/CFT models where perturbative approaches have suggested similar decay of interactions at large spin or angular momentum.

Future Directions

The paper's approach paves the way for further inquiry into several open areas. One potential direction is to investigate the implications of these universal large spin behaviors in non-perturbative regimes and explore possible deviations in CFTs of lower dimensions, where the arguments presented do not hold. Moreover, understanding how these results might inform or be informed by numerical bootstrap methods or holographic renormalization group (RG) processes could offer deeper insights into the dual nature of CFT correlators and AdS interactions.

Furthermore, the non-trivial behavior observed in SCFTs, along with the presence of infinite towers of operators with protected dimensions, invites a more comprehensive analysis of the interplay between R-symmetry and twist accumulation, especially in higher spin contexts.

Conclusion

This analysis of the analytic bootstrap in the context of AdS superhorizon locality advances our understanding of the intricate relations encoded in CFT OPEs and their dualities in AdS space. The universality of large spin behavior elucidated in this work holds significant value for both theoretical exploration in the realms of quantum field theory and holography and for practical computations within the framework of the AdS/CFT correspondence.