Universality of Long-Distance AdS Physics from the CFT Bootstrap

TL;DR

The paper demonstrates that long-distance AdS physics is universally encoded in the CFT bootstrap. By combining the higher-dimensional bootstrap with a detailed Virasoro-block analysis at large central charge, it shows that bulk Newtonian gravity, deficit angles in AdS$_3$, and BTZ black-hole physics are exactly mirrored in CFT spectra and OPE data. It establishes a concrete bridge between AdS locality and CFT data, including a thermal interpretation for heavy states in AdS$_3$ and a dense twist spectrum near BTZ thresholds. The work also provides a framework for extracting bulk dynamics non-perturbatively from boundary data, with clear avenues for refining the blocks and exploring finite-$c$ corrections.

Abstract

We begin by explicating a recent proof of the cluster decomposition principle in AdS_{d+1} from the CFT_d bootstrap in d > 2. The CFT argument also computes the leading interactions between distant objects in AdS, and we confirm the universal agreement between the CFT bootstrap and AdS gravity in the semi-classical limit. We proceed to study the generalization to 2d CFTs, which requires knowledge of the Virasoro conformal blocks in a lightcone OPE limit. We compute these blocks in a semiclassical, large central charge approximation, and use them to prove a suitably modified theorem. In particular, from the 2d bootstrap we prove the existence of large spin operators with fixed 'anomalous dimensions' indicative of the presence of deficit angles in AdS_3. As we approach the threshold for the BTZ black hole, interpreted as a CFT scaling dimension, the twist spectrum of large spin operators becomes dense. Due to the exchange of the Virasoro identity block, primary states above the BTZ threshold mimic a thermal background for light operators. We derive the BTZ quasi-normal modes, and we use the bootstrap equation to prove that the twist spectrum is dense. Corrections to thermality could be obtained from a more refined computation of the Virasoro conformal blocks.

Figures (7)

• Figure 1: This figure indicates the correspondence between a descendant operator/state in the CFT and a center-of-mass wavefunction in AdS. The relationship is entirely kinematical; it follows because the conformal group is the isometry group of AdS. A primary state would have its center of mass at rest near $\rho=0$, the origin of AdS in the metric of equation (\ref{['eq:AdSGlobalCoordinates']}).
• Figure 2: This figure shows two objects created by CFT operators ${\cal O}_A$ and ${\cal O}_B$ orbiting each other at large angular momentum, and therefore at large separation, in AdS. A major goal will be to show that such states exist and to describe their properties.
• Figure 3: One can only obtain an $s$-channel singularity in a scattering amplitude via an infinite sum of $t$-channel partial waves as $\ell \to \infty$. The same physical point, adapted to AdS/CFT, underlies the proof of cluster decomposition and the derivation of long-range forces from the CFT bootstrap.
• Figure 4: This figure depicts the AdS/CFT correspondence in global coordinates, emphasizing that AdS time translations are generated by the Dilatation operator, so that bulk energies correspond to operator/state dimensions in the CFT.
• Figure 5: This figure is suggestive of the relationship between certain $\ell \gg 1$ operators in the OPE of ${\cal O}_1$ and ${\cal O}_2$ and a '2-blob' state in AdS, corresponding to the two states created by the CFT primaries ${\cal O}_1(0)$ and ${\cal O}_2(0)$ in an orbit about each other at large separation $\kappa \sim \log \ell$. The existence and asymptotic dimension of these 2-blob operators at large $\ell$ in the CFT defines a cluster decomposition principle in AdS.