Conformal Bootstrap, Universality and Gravitational Scattering

TL;DR

The paper develops a non-perturbative holographic dictionary for AdS3/CFT2 by applying conformal bootstrap to irrational large-c CFTs with Virasoro symmetry. It uncovers a universal Liouville-like exchange algebra governing the interaction of infalling and outgoing particles near a BTZ horizon, with an R-matrix given by the quantum 6j-symbol and a scattering phase equal to the hyperbolic tetrahedron volume, matching the 2+1D gravitational amplitude. The authors connect Virasoro modular geometry to Teichmüller space quantization, show the spectral density aligns with Cardy entropy, and interpret scrambling behind the horizon via the exchange algebra, suggesting a robust framework for bulk reconstruction and potential generalizations to higher dimensions and matter couplings. Overall, the work provides a precise, universal CFT mechanism for capturing gravitational scattering and horizon dynamics in holography.

Abstract

We use the conformal bootstrap equations to study the non-perturbative gravitational scattering between infalling and outgoing particles in the vicinity of a black hole horizon in AdS. We focus on irrational 2D CFTs with large $c$ and only Virasoro symmetry. The scattering process is described by the matrix element of two light operators (particles) between two heavy states (BTZ black holes). We find that the operator algebra in this regime is (i) universal and identical to that of Liouville CFT, and (ii) takes the form of an exchange algebra, specified by an R-matrix that exactly matches with the scattering amplitude of 2+1 gravity. The R-matrix is given by a quantum 6j-symbol and the scattering phase by the volume of a hyperbolic tetrahedron. We comment on the relevance of our results to scrambling and the holographic reconstruction of the bulk physics near black hole horizons.

Figures (12)

• Figure 1: Penrose diagram describing an eternal BTZ black hole with an infalling and outgoing matter perturbation ShSt. The two particles collide near the horizon and affect each others' trajectory via a gravitational shock wave interaction. Alternatively (as shown in the figure) one may represent the effect of the shock wave as a shift in the location of the event horizon.
• Figure 2: Overview of our results. We study irrational 2D CFTs with a string theory dual on AdS${}_3$. If the CFT has large $c$, a sparse light spectrum Tom and only Virasoro symmetry, the AdS theory is expected to have two regimes in which gravity dominates: a high energy regime close to a black hole horizon, and a low energy regime at long distances. In the CFT, the dynamics in these regimes is dominated by the high entropy part of the spectrum, and the modular properties of conformal blocks are captured by Liouville theory. There exists a precise mathematical identification between 2+1 quantum gravity, defined as the quantization of its classical phase space, and Liouville modular geometry.
• Figure 3: The hyperbolic tetrahedron $T$. The dihedral angles at the six edges are determined by the six mass paramaters in (\ref{['massp']}) via ${l_i}/{2\pi} = \sqrt{8M_i}$.
• Figure 4: The scrambling of a signal (operator $A$) due to the a perturbation (operator $B$) at some earlier time $t_1<t_0$. An observer that measures the state can detect signal $A$ only if $A$ acts on the state from the left. Passing A through B produces a new intermediate channel with energy $\beta$, which for $t_0-t_1> t_{\rm crit}$ exceeds $\omega$. Signal A becomes scrambled: its coherent phase information get washed out by the large entropy region of the spectrum near $M+\beta$.
• Figure 5: The embedding of the Penrose diagram of fig 1 in the AdS${}_3$ cylinder. The left figure shows the two Poincaré wedges, connected at the corner points. The black hole horizon is indicated by the purple dashed line connecting the two corner points. The right figure shows the two trajectories of the infalling and out going particle $B=M_2$ and $A=M_4$.