A Cardy formula for off-diagonal three-point coefficients; or, how the geometry behind the horizon gets disentangled

TL;DR

The paper constructs an entanglement-insensitive kernel ${\cal F}(E_a,E_b,Δ)$ for the AdS3/CFT2 Wightman function, obtained by inverting Fourier and Laplace transforms of the boundary correlator and using Cardy-like density-of-states results. This kernel encodes the average squared off-diagonal matrix elements of a scalar primary and, in the large-c regime, yields explicit asymptotics that connect to eigenstate thermalization, including a clear χ-dependent structure with a peak at χ=0. With ${\cal F}$ in hand, the authors compute Wightman functions for doubled-CFT states with arbitrary entanglement patterns, showing how progressively disentangling the two CFT copies dismantles the geometry behind the horizon and can even destroy the geodesic interpretation, while leaving exterior regions largely unaffected. These results provide a quantitative bridge between entanglement structure, ETH-like matrix-element statistics, and the emergent bulk geometry behind black hole horizons, with potential extensions to traversable wormholes and non-thermal entanglement patterns.

Abstract

In the AdS/CFT correspondence eternal black holes can be viewed as a specific entanglement between two copies of the CFT: the thermofield double. The statistical CFT Wightman function can be computed from a geodesic between the two boundaries of the Kruskal extended black hole and therefore probes the geometry behind the horizon. We construct a kernel for the AdS3/CFT2 Wightman function that is independent of the entanglement. This kernel equals the average off-diagonal matrix element squared of a primary operator. This allows us to compute the Wightman function for an arbitrary entanglement between the double copies and probe the emergent geometry between a left- and right-CFT that are not thermally entangled.

Figures (8)

• Figure 1: Average of the square of matrix elements $\mathcal{F}(E,\chi=0,\Delta)$ of a scalar primary of dimension $\Delta$ as a function of the energy $E$ of the states (for states with the same energy), obtained from Eq. \ref{['eq:Fchi0']} with $\chi=0$ and $c=20$.
• Figure 2: (a), (b) and (c) show the average of the square of matrix elements $\mathcal{F}(E,\chi,\Delta)$ of scalars primary of various dimensions $\Delta$ as a function of the energy difference of the states. Computed numerically from eqs. \ref{['eq:F_definition']}, \ref{['eq:Fhat_line']} and \ref{['eq:DoS']} with $c=20$. (d) shows the tails of $\mathcal{F}(E,\chi,\Delta)$ near $\chi\sim 2E$ in logarithmic-linear scale. Dashed lines are a guide to they eye suggesting the exponential suppression observed in the context of ETH in one-dimensional systems Beugeling2015Mondaini2017.
• Figure 3: Sketch of a Penrose diagram with dashed lines indicating the symmetric space-like geodesics for $t=0$ and $t\neq 0$. The diagram on the left is for the BTZ black hole, the one on the right for the state we are investigating.
• Figure 4: Ratio of entanglement entropy of state with cutoff $\Lambda$ to BTZ entanglement entropy of the same $\beta$. $c=100$. The blue curve is $\beta/L=3/4$, the orange $\beta/L=1/2$ and the green $\beta/L=1/4$. The vertical lines correspond to a value of the cutoff where a saddle-point approximation predicts that the ratio should saturate.
• Figure 5: Normalized Wightman function with finite cutoff (left) and length difference from BTZ (right) as defined in eq. \ref{['eq:G_cutoff']} for fixed $\beta/L=1/2$. The central charge is $c=10$ and space separation is zero.