Ryu-Takayanagi Formula for Multi-Boundary Black Holes from 2D Large-$c$ CFT Ensemble

TL;DR

This work shows that the Ryu–Takayanagi entanglement entropy and the associated bulk geometries for multi-boundary black holes can be derived directly from boundary CFT data, using a 2D large-$c$ ensemble of OPE coefficients and their Gaussian moments. The norm and all RT phases emerge from statistical contractions that map to Liouville theory with ZZ boundary conditions, equating the CFT ensemble average to the exact gravitational path integral on the bulk geometry. Different RT phases correspond to distinct leading contraction patterns, naturally producing replica-wormhole–type transitions from the boundary statistics. This establishes a precise statistical-mechanical mechanism by which semiclassical gravity and holographic entanglement emerge from universal algebraic data in holographic CFTs, with the Liouville ZZ correspondence providing a concrete computational bridge. The framework extends to higher-genus multi-boundary configurations, offering a universal route to derive bulk entropy from boundary data without requiring full microscopic information about every CFT.

Abstract

We study a class of quantum states involving multiple entangled CFTs in AdS$_3$/CFT$_2$, associated with multi-boundary black hole geometries, and demonstrate that the Ryu-Takayanagi (RT) formula for entanglement entropy can be derived using only boundary CFT data. Approximating the OPE coefficients by their Gaussian moments within the 2D large-$c$ CFT ensemble, we show that both the norm of the states and the entanglement entropies associated with various bipartitions--reproducing the expected bulk dual results--can be computed purely from the CFT. All $\textit{macroscopic geometric}$ structures arising from gravitational saddles emerge entirely from the universal statistical moments of the $\textit{microscopic algebraic}$ CFT data, revealing a statistical-mechanical mechanism underlying semiclassical gravity. We establish a precise correspondence between the CFT norm, the Liouville partition function with ZZ boundary conditions, and the exact gravitational path integral over 3D multi-boundary black hole geometries. For entanglement entropy, each RT phase arises from a distinct leading-order Gaussian contraction, with phase transitions--analogous to replica wormholes--emerging naturally from varying dominant statistical patterns in the CFT ensemble. Our derivation elucidates how the general mechanism behind holographic entropy, namely a boundary replica direction that elongates and becomes contractible in the bulk dual, is encoded explicitly in the statistical structure of the CFT data.

Figures (17)

• Figure 1: Summary of various connections between CFT and its gravitational dual, established via averaging over OPE coefficients in the 2D large-$c$ CFT ensemble.
• Figure 2: The 2D hyperbolic cylinder can be obtained as a quotient of $H_2$, implemented by identifying the red curves on the upper half-plane. This construction can be extended via hyperbolic slicing to produce the 3D two-boundary BTZ black hole solution.
• Figure 3: The 2D three-boundary hyperbolic solution can be obtained as a quotient of $H_2$ generated by two Fuchsian elements, identifying the red and green curves on the upper half plane. This construction can be extended via hyperbolic slicing to produce the 3D three-boundary black hole solution.
• Figure 4: The 2D genus-one hyperbolic solution can be obtained as a quotient of $H_2$ generated by two Fuchsian elements, and can be extended to construct a 3D genus-one black hole solution using hyperbolic slicing.
• Figure 5: The Hartle–Hawking state associated with the BTZ black hole is obtained by slicing open the total path integral along the $\tau_E = 0$ slice, with the state defined on the red surface at the top. Its CFT dual is prepared by a path integral over the bottom grey surface and is defined on the blue curve.