It from ETH: Multi-interval Entanglement and Replica Wormholes from Large-$c$ BCFT Ensemble

TL;DR

This work furnishes a field-theoretic derivation of the Ryu-Takayanagi formula directly from a large-c BCFT ensemble, including RT-phase transitions and multi-interval entanglement, by linking universal BCFT data to Liouville theory and Karch-Randall brane holography. It introduces It from ETH, arguing that bulk spacetime and holographic tensor networks emerge from eigenstate-thermalization-like statistics of heavy BCFT/OPE data. The paper provides concrete constructions for multi-boundary black holes, computes entropies via two complementary channels, and validates them against bulk RT results, while offering algebraic signatures for replica wormholes in entanglement islands and black hole microstate counting. It also presents a holographic random BCFT tensor network that faithfully captures holographic entanglement structure and phase transitions. Overall, the framework unifies BCFT data, Liouville theory, and holographic geometry into a coherent, CFT-driven picture of bulk spacetime emergence.

Abstract

We provide a derivation of the Ryu-Takayanagi (RT) formula in 3D gravity for generic boundary subregion--including RT surface phase transitions--directly from the dual two-dimensional conformal field theory (CFT). Our approach relies on the universal statistics of the algebraic conformal data and the large-$c$ behavior of conformal blocks with Cardy boundaries involved. We observe the emergence of 3D multi-boundary black holes with Karch-Randall branes from entangled states of any number of CFT's with and without Cardy boundaries. The RT formula is obtained directly from the CFT in the high-temperature regime. Two direct applications are: $\textbf{1)}$ A simple derivation of the multi-interval entanglement entropy for the vacuum state of a single CFT; $\textbf{2)}$ A CFT-based detection of the emergence of replica wormholes in the context of entanglement islands and black hole microstate counting. Our framework yields the first holographic random tensor network that faithfully captures the entanglement structure of holographic CFTs. These results imply that bulk spacetime geometries indeed emerge from the eigenstate thermalization hypothesis (ETH) in the dual field theory in the large-$c$ limi--a paradigm we refer to as $\textit{It from ETH}$.

Figures (43)

• Figure 1: An example of the CFT state preparation path integral. The path integral on this 2D manifold generates an entangled state of three 2D CFT's living on the circles $A$, $B$ and $C$. The geometry of this 2D manifold is hyperbolic with $A$, $B$ and $C$ as the boundaries at infinity. There are three minimal area surfaces $L_{A}$, $L_{B}$ and $L_{C}$ and they are horizons of the bulk three boundary black hole geometry.
• Figure 2: Examples of the construction of various hyperbolic surfaces $\Sigma_{(g,n)}$ from quotienting $\mathbb{H}^{2}$ by appropriate Fuchsian subgroups $\Gamma$.
• Figure 3: a) The upper half plane with a conformal boundary. This is the open description. b) A half-infinite cylinder terminated at $t=0$. The circle at $t=0$ should be thought of as a state. This is the closed description.
• Figure 4: The AdS$_{3}$ bulk is foliated by AdS$_{2}$ slices. Each slice is a hyperbolic upper-half-plane $\mathbb{H}^{2}$, i.e. AdS$_{2}$.
• Figure 5: A demonstration of the geometry of positive tension Karch-Randall branes. For the cases with a small spherical cap (the red surface), the interior region of the cap is cut off. For a large spherical cap (the green surface), the exterior region is cut off. The gray regions are cut off. The vertical surface is the $y=0$ (zero time) slice in Fig. \ref{['pic:2BCFTt0slice']}.