Semiclassical 3D gravity as an average of large-c CFTs

TL;DR

This work proposes and tests a framework in which semiclassical gravity in AdS$_3$ emerges from an ensemble of large-$c$ CFT data constrained by bootstrap principles. By averaging OPE coefficients with Gaussian statistics around a sparse low-energy spectrum, the authors reproduce bulk gravity saddles, including multi-boundary wormholes, handlebodies, and their 1-loop corrections, with Liouville theory providing a unifying language for the averaged observables. They demonstrate explicit correspondences between averaged CFT correlators and gravity actions, show how random bulk couplings arise from wormholes (Coleman–Giddings–Strominger mechanism), and establish a comprehensive dictionary linking simple topologies, higher-genus surfaces, and their ensemble interpretations. The results offer a concrete realization of how coarse-grained CFT data can encode the statistical properties of bulk quantum gravity, with potential implications for understanding ensembles in higher-dimensional holography and the role of wormholes in quantum gravity. The methodology provides a robust bridge between bootstrap-based CFT data and semiclassical AdS$_3$ bulk physics, including precise matches at the classical and one-loop levels across a wide array of topologies.

Abstract

A two-dimensional CFT dual to a semiclassical theory of gravity in three dimensions must have a large central charge $c$ and a sparse low energy spectrum. This constrains the OPE coefficients and density of states of the CFT via the conformal bootstrap. We define an ensemble of CFT data by averaging over OPE coefficients subject to these bootstrap constraints, and show that calculations in this ensemble reproduce semiclassical 3D gravity. We analyze a wide variety of gravitational solutions, both in pure Einstein gravity and gravity coupled to massive point particles, including Euclidean wormholes with multiple boundaries and higher topology spacetimes with a single boundary. In all cases we find that the on-shell action of gravity agrees with the ensemble-averaged CFT at large $c$. The one-loop corrections also match in the cases where they have been computed. We also show that the bulk effective theory has random couplings induced by wormholes, providing a controlled, semiclassical realization of the mechanism of Coleman, Giddings, and Strominger.

Figures (7)

• Figure 1: Example 2-boundary (quasi-Fuchsian) wormholes. $(a)$ Spherical topology with 4 massive particles, which contributes to the product of CFT 4-point functions $G(x,\bar{x})G(x',\bar{x}')$. $(b)$ Torus with 1 massive particle, which contributes to the product of thermal 1-point functions $\langle O \rangle_{\tau} \langle O \rangle_{\tau'}$. $(c)$ Genus-2 wormhole, which contributes to the product of genus-2 partition functions $Z_{g=2}(\Omega) Z_{g=2}(\Omega')$. $(d)$ A more elaborate wormhole with higher genus and defects.
• Figure 2: If the mass of a conical defect is continued above the black hole threshold, it disappears, and results in two pieces of the boundary being glued together. This relates wormhole amplitudes to analytically continued Virasoro identity blocks. The figure shows how the saddlepoint changes as each additional conical defect is taken above threshold.
• Figure 3: $C_0$ is the crossing kernel for the identity operator. This equation also applies to subdiagrams inside more complicated blocks.
• Figure 4: A Maldacena-Maoz wormhole with genus $g=0$ and $n=3$ conical defects. Left: The quotient $\mathbb{H}_2/\Gamma$ and its reflection in the lower half-plane. Right: The wormhole is the quotient $\mathbb{H}_3/\Gamma$. The shaded region is a hyperbolic triangle, and two copies of the triangle form the fundamental domain for $\Gamma$ acting on $\mathbb{H}_2$. The red curves are the conical defects.
• Figure 5: $(a)$ Boundary of the Maldacena-Maoz wormhole with three defects, in the limit where one operator is taken to the black hole threshold $\Delta_2 \to \frac{c}{12}$. In this limit $\gamma_2$ is parabolic and the defect approaches the boundary of $\mathbb{H}_2$. $(b)$ The boundary when $\gamma_1$ is elliptic and $\gamma_2$ is hyperbolic has only one connected component, and there are only two defects. Thus the 3-defect Maldacena-Maoz wormhole is related by analytic continuation to a geometry with the topology of a ball and two defects.