Wormholes with Ends of the World

TL;DR

This work constructs a comprehensive 3D AdS gravity framework with conical defects, end-of-the-world branes, kinks, and punctures to realize ensemble averages over BCFT data. By mapping BCFT observables to bulk geometries via quadrupling, halving, and wedge tricks, the authors reproduce universal OPE asymptotics and show how g-function data emerge topologically from brane actions, while proving no two-disk wormholes exist for the ensemble-averaged g, consistent with a non-factorizing BCFT picture. They propose a general topological conjecture for the most general wormholes and provide substantial evidence through explicit examples, including second-moment computations for C, B, and D data and a BCFT Schlenker-Witten theorem. The results deepen the link between geometry, topology, and conformal bootstrap in AdS$_3$/CFT$_2$, and suggest powerful bulk tools for encoding BCFT ensembles and their higher moments. The framework paves the way for exploring ensemble dualities in more complex settings and for clarifying the role of defects and boundaries in holographic dualities.

Abstract

We study classical wormhole solutions in 3D gravity with end-of-the-world (EOW) branes, conical defects, kinks, and punctures. These solutions compute statistical averages of an ensemble of boundary conformal field theories (BCFTs) related to universal asymptotics of OPE data extracted from the 2D conformal bootstrap. Conical defects connect BCFT bulk operators; branes join BCFT boundary intervals with identical boundary conditions; kinks (1D defects along branes) link BCFT boundary operators; and punctures (0D defects) are endpoints where conical defects terminate on branes. We provide evidence for a correspondence between the gravity theory and the ensemble. In particular, the agreement of the $g$-function dependence results from an underlying topological aspect of the on-shell EOW brane action, from which a BCFT analogue of the Schlenker-Witten theorem also follows.

Figures (4)

• Figure 1: The upper half plane (shaded gray in the left panel) is a simple example of a 2D bordered surface. Under conformal transformation, it can be mapped to a unit disk (shaded gray in the right panel). From radial quantization, the partition function of such a disk can be regarded as the inner product $\langle B^{(a)}| {\mathbbm{1}} \rangle$, for some boundary state $|B^{(a)}\rangle$ living on the circular border.
• Figure 2: Closed Hilbert space (left) and open Hilbert space (right).
• Figure 3: $\mathbb{Z}_2$ quotient of Euclidean AdS$_3$ produces a manifold with an asymptotic disk boundary as well as a finite disk boundary that is a zero-tension EOW brane.
• Figure 4: Geometry of the 3D manifold near an asymptotic boundary ($z=0$). The EOW brane $Q$ (green) meets the boundary at an angle determined by the tension. The direction $\theta$ is orthogonal to the plane of the paper. If the tension were zero, the EOW would be located at $\rho=0$, which we label $\Sigma$ (blue horizontal line). The pink arc is where $\zeta=\epsilon$. The full geometry is cut off at $z=\epsilon$ (dashed vertical line). The region between $\Sigma$ and $Q$ is foliated by constant-$\rho$ slices, each locally AdS${}_2$ but with different constant curvature dependent only on $\rho$. The brane is cut off at $z=\epsilon$ or $\zeta=\epsilon\,{\mathrm{cosh}}\,\rho_*$.