Holography and wormholes in 2+1 dimensions

TL;DR

This work provides a concrete real-time holographic dictionary for a class of 2+1D wormholes, showing that the dual CFT state is entangled across multiple boundaries and that bulk topology behind horizons is encoded in boundary data via the Schwarzian of a filling map $J$. By coupling Euclidean handlebodies (Euclidean caps) to Lorentzian wormholes, the authors demonstrate how initial/final states, encoded by Teichmüller data, determine the bulk geometry through the boundary stress tensor, with $\langle T_{zz}\rangle=-S[J](z)$ in suitable coordinates. They compute 1- and 2-point functions, reveal how entanglement manifests across boundaries, and discuss how different fillings correspond to distinct states, including non-handlebody cases, offering a controlled framework to study information behind horizons. The results connect Teichmüller theory, Schwarzian derivatives, and Liouville action to holography, illustrating how topology and boundary data jointly fix the bulk in a setting with no local bulk degrees of freedom. These insights provide a tractable platform for exploring black hole information and the role of topology in holography, with clear extensions to rotating cases and higher dimensions.

Abstract

We provide a holographic interpretation of a class of three-dimensional wormhole spacetimes. These spacetimes have multiple asymptotic regions which are separated from each other by horizons. Each such region is isometric to the BTZ black hole and there is non-trivial spacetime topology hidden behind the horizons. We show that application of the real-time gauge/gravity duality results in a complete holographic description of these spacetimes with the dual state capturing the non-trivial topology behind the horizons. We also show that these spacetimes are in correspondence with trivalent graphs and provide an explicit metric description with all physical parameters appearing in the metric.

Figures (23)

• Figure 1: A wormhole spacetime with two outer regions corresponding to a Riemann surface of genus 2 with 2 boundary components.
• Figure 2: A fatgraph representing the wormhole spacetime sketched in figure \ref{['fig:sketch']}.
• Figure 3: On the left we sketched two fundamental domains in $H$. The boundaries are pairwise glued together as indicated by the arrows. After the gluing we find the Riemann surfaces shown on the right.
• Figure 4: The Schottky double of the Riemann surfaces of figure \ref{['fig:fundamentaldomain']} is constructed by gluing two copies of the fundamental domain to each other and identifying the boundaries. The line $\tau = 0$ is invariant and the Schottky double surface is symmetric under reflection in this line. The limit set $\Lambda(\hat{\Gamma})$ is a subset of the line $\tau = 0$ but is not shown here. It has to be removed from the $(\tau,x)$ plane before taking a quotient.
• Figure 5: The extension of the fundamental domain for $\hat{\Gamma}$ from the $S^2$ to $H^3$ is bounded by a set of hemispheres that should be pairwise identified. We recover $S$ as the surface given by $\tau = 0$.