The inside outs of AdS(3)/CFT(2): Exact AdS wormholes with entangled CFT duals

TL;DR

This work constructs an exact, infinite family of two-sided AdS$_3$ wormhole geometries by applying two independent solution-generating diffeomorphisms to the eternal BTZ black hole, yielding spacetimes with general boundary stress tensors on both ends. The dual CFT states are conformally transformed thermofield doubles obtained via unitary actions on each boundary, and bulk–boundary checks are performed across stress tensors, two-point functions, and entanglement entropy, demonstrating precise ER=EPR realized in explicit, computable examples. A holographic, horizon-based non-equilibrium entropy is also defined, showing a divergenceless flow and confirming that total entropy is preserved under conformal transformations. The results provide a concrete, controllable laboratory for exploring wormholes, quantum entanglement, and non-equilibrium dynamics in AdS$_3$/CFT$_2$, with implications for purifications, quenches, and higher-spin generalizations.

Abstract

We present the complete family of solutions of 3D gravity (Lambda<0) with two asymptotically AdS exterior regions. The solutions are constructed from data at the two boundaries, which correspond to two independent and arbitrary stress tensors T_R, \bar T_R, and T_L, \bar T_L. The two exteriors are smoothly joined on to an interior region through a regular horizon. We find CFT duals of these geometries which are entangled states of two CFT's. We compute correlators between general operators at the two boundaries and find perfect agreement between CFT and bulk calculations. We calculate and match the CFT entanglement entropy (EE) with the holographic EE which involves geodesics passing through the wormhole. We also compute a holographic, non-equilibrium entropy for the CFT using properties of the regular horizon. The construction of the bulk solutions here uses an exact version of Brown-Henneaux type diffeomorphisms which are asymptotically nontrivial and transform the CFT states by two independent unitary operators on the two sides. Our solutions provide an infinite family of explicit examples of the ER=EPR relation of Maldacena and Susskind [arXiv:1306.0533].

Figures (5)

• Figure 1: The (green parts of) the five figures on the right depict the five coordinate charts used in this paper to cover the eternal BTZ solution.The coordinate chart K5 is needed to cover the "bifurcation surface" where the past and future horizons meet (it is a point in the Penrose diagram). The leftmost diagram (in blue) represents the coordinate chart used in (\ref{['banados']}). Each of the coordinate charts is shown, for facility of comparison, within a Penrose diagram where the parts not within the chart are shown in gray.
• Figure 2: This figure shows the IR cut-off (\ref{['lamt-ir']}) in the new geometries. The effect of the SGDs, in the old (un-tilded) coordinates, is to deform the IR cut-off surfaces. The surface deformation on the right exterior is given by the change from (\ref{['lam-ir']}) to (\ref{['lamt-lam-ir']}); there is a similar surface deformation on the left exterior.
• Figure 3: A schematic illustration of metrics in our paper related by trivial and nontrivial diffeomorphisms (see the definition \ref{['def-nontrivial']}). The metrics (\ref{['EF']}), (\ref{['EF2']}), (\ref{['EF3']}) and (\ref{['EF4']}), represented by the blue lines, define the eternal BTZ geometry; they are all related by trivial diffeomorphisms, which either do not extend to the boundaries or when they do, they become identity asymptotically. The metrics (\ref{['newmetric']}), (\ref{['newmetric-2']}), (\ref{['newmetric-3']}) and (\ref{['newmetric-4']}), represented by the green lines, define our new solution characterized by the functions $G_\pm, H_\pm$. These are also all related by trivial diffeomorphisms, which satisfy the same criteria as above. The two sets however represent physically different metrics since they are related to each other by nontrivial diffeomorphisms; for instance, (\ref{['EF']}) and (\ref{['newmetric']}) are related by a diffeomorphism, schematically represented by their separation, which does not vanish (become identity) asymptotically.
• Figure 4: The figure on the right shows the location of the horizon on the right in the ${\tilde{\lambda}}, {\tilde{v}}, {\tilde{w}}$ coordinates. The figure on the left shows the location of the horizon on the left in the $\tilde{\lambda}_1, {\tilde{u}}, {\tilde{\omega}}$ coordinates. These are described by (\ref{['warped-horizon']}). These surfaces are diffeomorphic to the undeformed horizon (\ref{['lam-horizon']}) depicted in Figure \ref{['fig-ef']}. Although the horizon has an undulating shape in our coordinate system, the expansion parameter, measured by the divergence of the area-form, vanishes (see Eq. (\ref{['no-divergence']})).
• Figure 5: Time evolution of HEE. The red-line represents the linear growth of HEE for a region consisting of spatial half-lines of both sides of a constant 2-sided BTZ geometry. The blue-line represents the HEE growth of the region consisting of half-lines of both sides of the SGD transformed geometry, for $G(\tilde{t})=\tilde{t}+\frac{1}{6}\cos(3\tilde{t})$ and $H_1(\tilde{t})=\tilde{t}+\frac{3}{5}\sin(\tilde{t})$. The undulating curve can be explained in terms of the quasiparticle picture of Calabrese:2005in; the entanglement entropy departs from its usual linear behaviour as the quasiparticle pairs locally go out and back in to the entangling region as the region is subjected to a conformal transformation.