Aspects of AdS$_2$ Quantum Gravity and the Karch-Randall Braneworld

TL;DR

This paper analyzes quantum gravity in ($2$-dimensional) anti-de Sitter space using wedge holography with two Karch-Randall branes in an $AdS_3$ bulk. It shows that rigid branes yield a topological $2$-d Einstein–Hilbert theory, while brane fluctuations generate a dilaton gravity sector in which Jackiw–Teitelboim gravity emerges in special limits; these theories are proposed to have holographic duals in the form of one-dimensional quantum mechanics or random matrix theories. The authors compute holographic entanglement entropies for the defect system, reveal an infinite degeneracy of RT surfaces tied to conformal symmetry, and demonstrate how breaking this symmetry lifts the degeneracy and yields nontrivial dilaton profiles. They also connect the low-energy dynamics to a JT-like action minimally coupled to a massive scalar, discuss orbifold and tensionless special cases, and address an energy-spectrum puzzle by combining Schwarzian and topological sectors, suggesting a rich holographic structure for AdS$_2$ quantum gravity within wedge holography.

Abstract

In this paper, we use the Karch-Randall braneworld to study theories of quantum gravity in two dimensional (nearly) anti-de Sitter space (AdS$_2$). We focus on effective gravitational theories in the setup with two Karch-Randall branes embedded in an asymptotically AdS$_3$ bulk forming a wedge. We find the appearance of two-dimensional Einstein-Hilbert gravity (or the Lorenzian version of Marolf-Maxfield theory) when the branes are rigid but the emergence of a class of dilaton gravity models parameterized by the tensions of the two branes when brane fluctuations are accounted for. A special case of our result is Jackiw-Teitelboim (JT) gravity, which has been proven useful to address many important problems in quantum gravity. An important implication of our work is that these models have holographic duals as one-dimensional quantum mechanics systems. At the end, we discuss a puzzle regarding the energy spectrum and its resolution.

Figures (11)

• Figure 1: An illustration of wedge holography: the bulk geometry is a part of empty AdS$_{d+1}$ between two end-of-world Karch-Randall branes (black lines), meeting each other on the conformal boundary of AdS$_{d+1}$ at their common boundary (the red dot). The bulk shaded regions behind the branes are removed, leaving only a wedge in the bulk. Wedge holography states that the gravitational physics in the bulk wedge is dual to a conformal field theory living on the red dot (the defect).
• Figure 2: A constant-$t$ slice of AdS$_{d+1}$ foliated by global AdS$_d$ slices with two branes present. The defect is shown in red, and various candidate extremal surfaces (in order of decreasing area from left to right) are in green. For each surface, $\mathcal{R}$ and $\mathcal{I}$ are respectively the radiation region and island, which end orthogonally on both branes. The minimal, zero-area entanglement surface $r(\mu) = 0$ is shown on the right as a limit, cutting through the middle of the space.
• Figure 3: Embedding of a KR braneworld with two positive tension branes in the black string geometry. The dashed black curve is the black string horizon separating the exterior and interior regions. The dashed green curve connecting the two branes is a putative RT surface.
• Figure 4: The empty AdS$_{3}$ wedge consisting of two KR branes. Here we show both the global picture and the Poincaré patch (which is one-point compactified to the global picture). While the defect in the Poincaré patch appears to be a point (which only has the time coordinate).
• Figure 5: A typical brane configuration on a constant time $t=t_{0}$ slice on different sides of a bulk maximally extended BTZ black hole Equ. (\ref{['eq:sblackh']}). The red and dark blue dots are the defects on each side and they are in a thermofield double state. The blue lines are the conformal boundary which goes from $0$ to $2\pi$ in $\theta$, the black curves are the two Karch-Randall branes, the dashed black horizontal lines are the black hole horizon $r=r_h$ and the shaded regions are removed (we assume brane tension positive). The whole geometry should be identified along the two vertical black dashed lines on each diagram. We neglect the parts of the branes behind the black hole horizon which are easily seen to be timelike.