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Aspects of AdS2 quantum gravity and the Karch-Randall braneworld

Geng, Hao·2022·Journal of High Energy Physics
DOI 10.1007/jhep09(2022)024
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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 the theory considered by Marolf and Maxfield) 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.

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Figures

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 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).CC-BY · From the PDF
Figure 2 A constantt 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, R and I are respectively the radiation region and island, which end orthogonally on both branes. The minimal, zero-area entanglement surface r ( µ ) = 0 is shown on the right as a limit, cutting through the middle of the space.
Figure 2 A constantt 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, R and I are respectively the radiation region and island, which end orthogonally on both branes. The minimal, zero-area entanglement surface r ( µ ) = 0 is shown on the right as a limit, cutting through the middle of the space.CC-BY · From the PDF
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 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.CC-BY · From the PDF
Figure 4 The empty AdS 3 wedge consisting of two KR branes. Here we show both the global picture and the Poincar´ e patch (which is one-point compactified to the global picture). While the defect in the Poincar´ e patch appears to be a point (which only has the time coordinate).
Figure 4 The empty AdS 3 wedge consisting of two KR branes. Here we show both the global picture and the Poincar´ e patch (which is one-point compactified to the global picture). While the defect in the Poincar´ e patch appears to be a point (which only has the time coordinate).CC-BY · From the PDF
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. (3.12). 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 π in θ , 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.
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. (3.12). 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 π in θ , 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.CC-BY · From the PDF
Figure 6 The brane configuration in the ( x, z ) coordinate. The shaded regions is the removed region in the bulk. The two black curves are the two branes and they are part of circles centered at O 1 and O 2 . The dashed blue line is the image of the BTZ black hole horizon and we emphasize that at t = 0 there is no black hole interior in our maximally extended geometry.
Figure 6 The brane configuration in the ( x, z ) coordinate. The shaded regions is the removed region in the bulk. The two black curves are the two branes and they are part of circles centered at O 1 and O 2 . The dashed blue line is the image of the BTZ black hole horizon and we emphasize that at t = 0 there is no black hole interior in our maximally extended geometry.CC-BY · From the PDF
Figure 7 An array of AdS 3 wedges such that the adjacent branes for contiguous wedges are of equal tension. In other words, the orange branes and green branes are respectively of equal tension.
Figure 7 An array of AdS 3 wedges such that the adjacent branes for contiguous wedges are of equal tension. In other words, the orange branes and green branes are respectively of equal tension.CC-BY · From the PDF
Figure 8 Two wedges left over after modding out the translational symmetry. The Z 2 symmetry is the reflection with respect to the dashed vertical line.
Figure 8 Two wedges left over after modding out the translational symmetry. The Z 2 symmetry is the reflection with respect to the dashed vertical line.CC-BY · From the PDF
Figure 9 The empty AdS 3 wedge consisting of two KR branes. The bulk geometry is the AdS 3 global patch. The green curves are representatives of the degenerate RT surfaces calculating the entanglement entropy between the two defects (the blue dot and the red dot).
Figure 9 The empty AdS 3 wedge consisting of two KR branes. The bulk geometry is the AdS 3 global patch. The green curves are representatives of the degenerate RT surfaces calculating the entanglement entropy between the two defects (the blue dot and the red dot).CC-BY · From the PDF
Figure 10 The shape of the two branes are fluctuating. The orange curves are representatives of the bulk minimal area surfaces. The orange curve are of different length and we pick up the shortest one of them to calculate the entanglement entropy between the two defects (the blue dot and the red dot).
Figure 10 The shape of the two branes are fluctuating. The orange curves are representatives of the bulk minimal area surfaces. The orange curve are of different length and we pick up the shortest one of them to calculate the entanglement entropy between the two defects (the blue dot and the red dot).CC-BY · From the PDF
Figure 11 The shape of the two branes are fluctuating. The AdS 2 cutoff is induced by the AdS 3 cutoff (the red curve). The shaded regions are cutoff. Now the cutoff asymptotic boundary of the wedge are two intervals.
Figure 11 The shape of the two branes are fluctuating. The AdS 2 cutoff is induced by the AdS 3 cutoff (the red curve). The shaded regions are cutoff. Now the cutoff asymptotic boundary of the wedge are two intervals.CC-BY · From the PDF

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Metadata

Authors
Geng, H.
ORCID
Geng, Hao
Venue
Journal of High Energy Physics
Publisher
Springer Science and Business Media LLC
Published
2022-09-02
Volume
2022
Issue
9
ISSN
1029-8479
DOI
10.1007/jhep09(2022)024
Cited by
46
References
9
Language
en
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