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Gravity/ensemble duality

Bousso, Raphael, Wildenhain, Elizabeth·2020·Physical Review D
DOI 10.1103/physrevd.102.066005
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Image 1
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FIG. 2 Examples of holographic duality. Left: the solid bulk Md is dual to a holographic CFT d -1 on Md -1 (blue boundary). Right: in this example, Md -1 is a manifold with boundary, so the boundary theory is a BCFT d -1 and Md contains an end of the world brane EOW d . (Despite the appearance of a BCFT this is a ' singly holographic ' example. In Secs. IV and V we will consider a doubly holographic setting where the EOW d is a braneworld that localizes gravity and contains a holographic CFT d .)
FIG. 2 Examples of holographic duality. Left: the solid bulk Md is dual to a holographic CFT d -1 on Md -1 (blue boundary). Right: in this example, Md -1 is a manifold with boundary, so the boundary theory is a BCFT d -1 and Md contains an end of the world brane EOW d . (Despite the appearance of a BCFT this is a ' singly holographic ' example. In Secs. IV and V we will consider a doubly holographic setting where the EOW d is a braneworld that localizes gravity and contains a holographic CFT d .)CC-BY · From the PDF
FIG. 3 RT prescription, applied in the setting shown on the right of Fig. 2. The entropy of the boundary region Rd -1 is given by the generalized entropy of its entanglement wedge EW ð Rd -1 Þ . γ d is the quantum extremal surface.
FIG. 3 RT prescription, applied in the setting shown on the right of Fig. 2. The entropy of the boundary region Rd -1 is given by the generalized entropy of its entanglement wedge EW ð Rd -1 Þ . γ d is the quantum extremal surface.CC-BY · From the PDF
Image 4
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Image 5
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FIG. 6 Here we assume the gravity/ensemble interpretation, in the examples studied in Sec. II C (left) and Sec. II D (right). This resolves the state paradox. The RT prescription (yellow) computes the ensemble averaged entropy h S i of the full boundary (left) or of Rd -1 (right). The extrapolate dictionary (blue) yields the average state of the ensemble, h ρ i , in these regions.
FIG. 6 Here we assume the gravity/ensemble interpretation, in the examples studied in Sec. II C (left) and Sec. II D (right). This resolves the state paradox. The RT prescription (yellow) computes the ensemble averaged entropy h S i of the full boundary (left) or of Rd -1 (right). The extrapolate dictionary (blue) yields the average state of the ensemble, h ρ i , in these regions.CC-BY · From the PDF
FIG. 7 Hawking radiation escapes into an auxiliary system without gravity. The RT prescription can be applied to the boundary Md -1 , yielding the dark green entanglement wedge. A version of the RT prescription for AUX can be developed by requiring consistency with the analysis in Sec. II D. One finds that EW(AUX) (light green) includes AUX itself, and after the Page time, it also the bulk region I complementary to EW ð Md -1 Þ . The state paradox arises in AUX: the entropy must follow Hawking ' s rising curve for the island I to be part of EW(AUX), but with I included, RT yields the Page curve for AUX.
FIG. 7 Hawking radiation escapes into an auxiliary system without gravity. The RT prescription can be applied to the boundary Md -1 , yielding the dark green entanglement wedge. A version of the RT prescription for AUX can be developed by requiring consistency with the analysis in Sec. II D. One finds that EW(AUX) (light green) includes AUX itself, and after the Page time, it also the bulk region I complementary to EW ð Md -1 Þ . The state paradox arises in AUX: the entropy must follow Hawking ' s rising curve for the island I to be part of EW(AUX), but with I included, RT yields the Page curve for AUX.CC-BY · From the PDF
FIG. 8 Double holography without a bath. Md (purple surface) is the bulk dual of a holographic CFT d -1 (left) or BCFT d -1 (right) on Md -1 (dark green rim). So far this is identical to (a time slice of) the setups shown in Fig. 2. But we now assume that Md contains a holographic CFT d . This gives rise to a doubly-holographic bulk dual Md þ 1 (the solid interior). From the d þ 1 bulk perspective, Md is a Karch-Randall braneworld.
FIG. 8 Double holography without a bath. Md (purple surface) is the bulk dual of a holographic CFT d -1 (left) or BCFT d -1 (right) on Md -1 (dark green rim). So far this is identical to (a time slice of) the setups shown in Fig. 2. But we now assume that Md contains a holographic CFT d . This gives rise to a doubly-holographic bulk dual Md þ 1 (the solid interior). From the d þ 1 bulk perspective, Md is a Karch-Randall braneworld.CC-BY · From the PDF
FIG. 9 Examples of the doubly holographic entanglement wedge EW 2 ð Rd -1 Þ for a ð B Þ CFT d -1 region Rd -1 . As before, the light purple surface Md is the bulk dual of a holographic CFT d -1 (left) or BCFT d -1 (middle, right) on Md -1 (dark green rim). In each case, the doubly holographic entanglement wedge is bounded in part by the surface γ d þ 1 , shown in dark purple.
FIG. 9 Examples of the doubly holographic entanglement wedge EW 2 ð Rd -1 Þ for a ð B Þ CFT d -1 region Rd -1 . As before, the light purple surface Md is the bulk dual of a holographic CFT d -1 (left) or BCFT d -1 (middle, right) on Md -1 (dark green rim). In each case, the doubly holographic entanglement wedge is bounded in part by the surface γ d þ 1 , shown in dark purple.CC-BY · From the PDF
FIG. 10 A doubly holographic CFT d -1 on Md -1 is coupled to holographic bath: a CFT d on M d . The first holographic dual is Md ∪ M d , where Md contains the same CFT d coupled to gravity. The second holographic dual is Md þ 1 (solid interior). We consider a state which, in the first dual, corresponds to an evaporating black hole in Md with radiation escaping to M d . The von Neumann entropy of the radiation in the subregion R d ⊂ M d can be computed using the single or double RT prescription.
FIG. 10 A doubly holographic CFT d -1 on Md -1 is coupled to holographic bath: a CFT d on M d . The first holographic dual is Md ∪ M d , where Md contains the same CFT d coupled to gravity. The second holographic dual is Md þ 1 (solid interior). We consider a state which, in the first dual, corresponds to an evaporating black hole in Md with radiation escaping to M d . The von Neumann entropy of the radiation in the subregion R d ⊂ M d can be computed using the single or double RT prescription.CC-BY · From the PDF
FIG. 11 Entanglement wedges of the bath region R d before (left) and after (right) the Page time, when EW ð R d Þ has a disconnected island I . Each top figure is simply the bottom figure rotated around the axis. The ' squared ' entanglement wedge EW 2 is always connected. It can be found iteratively as EW ð EW ð R d ÞÞ , or in one step from Eq. (5.8) [42,43]. As in Sec. V C, γ d is a quantum extremal surface, but γ d þ 1 is an ordinary extremal surface.
FIG. 11 Entanglement wedges of the bath region R d before (left) and after (right) the Page time, when EW ð R d Þ has a disconnected island I . Each top figure is simply the bottom figure rotated around the axis. The ' squared ' entanglement wedge EW 2 is always connected. It can be found iteratively as EW ð EW ð R d ÞÞ , or in one step from Eq. (5.8) [42,43]. As in Sec. V C, γ d is a quantum extremal surface, but γ d þ 1 is an ordinary extremal surface.CC-BY · From the PDF
FIG. 12 Bulk dual of the state paradox. Left: we regard R d as a BCFT d subregion (top level). The homology rule following Eq. (5.8) applies: γ d þ 1 is allowed to end on the braneworld which here appears as an EOW brane. At late times, A ð γ d þ 1 Þ → 0 , resulting in the Page curve for S ð R d Þ . Right: we consider R d as a subregion of the CFT d on Md ∪ M d . The homology rule following Eq. (4.10) applies. The braneworld Md is now part of the boundary; since we are computing the entropy only for the region R d , γ 0 d þ 1 is not allowed to end on Md . A ð γ 0 d þ 1 Þ grows monotonically, resulting in Hawking ' s curve.
FIG. 12 Bulk dual of the state paradox. Left: we regard R d as a BCFT d subregion (top level). The homology rule following Eq. (5.8) applies: γ d þ 1 is allowed to end on the braneworld which here appears as an EOW brane. At late times, A ð γ d þ 1 Þ → 0 , resulting in the Page curve for S ð R d Þ . Right: we consider R d as a subregion of the CFT d on Md ∪ M d . The homology rule following Eq. (4.10) applies. The braneworld Md is now part of the boundary; since we are computing the entropy only for the region R d , γ 0 d þ 1 is not allowed to end on Md . A ð γ 0 d þ 1 Þ grows monotonically, resulting in Hawking ' s curve.CC-BY · From the PDF

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Metadata

Authors
Bousso, R., & Wildenhain, E.
Venue
Physical Review D
Publisher
American Physical Society (APS)
Published
2020-09-16
Volume
102
Issue
6
ISSN
  • 2470-0010
  • 2470-0029
DOI
10.1103/physrevd.102.066005
Cited by
101
References
19
Language
en
Licence
creativecommons.org/licenses/by/4.0/ (vor)
Funders
  • U.S. Department of Energy (DE-SC0019380, DE-AC02-05CH11231)
  • National Science Foundation (PHY1820912)
  • Berkeley Center for Theoretical Physics

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Bousso, R., & Wildenhain, E. (2020). Gravity/ensemble duality. Physical Review D. https://doi.org/10.1103/physrevd.102.066005