What’s the point? Hole-ography in Poincaré AdS
Eur. Phys. J. C
What's the point? Hole-ography in Poincaré AdS
Ricardo Espíndola 1 2
Alberto Güijosa 2
Alberto Landetta 2
Juan F. Pedraza 0
0 Institute for Theoretical Physics, University of Amsterdam , Science Park 904, Amsterdam 1098 XH , The Netherlands
1 Mathematical Sciences and STAG Research Centre, University of Southampton , Highfield, Southampton SO17 1BJ , UK
2 Departamento de Física de Altas Energías, Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México , Apartado Postal 70-543, Mexico City 04510 , Mexico
In the context of the AdS/CFT correspondence, we study bulk reconstruction of the Poincaré wedge of AdS3 via hole-ography, i.e., in terms of differential entropy of the dual CFT2. Previous work had considered the reconstruction of closed or open spacelike curves in global AdS, and of infinitely extended spacelike curves in Poincaré AdS that are subject to a periodicity condition at infinity. Working first at constant time, we find that a closed curve in Poincaré is described in the CFT by a family of intervals that covers the spatial axis at least twice. We also show how to reconstruct open curves, points and distances, and obtain a CFT action whose extremization leads to bulk points. We then generalize all of these results to the case of curves that vary in time, and discover that generic curves have segments that cannot be reconstructed using the standard hole-ographic construction. This happens because, for the nonreconstructible segments, the tangent geodesics fail to be fully contained within the Poincaré wedge. We show that a previously discovered variant of the hole-ographic method allows us to overcome this challenge, by reorienting the geodesics touching the bulk curve to ensure that they all remain within the wedge. Our conclusion is that all spacelike curves in Poincaré AdS can be completely reconstructed with CFT data, and each curve has in fact an infinite number of representations within the CFT.
-
Contents
1 Introduction and summary . . . . . . . . . . . . . .
2 Hole-ography at constant Poincaré time . . . . . . .
2.1 Closed curves . . . . . . . . . . . . . . . . . .
1 Introduction and summary
Twenty years from the inception of the AdS/CFT
correspondence [1–3], research is still being carried out to understand
how it achieves its grandest miracle: the emergence of a
dynamical spacetime out of degrees of freedom living on
a lower-dimensional rigid background. Over ten years ago,
a crucial insight in this direction was provided by Ryu and
Takayanagi [4], who argued that areas in the bulk
gravitational description are encoded as quantum entanglement in
the boundary field theory. More specifically, they proposed
that when the dynamics of spacetime is controlled by
Einstein gravity, the area A of each minimal-area
codimensiontwo surface anchored on the boundary translates into the
entanglement entropy S of the spatial region in the boundary
theory that is homologous to , via
A
S = 4G N .
Their proposal, originally conjectural and referring only to
static situations, was extended to the covariant setting in [5]
by taking to be an extremal surface, and later proved in
(
1
)
[6,7]. It has been generalized beyond Einstein gravity in
[8–18]. Many other notable developments have taken place,
including [19–33]. Useful reviews can be found in [34–36].
Another important step towards holographic
reconstruction was taken in [37], working for simplicity in AdS3,
where the extremal codimension-two surfaces are just
geodesics, and their ‘areas’ A refer to their lengths. It was
discovered in that context that one can reconstruct
spacelike curves C that are not extremal and are not anchored
on the boundary, by cleverly adding and subtracting the
geodesics tangent to the bulk curve. This procedure was
initially phrased in terms of the hole in the bulk carved out by
the curve, and was therefore dubbed hole-ography. It entails
two related insights. The first is that any given spacelike
bulk curve can be represented by a specific family of
spacelike intervals in the boundary theory, whose endpoints
coincide with those of the geodesics tangent to the bulk curve
(in a manner that embodies the well-known UV/IR
connection [38,39]). The second is that the length A ≡ AC of
the curve can be computed in the CFT through the
differential entropy E , a particular combination of the
entanglement entropies of the corresponding intervals, whose precise
definition is given below, in Eq. (
15
). The concrete relation
between these two quantities takes the form inherited from
(
1
), E = A/4G N .
Diverse aspects of hole-ography have been explored in
[40–51]. The works [37,45] carried out the hole-ographic
reconstruction of an arbitrary closed curve at constant time
in global AdS3 (and also on the BTZ black hole and on the
conical defect geometry). Upon shrinking a closed curve to
zero size at an arbitrary point in the bulk, a family of
intervals was obtained [45] describing a ‘poi (...truncated)