What’s the Point?

Hole-ography in Poincaré AdS

Ricardo Espíndola, Alberto Güijosa, Alberto Landetta and Juan F. Pedraza

Departamento de Física de Altas Energías, Instituto de Ciencias Nucleares,

Universidad Nacional Autónoma de México,

Apartado Postal 70-543, CDMX 04510, México

Mathematical Sciences and STAG Research Centre,

University of Southampton,

Highfield, Southampton, SO17 1BJ, UK

Institute for Theoretical Physics,

University of Amsterdam,

Science Park 904, 1098 XH Amsterdam, Netherlands

, , ,

Abstract

In the context of the AdS/CFT correspondence, we study bulk reconstruction of the Poincaré wedge of AdS via hole-ography, i.e., in terms of differential entropy of the dual CFT. 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

Twenty years from the inception of the AdS/CFT correspondence [1, 2, 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 of each minimal-area codimension-two surface anchored on the boundary translates into the entanglement entropy of the spatial region in the boundary theory that is homologous to , via

(1) |

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 [6, 7]. It has been generalized beyond Einstein gravity in [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18]. Many other notable developments have taken place, including [19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33]. Useful reviews can be found in [34, 35, 36].

Another important step towards holographic reconstruction was taken in [37], working for simplicity in AdS, where the extremal codimension-two surfaces are just geodesics, and their ‘areas’ refer to their lengths. It was discovered in that context that one can reconstruct spacelike curves 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 of the curve can be computed in the CFT through the differential entropy , 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), .

Diverse aspects of hole-ography have been explored in [40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51]. The works [37, 45] carried out the hole-ographic reconstruction of an arbitrary closed curve at constant time in *global* AdS (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 ‘point-curve’ of vanishing length. This could then be combined with the family for a second point, to compute the distance between the two points. This framework is thus able to extract the most basic ingredients of the bulk geometry, points and distances, from the pattern of entanglement in the state of the boundary theory.

In this paper we are interested in understanding how this entire story plays out on *Poincaré* AdS, where hole-ography faces a serious challenge.
The pure AdS geometry with coordinates and metric (4) is dual to the vacuum state of a CFT on 2-dimensional Minkowski spacetime, with coordinates . Hole-ography in this context has been examined before, at constant time in [41] and for curves with non-trivial time-dependence in [44]. Our motivation here is different, and its essence can be understood by looking at Fig. 1, which shows the Poincaré patch as a wedge within global AdS. The fact that Poincaré does not cover all of AdS implies that some curves within the Poincaré wedge can have a set of tangent geodesics whose endpoints fall outside of the wedge. Such geodesics cannot be associated with entanglement entropy in the Minkowski CFT. Their existence presents a challenge to the hole-ographic reconstruction program, because it leaves us without the means to encode in CFT language what should definitely be properties of the vacuum state.

*cannot be reconstructed*in the Minkowski CFT using the standard hole-ographic procedure.

There is one conceptual issue we should clarify. Since the global and Poincaré descriptions are related by a simple coordinate transformation (see Eq. (2.1)), it might seem that the success of hole-ography in reproducing curves, points and distances in global coordinates should automatically extend to Poincaré. The proper length of the closed curve is certainly invariant under coordinate transformations, and naively the same would seem to be true for the entanglement entropy, which on the gravity side is also a proper length, according to the Ryu-Takayanagi prescription (1). Indeed, the unregulated entropy (taking the length of the geodesic all the way to the AdS boundary) is invariant, but it is also divergent, so it cannot be used directly to compute . And as soon as we introduce a cutoff, we introduce coordinate dependence. This is truly a property of regulated entanglement entropy on the field theory side: its value depends on the regularization scheme, so it is not invariant under conformal/Weyl transformations (see e.g. [52, 53, 54, 55, 56, 57, 58]), which is what the bulk transformation from global to Poincaré amounts to in the CFT. As a result, equations involving cannot always be carried over directly from one set of coordinates to the other, which explains why it is important to study Poincaré hole-ography directly. This is what we set out to do in this paper, working first at constant time in Section 2, and then at varying time in Section 3.

In more detail, we begin by asking how to reconstruct closed curves, as opposed to the curves examined in [41, 44], which were infinitely extended, with a periodicity condition at infinity. A salient difference between the global and Poincaré settings, closely related to the geodesic incompleteness described two paragraphs above, is that in global AdS the boundary wraps all the way around the bulk. Given a closed curve, it is then easy to visualize how the sought-after family of CFT intervals will lead to geodesics that are tangent to each point on the curve.
In Poincaré, given a closed curve, the boundary does not wrap around it, so naively we would seem to be missing the intervals/geodesics that would be tangent to the portion of the curve that is farther away from the boundary. As explained above and seen in Fig. 1, this is allowed by the fact that Poincaré coordinates cover only a wedge of global AdS. But we know that the slice at Poincaré time completely coincides with the slice at global time , so at least in this case, there is no possibility for geodesics to be left out. In Section 2.1, our strategy will thus be to take the results of [45] for curves at and simply perform the required change of coordinates, to obtain the corresponding Poincaré description. Our conclusion is that arbitrary closed curves at can indeed be reconstructed, but with an important novelty: the dual family of intervals must run over the axis at least *twice*, for it is only on the second (or subsequent) pass(es) that we describe geodesics tangent to the more distant portion of the curve. Once we know how to do this at , invariance of the metric (4) under translations in will of course allow us to reconstruct curves and points on any other fixed- slice, independently of the value of .
(Translations in , on the other hand, will give us examples of curves at variable , which we examine in Section 3.)

In Section 2.2 we show that the differential entropy gives the correct length for a generic closed curve at constant time in Poincaré AdS: just like in the global case examined in [45], we find that . In this particular instance, then, no subtlety arises from the coordinate transformation. A subtlety does arise, however, when we analyze in Section 2.3 the hole-ographic description of open curves. It was found in [45] that in order to match the length of an open curve in global AdS, the differential entropy must be supplemented with a specific boundary function , given in (22). We find that the same is true in Poincaré, but the relevant boundary function, Eq. (21), is not the direct translation of its global counterpart. Nonetheless, it does continue to be true that can be described geometrically in the bulk, and has a specific interpretation in terms of entanglement entropy in the boundary theory. This is crucial in order for open curves to be reconstructed purely with CFT data. We combine with to define a ‘renormalized’ differential entropy , which directly matches the length of an arbitrary open curve, . A simple expression for in terms of boundary data is given in (30).

In Section 2.4 we shrink curves down to zero size to obtain the hole-ographic description of bulk points. We find that this can be done either with closed or open curves, but in the latter case we must take the slope to diverge at the endpoints of the curve, in order to still be left with a non-trivial collection of geodesics in the point limit. Following [45], we show that the families of CFT intervals that happen to be associated with points instead of finite-size curves can be obtained by extremizing an action based on extrinsic curvature, which in terms of field theory variables takes the form (43). We then verify in Section 2.5 that the distance between two arbitrary points can also be obtained from differential entropy. This can in fact be done in two different ways: using Eq. (64), which is essentially the same recipe as in [45], or Eq. (57), which is a generalization based on describing the points as open curves.

Moving on to the covariant case, in Section 3.1 we present, following [44], the basic formulas (3.1)-(70) that define the intervals and geodesics associated to an arbitrary (open or closed) spacelike bulk curve, whether or not it varies in time. The corresponding differential entropy plus boundary function is written down in (3.1), and contact is successfully made with the length of the bulk curve.

The main issue of the paper is then encountered in Section 3.2, where we show that any segment of a curve that violates condition (84) is nonreconstructible, in the sense that the geodesics tangent to it have at least one endpoint outside of the Poincaré wedge, and are consequently not associated to entanglement entropies in the CFT. Examples are given in Figs. 8-10. In Section 3.3 we discover that this challenge can be overcome by making use of a variant of hole-ography formulated previously in [44], where one is allowed to shoot from each point on the bulk curve a geodesic aimed in a direction that differs from the tangent by a null vector satisfying (87). We thus arrive at the central result of this paper: the statement that, contrary to appearances, hole-ography can successfully reconstruct any open or closed spacelike curve within Poincaré-AdS, in terms of differential entropy in the CFT on Minkowski spacetime.

In Section 3.4 we study again the limit where the size of the curve vanishes, emphasizing that there are infinitely many different ways to represent any given point in terms of a family of CFT intervals. As expressed in Eq. (96) and exemplified in Fig. 11, there is one family for each distinct choice of the path traced by the center of the intervals (or equivalently, the path traced by either one of the intervals’ endpoints). Generalizing the results of Section 2.4, we work out a covariant action whose extremization leads to any one of these families associated to a point. On the gravity side it is based on the normal curvature of the bulk curve, and in CFT variables it takes the form (3.4). In the final part of the paper, Section 3.5, we show that given two bulk points, the freedom to choose a representative family from the equivalence class associated to each point allows us to easily compute the distance between the pair imitating the constant-time procedure of Section 2.5.

In Appendix A we go back to the discrete versions (110)-(111) of differential entropy originally considered in [37, 41], to show that in the continuum limit they give rise to definitions that differ by a boundary term. This difference is negligible for the types of curves considered in [41, 44], but is important for our analysis of open curves in Sections 2.3 and 3.1. The definition (15) of differential entropy that we use in this paper arises directly from a discrete version that differs from (110) and (111), and belongs to the one-parameter family of alternative definitions given in (124).

There are various directions for future work. Along the lines of [41, 43, 44], we expect our results to extend to Poincaré AdS in higher dimensions, under the same assumptions of symmetry for the surfaces under consideration. On a different front, Poincaré AdS is a particular example of an entanglement wedge [63, 64, 65], with the special feature that it includes a complete global time slice, and therefore a full set of initial data for temporal evolution. A smaller entanglement wedge leaves some information out, and contains fewer complete geodesics, so it is interesting to ask whether or not it is possible again to reorient those geodesics that exit it to achieve complete hole-ographic reconstruction of any curve within the wedge. We will address this question in a separate paper [66].

Going beyond pure AdS, hole-ography is known to be restricted by the appearance of entanglement shadows [40, 47] and holographic screens [50]. It may be possible to circumvent the former obstacle using entwinement, a type of entanglement between degrees of freedom in the CFT that are not spatially organized [67, 45, 68, 69]. At least in the case where the gravitational description is three-dimensional, entwinement is associated with non-minimal geodesics, and it would be interesting to investigate whether the possibility of reorienting them by null vectors [44] affords hole-ography any additional coverage. Finally, the reconstruction program has focused recently on the description of local bulk operators that are integrated over extremal surfaces, which have been shown to be dual to blocks in the CFT operator product expansion [70, 71, 72, 73, 74, 75, 76, 77, 78]. A somewhat different approach to local operators has been pursued in [79, 80, 81, 82, 83, 84]. One would naturally like to understand in detail how hole-ography is related to these two approaches.

## 2 Hole-ography at constant Poincaré Time

### 2.1 Closed curves

To fix our notation, recall that the metric of global AdS can be written in different ways:

(2) | |||||

where is the AdS radius of curvature, , and the three different choices of radial coordinate are related through . With , and , the set covers the entire anti-de Sitter spacetime. The AdS boundary is at (). A gravitational theory on (2) is dual to a two-dimensional CFT defined on the boundary cylinder , parametrized by .

Defining

(3) | |||||

we bring the metric to Poincaré form,

(4) |

As is well-known, with and , these coordinates cover only the Poincaré wedge of AdS, i.e., the portion , of global AdS (see Fig. 1). Physically, these coordinates are associated with a family of bulk observers with constant proper acceleration , and the Poincaré horizon at () marks the boundary of the region with which they can interact causally. The AdS boundary is at . The dual CFT lives on the boundary Minkowski spacetime parametrized by .

Given a curve at fixed (fixed ) in global AdS, the associated family of tangent geodesics, or equivalently, CFT intervals, can be labeled as , where is the angular location of the interval’s center along the spatial , and is the interval’s (half-)angle of aperture. These are given by [45]

(5) | |||||

The endpoints of these geodesics/intervals are located at .

As explained in the Introduction, if we stick to the slice to begin with, we are assured that these same geodesics will provide full coverage of the bulk curve after translation to the Poincaré slice . We can determine them by using (2.1) to map the two angles to the -axis. The endpoint locations corresponding to will naturally be denoted . Halfway between these two endpoints lies the center of the interval,

(6) |

and its radius is

(7) |

We will let denote the direct translation of the center angle , which will serve then as a parameter that labels our intervals. As goes around the of the cylinder CFT, will run over the entire spatial axis of the Minkowski CFT. Notice that in general we expect .

Our one-parameter family of geodesics was parametrized with in the global setting, so after translation to Poincaré, we can naturally parametrize it with . The geodesic for each value of can be described with the pair , or equivalently, with . The latter description is sometimes more convenient. And instead of reporting our geodesics in parametrized form, , we can eliminate to obtain , which is certainly more intuitive, and directly analogous to the global expression reported in [45] in the form .

It will be instructive to consider first the simplest concrete example of a bulk curve: a circle which in global coordinates is centered at the origin, . It follows immediately from (5) that the family of geodesics tangent to this circle is simply , . Using (2.1) at , we can see that the resulting bulk curve in Poincaré AdS is also a circle,

(8) |

With the middle equation in (2.1) evaluated at , we can also translate the geodesic parameters . The result takes the form

(9) |

A representative sampling of these geodesics is plotted in Fig. 2.

As expected, we do find a tangent geodesic for each point on our circle. But there is an important novelty: the denominator in (9) vanishes at , with

(10) |

At each of these locations, one of the endpoints changes sign. For we have the expected ordering , but for other values of the endpoints are exchanged: as increases past , the value of crosses from to , while at , crosses in the opposite direction. The fact that the interval radius (13) diverges at these crossover points implies that the corresponding geodesic is becoming vertical, and the same is true then for the bulk curve itself, i.e., . At these points, starts to backtrack, as we pass from the lower to the upper half of the circle, or viceversa. This behavior is seen in Fig. 3, where we plot the endpoints as given by (9).

The main lesson here is that, as the parameter ranges from to , the interval midpoint covers this same range *twice*: once for the geodesics tangent to the lower part of our curve, which have , and a second time for the geodesics tangent to the upper part, which have on account of having their endpoints reversed.

This same lesson applies generally. Consider an arbitrary closed bulk curve (at constant ), described as , with some unspecified parameter. Since the curve is closed, the function must be non-monotonic, and we can find at least two values of where changes sign by crossing zero. At these points, the bulk curve becomes vertical, and the radius and one of the endpoints of the corresponding geodesic approach . The same would happen at points where vanishes without changing sign. The points where the closed curve is vertical () split the curve into consecutive segments. Some examples are shown in Fig. 4. We will demand, without loss of generality, that the sign of the parameter be chosen such that the point on the curve that is closest to the AdS boundary is on a segment where . We label this segment , and number the remaining segments consecutively in order of increasing . The edges of the th segment are naturally denoted . As in the case of the circle (where we had ), each connected segment will be associated with a family of geodesics whose centers run over the entire -axis. The sign of might or might not flip when moving from one segment to the next. We will refer to those segments where () as ‘positive’ (‘negative’).

Either by translating the global AdS results of [45], or by direct computation in Poincaré [44], one finds that the geodesics tangent to our curve have endpoints at

(11) |

Equivalently, they have midpoint

(12) |

and radius

(13) |

We have chosen the sign of the second denominator in (11) such that the positive segments of the curve () are associated with intervals whose endpoints are in the canonical order, , while the negative parts () correspond to intervals with reversed endpoints, . Through (13), this means that the designation as positive or negative, originally referring to an attribute of the bulk curve, also characterizes the sign of for the corresponding family of intervals in the CFT. Again, the main issue here is that, to fully wrap around our closed curve, we need not one but families of intervals whose midpoints run over the entire -axis.

Alternatively, we can think about this as decomposing the closed curve into *open* curves (), which join together at the places where the slope diverges. But if we adopt this perspective, the nontrivial question is whether the information from all families of geodesics can be smoothly combined to obtain a hole-ographic description for the entire closed curve, since we know from [45] that to obtain the length of open curves we need to add a surface term to the formula for differential entropy. We will address this question explicitly in Section 2.3.

Notice that (12) implies that . This shows that the center can backtrack if , which happens on the negative segments that we have discussed here, or if the curve is sufficiently concave, . The latter possibility had been pointed out in [41, 44, 45].

For use below, we note that (12) and (13) can be inverted [44] to give the bulk curve in terms of the boundary data,

(14) |

As an additional check, these same relations can be obtained by taking the zero-mass limit of the expressions worked out for the static BTZ black hole [62] in Eqs. (89)-(90) of [45]. In this limit, the BTZ metric reduces to Poincaré AdS with .

### 2.2 Differential entropy and the length of closed curves

The definition of differential entropy is most conveniently given in the form [44]

(15) |

Here and are the left and right endpoints^{1}^{1}1Notice that and only if . We will return to this point below. () of a family of intervals parametrized by an arbitrary parameter , and denotes the corresponding entanglement entropy. The definition (15) treats the right and left endpoints on a different footing, but as explained in [44], an integration by parts allows the role of and to be exchanged. The two alternative definitions differ by a boundary term, which can be neglected for the types of curves considered in [44], but will be important for our analysis of open curves in Section 2.3. In Appendix A we show that there is in fact a one-parameter family of possible definitions of differential entropy, arising from a corresponding ambiguity (124) in the discrete version of originally considered in [37, 41].

For convenience, from this point on we will rescale the entanglement entropy by a factor of , so that the Ryu-Takayanagi formula (1) reads . In terms of the central charge of the CFT, reporting in units of is the same as reporting it in units of [59]. For intervals at fixed time, as we are considering here, the entropy in the Minkowski space CFT is given by (see, e.g., [60, 41])

(16) |

where is a UV cutoff.^{2}^{2}2For comparison, in the case of global AdS, where the dual CFT lives on a cylinder, the entanglement entropy is (see, e.g., [45])

In the context of holographic entanglement, the authors of [41] were the first to study curves in Poincaré AdS at constant time. (Their analysis applies as well to codimension-2 surfaces in Poincaré AdS with planar symmetry, i.e., translationally-invariant under of the .) They restricted attention to curves that are infinitely extended along the direction, and moreover imposed periodic boundary conditions at . Under these conditions, they showed that the differential entropy (15) for the family of intervals tangent to the curve (surface) correctly reproduces its length (area).

We will now show that the same is true for the closed curves that we considered in the previous subsection. Their length is given by

(17) |

where is the induced metric. We want to check that this agrees with the differential entropy associated to the curve. The corresponding geodesics/intervals have endpoints located at (11). For ease of reading, we will phrase our discussion for the case (the closed curve has only one positive and one negative segment), but the extension to is immediate.

For the positive part of the curve (), the fact that means that the left and right endpoints are and . Using (16), (15) becomes

(18) |

For the negative part, and so the endpoints are reversed, and . Because of this, if we were to use (15) as it stands, we would get some additional minus signs, and would not be able to directly obtain the total length of the curve. But, for continuity in the family of intervals (crucial for the usefulness of differential entropy, and most clearly seen by referring back to the global AdS setup), the correct prescription is to depart from a literal reading of (15), and keep treating as the *right* endpoint of the interval. This ensures the appropriate cancelation of the final geodesics in the positive family against the initial geodesics in the negative family. Of course, for the logarithm in (16) to be real, we do need to use as its argument. We are then led again to (18), so this single expression applies for the entire closed curve. Periodicity then guarantees, just like it did for the infinite curves considered in [41], that surface terms can be ignored.

Let us now see explicitly the relation between differential entropy and length. Using (11) and (13), expression (18) can be easily seen to take the form

(19) | |||||

In the second line we have recognized that the first term precisely reproduces the length (17), while the others amount to a total derivative, and do not contribute. Inside the logarithm, we have chosen a particular value of the constant of integration, involving the UV cutoff . This choice will prove to be convenient in the next subsection.

### 2.3 Boundary terms and the length of open curves

We now move on to considering (still at ) an arbitrary *open* curve . It might or might not have points where , separating segments just like we discussed for closed curves (but now with ). The analysis in the preceding section directly establishes a relation between its length and the differential entropy for its associated family of intervals. This relation is again given by (19), with the sole difference that the integral now extends over a finite range,

(20) | |||||

In the second line we have given the name

(21) |

to the (now generally non-vanishing) boundary contribution.

Let us now try to gain some understanding on the form of (21). As we mentioned in the Introduction, the authors of [45] showed that, when considering an *open* curve in global coordinates, , with running from to , the differential entropy
does not directly reproduce the length . The two integrands differ by a total derivative. To obtain a match, one must add to a specific surface term , with

(22) |

where and are evaluated at the values corresponding to the bulk angle . (Alternatively, could be expressed as a function of the boundary angle .) Above their Eq. (12), the authors of [45] explain the geometric meaning of : it is the length of the arc of the geodesic that is contained in the angular wedge between and . Explicitly, this geodesic is described by

(23) |

and one can check that the length of its arc in the range of interest,

(24) |

indeed agrees with (22).

*A priori*, it is not obvious whether a similar interpretation can be given to the Poincaré boundary function (21), because, as we have explained before, entanglement entropy does not remain invariant when mapping from global to Poincaré AdS. In particular, the condition , which makes the global boundary function (22) vanish, does not translate into or .

Let us work this out for an arbitrary open curve . The geodesic tangent to the curve at the point labeled by is

(25) |

where the radius and the midpoint are given by (12) and (13), and are therefore held fixed for the present calculation. The length of the arc of this geodesic that runs from to is

(26) |

Notice that, in this last form, the length (26) looks rather analogous to the final version of (22), except for an overall minus sign which is due to the fact that in (20) we have chosen to define our with a sign opposite to that of [45]. Using (11) and the identity , this expression can be rewritten as

(27) |

which coincides with the second term of (21).

This agreement allows us to ascribe to the term (27) the entanglement interpretation developed for global AdS in Section 4.5 of [45]. The family of intervals/geodesics associated with our open curve ends at the bulk point . The final member of the family is centered at , and generally . We can add to the family the set of intervals whose center runs from to , with radii chosen such that the corresponding geodesics all go through , meaning that this addition does not enlarge our curve. (The added intervals belong to the family of the ‘point-curve’ , as will become clear in the next subsection.) After the addition, there is no longer any arc left for (26) to contribute, which means that the second term in (21), evaluated at , represents the extra differential entropy due to the added set of intervals. The same applies of course at the opposite endpoint of the the curve, .

Only the logarithm in (21) remains to be interpreted. But comparing with (16), we see that this term is half the entanglement entropy of the interval at (or ). We thus conclude that the entire formula (20) for the length of our open curve admits an interpretation based on entanglement in the CFT. In the bulk description, the interpretation is very simple: the boundary function (21) is the length of the arc of the corresponding geodesic, computed from the edge of our curve, at , all the way to the right endpoint of the geodesic, . Or, more precisely, to the regularized version of this endpoint,

(28) |

where the geodesic reaches the UV cutoff . This geometric interpretation is illustrated in Fig. 5.

For an open curve whose associated geodesics cover the entire -axis, such as the positive or negative semicircle that we analyzed in Section 2.1, (21) is logarithmically divergent (because both and diverge at the endpoints ). In that case, it is more convenient to reexpress as the integral over of a total -derivative, so that it can be subtracted directly from the *integrand* of in (15), to get a finite result. This takes us back to the total-derivative terms in the top line of (19), which can be rewritten in the the form

(29) | |||||

In the second line we have used (2.1) to express purely in terms of boundary data. Combining (29) with (18), we can define a ‘renormalized’ differential entropy

(30) |

From our previous analysis, this directly reproduces the length of an arbitrary open curve,

(31) |

As an example, consider the circle (8), shown in Fig. 2. In the language of Section 2.1, its lower half is a positive segment () and is labeled , whereas its upper half is a negative segment, denoted . These two semicircles are open curves described by

(32) |

with ranging between and . Their length is

(33) |

Notice that the length of the two semicircles is different, even though they do add up to the correct total, . This is due to the -dependence of the metric. Using (30), we find

(34) |

The reversal of sign for the negative semicircle is as expected from the convention adopted in the previous subsection and not implemented when writing down (33): for a negative segment, should run in the direction of decreasing . It is with this orientation that the full circle is traced by the original parameter or . Indeed, if we take this sign into account, we find that upon combining the two semicircles the contribution of the boundary function cancels, and we have

(35) |

### 2.4 Points

Now that we have a formula that computes the lengths of arbitrary closed or open bulk curves in terms of boundary entanglement entropies, we can shrink these curves as in [45], to obtain points. To describe a given point, we have two options. One is to start with a closed curve, which we will take for simplicity to have only one postive and one negative portion (e.g., a circle). Closed curves have the advantage of not needing any boundary terms, but require a family of intervals/geodesics covering the -axis at least twice. The other option is to start with an open (positive or negative) curve (e.g., a semicircle) whose slope diverges at the edges, so that (via (12) and (13)) the corresponding intervals/geodesics cover the entire -axis.^{3}^{3}3If we started instead with an open bulk curve whose slope is not divergent at the endpoints, then the range of covered by the corresponding CFT intervals would be finite, and when we shrink the size of the curve we would end up with nothing. In this case we do not have to deal with the double-valuedness of , but the price we pay is that we must include the boundary contribution (21).

Either way, upon shrinking the size of the curve all the way down to zero, we obtain the family of intervals (equivalently, ) whose associated geodesics all pass through the desired bulk point . These intervals can of course be determined directly from (25),

(36) |

where the family with the upper (lower) sign is needed to describe a positive (negative) open ‘point-curve’, and both families are needed to assemble a closed point-curve. If we wanted to, we could by convention always pick the positive branch of the square root in (36), which would amount to changing our notation to always insist on having . But when putting together the positive and negative segments to construct a closed point-curve, we would still need to use the appropriate signs, as discussed in the previous two subsections. Eq. (36) can be rewritten in terms of the intervals’ endpoints as

(37) |

It is interesting to ask what the special property is that allows the particular set of CFT intervals in (36) to be identified as describing a bulk point in AdS. This is important if we are attempting to reconstruct the bulk starting just from the boundary theory. By taking the first and second derivative of (36), we can see that our point-curves are solutions to the equation of motion

(38) |

This is then the analog of Eq. (21) in [45]. As explained there, it is natural to obtain a second-order differential equation, since there must be two integration constants, associated with the coordinates of the bulk point, . Incidentally, we might wonder why, to single out a point, we are prescribing here an infinite family of geodesics that pass through it, when it should suffice to specify just *two* such geodesics to locate the point where they intersect. Indeed, given only two intersecting geodesics (equivalently, two overlapping intervals in the CFT), we know the radii at the given midpoints, and , and these two data pick out a unique solution to (38), i.e., a unique family that covers the entire spatial axis and includes both of the geodesics that we started with. What we gain by thinking of the entire family instead of the original pair is that we can analyze the point-curve in parallel with any other bulk curve, and in particular verify that it has vanishing length by computing its differential entropy.

Following [45], we expect the equation of motion (38) to follow from an action principle based on extremizing the extrinsic curvature of closed curves. The idea is the following: in negatively curved spaces, the Gauss-Bonnet theorem states that

(39) |

for any closed curve such that , where is the length element along the curve, is the extrinsic curvature and is the Ricci scalar on the surface bounded by the loop. Evidently, if the loop shrinks to a point the second integral vanishes, and the inequality is saturated. Thus, we can find bulk points by extremizing the left-hand side of (39).

The extrinsic curvature is computed from

(40) |

where is a normal unit vector and is the induced metric on the curve. The scalar extrinsic curvature is computed by contracting with .

For an arbitrary (time-independent) closed curve, our proposed action , with Lagrangian , is found to take the form

(41) |

As we can see, this action contains second-order derivatives. Nonetheless, the Euler-Lagrange equations,

(42) |

simplify drastically, leading to and , respectively. The solution defines the bulk point , which serves as a consistency check of the functional (41).

In terms of boundary data, we can rewrite (41) as

(43) |

In the second form, the Lagrangian is independent of , so there is an associated conserved momentum,

(44) |

Solving for ,

(45) |

and plugging it back into (43) we obtain

(46) |

The equation for derived from (46) is trivially satisfied, so we can focus on (45) only. We can get rid of by writing (45) as

(47) |

which has solution

(48) |

If we identify the integration constants as and we recover equation (36), as expected. Consistent with this, if in (43) we choose and then extremize, we indeed recover the equation of motion (38).

### 2.5 Distances

We will now study how to compute the distance between two bulk points and , in terms of differential entropy. Let have coordinates , and similarly for . In this subsection we choose , and therefore denote the families of intervals in the CFT dual to our bulk points by and . For concreteness, we will take to be to the right of , . The geodesic that connects the two points, which we will denote , is centered at the point on the boundary that is ‘equidistant’ from and , in the sense that . The setup is illustrated in Fig. 6. Explicitly,

(49) |

and the radius of is

(50) |

The distance between and is given by the arclength along this geodesic. Using (25), this can be written as

(51) | |||||

where is the parametrization of , and refer to the left/right endpoints (at the AdS boundary) of the geodesic. Equation (26), which we used in our analysis of the boundary function , is a special case of (51), with and .

Expression (51) is what we want to reproduce using differential entropy. Notice that this formula computes the *signed* length between the two points, and satisfies .
We also note in passing that from (37) we know that

(52) |

for any point on the geodesic centered at , and using this we can rewrite the distance between and in the simplified form

(53) |

In defining and , if we regard each point as a vanishingly small *open* curve, then we pick only one sign in (36), and runs over the real axis once. Lengths in that case are determined using the ‘renormalized’ differential entropy (30), which includes the contribution of the boundary function (21). From (20), we know that for an arbitrary open curve ,
which implies that for open point-curves (which have ). Additionally, in the paragraph above (28) we learned that the boundary function has a simple geometric interpretation: as seen in Fig. 5, it is the distance between the edge of the curve which that geodesic is tangent to, and the regularized right endpoint of the geodesic centered at , .

Given these results, a strategy naturally suggests itself. To be able to extract information about the geodesic , we should compute the differential entropy not for the complete family , but for a truncation of it to the range , so that the final interval in the family is precisely the one associated with . We will denote the truncated version by , which we take to vanish for . The corresponding differential entropy will be denoted with the same symbol, . From what explained in the previous paragraph, we know that

(54) |

where refers to the regularized right endpoint, located at . To obtain the distance , we can combine this with a version of that is truncated to the complementary range , so that is now associated with the initial interval of the family. We will denote this truncation by . (In this notation, , and likewise for .) The corresponding differential entropy is

(55) |

Defining the combined family

(56) |

we find that its differential entropy is

(57) |

This serves as a formula for the desired distance between the two points in terms of entanglement entropy, save for the uncomfortable fact that the two remaining terms (which can also be expressed as distances) are both divergent: and