Bulk metric reconstruction from boundary entanglement

Shubho R. Roy1,* and Debajyoti Sarkar2,†1Department of Physics Indian Institute of Technology,

Hyderabad Kandi, Sangareddy, Telengana 502285, India2Albert Einstein Center for Fundamental Physics, Institute for Theoretical Physics,

University of Bern, Bern 3012, Switzerland

(Received 16 April 2018; published 27 September 2018)

Most of the literature in the bulk reconstruction program in holography focuses on recovering localbulk operators propagating on a quasilocal bulk geometry and the knowledge of the bulk geometry isalways assumed or guessed. The fundamental problem of the bulk reconstruction program, which isrecovering the bulk background geometry (metric) from the boundary CFT state is still outstanding. Inthis work, we formulate a recipe to extract the bulk metric itself from the boundary state, specifically, themodular Hamiltonian information of spherical subregions in the boundary. Our recipe exploits the recentconstruction of Kabat and Lifschytz [J. High Energy Phys. 06 (2017) 120.] to first compute the bulk twopoint function of scalar fields directly in the CFT without knowledge of the bulk metric or the equationsof motion, and then to take a large scaling dimension limit (WKB) to extract the geodesic distancebetween two close points in the bulk i.e., the metric. As a proof of principle, we consider threedimensional bulk and selected CFT states such as the vacuum and the thermofield double states. Weshow that they indeed reproduce the pure AdS and the regions outside the Rindler wedge and the BTZblack hole up to a rigid conformal factor. Since our approach does not rely on symmetry properties ofthe CFT state, it can be applied to reconstruct asymptotically AdS geometries dual to arbitrary generalCFT states provided the modular Hamiltonian is available. We discuss several obvious extensions to thecase of higher spacetime dimensions as well as some future applications, in particular, for constructingmetric beyond the causal wedge of a boundary region. In the process, we also extend the construction ofKabat and Lifschytz [J. High Energy Phys. 06 (2017) 120.] to incorporate the first order perturbativelocality for AdS scalars.

DOI: 10.1103/PhysRevD.98.066017

I. INTRODUCTION

AdSdþ1=CFTd duality [1–3] is an exact equivalencebetween quantum gravity theories in asymptotically Antide Sitter (aAdS) spacetimes and a nongravitational con-formal field theory (CFT) living in the conformal boundaryof the aAdS spacetime. Thus for the first time, thisremarkable duality furnishes a UV-complete and fullynon-perturbative description of quantum gravity of anykind in terms of a quantum field theory. Naturally this hasopened the doors to understanding the effects of fullquantum gravity and various puzzles and paradoxes ofgeneral relativity (namely, black hole information paradox,

structure of black hole interiors and gravitational singu-larities etc.), in terms of well-defined and in many cases,controllable field theory calculations. Since the inceptionof the duality conjecture, efforts have been made touncover how in general a quasilocal bulk (spacetime andmatter) emerges from the underlying CFT state andoperator spectrum at large N. A precise question thatone can ask is how do local quantum fields in the bulki.e., semiclassical curved space QFT arise from stronglycoupled conformal field theory. In the CFT this corre-sponds to limit of large N as well as large ’t Hooftcoupling. A major landmark result in this direction is the“HKLL smearing function” construction of [4–6] whichreconstructs local bulk (scalar) fields in terms of non-local(smeared out) boundary CFT primary operators withcompact support. At the level of free fields, whichcorresponds to the leading order planar limit of fieldtheory, it was shown that the resulting CFT smearedoperators indeed satisfy free bulk field equations inbackgrounds such as pure AdS, Rindler, BTZ etc.Since then, it has found several extensions including free

*sroy@iith.ac.in†sarkar@itp.unibe.ch

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higher spin fields in the bulk [7,8], connection with RGflows and dS [9,10], black hole information problem[11–14] etc. With HKLL it became technically much easierto study the dynamics of quantum fields in the classicalbulk spacetime, including their locality properties. It also,very organically, paves the way of incorporating theperturbative 1=N or 1=λ corrections [15] by demandingthat the bulk commutators obey bulk microcausality orderby order in 1=N [16–19].1 Thus the HKLL programprovides us a bottom-up view of the semiclassicalwindow of quantum gravity theories in terms of corre-lators of (smeared) operators. However, a huge limitationof the HKLL program is that one necessarily needs theknowledge of the bulk metric and field equations ofmotion to reconstruct it from the CFT. For the bulk to betruly emergent, the bulk metric and the equations ofmotion of bulk fields should be end products, notingredients of the bulk reconstruction recipe. This paperaims to extract the bulk spacetime metric itself from theboundary CFT data. This is a highly nontrivial issue sinceit is not at all clear that, even in principle, whether oneshould be able to construct the bulk metric given the stateand spectrum of operators, and in particular which CFTinformation is directly related to the bulk geometry. Forthis we note a parallel line of development in theholographic mapping, since the pioneering work ofRyu and Takayanagi [21], that of the boundary entangle-ment and the emergent bulk geometry. See e.g., [22–28]among many others. Recently [29] showed that theboundary entanglement structure can also be used toprovide an alternative derivation of the HKLL prescrip-tion, which does not require the knowledge of the bulkequations of motion or even the bulk metric! Thus, oneno longer needs any information about the bulk dynamicsin order to construct a bulk scalar field at any order of1=N or 1=λ expansion. This construction clears the pathfor us to recover the bulk metric itself as follows. Onefirst computes the supergravity (SUGRA) correlator oftwo bulk scalars directly in the CFT using [29] i.e.,without the need for knowing the bulk metric or the bulkequation of motion for the scalar fields. Subsequently, wetake the large conformal dimension limit, Δ → ∞ of thisCFT result, and we identify it with the WKB form of theSUGRA two-point function. From this identification, thebulk metric can be read off since in the WKB limit, thepropagator is expected to be exponential of the geodesiclength joining the two point. Following this sequence ofsteps, we show, that starting from the information ofboundary modular Hamiltonian one does reproduce themetric, but up to a conformal factor which can in generalbe spacetime dependent. We trace this subtlety back to thefield redefinition ambiguity of the bulk fields. However,

as we discuss at the end, with an additional intuitionregarding the bulk that comes up in the process, it is infact possible to obtain the bulk metric exactly. In additionwe also show that the prescription of [29] can beextended to obtain the bulk scalar, which is local at leastup to the first order of bulk perturbations.The plan of the paper is as follows. In Sec. II we briefly

review the work of [29], which uses the CFT modularHamiltonian data to recover the HKLL construction of thefree bulk scalar fields in the strictly planar limit. Inparticular, after discussing the essential ingredients thatgoes into the derivation of the free field in Sec. II A, weextend the formalism to recover the interacting, local (tocubic order) bulk scalar in this method. Then in Sec. III, weshow how to use this construction to evaluate the metriceverywhere in the bulk.2 These last two sections are themain parts of our results. In particular, in Sec. III B, weevaluate the bulk metric and show that we recover the AdSPoincare patch. Next in Sec. III C we point out that the fieldredefinition ambiguity in the work of [29] has an importanteffect for our construction of bulk metric. For translationalinvariant states, the metric can be reproduced exactly inunits of AdS radius i.e., when the conformal factor is aconstant. However, for general states, this makes the metricrecoverable up to a spacetime dependent, overall conformalfactor. Modulo this subtlety, Secs. III D and III E are thenrespectively devoted towards evaluating the global bulkmetrics for a CFTat zero and finite temperatures. In Sec. IIIF, we discuss how can this method be utilized to recover thebulk fields and consequently the metric, in the interior of ablack hole or causal horizon. Finally, we conclude inSec. IV, where we also discuss some possible directionsfor future studies.

II. RECOVERING BULK FIELDS USINGBOUNDARY MODULAR HAMILTONIAN

In this section, we will briefly review the construction of[29] which shows how from a CFT modular Hamiltonianassociated with a given boundary subregion, can oneconstruct a local bulk operator at any point inside thecausal wedge of the given subregion (see footnote belowfor some definitions). The situation is simplest inAdS3=CFT2 case, so we will focus on this case in whatfollows. However, the whole set up can be generalized toarbitrary dimensions. We will also only discuss the time-independent situation. For the covariant generalization,see [29].We start with reviewing some basic ideas which, in

hindsight, motivated this construction. Given a boundarysubregion of size R on a time slice, the correspondingentanglement entropy (EE) at the boundary is holo-graphically represented as the (one-quarter) area of

1Very recently, the same considerations also found an illumi-nating connection with conformal bootstrap [20].

2To avoid clutter, we will not always mention the phrase “up toa conformal factor”, but this is always implied.

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a minimal area surface homologous to R (known as Ryu-Takayanagi (RT) surface, γ) in Planck units [21]. We depictthe situation in Fig. 1. Computing the EE directly in fieldtheory is usually a very hard problem. It is given by

SEE ¼ −TrðρR log ρRÞ ¼AreaðγÞ4GN

; ð1Þ

where

ρR ¼ e−Hmod

is the reduced density matrix for the subregion R and Hmodis the associated modular Hamiltonian. The second relationin (1) comes from the RT proposal, which, as we can see,reduces the vastly complicated problem of taking loga-rithms of a density matrix to a relatively much simplerproblem of computing area of a geometric surface in somemanifold. The modular Hamiltonian or entanglementHamiltonian, is in general a very complicated, nonlocaloperator, but for spherical entangling surfaces, and espe-cially for CFT2 are quite easy to write down. To show howit works, let us consider the Poincare patch of AdS3,

3

ds2 ¼ l2

Z2ð−dT2 þ dZ2 þ dX2Þ; ð2Þ

with the dual CFT supported on Minkowski space,

ds2 ¼ −dT2 þ dX2: ð3Þ

Introducing lightcone coordinates

ξ ¼ X − T and ξ ¼ X þ T; ð4Þ

one can define the domain of dependence, D, of theboundary subregion as the region bounded by future andpast directed intersecting light rays starting from theendpoints of the subregion.4 Its upper tip yμ and lowertip xμ can be described using the light-front coordinates

ðu; uÞ ¼ ðx1 − x0; x1þ x0Þ and ðv; vÞ ¼ ðy1− y0; y1þ y0Þ:ð5Þ

If the endpoints of the subregion are y1 and y2 then

u ¼ y2; v ¼ y1; u ¼ y1; v ¼ y2: ð6Þ

In these coordinates the CFT vacuum modular Hamiltonianis written as [31]

Hmod ¼ HðRÞmod þHðLÞ

mod

¼ 2π

Zu

vdξ

ðu − ξÞðξ − vÞu − v

TξξðξÞ

þ 2π

Zv

udξ

ðv − ξÞðξ − uÞv − u

T ξ ξðξÞ: ð7Þ

Here Tξξ and so on are the CFT2 stress tensors. It can thenbe shown that a free bulk scalar Φð0Þ of mass m, whenintegrated over the minimal area RT surface mentionedabove, commutes with the boundary modular Hamiltonian.That is5

½Q;Hmod� ¼ 0 where Q ¼ Cbulk

8πGN

ZγdsΦ: ð8Þ

This is most easily obtained by writing down the bulk fieldΦ as a smeared CFToperatorOΔ a la HKLL and then usingthe commutation relations between the boundary stresstensors and the operator OΔ [4–6].

FIG. 1. Given a time slice in vacuum CFT (drawn on the leftpanel for pure AdS3=CFT2), denoted here by the red boundarycircle, one can compute EE at the boundary using the followingholographic RT prescription. As in the picture on the right panel,if we have a subregion of size R (denoted here by end points y1and y2 and in blue), then its entanglement with the rest of the CFTtime slice is given by the area of the minimal area surface γ(dashed line) in Planck units.

3We will denote the AdS radius by l, and reserve the symbol Rfor the subregion size on the boundary.

4For future reference, we point out that the bulk region whichis comprised of the intersection of the bulk causal future andcausal past of D is called the causal wedge (CW) of the region A.On the other hand, let us call the bulk region which is bounded bythe RT surface and A as RA, i.e.,

boundary ofRA ¼ ∂RA ¼ A ∪ γ:

Then, the bulk domain of dependence ofRA is known as the enta-nglement wedge (EW). In simple situations like ours, CW ¼ EW.However, they usually satisfy EW ⊇ CW. See e.g., [24,30].

5Below Cbulk is a bulk normalization factor and GN is theNewton’s constant. We have also dropped the zero from thesuperscript of Φ to avoid clutter. It signified that the bulk field isfree, which will always be the case throughout our discussionswith the exception of Sec. II B, where we will introduce the nextorder correctionΦð1Þ to free fieldsΦð0Þ which should be added forthe scalar field to be local at the cubic order of bulk perturbation.

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A. CFT constraints for bulk locality

Observations such as (8) play a crucial role in writingdown a local bulk field in AdS. In fact, this result isintuitively obvious, the RT surface acts as a bifurcationsurface for the flow under a modular Hamiltonian [26]. It isa generalization to what happens for a Rindler wedge.6 Fora Rindler wedge, the corresponding modular Hamiltonianis a boost generator which does not change the origin of theRindler horizon [32].However, one needs to work with a slightly modified

version of (8) in order to construct a bulk local operator.The point is, even though a bulk scalar Φ commutes withthe modular Hamiltonian when it is located on the RTsurface, it does not necessarily imply the opposite. Namely,there is an infinite set of bulk operators residing on the bulkregion complimentary to RA (of the bulk time slice ofFig. 1), which also commute with Hmod. In order to tacklethis, [29] defined an extended modular Hamiltoniangiven by

Hmod ¼ HðRÞmod þ HðLÞ

mod ð9Þ

where,

HðRÞmod ¼ 2π

Z∞

−∞

ðw − y1Þðy2 − wÞy2 − y1

TwwðwÞ

HðLÞmod ¼ 2π

Z∞

−∞

ðw − y1Þðy2 − wÞy2 − y1

Tw wðwÞ: ð10Þ

So one just extends the range of integrations compared tothe regular modular Hamiltonian defined in (7). As a result,its action within the boundary subregion is same as the onefor the usual boundary modular Hamiltonian. In fact, it canbe explicitly shown that [29]

½Hmod;ΦðZ; X; T ¼ 0Þ� ¼ 0; ð11Þ

provided that

Z2 − ðy1 þ y2ÞX þ y1y2 þ X2 ¼ 0: ð12Þ

If the metric is AdS3, (12) is simply the condition that thebulk point ðZ; X; T ¼ 0Þ lies on a spacelike geodesic whoseendpoints hit the boundary at ðT ¼ 0; y1Þ and ðT ¼ 0; y2Þ.However, (11) and (12) are much more powerful than

(8). The extended modular Hamiltonian is given by

Hmod ¼ Hmod;A −Hmod;Ac ; ð13Þ

where Hmod;A ¼ Hmod and Hmod;Ac are respectively themodular Hamiltonians for the subregion A and its

complimentary boundary region Ac. So, as we haveshown in Fig. 2, the commutation relation in (11) andthe associated equation (12) clearly suggests that twointersecting subregions naturally select7 a spatially local-ized bulk field at point P which commutes with theextended modular Hamiltonians of both the regions.These commutation conditions then serve as algebraicconstraints on the “bulk” fields Φ, where the bulk fieldsare written as

ΦðXÞ ¼Z

dt0dy0gðp; qÞOðq; pÞ: ð14Þ

At this stage (14) is simply an ansatz and gðp; qÞ are the“smearing functions” that we want to ultimately derive.Here q ¼ X − t0 þ iy0 and p ¼ X þ t0 þ iy0 and X iscurrently a free variable. Of course, the above complex-ification of bulk spatial coordinates from X to X þ iy0 ismotivated from earlier results of HKLL [5,6], but at thisstage, this is an educated guess.In what follows, we will represent this extended modular

Hamiltonians for regions bounded by ½y1; y2� and ½y3; y4� byH12

mod and H34mod and so on. These above ansatz and

definitions, along with

ðy2 − y1Þ½H12mod;Φðξ; ξÞ� ¼ 0

and ðy4 − y3Þ½H34mod;Φðξ; ξÞ� ¼ 0 ð15Þ

give us the correct smearing function with its support overthe boundary points given by the intersection of spacelikelight cone from the point P and the boundary. Moreprecisely, one finds the corresponding smearing functiongðp; qÞ ¼ Kðt; x; zjt0; y0Þ to be given by the d ¼ 2 versionof [29]8

FIG. 2. Shown here are two overlapping boundary subregionson the CFT2 time slice (red circle). The two subregions haveendpoints ½y1; y2� and ½y3; y4� (black and blue arcs respectively)and their overlapping region is from y1 to y4. As a result, theyintersect at point P. From the results of the previous subsection, abulk field at point P must commute with the modular Hamil-tonians of both these regions and is hence localized at P.

6In AdS, a Rindler wedge is a bulk causal wedge correspond-ing to a subregion which extends to half of the AdS boundary.

7This is true for AdS3. For AdSdþ1, we need d intersecting regi-ons to pinpoint a localized bulk operator.

8In our notation, for d ¼ 2, x; z; twill be replaced by uppercaseX, Z, T.

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Φð0Þðt; x; zÞ ¼ cΔ

Zdt0dd−1y0Kðt; x; zjt0; y0ÞOΔðtþ t0; xþ iy0Þ

¼ ΓðΔ − d2þ 1Þ

πd=2ΓðΔ − dþ 1ÞZt02þjy0j2<z2

dt0dd−1y0�z2 − t02 − y02

z

�Δ−dOΔðtþ t0; xþ iy0Þ: ð16Þ

This is precisely the HKLL expression [4–6]. However,now it is an outcome of a purely CFT calculations withoutusing any bulk metric or bulk equations of motion.

B. Perturbative locality using modular Hamiltonians

However, it turns out that the above-mentioned modularHamiltonian constraints are far more powerful and can beutilized to obtain the subleading 1=N corrections to bulklocality. This can be expected, as the modular Hamiltonianthat satisfy the constraints (15) can be written purely usingthe CFT2 conformal group generators in the following way:

H12mod ¼

2π

ðy2 − y1Þ½Q0 þ y1y2P0 þ ðy1 þ y2ÞM01�; ð17Þ

with

Q0 ¼ iðL1 − L1Þ; P0 ¼ iðL−1 − L−1Þand M01 ¼ iðL0 − L0Þ:

Thus, at this point, it is surprising that the resultingsolutions for Φ in (15) are only free fields Φð0Þ in the

bulk, as in (16).9 Indeed, it turns out that this is not true, andthe resulting set of solutions can at least be extended, so thatone also recovers CFT fields which mimic interacting (tofirst order) bulk scalars.The simplest way to see this is to try to solve the modular

constraints (15) with an ansatz general than (14). Inparticular, let us consider an ansatz

ΦðXÞ ¼Z

dt0dy0gðp; qÞOΔðq; pÞ

þXn

an

Zdt0dy0gnðp; qÞOΔn

ðq; pÞ; ð18Þ

with the only difference being an infinite sum of smearedconformal primaries with dimensions Δn and associatedcoefficients an which are Oð1=NÞ suppressed with respectto the leading counterpart.If we plug this ansatz in (15), then we get an additional

term in the constraint condition, similar to what we hadbefore, but which contain infinite number of terms.Namely, we have

ðy2 − y1Þ�H12

mod;Z

dt0dy0gðp; qÞOΔðq; pÞ�þXn

anðy2 − y1Þ�H12

mod;Z

dt0dy0gnðp; qÞOΔnðq; pÞ

�¼ 0; ð19Þ

and similarly for H34mod.

In general, it is hopeless to try to solve for all the unknown functions gðp; qÞ and gnðp; qÞ’s from just two sets ofequations. However, the derivation of

gðp; qÞ ¼ cΔ½Z2 − ðp − X0Þðq − X0Þ�Δ−2; with X0 ¼y1y2 − y3y4

y1 þ y2 − y3 − y4ð20Þ

and subsequently of (16), is independent of the choice of Δ and only relies on the fact that the boundary operator OΔ is aconformal primary [29]. Hence there is at least one set of solutions for (19), which gives the same functions gnðp; qÞ’s asgðp; qÞ with the only difference being the dependence onΔ. In other words, a consistent set of solutions for (19) is given by

gðp; qÞ ¼ cΔ½Z2 − ðp − X0Þðq − X0Þ�Δ−2 and gnðp; qÞ ¼ dΔn½Z2 − ðp − X0Þðq − X0Þ�Δn−2: ð21Þ

Above, dΔnis an overall dimension dependent factor which is undetermined along with the coefficients an. Therefore, the

solution for the bulk field now boils down to

Φðt; x; zÞ ¼ cΔ

Zdt0dd−1y0Kðt; x; zjt0; y0ÞOΔðtþ t0; xþ iy0Þ þ

Xn

andΔn

Zdt0dd−1y0KΔn

ðt; x; zjt0; y0ÞOΔnðtþ t0; xþ iy0Þ

ð22Þ

9We thank Gilad Lifschytz for discussions on this point.

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Using bulk microcausality and the free field smearingfunction, the second term above already looks like thecorrectionΦð1Þ that one needs to add to the free bulk field inorder to recover bulk locality at subleading order ofOð1=NÞ [16]. However, without the knowledge of thebulk metric, it is not possible to argue for bulk lightconesingularity structures and consequently bulk microcausal-ity. However, what we show in the next section is that thefree bulk field Φð0Þ is actually sufficient to determine thebulk metric up to an overall conformal factor, which in turn,completely specifies the bulk microcausality structures. So,we can use the constraints from bulk microcausalitysubsequently, to compute what the required correctionΦð1Þ is.

III. BULK METRIC FROM CFT DATA

In the previous subsection II A, we briefly reviewed how[29] extracted the smearing function appropriate for localbulk scalars in pure AdS3 without knowledge of the bulkgeometry (metric) or any bulk equation of motion. Sincethe end result is the same as those of [5,6], it is guaranteedthat these scalars will satisfy the correct bulk equations andmicrocausal commutation relations. In what follows wewant to build on the above construction to extract the bulkmetric structure purely from the boundary subregion data ofmodular Hamiltonians and CFT correlators.As a by-product of the construction of the previous

section, one finds that the correct smearing function isobtained only if the Z appearing in (16) above is given by(for d ¼ 2)

Z2 ¼ ðy1 þ y2ÞX0 − y1y2 − X20; ð23Þ

where

X0 ¼y1y2 − y3y4

y1 þ y2 − y3 − y4: ð24Þ

This is of course what is expected, since X0, Z above arenothing but the solution of the two minimal area surfaceswith endpoints yi and yiþ1 (i ¼ 1, 3), namely

ðX − yiÞðyiþ1 − XÞ ¼ Z2: ð25Þ

That is, they give the coordinates of the intersectionpoint P in Fig. 2.10 However, we can think of shiftingone of the endpoints of one of the subregions slightly (Fig. 3)and look for the resulting intersection point. For example, ifwe only shift y1 to y1 þ δy1, then the resulting intersectionpoint Q has new coordinates X0 and Z given by11

X0 ¼ X0 þyny2c

δy1 −yny3c

ðδy1Þ2 þ � � � and

Z ¼ Z

�1þ 2ys − ycðy3 þ y4Þ

2ðy1 − y3Þðy1 − y4Þycδy1 þ � � �

�; ð26Þ

where

yn ¼ ðy2 − y3Þðy2 − y4Þ; yc ¼ y1 þ y2 − y3 − y4

and ys ¼ y1y2 − y3y4: ð27Þ

Now as this CFT prescription gives us the correct AdS3smearing function, it is guaranteed that the two pointcorrelation function of two such smeared CFT fields Φ(starting from two-point function of CFT operators O) willfurnish the correct form for AdS3 supergravity (SUGRA)correlators,

hΦðx1; z1ÞΦðx2; z2ÞiCFT¼

Z ZKðx1; z1jxÞKðx2; z2jx0ÞhOðxÞOðx0ÞiCFT

¼ GSUGRAðx1; z1; x2; z2Þ; ð28Þ

just as it did for [6]. However, the crucial point to rememberis that it is now an end result of a CFT calculationwhich doesnot rely on any bulk dynamics.So, we can write down the resulting bulk two-point

function for two Φ fields located at P and Q ofFig. 3 and the corresponding answer is guaranteed to be[6,33,34]

hΦðX0; ZÞΦðX0; ZÞibulk ¼ cΔ1ffiffiffiffiffiffiffiffiffiffiffiffiffi

σ2 − 1p 1

ðσ þffiffiffiffiffiffiffiffiffiffiffiffiffiσ2 − 1

pÞΔ−1

:

ð29Þ

FIG. 3. We have perturbed one of the subregions (from y1 to y2)by a small amount, so that the new subregion is from y1 þ δy1 toy2. It then gives us a different intersection point Q where the bulkfield is localized. P and Q are of course, spacelike separated.

10We now see how the emergent radial direction Z in the bulkcomes about. In this framework, it simply coordinatizes thelocation of the field Φ in terms of the boundary coordinates as in(23). It makes the bulk, one higher dimensional as compared tothe CFT.

11The quantities to start with are symmetric under interchangesof y1 ↔ y2 and y3 ↔ y4. So, if e.g., we also perturb the other endpoint from y2 to y2 þ δy2, there will be an additional order δy2term, which is same as the one written above, but y1 and y2interchanged.

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Here cΔ is a constant factor dependent on CFT parameters12

and

σ ¼ Z2 þ Z2 þ ðX0 − X0Þ22ZZ

: ð30Þ

We have written it suggestively in this format as for AdS, σis the AdS invariant distance between two points X0, Z andX0; Z. Note that we have denoted the subscript of the bulkcorrelator above as just bulk, as we do not yet know what itsmetric is and this is what we want to find out.

A. Expanding Green’s function

Now to extract the bulk metric we will have to equate theCFT result (29) with the most general expression of bulk2-point correlators involving scalars [of some massm ¼ mðΔÞ], written in terms of the geodesic distance ina generally curved spacetime (endowed with a metric gMNwhich is hitherto unknown) in some limit, such as theWKB(large mass) where bulk supergravity correlators areexpected to have some universal form. Let us first repar-ametrize the bulk correlators in terms of a function L suchthat13

σ ¼ coshLlbulk

; ð31Þ

using which, we have

1

ðσ þffiffiffiffiffiffiffiffiffiffiffiffiffiσ2 − 1

pÞΔ−1

¼ e−L

lbulkðΔ−1Þ: ð32Þ

On the other hand,

ðσ2 − 1Þ ¼ 1

4e2L=lbulkð1þ e−4L=lbulkÞ − 1

2: ð33Þ

At this point, we can take various limits. To begin with,here we will be interested inΔ → ∞ limit (keeping L fixed)which signifies the stationary phase or WKB type approxi-mation. In this case, from (32), we see that the leadingbehavior of the bulk-bulk propagator becomes

hΦðX0; ZÞΦðX0; ZÞibulkjΔ≫1 ≈cΔffiffiffiffiffiffiffiffiffiffiffiffiffiσ2 − 1

p e−LΔlbulk ¼ c0Δe

− LΔlbulk :

ð34Þ

B. Metric for Poincare AdS

Equation (34) is quite well known as an expression ofbulk propagator in general spacetimes at the leading orderof geodesic distance [35,36] and evidently L is a geodesicdistance in the bulk. Thus our reparametrized expressioncan be used to extract the bulk metric structure. In fact,comparison of our (34) with the general results of [36]already tells us that the metric is AdS, as it is a well-knownexpression for asymptotically AdS spacetimes. However,we can make it more quantitative by writing (M, N denotebulk indices)

L ¼Z ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

gMNdxMdxNq

; ð35Þ

and then computing the metric using (30) and (31). Forexample, in general dimensions, the metric component gZZor gXX can be extracted by choosing the two points to havesame X and T or same Z and T coordinates respectively.The calculation of gTT will also follow the same procedure.Let us show the extraction of gZZ in detail. Using WKB

limit, we have already recognized the quantity L definedthrough (31) as the geodesic distance. But now, we willinstead consider the limit where the two bulk points areinfinitesimally close, as we discussed back in Fig. 3. This,along with (30), then gives us

σjX0→X0≈ 1þ 1

2

dZ2

l2bulkgZZ

which yields,

gZZ ¼ l2bulkZ2

:

Note that this way we can extract the local value of themetric arbitrarily deep into the bulk and similarly we can fixdifferent coordinates to extract out gTT and gXX. The final

result is of course jgTT j ¼ gXX ¼ gZZ ¼ l2bulkZ2 .

Note that our choice of σ back in (30) already helps usunderstand that the bulk metric is diagonal. In other words,after extracting the diagonal components as above, we cantry to solve for off-diagonal components for two closelyseparated points. This yields (here M ≠ N),

dZ2 þ dX2 − dT2

2Z2≈�gMM

l2bulkdx2M þ gMN

l2bulkdxMdxN

�:

If we now take two of the coordinates to be different and thelast one to be same, this clearly indicates that the off-diagonal elements vanish.

12This is not exactly true, as there is a subtle issue of fieldredefinitions and their effects on cΔ. This is often overlooked inliterature and we have discussed it below in Sec. III C. We shouldalso point out that the scalars on the left side of (29) does not havethe correct canonical bulk dimension. For example, in order torecover the correct bulk correlator, one needs to take into accountfactors of AdS radius.

13lbulk is a dimensionful length scale which simply existsbecause we know that we have an emergent radial direction.

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C. Field redefinition freedom

At this point, it is important to note a key feature of thisconstruction. As the subregion information of yi ’s aredirectly embedded in the bulk field’s smearing formula (16)and subsequently in the two-point function (29), it is clearthat the freedom of choosing an infinite pairs of subregionsin order to obtain a given bulk point is also inherent in sucha formalism. Moreover, the CFT constraints (15) are alsosatisfied if the bulk fields are multiplied with some arbitraryclassical function of spacetime. This later freedom wasalready noted in [29], which makes the coefficients such ascΔ appearing in (16) position dependent. This is preciselythe bulk field redefinition property (x collectively denotesall bulk coordinates)

ΦðxÞ → fðxÞΦðxÞ ð36Þof the above-mentioned smearing prescription. But, if weonly consider translational invariant states at the boundary,from the perspective of duality, the bulk states are alsoexpected to have the same invariance. This reduces thepossible sets of field redefinitions, in which f can be takenas a function of z alone. In fact, using the correctnormalization of bulk 1-point function as z → 0, it is easyto realize that the corresponding field redefinitions are justconstant scaling of the bulk scalar [29].14 In what follows,we will consider both situations where boundary transla-tional symmetry may or may not be broken and study itseffects on metric extraction. As noted in [29], theseredefinition ambiguities keep the bulk singularity structuresintact and hence by extension, do not affect microcausalprescriptions in higher orders of perturbation in 1=N(which only make use of the bulk singularity structures).In particular, we will point out that it makes the evaluationof the metric possible up to an overall conformal factor (See[37] for a similar conclusion but independent bulk argu-ments). Below we will make these statements quantitative.Let us first consider the situation with full generality, i.e.,

when the CFT state does not have any translation invari-ance. In this case, we can write the field redefinitionsacting as

hΦðxÞΦðxÞiCFT → fðx; LÞhΦðxÞΦðxÞiSUGRA: ð37ÞIn order to understand the effects of field redefinitions as

in (36), we again turn to our bulk two-point correlator asobtained within the CFT (29). We start by studying the largeΔ limit of this CFT correlation function which has beennoted in (34). Under (36), the correlator (34) changes to15

hΦðxÞΦðxÞiCFT ¼ f1ðxÞf2ðxÞe−ΔL=ffiffiffiffiffiffiffiffiffiffiffiffiσ2 − 1

p

¼ eF−ΔL=ffiffiffiffiffiffiffiffiffiffiffiffiσ2 − 1

p; where F ¼ logf1f2:

ð38ÞIn order to carry out the steps illustrated in Sec. III B above,we must be able to identify L as a geodesic distance. Butnow we see that due to an arbitrary function F, the requiredexponential behavior may be tampered. So, we will discussvarious choices of F below and see what are their effects onthe identification of L.Note that at the stage of (38), we are in the limit where

the product, ΔL → ∞ and fixed L. To this end we treatthree separate cases which cover all possible behaviors of F

F ∼Xn≥0

cnðxÞðΔLÞn; F ∼Xl≥1

clðxÞloglðΔLÞ;

F ∼Xm<0

cmðxÞðΔLÞm: ð39Þ

Here n ≥ 0, m < 0 and cn, cl and cm’s can be arbitraryfunctions of the bulk coordinates x. In order to sort out thephysical field redefinitions, we assume that we know thecorrect singularity structures of CFT correlators (includingtheir proper normalization) and we also expect that bulklightcone singularities will show up in appropriate limits.We also do not expect spurious singularities to show up inunexpected situations, e.g., when L → ∞ limit is taken in asuitable manner.(1) The first case with n > 1 is harmless as far as the

identification of L goes. They all come with a higherpower of L in the two-point correlator, whereas from[35,36], we know that the geodesic distance alwayscomes as a linear power of ΔL in the exponential.Thus for field redefinitions of this type, we do not

have any ambiguity in extracting the bulk metric.However, the ambiguity arise for the case of n ¼ 0,1, which we will discuss only at the end.

(2) In a similar way, we note that for nonzero clðxÞ, eF isa polynomial in ΔL. So once again, from thedistinctive exponential behavior we can alwaysidentify L as the geodesic distance. For example,we can take a derivative with respect toΔ in (38) andsubsequently take an L → 0 limit. This erases anypolynomial contribution that may be present afterthe field redefinition and distinguishes the exponen-tial behavior.

(3) The cmðxÞ’s are of course subleading terms in theWKB limit and do not contribute to the two-pointfunction at our required order.

(4) We now revisit the first case with n ¼ 0, 1. Indeed afield redefinition of type F ¼ c1ðxÞΔLþ c0ðxÞ isconsistent with the bulk singularity structure in theΔ → ∞ limit (alsowhen c1, c0’s are constants).

16 But

14Within HKLL, field redefinitions were also considered at theperturbative orders of 1=N [19].

15to avoid clutter, for the rest of this subsection we will putlbulk ¼ 1 in the exponent of the WKB correlator (34). We canalways restore it by looking at the scaling dimension of theexponent.

16The terms involving c0 is of course also subleading withrespect to the term involving c1.

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this precisely makes the definition of L ambiguousby an x-dependent factor, which in turn induces a x-dependent conformal factor in the evaluation of thebulk metric using (31), (35).We also note that it is not possible to get rid of such

an ambiguity by just demanding the boundarybehavior of the bulk correlators. For example,f1f2 could be f1f2 ¼ Necz

nΔL, such that asz → 0, we recover the correct CFT two-point func-tion with the required normalization factor N. Thesetypes of field redefintions, which still induces aconformal factor, are then invisible in the z → 0 limit.

To end the section, we briefly mention about the specialcase of translationally invariant states, which is simplyobtained from the results above for general state. Asmentioned before, in this case, we can consider the fieldredefinitions as multiplication by constant factors. Thus,from the point 4 discussed above, now the metric isobtained up to a constant rescaling of the conformal factor.This makes complete sense, as this is precisely our CFTignorance about the precise value of lbulk. In other words,for translationally invariant states, we can compute themetric exactly in the units of AdS radius lbulk.Now that we have obtained the bulk conformal metric,

we can finally fully reconstruct the local bulk fields at thesubleading order of boundary perturbations. Fortunately,the knowledge of the conformal metric is enough tounderstand bulk lightcone singularity structures and con-sequently that of bulk microcausality. Thus, we can getback to the expression we obtained at the end of Sec. II B,namely (22), and demand that the resulting bulk fieldsatisfies bulk microcausality. As shown in [16,19], this isnow sufficient to select the suitable conformal (double-trace) primaries that one requires to use in the infinite sumof (22).17

D. Extracting the metric for global AdS

In what follows, we will keep aside the issues of thefield redefinition and carry on the “observer dependent”construction for various other coordinate systems.Because we have now represented the required formulasof Sec. III B in terms of AdS covariant quantities, ourproof is valid for any other coordinate patches such asglobal AdS or Rindler patch. However, the situation isdifferent for BTZ BH as we have a physical singularitythere, even though the spacetime is locally AdS3. We willturn to these cases in the next subsections, but for nowwe deal with the next simplest example after the Poincare

AdS3 patch, which is the global AdS3 case. In thisscenario, if we define the function18

σðxjx0Þ ¼ cosðτ − τ0Þ − sin ρ sin ρ0 cosðΩ −Ω0Þcos ρ cos ρ0

; ð40Þ

then the above mentioned CFT techniques give us thebulk to bulk Green’s function (29). We can then proceedlike in the last section and define a quantity L via therelation (31). Once again, as soon as we take the WKBlimit, it enables us to interpret L as the geodesic distance.In particular, using e.g., Ω → Ω0 and ρ → ρ0, we get

σjρ→ρ0Ω→Ω0 ≈ 1þ dτ2

2l2bulkgττ ≈ 1 −

dτ2

2cosh2ρ: ð41Þ

Similarly, taking Ω → Ω0 and τ → τ0 and then τ → τ0 andρ → ρ0, we can respectively obtain

gρρ ¼l2bulkcos2ρ

and gΩΩ ¼ l2bulksin2ρ

cos2ρ:

Once more, the off-diagonal terms of the metric vanish.Thus we recover the global AdS metric

ds2 ¼ l2bulkcos2ρ

ð−dτ2 þ dρ2 þ sin2ρdΩ2d−1Þ; ð42Þ

as expected.

E. CFT at finite temperature

So far, we have only considered the CFT vacuum stateand its subregions in different boundary coordinate patches.But the above arguments and procedures already tell us thatthe calculation will go through similarly if we make acoordinate change to go to the Rindler patch of the AdSspacetime. This enables us to define a natural Rindler stateat the boundary which corresponds to an observer experi-encing a thermal bath. Ultimately the Rindler state can beupgraded to another boundary CFT state at a finite temper-ature, which corresponds to the presence of a (topological)BTZ black hole in the bulk [38]. In subsection III E 2 wewill discuss how to extract the exterior metric to the BTZgeometry, but first, in order to understand the subtletiesinvolving the BTZ case, we will warm up with the exampleof extracting the Rindler patch.

17However, we should note that the gauge dressings of scalars,which are important at subleading order in 1=N might also showup in certain gauges alongside the ΔL factor in the exponent of(34). It seems that choosing e.g., the holographic gauge [7], it ispossible to still extract the correct geodesic length L, even in thatcase.

18Here we have replaced the nomenclature of Z, X, T of theprevious section by τ, ρ and Ω and now the computation of [29]needs to be performed in these coordinate systems. The fact thatwe will obtain the correct smearing function and hence the correctbulk fields, are guaranteed due to the fact that the resultingexpression of the smearing function in [29] is fully covariant.

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1. Bulk metric outside the Rindler horizon

Once again, we start with the CFT procedure of con-structing bulk fields in terms of intersecting RT surfaces. Ashas been already pointed out in [29], this procedure gives

rise to the correct bulk operator in the background ofRindler AdS. For a bulk field in the (e.g.) right exterior of aRindler spacetime, it is obtained as [the coordinate con-ventions are given below in (47)]

ϕðt; r; ϕÞ ¼ cΔðrþÞZspacelike

dxdy limr0→∞

ðσ=r0ÞΔ−2ORindler;Rðtþ x; ϕþ iyÞ; ð43Þ

where as r0 → ∞, the AdS invariant distance becomes

σðt; r; ϕjtþ x; r0; ϕþ iyÞ ¼ rr0

r2þ

�cos y −

�1 −

r2þr2

�1=2

cosh x

�: ð44Þ

This is precisely what one obtains in the old HKLLprocedure [6]. Thus it is guaranteed that this CFT pre-scription will give us the necessary bulk to bulk two-pointfunction. However, if we only use a definite spectrum of agiven CFT (to which a given Rindler observer has an accessto) with their corresponding modular Hamiltonians, the

resulting bulk operator and hence the bulk metric will turnout to be located outside the Rindler horizon.19 We willbriefly visit the question of interior (of Rindler horizon)operator construction in Sec. III F.A simple way to extract the exterior Rindler metric is to

redo the whole calculation by defining [6,29]

σðt; r; ϕjt0; r0; ϕ0Þ ¼ rr0

r2þcoshðϕ − ϕ0Þ −

�r2

r2þ− 1

�1=2

�r02

r2þ− 1

�1=2

coshðt − t0Þ: ð45Þ

Because this gives the correct bulk scalar outside the rightRindler horizon, the corresponding two-point function isagain of the form (29). Clearly the two bulk points here arelocated at ðt; r; ϕÞ and ðtþ x; r0; ϕþ iyÞ respectively. Fromhere, we can extract the Rindler metric, but we can onlyaccess the outside part of the Rindler horizon.In this case, the geodesic distance L is once again

obtained from (31). In fact, the easiest way to understandthat the procedure will go through for exterior (to Rindlerhorizon) operators in Rindler and the above choice of σ in(45) is the correct choice, it is best to understand RindlerAdS as a coordinate transformed version from the globalpatch (42).20 The Rindler metric is given by21

ds2 ¼ −r2 − r2þl2bulk

dt2 þ l2bulkr2 − r2þ

dr2 þ r2dϕ2; ð47Þ

which are related to the global coordinates by the followingcoordinate transformations (see e.g., [39]):

iτ ¼ rþlbulk

ϕ; Ω ¼ rþl2bulk

ðitÞ and cos ρ ¼ rþr:

Indeed, the geodesic distance in Rindler (45) can beobtained from the geodesic distance in global AdS (40),by using the above coordinate transformations. In fact,the covariant relations (31) and the related constructions ofthe metric (outside the Rindler horizon) go throughunchanged.But we can also extract the metric quite straightfor-

wardly. Using (45) and (46) in (31), we find (using theexpression of geodesic distance in terms of the metric andlooking at diagonal components as before)

grr ¼l2bulk

ðr2 − r2þÞ; gϕϕ ¼ r2 and gtt ¼ −

r2 − r2þl2bulk

:

ð48Þ

21The coordinates t; ϕ are related to t;ϕ by

t ¼ rþtl2bulk

and ϕ ¼ rþϕlbulk

: ð46Þ

20In [6] the Rindler smearing function was obtained from thePoincare patch by going to dS via an analytic continuation andthen using the bulk equations.

19This is not true in general. Indeed, for generic spacetimes, theRT surface supported on one CFT can enter the causal, eventhorizon [30]. Because our construction relies on RT surfaces,rather than causal wedge, in those cases, a subhorizon metricconstruction is in principle possible.

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This is precisely the Rindler coordinate as shown in (47).22

2. Bulk metric outside the BTZ horizon

Once we are done with Rindler, there’s not much to bedone for the BTZ metric’s exterior. In fact, the BTZgeometry is obtained from the Rindler-AdS geometry bytaking the range of ϕ coordinate between 0 and 2π, solocally they are indistinguishable [6,38]. The key ingredientthat we need to show is that using the σ as defined in (45),the WKB connection between the BTZ correlator and theBTZ geodesic distance goes through.In other words, we know that for Rindler-AdS (RAdS)

and in the WKB approximation, the following is true:

hΦðt; r; ϕÞΦðt0; r0ϕ0ÞiAdS ¼ c0Δe−ΔLRAdS=lbulk :

Here we require that for BTZ, a similar relation holds in theWKB approximation as well:

hΦðt; r; ϕÞΦðt0; r0ϕ0ÞiBTZ ¼ c0Δe−ΔLBTZ=lbulk ð49Þ

for σ still given by (45). Of course, given this, the derivationof the metric is exactly similar as before (the only differ-ence being the above-mentioned periodicity of ϕ). WritingLBTZ in terms of BTZ metric components, we recover theBTZ metric.The way to see that (49) is correct is to note that the

BTZ correlator is given by the image sum of AdScorrelators [40]:

hΦðt; r; ϕÞΦðt0; r0ϕ0ÞiBTZ ¼X∞n¼−∞

hΦðt; r; ϕÞΦðt0; r0ϕ0 þ 2πnÞiAdS

¼X∞n¼−∞

cΔ1ffiffiffiffiffiffiffiffiffiffiffiffiffi

σ2n − 1p 1

ðσn þffiffiffiffiffiffiffiffiffiffiffiffiffiσ2n − 1

pÞΔ−1

; ð50Þ

where σn are given by

σn ¼rr0

r2þcoshðϕ − ϕ0 − 2πnÞ

−�r2

r2þ− 1

�1=2

�r02

r2þ− 1

�1=2

coshðt − t0Þ:

However, when we consider WKB approximation, i.e.,Δ → ∞, we see that the leading order (inΔ) contribution inthe right-hand side of (50) comes from σn¼0, which isnothing but σ defined in (45). All the σn terms for n ≠ 0(both positive and negative) are much larger and negligiblewhen taken to a very high power as in 1

ðσnþffiffiffiffiffiffiffiffiσ2n−1

pÞΔ−1

.

F. Bulk fields and metric inside the Rindler horizon

Finally, we make some brief comments regarding apossible interior operator construction in the modularHamiltonian approach, in particular, for Rindler/ thermo-field double (TFD) states. This question is at the very heartof the black hole firewall problem [41] and is still open.Although our lack of knowledge about modularHamiltonians for generic states renders such constructionmore subtle, its entanglement nature gives us a hope to gobeyond the causal wedge. For example, for Rindler, one cansimply recast the results of global or Poincare AdS inRindler coordinates, which clearly states that there are nophysical problems behind such a construction. But on theother hand, if we want to stick to the subregions that anygiven Rindler observer has access to, we cannot simply usethe global AdS result.In the bulk approach of constructing the local operators

[6], the problem appeared because as soon as one extendsthe bulk operator beyond the Rindler horizon (let us say forthe right Rindler exterior), the corresponding smearingregion goes beyond the right Rindler boundary region. Thiscompelled them to use the anti-podal map, which brings theextra extended region back to the left side of the Rindlerwedge and finally another analytic continuation in boun-dary time to bring it to the right.In our case however, the problem arises because there are

no two subregions on the same side of a Rindler patch (sayright side), for which the corresponding RT surface goesbeyond the Rindler horizon and intersects at a point (seehowever footnote 19). In such scenarios, we can consider

22We must mention some subtle issues with the field redefi-niton here, which is also applicable for the case of BTZ blackholes discussed afterwards. Even though the state is translationinvariant, the argument for constant field redefinitions is now a bitmore involved. This is because, using the new temperature scaleβ−1, one can expand the cΔðrþÞ in (43) in a series such as

cΔðrþÞ ¼ 1þ � � � þ cn

�zβ

�nþ � � � :

It then requires a bit more work to show that cΔðrþÞ is a givenfunction of Δ, d and rþ. However, since Rindler is pure AdS andthe left-hand side of (43) is a scalar, the right-hand side mustreduce to the right-hand side of the global AdS scalar uponmaking the Rindler to global coordinate transformations. Thiswill again lead to cΔ being a pure function of Δ; rþ and d.

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two intersecting RT surfaces stretching between two sidesof the full Rindler geometry, but located on the same timeslice [42]. This guarantees that they intersect at a singlepoint inside Rindler horizon. However, little is knownabout the field theory counterpart of such an entanglemententropy, let alone its associated modular Hamiltonian. Infact, this problem is a particular example of a more generalquestion of whether we can generically construct local bulkobservables inside the EW of a given boundary subregion,in terms of simple boundary operators located in thesubregion. As mentioned before, EW is usually biggerthan the causal wedge and such a “simple” construction isno longer possible [43,44].

IV. DISCUSSIONS AND OUTLOOK

In this work, we have provided a recipe to extract thebulk metric using the local (scalar) operator constructionfrom modular Hamiltonian data of boundary subregions[29]. We showed that, the knowledge of CFT data,associated with boundary modular Hamiltonian canindeed reproduce the bulk metric, but up to a conformalfactor. For the CFT states we considered, this conformalfactor turned out to be a constant. This is completelyexpected for such a pure CFT reconstruction, as CFT isblind to the AdS length scale and this ambiguity does notdo anything short of what is expected for a well-posedduality such as AdS=CFT. In order to reproduce the bulkmetric exactly, one requires some more information on thebulk-boundary connection, such as the Ryu-Takayanagiprescription for entanglement entropy mentioned before.This was e.g., the approach of [23,28]. In fact, a by-product of the modular Hamiltonian approach was that thefields Φ satisfied (15), only if their radial location wasgiven by (12). So, alternatively, we can consider (12) tobe the geodesic equation in the bulk, as from the point ofview of the bulk modular flow (which is dual to theboundary modular flow), it is the only bulk curve forwhich the bulk field does not modular transform. Such anadditional equation/information regarding bulk then pre-cisely computes the unknown conformal factor, therebycomputing the bulk metric exactly.However in this work, we have solely focused on

special states of the CFT, such as the vacuum or thermalstates. These states have a very high degree of symmetryand as a consequence the modular Hamiltonian is easy toconstruct. Also we worked with a two dimensionalboundary, for which the RT surface is one-dimensional.So here we make some comments when each of theseconditions are generalized. First we discuss the case ofhigher dimensions, i.e., AdSdþ1=CFTd. We expect thehigher dimension case to be a straightforward, thoughgeometrically more involved construction. RT surfacesbeing co-dimension 2 surfaces, will intersect on a co-dimension 3 surface. Thus one will need d number of RTsurfaces to intersect, so as to single out a point in the

(dþ 1) dimensional bulk, and further to solve d algebraicequations to reproduce the HKLL construction (and inturn to compute the metric). We leave this higher dimen-sional construction for future work.Second, we discuss the situation regarding the CFT

states. In this paper, we looked at vacuum state and thethermal state (thermofield double or TFD states) and wederived the bulk metric which is either pure AdS3 or theRindler wedge of AdS3 or the BTZ. Although this is niceand reassuring to see that the recipe works, these bulkmetrics are all too familiar to us and this work should beregarded more as a proof of principle. To realize the fullpotential of our approach, one needs to extend our results todetermine metrics for states of the CFT for which the bulkmetric is unknown. The main issue here is to findexpressions of modular Hamiltonians for CFT statesperturbatively close to the vacuum or thermal states. Seee.g., [45] (and references therein) for some progress in thisdirection. Although these states are perturbatively excitedwith respect to the CFT vacuum, these are not to beconflated with perturbative excitations in 1=N. The per-turbatively excited CFT states we are interested are the oneswhich cause a perturbation of the bulk metric. On the otherhand, CFT states which are perturbative in 1=N can bethought of as interacting multiparticle states on the unback-reacted bulk metric. Incidentally, the 1=N correction to thealternative HKLL construction [29] is also an interest-ing issue.There are several other avenues for future work. One

obvious direction is to look at the reconstruction of bulkgauge fields from boundary entanglement, and how bulkgauge redundancy is represented in the boundary modu-lar Hamiltonian. But perhaps the most interesting ques-tion is to find an extension of the recipe to reconstructlocal bulk fields and further the metric beyond the causalwedge, such as regions inside black hole horizons. Forthe two sided black holes, the construction should be astraightforward generalization of the Rindler horizoninterior, as discussed in Sec. III F. Entanglement amongsubregions having support on both boundaries of the twoRindler wedges/exterior regions code the interior localoperator or metric data. However for single sided blackholes, the region inside the horizon are outside thecausal wedge of the maximal boundary subregion. Sounless the RT surface somehow extends beyond thehorizon, the present recipe needs to be generalized forthese cases in order to reconstruct regions beyond thecausal wedge.

ACKNOWLEDGMENTS

We thank Dan Kabat for his perceptive comments,especially regarding the field redefinition ambiguity in[29], and for giving his valuable feedback on the draft. S. R.also thanks the Albert Einstein Center for FundamentalPhysics of the University of Bern for their hospitality,

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during which the project was conceived. D. S. thanks CityCollege of New York, KITP and Columbia university forhospitalities during work on this project. The work of S. R.is supported by IIT Hyderabad seed Grant No. SG/IITH/

F171/2016-17/SG-47 and his travel to Bern was supportedby IIT Block Grant. The work of D. S. is funded by theNCCR SwissMAP (The Mathematics of Physics) of theSwiss Science Foundation.

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