Local bulk physics from intersecting modular Hamiltonians

We show that bulk quantities localized on a minimal surface homologous to a boundary region correspond in the CFT to operators that commute with the modular Hamiltonian associated with the boundary region. If two such minimal surfaces intersect at a point in the bulk then CFT operators which commute with both extended modular Hamiltonians must be localized at the intersection point. We use this to construct local bulk operators purely from CFT considerations, without knowing the bulk metric, using intersecting modular Hamiltonians. For conformal field theories at zero and finite temperature the appropriate modular Hamiltonians are known explicitly and we recover known expressions for local bulk observables.


Introduction
Having a procedure for recovering bulk physics from the CFT is fundamental to our quest to understand quantum gravity using AdS/CFT [1]. The Ryu-Takayanagi (RT) relation [2] between the area of minimal surfaces anchored on the boundary (RT surfaces) and entanglement entropy in the CFT is one example of how bulk quantities can be computed from the CFT. The first law of entanglement entropy has been used together with the RT and HRT formulas [3] to derive the linearized Einstein equations in the bulk [4].
Natural bulk objects to consider are bulk field operators. Traditionally the program of constructing bulk field operators in the CFT starts from knowledge of the bulk metric. This information is used to compute smearing functions which provide the leading-order expression (in 1/N) for a local bulk field in terms of a single-trace primary scalar operator in the CFT [5,6,7].
Φ (0) is a CFT operator which reproduces the correct bulk 2-point function when inserted in a CFT correlator. However the expression for Φ (0) is not unique. Among the expressions we will be using are the complex coordinate representation (for Poincaré coordinates in AdS 3 ) [8] Φ (0) (Z, X, T ) = ∆ − 1 π and the representation in terms of mode functions [9].
1/N corrections to these expressions -for example to define CFT operators that will reproduce the expected bulk 3-point functions -can be obtained purely from the CFT using bulk locality as a guiding principle. Imposing bulk microcausality corresponds to canceling unwanted singularities in correlation functions. To achieve this it is necessary to correct the definition of the bulk field by adding to Φ (0) an infinite tower of higher-dimension multi-trace operators. To restore locality in n-point functions it is necessary to add CFT operators involving up to n − 1 traces, all smeared with the appropriate smearing functions as in (1). In this way the expressions relevant to bulk 3-point functions were obtained in [10,11], and the resulting bulk fields were shown to obey the correct bulk equations of motion. The expressions needed to reproduce bulk 4-point functions were obtained in [12] with the help of crossing symmetry. These results extend with some modifications to bulk fields with spin [13,14,15,16]. However throughout this program the starting point, that is the lowest-order smearing functions, were computed using knowledge of the bulk metric. 1 In this paper we obtain the zeroth-order bulk operator Φ (0) , up to a multiplicative coefficient, 2 purely from CFT considerations. The basic idea is that the modular Hamiltonian associated to a boundary region has the same action on bulk quantities as the associated bulk modular Hamiltonian [19]. The RT surface (the minimal bulk surface homologous to the boundary region) plays the role of a bifurcation surface in Rindler coordinates (the surface where the past and future Rindler horizons intersect). Just as the bifurcation surface is invariant under Rindler time evolution, the RT surface is invariant under the action of the bulk modular Hamiltonian. This means CFT objects which are localized in the bulk on an RT surface should commute with the corresponding boundary modular Hamiltonian.
To proceed it's convenient to define an extended modular Hamiltonian which generates a non-trivial flow everywhere in the bulk except on the RT surface. If a collection of RT surfaces intersect at a point in the bulk, then CFT quantities which commute with all of the corresponding extended modular Hamiltonians must be localized at the intersection point. We can impose this as a condition to construct local observables in the bulk. The construction is simplest in the case of 2-D CFT where RT surfaces are just lines.
Recently Φ (0) has been shown to be related to several natural CFT objects. In [20,21,22] the integral of Φ (0) over an RT surface was found to be related to a conformal block operator. 3 In [24,25,26] it was shown that Φ (0) creates a boundary cross-cap state in the CFT. One of the motivations of the present paper was to understand the relation between these different descriptions of Φ (0) . The construction developed here makes the connection much clearer.
An outline of this paper is as follows. In section 2 we review the modular Hamiltonian appropriate to a segment of the boundary and extend it outside this region in a natural way to obtain what we call the extended modular Hamiltonian. We use this to show that bulk operators on the RT surface commute with the modular Hamiltonian in a variety of contexts. In section 3 we turn the argument around and search for a CFT operator that commutes with the extended modular Hamiltonians associated with two different boundary segments. Provided the corresponding RT surfaces intersect we explicitly solve this condition and find the correct smearing function for a local operator that lives on the intersection. We do this for the vacuum state of the CFT on a line in three ways: using complexified coordinates, using a derivative expansion, and in momentum space. The latter recovers the Poincaré mode expansion of a bulk field from the CFT. We also do this for a CFT on a line at finite temperature and recover the complexified smearing function for a BTZ black hole. In appendix A we establish the relationship between conventional and extended modular Hamiltonians, in appendix B we show that the extended modular Hamiltonian has the same action on bulk operators which are off the RT surface as the bulk Rindler Hamiltonian, and in appendix C we study geodesics in BTZ. For convenience in all explicit computations we specialize to AdS 3 /CFT 2 .

Modular Hamiltonian in AdS 3
We work in AdS 3 / CFT 2 in Poincaré coordinates. The bulk metric is and the CFT metric is We introduce boundary light-front coordinates A space-like segment in a (1 + 1) dimensional CFT defines a causal diamond based on the segment. The diamond D(x, y) is defined through its upper tip y µ and its lower tip x µ , which we describe using the light-front coordinates If we choose a diamond on the boundary whose left and right tips lie on the T = 0 slice at points y 1 and y 2 then For a CFT in its vacuum state the modular Hamiltonian can be written explicitly [27].
We define coordinates η andη by which are solved by These Rindler-like null coordinates (η,η) cover the diamond. In terms of dimensionless time and space coordinates Under the change of coordinates ξ → η andξ →η we have 4 where For the change of coordinates in (11) one finds S(ξ, η) = S(ξ,η) = − 1 2 . Thus we can write the modular Hamiltonian (10) as In Poincaré coordinates a bulk geodesic γ on the T = 0 slice connecting the points y 1 and y 2 on the boundary is given by the semicircle Proper length along this geodesic is Using this and c =c = 3l 2G the last two terms in (16) can be seen to be This is just the RT term, i.e. the area of the minimal surface. So in fact The authors of [19] identified the left-hand side of (20) with the boundary modular Hamiltonian and interpreted H mod as a bulk modular Hamiltonian which generates bulk time evolution in the appropriate bulk Rindler wedge plus fluctuations of the RT surface (see also [28]). The computations of this paper will, among other things, confirm that H mod acts on CFT operators which represent bulk quantities in the manner expected for a bulk Rindler Hamiltonian.

CFT quantities invariant under the modular Hamiltonian
We start with the expression for a local bulk operator in Rindler coordinates [7], where the AdS 3 metric is Here l is the AdS radius, r + is the horizon radius and the region of integration is The Rindler operator in the CFT is normalized according to lim r→∞ r ∆ Φ (0) (r, φ, t) = O Rindler (φ, t).
We understand the analytic continuation to complex boundary coordinates to be defined by As r → r + the integration region becomes −∞ < x < ∞ and − π 2 < y < π 2 . Thus The integral over x sets ω to zero. Then using and defining we get This shows that zero frequency modes relative to the boundary Rindler Hamiltonian live on the bulk RT surface, and due to (20), that bulk objects on the RT surface commute with H mod . Setting k = 0 in (28) gives where the left-hand side is up to a constant the integral of the bulk field over the RT surface which serves as the horizon of the bulk Rindler wedge.
In fact (29) follows from results in the literature. The integral of a bulk field operator over an RT surface γ was identified in [20,22] with a particular CFT expression. In two dimensions, for a primary operator with dimensions h =h = 1 2 ∆ O , the appropriate identification was found to be where C blk and C O are normalization constants. To see what the right hand side of (30) represents, we make a conformal transformation ξ → η andξ →η as in (12) and define a Rindler operator O R as the conformal transformation 5 Then we see that which up to constants agrees with (29).
It will be useful below to show directly that Q commutes with H mod . In Lorentzian CFT the commutator of the energy-momentum tensor with a primary field is Now we can compute the action of the modular Hamiltonian on a CFT operator. We start with the modular Hamiltonian for a segment (y 1 , y 2 ) on the T = 0 time slice, The relation between O Rindler and O is usually taken to be O Rindler = lim r→∞ (rZ) ∆ O. See (39) in [8]. This normalization gives an extra factor of r ∆ + compared to (31), [H One can then easily check that Q(O; u,ū; v,v) commutes with H mod . In fact Q is the unique expression which commutes with both H In fact there are generalizations of (32). A mode of the boundary Rindler operator with zero frequency but non-zero momentum Q k ≡ O R (ω = 0, k) can be written as over the RT surface with a particular weight.

Extended modular Hamiltonian
In what follows we will want to compare the action of two modular Hamiltonians based on different segments of the boundary. To make this comparison it is very convenient to define what we call an extended modular HamiltonianH mod . 6 The extended modular Hamiltonian agrees with the usual modular Hamiltonian within its defining segment, but it extends in a natural way to be non-zero outside the segment. Thus the action ofH mod on operators inside the diamond D(x, y) based on the segment will be the same as the action of the usual modular Hamiltonian, but H mod andH mod act differently on operators outside the diamond.
A convenient definition of the extended modular Hamiltonian for a segment (y 1 , y 2 ) of the boundary at T = 0 is just Compared to the usual definition (33) all we've done is extend the limits of integration. This definition of the extended modular Hamiltonian has a natural interpretation. As we show in appendix A,H mod can be identified with the modular Hamiltonian for an interval A on the boundary minus the modular Hamiltonian for its complementĀ. 7 This has the nice feature thatH mod,A generates a non-trivial flow everywhere in the bulk, except on the RT surface associated with A which it leaves invariant. This means operators which commute withH mod must be localized on the RT surface. It follows from the definition that The action of the extended total modular HamiltonianH mod on a primary field is (40) Compared to the action of the usual modular Hamiltonian (34), the only change is that there are no step functions.
Let us now look at a local bulk operator in the Poincaré patch and show that it commutes with the extended modular Hamiltonian appropriate for a segment whose RT surface passes through the bulk point. The bulk operator in Poincaré coordinates can be written using the complexified smearing function as We understand the complexified spatial coordinate as corresponding to the formal expression We now define q = ξ + iy ′ and p =ξ + iy ′ , so the above expression is Now we can integrate by parts and after a little algebra we find that This is simply the condition that the bulk point (Z, X, T = 0) lies on a spacelike geodesic whose endpoints hit the boundary at (T = 0, y 1 ) and (T = 0, y 2 ). See (17).

Finite temperature
In this section we extend the previous discussion to treat a modular Hamiltonian which is not constructed from the ground state of the CFT. Instead we consider a CFT at finite temperature.
For a CFT on a line at finite temperature β −1 = r + 2πl 2 the modular Hamiltonian for a region (−R, R) is given by [31,32] with c = 2l 2 r + sinh r + R l 2 . The extended modular Hamiltonian for the same region is then given byH The action of the extended Hamiltonian on a primary scalar operator of dimension 2h is We want to show that this modular Hamiltonian commutes with a local bulk operator on the corresponding RT surface. The bulk operator has a representation using complexified coordinates as in (21), where both x, y are real and the region of integration is cos y > 1 − r 2 + r 2 cosh x. We understand the operator at complex boundary coordinates to be defined by Using (48) and (51) we get (q = lφ − l 2 x r + + i l 2 y r + and p = lφ + l 2 x r + + i l 2 y r + ) After integrating by parts and a little algebra one finds that As shown in appendix C, this condition is satisfied provided the bulk point (r, φ, t = 0) lies on a spacelike geodesic connecting the two boundary points (t = 0, lφ = −R) and (t = 0, lφ = R).

Bulk operators from intersecting modular Hamiltonians
We saw that a bulk operator Φ living on the RT surface associated with a segment of the boundary commutes with the modular Hamiltonian appropriate to that segment. Of course this does not imply that Φ is local in the bulk. But if there is another segment on the boundary whose RT surface intersects the RT surface of the first one at a point, then we can demand that Φ commutes with both modular Hamiltonians. In this case Φ must be a local bulk operator living on the intersection point.
To make a connection to other work, note that on a formal level the action of the extended modular Hamiltonian appropriate for the vacuum state of a CFT on a CFT primary given in (40) identifies it asH mod = 2π Here Q 0 , P 0 , M 01 are generators of the conformal group.
In this section we will solve (57) in coordinate space to recover the smearing function for a local bulk operator in the complex coordinate representation. We will do the same thing in momentum space and recover the bulk Poincaré modes which make up a local bulk operator. In addition we will solve the appropriate equations for a CFT at finite temperature and recover a local bulk operator in the BTZ background. This provides a new way of constructing the zeroth-order bulk operator and deriving bulk modes without knowing anything about the bulk geometry.
Note however that the conditions for bulk locality (57) only determine the bulk operator up to a coefficient. The coefficient could depend on bulk position, so in fact we can only generically recover Φ (0) up to a function of the bulk space-time coordinates. In states where the CFT has an unbroken spacetime translation symmetry the function can only depend on the bulk radial coordinate. In this case dimensional analysis fixes Φ (0) up to an overall constant. But in general locality is not enough to fix the function. Even with this freedom we get quite a lot of information. For example, given the two-point function of a local bulk operator with another local bulk or boundary operator we can identify the singularities and deduce the bulk causal structure. 8 Also the program of perturbatively correcting the zerothorder bulk operator to take interactions into account only relies on the singularity structure, so up to a multiplicative coefficient an interacting local bulk operator could be constructed. Moreover this multiplicative freedom cancels in any ratio of correlation functions involving a fixed bulk operator with any number of boundary operators, so one could determine these ratios unambiguously. As another example of an unambiguous quantity, along the way we will see that the construction generates the equations which describe bulk spacelike geodesics.

Recovering smearing functions for the vacuum state
We start with an ansatz for an object that commutes with the modular Hamiltonian where q = X − t ′ + iy ′ , p = X + t ′ + iy ′ . In the ansatz t ′ and y ′ are taken to be real and X is left as a free real variable. From (44) the action of the modular Hamiltonian for a segment (y 1 , y 2 ) on Φ is given by We take two such modular Hamiltonians with parameters (y 1 , y 2 ) and (y 3 , y 4 ) and demand It's convenient to first look at the difference of the equations in (60), After integration by parts this gives an equation for g(q, p), where X 0 = y 1 y 2 −y 3 y 4 y 1 +y 2 −y 3 −y 4 . The solution to this equation is where f is an arbitrary function. We now use this form and solve the equation [H 12 mod , Φ(X)] = 0. Following the same steps as before we get an equation for f The two parameters appearing in the solution X 0 , Z can be identified as the coordinates of the local operator in the bulk. Note, for example, that as y 1 , y 3 → y 2 we have Z → 0 and X 0 → y 2 . Comparing (66) to (17), note that we have recovered from the CFT the equation which describes a spacelike geodesic in the bulk.
For the integration by parts to work without any boundary terms we need the integration region to be bounded by Z 2 + (p − X 0 )(q − X 0 ) = 0. For this to be possible for real (t ′ , y ′ ) we see that we must have X = X 0 . So finally we get Since the vacuum state is translation invariant we expect correlation functions of local bulk fields to be translation invariant as well. From this we can deduce that the coefficient c ∆ is a function of Z only, which could be determined from a normalization condition such as In this way we have recovered the bulk operator written in the complex coordinate representation.

Derivative representation
Another possible representation for a bulk operator is In this case formally we can impose locality using the usual modular Hamiltonian and we do not need the extended modular Hamiltonian.
We wish to impose the conditions (60), that Φ commutes with two modular Hamiltonians. As before we start by looking at the difference of the two equations in (60) which gives ∞ n,m=0 Without loss of generality we take Using this in (69) and setting the coefficients of ∂ n ξ ∂ m ξ O(ξ = X 0 ,ξ = X 0 ) to zero gives (n − m)a nm = 0 (71) So in fact Φ must have the form gives the condition 0 . This implies the recursion relation whose solution is As before time and space translation invariance restrict a 0 to be a function of Z. The expression for a bulk operator in Poincaré coordinates (41) can be expanded in derivatives [21].
Comparing this to (72) and (76) we see that we have recovered the local bulk operator Φ(Z, X, T = 0).

Recovering bulk modes
In this section we wish to recover the momentum space representation for a bulk operator, i.e. the bulk modes. We start with the extended modular Hamiltonian for the segment (y 1 , y 2 ) Using (78) one finds We now look for operators Φ which commute with the extended modular Hamiltonians for two segments (y 1 , y 2 ) and (y 3 , y 4 ). We make the ansatz We first require that Φ satisfy the difference of the two equations in (81). This gives (83) Upon integration by parts we get an equation for g(k + , k − ), where X 0 = y 1 y 2 −y 3 y 4 y 1 +y 2 −y 3 −y 4 . The general solution to this equation is where f is an arbitrary function of k + k − .
Having imposed that Φ commutes with the difference (y 2 − y 1 )H 12 mod − (y 4 − y 3 )H 34 mod , we now require that Φ commute withH 12 mod itself. We start with the ansatz The first equation in (81) becomes a condition on f . After integration by parts and some algebra we find ( The solution is with ∆ = ν + 1 and Z 2 = (y 1 + y 2 )X 0 − y 1 y 2 − X 2 0 . Time and space translation invariance restrict c 0 to be a function of Z. Then (86) becomes 9 which is the bulk operator Φ(Z, X 0 , T = 0) given in (3).
If we had imposed the second equation in (81) we would have gotten the same result with Z 2 = (X 0 − y 3 )(y 4 − X 0 ). But in fact these two expressions for Z are the same. As long as the parameters (y 1 , y 2 , y 3 , y 4 ) assign a real value to Z we have a solution where the point (T = 0, Z, X 0 ) is the bulk point located at the intersection of the two boundary-anchored geodesics.

Time dependence
For completeness we show that the construction of local bulk operators based on intersecting modular Hamiltonians also captures the correct time dependence of bulk fields.
To do this we look at the modular Hamiltonian for a diamond that is shifted in time by an amount T . In light-front coordinates (7) such a diamond is characterized by The extended modular Hamiltonian acting in momentum space is given by We look for operators that commute withH total mod =H L mod +H R mod . We make an ansatz If we take two different boundary segments (y 1 , y 2 ) and (y 3 , y 4 ) at time T then the conditions we wish to impose are Taking the difference results in an equation for g(k + , k − ), with X 0 = y 1 y 2 −y 3 y 4 y 1 +y+2−y 3 −y 4 . The solution to this equation is with f an arbitrary function of k + k − . Thus our ansatz is now Having imposed the difference, the remaining condition is solved (after some algebra) by where Thus we've recovered the full Poincaré bulk mode, including its time dependence, purely from CFT considerations. This shows that we can get the complete zeroth-order expression for a local bulk field using intersecting modular Hamiltonians.
Starting with the general ansatz the condition [H mod,L,φ 0 , Φ] = 0 (108) becomes upon integration by parts 10 We first impose the condition [H mod,R,φ 0 =0 , Φ] = 0. To do this we setφ 0 = 0 andL =R in (109). Then using the method of characteristics, the most general solution to (109) is where f is an arbitrary function and c 0 is a constant. Since we also want Φ to obey we re-insert the solution (110) into (109). This now gives an equation for f (x). After some algebra the equation can be recast as 10 We will choose the region of integration to ensure that there are no boundary terms.
The parameter α can be seen to depend on only two parameters by defining and noting that We can set the free parameter φ in (105) to be φ = φ * so that g(p,q) becomes (116) The two parameters of the solutionφ * and coshφ * coshR can be identified as the coordinate parallel to the boundary and the radial coordinate, respectively, by looking at the limitL → 0, φ 0 →R. In this limitφ * →R and coshφ * coshR → 1 as expected. The region of integration is fixed by requiring that there are no boundary terms when we integrate by parts. This determines the region of integration to be cos y > coshφ * coshR cosh x On an equal-time geodesic stretching from −R toR, we show in appendix C that the boundary coordinate φ and the bulk coordinate r are related by Thus (116) is the smearing function for a bulk scalar operator Φ(r, φ * , t = 0) in a BTZ background [7]. In appendix C we show that φ * in (114) is just the φ coordinate where the bulk geodesics intersect. Thus again we have recovered the bulk space-like geodesics from the CFT.

Conclusions
In this paper we have shown that CFT operators which mimic local bulk operators commute with the modular Hamiltonian appropriate for a boundary segment whose RT surface passes through the bulk point. If two RT surfaces intersect at a point in the bulk then a bulk observable localized on the intersection must commute with both modular Hamiltonians. Turning this around, we used this as a new way to construct local bulk observables in the CFT, by constructing CFT quantities which commute with intersecting extended modular Hamiltonians. Along the way we recovered bulk space-like geodesics from the CFT.
The computations done in this paper were for AdS 3 / CFT 2 , but the generalization to higher dimensions is clear. The only complication is that AdS D requires D − 1 intersecting RT surfaces to define a bulk point.
It seems clear that at least in principle the construction can be carried out for CFT states which are not the vacuum. Indeed in this paper the finite temperature case was treated successfully. Explicit expressions may be difficult to obtain since we have little control over the modular Hamiltonian for non-vacuum states. Moreover in general the modular Hamiltonian will be non-local, so (unlike the examples treated in this paper) for generic states the approach will not lead to a system of local differential equations for the smearing functions. But in principle the same logic applies and should determine local bulk operators in the appropriately-deformed bulk background geometry.
Another, perhaps related, generalization of the construction in this paper would be to include interactions and make contact with the perturbative procedure developed in [10,11,12]. It would also be interesting to understand if there is a connection to the ideas proposed in [34,35].
In this paper we only considered scalar operators. It would be interesting to extend the construction to bulk fields with spin. For massive vector fields this seems straightforward. Bulk fields with gauge redundancy pose an additional challenge, since due to constraints they aren't local objects in the bulk even at the free field level. 11 Moreover even for bulk scalars gravitational dressing arises as an interaction effect, and once this is taken into account one cannot localize bulk scalar observables to the intersection of RT surfaces. This means that for free bulk gauge fields, and for interacting bulk scalars, one cannot simply demand that bulk observables commute with intersecting modular Hamiltonians. Whether there is an extension of the approach to deal with these issues is an interesting and important question.
The construction developed in this paper raises more speculative issues as well. For example it seems clear that the construction puts constraints on CFT states which are dual to classical bulk geometries. This comes about because a classical bulk geometry requires that an infinite family of equations, stating that different modular Hamiltonians commute with a smeared CFT operator, must all have a common solution. This restricts the form of the modular Hamiltonians and hence presumably the CFT states that can be dual to classical geometries. It would be interesting to make these restrictions more precise.

B Action ofH mod on operators off the RT surface
We make the ansatz where f solves This is the condition (87) that Φ 12 commutes with the extended modular Hamiltonian of the segment (y 1 , y 2 ). Now consider the extended modular HamiltonianH 34 mod for a different segment (y 3 , y 4 ). We wish to compute the commutator of this new modular Hamiltonian with Φ 12 . A simple computation gives [H 34 mod , Φ 12 (Z, X, T = 0)] = 2πi (y 4 − y 3 ) ((y 3 + y 4 − y 1 − y 2 )X − y 3 y 4 + y 1 y 2 ) × which is the correct action of the bulk Rindler Hamiltonian associated with the segment (X * − Z 0 , Z 0 + X * ), i.e. it generates a Rindler time translation.

C Geodesics in BTZ
The BTZ metric is ds 2 = − r 2 − r 2 + l 2 dt 2 + l 2 r 2 − r 2 + dr 2 + r 2 dφ 2 (127) We look for geodesics r(φ) which extremize the action dφ r 2 + l 2 r 2 − r 2 + dr dφ 2 (128) Since nothing depends explicitly on φ there is a constant of motion which we call r min . r min = r 2 r 2 + l 2 If we choose r(φ 0 ) = r min and require φ(r → ∞) = ±L/l the solution after a little algebra is Thus two geodesics, one stretching from −R to R and the other from φ 0 − L to φ 0 + L, intersect in the bulk at a point whose φ coordinate obeys tanh r + l φ = 1 sinh r + l φ 0 cosh r + l φ 0 − cosh r + l 2 L cosh r + l 2 R