Entanglement, holography and causal diamonds

Journal of High Energy Physics, Aug 2016

We argue that the degrees of freedom in a d-dimensional CFT can be reorganized in an insightful way by studying observables on the moduli space of causal diamonds (or equivalently, the space of pairs of timelike separated points). This 2d-dimensional space naturally captures some of the fundamental nonlocality and causal structure inherent in the entanglement of CFT states. For any primary CFT operator, we construct an observable on this space, which is defined by smearing the associated one-point function over causal diamonds. Known examples of such quantities are the entanglement entropy of vacuum excitations and its higher spin generalizations. We show that in holographic CFTs, these observables are given by suitably defined integrals of dual bulk fields over the corresponding Ryu-Takayanagi minimal surfaces. Furthermore, we explain connections to the operator product expansion and the first law of entanglemententropy from this unifying point of view. We demonstrate that for small perturbations of the vacuum, our observables obey linear two-derivative equations of motion on the space of causal diamonds. In two dimensions, the latter is given by a product of two copies of a two-dimensional de Sitter space. For a class of universal states, we show that the entanglement entropy and its spin-three generalization obey nonlinear equations of motion with local interactions on this moduli space, which can be identified with Liouville and Toda equations, respectively. This suggests the possibility of extending the definition of our new observables beyond the linear level more generally and in such a way that they give rise to new dynamically interacting theories on the moduli space of causal diamonds. Various challenges one has to face in order to implement this idea are discussed.

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Entanglement, holography and causal diamonds

Received: July Entanglement, holography and causal diamonds Jan de Boer 1 5 Felix M. Haehl 1 2 Michal P. Heller 0 1 3 Robert C. Myers 1 3 South Road 1 Durham DH 0 1 0 On leave from: National Centre for Nuclear Research , Hoz_a 69, 00-681 Warsaw , Poland 1 Science Park 904 , 1090 GL Amsterdam , The Netherlands 2 Centre for Particle Theory & Department of Mathematical Sciences, Durham University 3 Perimeter Institute for Theoretical Physics 4 Caroline Street North , Waterloo, Ontario N2L 2Y5 , Canada 5 Institute of Physics , Universiteit van Amsterdam We argue that the degrees of freedom in a d-dimensional CFT can be reorganized in an insightful way by studying observables on the moduli space of causal diamonds (or equivalently, the space of pairs of timelike separated points). dimensional space naturally captures some of the fundamental nonlocality and causal structure inherent in the entanglement of CFT states. For any primary CFT operator, we construct an observable on this space, which is de ned by smearing the associated one-point function over causal diamonds. Known examples of such quantities are the entanglement entropy of vacuum excitations and its higher spin generalizations. We show that in holographic CFTs, these observables are given by suitably de ned integrals of dual elds over the corresponding Ryu-Takayanagi minimal surfaces. Furthermore, we explain connections to the operator product expansion and the rst law of entanglement entropy from this unifying point of view. We demonstrate that for small perturbations of the vacuum, our observables obey linear two-derivative equations of motion on the space of causal diamonds. In two dimensions, the latter is given by a product of two copies of a two-dimensional de Sitter space. For a class of universal states, we show that the entanglement entropy and its spin-three generalization obey nonlinear equations of motion with local interactions on this moduli space, which can be identi ed with Liouville and Toda equations, respectively. This suggests the possibility of extending the de nition of our new observables beyond the linear level more generally and in such a way that they give rise to new dynamically interacting theories on the moduli space of causal diamonds. Various challenges one has to face in order to implement this idea are discussed. ArXiv ePrint: 1606.03307 AdS-CFT Correspondence; Conformal Field Theory; Gauge-gravity corre- - bulk spondence 1 Introduction 2 The geometry of causal diamonds in Minkowski space 2.1 Metric on the space of causal diamonds 2.2 The causal structure on the space of causal diamonds 3 Observables in a linearized approximation Dynamics on the space of causal diamonds Operators with spin and conserved currents Connection to the OPE Holographic description Euclidean signature Other elds 3.7 Two dimensions 3.1 3.2 3.3 3.4 3.5 3.6 4.1 4.2 5.1 5.2 5.3 5.4 5.5 6.1 6.2 6.3 6.4 4 Interacting elds on d = 2 moduli space Vacuum excitations Beyond vacuum excitations Evaluation of Wilson loops Pure gravity example Spin-three entanglement entropy 5 More interacting elds on d = 2 moduli space: higher spin case 6 Dynamics and interactions: future challenges De Sitter eld equations for higher spin entanglement entropy First law from Wilson loops Constraints Holographic dynamics in AdS3 Allowed quadratic local interaction terms on the space of causal diamonds Quadratic modi cations of the holographic de nition of Q(O) 7 Discussion A Geometric details A.1 Derivation of metric on the space of causal diamonds A.2 Conformal Killing vectors A.3 Moduli space of spacelike separated pairs of points { i { It has now been a decade since Ryu and Takayanagi [1, 2] discovered an elegant geometric prescription to evaluate entanglement entropy in gauge/gravity duality. In particular, the entanglement entropy between a (spatial) region V and its complement V in the boundary theory is computed as SEE(V ) =ext : A(v) v V 4GN That is, one determines the extremal value of the Bekenstein-Hawking formula evaluated on bulk surfaces v which are homologous to the boundary region V . In the subsequent years, holographic entanglement entropy has proven to be a remarkably fruitful topic of study. In particular, it provides a useful diagnostic with which to examine the boundary theory. For example, it was shown to be an e ective probe to study thermalization in quantum quenches, e.g., [3{6] or to distinguish di erent phases of the boundary theory, e.g., [7{9]. In fact, such holographic studies have even revealed new universal properties that extend beyond holography and hold for generic CFTs, e.g., [10{13]. However, holographic entanglement entropy has also begun to provide new insights into the nature of quantum gravity in the bulk. As rst elucidated in [14, 15], the RyuTakayanagi prescription indicates the essential role which entanglement plays in creating the connectivity of the bulk geometry or more generally in the emergence of the holographic geometry. In fact, this has lead to a new prescription to reconstruct the bulk geometry in terms of a new boundary observable known as `di erential entropy', which provides a novel prescription for sampling the entanglement throughout the boundary state [16{19]. The distinguished role of extremal surfaces in describing entanglement entropy has led to several other important insights. There is by now signi cant evidence that the bulk region which can be described by a particular boundary causal domain is not determined by causality alone, as one might have naively thought, but rather it corresponds to the socalled `entanglement wedge,' which in general extends deeper into the bulk, e.g., [20{23]. That is, the bulk region comprised of points which are spacelike-separated from extremal surfaces attached to the boundary region and connected to the corresponding boundary causal domain [ 22 ]. This entanglement wedge reconstruction in turn led to the insight that local bulk operators must have simultaneous but di erent approximate descriptions in various spatial subregions of the boundary theory, which resulted in intriguing connections to quantum error correction [24{26]. We also notice that while it is not at all clear that { 1 { a suitable factorization of the full quantum gravity Hilbert space corresponding to the inside and outside of an arbitrary spatial domain exists (there certainly is no obvious choice of tensor subfactors on the CFT Hilbert space), the RT prescription does provide a natural choice for such a factorization for extremal surfaces, and entanglement wedge reconstruction supports this point of view. It is therefore conceivable that a reorganization of the degrees of freedom which crucially relies on extremal surfaces will shed some light on the (non)locality of the degrees of freedom of quantum gravity, and this was in fact one of the original motivations for this work. One interesting result that was brought to light by holographic studies of the relative entropy [27] was the ` rst law of entanglement'. The relative entropy is again a general diagnostic that allows one to compare di erent states reduced to the same entangling geometry [28, 29]. For `nearby' states, the leading variation of the relative entropy yields a result reminiscent of the rst law of thermodynamics, i.e., SEE = hHmi ; where Hm is the modular or entanglement Hamiltonian for the given reference state 0 . While the latter is a useful device at a formal level [30], in generic situations, the modular Hamiltonian is a nonlocal operator, i.e., Hm cannot be expressed as a local expression constructed from elds within the region of interest. However, a notable exception to this general rule arises in considering a spherical region in the vacuum state of a CFT and in this case, the rst law (1.2) becomes SEE = hHmi = 2 Z B d d 1x0 R2 j ~x 2R ~x0j2 hTtt(~x0)i : Here B denotes a ball of radius R centred at ~x on a xed time slice, while hTtti is the energy density in the excited state being compared to the vacuum. Examining this expression holographically, the energy density is determined by the asymptotic behaviour of the metric near the AdS boundary, e.g., [31]. In contrast, through eq. (1.1), the variation of the entanglement entropy is determined by variations of the geometry deep in the bulk spacetime. Hence eq. (1.3) imposes a nonlocal constraint on perturbations of the AdS geometry which are dual to excitations of the boundary CFT. However, if one examines this constraint for all balls of all sizes and all positions, as well as on all time slices, this can be re-expressed in terms of a local constraint on the bulk geometry [32{34], namely, perturbations of the AdS vacuum geometry must satisfy the linearized Einstein equations! In terms of the boundary theory, the holographic results above point towards the utility of considering the entanglement entropy as a functional on the space of all entangling surfaces (or at least a broad class of such geometries) to characterize various excited states of a given quantum eld theory. In this regard, one intriguing observation [35] is that the perturbations of the entanglement entropy of any CFT naturally live on an auxiliary de Sitter geometry. In particular, the functional SEE(R; ~x), de ned by eq. (1.3), satis es the Klein-Gordon equation (1.2) (1.3) rdS m2 SEE = 0 ; { 2 { (1.4) in the following de-Sitter (dS) geometry: ds2dSd = L2 R2 dR2 + d~x2 : Note that the radius of the spheres R plays the role of time in dS space. The mass above is given by equation with an appropriate mass [35] | see section 3.2 below. The proposal of [35] was that this new dS geometry may provide the foundation on which to construct an alternative `holographic' description of any CFT. That is, it may be possible to reorganize any CFT in terms a local theory of interacting elds propagating in the auxiliary spacetime. We stress that here the CFT under consideration need not be holographic in the conventional sense of the AdS/CFT correspondence, and hence there is no requirement of a large central charge or strong coupling. Of course, the discussion in [35] only provided some preliminary steps towards establishing this new holographic dictionary and such a program faces a number of serious challenges. For example, the dS scale only appears as an overall factor of L2 in eq. (1.4) and so remains an undetermined constant. Of course, our experience from the AdS/CFT correspondence suggests that L would be determined in terms of CFT data through the gravitational dynamics of the holographic geometry and so here one faces the question of understanding whether the new auxiliary geometry is actually dynamical. Another challenge would be to produce a holographic description of the time dependence of quantities in the CFT, since the above construction was rmly rooted on a xed time slice. A natural extension is to consider all spherical regions throughout the d-dimensional spacetime of the CFT, i.e., all of the ball-shaped regions of all sizes and at all positions on all time slices. As described in [35], this extended perspective yields an auxiliary geometry which is SO(2; d)=[SO(1; d 1) SO(1; 1)] and the perturbations SEE can be seen to obey a wave equation on this coset. Further it was noted that this auxiliary space is 2d-dimensional and has multiple time-like directions. This new expanded auxiliary geometry is the starting point for the present paper. As we will describe, in the context where we are considering all spheres throughout the spacetime, it is more natural to think in terms of the causal diamonds, where each causal diamond is the domain of dependence of a spherical region. Following [36], our nomenclature will be to refer to the moduli space of all causal diamonds as generalized kinematic space, since it is a natural generalization of the kinematic space introduced there, i.e., the space of ordered intervals on a time slice in d = 2. Our focus will be to construct interesting nonlocal CFT observables on causal diamonds, similar to the perturbation SEE in 1We should note that for d = 2 essentially the same dS geometry appeared in [36], which used integral geometry to describe the relation of MERA tensor networks [37] to the AdS3/CFT2 correspondence. { 3 { eq. (1.3).2 Our objective will be two-fold: the rst is to examine if these new observables and the generalized kinematic space provide a natural forum to construct a complete description of the underlying CFT. The second is to investigate how the new perspective of the nonlocal observables interfaces with the standard holographic description given by the AdS/CFT correspondence. The remainder of the paper is organized as follows: section 2 contains a detailed discussion of the geometry of the moduli space of causal diamonds. In section 3 we de ne linearized observables associated with arbitrary CFT primaries. These observables are local elds obeying two-derivative equations of motion on the space of causal diamonds and they explain and generalize various known statements about the rst law of entanglement entropy, the OPE expansion of twist operators, and the holographic Ryu-Takayanagi prescription. From section 4 onwards, we focus on d = 2 and the question of extending the previous framework to nonlinearly interacting elds on the space of causal diamonds. Section 4 is concerned with a certain universal class of states, for which the entanglement entropy satis es a nonlinear equation with local interactions on the moduli space. Section 5 generalizes this discussion to higher spin theories. In particular, we construct a framework where the entanglement and its spin-three generalization are described by two nonlinearly interacting elds on the space of causal diamonds. Some challenges for the de nition of more general nonlinearly interacting elds are discussed in section 6. In section 7, we conclude with a discussion of open questions and future directions for this program of describing general CFTs in terms of nonlocal observables on the moduli space of causal diamonds, and also formulating the AdS/CFT correspondence within this framework for holographic CFTs. Appendix A discusses various geometric details and generalizations. Some of our conventions are xed in appendix B. Appendix C contains explicit computations to verify the AdS/CFT version of our generalized rst law. Note. While this work was in progress, the preprint [38] by Czech, Lamprou, McCandlish, Mosk and Sully appeared on the arXiv, which explores ideas very similar to the ones presented here. 2 The geometry of causal diamonds in Minkowski space In this section, we examine the geometry of the generalized kinematic space introduced in [35]. We begin by deriving the natural metric on this moduli space of all causal diamonds in a d-dimensional CFT. As noted above, this 2d-dimensional metric will turn out have multiple time directions, and in particular, has signature (d; d). We will also discuss how to intuit this signature geometrically in terms of containment relations between causal Metric on the space of causal diamonds Spheres are destined to play a special role in CFTs, as the conformal group SO(2; d) in d dimensions maps them into each other. The past and future development of the region 2As we review in appendix A.2, conservation and tracelessness of the stress tensor allows the modular Hamiltonian to be evaluated on any time slice in spanning the corresponding causal diamond. { 4 { ` c x w HJEP08(216) (2.1) (2.2) (2.3) timelike separated pair of points (x ; y ) is equivalent to specifying a spacelike (d null separated from both x and y , i.e., satisfying eq. (2.2). The alternative parametrization in terms of c = 12 (y +x ) and ` = 12 (y x ) will prove convenient in section 2.2. enclosed by a (d 2)-sphere form a causal diamond and hence the space of all (d 2)-spheres is the same as the space of all causal diamonds.3 Therefore a generic (d 2)-sphere can be parametrized in terms of the positions of the tips of the corresponding causal diamond. That is, given these positions, x and y , the (d 2)-sphere is the intersection of the past light-cone of the future tip and the future light-cone of the past tip, as shown in gure 1. Of course, these points are necessarily timelike separated,4 i.e., (x y)2 < 0 : The corresponding sphere comprising the intersection of the light-cones illustrated in the gure can be de ned as the set of points w which are null-separated from both x and y : (w x)2 = 0 and (w y)2 = 0 : Due to these considerations, in what follows we will interchangeably use the notions of spheres, causal diamonds, and pairs of timelike separated points. The generalized kinematic space is the moduli space of all causal diamonds. The easiest way to construct the metric on this space is to start with an (d + 2)-dimensional embedding space parametrized by coordinates Xb = (X ; X ; Xd) ; 3Implicitly, then we are assigning an orientation to the spheres, i.e., the interior is distinguished from the exterior. One could also consider unoriented spheres, which would amount to an additional Z2 identi cation in the coset given in eq. (2.11). See [35] for further discussion. 4Our notation here and throughout the following is that for d-dimensional vectors, (y x)2 = (y x) (y x) : { 5 { with = 0; ; d 1. Further this embedding space has a at metric with signature (2; d): where h ; i denotes the inner product with respect to the metric (2.4). It can be thought of as a set of all the points in the embedding space that can be reached by acting with SO(2; d) transformations on a unit timelike vector, e.g., on the vector (1; 0; : : : ; 0). Since any timelike vector in (2.4) is preserved by an SO(1; d) subgroup of the conformal group, (d + 1)-dimensional anti-de Sitter space is a coset space SO(2; d)=SO(1; d). The metric on this coset is induced by the embedding space metric (2.4). For example, the Poincare patch AdS metric hyperboloid (2.5): is obtained from the metric (2.4) upon using the following parametrization of the AdS Of course, the asymptotic boundary of AdS space is reached by taking the limit z ! 0. In the context of the AdS/CFT correspondence, SO(2; d) transformations leaving the embedding geometry (2.4) invariant become the conformal transformations acting on the boundary theory. Of course, this highlights the advantage of the embedding space approach. Namely, the SO(2; d) transformations act linearly on the points (2.3) in the embedding space. In the following, we will phrase our discussion in terms of the geometry of the CFT background being de ned by the boundary of the AdS hyperboloid (2.5) because we feel that it is an intuitive picture familiar to most readers. However, with only minor changes, the entire discussion can be phrased in terms of the embedding space formalism, e.g., [39{ 41], which can be used to consider any CFT and makes no reference to the AdS/CFT correspondence. Hence we stress that the geometry of the generalized kinematic space that emerges below applies for general d-dimensional CFTs. We now turn to the moduli space of causal diamonds in a CFT, which we construct using the language of cosets, in similar manner to that introduced above in discussing the { 6 { X T b X1 HJEP08(216) vector T b and to the spacelike vector Sb (the latter being hidden in the suppressed dimensions). The intersection of the d-plane with AdSd+1 yields the green minimal surface. Its boundary as the hyperboloid approaches the red lightcone de nes a (d 1)-sphere in the CFT. AdS geometry (2.5). In order to describe a sphere in a CFT, we choose a unit timelike vector T b and an orthogonal unit spacelike vector Sb, both of which are anchored at the origin of the (d+2)-dimensional embedding space. That is, we choose two vectors satisfying The sphere is now speci ed by considering asymptotic points in the AdS boundary that are orthogonal to both of these unit vectors, i.e., hT; Xi z!0 = 0 and hS; Xi z!0 = 0 : To explicitly illustrate this construction of a sphere in the CFT, let us consider the coordinates (2.7) yielding the Poincare patch metric (2.6). A convenient choice of the unit vectors is then T b = (0; 1; 0; : : : ; 0) Sb = (0; 0; 0; : : : ; 1) ! ! w0 = 0 ; w w = 1 : The expressions on the right denote the surfaces in the asymptotic geometry that are picked out by the orthogonality constraints (2.9), i.e., T b selects a particular time slice in the boundary metric while Sb selects a timelike hyperboloid. Of course, the intersection of these two surfaces then yields the unit (d 2)-sphere ij wiwj = 1 (on the time slice w0 = 0). Now a particular choice of the unit vectors, T b and Sb, picks out a particular sphere in the boundary geometry. Acting with SO(2; d) transformations, we can then reach all of the (2.8) (2.9) (2.10) { 7 { other spheres throughout the d-dimensional spacetime where the CFT lives. To determine the coset describing the space of all spheres, we must rst nd the symmetries preserved by any particular choice of the unit vectors. Given two unit vectors satisfying eq. (2.8), we have de ned a timelike two-plane in the embedding space. Hence the SO(2; d) symmetry broken to SO(1; d 1) transformations acting in the d-dimensional hyperplane orthogonal to this (T; S)-plane, as well as the SO(1; 1) transformations acting in the two-plane. Thus, in analogy with AdS coset construction above, the natural coset describing the moduli space of spheres, or alternatively of causal diamonds, in d-dimensional CFTs is M (d) Of course, this is precisely auxiliary geometry already described in [35]. The interpretation of the stabilizer group, which preserves a given sphere in the CFT, is as follows: the SO(1; d 1) factor of the stabilizer group is the subgroup of SO(2; d) comprising of (d 1)(d 2)=2 rotations and d 1 spatial special conformal transformations leaving a given sphere invariant. While it is obvious that the former transformations preserve spheres centred at the origin, it can also be veri ed that the latter do so as well. Further, let us note that these transformation also leave invariant the time slice in which the sphere is de ned. The additional SO(1; 1) represents a combination of special conformal transformations and translations, which both involve the timelike direction and leads to a modular ow generated by the conformal Killing vector K | see appendix A.2. The latter was constructed precisely in such a way to preserve a given spherical surface. We can also perform a simple cross-check at the level of counting dimensions. The moduli space of causal diamonds can parametrized by a set of 2d coordinates: x and y , i.e., the positions of the tips of the causal diamonds. Now, the number of generators of the isometry group SO(2; d) is (d + 2)(d + 1)=2, whereas for the stabilizer group SO(1; d 1) SO(1; 1) we have d(d 1)=2 + 1 = d(d + 1)=2 generators. The di erence between the two numbers matches the dimensionality of the space of causal diamonds, i.e., 2d, as it must. In the context of the AdS/CFT correspondence, we can remove the asymptotic limit from the orthogonality constraints (2.9), i.e., consider hT; Xi = 0 and hS; Xi = 0. These constraints now specify not only the sphere on a constant time slice of the AdS boundary (at z = 0), but the entire minimal surface anchored to this sphere. With the simple example of T b and Sb given in eq. (2.10), these constraints yield the unit hemisphere z2+ ij wiwj = 1 on the time slice w0 = 0. Of course, using the Ryu-Takayanagi prescription (1.1), the area of this surface computes the entanglement entropy of the region enclosed by the (asymptotic) sphere in the vacuum of the boundary CFT. M Let us now move to the object of prime interest for us, which is the metric on the coset (d) induced by the at geometry of the (d + 2)-dimensional embedding space. Towards this end, we parameterize motions in this generalized kinematic space by variations of the unit vectors T b and Sb. Of course, these are naturally contracted with the embedding space metric (2.4) and so the most general SO(1; d 1)-invariant metric can be written as: ds2 = T T hdT; dT i + SS hdS; dSi + T S hdT; dSi ; (2.12) { 8 { where T T , SS and T S are constant coe cients. Also requiring invariance under SO(1; 1) transformations, i.e., under boosts in the (T; S)-plane, requires that we set T S = 0 and SS = T T L2 | only the relative sign of SS and T T is determined by boost invariance but we choose SS > 0 here for later convenience. This then yields ds2 = L2 ( hdT; dT i + hdS; dSi) : (2.13) Next, we must impose the conditions (2.8) and (2.9) in the above expression to x the metric (up to an overall prefactor) in terms of geometric data in the CFT. This calculation is straightforward but somewhat tedious, and we refer the interested reader to appendix A.1 for the details. Our nal result for the metric on the coset M (d) given in (2.11) becomes: where x and y denote the past and future tips of the corresponding causal diamond, as illustrated in gure 1. This metric is the main result of the present section and the starting point for our investigations of the generalized kinematic space in the subsequent sections. Some comments are now in order: rst, it is straightforward to verify that this metric (2.14) is invariant under the full conformal group. Second, the pairs (x ; y ) appear as pairs of null coordinates in the metric (2.14). As a result, this metric on the coset (2.14) has the highly unusual signature (d; d). Third, it is amusing to notice that while AdS geometrizes scale transformations, the coset geometrizes yet another d 1 additional conformal transformations. Let us now discuss two special cases for which the general result (2.14) simpli es: lying on a given constant time slice, which we can always take to be t = 0. That is, we choose y0 = x0 = R and ~x = ~y and then we are considering spheres on the t = 0 slice with radius R and with ~x giving the spatial position of their centres. Constraining the coordinates x and y in this way, the coset metric (2.14) reduces to ds2 ~x=~y; y0= x0=R = L2 R2 dR2 + d~x2 ds2dSd : That is, we have recovered precisely the d-dimensional de Sitter space appearing in eq. (1.5) as a submanifold of the full coset M (d). Example 2: CFT in two dimensions. A second special case of interest is the restriction to d = 2. The metric on the coset in two dimensions has a structure of a direct product of two copies of two-dimensional de Sitter space. One can see this explicitly by introducing right- and left-moving light-cone coordinates, e.g., we replace the Minkowski coordinates ( 0; 1) with Then we may specify the two-dimensional causal diamonds, de ned by (x ; y ) above, in terms of the positions of their four null boundaries | see gure 3, (u; u) (x1 x0; x1 + x0) ; (v; v) (y1 y0; y1 + y0) : (2.15) (2.16) (2.17) { 9 { will provide a useful parametrization of the given diamond in section 3.7. Changing the endpoints corresponds to moving in the moduli space of causal diamonds parametrized by (u; u; v; v); thereby u is constant if x moves along the line = u, and so forth. Finally re-expressing the coset metric (2.14) in terms of these coordinates yields ds2 = 2L2 d=2 du dv (u v)2 + (u du dv v)2 2 1 nds2dS2 + ds2dS2 o : (2.18) Notice that the rst copy of de Sitter metric is only a function of the right-moving coordinates, whereas the second copy depends only on the left-moving coordinates. We chose the normalization on the right hand side of eq. (2.18) in such a way that L is the curvature scale in each de Sitter component and upon restricting to a timeslice (i.e., u = v x R and v = u x + R), eq. (2.15) obviously emerges. This way we can heuristically think of each of the two copies of dS2 in (2.18) as a copy of the geometry in eq. (2.15). Of course, the product structure found in the moduli space metric here has its origins in the fact that for two dimensions, the conformal group itself decomposes into a direct product, i.e., SO(2; 2) ' SO(2; 1) SO(2; 1), where the two factors act separately on the right- and left-moving coordinates. Hence the moduli space (2.11) of intervals in d = 2 CFTs becomes M (2) = SO(2; 1) where we recognize that each of factors corresponds to a two-dimensional de Sitter space. 2.2 The causal structure on the space of causal diamonds Given the metric (2.14) on the moduli space of causal diamonds, we are in the position to study the causal structure of this space. The essential feature of this causal structure comes from the fact that the space possesses d spacelike and d timelike directions. We start by writing the metric (2.14) in terms of the coordinates (2.19) (2.20) = u v = x c y + x 2 y 2 x : and ` Here, c denotes the position of the centre of the causal diamond or, equivalently, the centre of the corresponding sphere. Similarly, ` denotes the vector from the centre to the future tip of the causal diamond | see gure 1. The metric (2.14) then becomes First, we note that `2 < 0 from eq. (2.1), i.e., the tips of the causal diamond are timelike separated. Further, we observe that the tensor `22 ` ` is positive de nite again because ` is a timelike vector. This is easily veri ed by picking a frame where, say, ` / 0 . In such a frame, the metric (2.21) reduces to ` ` (dc dc d` d` ) : (2.21) (dc dc d` d` ) : (2.22) Therefore, the sign of ds2 is determined solely by the last factor in eq. (2.21) containing the di erentials. In particular, we can now see that c are the d spacelike directions in the space of causal diamonds, while ` are the d timelike directions. To make this precise, consider two in nitesimally close causal diamonds speci ed by their coordinates 1 = (c ; ` ) and 2 = (c + dc ; ` + d` ), we say that their separation is spacelike, timelike or null if ds2 (c ; ` ) is positive, negative or zero, respectively. From this, it is now easy to intuit the timelike, spacelike and null directions in the moduli space of causal diamonds as follows: (a) Moving the centre c of a causal diamond by an in nitesimal amount dc in any of the d directions of the background Minkowski spacetime of the CFT corresponds to moving in a spacelike direction in the coset space. Geometrically, this corresponds to translating the diamond without deforming it. (b) Moving any of the `relative' coordinates ` by some d` corresponds to a timelike displacement in the coset space. In the diamond picture, this corresponds to stretching the diamond in one of d independent ways while holding the centre of the diamond xed. (c) Null movements correspond heuristically to deforming the diamond by the `same' amount as it is translated in spacetime, as quanti ed by the condition ds2 = 0. These cases are illustrated in gure 4 for in nitesimal displacements. It is noteworthy that moving the centre of causal diamond in the time direction, i.e., with dc0, produces a spacelike displacement in the kinematic space. We return to discuss this point in section 7. Let us now give a slightly di erent perspective on the measure of distances on this moduli space. Consider two causal diamonds, speci ed by the coordinates of their tips, 1 = (x1 ; y1 ) and 2 = (x2 ; y2 ). The conformal symmetry ensures that there exists a natural conformally invariant measure of distance, namely, the cross ratio r(x1; y1; x2; y2) (y1 (y1 x2)2 (y2 x1)2 (y2 x1)2 x2)2 : (2.23) moves correspond to deformations of the diamond which leaves its centre xed, (c) null moves correspond to a combination of the previous two by the `same' amounts. As we will show the cross ratio paves the way to understanding the global causal structure of the moduli space of diamonds, however, rst we relate this expression to the previous discussion. Hence we translate it to the `centre of mass' coordinates and consider the two causal diamonds with 1 = (c ; ` ) and 2 = (c + c ; ` + ` ). Then the invariant cross ratio reads (2.24) (2.25) r( 1; 2) = (2` + ` + c)2(2` + ` c)2 16 `2 (` + `)2 = 1 + 1 2`2 2 `2 ` ` ( c c ` ` ) + : In the second line, we are expanding the cross ratio for in nitesimal displacements and the ellipsis indicates terms of cubic order in c and ` . Comparing to eq. (2.21), we see that causal diamonds that are very nearby r( 1; 2) ' 1 1 2L2 ds2 + : That is, for in nitesimal displacements, the cross ratio encodes the invariant line element (2.21) of the generalized kinematic space. Further, we observe that eq. (2.25) shows that timelike, spacelike and null displacements in this moduli space correspond, respectively, to r > 1, r < 1 and r = 1. Two other observations about the cross ratio in eq. (2.24): we note that the centre of mass coordinates c are Killing coordinates of the metric (2.21), i.e., the metric is independent of these coordinates. However, this feature also extends to nite separations, as is apparent from the rst line of eq. (2.24). That is, the position c of the reference diamond 1 is irrelevant for the distance to 2 and only the relative c appears in this expression. Similarly, dc = d` yields a null displacement in eq. (2.21) but two diamonds separated by nite displacements with c = ` are also null separated, i.e., it is straightforward to show that the rst line of eq. (2.24) yields r = 1 in this situation. dmax = Geometrically, similarly, c = c = ` corresponds to two diamonds whose past tips coincide (and ` corresponds to diamonds whose future tips coincide). We can go further and de ne an invariant `geodesic distance' function between two diamonds 1 = (x1 ; y1 ) and 2 = (x2 ; y2 ) in terms of the cross ratio as d( 1; 2) = < 8 As we will show in examples, this distance function computes geodesic distance between nitely separated diamonds, within the range of validity speci ed above. Note then that the corresponding cross ratio is greater than, less than or equal to 1 if two diamonds may be connected by a timelike, spacelike or null geodesic. However, the converse need not be true, i.e., , even if the cross ratio is positive, there may not be a geodesic connecting the corresponding diamonds | see further discussion below. Further, note that as r ! 1, the corresponding causal diamonds become in nitely timelike separated. However, there is a maximal spacelike separation that can achieved by following geodesics through the coset, i.e., at r = 0, we nd dmax = Equipped with the distance function (2.26), let us brie y comment on the structure of the cross ratio (2.23). We have the following interesting cases in general: (x1 y1)2 ! 0 or (x2 y2)2 ! 0: if one of the diamonds' volumes shrinks to zero,5 the cross ratio and the distance function both diverge, in particular, d( 1 ; 2) ! This is just the statement that zero-volume diamonds lie at the timelike in nity of the coset space M (d). y1 ! y2 or x1 ! x2: if either the past or future tips of two diamonds coincide, the cross ratio becomes one and the invariant distance d( 1; 2) vanishes, i.e., the diamonds become null separated. y1)2 ! 1 or (x2 y2)2 ! 1: if either of the diamonds' volumes grows to in nity, the cross ratio vanishes and the distance function reaches its maximal value, x2)2 ! 0 or (y2 x1)2 ! 0: if the future (past) tip of one causal diamond approaches the lightcone of the past (future) tip of the other diamond (as illustrated gure 5), the cross ratio vanishes and the corresponding separation again reaches the maximal value dmax = Let us comment further on the domain of validity of our geodesic distance function. As de ned in eq. (2.26), this function is well-de ned for r 0. However, as commented above, merely having r 0 does not ensure that the corresponding causal diamonds are connected by a geodesic. Further, certain pairs of causal diamonds will also yield r < 0. Examining eq. (2.23), we see that both factors in the denominator are negative by construction, i.e., 5The tips may not coincide in this limit rather they only need to be null separated. 2 leaves the red (green) shaded lightcone region, the geodesic distance d( 1 ; 2) becomes in nite, i.e., the diamonds are no longer geodesically connected. An example of this happening would be by moving the tip x2 along the arrow towards the lightcone of y1. the tips of each casual diamond must be timelike separated, and hence the sign of r is determined by the numerator. Let us consider beginning with two nearby diamonds, 1 and 2. Both (y1 x2)2 < 0 and (y2 have r x1)2 < 0 so that the cross ration is positive. As indicated by eq. (2.24), we will 1 in this situation. If we deform the second diamond away from 1 in a spacelike direction, (not necessarily along a geodesic), the cross ratio will decrease. As described above, if the future (past) tip of 2 reaches the lightcone of the past (future) tip of 1 the cross ratio and the corresponding distance vanishes | see gure 5. If we continue deforming in the same direction, one of the factors in the numerator is now positive and r becomes negative, e.g., pushing the future tip of 2 out of causal contact with the past tip of 1 gives (y2 x1)2 > 0. Now in this range of r, the distance function (2.26) is not de ned and there is no geodesic connecting the corresponding causal diamonds. Hence submanifold of con gurations where r ( rst) vanishes de nes the `maximum' range which the geodesics originating at 1 can reach in the kinematic space. Note that generically if 2 lies on this boundary where r = 0, then the two diamonds will not be connected by a geodesic. However there are exceptional con gurations with a vanishing cross ratio, which are connected. These are `antipodal' points in the kinematic space, which are in fact connected by multiple geodesics | see further discussion below. As noted above, this con guration yields to the maximal spacelike separation that can be reached along a geodesic, i.e., dmax = One can further deform 2 so that the two diamonds become completely out of causal contact with each other, i.e., both (y1 x2)2 > 0 and (y2 x1)2 > 0. In this case, the cross ratio passes through zero again to reach positive values. However, even though eq. (2.26) is well de ned for these diamonds, there will still be no geodesic connecting them. (a) spacelike (b) timelike (c) null all three diamonds are spacelike separated from each other. Case (b) shows three timelike separated causal diamonds. Finally, all diamonds in (c) are null separated. HJEP08(216) (two-dimensional) causal diamonds. In particular, note the cases (a) and (b) of that gure, which illustrate two statements that are generally true (in any number of dimensions): (i) If two causal diamonds are contained within one another, then they are timelike separated. (ii) If two causal diamonds touch in at least one corner, then they are null separated. Let us now return to the two examples which we identi ed as being of particular interest in section 2.1: xed time slice. If we compare diamonds 1;2 on a given time slice, we know from our previous discussion that we are restricting to a submanifold with the geometry of d-dimensional de Sitter space. Taking the time slice to be t = 0, we have c01 = c02 = 0 and `i1 = `i2 = 0. Using the same coordinates as before, xi ci and R `0 > 0, the cross ratio simpli es as rdSd (R1; ~x1; R2; ~x2) = (R1 + R2)2 + (~x1 ~x2)2 2 16 R12 R22 0 : (2.27) We observe the following causal relations between spatial spheres lying on a common time 1 if (~x1 ~x2)2 (R1 R2)2, i.e., one sphere is contained within the other. 1 if (~x1 ~x2)2 (R1 R2)2, i.e., the spheres overlap but neither is fully slice:6 rdSd rdSd contained within the other. rdSd = 1 if and only if (~x1 at least one point. ~x2)2 = (R1 R2)2, i.e., the spheres tangentially touch in 6We assume here that (~x1 ~x2)2 (R1 + R2)2, for otherwise the spheres would not be geodesically connected | see further discussion in the following. 1 r = r = 1 constant cross ratio r in dSd. The pink shaded \shadow" region is not connected to the diamond 1 by any geodesic. It can naturally be reached through geodesics starting at the antipodal of 1 . Note that rdSd ! 0 as (~x1 the two spheres become disjoint. ~x2)2 ! (R1 + R2)2, which corresponds to the point where It is straightforward to show that this de Sitter geometry is a `completely geodesic' submanifold of the full kinematic space (2.11). That is, all of the geodesics within dSd are also geodesics of M (d). Hence upon substituting eq. (2.27), it is sensible to compare eq. (2.26) to the geodesic distances in de Sitter space with the metric (2.15) and one can easily verify that d( 1 ; 2) reduces to the expected geodesic distances. To provide some intuition for our previous discussion, gure 7 illustrates representative geodesics emanating from a particular point in the dS geometry.7 We observe here that the cross ratio (2.27) never becomes negative for spheres restricted to a xed time slice, however, it does reach zero as noted above just as the spheres become disjoint. As illustrated in the gure, the boundary where r = 0 corresponds to the past and future null cone emerging 1 from the antipodal point to . Hence there are `shadow regions' in the dS space which cannot be reached along a single geodesic originating from this reference point. Note, however, that there are an in nite family of spacelike goedesics that extend from 1 to this antipodal point. Example 2: CFT in two dimensions. In our previous discusion, we showed that for d = 2, the coset factorizes into dS2 dS2, with the metric as in eq. (2.18). The cross ratio 7The planar coordinates used in eq. (2.15) and above actually only cover half of the de Sitter geometry. The surface R = 1 would correspond to a diagonal running across the Penrose diagram in gure 7. The gure and our discussion here assume a suitable continuation of the cross ratio to the entire geometry. Let us add here that the additional Z2 identi cation discussed in footnote 3 would here identify points by an inversion in the square in gure 7, as well as an inversion on the corresponding S d 2 at each point on the diagram, to produce elliptic de Sitter space. With regards to the minimal geodesic distances, this identi cation would essentially remove the right half of the square, e.g., there would no longer be any shadow regions. HJEP08(216) r also factorizes when written in the fu; v; u; vg coordinates: rdS2 dS2 ((u; v)1; (u; v)1; (u; v)2; (u; v)2) = rdS2 (u1; v1; u2; v2) rdS2 (u2; v2; u2; v2) ; (2.28) where the conformally invariant cross ratio for two points on the dS2 factor is given by the family of diamonds ( ) = (c1 ; and similarly with bars. Using this factorization of the cross ratio, one can then compute the geodesic distance on dS2 dS2 using eq. (2.26). We close this section with two explicit examples of simple geodesics on the full kine(d). First, consider some diamond 1 = (c1 ; `1 ). We wish to compare it with `1 ) for 0 < < 1. One can verify that parameterizes a timelike geodesic in the space of causal diamonds. As zero size and approaches a locus in the asymptotic past. Similarly, ! 0, the diamond shrinks to ! 1 follows a geodesic to future asymptotia. The geodesic distance in this case can be computed explicitly: Z = 0 q d( 1; ( 0)) = ds2 ( ( )) = cosh 1 1 + 0 2 p 0 : A second simple example corresponds to a class of null geodesics ( ) = (c ( ); ` ( )) denotes the a ne parameter along the geodesic. Here we begin by noting that because the center of mass coordinates are Killing coordinates for the metric (2.21), the following are conserved quantities along any geodesics in the kinematic space: P = P = p =1 L2 `2 L2 `2 (2.29) (2.30) (2.31) (2.32) (2.33) Further, the full geodesic equations for ` ( ) simplify greatly upon substituting @ c = and c0 = 0 t( ), which yields which consistently maintains the desired equality between @ c and above, c = ` corresponds to two diamonds whose past/future tips coincide and so these geodesics correspond to a simple monotonic trajectory through a family of causal diamonds where one tip remains xed. A simple example is given by choosing ` = 0 R( ) R = R1= = t ; where R1 is a constant determining the radius of the corresponding sphere at = 1. 3 Observables in a linearized approximation As discussed in the introductions, we are interested in trying to construct new nonlocal observables SO(x; y) with a (local) primary operator O in the CFT and associated to a causal 2 `2 2 `2 ` ` diamond with past and future tips, x and y. Our motivation in the present section is to construct extensions of the rst law of entanglement for spherical regions in the CFT vacuum. Again, as shown in eq. (1.3), the perturbations in the entanglement entropy is given by the expectation value of a local operator, the energy density, integrated over the region enclosed by the sphere. This result was used in [35] to show that such rst order perturbations obey a free wave equation on the corresponding kinematic space, i.e., d-dimensional de Sitter space. Moreover, a generalization of the rst law was constructed for a conserved higher spin current, which yields an analogous charge Q(s) de ned on the spherical region which also obeys a free wave equation on de Sitter space. Here, we would like to extend these results characterizing small excitations of the vacuum to arbitrary scalar primaries.8 We propose that a natural generalization of the rst law to arbitrary primaries takes the following form:9 SO(x; y) Q(O; x; y) = C O Z d d where the integral is over the causal diamond D(x; y) with past and future endpoints x; y, and O is the scaling dimension of the primary operator O. The constant C O is a normalization constant for which there is no canonical choice at the linearized level. Note that the integral above diverges for d 2, however, a universal nite term can still be extracted in this range. We return to this point in section 7. In the following, we will show that the quantity Q(O) has the following four properties: diamonds M (d), which was introduced in section 2. 1. Q(O) obeys a simple two-derivative wave equation (3.8) on the moduli space of causal 2. Q(O) reduces to a known `charge' associated with a spherical entangling surface in case that O is a conserved (higher spin) current [35]. 3. Q(O) can be interpreted as a resummation of all terms in the OPE of two operators of equal dimension which contain O and all its conformal descendants. It is therefore a natural building block of contributions to correlation function where two operators fuse into the O-channel. 4. In the case where the CFT has a holographic dual in the standard sense, Q(O) has a very simple bulk description. If is the bulk scalar that corresponds to O, we de ne Qholo(O; x; y) = Cblk Z 8 GN B~(x;y) d d 1 p u h (u) ; (3.2) where B~(x; y) is the minimal surface whose boundary @B~(x; y) matches the maximal sphere at the waist of the causal diamond in the boundary CFT, i.e., the intersection of the past light-cone of y with the future light-cone of x. We will show that 8We will brie y comment on non-scalar primaries later in this section; for two-dimensional conformal eld theories we will present results for general primaries in section 3.7. 9We are using the standard notation here that (y x)2 = (y x) (y x) and hence each of the three inner products in the kernel is negative. HJEP08(216) nonlinear equation for entanglement entropy valid in generic gravitational backgrounds but in the absence of other sources. It is also intriguing to notice that for space-like separated points, V itself is proportional to the geodesic distance between the two points, so that the constant curvature condition may have a natural meaning in that case as well. To test these ideas, one could for example check whether they apply to entanglement entropy in explicitly known non-trivial gravitational backgrounds such as black holes. We hope to return to these issues at some point in the future. Generalized twist operators. One open question is to provide a nonlinear generalization of observables introduced in section 3. Motivated by considerations of entanglement entropy, we are drawn to consider twist operators with regards to this issue. Recall that as was brie y reviewed in section 4, the entanglement entropy, as well as the Renyi entropies, can be evaluated in terms of twist operators in an n-fold replicated version of the CFT | see also [52, 55{57]. Further in higher dimensional CFTs, i.e., for d 3, the twist op erators n are codimension-two surface operators with support on the entangling surface. In [52, 80], it was argued that an e ective twist operator ~n is de ned if one considers correlation functions where the twist operator only interacts with other operators which are all from a single copy of the replicated CFT. In particular, one nds ~n = e (n 1)Hm where Hm is the modular Hamiltonian. This expression should apply for general geometries but, of course, the special case of a spherical entangling surface (in the CFT vacuum) is of interest here, where Hm is given by the local expression in eq. (1.3). This expression is particularly useful to investigate the limit n ! 1, which then yields ~n ' 1 (n 1)Hm + : In particular, this demonstrates that the modular Hamiltonian is the only nontrivial contribution in the OPE limit of the twist operator which survives in the n ! 1 limit. Ref. [81] suggested augmenting the twist operators with (the exponential of) a charge term which had the form of one of our new observables (3.17) with a spin-one conserved current. A similar extension [62] involving higher spin observables (3.35) was considered in the context of two-dimensional CFTs of the form discussed in section 5. Given these considerations, it is tempting to generalize eq. (7.12) to a family of `generalized twist operators' based on our nonlocal observables, e.g., (7.12) (7.13) (7.14) ~(O) = e Q(O) : We have included a numerical coe cient so that the linearized observable would emerge in a ` rst law'-like expression with the limit However, it is not immediately clear whether one can meaningfully construct the power series in implicit in the above de nition of ~(O). We hope to return to study this question and other issues for this possible nonlinear generalization of our nonlocal observables in the future. ! 0.36 36We have distinguished from the index n in eq. (7.12) since we need not consider the replicated CFT in de ning ~(O), i.e., it can be de ned in a single copy of the CFT. O < d O < d Universal constant? As noted in section 3, the integral in eq. (3.1) diverges for d 2 unless the expectation value vanishes at the boundaries of the causal diamond. That is, if hOi is nonvanishing somewhere, then eq. (3.1) diverges for causal diamonds over some region of the moduli space. However, we still expect that a universal nite term can be extracted from this expression in this situation. Examining eq. (C.3), where Q(O) is evaluated for a constant expectation value, we see that the result remains nite for 2. In fact, divergenes only arise for 2; d 4; . Hence our calculation O = d has implicitly analytically continued the expression to produce a nite result in the range 2. We expect that the same universal result could be produced if we explicitly introduced a short distance cut-o and focused on the cut-o independent constant term in the nal result. Further we expect for the special values of O where eq. (C.3) corresponds to the appearance of a logarithmic divergence whose coe cient would yield the universal contribution. These considerations would then put these universal contributions on the same footing as the constant F in the F -theorem [10, 11, 82, 83]. However, there are subtleties de ning F using entanglement entropy [84] and so as in that case, one might ask if a more robust de nition of Q(O) for the cases where eq. (3.1) contains divergences. Using the usual AdS/CFT dictionary, e.g., eqs. (C.5) and (C.6), it is straightforward O to see that analogous divergences appear in the holographic de nition in eq. (3.24). That is, the integral over the extremal surface in Qholo(O) will diverge for O d 2. Of course, the result in eq. (C.7) for a constant expectation value indicates that these divergences can again be avoided by a suitable analytic continuation or with a suitable regulator, i.e., the results there precisely match those in eq. (C.3). Hence the equivalence Q(O) = Qholo(O) survives for operators with d 2. However, the question of whether the wave equation (3.8) applies in this regime still requires more careful investigation. It is clear from the discussion above that our studies here have left open a variety of interesting questions and we hope to continue to study these in future research. Acknowledgments We would like to thanks Nele Callebaut, Alejandra Castro, Bartek Czech, Ben Freivogel, Diego Hofman, Veronika Hubeny, Aitor Lewkowycz, R. Loganayagam, Markus Luty, Miguel Paulos, Guilherme Pimentel, Mukund Rangamani, James Sully, Erik Tonni and Claire Zukowski for useful discussions and comments. This work is part of the research programme of the Foundation for Fundamental Research on Matter (FOM), which is part of the Netherlands Organisation for Scientic Research (NWO). Research at Perimeter Institute is supported by the Government of Canada through the Department of Innovation, Science and Economic Development and by the Province of Ontario through the Ministry of Research & Innovation. FMH is grateful to Perimeter Institute and UC Davis for hospitality while this work was in progress. RCM is also supported in part by research funding from the Natural Sciences and Engineering Research Council of Canada, from the Canadian Institute for Advanced Research, and from the Simons Foundation through the \It from Qubit" Collaboration. In this appendix, we consider various geometric details which are useful for the discussions in the main text. In particular, in the rst section, we discuss the details of the derivation of the precise form of the metric (2.14) on the moduli space of causal diamonds. In the second section, we discuss the moduli space for pairs of spacelike separated points, which arises naturally in a number of instances, e.g., two dimensions. Finally, in the last section, we elaborate on the form and properties of the conformal Killing vector which can be constructed to preserve the form of any given causal diamond. A.1 Derivation of metric on the space of causal diamonds In the following, we present further details in the derivation of the metric (2.14) on the moduli space of causal diamonds. Our approach is to continue working in the embedding space introduced in section 2.1, make a general ansatz compatible with the required symmetries, and subsequently impose conditions which x the free parameters. We remind the reader that the metric needs to be of the form (2.13), which we reproduce here for convenience: HJEP08(216) ds2 = L2 ( hdT; dT i + hdS; dSi) ; where the vectors T b and Sb still need to be fully determined, subject to the conditions in eqs. (2.8) and (2.9), i.e., (A.1) (A.2) (A.3) (A.4) (A.5) hS; Xi z!0 = hT; Xi z!0 = 0 : The form of the metric (A.1) was derived in section 2.1 by demanding SO(1; d 1) SO(1; 1) invariance. Let us start with the observation that for any metric of the form ds2 = N da2 + 2Ni da dmi + gij dmidmj ; where a is a Killing coordinate, i.e., none of the metric components depends on a, one obtains its SO(1; 1) coset by taking ds2 = gij NiNj =N 2 dmidmj ; where mi are the coordinates on the nal coset. In order to obtain the metric on the space of causal diamonds, we thus need to parametrize T b and Sb in terms of the corresponding mi-coordinates, which in our case are simply x and y specifying the tips of a causal diamond. We then need to evaluate eq. (A.1). The corresponding Killing coordinate will be that associated with the SO(1; 1) boost and this will allow us to use eq. (A.5) to explicitly write out the desired coset metric. As it turns out, the following parametrization of T b and Sb does the job for us: T b = (T 1 ; x x + y y ; T d) and Sb = (S 1 ; x x + y y ; Sd) : (A.6) In order to demonstrate this, let us start with the conditions (A.3), which by taking their two independent linear combinations can be recast as Cx(0) 2wx + Cx(2) w2 = 0 and Cy(0) 2wy + Cy(2) w2 = 0 ; (A.7) where Clearly, neither T b nor Sb can depend on w . As a result, demanding conditions (A.3) amounts to solving a set of 4 independent equations: Cx(0) = x2 and Cx(2) = 1 and Cy(0) = y2 and Cy(2) = 1 : Together with the three normalization conditions (A.2), eqs. (A.9) allow to solve for 7 out of 8 parameters specifying T b and Sb vectors (up to an irrelevant discrete choice of the vectors' orientations). The remaining real parameter corresponds to the boost freedom. Let us then solve eqs. (A.9) together with eqs. (A.2) for T d = 0. The solution reads We will regenerate the missing parameter by evaluating the metric (A.1) by performing a boost in the (T; S)-plane, (T 0)b = cosh a T b + sinh a Sb ; (S0)b = cosh a Sb + sinh a T b ; (A.11) Cx(0) = Cx(2) = Cy(0) = Cy(2) = y( T 1 + T d) + y(S 1 Sd) y(T 1 + T d) + y(S 1 + Sd) y x x y y x x y x(T 1 T d) + x( S 1 + Sd) y x x y x(T 1 + T d) x(S 1 + Sd) y x x y T 1 = p (x x = p (x y = p (x T d = 0 ; S 1 = p (x x = x ; y = y ; Sd = p (x x 2 y)2(1 y 2 1 + y2 y)2(1 1 + x2 x2)(1 y2) x2)(1 y2) y)2(1 x2)(1 y2) 1 + x2y2 y)2(1 x2)(1 y2) y)2(1 x2)(1 y2) : ; ; ; : ; ; ; ; (A.8) (A.9) (A.10) HJEP08(216) which preserves the conditions in eqs. (A.2) and (A.3). It is then a matter of tedious and rather unilluminating calculation to recast the metric in the form (A.4) and identify the corresponding gij and Ni. After using eq. (A.5), we are led to the desired metric on the SO(2; d)=[SO(1; d 1) SO(1; 1)] coset: ds2 = h dx dy = 4L2 y )(x y ) dx dy : (A.12) A.2 Conformal Killing vectors Given a causal diamond in Minkowski space, which is de ned by the positions of the future and past tips (y ; x ), there is a conformal Killing vector which preserves the diamond:37 w)2 (x w ) (x (A.13) From this expression, one can easily see that the vector vanishes at w = x and w = y , and when both (y w)2 = 0, i.e., when eq. (2.2) is satis ed. Hence the tips of the causal diamond and also the maximal sphere at the waist of the causal diamond are xed points of the ow de ned by K. Further, one sees that K is null on the boundaries of the causal diamond, i.e., when either (y w)2 = 0 or (x w)2 = 0. Finally, one can also observe that within the rest of the causal diamond K is timelike and future directed. Figure 10 illustrates the Killing ow both inside and outside of the causal diamond for a cross-section of the diamond. 37As usual, our notation here is that (y x)2 = (y x) (y x) . Working with standard `Cartesian' coordinates w chooses the frame where y = (R; ~x0) and x = ( R; ~x0) then the conformal Killing vector takes a recognizable form, e.g., [33] (R2 j ~x 2t (xi (A.14) Given this expression, one sees that the perturbation of the entanglement entropy in eq. (1.3) can be written in a covariant form as Z B d SEE = h T i K ; (A.15) higher spin observables constructed in section 3.2. where the integration runs over j~x form, we can regard the integrand is a conserved current which allows us to move the surface of integration to be any Cauchy surface spanning the associated causal diamond. That is, if we de ne J h T i K , it follows that r J = 0 because the stress tensor is conserved and traceless, i.e., r h T i = 0 = hT i, and because K is a conformal Killing R2 on the t = 0 time slice. However, in this K . Of course, similar statements apply for the We might note that in two dimensions using the null coordinates introduced in eqs. (2.16) and (2.17), the conformal Killing vector takes a particularly simple form: K = 2 and K = 2 (A.16) This allows us to re-express the observables (3.34) for d = 2 CFTs as Q(O; u; u; v; v) = C O 2 Z v u d K 2 h 1 Z v u d O( ; ) : (A.17) In the context of the AdS/CFT correspondence, the conformal Killing vector (A.13) extends to a proper Killing vector of the AdS geometry as follows: we describe the AdS geometry with Poincare coordinates ds2 = GMN dW M dW N = z2 dw dw ; (A.18) where we have introduced a (d + 1)-dimensional vector notation, e.g., we denote the bulk coordinates as W M = (w ; z). Hence we indicate the tips of the causal diamond in the boundary with Y M = (y ; 0) and XM = (x ; 0). With this notation, the bulk Killing vector becomes 2 X)2 (Y W )2 (XM W M ) (X W )2 (Y M where our notation here is that (Y X)2 = GMN (Y X)M (Y X)N . With this expression, one can easily verify that the tips of the causal diamond in the boundary are xed points of the Killing ow, as is the extremal surface where (Y W )2 = 0 and (X W )2 = 0. Further one can see that the Killing vector becomes null on the boundaries of the causal wedge in the bulk. One can also consider the analytic continuation of eq. (A.13) to Euclidean signature, which follows by simply replacing the Lorentzian inner product there by (y x)2 = (y x) (y x) . As discussed in section 3.5, there are two distinct moduli spaces to consider in Euclidean signature and associated conformal Killing vectors arise from di erent choices of the vectors x and y . If we choose real vectors, then x and y now de ne a pair of spacelike points and these points are the only xed points of the ow de ned by K .38 Hence this conformal Killing vector generates the SO(1; 1) symmetry in the coset SO(1; d+1)=(SO(d) SO(1; 1)), which corresponds to the moduli space of pairs of points discussed in section 3.5. The second distinct moduli space in Euclidean space is the space of all (d 2)dimensional spheres, which is described by the coset SO(1; d + 1)=(SO(1; d 1) SO(2)). In this case, the associated conformal Killing vector results from choosing `complex' vectors x and y . In particular, using the notation of eq. (2.20), we choose y x = c + ` = c + iR n = c ` = c iR n = (y ) (A.20) where n is an arbitrary unit vector in Rd. The conformal Killing vector then becomes w)2 ((y ) w ) w)2 (y (A.21) where we have introduced an extra overall factor of i to produce a real vector. Since w correspond to real positions, we cannot satisfy the equations w = y or w = (y ) . On the other hand, the equations (y w)2 = 0 and (y w)2 = 0 can be simultaneously solved by setting (c and n (c w) = 0 : (A.22) That is, the ow of the new vector Ke has a xed point on a (d 2)-sphere of radius R centred at w = c and lying in the (d 1)-dimensional hyperplane de ned by n (c w) = 0. Hence this new Killing vector generates the SO(2) symmetry in the coset describing the moduli space of (d 2) dimensional spheres in Rd. A.3 Moduli space of spacelike separated pairs of points that M Here we would like to consider the analog of our generalized kinematic space (2.11) for pairs of spacelike separated points in a d-dimensional CFT (with Lorentzian signature). Recall (d) was the moduli space of all causal diamonds, or equivalently of all spheres, or equivalently of all timelike separated pairs of points. Considering the space of spacelike separated points arises naturally in a number of instances, e.g., upon analytically continuing to a Euclidean signature, as discussed brie y in section 3.5. In fact, in two dimensions, 38That is, in Euclidean signature, the only solution of (x w)2 = 0 is w = x and hence we cannot simultaneously solve (y w)2 = 0. Note that if we were considering spacelike separated points but in Lorentzian signature, there would be the simultaneous solution of these two equations would de ne a spacelike hyperbola | see the following section. the space of spacelike hyperbolas: the intersection of lightcones of two spacelike separated points forms a spacelike hyperbola (dashed maroon curve) which lies in a timelike codimension-one hyperplane (shaded in yellow). a causal diamond can be de ned either in terms of a pair of timelike separated points or a pair of spacelike separated points.39 Hence it seems that d = 2 is a special case where the two moduli spaces are equivalent, i.e., the space of timelike separated pairs of points is the same geometric object as the space of spacelike separated pairs of points. Our nal conclusion here is that in fact this equivalence extends to CFTs in arbitrary dimensions! To understand this new moduli space, we begin by considering the intersection of the lightcones from a pair of spacelike separated points. As illustrated in gure 11, the intersection of the lightcones de nes a spacelike hyperbola lying in a xed timelike hyperplane (of codimension one). Hence in analogy to the previous discussion of kinematic space, we may say that the moduli space of pairs of spacelike separated points is equivalent to the moduli space of spacelike hyperbola. There is no obvious analog of the causal diamonds since for spacelike separated points, the two lightcones do not enclose a nite-volume region anywhere, as can be seen in the gure. Next we would like understand the coset structure of this moduli space by turning to the embedding space introduced in section 2.1. However, it is easiest to think in terms of a construction of the moduli space of spacelike hyperbolae in a d-dimensional CFT. A bit of thought shows that such a hyperbola will be described by choosing a pair of orthogonal unit vectors, T b and Sb, satisfying precisely the same conditions given in eqs. (2.8) and (2.9). This construction is again easily illustrated with the Poincare patch coordinates (2.7) where 39For example in gure 3, the causal diamond can be de ned in terms of the extreme points at the left and right corners, i.e., ( ; ) = (v; u) and (u; v). a convenient choice of the unit vectors is T b = (1; 0; 0; : : : ; 0) Sb = (0; 0; 1; : : : ; 0) w w 1 ; w1 = 0 : (A.23) The expressions on the right denote the surfaces in the asymptotic geometry that are picked out by the orthogonality constraints (2.9), i.e., Sb selects a particular timelike codimensionone hyperplane in the boundary while T b selects a spacelike hyperboloid. The intersection of these two surfaces then yields the (codimension-two) hyperbola (w0)2 d 1 X(wi)2 = 1 i=2 on the hyperplane w1 = 0 : (A.24) Now following the discussion of section 2.1, a particular pair of unit vectors, T b and Sb, speci es a particular hyperbola in the boundary geometry. We sweep out the rest of the moduli space by acting with SO(2; d) transformations, i.e., Lorentz transformations in the embedding space. However, the coset structure of the resulting moduli space of hyperbolae is then determined by the symmetries preserved by any particular choice of the unit vectors. However, since the constraints on the present unit vectors are precisely the same as in section 2.1, these symmetries are also the same and hence we arrive at the same coset as given in eq. (2.11), namely, SO(2; d) SO(1; d 1) SO(1; 1) : (A.25) At rst sight, this result may seem rather counterintuitive. Spacelike and timelike separated pairs of points are by de nition very di erent kinds of objects in Minkowski space and yet we found that in a d-dimensional CFT, the moduli spaces of such pairs are described by the same coset structure irrespective of whether the separation is spacelike or timelike. Further in the language of the embedding space, the two spaces are being described by precisely the same family of orthogonal unit vectors, i.e., pairs satisfying eqs. (2.8) and (2.9). Of course, this indicates that not only do we have two moduli spaces described by the same coset geometry (A.25) but that in fact we are considering one and the same moduli space from two di erent perspectives! In order to develop a better understanding of this counterintuitive result consider the following: the rst point to note is that our intuition about spacelike and timelike separated pairs of points is rmly rooted in at Minkowski space. However, recall that in the embedding space, the the Poincare patch coordinates (2.7) only cover a portion of the AdS hyperboloid (2.5) and some SO(2; d) transformations will take us out of this region, i.e., pairs of points maybe mapped beyond the corresponding Minkowski space in the asymptotic boundary. Hence it is more appropriate to think of working on global coordinates for the AdS geometry or transforming the CFT to the `cylindrical' background R Sd 1 (with R being the time direction).40 40With this transformation, we are actually extending the original Minkowski space to a geometry where the conformal group acts properly everywhere. (t = 0) S S (t = Rsph) x z (t = Rsph) S1. The point z is the antipodal point from the point x on the constant time slice containing this point and z has the maximal spacelike geodesic distance Rsph from x . Blue lines are past and future lightcones of x . The point x corresponds to the position where the future light cone of x rst self-intersects. The sphere S (indicated by black points) can be described as the intersection of past lightcone of y either with future lightcone of x , or alternatively with past lightcone of the antipodal point x . In the former case, S is characterized by a pair of timelike separated points, in the latter case by a pair of spacelike separated points. In the latter geometry, there are limits to how far apart the pairs of points can be.41 In particular for spacelike separated points, the maximum separation is Rsph where Rsph is the radius of curvature of the Sd 1, i.e., maximally separated pairs are antipodal pairs on the (d 1)-sphere | see gure 12. Similarly, the maximal separation for a timelike pair is 2 Rsph. For example, if the two points lie at the same pole on the sphere, then with this maximal time separation, the lightcones from these two points intersect at a point on the the opposite pole and hence the corresponding sphere has the maximal angular size, i.e., the sphere's proper size has actually shrunk to zero but the `enclosed' ball covers the entire Sd 1 . In fact, as illustrated in the gure, the null cones of these two maximally (timelike) separated points actually coincide.42 This leads to the observation that because 41As in at space, we measure the separation between points in R S d 1 as the (minimal) proper distance along geodesics connecting the points. 42In the embedding space, the two points considered here are actually coincident points on the boundary of the AdS hyperboloid (2.5). It is only when we consider the universal cover of the AdS hyperboloid (as we do implicitly here) that the points are separated. In particular, if we had been more precise we should have replaced the SO(2; d) group in the numerator of (A.25) by a suitable in nite cover in this case. of the compact structure of the Sd 1, when we choose any single point in the R by following the past and future null cones, we actually specify two families of preferred points. The rst being points lying at the same pole of the sphere at t = 2 nRsph where n is any integer (and we have assumed the initial point lies at t = 0, i.e., n = 0). The second family is points on the opposite pole lying at t = 2 (n+ 12 )Rsph where n is again any integer. This insight then allows us to understand the equivalence of the two spaces discussed above in very concrete terms. Consider the two timelike separated points designated x and y shown in gure 12. The future lightcone of x and the past light cone of y intersect on the sphere designated S. However, now consider the point x where the future lightcone of x ( rst) converges to a point on the opposite pole of the sphere. The pair x and y is now a spacelike separated pair of points. The past and future lightcones from these two points intersect at the spheres, S and S~, respectively. Now, in an appropriate conformal frame, where x and y are spacelike separated points in at Minkowski, these two spheres become the two branches of the corresponding spacelike hyperbola discussed above.43 However, the key point here is that in the R Sd 1 conformal frame, we can specify spheres either in terms of the intersection of the past and future lightcones of a pair of timelike separated points or in terms of the intersection of the past light cones from two spacelike separated points. Hence we recognize that moduli spaces of spacelike and timelike pairs in fact provide two di erent perspectives of the same geometric object! Given that the moduli spaces of spacelike and timelike pairs (on R Sd 1) are the same, it is interesting that the discussion in section 2.2 implies that the limit in which a timelike separated pair approaches a null separated pair of points is a limit that takes us to timelike in nity in the moduli space | see footnote 5. This is a consistency check in that it shows that there is no trajectory on the moduli space that carries one between timelike separation to spacelike separation. Of course, it would be interesting to further explore the implications of this equivalence. B B.1 Conventions for symmetry generators General de nitions Spinless case. Given the conformal symmetry generators Li(x), we de ne the second Casimir as the object C2 Cij Li(x)Lj (x) (where i; j = 1; ; (d + 1)(d + 2)=2) which acts on scalar primaries O in the CFT with dimension O such that: [C2; O(x)] = O(d O)O(x) : (B.1) In this appendix we discuss various realizations of C2 on objects which carry a representation of the conformal group: elds in AdSd+1 and functions on the moduli space of causal The conformal algebra in d dimensions is isomorphic to the group SO(2; d) Lorentz group of the embedding space (2.4). We write the action of SO(2; d) generators on primaries 43Each branch is topologically a (d 1)-sphere when we include the point at in nity. M P jO(x)i = i@ jO(x)i ; O) jO(x)i ; 2x O jO(x)i ; (B.2) (B.3) (B.4) (B.5) (B.6) where jO(x)i = O(x)j0i and the vacuum state j0i is annihilated by all of the generators. The SO(2; d) Lorentz generators Jab = Jba are hence represented by HJEP08(216) 1 2 1 2 J = M Jd = Q ) ; J d = D : These satisfy the algebra [Jab; Jcd] = i b(2c;d) Jad i a(2c;d) Jbd i b(2d;d) Jac + i a(2d;d) Jbc ; ab where (2;d) = diag( 1; 1; 1; : : : ; 1) is the embedding space metric. In terms of these Lorentz generators, we can represent the action of the Casimir on operators by C2 1 J abJab, which acts as a di erential operator whose eigenfunctions are the primary states: 2 C2jO(x)i 2 1 J ab(x)Jab(x)jO(x)i = O(d O) jO(x)i : Since SO(2; d) acts on the AdSd+1 hyperboloid in embedding space as standard Lorentz transformations, the above generators can also be represented as isometry generators of AdSd+1. This representation is given in embedding space coordinates by Jab = i(Xa@b Xb@a). In particular, the AdSd+1 Laplacian is represented by the combination 2 2 RAdS rAdS = 2 1 J abJab = C2 : Similarly, the action of the Casimir is represented on the moduli space of causal diamonds. Using the explicit representation (B.2), it is straightforward to verify the following relation between the second Casimir as a di erential operator acting on the space of causal diamonds, and the scalar Laplacian on the same space: 1 2 C2f (x; y) (J ab(x) + J ab(y))(Jab(x) + Jab(y))f (x; y) = L2 r2 f (x; y) ; (B.7) where f (x; y) is any function on the space of diamonds Laplacian on the moduli space of diamonds (2.14). = (x ; y ), and r 2 is the As an application of this, one can explicitly check that the kernel in eq. (3.4) is an eigenfunction of the Casimir as represented by eq. (B.7): C2 j y j y x j x j O d O(d O ) j y jj j y x j x j O d : (B.8) Generalization with spin. It is straightforward to generalize the above discussion to the case of primary operators with symmetric-traceless indices, O 1 ` . In this case, the eigenvalues of the conformal Casimir are [C2; O 1 ` (x)] = [ O(d O ) `(` + d 2)] O 1 ` (x) : (B.9) One can explicitly verify that eq. (B.7) then still holds for tensors instead of functions f (x; y). Most importantly, we nd that the kernel in our proposal (3.9) for the ` rst law'-like expression with spin satis es L2 r2 (G 1 ` (x; y; ) T 1 ` ( )) = [ O(d O ) `(` + d 2)] G 1 ` (x; y; ) T 1 ` ( ) HJEP08(216) where T 1 ` (e.g., T 1 ` = hO 1 ` i) is an arbitrary symmetric traceless tensor and we abbreviated the kernel as G 1 ` (x; y; ) y j x x O d ( y s 1 : : : s ` xjjy x )` : B.2 Two-dimensional case Let us brie y make the statements of the previous subsection more explicit in the case of two-dimensional CFTs (and AdS3, respectively). In this case, we can work in right- and left-moving coordinates ds2CFT2 = dt2 + dx2 = d d : In these coordinates, the non-zero generators (B.2) can be written as: M01jOi = i(L 1 L0)jOi ; L 1)jOi ; i(L 1 + L 1)jOi ; Q1jOi = i( 2 2 2 ) O)jOi i(L1 L1)jOi ; ( + ) O)jOi i(L1 + L1)jOi ; O)jOi i(L0 + L0)jOi : This de nes conformal generators Ln satisfying the usual de Witt algebra [Ln; Lm] = (n m)Ln+m ; m)Lm+n ; [Lm; Ln] = 0 ; (B.14) for n; m = 1; 0; 1. The conformal Casimir de ned in (B.5) reads as follows in terms of Ln: C2jOi = 2 C2 (d=2) + C2 jOi ; where we make the factorization into natural left- and right-moving Casimir operators (B.10) (B.11) (B.12) (B.13) (B.15) explicit by de ning C2 L20 + 1 2 C2 L20 + 1 2 (L1L 1 + L 1L1) and (L1L 1 + L 1L1) : (B.16) The above discussion concerned the action of conformal generators on CFT states. There is an analogous set of identities for AdS3 isometry generators. We work in Poincare coordinates in AdS3: ds2AdS3 = z2 RA2dS dz2 + d d : Using the general de nitions of section B.1, we then nd the following isometry generators 1 2 and L0 = and and similarly for Ln with interchanged. We then have that the combinations appearing in the Casimir C2, and its left- and right-moving parts de ned in (B.16), all correspond to the scalar Laplacian on AdS3: C2 (u) = 2 2 RAdS rAdS3 (u) ; C2 (d=2) (u) = C2 (d=2) (u) = 1 2 2 4 RAdS rAdS3 (u) : (B.19) C Relative normalization of CFT and bulk quantities In this appendix we demonstrate how to x the relative normalization between Q(O) as dened in eq. (3.1) and its holographic couterpart Qholo(O) in eq. (3.24). Our strategy will be to exploit the fact that the normalization can be determined in the limit of very small diaamond is located at 12 (x +y ) = 0, and we work on a time slice such that 12 (y monds, or equivalently with hOi = constant. For simplicity, we assume the centre of the dix ) = R 0 Consider rst the eld theory observable Q(O; x; y) in the limit x ! y, i.e., for a constant expectation value hOi throughout the causal diamond: Q(O; x; y) = CO hOi Z d 1 x)2 2 ( O d) To evaluate the integral, it is useful to parameterize the causal diamond as follows: 2 R; 2 R !~ y = (R; ~0 ) ; x = ( R; ~0 ) ; 1 2 ( O d)=2 2 ( O d)=2 d 2 2 (1 + ( 1)n) O n+1 2 d n 1 2 2 O d2+n+3 (C.3) Q(O; x; y) = CO hOi CO hOi 2 O+1 CO hOi 4 1=2 d 2 R O d 2 d 2 Z 1 1 O + 2 2 d 1 2 Z 1 0 d 2 1 R O O+2 d 2 2 O + 1 d 2 ( O d 2 X n=0 O 2 where !~ 2 Sd 2 is a unit vector that parameterizes the spacelike spherical slices. The full range ; 2 [ 1; 1] would cover the diamond twice. Considering the symmetries of the integrand in (C.1), we can e ectively integrate over the range 2 [ 1; 1] and (B.17) (B.18) (C.1) (C.2) where we binomially expanded the measure factor ( + )d 2 to perform a term-by-term integration. The nal line can be simpli ed slightly by substituting for the volume of a unit (d 2)-sphere, however, the present form is convenient for our d 2 = 2 d 1 2 = comparison below. Next, we compute Qholo(O) as de ned in eq. (3.24) using standard holographic techniques. In particular, we will work in Poincare coordinates z2 RA2dS dz2 dt2 + dr2 + r2 d 2d 2 : If one considers the dual eld (u) in a linearized approximation in this background, the asymptotic behaviour takes the following form: where (z ! 0; w ) = 0(w) zd O + 1(w) z O + = 0 and hOi = O d) 1 : d 1 2R`AdPdS1 (2 Here ( ) is the coupling to the operator in the boundary CFT and we set it to zero in the following.44 In keeping with the previous calculation, we also assume that hOi is constant, at least within the boundary region of interest. The boundary sphere in the previous calculation was chosen to be: t = 0 and r = R. The corresponding extremal surface in the bulk is the hemisphere: t = 0 and z2 + r2 = R2. We can parameterize this bulk surface with z = R sin and r = R cos where 0 Then keeping on the leading term in the asymptotic expansion of the bulk scalar, the computation of the observable Qholo(O) reads as follows: Qholo(O) = Cblk 1 `d 1 P RAddS1 Z = Cblk 2 2 hOi O d d 2 Z =2 0 d 1 2 cosd 2 sind 1 O+2 d 2 4 1=2 ( O ) (R sin ) O O 2 (2R) O ; where we have substituted `dP 1 = 8 GN and applied eq. (C.6) in the second line. We can now equate the two results (C.3) and (C.7) and thus x the relative normalization: Cblk = C O O+2 d 2 O Holographic computation for a free scalar in AdS3 We expect that the generalized rst law (3.34) provides the leading order contribution to a set of novel physical quantities in CFTs in an analogous way in which the entanglement 44Eqs. (C.5) and (C.6) present a standard set of holographic conventions, e.g., see [86], although perhaps not unique. Further we note that the choice = 0 means that we are only studying excitations the CFT ground state here. It would be interesting to extend the discussion in this paper to holographic RG ows where the boundary theory is deformed away from a conformal xed point. (C.4) (C.5) (C.6) 2 (C.7) (C.8) rst law provides the leading order perturbation of the vacuum entanglement entropy for excited states. In the present section we want to corroborate this proposal by providing the holographic dual of SO in a class of CFTs which admit a semi-classical gravity description. In section 3.4, we argued that Q(O) = Qholo(O) with an appropriate choice of the bulk normalization constant Cblk. The latter was xed above by comparing the two expressions in a situation where hOi was a constant. In the following, we explicitly demonstrate that the equivalence of the boundary and bulk expressions for a more nontrivial eld con guration. To do so, we focus on AdS3 with a free probe scalar eld dual to a primary operator O O=2 in a two-dimensional holographic CFT. In this case, the `sphere' of interest becomes an interval of length 2R, which for simplicity, we assume is centred at the origin on the t = 0 time slice. Further eq. (3.34) becomes R R2 2 2R 2 Z R R R2 2R 2 2 hO( ; )i : The holographic expression in eq. (3.24) reduces to an integral of the bulk scalar over the spatial geodesic connecting the endpoints of the interval in the boundary theory: Qholo(O) = C O P ( O=2)2 Z ( O 1) p h ( ) ; where we have used 8 GN = `P for d = 2 and substituted for the normalization constant Cblk using eq. (C.8). For our explicit computation, we pick a simple linearized perturbation by putting a delta-function source at a point ( 0; 0) on the boundary. The linearized solution is given by the usual bulk-boundary propagator (r; ; ) = z2 + ( 0)2 O ds2 = 1 z2 dz2 + d d Here, is an arbitrary constant measuring the strength of the source and we are using Poincare coordinates on AdS3 where the curvature radius is set to unity and w; w denote the null coordinates introduced in eq. (2.16), i.e., = x t and = x + t. For simplicity, we will assume that the source is spacelike separated from the interval, i.e., ( in the interval. 0)2 > 0 for any point = = x 2 [ R; R] The bulk geodesic spanning the boundary interval above may be parametrized by x = R cos and z = R sin : (C.13) (C.10) (C.11) (C.12) The line element along the geodesic is d = sin and then eq. (C.10) yields C O P C O C O 2`P O O O 1) 1) 1) Z 0 sin (2R) O ( 2O )4 O R2 sin2 + (R cos Z 0 j R 4R2 ( 02 R2) R sin sin O 1 0)(R cos 2 cos O 1 2 O=2 : 0j2 cos2 2 + jR + 0j2 sin2 2 where in a slight abuse of notation, we have de ned jR 0j 2 0) in the second HJEP08(216) line. The integral there can be found, e.g., in [87]. Note that the nal result can be split into right- and left-moving factors, which was not at all clear from the initial expression.45 Now let us now turn to the boundary computation. First we should extract the expectation value from the hOi from our linearized solution (C.11) for the bulk scalar. As we take z ! 0 in eq. (C.11), we immediately recognize the behavior of a normalizable mode Now applying eq. (C.6) with d = 2, we nd (z ! 0; ; ) = 0)2 O hO( ; )i = O P 1 Since this pro le factorizes into right- and left-moving contributions, upon substitution into eq. (C.9), we also nd a factorized answer: Q(O) = Q(O) = C O 2`P C O 2`P O O 1) 1) Z R R R2 2R 2 2 1 0) O ( 2O )4 O ( 02 4R2 R2)( 02 R2) 2 O=2 This integral can also be performed, e.g., see eq. (3.199) in [87] and one nds46 where as above, we are using jf ( )j2 = f ( )f ( ) in the notation of complex coordinates. which provides a perfect agreement with the holographic result in eq. (C.14). Since this is a linearized calculation, the agreement (3.25) readily extends to arbitrary eld con gurations that are generated by the insertion of sources that are spacelike separated from the interval of interest. Of course, eqs. (C.14) and (C.18) show that there are singularities that appear when the sources cross the lightcones of the endpoints of the interval, i.e., when the sources move into causal contact with the interval. It would be interesting to investigate further here to understand if Q(O) = Qholo(O) still applies in the latter situation. Following the general arguments in section 3.4, this is intimately related to the question of better understanding causal wedge reconstruction in the bulk. 45In the limit 02 R2, the expectation value is essentially constant across the interval | see eq. (C.16). Hence in this limit, the leading contribution above can be matched with that in eq. (C.7) with d = 2. 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Jan de Boer, Felix M. Haehl, Michal P. Heller, Robert C. Myers. Entanglement, holography and causal diamonds, Journal of High Energy Physics, 2016, 162, DOI: 10.1007/JHEP08(2016)162