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, 00681 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 ddimensional 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 onepoint 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 RyuTakayanagi 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 twoderivative equations of motion on the space of causal diamonds. In two dimensions, the latter is given by a product of two copies of a twodimensional de Sitter space. For a class of universal states, we show that the entanglement entropy and its spinthree 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
AdSCFT Correspondence; Conformal Field Theory; Gaugegravity 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
Spinthree 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 BekensteinHawking 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 spacelikeseparated 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 reexpressed 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
KleinGordon equation (1.2) (1.3)
rdS
m2
SEE = 0 ;
{ 2 {
(1.4)
in the following deSitter (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
ddimensional spacetime of the CFT, i.e., all of the ballshaped 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 2ddimensional and has multiple timelike 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 twofold: 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 twoderivative 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 RyuTakayanagi
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 spinthree 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 ddimensional CFT. As noted above, this 2ddimensional 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
lightcone of the future tip and the future lightcone 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 lightcones illustrated in the
gure can be de ned as the set of points w
which are nullseparated 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 ddimensional 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 antide 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 ddimensional 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 dplane 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 ddimensional 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 twoplane in the embedding space. Hence the SO(2; d) symmetry
broken to SO(1; d
1) transformations acting in the ddimensional hyperplane orthogonal
to this (T; S)plane, as well as the SO(1; 1) transformations acting in the twoplane. Thus,
in analogy with AdS coset construction above, the natural coset describing the moduli
space of spheres, or alternatively of causal diamonds, in ddimensional 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 crosscheck 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 RyuTakayanagi 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 ddimensional 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 twodimensional de Sitter space. One can see this explicitly by introducing
right and leftmoving lightcone coordinates, e.g., we replace the Minkowski coordinates
( 0; 1) with
Then we may specify the twodimensional 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 reexpressing 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 rightmoving
coordinates, whereas the second copy depends only on the leftmoving 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 leftmoving 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 twodimensional 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 zerovolume 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 wellde 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)
(twodimensional) 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 ddimensional 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., ddimensional
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 twoderivative 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 Ochannel.
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 lightcone of y with the future lightcone of x. We will show that
8We will brie y comment on nonscalar primaries later in this section; for twodimensional 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 spacelike 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 nontrivial 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 nfold 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 codimensiontwo 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 spinone conserved current. A
similar extension [62] involving higher spin observables (3.35) was considered in the context
of twodimensional 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 cuto and focused on the cuto 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
micoordinates, 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
crosssection 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 reexpress 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 ddimensional 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 codimensionone
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 nitevolume 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 ddimensional 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 (codimensiontwo) 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 ddimensional 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 selfintersects. 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 symmetrictraceless 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
Twodimensional case
Let us brie y make the statements of the previous subsection more explicit in the case of
twodimensional CFTs (and AdS3, respectively). In this case, we can work in right and
leftmoving coordinates
ds2CFT2 =
dt2 + dx2 = d d :
In these coordinates, the nonzero 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 rightmoving 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 rightmoving 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 termbyterm
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 semiclassical 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 twodimensional 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
deltafunction source at a point ( 0; 0) on the boundary. The linearized solution is given
by the usual bulkboundary 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 leftmoving 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 leftmoving 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. Note
that in this case,
0 = 2.
46Again in the limit 02
R2, the leading contribution in expression (C.18) agrees with eq. (C.3) for d = 2.
0)
O
O
(C.14)
(C.15)
(C.16)
(C.17)
(C.18)
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[1] S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT,
Phys. Rev. Lett. 96 (2006) 181602 [hepth/0603001] [INSPIRE].
[2] S. Ryu and T. Takayanagi, Aspects of holographic entanglement entropy, JHEP 08 (2006)
045 [hepth/0605073] [INSPIRE].
JHEP 11 (2010) 149 [arXiv:1006.4090] [INSPIRE].
[3] J. AbajoArrastia, J. Aparicio and E. Lopez, Holographic evolution of entanglement entropy,
[4] T. Albash and C.V. Johnson, Evolution of holographic entanglement entropy after thermal
and electromagnetic quenches, New J. Phys. 13 (2011) 045017 [arXiv:1008.3027] [INSPIRE].
[5] V. Balasubramanian et al., Thermalization of strongly coupled eld theories, Phys. Rev. Lett.
106 (2011) 191601 [arXiv:1012.4753] [INSPIRE].
[arXiv:1103.2683] [INSPIRE].
Phys. B 796 (2008) 274 [arXiv:0709.2140] [INSPIRE].
[7] I.R. Klebanov, D. Kutasov and A. Murugan, Entanglement as a probe of con nement, Nucl.
[8] T. Albash and C.V. Johnson, Holographic studies of entanglement entropy in
superconductors, JHEP 05 (2012) 079 [arXiv:1202.2605] [INSPIRE].
[9] R.G. Cai, S. He, L. Li and Y.L. Zhang, Holographic entanglement entropy in
insulator/superconductor transition, JHEP 07 (2012) 088 [arXiv:1203.6620] [INSPIRE].
[10] R.C. Myers and A. Sinha, Seeing a ctheorem with holography, Phys. Rev. D 82 (2010)
046006 [arXiv:1006.1263] [INSPIRE].
125 [arXiv:1011.5819] [INSPIRE].
[11] R.C. Myers and A. Sinha, Holographic ctheorems in arbitrary dimensions, JHEP 01 (2011)
conformal eld theories, Phys. Rev. Lett. 115 (2015) 021602 [arXiv:1505.04804] [INSPIRE].
[13] P. Bueno and R.C. Myers, Corner contributions to holographic entanglement entropy, JHEP
08 (2015) 068 [arXiv:1505.07842] [INSPIRE].
[14] M. Van Raamsdonk, Comments on quantum gravity and entanglement, arXiv:0907.2939
[15] M. Van Raamsdonk, Building up spacetime with quantum entanglement, Gen. Rel. Grav. 42
(2010) 2323 [arXiv:1005.3035] [INSPIRE].
[16] V. Balasubramanian, B.D. Chowdhury, B. Czech, J. de Boer and M.P. Heller, Bulk curves
from boundary data in holography, Phys. Rev. D 89 (2014) 086004 [arXiv:1310.4204]
[17] R.C. Myers, J. Rao and S. Sugishita, Holographic holes in higher dimensions, JHEP 06
(2014) 044 [arXiv:1403.3416] [INSPIRE].
[18] B. Czech, X. Dong and J. Sully, Holographic reconstruction of general bulk surfaces, JHEP
11 (2014) 015 [arXiv:1406.4889] [INSPIRE].
(2014) 149 [arXiv:1408.4770] [INSPIRE].
[19] M. Headrick, R.C. Myers and J. Wien, Holographic holes and di erential entropy, JHEP 10
[INSPIRE].
[INSPIRE].
[20] B. Czech, J.L. Karczmarek, F. Nogueira and M. Van Raamsdonk, Rindler quantum gravity,
[21] B. Czech, J.L. Karczmarek, F. Nogueira and M. Van Raamsdonk, The gravity dual of a
HJEP08(216)
entanglement entropy, JHEP 12 (2014) 162 [arXiv:1408.6300] [INSPIRE].
[23] X. Dong, D. Harlow and A.C. Wall, Reconstruction of bulk operators within the entanglement
wedge in gaugegravity duality, Phys. Rev. Lett. 117 (2016) 021601 [arXiv:1601.05416]
[24] A. Almheiri, X. Dong and D. Harlow, Bulk locality and quantum error correction in
AdS/CFT, JHEP 04 (2015) 163 [arXiv:1411.7041] [INSPIRE].
[25] E. Mintun, J. Polchinski and V. Rosenhaus, Bulkboundary duality, gauge invariance and
quantum error corrections, Phys. Rev. Lett. 115 (2015) 151601 [arXiv:1501.06577]
[26] F. Pastawski, B. Yoshida, D. Harlow and J. Preskill, Holographic quantum errorcorrecting
codes: toy models for the bulk/boundary correspondence, JHEP 06 (2015) 149
[arXiv:1503.06237] [INSPIRE].
[27] D.D. Blanco, H. Casini, L.Y. Hung and R.C. Myers, Relative entropy and holography, JHEP
08 (2013) 060 [arXiv:1305.3182] [INSPIRE].
[28] A. Wehrl, General properties of entropy, Rev. Mod. Phys. 50 (1978) 221 [INSPIRE].
[29] V. Vedral, The role of relative entropy in quantum information theory, Rev. Mod. Phys. 74
[30] R. Haag, Local quantum physics: elds, particles, algebras, Springer Science & Business
(2002) 197 [INSPIRE].
Media, Germany (2012).
[hepth/0002230] [INSPIRE].
[31] S. de Haro, S.N. Solodukhin and K. Skenderis, Holographic reconstruction of spacetime and
renormalization in the AdS/CFT correspondence, Commun. Math. Phys. 217 (2001) 595
[32] N. Lashkari, M.B. McDermott and M. Van Raamsdonk, Gravitational dynamics from
entanglement `thermodynamics', JHEP 04 (2014) 195 [arXiv:1308.3716] [INSPIRE].
[33] T. Faulkner, M. Guica, T. Hartman, R.C. Myers and M. Van Raamsdonk, Gravitation from
entanglement in holographic CFTs, JHEP 03 (2014) 051 [arXiv:1312.7856] [INSPIRE].
[34] B. Swingle and M. Van Raamsdonk, Universality of gravity from entanglement,
arXiv:1405.2933 [INSPIRE].
[35] J. de Boer, M.P. Heller, R.C. Myers and Y. Neiman, Holographic de Sitter geometry from
entanglement in conformal eld theory, Phys. Rev. Lett. 116 (2016) 061602
[arXiv:1509.00113] [INSPIRE].
[36] B. Czech, L. Lamprou, S. McCandlish and J. Sully, Integral geometry and holography, JHEP
10 (2015) 175 [arXiv:1505.05515] [INSPIRE].
[37] G. Vidal, Class of quantum manybody states that can be e ciently simulated, Phys. Rev.
Lett. 101 (2008) 110501 [INSPIRE].
bulk, JHEP 07 (2016) 129 [arXiv:1604.03110] [INSPIRE].
[38] B. Czech, L. Lamprou, S. McCandlish, B. Mosk and J. Sully, A stereoscopic look into the
Rev. D 82 (2010) 045031 [arXiv:1006.3480] [INSPIRE].
[41] M.S. Costa, J. Penedones, D. Poland and S. Rychkov, Spinning conformal correlators, JHEP
11 (2011) 071 [arXiv:1107.3554] [INSPIRE].
[42] S. Ferrara, A.F. Grillo, G. Parisi and R. Gatto, The shadow operator formalism for
conformal algebra. Vacuum expectation values and operator products, Lett. Nuovo Cim. 4S2
[43] D. SimmonsDu n, Projectors, shadows and conformal blocks, JHEP 04 (2014) 146
(1972) 115 [INSPIRE].
[arXiv:1204.3894] [INSPIRE].
11 (2011) 071 [arXiv:1107.3554] [INSPIRE].
[44] M.S. Costa, J. Penedones, D. Poland and S. Rychkov, Spinning conformal correlators, JHEP
[45] E. Hijano, P. Kraus, E. Perlmutter and R. Snively, Witten diagrams revisited: the AdS
geometry of conformal blocks, JHEP 01 (2016) 146 [arXiv:1508.00501] [INSPIRE].
[46] M.F. Paulos, Loops, polytopes and splines, JHEP 06 (2013) 007 [arXiv:1210.0578]
[INSPIRE].
[47] V. Balasubramanian, P. Kraus and A.E. Lawrence, Bulk versus boundary dynamics in
antide Sitter spacetime, Phys. Rev. D 59 (1999) 046003 [hepth/9805171] [INSPIRE].
[48] A. Hamilton, D.N. Kabat, G. Lifschytz and D.A. Lowe, Holographic representation of local
bulk operators, Phys. Rev. D 74 (2006) 066009 [hepth/0606141] [INSPIRE].
[49] I.A. Morrison, Boundarytobulk maps for AdS causal wedges and the ReehSchlieder property
in holography, JHEP 05 (2014) 053 [arXiv:1403.3426] [INSPIRE].
[50] R. Bousso, B. Freivogel, S. Leichenauer, V. Rosenhaus and C. Zukowski, Null geodesics, local
CFT operators and AdS/CFT for subregions, Phys. Rev. D 88 (2013) 064057
[arXiv:1209.4641] [INSPIRE].
[51] B. Freivogel, R.A. Je erson and L. Kabir, Precursors, gauge invariance and quantum error
correction in AdS/CFT, JHEP 04 (2016) 119 [arXiv:1602.04811] [INSPIRE].
[52] L.Y. Hung, R.C. Myers and M. Smolkin, Twist operators in higher dimensions, JHEP 10
(2014) 178 [arXiv:1407.6429] [INSPIRE].
[53] J. Gomis and T. Okuda, Sduality, 't Hooft operators and the operator product expansion,
JHEP 09 (2009) 072 [arXiv:0906.3011] [INSPIRE].
[54] E. Hijano, P. Kraus, E. Perlmutter and R. Snively, Semiclassical Virasoro blocks from AdS3
gravity, JHEP 12 (2015) 077 [arXiv:1508.04987] [INSPIRE].
[55] P. Calabrese and J.L. Cardy, Entanglement entropy and quantum eld theory, J. Stat. Mech.
06 (2004) P06002 [hepth/0405152] [INSPIRE].
JHEP 12 (2011) 047 [arXiv:1110.1084] [INSPIRE].
quantum integrable models and entanglement entropy, J. Statist. Phys. 130 (2008) 129
[57] L.Y. Hung, R.C. Myers, M. Smolkin and A. Yale, Holographic calculations of Renyi entropy,
[58] C. Holzhey, F. Larsen and F. Wilczek, Geometric and renormalized entropy in conformal
eld theory, Nucl. Phys. B 424 (1994) 443 [hepth/9403108] [INSPIRE].
[59] M.M. Roberts, Time evolution of entanglement entropy from a pulse, JHEP 12 (2012) 027
[arXiv:1204.1982] [INSPIRE].
conformal eld theory, arXiv:1604.02687 [INSPIRE].
[60] C.T. Asplund, N. Callebaut and C. Zukowski, Equivalence of emergent de Sitter spaces from
[61] M. Ammon, M. Gutperle, P. Kraus and E. Perlmutter, Black holes in three dimensional
higher spin gravity: a review, J. Phys. A 46 (2013) 214001 [arXiv:1208.5182] [INSPIRE].
[62] E. Hijano and P. Kraus, A new spin on entanglement entropy, JHEP 12 (2014) 041
[arXiv:1406.1804] [INSPIRE].
[63] J. de Boer, A. Castro, E. Hijano, J.I. Jottar and P. Kraus, Higher spin entanglement and WN
conformal blocks, JHEP 07 (2015) 168 [arXiv:1412.7520] [INSPIRE].
[64] M. Ammon, A. Castro and N. Iqbal, Wilson lines and entanglement entropy in higher spin
gravity, JHEP 10 (2013) 110 [arXiv:1306.4338] [INSPIRE].
04 (2014) 089 [arXiv:1306.4347] [INSPIRE].
[65] J. de Boer and J.I. Jottar, Entanglement entropy and higher spin holography in AdS3, JHEP
[66] A. Castro and E. Llabres, Unravelling holographic entanglement entropy in higher spin
theories, JHEP 03 (2015) 124 [arXiv:1410.2870] [INSPIRE].
CFT, JHEP 06 (2014) 096 [arXiv:1402.0007] [INSPIRE].
[67] S. Datta, J.R. David, M. Ferlaino and S.P. Kumar, Higher spin entanglement entropy from
[68] S. Datta, J.R. David, M. Ferlaino and S.P. Kumar, Universal correction to higher spin
entanglement entropy, Phys. Rev. D 90 (2014) 041903 [arXiv:1405.0015] [INSPIRE].
[69] B. Chen and J.Q. Wu, Higher spin entanglement entropy at nite temperature with chemical
potential, JHEP 07 (2016) 049 [arXiv:1604.03644] [INSPIRE].
AdS3/CFT2, JHEP 04 (2016) 107 [arXiv:1407.3844] [INSPIRE].
[70] J. de Boer and J.I. Jottar, Boundary conditions and partition functions in higher spin
[71] V.A. Fateev and A.V. Litvinov, Correlation functions in conformal Toda eld theory. I,
JHEP 11 (2007) 002 [arXiv:0709.3806] [INSPIRE].
329 [hepth/9112060] [INSPIRE].
[72] J. de Boer and J. Goeree, W gravity from ChernSimons theory, Nucl. Phys. B 381 (1992)
[73] G. Aldazabal, D. Marques and C. Nun~ez, Double eld theory: a pedagogical review, Class.
Quant. Grav. 30 (2013) 163001 [arXiv:1305.1907] [INSPIRE].
[74] D.S. Berman and D.C. Thompson, Duality symmetric string and Mtheory, Phys. Rept. 566
(2014) 1 [arXiv:1306.2643] [INSPIRE].
[75] O. Hohm, D. Lust and B. Zwiebach, The spacetime of double eld theory: review, remarks
and outlook, Fortsch. Phys. 61 (2013) 926 [arXiv:1309.2977] [INSPIRE].
onepoint functions in holographic states, JHEP 06 (2016) 085 [arXiv:1604.05308]
JHEP 07 (2016) 100 [arXiv:1512.01548] [INSPIRE].
HJEP08(216)
charged Renyi entropies, JHEP 12 (2013) 059 [arXiv:1310.4180] [INSPIRE].
theories on the threesphere, JHEP 06 (2011) 102 [arXiv:1103.1181] [INSPIRE].
(2011) 038 [arXiv:1105.4598] [INSPIRE].
JHEP 10 (2015) 003 [arXiv:1506.06195] [INSPIRE].
The Netherlands (1996).
plasmas, JHEP 05 (2013) 067 [arXiv:1302.2924] [INSPIRE].
Elsevier/Academic Press, Amsterdam The Netherlands (2007).
[6] V. Balasubramanian et al., Holographic thermalization , Phys. Rev. D 84 ( 2011 ) 026010 density matrix , Class. Quant. Grav . 29 ( 2012 ) 155009 [arXiv: 1204 .1330] [INSPIRE].
[22] M. Headrick , V.E. Hubeny , A. Lawrence and M. Rangamani , Causality & holographic [39] P.A.M. Dirac , Wave equations in conformal space , Annals Math . 37 ( 1936 ) 429 [INSPIRE]. [40] S. Weinberg , Sixdimensional methods for fourdimensional conformal eld theories , Phys. [76] L. Freidel , R.G. Leigh and D. Minic , Metastring theory and modular spacetime , JHEP 06 [77] M.J.S. Beach , J. Lee , C. Rabideau and M. Van Raamsdonk , Entanglement entropy from [78] J. Lin , M. Marcolli , H. Ooguri and B. Stoica , Locality of gravitational systems from entanglement of conformal eld theories , Phys. Rev. Lett . 114 ( 2015 ) 221601 [79] B. Czech , L. Lamprou , S. McCandlish and J. Sully , Tensor networks from kinematic space , [80] M. Smolkin and S.N. Solodukhin , Correlation functions on conical defects , Phys. Rev. D 91