#### Deriving covariant holographic entanglement

Received: September
Deriving covariant holographic entanglement
Xi Dong 0 1 2 5
Aitor Lewkowycz 0 1 2 3
Mukund Rangamani 0 1 2 4
Davis 0 1 2
CA 0 1 2
U.S.A. 0 1 2
Open Access 0 1 2
c The Authors. 0 1 2
0 Department of Physics, University of California
1 Princeton , NJ 08544 , U.S.A
2 Princeton , NJ 08540 , U.S.A
3 Jadwin Hall, Princeton University
4 Center for Quantum Mathematics and Physics (QMAP)
5 School of Natural Sciences, Institute for Advanced Study
We provide a gravitational argument in favour of the covariant holographic entanglement entropy proposal. In general time-dependent states, the proposal asserts that the entanglement entropy of a region in the boundary eld theory is given by a quarter of the area of a bulk extremal surface in Planck units. The main element of our discussion is an implementation of an appropriate Schwinger-Keldysh contour to obtain the reduced density matrix (and its powers) of a given region, as is relevant for the replica construction. We map this contour into the bulk gravitational theory, and argue that the saddle point solutions of these replica geometries lead to a consistent prescription for computing the eld theory Renyi entropies. In the limiting case where the replica index is taken to unity, a local analysis su ces to show that these saddles lead to the extremal surfaces of interest. We also comment on various properties of holographic entanglement that follow from this
AdS-CFT Correspondence; Classical Theories of Gravity; Gauge-gravity cor-
1 Introduction 2 Field theory construction 3
Gravitational construction
The time-dependent QFT wavefunctional
The reduced density matrix
Products of reduced density matrix and Renyi entropies
Review: time re ection symmetric case
Covariant generalization to time-dependent situations
Kinematics: setting up the bulk construction
Dynamics: equations of motion and extremal surfaces
A Bulk evaluation of the Renyi entropy
A.1 Evaluation of boundary term
A.2 Example of bulk integral
One of the intriguing aspects of the holographic AdS/CFT correspondence is the
geometrization of quantum entanglement. In a local QFT the amount of correlation between
degrees of freedom con ned in a spatial region, and those outside, is measured by the
entanglement entropy. While simple to state, this quantity is notoriously hard to compute
in all but a handful of circumstances. At a technical level its computation requires
determining the logarithm of a state-dependent operator (see below), which is challenging in an
Given this situation, it is rather remarkable that one has a rather simple way to
compute entanglement in holographic eld theories thanks to the AdS/CFT correspondence.
Speci cally, Ryu and Takayanagi (RT) [1, 2] argued, drawing analogy with the behaviour
of black hole entropy in gravitational theories, that the entanglement entropy ascribable
to a region A is given by solving a classical geometric problem in AdS. One is instructed
to nd a minimal area surface anchored on the boundary @A of the region of interest; its
area in Planck units measures the entanglement entropy S . The RT prescription was
originally given for static states; this extends trivially to more generally to states at a
moment of time re ection symmetry. However, the concept of entanglement being quite
fundamental is not limited to such situations alone. Indeed the notion of entanglement
entropy makes sense even when the state in question involves non-trivial temporal evolution.
The Hubeny-Rangamani-Takayanagi (HRT) proposal [3] mitigates this lacuna by arguing
that the correct extension of the RT prescription in time-dependent situations involves
consideration of codimension-2 extremal surfaces.
The primary intuition behind the HRT proposal was to ask what is the correct
covariant generalization of the RT prescription. As we know from other aspects of gravitational
physics, the principle of general covariance is a strong guiding principle, which in
conjunction with other physical requirements serves to almost always zero in on the (oftentimes
unique) dynamical construction. Demanding that the RT prescription admit a
covariant upgrade, along with agreement in a common domain of applicability, led HRT to the
extremal surface prescription.1
Whilst these prescriptions for computing entanglement entropy in the holographic
context are extremely simple, one would like to derive them from
rst principles using
nothing but the basic entries of the AdS/CFT dictionary. This has been achieved for the
RT prescription explicitly by Lewkowycz and Maldacena (LM) [6] who mapped the replica
construction usually employed to compute entanglement entropy in quantum
to the Euclidean quantum gravity path integral. In this context it is worth mentioning [7]
who argued for the RT prescription in the case of spherical regions in the vacuum of a CFT
(see also [8] for an initial attempt at a proof).
The argument forwarded by LM roughly proceeds as follows: given a density matrix
a measure of entanglement is provided by the von Neumann entropy S
In practice, owing to the technical complexities of taking the logarithm of an operator, one
instead computes the Renyi entropies S(q) = 1 1 q log Tr( Aq) for q 2 Z+ [9]. Analytically
A
continuing these Renyi entropies away from integral Renyi index q, one obtains the von
A =
Neumann entropy in the limit q ! 1.
When the density matrix is at a moment of time-re ection symmetry (or more simply
just time-translationally invariant) one can employ Euclidean path integral techniques.
One formally writes
A = e 2 KA which de nes the modular Hamiltonian K
the computation of the qth Renyi entropy as an evolution around an `Euclidean replica
circle' parameterized by
+ 2 q. This is entirely analogous to the computation
of the canonical partition function by evolving along the Euclidean thermal circle for a
period set by the inverse temperature before taking the trace. This analogy is in fact
exact in the case of the spherical regions A in the vacuum state of a CFT [7], for in this
case the reduced density matrix is unitarily equivalent to the thermal density matrix on
hyperbolic space (by a conformal mapping). The LM construction can be viewed as the
correct generalization of the global argument of [7] to situations where
thermal (or equivalently the modular Hamiltonian is not local).
On the bulk side the Renyi entropies are computed by evaluating the on-shell
gravitational action for the saddle point solution to the (Euclidean) quantum gravity path integral
1It should be noted that general covariance by itself is not strong enough [3]. For e.g., one can use causal
structures to motivate a di erent construction leading to the causal holographic information [4]. See also [5]
for a discussion of the merits of general covariance as a guiding principle in a closely related context.
can be distilled into:
with asymptotic AdS boundary conditions set to include a replica circle of size 2 q.2 If one
however assumes that the discrete Zq replica symmetry which is respected in the eld
theory remains unbroken by the bulk saddle, then one can consider instead the gravitational
action for the geometry obtained by a Zq quotient. This latter spacetime is singular; it has
generically a codimension-2 conical singularity (with defect angle 2q ), as the replica circle
is required to smoothly shrink in the original dual spacetime. The on-shell action is simply
q times that of the orbifolded geometry (with no contribution from the singular loci).
The main merit of this picture allows us to implement the analytic continuation away
from q 2 Z+ much more simply in the gravitational setting than in the eld theory. We
start with the geometry dual to the original density matrix
A and insert a conical defect
of opening angle 2q . Analyzing the local neighbourhood of the defect for q ! 1+ one learns
that a regular solution of gravitational equations of motion requires that the defect be a
codimension-2 extremal surface, i.e., the trace of its extrinsic curvature in both normal
directions vanishes. In the replica
direction this is a consequence of the time-re ection
symmetry, but in the spatial normal direction, this statement is the origin of the minimal
surface condition of RT. Once we have this basic statement, it then follows from di
eomorphism invariance that the variation of the gravitational path integral localizes on the
defect (as q ! 1) and computes (in Einstein gravity) the area of this codimension-2 surface
Let us now take stock of the discussion above. The essential ingredients used by LM
the entry in the AdS/CFT dictionary mapping eld theory partition functions to a
gravitational path integral with asymptotic boundary conditions [14],
the assumption of unbroken replica symmetry, and
the use of Euclidean quantum gravity techniques to evaluate the gravitational answer
as the on-shell action (in a saddle point approximation).
The rst two points are not quite speci c to situations with time-re ection symmetry
and ought to apply more generally, provided we understand the third, i.e., formulate the
computation of Renyi entropies in time-dependent situations carefully.
To appreciate this point let us rst ask what is involved in computing the matrix
element of a time-dependent density matrix
A(t). Let us rst start with the entire system
and construct the density matrix (which may be pure) on A [ A
is an operator on the Hilbert space, we are required to evolve both the state
c at time t. Consider the
2As emphasized by LM [6] the gravitational argument is quite general and transcends the speci c
application to deriving the RT prescription in the AdS/CFT context.
3The RT (HRT) prescription also needs to be equipped with a statement that the minimal (extremal)
surface is homologous to the boundary region A [10] (see also [11, 12]). This does follow from LM provided
that the extremal surface arises from a conical defect which for every q 2 Z+ admits a lift to a smooth
spacetime satisfying the boundary conditions of the replicated eld theory [13].
putation of the (a) density matrix and (b) its powers. We have explicitly shown the computation
of 3 in (b). The dots and lines in red correspond to an entangled initial state prepared in some
manner. This picture does not carry the spatial information necessary to ascertain the reduced
density matrices themselves, which is better understood from
gures 3. Note that in contrast to
the usual depiction of the Schwinger-Keldysh contour we will draw time running vertically.
with Ut1;t2 being the unitary operator that evolves the state forward from time t2 to t1, and
i being the initial state prepared in some manner at t =
4 As has been appreciated
by many authors in the past, from a path integral perspective, one is necessarily led to
doubling the degrees of freedom [15{17].
So we start with two copies of the eld theory and construct the instantaneous density
gure 1. Having done so we need to trace out the part of the system
corresponding to Ac. Since
A is a wedge observable [18] (see also [12]), it cannot depend
on which Cauchy slice we pick in the domain of dependence of the chosen region D[A].
In particular, we can think of tracing out A
c as setting boundary conditions on the past
of A's domain of dependence. Our choice of the Schwinger-Keldysh contour then allows
us to immediately compute the matrix elements of
gure 3 for a schematic
representation of this construction (which will be explained in section 2). Once we have
the matrix element of
, we simply string together q copies together cyclically. This is
e ciently done in the path integral construction by prescribing an appropriate
SchwingerKeldysh contour, which as can be guessed will involve 2q copies of our eld theory to
account for both doubling and replicating.
Once we have identi ed the eld theory algorithm for the computation of the Renyi
entropies, we can then ask what is the gravitational avatar of these Schwinger-Keldysh
contours. This question has been addressed in the AdS/CFT context a long time ago:
a prescription to compute real time correlation functions was rst given in [19] and was
subsequently derived from a Schwinger-Keldysh framework in [20]. More recently, [21, 22]
4Note that it is not essential that the initial state i is prepared at t =
1; it could well be prepared
nite ti (e.g. by a Euclidean path integral). However, in the following discussions we will set ti to
1 for linguistic simplicity.
DeWitt patch associated with a give Cauchy surface on the boundary. (b) Given a separation
of the boundary Cauchy surface into regions A and A
in the Wheeler-DeWitt patch admits a decomposition ~
c respectively, any bulk Cauchy surface ~ t
t = RA [ RcA. We also display the bulk
codimension-2 xed point locus e anchored on the entangling surface which approaches the extremal
gave a general prescription for computing real-time observables, focusing in particular on
answering the question of interest to us: what is the gravity dual of a Schwinger-Keldysh
To motivate their construction let us
rst realize one important fact about the
AdS/CFT map. For the boundary QFT, we are free to choose a background geometry
which we take to be a globally hyperbolic Lorentzian manifold B for this purpose. This
ensures that we have a nice foliation of B by Cauchy slices
t and can thus compute
observables at time t. A boundary slice
t however does not uniquely pick a corresponding
time slice in the bulk. Instead, it may continue into one of in nitely many bulk Cauchy
slices that all lie in the bulk region spacelike from
t itself, the Wheeler-DeWitt patch of
t, cf., gure 2. The gravity dual of the Schwinger-Keldysh contour [22] involves a bulk
geometry which coincides with the original spacetime up unto one of the bulk Cauchy slices
in the Wheeler-DeWitt patch and then reverses its path onto a mirror copy of the bulk
spacetime. Thus even the bulk spacetime is doubled in the process of giving a gravitational
construction of the real time contour, with the only proviso that there is an ambiguity in
the choice of the bulk slice in the Wheeler-DeWitt patch where we reverse the trajectory.
The ambiguity | being unphysical | must cancel out in physical observables; this was
shown in [22] to be the case for computing correlation functions of local operators.
Since we are interested in working with a piece A of the boundary Cauchy slice, we have
to generalize the above discussion to restrict attention to the past domain of dependence of
A on the boundary. What this amounts to is that we start with the part of the spacetime
which is relevant for computing (t) | this involves slicing the geometry dual to j i at
some Cauchy slice within the Wheeler-DeWitt patch of t and gluing back a second copy of
the same. However, while arbitrary bulk Cauchy slices are acceptable for the computation
of correlation functions in the state j i, for the computation of Tr( A
restrict to special bulk Cauchy slices in the Wheeler-DeWitt patch of
)q we will need to
t. Speci cally one
requires that the allowed bulk Cauchy slice contain the xed points under the Zq replica
Once we realize this we are in a position to set up the fully Lorentzian construction of
)q, involving as advertised 2q copies of the bulk spacetime glued together to re ect the
boundary Schwinger-Keldysh construction, which computes the Renyi entropies in a way
which is consistent with eld theory causality. One can then invoke the replica symmetry
to focus on a single unit of the Lorentzian forward-backward temporal evolution by taking
the Zq quotient of the resulting geometry. The question then reduces to ascertaining where
xed point set of the Zq replica symmetry, e, lies and what ensures regularity in
the q ! 1 limit. The locus e is the extension of the boundary
xed point set, viz., the
entangling surface @A, to the bulk and it has to be invariant with respect to unitaries both
c
in A and A . Furthermore, since these geometries correspond to the duals of the q ! 1
D[A] or D[Ac]. If it were, that would mean that A is in causal contact with A
the bulk [12]. The upshot is that e lies in the causal shadow [12]. The local analysis
in the neighborhood of e ends up being essentially the same as the one used in the LM
construction, except that the normal plane to the
xed point set has Lorentz signature.
We argue that this leads to the extremal surface condition of HRT when the bulk dynamics
is Einstein gravity (or an appropriate generalization along the lines of [23, 24] for theories
of higher derivative gravity). Requiring that the boundary causality be respected by the
bulk dynamics, we end up with a physical result consistent with the discussion in [12].
The outline of this paper is as follows. We begin in section 2 where we esh out the
details of the Schwinger-Keldysh contour relevant for the computation of Renyi entropies
in time-dependent (mixed) states of a quantum
eld theory. We then move on in section 3
with the gravitational analogue of this eld theory computation, reviewing in the process
the LM construction in [6] and the changes necessary for our argument. We demonstrate
that the HRT proposal involving extremal surfaces follows from the bulk analysis, and
sketch how the computation of the on-shell action in the Lorentzian context leads to the
desired area functional. Arguing that we have a method for computing all Renyi entropies
| at least in principle | turns out to be a bit more subtle, but possible | the technical
steps necessary are given in appendix A. In section 4 we end with a discussion of how
various aspects of the HRT proposal are manifested from our perspective.
Field theory construction
We will begin our discussion by presenting a eld theory construction of reduced density
matrices in time-dependent states using the Schwinger-Keldysh (aka in-in) formalism [15, 17].
As explained in section 1 our motivation is to use this prescription to set up a path integral
to compute Renyi entropies in non-trivial time-dependent states.
Without loss of generality let us consider a pure state of some d-dimensional local
QFT on a globally hyperbolic background geometry B
5 We will prepare an initial state
of this theory and evolve it up to some time t. The speci cs of the initial state will not
be important for our purposes; it could be constructed either by invoking some boundary
conditions in the far past or by slicing open a Euclidean path integral with possible sources.
Regardless of how the state is prepared, we will evolve it in real (Lorentzian) time, further
allowing ourselves the freedom to turn on arbitrary spacetime-dependent sources.
The time-dependent QFT wavefunctional
With this understanding let us attempt to write down a formal path integral expression
for the wavefunctional of the QFT. We rst note that the density matrix requires
information about both the state and its conjugate, which have to be separately evolved from
the initial state (perhaps with a time-dependent Hamiltonian in the presence of sources).
Schematically and working in the Schrodinger picture, one has therefore
=)
j (t)i = eiHt
h (t)j = h 0j e iHt
(t) = j (t)i h (t)j = eiHt
It is useful for the purpose of our discussion to convert this statement into a path
integral construction. A natural way to respect (2.1) is to incorporate a Keldysh contour [17]
that evolves the state j i forward and its conjugate h j backward. In the path integral
construction, the wavefunctionals are given as
t; 0(x) =
t; 0(x) =
R t2 dtL[ ] for the QFT Lagrangian L. It is then easy to write down the corresponding
stands for the entire collection of elds of the theory, and S[ ]tt21 is de ned as
path integral expression for (t) as:
= j i h j =
where the arrows indicate the direction of time evolution inherited from (2.1). As is well
appreciated in the literature, this can be viewed quite straightforwardly in terms of a
contour prescription for the path integral, cf., gure 3 for an illustration.6
The reduced density matrix
Having constructed the path integral representation for the total state of the system
at the instant of our choosing, we now want to use this to compute the entanglement
5A mixed state can be puri ed by introducing appropriate auxiliary degrees of freedom.
6To get physical answers we usually have to implement an appropriate i" prescription in case we encounter
singularities along the real time axis (which is the naive contour of integration); this is implicit in the
. The forward evolution for j i
t, while the backwards evolution for h j starts there. Gluing the two evolutions
t enables taking the trace of (t). We also depict the situation, where we open out
t to construct A(t). These cuts are introduced at t = t as described in the
class observables of A
A from the global state (t) on
entropy for a subsystem at this instant. Let us denote the achronal spatial section (a
Cauchy slice) of B at this instant as
t. The subsystem of interest will be a spatial region
t bounded by the entangling surface @A. We denote the complementary region as
Ac; clearly t = A [ Ac. Of interest to us is the reduced density matrix
t; subsequently we will want to compute various trace
The bipartition of
t allows the computation of the matrix elements of
that (t) has been obtained by combining two separate evolutions, one forward and one
backward. Gluing these two evolutions together along the Cauchy surface
(t) . At this point it is useful to upgrade the conventional pictures of
Schwinger-Keldysh contours to depict spatial information as well; see
gure 3. We now
A on t and introduce some additional boundary conditions to extract the reduced
To obtain the matrix elements of A(t), we introduce a cut along A in the path integral
by integrating over 0 only in region Ac; the cut however remains pinned at @A. We imagine
respectively. Since there are two temporal contours in the path integral we have two cuts,
which open up a slit in the folded geometry as depicted in
gure 3. On either opening of
the slit, we prescribe some boundary conditions for the elds
in the forward part of the contour and
(t = t )jA =
along A: (t = t+)jA =
in the backward part. The path
integral with these boundary conditions for the elds on the cuts across A de nes for us
the matrix element ( A)+ .
While our choice of boundary conditions to construct the reduced density matrix
disconnects the path integral contour across A, it remains geometrically connected across the
complementary region as we go through
t. If we want to compute Tr( A), we can simply
glue back the slit that we introduced and therefore recover the original (doubled) geometry.
7By trace class observables, we mean expectation values of operators localized in A (more precisely in
its past domain of dependence D [A], cf., footnote 8), i.e., Tr( A OA) where OA is a collection of operators
It will be important in what follows that the local geometry in the neighbourhood of the
entangling surface @A is at; in the Lorentzian context of interest we could say it is \locally
The construction of the matrix element of A involves picking a Cauchy slice. However,
we have some freedom here; the precise choice of
t is relevant only for the computation
of the density matrix itself. For computing trace class observables, we can use unitary
evolutions within causal domains of relativistic QFTs (see e.g., [12]) to pick other slices.8
Computation of traces involving
A only requires information about the temporal evolution
of the state in the past of the entangling surface. Thus we can push parts of the Cauchy
slice (and thus A, Ac) to the past as long as we do not modify the entangling surface. More
J [ t] with 0t \ t = @A is acceptable.9
Before we proceed to construct the geometries relevant for computing the Renyi
entropies, let us introduce some coordinates to describe the folded geometry for
coordinate chart is the one inspired by the fact that the local geometry in the vicinity of
@A is Rindler space (as opposed to a Cartesian chart around
t). To see this consider
(t; x; wi), i = 1; 2;
2. Take
the following example: in at spacetime B = R
d 1;1 we choose the Cartesian coordinates
t to be a constant time slice (say t = 0) and A to
be the half space de ned by positive x. In this case D [A] is the past half of the (right)
Rindler wedge of at space. The construction described above can be implemented in this
Let us therefore consider the Rindler chart:
ds2 =
dt2 + dx2 + dwi dwi =
The advantage of these Rindler coordinates is that we can simultaneously refer to all the
spacetime regions in question, by allowing
to be complex with a discrete imaginary part.11
Since we are dealing with the reduced density matrix, for x > 0, t
Consider then the ve coordinate patches covered by
m = 0; 1; 2; 3; 4 (see gure 4 for an illustration):12
m = 0: we use
A < 0 to coordinatize the domain D [A] below the fold at the
. One may think of these domain as the right Rindler wedge, with
boundary conditions + at A.
8We use standard relativistic notations: J [p] stands for the causal future/past of p and D [X] the
future/past domain of dependence of some set X.
9One can use this observation to deform
close to the null surface ND = @D [A] [ @D [Ac] and replace the wave functional on
wave functional on ND, ( D), obtained by Hamiltonian evolution. This perspective is useful for some
10In this highly symmetric example, one usually recognizes that it has a timelike Killing eld and exploits
the time-re ection symmetry. We phrase the statements herein without invoking this symmetry, therefore
enabling generalization to the time-dependent situation of interest.
11It is worth emphasizing that this is simply a convenient book-keeping notation. In particular, we are
not assuming any analytic properties on the complex
ve patches because we are describing the reduced density matrix in its entirety. When taking
trace we will end up identifying the patches under
t down all the way to a spacelike surface that lies arbitrarily
Figure 4. Local Rindler coordinates that we will use to describe the geometric construction of A
We have focused on the neighbourhood of the entangling surface and indicated the causal domains
and coordinates used therein (see text for details).
m = 1:
A + 2 i coordinatizes the Milne wedge J [@A] on the backwards segment
of the Schwinger-Keldysh contour.
in the backwards/forwards segment.13
m = 2:
A + i covers the past domain of dependence of the complementary
region Ac. Since we are describing
A, the two folds are glued at
. The forward
and backward domains of D [Ac] are distinguished because
A is positive/negative
m = 3: when
A + i 32 we are in the Milne wedge again J [@A], but this time
on the forward segment of the Schwinger-Keldysh contour.
part of the contour. This is the right Rindler wedge again, but on the backward part
of the Schwinger-Keldysh contour and we impose boundary conditions
It will be crucial to remember that when we compute Tr( A
) we glue the m = 0
0+ in the path integral computing the trace of the reduced density
matrix, since we want to think of the backward/forward parts of the right wedge as being
parametrized by a unique coordinate. We identify
+ 2 i. This is the geometric
encoding of the statement of @A being locally
at. One can equivalently phrase this
by invoking the standard Rindler interpretation of a local accelerated observer seeing a
The local geometry near @A for more complicated geometries is similar; all that we
require is that the normal bundle to @A admits Lorentzian sections. In an open
neighbourhood of the zero section i.e., at @A we can use the Rindler coordinates described above,
which provides a convenient way to keep track of the properties of the replicated geometry.
13Note that the signs are ipped with respect to the right wedge, as it is usual in Rindler.
⌧ 1
⌧ 2
⌧ 3
Figure 5. A schematic view of the replica Schwinger-Keldysh contour for computing ( A
elements. We have restricted attention to the neighourhood of the entangling surface for simplicity.
Products of reduced density matrix and Renyi entropies
At the end of the day, the reduced density matrix is a means for us to compute the
entanglement entropy via the replica trick. We recall the basic de nitions:
A =
S(q) =
) = lim S(q);
We have introduced the Renyi entropies, which help in implementing the replica trick via
the second equality in the rst line.
To compute the entanglement entropy we therefore need to compute powers of the
reduced density matrix ( A
)q. Since we have a path integral construction of A this is
easily done. Consider q copies of the geometry used to compute the reduced density matrix
indexed by I = 1;
; q. Computing the matrix product requires us to
cyclically glue these geometries together; this constructs the new
eld theory background
Bq. The construction involves identi cations along the cuts introduced at A: we identify
for the path integral, but also giving us a geometric way to obtain a Schwinger-Keldysh
contour which is relevant for the computation of ( A)q. In short one may view the thus
obtained path integral contour as the Lorentzian analogue of the replica construction, see
gure 5. The latter of course was useful in deriving the RT proposal from a gravitational
path integral [6]. Our task will be to abstract su cient lessons from the Lorentzian
construction in eld theory to implement the same in gravity.
It is worth noting in passing that in certain situations where we have a time reversal
t, we may simply evolve from t =
1 to t = 1 on each copy of the
replica construction. We may then analytically continue to Euclidean signature on each
copy. This is really the realm where the RT construction [1] and its Euclidean quantum
gravity derivation a la LM [6] is valid. While we will review this in section 3.1, our goal is to
unanchor ourselves from this special circumstance and derive the covariant proposal of [3].
Let us understand the geometry of interest, Bq. In the gluing construction, we have
initially 2q di erent time coordinates (two for each copy of the density matrix, and q density
matrices), which are glued together di erently in A and Ac respectively. Instead of using
di erent time coordinates, tI , for each I 2 f1;
; qg, it is useful to invoke the
inspired by the Rindler construction. Passing from t
I in each copy of the density
We now introduce a single coordinate
with an imaginary part supplied to take care of
this identi cation. Consider
A + m2 i where now m = 0; 1
; 4q. This coordinate
naturally allows for interpolation from 1 ! 2 !
that we take the trace. What this implies is that
q ! 1; the last identi cation ensures
+ 2 iq is the correct identi cation
for this coordinate along A to ensure that the density matrices are multiplied in the right
cyclic order. A pictorial depiction of this statement is presented in gure 5. This statement
is the Lorentzian analogue of the monodromy acquired by traversing sections of the normal
bundle across @A.
While a priori the choice of coordinates is just a matter of convenience, the
nates allow for a simple statement of the boundary conditions in the QFT. They should
be seen as a useful book-keeping device for the identi cations between the di erent
replicas. In particular, one is not performing any analytic continuation of the eld theory data
to complex times, as can be inferred by working directly with the Minkowski chart using
fxJ ; tJ g for each copy of the reduced density matrix.
The periodicity in
+ 2 iq is a statement about the gluing conditions in the
Schwinger-Keldysh path integral contour constructed above. The eld theory path integral
is done over the 4q temporal domains, which are conveniently encapsulated by the single
coordinate. We would like to reiterate what it means for elds to be periodic in
period 2 iq. In the QFT path integral we integrate over all allowed
eld con gurations.
domain of the elds consists of 4q disconnected horizontal lines in the complex
plane, each with imaginary part i 2 m, m = 0; 1;
1. We then impose boundary conditions at the asymptotic in nities of these horizontal lines. It is easiest to do this in the language of asymptotically incoming/outgoing modes. The coe cients of these mode must match between
= (m + 1) i
= m i + 1 and
1, for all m = 0; 1;
= (m + 12 ) i + 1, and between
= (m + 12 ) i
1. Note that in saying this, we are identifying = 2 iq
1 | this is what we mean by the periodicity
To recap: we are gluing q copies of the reduced density matrix, with region A identi ed
across the copies. This can be just as well stated in the fxJ ; tJ g coordinates, but the
coordinate is more useful for delineating the analogous boundary conditions in gravity.
While the general focus here is on the computation of the Renyi entropies themselves,
it will transpire that the gravitational computation is nicer for the derivative of the Renyi
entropy with respect to its index. De ne thus a related quantity [25], which we will call
the modular entropy :
S~(q) =
In writing this expression we have already assumed that we can analytically continue the
Renyi entropies away from the integer values of the index q. To our knowledge, this object
has not been considered before in the quantum information literature, but it is rather
natural. For instance, if we take
A to be of thermal (as for spherical domains for CFTs in
d 1;1), and view q to be a measure of the inverse temperature, then S~(q) is the thermal
entropy. The qth modular entropy is the the appropriate Legendre transform of the qth
Finally, there is a Zq symmetry relating the various replicas, with @A being the xed
point of its geometric action on Bq. In the rest of the paper, we will assume that this
symmetry is unbroken. As a consequence the one-point functions of our QFT should be
replica symmetric. In the
coordinate this corresponds to functions being strictly periodic
with a smaller period of 2 i. As we will see later, this point of view provides a particularly
straightforward route to understanding the boundary conditions for the dual gravitational
problem. At the end of the day we will require all elds in the bulk to be invariant with
respect to the replica symmetry. Coupled with the fact that we disallow any curvature
singularities, this serves to pick out the acceptable geometries, which satisfy Einstein's
equation, and provide the dominant contribution to the gravitational action.
Gravitational construction
We have thus far set up the problem of determining the matrix elements of the reduced
density matrix and its powers by invoking an appropriate Lorentzian Schwinger-Keldysh
contour prescription in the eld theory. Assuming that the eld theory in question is
holographic, with a semi-classical gravitational dual described by a di eomorphism invariant
local Lagrangian, we would now like to ask how to implement the aforementioned
computation in gravity. We will outline the construction below by rst asking what is the gravity
dual of the Schwinger-Keldysh contour of interest. This question was answered in [21, 22]
whose analysis will inspire us to provide a bulk prescription for the computation of the
reduced density matrix elements. Following this, assuming unbroken replica symmetry, we
will argue that the entanglement entropy of the said density matrix is computed in terms
of the area functional on a codimension-2 extremal surface in the geometry, thus deriving
the HRT prescription [3].
To set the stage for our discussion, we will rst quickly review the elements that
enter the derivation of generalized gravitational entropy [6], valid for Z2 time-re ection
Review: time re ection symmetric case
As explained in section 2, if we have a Z2 time-re ection symmetry t !
interested in computing the density matrix elements at the
xed point t = 0, then the
Schwinger-Keldysh path integral can be simpli ed. Exploiting the symmetry, we can simply
consider the evolution from t =
1 to t = 1, since the backward evolution is equivalent to
the forward one. This in turn allows us to simplify the computation by analytic continuing
to Euclidean time t ! i tE.
process of computing Tr( A
Once we have a Euclidean path integral for computing matrix elements of
)q is achieved by gluing the replicas in Euclidean space.
Equivalently, one is instructed to compute the partition function of the eld theory in a geometry
Bq with conical excess 2 q inserted at @A. This turns out to be a well-de ned gravitational
problem. All one needs is to construct a bulk geometry Mq whose conformal boundary
is Bq for q 2 Z+. While this serves to compute the Renyi entropies, in fact, we are
interested in computing the entanglement entropy, which is achieved by an analysis in the
limit q ! 1+. The key point of [6] is that the analytic continuation from integral q to
the vicinity of q
1 is much simpler in the gravitational context. We now review this
argument, splitting it into two convenient parts: a purely kinematic piece and one that
cares about the gravitational dynamics.
Kinematics. Let us rst discuss the case q 2 Z+. For integer q, the boundary manifold
Bq is a q-fold branched cover over B (branched at @A). Per se this provides a clean
boundary condition for the gravity problem as described above. However, we can exploit
that fact that the partition function has a Zq symmetry of Bq that exchanges the di erent
replicas. This is a symmetry owing to the cyclicity of the trace in the de nition of Renyi
Assuming as in [6] that this replica Zq symmetry extends to the bulk, we can take
geometry is not smooth and generically contains a codimension-2 xed point locus of the Zq
action.14 We will call this xed point set of the bulk eq | it will be part of the kinematic
data as we build up an ansatz for construction. Apart from being invariant under the Zq
symmetry exchanging the replicas, eq is the natural extension of @A into the bulk.
Let us now set up a bulk coordinate chart. First, consider a codimension-2
surface in the original spacetime M. We pick coordinates adapted to the surface: yi with
i = 1; 2;
1 parameterize tangential directions, while the normal directions are
coordinatized by x; tE. Expanding the metric in a derivative expansion around the surface,
ds2E = dx2 + dt2E + ( ij + 2 Kixj x + 2 Kitj tE) dyi dyj +
We have retained only the leading terms in the Taylor expansion about the surface located
i tE = r e i , where
+ 2 for regularity.15
If we introduce such local coordinates in the vicinity of eq as in (3.1), then the replica
symmetry implies that the action is invariant with respect to a global shift of the polar
coordinate in the normal plane , viz.,
+ 2 ; see gure 6 for an illustration. Near
eq this replica coordinate has to be identi ed under
+ 2 q. We can now use the
14There are some subtleties with this statement, for it is possible in certain situations that the xed point
set has `wrong' codimension; cf., [13] for a detailed discussion and examples. We will assume that we have
a family of replica symmetric geometries, parameterized by q, and smooth for q 2 Z+ which, as argued
cient to avoid any exotic scenarios.
15For convenience we are going to use the same notation for the normal bundle coordinates in the bulk
M and the boundary B. This is natural; as the xed point set e is the bulk extension of the entangling
extends in Mq into a
of the (Euclidean) normal bundle of this xed point set.
action on the boundary and the bulk. The region A terminates on the entangling surface @A, which
xed point locus eq. We use polar coordinates (r; ) to parametrize sections
smoothness of the covering space Mq,16 to infer that the local geometry near eq in the
ds2 = q2 dr2 + r2 d 2 + dst2ransverse + : : :
We have left implicit here the transverse part of the geometry which we will describe in
due course. The main point to note is the explicit q dependence. Its presence implies
backreaction; one cannot simply identify
+ 2 q in (3.1).
Exploiting the replica symmetry we can restrict our attention to a single fundamental
domain (or replica) of the Zq action in the bulk. Thence, the total action of the gravity
computation will be q times that of a single domain, viz.,
I[Mq] = q I[M^ q]
While the quotient space has a conical singularity with defect angle 2q , the covering space
is smooth; this observation will play a crucial role in setting up the boundary conditions.
The advantage of thinking about the orbifolded quotient space becomes manifest when
we think about computing the entanglement entropy which requires analytic continuation
16One might worry that the geometry is smooth in the bulk, but becomes singular as it approaches
the boundary due to the entangling surface. This singularity however can be dealt with by a suitable
regularization procedure. For example in some situations [7] we can use conformal mapping to send @A to
in nity and use a standard IR cut-o .
17Strictly speaking the geometry has a bration structure, whereby the normal bundle parameterized
by the (r; ) coordinates is non-trivially bred over the base. We have for simplicity dropped some of the
o -diagonal components in writing (3.2).
the strength of the opening angle at the conical defect in M^ q. Working in the orbifolded
space, we simply analytically continue q by dialing the strength of the singularity. This the
kinematic part of the analysis implies that we work in the q ! 1 limit, on a geometry with
a conical de cit of prescribed strength, with the same boundary conditions as the original
background geometry M.
Having set up the basic problem in the gravitational context, we now want
to gure out what con gurations dominate and thence compute their on-shell action. For
simplicity we will consider Einstein-Hilbert gravity here; generalizations to other classical
gravitational theories follow along the lines of [23, 24].
To enforce the boundary conditions in the gravitational solution, we examine the metric
close to eq. Consider a wave equation in the local coordinates of (3.2). It is easy to see that
ascertain which of these is admissible and thus give explicit boundary conditions, we invoke
two facts. Firstly, the replica symmetry requires a 2 periodicity for elds as functions of ,
restricting us to purely oscillatory functions and thereby xing ! 2 Z. Secondly, regularity
of the covering space implies that the elds have to admit an expansion in powers of rqe i ,
that rqe i will be the generic behaviour of the metric near the origin.18
From the above discussion we then learn that the most general ansatz for the geometry
near e compatible with our boundary conditions is:19
ds2 = (q2dr2 + r2 d 2) + ( ij + 2 Kixj rq cos
where fq is some analytic function of q such that fq(q
1) takes nonnegative even integer
values when q is a positive integer. This metric is smooth and Zq symmetric for integer q.
tions proportional to (q
1) Kra where Ka
Kiaj ij is the trace of the extrinsic curvature.
This divergent contribution cannot be compensated by modifying other components of the
metric. We are thence led to conclude that the equations of motion give us a constraint on
the allowed eq. The allowed codimension-2 surfaces are required to have vanishing trace
of the extrinsic curvature in the normal directions. Since we have a t !
with t = 0 ; Kx = 0 :
refer to the normal bundle of eq.
18The astute reader may worry that as a consequence we will have some components of the curvature
integrable in a suitable sense, as we shall see.
19Notation: Greek (lowercase) indices refer to the full spacetime, mid-alphabet lowercase Latin indices
refer to the tangent space of the
xed point set eq, and early-alphabet lowercase Latin indices
Let us record here for completeness that we have included in our metric ansatz (3.4)
potential contributions from
rst subleading orders (the g term).20 It may be veri ed that
this term by itself cannot cancel the divergences arising from the rq terms in the metric.
At this point we are almost done: once we know that one should restrict to EA as
in (3.5) all that is left is to compute the on-shell action of this geometry. It turns out to
be convenient to compute not the action itself, but its q-derivative @qI[M^ q] as explained
around (2.6). This perspective was explained in [6] and is based on the covariant phase
space approach used in the black hole context [27]. As shown in [25], there is in fact a
simple geometric prescription for this object @qI[M^ q] for any value of q, so we do not have
to set q = 1 in the following discussion.
The key point is to view @q as a variation of the bulk solution (and its boundary
conditions). Standard variational calculus says that any variation of the action can be
written as a combination of the equations of motion and boundary terms (using integration
by parts where necessary):
I[M^ q] =
For a typical variation that appears in a standard AdS/CFT calculation, this would
evalvariation of q, which instead changes the boundary condition near the xed point set eq.
the change of the action engendered by the replica index variation is localized at the xed
point locus and has no contribution from the asymptotic boundary of the spacetime. One
may therefore write
@qI[M^ q] =
where we have chosen to regulate the result by blowing up the singular locus to a tubular
neighbourhood. In other words the
x point set eq which was at r = 0 is now being
In the present case we will not actually evaluate this integral (which can be done given
the symmetries), but will follow an equivalent route. In the presence of a boundary for the
variational calculus to be well-de ned and give the correct equations of motion, we would
need to supply the correct boundary terms. While in our case the surface eq( ) is not really
a physical boundary, one may for purposes of evaluation imagine that it is and ascertain
the corresponding boundary terms. The advantage of this trick is that the on-shell action
will be given simply by evaluating these contributions. For Einstein-Hilbert gravitational
dynamics we evaluate the Gibbons-Hawking contribution from eq( )21
@qI[M^ q] =
Ibdy[M^ q] =
20We should note that such terms are sometimes desired in order to cancel subleading divergences in
higher derivatives theories [26].
The measure should hopefully be clear from the context.
21To prevent notational clutter, we drop the integration measure for simplicity in the formulae henceforth.
where K is the trace of the extrinsic curvature of the codimension-1 surface eq( ), evaluated
with the outward pointing normal vector. Again, this holds in the
Working in the local coordinates (3.4) in an open neighbourhood of eq, one
K = q1 , and thus we get22 for the modular entropy:
@qI[M^ q] =
which as q ! 1 gives us the RT formula.
Before moving to the Lorentzian case, some words of caution are in order. The orbifold
picture allows us to analytically continue the on-shell action I[Mq] to non-integer q. The
physical interpretation of the (parent space) solution for non-integer q is unclear, but these
geometries are just an intermediate step to compute the action. In this way, even if for
integer q these geometries do not have singularities, some components of the Riemann
tensor will go like
q 2 for q 2= Z. However, these components neither appear in the
equations of motion, nor the evaluation of the action, and are thus mostly harmless.
Covariant generalization to time-dependent situations
As we have extensively presaged in the earlier sections, in genuine time-dependent
circumstances, we cannot invoke the trick of passing to a path integral over a Euclidean manifold.23
Rather, we have a Schwinger-Keldysh or time-folded path integral which has to be dealt
with in Lorentzian spacetime. Indeed the eld theory construction of the density matrix
explained in section 2 requires evolution from the initial state up until the moment of
interest, say t, and then retracing one's footsteps back to the far past. This forward-backward
A [ Ac on the boundary B, as we only
evolution induces a kink at the Cauchy slice t
retain the part of the geometry to its past, i.e. J [ t].
In the bulk there will be an analogous fold along some Cauchy slice ~ t, with the proviso
that the bulk evolution will proceed only in the part of the spacetime to the past of ~ t,
to ~ t and then we evolve back to construct the bulk Schwinger-Keldysh contour.25 This
forward-backward evolution through ~ t, across which two copies of the bulk manifold are
glued together, is illustrated in
gure 7. On the Cauchy slice as we reverse the evolution,
we have to provide appropriate matching (boundary) conditions.26
As described in section 2, we may move
t itself as long as J [@A] is unmodi ed.
Let us say, for de niteness, we make a particular choice and stick with it w.l.o.g. This
23In the absence of time-re ection symmetry, the analytic continuation of t ! i tE will lead to a complex
manifold. Furthermore, analytic continuation to Euclidean signature would not work in generic cases where
the time-dependent sources are non-analytic.
24We will use a tilde to distinguish bulk Cauchy surfaces and causal sets from analogous quantities on
and then evolved up to t = 0 perhaps with sources, etc.
variation of the action doesn't contain boundary terms at ~ t.
26These boundary conditions guarantee that we have a well de ned variational principle and that the
RA A
the spacetime in question, which are glued across the part RA
c
with A . Taking the trace corresponds to gluing across the part RA of the bulk Cauchy slice
c of the bulk Cauchy slice associated
associated with A.
however does not single out a bulk Cauchy slice, since the boundary time coordinate does
not extend uniquely into the bulk. As remarked in section 1, ~ t can lie anywhere in the
Wheeler-DeWitt patch (part of the bulk spacelike to
moment we will take ~ t to be some representative in this Wheeler-DeWitt patch (as shown
for instance in gure 2); we will learn in the course of our analysis of potential restrictions
t) provided @ ~ t =
on bulk Cauchy slices.
Basically, we are extending the construction of the eld theory into the bulk in an
intuitive manner. Each piece of the bulk glued across ~ t corresponds to the state j i
(forwards) or its conjugate h j (backwards) in the gravitational description. This extension
of the eld theory Schwinger-Keldysh contour allows for computation of holographic real
time (in-in) correlation functions [22].
Having understood how to set up the Lorentzian problem in gravity, we now have
to face up to the harder question: \What is the dual of tracing out degrees of freedom
We will rst analyze the kinematic part of the construction and pick out the
over Ac can only be done if ~ t satis es some speci c properties.27
boundary conditions involved in computing Tr( A)q. We will then argue that the trace
It bears emphasizing that our construction has two distinct components:
(i). Constructing the dual of eld theory density matrix (t).
27It is worth contrasting this with the computation of correlation functions for which any ~ t in the
Wheeler-DeWitt patch of t is acceptable [22].
The rst involves the Schwinger-Keldysh framework, while the second involves identifying
the part of the bulk spacetime we trace over. The latter issue is already present in the
time-re ection symmetric case, should we view the discussion of section 3.1 in Lorentzian
Kinematics: setting up the bulk construction
Let us rst try to set up the gravitational problem and come up with an ansatz which
we can use to explore the bulk solution of relevance. From an operational point of view,
we need to understand how to translate the complicated path integral construction of the
boundary eld theory in terms of the bulk variables. The upshot will be the following: we
will rst ascertain what it takes to compute the Renyi entropies at integer q > 1 in the
bulk. Subsequently, we will argue that the computation can equivalently be done upon
taking a Zq quotient in a `single fundamental domain', which has the same asymptotics
existence of a xed point locus eq on ~ t.
The Renyi entropy, TrA( A
)q, is constructed in the boundary from q copies of the
density matrix itself on
t. This has q forward time-contours (or ket-folds) and q backward
time-contours (or bra-folds) glued together appropriately. The gluing is essentially dictated
by the split t = A[A
c which de nes TrA( A)q. Concretely, the part corresponding to A is
glued across from one density matrix to its immediate neighbour, while that corresponding
1. Local Rindler coordinates in the bulk, for q = 1. Consider the q = 1 geometry
used to construct the bulk analogue of TrA( A
corresponds to two J~ [ ~ t] segments glued across at ~ t. A priori, it is not clear how the
di erent regions A; Ac are encoded in the bulk. We will take a cue from the boundary
and assume that there is a simple extension of the split of the boundary Cauchy slice into
the bulk; even if this seems ad-hoc, the existence a codimension-2 surface implementing
this bipartitioning will be a consequence of the bulk replica symmetry, as we will discuss
later. We start with an ansatz that we can divide the bulk Cauchy slice into two regions
bulk extension of @A; see gures 2 and 7. While we do not know (as yet) how to determine
e, we can ascertain how the boundary statements are re ected in the bulk given such a
demarcation. We will later see that e is xed by analytic continuation of the Zq symmetric
) (equivalently Tr
(t) ). This geometry
xed points for the replicated geometries.
Of particular utility will be to understand how to extend the boundary coordinates
into the bulk. In particular, we should
nd an analogue of the
coordinate which helps
identify the boundary conditions. To get a sense of how to proceed, we can foliate the
causal development of RA in Rindler-like coordinates. We will use the fact that one can
naturally write the metric close to any codimension-2 surface (such as e) as:
ds2 = dr2
This metric is completely regular since we are just rewriting the original metric in adapted
the Cauchy slice ~ t. The meaning of the identi cation
@A. We show the local horizons of this surface and the boundary conditions imposed as we cross
+ 2 i is explained in the main text.
However, it would be useful to view this slightly di erently. For the Rindler-like
observer one encounters four horizons emanating from the codimension-2 surface; see gure 8.
As explained in section 2, in these coordinates a horizon crossing can be understood as
+ i 2 (along with r ! i 1r). The complex shift is a useful mnemonic to remember
the boundary conditions; the passage to complex values of
should not be viewed as a
fundamental necessity.28 It captures the local geometry in a neighbourhood of e e ciently.
Regularity demands that one should go back to the starting Rindler wedge (say the domain
of development of RA) after crossing four horizons. This is what we propose to encode as
+ 2 i. This is the Lorentzian analogue of the fact that we have a Euclidean time
circle with an appropriate size.
A more physical way to state the boundary conditions is the following: in A and
thence in RA, elds that approach @A and e, from within A and RA respectively, should
behave like local Rindler modes with an e ective temperature 21 . This local monodromy
condition calls for the choice
+ 2 i above; it is only in this restricted sense we talk
about complex shifts of the temporal coordinate.
If we have k = 1; 2;
; q domains in the bulk with coordinates frk; k; ykmg, then we
would have a similar story in each replica copy. The equivalent boundary conditions would
be to essentially demand that every time four horizons are crossed one goes to the next
ym . Given the assumption of bulk replica
28To reiterate, this can be clearly be seen in the Cartesian fx; tg coordinates, for there it is a simple swap
symmetry and that the boundary xed point @A extends naturally into a bulk xed point
e, it seems natural to think of the bulk as having this branched cover structure. We should
also understand under what conditions the metric is a smooth solution to the equations of
2. Construction of the replicated geometries. With this in mind, let us move on
to the construction of the bulk geometry Mq dual to Tr( A
copies of a folded geometry glued cyclically along part of t
)q. In the boundary, we have q
. We will now assume that the
partition function on this geometry Bq can be computed in the saddle point approximation
by a bulk geometry similar to those of [22]. As in the Euclidean case, we are going to
assume that the boundary replica Zq symmetry extends into the bulk. In this process,
the boundary xed point set @A also gets continued along some bulk codimension-2 locus
of xed points, which we will denote as eq.29 In each replica, this
construction lies on ~ t, a Cauchy slice whose boundary is
xed point locus by
t, and serves to demarcate
the surface into two parts. Given these, it seems natural to expect that this requires the
boundary branched cover structure inherent in the replica construction to be inherited by
the holographic map in the bulk. In this way, we have a branched geometry, whose action
should correspond to Tr( A)q. We also have a well-de ned codimension-2 surface which
extends @A to the bulk. In a suitable limiting sense, as described below, one should think
The previous discussion can be heuristically viewed as follows: we divide ~ t into two
pieces across eq as indicated and glue the components di erently across the multiple copies
of the density matrix viewed as geometries. That is, we now require that elds which
approach ~ t inside RA on the kth copy, pass onto the (k + 1)st copy as depicted for
gure 9. This cyclic gluing condition can equivalently be phrased as saying
that the elds which approach eq through RA feel a local temperature 2q ; this constrains
the mode functions for the elds.30 These should be thought of as prescribing the relevant
boundary conditions for our problem in gravity, thus naturally extending the eld theory
discussion of section 2; cf., gure 8. Of course, given these boundary conditions, the dual
geometry will be rather complicated. For one, it will be very di erent from the o -shell
picture of [8] and for another, for integer q > 1, we will not be able to say much more
beyond the simplest cases.
We have argued that eq
~ t, but since ~ t was a priori any bulk Cauchy slice inside
the Wheeler-DeWitt patch, we seem to have a large amount of freedom in the location of
the xed point set. One can x some of this ambiguity by invoking the causality properties
satis ed by entanglement entropy to argue that the xed point set eq must lie in the causal
shadow of D[A] [12]. This does not pin down the surface in any way, for the causal shadow
of a boundary domain of dependence generically is a bulk codimension-0 volume.
29If there is no bulk
xed point, we expect that the entropy is zero, as discussed in [6].
30In addition it determines the initial conditions for evolving
elds from one side to the other side of
the Rindler horizons (of each copy) for it gives appropriate matching conditions for the modes across the
replica copies cyclically.
copies of the spacetime to construct the replicated Schwinger-Keldysh path integral. The
identications of RcA are as described before in gure 7, while the identi cations of RA are across the
We have basically distilled the construction of Mq (similar to the Euclidean case)
to focus our attention on only one copy Mq
Mq=Zq of the replica geometry. This
geometry has a Zq
xed point set which, being spacelike, could be called an \instanton
conical singularity". More precisely it should be considered to be a codimension-2 S-brane
in spacetime [28] with tension set by q. Our task is now to understand what the constraints
are on eq (and also ~ t) that respect the boundary conditions of our problem. Thus having
used up the kinematic data to motivate an ansatz for the gravitational construction, we
now appeal to the dynamics of the bulk gravitational theory to provide constraints on eq,
using the fact that the covering space needs to satisfy the correct boundary conditions and
equations of motion near the xed points.
We have generalized the discussion of [22] beyond boundary contours where the gluing
doesn't have spatial dependence since the gluing conditions for the Renyi entropies are
di erent on the two sides of the entangling surface. We have done this by extending the
boundary contour into the bulk in the most straightforward manner that is compatible
with the replica symmetry: gluing purely Lorentzian segments and imposing the proper
boundary conditions. This is a natural extension of [22] which we assume without further
justi cation in what follows.
One of the features of the Schwinger-Keldysh construction is a redundancy built into
the construction. This can be understood from the ability to implement eld rede nitions
in the doubled theory, cf., [29]. This allows certain deformations of the contour which
nevertheless end up giving the same physical answers for observables (including the
onshell action). Readers may be familiar with a related statement in thermal eld theory,
where there is a one-parameter family of Schwinger-Keldysh contours, characterized by the
two Lorentzian contours separated by an arbitrary Euclidean distance, with the proviso
that the total contour be periodic in imaginary time with period . Though this argument
typically relies on the analyticity of thermal correlators, we cannot rule out in general a
deformed contour in the bulk which computes the Renyi entropies of interest.31
may perhaps have additional Euclidean segments, but the general expectation is that they
will also have the same on-shell action as the con guration that we favour with minimal
Euclidean excursions (just those necessary for a correct i" prescription). It would be
interesting to examine this issue further.
Dynamics: equations of motion and extremal surfaces
In section 3.2.1, we have used the kinematic data at hand to set up the problem. When all
the dust has settled, we have essentially reduced our attention to a fundamental domain M^ q
of the bulk under the replica Zq symmetry, namely a Schwinger-Keldysh double geometry
constructing the dual of the trace of the total density matrix Tr (t) with a Zq symmetric
xed point set, eq, localized on the Cauchy surface ~ t. The remaining task at hand is to
employ the bulk equations of motion, see what they imply for eq, and compute the on-shell
1. The extremality condition. We have described the boundary conditions that we need to satisfy in section 3.2.1. As in section 3.1 it is useful to switch to Rindler-like
coordinates fr; g for the normal bundle of eq in the bulk. In the following discussion, we
will focus on the forward segment of the Schwinger-Keldysh contour (
< 0).32 Analogous
to (3.4) in section 3.1, the metric is constrained by the Zq symmetry, boundary conditions,
and regularity for integer q to have the following expansion in the vicinity of eq:
ds2 = (q2dr2
where we denote the coe cients of the rq terms as Kiaj because in the q ! 1 limit they
give the extrinsic curvature.
With this ansatz for the geometry, we can now analyze the consequences of the
equations of motion. This is in fact quite easy, since the local geometry resembles the Euclidean
31The future gluing condition in the Schwinger-Keldysh contour is e ectively a projection of the nal
density matrix against the maximally entangled state in the doubled system. The latter is obtained as the
T ! 1 limit of the thermal density matrix. We thus can imagine a deformation wherein we glue a copy of
the Euclidean instanton corresponding to this limiting solution.
32Since the analysis is local below ~ t, we do not need to worry about the kink.
discussion. We have a deviation away from
at space (in Rindler coordinates) owing to
an instantonic brane source with tension set by q. The gravitational equations of motion
away from eq, however, do not care about this.
Indeed, evaluating the terms in the equations of motion for Einstein-Hilbert dynamics
in the bulk, we nd potentially divergent terms proportional to
1. Basically, the presence of the extrinsic curvature terms in (3.11) leads to
cannot be compensated for by any correction to the metric that respects the Zq symmetry
and boundary conditions.
One then learns that the trace of the extrinsic curvature in each of the normal directions
and spacelike (Kx) directions respectively, we can by taking suitable linear combinations
express this in terms of the null expansions which are more natural for codimension-2
spacelike surfaces in Lorentzian manifolds. De ning x
extremal surface condition postulated in [3], viz.,33
= p12 (x0
x1) we thus have the
Ka = 0
=)
=)
= p (K0
K1) = 0 ;
qli!m1 eq = EA ;
EA 2 M is extremal :
Having ascertained the dynamical constraint on eq in the limit q ! 1, let us return to
our earlier discussion. We originally argued in section 3.2.1 that eq should, by virtue of
the replica symmetry assumption, lie on the Cauchy surface ~ t which we pick to construct
the density matrix (t) for the entire system. As indicated in that context, the choice
of ~ t is restricted by the fact that it be spacelike to
t and @ ~ t =
t, but is otherwise
unconstrained. However, the dynamics indicates that not all such ~ t would be acceptable
in semiclassical saddle point solutions to the gravitational path integral. While an arbitrary
~ t in the Wheeler-DeWitt patch of the boundary Cauchy surface may be used a priori to
construct Tr (t), the semiclassical saddle point of the Lorentzian path integral for Tr( Aq)
1) only chooses those that pass through the extremal surface, see gure 2. More
pertinently, we conclude that Tr( Aq) can be constructed by the Lorentzian prescription
~ t. This restriction does not originate from the general Schwinger-Keldysh
construction, but rather is speci c to the process of tracing out the degrees of freedom
c
in A . More explicitly, it originates from the fact that we are e ectively introducing a
singularity along eq.
33Note here that Ki0j is the component of the extrinsic curvature in the timelike normal direction to a
codimension-2 surface (likewise Ki1j is the corresponding spacelike component) and should not be confused
with the extrinsic curvature for ~
t (which has a timelike normal), for which we use the symbol K when
2. The on-shell action. The computation of the on-shell action, once we realize that
the xed point locus of replica symmetry becomes the extremal surface in the q ! 1 limit,
proceeds in a similar manner as before, modulo a few small subtleties. The main di erence
is the fact that we have to work directly in Lorentzian signature, which means that the
regulated codimension-1 surface eq( ) would be more complicated. We will additionally have
some curvature components behave as rq 2. The functional form is similar to the Euclidean
case, but now the origin of the normal plane to eq is blown up in Lorentzian signature to a
codimension-1 null surface, which is the lightcone emanating from the origin. Fortunately,
these turn out to be mild singularities which do not contribute to the evaluation of the
The non-trivial computation here is that of the Renyi entropies, which are technically
more challenging than in the Euclidean case.
We have found it useful to compute the
quantity S~(q) introduced in (2.6) directly, but even this requires careful handling of an i"
prescription. We demonstrate in appendix A that this can in principle be done and provide
a few simple examples there. Presently we will give a sketch of how such a computation
Assuming that the extremal surface arises as a consequence of a well-de ned
variational principle as in (3.7), all that remains is to compute the boundary term. As before
the computation requires us to evaluate the Gibbons-Hawking term for Einstein-Hilbert
gravitational dynamics, cf., (3.8)
@qI[M^ q] =
We can proceed thus far without worrying about the change in the signature of the metric.
Now we have however to face up to the fact that the codimension-1 regulator surface eq( )
de ned as the hypersurface r =
comprises four distinct segments (two spacelike and two
timelike). The computation has to be done from scratch, because even under analytic
continuation this surface does not give us the r =
locus for the Euclidean problem in [6].
Note that the boundary terms at the Cauchy surface ~ t cancel out due the boundary
conditions inherent in the prescription of [22].
Despite these subtleties the evaluation of the boundary term works out to give the
expected result for the covariant modular entropy:
@qI[M^ q] = i
Thus we indeed obtain the area of the extremal surface as in [3] when we take the q ! 1
limit. Alternatively, the same result can be obtained by regularizing the singularity.
We have now a derivation of the extremal surface prescription of [3] for computing
holographic entanglement entropy in time-dependent states. We take the opportunity to
comment on several consequences of this construction.
The homology constraint.
As explained elsewhere [13] the RT and HRT proposals for
holographic entanglement entropy should respect the homology constraint. This requires
that there be a spacelike codimension-1 interpolating homology surface, whose only
boundaries are EA and A. The homology constraint is naturally incorporated in our construction.
The boundary conditions relevant for computing the qth Renyi entropy involves a
cutting and gluing in the bulk path integral. We have hitherto explained that our
construction naturally restricts the HRT surface to lie on a bulk Cauchy slice ~
This in particular implies that EA splits ~ t into two parts RA and RA
tjB =
c respectively with
the powers of the reduced density matrix elements are obtained by cutting open the path
integral along RA and sewing them cyclically. Speci cally, we need to identify RA
(I+1)+ to respect the ordering of the matrix elements.34 In e ect the basic construction
singles out a bulk codimension-1 region RA that serves to de ne how we carry out the
1 but even for nite q, in spite of the fact that the corresponding geometry will be
deformed signi cantly.
Entanglement wedge. In our analysis we have started by xing a boundary Cauchy slice
t and picked a de nite region on it. However, the computation of trace class observables
is insensitive to the particularities of the slice; we are free to deform this as long as J [@A]
remains unmodi ed [12]. Picking various deformations of A within its boundary domain of
dependence D[A], leaving the entangling surface untouched will in particular satisfy this
domain is the entanglement wedge WE [A] = D~ [RA].
The analogue of this freedom in the bulk corresponds to the choice of bulk Cauchy
slices ~ t, which lie pinned at EA. If we view RA, the piece of one such representative
Cauchy slice as the bulk analogue of A then as argued in [12] the corresponding bulk
In making these statements we are allowing ourselves the freedom to move A into the
future domain of dependence D+[A]. Strictly speaking, in our analysis we have always
t and A ab initio; this would allow access only to the past domain of
dependence on the boundary and correspondingly only the past half of the entanglement
wedge in the bulk.
The \dual" of tracing out.
We have traced out the degrees of freedom in the boundary
to implement the replica trick. Attempting to do it similarly in the bulk, we have seen that
one cannot do the replica trick on all Cauchy slices in the Wheeler-DeWitt patch. This
suggests that a dual picture for tracing out boundary degrees of freedom exists only if the
bulk Cauchy slice contains the extremal surface.
This observation is important in the context of the subregion-subregion duality in
holography. It has been argued by several authors [12, 30{34] that the entanglement wedge
is the natural bulk region to be associated with a boundary density matrix. Nevertheless,
34One can also state the prescription more completely by requiring that we cut open along ~ t and glue
one may wonder if the smaller causal wedge (which is more minimally de ned in terms of
the bulk causal structure) is not perhaps more fundamental. After all, local bulk operator
reconstruction seems to proceed more seamlessly within the causal wedge.35
that the extremal surface allows a decomposition of the bulk spacetime into four distinct
domains [12]: its future (which we are eschewing in our construction), its past (which
is included explicitly), and the entanglement wedges for the region and its complement
importantly allows the ability to decompose the bulk semiclassical Hilbert space. This is
not possible for the causal wedge (in fact worse still, the causal wedge is not even its own
domain of dependence [5]).
To illustrate this point, consider the eternal black hole viewed as a thermo eld double
state [36]: if we pick a bulk Cauchy slice that goes through the interior, the modes in the
interior do not have a nice interpretation in terms of left or right modes only. A Cauchy
surface that passes through the bifurcation surface on the horizon does not encounter this
issue. Generically causal wedges exemplify the former scenario, while entanglement wedges
by construction always conform to the latter.36
While the basic principles of the gravitational problem are general,
nal evaluation of the minimal surface condition and obtaining the on-shell action
1. Usually when we have multiple solutions to the equations of motion, the
saddle point analysis instructs us to pick the one with the least action (which translates
here to smallest area for EA). This statement is true for generic Renyi index q.
It is however important to note that the control parameter for the saddle analysis in
the computation of entanglement entropy, i.e., for q
1, encounters a further suppression
1. The true parameter is (q
1) ce . Requiring that this be large as q ! 1 is only
possible with an appropriate order of limits: we rst take ce ! 1 before taking q ! 1.
This point has been noted elsewhere, see for example [37].
If one goes to the opposite limit, (q
1, then there is no new saddle for
approximation and later set (q
Lorentzian replicated bulk geometries. In the case of local modular Hamiltonians,
one can understand explicitly how these Lorentzian geometries look like.
For example, if we consider a spherical entangling surface in the vacuum, [7] showed
that the density matrix was
= e
j =2 , with K the integral of the stress tensor over
t. The dual to D[A] can be thought as the exterior of the topological black hole. In this
way, [38] explained that the dual of the Renyi entropies was given by hyperbolic black hole
35This is true with the current technology modulo the subtlety that the HKLL procedure [35] involves a
nonstandard Cauchy problem.
36While for the eternal black hole the causal wedge and entanglement wedge for one whole connected
component of the boundary coincide, we can deform the black hole using shock-waves (cf., [12]) to separate
out the causal and entanglement wedges. In the latter scenario, a Cauchy slice is bipartitely divided across
the two boundaries by the entanglement wedge but not so by the causal wedge. The causal wedges for the
region and its complement fail to meet, being separated by a causal shadow domain.
at inverse temperature
= 2 q. Of course, this is a consequence of the density matrix
being thermal in the hyperboloid.
However, note that, as explained before, the boundary geometry to compute the Renyi
entropy should be thought as q geometries glued together. In this way, the dual geometry
should look like q asymptotic boundaries (corresponding to D[A] in each replica) which
are connected together through the interior of the black hole. This is clearly di erent from
a unique black hole. An explicit di erence is that while a black hole has four causally
disconnected regions (for a
observer), the geometry dual to the Renyi entropies would
Of course, the Euclidean picture is the same in the two cases and the Renyi entropies
coincide. However, they are geometrically di erent, while changing the temperature is
a boundary at a di erent temperature: Tr(
=2 q), the Renyi entropy is computed by
and have an explicit dual geometry. In other words, in one case we do Euclidean evolution
for 2 q and then evolve in Lorentzian time, while in the other we have q Euclidean segments
where we evolve 2 . At the end of each segment we evolve in Lorentzian time up to some
time t and back.
The above is an example of two di erent eld theory contours that end up having the
the same action, since they just di er with each other by some unitaries. As such one
might encounter many contours which end up giving the same on-shell action, but in our
discussion we have singled out the Zq symmetric contour which reduces to the canonical
Schwinger-Keldysh contour at q = 1.
Non-analytic geometries.
We have exercised care with not extending
domain except in a very particular sense described in section 3. To implement this, we
imposed local Rindler conditions (close to e) by expanding the elds in Rindler modes and
matching them across the horizons.
This was inspired by our desire for the construction to apply to generic non-analytic
metrics. In this way, even if we de ne the coordinate
patchwise, with a discrete imaginary
part, there is no analytic continuation involved; the discrete shifts in the imaginary part
simply corresponds to a labeling of domains.37
Nonetheless, it is quite common in gravity, for geometries that satisfy the equations of
motion, to be analytic almost everywhere. In such situations one could simply impose the
previous conditions by analytically continuing
to the entire complex plane and picking an
appropriate, continuous contour of integration there. Of course, one could also analytically
continue to Euclidean time (which would give us a complex metric in general).
Higher derivative gravitational dynamics.
The main thrust of our analysis has been
to derive the HRT proposal for holographic entanglement entropy, which is valid for strongly
coupled theories with large ce . If we start to include
e ects then we
37Recall that this is already useful in the context of Rindler geometry or a black hole spacetime where
we can label di erent domains in the maximally extended Lorentzian spacetime with a discrete imaginary
anticipate that the bulk dynamics is well described by a higher derivative gravitational
In the time-independent situation a prescription to incorporate such bulk dynamics was
given in [23, 24]. The kinematic part of the argument we have presented herein trivially
extends to these cases. We then have to work out the local analysis in the vicinity of the
xed point set eq for the given higher derivative theory. As in the Euclidean analysis we do
not expect that this local analysis will serve to pick out the functional whose variation gives
rise to the dynamical constraint on the xed point set eq; rather one has to work with the
full bulk dynamics. However, one can conclude that the functional we should evaluate once
we nd the surface of interest should be the one obtained in the aforementioned papers.
As discussed in [39, 40] the holographic entanglement entropy functionals serve as a
good starting point to examine the second law for higher derivative black hole entropy.
The discussion thus far has been con ned to the linear response regime of small amplitude
uctuations away from equilibrium. It would be interesting to examine whether one can
shed light on the non-linear second law using some of the machinery developed herein.
Quantum corrections.
A key part of our argument in section 3 was to implement the
Schwinger-Keldysh construction directly in the bulk spacetime. This naturally incorporates
a semiclassical separation of Hilbert spaces and thereby allows for a really transparent
interpretation of the quantum corrections. Following the arguments of [41], one can say that
the quantum corrections at rst subleading order O(ce 1), can be viewed as the
entanglement entropy of region RA in the bulk (suitably regulated). Similar statements can be
made for the boundary and bulk relative entropies. As a consequence one can nd a purely
bulk expression for boundary modular Hamiltonian as discussed recently in [33].38 It is
useful to recognize that the entanglement wedge naturally implements the Schwinger-Keldysh
contour in the bulk; consistent with the fact that the boundary bipartitioning induces a
corresponding one in the bulk (cf., the causal domain decomposition discussed in [12]).
Maximin construction. In [31], the covariant HRT construction involving extremal
surfaces was reformulated as a maximin construction. The primary motivation was a tool
in aid to proving strong-subadditivity of holographic entanglement entropy in the
timedependent situations, extending the initial result of [10] for static states. The idea was to
pick a bulk Cauchy slice ~ t corresponding to a given region A
a minimal surface on this slice, and then maximize the area of minimal surfaces across a
t on the boundary, nd
complete set of Cauchy slices inside the Wheeler-DeWitt patch of t
While the nal result of the maximin construction coincides with the extremal surface
prescription of HRT, from our point of view not all minimal surfaces on bulk Cauchy
slices respect the boundary conditions of the Schwinger-Keldysh construction. The replica
construction requires that only the slices that contain EA are admissible in the bulk path
integral. This does not however modify the discussion of strong-subadditivity. All it does
is restrict the set of Cauchy slices we need to consider for the maximin construction.
38It is also worth keeping in mind the comments made earlier regarding the entanglement wedge in this
To wit, one nds a common Cauchy slice
for two of the spacelike separated regions
appearing in the strong-subadditivity inequality (say A1 [ A2 and A1 \ A2) which can be
argued to exist [31] | so EA1\A2 , EA1[A2
. One then projects the extremal surfaces
for regions A1 and A2 which a priori lie on some other slice, onto
P EA2 respectively. The key point is that if the bulk theory satis es null energy condition
then area can only decrease under such projection, so Area(P EAi )
Area(EAi ). Since now
all the surfaces lie on a common Cauchy slice we can employ the local surgery argument
of [10] to learn that
Area(EA1\A2 ) + Area(EA1\A2 )
which establishes the strong-subadditivity result as desired.
Complex saddles. It has been suggested that complex extremal surfaces could
potentially play a role in the computation of holographic entanglement [37, 42]. The argument
relies on the fact that typical saddle point evaluations do often admit complex saddles; the
derivation of LM [6] could be interpreted in this manner. While the examples discussed
in the aforementioned papers are interesting, to our knowledge there is no clear boundary
we have some explicit boundary eld theory understanding such complex surfaces, while
seemingly present, break the time re ection symmetry and appear to be sub-dominant to
real saddle points [43].
Our take on the problem is rather di erent. In
eld theory we are instructed to
perform a real time computation for (t) and thence to manipulate it to construct the
reduced density matrix elements; cf., section 2. As described in detail, the construction
extends naturally into the bulk where we glue pieces of the state j i and its conjugate
h j across some bulk Cauchy slice ~ t, which contains the extremal surface EA. In other
words, the computation is phrased purely in Lorentzian terms and leaves no room for purely
The skeptical reader may argue that in situations with time re ection symmetry, we
could alternately use the Euclidean formulation of the problem as in section 3.1; that
computation could potentially be dominated by Euclidean surfaces which do not analytically
continue to real Lorentzian saddles. This viewpoint however obscures the temporal
ordering inherent in the construction of
A(t) leading to an inherent tension with causality,
which is not obviously resolved. While this issue deserves further investigation let us record
here that in general it remains unclear that potential time-re ection Z2 breaking saddles
arise from the q ! 1 limit in a replica construction. It would also be useful to ascertain
whether there are examples analogous to the ones discussed in [42] (i.e., absence of real
HRT extremal surfaces) in geometries with known unitary CFT duals.
Acknowledgments
It is a pleasure to thank Felix Haehl, Veronika Hubeny, R. Loganayagam, Juan Maldacena,
Don Marolf, Henry Max eld, Rob Myers, Tadashi Takayanagi, Mark Van Raamsdonk for
very useful discussions on various related issues.
XD and MR would like to thank the Aspen Center for Physics and KITP, Santa
Barbara, for hospitality during the course of this project, where their stays were supported
in part by the National Science Foundation (NSF) under grants PHYS-106629 and
PHY1125915 respectively. AL and MR would like to thank the Yukawa Institute for Theoretical
Physics, Kyoto for hospitality during the concluding stages of this project.
also like to thank the Perimeter Institute for Theoretical Physics. 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 and Innovation.
XD was supported in part by the NSF under Grant No. PHY-1316699, by the
Department of Energy under Grant No. DE-SC0009988, and by a Zurich Financial Services
Membership at the Institute for Advanced Study. AL was supported in part by the US
NSF under Grant No. PHY-1314198.
Bulk evaluation of the Renyi entropy
For integer q, we will have a well-de ned (smooth) action:
I[Mq] =
with @Mq simply the holographic boundary and the additional boundary term for the
Schwinger-Keldysh construction across the codimension-1 bulk Cauchy slice is explicitly
included.39 We can use the Zq symmetry to think about this as qI[M^ q], which will again
be a local integral in the bulk.
One should be able to evaluate in principle these partition functions. To do this, one
rst has to look for the solutions to the equations of motion with the boundary
condition (3.11) for the quotient spacetime M^ q. These solutions are completely real in the
Rindler wedges, but, for even q, they might present some imaginary phases in the Milne
Once one has a solution, after regularizing the bulk properly, one should be
Euclidean perspective in [38, 44], although the explicit evaluation of this action is far from
being trivial because (among other things) of the regularization of the bulk. While we
believe that a similar strategy ought to work in the Lorentzian case, we will not attempt
to implement such in the present work.
39We are going to assume that at nite q there are no relevant contributions from bulk singularities
40Using the coordinates in (3.11), the asymptotic boundary tells us that the Milne wedges are reached
+ i 2 and thus those components of the metric with rqe
might get nontrivial phases
We expect that if one is careful with how the boundary is regularized and exploits
correctly symmetries of the problem, one can get the Renyi entropies directly by integrating
the action. Due to the presence of light-like singularities, one has to be extremely careful
with how they are integrated. From the timefold perspective, we expect that one can
recover that the only contribution will come from these singularities, whereas the purely
real contribution will cancel. Said di erently we expect in the construction a complex
As an example, we consider the case of appendix A of [6], but for arbitrary Lorentzian
sources. We describe this explicitly in section A.2 once we have shown how to carry out
the evaluation of the boundary contribution to the on-shell action.
Evaluation of boundary term
are simple to evaluate, but they cancel pairwise because of the timefold. So we are left
with evaluating the contribution from the jump across the horizon, which we call eq
! 0 limit, the boundary is becoming vanishingly small, but it nevertheless leads to
a non-vanishing contribution owing to the boundary term which correspondingly diverges.
As a warm-up, let us consider the two-dimensional Rindler space: dr2
r2d 2 =
segment ejqump( ) joins a spacelike and a timelike surface. The extrinsic curvature will
blow up when this segment becomes lightlike. We are going to choose it so that this only
happens at a point, and we will employ an i" prescription when integrating the boundary
term added, not sure of how to say it, because our boundary is not actually physical.42
One can show that this contribution does not depend on the shape of segment and for some
initial and nal normal vectors n1, n2 it gives [45, 46]:
Ibdy =
8 GN ej1ump( )
K = cosh 1(n1 n2)
4Ibdy = i Ar4eGa(Ne1) .
When we have the quotient space metric (3.11): q2dr2
a little more subtle how to apply the previous argument. In order to compare the normal
the normal vectors in these coordinates. These coordinates are not really physical, but they
are useful to compare the normals. As we cross the horizon r ! i 1r and
+ i =2,
41We remind the reader that we are working in Lorentz signature and so the action is eiS.
42The choice of a positive i" sign guarantees that if this boundary was physical, the path integral does
43In order to keep the integrand xed across the whole surface (without changing the determinant of the
metric as it ips signature), we have to normalize the normal vectors to 1.
and n1 n2 = cosh 2iq , so
so if the initial normal is n1 = (cosh 0=q; sinh 0=q) then n2 = (cosh 0+i =2 ; sinh 0+i =2 )
q q
Ibdy[M^ q] = 4
Example of bulk integral
The contribution to the entropy of one interval in the vacuum from a time-dependent scalar
source was computed in [6] (see their appendix A). To do this, we rst consider a bulk
with boundary condition j@Mq =
contribution to the Renyi entropy. The action will be
1, and then compute the O( 2)
S~A(q)j 2 =
I[M^ q] = IEH[M^ q]
In the absence of the scalar term, the calculation will be that of [38]. The argument
from the main text tells us that the O( 2) contribution of S~(q) will be given by the area
xed point after accounting for the leading backreaction of the scalar eld,
but it was also shown in [6] that this is equivalent to the purely matter contribution for
the modular entropy:44
44We refer the reader to appendix A of [6] for more details.
where @~n denotes the derivative normal to the surface. Even though the area term is
purely gravitational, the other two denote di erent expressions for the variation of the area
purely in terms of the scalar solutions. While the integral looks imaginary, the presence
of a light-cone singularity in the integrand implies that we need to implement a proper i"
prescription in evaluating the integral. The result, as we shall see, will be a real answer for
would like to generalize this argument to the Lorentzian case, by inserting general
timedependent sources (not only time-dependent in the
direction as in [6]). In other words,
we want to consider a scalar pro le that depends on t in the Poincare patch:
However, in order to compute @qg, it is easier to work in the hyperbolic patch:
ds2 =
ds2 = f ( )
For technical reasons, it is easy to compute Renyi entropies in hyperbolic coordinates,
but di cult in Poincare coordinates. This means that if we want to compute the
contribution from the scalar action (A.5), this will be particularly simple when evaluating the bulk
explicit hyperbolic-to-Poincare coordinate transformation. If one wanted to compute the
boundary matter contribution from @Mq in (A.5), one would have to be careful with the
regularization of the manifold close to the boundary; we will leave this for the future.
Even though the evaluation of S~A(1)j 2 =
has to trivially give us the area
(because of the argument of [6]), we believe that this is a simple example of how the general
case works. More concretely, we just need to show that, if we analytically continue (i.e.,
tE ! it) the Euclidean calculation with the proper prescription, the answer is i times the
Euclidean one. Then the analysis of [6] tells us that we get the area.
Let us consider a scalar solution
ei!tf!(z).45 By expressing everything in Poincare
coordinates, we obtain:
iS~A(1)j 2 =
where for simplicity we have used l2 = x
t2, but we want to think about the previous
expression in terms of x; t; z.
Now, the question becomes a very simple one. We can integrate T~ in Euclidean
this is straightforward. The argument of appendix A of [6] shows that R
We only have to check that the previous Lorentzian integral gives i times the Euclidean
T~ (tE) = 4GAN .
integral. One can see this as follows:
rst integrate the previous expression with respect
to t. This will give a result involving just the integrals over x and z, which is insensitive to
the Lorentzian signature, which we perform by choosing an i" prescription. This leads to
the desired factor of i, and the nal integrand over x and z is i times the Euclidean one.
45Note that f (z) solves the scalar equations of motion, but we will not need its explicit form.
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