Deriving covariant holographic entanglement

Journal of High Energy Physics, Nov 2016

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 field 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 field theory Rényi entropies. In the limiting case where the replica index is taken to unity, a local analysis suffices 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 construction.

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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. 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Xi Dong, Aitor Lewkowycz, Mukund Rangamani. Deriving covariant holographic entanglement, Journal of High Energy Physics, 2016, 28, DOI: 10.1007/JHEP11(2016)028