Warped Entanglement Entropy
SSU-ITP-13/18
Warped Entanglement Entropy
Dionysios Anninos , Joshua Samani and Edgar Shaghoulian Stanford Institute for Theoretical Physics, Stanford University Department of Physics and Astronomy, University of California, Los Angeles
Abstract
We study the applicability of the covariant holographic entanglement entropy pro-posal to asymptotically warped AdS spacetimes with an SL (2 , R ) × U (1) isometry.We begin by applying the proposal to locally AdS backgrounds which are written asan R fibration over AdS . We then perturb away from this geometry by consideringa warping parameter a = 1 + δ to get an asymptotically warped AdS spacetime andcompute the dual entanglement entropy perturbatively in δ . We find that for large sep-aration in the fiber coordinate, the entanglement entropy can be computed to all ordersin δ and takes the universal form appropriate for two-dimensional CFTs. The warping-dependent central charge thus identified exactly agrees with previous calculations inthe literature. Performing the same perturbative calculations for the warped BTZblack hole again gives universal two-dimensional CFT answers, with the left-movingand right-moving temperatures appearing appropriately in the result. a r X i v : . [ h e p - t h ] D ec ontents AdS in fibered coordinates 73 Spacelike WAdS
144 Perturbative entanglement entropy 175 Summary and outlook 24A Geodesics 27B
AdS entanglement entropy via coordinate transformations 31 Understanding how the holographic principle works beyond the example of anti-de Sitterspace is a crucial and beautiful challenge which will elucidate the dynamics of quantumgravity in general backgrounds. A natural example is the geometry describing our universe,which is cosmological in nature, and more closely resembles an FRW/de Sitter type uni-verse. As another example, the geometry describing regions near the horizons of certainastrophysical black holes is not quite anti-de Sitter space but more closely resembles a slightdeformation thereof known as the NHEK/warped AdS geometry. There have been severalproposals for holographic descriptions of these and other non-AdS spacetimes [1, 2, 3], andthe story is still unfolding.In this paper we will focus on aspects of the warped AdS geometry and its putativeholographic description. As we will describe more concretely below, warped AdS is a de-formation of AdS that destroys the boundary asymptotics. The deformation preservesonly an SL (2 , R ) × U (1) subgroup of the original SL (2 , R ) × SL (2 , R ) isometry group ofAdS . From the point of view of the two-dimensional CFT dual to AdS , the warping ofAdS corresponds to an irrelevant chiral deformation (which does not die away in the ul-traviolet). Geometrically this manifests itself in the destruction of an asymptotically AdS boundary. Holographic considerations of this geometry began with [2]. Based on the ther-modynamic properties of asymptotically warped AdS black holes [4, 5, 6, 7], whose entropycould be written in a suggestive, Cardy-like fashion, it was proposed that it was dual to a1wo-dimensional conformal(-esque) field theory. Later work embedded and studied warpedAdS within string theory [8, 9, 10, 11, 12, 13, 14] and studied properties of two-dimensionalfield theories, dubbed warped CFTs, whose symmetry structure matches that of warpedAdS [15, 16]. Other work studied the wave equation, correlation functions and quasinormalmodes of fields in warped AdS [17, 18, 19, 20, 21]. Much of the work on warped AdS has so far focused on thermodynamic properties of the theory and its asymptotic symmetrystructure [22, 23]. In this paper we would like to focus instead on entangling propertiesof asymptotically warped AdS geometries. We do so by exploiting the simple holographicmanifestation of the entropy of entanglement of some state in a CFT as an extremal surfacein the bulk geometry dual to such a state, as described by [24], generalizing [25, 26]. Thoughentanglement entropy is a simple property of the quantum state, it has sufficient informationto independently verify features derived from the thermodynamics, such as central chargesand left- and right-moving temperatures. It can also provide additional insight into thenature of the dual as we will shortly discuss. We now move on to briefly review the warpedAdS geometry and the holographic entanglement entropy proposal before summarizing ourresults and giving an outline of the paper. AdS Consider AdS expressed as a real-line or circle fibration over a Lorentzian AdS base space.These geometries can be deformed with a nontrivial warp factor into the warped AdS spacetimes we will consider later. The (spacelike) warped AdS metric in global coordinateswith warp factor a ∈ [0 ,
2) is given by ds = (cid:96) (cid:18) − (1 + r ) dτ + dr r + a ( du + r dτ ) (cid:19) . (1.1)The coordinates range over the whole real line, { r, τ, u } ∈ R , although later we will considercompactifying u to recover a near-horizon extremal BTZ geometry. To obtain AdS , one sets a = 1. The conformal boundary in the case of a = 1 is the usual cylinder parsed by nullcoordinates and looks like a barber-shop pole; see Figure 1. The case a (cid:54) = 1 corresponds tospacelike warped AdS , which is the case we shall focus on in this paper. We will also com-ment on the timelike warped AdS case, whose base space is Euclidean AdS , in Section 4.3.For a (cid:54) = 1 there is no conformal boundary [27], although a generalized notion of “anisotropic In the literature, usually in the context of topologically massive gravity, one often sees an alternativeconvention in which the metric is characterized by parameters (cid:101) (cid:96) and ν related to our parameters by (cid:101) (cid:96) = (cid:96) ( ν + 3) / a = 4 ν / ( ν + 3). igure 1: This is the global AdS cylinder parameterized by the coordinates (1.1). The coordinates t g and θ g represent the usual global coordinates. We will primarily consider sticking to a region ofthe boundary with r = ∞ for simplicity. This figure is taken from [29]. conformal infinity” can be defined [28]. We will also consider the geometries in Poincar´e-likecoordinates with metric ds = 14 (cid:32) − (cid:96) dψ x + (cid:96) dx x + a (cid:18) dφ + (cid:96) dψx (cid:19) (cid:33) . (1.2)and coordinate ranges { ψ, x, φ } ∈ R .These spacetimes posses SL (2 , R ) × U (1) isometry for a (cid:54) = 1 and appear in a Penrose-likenear-horizon limit of extremal black holes. In the context of a trivial warp factor a = 1,these geometries are locally AdS , and we expect the HRT proposal to apply. We will seethat our results match field theory expectations, where the field theory is placed at zeroleft-moving temperature and finite right-moving temperature. This state of the field theoryhas not yet been considered in the holographic entanglement entropy literature, though it isclosely related to the extremal limit of the rotating BTZ black hole, considered in [24].For the case of nontrivial warp factor, the purported holographic duals of the spacetimeare referred to as warped CFTs and possess SL (2 , R ) × U (1) symmetry. This symmetry isautomatically enhanced to two infinite-dimensional local symmetries [15]: the left-moving SL (2 , R ) is enhanced to a left-moving Virasoro, while the right-moving U (1) is enhanced3o a left-moving U (1) Kac-Moody current algebra (indeed, the term WCFT is used for thecase that the U (1) is not enhanced to a full Virasoro, which is also possible). Not muchis known about these theories (a nontrivial example has only recently been suggested in[30]), but the symmetries can still be used to constrain properties that such a theory couldhave. This approach has been used successfully in reproducing a Cardy-like formula for theasymptotic growth of states in [16]. The bulk geometries are often considered in the contextof topologically massive gravity, but for simplicity we shall restrict ourselves to the casewhere they are solutions of three-dimensional Einstein gravity with matter fields, as studiedin [31, 8, 11]. In string theory, for example, the warped geometries can be constructed by ahyperbolic, marginal deformation of the SL (2 , R ) WZW model [32]. The use of entanglement entropy to study quantum field theories continues to surge due toits relevance to quantum gravity and condensed matter physics and its analytic tractability.Holographically, this has been studied with the Ryu-Takayanagi (RT) proposal [25, 26] forcomputing the entanglement entropy via geometric methods in the bulk. The proposal nowhas support for multiple intervals in asymptotically AdS bulk spacetimes [33, 34, 35] andspherical entangling surfaces in any dimension [36]. Strong arguments for the general caseare provided in [37] and essentially prove the conjecture. Quantum corrections have beenanalytically computed in [38], with a general prescription appearing in [39]. Prescriptions forgravitational theories with higher curvature corrections are given in [40, 41, 42, 43, 44] and,for higher spin theories, in [45, 46]. The covariant Hubeny-Rangamani-Takayanagi (HRT)proposal [24] has far less support, though it has passed nontrivial consistency checks [47, 48].It is natural to wonder how generally the proposal can apply. In this paper, we would liketo take a few steps toward understanding the issues of holographic entanglement entropy inwarped AdS spacetime and two-dimensional warped conformal field theory (WCFT ). Thespacetimes we will study are non-static and will therefore require the covariant proposal.Although these spacetimes are often studied as solutions of topologically massive gravity,here we will consider the case where they are supported by Einstein gravity plus matter,allowing us to use the usual HRT proposal.The goal of the HRT proposal in [24] is to obtain a holographic prescription for computingthe entanglement entropy for time-varying states in QFTs with bulk duals which are non-static, asymptotically AdS spacetimes. To describe the proposal, we consider a ( d + 1)-dimensional asymptotically AdS spacetime M with d -dimensional boundary ∂M , and we4onsider a field theory defined on this boundary. We choose a foliation of ∂M by spacelikehypersurfaces (time slices) (cid:99) M t . For each time t ∈ R , we write the slice (cid:99) M t as a union ofdisjoint sets A t and B t , and we can compute the entanglement entropy S AB ( t ) between thedegrees of freedom in the two regions for a given state (density matrix) of the full systemliving on (cid:99) M t .The HRT proposal is as follows: for each time t , determine the co-dimension 2 extremalsurfaces W t satisfying ∂W t = ∂A t . If there is more than one extremal surface satisfying theseboundary data, then choose the extremal surface W t, min with smallest area. Then we have S AB ( t ) = Area( W t, min )4 G ( d +1) N . (1.3)The question of which homology class to consider is interesting in the context of the covariantproposal [49], but we will not need to consider it here. This is the correct expression forEinstein gravity coupled to matter, which are the theories we will consider here, althoughsubleading corrections in G N (bulk quantum corrections) will depend on the bulk mattersupporting the geometry.It is worth noting a rather remarkable feature of the above proposal (1.3). In the contextof Einstein theories of gravity the entanglement entropy manifests itself in a purely geometricform at leading order in G N , as the area of an extremal surface. This universal featureis particularly surprising, given that entanglement entropy is a property of the particularquantum state under consideration, which is generally a functional of all the bulk matterfields and not just the metric. It is reminiscent of the universality of the Bekenstein-Hawkingentropy of a black hole, which also manifests itself as a geometric area in Einstein theoriesof gravity, regardless of the matter content that constitutes the black hole.In our use of this formula, we will keep the slice chosen on the boundary arbitrary butspacelike. Since we consider exclusively (2+1)-dimensional bulk geometries, this means thatour entanglement entropy answers will be phrased in terms of two distinct coordinate sepa-rations, which can then be chosen to give a particular spacelike slice. We present the answersin this way because it makes the split into left-moving and right-moving sectors transparent;see (2.24) for one such example. It is important to note that the HRT prescription (andindeed the original Ryu-Takayanagi prescription) is computing the entanglement entropybetween regions A t and B t defined by the unique geodesic along the boundary which con-nects the two points which define their separation. In other words, once one picks two pointson the boundary to connect by a bulk geodesic, there remains an ambiguity in choosing the5pacelike curves on the boundary which connect the two points and define the regions A t and B t . The holographic entanglement entropy prescription naturally picks the unique geodesicalong the boundary which connects the two points as defining the spatial regions A t and B t .To elaborate further, imagine applying the covariant proposal to the Poincar´e patch ofAdS by picking points on the boundary that are spacelike separated but arbitrary. Thelength of the regulated bulk geodesic connecting these two points, divided by 4 G N , is givenin terms of CFT quantities as S EE = c (cid:112) L x − L t ε . (1.4)To match with the universal 2D CFT answer, we conclude that the region being picked outon the boundary theory is the geodesic along the boundary which connects the two points,since this curve has length (cid:112) L x − L t . The fact that in this example the spatial lengthat fixed time gets replaced with the invariant Minkowskian length is a result of Lorentzinvariance. This will not be the case once we introduce a dimensionful scale, e.g. the radiusof the cylinder for global AdS or the temperature of a black hole. Although the validity of applying the HRT proposal to spacetimes with different asymptoticsis an interesting open question, in this paper we shall pursue a more modest goal. We willset up what is effectively a perturbation theory about the AdS point by considering warping a = 1+ δ and cutting off the WAdS spacetime deep in the interior, where it is AdS -like. Thiscan be understood as AdS/CFT in the presence of an infinitesimal, irrelevant deformation,a context in which holographic renormalization can be understood perturbatively in thedeformation [52, 53]. Thus, attacking the problem in this way puts our analysis on firmerfooting. We will see that such an approach gives sensible results, and in the regime of largeseparation in the fiber coordinate, the series can be summed to all orders in δ . The resultis precisely that of two-dimensional CFT, with c L = c R = 3 (cid:96)a/ G N . This exactly matchesan independent proposal for the central charge, deduced by demanding consistency withthe Cardy formula for two-dimensional CFTs [8]. We will also consider the warped BTZblack hole and again find universal CFT results which allow us to read off the left-movingand right-moving temperatures. Our central charge and temperatures altogether satisfy the We stress that calculations like the ones in [50], which consider a decoupled IR geometry, are stillunderstood as occurring in an asymptotically AdS spacetime, as discussed in [51]. For a specific implementation in Lifshitz backgrounds with z = 1 + ε , see [54]. spacetimes written as afibration over a Lorentzian AdS base space. In Section 3, we will deform these geometriesinto warped AdS and set up the problem of applying the HRT proposal to these spacetimes.In Section 4, we will complete the problem by performing a perturbative application of theHRT proposal to warped AdS geometries, where we will be perturbing around the locallyAdS geometries considered in Section 2. Finally, we will summarize and look toward futurework in Section 5. AdS in fibered coordinates We begin our story by considering AdS in fibered Poincar´e coordinates and fibered globalcoordinates. We will see that these coordinate systems are dual to states at zero left-movingand finite right-moving temperature, a feature reflected in the answer for the entanglemententropy. The geometry obtained by compactifying the fiber coordinate appears in a near-horizon limit of the extremal BTZ black hole. If the fiber coordinate remains uncompactified,the geometry is instead the near-horizon limit of a boosted extremal black string. AdS The metric (1.2) with a = 1 reduces to ds = 14 (cid:32) − (cid:96) dψ x + (cid:96) dx x + (cid:18) dφ + (cid:96) dψx (cid:19) (cid:33) . (2.1)We choose this parameterization since all coordinates and (cid:96) can be assigned dimensions oflength. Near the conformal boundary, the coordinates φ and ψ become null. We would liketo determine the affinely parametrized geodesics x µ ( λ ) = ( x ( λ ) , φ ( λ ) , ψ ( λ )). To do so, wenotice that this geometry has Killing vectors ∂ φ and ∂ ψ , corresponding to translations in φ and ψ , and these Killing vectors yield conserved quantities c φ = ˙ x · ∂ φ and c ψ = ˙ x · ∂ ψ .We will solve for the geodesics by using these conserved quantities and the affine constraint7 v = ˙ x µ ˙ x µ . This gives equations of motion c φ = x ˙ φ + (cid:96) ˙ ψ x , (2.2) c ψ = (cid:96) ˙ φ x , (2.3) c v = (cid:96) ˙ x + x ˙ φ ( x ˙ φ + 2 (cid:96) ˙ ψ )4 x . (2.4)The solutions to these equations are given in Appendix A.1.We want to compute the length of a geodesic beginning and ending near the conformalboundary at x ε ∼ ε /(cid:96) , where we have used the UV-IR relation to map our bulk IR cutoff x ε to a dual UV field theory cutoff ε [55, 26]. This follows from the quadratic relationshipbetween x and the usual Poincar´e coordinate z near the conformal boundary, x ∼ z . Sincewe chose our geodesics to be affinely parametrized, we can use the solution x ( λ ) to solve forthe cutoffs ± λ ∞ in the affine parameter defined by x ( ± λ ∞ ) = x ε . The regulated length isthen given by Length = √ c v (cid:90) λ ∞ − λ ∞ dλ = 2 √ c v λ ∞ , (2.5)which at the end will be c v -independent, as required by parameterization-invariance of thelength. Writing down the leading divergence of λ ∞ in terms of the conserved quantities c φ , c ψ and c v gives λ ∞ ≈ (cid:96) √ c v log c v (cid:96) (cid:113) c φ − c v c v cψ − cφ cψ ) (cid:96)ε , (2.6)we now attempt to trade the conserved quantities c φ and c ψ for spatial separations on theasymptotic boundary. This entails solving the equations L φ ≡ φ ( λ ∞ ) − φ ( − λ ∞ ) , L ψ ≡ ψ ( λ ∞ ) − ψ ( − λ ∞ ) , (2.7)to zeroth order in λ ∞ for c φ and c ψ in terms of L φ and L ψ . An important point about AdS solutions, which we state here to contrast with the warped AdS solutions of later sections,is that the “non-radial” coordinates (in this case φ and ψ ) asymptote to constant valuesas the affine parameter diverges. In other words, one can safely take the limit λ ∞ → ±∞ in either L φ or L ψ . The solutions to the geodesic equations of motion have two primarybranches, which we call the “cosh-like” and “sinh-like” branches. We will consider the“cosh-like” branch, defined by x ( λ ∞ ) = − x ( − λ ∞ ), although the “sinh-like” branch, defined8y x ( λ ∞ ) = x ( − λ ∞ ), can be handled analogously (see Appendix A.1 for details). Using c = 3 (cid:96)/ G N , we find S EE = c (cid:32) ε (cid:115) L ψ (cid:96) sinh (cid:18) L φ (cid:96) (cid:19)(cid:33) . (2.8)We will comment in the next section on what (cid:96) , the curvature scale, is doing in a field-theoryformula. Given that the geometry we are considering is simply a coordinate transformationof the usual Poincar´e patch on AdS , we could have gotten this answer by performing the ap-propriate transformations on the usual Poincar´e patch answer, ( c/
3) log(
L/ε ). This methodis easier since the Poincar´e patch is globally static, allowing us to use the time-independentproposal, and the geodesics are semicircles. To see how such an approach works, see Ap-pendix B. We will increasingly rely on using such coordinate transformations as we beginwarping the spacetime in the later sections.
We can suggestively rewrite the answer for the entanglement entropy as S EE = c (cid:114) L ψ (cid:96) sinh (cid:16) L φ (cid:96) (cid:17) ε = c L ψ ε + c (cid:18) (cid:96)ε sinh (cid:18) L φ (cid:96) (cid:19)(cid:19) . (2.9)This answer looks like the ground-state answer in the ψ direction and the finite-temperatureanswer in the φ direction, with the temperature being set by the curvature scale (cid:96) . Recallthat φ and ψ are null coordinates on the conformal boundary, so these correspond to theleft- and right-movers.To investigate the dual state corresponding to this bulk geometry, we can write out thebulk metric near the boundary in the Fefferman-Graham expansion [56]: ds = (cid:96) (cid:18) dρ ρ + h ij ( x i , ρ ) dx i dx j (cid:19) , h ij ( x i , ρ ) = g (0) ij ρ + g (2) ij + . . . . (2.10)In general, the boundary metric g (0) ij determines the trace and covariant divergence of g (2) ij through the equations of motion near the boundary asTr g (2) ≡ g (0) ij g (2) ij = − R [ g (0) ij ] , (2.11) ∇ i g (2) ij = ∇ j Tr g (2) , (2.12)9here the covariant derivative is with respect to the metric g (0) ij . The expectation value ofthe stress-energy tensor is then given by the variation of the renormalized on-shell actionwith respect to g (0) ij [57, 58], which in two boundary dimensions turns out to be (cid:104) T ij (cid:105) = (cid:96) πG (cid:16) g (2) ij − g (0) ij Tr g (2) (cid:17) . (2.13)For the usual Poincar´e patch, we identify g (2) ij = 0, so we see that (cid:104) T ij (cid:105) = 0. However, forthe Poincar´e fibered coordinates, since g (2) φφ = 1 / (cid:104) T φφ (cid:105) = (cid:96) πG = c π (2.14)with all other components vanishing (the tracelessness of the stress-energy tensor is preservedsince g (0) φφ = 0). Thus, we are not in the vacuum state of the dual theory and should not haveexpected to get the universal answer for the vacuum state, which in this case would havebeen S EE = c (cid:112) L ψ L φ ε (2.15)since lengths in the boundary metric are computed with ds = dφ dψ . In fact, we do get thevacuum answer for the ψ -movers, which agrees with (cid:104) T ψψ (cid:105) = (cid:104) T ψφ (cid:105) = 0. The φ -movers arein an excited state, which agrees with (cid:104) T φφ (cid:105) (cid:54) = 0. As it should, the bulk diffeomorphism thattakes one from Poincar´e coordinates to Poincar´e fibered coordinates induces a conformaltransformation on the boundary theory, and (2.14) is just what one obtains by conformallytransforming the vanishing stress-energy tensor from Poincar´e coordinates to Poincar´e fiberedcoordinates.Now that we have shown that the modes in the ψ direction are in their ground stateand the modes in the φ direction are excited, the expression for the entanglement entropyis becoming a bit clearer. To make the finite-temperature interpretation more precise, weconsider the metric (2.1) with compactified fiber coordinate: ds = 14 (cid:32) − (cid:96) dψ x + (cid:96) dx x + (cid:18) dφ + (cid:96) dψx (cid:19) (cid:33) , φ ∼ φ + 4 πr + . (2.16)This is precisely the geometry that appears in a Penrose-like near-horizon limit of the ex-10remal BTZ black hole ds = − ( r − r ) r (cid:96) dt + (cid:96) r ( r − r ) dr + r (cid:18) dφ − r (cid:96)r dt (cid:19) , (2.17)which has dimensionless J = M = r (cid:96) and S = 4 πr + in units where 8 G = 1. Definingleft-moving and right-moving energies as E L ≡ M − J = 0 , E R ≡ M + J, (2.18)and dimensionless left-moving and right-moving temperatures as T L ≡ (cid:96) ∂E L ∂S = 0 , T R ≡ (cid:96) ∂E R ∂S = r + π(cid:96) , (2.19)we see that the state dual to the background (2.1) is at zero left-moving temperature andfinite right-moving temperature. Though it is at zero Hawking temperature, the statisticaldegeneracy is explained by the Cardy formula and the nonvanishing right-moving tempera-ture: S = π c L T L + c R T R ) = π (cid:96) G N r + π(cid:96) = 4 πr + , (2.20)which matches the area of the horizon in coordinates (2.16) or (2.17). We have used 8 G N = 1to get to the final expression.Notice that our answer (2.9) applies for the geometry with compact fiber coordinate (2.1)as long as we consider small L φ . With the thermodynamic language developed above, wecan define (cid:101) φ = φ/ (2 r + ), (cid:101) ψ = r + ψ/π(cid:96) = ψ/β R and rewrite the second piece in (2.9) as c (cid:18) (cid:96)ε sinh (cid:18) L φ (cid:96) (cid:19)(cid:19) −→ c (cid:18) β R ε sinh (cid:18) π L (cid:101) φ β R (cid:19)(cid:19) , (2.21)where (cid:101) φ ∼ (cid:101) φ + 2 π and the first term in (2.9) remains unchanged. The UV-IR relation isfixed to match onto the ground state answer in the limit of small L (cid:101) φ . So we have seen thatthe entanglement entropy answer for the geometry with a compact fiber coordinate reflectsthe fact that the right-movers are at finite right-moving temperature.We pause for a moment to connect to an existing result in the literature, which is thecalculation of entanglement entropy in the state dual to the rotating BTZ black hole [24].11aking the extremal limit of their result, β R → ∞ , one finds S EE = c Lε + c (cid:18) β R ε sinh (cid:18) πLβ R (cid:19)(cid:19) (2.22)for purely spatial separation on the boundary. This is precisely our answer with L (cid:101) φ = L (cid:101) ψ = L . It seems that the IR limit we have taken to get to the geometry (2.16) has retained theentangling properties of the dual state.Now we would like to take the limit where the geometry decompactifies, i.e. r + /(cid:96) → ∞ ,since this allows us to recover our original geometry (2.1). Notice that in this limit, we aregoing from having two scales, (cid:96) and r + , to just one scale (cid:96) . Thus, all dimensionful parametersmust be measured relative to (cid:96) . Looking at the left-hand-side of (2.21), we see that thismeans that the argument of the sinh must remain fixed in this limit, since we want to keep L φ (in units of (cid:96) ) fixed. Expressed in terms of the CFT quantities on the right-hand-side of(2.21), we are taking β R small with L (cid:101) φ /β R fixed. We therefore retain the interpretation ofthe right-movers being at finite temperature in the decompactification limit. With compactfiber coordinate, the expression (2 .
9) can be understood in relation to the DLCQ limit whichfreezes the ψ -movers to their ground state [59].To aid with understanding taking arbitrary spacelike slices, we note here that the expres-sion for the length of an extremal geodesic in the rotating BTZ background for arbitraryspacelike separation on the boundary can be written: S EE = c (cid:20) β L β R π ε sinh (cid:18) π ∆ x L β L (cid:19) sinh (cid:18) π ∆ x R β R (cid:19)(cid:21) (2.23)= c (cid:20) β L πε sinh (cid:18) π ∆ x L β L (cid:19)(cid:21) + c (cid:20) β R πε sinh (cid:18) π ∆ x R β R (cid:19)(cid:21) (2.24)for x L = φ + t and x R = φ − t . Again, the contribution to the entanglement entropy splits upinto distinct contributions from the left- and right-moving sectors. This is analogous to howthe contribution to the thermodynamic entropy splits into left- and right-moving sectors inthe Cardy formula. The role of the lattice spacing ε will not be important for this argument. .2 Global fibered AdS The global fibered AdS metric is obtained by setting a = 1 in (1.1) to obtain ds = (cid:96) (cid:18) − (1 + r ) dτ + dr r + ( du + r dτ ) (cid:19) . (2.25)All coordinates are dimensionless while (cid:96) has dimensions of length. The coordinates u and τ become null near the part of the boundary reached by r → ±∞ , which is the region towhich we shall restrict our attention; see Figure 1 for the precise parameterization of theboundary cylinder in these coordinates. One can write conservation equations for affinelyparameterized geodesics just like in the Poincar´e fibered case. In this case, we label theconserved quantities corresponding to translations in τ and u by c τ and c u respectively,while c v = ˙ x µ ˙ x µ . After some manipulation, the conservation equations can be written asfollows: ˙ r = (cid:18) c v ( (cid:96)/ (cid:19) r − (cid:18) c u c τ ( (cid:96)/ (cid:19) r − c u − c τ − ( (cid:96)/ c v ( (cid:96)/ , (2.26)˙ τ = c u ( (cid:96)/ rr + 1 − c τ ( (cid:96)/ r + 1 , (2.27)˙ u = c τ ( (cid:96)/ rr + 1 + c u ( (cid:96)/ r + 1 . (2.28)Imposing the condition c v > c u − c v ( (cid:96)/ > c u − c v ( (cid:96)/ <
0, respectively. The solutionsare presented in Appendix A.2.We now wish to calculate the leading divergent piece of the length of these geodesics.The approach is identical to the previous section, so we will not repeat the details here.Using the UV-IR relation r ∞ ∼ ε − for dimensionless cutoff ε , we find λ ∞ ≈ ( (cid:96)/ √ c v log (cid:32) c v ( (cid:96)/ (cid:112) ( c u − c v ( (cid:96)/ )( c τ + c v ( (cid:96)/ ) 1 ε (cid:33) . (2.29)We can now trade in the conserved quantities c u and c τ for coordinate separations L u and L τ on the boundary and recover S EE = c (cid:32) ε (cid:115) sin (cid:18) L τ (cid:19) sinh (cid:18) L u (cid:19)(cid:33) . (2.30)13e will stick to L τ < π on the boundary to maintain spacelike separation between thetwo endpoints ( τ is a null coordinate that winds up the cylinder). Just as in the Poincar´efibered case (2.8) we see that the u -moving sector seems to be at finite temperature, withthe temperature scale set by (cid:96) (recall that our coordinate u is dimensionless), while the τ -moving sector is in its ground state. The appearance of the sine function is simply from thecompact U (1) of the global AdS cylinder. One can perform a Fefferman-Graham analysisby repeating the steps of Section 2.1.1, but the details are the same and we omit them here.The result for the “sinh-like” branch is similar: S EE = c (cid:32) ε (cid:115) cos (cid:18) L τ (cid:19) cosh (cid:18) L u (cid:19)(cid:33) . (2.31)Notice that the length remains well-defined when L u → L τ →
0, as it should sincethe geodesic is going through the bulk from r = −∞ to r = ∞ in this limit. We mentionthis branch due to its relevance to the metric (1.1) with compact fiber coordinate. Thisis the self-dual orbifold considered first in [60] and studied extensively in [59, 61]. Thegeometry is locally AdS and has an AdS factor, but a compact fiber coordinate causesthe two boundaries at r = + ∞ and r = −∞ to become disconnected, though they are causally connected through the bulk. The entanglement between the asymptotic boundarieswas computed via a reduction to AdS /CFT in [62]. Our answer can be used to computequantities like the holographic thermo-mutual information (HTMI) in these horizon-lessbackgrounds, as defined in [63], directly in AdS . WAdS We have seen in the previous sections how to apply the covariant HRT proposal to locallyAdS spacetimes written as a real-line fibration over AdS . The results agree with theuniversal CFT answers for a state at zero left-moving temperature and finite right-movingtemperature. We now move on to the case of nontrivial warping. We will set up the problemwith general warping parameter a (cid:54) = 1 and only specify our peturbative expansion aboutAdS with a = 1 + δ at a later point in our analysis. We consider the metric (1.1), and we determine the affinely parameterized geodesics x µ ( λ ) =( τ ( λ ) , u ( λ ) , r ( λ )) in this geometry. The metric has Killing vectors ∂ τ and ∂ u corresponding14o translations in τ and u , and they yield conserved quantities c τ = ˙ x · ∂ τ and c u = ˙ x · ∂ u ,respectively. Since we consider affinely parameterized geodesics, the square speed c v = ˙ x µ ˙ x µ along the geodesic is also conserved. The corresponding conservation equations are c τ = ( (cid:96)/ (cid:0) a r ( r ˙ τ + ˙ u ) − (cid:0) r + 1 (cid:1) ˙ τ (cid:1) , (3.1) c u = ( (cid:96)/ a ( r ˙ τ + ˙ u ) , (3.2) c v = ( (cid:96)/ (cid:104) − ( r + 1) ˙ τ + a ( r + 1) ( r ˙ τ + ˙ u ) + ˙ r (cid:105) r + 1 . (3.3)To solve these equations, it helps to manipulate them into the following form:˙ r = − (cid:18) c u (1 − a ) − ( (cid:96)/ a c v ( (cid:96)/ a (cid:19) r − (cid:18) c u c τ ( (cid:96)/ (cid:19) r − c u − a c τ − ( (cid:96)/ a c v ( (cid:96)/ a , (3.4)˙ τ = c u ( (cid:96)/ rr + 1 − c τ ( (cid:96)/ r + 1 , (3.5)˙ u = c τ ( (cid:96)/ rr + 1 + c u ( (cid:96)/ a r + 1 + c u ( (cid:96)/ (cid:18) − a a (cid:19) r r + 1 . (3.6)Equation (3.4) is now a decoupled, separable differential equation that can be integrated todetermine r ( λ ). The solution to (3.4) can then be plugged into equations (3.5) and (3.6),which can be integrated to obtain τ ( λ ) and u ( λ ), respectively. Notice also that setting a = 1in these equations gives the system of equations (2.26), (2.27), and (2.28). The equationsbecome a -independent in the limit c u = 0, though such a limit does not seem particularlyuseful for understanding warped AdS ; see Appendix A.3.1. The general solutions to theseequations can be found in Appendix A.3. We simply note here that the solution for u ( λ )has a piece that grows linearly with λ , unlike in the AdS case. This means that the relationbetween c u and L u will necessarily involve λ ∞ . This complicates the analysis, as we shall seeshortly.Let us focus on the “cosh-like” branch with 0 < a < c v = 1. This includes thesquashed and stretched cases. In our approach, we first write the length of the geodesic interms of the conserved quantities and the cutoff in the holographic coordinate r : λ ∞ = 1 √ c cosh − (cid:34) − c + 2 c r ∞ (cid:112) c + 4 c c (cid:35) , (3.7)where we have used the definitions in (A.17) and require c >
0. This expression holds forgeneral warping a as well as for the AdS case of a = 1 (the a -dependence is buried in c c ). Taking c r ∞ (cid:29) c and restoring the original constants of motion c u and c τ gives λ ∞ ≈ log (cid:20) c r ∞ √ c +4 c c (cid:21) √ c = log (cid:20) r ∞ a (1+ c u ) − c u √ ( − a + c u )( a (1+ c τ + c u ) − c u ) (cid:21)(cid:112) − /a ) c u , (3.8)where we have set c v = (cid:96)/ r ( λ ), τ ( λ ), and u ( λ ) are soluble, to write the answer for the length in terms of coordinate separations onan asymptotic boundary (instead of in terms of conserved quantities as done above) thereremains the task of inverting limits of those solutions to obtain the conserved quantities c u and c τ in terms of separations on the boundary L u and L τ . The equation for the τ coordinateis simply generalized from the AdS case: c τ = √ c cot (cid:18) L τ (cid:19) = (cid:114) c u ( a −
1) + a a cot (cid:18) L τ (cid:19) , (3.9)which holds as long as L τ < π . The new feature in these spacetimes, which is different fromasymptotically AdS spacetimes, is that the relation between c u and L u involves λ ∞ :2 (cid:18) − a (cid:19) c u λ ∞ + log c u + (cid:113) c u − c u a c u − (cid:113) c u − c u a = L u . (3.10)In other words, one cannot keep both c u and L u fixed as the cutoff is scaled large. Thisfollows directly from the linear divergence of u ( λ ) with λ , as would occur in an AdS × R background. One could at this point try to proceed by solving for c u in terms of L u and λ ∞ and plug c u and c τ into (3.8). This would then be an equation for λ ∞ that can be solved todetermine the length. Unfortunately, such an approach has two obstacles, one conceptualand one technical. The conceptual obstacle is that this would correspond to fully applyingthe HRT proposal in an asymptotically warped AdS spacetime, and it is unclear whethersuch a prescription makes sense. The technical obstacle (at least in this approach) is that(3.10) is a transcendental equation for c u . In the case of AdS × R , which can be realizedas the a → , the left-hand-side of the analog of (3.10) has only thepiece linear in λ ∞ and such a method can be carried out.In the next section, we will show that setting up a perturbative expansion about theAdS point by considering warping parameter a = 1 + δ will allow us to solve this equation One cannot in general be so cavalier in taking c r ∞ (cid:29) c without any restrictions on L u , since c dependson c u , which depends on λ ∞ . In our case, however, this can be consistently realized by taking r ∞ (cid:29) δ . Given that a nonperturbative application of the HRT prescription to asymptotically warpedAdS spacetimes is suspect, here we will try to infinitesimally perturb around the AdS pointand use the AdS/CFT dictionary, which presumably contains as one of its entries the HRTprescription. Deep in the IR, the geometry (1.1) is close to AdS , and it is only in the UVthat the nontrivial warping parameter begins to destroy the asymptotics. If we cut off ourspacetime before this happens, then we are at low enough energies where our analysis will beon firmer ground. Viewed in this way, we have a conformal field theory which we perturb byan infinitesimal, irrelevant operator. Holographic renormalization can then be understoodperturbatively in this infinitesimal source [52, 53]. We will find that in a certain limit wecan sum the perturbative expansion to all orders. The resulting answer takes the preciseform of a two-dimensional CFT and reproduces the warping-dependent central charge andleft- and right-moving temperatures postulated previously in the literature. We imagine that the warping parameter is close to 1, i.e. a = 1 + δ for | δ | (cid:28)
1. It is in thissense which we expand about the AdS point a = 1. Such a perturbative expansion will helpus solve (3.10) for c u order-by-order in δ . The solutions below follow a simple pattern ateach order, and though we list the general formulae for arbitrary order, we have technicallyonly checked that they are true to tenth order. Expanding c u = c u, + δ c u, + δ c u, + · · · , (4.1)we solve (3.10) to get c u, = ( (cid:96)/
2) coth L u , (4.2) c u, = 12 (( (cid:96)/ − λ ∞ + ( (cid:96)/
2) cosh L u ) coth L u L u , (4.3) c u,n = (cid:32) n − (cid:88) j =0 n (cid:88) i =0 λ i ∞ ( (cid:96)/ n − i k ( n ) ij cosh( jL u ) (cid:33) coth L u n L u n > , (4.4)17here the k ( n ) ij are calculable n - and λ ∞ -dependent constants. We require | δ n c u,n | (cid:28)| δ n − c u,n − | to assure convergence of our perturbative expansion. This can be satisfied bytaking L u (cid:38) , | λ ∞ δ | (cid:28) . (4.5) L u is being measured in units of (cid:96) . The latter condition ensures that we stay in an AdS -like part of the geometry and not get into the WAdS asymptotic. Notice that from thepoint of view of perturbing about AdS , this is an eminently sensible condition; regardlessof how small one takes δ , the geometry looks wildly different from AdS for sufficiently large λ ∞ , so we need to constrain their product. Incidentally, the curvature invariants of WAdS are all finite and continuously connected to the AdS case a = 1, so they are not a goodway to classify where to cut off the spacetime for a well-defined perturbation theory. Whencomputing the length, we keep only the leading divergent piece (in r ∞ ) at each order. Thisgives the following result for the entanglement entropy: S EE = (cid:96) G N (cid:20)(cid:18) δ coth L u (cid:19) log (cid:18) r ∞ sin L τ L u (cid:19)(cid:21) + (cid:96) G N ∞ (cid:88) i =2 δ i ( − i +1 coth L u i − L u (cid:20) log (cid:18) r ∞ sin L τ L u (cid:19)(cid:21) i (cid:32) i − (cid:88) j =0 c ij cosh( jL u ) (cid:33) . (4.6)The constants c ij are all positive. Notice that the zeroth order piece is precisely the answerfor AdS given in (2.30), as it should be. Unfortunately, the series does not seem simplysummable unless we take the scaling limit L u (cid:29)
1, in which case we use our knowledge ofthe c ij and sum the series to get S EE = (cid:96) G N (cid:20) (1 + δ ) log (cid:18) r ∞ sin L τ L u (cid:19)(cid:21) − (cid:96) G N e − L u (cid:20) − δ log (cid:18) r ∞ sin L τ L u (cid:19) + e − δ log ( r ∞ sin Lτ sinh Lu ) (cid:21) (4.7)to leading order in L u . Notice that the first two terms in the second line are suppressed by afactor of e − L u relative to the first line and can safely be dropped. Up to an overall constant,the last term in the second line can be written as (cid:18) sin L τ (cid:19) − δ (cid:16) r − δ ∞ e − L u (1+2 δ ) (cid:17) . (4.8)18or δ >
0, this is suppressed relative to the first line without further qualification and canbe dropped. For δ < − /
2, this term grows with L u and cannot be neglected. However, for − / < δ <
0, there is a competition between the factor containing r ∞ ∼ e λ ∞ (cid:29) L u (cid:29)
1. In this regime, for a given δ and r ∞ , one simply needs to choose L u sufficiently large ( L u (1 + 2 δ ) (cid:29) − δλ ∞ ) such that the resulting expression is dominatedby the expression in the first line.By combining these observations, we find that if δ > − /
2, then the leading behavior ofthe entanglement entropy in the large- L u regime is S EE = (cid:96) G N (1 + δ ) log (cid:32) ε (cid:115) sin L τ (cid:18) L u (cid:19)(cid:33) , (4.9)where we have used the UV-IR relation r ∞ ∼ /ε . Since we are sourcing an infinitesimalirrelevant operator and computing perturbatively, the UV-IR relation used should remainthat of AdS/CFT. We have also replaced the hyperbolic sine function with an exponentialfunction, since corrections are subleading in our expansion in e − L u . As usual, numericalfactors are absorbed into a redefinition of the cutoff ε .We see that for a = 1 + δ , the perturbative expansion in the large- L u gives simply thetwo-dimensional CFT answer of (2.30) upon identifying the coefficient of the logarithm with c/ c L = c R = 3 (cid:96) G N (1 + δ ) . (4.10)The equality of c L and c R is due to a lack of diffeomorphism anomaly, since we are workingin Einstein gravity. These are precisely the central charges of [8], conjectured by demandingconsistency with the Cardy formula (we will reproduce this check in Section 4.5). One ofthese central charges has been produced through an asymptotic symmetry group analysis[64]. Identifying the functional form of S EE with the AdS result (2.30) allows us to concludethat the dual state lives on a cylinder charted by null coordinates τ and u .It is important to keep in mind that the entanglement entropy computed in (4.9) isunderstood as an expansion to zeroth order in e − L u but to all orders in δ . This approach canin principle be extended to lower orders in L u , and the appearance of the logarithmic term inour general formulae suggests that the answer will remain roughly in the form of the CFT answer, except the logarithm will have an L u -dependent prefactor. This is consistent with To facilitate comparison with the notation of [8], one should take a → β and (cid:96) → (4 − β ) (cid:96) /
3. Noticethat as β →
4, which is the limit in which the central charge of [8] vanishes, there is an infinite rescalingthat allows our central charge to remain finite. u -dependence while keeping the τ -dependence the same. Our result at leading order in e − L u seems to suggest that warped CFTs behave like ordinary CFTs in the IR, for large L u . TheIR restriction is due to cutting off our spacetime deep in the bulk and is independent of thelarge L u restriction. The similarity to CFT jibes well with the fact that the deep interiorof the WAdS geometry is AdS -like. We will discuss the physical meaning of large L u inSection 4.4. We will also go beyond the small warping limit in Section 4.5 by arguing thatwarped CFTs are CFT-like generally, as long as one takes an infrared limit and studies large L u .Performing the same perturbative expansion in the case of Poincar´e coordinates wouldgive a result that can be obtained simply by coordinate transforming our current answer asin Appendix B.2, and it is given by S EE = (cid:96) G N (1 + δ ) log (cid:32) ε (cid:115) L ψ (cid:96) exp (cid:18) L φ (cid:96) (cid:19)(cid:33) . (4.11)This is again the appropriate answer at large L φ for a two-dimensional CFT, now on theMinkowski plane charted by null coordinates φ and ψ , as presented in (2.8). Taking thefiber coordinate L u large in the global coordinate system corresponds to taking the fibercoordinate L φ large in Poincar´e coordinates. In the case of Poincar´e coordinates, however,we can simultaneously take L ψ large if we want to consider a particular time slice L φ = L ψ .Due to the convergence of the perturbative expansion for any warping parameter a > / a in our formulae. In the limit of large separation in the fiber coordinate, we can match our results with thoseof two-dimensional CFT even at finite temperature. Since black holes in warped AdS aregiven by discrete quotients of the vacuum spacetime, they are locally warped AdS [65].This is analogous to BTZ black holes in AdS . Due to the local equivalence, we can exhibitlocal coordinate transformations that take us from the geometry with a black hole to the20eometry without a black hole. We will stick to the stretched case a > ds (cid:96) = 3 dt − a + dr r − r + )( r − r − ) + 6 √ − a ) / ( ar − √ r + r − ) dtdθ + 9 r (4 − a ) (cid:0) ( a − r + r + + r − − a √ r + r − (cid:1) dθ . (4.12)We will restrict to the stretched case a > S EE = (cid:96)aG N log (cid:32) r + − r − ε exp (cid:32)(cid:115) a (4 − a ) ∆ t + π ∆ θβ L (cid:33) sinh π ∆ θβ R (cid:33) , (4.13)with dimensionless temperatures β − L = T L = 32 π (4 − a ) (cid:18) r + + r − − a √ r + r − (cid:19) , (4.14) β − R = T R = 3( r + − r − )2 π (4 − a ) . (4.15)Due to the compactification of θ , there can exist many spacelike geodesics in this geometry,distinguished by their winding number and directionality. The expression ∆ θ refers to theseparation in a noncompact θ , i.e. without modding by 2 π . We can ignore the globaltopology by considering ∆ θ (cid:28) π . This is consistent with the large- L u limit taken in theprevious section, since that limit can be accomodated by taking ∆ t large. Adding windingwill only increase the length of the geodesic, so we see that our answer is valid in the regimeconsidered.In the case of AdS with a = 1, the coordinates are such that one picks a constant-timeslice by requiring ∆ t = 0. It is important to note that this case corresponds to the BTZ blackhole in a rotating coordinate system, and our answer for a = 1 is the universal CFT answerfor such a dual state. Since we are using a rotating coordinate system, it is not necessarythat the functional form of our answer precisely match the form of (2.24). The parameters Note that this would be more subtle if we considered the squashed case a <
1, since for large enough r winding in θ corresponds to a timelike direction and can decrease the length of the geodesic. L and β R give the inverse left-moving and right-moving temperatures of the BTZ blackhole in this frame, and we see that this match extends to the warped BTZ case as well; thedimensionful temperatures (4.14) and (4.15) match precisely with those of [65]. In Section4.5 we will show that these temperatures, combined with the central charge (4.10), satisfythe Cardy formula. Finally, implementing an appropriate homology constraint suffices toreproduce the thermodynamic black hole entropy in the limit where we consider the entireboundary density matrix without tracing out any degrees of freedom. We have produced the universal CFT results for states dual to spacelike warped AdS and thewarped BTZ black hole. However, as our formulae in the previous sections illustrate, noneof these states can be considered the vacuum state. The proposal in [16] is that the timelikewarped AdS geometry is a suitable candidate for the vacuum state in both topologicallymassive gravity and a specific string theory example that reduces to Einstein gravity plusmatter. The proposed vacuum geometry (which is in fact G¨odel space) can be written as ds (cid:96) = − dt − a + 3 dr r (4 − a + 3 r ) − ar √ − a ) / dtdθ + 3 r (4 − a − r ( a − − a ) dθ , (4.16)where θ is a compact coordinate with θ ∼ θ + 2 π . For a > r > (4 − a ) / a −
1) (see [66, 67] for a discussion). In our perturbativeapproach, we can take r ∞ δ (cid:28)
1, which is sufficient to excise the region with closed timelikecurves. Notice that we can get to this geometry by taking the warped BTZ black hole (4.12)and performing the identifications r + = 0 , r − = a − , t → it, θ → iθ . (4.17)If we replace the exponential function in (4.13) with a hyperbolic sine (i.e. start with aprecise match to 2D CFT instead of a match only at large fiber coordinate), then we canperform these identifications on the entanglement entropy result to get, for ∆ t = 0, S EE = (cid:96)a G N log (cid:18) sin( L θ / ε (cid:19) . (4.18)We see that this is the ground state answer for a two-dimensional CFT on a cylinder with acompact spatial coordinate θ . Unfortunately, this is merely illustrative because it runs afoul22f the requirement of large fiber-coordinate separation. The correct way to get the answer inour framework is to keep the entire expression (4.6) and perform the identifications necessaryto get to timelike warped AdS. From here, there does not appear to be a sensible regime inwhich the series can be summed and reduces to a two-dimensional CFT answer. We now discuss the meaning of the limit of large fiber-coordinate separation L φ on a fieldtheory calculation of entanglement entropy. Note that the limit is not necessarily a restrictionon the spatial size, ( L φ L ψ ) / , for which our result holds. The different spatial sizes lie onspacelike slices boosted with respect to one another. For example, large spatial sizes areaccommodated by taking L ψ ∼ L φ , which results in a “mostly spacelike” slice, whereassmall spatial sizes are accommodated by taking L ψ small, which makes the slice more null.Nevertheless, imposing large L φ without constraining the system size does impose a physicalrestriction on the reduced density matrix. Unlike the case of the vacuum state on theMinkowski plane, there is no Lorentz symmetry relating the different observers on theirdifferent spacelike slices. In the case of spacelike warped AdS , our result for S EE exhibitsthat there is a finite right-moving temperature turned on, which breaks Lorentz invariance.In the case of warped BTZ black holes, there is also a finite left-moving temperature. Thus,Lorentz transformations connecting different observers act nontrivially and lead to a differentreduced density matrix. On the other hand, for the vacuum state on the plane the answercan be boosted and replaced with the invariant Minkowskian interval, as shown in (1.4).The entanglement entropy in this case is only sensitive to the length of the spatial intervaland not the orientation of the spatial slice, whereas when Lorentz invariance is broken it issensitive to both. We have seen that the perturbative series we constructed converges for a > / a ∈ [0 , a ∈ [0 , /
2) includes the interesting case ofAdS × R , which can be reached by taking a → u → u/a in (1.1). Here we are referring to spacelike warped AdS in Poincar´e coordinates (1.2), where the fiber coordinateis denoted by φ , and there is no restriction on the separation in the other coordinate ψ . The dual state ison the Minkowski plane and has finite right-moving temperature. r ∞ ∼ /ε to hold nonperturbatively. In our perturba-tive approach we could make use of this UV-IR relation since we were working in the contextof AdS/CFT, where it is known to be true. Extending the requirement into the nonper-turbative regime is a natural choice. With it, we claim that our perturbative expansion issufficient to capture the nonperturbative dynamics entering into the entanglement entropy.A nontrivial check on this nonperturbative proposal is the Cardy formula. Our answersfor S EE allow us to read off left-moving and right-moving temperatures and the centralcharge. We now claim that all these results hold nonperturbatively. The central charge isgiven universally as c L = c R = 3 (cid:96)a G N . (4.19)For the warped BTZ black hole, our proposal allows us to identify the left-moving and right-moving temperatures as (4.14) and (4.15) nonperturbatively in a . These temperatures andthe central charge reproduce the entropy of the warped BTZ black hole through the Cardyformula: S = A G N = (cid:18) π(cid:96) G N (4 − a ) ( ar + − √ r + r − ) (cid:19) = π c L T L + c R T R ) . (4.20) We have taken the first steps toward understanding holographic entanglement entropy inthe context of asymptotically warped AdS spacetimes in Einstein gravity. We began byconsidering AdS as a real-line fibration over AdS , a coordinate system relevant to thestudy of extremal black holes. The calculation of the entanglement entropy indicated a stateat zero left-moving and finite right-moving temperature, as expected.Deforming the fibration by a nontrivial warp factor leads to the warped AdS geometries,appearing in the near-horizon limit of extremal Kerr black holes at constant polar angle. Toconnect with the HRT proposal in AdS/CFT, we constructed a perturbation theory about theAdS point with trivial warping. For a = 1 + δ , one can compute the length of the necessarygeodesic perturbatively in δ to all orders. The general answer is not particularly illuminating,except in the limit of large separation in the fiber coordinate. Recall that the U (1) isometryoriginating from translation invariance in the fiber coordinate is what is expected to enhanceto an infinite-dimensional U (1) Kac-Moody algebra in the boundary theory. In this limit,24he answer takes the universal form predicted by two-dimensional CFT. Interpreting ouranswer as a CFT answer allows us to read off the purported central charge of the dualtheory, which is given by c = 3 (cid:96)a/ G N . Since we are working in Einstein gravity, there is nodiffeomorphism anomaly and c L = c R = c . Furthermore, heating up the dual state with awarped BTZ black hole in the bulk again leads to universal two-dimensional CFT answers,with the left- and right-moving temperatures appearing appropriately in the entanglemententropy. Altogether, the central charge and left- and right-moving temperatures identifiedin this way satisfy the Cardy formula and thus reproduce the black hole entropy in thebulk. The central charge we have identified from the entanglement entropy calculationhas been previously produced in the literature [8] by demanding consistency with the Cardyformula. Our approach implements the covariant holographic entanglement entropy proposaland consistency with the Cardy formula is instead a promising output. Taking our results atface value, they seem to suggest that warped CFTs behave like ordinary CFTs in the IR; thismatches the intuition garnered from asymptotically warped AdS spacetimes in holography,since their deep interiors are AdS -like for small warping. Our perturbative expansion alsoshows that there exists nontrivial fiber-coordinate dependence at subleading order in theseparation of the fiber coordinate, suggesting that the full theory is not a standard conformalfield theory. How to implement a proposal for holographically computing entanglemententropy in asymptotically warped AdS spacetimes, without taking an IR limit, remains anopen question.The most immediate way one can make progress on the questions discussed in this paperis by studying the constraints of warped CFT on field-theoretic calculations of entanglemententropy. It has been shown in [16] that warped conformal invariance is strongly constrainingand allows one to reproduce a Cardy-like formula for the asymptotic growth in the densityof states by using the modular covariance of the partition function. As shown in [68], thecalculation of entanglement entropy in the vacuum and finite-temperature states of two-dimensional CFT can be conformally mapped to the calculation of a partition function.The constrained form of the partition function then allows one to write down the universalformulas for two-dimensional CFT. Such a procedure may prove fruitful in the case of warpedCFTs as well, although one of the primarily difficulties is due to warped CFTs not havingnatural Euclidean descriptions. Obtaining a universal entanglement entropy formula forsimple states of warped CFTs will allow one to determine if our holographic results areindicating the existence of a second hidden Virasoro algebra or if the infinite-dimensional U (1) is sufficient to constrain the answers in the way we have presented.25t is also interesting to see how far the analogy with two-dimensional CFT can be taken.For example, it is possible that for large separation in the U (1) coordinate, with an appro-priate IR limit, the field-theoretic calculation of entanglement entropy in a warped CFTreproduces the CFT result. A simpler question is the constraint on correlation functions: itcan be shown [15] that left-translation invariance, left-scale invariance, and right-translationinvariance constrain the vacuum two-point function of local operators φ i to be of the form (cid:104) φ i ( x − , x + ) φ j ( y − , y + ) (cid:105) = f ij ( x − − y − )( x + − y + ) λ i + λ j , (5.1)where λ i is the weight of the operator φ i . Furthermore, the symmetries are automaticallyenhanced to an infinite-dimensional left-moving U (1) Kac-Moody algebra and a left-movingVirasoro algebra. If the analogy to two-dimensional CFT is to be taken seriously, thesesymmetries should provide a constraint on f ij such that in the limit of large separation x − − y − and in an appropriate infrared regime the answer reduces to that of two-dimensionalCFT. Even if this simplification occurs, however, it does not imply that the theory can bedescribed by an ordinary two-dimensional CFT in this regime. The entanglement entropy inthe states we have considered and the vacuum two-point function give limited informationabout the theory and do not elucidate its full dynamics.Another home for the study of warped AdS and warped BTZ black holes is topologicallymassive gravity, a higher curvature theory of gravity. There exists a proposal for extendingthe holographic entanglement entropy proposal to this theory [44], although there has notbeen much work in this direction. Given our study of finite-temperature solutions, it isplausible that a proposal for topologically massive gravity which reproduces the CFT answerfor empty, warped AdS will also reproduce the correct answer for the warped BTZ blackhole, as shown in Section 4.2.We have seen that the method of holographically computing entanglement entropy, de-vised in AdS/CFT, can be adapted to the case of warped AdS holography. It providesfurther evidence that a sharp holographic correspondence can be developed in this context.The perturbative approach we implemented may be a promising way to study entangle-ment entropy in more general spacetimes continuously connected to AdS d +2 . It can also beadapted to the NHEK geometry, where one would like to independently deduce c L = 12 J .26 cknowledgements We would like to acknowledge useful conversations with Tatsuo Azeyanagi, Xi Dong, MichaelGutperle, Sean Hartnoll, Eliot Hijano, Diego Hofman, and Gim Seng Ng. This work hasbeen partially funded by DOE grant DE-FG02-91ER40654. E.S. is supported in part by NSFGrant PHY-0756174 and would like to thank KU Leuven and KITP for their hospitality whilepart of this work was performed. J.S. is supported in part by NSF Grant PHY-07-57702.D.A. would like to acknowledge the hospitality of the Aspen Center for Physics where partof this work was performed.
A Geodesics
A.1 AdS in Poincar´e fibered coordinates There are four solution branches for the geodesics in the background (2.1). To obtain thesesolutions, one solves equations (2.2) and (2.3) for ˙ ψ and ˙ φ in terms of x , plugs the resultback into (2.4), and then integrates the resulting equation to obtain x c, ± ( λ ) = (cid:96)c v c ψ c φ ± (cid:113) c ψ (cid:0) − (cid:0) c v − c φ (cid:1)(cid:1) cosh (cid:16) √ c v ( λ − λ (cid:96) (cid:17) , (A.1) x s, ± ( λ ) = (cid:96)c v c ψ c φ ± (cid:113) c ψ (cid:0) c v − c φ (cid:1) sinh (cid:16) √ c v ( λ − λ (cid:96) (cid:17) . (A.2)We typically refer to the solutions (A.1) as the “cosh-like” branch and to those in (A.2) asthe “sinh-like” branch. The differences between these branches will be clarified in the nextsection when we consider global coordinates. Note that in the process of obtaining these foursolutions, one made the assumption that c v − c φ < c v − c φ > x canthen be combined with the other conservation equations to find the following correspondingsolutions for φ : φ c, ± ( λ ) = (cid:98) φ c, ± + 2 (cid:96) coth − √ c v c ψ coth (cid:16) √ c v ( λ − λ (cid:96) (cid:17) c ψ c φ ∓ (cid:113) c ψ (cid:0) − (cid:0) c v − c φ (cid:1)(cid:1) , (A.3) φ s, ± ( λ ) = (cid:98) φ s, ± ∓ (cid:96) coth − √ c v c ψ ∓ c ψ c φ tanh (cid:16) √ c v ( λ − λ (cid:96) (cid:17) + (cid:113) c ψ (cid:0) c v − c φ (cid:1) , (A.4)27nd the following for ψ : ψ c, ± ( λ ) = (cid:98) ψ c, ± + (cid:96) √ c v (cid:16) c φ (cid:113) c φ − c v sinh (cid:16) √ c v ( λ − λ (cid:96) (cid:17) + (cid:0) c v − c φ (cid:1) sinh (cid:16) √ c v ( λ − λ (cid:96) (cid:17)(cid:17) c ψ (cid:16)(cid:0) c v − c φ (cid:1) cosh (cid:16) √ c v ( λ − λ (cid:96) (cid:17) + c v + 4 c φ (cid:17) , (A.5) ψ s, ± ( λ ) = (cid:98) ψ s, ± ± (cid:96) √ c v (cid:0) c v − c φ (cid:1) cosh (cid:16) √ c v ( λ − λ (cid:96) (cid:17) ± c ψ (cid:0) c v − c φ (cid:1) sinh (cid:16) √ c v ( λ − λ (cid:96) (cid:17) + 4 c φ (cid:113) c ψ (cid:0) c v − c φ (cid:1) . (A.6)At first glance one might be concerned about the continuity of the “cosh-like” branch solu-tions because of the presence of the function coth − . However, we see that the argument ofthe coth − is of the form α coth( βλ + γ ) where α , β , and γ are real, and this ensures thatthe overall solution is continuous. Similar remarks hold for the sinh branch. A.2 AdS in global fibered coordinates The geodesics of (2.25) are obtained by first solving (3.4) to obtain the following four solutionbranches: r ± , c ( λ ) = c u c τ c v ( (cid:96)/ ± (cid:112) ( c u − c v ( (cid:96)/ )( c τ + c v ( (cid:96)/ ) c v ( (cid:96)/ cosh (cid:18) √ c v ( (cid:96)/
2) ( λ − λ ) (cid:19) , (A.7) r ± , s ( λ ) = c u c τ c v ( (cid:96)/ ∓ (cid:112) − ( c u − c v ( (cid:96)/ )( c τ + c v ( (cid:96)/ ) c v ( (cid:96)/ sinh (cid:18) √ c v ( (cid:96)/
2) ( λ − λ ) (cid:19) . (A.8)The different branches can be clarified by considering the parametrization of the boundary inthese coordinates found in Appendix A of [29]. The two solutions in the “cosh-like” branchgive geodesics that go from r = + ∞ or r = −∞ back to r = + ∞ or r = −∞ , respectively,depending on the sign of the solution chosen. The two branches in the “sinh-like” solutioncorrespond to geodesics that go from r = + ∞ to r = −∞ or vice versa. In this paper weprimarily restrict attention to “cosh-like” branch solutions. If we define f c ( λ ) = e √ cv ( λ − λ (cid:96)/ h c , h c = (cid:112) ( c u − c v ( (cid:96)/ ) ( c τ + c v ( (cid:96)/ ) , (A.9) f s ( λ ) = e √ cv ( λ − λ (cid:96)/ hs , h s = (cid:112) − ( c u − c v ( (cid:96)/ ) ( c τ + c v ( (cid:96)/ ) , (A.10)28hen the corresponding solutions for τ can be written as τ ± ,c ( λ ) = (cid:98) τ ± ,c − cot − (cid:18) f c ( λ ) ± c τ c u f c ( λ ) + ( c τ − c v ( (cid:96)/ ) ( c u − c v ( (cid:96)/ )2 √ c v ( (cid:96)/
2) ( c u c τ − c v c τ ( (cid:96)/ ± c u f c ( λ )) (cid:19) , (A.11) τ ± ,s ( λ ) = (cid:98) τ ± ,s − cot − (cid:18) f s ( λ ) ∓ c τ c u f s ( λ ) + ( c τ − c v ( (cid:96)/ ) ( c u − c v ( (cid:96)/ )2 √ c v ( (cid:96)/
2) ( c u c τ − c v c τ ( (cid:96)/ ∓ c u f s ( λ )) (cid:19) . (A.12)The seeming discontinuity of the function arccot( y ) at y = 0 is not important since it canbe glued onto arccot( y ) + π there and continue smoothly to negative values of the argument.This can potentially introduce a shift of π into L τ , which is important and needs to betracked. Now if we define g c, ± ( λ ) = c u cosh (cid:18) √ c v ( (cid:96)/
2) ( λ − λ ) (cid:19) ± ( (cid:96)/ √ c v sinh (cid:18) √ c v ( (cid:96)/
2) ( λ − λ ) (cid:19) , (A.13) g s, ± ( λ ) = ( (cid:96)/ √ c v cosh (cid:18) √ c v ( (cid:96)/
2) ( λ − λ ) (cid:19) ± c u sinh (cid:18) √ c v ( (cid:96)/
2) ( λ − λ ) (cid:19) , (A.14)the solutions for u become u ± ,c ( λ ) = (cid:98) u ± ,c + 12 log (cid:32) (cid:0) c u − ( (cid:96)/ √ c v (cid:1) ((( (cid:96)/ c v + c τ ) g c, + ( λ ) ± c τ h c ) (cid:0) ( (cid:96)/ √ c v + c u (cid:1) ((( (cid:96)/ c v + c τ ) g c, − ( λ ) ± c τ h c ) (cid:33) , (A.15) u ± ,s ( λ ) = (cid:98) u ± ,s + 12 log (cid:32) (cid:0) c u − ( (cid:96)/ √ c v (cid:1) ((( (cid:96)/ c v + c τ ) g s, + ( λ ) ± c τ h s ) (cid:0) ( (cid:96)/ √ c v + c u (cid:1) ((( (cid:96)/ c v + c τ ) g s, − ( λ ) ∓ c τ h s ) (cid:33) . (A.16) A.3 Warped
AdS in global fibered coordinates To simplify our expressions for geodesics in the background (1.1) with a (cid:54) = 1, we define c = − c u − a c u − ( (cid:96)/ a c v ( (cid:96)/ a , c = 2( (cid:96)/ c u c τ , c = c u − a c τ − ( (cid:96)/ a c v ( (cid:96)/ a , (A.17)This illuminates the general form of (3.4):˙ r = c r − c r − c . (A.18) An easy way to not have to deal tracking constant shifts like this one is to do the naive calculation firstand obtain a function of the form sin(( L τ + c ) /
2) in the entanglement entropy answer for the “cosh-like”branch, with c an overall constant that has not been carefully tracked. Requiring the length to vanish when L τ → c = 0. c v > c + 4 c c . We ignore the case c + 4 c c = 0 which yields an exponentially decaying solution.For c + 4 c c >
0, there are two “cosh-like” branches r ± ,c and for c + 4 c c <
0, there aretwo “sinh-like” branches r ± ,s : r ± , c ( λ ) = c c ± (cid:115)(cid:18) c c (cid:19) + c c cosh (cid:0) √ c ( λ − λ ) (cid:1) , (A.19) r ± , s ( λ ) = c c ∓ (cid:115) − (cid:18) c c (cid:19) − c c sinh (cid:0) √ c ( λ − λ ) (cid:1) . (A.20)Comparing these solutions to (A.7) and (A.8), we find the same qualitative behavior in r ( λ )for both the warped and non-warped cases. Moreover, setting a = 1 in these warped solutionsyields precisely (A.7) and (A.8), as one would expect since the form of the r equation is leftunaltered by non-trivial warping. The form of the τ equation is unaltered by warping, so weexpect the corresponding solution to be of the same form. However, notice that the last termin (3.5) only appears in the case a (cid:54) = 1 where there is non-trivial warping, and this changesthe qualitative behavior of the solutions. In particular, manipulating the second and thirdterms allows one to write the u equation as˙ u = c u ( (cid:96)/ a − c u ( (cid:96)/ r r + 1 + c τ ( (cid:96)/ rr + 1 , (A.21)from which it becomes clear that integration of the first term with respect to λ leads to aterm in the solution for u that diverges linearly with λ . This is just like the AdS × R case. A.3.1 c u = 0A simple limit in which we can compute the length of the geodesic in terms of separations in τ and u at large r is for c u = 0. In this case, the equations of motion become a -independentand we are forced onto the “sinh-like” branch. The answer is then just given by the answerfor AdS in global fibered coordinates with c u = 0:Length ∼ log (cid:32)(cid:115) cos (cid:18) L τ (cid:19) (cid:96)ε (cid:33) . (A.22)It is not a problem that the argument of the log does not begin to vanish in the L τ = τ ( λ ∞ ) − τ ( − λ ∞ ) → r ( λ ∞ ) (cid:54) = r ( − λ ∞ ).30hus, the endpoints remain well-separated in the limit L τ → B AdS entanglement entropy via coordinate transformations For completeness, in this section we show how one can translate from entanglement entropyanswers in Poincar´e coordinates to global coordinates, or vice versa, by performing theappropriate coordinate transformations. The only point one needs to be careful about is themapping of the UV cutoff. As an illustrastive example, we will begin with showing how toget the answer in global coordinates from the answer in the Poincar´e patch, within which itis easiest to compute. We then show how to go from Poincar´e fibered coordinates to globalfibered coordinates. Such methods will come in handy when we go from warped AdS to thewarped BTZ black hole in Section 4.2. B.1 Global coordinates from Poincar´e patch
Recall that AdS can be defined as an embedded submanifold of R , defined by the constraint X − X − X + X = 1 (B.1)where X , X , X , X are the standard coordinates on R , . The embedding coordinates forglobal AdS are X = (cid:96) cosh ρ cos t g , X = (cid:96) sinh ρ sin θ g , (B.2) X = (cid:96) sinh ρ cos θ g , X = (cid:96) cosh ρ sin t g , (B.3)while for Poincar´e AdS they are X = 12 z ( z + (cid:96) + x − t ) , X = (cid:96)xz , (B.4) X = 12 z ( z − (cid:96) + x − t ) , X = (cid:96)tz . (B.5)31o get from global coordinates to Poincar´e coordinates we use the transformations X + X (cid:96) = cosh ρ = 14 (cid:96) z ( z + (cid:96) + x − t ) + t z ,X X = tan t g = 2 (cid:96)t ( z + (cid:96) + x − t ) ,X X = tan θ g = 2 (cid:96)x ( z − (cid:96) + x − t ) . The inverse transformations are given by (cid:96) X − X = z = (cid:96) cosh ρ cos t g − sinh ρ cos θ g ,(cid:96)X X − X = x = (cid:96) sinh ρ sin θ cosh ρ cos t g − sinh ρ cos θ g ,(cid:96)X X − X = t = (cid:96) cosh ρ sin t g cosh ρ cos t g − sinh ρ cos θ g . Using either set of relationships, we can see that for t = t g = 0 and z = 0, ρ = ∞ , we get x = (cid:96) sin θ g / (1 − cos θ g ). We want to show that L x /ε P = ( x − x ) /ε P , when written interms of L θ , is L x /ε P ∝ sin( L θ / /ε g . Our answer for the length of the curve in Poincar´ecoordinates is c (cid:18) L x z (cid:19) + c (cid:18) L x z (cid:19) = c (cid:18) L x √ z z (cid:19) (B.6)where we have picked two different endpoints z and z for the curve. Using z = (cid:96)e − ρ − cos θ g asthe asymptotic coordinate transformation between the coordinates gives L x √ z z = (cid:18) sin θ g, − cos θ g, − sin θ g, − cos θ g, (cid:19) (cid:112) (1 − cos θ g, )(1 − cos θ g, )2 e − ρ (B.7)= sin (cid:0) L θ (cid:1) e − ρ , (B.8)where we have picked ρ = ρ = ρ to fix to a constant cutoff surface. We then use e − ρ ∼ aL ∼ ε and recover S global = c L θ ε = c lπ/L ) ε (B.9)upon identifying L θ = θ g, − θ g, = 2 πl/L for total circumference L .32 .2 Global fibered from Poincar´e fibered We now map the Poincar´e fibered answer onto the global fibered answer. The coordinatetransformations between these two metrics are φ = (cid:96) log (cid:18) e σ cot( τ / − e σ cot( τ /
2) + 1 e u (cid:19) , ψ = cosh σ sin τ sinh σ + cosh σ cos τ , x = 1sinh σ + cosh σ cos τ , which asymptotically ( σ → ∞ ) become φ = (cid:96)u , ψ = tan( τ / , x = 2 e − σ τ . From these relations we see that L φ = (cid:96)L u , L ψ = tan( τ / − tan( τ / , and 1 √ x x = (cid:112) (1 + cos τ )(1 + cos τ )2 e − σ , since we will be assuming we are at different points x , x at the two ends of the curve inPoincar´e fibered coordinates, whereas for global fibered coordinates we will assume we areat the same σ coordinate at both endpoints of the curve. Our Poincar´e fibered answer wasfound to be S = c (cid:113) L ψ (cid:96) sinh L φ (cid:96) ε , (B.10)where to get here the cutoff relation x ∼ ε /(cid:96) was employed. Translating back, we find ε → √ ε ε = (cid:113) (cid:96) √ x x S = c (cid:115) (tan( τ / − tan( τ / (cid:112) (1 + cos τ )(1 + cos τ )2 e − σ sinh L u c (cid:114) sin( L τ /
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