Holography at an Extremal De Sitter Horizon
aa r X i v : . [ h e p - t h ] O c t Holography at an Extremal De Sitter Horizon
Dionysios Anninos and Thomas HartmanJefferson Physical Laboratory, Harvard University, Cambridge, MA 02138, USA
Abstract
Rotating maximal black holes in four-dimensional de Sitter space, for which the outerevent horizon coincides with the cosmological horizon, have an infinite near-horizonregion described by the rotating Nariai metric. We show that the asymptotic sym-metry group at the spacelike future boundary of the near-horizon region contains aVirasoro algebra with a real, positive central charge. This is evidence that quantumgravity in a rotating Nariai background is dual to a two-dimensional Euclidean con-formal field theory. These results are related to the Kerr/CFT correspondence forextremal black holes, but have two key differences: one of the black hole event hori-zons has been traded for the cosmological horizon, and the near-horizon geometry isa fiber over dS rather than AdS . Introduction
The cosmological horizon of de Sitter space resembles a black hole horizon in many ways,exhibiting a thermal spectrum and having entropy proportional to the horizon area [1].On the other hand, it is sharply different from a black hole horizon in the sense that it isobserver dependent, and its size does not decrease upon emission of Hawking radiationas it reabsorbs its own radiation. An important open problem is to understand whetherthere is a sensible theory of quantum gravity in a de Sitter background, and to accountfor its entropy from a microscopic point of view. There have been various attempts todo so in two and three dimensions [2, 3, 4], and in four dimensions [5, 6, 7]. The goal ofthis paper is to study the de Sitter horizon in the special case that it is coincident witha black hole horizon.In what follows, we consider rotating black holes in asymptotically de Sitter spacetimein four dimensional Einstein gravity. A rotating black hole in de Sitter space has aninner, outer and cosmological horizon. Extremal Kerr-de Sitter black holes are obtainedby bringing together the inner and outer horizons. The near horizon geometry of thisconfiguration is given by a generalization of the near-horizon extreme Kerr (NHEK)spacetime [8], which is a fibered product of two-dimensional anti-de Sitter space and thetwo-sphere with an SL (2 , R ) × U (1) isometry group.It was discovered in [9] that there exist consistent boundary conditions on NHEK forwhich the asymptotic symmetry group [10] enhances the U (1) isometry to a Virasoroalgebra with non-trivial central charge. This led to the conjecture that quantum gravityin the near horizon geometry is holographically dual to a 2d CFT. In support of thisconjecture, the central charge of the Virasoro algebra and the Frolov-Thorne temperature[11] were used in the Cardy formula to account for the black entropy from the viewpoint ofthe proposed dual CFT. This was extended to non-zero cosmological constant in [12, 13].For both positive and negative cosmological constant, the near horizon geometry is afiber over AdS .There is another extremal limit for rotating black holes in de Sitter space: the limitwhere the outer and cosmological horizons coincide. In this limit, the near horizongeometry becomes a fibered product of two-dimensional de Sitter space and a two-sphere,with the metric ds = Γ( θ ) (cid:20) − (1 − r ) dτ + dr − r + α ( θ ) dθ (cid:21) + γ ( θ )( dφ + krdτ ) . (1.1)The functions Γ , γ, α and k are given below in terms of the black hole parameters. Thisspacetime, which was extensively studied in [14], is known as the rotating Nariai ge-ometry and reduces to the Nariai geometry dS × S [15, 16, 17] in the non-rotatingcase. Observers in the rotating Nariai spacetime live inside a cosmological horizon whoseentropy corresponds to the entropy of the original horizon in the full geometry.We will see that the rules developed for the NHEK geometry are also applicable to therotating Nariai geometry. The asymptotic symmetry algebra of (1.1) at future spacelike1nfinity is the Virasoro algebra with real, positive central charge c L = 12 r c p (1 − r c /ℓ )(1 + r c /ℓ ) − r c /ℓ + 3 r c /ℓ , (1.2)where 3 /ℓ is the cosmological constant and r c is the cosmological horizon. We thereforeconjecture that quantum gravity in the rotating Nariai geometry is dual to a Euclidean 2dconformal field theory. Assuming the Cardy formula, this together with the left-movingtemperature allow for a holographic derivation of the cosmological horizon entropy. While the derivation of the asymptotic symmetry group and central charge is robust,this application of the Cardy formula is on a speculative footing. Unlike in AdS/CFTor Kerr/CFT, an understanding of how to map thermal states between the bulk andboundary is lacking.Despite the apparent similarities to Kerr/CFT, there are important differences be-tween NHEK and rotating Nariai. Asymptotically, the rotating Nariai spacetime is natu-rally foliated by a timelike “radial” coordinate and the foliations are spacelike. This is incontrast to the timelike foliations of the NHEK geometry, suggesting that the appropriateconformal field theory lives on a spacelike manifold and is in fact Euclidean very muchin the same vein as the dS/CFT proposal. Furthermore, it is unclear whether quantumgravity in the rotating Nariai geometry is unitary and thus application of the Cardyformula remains on a numerological footing much like it was in [18], where the entropy ofasymptotically dS conical defects was examined. Therefore we stress that this does notprovide a firm microstate counting of the de Sitter entropy. On the other hand, the factthat a naive application of Cardy’s formula indeed accounts for the cosmological entropyis a clear invitation for a solid explanation.The rotating Nariai/CFT duality proposed here shares some characteristics withKerr/CFT and dS/CFT, but is not continuously connected to either of these, and shouldbe considered a separate class of dualities. Adjusting the black hole mass to bring theinner black hole horizon toward the coincident outer and cosmological horizons, it seemspossible to interpolate between Kerr/CFT and Nariai/CFT. However, in the ultracoldlimit where all three horizons coincide, the near horizon geometry has no apparent CFTinterpretation. As in dS/CFT, the dual to rotating Nariai is Euclidean. In contrast, itis always two-dimensional, regardless of the dimensionality of the bulk spacetime.Various generalizations, applications, and tests of the rotating Nariai/CFT correspon-dence are possible. As a first step, the derivation of the asymptotic symmetry group isgeneralized to include electromagnetic charge in appendix B. The extension to higher-dimensional de Sitter space is not considered here but should be straightforward (see[12]). Since the near-horizon geometry in any number of dimensions is a fiber over dS ,we expect the CFT to always be two-dimensional. Another open question is how the cor-respondence applies to near-extremal black holes; progress along these lines in Kerr/CFThas been made in [19, 20, 21]. In this extremal (or maximal) limit, what we call the cosmological entropy is of course also the blackhole entropy. Nonetheless the cosmological nature of the double horizon leads to a dual CFT qualitativelydifferent from that dual to ordinary extreme black holes. not thermody-namically stable due to the incoming thermal radiation of the cosmological horizon thatcauses them to heat up [30, 31], thus eliminating the original suspicion.The outline is as follows. In section 2 we describe the geometry and thermodynam-ics of Kerr-de Sitter space and the rotating Nariai geometry. In section 3 we study theasymptotic symmetries of the rotating Nariai geometry and propose the rotating Nar-iai/CFT correspondence. In particular, we apply the Cardy formula to the CFT and findthat it reproduces the cosmological horizon entropy. Finally, we briefly discuss scalarsin the rotating Nariai geometry in section 4. In appendix B we show that the rotatingNariai/CFT correspondence generalizes to maximal dyonic Kerr-Newman-dS black holes.
Our story begins with a review of the Kerr-dS and rotating Nariai geometries as solutionsof Einstein gravity with a positive cosmological constant Λ = 3 /ℓ in four-dimensions: S grav = 116 π Z M d x √− g ( R − , (2.1)where we have set Newton’s constant G = 1. The Kerr-dS metric is ds = − ∆ ˆ r ρ (cid:16) d ˆ t − a Ξ sin θd ˆ φ (cid:17) + ρ ∆ ˆ r d ˆ r + ρ ∆ θ dθ + ∆ θ ρ sin θ (cid:18) ad ˆ t − ˆ r + a Ξ d ˆ φ (cid:19) (2.2)where ∆ ˆ r = (ˆ r + a )(1 − ˆ r /ℓ ) − M ˆ r, ∆ θ = 1 + a cos θ/ℓ (2.3) ρ = ˆ r + a cos θ, Ξ = 1 + a /ℓ . ˆ r = 0 and we denote them by r − , r + and r c , with r c ≥ r + ≥ r − . There are three extremal limits one can consider, allwith zero Hawking temperature: Taking the inner and outer horizon to coincide (extremallimit) such that r + = r − , taking the outer and cosmological horizon to coincide (Nariailimit) such that r + = r c , and taking the inner, outer and cosmological horizons to coincide(ultracold limit) such that r + = r − = r c . Each of these extremal configurations has adifferent near horizon geometry. The extremal limit r + = r − was studied in the context of Kerr/CFT in [12, 13]. Herewe instead focus on the Nariai limit r + = r c . The Penrose diagram of the geometry inthis limit is given in appendix A. The parameters a, M in the Nariai limit are a = r c (1 − r c /ℓ )1 + r c /ℓ (2.4) M = r c (1 − r c /ℓ ) r c /ℓ . The thermodynamic properties of the Kerr-de Sitter spacetime were obtained in [35, 36].The conserved charges associated to the ∂ ˆ t and ∂ ˆ φ Killing vectors are given by Q ∂ ˆ t = − M Ξ , Q ∂ ˆ φ = − aM Ξ . (2.5)Since slices of constant r are asymptotically spacelike, the charges are evaluated as in-tegrals over ˆ φ, θ at future spacelike infinity, and are “conserved” in the sense that theydo not depend on ˆ t . The entropy of the cosmological horizon is given by the usualBekenstein-Hawking relation S cosm = Area4 = π ( r c + a )Ξ (2.6)where we have set G = 1. The first law of thermodynamics for the cosmological horizonbecomes d Q ∂ ˆ t = T H dS cosm + Ω H d Q ∂ ˆ φ (2.7)where T H is the Hawking temperature of the cosmological horizon and Ω H is the angularvelocity at the horizon. In the Nariai limit r + = r c the variation of the entropy can beexpressed completely in terms of a variation of the angular momentum and takes thefollowing form: dS cosm = β L d Q ∂ ˆ φ (2.8) It is amusing to note that these near horizon geometries at fixed polar angle are all present intopologically massive gravity [32, 33, 34], where they were called warped AdS, warped dS, and warpedflat space respectively. β L = 1 /T L is the chemical potential associated to the angular momentum. Werefer to T L as the left moving temperature. Explicitly one finds T L = ( r c + a ) πar c Ξ (6 r c /ℓ + 3 r c /ℓ − r c + a )(1 + r c /ℓ ) . (2.9) The rotating Nariai geometry [14] is obtained when the cosmological and outer horizonscoincide, r c = r + . In this case, the Kerr-de Sitter solution becomes time dependentboth in the region outside the cosmological horizon as well as the region between thecosmological horizon and the inner horizon.We will take the Nariai limit r + → r c and the near horizon limit simultaneously.This is the Nariai analog of the near-NHEK limit of extremal black holes considered in[20, 19]. Define the non-extremality parameter τ = r c − r + r c . (2.10)For small τ , the Hawking temperatures of the outer and cosmological horizons are both T H ≈ bτ π , (2.11)where b = r c ( r c − r − )(3 r c + r − ) ℓ ( a + r c ) = r c ( − a /ℓ + 6 r c /ℓ )( r c + a ) . (2.12)The near-horizon coordinates are t = bλ ˆ t , r = r c − ˆ rλr c , φ = ˆ φ − Ω c ˆ t , (2.13)where Ω c = Ξ ar c + a (2.14)is the angular velocity of the cosmological horizon. Taking λ → , τ → τ /λ, t, r, φ held fixed, we find the rotating Nariai metric [14] ds = Γ( θ ) (cid:18) r ( r − τ ) dt − dr r ( r − τ ) + α ( θ ) dθ (cid:19) + γ ( θ )( dφ + krdt ) , (2.15)with φ ∼ φ + 2 π , r ∈ (0 , τ ), andΓ( θ ) = ρ c r c b ( a + r c ) , α ( θ ) = b ( a + r c ) r c ∆ θ , γ ( θ ) = ∆ θ ( r c + a ) sin θρ c Ξ , (2.16) k = − ar c Ξ b ( a + r c ) , ρ c = r c + a cos θ . + I − r=1 r=1r=1r=1r=-1r=-1 r=-1r=-1 Fig. 1:
Penrose diagram of dS with the cosmological horizons explicit. The left and right sidesare identified. The near horizon geometry is therefore a fiber over dS . The coordinate change r → τ r + 1) , t → τ t , φ → φ − kt (2.17)puts the dS base into the familiar static coordinates ds = Γ( θ ) (cid:18) − (1 − r ) dt + dr − r + α ( θ ) dθ (cid:19) + γ ( θ )( dφ + krdt ) , (2.18)with r ∈ ( − , U (1) × SL (2 , R ) generated by K = ∂ φ (2.19)¯ K = ∂ t ¯ K = r sinh t √ − r ∂ t + cosh t p − r ∂ r − k sinh t √ − r ∂ φ ¯ K = r cosh t √ − r ∂ t + sinh t p − r ∂ r − k cosh t √ − r ∂ φ . In the Penrose diagram shown in Fig. 1, static coordinates cover the patch inside thedotted lines. The same metric with r > Observers in the rotating Nariai geometry find This is equivalent to taking τ < r ∈ (0 , ∞ ) in the metric (2.15), which is obtained as the near-horizon limit of Kerr-dS starting outside the cosmological horizon rather than between the two horizons.This is the dS analog of the thermal “near-NHEK” coordinates [20]. r = ±
1. The entropy associated to thecosmological horizon of the rotating Nariai geometry is S Nariai = Area4 = π ( r c + a )Ξ , (2.20)which is precisely the cosmological entropy in the full Kerr-de Sitter geometry. Theboundaries reside at I ± and are spacelike for all values of θ . Note that constant r hyperslices are spacelike for large r and thus r is a timelike coordinate for large enoughvalues.The global coordinates defined bytan( η/
2) = tanh (cid:16) sinh − [ p − r sinh t ] (cid:17) (2.21)cos ψ = r (cosh t − r sinh t ) − / ϕ = φ + k (cid:18) sin( η + ψ )sin( η − ψ ) (cid:19) with ψ ∼ ψ + 2 π , η ∈ ( − π/ , π/
2) cover all of dS , with metric ds = Γ( θ ) (cid:18) − dη + dψ cos η + α ( θ ) dθ (cid:19) + γ ( θ )( dϕ + k tan ηdψ ) . (2.22)This coordinate change is useful to understand the dS Penrose diagram, but below wewill work in the coordinates (2.18).
For completeness we also include the near horizon geometry in the ultracold limit wherethe inner, outer and cosmological horizons coincide [14]. (We do not know of any wayto identify the ultracold geometry with a CFT.) In this limit, the parameter b ∝ r c − r − in (2.12) and (2.16) goes to zero, so we must rescale coordinates appropriately. Startingwith the static metric (2.18) on rotating Nariai, defining the ultracold coordinates r = ˜ r √ b , t = ˜ t √ b , (2.23)and taking b → ds = ˜Γ[ θ ] (cid:2) − d ˜ t + d ˜ r + ˜ α ( θ ) dθ (cid:3) + γ ( θ ) (cid:16) dφ + ˜ k ˜ rd ˜ t (cid:17) (2.24)with ˜Γ = b Γ , ˜ α = α/b , ˜ k = bk . (2.25)The ultracold geometry is a fibered product of two-dimensional Minkowski space andthe two-sphere. We can also obtain the solution as a fibered product of two-dimensionalRindler space and the two-sphere via a coordinate transformation. Which metric onMinkowski space naturally appears in the ultracold limit depends on how the limit istaken starting from the full Kerr-dS black hole.7 The Rotating Nariai/CFT Proposal
The Kerr/CFT correspondence proposes that quantum gravity in the NHEK geometryis dual to a thermal state in a 2d conformal field theory. The argument consists ofimposing a set of boundary conditions for all excitations of the geometry, finding theasymptotic symmetry group satisfying these boundary conditions and computing anyadditional central charges that may be present.We will now show that a similar argument allows us to relate quantum gravity in therotating Nariai geometry to a 2d CFT. One of the differences between the proposal wemake and that of Kerr/CFT is that the putative dual CFT lives on a spacelike manifoldgiven that constant r slices are in fact asymptotically spacelike. We take this as anindication that the dual CFT of the cosmological horizon is Euclidean. In this sense, theproposed duality lies along the same vein as the dS/CFT correspondence. Furthermore,given that there are two disconnected boundaries we are confronted with the question ofwhere the dual conformal field theory resides. We will not address this question whichhas yet to be fully understood in the Kerr/CFT case.The asymptotic symmetry group (ASG) of a theory is the set of allowed, nontriv-ial symmetries. A symmetry is allowed if it generates a transformation satisfying theboundary conditions, and nontrivial if it falls off at infinity slow enough for the asso-ciated conserved charge to be nonzero on some background. The ASG depends on theboundary conditions and on the action, which defines the conserved charges. Generallyfor a given action the window of consistent boundary conditions is small; if one choosesboundary conditions that are too loose, the conserved charges diverge, while if the bound-ary conditions are too tight, then all excitations are gauge-equivalent to the vacuum andthe theory is trivial. Boundary Conditions
We choose our boundary conditions in direct analogy to the NHEK boundary conditionsproposed in [9]. In the basis ( t, φ, θ, r ), h µν ∼ O r /r /r /r /r /r /r /r (3.1)always keeping in mind that r is asymptotically a timelike coordinate. Note that notall components of the perturbation are subleading with respect to the boundary metric.Below, we also impose a supplemental boundary condition that eliminates excitationsabove extremality. Asymptotic Symmetries
The boundary conditions are preserved by the diffeomorphisms ζ ǫ = ǫ ( φ ) ∂ φ − rǫ ′ ( φ ) ∂ r (3.2)8 ζ = ∂ τ (3.3)Expanding in a basis ǫ n = − e − inφ , the left-moving diffeomorphisms ζ ǫ give rise to a singlecopy of the Virasoro algebra i [ ζ n , ζ m ] = ( n − m ) ζ n + m , (3.4)with zero mode equal to the U (1) isometry, ζ = − ∂ φ . Charges
For each diffeomorphism there is an associated conserved charge which can be computedusing the Barnich-Brandt-Compere formalism [37, 38, 39]. In particular, one finds theconserved charges Q ζ ( h, g ) = Z I + k ζ [ h ; g ] (3.5)where k ζ is constructed from the Einstein equations, k ζ [ h ; g ] = 14 ǫ αβµν [ ζ ν D µ h − ζ ν D σ h µσ + ζ σ D ν h µσ + 12 hD ν ζ µ − h νσ D σ ζ µ + 12 h σν ( D µ ζ σ + D σ ζ µ )] dx α ∧ dx β (3.6)The integral is taken over the boundary of a constant- t slice at r → ∞ . To ensurea chiral spectrum and finite charges we impose the supplemental boundary condition Q ∂ τ ( h, g ) = 0.The Dirac bracket algebra which dictates the algebra of asymptotic symmetries isgiven by { Q ζ m , Q ζ n } = − i ( m − n ) Q ζ m + n + 18 π Z k ζ m [ L ζ n ¯ g ; ¯ g ] (3.7)Upon quantization, we transform the Dirac bracket algebra into a commutation relationallowing us to interpret the classical central charge as a quantum central charge of the dualconformal field theory. Replacing the classical charges Q ζ m by their quantum counterparts L m , we find the Virasoro algebra,[ L m , L n ] = ( m − n ) L m + n + c L
12 ( m − m ) δ m + n (3.8)with central charge c L = 3 | k | Z π dθ p Γ( θ ) α ( θ ) γ ( θ ) . (3.9)Explicitly, the central charge is c L = 12 ar c b ( a + r c ) , (3.10)which equals (1.2). Note that c L is real and positive.We therefore propose that quantum gravity in the rotating Nariai geometry is dualto a 2d CFT (or its chiral left-moving sector).9 ntropy Using the central charge (3.9) and temperature (2.9), we find that the Cardy formula forthe CFT entropy correctly reproduces the entropy of the cosmological horizon: S CF T = π T L c L = S cosm . (3.11)This is the appropriate temperature to use in the Cardy formula because it is the chemicalpotential conjugate to angular momentum, which is the zero mode of the Virasoro algebra.As discussed in the introduction, this application of the Cardy formula is speculative.There is no conclusive evidence that quantum gravity in a de Sitter background is in factunitary, given that it only appears as a metastable vacuum in string theory [40]. Fur-thermore, a positive central charge does not guarantee the unitarity of a two-dimensionalconformal field theory. Finally, it is not understood how the rotating Nariai geometrymaps to a thermal state in the CFT. Therefore, although satisfying, the above formulais somewhat numerological and requires further explanation. The agreement persistsstraightforwardly if we consider adding electric and magnetic charges, as is shown inappendix B.If the above picture is reasonable, it implies that we should treat the cosmologicalhorizon as any black hole horizon: a thermal state in the dual field theory.
We now briefly point out the difference in the behavior of scalars in the NHEK androtating Nariai geometries. In both cases, the scalar field can be separated into a productof spheroidal harmonics and radial functions, Ψ = R ( r ) S ( θ ) e − iωt + imφ . In the rotatingNariai metric (2.18), the wave equation for a scalar of mass µ is [41]1sin θ ∂ θ (∆ θ sin θ∂ θ S ) − (cid:18) ( m Ω c a sin θ − Ξ m ) ∆ θ sin θ + a µ cos θ − j lm (cid:19) S = 0 (4.1)( r − R ′′ + 2 rR ′ + (cid:18) ( ω + kmr ) r − j ℓm (cid:19) R = 0 (4.2)where ˜ j ℓm = r c ( j ℓm + r c µ ) b ( a + r c ) , (4.3)and j ℓm is a separation constant the must be determined numerically. The radial equationcan be solved exactly in terms of Whittaker functions, but we need only the large r behavior, R ( r ) ∼ r − ± β Nariai , β Nariai = 14 − k m − ˜ j ℓm . (4.4) We thank A. Strominger for discussions on this point. + β is the dimension of the dual operator, which is complex for genericmass and angular momentum.In NHEK, generalizing [8] to include a cosmological constant, R ( r ) ∼ r − ± β NHEK , β NHEK = 14 − k m + ˜ j ℓm . (4.5)Scalars in NHEK are equivalent to charged scalars in an electric field in AdS . The NHEKconformal dimension + β NHEK can be complex for certain values of the parametersand angular momentum, corresponding to the possibility of Schwinger pair productionin
AdS [42, 43].In rotating Nariai, on the other hand, the weight + β Nariai is always complex forlarge enough µ , as in the dS/CFT correspondence. The complexity of the conformalweight may indicate that the theory is non-unitary. A possibility for the theory to berendered unitary would be to consider the unitary principal series representation ratherthan the highest weight representation as in [44, 45]. The complexity of the conformalweight is related to cosmological particle production given that r becomes a timelikecoordinate for large enough values. Acknowledgements
It is a great pleasure to thank F. Denef and A. Strominger for useful conversations. Thiswork was supported in part by DOE grant DE-FG02-91ER40654.
A Penrose Diagram r c r c r c r c r n r n r n r n r − r − r c r c r − r − r c r c I + I − r c r c I − r n r n r n r c r c r c r c r n r − r c r c I + I − r n r c r c r n r n is the negative root of the equation∆ ˆ r = 0, and the left and right sides of the diagram are identified. B Charged Rotating Nariai
In this appendix we extend the computation of the asymptotic symmetry group andVirasoro central charge to include electromagnetic charge, along the lines of [13]. Thecharged rotating Nariai solution is found as a near horizon limit of the Kerr-Newman-deSitter black hole with coincident black hole and cosmological horizons, which is a solutionto Einstein-Maxwell gravity with a positive cosmological constant S grav = 116 π Z d x √− g (cid:18) R − − F (cid:19) (B.1)The metric with electric charge q e and magnetic charge q m is given by [14] ds = Γ[ θ ] (cid:20) − (1 − r ) dt + dr − r + α ( θ ) dθ (cid:21) + γ ( θ ) ( dφ + krdt ) (B.2)where Γ( θ ) = ρ c r c b q ( r + a ) , α ( θ ) = b q ( r c + a )∆ θ r c , γ ( θ ) = ∆ θ ( r c + a ) sin θρ Ξ (B.3)and φ is an angular coordinate with periodicity φ ∼ φ + 2 π . We have further defined ρ c = r c + a cos θ, b q = 6 r c /ℓ + 3 r c /ℓ + q /ℓ − r c ( r c + a )(1 + r c /ℓ ) , k = − ar c Ξ b q ( r c + a ) , (B.4)with q = q e + q m . The gauge field is given by A = f ( θ ) ( dφ + krdt ) (B.5)with f ( θ ) = ( r c + a )[ q e ( r c − a cos θ ) + 2 q m ar c cos θ ]2 ρ c Ξ ar c . (B.6)It is a straightforward extension of the non-charged case to find that in the limit where theblack hole and cosmological horizons coincide, the first law for the cosmological horizonreads dS cosm = β L d Q ∂ φ + Φ e d Q e + Φ m d Q m (B.7)where β L = 2 π | k | . The computation of the asymptotic symmetry group proceeds asbefore, resulting in a single Virasoro algebra with central charge c L = 12 r c p ( r c − r c /ℓ − q )(1 + r c /ℓ ) − r c /ℓ + 3 r c /ℓ − q /ℓ , (B.8)and a similar entropy matching formula S CF T = π c L T L = S cosm . (B.9)12 eferences [1] G. W. Gibbons and S. W. Hawking, “Cosmological Event Horizons, Thermodynamics,And Particle Creation,” Phys. Rev. D , 2738 (1977).[2] A. Strominger, “The dS/CFT correspondence,” JHEP , 034 (2001)[arXiv:hep-th/0106113].[3] S. Hawking, J. M. Maldacena and A. Strominger, “DeSitter entropy, quantum entan-glement and AdS/CFT,” JHEP , 001 (2001) [arXiv:hep-th/0002145].[4] M. Banados, T. Brotz and M. E. Ortiz, “Quantum three-dimensional de Sitter space,”Phys. Rev. D , 046002 (1999) [arXiv:hep-th/9807216].[5] T. Banks, “Some Thoughts on the Quantum Theory of de Sitter Space,”arXiv:astro-ph/0305037.[6] T. Banks, “More thoughts on the quantum theory of stable de Sitter space,”arXiv:hep-th/0503066.[7] T. Banks, B. Fiol and A. Morisse, “Towards a quantum theory of de Sitter space,”JHEP , 004 (2006) [arXiv:hep-th/0609062].[8] J. M. Bardeen and G. T. Horowitz, “The extreme Kerr throat geometry: A vacuumanalog of AdS(2) x S(2),” Phys. Rev. D , 104030 (1999) [arXiv:hep-th/9905099].[9] M. Guica, T. Hartman, W. Song and A. Strominger, “The Kerr/CFT Correspon-dence,” arXiv:0809.4266 [hep-th].[10] J. D. Brown and M. Henneaux, “Central Charges in the Canonical Realization ofAsymptotic Symmetries: An Example from Three-Dimensional Gravity,” Commun.Math. Phys. , 207 (1986).[11] V. P. Frolov and K. S. Thorne, “Renormalized Stress - Energy Tensor Near theHorizon of a Slowly Evolving, Rotating Black Holes,” Phys. Rev. D , 2125 (1989).[12] H. Lu, J. Mei and C. N. Pope, “Kerr/CFT Correspondence in Diverse Dimensions,”JHEP , 054 (2009) [arXiv:0811.2225 [hep-th]].[13] T. Hartman, K. Murata, T. Nishioka and A. Strominger, “CFT Duals for ExtremeBlack Holes,” JHEP , 019 (2009) [arXiv:0811.4393 [hep-th]].[14] I. S. Booth and R. B. Mann, “Cosmological pair production of charged and rotatingblack holes,” Nucl. Phys. B , 267 (1999) [arXiv:gr-qc/9806056].[15] H. Nariai, Sci. Re. Tohoku Univ. Ser. 1 , 160 (1950). H. Nariai, “On a newcosmological solution of Einstein’s field equations of gravitation,” General Relativityand Gravitation , 963 (1999). 1316] P. H. Ginsparg and M. J. Perry, “Semiclassical Perdurance Of De Sitter Space,”Nucl. Phys. B , 245 (1983).[17] V. Cardoso, O. J. C. Dias and J. P. S. Lemos, “Nariai, Bertotti-Robinsonand anti-Nariai solutions in higher dimensions,” Phys. Rev. D , 024002 (2004)[arXiv:hep-th/0401192].[18] R. Bousso, A. Maloney and A. Strominger, “Conformal vacua and entropy in deSitter space,” Phys. Rev. D , 104039 (2002) [arXiv:hep-th/0112218].[19] A. Castro and F. Larsen, “Near Extremal Kerr Entropy from AdS Quantum Grav-ity,” arXiv:0908.1121 [hep-th].[20] I. Bredberg, T. Hartman, W. Song and A. Strominger, “Black Hole SuperradianceFrom Kerr/CFT,” arXiv:0907.3477 [hep-th].[21] A. J. Amsel, G. T. Horowitz, D. Marolf and M. M. Roberts, “No Dynamics in theExtremal Kerr Throat,” arXiv:0906.2376 [hep-th].[22] O. J. C. Dias, H. S. Reall and J. E. Santos, “Kerr-CFT and gravitational perturba-tions,” arXiv:0906.2380 [hep-th].[23] F. Belgiorno, S. L. Cacciatori and F. D. Piazza, “Quantum instability forcharged scalar particles on charged Nariai and ultracold black hole manifolds,”arXiv:0909.1454 [gr-qc].[24] M. Cvetic and F. Larsen, “Greybody Factors and Charges in Kerr/CFT,” JHEP , 088 (2009) [arXiv:0908.1136 [hep-th]].[25] T. Hartman, W. Song and A. Strominger, “Holographic Derivation of Kerr-NewmanScattering Amplitudes for General Charge and Spin,” arXiv:0908.3909 [hep-th].[26] R. Bousso, “Positive vacuum energy and the N-bound,” JHEP , 038 (2000)[arXiv:hep-th/0010252].[27] R. Bousso, O. DeWolfe and R. C. Myers, “Unbounded entropy in spacetimes withpositive cosmological constant,” Found. Phys. , 297 (2003) [arXiv:hep-th/0205080].[28] E. Witten, “Quantum gravity in de Sitter space,” arXiv:hep-th/0106109.[29] N. Goheer, M. Kleban and L. Susskind, “The trouble with de Sitter space,” JHEP , 056 (2003) [arXiv:hep-th/0212209].[30] R. Bousso, “Quantum global structure of de Sitter space,” Phys. Rev. D , 063503(1999) [arXiv:hep-th/9902183].[31] R. Bousso and S. W. Hawking, “(Anti-)evaporation of Schwarzschild-de Sitter blackholes,” Phys. Rev. D , 2436 (1998) [arXiv:hep-th/9709224].1432] D. Anninos, W. Li, M. Padi, W. Song and A. Strominger, “Warped AdS BlackHoles,” arXiv:0807.3040 [hep-th].[33] D. Anninos, M. Esole and M. Guica, “Stability of warped AdS3 vacua of topologicallymassive gravity,” arXiv:0905.2612 [hep-th].[34] D. Anninos, “Sailing from Warped AdS to Warped dS in Topologically MassiveGravity,” arXiv:0906.1819 [hep-th].[35] M. H. Dehghani, “Quasilocal thermodynamics of Kerr de Sitter spacetimes and thedS/CFT correspondence,” Phys. Rev. D , 104030 (2002) [arXiv:hep-th/0201128].[36] A. M. Ghezelbash and R. B. Mann, “Entropy and mass bounds of Kerr-de Sitterspacetimes,” Phys. Rev. D , 064024 (2005) [arXiv:hep-th/0412300].[37] G. Barnich and F. Brandt, “Covariant theory of asymptotic symmetries, conserva-tion laws and central charges,” Nucl. Phys. B , 3 (2002) [arXiv:hep-th/0111246].[38] G. Barnich and G. Compere, “Surface charge algebra in gauge theories and thermo-dynamic integrability,” J. Math. Phys. , 042901 (2008) [arXiv:0708.2378 [gr-qc]].[39] G. Compere, “Symmetries and conservation laws in Lagrangian gauge theories withapplications to the mechanics of black holes and to gravity in three dimensions,”arXiv:0708.3153 [hep-th].[40] S. Kachru, R. Kallosh, A. Linde and S. P. Trivedi, “De Sitter vacua in string theory,”Phys. Rev. D , 046005 (2003) [arXiv:hep-th/0301240].[41] T. Tachizawa and K. i. Maeda, “Superradiance in the Kerr-de Sitter space-time,”Report of Sci. and Eng. Res. Lab, Waseda Univ. 92-9 (1992).[42] B. Pioline and J. Troost, “Schwinger pair production in AdS(2),” JHEP , 043(2005) [arXiv:hep-th/0501169].[43] S. P. Kim and D. N. Page, “Schwinger Pair Production in dS and AdS ,” Phys.Rev. D , 103517 (2008) [arXiv:0803.2555 [hep-th]].[44] A. Guijosa and D. A. Lowe, “A new twist on dS/CFT,” Phys. Rev. D , 106008(2004) [arXiv:hep-th/0312282].[45] D. A. Lowe, “q-deformed de Sitter / conformal field theory correspondence,” Phys.Rev. D70