JHEP04(2020)124 action q Springer Z April 2, 2020 June 17, 2019 April 21, 2020 : , : : January 18, 2020 , : 4 correspondence. 2 Accepted Received Published Revised /CFT c,d 3 1 permits the construction of an q > the four-dimensional geometry reduces Published for SISSA by https://doi.org/10.1007/JHEP04(2020)124 → ∞ and Per Sundell q c [email protected] quotient. The set of fixed points of the , q by means of a double Wick rotation. In this limit, the Z 3 Rodrigo Olea c [email protected] , . 3 spacetime, where the two original minimal surfaces can be mapped to 3 1906.05310 The Authors. Felipe Diaz, Conformal Field Theory, Models of Quantum Gravity, Conformal and holography c

a,b We model the back-reaction of a static observer in four-dimensional de Sitter , 3 [email protected] , whereupon the Gibbons-Hawking entropy arises as the corresponding modular 1 − q = E-mail: [email protected] Appelstraße 2, 30167 Hannover, Germany Departamento de Ciencias Universidad F´ısicas, AndresSazi´e2212, Bello, Piso 7, Santiago deDepartment Chile of Physics, Nanjing22 University, Hankou Road, Nanjing, China Department of Mathematics, UniversityDavis of CA California, 95616, U.S.A. Riemann Center for Geometry and Physics, Leibniz Universit¨atHannover, b c d a Open Access Article funded by SCOAP whose total central charge equals that computed usingKeywords: the dS W Symmetry, Gauge-gravity correspondence ArXiv ePrint: T entropy. We further observe that into the limit that of global dS the future and past infinitiesLiouville of theories dS on the minimal surfaces become boundary theories at zero temperature geometry. The introduction of aneffective action orbifold for parameter the bulkThe gravity effective theory action with corresponds support to onvacuum that each expectation of of value Liouville these of field minimal theory the surfaces. yields on a Liouville thermal a field. Cardy 2-sphere entropy The with that a intrinsic we reintrepret finite Liouville as theory a modular description free energy at temperature Abstract: spacetime by means ofconsists a of a singular pair of codimension two minimal surfaces given by 2-spheres in the Euclidean Cesar Arias, Liouville description of conicalGibbons-Hawking defects entropy in as dS modularand entropy, dS JHEP04(2020)124 2 ] that 1 spacetime only as 4 11 as the fundamental q Z / 4 5 9 7 3 ] arises from the entanglement between the 2 11 8 – 1 – 6 ], we model the back-reaction of a static observer in . That is, we think of observers that back-react with 5 q – Z 3 / 12 holography 3 4 3 4 15 central charge 2 1 limit and dS action. In this sense, we treat the orbifold dS 13 q q Z /CFT 3 One of the central ideas behind the above argument is that in order to measureFollowing any this idea and motivated by the conically singular geometries induced by spacetime via the quotient dS 4.2 dS 3.3 Central charge3.4 and Cardy formula Modular free energy and Gibbons-Hawking entropy 4.1 3D conformal boundaries from codimension two defects in 4D 2.1 Massless probe2.2 observers Massive observers and antipodal defects 3.1 Effective two-dimensional3.2 action On-shell correspondence with Liouville theory 4 point particles in three dimensionsdS [ the background geometry as inducingpoints codimesion of two the defects thatmanifold correspond on to which the the fixed gravity theory is formulated and think of the dS the dS interior. observer-dependent quantity, in particularhorizon, the one thermal has properties to ofother go the words, beyond the dS the observer cosmological standard back-reaction probe should be approximation of taken into a account. static observer. In The non-trivial topology of deboundaries Sitter and causally (dS) disconnected spacetime interior comprises regions.the two Gibbons-Hawking Recently, entropy disconnected it of spacelike has dS been spacetimepast argued [ [ and future conformalantipodal infinities and or, causally alternatively, disconnected from bulk the observers entanglement located between at two opposites Rindler wedges of 1 Introduction 5 Conclusions A Liouville theory 4 The large 3 Liouville theory description of a massive observer Contents 1 Introduction 2 Static observers in dS JHEP04(2020)124 (2.1) (1.1) . This ~ , the four- ∼ 4 Z 2 and the four- / γ 4 ` admits an intrinsic 4 1 limit as its massless radius . Taking the embedding 4 → 4 1. Consequently and by , geometry. q 1 4 ) predicts a Cardy entropy ]. → , M , where the two minimal sur- 2 3 1.1 12 q ) i – 6 dX ( . 4 2 4 ≤ i ` limit of the quotient dS G 3 X ≤ 1  + q 1 2 → ∞ boundaries inherit a central charge from the ) ) can be viewed as a four-dimensional timelike − 0 q 4 3 – 2 – 1 1 permits to build up a reduced two-dimensional 4, and considering the Minkowski metric correspondence [  dX ( 2 = q > − , and we shall think of the q ,..., c , with the Liouville coupling constant = 4 4 , /CFT = 0 G 1 3 action. Each of these minimal and tensionful surfaces have 4 µ 2 M q , Z 4 ds , 1 limit, the two dS ∈ M 1. As we shall argue, the resulting effective two-dimensional Euclidean µ massive observer → ∞ → X q q -dependent vacuum expectation value. q -dependent central charge arguably encodes degrees of freedom associated to a mas- q We conclude by observing that in the The correspondence between the effective action of a massive observer and the Liouville The aim of this note is to show that massive observers in dS In four dimensions, de Sitterhypersurface spacetime embedded (dS in five-dimensional Minkowskicoordinates space to be see, the total centralderived charge in of the the context two of boundaries the dS reproduces exactly the central2 charge Static observers in dS faces of the former canboundaries be of mapped the — latter. viafaces In a become this double boundary limit, Wick the theories rotationupon two (one — Liouville taking to for theories the the each on two boundary) theLiouville conformal at theory bulk on zero minimal the temperature. sur- corresponding minimal Moreover, surface in one higher dimension. As we shall means of the thermalthat Cardy equals formula, a modular theduces free central the energy charge Gibbons-Hawking whose ( area corresponding law. modular entropy correctly repro- dimensional geometry reduces to the global geometry of dS This sive observer which are not present in the massless limit acquires theory action links thedimensional gravitational Newton’s parameters, constant namelyrelation the results dS in a semiclassical central charge given by action functional with support onthe the set pair of of fixed codimension points twothe minimal of surfaces topology the that of define aof 2-sphere as in the the “worldvolume”tensionless Euclidean of limit geometry a and massive observer, theyaction whose can can massless be be limit identified formally with is thought a equivalent Liouville to theory the on a 2-sphere, in which the Liouville field static observer as a probe limit in which one recovers the original, non-singulardescription dS in terms ofintroduction a of two-dimensional an conformal orbifold field parameter theory. We shall argue that the a smooth limit of it. When such back-reaction is taken into account, we shall refer to the JHEP04(2020)124 (2.7) (2.5) (2.6) (2.2) (2.3) (2.4) , 4 , ≤ ) with manifest i static observer 3 t/` ≤ S π . 2 × 2 R , single cosh( i . This has the fix time ˆ y θ φ < r H , ≤ 2 2 = ˆ cos φ , i 0 Ω 2 d ξ 2 sin r ]. − θ 1 . π , 2 ,X is the metric on the unit 2-sphere. + ˆ is obtained by parametrizing the ` 2 ) 4 2 2 sin ≤ 2 2 4 ˆ ` r G p ` Ω π` 2 θ ˆ t/` d ˆ r ) can be alternatively solved by param- = = − d , = ≤ 1 4 1 2 0 ` 2.2 ) has the topology of and dS cosh( + S 2 = . quotient. This construction, which we shall 2 2.2 ˆ r ` ˆ t µ q d , φ , X – 3 – − ˆ = r < ` Z X ,X  2 µ r ) 1 π` ` 2 2 ≤ cos ˆ 2 ` r X ) describes the worldline of a θ p t/` − = ` < ξ < ` , 2.4 = 1 sin − 1 dS `  sinh( T = − , θ 3 ] given by = ,X ∞ 2 cos ) 2 2 ds ξ ˆ t/` = 0. The observer is causally connected with only part of the full θ , X − < t < r 2 ` radius. The hyperboloid ( sinh( cos 2 4 p ξ ˆ r − ∞ = = − denote the coordinates of the unit 2-sphere. The resulting line element 0 2 2 ` i X X y p is the dS hypersurface is defined by 2 = ` 4 0 1) symmetries. To begin with, we note that the constraint ( In the Euclidean vacuum, a static observer detects a temperature and a corresponding The time-independent metric ( , X (4 where modify the local geometry ofmassive the observer spacetime; we by propose means tonow of model briefly the a review, back-reaction singular has of been such spelled out in fulleterizing detail the in embedding [ coordinates as 2.2 Massive observers andThe antipodal above defects characterization ofa a static massless observer probe in object. dS spacetime Here, considers instead, the observer we as treat an observer as a massive object which topology of a 2-sphere and is located at ˆ Gibbons-Hawking entropy [ where the radial coordinate runs from 0 located at the originspacetime. ˆ Such region isits dubbed boundary the defines Rindler an wedge (or observer-dependent static cosmological patch) horizon of the observer, and where the ˆ 2.1 Massless probe observers The standard descriptionembedding of coordinates a as static observer in dS where O the dS JHEP04(2020)124 ) q ∈ Z and 2.9 / τ = 0, 4 (2.9) (2.8) } (2.10) (2.11) sector 2 ,π 0 2 | 0 S :=dS w 4 , respectively. θ < π/ c dS . defined in ( . , π , ). We interpret ≤ 2 2 S # ` ξ S 2 0 O 2 2 2.9 { R ` = 1 and ξ − dξ 2 1 ∈ = ,π − dξ and , respectively. 0 2. The global polar co- | 1 + π N , is N w π/ 2 R 4 = + O = 0)  g dt 2 θ =  dt 2 2 π, ξ τ of both northern and southern ` ξ , where the 2-sphere has radius  Θ). In the conformal time 2 = 2  2 S ` S ξ − interior: , τ, θ R dS 1 O 4  2 − ∪ •  g w geometry into that of a Thurston’s 1 2 := ( × N −  = 0 and Θ = 2 w S S 2 R S θ = − + O S R " space, with the extended radial coordinate  2 θ are located at + − 2 2 I I 1. The four-dimensional orbifold  – 4 – , spindle I N g N 2 q > , g R . ` R 2 ) + cos 1 = • 2 ∈ , with coordinates ( dφ 4 4 orbifold. This is done via the discrete identification N θ b g satisfy the holonomy conditions O q 2 2 θ dφ ). This geometry describes the worldline of two antipodal q θ = 0) Z 2 ) covers the union / sin 2 2.7 , ξ S + 2.8 + sin 2 = cos = 0 2 dθ θ w dθ ( = . The location of the two antipodal observers 2 } ` := ( ] defines the north and south poles by the points Θ = 0 line element, that we shall simply denote by π 4 , π N = ) has the warped product form ≤ [0 4 O ) covers the two Rindler wedges of the dS spindle g g 2.8 ∈ 2.8 denotes the radially extended dS < θ ]. The latter geometry has two antipodal conical singularities at the points , with an orbifold parameter 2], the future and past infinities ), as indicated in ( π 2 . Penrose diagram of dS q  2 13 2 [ π/ , which are precisely the locations of the two static observers ( π/ ) by performing a `, ` + + { , π , − 2 φ The azimuthal identification deforms the In order to incorporate the observers back-reaction, one next deforms the ( 2.8 = π/ ∼ = 0 ∈ and dS S − spindle θ is then endowed with the metric where the warp factor and the other). The foliationRindler ( wedges, as depicted in figure in ( φ ξ static observers which are causally disconnected (as any light ray can not be sent from one observer into The metric ( ` R coincides with the global north and south poles Θ = The resulting dS Figure 1 [ ordinates Θ The metric ( JHEP04(2020)124 and and (3.1) (2.15) (2.12) (2.16) (2.13) (2.14) )). In N N . O . h. 2.13 viz − √ y defects 2 d , S π ) as Σ = Z θ , h q

4 S −T c dS h , both endowed with the in- − := S √ S y ) and (Σ Σ 2 d , h , N N and Σ . Σ ,  ]  , 2 Z N 2 q 1 g . q 14 S [ =0 ij − = θ

h −T S ∼ = 1 4 q ,π   E S T 2 c dS =0 6 4 Σ geometry. ` 1). . These surfaces are the set of fixed points of θ – 5 – are codimension two minimal surfaces with an =

S 1 G 4 action and Σ − 4 − ∼ = := 4 ij S b g ) is a strong sign of the existence of an underlying q q R T N E N N Z = =  Σ Σ q g and Σ h 2.16 T − and Σ N √ , x N 4 S d Σ ) S (which corresponds to the analytic continuation of ( ∪ Σ 2 2 ∪ ” denotes Euclidean geometry. Hereafter, we shall drop this label N N Ω Z E d 1 corresponds to the tensionless limit in which one recovers the usual (Σ \ = Σ 4 → F c dS 4 q 1 πG 16 ] = action. Each of them contain the worldline of one of the massive observers 4 q Hence, by construction, Σ In terms of the gravity action and in order to have a well defined variational principle, The set of fixed points under the Z c dS [ , and they both have the topology of a 2-sphere in the Euclidean geometry, . The two Nambu-Goto terms are coupled through the tension I S S with induced metric the above, the labelwhen “ is clear from context. 3 Liouville theory descriptionIn of this a section, we massive construct observer mension an two effective minimal two-dimensional action surfaces withthe Σ support on the codi- O The localized stress energyfield tensor ( theory definedcorresponds on to the an two Eucliedan Liouville minimal theory surfaces. on a 2-sphere. As we shall next argue, this theory where the limit Einstein-Hilbert action on the smooth dS induced stress-energy tensor given by In the above, the supportΣ of the first integral excludes the location of the defects Σ Nambu-Goto terms with support on Σ duced metric In what follows, we shall refer to the submanifoldsthe (Σ two conical singularities are resolved by adding to the Einstein-Hilbert action a pair of observer, with a mass proportional to ( defines two antipodal, codimension two surfaces Σ these singularities as the response of the background geometry to the presence of a massive JHEP04(2020)124 ) ) τ, ξ 2.11 (3.8) (3.5) (3.6) (3.7) (3.2) (3.3) (3.4) ) and = ( 3.4 y , R , h ] S √ h . ) on the conically -equation) and by y [Σ 2 √ d φφ y NG 2 S 2.14 ” to indicate on-shell I , d ). This reduction holds , Σ ] by dimensional reducing ≈ Z S S  Σ ] + S 4 2 6 Z 2.13 S ` [Σ , 2 . q `  , qG [Σ ij . T ], we further assign to each of − 8 NG S d ] 1 I R 2 R [ S and Σ R − I [Σ θ h d S  ] := [Σ 2 N 2 ] + √ S g I ] = y eff N ] := 2 √ I S [Σ + S d ] is the intrinsic two-dimensional scalar x [Σ (and likewise the  4 h [Σ and Σ [Σ [ NG ] + d N Z d ). This gives = 3 cos NG θθ eff ) 2 R N N I 4 g S [Σ ij 2 d Σ = [Σ θ, φ 2 ` R qG ∪ ] + ,I I ) can be computed using the line elements ( ,I = 3 ) 4 eff 4 N ] – 6 – R Z I . R h , I N (Σ 3.4 θθ c dS 3.2 \ [ ≈ h √ 4 R [Σ ) ≈ − viz −→ ] y 2 √ in ( ] dim red 4 c 2 ` dS 4 y 4 bulk ] d NG d 3.7 2 I 4 2 c I dS consists of a bulk piece plus a pair of two-dimensional d [ I c dS N 1 [ πG c q Σ dS N [ ] = ] + Z Z Σ 16 E 4 total q / N bulk Z I 4 I − T bulk c 4 dS [Σ I [ 2 d 2 ` qG I ] := ] := 8 E total 4 := dS N I − 4 c ] = dS [Σ [ N c dS ] := NG [Σ bulk I N I eff I [Σ d 2 I denoting the Euclidean time), and τ down to two dimensions Due to the antipodal symmetry relating Σ The reduced Euclidean action Although the support of the bulk integral above excludes the location of the defects, bulk making use of the codimension two identity the defects half of the total inflow ( where the integral is(with over the two-dimensionalof submanifold curvature built coordinatized up by fromupon the imposing induced Einstein’s metric equations on the defects ( equalities.) and integrating out the spindle coordinates ( and such that the total on-shell action ( (From here and in what follows, we shall use the notation “ as to define an effectivethe action corresponding on Nambu-Goto each term, of the defects, which comprises the inflow ( we can define a “freeI energy inflow” from the bulk to Σ where the Euclidean integrals To begin with, wesingular recall manifold that theNambu-Goto total terms Euclidean gravity action ( 3.1 Effective two-dimensional action JHEP04(2020)124 ; π 2 (3.9) − ) with (3.12) (3.10) (3.11) ) differs 3.9 2.16 is the only , which in turns ~ i 2 . ∼ TT h ,  h 2 Φ ) closely resembles √ γ γ y 2 2 3.9 d , ). 0 N πµe , given by a broken phase Φ Σ ). It is important to point 3.10 γ S Z , as required for conformal 2 . γ A  ) by an overall factor of Φ + 4 ij q 1 + πµe δI R δh 2 1 -dependent central charge which and Σ − A.1 q − Q π − h 1 4 γ N , √  − = 0 = 4 Φ + =  j =Φ 1 G i q 1 Q ∂ 4 Φ h Φ ij CFT

i − ), upon giving to the Liouville field a fix + L ∂ ) corresponds precisely to the Liouville ac- 1 ] where the dimensional reduction of Einstein gravity I ij , h R  16 g N 3.9 ≈ h – 7 – 4  ,T ] g 1 √ G ). N ij y 4 √ 2 δI ) holds provided y δh [Σ d 2 h , 3.10 d N eff 2 ) = (Σ √ I 0 2 3.11 Σ , g Z Φ M 2 = 2 4 Z ) on the northern defect is given by Q ]: M 2 1 2 ij ) and ( ` grav qG 3.5 = 15 T − . That is 8 0 4 3.9 ) differs from the one given in ( 2 ]. This on-shell relation permits to establish the existence of an ] = ` qG S ≈ − γ = Φ 8 propagates when computing the operator product expansion ] ) is a two-dimensional Euclidean manifold and 3.10 [Σ i ) thus requires the normalization implemented in ( π N Φ; Φ 2 , g eff h 2 A.1 I g, [Σ − [ ]. L M eff I S I [Σ eff ) on the 2-sphere ( I while comparing ( 1 3.10 for The on-shell correspondence ( Indeed, the reduced effective action ( A similar idea has been previously discussed in [ We recall that in our conventions the definition of the gravitational stress energy tensor ( 2 1 The overall factor of produces a relative factor inthe the Liouville central action charge. ( Comparison of the effectiveto gravitational two-dimensional action Liouville ( theory ishorizons. proposed to describe the underlying degrees of freedom of black hole from the standard convention used in the CFT context: of Liouville theory. Asnumber we of shall compatibility now conditions see, thatwe the propose in effective turn encode field yield the theoretic degrees a description of encode freedom associated a to a massive observer. expectation value and similarly for effective field theory on each of the minimal surfaces, Σ out that the actionsuch ( a normalization istensor needed in order to uniformize the definitiontion of ( the stress-energy In the above, ( coupling constant of theand theory; further its defines strength theinvariance dictates background (for charge the a to classical brief be review and of quantum Liouville regimes theory see appendix 3.2 On-shell correspondenceWe with now Liouville observe theory that thethe Liouville structure theory of action the [ reduced effective action ( idem so that the effective action ( JHEP04(2020)124 . ) is q ] = 2 3.9 h − [ . ` (3.16) (3.17) (3.13) (3.14) (3.15) N R = 2 of Σ R 2 1 ensures the ) contribution − 2 context, where ` 2 /γ q > , to be greater than 1, then the central ) and = 2 (1 q  O R q ) provides a nontrivial /CFT q > 2.16 3 1 − , 1 3.15  = 0 , the bound 0 ) requires 1 -dependent and vanishes in the exp Φ − q which in turns defines (up to a γ γ 2  4 3.14 q . , there exists a + . ) is , space. 1 q ) it follows that positivity of − 4 /G γ 2 2 4 − 4 2 must satisfy the Liouville equation of 2 ) yields πγµe 6 ` 1 G ` γ ` γ G = 3 0 2 3.17  3.13 ∼ ≈ 4 + 8  Q 3.13  2 q 1 , in terms of which the central charge of the q 1 Q 1 2 Q γ πG ` Q 2 ), we straightforwardly find − 8 − – 8 – 1 1 = = ) and (   0 3.15 ). -dependence of the central charge of the boundary Φ = = q γ = 1 + 6 2 3.12 q c 3.15 c γ 1 for the orbifold parameter can be understood as a Q ] , µ 1, where thus 0, which according to ( in ( πγµe 17 is the trace of stress-energy tensor ( 1) ` q > .  ij − + 8 2 µ < 1 induces a T qG q γ ( and R q ij γ ∼ h Z Q γ 2 / 3 G 1. This is similar to what occurs in the AdS = = T 0 → Φ q ], with the Brown-Henneaux central charge being recovered by rescaling the 0, which indicates unitarity of the theory. in dimension four) the entropy of the dS 22 , where – > ). In addition, the expectation value Φ π R q 18 0 is only possible if c c 12 3.10 > Is it worth to notice that the central charge ( In what follows, we shall see that the semiclassical limit of ( Observe that the bound Compatibility of the equations ( = 2 i − ` T tensionless limit the bulk orbifold AdS theory [ Newton’s constant as This value of the central chargeh is consistent with the classicalthe conformal curvature anomaly of equation the correspondingcharge defect. Also, we note that since whose value can be computed in termsIndeed, of from the the gravity couplings semiclassical and the limit orbifold of parameter ( 3.3 Central chargeIn and the Cardy semiclassical regime formula to the Liouville central charge [ reality of the couplings link between the Liouville couplingtheory constant is defined, andfactor of the gravitational coupling consistency condition: one the2 one hand, fromone. ( On the other hand and remembering that and and ( motion for a constant field, which is given by where the first equality made use of the constant positive curvature as follows from matching the terms of the same order in derivatives of the metric in ( JHEP04(2020)124 ) qH − (3.21) (3.24) (3.18) (3.19) (3.20) (3.22) (3.23) )), the -dependent A.13 q ], the Cardy 25 whose derivative q ]). gives the Gibbons- F 28 /q ), we find the label left and right-movers , ] yields the Gibbons-Hawking , π 1 R 2 , ](see also [ . . 3.18 R 30 1 q T , = induces a Boltzmann factor exp( ρ qH 27 . , and 29  q − R 2 , q,R [ 4 e T c q 26 L 1 [ G c π` 2 qH − = 3 log tr = π R q −  ∼ q q 1 e L q 1 H 0 + = − L ∆ − T T = ) and by virtue of the thermal Cardy formula = tr 1 . 3 , q )) which makes the Liouville theory thermal and thus ).  Z – 9 – ,T q,L 1 ρ correspond to the temperature of the generalized ρ , ρ c q 3.17 = c 2 3.22 R 3.18 log 3  π T log = q 1 = tr 4 2 = − − ` G Cardy Z q ) in the Cardy formula ( q,R and , the orbifold parameter c = ] S := 1 ∼ L q − q = T q Cardy 3.20 F H 24 q c , S = q,L 23 c T modular free energy ) and ) can be understood as the modular free energy ) and ( ). Based on this simple observation, we shall next reinterpret the 3.17 ) applies. ) as 3.21 2.5 3.17 3.18 3.21 defined by the Rindler Hamiltonian is the (semiclassical) conformal dimension of the bound state (see ( R is given in ( 0 denoting the modular Hamiltonian ( πH q 2 c H − Hence, using ( For a non-chiral Liouville theory, we have Having obtained the central charge ( e Note that, by identifying -dependent entropy can be computed (as usual, 3 q = (with amounts to using the thermal Cardy formula ( Thus, we can write the modular partition function as in terms of which the modular free energy is given by 3.4 Modular freeThe energy Cardy entropy and ( Gibbons-Hawking entropy with respect to the dimesionless temperature area law. To this end, we define the modular Hamiltonian Note that minus the derivative ofHawking the entropy above entropy ( with respectCardy to entropy 1 ( where Hartle-Hawking vacuum of dSρ space. This is known to beCardy equivalent entropy to a thermal state where ∆ Cardy formula ( formula holds indimension the above extended zero. range Accordingly, in of our large case, since central charge and large gap in operator in the canonical ensemble [ a central charge and temperature). Indeed, based on the arguments of [ JHEP04(2020)124 (3.30) (3.31) (3.25) (3.26) (3.27) (3.28) (3.29) ]) and 4 ) has its S [ . E . Using the I 3.31 2 4 4 q − S G π` viz. action (given by  ) is equivalent to q exp( q 1 Z ≈ − 3.25 ] 1 ). 4  S [ . ]. The value of the latter is 3.21 Z q dS 1, in which one recovers the , ] = 2 Z S , 4 2 / 4 4 . S = → Z G [ π` S 2 [ 4 2 E 4 . q  I E G π` G , π` Π q ) on the 4-sphere defined by 1 ) log is the antipodal map that sends every  − / qI q q ). Observe that although ( ) ] q 2 ] = − 4 ]) q 4 q q∂ E N 4 ) comprises the contribution form both, 3.24 S 1 2 2.5 4 Z ] = − S S [ / 4  − q [ G 1 π` 4 → S Z ∪ R Z  3.27 [ S 4 ( [  E ] = E S S ≈ q 1 E 4 I = (1 – 10 – = R ] I S q q [ − q q ≈ Z ρ ), its value is independent of the modular parameter 1 ∂T F / ] := ( ∂F 4  4 1 2 tr 4 − S S ) (properly Euclideanized) restricted to a single Rindler [ -fold branched cover that we denote by [ S 3.24 ∂ = ∂q q E = Z 2 spacetime. S I q q 2.14 Σ e 4 q S = F ] q spacetime. Moreover, it naturally admits a 31 e S ] + log 4 4 q S [ Z denote the analytic continuation of the southern and northern Rindler log ]. It is important to point out that the 4-sphere ( N E 1 q 1 R − and ] = 4 E S S [ R q F Finally, we can compute the modular entropy The value of the modular free energy ( Next, we compute the modular free energy ( and hence this remains fix in the tensionless limit This is precisely the Gibbons-Hawkingorigin entropy in ( the modular freeq energy ( standard description of the dS which gives which corresponds exactly to the value of the Cardy entropy ( northern and southern defects. For a single defect (say the southern one), we thus have In the above, wethe have locality used of the thegiven gravity semiclassical by action the approximation to on-shell write valuewedge of (which ( we recall is given by a 4-sphere in the Euclidean geometry), it follows that Rindler wedge [ the analytic continuation ofcontinuation a of single global Rindler dS wedgeazimutal and identifications), also with equivalent a Calabrese-Cardy to formula the [ Euclidean Here, wedges (both given by 4-spheres),point and in Π the : southern Rindler wedge to the corresponding antipodal point in the northern JHEP04(2020)124 . π ). 0, 3 q 2 viz. → (4.3) (4.1) (4.2) 3 + 2.13 q , with shrinks . This φ ` dS ` `θ S ∼ ] and the φ := holography, ) reduces to has a higher . spindle, ) and ( , π` 4 2 , z and Σ 3 b g 2 [0 ) q , z N 4 Z 2.11 ∈ / t/` spacetime. CFT c dS 2 z / 3 S 3 . We will first argue q cosh( Z θ / geometry. We will further 4 3 cos ), that is 2 → ∞ h ξ q := dS ) by first identifying − θ , 4 , into four-dimensional Minkowski N 2 2 2.6 hyperboloid, defined by the hyper- ` h , Σ ` c dS sin ` q ) 3 iτ . p ` = = 2 z/` = = → q ) = 0. The remaining coordinates ( ` 1 3 3 2 4 spacetime with a radius equals to X X 3 . This corresponds to the zero radius limit ) can be mapped to the global foliation of dS q ,X of the dS + ( + cos • • Z – 11 – ) 2 / 3 2 4.2 , ) 2 1 iT , t 2 t/` S dz . The situation is illustrated in figure X holography → π` = z 3 3 + ( sinh( = → M g , 2 θ limit of the orbifold z ) 3 S 1 collapses to a single transverse direction, say (embedded in four dimensions). Σ at cos X S N S 2 q ξ → ∞ + ( limit provides a new mechanism to study dS Z θ , X 2 / . This operation sets − q q ) 2 2 0 and Σ S is equivalent to the zero radius limit of the and Σ limit of the spindle ` cos X q limit and dS S N ( p ξ → ∞ θ − ) (the latter being the time coordinate in q q = 0 and Σ Γ = = . z ∞ 3 0 2 , → ∞ , 0. In this limit, the two-dimensional geometry between the northern and 1 X X q → M −∞ ` ( . The large is the two-dimensional induced metric on the defects defined in ( 1 − ∈ h q t After taking the limit, the resulting geometry is In the above limit, the four-dimensional geometry of the manifold ( located at 2 := S N q where We futher observe that theThis line element is ( done viatime analytical continuation of the transverse coordinate surface equation spacetime and then taking parametrize the embedding dS southern defects Σ Σ the three-dimensional geometry ofcan global be dS seen directly from the embedding coordinates ( dimensional origin, namely, it isminimal inherited surfaces from Σ the Euclidean Liouville theory4.1 on the two 3D conformalThe boundaries from limit codimension two` defects in 4D Here, we consider the that this limit yields anpropose alternative that realization the of large thewhereby global the dS dual field theory defined on the two conformal boundaries of dS Figure 2 where the two-dimensional geometry between theto northern a and single southern defects transverse Σ dimension. The resulting geometry is4 that of global dS The large JHEP04(2020)124 of  (4.6) (4.7) (4.8) (4.9) (4.4) that I (4.10) 3 and the . This is  ∞ I ) in the limit − perspective one gravity — when ) — is described ∪ I 3 3 3.17 C limit, the Liouville < T < , + q . Hence, in the large I located at a polar angle −∞ θ ] which, in the context of → ∞ . ), the original codimension 3 12 T 4.3 ` . , . 2 and 4 2 2 ` = 4 , Ω G 6 ) (4.5) = 2.) Then ) i d 3 , ) θ − θ , ) , g I 3 ) = 3 − , sin )). Thus, recalling from section , S T/` sin G → −∞ I h + ( ( 3 ) 2 geometry can be thought of as the limit I (dS c 1 (Σ T ]. Indeed, in the large 4.3 dθ π` ( 3 S `G ∞ π 0 12 c [ R ) + →∞ = 2 −→ q 7−→ + = 4 − i – 12 – = are sent to the past and future infinities + cosh ) ) I spacetime. Under ( θ ) + 4 ( 4 i 2 becomes the global time Γ N = Vol( 3 c N ∪I b g θ G N h , Σ 4 z + dT 4 = denotes the metric on the unit 2-sphere: , (Σ G I sin S − c 2 2 h ∞ ) = c dS ) as denote the Liouville central charge ( c 1 ( π Ω (Σ = and Σ d S 2 4 4.5 = ` 3 G 3 S g c Vol( ) = , where S are respectively sent to 2 2 S Ω (Σ d ∞ c central charge → 2 there exist an Euclidean Liouville theory, from the dS and Σ h correspondence, predicts a non-thermal dual theory. S ). This average is given by ) = N 2 ) is defined as the average volumen of a meridian Γ 1 N 2 S /CFT (Σ 3 CFT ∞ and Σ / . Note that the four-dimensional Newton’s constant can be expressed in terms of c 3 N (after the double analytical continuation ( Accordingly, the total central charge of the composite boundary 3 → ∞ (see figure limit, they reincarnate as the past and future infinities of dS (In the above, we have used that q the three-dimensional one as where Vol( θ can be computed by means of ( where thermal theory, in agreementthe with dS the results established in [ should expects to haveconsistent some with Liouville-type the theory known on factformulated that each as the of two asymptotic the copies dynamics boundaries Chern-Simonsby of an theory pure Euclidean with dS Liouville gauge theorytheory group on SL(2 on each the minimal surfaces reaches its zero temperature limit becoming a non- where the minimal surfacesdS Σ on Σ q 4.2 dS The above maneuvers show that the global dS induced metric Clearly, this is the globaltwo foliation defects of Σ dS As a result, the compact coordinate JHEP04(2020)124 ] ]. S 32 12 1 in – (4.11) (4.12) (4.13) 6 → q . , viz 2 4 )), G π` ), in particular lead to correspondence [ 2  limit, the Cardy entropy 4.10 q 1 q 3.15 − 1 /CFT 3 , which comprises a free energy  . N ) and ( ) is 3 holography may emerge as the large G . The latter geometry exhibits two π` 4.7 ] = 2 2 q 3 3.14 S Z and Σ = / [Σ 4 , S 3 1, we have proposed the existence of an 3 dS ` G 3 Cardy 2 q q > S – 13 – = →∞ −→ S q c ] + ). The massless probe limit is defined by S N Σ action defines a pair of codimension two surfaces, Σ ). Making use of the thermal Cardy formula, we have ∪ spacetime. 2.9 [Σ q N 4 Z Σ q 3.17 S Cardy q ). Each of these two surfaces contains the worldline of one S ). ) seem to indicate that dS 2.12 := S 2.14 4.13 , defined in ( Σ orbifold geometry. ), corresponds to that of a Liouville theory on a 2-sphere with a fixed ∪ N q N O Σ 3.9 Z q / S ) and ( 4 and 4.11 S O , as indicated in ( N -dependent central charge ( The correspondence between the reduced action on the minimal surfaces and the Liou- By introducing an orbifold parameter The set of fixed points of the q limit of the dS vacuum expectation value of the Liouville field. ville theory action provides a non-trivialtheories. link These between the consistency couplings conditions, and displayedthe parameters in of ( both intrinsic field theoreticdimensional description conformal of each fielddimensional of theory. action the functional with minimal To supportinflow surfaces this on coming Σ in from end, dimensionally termsplus we reducing of the have the corresponding a four-dimensional built Nambu-Gotogiven two- Einstein-Hilbert in term up action, equation of an ( the effective surface. two- The resulting effective action, static observer and theyMoreover, both they have are the by topology construction minimalmust of surfaces be a in coupled 2-sphere the to in senseprinciple; the that the their cf. Einstein-Hilbert Euclidean area equation action geometry. functional ( in order to have a well defined variational antipodal conical singularitiesobservers, that we interpretwhich as one being recovers the created smooth by dS a pairand of Σ massive 5 Conclusions In this work, wede have Sitter modeled spacetime the back-reaction via of the a singular static quotient observer dS in four-dimensional The results ( q of the two defects correctly reproduces the thermodynamic(upon entropy using the of dimensional three-dimensional reduction dS of spacetime the Newton [ constant ( in accordance with the result derived inNote the that context this of result the is dS consistent with the fact that, in the large and therefore one finds that the total central charge ( JHEP04(2020)124 ). , it 1 − 3.17 inherit ). This T 4 3 dS comprises = 1). 4.7 q − q > ∪I Static + I ) and that we propose 1 → 3.17 , which is equivalent to the , as displayed in ( q q boundary ). Z N 3 / central charge for the boundary 4 4.5 2 , gives a modular free energy whose 1 − ). In this limit, the four-dimensional q /CFT T Z 3 N,S 2 / 4 = spacetime where the two minimal surfaces limit of the Liouville central charge ( ]. These theories are naturally formulated q 3 on Σ dS q 33 Defects (cod-2) Liouville theory – 14 – and another one Σ , as indicated in ( S 3 geometry and its defect/boundary field theory description. q ) which, upon identifying the modular parameter with limit of the quotient dS Z of dS / → ∞ → ∞ 4 − spindle (see figure q q 3.21 I is also equivalent to zero temperature limit of the Liouville q → ∞ Z / q 2 . and  I S 3 → ∞ + 3 I q dS Global are mapped, upon double analytical continuation, to the future and past Non-thermal N Boundaries (cod-1) . Different limits of the dS Liouville Theory on Regarding directions for future work, one may speculate that our construction be- Accordingly, the total central charge of the composite dS From the relation between the modular parameter and the temperature We finally studied the The above construction permits the interpretation of the Gibbons-Hawking entropy and Σ S topological field theories ofon the manifolds AKSZ with type multiple [ codimensions; boundaries Hilbert and spaces they arebulk assigned incorporate theory) to extended as boundaries objects well (encoding as ofber). boundary to various states defects In of (encoding the this defect states moduli labeled space, by the it codimension is num- natural to expect that the Hilbert spaces associated to can be directly computed byThe taking result the correctly reproduces large thefield value of theory. the dS longs to a broader scheme whereby (higher spin) gravity theories are formulated as quasi- follows that the limit theory on the minimalfrom surfaces. the minimal As surfaces a abe result, non-thermal summarized the Liouville in future theory. figure Schematically, and our past findings infinities can of dS two separate contributions, one from Σ zero radius limit ofgeometry the reduces to the globalΣ geometry of dS conformal boundaries of such observer inducesminimal a surface. conical This two-dimensional defect surfacein which encode terms their in of own turn field Liouvillecaptures theoretic is theory, the description the which degrees locus yields of freedom of the of codimension central the charge two observer ( (that are only visible when computed the Cardy entropythe ( inverse of the (dimensionless)modular temperature entropy equals the Gibbons-Hawking entropy. as representing microscopic degrees of freedom of the massive observer: the back-reaction Figure 3 JHEP04(2020)124 ~ ∼ (A.5) (A.1) (A.2) (A.3) (A.4) 2 γ Unab) . We plan 3 Regular grant , ,  ··· Φ is in part supported by ] and references therein. γ + 2 36 ) Fondecyt 2 2 – Ps z z πµe ( 34 − 1 ∂T z , 2 Φ + 4 + Φ) 2 R ) ∂ ) Q 2 ( 2 z . is a Universidad Andres Bello ( z 2 ( − − γ , T Q Φ + Φ 1 . In complex coordinates, the (holomorphic Fd 2 j + 2 z 1 ∂ ( − 1 γ Φ γ Q∂ i + – 15 – ∂ = 4 = → ij = 1 + 6 ) Q h 2 is partially supported by c γ zz 2014-0115. The work of z  2 T h limit, to the boundary Hilbert space of dS − c/ depends on the curvature of Σ, and the coupling Ro √ 1 q := Dpi y z µ 2 ( T ) be a two-dimensional Riemann surface. Liouville theory d 1151107. Σ , h would like to thank the hospitality of Lebedev Physical In- o Z ) = grant 2 π z 1 ( Fd 4 T ) = Let (Σ 1 z L ( I ). The work of T Conicyt Regular grant N ) acquires an extra overall minus sign. Dgi-Unab A.1 Scholarship holder, and his work is supported by the Direcci´onGeneral de Investi- Conformal invariance at the full quantum level sets the brackground charge would like to thank the hospitality of the Riemann Center for Geometry and Physics 1170765 and o to the central charge of the theory which is thus invariant under thepart shift of the) stress-enery tensor gives rise, via the operator product expansion controls the quantum effects.action When ( considering the theory on a Lorentzian manifold, the is an exact two-dimensional conformal field theory on Σ, defined by the action where the interaction parameter A Liouville theory In this appendix we collectclassical the limit. most For a relevant more results detailed of analysis Liouville see,Quantum field for theory instance, theory. [ and its semi- PhD gaci´on( N Fondecyt Acknowledgments Ca at LUH during thea completion of Riemann this Fellowship. project,stitute where during his the work was final partially stage supported of by this project. ferent limits of the moduliexample parameters. of such The a casedimension duality presented gives here in rise, would which then in theto be the Hilbert refine a large space and concrete of present a these codimension ideas two in defect a separate in work. four boundaries and defects are related via a (co)dimensional ladder of dualities involving dif- JHEP04(2020)124 37 (A.6) (A.7) (A.8) (A.9) (A.12) (A.13) (A.10) (A.11) ,  Class. Quant. Grav. ) and the bound state , 2Φ A.5 ··· πµe + . ) 2 2 0 ]. z z Φ + 4 , ( µ , − α R . 2 R 1 − =: ∆ , z c . ∂V ∈ 24 γ SPIRE ) 2 ) ]. z Φ + IN + Q α j ≈ → [ Φ( 1 4 ∂ 2 α ) 0 − ) ]. 2 2 Φ 2 ]. i e ∆ ≥ z z SPIRE Q ∂ ( iλ , λ ( 2 , µ α IN ij − λ α , – 16 – + [ V h ) = Φ 1 SPIRE 2 z 1  + z SPIRE 6 ∆ Q ( γ IN ( − h Three-Dimensional Einstein Gravity: Dynamics of Flat 2 IN α 1 2 ) scales to (1977) 2738 [ γ ∆ = Cosmological Event Horizons, Thermodynamics and √ V Q ≈ ][ y de Sitter Space and Entanglement = 1 4 c ) = 2 → A.1 , create the spectrum of primary operators of the theory. 2 d α z α Φ (1984) 405 D 15 ), which permits any use, distribution and reproduction in ( Σ Z ∆ = V ) 2 1 (1984) 220 153 Three-Dimensional Cosmological Gravity: Dynamics of Constant z 1 πγ ( ) is well approximated by 0. In this regime, the central charge ( 4 T The semiclassical limit of the theory is taken through the double 152 = → Phys. Rev. A.10 CC-BY 4.0 2 L , I γ arXiv:1901.04554 This article is distributed under the terms of the Creative Commons [ Annals Phys. , Annals Phys. , Particle Creation Curvature Space (2020) 015009 S. Deser and R. Jackiw, S. Deser, R. Jackiw and G. ’t Hooft, C. Arias, F. Diaz and P. Sundell, G.W. Gibbons and S.W. Hawking, [3] [4] [1] [2] References Open Access. Attribution License ( any medium, provided the original author(s) and source are credited. in the limit where conformal weight in ( Semiclassical limit. scaling under which the quantum action ( in terms of which determines the conformal dimensionthe of momentum primaries in terms of the background charge and The momenta of normalizable states labeled by the quantum number It follows that the operator product The vertex operators JHEP04(2020)124 ]. 08 ]. SPIRE ]. (2000) 192 ] B 440 Nucl. Phys. IN (2002) , , [ ]. JHEP Phys. Lett. SPIRE , , (1999) 213 SPIRE D 61 IN (2006) 941 IN ][ B 625 ][ SPIRE (1981) 207 (2002) 030 Phys. Lett. 148 IN (1986) 742 , B 454 04 ][ Annals Phys. , vol. 1, Princeton ]. , 56 103B hep-th/0106113 Phys. Rev. [ (2002) 050 , JHEP Nucl. Phys. SPIRE 10 , , IN Phys. Lett. ][ , arXiv:1011.1897 hep-th/9501127 [ [ Phys. Lett. (2001) 034 JHEP , , hep-th/0110031 ]. [ Theor. Math. Phys. 10 Phys. Rev. Lett. , Quantum information metric of conical , Conformal Invariance, the Central Charge Near Horizon Limits of Massless BTZ and SPIRE ]. A Multiboundary AdS orbifold and DLCQ JHEP IN (1995) 2133 , ][ (2011) 175012 (2002) 579 – 17 – arXiv:1804.08358 SPIRE 28 Quantum three-dimensional de Sitter space ]. ]. 19 D 52 [ IN Holographic particle detection ][ On the description of the Riemannian geometry in the ]. ]. SPIRE SPIRE Exciting AdS orbifolds ]. IN IN The Asymptotic dynamics of de Sitter gravity in Conformal anomaly from dS/CFT correspondence SPIRE ][ ][ ]. ]. ]. ]. SPIRE Phys. Rev. IN Three-Dimensional Geometry and Topology hep-th/9807216 , de Sitter gravity and Liouville theory IN SPIRE [ ][ (2018) 046008 String Sources in (2+1)-dimensional Gravity [ IN ][ SPIRE SPIRE SPIRE SPIRE ]. IN IN IN IN hep-th/0106191 Class. Quant. Grav. D 98 Class. Quant. Grav. [ Conformal description of horizon’s states , ][ ][ ][ ][ , Quantum Geometry of Bosonic Strings The dS/CFT correspondence SPIRE (1986) 186 Operator Content of Two-Dimensional Conformally Invariant Theories IN hep-th/9806119 hep-th/0311237 Statistical entropy of three-dimensional Kerr-de Sitter space Some aspects of the de Sitter/CFT correspondence (1999) 046002 Weyl symmetry and the Liouville theory [ [ [ ]. ]. hep-th/9906226 (2001) 145 Phys. Rev. [ B 270 , D 59 hep-th/0106247 [ SPIRE SPIRE hep-th/0206175 IN hep-th/9812056 hep-th/0511065 hep-th/0203268 IN defect Phys. and Universal Finite Size Amplitudes at Criticality holography: A Universal holographic(2004) description 023 of extremal black hole horizons Their CFT Duals 044007 [ [ [ [ Mathematical Series, Princeton University Press (2014). presence of conical defects 295 three-dimensions [ [ B 519 (1989) 352 (1998) 275 Rev. A. Strominger, S. Nojiri and S.D. Odintsov, M.-I. Park, M. Ba˜nados,T. Brotz and M.E. Ortiz, S. Deser and R. Jackiw, J.L. Cardy, H.W.J. Bloete, J.L. Cardy and M.P. Nightingale, J. de Boer, M.M. Sheikh-Jabbari and J. Simon, C.-B. Chen, W.-C. Gan, F.-W. Shu and B. Xiong, V. Balasubramanian and S.F. Ross, E.J. Martinec and W. McElgin, V. Balasubramanian, A. Naqvi and J. Simon, S.N. Solodukhin, R. Jackiw, W. Thurston and S. Levy, D.V. Fursaev and S.N. Solodukhin, A.M. Polyakov, S. Cacciatori and D. Klemm, D. Klemm and L. Vanzo, D. Klemm, [8] [9] [6] [7] [5] [23] [24] [21] [22] [18] [19] [20] [16] [17] [13] [14] [15] [11] [12] [10] JHEP04(2020)124 B ]. (1998) ] ]. 02 SPIRE , in IN J. Stat. Mech. Nucl. Phys. , SPIRE ][ , IN JHEP [ , Lect. Notes Phys. (1997) 1405 , ]. Prog. Theor. Phys. , hep-th/9304011 string theory (1994) R6031 A 12 The Geometry of the master D - (1989) 233 SPIRE 2 . gr-qc/9404036 IN [ D 50 ][ Quantum Gravity D B 324 2 gravity and ]. (1995) 1741 Phys. Rev. Wavefunction of a Black Hole and the D ]. , - Int. J. Mod. Phys. 2 , arXiv:1102.2098 Nucl. Phys. , SPIRE , D 51 arXiv:1405.5137 – 18 – IN SPIRE Universal Spectrum of 2d Conformal Field Theory in [ IN ][ Statistical entropy of de Sitter space ][ Holographic Entanglement Entropy Lectures on ]. ]. Phys. Rev. Entanglement entropy and quantum field theory , (2014) 118 ]. ]. 09 SPIRE Conformal Field Theory and ]. SPIRE IN IN [ hep-th/0405152 SPIRE ][ [ SPIRE IN JHEP SPIRE IN arXiv:1609.01287 ][ , IN ][ [ [ Geometry and Thermofields A Note on Hartle-Hawking vacua Notes on quantum Liouville theory and quantum gravity (1990) 319 , Boulder, U.S.A., 1–26 June 1992, pp. 277–469 (1993) [ Renyi Entropy and Free Energy ]. 102 (2004) P06002 gr-qc/9801096 (1989) 509 (2017) pp.1 [ SPIRE IN hep-th/9502010 gr-qc/9407022 Proceedings, Theoretical Advanced Study Institute (TASIto 92): Particles From Black Holes[ and Strings [ 321 Suppl. 0406 014 equation and topological quantum field theory [ 931 the Large c Limit Dynamical Origin of Entropy P.H. Ginsparg and G.W. Moore, J. Distler and H. Kawai, N. Seiberg, J.M. Maldacena and A. Strominger, M. Alexandrov, A. Schwarz, O. Zaboronsky and M. Kontsevich, J.C. Baez, M. Rangamani and T. Takayanagi, P. Calabrese and J.L. Cardy, R. Laflamme, A.O. Barvinsky, V.P. Frolov and A.I. Zelnikov, T. Jacobson, T. Hartman, C.A. Keller and B. Stoica, [36] [34] [35] [32] [33] [29] [30] [31] [26] [27] [28] [25]