Liouville description of conical defects in dS$_4$, Gibbons-Hawking entropy as modular entropy, and dS$_3$ holography

We model the back-reaction of a massive static observer in four-dimensional de Sitter spacetime by means of a singular $\mathbb Z_q$ quotient. The set of fixed points of the $\mathbb Z_q$ action consists of a pair of codimension two minimal surfaces given by 2-spheres in the Euclidean geometry. The introduction of an orbifold parameter $q>1$ permits the construction of an effective action for the bulk gravity theory with support on each of these minimal surfaces. The effective action corresponds to that of Liouville field theory on a 2-sphere with a finite vacuum expectation value of the Liouville field. The intrinsic Liouville theory description yields a thermal Cardy entropy that we reintrepret as a modular free energy at temperature $T=q^{-1}$, whereupon the Gibbons-Hawking entropy arises as the corresponding modular entropy. We further observe that in the limit $q\to\infty$ the four-dimensional geometry reduces to that of global dS$_3$ spacetime, where the two original minimal surfaces can be mapped to the future and past infinities of dS$_3$ by means of a double Wick rotation. In this limit, the Liouville theories on the two conformal boundaries become free bosons with background charge whose total central charge equals that computed using the dS$_3$/CFT$_2$ correspondence.


Introduction
The non-trivial topology of de Sitter (dS) spacetime comprises two disconnected spacelike boundaries and causally disconnected interior regions. Recently, it has been argued [1] that the Gibbons-Hawking entropy of dS spacetime [2] arises from the entanglement between the past and future conformal infinities or, alternatively, from the entanglement between two antipodal and causally disconnected bulk observers located at opposites Rindler wedges of the dS interior.
One of the central ideas behind the above argument is that in order to measure any observerdependent quantity, in particular the thermal properties of the dS cosmological horizon, one has to go beyond the standard probe approximation of a static observer. In other words, the observer back-reaction should be taken into account.
Following this idea and inspired by the conically singular geometries induced by point particles in three dimensions [3][4][5], one may treat the orbifold dS 4 /Z q as the fundamental spacetime manifold on which the gravity theory is formulated and think of the dS 4 spacetime only as a smooth limit of it. The orbifold parameter q > 1 thus amounts to describe the response of the geometry to the presence of a massive observer by inducing codimension two defects that contain the observers worldines, and where the q → 1 limit corresponds to the massless limit in which one recovers the original, non-singular dS 4 geometry.
The aim of this note is to show that massive observers in dS 4 admits an intrinsic description in terms of a two-dimensional conformal field theory. We shall argue that the introduction of an orbifold parameter q > 1 permits to build up a reduced two-dimensional action functional with support on the pair of codimension two minimal surfaces that define the set of fixed points of the Z q action. Each of these minimal and tensionful surfaces have the topology of a 2-sphere in the Euclidean geometry and they can be formally thought of as the "worldvolume" of a massive observer, whose massless limit is equivalent to the tensionless limit q → 1. As we shall argue, the resulting effective two-dimensional Euclidean action can be identified with a Liouville theory on a 2-sphere, in which the Liouville field acquires q-dependent vacuum expectation value.
The correspondence between the effective action of a massive observer and the Liouville theory action links the gravitational parameters, namely the dS 4 radius ℓ and the four-dimensional Newton's constant G 4 , with the Liouville coupling constant γ 2 ∼ . This relation results in a semiclassical central charge given by This q-dependent central charge arguably encodes degrees of freedom associated to a massive observer which are not present in the massless limit q → 1. Consequently and by means of the thermal Cardy formula, the central charge (1.1) predicts a Cardy entropy that equals a modular free energy whose corresponding modular entropy correctly reproduces the Gibbons-Hawking area law.
We conclude by observing that in the q → ∞ limit of the quotient dS 4 /Z 4 , the four-dimensional geometry reduces to the global geometry of dS 3 , where the two minimal surfaces of the former can be mapped-via a double Wick rotation-to the two conformal boundaries of the latter. In this limit, the interaction term in the Liouville theory action vanishes and the theory becomes a free boson with background charge. Moreover, upon taking the q → ∞ limit, the two dS 3 boundaries inherit a central charge from the Liouville theory on the corresponding minimal surface in one higher dimension. As we shall see, the total central charge of the two boundaries reproduces exactly the central charge derived in the context of the dS 3 /CFT 2 correspondence [6][7][8][9][10][11][12].
2 Static observers in dS 4 In four dimensions, de Sitter spacetime (dS 4 ) can be viewed as a four-dimensional timelike hypersurface embedded in five-dimensional Minkowski space M 1,4 . Taking the embedding coordinates to be X µ ∈ M 1,4 , µ = 0, ..., 4, and considering the Minkowski metric the dS 4 hypersurface is defined by where ℓ 2 is the dS 4 radius. The hyperboloid (2.2) has the topology of R×S 3 with manifest O(4, 1) symmetries.

Massless probe observers
The standard description of a static observer in dS 4 is obtained by parametrizing the embedding coordinates as where theŷ i denote the coordinates of the unit 2-sphere. The resulting line element where the radial coordinate runs from 0 ≤r < ℓ and dΩ 2 2 is the metric on the unit 2-sphere. The time-independent metric (2.4) describes the worldline of a single static observer located at the originr = 0. The observer is causally connected with only part of the full spacetime. Such region is dubbed the Rindler wedge (or static patch) of the observer, and its boundary defines an observer-dependent cosmological horizon H. This has the fix time topology of a 2-sphere and is located atr = ℓ.
In the Euclidean vacuum, a static observer detects a temperature and a corresponding Gibbons-Hawking entropy [2] given by

Massive observers and antipodal defects
The above characterization of a static observer in dS spacetime considers the observer as a massless probe object. Here, instead, we treat an observer as a massive object which modify the local geometry of the spacetime; we propose to model the back-reaction of such massive observer by means of a singular Z q quotient. This construction, which we shall now briefly review, has been spelled out in full detail in [1].
To begin with, we note that the constraint (2.2) can be alternatively solved by parameterizing the embedding coordinates as The resulting dS 4 line element, that we shall simply denote by g 4 , is (2.8) The metric (2.8) has the warped product form S 2 × w dS ± 2 , where the 2-sphere has radius ℓ and dS ± 2 denotes the radially extended dS 2 space, with the extended radial coordinate ξ ∈ (−ℓ, ℓ), as indicated in (2.7). This geometry describes the worldline of two antipodal static observers which are causally disconnected (as any light ray can not be sent from one observer into the other). The foliation (2.8) covers the union R N ∪ R S of both northern and southern Rindler wedges, as depicted in Figure 1. In order to incorporate the observers back-reaction, one next deforms the S 2 sector in (2.8) by performing a S 2 /Z q orbifold. This is done via the discrete identification φ ∼ φ + 2π q , with an orbifold parameter q > 1. The four-dimensional orbifold dS 4 := dS 4 /Z q is then endowed with the metric where the warp factor w = cos θ satisfy the holonomy conditions w| 0,π = 1 and w ′ | 0,π = 0, and (2.11) The azimuthal identification deforms the S 2 geometry into that of a Thurston's spindle [13].
The latter geometry has two antipodal conical singularities at the points θ = 0, π, which are precisely the locations of the two static observers (2.9). We interpret these singularities as the response of the background geometry to the presence of a massive observer, with a mass proportional to (q − 1).
The set of fixed points under the Z q action defines two antipodal, codimension two surfaces Σ N and Σ S , both endowed with the induced metric h = g 4 θ=0,π = g ± 2 .
In terms of the gravity action and in order to have a well defined variational principle, the two conical singularities are resolved by adding to the Einstein-Hilbert action a pair of Nambu-Goto terms with support on Σ N and Σ S [14] (2.14) In the above, the support of the first integral excludes the location of the defects Σ N and Σ S .
The two Nambu-Goto terms are coupled through the tension where the limit q → 1 corresponds to the tensionless limit in which one recovers the usual Einstein-Hilbert action on the smooth dS 4 geometry.
Hence, by construction, Σ N and Σ S are codimension two minimal surfaces with an induced stress-energy tensor given by The localized stress energy tensor (2.16) is a strong sign of the existence of an underlying field theory defined on the two minimal surfaces. As we shall next argue, this theory corresponds to an Eucliedan Liouville theory on a 2-sphere.

Liouville theory description of a massive observer
In this section, we construct an effective two-dimensional action with support on the codimension two minimal surfaces Σ N and Σ S . These surfaces are the set of fixed points of the Z q action.
Each of them contain the worldline of one of the massive observers O N and O S , and they both have the topology of a 2-sphere in the Euclidean geometry, viz.
with induced metric dΩ 2 2 (which corresponds to the analytic continuation of (2.13)). In the above, the label "E" denotes Euclidean geometry. Hereafter, we shall drop this label when is clear from context.

Effective two-dimensional action
To begin with, we recall that the total Euclidean gravity action (2.14) on the conically singular manifold dS 4 := dS 4 /Z q consists of a bulk piece plus a pair of two-dimensional Nambu-Goto terms where the Euclidean integrals Although the support of the bulk integral above excludes the location of the defects, we can define a "free energy inflow" from the bulk to Σ N and Σ S by dimensional reducing I bulk down to two dimensions as to define an effective action on each of the defects, which comprises the inflow (3.4) and the corresponding Nambu-Goto term, viz. 5) and such that the total on-shell action (3.2) (From here and in what follows, we shall use the notation "≈" to indicate on-shell equalities.) The reduced Euclidean action I 2d in (3.4) can be computed using the line elements (2.11) and integrating out the spindle coordinates (θ, φ). This gives where the integral is over the two-dimensional submanifold coordinatized by y = (τ, ξ) (with τ denoting the Euclidean time), and R = R[h] is the intrinsic two-dimensional scalar of curvature built up from the induced metric on the defects (2.13). This reduction holds upon imposing Einstein's equations ℓ 2 R θθ = 3g θθ (and likewise the φφ-equation) and by making use of the codimension two identity R ij = 3 cos 2 θ R ij .
Due to the antipodal symmetry relating Σ N and Σ S [1], we further assign to each of the defects half of the total inflow (3.7) so that the effective action (3.5) on the northern defect is given by

On-shell correspondence with Liouville theory
We now observe that the structure of the reduced effective action (3.9) closely resembles the Liouville theory action [15]: That is 11) 5 We recall that in our conventions the definition of the gravitational stress energy tensor (2.16) differs from the standard convention used in the CFT context: The overall factor of −2π propagates when computing the operator product expansion T T , which in turns and similarly for I eff [Σ S ]. This on-shell correspondence holds provided as follows from matching the terms of the same order in derivatives of the metric in (3.9) and (3.10). In addition, the expectation value Φ 0 must satisfy the Liouville equation of motion for a constant field, which is given by where the first equality made use of the constant positive curvature R = 2ℓ −2 of Σ N .
Compatibility of the equations (3.12) and (3.13) yields and Observe that the bound q > 1 for the orbifold parameter can be understood as a consistency condition: One the one hand, from (3.13) it follows that positivity of R[h] = 2ℓ −2 > 0 is only possible if µ < 0, which according to (3.14) requires q to be greater than one. On the other hand and remembering that Q = γ + γ −1 , the bound q > 1 ensures the reality of the couplings γ and ℓ in (3.15).
In what follows, we shall see that the semiclassical limit of (3.15) provides a nontrivial link between the Liouville coupling constant γ, in terms of which the central charge of the theory is defined, and the gravitational coupling ℓ 2 /G 4 which in turns defines (up to a factor of π in dimension four) the entropy of the dS 4 space.
Indeed, from the semiclassical limit of (3.15), we straightforwardly find This value of the central charge is consistent with the classical conformal anomaly equation where T = h ij T ij is the trace of stress-energy tensor (2.16) and R = 2ℓ −2 is the curvature of the corresponding defect. Also, we note that since q > 1, then the central charge c q > 0, which indicates unitarity of the theory.
Having obtained the central charge (3.17) and by virtue of the thermal Cardy formula [17,18] S Cardy Accordingly, in our case, since where ∆ 0 is the (semiclassical) conformal dimension of the bound state (see (A.13)), the Cardy formula (3.18) applies.
For a non-chiral Liouville theory, we have where c q is given in (3.17) and T L and T R correspond to the temperature of the generalized Hartle-Hawking vacuum of dS space. This is known to be equivalent to a thermal state ρ = e −2πH R defined by the Rindler Hamiltonian H R [20,21](see also [22]).
Hence, using (3.17) and (3.20) in the Cardy formula (3.18), we find the q-dependent Cardy entropy S Cardy Note that minus the derivative of the above entropy with respect to 1/q gives the Gibbons-Hawking entropy (2.5). Based on this simple observation, we shall next reinterpret the Cardy entropy (3.21) as modular free energy.

Modular free energy and Gibbons-Hawking entropy
The Cardy entropy (3.21) can be understood as the modular free energy F q whose derivative with respect to the dimesionless temperature T = q −1 [23,24] yields the Gibbons-Hawking area law.
To this end, we define the modular Hamiltonian Thus, we can write the modular partition function as Z = tr ρ q = tr e −qH , (3.23) in terms of which the modular free energy is given by Next, identifying the Cardy entropy (3.21) as the modular free energy (3.24), viz.
we can compute the thermal entropy This is precisely the Gibbons-Hawking entropy (2.5). It is important to point out that although (3.27) has its origin in the modular free energy (3.24), its value is independent of the modular parameter q and hence this remains fix in the tensionless limit q → 1, in which one recovers the standard description of the dS 4 spacetime.

3D conformal boundaries from codimension two defects in 4D
The limit q → ∞ is equivalent to the zero radius limit of the S 2 /Z q spindle, viz. ℓ q := q −1 ℓ → 0.
In this limit, the two-dimensional geometry between the northern and southern defects Σ N and Σ S collapses to a single transverse direction, say z := ℓθ, with Σ N located at z = 0 and Σ S at z = πℓ.
The situation is illustrated in Figure 2. Fig. 2: The large q limit of the spindle S 2 /Z q . This corresponds to the zero radius limit ℓ q → 0, where the two-dimensional geometry between the northern and southern defects Σ N and Σ S shrinks to a single transverse dimension. The resulting geometry is that of global dS 3 spacetime.
In the above limit, the four-dimensional geometry of the manifold ( dS 4 , g 4 ) reduces to the three-dimensional geometry of global dS 3 spacetime with a radius equals to ℓ. This can be seen directly from the embedding coordinates (2.6) by first identifying φ ∼ φ + 2π q and then taking q → ∞. This operation sets X 4 = 0. The remaining coordinates X 0 = ℓ 2 − ξ 2 cos θ sinh(t/ℓ) , X 1 = ℓ 2 − ξ 2 cos θ cosh(t/ℓ) , (4.1) parametrize the embedding dS 3 ֒→ M 1,3 of the dS 3 hyperboloid, defined by the hypersurface After taking the limit, the resulting geometry is where h is the two-dimensional induced metric on the defects defined in (2.11) and (2.13). We futher observe that the line element (4.2) can be mapped to the global foliation of dS 3 . This is done via analytical continuation of the transverse coordinate z ∈ [0, πℓ] and the time t ∈ (−∞, ∞) (the latter being the time coordinate in h), that is As a result, the compact coordinate z becomes the global time −∞ < T < ∞ and the induced metric h → dΩ 2 2 , where dΩ 2 2 denotes the metric on the unit 2-sphere: Clearly, this is the global foliation of dS 3 spacetime. Under (4.3), the original codimension two defects Σ N and Σ S are respectively sent to T → −∞ and T → ∞. Hence, in the large q limit, they reincarnate as the past and future infinities of dS 3 .

dS 3 /CFT 2 central charge
The above maneuvers show that the global dS 3 geometry can be thought of as as the limit where the minimal surfaces Σ S and Σ N are sent to the past and future infinities I ± of dS 3 (after the double analytical continuation (4.3)). Thus, recalling from Section 3 that on Σ N and Σ S there exist an Euclidean Liouville theory, from the dS 3 perspective one should expects to have some Liouville-type theory on each of the boundaries I ± . This is consistent with the known fact that the asymptotic dynamics of pure dS 3 gravity-when formulated as two copies Chern-Simons theory with gauge group SL(2, C)-is described by an Euclidean Liouville theory on I + ∪ I − [12].
In our current setup, the interaction term µ in the Liouville action vanishes in the large q limit, as can be readily checked from (3.14). Hence can be computed by means of (4.5) as where c ∞ (Σ N ) = c ∞ (Σ S ) = 3ℓ 2 G 4 denote the Liouville central charge (3.17) in the limit q → ∞. Note that the four-dimensional Newton's constant can be expressed in terms of the three-dimensional one as where Vol(S 1 ) is defined as the average volumen of a meridian Γ θ located at a polar angle θ (see Figure 2). This average is given by Vol(S 1 ) = Γ θ = 2πℓ sin θ = 4ℓ . (4.9) (In the above, we have used that π sin θ = π 0 dθ sin θ = 2.) Then and therefore one finds that the total central charge (4.7) is 11) in accordance with the result derived in the context of the dS 3 /CFT 2 correspondence [6][7][8][9][10][11][12].

Conclusions
In this work, we have modeled the back-reaction of a static observer in four-dimensional de Sitter spacetime via the singular quotient dS 4 /Z q . The latter geometry exhibits two antipodal conical singularities that we interpret as being created by a pair of massive observers, O S and O N , defined in (2.9). The massless probe limit is defined by q → 1 in which one recovers the smooth dS 4 spacetime.
The set of fixed points of the Z q action defines a pair of codimension two surfaces, Σ S and Σ N , as indicated in (2.12). Each of these two surfaces contains the worldline of one static observer and they both have the topology of a 2-sphere in the Euclidean geometry. Moreover, they are by construction minimal surfaces in the sense that their area functional must be coupled to the Einstein-Hilbert action in order to have a well defined variational principle; cf. Equation (2.14).
By introducing an orbifold parameter q > 1, we have proposed the existence of an intrinsic field theoretic description of each of the minimal surfaces in terms of a two-dimensional conformal field theory. To this end, we have built up an effective two-dimensional action functional with support on Σ S and Σ N , which comprises a free energy inflow coming from dimensionally reducing the four-dimensional Einstein-Hilbert action, plus the corresponding Nambu-Goto term of the surface. The resulting effective action, given in Equation (3.9), corresponds to that of a Liouville theory on a 2-sphere with a fixed vacuum expectation value of the Liouville field.
The correspondence between the reduced action on the minimal surfaces and the Liouville theory action provides a non-trivial link between the couplings and parameters of both theories.
These consistency conditions, displayed in (3.14) and (3.15), in particular lead to the q-dependent central charge (3.17). Making use of the thermal Cardy formula, we have computed the Cardy entropy (3.21) which, upon identifying the modular parameter with the inverse of the (dimensionless) temperature q = T −1 , gives a modular free energy whose modular entropy equals the Gibbons-Hawking entropy.
The above construction permits the interpretation of the Gibbons-Hawking entropy as representing microscopic degrees of freedom of the massive observer thought of as a defect described by a two-dimensional Liouville theory with central charge (3.17).
We finally studied the q → ∞ limit of the quotient dS 4 /Z q , which is equivalent to the zero radius limit of the S 2 /Z q spindle (see Fig. 2). In this limit, the four-dimensional geometry reduces to the global geometry of dS 3 spacetime where the two minimal surfaces Σ S and Σ N are mapped, upon double analytical continuation, to the future and past conformal boundaries I + and I − of dS 3 , as indicated in (4.5).
From the relation between the modular parameter and the temperature q = T −1 , it follows that the limit q → ∞ is also equivalent to zero temperature limit of the Liouville theory on the minimal surfaces. Moreover, the interaction term µ in the Liouville action vanishes and the theory becomes free, as can be seen from (3.14). As a result, the future and past infinities of dS 3 inherit from the minimal surfaces an effective theory that corresponds to a free boson with background charge Q (see Equation (3.15)). Schematically, our findings can be summarized as follows: q → ∞ Accordingly, the total central charge of the composite dS 3 boundary I + ∪ I − comprises two separate contributions, one from Σ S and another one Σ N , as displayed in (4.7). This can be directly computed by taking the large q limit of the Liouville central charge (3.17). The result correctly reproduces the value of the dS 3 /CFT 2 central charge for the boundary field theory.
Regarding directions for future work, one may speculate that our construction belongs to a broader scheme whereby (higher spin) gravity theories are formulated as quasi-topological field theories of the AKSZ type [25]. These theories are naturally formulated on manifolds with multiple boundaries and they incorporate extended objects of various codimensions; Hilbert spaces are assigned to boundaries (encoding boundary states of the bulk theory) as well as to defects (encoding defect states labeled by the codimension number). In this moduli space, it is natural to expect that the Hilbert spaces associated to boundaries and defects are related via a (co)dimensional ladder of dualities involving different limits of the moduli parameters. The case presented here would then be a concrete example of such a duality in which the Hilbert space of a codimension two defect in four dimension gives rise, in the large q limit, to the boundary Hilbert space of dS 3 .
We plan to refine and present these ideas in a separate work.

Acknowledgements
The work of Ca is supported by a Riemann Fellowship.

A Liouville theory
In this appendix we collect the most relevant results of Liouville field theory and its semiclassical limit. For a more detailed analysis see, for instance, [26][27][28] and references therein.
Quantum theory. Let (Σ, h) be a two-dimensional Riemann surface. Liouville theory is an exact two-dimensional conformal field theory on Σ, defined by the action where the interaction parameter µ depends on the curvature of Σ, and the coupling γ 2 ∼ controls the quantum effects. When considering the theory on a Lorentzian manifold, the action (A.1) acquires an extra overall minus sign.
Conformal invariance at the full quantum level sets the brackground charge which is thus invariant under the shift γ → γ −1 . In complex coordinates, the (holomorphic part of the) stress-enery tensor gives rise, via the operator product expansion