The two-sphere partition function in two-dimensional quantum gravity

We study the Euclidean path integral of two-dimensional quantum gravity with positive cosmological constant coupled to conformal matter with large and positive central charge. The problem is considered in a semiclassical expansion about a round two-sphere saddle. We work in the Weyl gauge whereby the computation reduces to that for a (timelike) Liouville theory. We present results up to two-loops, including a discussion of contributions stemming from the gauge fixing procedure. We exhibit cancelations of ultraviolet divergences and provide a path integral computation of the central charge for timelike Liouville theory. Combining our analysis with insights from the DOZZ formula we are led to a proposal for an all orders result for the two-dimensional gravitational partition function on the two-sphere.


Introduction
In this work we explore theories of two-dimensional quantum gravity coupled to conformal matter. Our approach rhymes with that of [1][2][3] in that we will rely heavily on the existence of a semiclassical limit. Although gravity in two-dimensions is a rather simple theory, when coupled to conformal matter it exhibits various features in common with a theory of gravity in a four-dimensional world. Particularly, the theory can have real semiclassical saddle point solutions [4], and it moreoever exhibits a direct analogue of the unbounded conformal mode rendering the Euclidean gravitation action an infamously subtle affair. The advantage of the two-dimensional approach, as has been explored and emphasised often in the past and also more recently, is the remarkable control one has over the gravitational path integral [5][6][7].

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The specific class of gravitational theories we consider are ones equipped with a positive cosmological constant Λ. We will couple these theories to two-dimensional conformal matter endowed with a large and positive central charge c m , thereby introducing a large number of field theoretic degrees of freedom. Examples of two-dimensional conformal field theories with c m 1 include tensor product theories, and holographic CFTs. 1 As we shall soon see, the large value of c m is crucially tied to the existence of a semiclassical picture. The technical object we will focus on will be the gravitational path integral over the matter and gravitational fields on an S 2 topology. We will offer a physical motivation for this shortly. The main goal of our work is to provide the systematics for a perturbative quantum loop expansion about the round two-sphere saddle in the large c m limit. We apply this to compute the saddle point approximation and the first few orders in the quantum loop expansion of the two-sphere partition function of interest. We work in the Weyl gauge whereby the gravitational path integral takes the form of a (timelike) Liouville theory [6,7]. From this perspective, our approach offers a path integral perspective for the semiclassical expansion of the timelike DOZZ formula proposed and explored in [3,[9][10][11][12]. Before embarking on our analysis we find it instructive to provide a physical derivation and motivation for the saddle point solutions of the gravitational theories under consideration.
Two-dimensional gravity coupled to conformal matter. The theory we will focus on consists of a two-dimensional matter CFT of central charge c m coupled to twodimensional gravity. The Euclidean path integral of interest is given by where we take Λ > 0, χ h is the Euler character of the genus h surface Σ h , and Z (h) CFT [g ij ] is the matter CFT partition function. The positivity of Λ suppresses large area configurations. Unless otherwise stated we will, for the most part, restrict our attention to S 2 topology.
Although the pure gravity theory has no classical solutions in two-dimensions due to the topological nature of the Einstein term, upon coupling to a matter theory the effective gravitational action including the contribution from Z (h) [4]. If we further require that the saddle be semiclassical, we must additionally take the central charge c m to be large. Exploiting that two-dimensional gravity is invariant under the two-dimensional diffeomorphism group we can fix the metric, at least within a small neighbourhood, to the Weyl gauge Here ϕ(x) is a real valued function encoding the Weyl factor of the physical metric. For vanishing genus we can further chooseg ij as the round metric on S 2 with area 4π, namely with θ ∈ (0, π) and φ ∼ φ + 2π. On an S 2 topology, and more specifically on an S 2 geometry, the form of Z

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absorbed into the bare gravitational couplings ϑ and Λ the form of Z (0) CFT [g ij ] is fixed by the conformal anomaly. Concretely, for genus zero [ϕ] , (1.4) where N is a normalisation constant, and S anomaly [ϕ] is 2 Consequently, in the semiclassical limit, the resulting action governing the Weyl mode is given by Interestingly, much like what happens for the conformal mode in higher dimensional Euclidean gravity, S eff has a wrong sign kinetic term for ϕ. Being semiclassical, S eff ignores contributions from the bc-ghost sector as well as the gravitational path integration measure. The reason is that these contributions, though present, will be subleading at large c m . They will constitute much of what follows in the later sections. The saddle point equations stemming from (1.7) admit a constant ϕ saddle with . (1.8) The saddle point solution corresponds to a round two-sphere of area ρ * = c m /6Λ. To leading order, the vanishing genus expression reads: where the constant is Λ independent, but will generally depend on c m . For ϑ 1, higher genus corrections to (1.1) are exponentially small as compared to the genus zero contribution. Thus, two-dimensional gravity coupled to a CFT with large positive central charge admits a classical (Euclidean) dS 2 solution with θ ∈ (0, π) and φ ∼ φ + 2π, which is the round two-sphere. Upon Lorentzian continuation φ → it, we are led to 2 More covariantly, the anomaly action is given by the Polyakov action whose form is Although the action is non-local, the degree of non-locality seems permissible in the context of twodimensional gravity. Here it stems from integrating out the massless matter content of the two-dimensional conformal field theory. We further note that if we allowgij to vary, N should be viewed as a functional of gij and the gravitational action acquires the structure of a WZW model (see for instance [14,15]).

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with t ∈ R. This is the static patch of dS 2 with the de Sitter horizon residing at θ = 0 and θ = π. Intuitively, the physical reason the matter CFT can support a positive curvature space with positive gravitational vacuum energy density Λ is due to the negative static patch energy E s.p. = −c m /12π (see for instance section 7 of [16]) associated to a matter CFT quantised on a fixed dS 2 background with de Sitter length (corresponding to a round twosphere of area ρ dS = 4π 2 upon Wick rotating to Euclidean signature). General relativity on a compact spatial slice with no boundaries enforces that the total energy vanishes, leading to the expression Λ − c m 24π 2 = 0 , (1.12) which is solved by = c m /(24πΛ), precisely corresponding to the saddle (1.8). According to Gibbons and Hawking [17,18] [19,20]. Indeed, for a CFT on the static patch of dS 2 with de Sitter length the entanglement entropy across the dS 2 horizon is [21] S ent = c m 6 log ρ dS + const , (1.13) where the constant is ρ dS independent while generally depending on c m . It should be clear then how to compare (1.13) with the logarithmic term in (1.9).
Outline. In the following sections we will explore the semiclassical expansion about the saddle (1.8) being more precise about the subleading corrections. We will discuss more detailed properties of Z grav [Λ] (1.1) at the quantum level perturbatively near the classical saddle. Corrections away from the standard CFT expression (1.13) are due to the interaction with gravity. In this sense, the analysis provides us a window into the effects of gravity on entanglement entropy in two-dimensions. In section 2 we discuss how to fix the Weyl gauge, including a careful treatment of the residual PSL(2, C) gauge symmetries of Z  (3.24). In section 4 we compare our results to a prediction of the exact sphere partition function stemming from an analytic continuation (4.8) of the DOZZ formula. We conclude with an outlook in section 5. The appendices provide important details leading to the results presented in the main text.

The Weyl gauge
In order to proceed, we will consider the problem in the Weyl gauge (1.2). In doing so, we employ a hypothesis of Distler-Kawai [6] and David [7] that fixes the path-integration measure over the Weyl mode ϕ. In this section, we consider the problem on a genus zero surface and provide a proper treatment of the residual gauge freedom.

Path integral in the Weyl gauge & timelike Liouville
In complex coordinates the two-dimensional metric in the Weyl gauge can be expressed as where our choice of fiducial metric takes the Fubini-Study form with area 4πυ. We have included the parameter b for future convenience. It will also be convenient to work with spherical coordinates such that the fiducial metric is given by with θ ∈ (0, π) and φ ∼ φ + 2π, capturing the two-sphere with area 4πυ.
In order to properly fix the Weyl gauge (2.1) we must introduce Fadeev-Popov bc-ghost fields. Upon integrating out the matter and bc-ghost fields, our resulting gravitational path integral on a genus zero surface is given by We have collected all dependence on the Weyl mode ϕ, including the contributions from the conformal anomalies of the matter and bc-ghost theories, into a single action which we have called S L [ϕ]. We have also used the general expression (1.13) to extract the υ dependence from the matter CFT and bc-ghost theory [13], where the constant A is Z According to the hypothesis of Distler-Kawai [6] and David [7], this action is given by the Liouville action [5]

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where the positive root for Q is a choice we can make given the redundancy Q → −Q , b → −b, and ϕ → −ϕ. On the other hand, b is chosen to be the solution admitting a semiclassical limit at large c m . It would be interesting to explore the other solution of b also. For c m > 25, Q and b are pure imaginary. To render the action real, we will consider the field ϕ to live on a purely imaginary contour. The resulting theory is known as timelike Liouville theory [5]. It is convenient to parameterise this theory as follows: where β = ib, q = −iQ (or similarly β = −ib and q = iQ) and ϕ has been continued to reside along the real axis. The reality properties of S tL more closely resemble those of the effective action (1.7) obtained from our gravitational considerations. The cost of having a real action is a kinetic term which is unbounded from below. This is the two-dimensional analogue of the unbounded conformal mode in higher-dimensional theories of Euclidean gravity.
In section 3 we study the semiclassical expansion of S tL . But before doing so, we must first discuss the residual gauge symmetries that remain upon fixing the Weyl gauge.

Residual gauge symmetries & further gauge fixing
The condition (2.1) does not fully fix the gauge and ensues further redundancies. For instance, the transformation is a redundancy of the parametrisation. As such, the resulting theory must be invariant under the above redundancy. Locally, conformal Killing vectors ofg ij are given by holomorphic maps (2.10) The above maps do not affect the form of the Weyl gauge (2.1) since they can be reabsorbed in a shift of ϕ(z,z). Of the space of holomorphic maps, we must select the subset of (2.10) which are normalisable under the norm on the space of diffeomorphisms ω i (z,z) [1,5] This leaves a PSL(2, C) subgroup of normalisable residual diffeomorphisms. Explicitly, these are the maps with a, b, c, and d in C, and their anti-holomorphic counterpart f (z). At the infinitesimal level, the transformations are given by linearly expanding near the identity a = d = 1 and where we must further impose δa = −δd to preserve ad − bc = 1 to linear order. The maximally compact SO(3) group of PSL(2, C) is given by restricting bb + dd = 1 , ab + cd = 0 , aā + cc = 1 . (2.14)
The presence of these residual transformations justifies why we must divide the partition function (2.5) by the volume of PSL(2, C) [24] vol PSL(2,C) = 2 This volume must be treated with care as it is infinite. The residual PSL(2, C) symmetry acts non-trivially on the Liouville field ϕ(z,z). We can study this in the semiclassical limit where β ≈ 1/q 1. For the sake of simplicity, we introduce Ω(z,z) satisfying (2.20) and the Liouville action (2.6) reads As shown in appendix B, it is relatively straightforward to check that in the semiclassical limit the transformation leaves S tL invariant. 4 Given the invariance of S tL , and assuming it persists at the quantum level, the path-integral over the Liouville field ϕ will produce a term proportional to the volume PSL(2, C). Thus, vol PSL(2,C) appears in both the numerator and denominator of the gravitational path integral (2.5) in the Weyl gauge.

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To fix this we must resort to further gauge-fixing. We can expand the Liouville field in a complete basis of real spherical harmonics with Ω a point on the round metric on S 2 with unit area, i.e. (2.4) with υ = 1, and the ϕ lm are real valued. We have normalised the Y lm (Ω) as Further conventions and properties for the spherical harmonics are given in appendix A.
Under an isometric map Ω → Ω (Ω) the Y lm (Ω) map to linear combinations of Y lm (Ω) with the same l. This is because for each l, Y lm (Ω) with m ∈ [−l, l] furnish irreducible representations of SO(3). As a gauge-fixing condition we follow [6] and impose that ϕ 1m = 0 with m = {−1, 0, 1}. 5 The reason this is a good gauge fixing procedure, as delineated in appendix C, is that the variation of δϕ under the three non-compact generators of PSL(2, C) is precisely equal to the l = 1 modes. A different gauge fixing procedure is discussed in [25]. This condition will remain unchanged under the action of the SO(3) subgroup of PSL(2, C), fixing three of the six parameters of PSL(2, C). Infinitesimally, these are given by (2.13) where we must now deform in directions α n that are outside of (2.17). A definition of α n and the gauge-fixing procedure is given in appendix C. Given the general transformation (2.22), the Fadeev-Popov determinant for the gauge choice ϕ 1m = 0 is and n ranges over the three non-SO(3) directions of PSL(2, C). SO(3) invariance fixes the form of the above to be independent of the ϕ 1m and ensures that ∆ FP takes the following structure As we delineate in appendix C, At this stage, all but the SO(3) isometry group of the original two-dimensional diffeomorphisms has been gauge fixed. Since this is a compact group, we can just divide out its volume explicitly. The volume of SO(3), as computed from the measure (2.16), is given by (2.28) 5 One might wonder whether this gauge condition is permissible and whether it suffers from Gribov ambiguities [26]. However, as pointed out in [6], in the semiclassical limit where we study small field fluctuations the issue is tamed. For instance, the sign of the Fadeev-Popov determinant (2.26) remains unchanged in the semiclassical q → ∞ limit.

JHEP09(2021)116 3 Timelike Liouville, semiclassically
We are now at a position to make sense of the genus zero timelike Liouville path integral in a semiclassical expansion about its real saddle. Our large parameter is taken to be 1/β, with β → 0 + , such that the remaining parameters are to be understood in terms of a large 1/β expansion. We recall the exact relations

Semiclassical saddle and small fluctuations
The classical equations of the Liouville action (2.8) are given by where −∇ 2 is the Laplacian with respect tog ij . The equations (3.3) admit a constant and real solution given by Due to the PSL(2, C) invariance of the Liouville action, in addition to the above solution there is a continuous family of solutions to (3.3) related to (3.4) by PSL(2, C) transformations. Upon fixing the gauge of the residual gauge symmetries (as discussed in section 2.2) we collapse the continuous solution space down to the constant saddle (3.4). As emphasised in [3], in addition to the real saddle point solution, (3.3) also admit complex solutions ϕ * + πin/β with n ∈ Z. In this section we will focus on the semiclassical expansion about the real saddle point solution.
The saddle point approximation to the path-integral (3.1) leads to (3.5) Comparing to (1.9) for υ = 1 and recalling (3.2), we see that the two agree to leading order in the semiclassical limit. It is also interesting to note that (3.5) bears some resemblance to the critical exponent type expressions encountered in spacelike Liouville theory [27][28][29].
Expanding the Liouville field ϕ = ϕ * + δϕ in (2.8) about the saddle (3.4) we find the action governing fluctuations (3.6) where we have kept terms up to order β 2 . Notice that the small correction in the mass is of the same order as the coefficient of the quartic interaction. We are interested in computing the path integral over the fluctuation field δϕ

7)
JHEP09 (2021)116 in a large 1/β expansion. Interestingly, since √g yields a factor of υ, the fluctuation action is independent of Λ and υ to all orders. But it will depend non-trivially on β, and it is this dependence we will proceed to explore. The path integration contour for (3.7) is over real values of the fluctuation field δϕ. As such, the wrong sign kinetic term renders the path integral ill-defined. A natural proposal, which we pursue here, is to rotate the contour of δϕ to pure imaginary values while keeping β fixed. This is the two-dimensional version of the proposal of Gibbons, Hawking, and Perry [30] for dealing with the unbounded conformal mode in Euclidean gravity. In rotating δϕ → −iδϕ the kinetic term becomes bounded, up to a zero mode which is Gaussian unsuppressed. To deal with this we will Wick rotate it back, as discussed in [31]. As we will soon see, the Wick rotation of the zero mode will produce an overall phase in Z tL [Λ]. On the other hand, though the continued perturbative theory has several imaginary couplings, we will argue that aside from the overall phase, the corrections are all real. One final remark is in order, before computing some contributions is in order. From (3.6) it looks like the fluctuation action is strongly coupled for some modes. This can be seen by expanding δϕ in spherical harmonics. The three l = 1 harmonics have an almost zero mode at the Gaussian level which is lifted at order ∼ β 2 . Fortunately, these almost zero modes are eliminated by the gauge-fixing condition for the residual gauge freedom.

One-loop contribution
The leading contribution beyond the saddle-point approximation stems from the Gaussian fluctuations. The quadratic action, after continuing δϕ → −iδϕ, is given by We note that although the kinetic term is positive, the mass-squared of the fluctuation is negative. 6 We now expand δϕ in a complete and orthonormal basis of real spherical harmonics, as in (2.23). In terms of these, the measure of path integration is taken to be (3.10) To render the measure local with respect tog ij we need to introduce an ultraviolet scale Λ uv with units of inverse area. The eigenvalues and degeneracies of the spherical Laplacian −∇ 2 on (2.4) are It may also be worth noting that the value of the tachyonic mass squared in the strict β → 0 + limit is −2 (in units of υ) which plays a special role in the representation theory of dS2 [32] as it furnishes the discrete series. In four-dimensional de Sitter, the discrete series is associated to gauge fields. In our case ϕ is a piece of the two-dimensional metric, itself a gauge field.

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For the case at hand, the constant mode fluctuation must be treated separately since it leads to an unsuppressed Gaussian. After the dust settles, we find that up to the Gaussian contribution (3.12) where the Gaussian one-loop contribution, including the Fadeev-Popov gauge fixing term at the quadratic level in the fields with (2.27), is given by (3.13) and Z saddle [Λ] is defined in (3.5). In (3.13), we keep only quadratic terms in the action (including those stemming from the Fadeev-Popov determinant) while keeping the β dependence exact. We now discuss the various pieces in (3.13). The first parenthesis in (3.13) comes from the zero-mode integral, defined by analytic continuation [31] as where the overall sign is ambiguous unless fixed by an additional principle. Integrating over the three l = 1 modes, including the δ-function from the Fadeev-Popov procedure, yields the second parenthesis in (3.13). We must also consider the five l = 2 modes separately, and these yield the third parenthesis in (3.13). The infinite product in (3.13) is divergent, but can be computed in a variety of ways. We consider the heat kernel method [33], which is covariant. Using a heat kernel regularisation scheme (see for example appendix C of [16]) where ν ≡ −2βq − 1/4, ∆ ± = 1/2 ± iν, and ζ(a, z) denotes the Hurwitz ζ-function. The heat kernel regularization amounts to where ε is a small parameter given by ε = e −γ E / √ πυΛ uv , and γ E denotes the Euler-Mascheroni constant. The 1/ε 2 divergence is local with respect tog ij and can be absorbed into an appropriate local counterterm built fromg ij . The parameterλ = υλ ∈ R + denotes an eigenvalue. Applying to (3.15) the relations [34]  with G(z) the Barnes G function, ζ (−1) = 1/12 − log A with A denoting Glaisher's constant, and logΓ(z) the logGamma function, we finally arrive at the small β expansion for (3.12) We note that the υ dependence in (3.18) is υ c L /6 , with The dependence υ c L /6 can be recognised to take the standard form (1.13) of a twodimensional conformal field theory on the two-sphere [13]. This is consistent with the proposal that timelike Liouville theory can be viewed as a two-dimensional conformal field theory [35][36][37]. Moreover, we observe that c L − 26 + c m = 0, a necessary condition to avoid any dependence on the redundancy (2.9).

Two-loop contribution
Beyond the Gaussian approximation, we must consider the contribution from higher loop Feynman diagrams. Given the perturbative action (3.6), it follows that to leading order beyond the Gaussian approximation we must consider the contribution from the cubic and quartic vertex whose couplings go as ∼ β and ∼ β 2 respectively. Given the constraints on specific angular momentum modes, it is convenient to express the computation in the space of spherical harmonics. As described in detail in appendix D, there are three connected diagrams that contribute. These are shown in figure 1. Two of the diagrams, a melonic diagram and a double-tadpole diagram, contain two cubic vertices. The third is a cactus diagram with a single quartic vertex. The double-tadpole diagram and the cactus diagram are logarithmically ultraviolet divergent. Nonetheless, due to the specific quartic and cubic couplings stemming from the Liouville action (3.6), the double-tadpoles and cactus diagrams exactly cancel each other and consequently we do not have to deal with any ultraviolet divergences. We obtain the finite result (2l 1 + 1)(2l 2 + 1)(2l 3 + 1) (l 1 (l 1 + 1) − 2)(l 2 (l 2 + 1) − 2)(l 3 (l 3 + 1) − 2)

Final two-loop expression & all-loop expectations
We can now assemble all the pieces and write down the expression up to two-loop contributions: The ratio a 1 /a 0 = −27/20π follows from (2.27), while the constant 'const' is defined in (3.19). The ellipses in (3.24) indicate subleading terms in odd powers of β starting at order O(β 3 ).
We end this section with some general observations and expectations about Z tL [Λ]: • Although the perturbative β expansion of Z tL [Λ] stems from a perturbative action (3.6) which becomes complex along the pure imaginary δϕ contour, Z tL [Λ] is pure imaginary to all orders in the small β expansion.
• The Fadeev-Popov determinant ∆ FP (2.26) contributes a term at the same order as the two-loop contribution.
• The dependence on υ and Λ in the pre-factor of (3.24) is exact and will receive no further corrections at subleading order.
• The dependence of Λ uv is such that Z tL [Λ] in (3.24) is dimensionless. At small β, the choice of Λ uv will affect the O(β) term in the parenthesis. Adding contributions from the matter and bc-ghost sectors will cancel the υ dependence but not the Λ uv dependence.
• There will be no ultraviolet divergences at higher orders in the quantum loop expansion. This is often a property of supersymmetric theories. Here it is due to the specific values of the interaction vertices in the perturbative action (3.6). In particular this indicates that if timelike Liouville theory is indeed a conformal theory, O β = e 2βϕ is an exactly marginal deformation.
It would be interesting to prove these expectations. In the next section we take a different route, and compare to an expression from the two-sphere partition function that is derived from the DOZZ formula [24,[39][40][41] for the three-point function applied to three exponential Liouville operators.

Sphere-partition function from (timelike) DOZZ
Having established a systematic semiclassical expansion of timelike Liouville theory we can ask how our expressions compare to other methods in the literature. The natural object to compare to is derived from the expectation value of three exponential operators O β = e 2βϕ . Assuming that timelike Liouville theory is conformally invariant at the quantum level [35][36][37], as evidenced for example by (3.18), the three-point function is fixed up to a coefficient C(β, β, β). To make contact with the Liouville partition function where we have used that [24] C 3 The integral over the three-points diverges logarithmically due to the regions of integration where two of the points collide. As is known from the critical string, these divergences are of the same magnitude as the divergent volume of PSL(2, C) such that the latter can be used to fix the three-points and avoid performing the integral altogether. It follows from (3.24) that the small β expansion of C(β, β, β) about the real saddle (3.4) is given by where we have taken the ultraviolet scaleΛ uv in the computation of the DOZZ formula to be linearly related to the ultraviolet scale Λ uv in the path-integral computation as Λ uv = sΛ uv . The β-independent dimensionless coefficient s affects the 1/β 5 coefficient.

Analytic continuation of spacelike Liouville?
There have been several proposals for C(β, β, β) in the literature. The proposal of [3,9,10] taken at face value leads to a vanishing result -C(β, β, β) = 0. Here, instead, we would like to compare our result (4.4) to the analytic continuation of the spacelike Liouville structure constant for the three-point function of O b = e 2bϕ . This is a special case of the DOZZ formula [24,[39][40][41] and is given by , (4.5) where γ(z) ≡ Γ(z)/Γ(1 − z) is a meromorphic function with poles when z is a non-positive integer and zeros when z is a non-vanishing positive integer. At least semiclassically, the DOZZ formula can be computed from the Liouville path integral [3]. To arrive at (4.5), which differs slightly from the standard expression in the literature, we must take into account some additional normalisation factors. In the literature the path integral approach to obtaining the DOZZ formula involves first mapping the spacelike Liouville theory on the sphere to the disk using (2.22). Concretely, upon transforming the spacelike Liouville action on the round two-sphere (2.2) with υ = 1 to the disk using the field redefinition [3] ϕ → ϕ − Q log 2 1 + zz , (4.6) a semiclassical comparison with (3.24), which is computed for the theory on the two-sphere, is only possible upon keeping the field-independent constants. Keeping all these terms, the spacelike Liouville action on a disk of radius κ 1 reads (4.7) where ds 2 d = dr 2 + r 2 dθ 2 with r ∈ (0, κ). Any remaining terms vanish as we take κ → ∞. Incorporating the field independent terms, leads to an additional factor in the DOZZ formula leading to the expression (4.5).
The resulting expression (4.5) allows [11,12] for a well defined analytic continuation b → ±iβ, Q → ∓iq. The two-sphere partition function Z DOZZ tL [Λ], as derived from the DOZZ formula (4.5) analytically continued to C(β, β, β), can be obtained by integrating C(β, β, β) three times with respect to Λ, and setting the integration constants to zero. The resulting expression -along the lines of [11] -reads where we recall that q = 1/β − β. In a small positive β expansion we find (4.9) where we have taken −1 = e −iπ . Given that the timelike Liouville action (2.8) can be obtained by continuing b to either positive or negative imaginary values, we interpret there to be an overall sign ambiguity in (4.9).

Comparison
At this stage we can compare the expressions (3.24) (with υ = 1) and (4.9). For convenience we recall both expressions We recall that s was implicitly defined in (4.4), and we further note that we have rescaled both expressions such that the coefficient of the 1/β term in the parenthesis is one.
• The Λ dependence between the two expressions agrees. In contrast, Z DOZZ

tL
[Λ] exhibits no Λ uv = sΛ uv dependence. We interpret this to mean that for Z DOZZ tL [Λ], dimensionful quantities are measured in units of some choice of ultraviolet cutoff scale. Any discrepancy between the path integral and DOZZ based on choices of Λ uv is encoded in s.
• The DOZZ formula has an overall factor 1 − e 2iπ/β 2 . We interpret this to mean that Z DOZZ tL [Λ] receives contributions from two saddle point solutions of (3.3). Particularly, the e 2iπ/β 2 term can be viewed as stemming from a complex saddle ϕ * + πi/β, where ϕ * is given by (3.4). Complex saddles were studied in detail in [3], where it was argued that they indeed contribute to the Liouville path integral. It is interesting that Z DOZZ tL [Λ] seems to require only one additional saddle.
• The leading order contribution to the small β expansion multiplying the classical saddle point approximation is 1/β for both expressions. From the perspective of our computation this stems from the O(β −3 ) term in the Fadeev-Popov determinant (2.26). This term was anticipated in [3].
• Both expressions have an overall sign ambiguity, but the real saddle contribution of Z DOZZ

tL
[Λ] differs from Z tL [Λ] by an overall factor of i. It is worth noting here that the defining recursion relation [41] satisfied by the DOZZ formula is ambiguous up to an overall rescaling, so in principle one could have defined the DOZZ formula with an overall i. From our perspective both this i as well as the overall sign ambiguity are associated to the unsupressed zero mode (3.14).
• Picking s = e This is the only remaining ambiguity (provided the measure over the Liouville field is indeed the flat measure [22,23]

Outlook
Two-dimensional quantum gravity coupled to conformal matter with a large c m is arguably the simplest gravitational theory interacting with matter which moreover admits semiclassical de Sitter vacua. (For large and negative c m one also finds semiclassical Euclidean dS 2 solutions upon fixing the area [1,[42][43][44].) As such, it constitutes a natural playground to test various proposals regarding quantum features for a de Sitter universe. In particular, one may hope for an exact expression of the Gibbons-Hawking path integral (1.1) at least for vanishing genus. The main result of this paper, building on [1,3], is a concrete procedure to systematically compute this path integral in a semiclassical expansion. We have applied this procedure up to two-loops resulting in the expression (3.24). Combined with insights from the DOZZ formula and its analytic continuation to timelike Liouville theory, and recalling (2.5), we arrive at the following conjecture for the exact genus zero partition function of gravity coupled to a two-dimensional conformal field theory: We further recall that q = 1/β − β, and that β is related to c m through (3.2). In arriving at (5.1) we have multiplied (4.8) by on overall factor of iA, and we are working in units of Λ uv . In the semiclassical limit β → 0 + , the above expression can be interpreted as stemming from two saddles of ϕ, each accompanied by their respective perturbative series: The perturbative series about each saddle have the same coefficients up to an overall sign. This is because the saddles themselves are structurally similar, both constant and merely differing by an imaginary shift. Normalising c 0 = 1, the first few coefficients are given by The real saddle corresponds to Euclidean dS 2 which is the round two-sphere. The contribution from this saddle is pure imaginary due to the overall i, a feature in common with odd-dimensional gravitational theories on a round sphere saddles [31]. The other saddle is a highly oscillatory complex function. As a result of the complex saddle, Z grav [Λ] is itself complex valued for generic β rather than pure imaginary. 8 This contrasts its formal definition as a real valued path integral over the timelike Liouville action (2.8) in an interesting way. For special values of β, the reality properties of Z (0) grav [Λ] simplify. We find that for n ∈ Z + ,

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A purely real Z (0) grav [Λ] requires a complex β, albeit one whose imaginary part can be made parametrically smaller than its real part. It would be interesting, and potentially resonate with other ideas in quantum gravity, if the consistency of our theory required β to take discrete values (from a timelike Liouville theory perspective see also [35,51]). A vanishing Z (0) grav [Λ] also follows form the timelike Liouville DOZZ formula proposed in [3], since it vanishes when evaluated for three O β = e 2βϕ operators. In particular we highlight that Z (0) grav [Λ] vanishes in the limit β → 1 − (5.4), followed by Λ → 0 + . One can view this limit as ending at the bosonic string with the 26 dimensional (c m = 25 plus c L = 1) flat target space. Our analysis therefore suggests a novel argument for the vanishing of the genus zero partition function of the critical bosonic string.
It is interesting to ask whether the real saddle accompanied by its infinite perturbative series expansion yields a function with finite radius of convergence. Somewhat remarkably, given that both saddles have precisely the same perturbative coefficients, we can divide (5.1) by (1 − e 2iπ/β 2 ) leading to the expressioñ The small β expansion ofZ is given by that of the real Euclidean dS 2 saddle alone whose perturbative coefficients are c n . It would be interesting to understand from a purely gravitational perspective whether we should include the additional saddle, as suggested from the DOZZ approach.
In terms of c m > 25, the Λ dependence at genus zero is given by This can be viewed as an all orders completion of (1.9) in the 1/c m expansion, where we understand the subleading terms as small corrections to the entanglement entropy across the dS 2 horizon due to the presence of dynamical gravity [52]. For instance, the first correction to the c m /6 coefficient in (1.9) is −19/6. It would be interesting to test these ideas through other more direct methods [53,54]. According to the picture of Gibbons and Hawking [17,18], the additional terms ϑ and f (0) (c m ) encode further contributions to the dS 2 entropy. Remarkably, an essentially equivalent formula to (5.7) holds for spacelike Liouville theory which is the theory one is led to when considering two-dimensional gravity coupled to a matter conformal field theory with c m < 1.
Finally, we comment on compact higher genus surfaces. 9 In even-dimensions the partition function of quantum field theory on a fixed background has a somewhat ambiguous overall normalisation (a discussion can be found in [55]). For the specific case of two dimensions one can absorb the constant by shifting the coefficient ϑ of the Euler character viewed as a local counterterm. On the other hand, it should be emphasised that the detailed structure of f (0) (c m ) encodes crucial information about the local field theoretic structure of the 9 It is also worth considering manifolds with boundaries such as the disk partition function (see for example [47,[56][57][58]).

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original theory. Further to this, the contribution from higher genus corrections renders the normalisation less ambiguous -we can only fix the physical coupling ϑ for a single genus at most. For higher genus, based on the reasoning of [6,7], it is reasonable to anticipate the form If ϑ 1, these contributions will be exponentially small corrections as compared to log Z (0) grav [Λ]. It would be very interesting to compute f (h) (c m ) in the large c m limit for all h. Unlike the case of genus zero, the higher genus contributions will not be organised in terms of an expansion around a real saddle point solution. Perhaps this is remedied by fixing the area of the surface [1,59].

A Spherical harmonics
We use real valued spherical harmonics throughout the paper. We denote by Y lm (θ, φ) the complex spherical harmonics defined by where P l,m is the associated Legendre function, and m ∈ [−l, l] with l ∈ N. Real spherical harmonics Y lm (θ, φ) can be obtained using the linear combinations The Wigner 3-j symbol is given by the Clebsch-Gordan coefficients and gives the integral of the product of three complex spherical harmonics

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3-j symbol relations. The Clebsch-Gordan coefficients satisfy various properties. In particular they obey the orthogonality relation

B Invariance under PSL(2, C)
We explain the invariance of the Liouville action In the fiducial metricg the Liouville action is given by Now we show how each term individually transforms under (B.2). We adopt the notation For the last term it is quite easy, although importantly this term has to be invariant under (B.2) only in the semiclassical limit. We have The first term in the Liouville action (B.5) transforms under (B.2) as

Melonic term.
Although position space is useful to assess the nature of the ultraviolet divergences, it is not so useful to evaluate the diagrams. It is convenient to go to momentum space for this purpose. To do this, we can substitute the expression (D.2) for G(Ω; Ω ) into the first term of (D.5) to obtain ≡ − 1 18π 2 c 1 (2π) 3

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Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.