The two-sphere partition function from timelike Liouville theory at three-loop order

While the Euclidean two-dimensional gravitational path integral is in general highly fluctuating, it admits a semiclassical two-sphere saddle if coupled to a matter CFT with large and positive central charge. In Weyl gauge this gravity theory is known as timelike Liouville theory, and is conjectured to be a non-unitary two-dimensional CFT. We explore the semiclassical limit of timelike Liouville theory by calculating the two-sphere partition function from the perspective of the path integral to three-loop order, extending the work in 2106.01665. We also compare our result to the conjectured all-loop sphere partition function obtained from the DOZZ formula. Since the two-sphere is the geometry of Euclidean two-dimensional de Sitter space our discussion is tied to the conjecture of Gibbons-Hawking, according to which the dS entropy is encoded in the Euclidean gravitational path integral over compact manifolds.


Introduction
Very little is known about de Sitter space at the quantum level. In particular, there is no S-matrix as for an asymptotically flat spacetime, or correlation functions as we encounter in the AdS/CFT dictionary (for a review see e.g. [1][2][3]). Because of the accelerated expansion, an observer in a de Sitter spacetime is surrounded by a cosmological horizon. Conjecturally a finite entropy is associated to this horizon, which in our Universe is of order S dS = 10 120 . Macroscopically, Gibbons and Hawking [4,5] conjectured that the entropy of a de Sitter universe is encoded in the path There are several questions about (1.1) which require further exploration. Firstly, no microscopic model for the de Sitter entropy is known. Moreover, performing the path integral (1.1) in general dimensions is a difficult task. Recent developments include [6][7][8][9][10][11][12][13]. In this paper, following the spirit of [6,[12][13][14][15], we restrict to a two-dimensional spacetime and explicitly calculate the gravitational path integral (1.1) in Weyl gauge [16,17]. While the path integral of two-dimensional quantum gravity is in general highly fluctuating, if coupled to a matter CFT with large and positive central charge c m , it admits a semiclassical two-sphere saddle. Vanishing conformal anomaly then implies that the Liouville central charge is large and negative, leading to timelike Liouville theory (TLT).
TLT is conjectured to be a CFT [6,18,19]. Unlike in (spacelike) Liouville theory, in TLT [6,18,19] the sign of the kinetic term is reversed. In Weyl gauge g = e 2βϕg the action of TLT on a two-sphere is given by In the above ϕ denotes the Weyl mode,g the fiducial metric with Ricci scalarR; Λ > 0 is the cosmological constant. Furthermore q = β −1 − β and the timelike Liouville central charge is given by c tL = 1 − 6q 2 . Restricting to genus zero, the path integral of interest is now We calculate three-loop corrections on top of the two-sphere saddle, extending the work in [6]. This allows us to explore the Gibbons-Hawking conjecture and could be useful to constrain a possible microscopic theory of a two-dimensional de Sitter universe.
Whereas the gravitational path integral (1.3) admits a semiclassical, β → 0, two-sphere saddle on top of which we can calculate loop corrections, there also exists a path integral independent approach toward the two-sphere partition function. Originally introduced for spacelike Liouville theory by Dorn-Otto [21] and Zamolodchikov-Zamolodchikov [22], the DOZZ formula captures the three-point function of the Liouville vertex operators. The DOZZ formula has been extended to TLT by Zamolodchikov and Kostov-Petkova [23][24][25][26] (see also [19]). Since however e.g. the timelike DOZZ formula in [19] for three area operators O β = e 2βϕ vanishes, contradicting the fact that the path integral yields a non-vanishing result, our comparison relies instead on analytically continuing the spacelike DOZZ formula, i.e.
where we also highlight the Λ dependence of the structure constant. Below we summarise the sphere partition functions obtained from the two approaches described above. Results.
• Analytically continuing C(b, b, b; Λ)| b→±iβ and thrice integrating with respect to Λ leads to the sphere partition function [6,27] Z DOZZ . This is conjectured to be the two-sphere partition function of TLT [27].
• On the other side, evaluating (1.3) on the round two-sphere saddle and calculating three-loop corrections on top of this saddle yields where a 1 = 27a 0 /(20π), γ E is the Euler-Mascheroni constant; Λ uv is the UV cutoff of our theory and 4πυ denotes the area of the two-sphere. Finally ⊖ denotes a "melonic" type of diagram, whereas "loops β 4 " comprises all three-loop diagrams. These diagrams appear at order O(β 4 ), and after taking into considerations all cancellations are diagrams of the form shown in figure 1. The gauge fixing of vol P SL(2,C) and the saddle create the expansion in odd powers of β in (1.6); It is immediately visible that Z DOZZ tL and Z tL do not agree with each other. From the path integral perspective, the conformal anomaly of the sphere, υ c tL /6 , is immediate, and the UV cutoff Λ uv combines with the area and the cosmological constant to render the sphere partition function dimensionless. These dimensions must be reverse-engineered in (1.5). Furthermore, whereas the forms of the small β expansions agree with each other, the coefficients do not. This was noted already in [6] and a conjectured solution was put forward. In this paper, by extending the loop corrections to third order, we could test this proposal. Taking scheme dependency into consideration and allowing the rescaling Λ uv → sΛ uv , we conjecture that for log s = − (2γ E + log π) the small β expansions of (1.5) and (1.6) agree not just to second but up to third order. Furthermore we observe a systematic cancellation of UV divergent diagrams as well as cancellations between UV finite diagrams. After the dust settles only the type of diagrams shown below survives at order O(β 4 ). These diagrams suggest a nice generalisation of the melonic type diagrams appearing at order O(β 2 ) to higher orders. Furthermore, we note that the agreement between(1.5) and (1.6) also at threeloop order provides further evidence that the semiclassical expansion leads to a loophole around Gribov-phenomena [28].
A major difference between (1.5) and (1.6) clearly remains. This is the appearance of the term . From a path integral perspective, it can be interpreted as the contribution of a second complex saddle. As we will explain in section 5, in the spirit of [29], both saddles are allowed from a path integral perspective.
Outline. The outline of this paper is as follows. In section 2 we introduce TLT and explain its main features. We explain the round two-sphere saddle and the small fluctuations thereof. For details of the calculations we refer to [6]. In section 3 we delve into the diagramatics. After introducing the propagator on the two-sphere we recap the two-loop diagramatics explored in [6]. Our main calculations are the three-loop contributions studied in section 3.3. In section 4 we summarise the two-and three-loop contributions, explain the cancellations of the UV divergences, and present the TLT sphere partition function. In section 5 we compare our result for the two-sphere partition function to the one obtained upon analytically continuing the spacelike DOZZ formula for three area operators. We discuss scheme dependency and the allowability of the complex saddle. Finally section 6 provides some concluding and speculative remarks.

Timelike Liouville theory
In this section we introduce the components of (1.3). We will not delve into any details, for which we refer to [6].
We fix the background fiducial metric ds 2 to be the Fubini-Study metric on the two-sphere with area 4πυ. If we denote by ϕ the Weyl mode we have ds 2 = e 2βϕ ds 2 , ds 2 = 4υ dzdz (1 + zz) 2 ≡ e 2Ω(z,z) dzdz . (2.1) The action of TLT on a two-sphere topology in Weyl gauge is given by Expanding the Weyl mode into a basis of real spherical harmonics on the two-sphere we define the measure in (2.4) Here Λ uv is the UV cutoff of the theory.

Fadeev-Popov gauge fixing, round saddle & small fluctuations
First and foremost we need to take care of the infinite volume of P SL(2, C) in (2.4). This infinite volume might suggest a vanishing sphere partition function, however the invariance of the Liouville action under [6] ϕ produces another volume of P SL(2, C) upstairs and thus yields an infinity over infinity situation in (2.4). In (2.8) f (z) is an element in P SL(2, C) and we defined Ω(z,z) in (2.1). We will fix the volume of P SL(2, C) by using a Fadeev-Popov approach. We follow [6,16] and set the three l = 1 modes δϕ 1,m , m ∈ {−1, 0, 1} to zero. This fixes three of the six parameters of P SL(2, C) and we are left with the finite volume of SO(3). Explicitly we obtain where m ∈ {−1, 0, 1} and α n denote directions in P SL(2, C)/SO(3). One can check that (2.9) is The classical equations of motion of (2.2) are where −∇ 2 is the Laplacian with respect tog ij . Equation (2.11) admits a constant and real solution given by 1 and we obtain the saddle point contribution of (1.3) (2.14) We now add a small fluctuation δϕ to ϕ * ϕ → ϕ * + δϕ . Using the expansion (2.5) the Laplacian∇ 2 for δϕ obeys the eigenvalue equatioñ Expanding (2.2) around (2.15) we observe that for l ≥ 2 we are dealing with an infinite number of unsuppressed Gaussian terms. To cure this we will follow [30] and rotate δϕ → ±iδϕ. The resulting Jacobian is an ultralocal contribution [31] and can be absorbed in the measure. We thus have where Z saddle we defined in (2.14) and the perturbative part is given by

One-loop contribution
To quadratic order in the fields and in the semiclassical small β limit the path integral (2.18) including the Fadeev-Popov determinant is given by We highlight that we also Wick-rotated the l = 2 mode in the Fadeev-Popov determinant (2.9) and consequently we have in the above expression Whereas the Wick rotation δϕ → ±iδϕ cured the unsuppressed Gaussians for l ≥ 2, from (2.19) we infer that it created a Gaussian unsuppressed l = 0 mode. We cure this by Wick rotating a single mode back δϕ 00 → ±iδϕ 00 [30,31] . Keeping track of the resulting Jacobian we arrive at In the above expression we have been treating the l = 0, 1, 2 and l ≥ 3 modes separately. In particular for l ≥ 3 we encounter an infinite product which we can evaluate using for example a heat kernel regularisation scheme. We obtain where ν ≡ −2βq − 1/4, ∆ ± = 1/2 ± iν, and ζ(a, z) denotes the Hurwitz ζ-function. Furthermore we find For details to the calculation of the Fadeev-Popov determinant and the heat-kernel analysis we refer to [6].

***
To conclude this section we discuss the effect of the Fadeev-Popov determinant (2.9). Expanding the contribution of the Fadeev-Popov determinant for small β and stripping off the factor a 0 q 3 we where for notational convenience we defined Expanding the exponential, we obtain to order O(β 4 ) In particular we note the first appearance of a 2 will be at order O(β 6 ) and that we absorbed the phases of the Wick rotation of the FP determinant into the coefficients a i (2.10).

Diagramatics
In this section we calculate the higher-loop corrections to (2.18). We start by explaining the propagator on the two-sphere, review the two-loop calculations of [6] and then delve into the three loop-contributions.

Propagators & spherical harmonics
First we note the propagator. For Ω, Ω ′ two points on the round two-sphere we have , Ω is a point on the round twosphere and Y l,m (Ω) denote the real spherical harmonics. Our conventions we explain in appendix A. In particular at coincidence where Ω = Ω ′ we have The last equality follows from the fact that G(Ω; Ω) is Ω independent. The above sum diverges logarithmically, as expected for coincident fields in two dimensions. (3.2) holds true also if we remove the l = 1 modes since for each l the spherical harmonics form an irreducible representation of SO (3). Finally for the l = 2 modes we also need to take into consideration the effects arising from the Fadeev-Popov determinant (2.19), thus shifting the "mass" in A l . We now examine the Feynman diagrams in (2.18). We first review the two-loop calculation in [6] and then delve into the three-loop contributions. Care must be taken since we need to remove the l = 1 modes and adjust the propagator for the l = 2 modes.

Two-loop contributions
We now discuss the path integral (2.18) in a small β expansion. We remind the reader of the relation q = β −1 − β. The leading two-loop contribution has already been calculated in [6]. We summarise the main results. To two-loop order we obtain three different types of diagrams which we denote as double tadpoles, cactus diagrams and melonic type diagrams (see fig. 2)  Due to the β dependency of Since the 3j symbol obeys the triangle condition we can evaluate the last two contributions explicitly.
Clearly then only the second contribution in (3.7) is logarithmically divergent in the UV whereas the other diagrams yield finite contribution.
We now start our journey calculating the three-loop contributions, thereby finding a contribution exactly and fully cancelling the logarithmically divergent contribution in (3.7).

Three-loop contributions
To three loop and at order O(β 4 ) we obtain from (2.18) the following seven contributions Here · denotes the expectation value with respect to the Gaussian integral (2.19). We explain the diagrams individually.
First we highlight that two-loop diagrams appear at order O(β 4 ) not only through the β dependency in A l (3.6) (see (3.7)) but also due to the fact that the couplings in (2.18) are of the form qβ a , a ≥ 2, which itself has a small β expansion. The first two diagrams in (3.8) are thus well known, adding melons, double-tadpoles and vector diagrams: where c 1 = 6 and c 2 = 9 are two combinatorial factors. So we find melons and double-tadpoles also at three-loop order. The second contribution in (3.8) are cactus diagrams 1 6π (3.10) Now we delve into the world of three-loop contributions. We have The combinatorial factors are c 1 = 60 and c 2 = 45; The sextic diagram evaluates to 1 45π Now we are entering more subtle ground. We find where we highlight that in the last sum the sum over l 5 runs over all positive integers subject only to the triangle condition of the 3j symbols (see appendix A). Finally we are dealing with the last two terms in (3.8). The evaluation of these diagrams is a bit more cumbersome and we refer to appendix B for the results. Graphically the above diagrams correspond to 2 Fig. 3: Diagrams in order of appearance starting at (3.11).

Sphere partition function
We now combine the 26 diagrams

Three-loop two-sphere partition function
Combining (2.14), (2.21), (2.28) as well as loops β 4 , the semiclassical two-sphere partition function in TLT at three-loop order is given by . (4.4) and c tL = 1 − 6q 2 is given by The partition function (4.3) clearly reflects the sphere anomaly of a two-dimensional CFT, providing independent evidence for timelike Liouville to be a conformal field theory. Numerically we obtain strong evidence (see appendix B) that at two-loop where we defined the melonic diagrams in (3.4). We thus end up with

(4.7)
We finish this section by summarising the main ingredients of (4.7) • The ±i ambiguity arises after Wick rotating the unsuppressed l = 0 mode backwards.
• The υ dependency is reminiscent of the sphere anomaly of a two-dimensional CFT providing independent evidence that TLT is indeed a two-dimensional CFT.

Comparison to (timelike) DOZZ
The goal of this section is to compare (4.7) to the sphere partition function obtained upon analytically continuing the DOZZ formula for three area operators O β = e 2βϕ .

Sphere partition function from (timelike) DOZZ
Another possibility to obtain the sphere partition function exploits the DOZZ formula of three area operators. The three-point function structure constant has originally been introduced for spacelike Liouville theory. Whereas the analytic continuation ϕ → ±iϕ, b → ∓iβ and Q → ±iq from the spacelike to the timelike Liouville action is an admissible process, it is in general not for the threepoint function. Care must be taken because of the pole structure of the DOZZ formula. It seems however that the spacelike DOZZ formula for three area operators O b = e 2bϕ admits a well defined analytic continuation. Since additionally the timelike DOZZ formula in [19] for three area operators O β = e 2βϕ vanishes, contradicting a non-vanishing sphere partition function from a path integral perspective we will proceed with our comparison using C(b, b, b; Λ)| b→±iβ . We thus consider To make contact with the Liouville partition function Z tL [Λ] in (1.3) we note that [27] − where we have used that [22] This leads to [6,27] Z DOZZ In the semiclassical β → 0 + limit we obtain where we have taken e −iπ = −1. This we compare with (4.3). However we quickly realise that (5.5) bears two subtleties, which we can phrase in terms of two questions: • What is the regularisation scheme of the DOZZ formula?
• What is the meaning of e It is clear that the challenge consists in evaluating the diagrams in (4.2), in particular the last contributions in loops β 4 stemming from We rely on a numerical evaluation of the 3j symbols in (4.2) closely approaching the exact result (5.7) (see appendix B for some numerical evidence). We believe that proving (5.7) exactly is solely a numerical issue. 5 To approach the second question we allow ourselves a short detour. One possible interpretation of the exponential in e 2iπ β 2 is as a second complex saddle in the path integral with ϕ c = ϕ * + πi/β. Whereas in principle the gravitational path integral allows for the integer indexed family of complex constant solutions ϕ c, * = ϕ * +πin/β, n ∈ Z (2.11) the DOZZ formula dictates that only the complex saddle with n = 1 should be included in the path integral. It is not clear to us why this specific saddle needs to be included. Since (2.1) ds 2 = e 2βϕc ds 2 = e 2βϕ * +2πi ds 2 = e 2βϕ * ds 2 , (5.9) it does not affect the metric and hence following [29] both saddles are allowed.
Finally we note that the agreement between (5.5) and (4.3) provides further evidence that the semiclassical expansion provides a loophole around Gribov-phenomena [28].

Outlook
All-loops path integral. The analytically continued DOZZ formula provides the all-loop conjectured sphere partition function of two-dimensional quantum gravity. Provided we take into consideration the scheme dependency of the regularisation scheme (5.6) its semiclassical expansion agrees with the path integral expansion. Importantly however the DOZZ formula predicts the inclusion of one additional complex saddle in the path integral picture. It would be interesting to understand why the DOZZ formula picks one and not multiple additional complex saddles. Furthermore we observe that at two-and three-loop all UV divergences in the path integral cancel and that the surviving diagrams are some sort of generalised melonic type diagrams ( fig. 7). It would be interesting to understand if this is the general structure of diagrams in TLT on the two-sphere.
Besides being non-unitary timelike Liouville theory provides a well defined UV finite QFT on the sphere. It would be interesting to understand if there is a symmetry argument that could explain the cancellations of the UV divergent and UV finite diagrams observed in loop calculations. Maybe understanding the set-up in position space could be helpful for this also. Higher genera. Whereas timelike Liouville theory provides an unconstrained sphere saddle on a genus zero surface, as compared to (spacelike) Liouville theory which does so only upon fixing the area of the physical metric [13,20], upon increasing the genus of the Riemann surface spacelike and TLT switch roles. Since the Ricci scalar is negative for h ≥ 2 TLT now only admits a saddle upon fixing the area of the physical metric.
A microscopic model for timelike Liouville? Although various attempts of a microscopic model for timelike Liouville theory have been discussed (see e.g. [35]) none of them has so far been able to provide sufficient evidence. It would be interesting to see whether the comparison of the sphere partition function of timelike Liouville theory with the sphere partition function of spacelike Liouville theory [13,20] can provide some evidence for a microscopic model. This model might be more subtle than a matrix model.

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 3j symbol is given by the Clebsch-Gordan coefficients and gives the integral of the product of three complex spherical harmonics 3j symbol relations. The Clebsch-Gordan coefficients satisfy various properties. In particular they obey the orthogonality relation The 3j symbol is non-vanishing iff The latter condition we refer to as the triangle condition.