Instantons and the Hartle-Hawking-Maldacena Proposal for dS/CFT

We test the Maldacena proposal for the Hartle-Hawking late time quantum state in an asymptotically de Sitter universe. In particular, we calculate the on-shell action for scalar instantons on the southern hemisphere of the four-sphere and compare the result with the renormalized on-shell action for scalar instantons in EAdS$_4$. The two results agree provided the corresponding instanton moduli as well as the curvature radii are analytically continued. The instanton solutions in de Sitter are novel and satisfy mixed boundary conditions. We also point out that instantons on $S^4$ calculate the regularized volume of EAdS$_4$, while instantons on EAdS$_4$ calculate the volume of $S^4$, where the boundary condition of the instanton in one space is identified with the radius of curvature of the other. We briefly discuss the implications of the above geometric property of instantons for higher-spin holography.


Introduction
The task of extending the holographic principle to an explicitly time-dependent, cosmological, setting proves to be as non-trivial as it is interesting.A particularly simple approach is Maldacena's proposal [1] (see also [2]) for the evaluation of the Harte-Hawking (HH) late time quantum state in an asymptotically de Sitter universe 1 .The HH state is specified [7] by a Euclidean wave function of the schematic form: where ∂M is a 3-dimensional spacelike hypersurface near the future infinity of an asymptotically de Sitter space-time with radius ℓ dS .We have collectivelly denoted the bulk fields as φ.
Then, Maldacena's proposal entails that the HH state can be obtained if one calculates the corresponding renormalized on-shell action on Euclidean AdS 4 (EAdS 4 ) with radius ℓ AdS , and performs the analytic continuation ℓ AdS → iℓ dS .Since the EAdS 4 on-shell action is reasonably well defined-it gives the partition function of a Euclidean 3-dimensional CFT-Maldacena's proposal gives a way to make sense of and calculate the HH wave function from AdS/CFT.
When analytically continued to Lorentzian signature, (1.1) gives the Bunch-Davies wave function in de Sitter space.AdS/CFT works best when there is an explicit string theory realisation of the bulk physics.At present there is no completely satisfactory string theory description for gravity with a positive cosmological constant, but an alternative and more adventurous set up for a concrete realisation of dS/CFT was proposed in [8].This is based on Vassiliev's higher-spin (HS) theory2 which provides the only known consistent classical description of interacting higher-spin gauge fields in a de Sitter background.Its suggested holographic dual is the Euclidean Sp(N ) vector model with anti-commuting scalars and it is a free CFT 3 when the higher-spin symmetry is unbroken.Since the quantum properties of free theories are known, the HS version of dS/CFT offers some hope of understanding quantum gravity (plus HSs) on dS 4 .Moreover, an analytic continuation is also at work here, as N → −N , where N ∼ ℓ 2 dS /G N , maps the Sp(N ) anti-commuting vector model to usual (commuting) O(N ) vector model which is believed to be the holographic dual of HS theory on AdS 4 .Notice that Newton's constant G N is held fixed.
The Hartle-Hawking-Maldacena ('HHM') proposal has been tested in a number of cases.The most explicit check in [10] involves pure gravity only: the on-shell action on Euclidean de Sitter (the four-sphere) is shown to analytically continue to the finite part of the volume of EAdS 4 , after dropping counterterms which are argued to give only imaginary phases which do not contribute to the measure |Ψ HH | 2 [1,10].It has also been tested in the presence of scalars and gauge fields by showing that their generating functionals for two-point functions in EAdS 4 and dS 4 are related by suitable analytic continuations [2,11,12].However, the HHM proposal and the relevant analytic continuation has not been tested for cosmological theories with matter fields and generic potentials.This would entail a discussion of exact non-trivial solutions which come with a moduli space, and then one wonders how the latter transforms under the analytic continuation in the HHM proposal.Also, it is important to know how boundary deformations (such as multiple trace deformations) of the CFT dual to EAdS 4 carry over to de Sitter space, and whether some complexification of the deformation parameter is involved.Finally, one may ask whether there exists an analytic continuation at the level of fields rather than particular solutions.
In this short note we study exact solutions in a scalar theory conformally coupled to gravity and check that the HHM proposal works, namely the HH state is given by an analytic continuation of the holographic EAdS 4 partition function.These solutions are instantons: they are zero energy, exact solutions of the Euclidean equations of motion with finite action.Such solutions for scalar fields exist in EAdS 4 [13,14] and, as we show here, also in 4-dimensional de Sitter space.Further investigation of these solutions is relegated to a companion paper [15].We test the HHM proposal for these solutions and find exact agreement, provided the EAdS 4 radius as well as the moduli of the solution and their boundary conditions are analytically continued.We also note there exists a simple geometric description of our results.In particular, the renormalized on-shell action of scalar instantons on EAdS 4 evaluates the volume of the four-sphere, while the corresponding on-shell action of instantons on S 4 evaluates the regularized volume of EAdS 4 .In both cases, the instanton moduli serve as regulators of the corresponding volume forms.The above geometric description allows us to interpret the on-shell action of scalar instantons as the free-energy of a theory on S 3 and this in turn may have implications for the holography of HS theories.
2 Instanton Solutions in Euclidean AdS 4 and dS 4

The HH state and holographic partition functions
A central object for the holographic principle is the properly renormalized bulk on-shell action.This can be evaluated as a functional of the boundary conditions for a suitably regular solution of the bulk equations of motion.In this case, depending on the choice of boundary conditions it yields either a generating functional for quantum correlation functions or an effective action for a putative CFT living on the boundary.
However, given a particular solution of the bulk equations of motion, namely one where the bulk fields assume fixed boundary values, the renormalized bulk on-shell action evaluates the free energy of the boundary theory and therefore the partition function as: For example, the renormalized Einstein-Hilbert action using the Poincaré patch of EAdS 4 gives zero, while the corresponding quantity evaluated using a global parametrization of the bulk metric with conformal boundary metric S 3 yields the non-zero free energy of a CFT on S 3 .In the presence of bulk matter fields, things are less clear as exact solutions of the corresponding nonlinear equations of motion are uncommon.Notable exceptions are instantons in EAdS 4 [13,14].In particular, as we we review in Appendix A, for conformally coupled scalars in EAdS 4 with boundary behaviour φ(z, x) → z φ (0) ( x) + z 2 φ (1) ( x), where φ (0) ( x) takes a fixed form, the bulk on-shell action S on-shell [φ (0) ] = − ln Z0 gives the logarithm of the partition function Z0 of the dual boundary CFT, namely the CFT having in its spectrum the operator O 1 of dimension ∆ = 1 and O 1 ∼ φ (0) .Moreover, having this result one can calculate by a Legendre transform the partition function of the usual boundary CFT, namely the one having an operator O 2 of dimension ∆ = 2 and O 2 ∼ φ (1) .In [14] the partition function Z0 was interpreted as giving the probability for the nucleation of the instanton vacuum in the boundary theory.
Regarding now the HH state for an asymptotically dS 4 universe, a simple example arises in the absence of matter fields when it is given by the on-shell value of the EH action with a positive cosmological constant.After the analytic continuation of dS 4 to the 4-sphere, the result is finite and proportional to the volume of S 4 .It is then easily seen [10] that this is analytically continued to the finite part of the holographic partition function on S3 .As in the EAdS 4 case, non-trivial results for the HH state in the presence of bulk matter are uncommon.In this note we will improve on this situation by providing results for dS 4 instantons which, as we will see, are intimately related to the usual EAdS 4 ones.Our main aim is therefore to evaluate both the HH state and the holographic partition functions of systems involving gravity and matter fields and to test whether they are still related by analytic continuation.In doing so, we will obtain some interesting new results.

Action and solutions
In the context of AdS/CFT there are few examples where particular exact solutions of the bulk equations of motion of gravity coupled to scalars are known.Exact solutions are needed in order to calculate the exact partition function of the CFT.A non-trivial example was studied in [13,14].In this paper we write down similar solutions for de Sitter space and use them to test the HHM proposal.
The Euclidean bulk action we will consider contains the Einstein-Hilbert term, a conformally coupled scalar with a φ 4 potential, and the Gibbons-Hawking term modified by a coupling to the conformal scalar.It should be noted from the outset that for Λ < 0 and the special value of the quadratic coupling λ = λ cr = 2κ 2 /ℓ 2 AdS , this action is a consistent truncation of N = 8 sugra to the diagonal of the Cartan subgroup U(1) 4 of the SO(8) gauge group [14,16].We will encounter this special value of λ later on.
As usual, we regulate the theory introducing a large distance cutoff ǫ which we send to zero at the end of the calculation [17] 3 : where κ 2 = 8πG N and the cosmological constant can be either positive or negative.The difference between a positive and a negative cosmological constant Λ = ∓ 3 ℓ 2 is in the location and orientation of the regulated conformal boundary ∂M ǫ and in the counterterms: For the Euclidean dS case, no counterterms are needed because the wave function (1.1)only includes contributions from configurations that are asymptotically regular.The instanton solutions are constructed using the Weyl invariance of the matter part of the action, hence it is useful to use global coordinates that are conformal to I × ∂M , where I is a (finite or infinite) interval.In the cases at hand ∂M = S 3 .In EAdS 4 we will use conformal cylinder coordinates: For the purposes of the HHM proposal we will work with half Euclidean de Sitter, i.e. the southern hemisphere of S 4 : Lorentzian de Sitter is obtained by Wick rotating back r → −iℓ dS τ .The relevant solutions of the scalar sector [15] are obtained by solving the Klein-Gordon equation: together with the requirement that the stress-energy tensor vanishes.The latter requirement turns out to give [19]: The solutions on EAdS 4 and S 4 are then given by: where i = 1, . . ., 4 and Ω i is a unit vector normal to the three sphere.ε can take the values ε = ±1.The requirement that these are solutions to the equation of motion (2.6) renders the moduli space non-trivial, 12 ℓ 2 dS − a 2 0 .In the next subsection we will see the reason for this: b 5 and a 0 are not moduli of the solution; they parametrize boundary conditions instead.From the boundary point of view, b 5 is the marginal coupling of a triple-trace deformation of the CFT.Therefore the moduli space of the solutions is EAdS , which is the condition required for regularity of the bulk solution.At the critical value λ AdS 12 = b 2 5 the effective potential of the dual field theory becomes unbounded from below, which was interpreted as an instability of the dual theory against marginal deformations, decaying via quantum tunneling [14].

Boundary conditions
Not all of the parameters a I , b I (I = 0, . . ., 5) are moduli as we now show.Two of them are boundary conditions relating the leading and subleading modes of the scalar: The leading and subleading terms in the expansion of the field are related by: where α = b 5 for EAdS 4 and α = a 0 for S4 − .Thus, b 5 , a 0 parametrize marginal triple trace deformations that change the boundary conditions from Dirichlet to mixed [14] (see also [19]).

On-shell action and the HH wave function
We stress that, given the action (2.2), the solutions (2.8) are exact: they have zero stressenergy tensor hence the EAdS/dS background stays unmodified.Thus we can compute the on-shell effective action including its finite part 4 .For simplicity we set the spherical modes a i = b i = 0 and get: where in the last line we used (2.9).In the special case of a Neumann boundary condition b 5 = 0, which requires negative coupling λ AdS < 0, we reproduce the result in [13], Notice that this is a positive quantity.The case λ AdS > 0 was described in [14] where it was found that the instanton solution signals an instability of EAdS against deformations by mixed boundary conditions that can be described via a Coleman-de Luccia scenario.The decay rate calculation was presented in [14,20].
The result of the corresponding calculation (1.1) for the de Sitter case is: and again we used the condition for the moduli space (2.10).If we take a 0 = 0, we get 12 ℓ 2 dS which can only be the if case λ S 4 < 0. In that case, the on-shell value of the action is:  2.15) we see that the first, pure gravity, term matches under the analytic continuation ℓ AdS → i ℓ dS .This is Maldacena's result in [10].The result is nontrivial in part because the EAdS 4 needs to be regularized and renormalized, whereas the S 4 − calculation of the HH wave function is completely finite.In order to match the matter contributions, however, in the second term we need to analytically continue the couplings as well.This analytic continuation from EAdS 4 to S 4  − is an invertible map γ: (2.17) It follows that the coupling constant does not change, i.e. γ(λ AdS ) = λ dS .Notice that the fact that the moduli have to be analytically continued is a consequence of the prescription to analytically continue the EAdS 4 radius of curvature.Then the two expressions exactly match.
In contrast to [10,12], we did not need to write absolute value bars around the HH wave function (2.15) because this is the full semi-classical result (1.1) which is real in the Euclidean.This result can be directly compared after analytic continuation, as we have seen, to the Euclidean AdS/CFT partition function because the latter is finite-we took into account the correct counterterms, thereby rendering a result that can be precisely matched without the need to take the real part.
The complexification of the moduli can be understood from the fact that they are dimensionful quantities, to be measured in units of the radius.Defining dimensionless moduli y I = a I /ℓ dS , z I = b i /ℓ AdS , I = 0, . . ., 5, we find the the moduli spaces can be represented as: (2.18) These are all real quantities on both sides.The moduli space is O(1,5) invariant: with η IJ the O(1,5) Minkowski metric.The analytic continuation is then simply an SO(1,5) map of the moduli space onto itself: and δ 2 = 1.
We note here that in order to enforce the boundary conditions (2.12) from the bulk equations of motion, one needs to add a further term to the action: This of course agrees, after the analytic continuation (2.17), with the term one gets in the dS 4 case, − 2π 2 ℓ 2 dS a 0 3a 3

Geometric interpretation
The 3-sphere partition function Z S 3 for a three-dimensional CFT measures its number of degrees of freedom.Moreover, it has been argued that for unitary CFTs the corresponding free energy is given (in suitably chosen units) by which is positive and satisfies an F -theorem, namely it decreases along RG flows from the UV to the IR [21,22].Holographically, the partition function is usually calculated using the bulk gravitational action on EAdS 4 with all other matter fields set to zero.The result is the first term in (2.14) and it is proportional to the dimensionless ratio ℓ 2 AdS /κ 2 .One may then wonder what the physical interpretation is of the second term in (2.14), which corresponds to the contribution of the bulk scalar fields.Notice that for λ AdS < 0 this term is also positive.A similar question may be asked for the result (2.16), namely whether this can also be interpreted as a partition function of a CFT 3 on a 3-sphere.Since (2.16) is not always positive, such a CFT 3 need not be unitary.
To this end, we will point out that the instanton contributions in (2.14) and (2.16) also arise from bulk gravitational actions, and hence can be interpreted as F -functions on the 3-sphere.This follows [23] from the well-known representation of the conformal factor of a metric as a scalar field with quartic self interaction.Indeed, consider the conformally related metrics One can then show that where we have defined We then see that the critical value for λ mentioned in the previous section arises when we related the scalar action to a gravitational action.Now, if we calculate the on-shell value of the instanton action I (g,φ) , we find: which is minus the on-shell value for the Einstein-Hilbert action I h , since R[h] = 4Λ h .The crucial observation now is that the instanton solutions φ inst on either S 4 − and EAdS 4 , with the moduli set to specific values, correspond exactly to the conformal factor relating the two metrics.Hence, the on-shell action of instantons on EAdS 4 corresponds to the volume of (half) S 4 and conversely, the on-shell action of instantons on the half S 4 corresponds to the volume of EAdS 4 , where the instanton deformation parameter b (which corresponds to a 0 in the previous section) regulates the volume of EAdS 4 .
Let us see how this arises.On the half S 4 with curvature radius ℓ and metric the instanton solutions are given by where λ S 4 > 0. It is important to note the presence of the instanton modulus b which is in principle unrelated to ℓ.In particular, since the range of the radial coordinate is ρ ∈ [0, ℓ), if we consider b > ℓ the solution is everywhere regular.
Next we notice that half S 4 and EAdS 4 are conformally related metrics.In particular where on the right EAdS 4 has a generally different radius b which is set equal to the instanton modulus.Hence, the calculation of the on-shell action for instantons on half S 4 -including an appropriate boundary term as in (3.3)-boils down to the calculation of the regularized volume of EAdS 4 .Notice that it is the presence of the instanton modulus b > ℓ that gives rise to a particular regularization of the volume of the EAdS 4 space with radius ℓ.
Explicitly, we obtain This diverges as α → 1, which we shall interpret presently.
Next we consider the instanton solutions on EAdS 4 with radius ℓ.These have the form with λ AdS < 0 and b the modulus which we take here again to be b > ℓ.The on-shell action would give part of the volume of S 4 with radius b.We find where here λ AdS < 0.
The association of the on-shell instanton actions with an EH action leads now to a remarkable correspondence.Namely, it is not hard to see that (3.9) and (3.11) are mapped to each other by the transformation b → i b.This is natural, if we recall that b is the radius of both the S 4 and EAdS 4 of the associated EH actions.Thus, we have shown that the on-shell instanton action can be viewed as a partition function on a 3-sphere of radius b, arising from a bulk gravitational action with Newton's constant κ, if we identify the coupling λ ∼ κ 2 /b 2 .
We notice here that the instanton (3.7) is of the type (2.8) under the following identification: as is easy to verify doing the coordinate transformation.We see that the analytic continuation b → ib indeed corresponds to ℓ dS → −i ℓ AdS .The limit α → 1 of the above action, in which the instanton deformation parameter approaches the curvature radius, corresponds to the limit a 2 5 → 0 − and a 2 0 → λ S 4 12 ℓ 2 dS + which is the critical value at which the moduli space shrinks to zero curvature radius.In this limit, the instanton part of the action (2.15) diverges as 1/a 3 5 , which is precisely the divergence of the EAdS 4 volume5 .Thus, the instanton is computing the EAdS 4 volume, and the boundary deformation parameter a 0 regulates this volume.The critical value of the deformation parameter a 2 0 = λ S 4 12 ℓ 2 dS , for which we get the correct divergence, corresponds, via the analytic continuation (2.17), to the critical value at which the dual boundary theory becomes unstable.

Discussion and conclusions
In this note we have tested the HHM proposal in the case of scalar instantons.We have calculated the on-shell action for instantons on half-S 4 , which yields the late-time HH state, and compared it with the on-shell action for instantons on EAdS 4 .The results match under the HHM prescription of analytically continuing the curvature radii.Additionally, we have found that it is also necessary to analytically continue the boundary condition, which corresponds to a marginal triple trace deformation.This provides new evidence that the HHM proposal works for exact, but non-trivial configurations.
Our goal was to check the HHM proposal but along the way we got a number of results which we believe are relevant for the HS proposals in de Sitter space, in particular for the Sp(N ) vector model that has been conjectured to be dual to it.On the EAdS side, the instanton modulus b 5 corresponds to the coefficient of a marginal triple trace deformation (2.21) for an operator of dimension 1.Under the analytic continuation to dS space this is again a triple trace deformation, and the free energy has been computed.The free energy as a function of the boundary deformation parameter presents zeroes which usually signal an instability of the theory.The presence of instabilities seems to be connected to the fact that our free energy result (2.16) may be negative, hence it corresponds to a non-unitary CFT 3 .Similar behaviour has been found in [12].It would also be interesting to work out the relationship of this result with holographic stochastic quantisation [23,24].
We also found an interesting geometric realization of the same computation, in which the S 4 instantons (for particular values of the moduli) are seen to compute the regularized volume of EAdS 4 , and the EAdS 4 instantons are seen to compute the volume of the foursphere.The regulator of the EAdS 4 volume is a 0 , with the divergence appearing precisely for the critical value a 2 0 = ℓ S 4 12 ℓ 2 dS .This might imply that a sector of the Sp(N ) model with this marginal deformation is dual to a pure gravitational theory with no scalars, and hence signal a duality between Sp(N ) models with different values of the deformation parameter.This is reminiscent of similar scenarios as e.g. the dualities in [25,26].
As stressed in [13,27], instantons of the type found here seem to have special holographic properties because they parametrize different Weyl vacua of the theory.The exact bulk action can be calculated and compared to the boundary effective potential [14,19].It was pointed out in [13] that the boundary values of bulk instantons are also solutions of the equations of motion of a three-dimensional conformally coupled scalar field theory on the boundary, which was conjectured to be the effective action for an operator of dimension 1.In [14,19] it was found that this action in fact agrees with the effective action near the critical point, which can be calculated by different methods.Given the robust structure of the instanton solutions it is unlikely that this is a mere coincidence.Near the critical point, as we have seen, the instantons in fact just calculate the volume.We point out here that a similar boundary effective action description applies to the de Sitter instantons as well.
Finally, we wish to point out that our instanton solutions are intimately related to the SO(4) and SO(3, 1) invariant solutions of 4-dimensional HS theory found in [28], which are also related [14] to a consistent truncation of N = 8 gauge supergravity down to a single scalar of the SO(8) group.In that sense, our results should also provide the partition functions of both the above theories, at the scalar instanton vacua.We leave a more detailed analysis of this intriguing point of view for future work.
to evaluate W [ φ(0) ].According to (A.9) this would give us the partition function Z0 of the dual boundary theory as W [ φ(0) ] = Γ[− φ(0) ] = ln Z0 (A.13) On the other hand, given W [ φ(0) ] we are able using (A.3) to obtain the value of the effective action of the boundary theory at its non-trivial vacuum A * , as Γ[A * ] = W [ φ(0) ] − A * φ(0) = ln Z 0 (A.14) ) and (2.10) we see that the moduli spaces of instantons on EAdS 4 and dS 4 are themselves EAdS 4 or dS 4 depending on the sign and values of the quartic couplings λ AdS and λ dS ; more specifically, the curvatures depend on the particular combinations λ AdS 12 ℓ 2 .16) which after analytic continuation ℓ S 4 → −iℓ AdS , λ S 4 → λ AdS agrees with (2.14), as it should.Notice that this is negative for λ S 4 < −2κ 2 /ℓ 2 S 4 and vanishes for λ S 4 = −2κ 2 /ℓ 2 S 4 , which coincides with the critical value arising in the consistent truncation of N = 8 sugra, after the analytic continuation of the dS 4 radius.