Higher spin de Sitter holography from functional determinants

We discuss further aspects of the higher spin dS/CFT correspondence. Using a recent result of Dunne and Kirsten, it is shown how to numerically compute the partition function of the free Sp(N) model for a large class of SO(3) preserving deformations of the flat/round metric on ℝ3/S3 and the source of the spin-zero single-trace operator dual to the bulk scalar. We interpret this partition function as a Hartle-Hawking wavefunctional. It has a local maximum about the pure de Sitter vacuum. Restricting to SO(3) preserving deformations, other local maxima (which exceed the one near the de Sitter vacuum) can peak at inhomogeneous and anisotropic values of the late time metric and scalar profile. Numerical experiments suggest the remarkable observation that, upon fixing a certain average of the bulk scalar profile at \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$ {{\mathcal{I}}^{+}} $\end{document}, the wavefunction becomes normalizable in all the other (infinite) directions of the deformation. We elucidate the meaning of double trace deformations in the context of dS/CFT as a change of basis and as a convolution. Finally, we discuss possible extensions of higher spin de Sitter holography by coupling the free theory to a Chern-Simons term.


JHEP02(2014)007 1 Introduction
A natural object to consider when studying an asymptotically (approximately) de Sitter spacetime, such as the universe during the inflationary era, is the wavefunction [1] as a function of small fluctuations of the bulk fields. For a free massless scalar φ (such as the inflaton in slow roll inflation) in the Bunch-Davies vacuum state 1 |E [2][3][4][5][6][7][8][9][10] in a fixed four-dimensional de Sitter background: one finds the late time Hartle-Hawking Gaussian wavefunctional at η = η c → 0 [11]: where ϕ k are the Fourier components of the late time profile ϕ( x). Such a Gaussian wavefunction gives rise to the scale-invariant fluctuations of the cosmic background radiation. Understanding the behavior of such a wavefunction for a large range of values for its arguments, which include the metric, and with the inclusion of quantum corrections is a basic problem in quantum cosmology. The perturbative Bunch-Davies wavefunction (1.1) was noticed to be a simple analytic continuation [11] of the partition function of a free massless field in a fixed Euclidean anti-de Sitter space. These observations, coupled with the correspondence between antide Sitter space and conformal field theory, motivate the proposal that at late times (or large spatial volume) Ψ HH is computed by a statistical (and hence Euclidean) conformal field theory, in what has come to be known as the dS/CFT conjecture [11][12][13]. 2 In its weakest form dS/CFT conjectures that the Taylor  (1. 3) The correlators O( x 1 ) . . . O( x n ) CF T , where the operator O has been rescaled by an appropriate η c dependent factor, compute late time bulk correlation functions with future boundary conditions [64]. The bulk late time profiles ϕ( x) are taken to be infinitesimal such that Ψ HH [ϕ( x), η c ] is merely a generating function of late time correlators about ϕ( x) = 0. In its strongest form, the claim is that the CFT is a non-perturbative definition of Ψ HH for finite deviations away from the pure de Sitter vacuum and at finite N . Particularly, Ψ HH is computed by the partition function of the putative CFT with sources turned JHEP02(2014)007 on. The single-trace operators are dual to the bulk de Sitter fields. Abstractly speaking, if we could write down a complete basis B (which may include information about topology, geometry and matter) for the Hilbert space of our theory, Ψ HH would be computing the overlap of the Hartle-Hawking state |E with a particular state |β ∈ B. One could also consider computing the partition function with sources for more general multi-trace operators turned on. As we shall see this computes the overlap of |E with states which are not sharp eigenstates of the field operator or its conjugate momentum. More dramatically, if there are single-trace operators which are irrelevant they may correspond to an exit from the de Sitter phase (see for instance [32]).
de Sitter space arises as a non-linear classical solution to four-dimensional higher spin gravity [33][34][35][36][37][38]. This theory has a tower of light particles with increasing spin, including a spinless bulk scalar with mass m 2 2 = +2 and a spin-two graviton. The scalar potential has a minimum about the pure de Sitter solution and the kinetic terms of the higher spin particles carry the right signs and are canonical. Hence the de Sitter vacuum is perturbatively stable and free of tachyons and ghosts in this theory. Beyond perturbation theory, the late time Hartle-Hawking wavefunctional of asymptotically de Sitter space in higher spin gravity (at least when the topology at I + is sufficiently simple) is conjectured to be computed by the partition function of the Sp(N ) model [39] (see also [40,41]), with N ∼ ( / p ) 2 . This model comes in two flavors, either a free theory of N anti-commuting scalar fields transforming as vectors under the Sp(N ) symmetry or as a critical theory obtained from the Sp(N ) model by a double trace deformation [42]. The partition function of the critical model (at least at large N ) as a functional of the sources of the single-trace operators computes the wavefunction in the ordinary field basis. On the other hand, the free theory computes the wavefunction in a slightly modified basis. The quantum mechanical analogue of this basis is given by eigenstates of the Hermitian operatorς = (βx − αp) with α, β ∈ R. Given the wavefunction in the coordinate basis ψ(x) we can compute in thê ς-basis by performing the transform: dς e − iςx α ψ(ς). (1.4) Normalizability in thex-basis implies normalizability in theς-basis and vice-versa. However, a node-less ψ(x) will not necessarily give a node-less ψ(ς). The free Sp(N ) model computes the late time Hartle-Hawking wavefunction of the bulk scalar in the eigenbasis of the late time operator [43]: whereφ is the bulk scalar field operator andπ φ is the field momentum density operator. We have taken |η c | 1 as a late time cutoff and the combination g ij /η 2 c represents the spatial metric at this time in Fefferman-Graham gauge. It is somewhat remarkable that there exists a basis for which the wavefunctional is computed by a free dual theory given that in the ordinary field basis it is computed by a strongly coupled theory. The partition function of the free Sp(N ) theory on an R 3 topology is an explicit resummation of the correlation JHEP02(2014)007 functions and there are no non-perturbative phenomena. The remaining coordinates of the wavefunctional, which are late time profiles of the higher spin fields including the bulk graviton, are computed in the ordinary field basis for both the free and critical models.
The wavefunctionals for the bulk scalar in the two different bases are related by a functional version of the transform (1.4). As shown in [43], in the large N limit performing the transform amounts to finding the saddles of a functional equation. One of these saddles (not necessarily the dominant one) gives the field basis wavefunction whose perturbative expansion agrees with that computed perturbatively in the bulk about the pure de Sitter solution. In most of what follows, we will perform computations in theσ-basis, which amounts to the calculation of a functional determinant. This is a considerably hard object to compute and we will limit ourselves to situations where we have conditioned all bulk fields except the metric g ij and the scalar φ to have vanishing late time profiles. We should emphasize that this amounts to a very sharp conditioning and ultimately the situation could be altered by allowing higher spin deformations. 3 The main body of this paper is dedicated to the study of the partition function of the Sp(N ) model for a large class of SO(3) preserving deformations of the bulk scalar and graviton. That is to say we study the late time wavefunction for bulk graviton and scalar configurations with late time profile on an R 3 topology: where dΩ 2 2 is the round metric on an S 2 whose SO(3) symmetry is the one preserved. We also study the analogous problem on an S 3 topology, which amounts to a simple conformal transformation of the R 3 case, where the late time profiles take the form: (1.7) This allows us to examine the behavior of the wavefunction of higher spin de Sitter space for inhomogeneous and anisotropic deformations which extend the rather uniform and homogenous deformations studied in [43]. In [43] it was found that the wavefunction in thê σ-basis diverges as a function of uniform mode of the bulk scalar on the round metric on S 3 . One of the questions we would like to understand is whether such non-normalizabilities persist for less 'global' late time configurations such as the above SO(3) deformations. The above metric deformations are conformally equivalent to the flat/round metric on R 3 /S 3 . We find particularly striking numerical evidence, discussed in section 5, that upon fixing the uniform mode of the bulk scalar on S 3 (in the conformal frame where it is endowed with the standard metric) all other directions of a general SO(3) late time deformation are normalizable. We have also analyzed a different geometric deformation, this time homogeneous but anisotropic, which does not keep the metric in the same conformal class. This is a squashing deformation α of the round metric on S 3 expressed as an S 1 fiber over an S 2 : along with a uniform late time profile for the bulk scalar. When α = 0 the metric (1.8) reduces to the standard metric on S 3 . Once again, so long as the scalar profile is kept fixed we find that the wavefunction is bounded in the α direction. We begin by briefly reviewing the Sp(N ) model in section 2. In section 3, using technology developed in [45], we compute the wavefunction (in theσ-basis) for some Gaussian radial deformations of the bulk scalar. This amounts to computing a functional determinant of a scalar field with a radially dependent mass term. In section 4 we compute the wavefunction for a radial deformation of the geometry, in the presence of a radial mass deformation, that takes it from the flat metric on R 3 to a general form ds 2 = dr 2 +r 2 f (r) 2 dΩ 2 2 . In section 5 we compute the wavefunction for several harmonics on the three-sphere and linear combinations thereof and note that the wavefunction seems to diverge only when the zero harmonic becomes large and negative. We then discuss the behavior of the wavefunction on a squashed sphere with a uniform profile of the late time bulk scalar in section 6. In section 7 we make some general remarks about double trace deformations. We end by speculating on possible extensions of higher spin holography in section 8. Most of our calculations can carry over to the O(N ) model and its AdS 4 dual in higher spin gravity [46][47][48][49].

Wavefunctionals and the free Sp(N ) model
We wish to study the Hartle-Hawking wavefunctional of an asymptotically de Sitter higher spin gravity for deformations of the bulk scalar and graviton away from the pure de Sitter solution. We will restrict the topology of space to be R 3 or S 3 and allow only deformations that decay sufficiently fast at infinity. In this section, we remind the reader of the Sp(N ) theory and discuss how to compute its functional determinant for certain SO(3) invariant radial deformations. One motivation to do so is to understand the behavior of the wavefunction of higher spin de Sitter space for mass deformations that are more 'localized' than those studied in [43] which were uniform over the entire S 3 . It is also worth noting that computing analogous pieces of the Hartle-Hawking wavefunction for a simple toy model of Einstein gravity coupled to a scalar field with a simple potential would require significant numerical work even in the classical limit, let alone at finite N . One would have to find a complex solution of the Euclidean equations of motion that caps off smoothly in the interior and has the prescribed boundary values at large volume, and compute its on-shell action.

Wavefunctional
Recall that the action of the free Sp(N ) model on a curved background g ij with a source, m(x i ), turned on for the J (0) = Ω AB χ A χ B ≡ χ · χ operator (dual to the bulk scalar) is given by: (2.1) where N should be even. The fields χ A are anti-commuting scalars that transform as Sp(N ) vectors and Ω AB is the symplectic form. Notice that due to the presence of the JHEP02(2014)007 conformal coupling, the action is invariant under a local Weyl transformation of the metric: g ij → e 2W (x i ) g ij , so long as we also rescale the source as m(x i ) → e −2W (x i ) m(x i ). From the bulk point of view this amounts to performing a coordinate transformation η = e −W (x i ) η, as can be seen by studying the Starobinski-Fefferman-Graham expansion [50][51][52] near η = 0: is not the coefficient of the slowest falling power of η for bulk scalar. At the linearized level, the ν(x i ) profile is dual to the vev of the J (0) operator in the presence of an infinitesimal m(x i ) source.
Computing the late time Hartle-Hawking wavefunctional in theσ-basis with σ = m(x i ) amounts to computing the partition function of the Sp(N ) theory with finite sources turned on. Given that it is a Gaussian theory, we can integrate out the anti-commuting χ A fields and find: where: In the case of a metric g ij = e 2W (x i ) δ ij that is conformally equivalent to the flat metric on R 3 , it is convenient to compute the functional determinant in the conformal frame where g ij is the flat metric. This amounts to rescaling the source to: We will use this fact in section 4.

Functional determinant for radial deformations
We have seen that for conformally flat metrics our problem reduces to computing a functional determinant: where ∇ 2 R 3 is the Laplacian of the round metric on R 3 , namely ds 2 = dr 2 + r 2 dΩ 2 2 . The above object is badly divergent unless we regulate it somehow. We will regulate it using a heat kernel or zeta function approach, both of which give the same answer. In fact, this precise problem has been studied by Dunne and Kirsten in [45] for functionsm(x i ) which only depend on the radial coordinate, i.e.m(x i ) =m(r), and which vanish sufficiently fast at infinity. It was shown that the zeta function regulated determinant is given by the following sum:

JHEP02(2014)007
In the above, the factor (2l + 1) originates from the degeneracy of eigenfunctions on a two-sphere and T (l) (r) solves the equation: with boundary conditions T (l) (0) = 1 and dT (l) (0)/dr = 0. The parameter µ 2 ∈ R is a constant mass parameter that we will set to zero. The derivation of the above formula employs the Gelfand-Yaglom theorem [53], which expresses the regulated functional determinant of a one-dimensional Schrödinger operator in terms of a single boundary value problem. The problem of computing the logarithm of a ratio of functional determinants for purely radial operators reduces to an infinite number of Gelfand-Yaglom problems, one for each l, whose solutions need to be summed (this is the first piece on the right hand side of (2.7)) and regularized (this is the second piece on the right hand side of (2.7)). The applicability of the formula requiresm(r) to vanish faster than r −2 at infinity, and these are the only types of deformations for which we will compute the wavefunction in the latter sections. When implementing the above formula we must sum up to a certain cutoff l = l max which we take to be l max = 45. A discussion of how the error decreases with l max is given in appendix A.

Simple examples of radial deformations
The purpose of this section is to exploit the general formula (2.7) for a simple set of radial functions. By studying Ψ HH as a functional ofm(r) we can identify some qualitative features already observed in [43], such as regions where the wavefunction oscillates and grows exponentially, as well as some new ones. Furthermore, we can study its dependence on more detailed features of the localized deformation. The zeroes of the wavefunction in theσ-basis occur only when the effective potential V eff (r) = l(l + 1)/r 2 +m(r) of the differential operator −∇ 2 R 3 +m(r) is negative for some range of r. If the effective potential were positive for all r it could not have vanishing eigenvalues, and hence the wavefunction could not vanish. Thus, we expect all oscillations of the wavefunction in theσ-basis to occur in directions wherem(r) is negative for some range of r.
Assessing the magnitude of the wavefunction as a functional ofm(r) is a more complicated task. We observe that the wavefunctional acquires increasingly high local maxima between its oscillations only in regions where the quantity Im ≡ ∞ 0 dr rm(r) appearing in (2.7) becomes large and negative.
It is important to note that because we are working with the flat metric on R 3 , which has no scale, our functional determinants will have an associated scaling symmetry given by r → r/λ andm(r) →m(r/λ)/λ 2 . We should thus fix the scaling when studying the functional determinant/wavefunction.

Single Gaussian
We first considerm(r) to be given by a general single Gaussian profile:  See the left panel of figure 1 for an illustration of this deformation. Using equation (2.7) we can explore |Ψ HH (λ, A)|, where from now on whenever we write Ψ HH it is implied as a late time wavefunctional. Using the scaling relation r → r/ λ andm(r) →m(r/ λ)/ λ 2 we can set λ = 1. In figure 2 we show a plot of the functional determinant. We immediately notice the same qualitative feature that was present for the constant mass deformation on a round S 3 (displayed later on in figure 12). Namely, it oscillates and grows exponentially in the negative A direction. This is somewhat expected since our deformation is qualitatively similar to the mass deformation on the flat metric on R 3 one gets by the conformal transformation of a constant mass on S 3 (see appendix A). In particular, all oscillations occur for A < 0 and the magnitude of the local maxima increases for increasing |A| for fixed λ.

Gaussian ring
We can also study the functional determinant of a profile of the type: which describes a Gaussian-like ring peaked around r ∼ 1/2 a + √ a 2 + 4λ 2 . Erf[x] denotes the error function. See the middle panel of figure 1 for an illustration of this deformation. The factor of r 2 is included to ensure that the profile is continuously differentiable near the origin. Again, using the scaling relation r → r/ λ andm(r) →m(r/ λ)/ λ 2 we either fix the value of λ, |a| or |A|. We show an example in figure 3 where we have fixed the value of a.

Double Gaussian
As a third example we consider a double Gaussian profile: See the right panel of figure 1 for an illustration of this deformation. An example of |Ψ HH (λ i , a i , A i )| 2 with a 1 = 0 is shown in figure 4. Once again we observe a pattern of maxima encircled by regions where the wavefunction squared vanishes identically. Furthermore, the wavefunction grows for increasingly negative values of A 1 and A 2 .

Radial deformations of flat R 3 and pinching limits
In this section, we introduce and study a class of SO(3) preserving deformations of the flat metric on R 3 . We show that they are conformally equivalent to the flat metric on R 3 . Thus, the wavefunction can only depend on such deformations of the metric if we also turn on a radial mass m b (r). Turning on such a mass, we can then perform the symmetry transformation discussed at the end of section 2.1 to get a mass deformation m(r) on the flat metric on R 3 . We will pick a functional form that is related by the symmetry transformations of 2.1 to a constant mass deformation on a deformed threesphere geometry (see appendix A for details). Depending on the sign of a parameter, the deformed three-sphere will look either like a peanut or an inverse peanut, i.e. like bulbous pears inverted relative to one another and conjoined on their fatter ends. The partition function that we compute can then also be understood as the answer for the partition function on this deformed three-sphere with a constant mass deformation. The pinching limit will be when the waist of the peanut-shaped geometry vanishes. We emphasize that how we perceive these deformations of the late time metric depends greatly on what we decide are natural constant time slices, since there always exists a conformal frame where the late time metric is the flat one. Ideally, it would be useful to analyze a qualitatively similar geometric deformation that would take the original geometry outside its conformal class, but we must restrict to the former case in this section since we will be constrained by considering SO(3) preserving deformations. Section 6 will go beyond this restriction by considering a new conformal class.

Balloon geometry
Consider the following class of SO(3) preserving metrics defined on R 3 : with r ∈ [0, ∞). Consider a family of smooth functions f ζ (r) with ζ ≤ ζ * for positive ζ * that tend to unity both at large r and near r = 0. We require that f ζ (r) vanishes at some r = r * for the critical value ζ = ζ * . We furthermore impose that: For positive ζ, the geometries described by (4.1) can be pictured as a two-sphere whose size at some finite ζ < ζ * grows, then shrinks and subsequently grows again. If f ζ (r) = 1 the geometry is of course nothing more than the flat metric on R 3 . As we approach ζ = ζ * , the size of the two-sphere tends to vanish at r = r * eventually pinching the geometry into a warped three-sphere and a slightly deformed metric on R 3 . The choice (4.2) ensures there are no conical singularities at the pinching point. It would be of interest in and of itself to study geometries with conical singularities (see for example [54]). As a concrete example, we will take: The parameters a and λ are chosen and γ is tuned to obey the condition (4.2). Though ζ * is not an independent parameter it is useful to isolate in the expression. A schematic representation of this deformation, for positive ζ, is presented in figure 5.

Conformal flatness of balloon geometry
It is important to note that the geometry (4.1) is conformally flat. This can be shown in a straightforward fashion. Consider a coordinate transformation r = g(x). It immediately follows that if the following ordinary differential equation: has a smooth solution for g(x) whose derivative is positive for all x > 0 then our metric becomes: Though we cannot solve the non-linear o.d.e analytically, we can easily evaluate it numerically and confirm for several cases that g(x) satisfies the necessary requirements. Hence, our metric (4.1) is indeed conformally equivalent to the flat metric on R 3 . In figure 6 we give a numerical example of this. This result is already of some interest even for the case of ordinary Einstein gravity. It informs us that upon conditioning that all other fields vanish at late times, the abso-

Wavefunctions and balloon geometries
We now examine what happens to the functional determinant as we vary the waist parameter ζ for an example. We will also turn on a mass that would correspond to a uniform mass m on the deformed three-sphere (as discussed further in appendix A.3), which upon the conformal transformation discussed there becomes: This is the mass deformation on the balloon geometry. The final deformationm(x), to be used in the Dunne-Kirsten formula, is obtained by performing a conformal rescaling of the balloon geometry to the flat metric on R 3 : ds 2 = dx 2 + x 2 dΩ 2 2 . This requires a conformal rescaling of m b (r) to:m (4.7) Thus, we will study the functional determinant as a function of m and ζ. In figures 7 and 8 we display our numerical results. As expected, at m = 0, nothing changes as we vary ζ since the balloon geometries are conformally flat. However, when we turn on m = 0 the wavefunction becomes sensitive to changes in ζ. Interestingly, decreasing the girth of the throat while keeping everything else fixed is favored, at least near m = 0. Thus, though supressed exponentially with respect to the local maximum of the wavefunction at m = 0, the wavefunction does not vanish in the pinching limit. It is tempting to speculate that such pieces of the wavefunction might be connected to the fragmentation picture of [55].

Spherical harmonics and a conjecture
In this section, we present numerical evidence that when mapping the problem back to the three-sphere (using the discussion in appendix A), all profiles give a normalizable wavefunction upon fixing their average value over the whole three-sphere. Thus, it is conceivable that the only divergence of the wavefunction occurs precisely for large and negative values of a uniform profile over the whole three-sphere [43]; a single direction in an infinite dimensional configuration space! For instance, as we shall show below, by mapping the Gaussian profile (3.1) to the three-sphere and removing the zero mode from its expansion in terms of three-sphere harmonics, the resultant profile produces a normalizable wavefunction as a function of its amplitude.

Three-sphere harmonics
We now study some examples of SO(3) invariant deformations which correspond to harmonics of the three-sphere conformally mapped back to R 3 . These harmonics are the eigenfunctions of the Klein-Gordon operator on the three-sphere with metric ds 2 = dψ 2 +sin 2 ψ dΩ 2 2 . The k th harmonic (independent of the S 2 coordinates) is given by: As explained in appendix A, to evaluate the partition function for this deformation using the method of Dunne and Kirsten we must first perform the coordinate transformation ψ(r) = 2 cot −1 r −1 , and then scale the deformation by the inverse of the conformal factor that maps the three-sphere metric to the metric on R 3 . The final radial deformation which is to be used in the Dunne-Kirsten formula is: Taking k = 0 corresponds to the zero mode which has been previously studied in [43] and was found to be oscillatory and divergent as the coefficient c k goes to large negative values.
In figures 9 and 10 we plot the partition functions for higher harmonics as a function of the coefficient c k . We notice that they are all well-behaved and normalizable, at least in the range we have explored. This motivates us to consider deformations which are linear combinations of spherical harmonics. Having looked at a deformation that is the linear combination of the zero mode with the first harmonic, as well as a deformation that is the linear combination of the first harmonic with the second harmonic, we notice that the partition function is not divergent so long as the coefficient of the zero mode is kept fixed. Postponing a more systematic study for the future, here we simply consider a modified version of the single Gaussian deformation given in (3.1). Previously, we had found that as the overall coefficient of the profile becomes large and negative, the partition function diverges. The new profile we will study is obtained by mapping the Gaussian profile to the three-sphere, subtracting off its zero mode, and mapping it back to R 3 . Notice that the Gaussian profile mapped to the three-sphere is constructed from an infinite number of harmonics, and here we are subtracting the piece that seems problematic from our analysis where r(ψ) = tan( ψ 2 ) andm(r) = Ae −r 2 −4a 1 /(r 2 +1) 2 . In this case the integral can be done explicitly and we can solve for the coefficient a 1 analytically. The final form of the single Gaussian radial deformation orthogonal to the zero mode of the three-sphere becomes: We plot the functional determinant as a function of A in figure 11. Interestingly, the partition function is once again well-behaved for large values of A. An analogous analysis for the balloon geometries, where we fix the zero-mode on the conformally related three-sphere, also results in the boundedness of the partition function in the ζ direction (ζ defined in (4.3)). The above results motivate the conjecture: The partition function of any SO(3) symmetric "radial" deformation for which the three-sphere zero mode harmonic is fixed is bounded.
Notice that the three-sphere is chosen as the geometry in the conformal class on which to fix the uniform profile. show that fixing the uniform profile on the peanut is not sufficient to ensure that the partition function is bounded (though fixing some non-uniform profile would suffice). This is simply because keeping the uniform mass profile fixed to some negative value while taking ζ large and negative, which corresponds to the conformally related sphere getting fatter at the waist, implies that the uniform profile on the sphere is getting large and negative. Consistent with the rest of our observations, we see that the partition function is unbounded in this direction. Furthermore, the next section will provide evidence for a more general conjecture that would extend beyond the conformal class of the sphere.

Three-sphere squashed and massed
In this section, we would like to briefly revisit and extend some of the observations of [43] for a constant mass deformation on an S 3 . There, it was observed that in theσ-basis the wavefunction on an S 3 with a uniform mass deformation oscillated and diverged at large negative m S 3 . We explore in this section a new direction which is the squashing parameter of the round metric on the three-sphere (in the presence of a non-zero uniform scalar profile) and its effect on the zeroes and maxima. Unlike the previous SO(3) preserving deformations which were inhomogeneous, this SO(3) × U(1) preserving deformation is homogeneous yet anisotropic. Furthermore, the squashed three-sphere is not conformal to the ordinary round three-sphere. In fact, squashed spheres with different values of the squashing parameter belong to distinct conformal classes. Part of our motivation is to provide further evidence that the zeroes of the wavefunction are extended and that the local maxima of the wavefunction (other than the pure de Sitter one) will no longer necessarily peak about homogeneous and isotropic geometries. Additionally, and in the same spirit as the observations made in section 5, we find that upon fixing the value of the uniform profile over the whole squashed three-sphere the wavefunction is normalizable in the squashing direction.

Squashed and massed
Consider turning on a constant mass m S 3 for the free Sp(N ) model on the round metric on an S 3 whose radius a is fixed to one unless otherwise specified. The partition function JHEP02(2014)007 is given by [43]: The analyticity of Z f ree [m S 3 ] in the complex m S 3 -plane is ensured by the uniform convergence of the (regularized) infinite product of analytic eigenvalues which defines it. For physical applications we restrict to real m S 3 and note that this implies that the Taylor series expanded about any m S 3 = m 0 S 3 converges to the partition function above. Furthermore, the partition function has a zero if and only if one of the product eigenvalues in the functional determinant vanishes, which can only happen for m S 3 < 0. As shown in figure 12, this wavefunction grows exponentially and oscillates in the negative m S 3 direction. 4 It is worth noting that the m S 3 dependent part of the phase of the wavefunction vanishes. The real part of log Z f ree /N goes as |m S 3 | for large negative m S 3 and −(m S 3 ) 3/2 for large positive m S 3 .
It is worth studying what happens to the zeroes and local maxima of Z f ree in the presence of an additional deformation. A computationally convenient deformation is to squash the round metric on S 3 into that of a squashed sphere, which is a homogeneous yet anisotropic geometry. In this sense, this deformation is complementary to the inhomogeneous deformations we have been studying so far. We review the metric and eigenvalues of the squashed sphere with squashing parameter ρ in appendix B. Our method of regularization is a straightforward extension of heat kernel techniques used in section 3.2 of [43], and details can be found therein. In figure 13 we present a plot of the wavefunction as a function of the mass m S 3 and the squashing parameter ρ (the round metric on S 3 occurs at ρ = 0). 4 Divergences of the Hartle-Hawking wavefunctional have been discussed in other circumstances such as Einstein gravity coupled to a scalar field with a quadratic scalar potential and vanishing cosmological constant [56] or the wavefunction of dS3 on a toroidal boundary [57]. Their physical interpretation remains to be understood. For example, it might be a result of very sharp conditioning or an indication of an instability. We find that the local maxima are in general pushed away from ρ = 0 and the zeroes of the wavefunction are extended to enclose the local maxima. The fact that the zeroes are extended (codimension one in the (ρ, m S 3 ) plane) is perhaps not surprising given that the zeroes in figure 12 arise from Ψ HH [m S 3 ] (which is purely real) changing sign. This feature should not disappear, at least for small perturbations to the equation which determines Ψ HH [m S 3 ]. Furthermore, the maxima of figure 12 are pushed even higher when squashing is allowed. This can be seen in the right hand side of figure 13, where one notices that the second local maximum of Ψ HH [m S 3 ] at ρ = 0 is pushed further up to some ρ > 0. This observation strongly suggests that away from the origin, and under such extreme conditioning of the late time profiles of the bulk fields, the wavefunction peaks in regions where the metric and perhaps even the higher spin fields are highly excited. On the other hand, the perturbative de Sitter saddle centered at the origin remains a true local maximum.

Basis change, critical Sp(N ) model, double trace deformations
In this section we further discuss the transformation to the field basis and clarify the role of double trace deformations. We would like to comment on the transform to the bulk field basis, which gives us the wavefunctional as a functional of ν ≡ √ N σ (with ν defined in (2.2) and σ being related the source of the single-trace operator dual to the bulk scalar), in the large N limit.
As discussed in [43], one has to consider the free theory deformed by a relevant double trace operator f (χ·χ) 2 /(8N ). We also keep a source −if σ turned on for the single-trace χ·χ operator. The parameter f ∈ C has units of energy. Performing a Hubbard-Stratonovich transformation by introducing an auxiliary scalar field σ we find: Integrating out the χ A fields, the partition function becomes: , we see that the above expression is a Fourier type transform of theσ-basis wavefunction. In order for this to become the actual basis changing transform (1.4) to the eigenbasis of the field operator, the constant f must be taken to infinity. The f → ∞ limit (where we are keeping the size of the three-sphere fixed) corresponds to sending the ultraviolet cutoff (of the infrared fixed point theory) to infinity, in the same sense as [58], or roughly speaking it corresponds to the late time limit in the bulk. 5 We begin by performing a perturbative analysis for infinitesimal deformations of the free Sp(N ) theory at large N on an R 3 .

Perturbative analysis on R 3
For the sake of simplicity, we will put the theory on the flat metric on R 3 , akin to studying perturbations in a small piece of I + . 6 For reasons that will be clear momentarily we choose f = i|f | to be pure imaginary. 5 Another way to think about this is keeping f fixed and taking the large size limit of the three-sphere that the CFT lives on. This is because the dimensionless quantity is |f |a where a is the size of the sphere. Indeed at late times the three-sphere grows large. 6 The parallel story in anti-de Sitter space has been studied extensively [59][60][61][62]. From the bulk perspective in planar AdS4: ds 2 = 2 A (dz 2 + d x 2 )/z 2 , at least perturbatively about the empty AdS4 vacuum, the finite fA ∈ R double trace deformed theory computes correlation functions of the bulk scalar quantized with mixed boundary conditions. Near the boundary z → 0 of AdS4, the bulk scalar with mass m 2 2 A = −2 behaves as φ(z, x) ∼ α( x)z 2 + β( x)z with α( x) = fAβ( x). This boundary condition is different from the conformally

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We are interested in setting up a perturbative expansion about the σ ∼ 0 Gaussian peak of Z f ree [σ]. From (7.2) we can compute the two point function of O ≡ χ · χ at large N in the double trace deformed Sp(N ) theory on an R 3 . We do this by taking two variational derivatives of the logarithm of (7.2) with respect to σ(x i ) and evaluating at σ(x i ) = 0: where k is the magnitude of the momentum. For k |f | this becomes the two-point function of the free Sp(N ) model, whereas for k |f | this becomes the two-point function of the critical Sp(N ) model. Expanding for large |f | we find: Notice that the real part of the two-point function (7.4) is negative. Recalling (1.3), this is the Gaussian suppression of the Hartle-Hawking wavefunction near the de Sitter vacuum. Also, the local momentum independent term has become a phase for pure imaginary f . We can compare this to the bulk Hartle-Hawking wavefunction for a free m 2 2 = +2 scalar in planar coordinates computed in (C.13) of the appendix. This allows us to (roughly) identify |f | −1 with the late time cutoff |η c | at large |f |. In a similar way we can compute the rest of the perturbative correlators of the critical Sp(N ) theory [42]. Beyond such perturbative analyses, we must resort to a saddle point approximation which we now proceed to.

Large N saddles for uniform S 3 profiles
We now put the theory on the round metric on S 3 . At large N , we can evaluate (7.2) by solving the saddle point equation (for σ = σ( σ) and σ uniform over the whole three-sphere): For a given solution Σ i of (7.5), we can evaluate Z crit [ σ, Σ i ]. For example, there is a solution Σ 0 where σ ∼ 0 when σ ∼ 0 (with f → ∞). It is the piece of the wavefunctional evaluated from this saddle near σ = 0 that reproduces the dS invariant perturbation theory in the Bunch-Davies state (about the pure de Sitter vacuum).
In figure 15 we plot Z crit for the first few Σ i at large f = i|f |. These have σ = 0 near the subsequent zeroes of cot π 2 √ 1 − 4σ in (7.5). Notice that for all large N saddles Z crit [ σ] is peaked at σ = 0 but the saddles coming from the more negative σ peaks contribute more near σ = 0.
invariant one which sets either α( x) or β( x) to zero, corresponding to the free or critical O(N ) models respectively. In de Sitter space we would consider a wavefunction of a scalar of mass m 2 2 = +2 computed by imposing future boundary conditions [11,63,64] φ(η, x) ∼ α( x)η 2 + β( x)η (with α( x) = fDβ( x)) and Bunch-Davies conditions φ ∼ e ikη for k|η| 1. At the level of perturbation theory this is computed by continuing the Euclidean AdS4 partition function by z → −iη, A → i and fA → ifD (where fD ∈ R for the 'normalizable' profile of the scalar field to be real). Away from the large N limit we must compute the integral in (7.2) without resorting to a saddle point approximation. If we restrict to uniform σ and σ, we note that as we increase f = i|f | more and more of the growing negative σ peaks in Z f ree [σ] contribute to the integral before it is cutoff by the rapid oscillations due to the iN σ 2 /|f | piece. One can check that the integral grows for large and negative σ upon fixing |f |.
Some of the saddles in the large N limit should correspond to classical (complex) bulk solutions with a uniform late time profile of the scalar on the round metric of the threesphere. Some of these solutions labelled by a continuous parameter, which only involve the bulk metric and scalar, were found by Sezgin and Sundell in [44]. It is worth noting that though it may sound confusing that there are bulk solutions that have only the metric and scalar turned on (since all higher spin fields interact on an equal footing), this is natural from the CFT since turning on an SO(4) symmetric source for J (0) = χ · χ need not source the traceless higher spin currents due to symmetry reasons. The metric is non-vanishing since in effect we have also turned on a source for it by having the round metric on S 3 at the boundary. At finite N , all these saddles mix quantum mechanically. Each Ψ HH [ σ, Σ i ] comes with a phase, so one should be careful when summing contributions from different saddles. Finally, it would be extremely interesting to understand the Lorentzian cosmologies associated to the wavefunction using the ideas developed in [65,66] (see also [67]). In order to have a classical cosmology (or an ensemble of such cosmologies) we must ensure that the wavefunction takes a WKB form with a phase oscillating much more rapidly than its absolute value. At least in the large N limit, and for large |f |, this is ensured by the first term in (7.2).

Double trace deformations as convolutions
We can also consider keeping f finite and real. This defines a double-trace deformed field theory, in and of its own right, whose partition function Z (f ) [σ ] can be computed in the large N limit. We also keep a uniform (on S 3 ) source σ turned on for the single-trace χ · χ operator. This partition function is no longer computing overlaps between the Bunch-Davies vacuum and some late time field configuration which is an eigenstate of the field operatorφ. Given that Z f ree [σ] = σ|E = Ψ HH [σ], we see that Z (f ) [σ ] is computing JHEP02(2014)007 instead a convolution of the wavefunction in theσ-basis: 7 We can also view Z (f ) [σ ] as computing the overlap of the Hartle-Hawking state with the state: Notice that though the integral itself is convergent for finite f , the resulting function Z (f ) [σ ] will grow exponentially at large negative σ . One could also consider more generally a complex valued f ∈ C which would correspond to a kind of windowed Fourier transform.

Euclidean AdS 4 with an S 3 boundary
The situation can be contrasted with the case of Euclidean AdS 4 (with an S 3 boundary) [62].
8) The first term in front of the integral is local in the limit f A → +∞ and we can remove it by adding a counterterm. To ensure convergence of the integral we must choose an appropriate contour, which in this case is given by σ running along the imaginary axis (see for example [43]).

Extensions of higher spin de Sitter holography?
So far, our discussion was restricted to the minimal bosonic higher spin theory. A natural question that arises, particularly given possible interpretational issues of the wavefunction such as its (non)-normalizability, is whether this theory is part of a larger framework. We briefly discuss possible extensions of higher spin de Sitter holography, inspired by the analogous situation in anti-de Sitter space.

AdS 4
There exist parity violating deformations of the bulk equations of motion which deform the original Vasiliev equations (in anti-de Sitter space) to a one-parameter family [34,37,38]. It was proposed that the dual description is given by coupling the theory to a Chern-Simons theory with level k, at least for simple enough topologies. The new parameter is given by the 't Hooft coupling λ = N/k which is small when the dual is higher spin gravity. Such a theory was shown in the infinite N limit with fixed small λ [68][69][70] to have a spectrum of single-trace operators which is precisely that of the free U(N ) model, namely a tower of higher spin currents which are conserved (up to O(1/ √ N ) corrections), in accordance with a bulk higher spin theory.
As discussed in [69,71] in the context of anti-de Sitter higher spin gravity, one can also endow the bulk higher spin fields with U(M ) Chan-Paton factors, such that the fields all lie in the adjoint representation of the U(M ). In the dual theory, this corresponds to adding a U(M ) flavor symmetry to the U(N ) model. If the flavor symmetry is weakly gauged, which can be achieved through the procedure described in [72], then one can form single-trace operators of the form Tr (ABAB . . . AB). The fields A and B transform in the ( , ) and ( , ) of the U(N )×U(M ) gauge group respectively. As M/N increases the 'glue' between each TrAB (which are dual to higher spin fields in the bulk) becomes stronger. From the bulk point of view, it was suggested that the higher spin fields, now endowed with additional U(M ) interactions, form bound states with binding energy that increases as we crank up M/N . These bound states would correspond to the Tr (ABAB . . . AB) operators. An appropriately supersymmetrized version of this story [69,71] was conjectured to connect the higher spin (supersymmetric) theory to the ABJ model [73] where such long string operators are dual to bulk strings.

dS 4
It is convenient in our discussion for the bulk to contain a spin-one gauge field in its spectrum. We thus consider the non-minimal higher spin model with even and odd spins JHEP02(2014)007 whose dual (at least at the level of correlation functions on R 3 ) is a free/critical U(N ) theory with N anti-commuting scalars transforming as U(N ) vectors. We refer to this as the U (N ) model.
Given that the parity violating deformations of the Vasiliev equations in AdS retain the original field content and reality conditions, it seems natural that they are present for the de Sitter theory as well. Thus, one might consider that such theories are dual to parity violating extensions of the U (N ) theory obtained by adding a level k Chern-Simons term to the free anti-commuting complex scalars. The Lagrangian of this theory is: The fields A a i are (possibly complexified) U(N ) gauge fields, the χ A fields are anti-commuting complex scalars transforming in the fundamental of the U(N ) gauge symmetry, and the ∇ i derivative is covariant with respect to U(N ) gauge transformations.
Though classically this theory is conformally invariant, this need not be the case when we include loops, as the β-function might be non-vanishing. The ordinary U(N ) Chern-Simons theory coupled to a vector of charged scalars has two exactly marginal deformations at infinite N as in [68,74,75]. These are the 't Hooft coupling λ ≡ N/k and the coupling constant λ 6 of the triple trace interaction (χ AχĀ ) 3 . Though the Chern-Simons-U (N ) theory (8.1) is non-unitary, it is conceivable that it is also has a vanishing β-function at large N [76]. This theory would also have an unchanged spectrum of single-trace operators at large N and small λ by the same arguments as those in [68,77]. 8 One can also consider adding U(M ) Chan-Paton factors to the bulk higher spin de Sitter theory. This merely requires tensoring the * algebra with that of M × M matrices. Once again, this will not affect the reality conditions on the higher spin fields and they will all transform in the adjoint of the U(M ). This corresponds to adding a U(M ) flavor symmetry to the U (N ) vector model, which can be weakly gauged (see appendix D for a discussion). The single-trace operators Tr (ABAB . . . AB) have increasingly real conformal weight. From the point of view of dS/CFT this would imply that the bulk theory has a tower of tachyonic bulk fields since the conformal weight of a bulk field goes as ∆ ± ∼ 3/2 ± 9/4 − m 2 2 . One might suspect that these will be the continuations of the higher spin bound states previously discussed for the anti-de Sitter case. Thus we see that even though the fundamental constituents (i.e. the higher spin particles) of such an extension 8 At finite k, the partition function on a non-trivial topology such as M = S 1 × Riemmg will grow as [78] (see also [79]): ZCF T ∼ exp (g − 1)N 2 log k + O(N ) where g is the genus of the M. This drastically favors higher topologies if interpreted as a probability, but it is unclear whether and how one should compare topologies and what the correct normalization for ΨHH is. It is interesting that at finite k one might also encounter monopole operators. In ABJM, such operators are dual to D0-branes in the bulk. It is unclear how they should be understood in the context of de Sitter space and higher spin gravity. For instance, they have a conformal weight that goes like k, which might suggest taking k to be complex or imaginary [80] in the de Sitter case. It is also worth noting that the potentially infinite wealth of topological data at I + might be at odds with the finiteness of de Sitter entropy [81]. In Einstein gravity adding too much topology at I + often results in bulk singularities [82]. The issue of topology in the context of dS/CFT is further discussed in [83].

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of higher spin de Sitter gravity are not pathological (at least at the level of perturbation theory), they may form configurations which resemble tachyonic fields in de Sitter space. It is of interest to understand whether the late time behavior of such a theory can ever be asymptotically de Sitter [76]. From the CFT point of view these are highly irrelevant operators which are not conserved currents. For there to be a late time de Sitter phase, one would require that turning on such irrelevant deformations can flow the theory to a UV fixed point.

Acknowledgments
It is a great pleasure to thank Tarek  A Conformal transformation from round S 3 to flat R 3 In this appendix, we remind the reader of the conformal relation between the three-sphere and the three-plane.

A.1 Coordinate transformation and Weyl rescaling
Consider the metric of the three-sphere: Upon a coordinate transformation, ψ(r) = 2 cot −1 r −1 , the above metric maps to: which is conformally equivalent to the flat metric on R 3 . According to our discussion in section 2.1, a constant mass source m S 3 in the free Sp(N ) theory on a three-sphere corresponds to the free Sp(N ) theory on the flat metric on R 3 with the following source for the χ · χ operators:

A.2 Numerical error
As a check on our numerics, we display in figure 17 a plot of the functional determinant using the Dunne-Kirsten method for the radial mass given in (A.3) laid over the analytic result for the three sphere written in (6.1). In the right hand side of the figure we display JHEP02(2014)007 the error between the two as a function of the cutoff value of l for the sum in (2.7). For a maximum cutoff l max = 200 the partition function is within 3 percent of the exact answer at m S 3 = −2.2.

A.3 Balloon Geometries
We can use the same conformal transformation between the round metric on S 3 and the flat metric on R 3 to show the conformal equivalence of the balloon geometry on an R 3 topology discussed in section 4 and a different geometry on an S 3 topology: ds 2 = dr 2 + r 2 f ζ (r) 2 dΩ 2 2 = 2 cos 2 (ψ/2) −2 (dψ 2 + sin 2 ψ f ζ (ψ) 2 dΩ 2 2 ) .

(A.4)
Upon a conformal rescaling, we can see that this is just a deformed three-sphere: d s 2 ≡ 2 cos 2 (ψ/2) 2 ds 2 = dψ 2 + sin 2 ψ f ζ (ψ) 2 dΩ 2 2 . (A.5) The function f ζ (ψ) serves to add a waist to the three-spheres along the ψ direction, with the parameter ζ controlling whether the sphere tapers or bulges. Thus, the mass deformation chosen in section 4 for the balloon geometry corresponds to a constant mass deformation on its conformally related deformed sphere. When mapping the balloon geometry to flat space to apply the Dunne-Kirsten method for computing the partition function, we should understand that the answer we recover is also the answer for the partition function on this deformed sphere with a constant mass. The pinching limit of the balloon geometry maps into a pinch that tears the peanut-like geometry into two warped spheres.
C Perturbative Bunch-Davies modes for m 2 2 = +2 We briefly review the Bunch-Davies zero modes of a free conformally coupled scalar with m 2 2 = +2 in a fixed global dS 4 background: The action of the scalar is given by: Assuming φ = φ(τ ) does not depend on the three-sphere coordinates, it is governed by the equation of motion: 2 φ(τ ) + 3 tanh τ φ (τ ) + φ (τ ) = 0 . (C. 3) The general solution to the above equation is given by: φ(τ ) = sech 2 τ c 2 + c 1 sinh τ . (C.4) The positive frequency modes of the Bunch-Davies vacuum are those which are regular on the lower Euclidean hemisphere θ ∈ [−π/2, 0] obtained by continuing the global metric by τ → −iθ. 9 This fixes c 1 = ic 2 resulting in: φ BD (τ ) = c 2 sech 2 τ 1 + i sinh τ . (C.5) We can expand the Bunch-Davies solution at late times to find: φ BD (τ ) ∼ 2 c 2 2 e −2τ / + i e −τ / . (C.6) 9 Notice that the continuation τ / → iθ with the same regularity condition would lead to a wavefunctional C.11 that diverges with the late-time profile of the scalar. It is thus an inappropriate choice for a perturbatively stable vacuum state.

C.1 Continuation from Euclidean AdS 4
Another way to see this is by continuing the perturbative solutions of a conformally coupled free scalar with mass m 2 2 A = −2 in a fixed Euclidean AdS 4 background: which are smooth in the interior. These scalars obey the equation: The smooth solution in the interior is given by: