Antipodally symmetric gauge fields and higher-spin gravity in de Sitter space

We study gauge fields of arbitrary spin in de Sitter space. These include Yang-Mills fields and gravitons, as well as the higher-spin fields of Vasiliev theory. We focus on antipodally symmetric solutions to the field equations, i.e. ones that live on"elliptic"de Sitter space dS_4/Z_2. For free fields, we find spanning sets of such solutions, including boundary-to-bulk propagators. We find that free solutions on dS_4/Z_2 can only have one of the two types of boundary data at infinity, meaning that the boundary 2-point functions vanish. In Vasiliev theory, this property persists order by order in the interaction, i.e. the boundary n-point functions in dS_4/Z_2 all vanish. This implies that a higher-spin dS/CFT based on the Lorentzian dS_4/Z_2 action is empty. For more general interacting theories, such as ordinary gravity and Yang-Mills, we can use the free-field result to define a well-posed perturbative initial value problem in dS_4/Z_2.


A. Motivation from dS/CFT
The AdS/CFT correspondence [1,2] offers a non-perturbative model of quantum gravity and a concrete realization of the holographic principle. The correspondence relates a gravitational theory in a (locally, asymptotically) anti-de Sitter space with a conformal quantum field theory (CFT) on its boundary at spatial infinity. A field in the AdS bulk has two possible falloff behaviors at infinity. These are the asymptotic analogs of Neumann and Dirichlet boundary conditions. In the CFT, these two types of boundary data correspond to operators and their conjugate background fields.
The observed positive value of the cosmological constant implies that de Sitter space dS 4 is the more realistic of the maximally symmetric spacetimes. De Sitter space is also an ideal theoretical laboratory for quantum gravity in the presence of causal horizons. Unfortunately, the theoretical understanding of dS is somewhat behind the AdS and flat cases. Conformal infinity in dS 4 consists of two spacelike 3-spheres I ± , one in the infinite past and the other in the infinite future. One would like to know more about field asymptotics at I ± and their physical meaning. A related and more ambitious goal is to formulate dS/CFT -a version of AdS/CFT for positive cosmological constant.
dS/CFT was first considered in [3], with emphasis on the dS 3 /CFT 2 case. A concrete proposal for the physically relevant dimensions dS 4 /CFT 3 was made in [4], by analytically continuing a suitable version of AdS/CFT [5][6][7]. The duality proposed in [4] relates an Sp(N) vector model on the boundary with higher-spin (Vasiliev) gravity [8] in the bulk.
Vasiliev gravity is an interacting theory with an infinite tower of gauge fields of arbitrarily large spin [9,10], including the spin-2 graviton. The theory is believed to be non-local at cosmological scales. At least perturbatively, it does not reduce to General Relativity in any limit. It is nevertheless worth studying, both as a marvel of mathematical physics and as the only concrete proposal for holography in dS 4 .
In the early discussions of dS/CFT, some basic questions arose. Since the boundary of de Sitter space is composed of two disjoint pieces I ± , on which manifold does the dual CFT live? What is the bulk interpretation of the CFT correlators? On these issues, the dS 4 /CFT 3 proposal of [4] follows the "Hartle-Hawking-Maldacena" paradigm developed in [11,12]. Only I + plays an explicit role in the duality. The CFT partition function Z CFT (as a function of background fields) is equated by the duality to a preferred wavefunction over the bulk field asymptotics on I + . This is the Hartle-Hawking wavefunction, obtained by a path integral over Euclidean modes. Schematically: The Euclidean modes used to calculate Ψ HH can be expressed either as real field configurations on a Euclidean AdS (i.e. hyperbolic) space bounded by I + , or as complex configurations on dS with positive frequency in the Bunch-Davies sense.
In [13][14][15], a different kind of dS/CFT was considered. The idea is to identify antipodal points in dS 4 , yielding the so-called "elliptic" de Sitter space dS 4 /Z 2 . Past and future infinity are now identified into a single 3-sphere I id , which every observer can both see and affect. One can then imagine a CFT on I id , which calculates "transition amplitudes", i.e.
Lorentzian path integrals, between a state on I − and the same state on I + .
In [13], the antipodal identification was motivated by the information puzzles concerning cosmological horizons. In particular, dS 4 /Z 2 provides a radical realization of horizon complementarity [16,17]: the two sides of each horizon are literally the same. In addition, it was argued in [13] that in dS 4 /Z 2 , the Hilbert space (as opposed to the transition amplitudes) is observer-dependent, and thus not invariant under the full de Sitter group. This may resolve the puzzle of the finite de Sitter entropy. In this paper, we will not deal with these aspects of dS 4 /Z 2 , but we list them here as further motivation for studying this spacetime.
The present paper's main goal was to explore the idea of dS/CFT for antipodally symmetric transition amplitudes, using the concrete bulk theory from [4], i.e. Vasiliev gravity.
Our conclusion is that such a dS/CFT would be empty, at least when the bulk fields are perturbative and classical. Specifically, we find that the boundary n-point functions of (Lorentzian) Vasiliev gravity on dS 4 /Z 2 all vanish. More precisely, the n-point functions with boundary conditions that preserve the higher-spin symmetry vanish, while the n-point functions with other boundary conditions are ill-defined. This result stems from the fact that interactions in Vasiliev theory can be evaluated at any single point in spacetime [18][19][20][21], while the boundary-to-bulk propagators in dS 4 /Z 2 are distributions that vanish almost everywhere.
On our way to the above conclusion, we present results of more general interest on antipodally symmetric gauge fields in dS 4 , or, equivalently, on gauge fields in dS 4 /Z 2 . We summarize these below, along with the plan of the paper. When discussing spin-s gauge fields, we will often include the scalar with mass m 2 = 2 (in units of the de Sitter radius) as the spin-0 case. Such a field appears in Vasiliev gravity alongside the higher-spin gauge fields.

B. Plan of the paper
The paper is structured as follows. In section II, we review the bulk geometry of real and complex dS 4 in the ambient R 4,1 formalism. We discuss the antipodal map, stressing that it is an operation of the CT type. We review the definitions and field equations for free spin-s gauge fields in dS 4 . We then review the twistor space of dS 4 , presented as the spinor space of R 4,1 , and its relation to SO(3, 1) spinor fields. This leads to a review of the free gauge field equations in spinor form. In section III, we review the asymptotic geometry of dS 4 and the asymptotic boundary data for spin-s gauge fields.
In section IV, we prove our first main result: antipodally symmetric gauge fields in dS 4 have just one type of boundary data (electric/magnetic) non-vanishing on I, depending on the sign of the antipodal symmetry. There is a similar result for m 2 = 2 scalars vis.
Dirichlet/Neumann boundary data, which was already noted in [22] (in addition, the spin-2 result was almost stated in [23]). We prove these statements using the free equations' conformal symmetry, along with smoothness through I ± in the ambient R 4,2 picture. We then justify the smoothness assumption by presenting a spanning set of solutions that satisfy it explicitly. These solutions are bulk 2-point functions, with the second point on the EAdS 4 in the imaginary future/past.
In section V, we review the geometry of elliptic de Sitter space dS 4 /Z 2 , and reinterpret the previous section's result in terms of boundary two-point functions in dS 4 /Z 2 . The result then states that the 2-point functions of gauge fields are always vanishing or ill-defined, depending on the choice of boundary conditions. In section VI, we use the structure of the free equations to formulate a well-posed perturbative initial value problem for gauge fields in dS 4 /Z 2 , with arbitrary parity-conserving interactions. In section VII, we return to free fields, and present the boundary-to-bulk propagators for gauge fields in dS 4 /Z 2 . In the scalar case, we find both the Neumann and Dirichlet propagators; for the s > 0 gauge fields, we find the magnetic propagators, both as gauge potentials and as field strengths.
In section VIII, we turn to Vasiliev gravity. We focus on the type-A and type-B versions of the theory, since the parity-violating versions cannot be defined on dS 4 /Z 2 . We find the propagators for the zero-form master field in dS 4 /Z 2 with boundary conditions that preserve the higher-spin symmetry. We then plug these propagators into the n-point function calculations of [18,20,21] We define de Sitter space dS 4 as the hyperboloid x µ x µ = 1 in the 4+1d Minkowski space R 4,1 . In this "ambient formalism", the de Sitter isometry group SO(4, 1) is identified with the rotation group in R 4,1 . We denote tensors in R 4,1 with indices (µ, ν, . . . ), which are raised and lowered by the flat metric η µν with mostly-plus signature. The 3+1d tangent space to dS 4 at a point x µ is picked out from the 4+1d vector space by the projector P ν µ (x) = δ ν µ − x µ x ν . We use the same indices for tensors in R 4,1 and dS 4 , with the understanding that the latter are restricted to the span of P ν µ (x). In this language, the intrinsic metric of dS 4 is g µν (x) = P µν (x). Covariant derivatives in dS 4 are defined in terms of flat derivatives in R 4,1 as: The d'Alembertian is defined as = ∇ µ ∇ µ . The commutator of covariant derivatives takes the form: Every point x µ ∈ dS 4 has an antipodal point −x µ . The tangential projector P ν µ (−x) is the same as P ν µ (x), so that tensors at the two points can be directly compared. We say that a field w µ 1 ...µ k (x) on dS 4 is antipodally even/odd when it goes into +/− itself under the diffeomorphism x → −x. In our tensor notation, this implies w µ 1 ...µ k (−x) = ±(−1) k w µ 1 ...µ k (x) for antipodally even/odd fields. With this definition, the dS 4 metric and covariant derivative are antipodally even.
The Levi-Civita tensor in dS 4 is obtained from the one in R 4,1 through ǫ µνρσ = ǫ µνρσλ x λ .
Under the antipodal map, ǫ µνρσ flips sign. It follows that the antipodal map sends self-dual fields into anti-self-dual ones, and vice versa.
In [13], it was argued that the antipodal map should involve a complex conjugation of dynamical fields, because a symmetry of the form w µ 1 ...µ k (−x) = ±w * µ 1 ...µ k (x) ensures that antipodal points carry opposite charges. As noted in [24], this is incorrect: it is the symmetry without complex conjugation that leads to opposite charges. Furthermore, the relation is not. We conclude that in the standard C,P,T classification of discrete symmetries, the antipodal map in dS 4 is of the CT type. Indeed, the map interchanges past and future (hence the T), does not involve complex conjugation of fields (hence the C to revert the conjugation due to the T), and flips the spacetime orientation as captured by ǫ µνρσ (hence no P that would revert the orientation flip due to the T). In fact, the antipodal map is CT in de Sitter space of any even spacetime dimension. In odd dimensions, the map is CPT, since the Levi-Civita tensor in that case is antipodally even. This distinction is contrary to the claim in [13] that the map is always CPT.
In addition to the real spacetime dS 4 , we will make use of its complexification dS 4,C . This is defined as the submanifold x µ x µ = 1 in the complex space C 5 . Two slices of interest in dS 4,C are the imaginary past and future spaces: The H ± are 4d Euclidean anti-de Sitter (i.e. hyperbolic) spaces.

B. Free gauge fields and field equations -tensor form
A spin-s gauge field strength is a rank-2s tensor ϕ µ 1 ν 1 µ 2 ν 2 ...µsνs (x) that is antisymmetric in each pair of indices µ k ν k and symmetric under the interchange of any two such pairs.
In addition, all traces vanish, as does the antisymmetrization over any three indices. The cases s = 1, 2 correspond respectively to a Maxwell field strength F µν and a (linearized) Weyl tensor C µ 1 ν 1 µ 2 ν 2 . As discussed in the Introduction, we consider an m 2 = 2 scalar as an "honorary" gauge field with s = 0. For s > 0, the field strength ϕ µ 1 ν 1 ...µsνs decomposes into two pieces: one that is left-handed (anti-self-dual) in every µ k ν k pair, and one that is right-handed (self-dual).
For s = 0, the field ϕ(x) satisfies the Klein-Gordon equation: This is the field equation ( − R/6)ϕ = 0 for a massless conformally coupled scalar (in our dS 4 space with unit radius, the Ricci scalar is R = 12).
For s = 1, we have the two free Maxwell equations: For s ≥ 2, the analog of the first equation in (6) is sufficient: To describe an interacting theory, the field strengths ϕ µ 1 ν 1 µ 2 ν 2 ...µsνs (x) are not enough.
Instead, one needs to work with gauge potentials h µ 1 µ 2 ...µs (x) [9]. These are totally symmetric rank-s tensors, which for s ≥ 4 have a vanishing double trace: h νρ νρµ 5 ...µs = 0. For s = 0, we can define the "potential" h(x) to coincide with the "field strength" ϕ(x). For s = 1, 2, the potentials correspond respectively to a Maxwell potential A µ and a metric perturbation h µ 1 µ 2 . The free field equations for h µ 1 ...µs (x) in dS 4 take the form [10]: For s = 0, this reduces to eq. (5). For s > 0, the field equations respect a gauge symmetry: where the gauge parameter θ µ 1 ...µ s−1 is a symmetric traceless tensor. There are enough degrees of freedom in θ µ 1 ...µ s−1 to enforce transverse gauge ∇ µ 1 h µ 1 µ 2 ...µs = 0. For s ≥ 2, one can use the remaining free initial data in θ µ 1 ...µ s−1 to enforce traceless gauge h ν νµ 3 ...µs = 0, which is then consistently evolved by the field equation (8) [25]. In transverse traceless gauge, the field equation simplifies to: This definition coincides with the standard terminology for s = 1, 2, up to normalizations. In particular, if we take A µ ≡ h µ and h µν to be the Maxwell potential and metric perturbation, then the Maxwell field strength and Weyl tensor read: In the definition (11), the s covariant derivatives are effectively symmetrized, since any derivative commutators yield trace pieces via (3). The correct index symmetries for a field strength directly follow. ϕ µ 1 ν 1 ...µsνs is gauge-invariant, as can be seen by plugging the gauge variation (9) into its definition. Indeed, the µ k ν k antisymmetrizations and the derivative in (9) reduce the gauge variation to derivative commutators, which again become trace pieces due to (3).
(10), any expression in h µ 1 ...µs with index contractions can be reduced to an expression with fewer derivatives. In particular, ∇ µ 1 ϕ µ 1 ν 1 ...µsνs reduces to an expression with fewer than s derivatives. But we've seen that under eqs. (10), any gauge invariant with fewer than s derivatives must vanish. The field equation (7) is thus established.

C. Spinors and twistors in dS 4
The de Sitter group SO(4, 1) has a unique spin-1/2 representation, with 4-component Dirac spinors. This is the twistor space [26,27] of dS 4 , though we will mostly use the word "spinor", in keeping with the R 4,1 perspective. We use indices (a, b, . . . ) for the SO(4, 1) spinors. The spinor space has a symplectic metric I ab , which is used to raise and lower indices via ψ a = I ab ψ b and ψ a = ψ b I ba , where I ac I bc = δ a b . Tensor and spinor indices are related through the gamma matrices (γ µ ) a b , which satisfy the Clifford algebra {γ µ , γ ν } = −2η µν . These 4+1d gamma matrices can be realized as the usual 3+1d ones, with the addition of γ 5 (in our notation, γ 4 ) for the fifth direction in R 4,1 . These matrices can be represented in 2 × 2 block notation as: where σ k with k = 1, 2, 3 are the Pauli matrices. The γ µ ab are antisymmetric and traceless in their spinor indices. We define the antisymmetric product of gamma matrices as: The γ µν ab are symmetric in their spinor indices. Useful identities include: We can use γ ab µ to convert between 4+1d vectors and traceless bispinors as: Similarly, we can use γ ab µν to convert between bivectors and symmetric spinor matrices: Further details may be found in [24].
When we choose a point x ∈ dS 4 , the Dirac representation of SO(4, 1) becomes identified with the Dirac representation of the Lorentz group SO(3, 1) at x. It then decomposes into left-handed and right-handed Weyl representations. The decomposition is accomplished by the pair of projectors: These serve as an x-dependent version of the familiar chiral projectors in R 3,1 . Given an SO(4, 1) spinor ψ a , we denote its left-handed and right-handed components at x as As in our treatment of tensors, it is possible to use the (a, b, . . . ) indices for both SO(4, 1) and SO(3, 1) Dirac spinors. In addition, at a point x ∈ dS 4 , it will be convenient to use left-handed (α, β, . . . ) and right-handed (α,β, . . . ) Weyl indices, which are taken to imply P L (x) and P R (x) projections, respectively. Thus, for a Dirac spinor ψ a , we have the projections ψ α L (x) and ψα R (x). In this scheme, the matrices P L ab (x) and P R ab (x) serve as the spinor metrics ǫ αβ and ǫαβ for the two Weyl spinor spaces.
For a vector v µ in the 3+1d tangent space at a de Sitter point x, the bispinor v ab can be decomposed into Weyl components v αβ = −vβ α . For such vectors, we therefore have: The power of this formalism is that the SO(4, 1) spinors are flat, just like the SO(4, 1) vectors. We can therefore transport them freely from one de Sitter point to another. What changes from point to point is the spinor's decomposition into left-handed and right-handed parts. As a special case, the identity P ab R (−x) = P ab L (x) defines an isomorphism between left-handed spinors at x and right-handed spinors at −x. This is consistent with the fact that self-duality signs get flipped by the antipodal map.
Covariant derivatives for Weyl spinors in dS 4 can be constructed from the 4+1d flat derivative, in analogy with the tensor formula (2):

D. Free gauge fields and field equations -spinor form
A field strength tensor with the appropriate index symmetries can be translated into a totally symmetric rank-2s spinor as: where the only non-vanishing components of ϕ a 1 ...a 2s are the totally left-handed ϕ α 1 ...α 2s and the totally right-handed ϕα 1 ...α 2s . For s > 0, the field-strength spinors satisfy the field equations: A gauge potential h µ 1 ...µs in traceless gauge can be translated into spinor form as: where h α 1 ...αsα 1 ...αs is symmetric in both its dotted and undotted indices. We will not consider here the extension to half-integer spins. The field equations and gauge conditions (10) translate directly into spinor language. On the other hand, the relation (11) between potentials and field strengths simplifies considerably. It can be formulated succinctly in spinor language as:

III. PRELIMINARIES: GEOMETRY AND GAUGE FIELDS AT I ±
In this section, we outline the asymptotic geometry of dS 4 , along with the appropriate boundary data for the gauge fields. For the latter, we will use the conformal properties of the field equations (5)- (7).

A. Asymptotic geometry
The asymptotic boundary of dS 4 is a pair of spacelike conformal 3-spheres -one in the infinite past (I − ), and the other in the infinite future (I + ). In the flat 4+1d picture, these can be viewed as the 3-spheres of past-pointing and future-pointing null directions in R 4,1 .
The antipodal map interchanges I − and I + , in such a way that the lightcone of a point on I − refocuses at the antipodal point on I + . A bulk point x is said to "approach infinity" when the unit vector x µ is highly boosted, i.e. when its components become very large. This condition is not invariant under the de Sitter group SO(4, 1), but that is to be expected: the statement that a point is "very far away" cannot be invariant under large translations.
I − and I + can be assigned an orientation by contracting the bulk Levi-Civita tensor ǫ µνρσ with the future-pointing or past-pointing timelike normal ±n µ (where we take n µ to be the future-pointing choice). To avoid a preferred global time direction, we must use −n µ at I − and +n µ at I + , or vice versa. This choice of normals is antipodally even, while ǫ µνρσ is antipodally odd. Therefore, in this scheme, the antipodal map reverses the orientation of I ± . In this sense, I − and I + have opposite orientations.
To include I ± in the spacetime manifold, we perform a conformal completion: we choose a time coordinate z that vanishes on I ± , such that the conformally rescaled metric z 2 g µν is regular at z = 0. We can then define a metric on I ± as: Since we are free to multiply z by any function of the spatial coordinates, the metric (25) is only defined conformally. To remove any ambiguity between I − and I + , we choose the z coordinate to be antipodally odd, such that I − and I + correspond to z → 0 − and z → 0 + , respectively.
B. Boundary data for the scalar field (s = 0) We now turn to the issue of appropriate boundary data for our fields on I ± . We begin with the m 2 = 2 scalar ϕ(x), satisfying the field equation (5). As already mentioned, eq.
(5) can be written as: where R is the Ricci scalar. Eq. (26) is invariant under the conformal rescaling g µν → z 2 g µν , where ϕ has conformal weight 1 (we say that a quantity has conformal weight ∆ if it scales as z −∆ under g µν → z 2 g µν ). Since the metric z 2 g µν is regular at I ± , we conclude that the rescaled field z −1 ϕ can be Cauchy-evolved from I ± . We can therefore define configuration and momentum fields on I ± as: See [28] for the dS/CFT perspective on these definitions. The chosen sign factors ensure that an antipodally even/odd ϕ(x) induces the same symmetry on φ(x) and π(x). As fields on I ± , φ(x) and π(x) have respective conformal weights 1 and 2 under rescalings of the metric (25). The weights add up to 3, as appropriate for canonical conjugates in a 3-dimensional CFT. A solution to the field equation (5) is uniquely determined by the boundary data C. Boundary data for gauge fields (s ≥ 1) We now turn to the gauge field strengths ϕ µ 1 ν 1 ...µsνs (x) with field equations (6)- (7). On spatial slices of dS 4 , for which I ± are limiting cases, we can decompose ϕ µ 1 ν 1 ...µsνs into electric and magnetic components with respect to the future-pointing unit normal n µ . Apriori, every µ k ν k index pair can be decomposed separately. However, due to the index symmetries of ϕ µ 1 ν 1 ...µsνs , a simultaneous Hodge dual on any two pairs yields the original field with a minus sign: Thus, pieces of ϕ µ 1 ν 1 ...µsνs with an even (odd) number of magnetic µ k ν k pairs are all equivalent to the piece with zero (one) such pairs. We can therefore decompose ϕ µ 1 ν 1 ...µsνs into electric and magnetic parts as follows: The tensors (29) are purely spatial, totally symmetric and traceless.
On a spatial slice, the field equations (6)-(7) decompose into constraints and dynamical equations. The constraint equations read: where D µ is the spatial covariant derivative. The dynamical equations evolve the ..µs (x)} values on a spatial slice into a spacetime solution.

A. Results
In this section, we prove the following results, which relate the antipodal symmetry of free gauge fields with their asymptotic behavior: With regard to Theorem 2, we note that an antipodally even/odd field strength can always be derived from a gauge potential with the same symmetry: start with any gauge potential for the given field strength, and take its antipodally even/odd piece; the remaining piece is necessarily pure gauge.
Theorems 1-2 are not surprising, once one realizes that the fields in question propagate along lightrays. Since the lightcone from a point on I − refocuses at its antipode on I + , it is natural for the boundary data on I ± to be antipodally symmetric. What remains to be established is the sign of this antipodal symmetry, which depends on the type of boundary data.
Once we know the antipodal symmetry of each type of boundary data, it is easy to demonstrate the bulk antipodal symmetry of solutions with only this type of boundary data non-vanishing. Indeed, since the field equations are antipodally symmetric, the solution's antipodal image is also a solution. But this will have the same boundary data as the original solution on e.g. I − , up to an overall sign. The uniqueness of Cauchy evolution then implies that the solution must coincide (up to sign) with its antipodal image.
It remains, then, to map each type of boundary data to its antipodal symmetry on I ± .
In section IV B, we accomplish this by assuming smoothness through I ± on conformally compactified dS 4 . We will justify this assumption through explicit solutions in sections IV C-IV D.
B. Proof from smoothness on conformally compactified dS 4 We've seen that in order to prove theorems 1-2, it is enough to map each type of boundary data to its antipodal symmetry on I ± . We will now do this, using the conformal symmetry of the field equations. The conformal group SO(4, 2) in 3+1d spacetime can be realized as the group of rotations in 4+2d flat space. In this realization, our dS 4 is a section of the lightcone in R 4,2 . Specifically, a point x ∈ dS 4 , which in R 4,1 is described by the unit spacelike vector x µ , is associated with the following null vector in R 4,2 : where the index A takes the values A = −1, 0, 1, 2, 3, 4, and the new A = −1 direction is timelike. Other sections of the lightcone describe conformally related 3+1d metrics.
Forgetting the particular section (33), one can define conformally compactified dS 4 as the projective lightcone in R 4,2 , i.e. the space of nonzero null X A modulo rescalings X A → zX A .
Fields on dS 4 can be written as functions of the null vector ℓ A . Fields with conformal weight ∆ will scale as z −∆ under X A → zX A . The asymptotic 3-sphere I − gets mapped into an ordinary 3d surface on the projective lightcone in R 4,2 . Crucially, I + gets mapped into the same surface, so that antipodal points on I ± (but not elsewhere) become identified.
The I ± become singular and distinct from one another only in the particular section (33).
After rescaling X A → zX A , with z the time coordinate from section III, the points of I ± take the form: where the ℓ µ are future-pointing null vectors forming a section of the lightcone in R 4,1 .
Now, consider a solution ϕ(x) to the free scalar equation (5) in dS 4 . Assume that in conformally compactified dS 4 , this solution is regular on the surface corresponding to I ± .
Taking into account the rescaling between the conformal frames (33)- (34), this means that ϕ(x)/z is regular on the section (34). Now, as we've seen, the antipodal map between the I ± is trivial on the section (34). This implies that ϕ(x)/z, as well as ∂ z (ϕ(x)/z), are antipodally even. The antipodal symmetries in Theorem 1 can now be read off from the definitions (27).
Similarly, consider a free field-strength solution ϕ µ 1 ν 1 ...µsνs (x) on dS 4 with s ≥ 1. Assume again that in conformally compactified dS 4 , the solution is regular on the surface corresponding to I ± , i.e. on the section (34). To draw conclusions about our tensor field in dS 4 , we must take its components with respect to directions that are smooth through (34). The directions tangential to I ± with this property are antipodally even, but the normal direction (e.g. the future-pointing one) is antipodally odd. Taking into account the appropriate conformal weight, we conclude that components of z s−1 ϕ µ 1 ν 1 ...µsνs (x) with an even/odd number of normal indices are antipodally even/odd. The antipodal symmetries in Theorem 2 can now be read off from the definitions (29), (31).
It remains to justify the assumption of smoothness through I ± on conformally compactified dS 4 . In the next subsections, we show that this property is indeed satisfied by a spanning set of solutions to the free field equations.

C. Smooth solutions on conformally compactified dS 4 : scalar field
We begin with the scalar case. Our solutions for ϕ(x) will be parametrized by a point ξ in the imaginary future slice H + or the imaginary past slice H − of complexified de Sitter space; see eq. (4). In the 4+1d language, ξ is encoded by an imaginary timelike vector ξ µ such that ξ µ ξ µ = 1 and Im ξ 0 ≷ 0, respectively. We then consider the solution: where x · ξ ≡ x µ ξ µ . This is just a bulk 2-point function between the points x and ξ [29].
The denominator in (35) is proportional to the squared distance between x µ and ξ µ in the complexified R 4,1 : Since lightrays in dS 4 are also lightrays in R 4,1 , the solution (35)  Finally, it's easy to see that the solutions (35) are regular on I ± in conformally compactified dS 4 . This follows directly from the fact that I ± doesn't lie on the lightcone of the imaginary point ξ. Explicitly, the solutions can be written in the 4+2d language as: where X A = (1, x µ ) and Ξ A = (1, ξ µ ). The solutions are now manifestly SO(4, 2)-covariant with conformal weight 1, and regular on the asymptotic section (34).

D. Smooth solutions on conformally compactified dS 4 : gauge fields
We now turn to spin-s gauge field strengths with s ≥ 1. We will again use 2-point functions between the measurement point x ∈ dS 4 and a point ξ ∈ H ± in the imaginary future/past. To write the solutions compactly, we will use the spinor language of sections II C-II D. Our 2-point functions will have opposite handedness at x and ξ. In principle, they can be derived from the bulk-to-bulk gauge-potential propagators, given e.g. in [30].
However, the end result is much simpler than the calculation, so we present the field-strength solutions directly.
To encode the polarization, we will use a Weyl spinor M at the point ξ.
where M α L (x) is the projection P L α b (x)M b of M a onto the left-handed spinor space at x. The scalar solution (35) is contained in (38) as the s = 0 case. One can verify that (38) indeed solves the field equation (22), using the relations: Field strengths with the opposite handedness can be obtained by interchanging the P L and P R projectors, or, equivalently, by substituting x → −x. As in the scalar case, the solutions  We can represent them in terms of the 4+1d matrices (13) as: The γ A ab and γ ab A span the spaces of lower-index and upper-index bispinors. Using the representation (33) for a point x ∈ dS 4 , we can now write the chiral projectors (18) as: The left-handed field strength solutions (38) can now be written in 4+2d language as: and similarly for the right-handed solutions. The expression (42) is manifestly SO(4, 2)covariant with conformal weight 1, and regular on the asymptotic section (34). The conformal weight 1 − s of ϕ µ 1 ν 1 ...µsνs is recovered in the translation to 3+1d tensors.
We have thus demonstrated that a spanning set of free-field solutions in dS 4 is regular through I ± on the conformal compactification. This concludes the proof of Theorems 1-2 on the relation between antipodal symmetry and asymptotics. Note that the solutions (35) and (38) are not themselves antipodally symmetric. However, they can be combined with their antipodal images to form spanning sets of antipodally even/odd solutions. These will be closely related to the boundary-to-bulk propagators of section VII. There are two ways to construct tensor fields on dS 4 /Z 2 . Formally speaking, the fields can take values in two different line bundles over dS 4 /Z 2 , which we will call the even bundle and the odd bundle. The even bundle is the trivial bundle of real/complex numbers at each point. The odd bundle is topologically non-trivial, such that the fiber (with all the field values in it) flips sign upon traversing an incontractible cycle. Clearly, antipodally even/odd fields on dS 4 correspond to even/odd fields on dS 4 /Z 2 . In particular, the dS 4 /Z 2 metric is an even field, while the Levi-Civita tensor is an odd one.
To be well-defined, interacting field equations in dS 4 /Z 2 must be such that powers of odd dynamical fields go together with powers of ǫ µνρσ . Thus, dS 4 /Z 2 only supports field theories that conserve P (and therefore CT), where the even/odd fields have even/odd intrinsic parity.
Thus, solutions in dS 4 /Z 2 correspond to dS 4 solutions where the parity-even (parity-odd) fields are antipodally even (antipodally odd). The restriction to CT-preserving theories is not surprising: recall from section II A that the antipodal map in dS 4 is an operation of the CT type.
The conformal boundary of dS 4 /Z 2 is a single 3-sphere I id , resulting from the antipodal identification of I − and I + . While I id is of course orientable, it does not inherit a preferred orientation from the bulk. In particular, we've seen in section III A that I ± are oppositely oriented (unless one chooses a preferred time direction, which is impossible in the dS 4 /Z 2 context). As with the bulk, one can define even and odd fields intrinsically on I id , where the even/odd distinction refers both to the fields' intrinsic parity and to their antipodal symmetry on I ± . There are incontractible cycles in dS 4 /Z 2 that connect a point on I id to itself, via the bulk. Odd fields on I id flip their sign upon traversing such a cycle.
Having understood the geometry of dS 4 /Z 2 and its boundary I id , we can reformulate the statements of Theorems 1-2 as follows: 1. An even/odd scalar field on dS 4 /Z 2 that satisfies the field equation ( − 2)ϕ = 0 has vanishing Dirichlet/Neumann boundary data φ/π (and is determined by the boundary data of the other type).
This can be further reformulated in terms of 2-point functions on I id : 1. An even/odd scalar field on dS 4 /Z 2 with field equation ( − 2)ϕ = 0 has a vanishing Neumann/Dirichlet 2-point function on I id , while the 2-point function of the other type is ill-defined.
2. An even/odd spin-s gauge field on dS 4 /Z 2 with field equations (6)- (7)  However, in the special case of Vasiliev gravity, we will see in section VIII that the n-point functions have the same singular behavior for n > 2 as they do for n = 2.

VI. INITIAL VALUE PROBLEM FOR INTERACTING THEORIES IN dS 4 /Z 2
We will now use the free-field result of section IV to formulate a well-defined initial value problem for interacting gauge fields in dS 4 /Z 2 . It is helpful to first formulate and prove the statement in terms of antipodally symmetric fields on dS 4 . The dS 4 /Z 2 statement will be given at the end of the section. As we recall from section V, only interactions that preserve P (and thus CT) respect the antipodal symmetry. For such theories, we find the following result in dS 4 : The theorem covers a wide range of theories, including: • Theories of m 2 = 2 scalars and Maxwell/Yang-Mills fields on a fixed dS 4 metric.
• General Relativity with Λ > 0, coupled to any of the above matter fields.
Note that the same field can sometimes be taken as either parity-even or parity-odd (though one must make a choice when going over to dS 4 /Z 2 !). This is the case for scalar fields with an even potential in standard s ≤ 2 theories, as well as for Maxwell fields.
We now turn to prove the theorem. In the interest of readability, we only present the proof for the case where all fields are parity-even. The field equations can then be written with no ǫ µνρσ factors, and the boundary data prescribed in the theorem is {π, B µ 1 ...µs }. The proof with parity-odd fields is analogous. Now, assume that the statement holds for the first n − 1 orders in perturbation theory.

Proof of part (a).
Let us fix the solution at these orders to the unique antipodally even one. The field equations for the n'th-order fields are just the linear equations (8), but with source terms on the righthand side. These source terms must be constructed covariantly out of the lower-order fields, the background metric g µν and the background covariant derivative ∇ µ . These objects are all antipodally even; therefore, the source terms constructed from them are also antipodally even. Now, consider an arbitrary solution to the n'th-order equations, e.g. the one with vanishing {φ, π, B µ 1 ...µs , E µ 1 ...µs } on I − . The antipodal image of this is also a solution, due to the symmetry of the source terms. Since the equations are linear, we can take the average of the two antipodal images, producing an antipodally even solution (which does not yet satisfy the required boundary conditions). The general n'th-order solution can be obtained from this by adding a solution to the free field equations. Now, by Theorems 1-2, the free antipodally even solutions are in one-to-one correspondence with {π, B µ 1 ...µs } boundary data.
Thus, we can add a unique free even solution that will fix {π, B µ 1 ...µs } to the required values.
The remaining freedom is to add an antipodally odd free solution, which by Theorems 1-2 will have vanishing {π, B µ 1 ...µs }. However, such an addition would spoil the antipodal symmetry. We conclude that the antipodally even solution with given {π, B µ 1 ...µs } values is unique.
In dS 4 /Z 2 , Theorem 3 becomes the statement of an initial value problem: • Consider a field theory as in Theorem 3. Fix a configuration of π/B µ 1 ...µs (φ/E µ 1 ...µs ) boundary data on I id for the parity-even (parity-odd) fields. Then there exists a unique bulk solution in dS 4 /Z 2 .

VII. BOUNDARY-TO-BULK PROPAGATORS IN dS 4 /Z 2
In this section, we present boundary-to-bulk propagators for gauge fields in dS 4 /Z 2 . In the scalar case, we present both the antipodally even (i.e. Neumann) and the antipodally odd (i.e. Dirichlet) propagators. For gauge fields with spin s > 0, we present only the antipodally even (i.e. magnetic) propagators. We will use these propagators in our treatment of Vasiliev gravity in section VIII.

A. Scalar propagators
The boundary-to-bulk propagators for scalars in EAdS 4 are well-known [2,31]. For our m 2 = 2 case, they read, in the 4+1d language: Here, x µ is a unit timelike vector in R 4,1 representing the bulk point in x · ℓ → x · ℓ ± iε yield the positive-frequency and negative-frequency propagators in the Bunch-Davies sense. These two prescriptions are antipodal images of each other. We can therefore obtain antipodally symmetric propagators by superposing them. In accordance with Theorem 1, the antipodally even/odd propagators should have a delta-function for the Neumann/Dirichlet boundary data, with the other type of boundary data vanishing. Since the field only propagates along lightrays, this implies that the propagators must vanish away from the lightcone x · ℓ = 0 of the boundary point. Thus, the propagators in dS 4 /Z 2 take the form: The bulk point x µ can be parametrized as: x µ = sinh η e µ 0 + cosh η (cos χ e µ 4 + sin χ (cos θ e µ 3 + sin θ(cos φ e µ 1 + sin φ e µ 2 ))) .
As a hypersurface approaching e.g. I + , we choose a constant-η slice with η → ∞. This is a 3-sphere with radius cosh η. It becomes a unit 3-sphere upon rescaling with e.g. z = 1/ sinh η at z → 0 (this choice of z has the correct signs at I ± ). The scalar product in the delta functions (44)-(45) reads: Since this has no (θ, φ) dependence, we can write the volume element on the 3-sphere as: where the arrow denotes the z → 0 limit. Integrating the delta functions in (44)-(45) on the (χ, θ, φ) 3-sphere, we get: where in (51) we kept only the leading term in z. This shows that the propagators (44)-(45) yield normalized delta functions for the π/φ boundary data, respectively.

B. Gauge potential propagators
We now turn to the magnetic propagators for the spin-s gauge potentials. In addition to the boundary point encoded by ℓ µ , the propagator must now depend on a (symmetric, traceless) polarization tensor on I. Without loss of generality, we can take this to have the form λ µ 1 . . . λ µs , where λ µ is a (complex) null vector on I. In the 4+1d picture, this means that λ µ is a null vector orthogonal to ℓ µ , defined up to multiples of ℓ µ . This data can be neatly encoded in a totally null bivector M µν = ℓ µ ∧ λ ν , which has the properties: In EAdS 4 , the propagators have been worked out in [25]. In our language, they read: These are solutions to the free field equations (10) in transverse traceless gauge. Note that M µν x ν is automatically in the tangent space of dS 4 , i.e. orthogonal to x µ .
As in the scalar case, when translating the propagators (53) to dS 4 /Z 2 , we must choose the delta-function-like combination of the two x · ℓ → x · ℓ ± iε prescriptions. This gives: where C(s) is a normalization factor, and δ (2s) is the 2s-th derivative of the delta function.
The propagator (54) is antipodally even, and therefore has purely-magnetic boundary data.
At the boundary, the tangential components of the propagator (54) scale as z 2−2s (in a coordinate basis on I); the coefficient is λ µ 1 . . . λ µs times a delta function at the point encoded by ℓ µ . In a previous version of this manuscript, a wrong value was given for the normalization coefficient C(s) that leads to a normalized delta function. A more careful analysis has been carried out in [38], yielding the value: where the double factorial (2s − 3)!! is defined as 1 for s = 1 and 1 · 3 · 5 · · · · · (2s − 3) for s ≥ 2.

C. Gauge potential and field strength propagators in spinor form
To derive the field strengths from the gauge potential propagators (54), it is helpful to first rewrite them in spinor form. The polarization bivector M µν gets translated into a 4+1d spinor M a via: One can then show that the vector M µν x ν becomes: where we recall that M α L (x) and Mα R (x) are the projections of M a onto the left-handed and right-handed spinor spaces at x. The gauge-potential propagator (54) then becomes: The left-handed and right-handed field strength propagators can now be derived as in (24) to give: where the individual terms on the second lines are the propagators with positive/negative frequency in the Bunch-Davies sense. The derivation of (59) from (58) follows from the relations: along with their counterparts of opposite chirality.

A. Choice of theory and boundary conditions
We now turn to discuss Vasiliev's higher-spin gravity in dS 4 /Z 2 . The theory comes in a variety of versions. In this paper, we focus on purely bosonic ones. As discussed in section V, the dS 4 /Z 2 context further restricts us to parity-invariant theories. This leaves us with just four possibilities, distinguished by two binary choices. The first choice is between a minimal theory (even spins only) and a non-minimal one (both even and odd spins). We will treat these two options simultaneously, with the minimal n-point functions forming a subset of the non-minimal ones. The second choice is between type-A (parity-even scalar field) and type-B (parity-odd scalar field); the s > 0 gauge fields are always parity-even.
The dS/CFT model of [4] uses the minimal type-A theory. Here, we consider all four of the parity-invariant bosonic versions.
Having chosen the bulk theory, one can work with different choices of boundary conditions. In ordinary (A)dS, the possibilities are as follows [32]. are precisely the boundary conditions that preserve the higher-spin symmetry [33], and that correspond to free boundary theories in AdS/CFT [32].
In the following, we will focus on this particular choice of boundary conditions, and argue that not only the 2-point functions, but all the n-point functions vanish. This means that the opposite types of boundary data (electric for the gauge fields, Dirichlet/Neumann for the type-A/type-B scalar) continue to vanish at all orders in perturbation theory. As a consequence, the n-point functions with these data as boundary conditions are ill-defined.
The same conclusion applies to mixed boundary conditions that fix a combination of E µ 1 ...µs and B µ 1 ...µs .

B. Higher-spin framework
In Vasiliev gravity, one augments spacetime with an internal twistor space. In standard treatments, this twistor space is rigidly decomposed into left-handed and right-handed spinor spaces. Such a formalism is well-suited for calculations in Poincare coordinates, but it is not covariant under the full SO(4, 1) de Sitter group. Moreover, it cannot be used in dS 4 /Z 2 , since the latter is non-orientable. A simple alternative is to identify the internal twistor space with the global space of SO(4, 1) spinors from section II C. The price is that the decomposition into left-handed and right-handed spinors is now x-dependent, governed by the P L/R (x) projectors from (18) (and is only possible locally in dS 4 /Z 2 , since the P L/R are interchanged by the antipodal map). A formalism for such generalized gauges is given in [34,35]. It involves a "compensator field" which reduces the symmetry of the internal space from SO(4, 1) to SO(3, 1). In our case, this role is played by the radius-vector x µ .
The detailed framework is as follows. The higher-spin algebra is generated by twistor variables Y a , subject to the star product: As discussed above, we use a gauge where the space of the Y a is identified with the global SO(4, 1) spinor space in pure dS 4 (or dS 4 /Z 2 ). In this gauge, the background higher-spin connection Ω vanishes. Instead, we have the frame one-form Σ(x) = T µν dx µ x ν , which encodes the translation generators at the point x.
Our treatment of perturbations around dS 4 /Z 2 will focus on the zero-form master field B(x, Y ), which encodes the field strengths for all spins along with their spacetime derivatives.
It will suffice to work with B(x, Y ) at the linearized level. In our Ω = 0 gauge, the free field equation for B reads simply: The master field B(x, Y ) is parity-even in the type-A theory, and parity-odd in the type-B theory. This is despite the fact that the s > 0 component gauge fields are parity-even in both cases: in the type-B case, there is a handedness-dependent sign factor in the translation between the master field and the component fields. In dS 4 /Z 2 , the intrinsic parities are translated into antipodal symmetry signs: B(x, Y ) is antipodally even/odd in the type-A/type-B theory, even though the s > 0 component fields are always antipodally even.

C. Master-field propagators
The scalar propagators (44)-(45) and the field strength propagators (59) can be embedded (up to normalizations) into a pair of master fields that satisfy eq. (63): B(x, Y ; ℓ) ∼ 1 (x · ℓ + iε) 2  should really be defined as the limit of a sequence of non-singular fields, which do not vanish anywhere. However, the conclusion remains intact: in the limit, the propagators away from the lightcone become arbitrarily small, and one still gets zero when plugging them into the 3-point function calculation of [18].
Similarly, in the n-point function calculations of [20,21], the result is obtained as a multilinear functional of the B master-field propagators at an arbitrary point. Since our dS 4 /Z 2 propagators vanish away from the boundary sources' lightcones, we conclude that all the n-point functions vanish.

IX. DISCUSSION
In this paper, we studied the relations between asymptotic boundary data, parity and antipodal symmetry for gauge fields in dS 4 . We constructed a perturbatively well-posed initial value problem at the conformal boundary of elliptic de Sitter space dS 4 /Z 2 . The results apply to realistic theories such as Yang-Mills and General Relativity, as well as to Vasiliev's higher-spin gravity. The latter features as the bulk theory in a family of AdS/CFT dualities, which appear particularly suited for reformulation with a positive cosmological constant. We explored the possibility of a dS/CFT duality that calculates Lorentzian higherspin "transition amplitudes" in dS 4 /Z 2 . We found that this notion is empty, since the n-point functions are all either zero or ill-defined, depending on the choice of boundary data. The same is true for 2-point functions in any theory of free or interacting gauge fields. However, the conclusion for the higher n-point functions seems to be special to Vasiliev gravity. For instance, using the propagator (44), one can compute the 3-point function for an m 2 = 2 scalar with a simple ϕ 3 interaction, and the result is finite.
Our proof for the vanishing of the higher-spin n-point functions is only as good as the n-point function calculations of [18,20,21]. In the first of these references, only the 3-point function is computed. In the other two, one employs an indirect argument based on higherspin symmetry, which is only conjectured to agree with the explicit solution of Vasiliev's field equations. When more complete calculations appear, it will be possible to test our result against them.
As discussed in the Introduction, elliptic de Sitter space remains a fascinating testing ground for ideas in quantum gravity, in particular horizon complementarity. There is much to understand about fields in this spacetime. For example, it appears that they cannot be quantized globally, but only after choosing an observer with his associated cosmological horizons. We pursue these issues in a separate work [37]. Our null result for the higher-spin correlators fits neatly into this picture, as another piece of evidence that one cannot do global physics in dS 4 /Z 2 .