FRW and domain walls in higher spin gravity

We present exact solutions to Vasiliev's bosonic higher spin gravity equations in four dimensions with positive and negative cosmological constant that admit an interpretation in terms of domain walls, quasi-instantons and Friedman-Robertson-Walker (FRW) backgrounds. Their isometry algebras are infinite dimensional higher-spin extensions of spacetime isometries generated by six Killing vectors. The solutions presented are obtained by using a method of holomorphic factorization in noncommutative twistor space and gauge functions. In interpreting the solutions in terms of Fronsdal-type fields in spacetime, a field-dependent higher spin transformation is required, which is implemented at leading order. To this order, the scalar field solves Klein-Gordon equation with conformal mass in (anti) de Sitter space. We interpret the FRW solution with de Sitter asymptotics in the context of inflationary cosmology and we expect that the domain wall and FRW solutions are associated with spontaneously broken scaling symmetries in their holographic description. We observe that the factorization method provides a convenient framework for setting up a perturbation theory around the exact solutions, and we propose that the nonlinear completion of particle excitations over FRW and domain wall solutions requires black hole-like states.


Introduction
Vasiliev's theory in four dimensions [1] has so far been studied mainly around its maximally symmetric anti-de Sitter vacuum. The perturbations around the anti-de Sitter spacetime describe an unbroken phase of the theory, with spectrum given by infinite towers of massless fields, corresponding to conserved higher spin currents of dual free conformal field theories in three dimensions [2,3,4]. Higher spin gravity is well known to admit a cosmological term of positive sign and de Sitter vacuum solution as well. It has been proposed that the parity invariant minimal version of higher spin dS 4 gravity is holographically dual to the three dimensional conformal field theory of an Euclidean Sp(N ) vector model with anticommuting scalars residing at the boundary of dS 4 at future timelike infinity [5]. For further developments in this direction, see [6,7,8,9,10,11,12]. These studies mostly exploit the higher spin symmetries.
On the other hand, a detailed bulk description of the early universe physics, including the inflationary era, requires understanding of accelerating solutions of Vasiliev theory and cosmological perturbations around them. Such solutions have isometries forming a subgroup of the de Sitter spacetime symmetries.
Higher spin gauge symmetries can be broken by quantum [13] as well as classical effects. In the latter case, a simple mechanism is to replace the maximally symmetric vacuum by vacua with six Killing symmetries forming a Lie algebra g 6 , as summarized in Table 1 1 . These correspond to the isometries of domain walls, FRW-like solutions and quasi-instantons 2 . While we shall leave to a future work an analysis of the the holographic aspects of the exact solutions that we present here, we propose to interpret the domain walls as bulk duals of vacua of three-dimensional massive quantum field theories arising through spontaneous breaking of conformal (higher spin) symmetries; for a relatively recent study of spontaneous breaking of scale invariance in certain CFTs in D = 3, see [16].
In this paper, we shall use a solution generating technique [17,18,19] to build g 6 -invariant solutions to Vasiliev's bosonic theory with non-vanishing (positive or negative) cosmological constant from gauge functions, representing large gauge transformations that alter the asymptotics of the gauge fields, and g 6 -invariant scalar field profiles in the maximally symmetric background. Solutions of Vasiliev's equations with g 3 , g 4 and g 6 symmetries, which are subgroups of the AdS 4 symmetry group, were constructed only at the linearized level in [14] (see 1 The various vacua possess unbroken higher spin symmetries; the unbroken symmetry algebra of the g6invariant vacua is the intersection of the enveloping algebra of g6 with the unbroken symmetry algebra of the maximally symmetric vacua (see [14] for the case of an so(1, 3)-invariant solution), which is given by the quotient of the enveloping algebra of the (A)dS4 Killing symmetry algebra over the two-sided ideal given by the singleton annihilator. 2 The quasi-instantons are Lorentzian counter parts of instantons, which can be viewed as the results of gluing together a domain wall and a FRW-like geometry [14,15]. [19] for a review) by using a different technique. The fully non-linear solutions presented in this paper are instead obtained by using a different method based on a holomorphic factorization ansatz, and in what we refer to as the holomorphic and L-gauges, described in Section 3. In furnishing an interpretation of the solution in terms of Fronsdal-type fields in spacetime, however, a higher spin transformation needs to be implemented order by order in weak fields to reach what we refer to as the Vasiliev gauge, also discussed in Section 3. We have implemented this gauge transformation only at leading order in this paper, leaving the computation of higher order terms to a future work. As we shall see in Section 5, an important advantage of the method we have used to obtain the exact solutions in the holomorphic gauge is the validity of linear superposition principle in constructing solutions, thus facilitating the description of fluctuations around an exact solution. Even though we leave to future work the analysis of a cosmological perturbation theory around our solutions, an inspection of the star product algebra among the master field will lead us to propose that the nonlinear completion of particle excitations over FRW and domain wall solutions requires black hole-like states (see [18] for the study of scalar particle fluctuations over higher-spin black hole modes).
Among all solutions we have found, we shall, in particular, take a closer look at the FRWlike solution with iso(3) symmetry and positive cosmological constant. We will provide a perturbative procedure for obtaining the solutions in the Vasiliev gauge mentioned above, to any order in a suitable perturbation parameter that breaks the de Sitter symmetry to iso(3).
On the solutions, the scalar field, whose value is vanishing in de Sitter vacuum, is turned on at first order in the symmetry-breaking parameter, and the metric gets corrected at the second order. Moreover, at linear order the fields with spins s > 2 vanish in the background solution.
Whether they arise in higher orders remains to be determined. At linear order the scalar field behaves similarly to a conformally coupled scalar field in dS 4 . In Section 5, we shall compare its behaviour with that of the inflaton in the standard cosmological scenarios.
The FRW-like solutions are intriguing because if higher spin fluctuation fields are suppressed by the background, then they may yield cosmologically viable models based on Vasiliev's theory, opening up a new window for embedding the standard models of particles and cosmology into higher spin theory, which may be viewed as the unbroken phase of string theory in which the string is tensionless [21,2,22,24,23]. This setting will inevitably involve the coupling of an infinite number of (massive) higher spin multiplets. One may envisage a scenario in which their presence will play a role in the resolution of the initial singularity, and near the end or after the inflation when the breaking of higher spin symmetry is expected to take place. A much bolder proposal would be the consideration of only massless higher spin theory with its matter couplings furnished through the Konstein-Vasiliev or supersymmetric extension of Vasiliev theory (see [26] for a survey). Such a proposal is motivated by the high degree of symmetry that may yield a UV finite theory, and by the availability of a mechanism [13] for breaking of higher spin symmetries by quantum effects without the need to introduce fields other than those present in the theory, whose spectrum consists of the two-fold product of the singleton representation of the AdS 4 group. Thus it is natural to consider the (matter coupled) higher spin theory as the candidate for a tensionless limit of string theory, in which all the massive trajectories are decoupled completely, and to investigate its consequences for the early universe physics. There are very powerful no-go theorems that forbid accelerating spacetimes in string theory in its tensile phase (see [27] and references therein), inviting the considerations of non-perturbative and string loop effects in a full-fledged formulation of string field theory, and finding its vacuum solutions. On the other hand, higher spin theory can be viewed as a much simpler version of string field theory, in which finding asymptotically de Sitter vacua is a more amenable problem.
The introduction of matter and higher spin symmetry breaking remain a largely uncharted terrain. These aspects are expected to play key roles either for reheating in an inflationary scenario or an analogous mechanisms in non-inflationary scenarios. In the simplest inflation model in standard cosmology, Einstein gravity and a single real scalar field with a suitable potential dominate the early inflationary phase. Here we instead envisage a scenario in which the Einstein plus scalar system is replaced by the bosonic Vasiliev higher spin theory, which consists of a coupled set of massless fields with all integer spins s = 0, 1, 2, 3, ..∞. One can then try to employ the well-known mechanism whereby rapidly inflated fluctuation modes with wavelengths larger than the Hubble length freeze and subsequently re-enter the cosmological horizon after inflation has ended. Assuming that higher spin symmetry breaking and reheating take place at around the same time, one can compute the effects of higher spin fluctuations on the CMB observations at large scales. In these scenarios, it is important to keep in mind that while the higher spin modes may dissipate in time, their couplings to and mixing with the gravitational field may have observable effects. Some studies have already been done along these lines, see e.g. [28,29,30], but based on assumptions on higher spin dynamics not born out of Vasiliev's theory. Let us also note that the analog of the so(1, 3) invariant solution, referred to as the "instanton" solution in Table 1, was obtained as an exact solution for Λ < 0 in [15] and for Λ > 0 as well in [20]. In the case of Λ < 0, a cosmological implications of the solution has been discussed in [15] where it has been argued that it leads to a bouncing cosmology, in some respects reminiscent of the work of [31] based on supergravity considerations 3 .
This paper is organized as follows: In Section 2, we review Vasiliev's higher spin gravity equations. They are formulated in terms of master one-form A and master Weyl zero-form Φ which live on a base manifold X 4 ×Z 4 with coordinates (x µ , Z α ) where Z 4 is a non-commutative real four manifold. The master fields also depend on the coordinates of the fiber space Y 4 with coordinates Y α . In Section 3, we describe the construction of the exact solutions with g 6 symmetries. For the reader's convenience we summarize the solutions here. The master fields are the zero-form Φ and one-form A whose components are displayed in (2.11). In holomorphic gauge, Φ ′ is given in Table 1, and A ′ α in (2.88) and (2.93). In the L-gauge, Φ (L) is given in (3.53) and (3.12), A In Section 4, we examine the regularity of the Weyl zero-form. The scalar field profiles φ(x) are described in a unified manner by using stereographic coordinate system. In studying their regularity, one needs to distinguish between singularities that are gauge artifacts and genuine singularities in the full (x, Y, Z) space, sometimes referred to as the correspondence space.
To this end, one needs to study the solution Φ(x, Y, Z) for the Weyl zero-form, and associated higher spin invariant and the on-shell conserved zero-form charges, as we shall discuss further in Section 4. In Section 5, we take a closer look at the iso(3) invariant solution and compare with the standard cosmological backgrounds. In Section 6, we summarize our results and comment on future directions. Frequently used symbols and notation are summarized in Appendix A.

Bosonic Vasiliev model
In what follows, we review the basic properties of Vasiliev's equations [35] and their classical solution spaces, including boundary conditions in spacetime and twistor space suitable for asymptotically (anti-)de Sitter solutions. For a recent review of the exact solutions see [19].

Non-commutative space
Vasiliev's theory is formulated in terms of horizontal forms on a non-commutative fibered space C with four-dimensional non-commutative symplectic fibers and eight-dimensional base manifold equipped with a non-commutative differential Poisson structure. On the total space, the differential form algebra Ω(C) is assumed to be equipped with an associative degree preserving product ⋆, a differential d, and an Hermitian conjugation operation †, that are assumed to be mutually compatible in the sense that if f, g, h ∈ Ω(C), then where |f | denotes the form degree of f . We shall also assume that 4 It is furthermore assumed that Ω(C) contains a horizontal subalgebra, Ω hor (C), consisting of equivalence classes defined using a globally defined closed and central hermitian top-form on the fiber space, and whose product, differential and hermitian conjugation operation we shall denote by ⋆, d and † as well.
The base manifold is assumed to be the direct product of a commuting real four-manifold X 4 with coordinates x µ , and a non-commutative real four-manifold Z 4 with coordinates Z α ; the fiber space and its coordinates are denoted by Y 4 and Y α ′ , respectively. The non-commutative coordinates are assumed to obey 5) and the differential Poisson structure is assumed to be trivial in the sense that The star product is defined in (A.1). The non-commutative space is furthermore assumed to have a compatible complex structure, such that where the complex doublets obey The horizontal forms can be represented as sets of locally defined forms on X 4 × Z 4 valued in oscillator algebras A(Y 4 ) generated by the fiber coordinates glued together by transition 4 More generally, † • † can be an automorphism of Ω(C), which is of relevance, for example, in the case of models in de Sitter signature with fermions.
functions. Assuming the latter to be defined locally on X 4 yields a bundle over X 4 with fibers given by the differential graded associative algebra Ω(Z 4 ) ⊗ A(Y 4 ), whose elements can be given represented using symbols defined using various ordering schemes, which correspond to choosing different bases for the operator algebra. In what follows, we shall assume that it is possible to describe the field configurations using symbols defined in the Weyl ordered basis, which is manifestly Sp(4; R)×Sp(4; R) ′ invariant, as well as the normal ordered basis consisting of monomials in 5 with a-and b-oscillators standing to the left and right, respectively, which breaks Sp(4; R) × Sp(4; R) ′ → (Sp(4; R) × Sp(4; R) ′ ) diag . Equivalently, we shall assume that the elements in have well-defined symbols in normal order, which can be composed using the Fourier transformed twisted convolution formula (A.1), and that they can furthermore be expanded over the Weyl ordered basis of A(Y 4 ) with coefficients in Ω(Z 4 ), using the aforementioned star product.
As for the fiber algebra A(Y 4 ), it is assumed to be an associative algebra closed under the star product and the hermitian conjugation operation † defined above. As we shall describe in more detail in Sections 2.4 and 5.3, the algebra A(Y 4 ) will furthermore be assumed to contain certain nonpolynomial elements and distributions playing a role in constructing higher spin background and fluctuation fields 6 .

Master fields
The model is formulated in terms of a zero-form Φ, a one-form and a non-dynamical holomorphic two-form with Hermitian conjugate J = (J) † , where b is a complex parameter and κ := κ y ⋆ κ z , κ y := 2πδ 2 (y) , are inner Klein operators obeying 14) 5 The normal order reduces to Weyl order for elements that are independent of either Y or Z. 6 In order to construct higher spin invariants playing a role as classical observables, the algebra needs to be furthermore equipped with a trace operation that provides it with a Hilbert space structure or other suitable inner product structure for any zero-form f , where π y and π z are the automorphisms of Ω(Z 4 ) ⊗ A(Y 4 ) defined in Weyl order by π y : (x; z,z; y,ȳ) → (x; z,z; −y,ȳ) , π z : (x; z,z; y,ȳ) → (x; −z,z; y,ȳ) , (2.15) and π y • d = d • π y and π z • d = d • π z . It follows that for any form f , idem J andπ := πȳ • πz.

Kinematic conditions
Higher spin gravities consisting of Lorentz tensor gauge fields can be obtained by imposing the integer-spin projection Models in Lorentzian spacetimes with cosmological constants Λ are obtained by imposing reality conditions as follows [20]: that is, the real form of the sp(4; C) realized in terms of bilinears in Y α is chosen by the Hermitian conjugation operation ρ • †; the consistency follows fromπ(f ) = (π(f † )) † and the fact that if Λ > 0, then (ρ • †) 2 ≡ π •π, which reduces to the identity modulo the integer-spin projection.

Equations of motion
Introducing the curvature and twisted-adjoint covariant derivative defined by respectively, one has the Bianchi identities where ordinary and π-twisted star commutators respectively. The Vasiliev equations of motion are given by which are compatible with the kinematic conditions and the Bianchi identities, implying that the classical solution space is invariant under the following infinitesimal gauge transformations: for parameters obeying the same kinematic conditions as the connection, viz. (2.23)

Component form
Decomposition of the equations of motion under the coordinate basis ( ∂ µ , ∂ α ) for the tangent space of the base manifold, yields the Vasiliev equations using π(A) =π(A),π(dz α ) = dz α and π(dzα) = dzα, and the one-form components obey the following kinematic conditions: (2.30)

Deformed oscillators
Alternatively, introducing the equations of motion involving twistor-space derivatives can be written as that is, the master fields (S α , Sα) define a covariantly constant set of Wigner-deformed oscillators with deformation parameter given by Φ. The deformed oscillators obey reality conditions and integer-spin conditions as follows: (2.36) Besides being useful in constructing exact solutions, observables and exhibiting certain discrete symmetries, the deformed oscillators facilitate the casting of the equations of motion into a manifestly Lorentz covariant form.

Discrete symmetries
The equations of motion and the gauge transformations exhibit the following discrete symmetries: i) Holomorphic parity transformation (Φ, A; ǫ) → (π(Φ), π(A); π(ǫ)) ; (2.37) ii) Deformed oscillator parity transformation iii) Vectorial parity transformation (Φ, A; ǫ) → (P (Φ), P (A); P (ǫ)) , (2.39) where P is the star product algebra automorphism P (y α ,ȳα; z α ,zα) := (ȳα, y α ;zα, z α ) , from which it follows that P • † = † • P and P • π =π • P From it follows that (iii) exchanges a solution to the equations with parameter b to a solution to the equations with parameterb. In particular, ifb = ±b, then one can extend P to where P ′ is an internal parity map acting on the component fields, and project the spectrum of the theory by demanding which thus correlates the internal parity with the vectorial parity in twistor space.

Manifest Lorentz covariance
To cast the equations on a manifestly Lorentz covariant form, one introduces the field-dependent generators [35,36] M (tot) αβ and redefines where (ω αβ µ (x), ωαβ µ (x)) is a bona fide canonical Lorentz connection on X 4 , after which the equations of motion involving spacetime derivatives can be re-written on the following manifestly Lorentz covariant form 7 [37,17,38]: where with The Lorentz connection is defined, as usual, up to tensorial shifts, that can be fixed by requiring that the projection of W onto M

Vacuum solutions
Flat connections. The equations of motion admit solutions where Ω is a locally defined one-form on If Ω ∈ Ω(X 4 ) ⊗ A(Y 4 ), then there exists locally defined gauge functions L on X 4 such that that we shall refer to as vacuum connections, as they preserve higher symmetries with rigid , on the other hand, contains flat connections constructed from projector algebras that cannot be described using gauge functions and that break some of the vacuum symmetries [14].
Maximally symmetric spaces. The (A)dS 4 vacua are described by gauge functions valued in the real form G 10 of Sp(4; C) selected by the reality condition introduced above. Thus, G 10 refers to AdS group for λ 2 > 0 and dS group for λ 2 < 0, with the commutation rules for the g 10 algebra given by (2.57) and they can be realized in terms of the Y -oscillators as Introducing coset elements the Maurer-Cartan form decomposes into a frame field and a Lorentz connection as follows: where thus In these bases, the flatness condition reads where the Riemann two-form The full equations of motion can be solved in two dual fashions, one involving normal ordered scheme and perturbatively defined Fronsdal fields, and the other based on a topological field theory approach, which we describe below.

Normal ordered perturbation scheme
In the normal order, defined by the star product formula (A.1), the inner Klein operators become real analytic in Y and Z space, viz.
Assuming that the full field configurations are real-analytic on Z 4 for generic points in X 4 , one may thus impose initial conditions Assuming furthermore that A α | C=0 is a trivial flat connection on Z 4 , that one may choose to be A α | C=0 = 0, and choosing a homotopy contractor for the de Rham differential on Z 4 , which entails imposing a gauge condition on A α , one may solve the constraints on D α Φ, F αβ and F αµ on Z 4 in a perturbative expansion of the form: . . , C) and A (n) (a µ ; C, . . . , C), and the latter is linear in a µ . These quantities are real-analytic in Y 4 × Z 4 provided that C and a µ are real analytic in Y -space and all star products arising along the perturbative expansion are well-defined.
From the Bianchi identities, it follows that the remaining equations, that is, F µν = 0 and D µ Φ = 0, are perturbatively equivalent to F µν | Z=0 = 0 and D µ Φ| Z=0 = 0, which form a perturbatively defined Cartan integrable system on X 4 for C and a µ .
To Lorentz covariantize, one imposes and substitutes . . , C). Due to the manifest Lorentz covariance, the quantities F µν | Z=0 and D µ Φ| Z=0 depend on the Lorentz connection only via the Lorentz covariant derivative ∇ and the Riemann two-form (r αβ , rαβ); it follows that where w can be chosen to not contain any component field proportional to y α y β andȳαȳβ.
Perturbatively defined Fronsdal fields. Expanding the differential algebra around the (anti-)de Sitter vacuum and assuming that the homotopy contraction in Z-space is performed such that referred to as the Vasiliev gauge [40], the resulting linearized system on X-space provides an unfolded description of a dynamical scalar field and a tower of spin-s Fronsdal fields where we use the convention that repeated indices are symmetrized.
At the nonlinear level, the Cartan integrable system on X-space provides a deformation of the equations of motion for these fields, which is consistent as a set of partial differential equations but that depends on the choice of initial data for Φ and W µ as well as the gauge for A α (which enters via the homotopy contractor in Z space). Whether there exists a choice that yields a formulation of higher spin gravity in X-space that lends itself to a standard path integral formulation remains an open problem 8 .

Topological field theory approach
Alternatively, one may treat the system as an infinite set of topological fields on X 4 × Z 4 packaged into master fields valued in A(Y 4 ) represented by symbols in Weyl order, that is, as expansions in terms of the generators of A(Y 4 ) star multiplied by differential forms on Ω(X 4 × Z 4 ), referred to as mode forms.
The field configurations are assigned a bundle structure, whereby a projection of A is assumed to define a connection valued in a Lie subalgebra of A(Y 4 ). The complementary projection of A, referred to as the generalized frame field, together with the Weyl zero-form Φ are taken to belong to adjoint and twisted adjoint sections, respectively, over X 4 × Z 4 , which is treated as a base manifold; the two-forms J and J by their definition belong to twisted adjoint 8 To our best understanding, the standard classical Noether procedure breaks down [42], while there exists a quantum effective action in X4 for asymptotically (anti-)de Sitter boundary conditions. In order to obtain a path integral measure, one may instead follow the approach proposed in [43,37,44].
sections. The bundle connection is assumed to act faithfully on the sections, and the bundle curvature is assumed to be an adjoint section, as required by the equations of motion.
As for boundary conditions, the base manifold is assumed to be compact, and the sections, i.e. their mode forms, are assumed to be bounded away from a set of marked points representing boundaries. In a generic coordinate chart U ⊂ X 4 , the sections are described by an integration constant for the Weyl zero-form, a flat connection on Z 4 and a gauge function on U × Z 4 . At the marked points, the initial data instead consists of prescribed singularities in the generalized frame field and related fall-off in the Weyl zero-form (including the physical scalar field). For asymptotically (anti-)de Sitter solutions, we take X 4 to have the topology of S 1 × S 3 with a marked S 1 , such that, to all orders in classical perturbation theory, the leading terms in the master fields at the marked S 1 × Z 4 describe a set of free Fronsdal fields, which one may view as a condition at the boundary of (A)dS 4 times Z 4 . To impose boundary conditions on Z 4 , we assume that Ω(Z 4 ) i) is closed under star products, which can be achieved by taking the Fourier transforms of the zero-forms in Ω(Z 4 ) to be L 1 in momentum space (i.e. to be expandable in terms of plane waves that generate a twisted abelian group algebra), which requires the zero-form sections on Z 4 to be bounded at Z = 0; ii) has a graded trace operation given by integration of the top-forms on Z 4 , which requires these to fall off at Z = ∞ so as to belong to L 1 (Z 4 ).
Finally, A(Y 4 ) is taken to be a set of operators in a quantum-mechanical system equipped with a (possibly regularized) trace operation that is dual to the boundary conditions at the marked The above geometries can be characterized by functionals, playing the role of classical observables (including on-shell actions), given by combined traces over A(Y 4 ) and integrations over X 4 × Z 4 (possibly with insertions of delta functions localized to submanifolds). These gauge transformations that leave these functionals invariant are referred to as proper, or small, gauge transformations, as opposed to large gauge transformations that alter the asymptotics of the fields and hence the value of the observables. The resulting moduli space is thus sliced into (proper) gauge orbits labelled by the observables, each of which defines a microstate of the theory 9 .
9 A subset of the classical observables are extensive; keeping these fixed defines a higher spin ensemble consisting of a large number of microstates. In [45,38], it has been proposed that the extensive variables are the zero-form charges [14,37], and in [46] it has been proposed that a complete set of classical observables in the case that X4 has trivial topology is given by the space of twisted open Wilson lines in Z4. According to these proposals, the rigid symmetries of the vacuum should leave the extensive variables invariant while acting nontrivially on the microscopic variables; for related remarks, see [46,18].

Gauge functions
In the topological field theory approach, solution spaces are obtained starting from a reference at a base point p 0 ∈ X 4 , constructed from an integration constant C ′ for Φ ′ at, say, Z = 0, and an flat connection on Z 4 , that we shall trivialize in most of what follows. Moduli associated to the connection and generalized frame field on X 4 are then introduced by means of a large gauge transformation α ), where M α ∈ C 4 (for details, see [37,38,46] µ ) asymptote to configurations describing free Fronsdal fields 10 in accordance with the central on mass-shell theorem close to the marked S 1 × Z 4 .
We shall refer refer to (a) and (b) as dual boundary conditions, as (a) requires factorization of the master fields in Weyl order, whereas (b) requires normal order. We thus propose to fix H (n) by requiring i ) Manifest Lorentz covariance and real analyticity in Y of the normal ordered symbols of (Φ (G) , A (G) ) at the origin of Y 4 × Z 4 , so that the field configurations are expandable in terms of Lorentz tensorial component fields on X 4 defined by Taylor expansion in Y at ii) the Weyl ordered symbols of (Φ (G) , V (M )) to be traceable over Ω(Z 4 ) ⊗ A(Y 4 ), for there to exist higher spin invariants playing the role of classical observables; iii) Perturbatively stable asymptotic Fronsdal fields in weak-coupling regions of X 4 (where the Weyl zero-form goes to zero), for the classical observables to admit perturbative expansions in terms of parameters related to sources for weakly coupled higher spin gauge fields.
The following additional remarks are in order: 1. Zig-zagging self-consistency: At nth order, the quantity Φ (G,n) is a functional of H (n ′ ) with 1 n ′ n − 1 and initial data C ′(n ′ ) with 1 n ′ n, which means that condition (a), which must hold for finite Z, is in effect a non-trivial admissibility condition on the Y -dependence of the initial data C ′ , i.e. on A(Y 4 ).
2. Residual small gauge transformations: The above conditions do not determine the hs 1 (4) part of H (n) , which is real analytic in Y , and which can thus be used for small gauge transformations inside the bulk.

Deformed oscillators:
Although the master fields S (G) are not sections, one can require that Φ (G) and the twisted open Wilson loops V (M )) form an associative algebra with traces, which can be use to construct a complete set of higher spin invariant observables that one may think of as substitutes for the standard ADM-like charges that can be used to define higher spin ensembles in unbroken phases; for further details, see [18] where ǫ ′ are constant parameters stabilizing Ψ, viz.
Conversely, given a set of symmetries forming a Lie algebra g, spaces of g-invariant solutions can be found by solving the linear constraint (2.84) on Ψ together with the conditions that Ψ belongs to an associative algebra that is left invariant under star multiplication by the inner Klein operators, i.e. Ψ ⋆ κ y and Ψ ⋆ Ψ should belong to the algebra, which is the approach that we shall employ.
In summary, the dual boundary conditions are physically well-motivated and non-trivial; in this paper, we shall focus on their implementation at the linearized level, leaving higher orders, starting with the issue of whether Φ (G,2) obeys (a), for a forthcoming publication including various types of boundary conditions.

A universal particular solution in holomorphic gauge
For all vector fields v tangent to X-space, we have ı v A ′ = 0, and hence ı v dA ′ = 0 and ı v dΦ ′ = 0, i.e. and Thus, prior to switching on the gauge function G, we need to find a particular solution to the above system subject to a generic zero-form initial datum. To this end, we observe that the where thus both Ψ and Ψ ⋆ κ y are assumed to be elements in A(Y 4 ), and solves the fully non-linear equations provided that and that One class of solutions is given by [17] where we have introduced a spinor frame (u + α , u − α ) obeying and z ± is defined in (A.4). The introduction of these variables is required in order to integrate the delta function in Weyl order.
Thus, in order to construct solution spaces with desired physical properties, we need to expand Ψ over suitable subalgebras of A(Y 4 ); for the cases of particle fluctuation modes and black hole-like generalized Type D modes, see [17,18]. In what follows, we shall examine a new type of subalgebras related to solutions with six Killing symmetries inside the isometry algebra of (A)dS 4 .

Construction of the exact solutions with six symmetries
In this section, we shall begin by describing the factorization method that will be used to construct the solutions. We shall than construct domain walls (DW), instantons 12 (I) and FRW-like solutions (FRW) given by foliations of a four-dimensional spacetime M 4 with threedimensional foliates M 3 that are maximally symmetric metric spaces, we shall first choose embeddings of the corresponding six-dimensional isometry algebras g 6 into the ten-dimensional isometry algebra g 10 of the vacuum solution. We then switch on g 6 -invariant Weyl zero-forms and gauge functions.
3.1 Initial data for Weyl zero-form with six Killing symmetries

Unbroken symmetries
In order to describe foliations with maximally symmetric foliates, we embed g 6 into g 10 as follows [14]: 2) and the representatives of the cosets so(3, 1)/so(2, 1) for ǫ = 1, and the coset so(3, 1)/so (3) for where we have introduced the parameter ǫ. The resulting symmetry algebra reads as follows 14 : giving rise to the cases listed in Table 1. 12 The instantons break all transvection isometries of the vacuum solution, i.e. g6 coincides with the Lorentz algebra of the vacuum solution. 13 One can always choose α 0 by redefining L a r , after which one may take β 0 using the global Z2-symmetry generated by the π-map, which exchanges β with −β.
14 The symmetry of the full solutions is generated by the parameters given by the conjugation of the linearized symmetry parameters by the gauge function.

23
Type Table 1: g 6 -invariant M 3 -foliations arising in the bosonic models, with I standing for instantons, and FRW k and DW k , respectively, standing for FRW-like solutions (ǫ = −1) and domainwalls (ǫ = +1) with foliates with curvatures of sign k = sign(ǫα 2 − λ 2 β 2 ). The embeddings of g 6 into the isometry algebra of the (A)dS 4 vacua are governed by a vector L a with L 2 = ǫ and two real parameters α, β > 0. The last column contains the corresponding g 6 -invariant initial data for the Weyl zero-form. Two families of foliations with k = −1 interpolate between the cases with k = 0 and the instantons.

Invariant Weyl-zero form integration constant
Imposing g 6 -invariance of zero-form initial data, viz.
it follows from the first condition that and from the second condition that where we have used (3.9)

Regular presentation
To solve (3.7), we Laplace transform Φ ′ as This gives a characteristic equation for η solved by that are either real or purely imaginary. Thus, for β > 0 we have 12c) The small contours C encircle the poles of Φ ′ counterclockwise. The corresponding solutions for Φ ′ are listed in Table 1

Limits
The flat solutions with k = 0 arise from the so(1, 3)-invariant families in the limit 14) fixed. In the limit β → 0, one has η + → 0 and η − → ∞, and hence and C is a small contour encircling η = 0 counterclockwise 15 .

Regularization of star products
In order to compute we use the lemma 17) and the regularization procedure spelled out in [17], viz.
which suffices to handle the cases with k = 0. In the case of g 6 = iso (3), we have 15 Alternatively, the decoupling of the η+-mode in the limit β → 0 can be achieved using a twisted-adjoint G10 conjugation of Φ ′ that annihilates the η+-mode using the regularization procedure. and the case of g 6 = iso(1, 2) is similar 16 . It follows that where C 2 is a constant given by ǫk = −1 : where the last case contains the instantons, for which C 2 = ν 2 .

Twistor space connection in holomorphic gauge
The general solution in integral form. In the expression for the twistor space connection, it is convenient to express the hypergeometric function in an integral representation as follows Thus, expanding in powers of the deformation parameters, we find that all odd terms are linear in Ψ, while all even terms are (y α ,ȳα)-independent, viz.
where k = 1, 2, . . ., and (3.28) 16 Crucial for the regularization is that, after evaluating (any) one of the two integrals, the exponential that results from the star product e −4ηλ −1 P ⋆ e 4ξλ −1 P becomes independent of the other auxiliary contour-integral variable. Supposing, for concreteness, that one evaluates the integral over ξ first, as above, the calculation shows that the only assumption one uses in this kind of regularization is that |ξ + 1| << |η + 1| << 1, in order to keep ξ = −1 as the only pole encircled by the contour [17]. where Let us proceed by looking into the internal connection order by order in its perturbative expansion.
First order. The linearized twistor space connection is given by For its basic distributional properties, see remark made above. Clearly, in Weyl order, the linearized twistor space connection is not real-analytic at the origin of Z-space; whether it becomes real analytic in normal order depends on the details of Ψ, as we shall analyze in more detail below.
Second order. We have the second order (3.32) Thus we can split the integral into two pieces as follows: where which is convergent for all real w. For w > 0, we can integrate by parts and rewrite it as where the exponential integral This function can be extended from the positive real axis to a complex function that is analytic away from the negative real axis, where is has a Taylor expansion given by where γ E is the Euler-Mascheroni constant; we note that w d dw E 1 (w) = − exp(−w). Thus, continuing I > 2 (w) to complex w, and adding I < 2 (−w), we find which can be rewritten as where R 1,2 are real analytic at w = 0: Therefore, in summary the second order correction (A ′ α ) (2) is independent of Y and bounded in Z, though it is not real analytic at Z = 0.

Master fields in L-gauge
We recall that starting from the particular solution obtained in the the holomorphic gauge, which incorporates the zero-form initial data, gauge inequivalent solutions can be reached by means of large gauge transformations generated by gauge functions G defined locally on patches.
In the case of asymptotically (anti-)de Sitter spacetimes, we use G = L ⋆ H, where L is the vacuum gauge function, which brings the master fields to what we refer to as the L-gauge, after which H is constructed order by order by imposing the dual boundary conditions (a) and (b) specified in Section 2.4.2. Finally, the patches are glued together using transition functions belonging to a structure group.

Weyl zero-form
The Weyl zero-form in L-gauge is given by (3.41) Substituting Φ ′ = Ψ ⋆ κ y according to the ansatz (2.88) we get We first compute where we have used (3.10). The L-conjugate of Ψ is given by where form a new set of canonical coordinates in which The matrices K and L are computed in stereographic and planar coordinate systems in Appendix C.2 and C.4, respectively. It follows that indeed where C is the constant in (3.20), as can be seen using the lemma whereb a = iηL a , followed by contour integration. Going back to the original canonical coordinates for Y 4 , we have where

50)
Provided that A is invertible, we can write We thus find which is readily computed with the result with Φ ′ (η) from (3.12). The resulting Weyl zero-forms consist of scalar field profiles, that we shall analyze in more detail in Section 4 using stereographic coordinates, and in Appendix C using adapted coordinate systems.

Twistor space connection at even orders
The even order terms are the same in the holomorphic gauge and the L-gauge, as they are independent of Y . From (3.27), the sum of all even orders is given by where We note that (A (L) α ) (even) is independent of X and Y , and bounded in Z-space.

Twistor space connection at odd orders
In the L-gauge, the sum of all odd-order terms from (3.26) is given by where the generating function and where the extended generating function Next we perform the Gaussian star product (3.60) It follows that (A (L) α ) (odd) is real analytic in Z-space, which simplifies the construction of H, and that it has singularities in Y -space, stemming from the divergence at τ = +1 that arises as a result of the above Gaussian integration. Thus, we need to demonstrate that the latter singularities go away upon switching on H. From (3.58), (3.59) and (3.60) we find where the generating function consists of even orders in deformation parameters, and its η dependence enters via A andỹ α .

Spacetime connection
The spacetime connection is simply given by

Patching
The expressions given so far are defined in the region of validity of the gauge function L, that is, for λ 2 x 2 < 1, whereas a global formulation on the the vacuum manifold M

65)
i.e. this particular configuration can be implemented for any choice of structure group, that is, in any topological phase of the theory. In these types of configurations, it follows from the reflection symmetry that any singularity in the master fields that arises inside U ± cannot be removed using patching.

The Weyl zero-form
At first order we observe that

Twistor space connection
At the linearized level, the role of H (1) is to ensure that is real analytic in Y 4 × Z 4 and obeys the Vasiliev gauge condition, is real analytic in Z 4 , it follows that where L Z = {q, ı Z } is the Lie derivative along the Euler vector field whose invserse can be represented (on real analytic functions) as where t L Z acting on differential forms implements the diffeomorphism z α → tz α . Thus, to which the initial datum H (1) | Z=0 does not contribute, and we have taken into account the holomorphicity in Z space. Writing it follows that and and where the dependence on the auxiliary spinor frame u ± α has dropped out. Indeed, the above result agrees with that found working directly in normal order, viz.

Spacetime connection
In the Vasiliev gauge, the linearized spacetime connection where the background covariant derivative is real analytic on Y 4 × Z 4 and independent on the auxiliary spin frame, it follows that these properties hold true as well for the Z-dependent part of U (G,1) . As for its Z-independent part, viz.

Comments on residual symmetries, factorization and Vasiliev gauge
We recall from Section 2.4.2 that as far as symmetry considerations are concerned in finding exact solutions, these are facilitated by the the combined use of gauge functions and the holomorphic factorization method employed in (2.88), which ensures that the symmetries of the initial datum Ψ(Y ), that can be imposed by means of undeformed generators, remain symmetries of the full master fields. We would like to contrast this approach to that followed in [14], where an exact so(1, 3)-invariant solution was constructed for Λ < 0 using the vacuum gauge function L followed by requiring the primed twistor space configuration to be invariant under the full field-dependent Lorentz generators (instead of using the holomorphic factorization method). In the same paper, the six-parameter symmetry groups considered here were also examined, but as the factorization method was not used, the imposition of symmetry conditions involving translations became problematic at the nonlinear level, and FRW-like and domain wall solutions were given explicitly at the linearized level, in agreement with Table 1, and shown to exist at the second order of classical perturbation theory. It would be interesting to pursue the latter construction to the second order and compare it to the second order expressions obtained in the current paper in L-gauge.
The factorization method implies, however, that the linearized master gauge fields are not real analytic in (y α ,ȳα) in L-gauge, but as we have seen, these singularities can be removed by going to Vasiliev gauge by means of a large gauge transformation. It remains to be shown whether this procedure can be imposed to order by order in weak field perturbative expansion by imposing dual boundary conditions as discussed in Section 2.4.2.
We conclude this section by explaining technically the reason for being able to impose equally the first of the conditions (3.6) via the full Lorentz generator (2.44). First, defining c., one can show [34,35,36] that the fully nonlinear completion of the Lorentz generator is exactly such that, on the solutions of the Vasiliev equations, , Φ] π = 0. It is then clear that, on a purely Y -dependent Weyl zero-form, such as that of our Ansatz (2.88), the action of the Lorentz generators reduces to the one of their zeroth-order, purely Y -dependent piece, and we can therefore impose so(3)symmetry as in (3.6). In general, however, imposing invariance conditions on the master fields under subalgebras that include translations can only be done perturbatively, as it was suggested in [14], since no fully non-linear completion of the P a is known. This limitation is not present on the subspace of the full solution space captured by the factorized Ansatz (2.88), where Φ is first-order-exact.
The virtue of the factorization (combined with the gauge function method) is that it gives us the possibility of solving exactly for the Z-dependence of the master fields irrespectively of the initial datum Φ ′ (Y ), thereby dressing a solution of the linearized twisted-adjoint equation into a full solution of the Vasiliev equations. In particular, due to the factorized form, the equations for S ′ α reduce to (2.92), that do not involve Φ ′ and can be therefore solved once and for all (via the methods developed in [17,18]). This allows us to impose the g r symmetries at full level, since the action of symmetry parameters ǫ(Y ) is sufficient to impose symmetry conditions on the full solution space (2.88). Indeed, the symmetries ǫ(Y ) of Φ ′ (that is, the parameters

Regularity of full master fields on correspondence space
In this section, we examine the scalar field profiles and the Weyl zero-form using the stereographic coordinates (see Appendix B and C for details), which facilitate a uniform treatment of all cases. We first study the linearized scalar field profile and then turn to the analysis of the regularity of the full Weyl zero-form in the correspondence space, by which we mean the twistor space extended spacetime with coordinates (x, Y, Z).

Linearized scalar field profile
In stereographic coordinates, the linearized scalar field is given by which can be re-written as where R is the reflection map. For β > 0 and k = 0, we have where η ± = −γ ± ǫ + γ 2 , with γ = iα λβ and ǫ = L a L a , such that that R a (b ± ) = −b a ∓ , and ν + := µ and ν − =μ for ǫk = −1.
The iso-scalar surfaces S i (c i ) (i = ±) are defined by which are g 6 invariant, and complexified for ǫk = −1; in particular, S i (0) is the boundary.
Away from the boundary, we have It follows that as can be seen first by eliminating h 2 , which yields that must hold identically for all x a modulo η − b a , and η + η − = −ǫ, provided that c ± obey the relation in (4.9). Taking the limit |c ± | → ∞, it follows from (4.9) that if k = 0, so that η + = η − , then S + (∞) ∩ S − (∞) = ∅ away from the boundary; conversely, requiring S + (∞) ∩ S − (∞) = ∅ implies that 11) i.e. the two singular surfaces coincide on a two-dimensional subspace of the boundary. If k = 0, then η + = η − , and it follows from (4.9) that S + (c) = S − (c) for all c. Moreover, if ǫk = −1, is the cone

(Ir)regularity of Weyl zero-form
As the description of the solutions in terms of Fronsdal fields is reliable only at weak coupling, we resort to the full master fields close to the surfaces S ± (∞). The Weyl zero-form (see (3.42), (3.44) and (3.49)) is given by (4.14) from which it follows that At the light conesx 2 ± = 0, away from the apexes, we can treat the case ǫk = +1 by writing where τ = ǫτ † is a linear coordinate along the lightcone and u (±) α is a normalized x-dependent spin frame, viz.
α ) † . It follows from (4.12) that 18) and hence which provides a normalization of the real analytic delta function in two variables, corresponding to the delta sequence lim a→0 a exp(iast) = 2πδ(s)δ(t) , (4.21) and staying away from the apex, i.e. taking τ = 0, we find that (4.22) whose apex limit is indeed in agreement with (4.15), viz. (4.23) In general, one needs to distinguish between singularities that are gauge artifacts and genuine singularities showing up in higher spin invariants. In particular, the zero-form charges of [14,37] are higher spin invariants built directly from the Weyl zero-form and the twistor space connection; in the present case, the simplest such charges takes the form which are formally de Rham closed on X 4 on shell. We have checked that indeed while it remains to compute Φ ; if the latter quantity vanishes, which is our expectation, then the Weyl zero-form Φ is regular on M As for k = 0, on the other hand, we can treat the caseν = 0 by taking a limit, leading to that indeed diverges as τ → 0, in agreement with the separate analysis at the apex given above, and yields divergent values for I n . Thus, as I n vanishes away from the conex 2 ± = 0, it follows that if k = 0, then the zero-form charges are not smooth on M  where Z 2 = {e, γ} is defined by  vanishes. Since the metric in this case is de Sitter, we expect that the solution for the metric will be a deformation from de Sitter metric starting at second order in the deformation parameter.
As for the fields with spins s > 2, they vanish in the background solution at lowest order in the deformation parameter. Whether they arise in higher orders remains to be determined.

Comparison with standard cosmological backgrounds
In order to compare with standard inflationary models, let us quickly summarize the behavior of the inflaton. In standard slow roll inflation, one studies a solution for which the metric is close to de Sitter, with deviations parametrised by the slow-roll parameters 17 where H :=ȧ/a is the Hubble parameter, and a dot denotes a derivative with respect to cosmic time t. The metric is de Sitter when H is constant, and by definition inflation happens when a > 0 or ǫ < 1. The solutions for the scalar field and the metric to next-to-leading order in slow-roll, and valid for t − t * φ/φ, are given in terms of conformal time τ by where at some fixed time τ * we have imposed that the scale factor and the scalar field take on some given values a * and φ * . For the simplest potential V (φ) = m 2 φ 2 /2, the slow-roll parameters are ǫ v = 2φ −2 and η v = 2φ −2 , with φ being the background solution. The slow-roll approximation then requires φ ≫ 1 in Planck units. It follows that φ * = 6H/m, which means that we must have m ≪ H. If one perturbs around this background, writing φ(x) =φ(τ )+ϕ(x), one obtains in Fourier space after choosing the Bunch-Davies vacuum ϕ(k, τ ) = e i(2κ+1)π/4 − πH 4 τ 3 8ǫ where we see that the slow-roll parameters appear in the index of the Hankel functions. These solutions are long-lived in the sense that ϕ ∼ τ 3/2−κ at late times τ → 0. For a recent review, see [49].
In the approaches to cosmology with higher spin fields in [28,29,30], the scalar field is taken to be an inflaton, whose background behaves as described above, and the background metric field is taken to be de Sitter spacetime; all higher spin fields are taken to vanish in the background. However, these approaches are not derived from a theory with higher spin symmetry.
In comparing the background solutions summarised above with ours, we note that the linearised Vasiliev scalar field satisfies the equation ( (0) + 2λ 2 )φ = 0, subject to iso(3) invariance as discussed earlier. As such it has a potential with m ∼ H in the notation employed above, which is incompatible with the slow-roll approximation. Indeed, it does not have the shape of the solution for standard slow-roll inflation (5.6), and even goes to zero at late times. 17 These are the slow-roll parameters defined in terms of the Hubble parameter and the derivatives of the field.
The slow-roll parameters defined in terms of the potential ǫv := V (φ) are related to these by ǫ ≈ ǫv and η ≈ ηv − ǫv to leading order in slow-roll.
Even if the scalar behaves differently from the standard slow-roll inflaton at the level we are working, our solution is still inflationary in the sense that the metric is close to de Sitter.
The time-dependent deformation away from de Sitter may in principle lead to an end of the accelerated phase. This would requireä < 0 which is far from de Sitter, and may in principle be achieved by summing all orders in the deformation parameter.
Though the calculation of the fluctuations in higher spin gravity is beyond the scope of this paper, let us discuss their expected behavior by considering fluctuations of a conformally coupled scalar field around de Sitter. CMB observations have fixed primordial fluctuations sourced by scalar fluctuations, to be nearly scale-invariant (in this context this is defined as their 2-point function in Fourier space behaving as 1/k 3 ). They are also observed to have a larger amplitude than fluctuations sourced by the graviton. This is different from what would be generated by a conformally coupled scalar field: The behaviour of its 2-point function in Fourier space in the limit τ → 0 goes like 1/k. Furthermore, the amplitude of this 2-point function is suppressed by positive powers of τ , so one can say that they are "short lived" and suppressed with respect to the fluctuations of a massless graviton (which go as τ 0 in that limit, and are thus "long lived"). We expect that the corrections to this behaviour of the scalar field during inflation will be suppressed by the deformation parameter.
We can envisage two mechanisms by which the behavior of the scalar in higher spin theory may be "long lived". One possibility is that an exact FRW-like solution, i.e. to all orders in the deformation parameter, may lead to a behavior of the metric for which it takes an infinite proper time to reach a critical value of the conformal time. Whether this leads to long-lived scalar fluctuations remains to be seen. Another possible mechanism is to consider a coupling with a massive higher spin multiplet that contains a massive scalar with conformal dimension zero. Indeed, this arises in 6-fold product of the fundamental representation of de Sitter group.
A long-lived scalar field would arise in this scenario even though the coupling of massive higher spin multiplets with Vasiliev higher spin gravity is a formidable task which has hardly been studied so far. We should also require the scalar two-point function to be approximately, but not exactly, scale invariant. Since our solution is close to de Sitter, but not exactly, such behaviour can emerge.
Assuming that one resolves the problem described above, the amplitude of fluctuations produced should agree with observations, in particular the CMB data. Clearly, since observations are made at very late times, when the characteristic energy scales are small, higher spin symmetry should be broken. In a conservative scenario, one may assume that at such low energies physics is well described by the Standard Model coupled to gravity in a gravitational background inherited from inflation. However, it remains to be seen whether the details of the higher spin symmetry breaking gives rise to novel interactions in the effective action. For inflation to be described by the unbroken phase of higher spin gravity, the symmetry breaking should happen at energy scales smaller than ∼ 10 15 GeV according to the upper bounds on graviton (tensor mode) fluctuations 18 [41]. If the dependence of the graviton two-point function deviates from the H 2 /M 2 pl behavior significantly, this scale will change accordingly.

Towards perturbation theory around exact solutions
Given the g 6 -invariant solutions constructed above, it is natural to study fluctuations around them. This can be facilitated using the factorization method, with zero-form initial data and treating Ψ bg exactly while keeping only the first order in Ψ fl . In what follows, we shall make the stronger assumption that Ψ bg , Ψ fl ∈ A(Y 4 ), i.e. we assume that both background and fluctuations belong to the same algebra, such that Ψ ⋆n ∈ A(Y 4 ), which can then be expanded separately in background as well as fluctuation parameters.
Thus, in order to construct a concrete model, we need to choose A(Y 4 ) in accordance with the dual boundary conditions in twistor space and spacetime. As a concrete example, let us take Λ < 0, and consider fluctuations around i.e. the (unique) FRW-like solution in the case of negative cosmological constant. On physical grounds, we take A(Y 4 ) to consists of deformations of the cosmological background, which correspond to spacetime mode functions that cannot be localized 19 , and particle and black hole-like states, corresponding to localizable spacetime mode functions. Thus, given, respectively, by the orbits of the higher spin algebra hs 1 (4) (obtained by repeated action with constant hs 1 (4) parameters) of Ψ FRW (AdS) − , denoted by A nl , and the identity operator; the massless scalar particle ground state (with anti-de Sitter energies ±1 and vanishing spin); and the black hole-like solution with vanishing anti-de Sitter energy and spin [48,17]. The higher spin algebra hs 1 (4) is simply the algebra of even order polynomials in Y α with respect to the 18 In particular, the amplitude of the two-point function of massless graviton (tensor mode) fluctuations should be smaller than ∼ 10 −11 . If the two-point function is similar to that of standard inflation, which goes as H 2 /M 2 pl , this means that H 10 −4 M pl . 19 As for holographic interpretations, while the particle and black hole-like states can be mapped to operators of dual conformal theories, it is natural to associate the non-localizable modes to operators in a phase of the boundary field theory in which conformal invariance is spontaneously broken; for the case of the flat domainwall in anti-de Sitter spacetime, see [16].
commutation rule (2.5). Using the regular presentation (see Eq. (3.17) and Appendix E), the star products of these ground states are well-defined, leading to the following fusion rules: where we note the interesting facts that the non-localizable modes and the localizable modes form two self-interacting subsystems, and that non-localizable modes undergo stimulated decay to localizable modes. The system of self-interacting system of particles and black holes has been studied in [18], where a fully non-linear solution space was obtained by superposing rotationally invariant scalar particle and black hole-like states, which form a subalgebra of A pt ⊕A bh spanned by projectors and twisted projectors. Put into equations, letting E denote the energy operator, one has Ψ pt := µ=±1,±2,... µ n P n ⋆ κ y , µ −n = µ * n , (5.13) where P n (E) = 2(−1) n−1 ε and with C(ε) and C(±i) being small contours encircling ε and ±i, respectively. Using the star product lemmas in Appendix E and contour integration techniques, it follows that P n ⋆ P m = δ m,n P n , (P n ) † = P n , P n ⋆ κ y ⋆κȳ = (−1) n P n , (5.17) and indeed 18) in accordance with the fusion rules given above.
The following remarks are in order: 1. The particle states form Hilbert spaces with hs 1 (4)-invariant sesqui-linear forms that are isomorphic to direct product of two singletons, the black hole-like states belong to a real vector space with hs 1 (4)-invariant Euclidean bilinear forms that are isomorphic to the direct product of a singleton and an anti-singleton [18,33]; it remains to be seen whether the non-localizable modes admit any such first-quantized interpretation.
2. The above considerations apply to other g 6 invariant solutions with Λ < 0 as well, while for Λ > 0 the star product realization of particle and black hole-like states need further study.
3. In the case of Λ > 0, we let A nl stand for the hs 1 (4) orbit generated from the so(4)-invariant and the identity operator, and A pt for the orbit of the iso(3) invariant solutions. Using Eq.
(3.17) and the regular presentation, it follows that Thus, if it is possible to equip A pt with a basis plane waves normalized on Dirac delta functions that permits an interpretation in terms of localizable particle states such that the two first fusion rules remain intact, while possibly A pt ⋆ A pt may become nontrivial, then we would have a mechanism in the case of Λ > 0 analogous to that given above in the case of Λ < 0.

Conclusions
We have constructed classes of exact solutions of Vasiliev's bosonic higher spin gravities with Killing symmetries given the enveloping of six-dimensional subalgebras of the (anti-)de Sitter symmetry algebras. In order to construct the solutions we have used the fact that Vasiliev's equations form a integrable system on an enlargement of spacetime by an internal non-commutative twistor space. As the integrability is of Cartan type, we can solve the integrable system transforming a particular holomorphic solution (Φ ′ , A ′ α ) in twistor space into a physical solution (Φ (G) , A α , ∂ µ ) ⋆ G using a gauge transformation generated by a gauge function G = L ⋆ H that is large in the sense that it alters the asymptotic behaviour of the master fields in both spacetime and twistor space. The resulting chain of maps take the following form: where L is a vacuum gauge function and H is a field dependent gauge transformation. The role of L is to switch on the dependence of the fields on the spacetime coordinates and to create a finite region of spacetime in which are real analytic in the twistor Z space. The latter property permits the perturbative construction of H, whose role is to create asymptotic Fronsdal fields. The symmetries of the solution are encoded into the particular solution, which is chosen to be invariant under parameters in the enveloping algebra generated from the six-dimensional symmetry Lie algebra g 6 , viz.
where ǫ ′ are constants built from star products of the generators of g 6 . As a result, the solutions in L-gauge and the physical gauge are invariant under gauge transformations generated by the rigid gauge parameters ǫ (L) = L −1 ⋆ ǫ ′ ⋆ L and ǫ (G) = H −1 ⋆ ǫ (L) ⋆ H, respectively. In the holomorphic and L-gauges, we have given the master fields to all orders, involving an expression for the twistor space connection given by two parametric integrals. In the physical gauge, we have given the solution to first order, and proposed a perturbative scheme for continuing to higher orders based on dual boundary conditions in spacetime and twistor space. It remains to push the gauge function method to higher orders of perturbation theory in the physical gauge, which we hope to report on in a future work. We expect this to generate physically interesting domain wall solutions and FRW-like solutions.
A strong motivation for this work has been the prospects for a higher spin cosmology by a direct approach based on finding its accelerating solutions and studying the cosmological perturbations around them. As a first step in this direction, we have constructed the FRW- In a realistic higher spin cosmology, matter couplings and internal symmetry will need to be introduced. The requirement of higher spin symmetry puts severe constraints in doing so.
The Vasiliev higher spin theory we have considered here is a universal sector of any higher spin theory, just as the graviton, dilaton and Kalb-Ramond two-form potential form a universal sector of any string theory. Assuming that the universal higher spin gravity sector dominates the physics of the inflation, it has the advantage of being unique, thereby avoiding the excessive freedom in choosing field content, interactions and parameters. For example, in the favored approach to standard inflationary scenario, Einstein gravity is coupled to a real scalar with a potential that is picked by hand to satisfy suitable 'slow-roll' conditions. Moreover, the origin of the scalar field in a fundamental theory is not known. In the higher spin theory based inflation scenario envisaged here, however, the scalar field is necessarily part of the spectrum for the consistency of the higher spin theory, and the inflation is not driven solely by the energy stored in a slowly varying scalar field. Indeed, there is a frame in which the only contact term for the scalar field is a mass term [4]. Note, however, that the theory comes with infinite derivative couplings even at a given order in weak fields. Given that there is no mass scale at our disposal to argue that those couplings will be suppressed, they are all equally important. Thus, the inflationary solution to the higher spin theory will be driven by the higher spin invariant, higher derivative couplings.
While the problem of matter couplings and breaking of higher spin symmetry will need to be ultimately attended to, at present the more pressing problems to tackle seem to be the determination of the higher order terms in the FRW background in Vasiliev gauge, carrying out the cosmological perturbation theory along the line described in Section 5.3 and seeking possible holographic interpretation of the results.
We use the convention in which (σ a ) αα = (1, σ) where σ are the Pauli matrices, and (σ)α α is complex conjugate of (σ a ) αα . Furthermore we define (σ ab ) αβ = σ [a is defined by where f * denotes the pull-back operation, that is, in terms of coordinates x µ on M 4 , one has

B.2 Killing vector fields
The symmetry algebra g 6 of the solutions under consideration is embedded via (3.1) into the isometry algebra g 10 of (M 4 , (ds 2 4 ) (0) ). The latter is inherited from the isometry algebra of (R 5 , ds 2 5 ), namely as its subalgebra where f * denotes the push-forward operation, that is , in terms of an intrinsic coordinate x µ at p. In the global coordinate basis, inducing the intrinsic Killing vectors fields It follows that the intrinsic Killing vector fields associated with the g 6 generators M rs and T r defined in (3.1) are given by Thus, under the g 6 transformations defined in (3.1), we have with constant parameters (ξ rs , ξ r ).
such that where U +,+ ∩ U −,− = ∅, with equivalent expressions using U ±,∓ . Adding U ±,∓ one obtains an atlas with nontrivial transition functions given by then τ → 0 − at the future (or past) boundary, and the metric takes the form In terms the inverse conformal time, which can be extended from R − to R, the transition function between U +,+ and U −,− is given by In the case of AdS 4 , the conformal radius obeys z → 0 + at the boundary, and the metric takes the form in each Poincaré patch; the two Poincaré patches can be glued together using

B.6 Stereographic coordinates.
A convenient set of coordinates, that facilitate a unified description of all solutions, are the stereographic coordinates {x a ± } a=0,1,2,3 that arise via the parameterization with inverse where U ± denotes the two stereographic coordinates charts. Each chart covers one half of (A)dS 4 , and can be continued smoothly into λ 2 x 2 ± < −1; the resulting transition function is given by where the reflection map The charts can also be extended (non-smoothly) into λ 2 x 2 ± > 1 as follows: as a point p ∈ (A)dS 4 approaches a point p 0 the subspace λ 2 x 2 ± (p 0 ) = 1 from the inside, i.e. λ 2 x 2 ± (p) < 1, the reflected image R(p) approaches the point R(p 0 ) with x µ ± (R(p 0 )) = −x µ ± (p 0 ) from the outside, i.e.
λ 2 x 2 ± (R(p)) > 1. Thus, one may cover all of (A)dS 4 using a single stereographic coordinate, that we shall take to be x a ≡ x a + , defined on four-dimensional Minkowski space minus the subspace λ 2 x 2 = 1. The boundary is given two copies of the surface λ 2 x 2 = 1; an outer sheet with normal pointing inwards and an inner sheet with normal pointing outwards.
In the AdS 4 case, the surface λ 2 x 2 ± = 1 has the topology of dS 3 ∼ = R × S 2 , while its twosheeted counterpart can be glued together using the reflection map into a single surface with finally, cross back to R(p 3 ) = p 1 , thereby closing a time-like curve.
In the dS 4 case, the boundary consists of two two-sheeted surfaces; one with x 0 > 0 and another one with x 0 < 0. Using the reflection map, these four sheets, each of which thus has the topology of an hyperbolic three-plane, form two pairs, each of which can be glued together into a three-sphere, i.e.

E.2 Fusion rules
Denoting the generators of the complexified Weyl algebra W by (I, u, v), where I is central and [u, v] ⋆ = I, we factor out the ideal generated by I − Id W , and set = 1, leading to a graded associative algebra degree map given by the monomial degree and osp(1|2) subalgebra (u, v; u ⋆ u, 1 2 {u, v} ⋆ , v ⋆ v). Letting w := 1 2 {u, v} ⋆ , g ξ := exp ⋆ (ξw) , ξ ∈ C , (E.6) one has g ξ ⋆ g ξ ′ = g ξ+ξ ′ . Going to Weyl order, one finds the symbols g ξ = 1 cosh ξ exp[tanh(ξw)] , (E.7) which are real analytic except for ξ ∈ (Z + 1 2 )πi in which case they are phase space delta functions defined using delta sequences. It follows that E η := exp(−2ηw) , (E.8) whose star product we extend to all values of η using the closed contour regularization scheme defined in Section 3.1.5. In particular, for η = ±1 we recover the Fock space and anti-Fock space ground state projectors P σ := 2E σ , σ = ±1, thus obeying u ⋆ P + = v ⋆ P − = 0 and (P σ ⋆ P σ ′ )| reg = δ σ,σ ′ P σ . (E.9) The g 6 -invariant solutions make use of the elements E ±i , which thus obey (E σi ⋆ E σ ′ i )| reg = 1 2 δ σ,−σ ′ , (E.10) that is, they close on the identity. We note that E ±i = 1 2 g ±π/4 , that is, the regularization amounts to discarding the non-real analytic group elements g ±π/2 . We also remark that E η gives rise to an Env(osp(1|2)) orbit obtained by left-and right-action by polynomial elements in osp(1|2), that is, by u and v; for η = ±1 these are simply the algebras of endomorphisms of the Fock and anti-Fock spaces. Taking instead η = ±i and restricting to even elements, the resulting Env(sp(1|2)) orbits of E ±i are of use in considering fluctuations around the g 6invariant solutions; see Section 5.3 for an outline.