Group manifold approach to higher spin theory

We consider the group manifold approach to higher spin theory. The deformed local higher spin transformation is realized as the diffeomorphism transformation in the group manifold M. With the suitable rheonomy condition and the torsion constraint imposed, the unfolded equation can be obtained from the Bianchi identity, by solving which, fields in M are determined by the multiplet at one point, or equivalently, by (Wμ[α(s − 1),b(0)], H) in AdS4 ⊂ M. Although the space is extended to M to get the geometrical formulation, the dynamical degrees of freedom are still in AdS4. The 4d equations of motion for (Wμ[α(s − 1),b(0)], H) are obtained by plugging the rheonomy condition into the Bianchi identity. The proper rheonomy condition allowing for the maximum on-shell degrees of freedom is given by Vasiliev equation. We also discuss the theory with the global higher spin symmetry, which is in parallel with the WZ model in supersymmetry.


Introduction
Group manifold approach provides a natural geometrical formulation for supergravity [1][2][3][4]. The starting point is the supergroup Osp(1/4) or Osp(1/4). Supergravity field and matter field are vielbein 1-form ν Ā M and 0-form H on the group manifold M, A,M = 1, · · · , dim Osp(1/4). Local super Poincaré transformation is realized as the diffeomorphism transformation on M. The curvature R Ā MN for the 1-form can be defined, on which, the rheonomy condition is imposed [1][2][3][4]. The condition requires that R Ā MN can be algebraically expressed in terms of its purely "inner" components R A µν with µ, ν = 1, 2, 3, 4 the indices in a four-dimensional submanifold M 4 . Namely, where r Ā MN | µν B and r A CD | ab B are constant holonomic and anholonomic tensors. a, b = 1, 2, 3, 4. The rheonomy condition ensures that fields on the whole M are determined by fields on to the on-shell super Poincaré transformation of the 4d fields. The equations of motion in M 4 are obtained by plugging the rheonomy condition into the Bianchi identity. Instead of imposing the rheonomy condition, one can also construct the extended action, which is the integration of some 4-form on a 4d submanifold M 4 . Variation of the action with respect to both fields and M 4 gives the rheonomy condition as well as the 4d equations of motion.
In this paper, we will reformulate the group manifold method, adding an infinite number of auxiliary fields so that the final system is equivalent to the unfolded dynamics approach which is convenient for higher spin theory [5]. For simplicity, we will consider the minimal bosonic 4d HS algebra ho(1|2 : [3,2]) with spin s = 0, 2, · · · [6]. The corresponding group manifold is denoted as M. Fields are 1-form W ᾱ M and 0-form H on M with the curvature 2-form and the 1-form dH = H α W α .
The whole dynamics is encoded in functions (f α βγ , h α ), which should satisfy the Bianchi identity and also give the correct free theory limit. With the unfolded equation plugged in, the Bianchi identities are polynomials of (R , H c 1 ···cn ) acting as the 4d equations of motion. The procedure is simple in supergravity but is extremely complicated in higher spin theory. Instead of fixing (f α βγ , h α ) and getting the 4d equations of motion by solving the Bianchi identity, one can first identify the on-shell degrees of freedom, for example, Φσ ∼ Φ [a(s+n),b(s)] in the twisted-adjoint representation of the higher spin algebra, then find the suitable (f α βγ , h α ) so that the Bianchi identity is satisfied for the arbitrary Φσ.
{H c 1 ···cn , n = 0, 1, · · · } ∪ {R [a(s−1),b(s−1)] ab;c 1 ···cn , s = 2, 4, · · · , n = 0, 1, · · · } (1. 8) and {Φ [a(s+n),b(s)] , s = 0, 2, · · · , n = 0, 1, · · · } have the same number of indices. With the 4d equations of motion imposed on (1.8), the two may contain the same number of degrees of freedom. Written in terms of Φσ, the unfolded equation becomes It remains to find (f α βγ , Fα β ) satisfying the Bianchi identity and also giving rise to the correct free theory limit. 1 Vasiliev theory gives the elegant solution to this problem [7][8][9]. By solving the Z part of the Vasiliev equation order by order, one may finally get the required (f α βγ , Fα β ) [10]. For supersymmetry, it is also possible to study the dynamics of the 0-form matter on group manifold with the fixed background such as the WZ model. The component expansion of the 0-form on superspace gives the spin 0 and 1/2 fields in 4d. The allowed gauge transformation is the global super Poincaré transformation, which is the diffeomorphism transformation on M preserving the background. For higher spin theory, one can similarly consider the 0-form H on M with (1.10) W α 0 describes the background with the vanishing curvature. The system has the global HS symmetry. The component expansion of H on M gives the spin s = 0, 2, · · · fields R s a 1 ···as,b 1 ···bs . On the other hand, the linearized Vasiliev equation for the 0-forms on background W α 0 is dΦα = kα βγ ΦγW β 0 , (1.11) 1 As is shown in appendix C, there are (f α βγ , Fα β ) satisfying the Bianchi identity but failing to give the correct free theory limit. It is unclear whether the two requirements can uniquely fix (f α βγ , Fα β ) (up to a field redefinition) or not.

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which is also invariant under the global HS transformation. kα βγ is the constant. With Φ ≡ Φ [a(0),b(0)] = H, from dΦ = k βγ ΦγW β 0 , we have H β = k βγ Φγ. R s a 1 ···as,b 1 ···bs can then be taken as the Weyl tensor of the linearized HS theory. With the space extended from AdS 4 to M, 0-forms in the linearized Vasiliev theory get the interpretation as the derivatives of a single 0-form H on M.
The rest of the paper is organized as follows. In section 2, we construct a symmetric space M with the higher spin transformation group the isometry group. In section 3, we consider the theory with the local higher spin symmetry. The discussion and conclusion are given in section 4.
2 Symmetric space from the higher spin algebra We will consider the minimal bosonic higher spin theory in AdS 4 with the coordinate u µ , µ = 1, 2, 3, 4. The related HS algebra is ho(1|2 : [3,2]) with the basis With a i , b i = 1, 2, 3, 4, basis of ho(1|2 : [3,2]) can be rewritten as be the basis of a[E],  [3,2])]/E. In the following, we will give a construction based on the operators and the conserved charges of the quantum higher spin theory in AdS 4 . For earlier work on space with the tensor coordinates, see [11,12].
The higher spin algebra is decomposed as The metric on the coset space M = G[ho(1|2 : [3,2])]/E is defined in group theory. Alternatively, we can use the operator O(z) to get the same result. There is a one-to-one correspondence between T z (M ) = {v M ∂ M |M = 1, · · · , dim M } and K(z). For the given (2.9) {k M (z)} compose the basis for K(z), from which, one can define a special set of the The metric on T z (M ) can be induced from K(z), i.e.

Theory with the local higher spin symmetry
In section 2, the background in M is fixed to be the intrinsic geometry with dW α , H) in AdS 4 . We then discuss the relation between the unfolded equation in group manifold approach and the unfolded equation in Vasiliev theory. We will also make a comment on theory with the global higher spin symmetry.
3.1 Higher spin theory on group manifold and the rheonomy condition wheref α βγ is the deformed structure constant. The Bianchi identity is where ∂ γ = WM γ ∂M . In addition, we can add the 0-form matter field H on M, (3.6) can be rewritten as which is the deformed local higher spin transformation.
The algebra is closed with the deformed structure constantf α βγ . 5 Here, W ᾱ M is invertible, which is general enough to account for the 4d HS theory, in which, the relevant field is W α µ . Let

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allow as many on-shell degrees of freedom as possible. In this sense, (3.27) determines both (r α βγ , h γ ) and the 4d equations of motion.
To guarantee the local Lorentz invariance, in (3.14), R α (ab)β = H (ab) = 0. Since The initial value is (R α ab , R α ab;c 1 , · · · , H, H c 1 , · · · ) at one point, it is desirable to express it in terms of (W α µ , H) as well as its 4d derivatives at that point.

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The local gauge transformation of (W α µ , H) in AdS 4 is gives the local gauge transformation rule of the matter-gravity coupled system (W α µ , H) in AdS 4 . Since . This is the rheonomy in higher spin theory. As is shown in section 2, although the space is M with the infinite dimension, the physical Hilbert space is still the same as the 4d higher spin theory. Imposing the rheonomy condition is a way to project out the physical degrees of freedom.

Group manifold approach to supergravity
In this subsection, we will give a review of the group manifold approach for supergravity [1][2][3][4]. Some modification is made so that supergravity is treated in the same way as the above discussed higher spin theory.
For N = 1 supergravity in R 3,1 , the coordinate in group manifold is (x µ , x µν , θ χ ), the associated 1-form is ν A = (ω ab , e a , ψ α ), 6 and the 0-form matter field is H. We have 36) which, when written in terms of the components, are local Lorentz transformation, the 4d diffeomorphism transformation and the supersymmetry transformation respectively. With ǫ A = ξM ν Ā M , (3.36) can be rewritten as

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Until now, no dynamics is involved at all. The dynamical information is brought by imposing the suitable constraints on R A BC and H A . Here, the constraints that will be imposed are (b) Rheonomy condition and the torsion constraint: (a) is imposed so that the local Lorentz transformation is undeformed. In (b), the rheonomy condition requires that the lower index of the independent fields can only contain a so that the whole dynamics in group manifold is determined by that in a 4d submanifold; torsion constraint requires that the upper index cannot be a so that ω ab can be solved in terms of the rest fields. There are two possibilities. In (i), the final dynamical fields are (e a µ , ψ α µ , H, H α ) in M 4 , which is the situation for N = 1 supergravity coupled to the WZ matter. In (ii), the dynamical fields are (e a µ , ψ α µ , H) in M 4 like that in higher spin theory. r A BC and h A are polynomials, the coefficients of which should be selected so that some scaling relation is respected [3]. The weight of t A is denoted as w The 0-forms H A and R A BC have the weight −w(A) and w(A) − w(B) − w(C) as follows Especially, (R cd ab , R α ab , H, H a , H α ) have the weight (−2, −3/2, 0, −1, −1/2). (ii) cannot satisfy the scaling relation thus should be ruled out. For (i), with the H A odd terms dropped, the most general form of r A BC is where r * * * | * * * = r * * * | * * * (H) are functions of H since H has the weight 0. r * * * | * * * should be a Lorentz invariant to preserve the local Lorentz invariance. Although the torsion JHEP10(2015)019 constraint is also imposed, R a AB does not need to vanish, see for example [16]. However, if H α = H a = 0, R a AB = 0, so in pure supergravity case, we do have R a AB = 0. Due to the scaling relation, the rheonomy condition is greatly simplified. For supergravity in AdS 4 with the symmetry group Osp(4|1), a constant L with the weight 1 is involved. L → ∞ gives the flat space limit, so only the L −n terms with n ≥ 0 are allowed in rheonomy condition. (3.41) remains valid.
(3.41) should satisfy the Bianchi identity come out. If we use the on-shellR cb ab andR β cd satisfying (3.48) to parameterize r A BC , (3.47) will hold automatically. This is in analogy with Vasiliev theory, with R α βγ parametrized by the 0-form Φα in the twisted-adjoint representation of the higher spin algebra, the Bianchi identity is satisfied for the arbitrary Φα.
Written as the unfolded equation, compose the complete supersymmetry multiplet.
In addition to the 1-form ν A , the 0-form multiplet is introduced, forming the representation of the deformed local super Poincaré transformation. The physical interpretation of the 0-form is the curvature and the matter field plus their derivatives. This is in the same spirit as the higher spin theory. Different from the higher spin theory, rheonomy condition (3.41) only contains R cd ab , R α ab , H a , H α , so the infinite length 0-form multiplet does not enter into the 4d equations of motion. As a result, the equations of motion for (e a µ , ψ α µ , H, H α ) do not contain the higher order derivatives. One may similarly make a robust requirement R α βγ = r α βγ (R σ ab , H) and H γ = h γ (R σ ab , H) in higher spin theory. However, such (r α βγ , h γ ) may only allow the trivial solution R σ ab = H = 0 when the Bianchi identity is imposed, no matter how coefficients in (r α βγ , h γ ) are adjusted. Again, With the on-shell (e a µ , ψ α µ , H, H α ) given on M 4 , (ν A , H) on the whole group manifold can be determined up to a gauge transformation.
The dynamics is entirely encoded in function r A BC (R ad bc , R β bc , H, H α ). By setting H to 0, we obtain the pure supergravity situation. Alternatively, one can consider the dynamics of the 0-form matter on the fixed supergravity background by setting r A BC to 0. Witĥ ν A 0 describes the intrinsic geometry of the group manifold. The allowed gauge transforma- ǫ A 0 generates the global super Poincaré transformation on group manifold.

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Evolution along (ad) direction is a Lorentz transformation. One cannot assume H α is the function of (H, H c 1 , H c 1 c 2 , · · · ), since the scaling relation is not respected. Let α = (λ,λ), one can at most require Hλ = 0, which is the chiral constraint for superfield.

Imposing the torsion constraint in higher spin theory
Back to higher spin theory, a further reduction of (3.33) can be made by imposing the following torsion constraint Namely, in (3.14), σ is restricted to [a(s − 1), b(s − 1)] with s = 2, 4, · · · . In (3.29), the number of equations is equal to the number of degrees of freedom of (R α ab,c 1 ···cn , H c 1 ···cn ) but the number of unknowns is equal to the degrees of freedom of (R In fact, at least in free theory limit, imposing the torsion constraint R which is the local gauge transformation rule of (W [a(s−1),b(0)] µ , H) in AdS 4 . In free theory limit, it is that will finally appear in equations of motion and the gauge transformation. One may expect in interacting case, the final dynamics is also expressed in terms of some h µ 1 ···µs , which can be a more complicated combination of W a 1 ···a s−1 ,0···0 µ . The frame-like formulation reduces to the metric-like formulation.

Relation with the unfolded equation in Vasiliev theory
With Φα representing the on-shell degrees of freedom of (R

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It remains to find the suitable (f α βγ , Fα γ ) with the Bianchi identity (3.71) satisfied for the arbitrary Φα. Under the field redefinition Φα → ϕα = ϕα(Φσ), Fα β → ∂ϕα ∂Φσ Fσ β . In Vasiliev theory, Φα ∼ Φ [a(s+n),b(s)] is in the twisted-adjoint representation of the higher spin algebra.   , H c 1 ···cn ) that has the physical meaning. We are free to make a change of the variables ϕα = ϕα(Φσ) to use ϕα to parameterize (R The nonlinear higher spin theory should also have the proper free theory limit that is equivalent to Fronsdal theory [18,19]. In free theory limit, the equations of motion in (3.60)-(3.68) become where t < s − 1. (3.85) is also called the "central on-mass-shell theorem" [20,21]. In Vasiliev theory, R α βγ satisfies (3.85) at the first order of the Φα expansion. Since the adjoint representation and the twisted-adjoint representation are related by a Klein transformation which is invertible, we may try to use Φ α to parameterizef α βγ as is in (3.74). If we further make a restriction that (3.75) can be written as

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with H(W, Φ) and F (W, Φ) polynomials of W = W α t α and Φ = Φ α t α , the solution for (3.76) can be easy fixed, which is given in appendix C. Although the Bianchi identity is satisfied for the arbitrary Φ α , (3.85) does not hold at the first order of the Φ α expansion, so the theory does not have the correct free theory limit. Satisfying the Bianchi identity for the on-shell (R , H c 1 ···cn ) and giving rise to the correct free theory limit are two requirements for (f α βγ , h γ ). It is unclear whether the requirements can uniquely fix (f α βγ , h γ ) or not. Starting from the the rheonomy condition (3.14) in section 3.1, one may get (3.30) with no torsion constraint imposed on W α µ . The torsion constraint is just (3.85), or concretely, and also guarantee the correct free theory limit. In this case, having the correct free theory limit and satisfying the torsion constraint are the same thing. If there is such (3.14), for which the Bianchi identity on (R α ab;c 1 ···cn , H c 1 ···cn ) reduces to the 4d equations of motion, then by setting R Finally, we will have a heuristic discussion on the group manifold approach to conformal HS theory. 3d conformal HS algebra and AdS 4 HS algebra are the same, so the corresponding group manifold is also M. The equations are still The submanifold of interest is not AdS 4 but ∂AdS 4 ⊂ ∂M. The solution of the unfolded equation in M is determined by the value of the 0-form multiplet at one point. In previous discussion, this point is selected at the bulk of AdS, but now, it should live at ∂AdS. The generated solution will remain at the near boundary region, since an infinite evolution is needed to move from the boundary to the bulk. The rheonomy conditionf α βγ =f α βγ (Φσ) and Fα γ = Fα γ (Φσ) in Vasiliev theory may undergo a reduction at the boundary with the role of Φσ played by a smaller set of 0-forms so that the solution at the near boundary region is determined by the dynamical fields in 3d.
There is a conjecture that the conformal HS theory at ∂AdS d+1 is related to the HS theory in AdS d+1 with the action of the conformal HS fields for even d equals to the logarithmically divergent term of the action of HS fields in AdS d+1 [23,24]. In [15], the unfolded equation for a 3d conformal HS theory coming from the boundary limit of the AdS 4 Vasiliev theory was considered. It was shown that at ∂AdS 4 , R α ab = 0 only when t α = K i 1 ···i s−1 . The condition could make W α m expressed in terms of the dynamical field W i 1 ···i s−1 m without imposing constraints on the latter. Correspondingly, in (3.56), the JHEP10(2015)019 independent 0-forms are (R α ij , R α ij;k 1 , · · · , H, H k 1 , · · · ) for t α = K i 1 ···i s−1 , s = 2, 4, · · · . This is consistent with the fact that in odd dimensions, the conformal HS theory is trivial with no equations of motion imposed on dynamical fields [22,24,25].
On the other hand, in even dimensions, dynamical fields should satisfy Fradkin-Tseytlin equation [26]. The unfolded system of Fradkin-Tseytlin equation was formulated in [27,28], where the 0-form multiplet is Weyl module generated by Weyl tensor, which, according to the terminology of [22], is the ground field strength. Equivalently, the 0-forms in (3.56) should now be taken as (R , · · · , H, H k 1 , · · · ). In free theory limit, R α ij = 0 if ∆ α < 0, R α ij with ∆ α > 0 can all be expressed in terms of the derivatives of the Weyl tensor R . This is somewhat different from [15] for 3d, where R [i(s−1),j(s−1)] ij = 0. It is interesting to consider the 4d conformal HS system arising from the boundary reduction of the Vasiliev equation in AdS 5 in analogy with [15]. In free theory limit, the obtained equation is expected to give the unfolded system of Fradkin-Tseytlin equation [27,28]. The boundary value of the AdS d+1 HS fields was considered in [25,29] in the ambient approach, where it was shown that for even d there is an obstruction for the bulk extension unless the conformal HS fields at ∂AdS d+1 satisfy the Fradkin-Tseytlin equation. In this case, the near boundary expansion of the on-shell AdS field (see, for examaple, [30]) does not have the logarithm term, which is required in the unfolded formalism, which in the minimal version does not allow for logarithmic terms to cancel the obstruction.
The unfolded equation for the 4d HS theory is invariant under the local Lorentz transformation SO(3, 1), i.e. R α (a,b)γ = 0. In [15], it is possible to impose the suitable boundary condition so that R α (0,4)i = 0. If the conclusion can be extended to R α (0,4)γ = 0, then the dilatation is unformed. Moreover, the original HS theory already have the undeformed SO(3, 1) local Lorentz transformation, so by a naive counting, it seems that the inhomogeneous Weyl group IW generated by {D, K i , L i,j } can be undeformed at the boundary. In this case, the evolution along the t 0,4 direction is a conformal (gauge) transformation and the dynamics is reduced from 4d to 3d. It remains to see whether there are consistent nonlinear unfolded equation for the conformal HS theory meeting this requirement. At least, the 3d local Lorentz transformation is undeformed.

The extended action principle for higher spin theory
In group manifold approach to supergravity, instead of imposing the rheonomy condition directly, one may construct the extended action whose variation gives both the rheonomy condition and the 4d equations of motion [3].
For example, in N = 1 supergravity, the extended action is of the form where M 4 is a 4d submanifold of the superspace M , 8 and L (4) is a local Lorentz invariant 4-form in M constructed from ν A via the exterior differentiation and the exterior product.

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Variation of S with respect to both ν A and M 4 gives A is a 3-form that should vanish all over M . K  where L (4) is a 4-form invariant under the local Lorentz transformation.
We need to find the configuration W α on M with K σ = 0 is on-shell gauge invariant. Off-shell higher spin invariance has the further requirement dL (4) = 0 [3]. Although the on-shell gauge invariance is automatically guaranteed, for the generic L (4) , K In the simplest situation, if with κ constants, then (3.97) (3.97) imposes a set of linear relations among R α βγ , which, when plugged into the Bianchi identity, may only allow the trivial solution R α βγ = 0. The more general form of L (4) is including an infinite number of derivatives. K σ[αβγ] should automatically vanish for the arbitrary Φσ if it is the action from which, the Vasiliev equation comes out. However, it is too complicated to fix the exact form of (3.98).

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3.6 Dynamics of 0-form matter on group manifold with the fixed background (3.60)-(3.68) describes the coupling of the spin 0 matter H and the spin 2, 4, · · · gravity field W α . Under the local gauge transformation, which is the deformed higher spin transformation as well as the diffeomorphism transformation on M, spin 0, 2, 4, · · · fields mix with each other. To describe the dynamics of the 0-form matter on M with the fixed background, the matter-gravity coupling must be turned off. One may let r α βγ = 0, then W α 0 gives the intrinsic geometry of the group manifold M discussed in section 2. The equations of motion reduce to (3.99) The allowed gauge transformation parameter ǫ α 0 should satisfy generating the global higher spin transformation on M.
The next step is to impose the suitable rheonomy condition and derive the unfolded equation so that the solution on M is determined by the (on-shell) fields in lower dimensions. In the following, we will consider two kinds of the rheonomy conditions which will make the final dynamics reduce to 4d and 3d respectively. The former gives a system equivalent to the linearized Vasiliev theory expanded on the background W α 0 , which also has an abelian local gauge symmetry invisible if we only focus on the equation for curvature. The latter comes from the 3d free massless scalar field theory at ∂AdS 4 . Since the 3d scalar forms the representation of the HS symmetry, it is possible to extend the scalar from 3d to (the boundary region of) M with the global HS transformation realized as the isometry transformation.

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which is the linearized version of the Vasiliev equation (3.70) expanded on background W α 0 withW α the fluctuation on it. kα βγ = ρα α ρ γ γ f α βγ is a constant. kα βγ Φγ is the lowest order term of Fα β (Φσ) = kα βγ (Φ Φσ )Φγ in (3.69). R α 1βγ is the first order term of the polynomial f α βγ (Φσ) in (3.69), i.e.f α which is indeed the case due to the vanishing of the the first order part of the left hand side of (3.71).
There is also a residue local HS transformation which is invisible if we only focus on the equation for the 0-form. Intuitively, it seems that the global HS transformation forW α and R α 1ρσ should be δW α = f α βγ ǫ β 0W γ and δR α 1ρσ = f α βγ ǫ β 0 R γ 1ρσ , which, however, is not consistent with the transformation law of Φα. The global HS transformation is a diffeomorphism transformation other than a gauge transformation.
In the interacting theory, ∂ β Φα =kα βγ (Φσ)Φγ, Φγ, where W a 0 µ and W (3.126) R α 1aβ (Φσ) = r α aβ |σΦσ with r α βγ |σ the constant. Φσ can be expressed in terms of the 4d derivatives of Φ [a(s),b(s)] , which, in turn, is determined byW α µ and thusW where Φγ can be written in terms of the 4d derivatives of Φ [a(s),b(s)] via the relation D µ Φα = kα aγ ΦγW a 0 µ . It is well-known that the linearized Vasiliev theory is global HS invariant. By extending the space from AdS 4 to M, the linearized Weyl module of the free higher spin theory can be compactly interpreted as ∂ α H, the outer derivatives of a single scalar field H on M.

The 3d global HS invariant system
The above 4d global HS invariant theory also has a local gauge symmetry. The genuine global HS invariant system without the local gauge symmetry is the 3d massless free scalar field theory living at ∂AdS 4 . In 3d free CFT, let φ be the operator for the dimension 1/2 massless scalar and consider φ(X) = g(X)φ(0 ′ )g(X) −1 , ∀ g(X) ∈ G[ho(1|2 : [3,2])]. In contrast to O(0) in the bulk, φ(0 ′ ) is at the origin of ∂AdS 4 , so for the finite X, φ(X) is still at the near boundary region of M with X the coordinate. Scalar field at the near boundary region of M also forms the representation of G[ho(1|2 : Generically, in 3d free CFT of the scalar φ, we have the relation where i k = 1, 2, 3, ρ is the constant, because ho(1|2 : [3,2]) can be realized as the quotient of the enveloping algebra of so(3, 2) [13] (for HS algebra of any classical Lie algebras and in particular, sp 2N , so N and sl N , see [14]). As a result, the relation

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is valid everywhere at the boundary region of M for the constant ρ. 3d equations of motion for φ are also implicitly imposed by (3.130). The derivatives of φ in outer space can be expressed in terms of the derivatives of φ in inner space (∂AdS 4 ). This is not possible in (3.102), because the scalar field in AdS 4 cannot form the representation of the HS symmetry. One must introduce the higher spin fields, which, in (3.102), is reflected by ∂ 0c 1 ···cr,c r+1 ···c 2r+1 . Return to (3.99)-(3.101) and restrict to the near boundary region with H replaced by φ. From the rheonomy condition one may get the unfolded equation where φ i 1 ···i k α is the linear combination of {φ, φ i 1 , φ i 1 i 2 , · · · } with the constant coefficients. The Bianchi identity is satisfied. From the on-shell {φ, φ i 1 , φ i 1 i 2 , · · · } at one point, or equivalent, the on-shell φ in ∂AdS 4 , φ in the near boundary region of M can be determined. (3.132) is invariant under the global HS transformation In conclusion, to construct a theory with the global HS symmetry, we may try to find an unfolded equation like (3.108) and (3.132) for a 0-form multiplet on M with the background geometry W α 0 . The equation should be integrable with the only dependence on M comes from the 0-form and the 1-form W α 0 . Therefore, it is of course diffeomorphism invariant. The global higher spin transformation is a special diffeomorphism transformation preserving W α 0 .

Discussion
In supergravity, the rheonomy condition is simply R A BC = r A BC (R cd ab , R α ab , H, H α ). Nevertheless, the most generic rheonomy condition in group manifold approach takes the form of (1.4) and (3.38) with all orders of derivatives included. If we make a similar trunca- , H) in higher spin theory, then with r α βγ plugged into the Bianchi identity, we will get the 4d equations of motion, which, when expressed in terms of (W } at that point. Merely based on group manifold approach without the knowledge of Vasiliev theory, we will finally arrive at (3.60)-(3.68) and then face the problem of finding the proper rheonomy condition that could solve the Bianchi identity, allow for the maximum on-shell degrees of freedom and have the correct free theory limit. It is Vasiliev theory that gives the solution meeting all these requirements. A question is whether there are other solutions or not. In appendix C, we give a rheonomy condition (for the bosonic higher spin theory) satisfying the Bianchi identity with the on-shell degrees of freedom {Φ α }. However, the correct free theory limit is not recovered and the local Lorentz transformation is deformed.
In superspace with the fixed background geometry, the local super Poincaré symmetry reduces to the global super Poincaré symmetry. With the chiral constraint imposed, the component expansion of the scalar superfield in superspace gives the spin 0 and 1/2 fields (H, H α ) in M 4 . For higher spin theory, one can fix the background geometry of M and then study the scalar field H in M with the global higher spin symmetry. The component expansion of H gives the spin 0, 2, 4, · · · fields (H, , · · · ) in AdS 4 , which, however, are not the gauge fields but the linearized Weyl tensors of the free HS theory, since the massless gauge fields are not the Lorentz tensor. Restricted to the near boundary region of M, it is also possible to impose the rheonomy constraint so that the component expansion of H only gives the spin 0 field H in ∂AdS 4 . This is because although the 4d spin 0, 2, 4, · · · fields all together form the representation of the HS symmetry, the 3d spin 0 field alone forms the HS representation.

B CFT realization of the spin s linearized Riemann tensor operator in AdS 4
In radial quantization of the 3d O(N ) vector model, for each s = 0, 2, · · · , there is an unique primary operator O i 1 ···is (0 ′ ) with spin s.
{O(0 ′ ), O i 1 i 2 (0 ′ ), · · · } is the higher spin multiplet. The action of the generic Q α on the spin 0 primary operator O(0 ′ ) = a + a + can be decomposed as Let us construct the SO(3, 1) tensor operator with spin s in the sense of (B.1) in AdS bulk. Such operator does not represent the spin s gauge field which is not a tensor, but rather the field strength of it. The spin 0 operator O(0) is already given by (A.4). For operators with the higher spin, consider
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