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 $\textbf{M}$. 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 $\textbf{M}$ are determined by the multiplet at one point, or equivalently, by $(W^{[a(s-1),b(0)]}_{\mu},H)$ in $AdS_{4}\subset \textbf{M}$. Although the space is extended to $\textbf{M}$ to get the geometrical formulation, the dynamical degrees of freedom are still in $AdS_{4}$. The $4d$ equations of motion for $(W^{[a(s-1),b(0)]}_{\mu},H)$ 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]. The starting point is the supergroup. Supergravity field and matter field are vielbein 1-form and 0-form on the group manifold M. Local super Poincare transformation is realized as the diffeomorphism transformation on M. Curvature for the 1-form can be defined, on which, the rheonomy condition is imposed. The condition ensures that fields on the whole M are determined by fields on a four-dimensional (4d) submanifold M 4 . So the final dynamics is still in M 4 , on which the diffeomorphism transformation of M reduces to the on-shell super Poincare 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 4form 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 [3]. For higher spin theory in group manifold M of the algebra ho(1|2 : [3,2]), the fields are 1-form W ᾱ M and 0-form H with the corresponding curvature 2-form and the 1-form dH = H α W α , (1.2) whereM = 1, 2, · · · , dim ho(1|2 : [3,2]), α ∼ [a(s − 1), b(t)] is in the adjoint representation of ho(1|2 : [3,2]), f α βγ is the structure constant of ho(1|2 : [3,2]). The deformed higher spin transformation is the diffeomorphism transformation on M. The rheonomy condition iŝ , · · · , H, ∂ ν 1 H, · · · ), (1.5) where ∂ µ is the derivative on AdS 4 . So equivalently, with (W , · · · , H, H c 1 , · · · ) forms the complete representation on-shell. 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. The Bianchi identity, with the unfolded equation plugged in, 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),b(s+n)] 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. The off-shell {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. 6) and {Φ [a(s),b(s+n)] , s = 0, 2, · · · , n = 0, 1, · · · } have the same number of indices. The two may contain the same number of degrees of freedom after the 4d equations of motion are imposed on (1.6). Written in terms of Φσ, the unfolded equation becomes It is still difficult to find (f α βγ , Fα β ) satisfying the Bianchi identity and also giving rise to the correct free theory limit 3 . The Vasiliev theory gives the elegant solution [4,5,6]. By solving the Z part of the Vasiliev equation order by order, one will finally get the consistent (f α βγ , Fα β ) [7]. In group manifold approach, ho(1|2 : [3,2]) has a subalgebra so (3,2), so higher spin theory is finally defined in a 4d submanifold with the topology of AdS 4 . For supergravity, we may start from the uncontracted Osp(1/4) group with SO(3, 2) a subgroup, or the contracted Osp(1/4) with Poincare group a subgroup. The corresponding supergravity theories are defined in AdS 4 and the Minkowski space M 4 respectively. Unfortunately, there is no contracted version of ho(1|2 : [3,2]). As a result, the flat space higher spin theory cannot be constructed here. This is consistent with the no-go theorem [8,9] as well.
With H set to 0, we get the dynamics for gauge fields with s = 2, 4, · · · like the pure supergravity system. 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 allowed gauge transformation is the global super Poincare 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 The corresponding rheonomy condition, which is in parallel with the chiral constraint in WZ model, is 1 ···dt,e 1 ···e t+1 ] = m [0a 1 ···as,b 1 ···b s+1 ],[0d 1 ···dt,e 1 ···e t+1 ] , (1.9) where h α and m [0a 1 ···as,b 1 ···b s+1 ],[0d 1 ···dt,e 1 ···e t+1 ] are functions of (1.10) is in one-to-one correspondence with the spin s = 0, 2, 4, · · · particles carrying the arbitrary on-shell momentum in AdS 4 and forms the complete higher spin multiplet. Again, in AdS 4 , (1.10) and the 4d fields are equivalent. We will give an explicit form for (1.9). 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 discuss the theories with the global higher spin symmetry. In Section 4, we consider the theory with the local higher spin symmetry. The discussion and conclusion are given in Section 5.
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 corresponding algebra is ho(1|2 : [3,2] With a i , b i = 1, 2, 3, 4, bases of ho(1|2 : [3,2]) can be rewritten as be the bases of a[E], be the bases of K, ho(1|2 : [3,2] With the group given, it is a standard procedure in mathematics to construct the group manifold M for G[ho(1|2 : [3,2])] and the symmetric space M for G[ho(1|2 : [3,2])]/E. In the following, we will give a construction based on the operators and the conserved charges in the quantum higher spin theory in AdS 4 . For earlier work on space with the tensor coordinates, see [10,11].
In higher spin theory, there are conserved charges is the operator for the spin s field in AdS 4 . We will focus on O 0 (u) ≡ O(u) which is the operator for the spin 0 field. Let 0 be a point in the bulk of AdS 4 , for example, , then the orbit generated by SO (3,2) gives the operators in AdS 4 .  [3,2])]/E. It remains to determine the subalgebra a[E]. Although the direct quantization of the higher spin theory in AdS 4 is still not available, its CFT dual is quite simple. In appendix A, the CFT realization of O(0), or more accurately, O + (0), is given. It is shown that the charge Q 0···0a 1 ···a 2k−1 ,b 1 ···b s−1 corresponding to (2.3) commutes with O(0). So a[E] here is indeed the same as (2.3).
The metric on the coset space M = G[ho(1|2 : [3,2])]/E is given in group theory. Alternatively, we can use the operator O(z) to give 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 ∂ M , {k M (z)} compose the bases for K(z), from which, one can define a special set of the coordi- The metric on T z (M) can be induced from K(z), i.e. (2.13) (2.14) Let k α (Z) = WM α (Z)kM (Z) and suppose ∂N kM We will assume k α (Z) = g(Z)k α g −1 (Z), which is always possible for the suitably selected (2.18) there will be (2.21) Just as the chiral constraint relates ∂θ with ∂ µ , here, ∂ 0···0a 1 ···as,b 1 ···b s+k is determined by |0 are all in the 1-particle Hilbert space of the higher spin theory 4 , for which compose the complete bases. In where α(a 1 · · · a s , b 1 · · · b s+k ) are constants to be determined.

Theory with the global higher spin symmetry
We have constructed the space M and the operator O(z) on it. ∀ g ∈ G[ho(1|2 : [3,2])], gO(z)g −1 = O(z g ) induces the global higher spin transformation z → z g , which is the isometric transformation on M. This is exactly in parallel with supersymmetry. Consider a supersymmetric theory with the supersymmetry generators P µ , Q andQ.
gives the operators in superspace M with the coordinate (x, θ,θ). Under the supersymmetry transformation Superalgebra is realized as the killing vector fields in superspace, for which, the supersymmetry transformation is the isometric transformation. Similarly, for higher spin theory, ∀ t σ ∈ ho(1|2 : [3,2]), there is a corresponding killing field Λ σ in M. {Λ σ } compose a Lie algebra which is isomorphic to ho(1|2 : [3,2]). Under the G[ho(1|2 : [3,2])] transformation, where One can also put the scalar field H(z) on M, which is the higher spin counterpart of the superfield.
Under the global higher spin transformation, we have (3.11) where∂ A H(z) = H A (z). Generically, forms the linear representation of the higher spin algebra. However, such representation is highly reducible. As is shown in the previous section, not all of D · · · DO are linear independent. The linear independent basis can be selected as on H(z), which is invariant under the global higher spin transformation. The imposing of the constraint (3.13) is necessary. Although the space is extended from AdS 4 to M, the dynamics is still in 4d. The operators O(z) on M are all in the 1-particle Hilbert space of the 4d higher spin theory, so they are highly linear dependent. Rheonomy constraint simply picks out the independent degrees of freedom.
The coordinate z on M can be decomposed as z M = (u µ , y i ) with u µ and y i the coordinates on the AdS 4 fiber and the base space. In analogy with supersymmetry, from H(u, y), one can write the most generic action that is invariant under the global higher spin transformation where dudy g(u, y) = du ′ dy ′ g(u ′ , y ′ ) because (u, y) → (u ′ , y ′ ) is the isometric transformation of M. H(u, y) → H(u ′ , y ′ ) = H ′ (u, y) is the rigid higher spin transformation for the scalar field H(u, y). H(u, y) can be expanded in terms of H [0a 1 ···as,b 1 ···b s+1 ] (u, 0), from which, the higher spin transformation law H [0a 1 ···as,b 1 ···b s+1 ] (u, 0) → H ′ [0a 1 ···as,b 1 ···b s+1 ] (u, 0) can be induced. Integration over y gives the Lagrangian in AdS 4 , which is invariant under . However, unlike supersymmetry, it is not quite convenient to do this, since the y integration will give the infinity.

Theory with the local higher spin symmetry
In previous discussion, the background on M is fixed to be the intrinsic geometry with The allowed diffeomorphism transformation is the global higher spin transformation preserving W α 0 . To have the local higher spin symmetry, the 1-form W α on M should be dynamical. We will study the dynamics of the 1-form W α and the 0-form H on M. With the suitable rheonomy condition and the torsion constraint imposed, , H) on AdS 4 . We then discuss the relation between the unfolded equation in group manifold approach and the unfolded equation in Vasiliev theory.

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 as follows The group manifold M is necessarily involved in the definition of R α βγ and H α . (4.3) and (4.5) are defined in M as well.
The definition (4.2) and (4.4) is invariant under the diffeomorphism transformation generated by ξM , (4.6) can be rewritten as which is the deformed local higher spin transformation.
The algebra is closed with the deformed structure constantf α βγ . If for some Λ, (4.10) The evolution along the (ab) direction is a local Lorentz transformation, so the group manifold M effectively reduces to the coset space M = G[ho(1|2 : [3,2])]/SO (3,1). Recall that in Section 2, we have discussed the coset space M = G[ho(1|2 : [3,2])]/E. For M to reduce to M, there must be R α Qγ = 0 so that the local gauge transformation generated by ǫ Q is undeformed. However, at least in Vasiliev theory, R α (ab)γ = 0 is valid but R α Qγ = 0 does not necessarily hold.
To guarantee the local Lorentz invariance, in (4.13), R α (ab)β = H (ab) = 0. Since where ∂ (ab) R α ρσ , ∂ (ab) H α , ∂ (ab) R β de;c 1 ···cn and ∂ (ab) H c 1 ···cn are all standard local Lorentz transformations, the coefficients in r α ρσ and h α must be Lorentz invariants. In fact, (4.27) are also included in (4.25), so the Lorentz invariance of r α ρσ and h α is also the requirement of the Bianchi identity if R α (ab)β = H (ab) = 0. We only considered the equation (4.23) on group manifold M, since in that space, the diffeomorphism transformation and the local gauge transformation are in one-to-one correspondence. As the universal property of the unfolded equation [13], 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.
r and h are functions of (R α ab;c 1 ···cn , H c 1 ···cn ). In (4.28), the unknowns are (R α ab;c 1 ···cn , H c 1 ···cn ), while the number of equations is the same as the number of the degrees of freedom of (R α µν;λ 1 ···λn , H λ 1 ···λn ), where µ, ν = 1, 2, 3, 4. (4.11) and (4.12) also impose constraints on the off-shell (R β ab;c 1 ···cn , H c 1 ···cn ) to make it have the same number of degrees of freedom as (R β µν;λ 1 ···λn , H λ 1 ···λn ), so in principle, from (4.28), (R α ab;c 1 ···cn , H c 1 ···cn ) can be solved in terms The local gauge transformation of (W α µ , H) in AdS 4 is With (4.29) plugged in (4.30), gives the local gauge transformation rule of the matter-gravity coupled system (W α µ , H) in . 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]. 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 the group manifold is (x µ , x µν , θ χ ), the associated 1-form is ν A = (ω ab , e a , ψ α ) 5 , and the 0-form matter field is H. We have 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 , (4.35) can be rewritten as 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 (a) Factorization condition R A (ab)C = 0 = H (ab) ; (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 [1]. The weight of t A is denoted as w(A), w(a) = 1, w(ab) = 0, 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 constraint is also imposed, R a AB does not need to vanish, see for example [14]. 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. (4.39) is then unmodified.
(4.39) should satisfy the Bianchi identity In pure supergravity situation with H = 0, r A BC becomes R bc aα = r bc come out. If we use the on-shellR cb ab andR β cd satisfying (4.45) to parameterize r A BC , (4.44) will hold automatically. This is in analogy with the 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.
In addition to the 1-form ν A , the 0-form multiplet is introduced, forming the representation of the deformed local super Poincare 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, the rheonomy condition (4.39) only contains R cd ab , R α ab , H a , H α , so the infinite dimensional 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. Withf A BC = f A BC , (4.33) and (4.34) reduce to ν A 0 describes the intrinsic geometry of the group manifold. The allowed gauge transformation parameter ǫ A 0 generates the global super Poincare transformation on group manifold.
Evolution along (ad) direction is the 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 the higher spin theory, a further reduction of (4.32) can be made by imposing the following torsion constraint , · · · , H, H c 1 , · · · ). (4.53) Namely, in (4.13), σ is restricted to [a(s − 1), b(s − 1)] with s = 2, 4, · · · . In (4.28), 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 , · · · , H, ∂ ν 1 H, · · · ), (4.55) which is the local gauge transformation rule of (W , · · · , H, H c 1 , · · · ), (4.60) The input is (f α βγ , h γ ), from which, all the rest equations are determined. The left hand side of the 4d equations of motion (4.64)-(4.65) are polynomials of (R , H c 1 ···cn ). In supergravity situation, the procedure is quite simple as is demonstrated in Section 4.2. In higher spin theory, the more direct way is to first determine the on-shell degrees of freedom Φα. Then with the off-shell (R

Relation with the unfolded equation in Vasiliev theory
With Φα representing the on-shell degrees of freedom of (R   and {Φ [a(s),b(s+n)] , s = 0, 2, · · · , n = 0, 1, · · · }. The two have the same number of indices, but the former is the off-shell field while the latter is the on-shell field. With the 4d equations of motion imposed on (4.69), the two may contain the same number of degrees of freedom. Fields in the twisted-adjoint representation and the adjoint representation are related via the action of the Klein operator where ∂ c Φα =kα cγ (Φσ)Φγ is used.  [a,0] and taking the L → ∞ limit, the so(3, 2) algebra reduces to the Poincare algebra. However, for higher spin algebra, the similar flat space limit is not well-defined. For It is only when f α 3 α 1 α 2 = 0 for w(α 3 ) > w(α 1 ) + w(α 2 ) the algebra has the well-defined flat space limit. This is the case for so(3, 2) subalgebra but does not hold in general in ho(1|2 : [3,2]). Unlike Osp(4|1) which has a contracted version Osp(4|1), ho(1|2 : [3,2]) does not have the well-defined contraction with so(3, 2) reducing to the Poincare algebra. As a result, in group manifold approach, supergravity can be defined in AdS 4 as well as the 4d Minkowski space, but the higher spin theory can only be constructed in AdS 4 . The theory has the unique vacuum with R α βγ = 0. The corresponding geometry is discussed in Section 2. M has a fiber bundle structure with the AdS 4 fiber attached at each point of the base space. and H c 1 ···cn have the weight w(α) − w(β) − w(γ), −w(γ), −n − 2 and −n respectively. In Section 4.2, the scaling relation makes the rheonomy condition for supergravity in R 3,1 simplify a lot. For supergravity in AdS 4 , the weight 1 parameter L is involved. To have the well-defined flat space limit, only the L −n terms are allowed, so the rheonomy condition remains the same. For higher spin theory in AdS 4 , if we require the well-defined flat space limit so that only the L −n terms are included in (4.53), then with the Bianchi identity imposed, (4.53) may only have the trivial solution R [a(s−1),b(s−1)] ab = H = 0 no matter how its coefficients are adjusted. To allow for the nontrivial on-shell degrees of freedom, L n terms must be added which will then give an ill-defined flat space limit. Since both L −n and L n terms are allowed, the scaling relation is not so powerful to restrict the form of (4.53).
Finally, higher spin theory should have the proper free theory limit that is equivalent to Fronsdal theory [16,17]. In free theory limit, the equations of motion in (4.57)-(4.65) become where D is the standard covariant derivatives for W cd and especially, For the theory to have the correct free theory limit, there will be so that (4.78) becomes where t < s − 1. (4.83) is also called the "central on-mass-shell theorem" [18,19]. In Vasiliev theory, R α βγ satisfies (4.83) 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 (4.71). If we further make a restriction that (4.72) can be written as with H(W, Φ) and F (W, Φ) polynomials of W = W α t α and Φ = Φ α t α , the solution for (4.73) can be easy fixed, which is given in appendix C. Although the Bianchi identity is satisfied for the arbitrary Φ α , (4.83) 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 (4.13) in Section 4.1, one may get (4.29) with no torsion constraint imposed on W α µ . The torsion constraint is just (4.83), or concretely, (W σ µ , ∂ ν 1 W σ µ , · · · , H, ∂ ν 1 H, · · · ) = 0, for t = s − 1, (4.86) which will make W α µ reduce to W [a(s−1),b(0)] µ 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 (4.13), 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 [a(s−1),b(t)] ab,c 1 ···cn to 0 for t = s − 1, we will get (4.53) satisfying the Bianchi identity for the on-shell (R [a(s−1),b(s−1)] ab;c 1 ···cn , H c 1 ···cn ) and having the right free theory limit. (4.83) holds exactly in this situation.

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 [1].
For example, in N = 1 supergravity, the extended action is of the form where M 4 is a 4d submanifold of the superspace M, 6 and L (4) is a local Lorentz invariant 4-form in M constructed from ν A via the exterior differentiation and the exterior product. 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 ρ α M N = R α M N . For higher spin theory, if the extended action exists, it will take the form 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 [1]. Although the on-shell gauge invariance is automatically guaranteed, for the generic L (4) , K (3) σ = 0 only has the trivial solution R α βγ = 0, so the question is whether there is L (4) for which, the related K (3) σ = 0 has the nontrivial solution or not. In supergravity, having the nontrivial solution also puts the severe constraint on S.
In the simplest situation, if with κ the constant, then 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 come out. However, it is too complicated to fix the exact form of (4.95).

4.6
The dynamics of the 0-form matter on group manifold with the fixed background (4.57)-(4.65) describes the coupling of the spin 0 matter H and the spin 2, 4, · · · gravity fields 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. The pure gravity system can be obtained by setting H to 0. On the other hand, to describe the dynamics of the 0-form matter on M with the fixed background, the matter-gravity coupling must be turned off and so, r α βγ should be modified. One may assume 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 are simplified to The allowed gauge transformation parameter ǫ α 0 should satisfy generating the global higher spin transformation on M According to the previous decomposition α = (A, Q), we may further let ∂ Q H = H Q = 0 and then The evolution along the Q direction is a gauge transformation. The rest Bianchi identity is (4.100) We will assume The reason is that H on AdS 4 cannot form the nontrivial representation of the higher spin algebra. Similarly, in supersymmetry, H α cannot be expressed in terms of (H, H c 1 , · · · ) not only due to the scaling relation, but also because H on M 4 cannot form the nontrivial supersymmetry representation.
Although the discussion in higher spin theory is a direct extension of the supersymmetry theory, it looks strange there are global higher spin invariant theories for 0-form fields with spin 0, 2, 4, · · · on the fixed AdS background. A natural expectation is that the theory may be a gauge-fixed version of the traditional higher spin theory with fields h µ 1 ···µs for s = 0, 2, 4, · · · . The gauge-fixing makes the local higher spin symmetry reduce to the global one. In fact, H [0a 1 ···as,b 1 ···b s+1 ] is related with [Q [0a 1 ···as,b 1 ···b s+1 ] , O] which are indeed the operators in the gauge-fixed higher spin theory in AdS 4 [12]. For supersymmetry, N = 1 supergravity multiplet with spin (3/2, 2) can couple with WZ matter with spin (0, 1/2). It is unclear whether the higher spin version of the supergravity-WZ matter coupled system makes sense or not. The theory describing the coupling between the spin (2, 4, · · · ) gauge fields and the spin (0, 2, 4, · · · ) matter fields may contain two copies of fields with the same spin.
Finally, we need to determine the exact form of h A and m [0a 1 ···as,b 1 ···b s+1 ],[0d 1 ···dt,e 1 ···e t+1 ] satisfying ∂ A h B = ∂ B h A . This is done in Section 2, where (2.21) and (2.24) are obtained. The Bianchi identity is satisfied since O(Z) is a scalar field on M.
The complete h α is exhausted by h Q = 0 and (4.106). It is also reasonable to add h 0···0a 1 ···as,b 1 ···b s+t with s even, t even. h andh together form the twisted-adjoint representation of the higher spin algebra. One may assume (4.108) The relation (4.106) is obtained from the operator O(Z) on M. We may get the similar relation from Vasiliev theory. Withf α βγ = f α βγ , the unfolded equation in Vasiliev theory reduces to In particular, · · · · · · (4.113)  In the interacting theory, ∂ β Φα =kα βγ (Φσ)Φγ, Φγ, in a complicated way.

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 (1.3) and (4.37) with all orders of derivatives included. If we make a similar truncation R α βγ = r α βγ (R [a(s−1),b(s−1)] ab , 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 [a(s−1),b(0)] µ , H), do not contain the derivatives higher than two. However, it is quite likely that such equations may only have the trivial solution R [a(s−1),b(s−1)] ab = H = 0 no matter how the functions r α βγ are adjusted. To allow for the nontrivial on-shell degrees of freedom, higher derivatives must be included so that R α βγ at one point is effectively determined by (W , H) will also contain an infinite number of the higher derivative terms which makes the theory nonlocal.
To write the unfolded equations (4.46) and (4.61)-(4.63), the infinite 0-form multiplets are necessarily involved in both supergravity and higher spin theory, since the solution on the whole M, including M 4 /AdS 4 , is characterized by the on-shell 0-form multiplet at one-point. For higher spin theory, the on-shell (R Merely based on the group manifold approach without the knowledge of the Vasiliev theory, we will finally arrive at (4.57)-(4.65) 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 the Vasiliev theory that gives the solution meeting all these requirements. A question is whether there are other kinds of the on-shell degrees of freedom. One may consider Φ α in the adjoint representation, but Φ α is related with Φα by an invertible Klein transformation thus is equivalent to Φα. The rheonomy condition satisfying the Bianchi identity for the arbitrary Φ α /Φα is not unique. At least in appendix C, there is such an example (for the bosonic higher spin theory). However, the correct free theory limit is not recovered and the local Lorentz transformation is deformed there.
In superspace with the fixed background geometry, the local super Poincare invariance reduces to the global super Poincare invariance. With the chiral constraint imposed, the component expansion of the scalar superfield on superspace gives the WZ fields (H, H α ) on M 4 . For higher spin theory, one can fix the background of M and then study the scalar field on M with the global higher spin symmetry. It turns out that with the suitable rheonomy constraint imposed, the dynamics of the scalar on M is determined by s = 0, 2, 4, · · · fields (H, , · · · ) on AdS 4 . We get the global higher spin invariant theory for scalar fields with spin 0, 2, 4, · · · on the fixed AdS 4 background, which may be related with the gauge-fixed version of the standard higher spin theory.