Conformal Coupling of Higher Spin Gauge Fields to a Scalar Field in $AdS_{4}$ and Generalized Weyl Invariance

The higher spin interaction currents for the conformally coupled scalar in $AdS_{4}$ space for both regular and irregular boundary condition corresponding to the free and interacting critical point of the boundary O(N) sigma model are constructed. The explicit form of the linearized interaction of the scalar and spin two and four gauge fields in the $AdS_{D}$ space using Noether's procedure for the corresponding spin two and four linearized gauge and generalized Weyl transformations are obtained.


Introduction
In contrast to the case of usual AdS 5 /CF T 4 correspondence [1] where the strong coupling regime of the boundary theory corresponds to the weak coupled string/supergravity theory on the bulk, the AdS 4 /CF T 3 correspondence of the critical O(N) sigma model [2] operates at small t'Hooft coupling λ and the corresponding bulk theory is described as a theory of arbitrary even high spins. So it is a theory of Fradkin-Vasiliev type [3]. This case of AdS/CF T correspondence is also of great interest because here dynamical considerations and calculations both in AdS d+1 and CF T d cases are essential on account of the absence of supersymmetry and BPS arguments and because in this case of correspondence perturbative expansions with small coupling constants are mapped on each other. So the essential point of HS(4) and d=3 O(N) sigma model correspondence is that both conformal points of the boundary theory i.e. unstable free field theory and critical interacting point, in the large N limit correspond to the same higher spin theory. Moreover as we have learned from [2] these two points are connected on the boundary by a Legendre transformation which corresponds to the different boundary condition (regular dimension β = 1 or shadow β = 2) in the quantization of the bulk scalar field. This quantization of the free scalar field in the AdS with different boundary conditions and corresponding multi-trace deformation of the boundary theory were investigated and explored in many papers, we will refer just to the articles [4] most interesting for us.
In this article we explore these two different boundary conditions of the scalar field from the point of view of the linearized interaction of the scalar field with the high spin fields in d = 4 AdS space. In the section 2 we extend our previous consideration [5] to the case of the β = 2 boundary condition. We show that if the β = 1 case corresponds to the interaction of HS(4) gauge fields with the conformal (traceless) conserved higher spin currents constructed from scalar field, then the β = 2 case we can describe in non-contradictory fashion with the AdS/CF T correspondence using the non-conformal double-traceless currents and gauge fields in Fronsdal's formulation [6]. In the last two sections we explicitly construct a linearized interaction Lagrangian of the conformal scalar field with the spin two and four gauge field using Noether's procedure for gauge and generalized Weyl invariance (some consideration of nonlinear gauge invariant coupling of the scalar field on the level of equation of motion one can find in [7]). We show that the interaction of the scalar with the spin four Fronsdal gauge field can be constructed in a non unique way due to the existence of the gauge invariant combinations of gauge field itself (analogue of Ricci scalar for higher spins). But this ambiguity can be fixed in unique fashion by gauging the analogue of scale invariance in the higher spin case. This is a symmetry with the local tensor parameter permitting to gauge away the trace of a double traceless gauge field and leading to the tracelessness of the corresponding spin four current.

Conformal and Fronsdal higher spin currents
and AdS 4 /CF T 3 In our previous article [5] we considered coupling of the HS(d + 1) gauge field with a current constructed from a scalar field in fixed AdS d+1 background. Using the ansatz including the AdS d+1 curvature corrections we have shown there that all ambiguity in the construction of a spin ℓ traceless current from the conformally coupled scalar field reduces to the ambiguity of the set of leading coefficients A p in the expansion of the current 1 The tracelessness condition fixes relations between B p , C p and A p in the following way [5] The unknown A p we can fix in two different ways: The first possibility is to use the conservation condition for the current. This leads to the recursion relation with the same solution for the A p coefficients [8] as in the flat D = d + 1 dimensional case For the important case D = 4 this formula simplifies to 1 For the investigation of the conservation and tracelessness conditions for general spin ℓ µ1µ2...µ ℓ we contract it with the ℓ-fold tensor product of a vector a µ The result of our previous consideration [5] was the following: the curvature corrections do not change flat space tracelessness and conservation conditions between leading coefficients and therefore the solution (5) remains valid. The other way to fix this ambiguity is a boundary CF T 3 consideration. The result is [5] A Here β is the dimension of the scalar field of the boundary CF T 3 , and C ℓ is the normalization constant of the three point function of two scalars and the spin ℓ current in CF T 3 (We define also the Pochhammer symbols (z) n = Γ(z+n) Γ(z) .). The expression (7), for β = 1, is in agreement with the previous one (6) obtained from AdS 4 consideration, if we will normalize in (7) C ℓ = 2 ℓ ℓ!. It means that the β = 1 point of boundary CF T 3 (free field conformal point of O(N) vector model) we can describe as a conformal HS(4) model in AdS 4 . In other words in this case we have to operate in dual higher spin theory with the linearized interaction where the corresponding current is conserved and traceless For β = 2 we have to change the constraints imposed on (1). For that we turn from conformal higher spins to Fronsdal's [6] formulation where gauge fields and currents are double traceless only and the conservation condition looks a little bit different from the usual one due to the double-tracelessness of the gauge field and current. Then we can realize the double-traceless current Ψ (ℓ) using two traceless (but not conserved) currents J (ℓ) , Θ (ℓ−2) with the same dimension ℓ+2β +O( 1 N ) on the boundary [9]. It means that the expansions for these fields start from the following series The Fronsdal field Ψ (ℓ) we can present then as The conservation condition (12) in this representation is or .
From this we can read off a restriction on the coefficients in (13) and (14) p For D = 4 we get Then after using (7) for β = 2 we obtain The solution of this equation fulfilling the boundary conditions The latter sum can be proceeded using Pascal's formula for binomials. The result is very elegant So we show that in contrast to the β = 1 case where the interaction includes the traceless conformal higher spin currents, the β = 2 boundary condition necessitates the interaction with the double trace higher spin currents. The connection between these two types of interaction can be described adding local Weyl (in the spin two case) and generalized "Weyl" invariants realizing the conformal coupling of the scalar with the higher spin fields.
3 Linearized spin two gauge and conformal scalar field interaction (ℓ = 2) The well known action for the conformally coupled scalar field in D dimensions in external gravity is In this section we restore the linearized form of this action in fixed AdS background using a gauging procedure both for the gauge and Weyl symmetry on the linearized level. We do this derivation just for methodical reasons because the final nonlinear answer is known (25). But we would like to extend this consideration to the higher spin case and try to elaborate a linearized construction which works in the case ℓ = 4 where the final answer is unknown. We start from the massive free scalar action in the fixed AdS external metric 2 For getting an interaction with linearized gravity using the gauging procedure we have to variate S 0 with respect to δ 1 and solving (we assume that ε µ and h µν have the same infinitesimal order) the equation we immediately find the following cubic interaction linear in the gauge field 2 We will use AdS conformal flat metric, curvature and covariant derivatives commutation rules of the type Note that here we used many times partial integration which means that we admit that all fields or at least parameters of symmetry are zero on the boundary, otherwise we would have to check all symmetries taking into account some boundary terms and their variations also. It is clear that for constructing the local interaction on the bulk we can use partial integrations without watching the boundary effects. So we see that gauge invariance in this linear approach does not fix the free parameter λ and the corresponding spin two Noether current (energy-momentum tensor) is conserved but not traceless. But we can fix this problem having noted that there is one more gauge invariant combination of two derivatives and one h µν field It is of course the linearized Ricci scalar-but at this moment it is important for us that there is only one gauge invariant combination of h (2) µν (z) , two scalars φ(z) and two derivatives which we can add to our linearized action with one more free parameter. So finally we can write the most general gauge invariant action in this approximation of the first order in the gauge field Then we search for the additional local symmetry permitting to remove the trace of the gauge field h µν and therefore leading to the traceless conformal spin two current. The natural choice here is of course Weyl invariance and we will define local Weyl transformation in linear approximation in the following way where ∆ is the conformal weight (one more additional free parameter to fit) of the scalar field. The important point here is that when we impose on the gauge invariant action (35) conformal (Weyl) invariance (36) we obtain the condition with the unique solution for all free constants So finally we come to the gauge and conformal invariant action where which is of course the linearized action (25) and can be obtained from that after expansion near to the AdS D background G µν (z) = g µν + h

Solution for spin four
Now we start from action (41) to apply Noether's method for the following transformation of the scalar field with a traceless third rank symmetric tensor parameter First of all we have to calculate δ 1 S 0 . For brevity we introduce the notation (and in a similar way for any other tensor) Then after variation of (41) we obtain We see that we can introduce an interaction with the spin four gauge field h (4) µναβ in the minimal way if we will deform the transformation law for the spin two field. The solution for the equation So we obtain the following gauged action with linearized interaction with both spin two and spin four gauge fields and linearized usual Weyl invariance S GI (φ, h (2) , h (4) ) = S W I (φ, h (2) ) + S Ψ (4) 1 (φ, h (4) ) (51) δ 0 h (4)µνλα = 4∇ (µ ǫ νλα) , δ 0 h (2)µν = 2∇ (µ ε ν) + 2∇ (µǫν) + 2σg µν (52) where S W I (φ, h (2) ) can be read from (40)-(43) and we note that on this linearized level usual Weyl transformation does not affect the spin four part of the action but the spin four gauge transformation deforms the gauge transformation for spin two gauge field. Now we turn to the construction of the conformal invariant coupling of the scalar field with the spin four gauge field in a similar way as in the case of spin two. For this we note first that here we can construct also the gauge invariant combination of two derivatives and h (4)µναβ . This is the following traceless symmetric second rank tensor This is the analogue of the Ricci scalar in the spin four case and we can construct using this tensor two additional gauge invariant combinations of the same order.