On the Local Frame in Nonlinear Higher-Spin Equations

Properties of the resolution operator ${\rm d}_{loc}^*$ in higher-spin equations, that leads to local current interactions at the cubic order and minimally nonlocal higher-order corrections, are formulated in terms of the condition on the class of master fields of higher-spin theory that restricts both the dependence on the spinor $Y$, $Z$ variables and on the contractions of indices between the constituent fields in bilinear terms. The Green function in the sector of zero-forms is found for the case of constituent fields carrying helicities of opposite signs. It is shown that the local resolution ${\rm d}_{loc}^*$ differs from the conventional De Rham resolution ${\rm d}_Z^*$ by a non-local shift.


Introduction
Nonlinear field equations for massless fields of all spins in four dimensions were found in [1,2]. The most symmetric vacuum solution to these equations describes AdS 4 . Due to the presence of AdS 4 radius as a dimensionful parameter, higher-spin (HS) interactions can contain infinite tails of higher-derivative terms. This can make the theory not local in the standard sense, raising the question which field variables lead to the local or minimally nonlocal setup in the perturbative analysis.
Geometric origin of the dimensionful parameter ρ has an important consequence that any HS gauge theory with unbroken HS symmetries does not allow a low-energy analysis because a dimensionless derivative ρ ∂ ∂x where ρ = λ −1 is the AdS radius, that appears in the expansion in powers of derivatives, cannot be treated as small. This is because the rescaled covariant derivatives D = ρD, which are non-commutative in the background AdS spacetime of curvature ρ −2 , have commutator of order one, [D , D] ∼ 1. As a result, all terms with higher derivatives may give comparable contributions.
Another important feature of HS theory is that it describes interactions of infinite towers of HS fields while HS symmetry transforms a spin-s field to fields of other spins. In particular, HS symmetries transform spin two to higher spins. As emphasized in [3], this has a consequence that such concepts of Riemannian geometry as space-time point and dimension are not necessarily applicable to interacting HS theories.
The importance of the proper definition of locality was originally emphasized in [4] where it was shown that by a seemingly local field redefinition it is possible to get rid of the currents from the nonlinear HS field equations in AdS 3 . The question what is a proper definition of a weakly local field redefinitions in HS nonlinear theories was addressed recently in several papers [5,6,7,8,9,10], in particular, in the context of the derivation of current interactions of massless fields from the nonlinear HS field equations of [2]. In [5] a proposal was put forward on the part of the problem associated with the exponential factors resulting from so-called inner Klein operators while the structure of the preexponential factors was only partially determined. In this paper we make a step towards completing the definition of locality in the setup of HS equations of [2].
Conclusions of the papers [6,8,10] and [9] were somewhat opposite. The authors of [6,8,10] argued that no preferred frame of field variables in HS theory exists, bringing current interactions to the proper form with the conclusion that using field redefinitions exhibiting the same asymptotic behaviour it is possible to obtain current interactions with arbitrary charges in front of different currents, including zero charges which means no interactions. In the absence of a further selection criterion such a conclusion would imply difficulties of the physical interpretation of the HS equations of [2].
The analysis of [6,8,10] was performed in the one-form gauge sector of equations of [2]. On the other hand, in [9] we considered the problem in the zero-form sector of the HS equations which is simpler in many respects being the same time fully informative, finding a simple field redefinition that brings the quadratic corrections to free field equations following from the nonlinear HS equations to the canonical local current form in agreement with the unfolded form of current interactions found earlier in [11]. In [12] these results were extended to the one-form sector. The results of [9] were then shown in [13] (for a related particular result see also [14]) to lead to correct AdS/CF T predictions at the boundary of AdS 4 thus resolving some of the puzzles of the analysis of HS holography conjectures of [15,16] encountered in [17,18,19] (and references therein).
The resulting couplings for different HS currents were expressed unambiguously in [9,12] via the coupling constant of the nonlinear HS equations. In this paper it will be shown that the choice of field variables of [9] is distinguished by the condition that the associated higherorder corrections in HS theory are minimally nonlocal. To see this it is necessary to take into account the dependence on the spinor variables Z A in addition to that on Y A considered in [6,8,10], that is insufficient to control higher-order locality. It will be shown that upon a proper redefinition of the resolution underlying the procedure of solving the nonlinear HS equations from the conventional one d * Z to d * loc the results of [9] come out directly with no need of further field redefinitions. This implies that the De Rham resolution d * Z is related to the local one d * loc by a nonlocal field redefinition. In this paper we determine explicit form of d * loc in the sector of zero-forms and to the lowest order. Extension of these results to forms of higher degrees and to higher orders will be given elsewhere [20].
It should be stressed that, as explained in more detail below, the freedom in the definition of the resolution parametrizes the freedom in the choice of the homogeneous part of the solution to the differential equations in spinorial Z variables in the process of solving nonlinear HS equations. As usual, this freedom has to be fixed by appropriate boundary conditions. In quantum mechanics proper boundary conditions for solutions to the Schrodinger equation are fixed from the normalizability condition. (Discarding the normalizability condition may lead to meaningless conclusions, making it impossible to find a physical spectrum in a problem in question.) In HS theory, the appropriate choice of the boundary conditions is determined by the minimal nonlocality condition (which in the cases considered so far implies locality of HS vertices). Since, as explained below, this demands the solution to belong to a proper functional class, it is natural to speculate that, eventually, this functional class can be determined by a certain normalizability condition. The results of this paper provide a starting point for exploring this problem setting.
The rest of the paper is organized as follows. In Section 2 we recall unfolded formulation of free massless fields in four dimensions. The form of nonlinear HS equations is sketched in Section 3. In Section 4 details of the perturbative analysis of nonlinear equations with some emphasize on the homotopy technique are presented. The form of the local resolution d * loc leading to proper current interactions is found in Section 5. The structure of the unfolded equations describing current interactions of massless fields of all spins in the zero-form sector is recalled in Section 6.1. Green function for the zero-form sector of HS equations is found in Section 6.2 for the case of opposite helicity signs of the constituent fields of the current. Conclusions and perspective are briefly discussed in Section 7. Appendix contains details of the derivation of the Green function.

Free fields
In the frame-like formulation, the infinite set of 4d Fronsdal [21,22] massless fields of all spins s = 0, 1, 2 . . . is described by a one-form ω(Y ; K|x) = dx n ω n (Y ; K|x) [23,24] and More precisely, to describe massless fields, the one-form ω(Y ; K|x) should be even in k,k (i.e., i = j in (2.1)) while the zero-form C(Y ; K|x) should be odd (i.e., i + j = 1). The Central on-shell theorem states that unfolded system of field equations for free massless fields of all spins has the form [25] ω 1 describes first-order fluctuations of the HS gauge fields. Background AdS 4 space of radius λ −1 = ρ is described by a flat sp(4) connection w = (ω 0αβ , ω 0αβ , h αβ ) containing Lorentz connection ω 0αβ , ω 0αβ and vierbein h αβ that obey the equations where, here and after discarding the wedge product symbols,

9)
R αβ := dh αβ + ω 0αγ h γβ + ω 0βδ h αδ . C(y, 0; K|x) and C(0,ȳ; K|x) describe gauge invariant combinations of derivatives of the gauge fields of spins s ≥ 1 and the matter fields of spins s = 0 or 1/2. For s = 1, 2, C(y, 0; K|x) and C(0,ȳ; K|x) parameterize Maxwell and Weyl tensors respectively. Those associated with higher powers of auxiliary variables y andȳ describe on-shell nontrivial combinations of derivatives of the generalized Weyl tensors as is obvious from equations (2.4), (2.6) relating second derivatives in y,ȳ to the x derivatives of C(Y ; K|x) of lower degrees in Y . Higher derivatives in the nonlinear system result from the components of C(Y ; K|x) of higher degrees in Y . AdS geometry induces filtration with respect to spacetime derivatives rather than gradation as would be the case for massless fields in Minkowski space free of a dimensional parameter. System (2.3), (2.4) decomposes into subsystems of different spins, with a massless spin s described by the one-forms ω(y,ȳ; K|x) and zero-form C(y,ȳ; K|x) obeying the homogeneity conditions ω(µy, µȳ; K | x) = µ 2(s−1) ω(y,ȳ; K | x) , C(µy, µ −1ȳ ; K | x) = µ ±2s C(y,ȳ; K | x) , (2.12) where + and − signs correspond to selfdual and anti-selfdual parts of the generalized Weyl tensors C(y,ȳ|x) with helicities h = ±s.

Nonlinear higher-spin equations in AdS 4
The master fields of the construction of nonlinear equations of [2] consist of the zero-form B(Z; Y ; K|x), space-time one-form W (Z; Y ; K|x) and an additional spinor field S A (Z; Y ; K|x).
HS equations determining dependence on the variables Z A in terms of "initial data" are formulated in terms of the associative star product * acting on functions of two spinor variables where C AB = (ǫ αβ ,ǭαβ) is the 4d charge conjugation matrix and U A , V B are real integration variables. 1 is a unit element of the star-product algebra, i.e., f * 1 = 1 * f = f . Star product (3.3) provides a particular realization of the Weyl algebra The Klein operators satisfy with w α = (y α , z α , θ α ),wα = (ȳα,zα,θα). These relations extend the action of the star product to the Klein operators.
The nonlinear HS equations are [2] dW where F * (B) is some star-product function of the field B. The simplest choice of the linear function where η is a complex parameter leads to a class of pairwise nonequivalent nonlinear HS theories. The cases of ϕ = 0 and ϕ = π 2 correspond to so called A and B HS models distinguished by the property that they respect parity [26].
The left and right inner Klein operators κ := exp iz α y α ,κ := exp izαȳα , (3.14) which enter Eq. (3.11), change a sign of undotted and dotted spinors, respectively, Decomposing fields with respect to the Klein operator parity, , HS gauge fields are W + , S + and B − while W − , S − and B + describe an infinite tower of topological fields with every AdS 4 irreducible field describing at most a finite number of degrees of freedom. (For more detail see [2,27]). They can be treated as representing an infinite set of coupling constants in HS theory. In this paper all of these fields are truncated away.
The perturbative analysis goes as follows. Suppose that an order-n solution has been found Then Eq. (3.10) gives Using that S 0 has a trivial star-commutator with the Klein operators K, an elementary computation gives is the Z-space exterior derivative. As a result, Eq. (4.7) is equivalent to Since B is a zero-form, in the first nontrivial order this gives i.e., the zero-form C(Y ; K|x) of the free theory appears as Z-space De Rham cohomology. Analogously, Eq. (3.11) reads as In the lowest order this gives the equation In the Z− space sector, perturbative analysis consists of solving repeatedly the equations of the form Formal consistency of HS equations (3.11) and (3.10) guarantees that implying formal consistency of Eq. (4.14). However, it admits a solution only if J is d Z -exact. Analysis of the equations involving space-time one-forms is analogous. Firstly one resolves all equations that contain d Z . The remaining equations in d Z -cohomology produce dynamical equations on the dynamical fields ω and C which reproduce Central on-shell theorem (2.5), (2.6) along with all nonlinear corrections.

Homotopy trick
Let us now recall the standard homotopy trick. Let d be a differential (later on to be identified with d Z ) obeying d 2 = 0 (4.16) as well as a homotopy operator ∂ ∂ 2 = 0.  In this case the projectorĥ to KerAĥ 2 =ĥ (4.21) can be defined to obey Also we can introduce the operator A * such that This allows us to define the resolution operator Hence where an exact part dǫ and g ∈ H (d) remain undetermined. The simplest choice of the homotopy operator for the exterior differential d = d Z is This gives Eq. (4.23) is checked by using KerA consists of Z, θ-independent functions and thus, by Poincaré lemma, relation (4.20) becomes an exact equality In this construction, the freedom in the choice of the homotopy operator ∂ affects the homogeneous solution to equation (4.26), i.e., dǫ + g. Since dynamical fields C and ω are valued in the d Z cohomology, this implies that going to a different ∂ may imply a field redefinition. Alternatively, one can directly redefine the resolution d * by adding a closed form.
Though the conventional resolution operator d * Z looks simple and natural, it is known since [4] to lead to nonlocalities at the nonlinear order. In the next section we identify a resolution operator d * loc that leads directly to the local setup in the process of solving HS equations in the lowest nonlinear order. Its form is deduced from the results of [9]. In accordance with the general analysis the difference between the two approaches effectively results in a field redefinition associated with the difference between cohomological g-terms in the respective formulae (4.28). 1 Since d * loc gives a local result while d * Z leads to a nonlocal one, they should differ by nonlocal cohomological g-terms in (4.28). One can argue following the authors of [6,8] that one can equally well choose another resolution operator d * ′ loc = d * loc , that might lead to another local result. The weak point of this argument is however due to insufficient representativity of the lowest-order analysis of consistency insensitive to the specific coefficients in front of different vertices. Things do change drastically in the higher orders. As will be explained in the next section the choice of d * loc associated with the solution found in [9] is singled out by the condition that higher-order corrections remain minimally nonlocal (if nonlocal at all) which phenomenon is properly captured by the lowest-order corrections once the dependence on Z variables is taken into account. The analysis of [9] was based on the separation of variables taking into account that, in accordance with the fact that the left and right parts of d Z form a bicomplex, in the left(right) sector the dependence on the right(left) spinors remains unchanged in the lowest order being governed by the original star product. At this condition the solution of [9] is the only one leading to local current interactions in the Y, x sector.

Perturbative analysis of locality
To see what distinguishes between the local frame of HS equations of [9] and other possible frames let us consider in more detail the perturbative analysis in the sector of zero-forms starting from De Rham resolution d * Z . The first step is to evaluate the first correction to S, i.e., S 1 . Using (3.15) and (3.16), Eq. (4.31) gives Resolution of (4.10) with the homotopy operator ∂ Z (4.29) gives for B 2 in the η-sector where we use notation with the convention that θ(τ )δ(τ ) = δ(τ ) .
Equivalently, this expression can be written in the differential form symmetric with respect the first and second factors of C It is important that since τ 4 ≥ 0, and hence τ 3 + τ 4 ≥ τ 3 , the expansion coefficients in powers of ∂ 1α ∂ α 2 are larger than those of y α ∂ iα , which as will be explained later, indicates nonlocality.
Let us now use Schouten identity expressing the fact that antisymmetrization over any three two-component indices is zero to transform B 2η to a different form. To this end, we observe that (5.14) Using these relations and integrating by parts we obtain Comparison of this with (5.7) allows us to represent B 2η in the form The terms with δ(τ 3 ) and δ(τ 4 ) have different meaning. The part with δ(τ 3 ) is z-independent. Indeed, denoting it ∆C 2η we obtain using (3.15) and (3.16) Since ∆C 2η is in the d Z cohomology, according to (4.28) we can define a new resolution operator d * has the form 2η can be represented in the integral form Formula (5.18) just describes the nonlinear shift found in [9] to reduce the nonlocal bilinear corrections to the local form directly in the sector of x, y-variables.
We observe that B loc 2η (5.22) has the remarkable property that the coefficient in front of the term responsible for the index contraction between the first and second factors of C equals to those in front of the y variable in the arguments of C, which is analogous to the star product of Z-independent functions.
The results of this paper prove that this is the only option consistent with locality. Indeed, we have shown that the dependence on y in the arguments of C in (5.22) contains the same dependence on the homotopy parameter as u, that determines the contractions of spinorial indices. This has to be compared with the original (non-local) source (5.4) where the coefficients in front of u (or v) determining contractions are larger that those in front of y. Clearly, since τ 4 ≥ 0 the solution (5.22) is strictly minimally nonlocal, having the same type of nonlocality as the original star product in y variables.
The following comments are now in order. An important feature of B loc 2η (5.22) is that it contains the rightmost star-product factor k * κ. Because, by (3.15), (3.16), star product with κ exchanges y and z variables, Eq. (5.22) can be equivalently rewritten in the form In this form the coefficient in front of u that governs contractions is dominated by that in front of z α y α in the exponential, as well as the coefficients in front of z in the arguments of C. We observe that the relevant terms disappear at Z = 0, not allowing to distinguish between proper and improper nonlinear corrections in attempt to analyze the issue of locality at Z = 0 as in [6,8]. This feature is in agreement with the well-known fact that the overall coefficients in front of different currents are not determined by the lower-order consistency and hence one can seemingly freely go from one set of coefficients to another by a nonlocal field redefinition of the original variables which are those associated with the Y dependence at Z = 0. Such arguments led some of the authors of [6,8] to claims that it is impossible to compute current vertices from the HS equations of [2]. (See e.g. [28] for such interpretation of conclusions of [8]). In fact, the meaning of the results of [8] is that the authors were using a bad luck ad hoc assumption that the problem can be analyzed with the help of the conventional resolution operator d * Z (4.35). To make a proper choice, higher-order effects have to be taken into account.
Indeed, our results imply that the setup of [9] not only leads to the local result in the first nontrivial approximation but, most significantly, it will lead to the minimally nonlocal setup in the higher orders. To see this it is important to have expressions for the bilinear corrections that account for the full dependence on both Y and Z. The computation of higher-order corrections will involve star products of the expressions like B loc 2η (5.22) or B 2η (5.4) with B 2η being more nonlocal than B loc 2η . Also let us note that the factor of k * κ is central and involutive. Hence it will cancel in particular the similar factor on the r.h.s. of (3.11) in the next order with the effect that no exchange of y and z variables will occur in (some of) the higher orders, i.e., the minimal order of nonlocality will be visible directly in the physical Y -space in the higher-order terms. From this perspective our approach is somewhat similar to the derivation of cubic HS vertices by Metsaev in [29] where the proper form of cubic HS vertices that precisely corresponds [12,30] to that resulting from the application of the resolution d * loc to equations of [2] was deduced from the higher-order analysis. One of the main results of this paper is the identification of the proper resolution operator d * loc (5.18) that directly leads to the formulation of the HS theory in the local (or minimally nonlocal in the higher orders) setup for the full system of nonlinear HS equations. So far, d * loc was found only in the lowest order of the 0-form sector. The goal is to find its full fledged extension to all orders and all types of differential forms. This is the ongoing project [20].
Completion of the latter project will also imply the completion of the program of [5] of establishing a proper class of star-product functions associated with the minimally nonlocal setup in HS equations. Indeed, in [5] the proper dependence on Z and Y variables was established for the expressions like with φ(W ; U; τ ) regular in W and U and integrable in τ . Being accompanied by the factor of τ and 1 − τ , the dependence on Z and Y on the r.h.s. of (5.24) trivializes at τ → 0 and τ → 1, respectively. In [5] the space of functions (5.24) called V 0,0 was extended to the spaces V k,l of such star-product elements (5.24) that φ(W ; U; τ ) scales as τ k at τ → 0 and (1 − τ ) l at τ → 1. More precisely, we allow (poly)logarithmic dependence on τ and 1 − τ at τ → 0 and τ → 1, respectively, with the convention that it does not affect the indices k and l. In [5] V k,l with both positive and negative k and/or l were considered. The problem not considered in [5] was which restrictions on the inner structure of φ(W ; U; τ ) have to be imposed to respect locality or minimal nonlocality. In particular, a question to be addressed is what is the proper dependence on τ i in the expressions like 25) The answer combining the results of [5] with those of this paper is where the new restriction is the last one dominating the dependence on τ 5 by that on the parameter t in front of the factor of z α y α in the exponential. Let us stress that not only solutions have to be of the form (5.25), (5.26) but also gauge transformation and field redefinitions (as explained in [5], the latter should even obey stronger conditions due to further restrictions on the measure ρ(τ )). In any case, the Zindependent field redefinition (5.17) has t = 1 and hence does not belong to the proper class. Moreover, that τ 5 ≤ 0 for Z-independent functions obeying (5.26) implies that they should be distributions supported at τ 5 = 0 hence being represented by a finite number of delta-function derivatives δ n (τ 5 ). In turn this means that the allowed class of Z-independent field redefinitions is genuinely local. Direct analysis of the next section also demonstrates that d * Z and d * loc are not related by a local field redefinition.

Cohomology shift and Green function
In this section we show that the cohomology shift (5.17) relating d * Z and d * loc is nonlocal. To this end we first recall in Section 6.1 the structure of the nonlinear corrections to dynamical equations in the sector of Y variables (i.e., d Z cohomology) resulting from d * Z , computing the zero-form Green function in Section 6.2.

d * Z -induced nonlocal deformation in the zero-form sector
The lowest-order deformation of free equations (2.3), (2.4) has the form dω + ω * ω + L(w, ω, C) + Γ cur (w, J) = 0 , (6.1) where L(w, ω, C) is at most linear in both ω and C while the two-form Γ cur (w, J) and one-form H cur (w, J) are some functionals of the background fields w and the current J As a consequence of equations (2.4) on C, so defined current obeys the current equation αȳ 2β − ∂ 1α∂1β + ∂ 2α∂2β J(y 1 , y 2 ;ȳ 1 ,ȳ 2 ; K|x) = 0 (6.4) at the convention that derivatives ∂ 1α (∂ 1α ) and ∂ 2α (∂ 2α ) over the first and second undotted(dotted) spinorial arguments of J are defined to anticommute with k(k). The star product in (6.1), (6.2) results from the restriction of (3.3) to Z-independent functions. Gauge invariant current interactions are associated with Γ cur (w, J) and H cur (w, J) linear in J. Other ω-dependent terms bilinear in fluctuations, describe gauge non-invariant interactions which do not contribute if the spins s 1 and s 2 of two fields entering the bilinear terms on the right-hand sides of (6.1) and (6.2) and spin of the current s J identified with the spin of the field contributing to the linear part of (6.1) and (6.2) (i.e., by its definition, the current of spin s j contributes to the nonlinear corrections to the equations on the spin-s J field) obey the condition s J ≥ s 1 + s 2 . (6.5) (For the derivation of (6.5) directly from Eq. (6.2) see [31]). The final result of [9] in the 0-form sector is This corresponds to H cur (w, J) in (6.2) with the minimal number of derivatives which is finite for any spins s 1 , s 2 and s J . The deformation H(w, J) resulting from the nonlinear equations of [2] by virtue of the resolution operator d * Z is [6,9] H(w, J) = H η (w, J) + Hη(w, J) , τ )y + s, τ t,ȳ +s;ȳ +t; K)] * k , (6.9) τ )s,t)J(y + s, y + t; (1 − τ )ȳ +s, τt); K] * k . (6.10) The integration over S and T in (6.9), (6.10) brings infinite tails of contracted indices, inducing by (2.4) and (2.6) an infinite expansion in higher space-time derivatives of the constituent fields. Hence, H (6.9), (6.10) differs from the conventional current interactions (6.6) which, being free of the integration over s α and t α , contains a finite number of derivatives for any s 1 , s 2 and s J .
To reproduce standard current interactions from those resulting from d * Z one has to find a field redefinition with Φ linear in J, bringing H (6.8) to the form (6.6), i.e., The proper field redefinition found in [9] is Φ = ∆C 2η + ∆C 2η with ∆C 2η (5.17) and ∆C 2η being its complex conjugate. It should be stressed that formula (6.12) is valid for any spins s 1 , s 2 and σ J independently on whether the condition (6.5) holds or not.

Green function
To discuss locality properties of the field redefinition (6.11) it is useful to consider Green function that removes the current interactions. The goal is to find a solution to the equation Since H η cur (w, J) and Hη cur (w, J) describe an arbitrary local current with certain coefficients the resulting solution G C (J) describes Green function applied to J. Let us stress that this problem setting is only applicable at the condition (6.5), when the HS connections ω do not contribute. So, we will assume that (6.5) is true. Since the problem is linear, the terms proportional to η andη can be found separately. We consider the part G η (J) linear in η.
The term Gη(J) withη can be obtained by complex conjugation. Let us look for G η (J) in the most general Lorentz covariant form , (6.14) where We use convention that both y α and y α i along with the respective derivatives anticommute with the Klein operator k inside J, whileȳα andȳα i anticommute withk.
Using (2.4) and (6.4) it is not difficult to obtain To reproduce H η cur (w, J) (6.8) we have to demand that all terms on the r.h.s. of (6.21) should be zero except for those containing y α∂ iβ . However, demanding this, one should take into account that antisymmetrization over any three two-component indices gives zero. This yields the relations allowing to add the following expression to DG η (6.21) , (6.20) where a, b i ,ā andb i are arbitrary functions of N i , M, N i and M . As a result, Now we can solve the equation ignoring relations (6.19). However, this equation admits a solution only at the condition (6.5) since otherwise it is simply inconsistent as long as the contribution of the HS connections ω is not taken into account.
To project the currents to appropriate helicities h J , h 1 and h 2 , where the function φ should be chosen appropriately. Firstly, we observe that the operators do not affect the helicities since they commute with the operators (6.23).
Ansatz that is consistent with condition (6.5), allowing to solve (6.22), is (6.25) where, assuming for definiteness that s 1 ≥ s 2 (the opposite case can be considered analogously) This implies, in particular, 2s 1 ≥ t ≥ 2s 2 (6.27) and The complex conjugated Ansatz solves for Hη cur (w, J). In this paper we only consider the case of opposite helicities h 1 h 2 ≤ 0 which is sufficient for our purposes. It would be interesting to extend the obtained results to helicities of equal signs as well. In particular, this is useful for the analysis of holography along the lines of [13].
It is also convenient to use the following Ansatz Plugging this into (6.22) gives the following system of differential equations in the variables where Here ψ 1 (x) are functions of x i which should reproduce H η cur (w, J) (6.8). ψ 1 (x) are demanded to be independent of z since, containing no integration over s and t H η cur (w, J) (6.8) contains no contractions of undotted indices. In Appendix we shall see that it is enough to demand this to derive the proper form of ψ 1 (x) that corresponds to H η cur (w, J) by solving the system (6.35)-(6.43). The final result is On the other hand, using the generalized beta-function formula [12] the field redefinition (5.18) can be rewritten in the form Φ η (J) = η ∞ r 1,2 ,r 1,2 ,p,p=0 . (6.47) We observe that the Green function (6.45) differs from the field redefinition (5.18) only by the factors of θ restricting the field redefinition to the region (6.5), (6.26), (6.27).
The fact that in the allowed region of helicities the Green function (6.45) and field redefinition (6.47) coincide is not accidental. It is a consequence of the property that, as is easy to see, the contribution (6.9) resulting from application of the resolution d * Z to the nonlinear equations is zero in this sector. Hence, in this sector, the field redefinition (6.47) must have the form of the Green function applied to the resulting local current. This means that the field redefinition (5.18) is essentially nonlocal, i.e., in agreement with the conclusions of Section 5 the resolution operators d * Z and d * loc are locally nonequivalent.

Discussion
We have identified the proper resolution operator in the space of spinorial Z A -variables, that leads to local first nonlinear correction to HS equations. It is shown to correspond to certain class of functions of the type identified in [5] extended to the terms accounting contractions between indices of the field product factors. As shown in [13], the current interactions of [5] resulting from this resolution operator properly reproduce the anticipated HS holographic results. It should be stressed that the distinguished role of the resolution d * Z becomes manifest only if the dependence on Z A affecting the higher-order nonlinear corrections to HS equations is taken into account. It is however less obvious at Z A = 0 in which case relevant terms in the deformation trivialize.
Naive interpretation of our results might be that the field redefinition (6.47) relating the simplest resolution operator d * Z in the HS theory to d * loc is not allowed, being nonlocal. The proper interpretation however is just opposite: to reach a minimally nonlocal setup in HS theory (which is fully local to the order in question), one has to use the resolution d * loc with no reference to d * Z at all. It is d * loc that fulfils locality compatible boundary conditions in the process of solving HS equations with respect to Z A -variables. Hence, the proper interpretation is that d * Z is related to the local resolution d * loc by a nonlocal field redefinition making the d * Z setup improper from the locality perspective. This raises the question of the proper definition of the resolution operator d * loc at the higher orders and its extension to the sector of one-forms. These issues will be considered in [20]. The analysis of higher orders is also interesting in the context of conclusions of the recent paper [28] claiming that the level of nonlocality in HS gauge theories is somewhat extreme. This remains to be analyzed carefully from several perspectives, however.
One is that the space-time derivatives in AdS do not commute, having the commutator of order one in dimensionless units [λ −1 D , λ −1 D] ∼ 1. This raises the question in which ordering prescription the properties of the functions of the covariant D'Alambertian f ( ) are analyzed. As is well known, going from one ordering to another may significantly affect analytic properties of the function in question. For instance, being exponentials in the HS star product (3.3), inner Klein operators have a form of distributions in the Moyal-Weyl star product [32] which property in fact highlights the distinguished role of the HS star product (3.3) in the HS gauge theory.
Another is that in presence of an infinite tower of HS states even local field redefinitions at the level of quadratic corrections may induce nonlocal contributions at higher orders. This phenomenon has to be properly taken into account in the locality analysis of the cubic corrections to the HS field equations.
Also it should be stressed that the relation between the form of local corrections in terms of spinor Y A -variables and that in terms of space-time derivatives via (2.6) acquires nonlinear corrections. As a result, analysis of the problem in terms of spinors may have much simpler form, simultaneously providing a distinguished ordering prescription mentioned above.
Finally, as a byproduct of our consideration we have found explicit expression for the zero-form Green function in the case of the constituent fields of opposite helicity signs. It would be interesting to extend these results to the helicities of the same sign as well as to the sector of one-forms.