Spin-Locality of $\eta^2$ and $\bar\eta^2$ Quartic Higher-Spin Vertices

Higher-spin theory contains a complex coupling parameter $\eta$. Different higher-spin vertices are associated with different powers of $\eta$ and its complex conjugate $\bar \eta$. Using $Z$-dominance Lemma, that controls spin-locality of the higher-spin equations, we show that the third-order contribution to the zero-form $B(Z;Y;K)$ admits a $Z$-dominated form that leads to spin-local vertices in the $\eta^2$ and $\bar \eta^2$ sectors of the higher-spin equations. These vertices include, in particular, the $\eta^2$ and $\bar \eta^2$ parts of the $\phi^4$ scalar field vertex.


Introduction
Higher-spin (HS) gauge theory is a theory of an infinite set of gauge fields of all spins. Since gauge invariant HS interaction vertices contain higher derivatives of degrees increasing with spin [2,3,4,5], HS gauge theory is not a local field theory in the usual sense. Some arguments that HS gauge theory has to be essentially non-local were given in [6] based on the holographic correspondence with the boundary (critical) sigma-model, conjectured by Klebanov and Polyakov [7] (see also [8]).
To be free from the assumptions of holographic correspondence it is important to analyze the issue of (non)locality of HS gauge theory directly in the bulk. Based on the nonlinear HS equations of [9,10] such analysis was performed in [11,12,13,14,15] in different sectors of the theory at some lowest orders. All vertices derived in these papers turned out to be spin-local 1 including some of the quintic vertices in the Lagrangian counting. Also somewhat different arguments pointing out at locality of the HS gauge theory were presented in a recent paper [17].
The obtained vertices agree with holographic prediction at cubic order [18,19]. However, the bulk vertices derived from nonlinear HS equations so far did not contain the φ 4 vertex for the spin-zero field φ. On the other hand, it is this vertex [20] that was argued to be highly nonlocal [6] in the HS theory holographically dual to the boundary sigma-model [7]. The degree of non-locality prescribed by the analysis of [6] led the authors to a conclusion of a fundamental failure for HS holographic reconstruction programme beyond cubic order. Still, the same vertex was also analyzed in [21] concluding that the non-locality if present is of a very special form.
The aim of this paper is to carry out holographically independent approach to the locality problem. We do it by extending the analysis of locality of [13,1,14,15,16] to the vertices of order C 3 in the sector of equations on the zero-forms C that contain in particular the φ 4 vertex of interest in the form of φ 3 contribution to the field equations. Note that the vertices studied in this paper include an AdS 4 extension of those obtained by Metsaev in [22].
In this paper we give general arguments based on the so-called Z-dominance Lemma of [1] that the holomorphic, i.e., η 2 and antiholomorphicη 2 vertices in HS gauge theory must be spin-local, where η is a complex parameter in the HS equations. We explicitly demonstrate by direct calculation that every individual contribution to the (anti)holomorphic part of a quartic vertex acquires a form that results in a complete spin-locality of the entire piece. While our analysis is sufficient to see that the result is local it does not give directly the manifestly local form of the remaining local vertex. The derivation of the latter, which uses partial integrations and Schouten identities, we leave for the future. The analysis of the mixed ηη vertex that is the only remaining sector in the analysis of locality of the φ 4 sector, needs somewhat different tools and is beyond the scope of this paper.
The paper is organized as follows. In Section 2, the necessary background on HS equations is presented. Section 3 contains brief recollection on the so called limiting shifted homotopy and the interpretation of the Z-dominance lemma via space H + of star-product functions. In Section 4, we collect expressions for the holomorphic vertices obtained from the generating equations. Discussion of the obtained results and problems yet to be solved is placed in Section 5. Some useful formulas are collected in Appendix A. Appendices B and C contain the detailed derivation of the third-order contribution B ηη 3 to the zero-form and second-order contribution W ηη 2 to the one-form fields, respectively.
Star product is defined as follows The form of the vertices on r.h.s. of (2.1) and (2.2) is determined by the consistency condition with d 2 x = 0. This determines the vertices up to field redefinitions ω ′ = F (ω, C) , where F (ω, C) is linear in the one-form ω, while both F (ω, C) and G(C) can be nonlinear in C. Indeed, a field redefinition in the consistent system produces another consistent system.
Since ω and C contain all on-shell nontrivial derivatives of the Fronsdal fields, non-linear field redefinitions (2.6) may contain infinite tails of higher derivatives thus being non-local though having particular quasi-local form expandable in power series in terms of components of (2.3) and (2.4) So, if system (2.1), (2.2) is local or spin-local (for more detail see [16]) in some specific choice of variables it may lose this property in other variables. Other way around, if system (2.1), (2.2) is non-local in some set of variables, this does not necessarily imply that there is no other set of variables making the system spin-local. Direct computation of vertices consistent with a given locality requirement from compatibility conditions is technically involved. An alternative scheme making the derivation of vertices much easier is based on the generating system of [10], that has the form where master fields W , S and B depend on space-time coordinates x and commuting spinor coordinates Y A and Z A = (z α ,zα). In what follows the x-dependence is implicit. In addition there is also a dependence on discrete involutive elements K = (k,k) such that and analogously fork. Star product * acts on functions of Y and Z according to where sp(4) indices A, B, . . . are raised and lowered by the antisymmetric form ǫ AB = −ǫ BA as follows X A = ǫ AB X B and X A = X B ǫ BA . Master fields are differential forms with respect to space-time differential dx ν and auxiliary spinor differential θ A = (θ α ,θα) satisfying is a zero-form, while W = W µ dx µ is a one-form in space-time differential and S = S α θ α +Sαθα is a one-form in auxiliary spinor differentials. Finally, γ andγ are central two-forms attributed to the so called Klein operators To see that they are central [10], one should use that θ 3 =θ 3 = 0 and that the star-product elements κ := exp(iz α y α ) andκ := exp(izαȳα) have the properties analogous to (2.12) with respect to star products with f (Y, Z) but commute with the Klein operators and differentials θ α andθα.

Perturbation theory
A proper HS vacuum is the following exact solution of (2.7)-(2.11) The flat connection ω can be chosen to describe AdS 4 . Since vacuum value of S 0 is non-trivial it is going to generate via star (anti)commutators PDEs in Z for master fields. Indeed, consider equation (2.11) in the first order Using star product (2.13) one can check that (2.20) It means that B 1 field is Z -independent which is the generating function for HS curvatures (2.4). Lower index in B 1 , shows the order of the expression in the C-field. Correspondingly, B 2 is of second order in C and so on The same rule applies to S and W , i.e., To solve for Z-dependence of master fields at each perturbation order one has to solve equation of the form where J originates from the lower-order terms and f is either B, S or W . Let us note that it is vacuum solution of the auxiliary field S 0 (2.17) manifesting itself in operator d Z that allows one to capture the field redefinition ambiguity as its kernel. Different solutions can be obtained by one or another contracting homotopy operator △ f = △J (2.25) resulting from the standard homotopy trick. For an operator ∂, that should be also nilpotent, ∂ 2 = 0, one considers operator If N is diagonalizable one can introduce the almost inverse operator N * . Since all the d Zcohomologies are in the kernel of N one rewrites the solution to (2.24) as The simplest choice for ∂ is Then N turns out to be an Euler operator and thus easily invertible This choice of ∂ leads to contracting homotopy operator referred to as the conventional homotopy operator in [14]. Using (2.10) and (2.17) one finds in particular − 2id z S η 1 = iηC * γ = iηθ α θ α e izαy α C(−z ,ȳ)k . (2.31) Using now (2.30) we find solution for S η 1 in the form Different choices of homotopy operators represent gauge and field redefinition ambiguity. A particular class of the so-called shifted homotopies can be defined by considering Z A − Q A instead of Z A with some Z-independent Q A . Local properties of HS vertices crucially depend on properties of the chosen homotopy operators. A class of homotopy operators consistent with locality requirement based on shifted homotopies at non-trivial interaction level was proposed in [15]. Let us also note that another way of fixing the Z-dependence based on the so-called gauge function method which applies for AdS vacuum is reviewed in [25].

Notation
Let us set up our notation. Derivative with respect to holomorphic argument of the C-fields is denoted as ∂ i α where index i indicates position of the C-field in expression that contains several C's as seen from left to right. Derivative with respect to holomorphic argument of ωfield is denoted as ∂ ω α .
Whenever arguments of C's or ω are not written explicitly we assume the exponential form. This means the following: suppose one has ωCCC. Then it should be understood as ω(y ω ,ȳ) * C(y 1 ,ȳ) * C(y 2 ,ȳ) * C(y 3 ,ȳ) , (2.33) where * denotes the star product with respect to the barred variables. Derivatives ∂ ω and ∂ i act as followed by all the auxiliary variables set to zero, i.e., To make contact with the p, t notation of [14] and [15] note that where q α is a z-independent spinorial shift parameter while β ∈ (−∞, 1) is a free parameter. For simplicity we confine ourselves to the holomorphic sector of undotted spinors which is of most interest in this paper. (Antiholomorphic sector of dotted spinors is analysed analogously.) This operator satisfies the following resolution of identity is the projector on d z -cohomology.
We say that function f (z, y) of the form belongs to the space H + if there exists such real ε > 0, that Note that the definition of space H + is relaxed compared to that of space H +0 of [16] because it does not require any specific behaviour of the φ at T → 1. Nevertheless in our calculations sometimes it is convenient to use specific form degree relations of [16] that describe star products for the forms of specific degrees p , p ′ , belonged to spaces H 0+ p and H +0 p ′ . There are two main options that appear in the computations below to satisfy (3.5): with some δ 1,2 > 0. (Note that according to [16] the poles in T in (3.6) are fictitious being cancelled by the T -dependence of z-and θ-dependent terms.) Space H + can be represented as the direct sum where H + p are spanned by the degree-p functions in θ with kernels that satisfy (3.5). Equations of motion (2.1), (2.2) resulting from nonlinear system (2.7)-(2.11) have r.h.s.'s independent of Z A and θ A since they belong to the sector of zero-forms in θ and are d Zclosed as a consequence of equations (2.7)-(2.11) resolved at the previous stages. On the other hand, various terms contributing to the r.h.s.'s of equations (2.1), (2.2) as a result of solution of equations (2.7)-(2.11) are of the form (3.4). In particular, each of these terms is usually Z-dependent. While r.h.s.'s of (2.1), (2.2) are Z-independent as a consequence of equations (2.7)-(2.11), the fact that the sum of all of them is Z-independent is not obvious, demanding an appropriate partial integrations over homotopy parameters that appear at various stages of the order-by-order analysis of nonlinear HS equations. After all, functions (3.4) can be Z-independent only if they have a distributional measure supported at T = 0, i.e., after appropriate partial integrations the measure contains a factor of δ(T ). Such a measure has dimension −1 in T . If a function contains an additional factor of T ε , it cannot contribute to the Z-independent answer. This just means that functions of the class H + 0 cannot contribute to the Z-independent equations (2.1), (2.2). This is the content of Z-dominance Lemma of [1]: any terms in φ(w, u, θ, T ) dominated by a positive power of T do not contribute to the dynamical equations (2.1), (2.2). Application of this fact to locality is straightforward once it is shown that all terms containing infinite towers of higher derivatives in the vertices of interest belong to H + 0 and, therefore, do not contribute to HS equations (2.2). This is what is shown in this paper.
A related fact is that the space H + exhibits special properties under the action of the limiting shifted homotopy △ q,β at β → −∞ shown in [15] to lead to local HS interactions. Namely, (i) being applied to the function from H + 1 , it gives function from H + 0 [16], and (ii), the limiting projector h q,β (3.3) acts trivially on functions from H + 0 , i.e., These properties allow us to discard all terms that belong to H + , in the analysis of the ωC 3 vertex in equation (2.2). Indeed, consider equation where Solving equation (3.10) with the aid of the △ q,β homotopy in the β → −∞ limit we obtain where sign ≈ implies equality up to terms from H + .

General structure of equations
Dynamical equations up to the third order in the zero-forms C can be schematically put into the form Here W η 1 , W ηη 2 and B η 2 , B ηη 3 are master fields of the corresponding orders from expansions (2.22), (2.23) which are to be obtained from the generating system via solving equation of the type (2.24). Each term on the r.h.s. of this equation depends both on Y and on Z. These vertices can be decomposed into two parts where unlike the whole Υ ηη (ω, C, C, C) its two contributions on the right of (4.3) can be zdependent.
In this paper we compute the Υ ηη (ω, C, C, C) part of the vertices. This part turns out to be free from contractions between C-fields because such terms belong to H + 0 . Consistency of equations (2.7)-(2.11) guarantees that Υ ηη (ω, C, C, C) is Z-independent and, according to Z-dominance Lemma it can be realized only as δ(T ) in the kernel. Hence Z-independent expression for Υ ηη (ω, C, C, C) must be free from contractions which implies spin-locality of the resulting HS equations.
In this section we present final expression for Υ ηη (ω, C, C, C) obtained from the generating system (2.7)-(2.11) using the perturbation scheme up to the third order in C-field. Details of their derivation are presented in Appendices B and C. The vertices in (4.4) are composed from the following terms The expression for B η loc 2 has the form [13] B η loc where we use a short-hand notation [14] and local cohomology (i.e., Z-independent) shift δB η The expressions W η 1 ωC and W η 1 Cω were obtained in [14], having the form Note that the terms with d x contribute to the third order via the second-order contribution (4.14) Here Υ η ωCC , Υ η CωC and Υ η CCω are vertices obtained in [14] and δΥ η ωCC , δΥ η CωC , δΥ η CCω result from the local field redefinition of B η 2 (4.11) giving

The fields
The expressions for B ηη 3 and W ηη 2 derived in Appendices B and С are From now on we skip antiholomorphic (barred) variables for brevity. More precisely, CCC on the r.h.s. of (4.18) is to be understood as C(y 1 ,ȳ) * C(y 2 ,ȳ) * C(y 3 ,ȳ) y i =0 , ωCC on the r.h.s. of (4.19) as ω(y ω ,ȳ) * C(y 1 ,ȳ) * C(y 2 ,ȳ) yω=y i =0 etc. Expressions (4.18)-(4.21) are spin-local because the exponential factors in all of them are free from terms ∂ iα ∂ α j describing contractions between higher components of the zero-forms C(Y ) bringing higher-derivative vertices for fields of particular spins. So are the terms induced by these expressions in vertices (4.5)-(4.8). Indeed, differentiating B ηη 3 one should use only firstorder part from r.h.s. of (4.14) which does not bring contractions between C-fields, similarly star product with ω does not bring contractions due to (A.3), (A.4). On the other hand, though star product of W ηη 2 with C does bring contractions between the fields C, all of them result from the z-dependent terms in the exponentials (4.19)-(4.21) that carry at least one power of T . Such terms contain an additional factor of T in front of the contraction terms ∂ iα ∂ α j thus belonging to H + . Hence all the contributions to the vertex (4.2) induced from B ηη 3 and W ηη 2 are spin-local modulo terms in H + .

B 2 driven terms
The terms resulting from d x differentiation of B 2 and its multiplication with W 1 by virtue of (4.12), (4.13), (4.14) and (A. 2) give

W 2 driven terms
Terms resulting from star product with W ηη 2 are (4.41) In the end of this section let us stress again that all terms on the r.h.s. of vertex (4.2) are free from C-field contractions ∂ iα ∂ α j in the exponentials, hence being spin-local. This is the central result of this paper.

Conclusion
In this paper we have analyzed the ωC 3 vertices in the equation for the zero-form C (2.2) in the holomorphic η 2 sector, showing that these vertices are spin-local in the terminology of [16]. In particular, they contain the holomorphic part of the φ 4 vertex in the Lagrangian nomenclature for a spin-zero scalar field φ. This is another step in the analysis of locality of HS gauge theory performed in [14,15]. To complete the analysis of spin-locality of the HS gauge theory at quartic order it remains to extend these results to the mixed ηη sector. This problem differs in some respects from the (anti)holomorphic one and will be analyzed elsewhere.
On the other hand, there are remaining problems even in the holomorphic sector left unsolved. The most important one is to find explicit Z-independent local form of the holomorphic vertex ωC 3 . The naive attempt to set Z = 0 in the vertex obtained in this paper does not necessarily lead to correct result since the omitted terms in H + 0 are needed for consistency of the equations and may contribute to the sector of equations. Indeed, setting Z = 0 corresponds to the application of the conventional homotopy projector which does not eliminate the part of the vertex in H + . Let us stress again in this regard that the elaborated technique based on dropping off terms from H + 0 turns out to be highly efficient for checking out spin-locality. To obtain explicit form of these vertices there are two alternative ways of the analysis.
One is to eliminate the Z-dependence from the vertex by direct partial integration. This can be technically involved and not at all obvious mainly due to the need of using Schouten identities.
Another is to apply the limiting shifted homotopy with appropriately chosen shift. Since the choice of homotopy shift and hence cohomology projector (3.9) affects field redefinitions that can themselves be non-local the art is to find a shift that does not spoil locality of a spin-local vertex by Z-dominance Lemma. This is an interesting problem for the future.
To summarize, the results of this paper indicate that equations of motion of HS gauge theory have a tendency of being spin-local. At this stage it is crucially important to see whether this property extends to the mixed ηηωC 3 sector of equation (2.2) which is the most urgent problem on the agenda.

Appendix A. Useful formulas
Useful multiplication formula for the star product of functions of the form is [26] f 1 * f 2 (z, y) For instance, if one function is z-independent the following formulas are handy in star-product computation Second-order zero-form vertices are [14] Υ η ωCC = − Computation of B ηη 3 goes as follows. Equation for B ηη 3 from (2.11) is An important observation of Section 6.2 of [15] based on the technique of re-ordering operators O β f (z, y) was that if S ηη 2 is computed using B η loc 2 (4.9), then the contribution to the vertices Υ ηη (ω, ω, C, C) from S ηη 2 vanishes at β → −∞. Proceeding analogously one can see that contribution to the vertices Υ ηη (ω, C, C, C) from such S ηη 2 also vanishes at β → −∞. Hence to find the part of B ηη 3 that contributes to Υ ηη one has to solve the equation S η 1 is given by (2.32). Replacing the integration over simplex in (4.9) by the integration over unit square by changing the variables as follows and then performing partial integration with respect to τ 1 using star-product formula (A.2) and dropping the terms from H + 1 we obtain (B.5) Recall that we use notation with hiddenȳ variables: Since only small values of contribute to Υ ηη one needs to consider two triangle regions of the init square in (τ 1 , t) coordinates. Only the lower triangle with small t and τ 1 contributes because the upper-one gives T 3 in the pre-exponential thus belonging to H + 1 . The following change of variables is handy in the further analysis Adding the terms from H + , which do not affect the HS field equations, one can reach further simplifications. For instance, one can add (Recall that sign ≈ means that equality is up to terms from H + .) In (B.5) it is convenient to introduce new variables They form a simplex since ρ 1 + ρ 2 + ρ 3 = 1. The inverse formulas are In these new variables, the commutator takes the form Note that z-dependence is the same in the both terms. Introducing an additional integration parameter ξ, the y-dependence can be uniformized as follows where the following notations are used Evaluating the derivative with respect to ξ taking into account that along with the Schouten identity the expression for the commutator can be rewritten in the form Since the second (boundary) term belongs to H + 1 and thus contributes to Υ ηη + the part of B ηη 3 that contributes to Υ ηη can be chosen in the form (4.18).
Appendix C. W ηη 2 A particular solution for W ηη 2 was found in [15] where it was used in the computation of vertices Υ ηη (ω, ω, C, C). However, this solution turns out to be technically inconvenient for the analysis modulo H + subspace. In this section we apply the approach proposed in the previous section that allows us to single out the H + 0 part from W ηη 2 . Vertices Υ ηη (ω, ω, C, C) computed with W ηη 2 given by (4.19)-(4.21) may differ from those of [15] by a local field redefinition. To compute W ηη 2 consider the equation As mentioned in Appendix B contribution to the vertices Υ ηη (ω, C, C, C) from S ηη 2 vanishes at β → −∞. The remaining equation to be solved is Here S η 1 is given by (2.32) while W η 1 consists of two parts (4.12) and (4.13). To calculate d x S η 1 one needs second-order zero-form vertices Υ η (ω, C, C) obtained in [14] with additional shifts δΥ η (ω, C, C) generated by the local shift δB η 2 (4.11). These are given by (4.15)-(4.17).
Computing S η 1 * W η 1 Cω and dropping terms from H + 1 one obtains discarding barred variables as in (B.6) Taking into account that only small values of τ 1 • t contribute to Υ ηη one can change integration variables as in (B.9) to obtain In the simplex variables (B.10), (B.11) this expression can be rewritten as To compute d x S η 1 one has to use vertex (4.16) As in Appendix B, y-dependence in (C.3) can be uniformized with the help of the new integration parameter ξ. Using new notation for brevity one has from (C.3) taking into account (C.6) and (C.7) Evaluating the derivative over ξ and taking into account that along with the Schouten identity the pre-exponential part can be written in the form and thus r.h.s. of (C.3) can be put into the form After integrating by parts with respect to T in the first term, the resulting boundary term belongs to H + (cf. the second case of (3.6)) and hence can be discarded. This brings equation for W ηη 2 CCω to the form This allows us to choose the part of W ηη 2 CCω , that contributes to Υ ηη , in the form CCω. (C. 16) Finally, one can change the integration variables to rewrite W ηη 2 CCω in the form of the integral over a four-dimensional simplex according to (C. 17) In these variables W ηη 2 CCω acquires the form (4.21).

C.2 W ηη 2 CωC
Equation for this part of the connection is Star product S η 1 * W η 1 ωC can be computed by (A.2). Discarding terms in Υ ηη + and omitting the barred variables, S η 1 * W η 1 ωC takes the form Since only small values of T contribute to Υ ηη we consider only lower triangle of the init square in (τ 1 , t) and perform the same change of variables as in (B.9). Using the simplex variables (B.10), (B.11) the result can be re-written as Analogously, for star product W η 1 Cω * S η The d x S η 1 part computed using vertex (4.17) is It is natural to group the r.h.s. of (C.18) in the following way Introducing new notations for brevity the r.h.s. of (C.18) can be written as an integral of a total derivative analogously to (C.12) by virtue of Schouten identity one has Therefore the Z 1 -dependent part from the r.h.s. of (C.25) can be rewritten in the form Integrating by parts in the last term, the resulting boundary term contributes to Υ ηη + . Hence Analogously, since ∂ ω α ∂Z 2 ∂z α , θ β z β ∂Z 2 ∂ξ = θ β z β ρ 3 T z + i∂ 1 + i∂ 2 α ∂ ωα − i ρ 1 (z γ ∂ ωγ ) T (ρ 1 + ρ 2 ) d z Z 2 + i ρ 1 θ β ∂ ωβ ρ 1 + ρ 2 ∂Z 2 ∂T (C.30) and, therefore, the Z 2 -dependent part of the r.h.s. of (C.25) yields Modulo terms from Υ ηη + this equals to As a result, the r.h.s. of (C.25) acquires the form ρ i ρ 3 T z+i∂ 1 +i∂ 2 α ∂ ωα e Z 2 − e Z 1 CωC.