Manifest form of the spin-local higher-spin vertex ΥωCCCηη\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varUpsilon ^{\eta \eta }_{\omega CCC}$$\end{document}

Vasiliev generating system of higher-spin equations allowing to reconstruct nonlinear vertices of field equations for higher-spin gauge fields contains a free complex parameter η\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\eta $$\end{document}. Solving the generating system order by order one obtains physical vertices proportional to various powers of η\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\eta $$\end{document} and η¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\bar{\eta }}$$\end{document}. Recently η2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\eta ^2$$\end{document} and η¯2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\bar{\eta }}^2$$\end{document} vertices in the zero-form sector were presented in Didenko et al. (JHEP 2012:184, 2020) in the Z-dominated form implying their spin-locality by virtue of Z-dominance Lemma of Gelfond and Vasiliev (Phys. Lett. B 786:180, 2018). However the vertex of Didenko et al. (2020) had the form of a sum of spin-local terms dependent on the auxiliary spinor variable Z in the theory modulo so-called Z-dominated terms, providing a sort of existence theorem rather than explicit form of the vertex. The aim of this paper is to elaborate an approach allowing to systematically account for the effect of Z-dominated terms on the final Z-independent form of the vertex needed for any practical analysis. Namely, in this paper we obtain explicit Z-independent spin-local form for the vertex ΥωCCCηη\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varUpsilon ^{\eta \eta }_{\omega CCC}$$\end{document} for its ωCCC\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega CCC$$\end{document}-ordered part where ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega $$\end{document} and C denote gauge one-form and field strength zero-form higher-spin fields valued in an arbitrary associative algebra in which case the order of product factors in the vertex matters. The developed formalism is based on the Generalized Triangle identity derived in the paper and is applicable to all other orderings of the fields in the vertex.


Introduction
Higher-spin (HS) gauge theory describes interacting systems of massless fields of all spins (for reviews see e.g. [1,2]). Effects of HS gauge theories are anticipated to play a role at ultra high energies of Planck scale [3]. Theories of this class play a role in various contexts from holography [4] to cosmology [5]. HS theory differs from usual local field theories because it contains infinite tower of gauge fields of all spins and the number of space-time derivatives increases with the spins of fields in the vertex [6][7][8][9]. However one may ask for spin-locality [3,[10][11][12] which implies space-time locality in the lowest orders of perturbation theory [11]. Even though details of the precise relation between spin-locality and space-time locality in higher orders of perturbation theory have not been yet elaborated, from the form of equations it is clear that spin-locality constraint provides one of the best tools to minimize the space-time non-locality. Moreover demanding spin-locality one actually fixes functional space for possible field redefinitions that is highly important for the predictability of the theory.
A useful way of description of HS dynamics is provided by the generating Vasiliev system of HS equations [13]. The latter contains a free complex parameter η. Solving the generating system order by order one obtains vertices proportional to various powers of η andη. In the recent paper [14], η 2 and η 2 vertices were obtained in the sector of equations for zeroform fields, containing, in particular, a part of the φ 4 vertex for the scalar field φ in the theory. Though being seemingly Z -dependent, in [14] these vertices were written in the Zdominated form which implies their spin-locality by virtue of Z -dominance Lemma of [15]. In this paper we obtain explicit Z -independent spin-local form for the vertex Υ ηη ωCCC starting from the Z -dominated expression of [14]. The label ωCCC refers to the ωCCC-ordered part of the vertex where ω and C denote gauge one-form and field strength zero-form HS fields valued in arbitrary associative algebra in which case the order of the product factors in ωCCC matters.
There are several ways to study the issue of (non)locality in HS gauge theory. One is reconstruction the vertices from the boundary by the holographic prescription based on the Klebanov-Polyakov conjecture [4] (see also [16,17]). Alternatively, one can analyze vertices directly in the bulk starting from the generating equations of [13]. The latter approach developed in [11,12,14,15,18] is free from any holographic duality assumptions but demands careful choice of the homotopy scheme to determine the choice of field variables compatible with spin-locality of the vertices. The issue of (non)locality of HS gauge theories was also considered in [19,20] with somewhat opposite conclusions.
From the holographic point of view the vertex that contains φ 4 was argued to be essentially non-local [21] or at least should have non-locality of very specific form presented in [22]. On the other hand, the holomorphic, i.e., η 2 and antiholomorphicη 2 vertices, where η is a complex parameter in the HS equations, were recently obtained in [14] where they were shown to be spin-local by virtue of Z -dominance lemma of [15]. The computation was done directly in the bulk starting from the non-linear HS system of [13].
In this formalism HS fields are described by oneforms ω(Y ; K |x) and zero-forms C(Y ; K |x) where x are space-time coordinates while Y A = (y α ,ȳα) are auxiliary spinor variables. Both dotted and undotted indices are twocomponent, α,α = 1, 2, while K = (k,k) are outer Klein operators satisfying k * k =k * k = 1, where θ andθ are anticommuting spinors in the theory.
HS equations have remarkable property [23] that they remain consistent with the fields W and B valued in any associative algebra. For instance W and B can belong to the matrix algebra Mat n with any n. Since in that case the components of W and B do not commute, different orderings of the fields should be considered independently. (Mathematically, HS equations with this property correspond to A ∞ strong homotopy algebra introduced by Stasheff in [24][25][26].) For instance, holomorphic (i.e.,η-independent) vertices in the zero-form sector can be represented in the form where the subscripts of the vertices Υ refer to the ordering of the product factors. The vertices obtained in [14] were shown to be spin-local due to the Z -dominance Lemma of [15] that identifies terms that must drop from the r.h.s.'s of HS equations together with the Z -dependence. Recall that spin-locality implies that the vertices are local in terms of spinor variables for any finite subset of fields of different spins [18] (for more detail on the notion of spin-locality see [18]). Analogous vertices in the one-form sector have been shown to be spin-local earlier in [12].
The main achievement of [14] consists of finding such solution of the generating system in the third order in C that all spin-nonlocal terms containing infinite towers of derivatives in y(ȳ) between C-fields in the (anti)holomorphic in η(η) sector do not contribute to η 2 (η 2 ) vertices by virtue of Z -dominance Lemma. Thus [14] gives spin-local expressions for the vertices Υ ηη (ω, C, C, C) which, however, have a form of a sum of a number of Z -dependent terms. To make spin-locality manifest one must remove the seeming Z-dependence from the vertex of [14]. Technically, this can be done with the help of partial integration and the Schouten identity. The aim of this paper is to show how this works in practice.
Since the straightforward derivation presented in this paper is technically involved we confine ourselves to the particular vertex Υ ηη ωCCC (1.4). Complexity of the calculations in this paper expresses complexity of the obtained vertex having no analogues in the literature. Indeed, this is explicitly calculated spin-local vertex of the third order in the equations, corresponding to the vertices of the fourth (and, in part, fifth) order for the fields of all spins. The example described in the paper explains the formalism applicable to all other orderings of the fields in the vertex that are also computable. So, our results are most important from the general point of view highlighting a way for the computation of higher vertices in HS theory that may be important from various perspectives and, in the first place, for the analysis of HS holography. It should be stressed that the results of [14] provided a sort of existence theorem for a spin-local vertex that was difficult to extract without developing specific tools like those developed in this paper. In particular, it is illustrated how the general statements like Z -dominance Lemma work in practical computations. Let us stress that at the moment this is the only available approach allowing to compute explicit form of the spin-local vertices for all spins at higher orders.
The rest of the paper is organized as follows. In Sect. 2, the necessary background on HS equations is presented with brief recollection on the procedure of derivation of vertices from the generating system. Section 3 reviews the notion of the H + space as well as the justification for a computation modulo H + . In Sect. 4, we present step-by-step scheme of computations performed in this paper. Section 5 contains the final manifestly spin-local expression for Υ ηη ωCCC vertex. In Sects. 6-10 technical details of the steps sketched in Sect. 4 are presented. In particular, in Sect. 7 we introduce important Generalised Triangle identity which allows us to uniformize expressions from [14]. Conclusion section contains discussion of the obtained results. Appendices A, B, C and D contain technical detail on the steps listed in the scheme of computation. Some useful formulas are collected in Appendix E.

Perturbation theory
Starting with a particular solution of the form (2.14) Hence, Eq. (2.13) yields (2.15) The Z -independent C-field that appears as the first-order part of B is the same that enters Eqs. (1.2) and (1.3). The perturbative procedure can be continued further leading to the equations of the form where Φ k is either W , S or B field of the k-th order of perturbation theory, identified with the degree of C-field in the corresponding expression, i.e., To obtain dynamical equations (1. Note that this definition does not demand any specific behaviour of φ at T → 1 as was the case for the space H +0 of [18]. In the sequel we use two main types of functions that obey (3.2): with some δ 1,2 > 0. (Note that the second option with δ 2 > 0 can be interpreted as the first one with arbitrary large δ 1 . Here step-function is denoted as ϑ to distinguish it from the anticommuting variables θ .) Space H + can be represented as the direct sum where φ(w, u|θ, T ) ∈ H + p are degreep forms in θ satisfying (3.2).
All terms from H + on the r.h.s. of HS field equations must vanish by Z -dominance Lemma [15]. Following [14] this can be understood as follows. All the expressions from (2.17) have the form (3.1) and the only way to obtain Zindependent non-vanishing expression is to bring the hidden T dependence in φ(T z, y|T θ, T ) to δ(T ). If a function contains an additional factor of T ε or is isolated from T = 0, it cannot contribute to the Z -independent answer which is the content of Z -dominance Lemma [15]. This just means that functions of the class H + 0 cannot contribute to the Z -independent equations (1.3). Application of this fact to locality is straightforward once this 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. This is what was in particular shown in [14].

Notation
As in [14] we use exponential form for all the expressions below where by ωCCC we assume with * denoting star-product with respect toȳ. Derivatives ∂ ω and ∂ j act on auxiliary variables as follows After all the derivatives in y ω and y j are evaluated the latter are set to zero, i.e., In this paper we use the following notation of [14]: The η 2 C 3 vertex in the equations on the zero-forms C resulting from equations of [13] is (3.10) Recall, that, being Z -independent, Υ ηη is a sum of Zdependent terms that makes its Z -independence implicit.
As explained in Introduction, Υ ηη can be decomposed into parts with different orderings of fields ω and C. In this paper we consider (3.11) Since the terms from H + do not contribute to the physical vertex such terms can be discarded. Following [14] equality up to terms from H + referred to as weak equality is denoted as ≈ .
In this paper Z -dependence of Υ ηη ωCCC (Z ; Y ) is eliminated modulo terms in H + by virtue of partial integration and the Schouten identity. As a result, where Υ ηη ωCCC (Y ) is manifestly spin-local and Z -independent. Since H + 0 -terms do not contribute to the vertex by Zdominance Lemma [15] Our goal is to find the manifest form of Υ ηη ωCCC (Y ).

Calculation scheme
The calculation scheme is as follows.
Using partial integration and the Schouten identity we transform Eqs. We observe that the r.h.s.'s of Eqs. (6.1)-(6.4) can be rewritten modulo cohomology and weakly zero terms in a form of integrals dΓ over the same integration domain I where the integrand contains an overall exponential function E (4.8) the integral over I is denoted as By collecting all resulting Z -independent terms we finally obtain the manifest expression for vertex Υ ηη ωCCC , being a sum of expressions (5.2)-(5.12).

Main result Υ ηη ωCCC
Here the final manifestly Z -independent ωCCC contribution to the equations is presented. Vertex with J i given in Eqs. (5.2)-(5.12). Note that the integration regions may differ for different terms J j in the vertex, depending on their genesis. Firstly we note that B Recall that E and dΓ are defined in ×EωCCC, (5.12) Let us emphasize, that neither exponential function E (4.4) nor the exponentials on the r.h.s.'s of Eqs. (5.10)-(5.12) contain ∂ i α ∂ k α terms. Hence, as anticipated, all J j are spin-local.
The case of (5.4)-(5.8) is a bit more tricky. By partial integration one obtains from (5.4)-(5.6) Using that, due to the factor of δ(1 − ρ i ), for positive ρ i it holds , (5.15) one can make sure that each of the expressions with poles in the pre-exponential in Eqs. (5.7), (5.8) and (5.13) can be represented in the form of a sum of integrals with integrable pre-exponentials. For instance, the potentially dangerous factor in (5.8), by virtue of (5.14) and (5.15) satisfies . (5.16) Each of the terms on the r.h.s. of Eq. (5.16) is integrable, because integration is over a three-dimensional compact area ρ i = 1 in the positive quadrant. For instance consider the first term. Swopping ρ 4 ↔ ρ 2 one has which is integrable. Analogously other seemingly dangerous factors can be shown to be harmless as well.

To z-linear pre-exponentials
Step II of the calculation scheme of Sect. 4 is to transform r.h.s.'s of Eqs. (3.13)-(3.17) to Z -independent terms plus terms with linear in z pre-exponentials (modulo H + ).
To this end, from (A.10) one straightforwardly obtains that where J 1 is the cohomology term (5.2). Analogously, with J 2 (5.3). Using the Schouten identity and partial integration one obtains from Eqs. ×ωCCC. (6.5)

Generalised triangle identity
Here a useful identity playing the key role in our computations is introduced. For any F(x, y) consider with arbitrary τ, ξ -independent P and a i . Let G(x, y) be a solution to differential equation Note that there is a factor of (a 1 − a 3 ) α (a 3 − a 2 ) α equal to the area of triangle spanned by the vectors a 1 , a 2 , a 3 on the r.h.s. of (7.3). This identity is closely related to identity (3.24) of [11], that, in turn, expresses triangle identity of [27]. Hence, (7.3) will be referred to as Generalised Triangle identity or GT identity.
Note that, for appropriate G partial integration on the r.h.s. of (7.3) in τ gives z-independent (cohomology) term plus H + -term. Namely, The second term on the r.h.s. belongs to H + if G is of the form (3.1) satisfying (3.2).
To prove GT identity let us perform partial integration on the r.h.s. of (7.1) with respect to ξ i . This yields The Schouten identity yields (7.6) One can observe that whence it follows (7.3). A useful particular case of GT identity is that with (7.8)
Note that different terms of F j will be considered separately in what is follows. For the future convenience the underbraced terms are re-numerated, being denoted as F j,k , where j refers to F j while k refers to the respective underbraced term in the expression for F j . For instance, EωCCC, ×ωCCC.

(8.5)
Note that Let us emphasise that, by virtue (E.1), each F j is of the form (4.1) as expected.

Final step of calculation
Here this is shown that the sum of the r.h.s.'s of Eqs. (9.2)-(9.4) gives a Z -independent cohomology term up to terms in H + . More in detail, the expression G 1 + G 2 + G 3 of the form (9.5) consists of two types of terms with the pre-exponential of degree four and six in z, y, t, p 1 , p 2 , p 3 , respectively. That with degree-four pre-exponential separately equals a Z -independent cohomology term up to terms in H + . This is considered in Sect. 10.1. The term with degree-six preexponential is considered in Sect. 10.2. As a result of these calculations J 6 (5.7) and J 7 (5.8) are obtained.

Degree-four pre-exponential
Consider the sum of expressions with z-dependent degreefour pre-exponential from Eqs. (9.2)-(9.4), denoting it as S 4 . Partial integration yields where the cohomology term J 6 is given in (5.7). It is not hard to see that the integrand of the remaining term is zero by virtue of the Schouten identity.

Degree-six pre-exponential
Terms of this type either appear in (9.2), (9.3) via differentiation in ρ j or in (9.4) via differentiation in ξ j . Denoting a sum of these terms as S 6 we obtain ×E y + P α (y +t) α ωCCC (10.2) Recall that the integral measure dΓ (4.9) contains the factor of δ(1 − 3 1 ξ i ). Hence taking into account the factor of δ(ξ 3 ) on the r.h.s. of (10.2) the dependence on ξ 2 , ξ 3 can be eliminated by the substitution ξ 2 → 1−ξ 1 , ξ 3 → 0. Then we consider separately the terms that contain and do not contain ξ 1 in the pre-exponentials. As shown in Appendix D, those with ξ 1 -proportional pre-exponentials give J 7 (5.8) up to H + , while those with ξ 1 -independent pre-exponentials give zero up to H + .

Conclusion
In this paper starting from Z -dominated expression obtained in [14] the manifestly spin-local holomorphic vertex Υ ηη ωCCC in the Eq. (1.3) is obtained for the ωCCC ordering. Besides evaluation the expression for the vertex, our analysis illustrates how Z -dominance implies spin-locality.
One of the main technical difficulties towards Z -independent expression was uniformization, that is bringing the exponential factors to the same form, for all contributions (3.13)-(3.17) with the least amount of new integration parameters possible. Practically, some part of the uniformization procedure heavily used the Generalized Triangle identity of Sect. 7 playing important role in our analysis.
Let us stress that spin-locality of the vertices obtained in [14] follows from Z -dominance Lemma. However the evaluation the explicit spin-local vertex Υ η 2 ωCCC achieved in this paper is technically involved. To derive explicit form of other spin-local vertices in this and higher orders a more elegant approach to this problem is highly desirable.
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Appendix A: B ηη 3 B ηη 3 modulo H + terms from [14] is given by where dΓ is defined in (4.9), Performing partial integration with respect to T twice we obtain Noticing that and performing partial integration with respect to ρ 1 and ρ 3 we obtain Observing that and using the Schouten identity after partial integration with respect to ξ 1 we obtain The δ(T )-proportional term gives rise to J 1 (5.2) and J 2 (5.3).
Using that E (4.2) does not depend on ξ 3 , swapping ξ 3 ↔ ξ 1 in the remaining part of (6.5), then swapping ξ 3 ↔ ξ 2 in the remaining part of (6.4), one then can apply GT identity (7.8) to the sum of the two obtained terms . As a result, Eqs. (6.4), (6.5) yield where the terms in the second row of formula (B.1) result from applying GT -identity. Rewriting the underlined part as the result of differentiation with respect to T and performing partial integration one obtains Eq. (8.4) plus the cohomology term J 8 (5.9).
Uniformization of the sum of δ(ρ 1 )-proportional terms on the r.h.s.'s of (6.2) and (6.1) is done with the help of GT identity (7.8) as follows. Denoting one can see that partial integration in T yields The sum of (B.3) and (B.4) gives with J 9 (5.10) and J 10 (5.11). By virtue of GT identity (7.8) the first term weakly equals J 11 (5.12). Finally, Eq. (B.5) yields (3.13). This is convenient to change integration variables, moving from the integration over simplex to integration over square. As a result Partial integration with respect to T yields (B.9) By virtue of evident formulas (B.10) After partial integrations in τ 1 ,τ 2 and σ 1 one obtains After a simple change of integration variables the underlined term on the r.h.s. of Eq. (B.12) cancels the r.h.s. of Eq. (B.6).
Performing integration with respect to τ 2 in the remaining part of (B.12), after the following change of the integration variables
since, using the Schouten identity, one can see that the preexponential of the integrand on the r.h.s. of (D.3) equals zero.