Differential contracting homotopy in higher-spin theory

A new efficient approach to the analysis of nonlinear higher-spin equations, that treats democratically auxiliary spinor variables ZA and integration homotopy parameters in the non-linear vertices of the higher-spin theory, is developed. Being most general, the proposed approach is the same time far simpler than those available so far. In particular, it is free from the necessity to use the Schouten identity. Remarkably, the problem of reconstruction of higher-spin vertices is mapped to certain polyhedra cohomology in terms of homotopy parameters themselves. The new scheme provides a powerful tool for the study of higher-order corrections in higher-spin theory and, in particular, its spin-locality. It is illustrated by the analysis of the lower order vertices, reproducing not only the results obtained previously by the shifted homotopy approach but also projectively-compact vertices with the minimal number of derivatives, that were so far unreachable within that scheme.


Introduction 1.General background
Higher-spin (HS) gauge theory is a nonlinear theory of HS massless fields of spins s > 2 originally identified as gauge fields by Fronsdal [1] at the free field level.At the nonlinear level, HS gauge theory admits a natural formulation in the Anti-de Sitter space [2].The HS gauge theory is organized by the symmetry principle resulting from gauging the infinitedimensional HS algebra found in [3].
That HS symmetry is infinite dimensional has several important consequences.First of all this implies that HS gauge theory describes infinite towers of fields of unlimited spins as was first noticed in [4]. 1 By construction, HS symmetries are extensions of the conventional space-time symmetries (usually in (A)dS d ) with the generators t i .The full set of generators T A contains space-time generators t i and HS generators T HS A T A = (t i , T HS A ) . (1.1) The generators t i form a space-time algebra s, The commutation relations between space-time and HS generators have the structure Since in Cartan formulation of gravity, the gauge fields associated with t i describe a spin two vielbein and Lorentz connection, (1.3) implies that HS symmetry transforms the graviton to a HS field.Hence, the metric tensor is not an invariant concept in presence of HS symmetries.
Since the metric tensor is a tool for measuring a distance between space-time points, this elementary observation suggests that the issue of locality may be nontrivial in the HS gauge theory.Practically, this is manifested as follows.Being consistently formulated in AdS background [2], HS gauge theories contain higher derivatives of degrees increasing with spins in interactions [9,4,2], and involve infinite towers of fields of unlimited spins [4,3].The fact that HS cubic vertices contain higher derivatives with the number of derivatives increasing with spin was originally found in the light-front formalism in the seminal paper by Anders Bengtsson, Ingemar Bengtsson, and Lars Brink [9].It should be stressed that any vertex for a finite subset of spins is local, containing a finite number of derivatives depending on the spins in the vertex, which property is called spin-locality [10].
Being one of the founders of String Theory, Lars Brink influenced a lot HS theory as well.Having a lucky opportunity to discuss fundamental physics with Lars in person, I can confirm that both of us shared an opinion that String Theory is closely related to HS theory and, most likely, is a spontaneously broken realization of the latter.Though the realization of this program (see, e.g., [11]- [13]) is still incomplete, a better understanding of the underlying symmetries was reached recently in [14], that gives a hope to find a solution to this fundamental problem relatively soon.It is worth noted that the construction of [14] is based on the algebra of observables of the N-body Calogero model, introduced in collaboration with Lars Brink, Hans Hansson and Semyon Konstein in [15,16], and its further multiparticle extension [14] to the Coxeter group B 2 , while the simplest HS algebra [5,7] corresponds to the two-body Calogero case of A 1 = Z 2 .
Another fundamental problem in HS theory also closely related to the studies of Lars and collaborators is to understand the degree of non-locality of HS theory still debatable in the literature within various formalisms.One of the most popular and seemingly simplest ones is based on the holographic correspondence [17]- [19].Klebanov-Polyakov conjecture suggested that HS theory is holographically dual to a 3d vector boundary sigma model [20].(See also [21]; further generalizations were worked out to supersymmetric [22,23] and Chern-Simons extensions [24,25].)Though Klebanov-Polyakov conjecture obeys kinematic constraints of the linearized holography expressed by the Flato-Fronsdal theorem [26] implying the relation between currents (tensor products) built from free 3d conformal fields and free massless fields in AdS 4 , attempts to reconstruct HS interactions in the bulk led the authors of [27]- [30] to the conclusion that HS gauge theory must be essentially non-local beyond the leading order.
Alternatively, HS gauge theory can be studied directly in the bulk.Apart from the standard Noether procedure [4] and its further extension by the BV-BRST formalism (see e.g.[31]- [41]) typically efficient at the lower orders, there are two important tools to simplify the problem.One is the light-cone formalism initiated by Lars Brink and collaborators in [9] and further developed by A. Bengtsson [42], Metsaev [43]- [47], and others (see e.g.[48]).Since HS holography is weak-weak [20,49] it is important to study HS gauge theory both holographically and in the bulk independently.
An efficient covariant approach of [50] makes it possible to reconstruct on-shell HS vertices order by order in the so-called unfolded formalism.This way, using shifted homotopy approach, in [10], [51]- [55] some higher-order vertices in HS theory were reconstructed that all have been shown to be spin-local.(For the closely related though somewhat different approaches for the reconstruction of the so-called holomorphic HS vertices see also [56,57]. 2 ) In [58] a sufficient condition was formulated that guarantees equivalence of space-time and spinor (twistor) spin-locality, requiring the vertices to be not only spin-local in the spinor space but also "projectively-compact".It was also noted in [58] that the vertices found earlier in [59,60] belong to the projectively-compact class while those found in [54] do not.The difference between the two types of vertices is spin-local: vertices of [59,60] contain the minimal possible number of space-time derivatives while those of [52] contain local improvement terms.Naively, that makes no difference.That would indeed be true in the models with a finite number of fields but may not be true in the HS theories due to infinite summations over spins [58].So far it was not clear how to reproduce the vertices of [59,60] within the shifted homotopy formalism of [54] without a by hand local field redefinition.
The formulation of [50] is complete in the sense that it represents any solution to the problem including all possible field redefinitions.This has both advantages and disadvantages.The disadvantage is that equations of [50] do not directly lead to a local or minimally non-local form of the equations whatever it is.In other words, to proceed one has to work out appropriate additional conditions that single out a proper form of HS field equations.This is somewhat analogous to the Schroedinger equation in QM: most of its solutions have no relation to physics.One has to, first, impose an additional condition that the wave-function must belong to L 2 and then find appropriate solutions.Analogously, in the analysis of HS gauge theory based on the equations of [50] a list of conditions has been identified in [10] that reduce the degree of non-locality of the resulting vertices.As a result, it was shown that all lowest-order vertices are properly reproduced by the equations of [50].These results were then extended [54] to a class of non-linear vertices of the types discussed in [28].
Within the shifted homotopy technique, different versions of the HS theory associated with different choices of field variables result from different shifted homotopy procedures that depend on the homotopy parameters τ and so-called shift parameters ρ, β, σ, . ... Being efficient in the lowest orders, the original shifted homotopy approach is less efficient at higher orders.In particular, the higher-order locality condition seemingly suggests [61] that higher-order shift homotopy parameters must depend on those that appeared at the previous orders.Naively, this is impossible since the homotopy parameters that appeared at the lower orders have been already integrated.Nevertheless, as shown in this paper there exists an extension of the formalism in which the homotopy and shift parameters can be treated as local coordinates in a larger space M, that extends space-time M, while the HS vertices are understood as integrals of the differential forms in M. (Note that dim M increases with the order of the vertex and it is usually convenient to realize M as a polyhedron rather than a smooth manifold.)The proposed formalism is invariant under diffeomorphisms in M that gives a simple explanation to some of the obscure relations in the previous approach.Most remarkably, however, it trivializes the role of the Schouten identities in the theory that was a source of headache in the standard formalism like that of [55].Let us mention that the idea of the approach of this paper has much in common with the approach of Gelfond and Korybut [55] where HS vertices were reconstructed without direct use of the homotopy technique.Numerous nice properties of the proposed formalism result from a specific Ansatz for HS fields that simplifies the analysis drastically.
The aim of this paper is to explain main elements of the suggested formalism applying it to the lower-order vertices.Even at this level the developed formalism allowed us to obtain a new result deriving directly (i.e., without by hand field redefinitions as in [59]) the HS current vertex in the projectively-compact spin-local form of [58] with the minimal number of space-time derivatives.

Higher-spin equations
The nonlinear equations of [50] Similarly for k.HS star product is defined as follows Indices are raised and lowered with the aid of the symplectic form ǫ AB = −ǫ BA as follows: X A = ǫ AB X B and X A = X B ǫ BA .(Nonzero elements of ǫ AB are ǫ αβ and ǫ α β .)There are two types of anticommuting differentials in (1.4)-(1.8),namely space-time dx n and spinor θ A = (θ α , θ α).Master fields have different gradings with respect to these differentials, namely W = W n dx n and S = S α θ α + S α θ α are one-forms in dx n and θ A , respectively, while B is a zero-form.All products are wedge, with the wedge symbol implicit.θ-differentials have the following commutation rules turn out to be central with respect to the star product, i.e., Following [50], to analyse equations (1.4)-(1.8)perturbatively one starts with the vacuum solution Plugging it into (1.4)-(1.8)and using that the graded star-product commutator yields we find that W 0 should be Z-independent, W 0 = ω(Y ; K|x), and satisfy (1.4).Similarly, at the next order one gets B 1 = C(Y ; K|x) from [S 0 , B 1 ] * = 0 and C satisfies (1.6).This way we find the first terms on the r.h.s. of unfolded equations of the form originally proposed in [62] ) As in [62], our perturbative expansion is in powers of the zero-forms C with the one-forms ω treated as being of order zero.

Conventional contracting homotopy analysis
In the perturbative analysis of the nonlinear HS field equations one starts with the Sdependent equations (1.5), (1.7), (1.8) on the auxiliary spinor coordinates Z A the dependence on which eventually determines the form of the space-time HS equations (1.16), (1.17) resulting from (1.4), (1.6).As a consequence of (1.15) the S-dependent equations have the form Consistency of the HS equations implies that g(Z; Y ) is d Z -closed at every perturbation order, As a result, equation (1.18) can be solved in the form where h is in d Z -cohomology and conventional contracting homotopy △ 0 is defined by the Poincaré formula Though looking most natural, at ǫ = h = 0 it does not lead to the local (more precisely, spin-local [53]) frame beyond the free field level in which case it was used in the perturbative analysis of HS equations since [50].Generally, the undetermined exact part d Z ǫ and h describe solutions to the homogeneous equation (1.18) with g = 0.The freedom in ǫ is gauge while h(ω, C) with HS fields ω(Y ; K|x) and C(Y ; K|x) induces nonlinear field redefinitions.
Shifted homotopy operator results from the shift Z A → Z A +a A with some Z-independent a A .Shifted contracting homotopy △ a and cohomology projector h a act as follows [51] (1.23) The form of the resulting equations at ǫ = h = 0 depends on the choice of △ a .Transition from one contracting homotopy to another affects both ǫ and h.The problem is to identify a homotopy procedure that leads to the spin-local form of the field equations.
As shown in [51], in the lowest nonlinear order locality is achieved with the aid of an appropriate shifted homotopy operator.This shift is to large extent determined by the general analysis of [10].Further analysis shows [61], that the extension of these results to the higher orders is not straightforward, suggesting the shifts at the higher orders to depend on the homotopy integration parameters like t in (1.22) introduced at lower orders, that looks impossible because the integration in (1.22) has been completed.
The aim of this paper is to work out a novel differential homotopy procedure allowing to make homotopy parameter-dependent shifts postponing the integration to the very last step of the evaluation of corrections to HS equations.Apart from providing a powerful extension of the usual shifted homotopy procedure, the proposed scheme leads to a remarkable new representation for the functions resulting from the application of the differential contracting homotopy to nonlinear HS equations bringing the r.h.s. of equation (1.18) to a uniform form, that reduces its analysis to a certain polyhedra cohomology problem with respect to the homotopy integration parameters.This technique provides a far going generalization of the idea of [55] to represent the r.h.s. of (1.18) in the manifestly exact form by virtue of partial integrations and Schouten identity.The remarkable byproduct of our approach is that the form of the resulting class of functions accounts both of these ingredients automatically.
The proposed scheme will be shown to have a number of remarkable properties leading to a significant improvement of the known results at the first and second orders in the zeroforms C(Y ), that reproduces the projectively-compact spin-local form of the lower-order vertices originally obtained by hand (direct field redefinition) in [59] and so far unavailable within the shifted homotopy approach of [51]- [54].Extension of these results to higher orders will be presented elsewhere.
The rest of the paper is organized as follows.Definition of the differential homotopy is given in Section 2 where also the fundamental Ansatz that results from the application of the differential homotopy approach and captures various shift parameters of [51,52] is presented and the duality between the HS homological problem and that in the homotopy parameter space is established.The star-multiplication formulae analogous to the star exchange formulae of [51] are presented in Section 3. Special properties of the fundamental Ansatz for particular values of the homotopy parameters are discussed in Section 4. The differential homotopy form of the nonlinear system of HS equations is presented in Section 5. Examples of application of the proposed approach to the derivation of lower-order expressions for HS fields and equations are collected in Section 6.General lessons of the lower-order analysis are summarized in Section 7 including the Ansatz for the projectively-compact vertices.Conclusions and perspectives are Section 8.

Differential homotopy 2.1 Total differential
The homotopy parameter t in (1.22) and its further extension t a are interpreted as additional coordinates of some manifold M with the total differential where θ A and dt a are anticommuting differentials.It is assumed that M = M Z × M with some compact homotopy space M and M Z = R 4 .Integration in M will be over M or, more generally (though equivalently), a dim M-dimensional compact cycle in M.
Equations to be solved at every perturbation order are of the form with some d-closed g, dg(Z, t, θ, dt) = 0 .
Homotopy coordinates t a , a = 1, . . .n vary in a compact domain of R n : functions like f and g contain θ-and δ-functions like θ(t)θ(1 − t) confining integration to a compact M. Physical fields and equations in HS theory are supported by the d cohomology which, in the case of compact M relevant to our analysis, is carried by the Z A , θ A -independent integrals over M with d t cohomology represented by the volume integral.(H 0 (d t ) plays no role in our analysis while H p (d t ) = 0 at 0 < p < dim M.) The idea of differential homotopy is based on the removal of the integrals in the equations where g weak does not contribute to the integral because its degree deg g weak as a form in M differs from dim M. Setting 2) are invariant under diffeomorphisms in the total space with local coordinates Z A , t a and differentials θ A , dt a .Moreover, sometimes it is convenient to extend the setup by the coordinates like U A and V A of the non-compact non-commutative space (1.10),

Integration
To avoid a sign ambiguity due to (anti)commutativity of differential forms we recall that every differential expression is in the end accompanied with a set of integrals on the left . . .
We take the convention that The latter formula will also be used other way around inserting a unity in the form of integral of a delta-function of a new variable.
To simplify formulae we will use a shorthand notation . . .

Fundamental Ansatz
Being originally designed to capture a most general class of solutions of the HS equations, the differential homotopy approach turns out to be useful in many more respects as we explain now starting from the simplest case of the lowest order holomorphic deformation linear in η.
As shown in Section 6, direct lower-order computation in the holomorphic sector within the differential homotopy approach yields expressions of the following remarkable form where dΩ 2 := dΩ α dΩ α , (2.12) and with some integration parameters σ i , G l (g) := g 1 (r 1 ) . . .g l (r l )k , (2.17) g i (y) are some functions of y α (e.g., C(y) or ω(y)).Antiholomorphic variables ȳ α, Klein operators K, space-time coordinates x in the arguments of C(Y ; K|x) and the antiholomorphic star product * between the product factors g i are implicit).Analogously to (2.6) in the sequel we will assume that where du α , dv α , dp α i and dr α i are anticommuting differentials treated on equal footing with θ A , dτ , dσ i , dρ and dβ.In particular, We use such a normalization that ) where the dependence on u α , v α and p α i on the l.h.s. is implicit and the terms with du α , dp α i and dv α have been discarded since they do not contribute upon multiplication by the measure factor d 2 ud 2 v l i=1 d 2 p i d 2 r i in (2.20).(Note that though such terms can be treated as weak they are a kind of trivial never contributing to the final result as long as the measure factor d 2 ud 2 v l i=1 d 2 p i d 2 r i is preserved at all stages.)It is convenient to assume that the integration over homotopy parameters like τ , σ i , ρ, β is over R m of an appropriate dimension m while µ(τ, σ, β, ρ) has a compact support M determined by δ and θ functions of some combinations of the homotopy parameters.We shall see that in practice µ(τ, σ, β, ρ) is supported by certain polyhedra.
Note that, by virtue of Taylor expansion 16) can be rewritten in the differential form with r α i integrated out to give at r α j = 0 upon differentiation.In the sequel, this differential form will be used as well.
It should be stressed that M is demanded to be compact to guarantee that the generating functions like (2.11) are entire in p jα , i.e., derivatives with respect to spinor variables which, by virtue of the unfolded HS equations discussed in Section 6, in turn implies that the resulting space-time HS equations are expandable in space-time derivatives.
The differentials dΩ α and therefore dΩ α dΩ α contain various combinations of θ α and dt a = (dτ, dσ i , dρ, dβ).Hence not all of the terms in dΩ 2 contribute to the integrated expression.In particular, dΩ α dΩ α always contains the θ α θ α term.To put it differently, the measure µdΩ 2 may contain weak terms that do not contribute upon integration.A related comment is that µ(τ, σ, β, ρ) in (2.20) may not contain the differentials for all its arguments.We will come back to this issue in Section 5.
The parameters σ i , β and ρ are related to the shift parameters of [51,52].The difference is that in our approach these are integration variables while in [51,52] parameters like σ i and β were taking definite values, say, σ 0i and β 0 .In fact, this difference is not too important since in the new approach one can choose µ(t a ) to contain the factors of dρδ(ρ − ρ 0 ) and/or dβδ(β − β 0 ).Other way around, the results of [51,52] at a given perturbation order can be integrated over the homotopy parameters like β 0 and ρ 0 , that appear at the same order.
The most important feature of the Ansatz (2.11) is that the homotopy parameters τ , σ i , β and ρ contribute either via the measure µ(τ, σ, β, ρ, u, v, p, r) or via Ω (2.13).As we explain now, this reduces the cohomological HS problem to that on the homotopy space M.

Homology map
Formula (2.11) has the remarkable property that its Ω-dependent part is d-closed, This is because the one-forms dΩ α are anticommuting and hence, analogously to θ 3 = 0, (dΩ) 3 = 0 (2.25) since α = 1, 2. (Analogously the terms with dr i and dp i do not contribute because (dr i ) 3 = (dp i ) 3 = 0.) As a result, the differential d, that acts on f µ (2.11), effectively acts only on the measure µ on the r.h.s. of (2.20), where N is the number of integration parameters τ, σ i , β, ρ.This maps the original homological problem in terms of Z variables to that on the space of the measure factors µ(t i ) with no reference to the spinor variables.The great advantage of this formalism is that there is no need to use the Schouten identity in practical computations: the only formula where Schouten identity in the form (2.25) manifests itself is (2.26).(The necessity of accounting the Schouten identity in the formalisms available so far was one of the hardest complications.)By virtue of (2.26), equation (2.2) amounts to where ∼ = denotes the weak equality up to possible weak terms, that do not contribute under the integrals in f dµ f and g µg because the number of differentials does not match the number of the integration variables.That g in (2.2) is d closed implies that µ g must be weakly d-closed, dµ g ∼ = 0 . (2.29) In most cases this implies that µ g ∼ = dh g (2.30) Being analogous to the approach of [55], where the equations d Z f = g were resolved explicitly in terms of the preexponent factors that however resulted in a rather involved procedure due to the lack of the formula (2.26) and necessity to resolve the mess resulting from the straightforward application of the Schouten identity, the new approach is incomparably simpler as will be illustrated in the examples of Section 6.The same time it is far more general allowing a broader class of differential homotopy procedures.
Since the differentiation of g ′ (2.32)only hits l(τ ′ ), we obtain where with the notation often used in the sequel, D(a) := daδ(a) .

Star multiplication formulae
The so-called star-exchange formulae of [51] for the star multiplication of the result of the action of the contracting homotopy operator with a Z-independent function greatly simplify the perturbative analysis of HS theory.The differential homotopy approach proposed in this paper admits their useful analogue presented in this section.Note that the Ansatz with parameters σ i , ρ and β captures all types of shifted homotopy of [51,52].
For convenience, we confine ourselves to the holomorphic sector with Z and Y variables carrying only undotted indices.The antiholomorphic sector of dotted indices can be treated analogously.
For f µ (2.11) the star products with a Z-independent function ϕ(y) yield where Ω α still has the form (2.13) with p +α now including p ϕ ≡ p 0α , and with Ω α of the form (2.13) with p k+1 ,α ≡ p ϕ α .Let us explain for instance how formula (3.2) is derived.Star product (1.10)yields ) where ) Note that the sign of the argument of ϕ(−y − t) is changed because the Klein operator k has been moved to the right.Rewriting Ω α in the form and using that since ds α does not contribute to (3.3), we obtain after renaming Finally, setting s α = p l+1α , shifting v α → v α + p l+1α , and renaming Ω ′ α → Ω α we recover (3.2).Formula (3.1) is derived analogously.
(3.13) (It is convenient to assume that the label ϕ of p ϕα is between l and l + 1.) At the condition that, in agreement with the results of [52], ) where the spinor integration variables p α i , r α i , u α and v α are implicit, and Later on we will see that the extension of P (σ l , σ l+1 , σ l+2 ) to an arbitrary string of arguments also plays a role in the analysis.In these terms, P (σ l , σ l+1 , σ l+2 ) is P 2 (σ l , σ l+1 , σ l+2 ).
To summarize, the remarkable property of formulae (2.11), (2.13) is that they preserve their form under the star product with Z-independent functions.

Special properties
Written in terms of differential forms, formula (2.11) is invariant under diffeomorphisms in the homotopy variables t a → t ′a (t).In most cases such diffeomorphisms affect the form of Ω α (2.13) and, hence, the exponent (2.16) in (2.11).These can be applied of course, but it does not help just affecting the form of the expression as a whole.However, remarkably, the dependence on all coordinates σ i , β, ρ disappears from Ω α at τ = 1 and the dependence on ρ disappears from the exponent in (2.11) since This has two consequences allowing to perform any change of variables in t a at τ = 1 and in ρ at any τ without affecting the exponent (2.16).Let us consider these important cases in more detail.

τ = 1
Let t a denote all homotopy coordinates except for τ .(For simplicity we discard the sector of antiholomorphic variables.)Let the measure factor µ(τ, t) be of the form In that case, the t a -dependent terms drop out from the exponent in (2.11) while the t a , dt a -dependent terms drop out from dΩ α dΩ α with Ω α (2.13), This allows us to make any change of variables t a which only affect the measure µ.Such changes of variables can be done freely at any stage of the computation.

ρ
That the dependence on ρ drops out from the exponent in (2.11) is the manifestation and further extension of the phenomenon found in [51,63] at β = 0 of the independence of vertices bilinear in C on the uniform shifts in the variables of those references which is equivalent to in the variables of this paper.In [51,63] the parameter ρ was a constant while in this paper it is a variable, i.e., a coordinate in M. Though the dependence on ρ drops out from the exponent in (2.11), it contributes to dΩ 2 in a very special way.Indeed, let us set ) Using the differential realization of p jα (2.23), this yields with E(Ω) := exp i w β (y From (4.5) it follows that and, hence, Since any change of variables of the form does not affect the exponent, it can be freely performed only affecting the form of the factor µdΩ α dΩ α to relate seemingly different expressions.This property will play instrumental role in the analysis of Section 6.3.1 of the form of field equations guaranteeing equivalence of the spinor and space-time spin-locality according to the criterion of [58].

β
An interesting feature of the β-dependence of the shifted homotopy procedure is that it does not affect the lower-order analysis as was explained in [52] in terms of certain star-product re-ordering inspired by the analysis of [64,65].Here we explain it in a somewhat different way within the approach of this paper.Consider the β-dependent part of the Ansatz (2.11), with Ω α (2.13) and µ(τ, σ, β, ρ, u, v) = µ(τ, σ, β, ρ)d 2 ud 2 v .This yields where Note that λ > 0 in the allowed domain of the variables τ and β: The term linear in v α in the exponent can be removed by a change of integration variables ) This yields or, upon the redefinition, with ) Finally, making the rescaling we recast X into the form ) Equivalently, taking into account that the ρ-dependent term does not contribute to the exponent, ) Since Ω 0 α ′′ (4.30) has the same form as Ω α at β = 0, β can only affect the vertex via the preexponential factor, dΩ 2 µ(τ, λσ ′ , β, λρ ′ , u, v) with Let us stress that the exterior algebra formalism underlying the proposed approach allows us to make any changes of variables τ, γ i , ρ, β and z α .This is different from the analysis of [52] where the rescaling analogous to (4.26) was performed with the factor of 1 − β treated as a constant parameter rather than 1 − (1 − τ )β as in this paper.These rescalings coincide at τ = 0 and hence give rise to the same physical vertices at τ = 0.However, the approach of this paper allows one to trace the β-dependence effects at the intermediate steps as well.

Differential homotopy form of HS equations
A remarkable novel feature of the proposed approach is that it treats democratically spacetime coordinates x n , auxiliary spinor variables Z A , homotopy parameters t a and even coordinates of the non-commutative star-product space like U A and V A .This allows us to rewrite the field equations in the differential homotopy form stripping away the integrations over homotopy parameters t a and non-commutative coordinates U A , V A , extending the differential in the Z A space to that in the Z A , t a , U A , V A , where t a and U A , V A denote the whole set of the homotopy parameters and noncommutative coordinates, respectively.For brevity, the dependence on the noncommutative coordinates is implicit in the rest of this section.
Thus, in addition to the space-time differential d x := dx n ∂ ∂x n , we introduce total exterior differential d (2.1).

Differential homotopy HS equations
The transition to the differential homotopy setup with space-time coordinates being on equal footing with the homotopy parameters demands slight modification of the form of the HS equations.The idea is that upon integration over homotopy parameters one has to recover the original system (1.4)- (1.8).For that purpose we let the field B contain the homotopy differentials dt a and Z A -differentials θ A as well as space-time differentials that enter via the one-form ω.Hence we set Analogously W is extended by the components being space-time differential forms of higher degrees.
It should be stressed that the new added fields will enter under the d differential and hence will not contribute to the field equations at Z A = θ A = 0 upon integration over the homotopy parameters.Thus, the differential homotopy system that we introduce now is equivalent to the original integral one (1.4)-(1.8).
Namely, introducing notation where the term θ A Z A is explicitly introduced to incorporate d Z into d, the differential homotopy system of HS equations takes the form where the sign ∼ = implies weak equality up to terms that do not contribute under integration.There are three types of weak terms, that appear in the practical analysis.Firstly, these are the terms in the space V < with the number of the homotopy differentials dt a smaller than that of homotopy parameters to be integrated.
Secondly, these are the terms in the space V > with the form degree greater than the number of homotopy parameters to be integrated.Formally, such terms vanish because their form degree exceeds that of the volume form.However, it is useful to keep them because they may become relevant at higher orders after the number of homotopy parameters (i.e., the dimension of the homotopy space) is increased within the Ansatz of Section 2.3.
Thirdly, these are d-exact terms in V E v = dφ . (5.5) Note that this is equivalent to the space of d t -exact terms since the difference dφ The ω-dependent part of B now contributes into the d x sector of the equation on the zero-form C that acquires the form (5.6) Here the additional term with B1 enters either under the exterior differential in the homotopy parameters sector, that does not contribute upon integration, or under d Z differential which terms must cancel out in the Z-independent (i.e., d Z -cohomological) sector of the dynamical equations.Thus the modification of the B sector (5.1) does not affect the form of integrated equations.Nevertheless, it still plays an important role in the analysis of HS equations within the differential homotopy approach, being demanded by the formal compatibility of the equations (5.3) and (5.4).

Bulk and boundary terms in the homotopy space
In the practical analysis it is convenient to distinguish between the bulk and boundary terms in the space of homotopy parameters.The bulk terms are those that are integrated over the interior of the square formed by the homotopy parameters 0 < τ < 1 and 0 < τ < 1 that stand in front of z α or z α in Ω (2.13) or its conjugate.The boundary terms are those that are supported by the boundary of the square, i.e., τ and/or τ equals to 0 or 1 containing a factor of D(τ ) or D(τ −1) and/or D(τ ) or D(τ −1).Note that the cohomological contribution to the dynamical HS equations is located at τ = τ = 0.For the degenerate square with no homotopy parameters τ and τ the corresponding fields are treated as bulk.These are the HS fields ω(Y ; K|x) and C(Y ; K|x).
A useful form of the homotopy-differential equations is where V B and V W are homotopy boundary terms.For instance, the r.h.s. of (5.4) is a boundary term because γ and γ (1.12) are supported at τ = 1 and τ = 1, respectively.In practice we set (5.9) Now we observe that γ can be represented in the form (2.11) with p iα = 0 where the first term is d-exact, while the second one yields a weakly zero result upon multiplication by dΩ 2 (2.13) since there are no σ i in that case.Thus, the boundary terms with γ and γ can be represented in the manifestly d-exact form modulo weakly zero terms.As a consequence of the compatibility conditions for the equations (5.7), (5.8), (5.12) Note that V W (5.9) obeys these conditions as a consequence of (5.7) and since γ and γ are central.
The gauge transformations, that leave invariant equations (5.7), (5.8), (5.11) and (5.12) ) where the gauge parameters ǫ W and ǫ B are associated with the fields W and B, respectively.Note that the gauge transformations with the parameters ǫ B are not present in the original integral formulation of [50] (1.4)-(1.8)where the field B is a zero-form.In the differential formulation of this paper with B being a differential form both in M and in space-time, the gauge symmetry parameters ǫ B are responsible for the addition of exact forms, that do not affect the final integrated result.

Lower-order computations
In this section we illustrate the differential homotopy machinery by the lower-order analysis of the HS equations.Most of the results obtained in this section were reached before within the shifted homotopy analysis of [51] with the only important exception of the reconstruction of the projectively-compact spin-local form of the HS equations with the minimal number of space-time derivatives, obtained originally "by hand" in [59].The aim of this section is to illustrate how the proposed approach works in practice.The higher-order analysis within the differential homotopy scheme will be presented elsewhere.Note that in this section we will use the differential version (2.23) for the variables p α and r α .

Vacuum
As a vacuum solution to equations (5.7), (5.8) we choose as usual [50] (see [66] for a review) where ω(Y ; K|x) describes a flat HS connection that verifies For spin-two fields ω(Y ; K|x) bilinear in Y A ω(Y ; K|x) = ω AB (x)Y A Y B these equations describe AdS 4 geometry in terms of the flat sp(4) connection ω AB (x).

First order 6.2.1 B 1
In the lowest order, B 1 is a bulk field on the degenerate homotopy square.Since the latter has no boundary, equation (5.7) on B 1 yields where D x is the covariant derivative in the adjoint representation of the star-product algebra, The part of equation ( 6.3) of degree zero as a space-time form implies that the dxindependent part of B 1 is d-closed.The gauge transformations (5.15) allow one to gauge away the d-exact part.Choosing B 1 = C(Y ; K|x) in d cohomology H 0 (d) the dx-dependent part of equation ( 6.3) yields the standard form of the linearized HS equations [62], To reproduce the standard linearized HS equations of [62], where the η deformation terms contain the zero-form C(y, ȳ; K|x) at y = 0, i.e., C(0, ȳ; K|x), S η 1 has to be free from the terms y α p α in the exponent at τ = 0. (This is because the physical sector of the HS equations is carried by Z-independent terms from d Z cohomology concentrated at τ = 0, which is the content of the Z-dominance Lemma [51] (see also [67]).) Taking into account the form of Ω α (2.13) (note that p +α = p α for the first order with a single p α ), we observe that Hence, the relevant part of the exponent at τ = 0 takes the form The condition that it is free from p α y α demands σ = 0. Thus, we have to look for S η 1 in the form (2.11) with l(τ ) (2.33) and µ(σ, ρ, β) = D(σ)μ(ρ, β) .(6.12) Analogously, S 1 has to obey equation (6.7).Since γ and γ have the form (1.12) this is possible if d hits only l(τ ) and the term with D(τ ) is weak.This demands μ(ρ, β) and μ(ρ, β) be d-closed, which is not a restriction since any two-form holomorphic in ρ and β or antiholomorphic in ρ and β is d-closed.
As a result, we can write where the term with D(τ ) is weakly zero.Indeed, consider .17) where Ω 0α := βv α − ρ(y α + u α + p α ) .(6.18) In this case dΩ α 0 dΩ 0α is a two-form in β and ρ which, together with the two-form μ(ρ, β), gives a four-form in the two-dimensional space, hence being weakly zero.Though one might think that the weak terms have to be discarded, it is useful to keep them since they induce non-zero contribution at further computations respecting the form of the Ansatz (2.11).In particular, as explained in Section 6.2.4, the weak term V S η 1 eventually determines the form of the linearized field equations.
To summarise, S 1 and V S 1 , that solve (5.8) within the differential homotopy approach in a way compatible with the standard form of the free unfolded HS equations of [62,66], read as The conjugated expressions are V S η 1 := iη τ ρ βσ ū2 v2 The closed two-forms µ(ρ, β) and μ(ρ, β) are normalized to obey being otherwise arbitrary.The conventional solution for S η 1 used originally in [50] (for review see [66]) is a particular case with μ(ρ, β) = D(β)D(ρ) (6.28) implying zero y− and ∂ ∂y − shifts in terms of the shifted homotopy of [51].

W 1
To find W 1 , we have to solve the equation with D x (6.4).W 1 contains four η, η-dependent pieces of all possible kinds: where the subscripts ωC and Cω refer to the order of the product factors while the superscripts η and η distinguish between the terms proportional to η and η, resp.Note that, to simplify formulae, we use convention of [62] where ω contains both zero-order vacuum part and the first-order fluctuational part.Beyond the vacuum approximation it is not demanded to obey (6.2), however.
Using that d x C = −{ω , C} * + . . ., where ellipses denotes higher-order corrections, to evaluate D x S 1 it is convenient to use formulae (3.15), (3.16) which, taking into account that the one-form ω anticommutes with the differentials of the homotopy coordinates and, hence, with the three-form µ(σ, β, ρ) (6.12), yield (We use notations ( * ) * for the star product in the (anti)holomorphic sector, respectively.The integration variables u α , v α , ū α and v α are implicit in the sequel.)This yields (6.34)where is a one-form in dσ, dσ ω , dβ, dρ.As a result, the measure factor is a five-form in these four differentials, hence being zero.Other way around, the term does not contain the differentials dσ, dσ ω , dβ, dρ at all.Hence, in that case, the measure is a three-form in four variables that does not contribute to the integral as well.This analysis illustrates the important point that whether an expression is non-trivial or weakly zero depends on how many homotopy parameters like dσ, dσ ω , dβ, dρ enter.Since some of them enter implicitly via dΩ α , an expression can be weakly zero in presence of, say, σ ω , being nontrivial at the previous step without ω, or other way around.This is why we prefer to keep track of all boundary terms no matter whether they are weakly zero or not.

Linearised field equations in the one-form sector
The linearised equation on ω, that results from (5.8), yields where W 1,2 is a space-time two-form and V Dxω is weakly zero.Consider the holomorphic sector.Using (3.1), (3.2), (3.11) and (3.13), it is easy to obtain (6.50)Here the three terms in dP Analogously, (6.51)The contribution to the ωCω sector consists of four terms where the first two result, respectively, from W 1 | ωC * ω and ω * W 1 | Cω while the last two come from W 1 | ωdxC and W 1 | dxCω with d x C = −[ω , C] * + . . .where ellipses denotes higher-order terms that do not contribute to the order in question.This yields To see that the expressions (6.52) and (6.53) are indeed equivalent, one has to take into account inequalities that follow from the fact that σ = 0 due to the factor of D(σ) and 1−β > 0, that trivialize some of the theta-function factors like, for instance, θ(σ+1−β) = 1 .Note that such inequalities allow us to extend the P 3 factors in the ω 2 C and Cω 2 sectors to P 4 as follows: This makes it possible to write D x W 1 in the form Using that μ(ρ, β) is d-closed this allows us to find W η 1,2 modulo terms containing d(l(τ )) = D(τ ) + D(1 − τ ).The terms with D(1 − τ ) are weakly zero while those with D(τ ) give the contribution to the r.h.s. of the field equations.All in all this yields k in the holomorphic sector and k in the antiholomorphic.The field equations on the HS gauge fields are k .
The r.h.s. of (6.61) is in d Z cohomology because of the factors of D(τ ) or D(τ ).It is nonzero (not weak) since the pre-exponential factor is a six-form integrated over a sixdimensional homotopy space.On the other hand, V η Dxω and V η Dxω are weak because the factors of D(1 − τ ) or D(1 − τ ) eliminate the differentials of homotopy coordinates from dΩ 2 or d Ω2 leaving a four-form integrated over the six-dimensional homotopy space, which is zero.
Equation (6.61) reproduces the standard form of Central On-Shell Theorem at μ(ρ, β) (6.28).Indeed, from the fundamental Ansatz (2.11), (2.13), (2.16) and the fact that (6.61) contains D(σ) in the η sector and D(σ) in the η sector it follows that the r.h.s.contains C(0, ȳ; K|x) and C(y, 0; K|x) in these sectors, respectively.This is the characteristic feature of the standard form of the Central On-Shell Theorem that fixes it uniquely by formal consistency.Further details demand explicit evaluation of the various types of terms in (6.61) analogous to the standard computation within the conventional formalism (see e.g.[66] and references therein).
A related question is what is the effect of the specific choice of μ(ρ, β)?The answer is that exact μ(ρ, β) do not contribute upon integration.In the case of ρ, this is easy to see.Indeed, since the two-form μ(ρ, β) contains dρ, this is the only place where dρ contributes.As a result, neither dρ nor ρ appears anywhere else and, hence, the terms dρ ∂ ∂ρ φ(ρ, β) give rise to the exact integrand in (6.61).It turns out that the β dependence of μ(ρ, β) also does not contribute at the lowest order which is consistent with the analysis of Section 4.3.
Another question is to which extent the form of the Central On-Shell Theorem can be affected by the application of the general homotopy of Section 2.5.It is not hard to make sure that the application of the general homotopy does not affect the field equations except that it can add an exact form not contributing upon integration over homotopy parameters.So far, in the analysis of this section, the generalization of Section 2.5 could only affect the form of S 1 and W 1 .As far as we can see, such a modification would affect the form of the linearized HS equations (6.61) that is not allowed as spoiling the meaning of the components of the zero-form C(Y ; K|x) in terms of derivatives of the HS gauge fields ω(Y ; K|x).The second-order part of B, is determined by the equations with S η 1 (6.11) and S η 1 (6.13).An elementary computation using (3.1) and (3.2) yields (6.66)This expression can be represented in several different forms with the first two insensitive to the ρ-dependence: (6.68) Solving (6.63) with the aid of (6.67) and (6.68) yields, respectively, the following results: and the weak terms (6.72) B 2con and B 2sh can be easily recognized to reproduce those resulting, respectively, from the conventional (unshifted) homotopy, that leads to the non-local field equations, and the shifted homotopy of [10], that leads to the spin-local HS equations.It should be stressed that spin-locality of the vertex associated with B η 2sh is due to the factor of D(σ 2 − σ 1 − (1 − β)) on the r.h.s. of (6.70) that guarantees the cancellation of the terms with p 1α p α 2 in the exponent at τ = 0. (For more detail see Section 6.3.2.)However, being spin-local, the resulting vertex is not projectively-compact [58] which means that it is not of the lowest order in derivatives and most important, can induce spacetime non-local terms at higher-orders.Though the proper projectively-compact HS vertex was found in [59] by hand field redefinition, so far it was not known how to reach this result by a systematic homotopy method.One of the results of this paper is the elaboration of such a method, that is based on the extension of the approach by the parameter ρ.
To this end, setting in (6.66) yields where This yields where it is used that E(Ω) is ρ-independent and 1−β−ρ = − ρ .
Taking into account that the measure factors in (6.77) contain dρ ′ , we observe that and, hence, using relations analogous to (4.5), (4.6), (4.9) and (4.10), Then (6.63) gives B η 2pc that leads to a projectively-compact spin-local vertex, where the vanishing term with d(l(τ ))dτ has been omitted.The term V η dB 2pc is weakly zero because D(τ )dΩ 2 and D(1 − τ )dΩ 2 contain, respectively, too many and not enough differentials versus the number of integration variables.
The antiholomorphic expressions are We shall see in the next section that the nontrivial contribution to the field equations results from the weak term (6.81).The key property of B η 2pc is that the resulting vertices are projectively-compact, that eventually both minimizes the number of space-time derivatives and guarantees equivalence of the space-time and spinor spin-locality of the equations at the higher order.As explained in Sections 6.3.2 and 7.2, this follows from the specific form of B 2 containing the differentials dσ i and dρ either via their sum dσ i + dρ or via dΩ 2 .
Note that the dΩ 2 -independent term in (6.79) in our construction is a counterpart of the shift δB 2 found in [59] to reach the vertex with the minimal number of derivatives.

Equations DC = J
The d x C part of equation (5.7) has the form where V DxB 2 is the weak part that does not contribute upon integration over homotopy parameters and D x C is the free part of the equations.With the aid of (6.45), (6.47), (3.15)-(3.17)and (6.80) this yields in the holomorphic ωCC sector where the arguments like ωC 1 or ωC 1 C 2 indicate the order of the product factors in the respective expressions resulting from insertion of the factors of ω via formulae (3.15)-(3.17).
To cancel the d-exact term we set Now we observe that the first term with S η 1 * C cancels against the analogous term from dB η 2pc (6.63).The term C * S η 1 from dB η 2pc does not contribute because, by (3.1), it is proportional to D(β − 1 − σ 1 ) hence giving zero since P (β − 1, σ 0 , β − 1) = 0 in agreement with the property (3.20) that the arguments of P (a 1 , a 2 . ..) are ordered, a i ≤ a i+1 .As a result, the only contributing term results from the weak term (6.81) in dB 2pc (6.63) The term (6.81) consists of two parts.The one with D That with D(τ ) is cohomological (i.e., Z, θ-independent).It determines the nonlinear correction J η pc ωCC to the HS equations, Other two orderings are analysed analogously.The bulk terms in τ cancel up to d-exact terms.(Note that in the sector CωC the bulk terms between [W 1 , C] * and P dB 2 cancel pairwise.)The final result has concise symmetric form Note that the ordering of the arguments of the kernels P in (6.91), (6.92) matches that of the factors ω and C 1,2 .Though these parts of the formulae are symmetric with respect to the exchange of ω and C 1,2 , the overall factor distinguishes between ω and C 1,2 in a way prescribed by the proper form of the Central on-Shell Theorem and projectively-compact spin-locality of the currents J. Also note that some of the θ-functions in the definition of P (3.20) may trivialize upon multiplication by the overall measure factor.For instance, one can see that the first term in (6.91) reproduces (6.89).
The expressions for V η DxB 2 and V η DxB 2 are weak (do not contribute) since they contain only two differentials for the four variables ρ, σ 1,2 and σ ω because the factor of D(1 − τ ) annihilates the part of dΩ α that contains dρ, dσ 1,2 and dσ ω .On the other hand, the currents J η are non-zero because in this case dΩ 2 just brings two more differentials in dρ, dσ 1,2 and dσ ω .(Analogously in the antiholomorphic sector.) It should be stressed that the weak term in (6.63) contributes to the field equation for C just because the latter contains an extra homotopy parameter σ ω associated with the one-from ω.In other words, while in (6.63) the measure contained a four-form in the threedimensional homotopy space, in the field equations it is still a four-form which however is integrated over a four-dimensional homotopy space with the extra coordinate σ ω .This phenomenon illustrates a general feature of the proposed formalism making it useful to keep track of the weak terms that may contribute at the later stage.
Let us now consider more in detail the structure of the current J η to show that it is spinlocal.(Recall that spin-locality [53] means that the restriction of the vertex to any finite subset of fields is local.The vertex is not local in the usual sense because * ( * ) products in the η(η) sector mix vertices of fields of different spins.)Since locality is determined by the terms bilinear in p i in the exponent (2.16), it suffices to keep there only the y-independent terms at τ = 0:

.96)
Neglecting the β-dependent Jacobian, upon integration over u α and v α this yields in agreement with [10,54] this implies the absence of p 1α p α 2 in the exponent and, hence, spin-locality.That the constructed vertex is projectively-compact is explained in Section 7.2.

General lessons
Let us highlight some general features of the developed formalism illustrated by the analysis of the previous section.

Cancellation of the bulk homotopy terms
As we have seen, the bulk terms in the homotopy space with τ ∈ (0, 1) must cancel out in the final result leaving only the boundary terms located at τ = τ = 0.This fact was known before as the Z-dominance Lemma [51] (see also [67]).)The novel feature of the proposed formalism is that now the bulk homotopy terms cancel out pairwise without detailed computations that in the previously available formalisms demanded essential use of the Schouten identity.This remarkable simplification is due to the fundamental Ansatz (2.11), (2.13) that trivializes the role of the Schouten identity.
Let us stress that the final result for the dynamical field equations must belong to d Zcohomology in the original integral formulation of Section 1.2.Indeed, one can see that d Z of the l.h.s. of equations (1.4) and (1.6) yields zero once equations (1.5), (1.7), (1.8) have been solved.This gives a perturbative proof that equations (1.4) and (1.6) are Z-independent.
In the differential homotopy approach of this paper the analogous statement is that the l.h.s. of equations (5.3) and (5.4) in the physical sector must be Z, θ independent upon integration over the homotopy space M, which in turn implies that such terms must be dexact.These however are cancelled by the terms like W 1,2 of Section 6.2.4 or B 2ω of Section 6.3.2.Thus all bulk terms in physical HS equations have to cancel out while the nontrivial contribution is concentrated at τ = τ = 0. (The terms at τ = 1 and/or τ = 1 are weakly zero in physical equations.)

Projectively-compact spin-locality
Generally, spin-locality in the spinor space of Y variables does not automatically imply spacetime spin-locality at higher orders.As was explained recently in [58], there exists however a class of projectively-compact spin-local vertices for which this is true.The characteristic feature of such vertices, which, in fact, contain a minimal possible number of derivatives [59], is that they contain contraction between HS connection ω(Y ) and the Y -variables.In terms of vertices this demands them to contain the contraction y α p α ω or ȳ α p α ω .The approach of this paper gives hints how to control the projective compactness of the resulting spin-local vertices.To illustrate this point let us show that the vertices (6.91) do indeed share this property.
Thus, the form of B 2 (6.80) leads to the field equations containing an overall factor of y α p ωα .Our analysis shows that, more generally, to reach projective compactness of the vertex of this type at higher orders the prexponent factors have to be of the form i D(σ C i + ρ + . ..) (7.2) in which case dρ and dσ ω will contribute via dΩ 2 along with the factor of y α p ωα .This simple observation gives a hint what is a proper form of the general projectively-compact vertices in the proposed formalism.It would be interesting to understand better its relation with the shift symmetry condition of [67].

Conclusion
In this paper, a new approach to the analysis of HS equations of [50] is proposed with the homotopy parameters, that result from the Poincaré resolution of the differential equations, treated democratically with the space-time coordinates and non-commutative coordinates of the auxiliary spinor space.The geometric form of the approach is very suggestive: extra coordinates associated with the homotopy parameters parameterize compact polyhedra embedded into a multidimensional hypercube.From that perspective, they are reminiscent of the coordinates of the extra compact spaces like Calabi-Yau in String Theory, etc.Since the original unfolded approach to HS theory of [62,50] is formulated entirely in terms of the exterior algebra machinery, the formulation is coordinate independent, allowing in particular to reformulate the theory in terms of smooth manifolds rather than polyhedra.The differential homotopy approach shares two seemingly conflicting features: it is both far more general and far simpler than the standard shifted homotopy approach of [51].One of its great advantages is that it trivializes the complicated issue of accounting the Schouten identity.The same time, the proposed approach allowed us to derive HS current interactions in the projectively-compact spin-local form, that, as shown in [58], implies equivalence of the spinor and space-time spin-locality.(Though this form of HS vertices was obtained by hand field redefinitions in [59], so far it was unreachable by systematic homotopy approaches.)Moreover, the differential homotopy approach will be used in [68] to derive moderately nonlocal ηη vertices, establishing some bound on non-locality of HS gauge theory.This raises two related interesting problems.One is to compare the level of non-locality of HS theory found in [68] with that derived by Sleight and Taronna [28] from the Klebanov-Polyakov holographic conjecture [20].Another one is to look for better schemes that may further reduce the level of non-locality of HS theory at higher orders.
Another remarkable output of the analysis of this paper is that, as explained in Section 7.1, in the process of computation of the terms that contribute to field equations most of them (so-called homotopy bulk terms) cancel out by using general HS equations without going into their detailed form.To apply this mechanism to higher-order computations, where it is anticipated to be particularly efficient, it is desirable to work out its extension to multilinear combinations of the elementary fields.We hope to come back to this interesting problem in the future.
We believe that the results of this paper highlight the beauty and deepness of HS gauge theory to the extent making its study extremely interesting and exciting.The generality of the proposed approach gives a hope to uncover its relationship with other powerful tools in field theory like, for instance, the harmonic superspace approach to supersymmetric models [70] extended recently to HS multiplets [71]- [73].
Let us stress that the differential homotopy approach tested in this paper within the AdS 4 HS theory is applicable to other HS models like 3d HS gauge theory of [8] and HS gauge theory in any dimension of [69].The form of the equations is still (5.3),(5.4) while the procedure is exactly the same as in the 4d theory considered in this paper though the master fields may depend on somewhat different auxiliary variables.Namely, one introduces the homotopy space M, strips out integrations and adds additional components of fields that contribute under the total differential hence not affecting the final integrated results.Since the extension to other HS models is straightforward we will not discuss it in more detail here.Analogously, the differential homotopy approach can be applied to the multi-particle Coxeter HS theory [14] anticipated to shed light on the relation between HS theory and String Theory, one of the favorite ideas of Lars Brink.

t 1
) formulae (3.1), (3.2), (3.11) and (3.13) have the following important consequences used in Section 6 that, in fact, relate the Hochschild cohomology of the HS problem to the De Rham cohomology of the measure factors of µ on the compact polyhedra,
.75) Now, we make a change of variables of the class (4.11) in the term with D(σ 2 − ρ) ρ → ρ ′ = 1 − β − ρ .(6.76) reproduce field equations on dynamical HS fields in any gauge and field variables.Apart from space-time coordinates x n with De Rham differential d x := dx n ∂ ∂x n , master fields W (Z; Y ; K|x), S(Z; Y ; K|x) and B(Z; Y ; K|x) depend on spinor variables