Homotopy Operators and Locality Theorems in Higher-Spin Equations

A new class of shifted homotopy operators in higher-spin gauge theory is introduced. A sufficient condition for locality of dynamical equations is formulated and Pfaffian Locality Theorem identifying a subclass of shifted homotopies that decrease the degree of non-locality in higher orders of the perturbative expansion is proven.


Introduction
Nonlinear field equations for 4d massless fields of all spins were found in [1,2]. The most symmetric vacuum solution to these equations describes AdS 4 . Due to the presence of AdS 4 radius as a dimensionful parameter, higher-spin (HS) interactions can contain infinite tails of higher-derivative terms. This can make the theory non-local in the standard sense, raising the question which field variables lead to the local or minimally non-local setup in the perturbative analysis. Recently, in [3,4,5] it was shown how nonlinear HS equations of [2] reproduce current interactions in the lowest order in interactions. It has been then checked in [6,7,8] that the results of [3,4] properly reproduce the holographic expectations thus resolving some of the puzzles of the analysis of HS holography conjectures of [9,10] encountered in [11,12,13] (and references therein).
The derivation of [3,4] was based on the separation of variables (holomorphic factorization) in the zero-form sector of the 4d HS theory. So far the perturbative analysis of HS equations was based on the conventional homotopy operator technics proposed in [2]. In [5] it was explicitly checked that, in agreement with [11,14], the field redefinition that brings the results obtained by virtue of the conventional homotopy to the correct local form is nonlocal. Moreover, in [5] it was shown that from the perspective of the full nonlinear HS equations the field redefinition found in [3] has distinguished properties indicating that it leads to minimal order of non-locality in the higher orders. However it was not clear how the homotopy technics should be modified to lead directly to the correct local results in the perturbative analysis of HS equations with no reference to field redefinitions.
The main aim of this paper is to generalize the conventional homotopy technics in such a way that it will give immediately correct local results in the lowest order. Based on this generalization we prove a theorem showing how to choose the proper class of homotopy operators to decrease the level of non-locality of HS equations in higher orders as well. In fact, this result allows us to speculate that, by virtue of a proper choice of homotopy operators, HS contact interactions can be brought to the local form in all orders in interactions for every fixed set of spins in a vertex (see also Section 8). Let us stress that, since HS theory describes interactions of infinite towers of HS fields some sort of non-locality in HS theory survives when all spins are taken into account.
Note that what is interpreted as locality in this paper is probably better to call spin locality as it refers to the form of expressions in the sector of spinor variables underlying the unfolded formulation of HS equations of [1,2]. Its relation to the conventional definition in terms of space-time derivatives is via unfolded equations as we briefly recall now.
(1.1) More precisely, to describe massless fields, the one-form ω(Y ; K|x) should be even in k,k while the zero-form C(Y ; K|x) should be odd. Thus, massless fields are doubled Unfolded field equations for free massless fields of all spins in the AdS 4 are [17] (1.7) Background AdS 4 space of radius λ −1 = ρ is described by a flat sp(4) connection w = (w αβ , wαβ, h αβ ) containing Lorentz connection w αβ , wαβ and vierbein h αβ that obey where H αβ := h αα h βα and Hαβ := h αα h αβ are the frame two-forms.
In the massless sector, system (1.3), (1.4) decomposes into subsystems of different spins, with a spin s described by the one-forms ω(y,ȳ; K|x) and zero-forms C(y,ȳ; K|x) obeying ω(µy, µȳ; K | x) = µ 2(s−1) ω(y,ȳ; K | x) , C(µy, µ −1ȳ ; K | x) = µ ±2s C(y,ȳ; K | x) , (1.9) where + and − correspond to helicity h = ±s selfdual and anti-selfdual parts of the generalized Weyl tensors C(y,ȳ|x). For spins s ≥ 1, equation (1.3) expresses the Weyl 0-forms C(Y ; K|x) via gauge invariant combinations of derivatives of the HS gauge connections. More precisely, the primary-like Weyl 0-forms are just the holomorphic and antiholomorphic parts C(y, 0; K|x) and C(0,ȳ; K|x) which appear on the r.h.s. of Eq. (1.3). Those associated with higher powers of auxiliary variables y andȳ describe on-shell nontrivial combinations of derivatives of the generalized Weyl tensors as is obvious from Eqs. (1.4), (1.6) relating second derivatives in y,ȳ to the x derivatives of C(Y ; K|x) of lower degrees in Y . Hence higher derivatives in the nonlinear system hide in the components of C(Y ; K|x) of higher orders in Y . To see whether the resulting equations are local or not at higher orders one has to inspect the dependence of vertices on the higher components of C(Y ; K|x).
At the linearized level, Eq. (1.6) implies that ∂ ∂x is equivalent to ∂ 2 ∂y∂ȳ . Hence, at this level the analysis of spin locality in terms of y,ȳ variables is equivalent to that in terms of spacetime derivatives. However in higher orders Eq. (1.6) acquires nonlinear corrections. This makes the relation between spin locality in terms of y,ȳ variables and space-time locality less straightforward. Since the spinor sector of HS equations is of fundamental importance all concepts in HS theory including locality have to be originally defined in these terms. Therefore, we regard the spin locality of the HS theory as the fundamental concept. Relation to the space-time locality at higher orders is not straightforward being somewhat analogous to the effect of current exchange contribution in the space-time formulation.
The rest of the paper is organized as follows. The form of nonlinear HS equations is sketched in Section 2. Perturbative analysis of HS equations in terms of homotopy operator technics is recalled in Section 3. In Section 4 we introduce modified homotopy operators appropriate for the analysis of locality of HS equations. In Section 5 we prove Z-dominance Lemma providing a sufficient criterion for the locality of nonlinear corrections to dynamical field equations. Pfaffian Locality Theorem providing a criterion for the choice of homotopy decreasing the degree of non-locality in higher orders of interactions is proven in Section 6. A further generalization of the homotopy techniques useful for the analysis of field redefinitions relating different homotopies is presented in Section 7. Section 8 contains brief conclusions.
2 Nonlinear higher-spin equations 4d nonlinear HS equations [2] have the form (wedge symbol is omitted). W and B are fields of the theory which depend both on spacetime coordinates x n and on twistor-like variables Y A = y α ,ȳα and Z A = z α ,zα . (A = 1, . . . 4 is a Majorana spinorial index while α = 1, 2 andα = 1, 2 are two-component ones. The latter are raised and lowered by ε αβ = −ε βα , ε 12 = 1: A α = ε αβ A β , A α = A β ε βα and analogously for dotted indices.) The Y and Z variables provide a realization of HS algebra through the following noncommutative associative star product * acting on functions of two spinor variables where C AB = (ǫ αβ ,ǭαβ) is the 4d charge conjugation matrix and U A , V B are real integration variables. 1 is a unit element of the star-product algebra, i.e., f * 1 = 1 * f = f . Star product (2.4) provides a particular realization of the Weyl algebra The Klein operators satisfy relations analogous to (1.1) with y α → w α = (y α , z α , θ α ), yα →wα = (ȳα,zα,θα), which extend the action of the star product to the Klein operators. Decomposing master-fields with respect to the Klein-operator parity, A ± (Z; Y ; K|x) = ±A ± (Z; Y ; −K|x), HS gauge fields are W + , S + and B − while W − , S − and B + describe an infinite tower of topological fields with every AdS 4 irreducible field describing at most a finite number of degrees of freedom. (For more detail see [2,19]).
F * (B) is some star-product function of the field B. The simplest choice of the linear function F * (B) = ηB,F * (B) =ηB , where η is a complex parameter η = |η| exp iϕ, ϕ ∈ [0, π) , leads to a class of pairwise nonequivalent nonlinear HS theories. The cases of ϕ = 0 and ϕ = π 2 correspond to so called A and B HS models that respect parity [18].
3 Perturbative analysis and homotopy operator

Vacuum
Perturbative analysis of Eqs. (2.1), (2.2) assumes their linearization around some vacuum solution. The simplest choice is where w(Y |x) is some solution to the flatness condition Since S 0 has a trivial star-commutator with the Klein operators, a simple computation gives

Homotopy trick
To eliminate Z-variables one has to repeatedly solve equations of the form

Consistency of HS equations guarantees that
For diagonalizable A, the standard Homotopy Lemma states that cohomology In this case the projector h to KerA h 2 = h (3.14) and the operator A * can be defined to obey The resolution operator gives the resolution of identity allowing to find a solution to the equation where an exact part d Z ǫ and g ∈ H d Z remain undetermined.

Perturbative expansion
HS equations reconstruct the dependence on Z A in terms of the zero-form C(Y ; K|x) ∈ H d Z and one-form ω(Y ; K|x) ∈ H d Z representing the d Z -cohomological parts of B and W ′ , where zero-forms B j (Y, Z; K|x) and one-forms W j (Y, Z; K|x) are of order j in ω and C and obey The perturbative analysis goes as follows. Suppose that an order-n solution where ellipsis denotes higher-order terms, A(C, ω)| k is the order-k part of A(C, ω) in ω and C, and From (3.23) it follows by virtue of (3.21), (3.22) using (3.6) Let us stress that being of order m in C and ω, X W m and X B m contain B j and W j with j < m. Also it is used that the order-n parts of dynamical equations are of the form where the two-forms J 0 j ∈ H d Z and one-forms J 1 j ∈ H d Z are of order-j in ω and C. Since, acting trivially on ω and C, d Z does not mix different perturbation orders, equations (3.24) and (3.25) are d Z -closed separately at any m. This allows one to use different homotopy operators for any m in each of these equations.

Shifted homotopy
The conventional homotopy operator and resolution were used in the perturbative analysis of HS equations since [2]. Though being simple and looking natural, they are known to lead to nonlocalities beyond the free field level [11,14,5]. An obvious freedom in the definition of homotopy operator (4.1) is to replace Resolution △ a and cohomology projector h a act on a φ(Z, Y, θ) as follows △ 0 is the conventional resolution (4.2). The resolution of identity has standard form For instance, one can set a A = cY A with some constant c. 1 Naively, this exhausts all Lorentz covariant choices for a A . This is however not the case since a A can also be composed from derivatives with respect to arguments of ω(Y ; K) and C(Y ; Various terms on the r.h.s. of HS field equations contain ordered products An important feature of system (2.1), (2.2) noticed originally in [17] even before this system was obtained in [2] is that it remains formally consistent if the fields W and B are valued in any associative algebra, for instance, in the matrix algebra Mat N (C). As a result, the terms corresponding to different sequences of a i = 1 or 0 like {0, 1, 0, 0, 1, 0, . . .} etc , referred to as a, are separately d z -closed. Then the homotopy operators (4.3) are allowed to be different for different a. The simplest option is where ∂ iA is the derivative with respect to the argument of the i th factor Φ a i (Y i ; K). It is important that the modification via derivative homotopy shift affects locality when two or more arguments are available, i.e., only at the nonlinear level. This construction provides a broad extension of the class of allowed homotopy operators. In fact it can be further extended by letting the coefficients c j (a) be arbitrary functions of the covariantly contracted combinations ∂ jA and Y A . Practical analysis shows however that the simplest extension with constant c j (a) is sufficiently general.

Z-dominance Lemma
The following evident formula shows that to control locality it suffices to consider the exponential parts of the operators acting on ordered products C(Y 1 ; K)C(Y 2 ; K) . . . C(Y n ; K) focusing on the derivatives p j (p j ) with respect to y j (ȳ j ). To simplify analysis it is convenient to define p j andp j as respecting the chain rule in a way insensitive of the dependence of C(Y 1 ; K) on K. (Formally this can be achieved following [22,23] by introducing additional Clifford elements that anticommute with the Klein operators.) For each factor of C(Y j ; K) p j α is defined as the left derivative, i.e., p α (C ij (Y )k ikj ) = −i ∂ ∂y α (C ij (Y ))k ikj . Also it is useful to keep track of the extra degree of Klein operators originating from the operators γ (2.3). Hence, general exponential representation has the structure 4) where parameters T ∈ C, A, B ∈ C n , P ij = −P ji ∈ C n × C n and p = 0, 1 may depend on the integration homotopy parameters.
For instance, in [5] it was shown that the second-order correction to B(Z; Y ; K) that eventually leads to local HS equations in the zero-form sector is H loc k , (5.5) where * is the star product with respect to barred variables, and B loc 2η is complex conjugated to B loc 2η . What we would like to explain now is that from (5.5) it immediately follows that the equations of motion in the sector of physical fields are local rather than being a priori minimally nonlocal according to the argument of [5].
Indeed, the first nontrivial correction to the field equations in the zero-form sector is Here C and ω are Z A -independent and hence the correction H cur (w, J) must be Z-independent as well. This happens because consistency of the equations guarantees that the corrections belong to the d Z -cohomology. Though in practical computations it is sometimes convenient to set Z = 0 to simplify the derivation of the explicit form of the corrections to field equations, this is not necessary since the Z-dependence should drop out anyway as a consequence of the previously solved equations implying that d Z of the both sides of equations is zero. Practically, this works as follows. The Z-dependent term in the exponential contains the integration homotopy parameter τ 3 . The fact, that the r.h.s. of (5.8) must be Z-independent implies that the integral over τ 3 on the r.h.s. of (5.8) must reduce to integration over such a total derivative that the final result is located at the lower integration limit τ 3 = 0. This however means that not only the z-dependent term iτ 3 z α y α in the exponential in (5.8) disappears but the term τ 3 ∂ 1α ∂ α 2 must disappear as well, so that the final expression will contain at most a finite number of contractions in the preexponential with respect to the undotted variables, leading to a local result.
Generally, we arrive at the following Z-dominance Lemma Lemma 1: All terms in the exponential representation (5.4) dominated by the coefficients in front of the Z-dependent terms T (τ ) and A i (τ ) do not contribute to the field equations on the d Z -cohomology-valued dynamical fields.
This simple lemma allows us to show that the level of non-locality of HS equations can be decreased in higher orders by an appropriate choice of shifted homotopy operators (4.3). Let us stress that Z-dominance Lemma 1 applies to each term in expressions containing linear combinations of a finite number of exponentials (5.4) as is most easily seen by rewriting the z-dependent part of the exponentials in the form (5.10) allowing to rewrite a sum of integrals of different exponentials as a sum of terms in the integration measure in front of a single exponential factor exp i(tz γ y γ − t j p j γ z γ ).

Pfaffian Locality Theorem
Here we prove Pfaffian Locality Theorem (PLT) stating that, in the holomorphic sector, there exists such a choice of the shifted resolutions that the matrix P ij of (5.4) is degenerate.
In the second order this implies that P ∂ ∂y i = 0 and, hence, J 2 is local in agreement with [3,5]. In higher orders PLT implies at least the decrease of the level of non-locality indicating however that it can be decreased further.
PLT heavily relies on the properties of star product (2.4) and shifted homotopies. In our analysis we focus on the dependence on the zero-forms C discarding the dependence on ω and w that do not affect spin-locality in the 4d HS theory.
It is useful to restrict the representatives of the exponential classes (5.4) as follows.
Perturbative analysis implies that corrections to dynamical equations at any perturbation order can be constructed inductively, starting from γ (2.3), C(y,ȳ|x) and ω(y,ȳ|x) via application of the star product, shifted resolutions, cohomology projectors, operators D w (3.6) (not affecting locality) and d x . By Structure Lemma 6 proven in Sections 6.1-6.3, with the proper choice of homotopies (6.14), (6.16), these operations respect the classes E j n . This allows us to decrease the level of non-locality in the higher order corrections to dynamical equations in the zero-form (anti) holomorphic sector. Indeed, in this case exponential parts of the order-n deformations J 1 n (Y ; K|x) to the field equations in the zero-form sector (3.28) are of the form (5.4) with the parameters obeying (6.4). According to Lemma 1 the coefficients T and A i trivialize in the d Z -cohomology. Hence (6.4) yields in the d Z -cohomology proving Pfaffian Locality Theorem: the shifted homotopy can be chosen in such a way that the matrix P ij be degenerate with the null-vector (6.6) in the (anti)holomorphic sector of the dynamical field equations in the zero-form sector. In even interaction orders condition (6.6) is essential. For instance, from Section 6.2 it follows, that to obtain a local form of dynamical equations to the second order in the holomorphic sector, it is necessary to take the shifted resolution operator with arbitrary parameters α and β. Details of the derivation of the local form of equations are presented in [21] where it is also proven that the resulting equations coincide with those of [4] up to β-dependent local field redefinitions. In odd orders the antisymmetric matrix P ij (τ ) is automatically degenerate. However, the additional information following from (6.6) is that it admits a null vector independent of the homotopy parameters τ on which P ij (τ ) depends. Though we do not know yet whether condition (6.6) increases the degree of degeneracy of P ij (τ ) further or not, its special structure implying that the vertex depends on some linear combinations of helicities associated with different fields suggests that the level of non-locality of the resulting vertices which are rather unusual from the QFT perspective is likely to be further reducible. We interpret this as a possible indication that HS interactions may admit a spin-local form in all orders.

d x mapping
It remains to consider the mapping generated by d x that acts nontrivially on the dynamical fields ω and C. The action on w does not affect locality. By Eq. (3.28) Note that the lower label j of J i j (Y ) equals to the total degree in the dynamical fields, while the respective k-equipped exponentials depend on the degree in C. For the future convenience we set where j ω and j c are the degrees of J i j (Y ) in ω and C, respectively. J 1,2 j do not depend on z. Hence the k-equipped exponential Ep jc (5.4) of J 1,2 jω+jc (Y ) is Ep jc ( B, P |y) := Ep jc (0, 0, B, P |0, y) . (6.20) Since d x ω(Y ; K|x) (6.18) contributes to the sector of two-forms it does not affect field corrections B (n) (3.22) and dynamical equations on the zero-form C(Y ) (3.28) for which we obtain schematically . The resulting mapping S i, Ep jc (E p n ) generated by d x (6.22) for any i and j c is where parameters are defined straightforwardly via Eq. (5.4). For instance, . . , p i−1 ,p 1 , . . . ,p jc , p i+1 , . . . , p n }.
satisfying conditions (6.2) for a = 0 and (6.4) for a = 1. Analogously, one can make sure that from (6.26) it follows that the parameters P ′ kj on the r.h.s. of (6.23) satisfy the respective conditions (6.2) for a = 0 and (6.4) for a = 1.
Let us stress that otherwise, if Ep jc (6.20) is even, the resulting k-equipped exponential E p+p n+jc−1 (T ′ , A ′ , B ′ , P ′ , p ′ ) (6.23) in general has no definite parity. By induction over perturbation orders, Lemmas 2-5 provide following Structure Lemma Lemma 6: If the perturbative analysis in the one-form sector contains shifted resolutions △ satisfying (6.16), while that in the zero-form sector contains shifted resolutions satisfying (6.14), then all B j generate odd k-equipped exponentials, while all space-time zero-and one-form components of W j , not containing terms resulting from d x ω(Y ) (6.18), generate even k-equipped exponentials in the holomorphic sector.
Antiholomorphic sector analysis is analogous up to swap of dotted and undotted spinors.

Covariant derivative homotopy
Now we extend the homotopy trick to the full covariant derivative containing both xdependent and Z-dependent parts. This consideration extends that of [20] to arbitrary background connection and shifted resolutions.

Generalized spectral sequence
Consider a system of the form (d Z − D) f = J , (7.1) where d Z and D (in the end to be replaced by − i 2 D (3.6)) obey the standard spectral sequence relations Our aim is to invert operator d Z − D in a form allowing to use different elementary resolutions △ a in different sectors. Namely, consider a set of resolutions (4.4) (in what follows resolution set) with the respective cohomology projectors h i . Let Π k be a projector to degree-k differential forms in θ A . Evidently, one can make sure straightforwardly that the following resolution of identity is true where The resolution of identity (7.10) allows one to solve equation (7.1) in the form where g is a restriction of f to H d Z . From (7.9) it follows that generalized cohomology operator H is a projector. Indeed, since H (7.9) is in d Z -cohomology then by virtue of (7.6), Π j H = δ j 0 H, hence for any two resolution sets R and R ′ (7.

Descendant relations
Formula (7.10) admits a chain of interesting extensions. As a first step one can make sure that ∆ (1) (7.11) obeys where These relations can be inductively extended to where Let {d Z , △} + h = 1 be a resolution of identity with some △ and cohomology projector h. Hence from (7.20) along with the following evident identities it follows that H = h H on cohomologies, i.e., with H (1) (7.15). One can also see that By virtue of (7.18), for any two resolution sets R(△) and R ′ (△ ′ ) (7.4) if m < n .

(7.24)
By virtue of (7.2), (7.9) H acquires the following useful form Analogously, from (7.8) it follows The following redefinitions leave resolution of identity (7.10) invariant where A, B and C are arbitrary while F belongs to d Z -cohomology. (Note that F and G should obey some additional conditions for H ′ to satisfy (7.13).) In what follows these will be referred to as A-redefinition, B-redefinition, etc . The meaning of A-redefinition and F -redefinition is that in terms of solution (7.12) to equation (7.1) they induce the field redefinitions By virtue of (7.32), (7.34), taking into account (7.22), this leads to the following equations By virtue of such redefinitions one can map any generalized resolution ∆(R) and projector H(R) depending on R (7.4) to ∆(R ′ ) , H(R ′ ) depending on R ′ = R \ △ 0 , i.e., R ′ = {△ 1 , △ 2 , . . .} . Indeed, using F -redefinition and A-redefinition (7.29) with as well as B-and G-redefinition (i.e., omitting terms of the form (...)(D − d Z ) that vanish on the field equations), one obtains from (7.26) and (7.28) Formula (7.33) giving exact form of the field redefinition (7.30) relating one resolution set to another is very useful in practical analysis.

Conclusion
In this paper we explain how to extend the class of homotopy operators in HS theory to make it possible to systematically analyze locality of interactions derived from nonlinear HS equations. It is shown that a number of available homotopy operators increases quickly with the order of nonlinearity, containing in particular a subclass of homotopy operators that lead directly to the known lower-order local results as shown explicitly in [21]. Also we prove a Z-dominance Lemma giving a sufficient condition controlling locality of field equations on dynamical fields and Pfaffian Locality Theorem (PLT) showing how to choose generalized homotopy operators to reach that the Pfaffian matrix of derivatives acting on spinor variables of different fields in multilinear corrections degenerates. As shown in [21], the choice suggested by PLT in the case of bilinear corrections leads to the local results of [3,5]. In the higher orders the PLT allows us to choose homotopy operators in such a way that the level of higher-order non-locality gets decreased, which result is somewhat analogous to conclusions of [24]. The results of this paper are applicable not only to the 4d HS theory of [2] but also to 3d HS theory [22] and Coxeter HS theories proposed recently in [25].
To appreciate this result, the following comments have to be taken into account. The structure of the remaining non-local higher-order interactions obtained by virtue of the homotopy operators satisfying PLT is very special, containing some linear combinations of helicities associated with different fields in a vertex as prescribed by (6.6). Such vertices are rather unusual from the QFT perspective and are anticipated to be further removable by an appropriate homotopy choice. A related comment is that conditions of PLT leave a lot of freedom in the choice of shifted homotopy operators in higher orders to be used to further reduce the level of non-locality of HS interactions. Hopefully, there may exist a specific homotopy choice leading to spin-local higher-order nonlinear corrections at any order.
The conjecture that contact HS interactions can be spin-local should not be interpreted as the claim that all HS interactions are space-time local. Most likely they are not due to the spin-current exchange phenomenon. Indeed, what is proven in our formalism is that contractions with respect to spinorial variables of C(Y ; K|x) are suppressed. However, their relation to the space-time derivatives is direct only at the level of free equations (1.6) which, however, receive nonlinear corrections at higher orders. As a result, the relation between space-time and spinor derivatives of C(Y ; K|x) becomes nonlinear and also involves higher derivatives. Due to summation over different spins this may eventually lead to further x-space non-localities. This mechanism is somewhat analogous to the current exchange mechanism in QFT. Our results indicate that contact HS interactions may be spin-local in all orders.
It should be stressed that the results of this paper are heavily based on the specific form of HS equations (2.1), (2.2) and, in particular, of the star product (2.4). Extension of the analysis of this paper to other cases including the sector of one-forms ω and mixed (nonholomorphic) sectors needs application of the remarkable properties of the shifted homotopy formalism elaborated in [21] where a number of examples of its applications are presented.