Higher-spin fermionic gauge fields and their electromagnetic coupling

We study the electromagnetic coupling of massless higher-spin fermions in flat space. Under the assumptions of locality and Poincaré invariance, we employ the BRST-BV cohomological methods to construct consistent parity-preserving off-shell cubic 1 − s − s vertices. Consistency and non-triviality of the deformations not only rule out minimal coupling, but also restrict the possible number of derivatives. Our findings are in complete agreement with, but derived in a manner independent from, the light-cone-formulation results of Metsaev and the string-theory-inspired results of Sagnotti-Taronna. We prove that any gauge-algebra-preserving vertex cannot deform the gauge transformations. We also show that in a local theory, without additional dynamical higher-spin gauge fields, the non-abelian vertices are eliminated by the lack of consistent second-order deformations.


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Contents Consistent interacting theories of massless higher-spin fields in flat space are difficult to construct. Severe restrictions arise from powerful no-go theorems [1][2][3][4][5], which prohibit, in Minkowski space, minimal coupling to gravity, when the particle's spin s ≥ 5 2 , as well as to electromagnetism (EM), when s ≥ 3 2 . However, these particles may still interact through gravitational and EM multipoles. Indeed, N = 2 SUGRA [6,7] allows massless gravitini to have dipole and higher-multipole couplings, but forbids a non-zero U(1) charge in flat space. Gravitational and EM multipole interactions also show up, for example, as the 2 − s − s and 1 − s − s trilinear vertices constructed in [8,9] for bosonic fields. 1 These cubic vertices are but special cases of the general form s − s ′ − s ′′ , that involves massless fields of arbitrary spins. Metsaev's light-cone formulation [10,11] puts restrictions on the number of derivatives in these vertices, and thereby provides a way of classifying them. For bosonic fields, while the complete list of such vertices was given in [12], Noether procedure has been employed in [13][14][15] to explicitly construct off-shell vertices, which do obey the number-of-derivative restrictions. Also, the tensionless limit of string theory gives rise to a set of cubic vertices, which are in one-to-one correspondence with the ones of Metsaev, as has been noticed by Sagnotti-Taronna in [16,17], where generating functions for off-shell trilinear vertices for both bosonic and fermionic fields were presented.
In this paper, we consider the coupling of a massless fermion of arbitrary spin to a U(1) gauge field, in flat spacetime of dimension D ≥ 4. Such a study is important in that fermionic fields are required by supersymmetry, which plays a crucial role in string theory, which in turn involves higher-spin fields. This fills a gap in the higher-spin literature, most of which is about bosons only (with [11,[16][17][18][19] among the exceptions). We do not consider mixed-symmetry fields, and restrict our attention to totally symmetric Dirac fermions ψ µ 1 ...µn , of spin s = n+ 1 2 . For these fields, we employ the powerful machinery of BRST-BV cohomological methods [20,21] to construct systematically consistent interaction vertices, 2 with the underlying assumptions of locality, Poincaré invariance and conservation of parity, and without relying on other methods. The would-be off-shell 1 − s − s cubic vertices will complement their bosonic counterparts constructed in [9].
The organization of the paper is as follows. We clarify our conventions and notations, and present our main results in the next two subsections. In section 2, we briefly recall the BRST deformation scheme [20,21] for irreducible gauge theories. With this knowledge, we then move on to constructing consistent off-shell 1−s−s vertices in the following three sections. In particular, section 3 considers the massless Rarita-Schwinger field, while section 4 pertains to s = 5 2 , and section 5 generalizes the results, rather straightforwardly, to arbitrary spin, s = n + 1 2 . In section 6, we prove an interesting property of the vertices under study: an abelian 1 − s − s vertex, i.e., a 1 − s − s vertex that does not deform the original abelian gauge algebra, never deforms the gauge transformations. Section 7 is a comparative study of our results with those of Metsaev [11] and Sagnotti-Taronna [16,17], where we JHEP08(2012)093 explicitly show their equivalence. Section 8 investigates whether there are obstructions to the existence of second-order deformations corresponding the non-abelian vertices, i.e., if they are consistent beyond the cubic order. We conclude with some remarks in section 9. Two appendices are added to present some useful technical details, much required for the bulk of the paper.
We see that the (n − 1)-curl of the fermionic field appears in the vertex.
• The 2n-derivative vertex is abelian. It exists only for D > 4, and is gauge invariant up to a total derivative. It involves the curvature tensor, and takes the form It can be interpreted as a Chern-Simons term. To see this, let us start with the spin-3 2 case. For any choice of spinor indices a, b, the expressionΨ a ∧ Ψ b defines a closed 4-form, d Ψ a ∧ Ψ b = 0. Here Ψ b = 1 2 Ψ b αβ dx α ∧ dx β and a similar expression holds forΨ a . The associated Chern-Simons 5-form and summing over the spinor indices give a scalar, which is gauge invariant up to a total divergence. This understanding of the 2n-derivative vertex explains why it exists only in D ≥ 5. For higher-spin fields, the curvatures are not exterior forms since they are described by mixed-symmetry Young tableaux. However, the contracted expression is a closed 4-form, and the construction proceeds then in the same way.
• The non-abelian cubic vertices generically get obstructed, in a local theory, at second order deformation. In special cases, such vertices may extend beyond the cubic order, if additional dynamical higher-spin gauge fields are present.

The BRST deformation scheme
As pointed out in [20,21], one can reformulate the classical problem of introducing consistent interactions in a gauge theory in terms of the BRST differential and the BRST cohomology. The advantage is that the search for all possible consistent interactions becomes systematic, thanks to the cohomological approach. Obstructions to deforming a gauge-invariant action also become related to precise cohomological classes of the BRST differential. In what follows, we briefly explain the BRST deformation scheme.

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Let us consider an irreducible gauge theory of a collection of fields {φ i }, with m gauge invariances, δ ε φ i = R i α ε α , α = 1, 2, . . . , m. Corresponding to each gauge parameter ε α , one introduces a ghost field C α , with the same algebraic symmetries but opposite Grassmann parity (ǫ). The original fields and ghosts are collectively called fields, denoted by Φ A . The configuration space is further enlarged by introducing, for each field and ghost, an antifield Φ * A , that has the same algebraic symmetries (in its indices when A is a multi-index) but opposite Grassmann parity.
In the algebra generated by the fields and antifields, we introduce two gradings: the pure ghost number (pgh) and the antighost number (agh). The former is non-zero only for the ghost fields. In particular, for irreducible gauge theories, pgh(C α ) = 1, while pgh(φ i ) = 0 for any original field. The antighost number, on the other hand, is non-zero only for the antifields Φ * . The ghost number (gh) is another grading, defined as gh = pgh − agh.
On the space of fields and antifields, one defines an odd symplectic structure which, by virtue of the Noether identities and the higher-order gauge structure equations, satisfies the classical master equation In other words, the master action S incorporates compactly all the consistency conditions pertaining to the gauge transformations. This also plays role as the generator of the BRST differential s, which is defined as Notice that S is BRST-closed, as a simple consequence of the master equation. From the properties of the antibracket, it also follows that s is nilpotent, Therefore, the master action S belongs to the cohomology of s in the space of local functionals of the fields, antifields, and their finite number of derivatives. As we know, the existence of the master action S as a solution of the master equation is completely equivalent to the gauge invariance of the original action S (0) [φ i ]. Therefore, one can reformulate the problem of introducing consistent interactions in a gauge theory 3 This definition gives Φ A , Φ * B = δ A B , which is real. Because a field and its antifield have opposite Grassmann parity, it follows that if Φ A is real, Φ * B must be purely imaginary, and vice versa.

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as that of deforming the solution S of the master equation. Let S be the solution of the deformed master equation, (S, S) = 0. This must be a deformation of the solution S 0 of the master equation of the free gauge theory, in the deformation parameter g, The master equation for S splits, up to O(g 2 ), into Eq. (2.7) is fulfilled by assumption, and in fact S 0 is the generator of the BRST differential for the free theory, which we will denote as s. Thus, eq. (2.8) translates to i.e., S 1 is BRST-closed. If the first-order local deformations are given by S 1 = a, where a is a top-form of ghost number 0, then one has the cocycle condition Non-trivial deformations therefore belong to H 0 (s|d) − the cohomology of the zeroth-order BRST differential s, modulo total derivative d, at ghost number 0. Now, if one makes an antighost-number expansion of the local form a, it stops at agh = 2 [9,[34][35][36][37][38], a = a 0 + a 1 + a 2 , agh(a i ) = i = pgh(a i ). (2.12) For cubic deformations S 1 = a, it is indeed impossible to construct an object with agh > 2 [9]. The result is however more general and holds in fact also for higher order deformations, as it follows from the results of [34][35][36][37][38][39]. The significance of the various terms is worth recalling. a 0 is the deformation of the Lagrangian, while a 1 and a 2 encode information about the deformations of the gauge transformations and the gauge algebra respectively [20,21]. Thus, if a 2 is not trivial, the algebra of the gauge transformations is deformed and becomes non-abelian. On the other hand, if a 2 = 0 (up to redefinitions), the algebra remains abelian to first order in the deformation parameter. In that case, if a 1 is not trivial, the gauge transformations are deformed (remaining abelian), while if a 1 = 0 (up to redefinitions), the gauge transformations remain the same as in the undeformed case.
The various gradings are of relevance as s decomposes into the sum of the Koszul-Tate differential, ∆, and the longitudinal derivative along the gauge orbits, Γ: (2.13) ∆ implements the equations of motion (EoM) by acting only on the antifields. It decreases the antighost number by one unit while keeping unchanged the pure ghost number. Γ acts

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only on the original fields and produces the gauge transformations. It increases the pure ghost number by one unit without modifying the antighost number. Accordingly, all three ∆, Γ and s increase the ghost number by one unit, gh(∆) = gh(Γ) = gh(s) = 1. Note that ∆ and Γ are nilpotent and anticommuting, Given the expansion (2.12) and the decomposition (2.13), the cocycle condition (2.11) yields the following cascade of relations, that a consistent deformation must obey Note that a 2 has been chosen to be Γ-closed, instead of Γ-closed modulo d, as is always possible [34][35][36][37]. We now analyze the conditions under which a 2 and a 1 are non-trivial.
• Non-triviality of the deformation of the gauge algebra: The highest-order term a 2 will be trivial (i.e., removable by redefinitions) if and only if one can get rid of it by adding to a an s-exact term modulo d, sm + dn. Expanding m and n according to the antighost number, and taking into account the fact that m and n also stop at agh = 2 since they are both cubic, one finds that a 2 is trivial if and only if a 2 = Γm 2 + dn 2 . We see that the cohomology of Γ modulo d plays an important role. The cubic vertex will deform the gauge algebra if and only if a 2 is a non-trivial element of the cohomology of Γ modulo d. Otherwise, can always choose a 2 = 0, and a 1 = Γ-closed [34][35][36][37]. Note that since a 2 is a cocycle of the cohomology of Γ modulo d, which can be chosen to be Γ-closed [34][35][36][37], one can investigate the general form of a 2 by studying the elements in the cohomology of Γ that are not d-exact.
• Non-triviality of the deformation of the gauge transformations: We now assume a 2 = 0. In this case, a 1 can be chosen to be a non-trivial cocycle of Γ. The vertex deforms the gauge transformations unless a 1 is ∆-exact modulo d, a 1 = ∆m 2 + dn 1 , where m 2 can be assumed to be invariant [34][35][36][37][38][39]. In that instance, one can remove a 1 , and so one can take a 0 to be Γ-closed modulo d: the vertex only deforms the action without deforming the gauge transformations. The cohomology of ∆ is also relevant in that the Lagrangian deformation a 0 is ∆-closed, whereas trivial interactions are given by ∆-exact terms.
Finally, while the graded Jacobi identity for the antibracket renders (S 1 , S 1 ) BRSTclosed, the second-order consistency condition (2.9) requires that this actually be s-exact: (2.18) This condition determines whether or not, in a local theory, a consistent first-order deformation gets obstructed at the second order. Such obstructions are controlled by the local BRST cohomology group H 1 (s|d).

EM coupling of massless spin 3/2
In this section we construct parity-preserving off-shell 1 − 3 2 − 3 2 vertices by employing the BRST-BV cohomological methods. The spin- 3 2 system is simple enough so that one can implement the BRST deformation scheme with ease, while it captures many non-trivial features that could serve as guidelines as one moves on to higher spins.
The starting point is the free theory, which contains a photon A µ and a massless Rarita-Schwinger field ψ µ , described by the action which enjoys two abelian gauge invariances: For the Grassmann-even bosonic gauge parameter λ, we introduce the Grassmann-odd bosonic ghost C. Corresponding to the Grassmann-odd fermionic gauge parameter ε, we have the Grassmann-even fermionic ghost ξ. Therefore, the set of fields becomes For each of these fields, we introduce an antifield with the same algebraic symmetries in its indices but opposite Grassmann parity. The set of antifields is Now we construct the free master action S 0 , which is an extension of the original gaugeinvariant action (3.1) by terms involving ghosts and antifields. Explicitly, Notice that the antifields appear as sources for the "gauge" variations, with gauge parameters replaced by corresponding ghosts. It is easy to verify that (3.5) indeed solves the master equation (S 0 , S 0 ) = 0. The different gradings and Grassmann parity of the various fields and antifields, along with the action of Γ and ∆ on them, are given in table 1. For the spin-3 2 field the Fronsdal tensor is S µ = i / ∂ψ µ − ∂ µ / ψ = −iγ ν Ψ µν , i.e., the γ-trace of the curvature. The cohomology of Γ is isomorphic to the space of functions of • The undifferentiated ghosts {C, ξ}, • The antifields {A * µ , C * ,ψ * µ ,ξ * } and their derivatives, • The curvatures {F µν , Ψ µν } and their derivatives.

Gauge-algebra deformation
The next step is to consider, for the first-order deformation, the most general form of a 2 − the term with agh = 2, that contains information about the deformation of the gauge algebra. a 2 must satisfy Γa 2 = 0, and be Grassmann even with gh(a 2 ) = 0. Besides, we require that a 2 be a parity-even Lorentz scalar. Then, the most general possibility is

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which is a linear combination of two independent terms: one contains both the bosonic ghost C and the fermionic ghost ξ, while the other contains only ξ but not C. The former one potentially gives rise to minimal coupling, while the latter could produce dipole interaction. This can be understood by first noting that the corresponding Lagrangian deformation, a 0 , is obtained through the consistency cascade (2.15)-(2.17). From the action of Γ and ∆ on the fields and antifields, it is then easy to see that the respective a 0 would contain no derivative and one derivative respectively.

Deformation of gauge transformations
Next, we would like to see if a 2 can be lifted to certain a 1 , i.e., with the given a 2 , if one could solve eq. (2.16) to find an a 1 . Indeed, one finds that 4 Therefore, in view of eq. (2.16), one must have where the ambiguity,ã 1 , belongs to the cohomology of Γ. Its most general form will bẽ where X, Y and Z may contain derivatives and spinor indices.

Lagrangian deformation
We note that ∆a 1 must be Γ-closed modulo d, since Condition (2.17), however, requires that ∆a 1 be Γ-exact modulo d. The ∆-variation of neither of the unambiguous pieces in a 1 is Γ-exact modulo d, and the non-trivial part must be killed by ∆ã 1 , if (2.17) holds at all. But such a cancelation is impossible for the first piece, i.e., the would-be minimal coupling, simply becauseã 1 contains too many derivatives. Thus, minimal coupling is ruled out, and we must set g 0 = 0. Then, we have The second term on the right hand side is in the cohomology of Γ, and must be canceled by ∆ã 1 . To see if this is possible or not, we make use of the identity

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to rewrite the term as Notice that, we have rendered the second line in the first step Γ-exact modulo d, by virtue of the Bianchi identity, ∂ [µ F νρ] = 0. We plug eq. (3.12) into (3.10) to obtain Now, the most important point is that, the terms in the second line of the above expression are ∆-exact, such that it is consistent to set This is tantamount to setting which, of course, is in the cohomology of Γ. Then, eq. (3.13) reduces to so that we have a consistent Lagrangian deformation a 0 . To summarize, we have

Abelian vertices
Now that we have exhausted all the possibilities for a 2 , any other vertex can only have a trivial a 2 . In this case, as we will show in section 6, one can always choose to write a vertex as the photon field A µ contracted with a gauge-invariant current j µ , where the divergence of the current is ∆-exact: so that one has a 1 = M C. If, however, M happens to be ∆-exact modulo d in the space of invariants, one can add a ∆-exact term in a 0 , so that the new current is identically conserved [34][35][36][37]. In the latter case, the vertex does not deform the gauge symmetry at all.

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Now the most general vertex of the form (3.18) contains the current whose divergence is required to obey the condition (3.19). Here X may contain Dirac matrices as well as derivatives. It is not difficult to see if X contains more than one derivatives, a 0 is ∆-exact modulo d, i.e., trivial. First, if X contains the Laplacian, , the contribution is always ∆-exact, by the EoM Ψ µν = 0. We can also forgo the Dirac operator, / ∂, because by using the relation / ∂γ µ = 2∂ µ − γ µ / ∂, one can always make / ∂ act on the curvature to get ∆-exact terms, thanks to the EoM / ∂Ψ µν = 0. Therefore, any derivative contained in X µναβλ must carry one of the five indices. Given the EoM ∂ µ Ψ µν = 0, the antisymmetry of the field strength Ψ µν , and the commutativity of ordinary derivatives, the only potentially non-trivial way to have more-than-one derivatives is But algebraic manipulations show that this vertex is actually ∆-exact modulo d, i.e., trivial.
To see this, we use Ψ αν = ∂ α ψ ν − ∂ ν ψ α , and rewrite (3.21) as While the first term is identically zero, in the second term, one can use the 3-box rule, Here, the last two terms are ∆-exact, whereas in the first term a double integration by parts gives A λ , which is equal to ∂ λ (∂ · A) by the photon EoM. Then, one is left with Upon integrating by parts w.r.t. ∂ λ , this indeed becomes ∆-exact modulo d, The only possibilities are therefore that X contains either no derivative or one derivative. For the former case, we have the candidate X µναβλ = −2η µν|αβ γ λ . This gives which is obviously gauge invariant: ΓM = 0. However, explicit computation easily shows that M is actually ∆-exact modulo d. Therefore, one can render the current identically conserved by adding a ∆-exact term. In fact, in view of identity (3.11), our candidate j µ is Then, it is clear from the identity This vertex does not deform the gauge symmetry, and is gauge invariant up to a total derivative. Note that the vertex does not exist in D = 4, because of the presence of γ µναβλ . This is in complete agreement with Metsaev's results [11]. Finally, we are left with the possibility of having just one derivative in X, which would correspond to a 3-derivative vertex. The only candidate is up to ∆-exact terms, thanks to the identity (3.11). We have In this case too, our candidate current reduces on-shell to an identically conserved one, so that the vertex actually does not deform the gauge symmetry. To see this, we use the Bianchi identity ∂ λ Ψ µν = −∂ µ Ψ νλ + ∂ ν Ψ µλ , to write the vertex as Thanks to the EoM ∂ µ Ψ µν = 0, up to ∆-exact terms, the current reduces to the total derivative of a fermion bilinear, which is identically conserved: Upon integration by parts, this is just a 3-curvature term (Born-Infeld type), This exhausts all possible 1 − 3 2 − 3 2 vertices. The results are summarized in table 2. Here we parenthetically comment about the nature of the abelian vertices. As it turned out, the vertices that do not deform the gauge algebra do not deform the gauge transformations either. In other words, if a 2 is trivial, so is a 1 . This is not accidental at all. In fact, in section 6 we are going to show that, for a massless particle of arbitrary spin s = n+ 1 2 coupled to a U(1) vector field, the cubic couplings that do not deform the gauge algebra actually do not deform the gauge transformations and hence only deform the Lagrangian.

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4 Massless spin 5/2 coupled to EM Now we move on to constructing parity-preserving off-shell cubic vertices for a spin- 5 2 gauge field, which is a symmetric rank-2 tensor-spinor ψ µν . The original free action is where the tensor R µν is related to the spin-5 2 Fronsdal tensor, S µν , as follows.
Here the photon gauge invariance is as usual, while the fermionic part is gauge invariant under a constrained vector-spinor gauge parameter, ε µ , Then, the corresponding Grassmann-even fermionic ghost, ξ µ , must also be γ-traceless: and so will be its antighost. The set of fields and antifields under study are given below.
The free master action, S 0 , takes the form Properties of the various fields and antifields are given in table 3. Note that the spin-5 2 curvature tensor is the 2-curl (see appendix A for its properties), The cohomology of Γ is isomorphic to the space of functions of (see appendix B) • The undifferentiated ghosts {C, ξ µ }, and the γ-traceless part of the 1-curl of the spinorial ghost ξ • The antifields A * µ , C * ,ψ * µν ,ξ * µ , and their derivatives, • The curvatures F µν , Ψ µ 1 ν 1 |µ 2 ν 2 , and their derivatives, • The Fronsdal tensor S µν , and its symmetrized derivatives. Table 3. Properties of the various fields & antifields (n = 2).

Non-Abelian vertices
The set of all possible non-trivial a 2 's falls into two subsets: Subset-1 contains both the bosonic ghost C and the fermionic ghost ξ µ , while Subset-2 contains only ξ µ but not C.
One can easily verify that other possible rearrangements of derivatives or other possible contractions of the indices, e.g., by γ-matrices, all give trivial terms, thanks to the γtracelessness of the fermionic ghost and its antighost. Here, the term Cξ * µ ξ µ corresponds to potential minimal coupling, while the other candidate a 2 's to multipole interactions.
To see which of the a 2 's can be lifted to a 1 , let us solve eq. (2.16). A computation, similar to what leads one from eq. (3.6) to eq. (3.7), shows that both the elements in Subset-1 enjoy such a lift, thanks to the relations (B.9)-(B.10) among others. Explicitly, and similarly for the hermitian conjugate terms. Hereã 1 is the usual ambiguity: Γã 1 = 0. To see whether these could further be lifted to a 0 's, we write It is important to notice that, up to total derivatives, the ∆a 1 's have an expansion in the basis of undifferentiated ghosts, ω I = {C, ξ µ }. Because Γ(∆ã 1 ) = −∆(Γã 1 ) = 0, the coefficients α I in the expansion of the ambiguity will be Γ-cocycles, i.e., they will be "invariant polynomials". Clearly, this is not the case for the unambiguous pieces; in fact, JHEP08(2012)093 their expansion coefficients β I are not even cocycles of H 0 (Γ|d). 5 Schematically, Now, Γa 0 is a pgh-1 object that can be expanded, up to a total derivative, in the basis of ∂ µ C, ∂ (µ ξ ν) . Then, obviously, one can also expand it in the undifferentiated ghosts ω I : One can plug the respective expansions (4.10) and (4.11) for ∆a 1 and Γa 0 into the consistency condition (2.17), and then take a functional derivative w.r.t. ω I = {C, ξ µ } to find that But if this is true, then Γ α I + β I = d(. . .), which is in direct contradiction with the properties of α I and β I , given in (4.10). 6 The conclusion is that none of the a 2 's in Subset-1 can be lifted all the way to a 0 . It is important to notice that this obstruction originates from the very nature of the a 2 's themselves, namely each of them contains both the ghosts. For Subset-2, the analysis simplifies because only one term, C * ξ (1) µν ξ (1)µν , with the maximum number of derivatives, can be lifted to an a 1 . For the other term we have , which is the sum of a non-trivial and a trivial element in the cohomology of Γ, the right hand side of eq. (4.13) cannot be Γ-exact modulo d. Therefore, the candidate C * ξ µ ξ µ is ruled out. However, one finds that thanks to the relation (B.10). Thus, indeed, C * ξ (1) µν ξ (1)µν gets lifted to an a 1 : To see if this a 1 can be lifted to an a 0 , we compute its ∆ variation, This equation bears striking resemblance with its spin-3 2 counterpart eq. (3.10). We recall that, in the latter, cancelation of non-Γ-exact terms was possible by the insertion of identity (3.11) in the contraction of curvatures, the Bianchi identity ∂ [µ F νρ] = 0, and the fermion EoMs in terms of curvature, γ µ Ψ µν = 0, γ µν Ψ µν = 0. In the present case as well, as shown in appendix A, the fermion EoMs can be written as the γ-traces of the curvature, JHEP08(2012)093 γ µ Ψ µν|αβ = 0, γ µν Ψ µν|αβ = 0. Therefore, the non-Γ-exact terms from the unambiguous piece in (4.16) can indeed be canceled by the ∆ variation of a Γ-closed ambiguity, Thus, we have a lift all the way to a 0 , the latter being a 3-derivative non-abelian vertex

Abelian vertices
In this case, all the statements (3.18)- (3.19) hold, and the current in the vertex, a 0 = j µ A µ , is an invariant polynomial, which takes the most general form Notice that the Fronsdal tensor, although allowed in principle, cannot appear in the current simply because it would render the vertex ∆-exact. In view of the spin-5 2 EoMs and the symmetry properties of the field strength, one can show, like in section 3.4, that X µ 1 ν 1 α 1 β 1 λµ 2 ν 2 α 2 β 2 can contain at most one derivative, which must carry one of the indices.
The rank-n tensor-spinor R µ 1 ...µn appearing in the spin-s EoMs is an arbitrary-spin generalization of (4.2); it is related to the Fronsdal tensor as The cohomology of Γ has already been given in section 2.1, with the details appearing in appendix B. One can immediately write down the set of all possible non-trivial a 2 's. Again, they fall into two subsets: Subset-1 contains both the bosonic ghost C and the fermionic ghost ξ µ 1 ...µ n−1 , while Subset-2 contains only ξ µ 1 ...µ n−1 but not C.
• Subset-1 = Cξ * (m) Here, 0 ≤ m ≤ n − 1. As a straightforward generalization of the spin-5 2 case, one finds that each element in Subset-1 gets lifted to a 1 : Now, one can compute ∆a 1 and expand it in the basis of pgh-1 objects in the cohomology of Γ, namely ω I = C, ξ  the expansion coefficients for the unambiguous piece and the ambiguityã 1 , again one can conclude that none of these a 1 's can be lifted to an a 0 . On the other hand, for the elements of Subset-2, one notices that Then, in view of eq. (B.9)-(B.10), it is clear that the right side of the above equation is Γ-exact modulo d only for m = n − 1. This rules out, in particular, the would-be minimal coupling corresponding to m = 0. Therefore, one is left with the lift: whose ∆-variation is given by ∆a 1 = Γ ψ (n−1) µ 1 ν 1 |...|µ n−1 ν n−1 µn F µn νn ψ (n−1)µ 1 ν 1 |...|µ n−1 ν n−1 νn + ∆ã 1 + d(. . .) In view of eq. (3.10) and (4.16), pertaining respectively to the spin-3 2 and spin-5 2 cases, and the subsequent steps, we realize that it is possible to cancel the non-Γ-exact terms in (5.8) by inserting identity (3.11) in the contraction of curvatures, thanks to the Bianchi identity ∂ [µ F νρ] = 0, and to the fermion EoMs in terms of curvature (see appendix A), γ µ 1 Ψ µ 1 ν 1 |...|µnνn = 0, γ µ 1 ν 1 Ψ µ 1 ν 1 |...|µnνn = 0. In other words, ∆a 1 is rendered Γ-exact modulo d by an appropriate choice of the ambiguityã 1 , so that one finally has This (2n − 1)-derivative non-abelian vertex contains the (n − 1)-curl of the fermionic field. For an abelian vertex, a 0 = j µ A µ , the gauge-invariant current does not contain the Fronsdal tensor nor its derivatives, since their presence would render the vertex ∆-exact. Again, non-triviality of the abelian deformation allows only two possible values for the number of derivatives in the vertex: 2n and 2n + 1. The off-shell vertices can be obtained exactly the same way as for spins 3 2 and 5 2 , considered in sections 3.4 and 4.2 respectively. A summary table 6 for all possible 1 − s − s vertices is presented.

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6 Abelian vertices preserve gauge symmetries Abelian vertices are those that do not deform the gauge algebra, i.e., they can only have a trivial a 2 . For such a vertex, it is always possible to choose a 1 to be Γ-closed [34][35][36][37]: If this gets lifted to an a 0 , one has the cocycle condition (2.17), For the 1 − s − s vertices under study, one can always write a vertex as the photon field A µ contracted with a current j µ , which is a fermion bilinear: One can always choose the current such that it satisfies To see this, let us note that the a 1 corresponding to (6.3) has the general form where M and P µ 1 ...µ n−1 belong to H(Γ), with pgh = 0, agh = 1, and a ′ 1 stands for expansion terms in the ghost-curls. Given (6.3) and (6.5), the condition (6.2) reads Γ (j µ A µ ) + ∆M C + ∆P µ 1 ...µ n−1 ξ µ 1 ...µ n−1 −ξ µ 1 ...µ n−1 ∆P µ 1 ...µ n−1 + ∆a ′ 1 + db 0 = 0. (6.6) Now, P µ 1 ...µ n−1 consists of two kinds of terms: one contains the antifield A * µ and its derivatives, and the other contains the antifield ψ * ν 1 ...νn and its derivatives. The former one also contains (derivatives of) the Fronsdal tensor S ν 1 ...νn , or (derivatives of) the curvature Ψ ρ 1 ν 1 |...|ρnνn , while the latter one contains (derivatives of) the EM field strength F µν . One can choose to get rid of derivatives on A * µ and F µν by using the Leibniz rule, where Γp µ 1 ...µn = 0, and the P 's are operators acting to the right. Notice that the quantity in the parentheses in the first line is both Γ-closed and ∆-exact. 7 One can take the ∆variation of (6.7), and then add a total derivative in order to cast it in the form , µ 1 ...µ n−1 + ∂ µn ∆q µ 1 ...µn , (6.8)

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where, ΓQ [µν], µ 1 ...µ n−1 = 0, Γq µ 1 ...µn = 0. Therefore, we havē ..µn +d(. . .). (6.9) The second term on the right side is Γ-closed, and can be broken as a Γ-exact piece plus terms involving the ghost-curls. The latter can always be canceled in the cocycle condition (6.6) by appropriately choosing the similar terms coming from a ′ 1 . Thus, The ∆-exact term added to the original vertex j µ A µ is trivial, and therefore can be dropped. Now we are left with Taking functional derivative w.r.t. C produces part of the sought-after conditions (6.4), Therefore, by field redefinitions, the current can always be made gauge invariant: This completes the proof of (6.4). Then, from (6.2), one obtains the lift: Now we will show that M must be ∆-exact modulo d. We recall that M belongs to the cohomology of Γ, with pgh = 0, agh = 1. It will contain (derivatives of) the fermionic antifield, and (derivatives of) the Fronsdal tensor S ν 1 ...νn or the curvature Ψ ρ 1 ν 1 |...|ρnνn . However, one can choose to have no derivatives of the antifield by using the Leibniz rule. Thus M has the most general form where Γm µ = 0, and the operators M 's act to the right. The first term in the parentheses is manifestly ∆-exact, while the second one must contain either a γ-trace and or a divergence JHEP08(2012)093 of the curvature, which are ∆-exact too (see appendix A). Therefore, M must be ∆-exact modulo d. This means that a 1 , given in (6.16), can be rendered trivial by adding a ∆-exact piece in a 0 [34][35][36][37], and so the vertex will be gauge invariant up to a total derivative: In other words, one can always add a ∆-exact term in a 0 , so that the new current is identically conserved [34][35][36][37]: Thus we have proved that no abelian vertex can deform the gauge transformations.

Comparative study of vertices
We have found that the possible number of derivatives in a 1 − s − s vertex, with s = n + 1 2 , is restricted to the values: 2n − 1, 2n, and 2n + 1. Moreover, the 2n-derivative vertex exists only in D ≥ 5. These are in complete agreement with Metsaev's light-cone-formulation results [11]. While the light-cone vertices are maximally gauge fixed, the corresponding covariant on-shell vertices were also written down in [11] for lower spins, from previously known results. These on-shell vertices are partially gauge fixed, with the gauge choice being the transverse-traceless gauge (TT gauge), ∂ µ 1 ψ µ 1 ...µn = 0, γ µ 1 ψ µ 1 ...µn = 0, ∂ µ A µ = 0. We will find that our off-shell vertices reduce in the TT gauge to the on-shell ones given in [11]. So do the Sagnotti-Taronna (ST) off-shell vertices [16,17], as we will see. If two vertices are shown to match in a particular gauge, say the TT gauge, the full off-shell ones must be equivalent, i.e., differ only by terms that are ∆-exact modulo d. Still, for the simplest case of spin 3 2 , we will make explicit the off-shell equivalence of the ST vertices with ours. For s ≥ 5 2 , we match our vertices with the ST ones in the TT gauge. The ST off-shell vertices, when read off in the most naive way, contain many terms, and it is not straightforward at all to see that some of them actually vanish in D = 4. In comparison, the off-shell vertices we present for arbitrary spin are rather neat in form, and the absence of some of them in D = 4 is obvious from inspection.
We will denote a p-derivative off-shell vertex of ours as V (p) , and its Sagnotti-Taronna counterpart as V TT . 8 8 Both the spins m + 1 2 and m + 3 2 will respectively have one vertex with 2m + 1 derivatives. Our notation should not cause any confusion, as we will be considering one spin at a time.

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in order to be able to pass γ µ past / F in both the terms in the parentheses on the right hand side of eq. (7.11). As a result, we will obtain, among others, the term 1 2/ ψ / F / ψ, in which we replace / F = / ∂ / A − ∂ · A. Now in all the resulting terms we perform integrations by parts such that no derivative acts on the photon field. The final result is This is manifestly ∆-exact modulo d, which proves the equivalence of the off-shell vertices: ST . (7.14) Next, we consider the 2-derivative vertex, One can use the definition Ψ µν = 2∂ [µ ψ ν] to rewrite it as In the 1st term, we can use the 3-box rule, already given in section (3.4), and perform a double integration by parts in order to have a acting on the photon field. In the 2nd term on the right hand side of (7.15) one can integrate by parts w.r.t. any of the derivatives. When the derivative acts on the photon field, one can use ∂ µ A ν = F µν + ∂ ν A µ to rewrite it in terms of the field strength. The result is Now, in the last term of the first line we perform integration by parts so that no derivative acts on the photon field. On the other hand, the last term in the second line is ∆-exact, and therefore can be dropped. Thus we are left with As one reads off the 2-derivative ST vertex, it gives

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As we mentioned already, in this form it is not evident at all that this vertex vanishes for D = 4. Let us integrate by parts the 2nd and 3rd terms in the first line of eq. (7.20), w.r.t. ∂ µ . The 2nd term in the second line and the 1st term in the third line contain the gradient of ∂ · A; we integrate by parts the gradient in both these terms. Thus we have Notice that the 2nd, 4th and 5th terms combine into a total derivative. One can rewrite the 1st and 3rd terms in terms of the photon field strength by using ∂ µ A ν = F µν + ∂ ν A µ . Also, one can extract a ∆-exact piece, by using EoMs: / ∂ ψ µ − ∂ µ / ψ = 0, in the term containing (∂ · A). This leaves us with Again, we integrate by parts the last term of the first line, so that no derivatives act on the photon field. In the second line as well we perform integration by parts to have 2 derivatives acting on the photon field. This finally gives It is clear that, in the TT gauge, both the off-shell vertices (7.19) and (7.23) reduce to which is nothing but the 2-derivative on-shell vertex given in [11]. The equivalence of the two off-shell vertices is also evident as, upon subtracting (7.23) from (7.19), we have Finally, we consider the vertex with 3 derivatives, which reads Integration by parts w.r.t. ∂ µ , appearing in the 1st term inside the first parentheses, gives Here the 2nd term inside the parentheses on the right side is identically zero, while the term containing ∂ µ F µν is ∆-exact. We use the 3-box rule (7.17) in the penultimate term.

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Also we integrate by parts w.r.t. ∂ ν in the last term, and it produces a ∆-exact piece, containing ∂ ν F µν , that we discard. The result is Now, one can perform a double integration by parts in the last term in the brackets in order to have F µν , which gives a ∆-exact piece, so that we finally have On the other hand, the 3-derivative off-shell ST vertex contains as many as 14 terms: Here we will perform a number of integrations by parts. In the first line, we integrate by parts the 1st term w.r.t. ∂ α , the 3rd w.r.t. ∂ µ , and the 4th w.r.t. ∂ ν . In the second line, the 1st, 2nd and 4th terms are integrated by parts respectively w.r.t. ∂ ν , ∂ µ and ∂ α . In the third line, this is done only on the 3rd term w.r.t. ∂ α . Finally, in the fourth line, the 1st and 3rd terms are integrated by parts w.r.t. both ∂ µ and ∂ α , while the 2nd one only w.r.t. ∂ α . Dropping total derivatives, the result is Let us rewrite the first two terms in the first line in terms of the photon field strength by using ∂ µ A ν = F µν + ∂ ν A µ , and use the 3-box rule (7.17) in the additional terms. Also, we notice that the last line in (7.30) reduces exactly to the 2nd term on the second line, up to a total derivative. Then, the vertex reads Clearly, the 2nd term in the second line and the entire third line are ∆-exact, while, modulo ∆-exact pieces, the 1st term in the second line can have ψ ν replaced by ∂ ν ∂ · ψ. The JHEP08(2012)093 latter result can be combined with the 2nd term in the first line to give where we have reached the second step by performing integration by parts w.r.t. ∂ µ in the 2nd term of the first step, and dropping ∆-exact terms containing ∂ µ Ψ µν . In the 2nd term of the second step, one can write Ψ µν = 2∂ [µ ψ ν] , and integrate by parts to obtain, among others, ∆-exact terms containing ∂ µ F µν , which can be dropped. The result is where we have made the rescaling A µ → − 1 4 A µ , for convenience of comparison with our vertex V (3) . One finds that both the vertices reduce in the TT gauge to which indeed is the 3-derivative on-shell vertex reported in [11]. In view of eq. (7.29) and (7.33), one also finds that the two vertices differ by ∆-exact terms: This shows the equivalence of the full off-shell vertices.

1-s-s vertices: s ≥ 5/2
For the sake of simplicity, from now on we restrict our attention to on-shell equivalence of vertices. As we already mentioned, if two vertices match in some gauge, say the TT one, they should also be off-shell equivalent. With this end in view, we read off the ST vertices [16,17], which would generally contain a bunch of terms to begin with even in the TT gauge. However, one can perform integrations by parts to see that actually the on-shell vertices are extremely simple, containing no more than a few non-trivial terms. For example, one can take the 3-derivative 1 − 5 2 − 5 2 ST vertex in the TT gauge, and integrate by parts in order to have one derivative on each field. The result is simply where ∼ means equivalence in the TT gauge up to an overall factor. In the 2nd term we integrate by parts to avoid derivatives on the photon field. We get

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One can make use of the Clifford algebra to write , in the 2nd term on the right hand side of eq. (7.37), and then integrate by parts w.r.t. this derivative. Dropping some ∆-exact terms in the TT gauge, we get Because ∂ · A = 0 in our gauge choice, we can write (∂ ρ A σ ) γ σ γ ρ = − 1 2 / F , by making use of the identity γ σ γ ρ = η σρ − γ σρ . Therefore, we are left with We would like to see how this compares with our 3-derivative 1 − 5 2 − 5 2 vertex, The same steps as took us from eq. (7.3) to eq. (7.5) for the spin-3 2 case lead to Now, one can rewrite the fermionic 1-curl in terms of the original field. There will be terms that have at least one pair of mutually contracted derivatives: one acting onψ µ and the other on ψ µ . For such terms one can make use of the 3-box rule (7.17) to see that they are trivial in the TT gauge. Up to a trivial factor, one then has From eq. (7.39) and (7.42), we see that the two vertices are indeed on-shell equivalent. Let us move on to the 4-derivative 1 − 5 2 − 5 2 vertex. The ST one is found to be whereas our one is given by We rewrite the curvature in terms of the spin-5 2 field. Among the resulting terms those with contracted pair(s) of derivatives are, again, trivial in the TT gauge, thanks to the 3-box rule (7.17). The other terms clearly add up to reproduce the expression (7.43). Therefore, For spin 5 2 , the only other vertex is the 5-derivative one. The ST one reads

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On the other hand, we have the 5-derivative Born-Infeld type vertex: The off-shell equivalence can be understood in view of eq. (3.27)- (3.29), which pertain to spin 3 2 . In the equivalent vertex, again, we rewrite the fermionic curvature in terms of the spin- 5 2 field, and massage the resulting terms the same way as was done for V (4) . Thus, up to overall factors, we reproduce on-shell (7.46), so that For arbitrary spin, s = n + 1 2 , the story is very similar, and there are no further complications other than cluttering of indices. One can write down the ST vertices in the TT gauge from eq. (A.16) of [16,17]. They turn out to be Their similarity with the lower-spin counterparts is obvious. Indeed, setting n = 2 produces exactly the respective 1 − 5 2 − 5 2 vertices given in eq. (7.37), (7.43) and (7.46). One can massage the (2n−1)-derivative vertex, in particular, the same way as its spin- 5 2 counterpart to obtain an arbitrary-spin generalization of eq. (7.39), namely Our arbitrary-spin vertices are also straightforward generalizations of their lower-spin examples. In view of the spin-5 2 counterparts, eq. (7.40), (7.44) and (7.47), one can write Again, one can use the 2nd identity in (7.4) to rewrite the F +µν in the first vertex, and express the fermionic (n − 1)-and n-curls in all the vertices (7.53)-(7.55) in terms of the original field. The terms with contracted pair(s) of derivatives are, as usual, subject to the 3-box rule (7.17), and hence trivial in the TT gauge. One finds that our vertices indeed reduce on shell respectively to (7.52), (7.50) and (7.51). This proves the on-shell (and therefore off-shell) equivalence of the 1 − s − s vertices: ST , p = 2n − 1, 2n, 2n + 1. We recall that consistent second-order deformation requires (S 1 , S 1 ) to be s-exact: For abelian vertices, this antibracket is zero, so that the first-order deformations always go unobstructed. Non-abelian vertices, however, are more interesting in this respect.
We can see that there is obstruction for the non-abelian vertices we have obtained, which do not obey eq. (8.1). We prove our claim by contradiction. If eq. (8.1) holds, then the most general form of the antibracket evaluated at zero antifields is Note that M is obtained by setting to zero only the antighosts in S 2 . Furthermore, the equality (8.2) holds precisely because S 2 is linear in the antifields. The Γ-variation of (8.2) is therefore ∆-exact: It is relatively easier to compute the left hand side of (8.3) for our non-abelian vertices. For spin 3 2 , we recall that To compute the antibracket of S 1 = (a 2 +a 1 +a 0 ) with itself, we notice that a field-antifield pair shows up only in a 1 , and between a 0 and a 1 , so that it reduces to (S 1 , S 1 ) = 2 a 0 , a 1 + a 1 , a 1 .
Now, the second antibracket on the right hand side necessarily contains antifields, while the first one will not contain any. Thus we have Notice that, while the unambiguous piece in a 1 contains the antifield A * µ , the ambiguity,ã 1 , contains instead the antifieldψ * µ . Correspondingly, a 0 , a 1 will contain two distinct kinds of pieces: 4-Fermion terms and Fermion bilinears. Explicitly, second line of eq. (8.8), originating from a 0 , ã 1 . It is easy to see that their Γ-variation is not ∆-exact. We conclude that the non-abelian 1 − 3 2 − 3 2 vertex gets obstructed beyond the cubic order. The proof for arbitrary spin will be very similar.
Notice that the vertexψ µ F +µν ψ ν is precisely the Pauli term appearing in N = 2 SUGRA [6,7]. The theory, however, contains additional degrees of freedom, namely graviton, on top of a complex massless spin 3 2 and a U(1) field. It is this new dynamical field that renders the vertex unobstructed, while keeping locality intact. If one decouples gravity by taking M P → ∞, the Pauli term vanishes because the dimensionful coupling constant is nothing but 1/M P [6,7]. One could integrate out the massless graviton to obtain a system of spin-3 2 and spin-1 fields only. The resulting theory contains the Pauli term, but is necessarily non-local. Thus, higher-order consistency of the non-abelian vertex is possible either by forgoing locality or by adding a new dynamical field (graviton).

Remarks & future perspectives
In this paper, we have employed the BRST-BV cohomological methods to construct consistent parity-preserving off-shell cubic vertices for fermionic gauge fields coupled to EM in flat space. We have shown that consistency and non-triviality of the deformations forbid minimal coupling, and pose number-of-derivative restrictions on a 1 − s − s vertex, in accordance with Metsaev's light-cone-formulation results [11].
The vertices either deform the gauge algebra or, when they do not deform the gauge algebra, turn out not to deform the gauge transformations either and to deform only the Lagrangian. The non-abelian ones get obstructed in a local theory beyond the cubic order in the absence of additional higher-spin gauge fields.
Our off-shell cubic vertices are equivalent to the string-theory-inspired ones of Sagnotti-Taronna [16,17]. Note that in [16,17] there appears just one dimensionful coupling constant, which can be set to unity. Then, each cubic vertex will come with a fixed known numerical coefficient. This is apparently in contrast with our results, where each of the three 1 − s − s vertices has an independent coupling constant. However, it is well known that higher-order consistency requirements may impose restrictions on the cubic couplings by relating them with one another [34][35][36][37]. Because the consistency of string theory is not limited to the cubic order, then it should not come as a surprise that the cubic vertices it gives rise to have no freedom in the coupling constant. At any rate, string theory may not be the unique consistent theory of higher-spin fields. If this is true, other possible choices of the cubic couplings would pertain to other consistent theories.
The number of possible 1−s−s vertices for fermions differ from that for bosons. While in both cases there is only one non-abelian vertex, fermions have, beside the usual Born-Infeld type 3-curvature term, another abelian vertex in D ≥ 5, which is gauge invariant up to a total derivative. In this respect, fermionic 1 − s − s vertices are strikingly similar in nature to the bosonic 2 − s − s ones. The latter include one non-abelian (2s − 2)-derivative vertex, a 2s-derivative abelian one that is gauge invariant up to a total derivative and exists in D ≥ 5, and a Born-Infeld type abelian one containing 2s + 2 derivatives. This could be seen by using either the light-cone method [10] or the cohomological methods [8,9].

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For gravitational coupling, spin 3 2 has no consistency issues, but consistent deformations of the free theory uniquely lead one to N = 1 SUGRA [39], under certain reasonable assumptions. Fermionic gauge fields with higher spin, s ≥ 5 2 , and their coupling to gravity are more interesting, for which one can also employ the BRST-deformation technique [40]. Another interesting avenue to pursue are the mixed-symmetry fields.
It is instructive to consider the EM coupling of massive higher-spin fields in flat space, which has been discussed by various authors. 9 If Lorentz, parity and time-reversal symmetries hold good, a massive spin-s particle will have 2s + 1 EM multipoles [51]. This immediately sets for the possible number of derivatives in a 1 − s − s vertex an upper bound, which remains the same in the massless limit. The assumption of light-cone helicity conservation in D = 4 uniquely determines all the multipoles [51]. However, only the highest multipole survives in an appropriate massless chargeless scaling limit. This observation is in harmony with our results, since any of our lower-derivative vertices either vanishes in 4D or is not consistent by itself in a local theory.
On the other hand, causal propagation of a charged massive field may call for certain non-minimal terms. Indeed, for a massive spin 3 2 in flat space, causality analysis in a constant external EM background [52] or in the case of N = 2 broken SUGRA [53][54][55][56] reveals the crucial role played by the Pauli term,ψ µ F +µν ψ ν . In the massless case, as we have seen, the same term arises as the unique non-abelian deformation of the gauge theory. These facts go in favor of the gauge-invariant (Stückelberg-invariant) formulation, adopted in [47][48][49][50], for constructing consistent EM interactions of massive higher spins.
Our non-abelian vertices are seen to be inconsistent beyond the cubic order in a local theory. Such obstructions are rather common for massless higher-spin vertices in flat space, and some could not even be removed by the inclusion of an (in)finite number of higher-spin fields, as has been argued in [57]. Non-locality may therefore be essential. In fact, as noticed in [58], evidence for non-locality shows up already at the quartic level. The geometric formulation of free massless higher spins also hints towards the same, as they generically yield non-local EoMs [59,60] if higher-derivative terms are not considered [61,62].
If one has to give up locality, what becomes relevant for studying higher-spin interactions is a formulation that does not require locality as an input, e.g., the old S-matrix theory, or perhaps the more powerful BCFW construction [63] and generalizations thereof. The latter seem promising for the systematic search of consistent interactions of massless higher-spin particles in 4D Minkowski space [64][65][66][67].
There are certain technical difficulties in extending the applicability of the BRST-BV cohomological methods to constant curvature spaces. For AdS space, in particular, those could be avoided by using the ambient-space formulation [68][69][70]. If so, one would be able to construct off-shell vertices for AdS, and compare them with the recently-obtained results of [71][72][73][74][75][76]. This would be one step towards finding a standard action for the Vasiliev systems [77][78][79][80], which are a consistent set of non-linear equations for symmetric tensors of arbitrary rank in any number of dimensions. We leave this as future work. (conventions 4.4511.06 and 4.4514.08) and by the "Communauté Française de Belgique" through the ARC program.
These symmetries actually hold good for any m-curl of the field, m ≤ n. Another important property of the curvature is that it obeys the Bianchi identity The (Weinberg) curvature (A.1) and the EM field strength F µν are useful in casting the EoMs into a variety of forms, which, among others, can help one identify ∆-exact pieces. First, we write down these various forms for the photon field. Next, we do the same for spin 3 2 , explaining as well how to derive them and writing them explicitly as ∆-variations. Then we move on to spin 5 2 , and finally to arbitrary spin.

A.1 The photon
The original photon EoMs are given by One can take its 1-curl to obtain

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From the definition (1.1) of the Fronsdal tensor, it is easy to see that whose contraction with γ ν 1 on the left gives Also, a curl of eq. (A.24) yields Finally, from eq. (A.24) and (A.25) one obtains

B The cohomology of Γ
This appendix is devoted to clarifying and providing proofs of the statements about the cohomology of Γ appearing in section 2.1. We recall that the action of Γ is defined by Γψ ν 1 ...νn = n∂ (ν 1 ξ ν 2 ...νn) . (B.1b) Note that the non-trivial elements in the cohomology of Γ are nothing but gauge-invariant objects that themselves are not gauge variations of something else. Here we consider one by one all such elements enlisted in section 2.1. In the process, we also prove the statements made towards the end of section 2.1 about some Γ-exact terms.

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In particular, when m = n, we have the Γ-variation of the curvature, which vanishes: Γ Ψ µ 1 ν 1 |...|µnνn = 0. (B.5) Notice that the Γ-closedness of the curvature does not require any constraints on the fermionic ghost. That the curvatures are not Γ-exact simply follows from the fact that these are pgh-0 objects, whereas any Γ-exact piece must have pgh > 0. Therefore, the curvatures are in the cohomology of Γ, and so are their derivatives.
We have seen that only the highest curl (n-curl) of the spinor ψ ν 1 ...νn is Γ-closed, while the lower curls are not. The key point is the commutativity of partial derivatives, and clearly, any arbitrary derivative of the field will not be Γ-closed in general. However, some particular linear combination of such objects (or γ-traces thereof) can be Γ-closed under the constrained ghost. The latter possibility is exhausted by the Fronsdal tensor and its derivatives, which we will discuss later.

B.3 The ghosts & curls thereof
The undifferentiated ghosts {C, ξ µ 1 ...µ n−1 } are Γ-closed objects simply because Γ does not act on them. Also they cannot be Γ-exact, thanks to eq. (B.1), which tells us that any Γ-exact piece must contain at least one derivative of any of the ghosts.
Any derivatives of the ghosts will also be Γ-closed. Some derivatives, however, will be Γ-exact, and therefore trivial in the cohomology of Γ. One can immediately dismiss as trivial any derivative of the bosonic ghost C, because ∂ µ C = ΓA µ from eq. (B.1a).
Derivatives of the fermionic ghost 10 are more subtle. One can show that only the γtraceless part of the curls of the ghost {ξ (m) µ 1 ν 1 |...|µmνm ν m+1 ...ν n−1 , m ≤ n − 1} are non-trivial elements in the cohomology of Γ. First, one can convince oneself step by step why only the ghost-curls are interesting. In the simplest non-trivial case of n = 2, we see that For n = 3, we find It is easy to generalize this to the arbitrary spin case, for which we obtain Although any m-curl, ξ (m) µ 1 ν 1 |...|µmνm||ν m+1 ...ν n−1 , is in the cohomology of Γ, its γ-trace is always Γ-exact. In fact, the latter vanishes when the γ-matrix to be contracted carries one of the unpaired indices {ν m+1 , . . . , ν n−1 }, thanks to the γ-tracelessness of the ghost. If the index contraction is otherwise, the same constraint gives Comparing eq. (B.11) with (B.12), it is clear that γ µ 1 ξ (m) µ 1 ν 1 |...|µmνm ν m+1 ...ν n−1 is Γ-exact. Therefore, one can exclude the γ-traces of ghost curls from the cohomology of Γ.

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which vanishes if the ghost is γ-traceless. Being a pgh-0 object, S µ 1 ...µn can also not be Γ-exact. So, the Fronsdal tensor and its derivatives belong to the cohomology of Γ. However, in view of eq. (A.24) and (A.26), the two highest curls of the Fronsdal tensor boil down to objects we have already enlisted in subsection B.1, and therefore need not be considered separately. These equations are generalizations of the Damour-Deser relations [81] (see also [61,62]). For spin 5 2 , in particular, they make it sufficient to consider only symmetrized derivatives of the Fronsdal tensor.
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