Off-Shell Fields and Conserved Currents

We study interactions of higher-spin massless fields $\varphi$ with conserved currents multilinear in the off-shell matter fields $\phi$. Specifically, we focus on the 3d case where a slight modification of the $\sigma_-$-cohomology technique developed earlier is directly applicable to control nontriviality of the interaction vertices at the convention that the vertices removable by a local field redefinition of a higher-spin field or having schematic form $F(\varphi) G(\phi)$, where $F(\varphi)$ is a gauge invariant field strength of a free higher-spin field, are called deformationally trivial. It is demonstrated how the $\sigma_-$-cohomology approach can be applied to the analysis of nonlinear vertices. Generally, deformationally trivial vertices are not $\sigma_-$-closed while the deformationally non-trivial ones must be in $H(\sigma_-)$. It is shown that, at least in the $3d$ case, the relevant cohomology group $H^1(\sigma_-)=0$ and, hence, no deformationally non-trivial off-shell vertices exist. On the other hand, there exists an infinite class of deformationally trivial vertices, that includes the vertex recently proposed for spin three. Our analysis goes beyond the higher-spin vertices allowing to show that, at least in three dimensions, nonlinear combinations of the off-shell scalar fields and their derivatives cannot obey non-trivial equations.

It is well known how to construct interactions of higher-spin (HS) massless fields with conserved currents J(φ) bilinear in the fluctuating fields φ of various spins (see e.g.[2,3,4]).In that case, the HS currents J(φ) are conserved on shell, i.e. at the conditions that the fields φ obey their free field equations.
In the paper [5] a conserved spin-three current T µνλ [φ] trilinear in the off-shell scalar field φ was found within the BV-formalism.An off-shell field is a field that obeys no equations of motion.Nevertheless, multilinear combinations of derivatives of such fields, which we call composite fields, still can satisfy some partial differential equations.For instance, such equations may be associated with the gauge invariance.In particular, equations for the spin-three current T µνλ , built from an off-shell scalar φ, found in [5] 1 , imply the gauge invariance of the quartic spin-three spin-zero vertex under the gauge transformations where ξ µν is a traceless gauge parameter, = ∂ µ ∂ µ , and η µν is the Minkowski metric in d dimensions.This implies the current conservation equation Note that, though in our case the field φ is off-shell, the same gauge-invariant vertex exists for an on-shell scalar field with the standard kinetic term in the Lagrangian.Analogous interaction vertices both in Minkowski and AdS space were studied more generally in [6,7,8].According to these works such vertices must be either on-shell trivial (i.e., removable by a local field redefinition), or composed from the gauge invariant field strengths in which case the gauge transformations remain undeformed (Abelian).Both types of such vertices will be called deformationally trivial to stress that they do not induce a deformation of the HS gauge symmetry.Note that cubic vertices for arbitrary spins in 3d Minkowski space were classified in [9].
In this paper we develop a technics allowing to check whether there exist composite fields built from the off-shell fields, that obey some non-trivial partial differential equations including the conservation conditions like (1.6).This problem will be analyzed in terms of the σ − -cohomology technique [10] allowing to control the dynamical content of the theory in question.Note that this approach is free from the additional conjectures on the possibility to use non-local inverse operators as in [6,7,8].
Those components T pr (x) of the (composite) fields, that cannot be expressed via derivatives of the other components, are called dynamical (primary).Primary components of the fields described by the p-forms are represented by the σ − -cohomology H p (σ − ) [10].The primary components may or may not obey some differential equations represented by H p+1 (σ − ).The descendants T dec = L(∂)T pr result from the action of various differential operators on the primary components.There are always some descendant fields that obey certain differential equations (e.g., Bianchi identities) expressing the properties of the operators L(∂).Such equations are in a certain sense trivial expressing the properties of L(∂) rather than T pr (x).These are not represented by any σ − -cohomology.Specifically, in the HS theory, there are two types of the operators L(∂) leading to the gauge invariant current interactions.These are L F (∂) associated with the l.h.s. of the Fronsdal equations and L W (∂) associated with the gauge invariant combinations of the Fronsdal fields like Faradey tensor in the spin-one case or Weyl tensor in the spin-two case.The terms with L F (∂) are removable by appropriate field redefinitions in the free action.Those with L W (∂) describe interactions with the gauge invariant field strengths.These are not removable by a local field redefinition but do not affect the gauge transformation law, hence called deformationally trivial.
Specifically, the σ − -cohomology analysis of the so-called higher-rank fields to be used in this paper, has been elaborated in [4] for the 3d case.In these terms, a 3d off-shell scalar field can be realized as a rank-two field while its cubic combination is a rank-six field.However, the approach developed in [4] for fields of any rank operates with the lowest weight vectors of the Howe-dual algebra (for more detail see Section 4) while, as discussed in Section 3, the off-shell scalar fields carry zero weights, that requires some modification of the scheme.Our final result shows that the σ − -cohomology representing possible non-trivial partial differential equations (in particular, conservation conditions) obeyed by cubic combinations T µνλ [φ] of the 3d off-shell scalar fields φ is empty.In turn, in agreement with [6,7,8], this implies that T µνλ [φ] cannot be a primary conserved current.Hence, it has to be an improvement while the vertex (1.4) must be removable by a local field redefinition.The explicit form of the latter is found in Section 5.
The aim of this paper is to illustrate how the σ − -cohomology technique can be used for the analysis of (non-)triviality of vertices.As a consequence it will be shown that, at least in three space-time dimensions, there are no primary composite fields, that obey non-trivial equations (including, the conservation conditions) built from any number of the off-shell scalar fields and their derivatives as well as from HS conserved currents.The rest of the paper is organized as follows.The σ − -cohomology technique is recalled in Section 2. In Sections 3 and 4, the analysis of the σ − -cohomology groups H 0 (σ − ) and H 1 (σ − ) is presented for the composite fields built from the 3d off-shell scalar fields and HS conserved currents.In Section 5, our results are compared against those of [5].Section 6 contains conclusions.Some technical details of the σ − -cohomology analysis are in Appendix.

σ − -cohomology
In this section, we briefly sketch relevant aspects of the HS theory and σ − -cohomology approach referring for more detail to [11].
The Sp(2M)-invariant equations, that describe 3d and 4d massless fields at M = 2 and M = 4, respectively, are [12] where y α are auxiliary commuting variables, dX αβ are anticommuting differentials for symmetric matrix space-time coordinates X αβ = X βα and α, β = 1, . . ., M. In this paper we confine ourselves to the case of M = 2 associated with a 3d massless theory.Let us recall some terminology.A field is called auxiliary if it is expressed in terms of other fields and their derivatives by virtue of equations of motion or can be gauge fixed to zero by some shift (Stueckelberg) gauge symmetry.The dynamical sector consists of the fields that are neither expressed in terms of the other fields nor are pure gauge.Thus, it makes sense to focus on the dynamical fields and their equations of motion.For instance, Fronsdal fields are of that type.
It can be shown [10] that dynamical fields and their field equations associated with the equations (2.1) are contained in zeroth cohomology H 0 (σ − ) and first cohomology H 1 (σ − ), respectively.Recall that where | p implies restriction to the space of p−forms.That dynamical fields are in zeroth cohomology is fairly simple.Indeed, it is obvious from (2.1) that the fields, that are not in ker σ − , are expressed in terms of space-time derivatives of the fields from ker σ − .
The fields (2.2) describe a so-called rank-one system.The rank-r fields are where i k = 1 . . ., r.The rank-two system with was studied in [13] where it was shown, in particular, that if the operator has zero eigenvalue then the corresponding fields are off-shell scalars.We will consider them as off-shell fields from which the current (1.3) is built.
In our case the relevant operator σ − is (2.7) We are interested in the trilinear combinations of scalar off-shell fields because the current (1.3) is cubic in them.Each of the three pairs of variables y ±,i is associated with its own scalar field, that in addition obeys the restriction This is the same restriction as (2.5) in the rank-two case.
Since the operators h j commute with σ − , the fields corresponding to the different eigenvalues of h j form different representations.Hence, each eigenvalue corresponds to its own subsystem.In [13] it was shown that, generally, such systems describe conserved currents of various spins, that include currents of spin zero satisfying no conservation conditions thus being off-shell.So, we analyze a three-linear current built from the spin zero fields, that obey (2.8).
In the end of this section let us recall the Poincaré homotopy trick conventionally used to calculate the cohomology group.Let D be a linear operator acting in a Hilbert space V , such that D 2 = 0. Let there also be a nilpotent operator D * , (D * ) 2 = 0.The homotopy operator is ∆ := {D, D * } .As shown in [4] for H p (σ − ) with any p formula (2.12) is an equality, H(D) = ker ∆ ∩ ker D.
3 H 0 (σ − ) Here we analyze H 0 (σ − ) with σ − (2.7), where the index i ranges from 1 to 3 because we are interested in the cubic combinations of the off-shell scalar fields.The variables y ±,i with different i are associated with different off-shell scalar fields.Let us rewrite σ − in the form that commute with σ − , [σ − , τ mn ] = 0 .
In the original index notations τ mn read as ) ) ) with mutually commuting Cartan elements Introducing operators y α i and sp(2M) can be represented in the form Being mutually commuting, algebras o(3, 3) and sp(2M) form a Howe dual pair [14].
The operator σ * − conjugated to σ − is defined analogously to [4]: An elementary computation yields where 1 2 τ mn τ mn is the quadratic Casimir of o(3, 3), To calculate the part of H 0 (σ − ) associated with the off-shell scalar fields we (i) choose a Chevalley basis in o(3, 3), consisting of raising e i , lowering f i and Cartan operators h i , (ii) find the lowest vectors annihilated by the lowering operators, (iii) act by the raising operators on the lowest vectors to obtain the vectors annihilated by the Cartan operators Once |vac ∈ ker ∆, from the commutativity of o(3, 3) and sp(2M) it follows that any vector F (e 1 , e 2 , e 3 )|vac also belongs to ker ∆.Hence all solutions of (3.22) also belong to The Chevalley decomposition is (α i are simple roots)

.23)
In this basis the commutation relations are where a ij is the Cartan matrix, In the case of interest it reads as and the Chevalley basis is where µ i are the lowest weights.The idea is to analyze the problem for general µ i , setting µ i = 0 in the end.
According to the method of characteristics, the general solution of a system of first-order equations is a function of elementary solutions.The elementary solutions to the first three equations (3.29) are The proof that these are indeed elementary solutions will be given in Appendix A.
The solution G 1 is obvious because there are no y α +,2 derivatives in (3.29).The solution G 2 is irrelevant because it does not belong to the kernel of σ − while G 1 does hence belonging to H 0 (σ − ).Now we are in a position to analyze how the raising operators act on this solution.As mentioned above, functions of elementary solutions also satisfy the set of equations.So we will take for example ψ Searching through all the possible combinations of raising operators, we find the solutions, annihilated by the Cartan operators: e 2 e 1 e 3 e 1 ψ α( 2) A linear combination of these solutions can be written as also belongs to H 0 (σ − ).Thus, checking all possible combinations of the raising operators acting on the vacuum solutions (y α +,2 ) n , new functions, annihilated by Cartan operators, can be found, associated with the off-shell scalar field , In the case of H 1 (σ − ) we consider the lowest weight sector following [4], where this problem has been solved.Note that in [4] τ mn were representing algebra o(6) rather than o(3, 3) as in this paper.However, for the finite-dimensional modules, the representation theory of orthogonal Lie algebras is equivalent (up to possible (anti)selfduality conditions irrelevant to the analysis of this paper) to that of the pseudo-orthogonal ones.This allows us to use the results of [4] directly.In [4], the general case of sp(2M) and o(r) was considered while in our problem we confine ourselves to M = 2, implying d = 3, and r = 6, implying that the current is trilinear in the off-shell scalars.
The problem is to solve the homotopy equation (2.13) with the homotopy operator (3.18) expressed as a sum of quadratic Casimirs of o(r) and gl(M).The representation is described in terms of the gl(M) Young diagrams (YD) which determine the symmetry of the l.h.s.'s of equations of motion and dynamical fields.These gl(M) YD are associated with traceless tensors with respect to the color indices a, b in (3.1) that have the same symmetry properties as the gl(M) YD.We say that the respective traceless symmetric o(r) tensors are described by the traceless symmetric YD.
In [4] it was shown that H 1 (σ − ) consists of the traceless tensors described by the YD Y 1 (n 1 , . . ., n m , 2, . . ., 2 r+1−2m−q , 1, . . ., 1 q ), where 2m + q r.In our case of r = 6 these are 1) m=0, q=0. . .6, . . ., According to the two-column theorem (see, e.g., [15]) traceless tensors corresponding to Young diagrams, in which the sum of the heights of the first two columns exceeds r, vanish.This implies that all these diagrams are trivial, which manifests the comment from [4] that for M < r many diagrams trivialize.In turn, this implies that, in the case of interest, H 1 (σ − ) = 0. Therefore, there are no non-trivial equations obeyed by the trilinear combinations of the off-shell scalar fields.
Naively, this contradicts the results of [5].Before explaining the origin of this seeming disagreement in Section 5 let us make the following comment.It is easily observed that such YD are trivial also for higher ranks (r > 6), which stands for arbitrary nonlinear combinations of currents.Indeed, the sum of lengths of the first two columns l of diagrams Y 1 (n 1 , . . ., n m , 2, . . ., 2 r+1−2m−q , 1, . . ., 1 q ) is l = 2r + 2 −2m−q.Taking into account that 2m+ q r it is obvious that l > r.Thus, all these diagrams and corresponding equations on the multilinear composite fields must be trivial.It also should be mentioned that H 1 (σ − ) describes the equations of motion for fields of arbitrary spin.According to [13], the subsystems associated with different eigenvalues of operators h i (2.8) describe conserved currents of different spins.That H 1 (σ − ) is empty means that actually there are no dynamical equations not only for multilinear combinations of scalar currents, but also for multilinear combinations of spin-s conserved currents in three dimensions.

Spin three example
Thus it is shown that composite fields in question obey no differential equations (to single out the current of definite spin, for instance spin 3, one has to restrict H 0 (σ − ) to the polynomials of the corresponding degree in y α ±,i ).On the other hand, the current (1.3) found in [5] does obey the conservation condition.This seeming contradiction can be resolved however if the current (1.3) is actually deformationally trivial in which case the action term (1.4) can be compensated by a local field redefinition of the free Fronsdal action.
(5.3)Note that T dec µνλ (5.2) is a descendant according to the terminology of Section 1.This implies that there exists such a local transformation of the spin-three field ϕ µνλ that eliminates the vertex (1.4).Indeed, by substituting ϕ µνλ → ϕ µνλ + δϕ µνλ (5.4) into the Fronsdal action and using that, as one can easily check, it is a symmetric functional, one observes that the substitution (5.4) induces the vertex (1.4).
Clearly, there exists an infinite number of such transformations and corresponding nonlinear conserved currents, because an arbitrary combination of derivatives and fields P α 1 α 2 ... [∂ α ; φ] would produce a current in the same way as for T dec µνλ .Being removable by local field redefinitions all such vertices are dynamically trivial, however, which conclusion is in agreement with the results of [6,7,8].
Also let us stress that the resulting currents, including that of (5.3), are of the form L F (∂)T pr discussed in Introduction, where L F (∂) is the Fronsdal operator on the RHS of (5.1).

Conclusion
In this paper, interaction vertices of spin-s massless fields and composite fields identified with nonlinear combinations of other fields φ, that may obey no field equations thus being off-shell, are considered.It is illustrated how the σ − -cohomology technique can be applied to the analysis of the interaction vertices.To this end, composite fields are interpreted as primaries of the tensor products of the unfolded modules associated with φ.It is explained that, in general, there may be two types of vertices.The deformationally trivial ones, that do not demand the deformation of the transformation law, are associated with the descendants of the unfolded vertex modules.The non-trivial vertices, associated in particular with usual conserved currents, are in the σ − -cohomology.
For simplicity we focus on the case of three dimensions, though some general conclusions are made for any dimension.The σ − -cohomology technique, which allows one to control the dynamical sector of the theory in question, was applied to find out whether composite fields obey any non-trivial equations of motion as well as the nontriviality of their interaction vertices.As explained in Section 2 the problem amounts to the solution of the homotopy equation with the restriction (2.8) that the eigenvalues of the operators h i are zero.The key observation, that reduces the problem to the Lie algebra representation theory, is that h i form the Cartan subalgebra of the o(3, 3) Howe dual [14] to the space-time symmetry algebra.It is explained in detail how to calculate H 0 (σ − ) in order to find the independent composite fields in the theory.Howe duality plays instrumental role in this analysis.
Using the results of [4] we have shown that the first cohomology is empty, H 1 (σ − ) = 0, which means that there are no non-trivial equations of motion obeyed by the composite fields.In particular, this implies that the conserved current of [5] must be deformationally trivial, that was checked directly in any dimension.Moreover, that H 1 (σ − ) = 0 implies that no nontrivial equations of motion exist for all (not only trilinear) multilinear combinations of spin-s currents with spin-zero current identified with the off-shell scalar field.Thus, all possible interaction vertices between spin-s fields and multilinear currents must be deformationally trivial at least in three dimensions.In agreement with the earlier papers [6,7,8], we anticipate this result to be true in any dimension.Note however that our analysis is free from the conjectures of [6,7,8] on the possibility to use certain non-local operators in the proof.Another comment is that our results go far beyond the specific case of the conservation conditions related to HS dynamics, stating that primary components of the composite combinations of the off-shell fields and their derivatives cannot obey any nontrivial partial differential equations at least in three dimensions.
In the first order in y