Unified formulation for helicity and continuous spin fermionic fields

We propose a unified BRST formulation of general massless fermionic fields of arbitrary mixed-symmetry type in $d$-dimensional Minkowski space. Depending on the value of the real parameter the system describes either helicity fields or continuous spin fields. Starting with the unified formulation we derive a number of equivalent descriptions including the triplet formulation, Fang-Fronsdal-Labastida formulation, light-cone formulation and discuss the unfolded formulation.


Introduction and summary
Continuous spin fields provide an interesting example of field theory systems with an infinite number of physical degrees of freedom [1,2] (for recent review see [3]). Grouptheoretically, continuous spin particles are unitary representations of Poincaré algebra iso(d−1, 1), induced from unitary representations of the stability subalgebra iso(d−2) ⊂ iso(d − 1, 1). 1 Contrary to the standard helicity fields where one induces from finitedimensional unitary representations of the little algebra o(d − 2), the continuous spin representations correspond to inifinite-dimensional iso(d − 2)-modules.
A continuous spin parameter denoted by real number µ is an eigenvalue of the squared iso(d − 2) momentum or of the quartic iso(d − 1, 1) Casimir operator [5]. Remarkably, the standard mass parameter associated to the quadratic iso(d − 1, 1) Casimir operator is zero, m = 0, so that the continuous spin fields are massless fields simultaneously characterized by a dimensionful parameter µ. 2 Since the original Wigner's equations for the continuous spin fields were proposed [1,2], several interesting descriptions were developed both at the level of equations of motion [7][8][9][10][11] and of the action functional [12][13][14][15][16][17][18][19][20]. A characteristic feature of these formulations is that the space of fields is infinite-dimensional [7] in accord with inifinite dimensionality of the respective little group representation.
The relation between the continuous spin and usual helicity fields becomes manifest within the Schuster-Toro formulation [12,15]. In this approach a single continuous spin field is described by an infinite collection of Fronsdal tensors with ranks running from zero to infinity, making it quite similar to the standard interacting higher spin theory [21,22] (for a review see e.g. [23]) whose free limit is an infinite tower of the helicity spin fields. 3 In particular, for vansiihing continuos spin parameter µ = 0 Schuster-Toro system decomposes into an infinite collection of decoupled Fronsdal spins fields with s = 0, 1, 2, ..., ∞. In this regard, the continuous spin field is somewhat similar to the standard Minkowski space massive field which also decomposes into a collection of massless ones in the zero mass limit, see e.g. [26]. One can also draw an analogy with a generic massless higher spin field in AdS that in the flat limit decomposes into a collection of Minkowski space massless fields [27].
A rather concise unified BRST formulation of the bosonic continuous spin fields that explicitly manifests all of the above features has been proposed recently in [10]. It is based on a constrained system which is a deformation of the one employed in studying mixed symmetry helicity fields [28]. This formulation is suitable for analyzing the content of the system through studying its BRST cohomology. In particular, in 1 See e.g. [4] for a review of the Poincaré representations relevant in the present context. 2 In this respect, the original term "continuous spin" is somewhat misleading because such systems behave like massive ones. For instance, they can be obtained through a dimensional reduction of the standard higher spin massive systems where the mass m → 0 and the spin s → ∞, while a combination ms is kept finite [6,7]. 3 Note that continuous spin fields themselves can consistently interact with massive higher spin fields, at least in the cubic order [24,25]. this way it was shown that with the naive choice of the functional class in the sector of auxiliary variables the system is pure gauge, i.e. there are no degrees of freedom. Nevertheless, it turns out that with the proper choice of the functional class the system indeed describes proper degrees of freedom [10].
An additional attractive feature of the unified formulation is that its different reductions reproduce various other equivalent formulations including the metric-like [7,12] and frame-like [16,17,29] formulations as well as the light-cone formulation [24].
In this paper we propose a fermionic extension of the unified BRST formulation of both the helicity and continuous spin fields. 4 Just like in the bosonic case underlying the formulation is the constrained system whose constraints belong to a subalgebra of osp(1|2n) superalgbera in the representation where osp(1|2n) and o(d − 1, 1) form a reductive dual pair in the sense of Howe [46].
The paper is structured as follows. There are two main parts divided between the helicity and the continuous spin cases. In Section 2 we describe o(d − 1, 1)-osp(1|2n) bimodule (which is also a Poincaré one) on the of functions of auxiliary (anti)commuting variables, which serves as a representation space of the constrained system. In Section 3 we formulate one-parameter constraint system such that the helicity and the continuous spin fields correspond to different values of the parameter. In two main Sections 4 and 5 we build the triplet, metric-like, light-cone formulations for respectively helicity and continuous spin fermionic fields. The analysis of BRST cohomology is performed in Section 6. Two Appendices discuss various aspects of the space-time and symplectic (super)algebras.
2 Algebraic preliminaries 2.1 Spinor-tensor fields Let us introduce Grassmann even variables a a I andā J b , where a, b = 0, ..., d − 1, I, J = 0, ..., n and Grassmann odd variables θ a satisfying the canonical commutation relations where η ab = (− + · · · +) is the Minkowski tensor. These variables generate the associative algebra which is then promoted to the operator algebra of a quantum constrained system.
Consider the space P d n (a I ) = S ⊗ C[a I ], where S is the Dirac representation of the Clifford algebra generated by θ a and C[a I ] is the space of polynomials in a a I . In terms of components elements of P d n (a I ) have the form where m I ≡ (m 0 , ..., m n ) are arbitrary non-negative integers, e α is a basis in S, and α = 1, ..., 2 [d/2] is the Dirac spinor index. It is also useful to regard P d n (a I ) as the space of polynomial functions in a a I with values in S. The associative algebra generated by a a I ,ā J b and θ a can be represented on the linear space P d n (a I ) in a natural way if one defines the action of the generators according to where the gamma-matrices (γ a ) α β are defined in terms of the basis e α in S as θ a e β = (γ a ) α β e α .

Lorentz algebra and orthosymplectic superalgebra
The Lorentz algebra so(d − 1, 1) can be embedded as a Lie subalgebra into the above operator algebra by postulating This also defines a representation of so(d−1, 1) on P d n (a I ). It follows that the expansion coefficients in (2.2) transform as Lorentz spinor-tensors.
Simultaneously, the orthosymplectic superalgebra osp(1|2n + 2) can also be embedded into the operator algebra and hence is also represented on P d n (a I ). The even basis and the odd basis elements are given respectively by and with the graded commutation relations given in Appendix A. The space P d n (a I ) is now so(d − 1, 1) − osp(1|2n + 2) bimodule. The two algebras mutually commute forming a reductive dual pair [46].

Poincaré algebra
The Poincaré algebra iso(d − 1, 1) can be realized on the same set of oscillators. To this end we split the original variables as a a 0 ≡ x a , a a I ≡ a a i , I > 0 with i = 1, ..., n. Then, translations and Lorentz rotations are given by and naturally act in the space P d n (x, a) of smooth functions in x a with values in P d n (a i ). We also introduce special notation for some of sp(2n + 2) even basis elements and for odd basis elements In particular, from the osp(1|2n + 2) graded commutation relations we have { D, D} = 2 meaning that the Dirac operator D squares to the Klein-Gordon operator .

One-parameter family of constrained systems
We claim that both helicity and continuous spin fermionic fields can be uniformly described by a one-parameter system of constraints which are (deformed) generators of a subalgebra of osp(1|2n + 2). The constraints are imposed on a spinor-tensor field ψ ∈ P d n (x, a). The constraint algebra is generated by the Dirac constraint the gamma-trace conditions and the spin weight and Young symmetry conditions: Here, ν ∈ R, spin weights s m ∈ N, and Γ is the extra Clifford element satisfying {Γ, θ a } = 0 and Γ 2 = 1.
The additional constraint is implemented in a dual way through the equivalence relation determined by the following gauge transformation law Here, µ ∈ R and χ i ∈ P d n (x, a) are the gauge parameters satisfying relations following from the gauge invariance of the differential/algebraic constraints (3.1)-(3.3).
The complete set of constraints also involves which are consequences of (3.1) and (3.2). Indeed, In what follows it is also useful to split the constrants into differential ones that necessarily involve space-time derivatives ∂ a and the algebraic constraints that involve only a a i and θ a auxiliary variables. A few comments are in order.
• At µ, ν = 0 the system enjoys extra reducibility which can be removed by imposing in addition N 1 ψ = s 1 ψ and N 1 k ψ = 0, where k = 2, . . . , n and s 1 ∈ N. Then, the resulting constraint system describes fermionic helicity fields (see Section 4). For µ, ν = 0 the additional constraints are not consistent with the gauge transformations.
• Definition of the Clifford element Γ depends on the parity of d. More precisely, for even d the Γ can be chosen as the "fifth gamma" that is Γ can be realized in terms of the original Clifford algebra (2.1) and its module. In odd d it is not the case and the Γ extends the original Clifford algebra to {θ A , θ B } = 2η AB , where A = (a, d + 1), and η AB = (η ab , +), and θ d+1 ≡ Γ. In this case the spinor representation also gets extended and hence the spectrum of fields is duplicated. However, the extended Clifford algebra is even dimensional and therefore there is a new "fifth gamma" Γ = iΓ d+1 Γ that can be used to project out a half of spinor components via the standard P ± = 1 2 (1± Γ). Throughout the paper we keep d even unless otherwise indicated.
• The values of the quadratic and quartic Casimir operator of the Poincaré algebra evaluated on the subspace (3.1)-(3.4) are given by (see Appendix B) At µ, ν = 0 we find out that the above constrained system describes massless fields characterized by the continuous spin parameter µν. Thus, we indeed have one-parameter constraint system. At µ, ν = 0 the constraint system describes a collection of higher spin massless fields with arbitrary half-integer helicity. In this case the eigenvalue in (3.7) is zero what exactly matches vanishing eigenvalue of the quartic Casimir operator in the helicity case.
For general parameters µ, ν we fix the functional class in a a i to be that of formal power series in a a i such that a decomposition of a given element ψ with respect to traces, i.e.
is such that all coefficients are polynomials of bounded order (that means that for a given ψ there exists N ∈ N such that any ψ k ij... is of order not exceeding N ). This functional class was introduced in [10] in the context of bosonic continuous-spin fields. Note that in contrast to [10] now we are concerned with series with coefficients in S. Equivalently, one can characterize the functional class using the gamma-trace decomposition where all coeffcients are also required to be polynomials in a a i of finite order .

Fermionic helicity fields
In this section we explicitly study the constraint system (3.1)-(3.4) in the helicity case, µ, ν = 0. As we noted before, the constraints can be augmented by adding more algebraic conditions so that the resulting system describes a single massless half-integer spin field. The augmented system still contains Dirac equation while the complete set of algebraic constraints now reads as The spin weight conditions imposed on each type of auxiliary variables constrain functions ψ to be homogeneous polynomials in a i . The Young symmetry and gammatracelessness conditions are the standard irreducibility conditions for the o(d − 1, 1)representation of spin The gauge transformations read where the gauge parameters χ i satisfy the same constraints as the fields ψ except for the spin weight and Young symmetry constraints which are replaced by Note that the Klein-Gordon operator, the divergence and the trace conditions are imposed by virtue of

Simplest BRST formulation
Let us introduce the anticommuting ghost variables b i of ghost number gh(b i ) = −1.
The gauge symmetries can be realized via the BRST operator which acts on the space of functions Ψ(x, a|b) regarded as functions in x a taking values Homogeneous components of Ψ in b i carry definite ghost degree and are introduced according to: The spinor-tensor field ψ above is identified with the ghost 0 component Ψ (0) , gauge parameters are in Ψ (−1) component, and the order k reducibility components are in the ghost degree −k component. The function Ψ is subject to the BRST invariant extention of the constraints (4.2), where The component form of these constraints reproduce that for fields and gauge parameters (4.2) and (4.4). Starting with the above BRST formulation one can systematically rederive unfolded formulation [31,38] of mixed-symmetry fermionic helicity fields in Minkowski space. Indeed, according to [47] the set of fields of the unfolded formulation is given by cohomology of Q evaluated in the subspace (4.7). Strictly speaking in so doing one should also replace x a with formal coordinate y a and consider elements that are formal series in y a . Moreover, the nilpotent differential determining the unfolded equations and gauge symmetries is just the differential induced by dx a ( ∂ ∂x a − ∂ ∂y a ) + Q in the cohomology of the second term. Note that the first term has an interpretation of the flat connection of the Poincaré algebra. The procedure is a straightforward generalization of the derivation [28] of the unfolded formulation for generic bosonic helicity fields.

Extended triplet formulation
Let us impose all the differential constraints via BRST operator, while all the algebraic constraints or, more precisely, their appropriate BRST invariant extensions we impose directly in the representation space. In the case of integer spin field this reproduces [28,47] the triplet formulation discussed previously in [48][49][50][51][52][53]. As we are going to see for half-integer spin field this gives the extended description from which the familiar triplet formulation of [49,50] can be obtaned by eliminating auxiliary fields and solving constraints.
The extended triplet BRST operator for fermionic helicity fields is given by where in addition to ghost variables b i we introduced new anticommuting ghost variables c 0 , c i , i = 1, ..., n and commuting ghost variable α with ghosts numbers gh(c 0 ) = gh(α) = 1. As α is a commuting variable there is an ambiguity in the functional class to work with. We choose functions Ψ(x, a|α, c 0 , c, b) to be polynomials in α.
BRST operator (4.9) is defined on the subspace singled out by the following BRSTinvariant extended constraints where Note that the BRST operator (4.9) is well-defined and is nilpotent on the entire representation space and not only on the subspace (4.10).

Homological reduction and the triplet formulation
The triplet formulation can be used as a starting point to obtain various other dynamically equivalent formulations including the metric-like formulation and the light-cone formulation. In so doing it is convenient to employ the method of homological reduction developed in [47] (see also [54]) and applied earlier to bosonic mixed-symmetry fields in Minkowski space [10,28] in a similar framework. Let us briefly recap the main ingredients of the homological reduction method. Suppose we have a linear gauge theory theory (H, Ω) defined in terms of the Ω acting on the representation space H graded by the ghost number. Let H be split into three subspaces: H = E ⊕ F ⊕ G in such a way that a linear operator It turns out that all the fields associated with F and G are generalized auxiliary fields that are usual auxiliary fields and Stueckelberg fields as well as the associated ghosts and antifields. Generalized auxiliary fields can be eliminated, resulting in an equivalent formulation (E, Ω) of the same theory. The reduced BRST operator can be expressed explicitly as where EE

Ω and
GE Ω are the respective components of Ω. In applications a triple decomposition of H is often determined by a certain piece of Ω. More specifically, suppose that H admits an additional grading such that Ω decomposes into homogeneous components as follows Ω = Ω −1 + Ω 0 + Ω 1 + . . . . (4.14) Then, the lowest grade part Ω −1 of the BRST operator is nilpotent and defines the triple decomposition according to Note that the subspaces G ⊂ H and G ⊕ E ⊂ H are defined by Ω −1 unambigously while the embedding of F and E into H is defined up to an ambiguity. The reduced operator Ω can be explicitly expressed [47] in terms of the inverse of GF Ω −1 . Typically though not always one is interested in local gauge field theories in which case one requires that generalized auxiliary fields can be eliminated algebraically. In our case, it means that additional gradings give rise to Ω −1 that do not involve x-differential pieces of the triplet BRST operator.
Triplet formulation. The triplet and metric-like formulations can be obtained from the extended triplet formulation through the homological reduction by taking as additional degree the homogeneity in c 0 . BRST operator (4.9) then decomposes as Ω = Ω −1 + Ω 0 + Ω 1 with and, therefore, we can reduce the theory to the cohomology H(Ω −1 ).
Because Ω −1 is algebraic it is enough to compute cohomology in the space of xindpendent elements. Let us decompose a generic element as 1 (a, b, c, α). The cocycle and the coboundary condition take the form It is easy to check that in the space of polynomials in α the first equation implies φ 1 = 0 while φ 0 can be assumed to be at most linear in α thanks to the second equation. Moreover, each equivalence class has a unique representative that is at most linear in α.
To summarize, the cohomology of Ω −1 is concentrated in degree 0 and can be realized as a subspace E of c 0 -independent elements that are at most linear in α. The space of fields with values in this subspace is equipped with the induced BRST operator Ω which in this case is simply Ω 0 defined on the equivalence classes. In terms of representatives which are at least linear in α it is given by The second term arises from α 2 Dφ 1 which does not belong to E and hence one needs to pick another representative of the same equivalence class. To see what are the equations of motion encoded in Ω let us restrict ourselves to totally symmetric fields. The field of ghost degree 0 is then given by where components ψ, χ, λ form the fermionic triplet [49]. The equations of motion ΩΦ = 0 take the form of the fermionic triplet equations Let us now consider the case of mixed-symmetry fields. Elements of E of ghost numbers 0 and −1 are given respectively by From the algebraic constraints (4.10) it follows that the lowest components ψ and i satisfy the triple trace conditions as well as the Young symmetry and spin conditions: One concludes that these are precisely the original Fang-Fronsdal-Labastida spinortensor fields and their associated gauge parameters [30,55,56].
To derive the metric-like equations we only need the analog of the first equation in (4.20), which reads as The BRST-extended gamma-trace conditions (4.10) implies χ |i = −Υ i ψ thereby giving the reduced equations of motion This is the Fang-Fronsdal-Labastida equations for mixed-symmetry fermionic helicity fields [30,40,55]. Note that, just like the standard Dirac equation, the Fang-Fronsdal-Labastida equation can be squared, resulting in which is the Labastida equations for mixed-symmetry bosonic helicity fields [56]. Here we made use of DD † i Υ i ψ = ψ which is the result of acting by D on (4.30). By construction, the reduced equations (4.30) are invariant with respect to the gauge transformations where the gauge fields and parameters satisfy the algebraic conditions (4.27)-(4.28).

Triplet Lagrangian
The triplet BRST formulation in terms of E-valued fields and BRST operator Ω can be made Lagrangian by observing that E is equipped with a natural non-degenerate inner product. Let us consider the operator algebra generated by a a i , c i , b i , as well as their canonically conjugated variables denoted byā i a ,c i ,b i , and consider the following involution †: 33) where A, B are generic elements of the algebra. Note that this is involution is compatible with the notations for D i , D † i employed before. Consider the following Fock space generated by these operators from the following vacuum  The above involution uniquely determines the inner product on the Fock space which makes † into conjugation. Tensoring this Fock space with S (see section 2.1) and equipping S with an inner product such that (θ a ) † = −θ a we end up with the space equipped with an inner product , . Finally, the formal inner product on E is taken to be φ, ψ ≡ φ 0 + αφ 1 , ψ 0 + αψ 1 : It is straightforward to see that Ω is formally symmetric with respect to the inner product. Indeed, the only nontrivial part is to show that α † = α which is clear from the following explicit expressions

Light-cone formulation
Starting from the triplet BRST operator (4.9) and eliminating unphysical degrees of freedom by means of the homological reduction we arrive at the standard light-cone formulation for half-integer spin massless fields (see, e.g., [57]). This is done by employing the approach developed in [28,58] (see also [59] for an earlier important contribution). As usual, the light-cone coordinates are introduced as (x + , x − , x m ), m = 1, ..., d−2. The light-cone description of the Clifford elements is more tricky. Let us represent θ + and θ − on Grassmann algebra C[θ + ] generated by θ + as θ + and 2 ∂ ∂θ + , respectively. Then, consider the representation of Clifford algebra with generators θ + , θ − , θ m as a tensor product of C[θ + ] and irreducible representation of the Clifford algebra generated by θ m . In this way, we realize the representation of Clifford algebra with generators θ + , θ − , θ m as polynomials in θ + with coefficients in o(d − 2) spinors ψα, where Dirac spinor indexα = 1, ..., 2 [d/2]−1 . In other words, the light-cone spinor is half of the original spinor.
To do the light-cone reduction of the fermionic triplet formulation we introduce the grading (this is a generalization of the one employed in [28,58,59]) The operator (4.9) decomposes into the homogeneous degree components as Ω = Ω −1 + Ω 0 + Ω 1 + Ω 2 + Ω 3 , where , , (4.39) We assume p + = 0. Then, we observe that Ω −1 is just the de Rham differential on the superspace so that the only non-vanishing cohomology H(Ω −1 ) is in degree 0 and can be identified with the subspace E of elements ψα = ψα(x|a m , c 0 ) depending on the spacetime cooridinates, transverse auxiliary variables, and ghost c 0 taking values in the representation of the Clifford algebra generated by transverse θ m (in what follows we omit the spinor index). The cohomology of Ω −1 is concentrated in one degree so that the reduced operator Ω is given by (see e.g. [58]) Ω = c 0 .  Thus, the field content is given by the spinor-tensors with transversal components only and subject to the light-cone condition p 2 = 0 and the algebraic conditions (4.41). The iso(d − 1, 1) generators in the light-cone basis are split into two groups of kinematical G kin = (P + , P m , M +m , M +− , M mk ) and dynamical G dyn = (P − , M −k ) generators. After the reduction to the Ω −1 -cohomology both types of generators give rise to the reduced generators G kin and G dyn defined on the subspace E. While G kin retain its form upon the reduction, the explicit expressions for G dyn are given by

Continuous spin fermionic fields
In this section we turn to the continuous spin fermionic field system introduced in section 3. Based on our analysis of the standard fermionic fields in section 4 we propose the deformed triplet formulation and describe its metric-like and light-cone reductions.

(5.2)
Recall that µ i = δ i1 µ and ν i = δ i1 ν so that the deformed triplet operator (5.1) differs from the undeformed triplet operator (4.9) only by the term µ ∂ ∂b . Also, just as in the helicity case, (5.1) retains the property of being nilpotent on the whole space, not only on the subspace (5.2).

Metric-like formulation
Similarly to the helicity case of Section 4.3 the representation space can be endowed with an additional grading with respect to the ghost c 0 so that the lowest component Ω −1 of the deformed BRST operator (5.1) remains the same, while the deformation term enters Ω 0 . Furthemore, one can check that H(Ω −1 ) cohomology remains unchanged except that now the entire subspace is singled out by the deformed constraints (5.2) rather than the undeformed ones.
Repeating the same steps as in the helicity case we obtain the reduced equations of motion which are invariant with respect to the gauge transformations where both fields and parameters are subjected to the modified trace conditions where we introduced the notation for the deformed gamma-trace operator Υ i = Υ i + ν i Γ. In the case of spin-1 2 continuous spin field (i.e. n = 1 so that there is only one commuting auxiliary variable a m 1 ) the equation (5.3) reproduces the field equations obtained in [7,14].
It is worth noting that the equation (5.3) can be squared to yield which is the bosonic continuous spin metric-like equation [10].
To conclude this section we describe the field space in the case of spin-1 2 continuous spin field. The corresponding metric-like fields were previously considered in [17,18]. Solving the deformed constraints (5.5) we find that both the fields and parameters can be equivalently represented as infinite chains of the Fang-Fronsdal tensors Technically, the above decompositions are obtained by substituting (3.9) into (5.5) and solving the recurrent equations for expansion coefficients. Using (5.7) in the metric-like equations (5.3) and the gauge transformations (5.4) gives rise to the Schuster-Toro type equations invariant with respect to the µ-deformed gauge transformations [18].

Light-cone formulation
We start from the BRST operator (5.1) and use grading (4.38). In the considered functional class (3.9) any element has a finite grading because deg Υ i = 0 and by assumption the degree of coefficients in (3.9) is bounded so that we can use the homological reduction technique. The operator (5.1) decomposes into the homogeneous degree components as Ω = Ω −1 + Ω 0 + Ω 1 + Ω 2 + Ω 3 , where The deformation term µ ∂ ∂b is contained in Ω 1 only, and, therefore, the reduced BRST operator is the same as in the helicity case, The light-cone algebraic constraint following from (5.2) read  thereby forming the iso(d − 2) algebra. Let us evaluate the first two Casimir operators of the iso(d − 2) algebra on the subspace (5.10) following the analogous considerations in Section 3. We find 6 Weyl and gauge modules A linear gauge system is essentially determined by the space of gauge inequivalent formal solutions to the equations of motion, known as Weyl module, and the space of (higher-order) global reducibility parameters, known as gauge module. These spaces are usually considered as modules over the space-time global symmetry algebra. In particular, if the gauge module vanishes the system is non-gauge, i.e. all the gauge symmetries are Stueckelberg-like. Note also, that if the gauge module vanishes and the space-time global symmetries (e.g. Poincaré or AdS or conformal) act transitively, the system is entirely determined by the Weyl module structure. This property is manifest in the unfolded approach. 5 We are now interested in the gauge and Weyl modules of the fermionic (continuous) spin system. To this end we extend the analysis of [10,28] to the case of fermionic fields. To study formal solutions in this section we replace space-time coordinates x a by formal coordinates y a . In particular, it is implicitly assumed that in all the expressions for fields, parameters, operators, etc. x a and ∂ ∂x a are replaced with y a and ∂ ∂y a respectively. Moreover, instead of smooth functions in x a we work with formal power series in y a so that the relevant space is that of formal series in y a and a a i with coefficients in S such that for a given element the coefficients of the trace-decomposition are polynomials in a a i . The gauge and Weyl modules can be defined as the cohomology H k (Q, H 0 ) of the continuous spin generalization where i = 1, . . . , n and m, k = 2, . . . , n for continuous spin case µ, ν = 0 and m, k = 1, . . . , n for the helicity one (µ, ν = 0). The Weyl module is the zero ghost number cohomology H 0 (Q, H 0 ), the gauge module is a collection of modules identified with negative ghost degree cohomology H k (Q, H 0 ) at k < 0 [28,47,62,63].
To compute Q-cohomology we realize the space H 0 defined by (6.2) as a subspace of the tensor product where ψ takes values in C. Indeed, H 0 is just a subspace of (6.3) singled out by As a next step we note that it is enough to compute Q-cohomology in H. Indeed, H can be represented as a direct sum of H 0 and the complementary subspace H 1 in such a way that Q preserves both subspaces. As H 1 one can take a subspace of elements that can be represented as (γ · a)α + (γ · y)β for some α, β ∈ H 0 (the dot denotes summation over Lorentz indices). As representatives of Q-cohomology in H 0 one can take those representatives of Q-cohomology in H that belong to H 0 . Finally, the action of Q on S ⊗ G originates from the action of Q on G because Q does not affect S and hence Q-cohomology in H is just a tensor product of S with the Q-cohomology in G. In its turn Q-cohomology in G are known for both helicity fields [28] and continuous spin fields [10].

Q-cohomology for helicity fermionic fields
For µ, ν = 0 space G defined by (6.3) is precisely the representation space involved in describing bosonic helicity fields of general symmetry type. Let us spell out the explicit description [28] of the representatives of Q-cohomology classes. Introduce the following subspaces M k ⊂ G, k = 0, . . . , n − 1 . . , n − 1 .

(6.5)
For any k = 1, . . . , n − 1 each cohomology class from H −k (Q, G) has a unique representative belonging to M k . Taking into account the above characterization of the cohomology classes one concludes: Note that the last equality makes sense as M k is naturally a subspace in G.

Q-cohomology for continuous spin fields
In the case µ, ν = 0 the subspace G is again a relevant subspace and cohomology H k (Q, G) was studied in [10], where it was shown that H k (Q, G) = 0 for k < 0 and H 0 (Q, G) = 0 with our choice of the functional class. Because of it follows that H k (Q, H) = 0 at k < 0. In particular, we conclude that fermionic continuous spin fields are also not gauge fields, i.e. the gauge fields present in the formulation are actually Stueckelberg ones. The Weyl module for the bosonic continuous spin fields is given by H 0 (Q, G) and is non-vanishing. Let us show that the same applies to H 0 (Q, H 0 ). Let ψ be a nontrivial representative of H 0 (Q, G) then one can choose ξ ∈ S such that ψ 0 = (ψ ⊗ ξ) ∩ H 0 is also non-vanishing. Furthermore, Qψ 0 = 0 because Q preserves both H 0 and H 1 .
Acknowledgements. We are grateful to R. Metsaev

B Casimir operators
The quadratic and quartic Casimir operators of the iso(p, q) algebra are C 2 iso(p, q) = P a P a ≡ P 2 , C 4 iso(p, q) = M ab P b M ac P c − 1 2 M 2 P 2 , (B.1) where P a stands for translation and M ab for rotation generators, respectively. In what follows we express (B.1) in terms of osp basis elements.
Regular spinor-tensor representation. Let iso(d−1, 1) basis elements P a , M ab , a, b = 0, ..., d − 1 act as Note that dropping terms in curly brackets we get the bosonic Casimir operator. Also, the above osp(1|2n + 2) representation holds for any iso(k, l) with k + l = d.