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BRST approach to Lagrangian formulation for mixed-symmetry fermionic higher-spin fields

We construct a Lagrangian description of irreducible half-integer higher-spin representations of the Poincare group with the corresponding Young tableaux having two rows, on a basis of the BRST approach. Starting with a description of fermionic higher-spin fields in a flat space of any dimension in terms of an auxiliary Fock space, we realize a conversion of the initial operator constraint system (constructed with respect to the relations extracting irreducible Poincare-group representations) into a first-class constraint system. For this purpose, we find auxiliary representations of the constraint subsuperalgebra containing the subsystem of second-class constraints in terms of Verma modules. We propose a universal procedure of constructing gauge-invariant Lagrangians with reducible gauge symmetries describing the dynamics of both massless and massive fermionic fields of any spin. No off-shell constraints for the fields and gauge parameters are used from the very beginning. It is shown that the space of BRST cohomologies with a vanishing ghost number is determined only by the constraints corresponding to an irreducible Poincare-group representation. To illustrate the general construction, we obtain a Lagrangian description of fermionic fields with generalized spin (3/2,1/2) and (3/2,3/2) on a flat background containing the complete set of auxiliary fields and gauge symmetries.


Introduction
The study of various aspects of higher-spin (HS) field theory has attracted a considerable attention for a long time due to the hope of discovering new possible approaches to the unification of the fundamental interactions. Higher-spin field theory is closely related to superstring theory, which operates with an infinite tower of bosonic and fermionic higher-spin fields. The problem of a covariant Lagrangian description of fields with an arbitrary spin propagating on flat [1]- [15] and (A)dS [16]- [27] backgrounds as well as the problem of constructing an interacting higher-spin field theory are in the permanent focus of research (for reviews and more references, see, e.g., [28]). One of the attractive features of investigating higher-spin gauge theories in AdS spaces is due to a possible relation of this study to the tensionless limit of superstring theory on the AdS 5 × S 5 in Minkowski space of any dimension, corresponding to a unitary irreducible Poincare-group representation with a Young tableaux having two rows of length n 1 , n 2 (n 1 ≥ n 2 ).
The paper is organized as follows. In Section 2, we formulate a closed Lie superalgebra of operators, based on the constraints in an auxiliary Fock space that determines an irreducible representation of the Poincare group with a generalized spin s = (n 1 +1/2, n 2 +1/2). In Section 3, we construct a Verma module, being an auxiliary representation for a rank-2 subsuperalgebra of the superalgebra of the initial constraints corresponding to the subsystem of second-class constraints. This representation is then realized in terms of new (additional) creation and annihilation operators in Fock space. Note that a similar construction for bosonic HS fields in a flat space has been presented in [46]. In Section 4, we carry out a conversion of the initial system of first-and secondclass constraints into a system of first-class constraints in the space being the tensor product of the initial and new Fock spaces. Next, we construct a BRST operator for the converted constraint superalgebra. The construction of an action and of a sequence of reducible gauge transformations describing the propagation of a mixed-symmetry fermionic field of an arbitrary spin is realized in Section 5. We demonstrate that the Lagrangian description of a massive half-integer mixedsymmetry HS field in a d-dimensional Minkowski space can be deduced not only by using the same algorithm, with allowance for the presence of 4 additional second-class constraints instead of the respective first-class ones with first-order derivatives, but also by using dimensional reduction for a massless HS field theory of the same type in a (d+1)-dimensional flat space. In Section 6, we sketch a proof of the fact that the resulting action reproduces the correct conditions for a field that determine an irreducible representation of the Poincare group with a fixed s = (n 1 + 1/2, n 2 + 1/2) spin. We illustrate the general formalism by a construction of gauge-invariant Lagrangian actions for massless and massive spin-(1 + 1/2, 1/2) and spin-(1 + 1/2, 1 + 1/2) fields in Section 7. In Conclusion, we summarize the results of this work and outline some open problems.
In addition to the conventions of [9,44,46], we use the notation ε(A), gh(A) for the respective values of Grassmann parity and ghost number of a quantity A, and denote by [A, B} the supercommutator of quantities A, B, which in the case of definite values of Grassmann parity is given by [A , B} = AB − (−1) ε(A)ε(B) BA.
In order to describe all the irreducible representations simultaneously, it is convenient to introduce an auxiliary Fock space H generated by creation and annihilation operators a i+ µ , a j µ with additional internal indices, i, j = 1, 2, [a i µ , a j+ ν ] = −η µν δ ij , δ ij = diag (1,1) . (2.6) The general state (a Dirac-like spinor) of the Fock space has the form providing the symmetry property of Φ (µ)n 1 ,(ν)n 2 (x) under the permutation of indices of the same type. We refer to the vector (2.7) as the basic vector.
(2.12) 2 Throughout the paper, we use the mostly minus signature η µν = diag(+, −, ..., −), µ, ν = 0, 1, ..., d − 1, and the Dirac matrices satisfy the relations {γ µ , γ ν } = 2η µν . 3 For more details, see [44]. The quantitiesγ µ ,γ may be viewed as intermediate objects as compared to γ µ . Indeed, the final Lagrangian description in terms of ghost-independent or spin-tensor forms depends only on the standard Grassmann-even matrices γ µ , and does not depend onγ µ ,γ, because the latter enter the Lagrangian only in even degrees, and also due to the homogeneity of the reducible gauge transformations w.r.t.γ, as shown by the examples in Section 7.
We can now define Grassmann-odd constraints, t 0 = −ıγ µ ∂ µ , t i =γ µ a i µ , (2.13) related to the operators (2.8) as follows: (2.14) We next define an odd scalar product: The operators t 0 , t i , t in (2.9) and (2.13), with t i+ =γ µ a i+ µ and t + = a 2+ µ a 1µ being Hermitian conjugate, respectively, to t i , t with reference to the scalar product (2.15), generate an operator Lie superalgebra composed of the operators 21) which is invariant under Hermitian conjugation. The operators (2.16)-(2.21) form a superalgebra given by Table 1, with an omission of the Poincare-group Casimir operator l 0 being the central charge of this algebra, where the quantities We call this algebra the half-integer higher-spin symmetry algebra in Minkowski space with a Young tableaux having two rows. From the viewpoint of constraint system theory, the above superalgebra is a system of constraints, except for the operators g k 0 , being non-degenerate in H. These operators, as follows from Table 1, determine an invertible operator supermatrix of commutators for the subsystem of second-class constraints, {t k , t + k , l ij , l + ij , t, t + }, with the other constraints, t 0 , l 0 , l i , l + i , being firstclass ones.
A conversion of this constraint system {o I }, including the operators g k 0 , into a first-class constraint system {O I } by means of an additive composition of o I with certain operators o ′ I depending on some new creation and annihilation operators, o I → O I = o I + o ′ I , can be effectively realized only for the subsuperalgebra of the entire symmetry superalgebra that contains the subsystem of second class-constraints and g k 0 . The only requirement, as shown in [46], is that each of the Hermitian operators g k 0 should have a linear dependence on an arbitrary parameter h k , whose values are to be determined later.

Auxiliary Representation for the Superalgebra with Second-class Constraints
In this section, we describe the method of Verma module construction for the Lie superalgebra with second-class constraints alone. Having denoted {o a } = {t k , t + k , l ij , l + ij , g k 0 }, o a ∈ {o I }, as the basis elements of the above superalgebra, and using the requirements that o a , o ′ a must supercommute, {o a , o ′ a ] = 0, and that the converted constraints must be in involution, {O a , O b ] ∼ O c , we find that the superalgebra of the additional parts o ′ a is uniquely determined by the same algebraic conditions as those for the initial constraints. In this case, it is unnecessary to convert the subsystem of the initial first-class constraints not entering {o a }, and therefore they remain intact.
Following [44] and the general method of Verma module construction for mixed-symmetry integer-spin HS fields [46], let us denote E α ≡ (t k ; l ij , t) = (E α 0 ; E α 1 ), (α 0 > 0, α 1 > 0) for i ≤ j, and define The quantities g i 0 , E α 1 , E −α 1 are the Cartan generators, positive and negative root vectors, except for α 1 = 2 (see footnote 4) of the subalgebra so (3,2) in the superalgebra of second-class constraints, and the odd generators E α 0 , E −α 0 supplement the basis so (3,2) up to that of the above superalgebra. The quantities g ′i 0 , E ′α , E −′α and H i , E α , E −α have the same identification respectively for the additional and enlarged operators of the symmetry superalgebra.
Consider the highest-weight representation of the superalgebra of the additional parts with the highest-weight vector |0 V annihilated by the positive roots and being the proper vector of the Cartan generators: Following the Poincare-Birkhoff-Witt theorem, the basis space of this representation, called the Verma module in the mathematical literature [47], is given by the vectors where n 0 k = (n 0 1 , n 0 2 ), n ij = (n 11 , n 12 , n 22 ), n 0 1 , n 0 2 = 0, 1, n ij ∈ N 0 . Note that the restriction for the values of n 0 k in (3.3) is due to the identities Using the commutation relations of the superalgebra given by Table 1 and the formula for the product of graded operators, we can calculate the explicit form of the Verma module. In (3.5), we have introduced generalized coefficients for a number of graded combinations, C (s)n k , that coincide with the standard ones only for the bosonic operator B: C (0)n k = C n k = n! k!(n−k)! . These coefficients are defined recursively, by the relations and possess the properties C (s)n k = C (s)n n−k . The corresponding values of C (1)n k are defined, for n ≥ k, by the formulae , (3.8) which follow by induction, and in which [a] stands for the integer part of the number a. For our purposes, due to n 0 Then, following [48] and making use of the mapping n 0 k , n ij , n V ↔ n 0 k , n ij , n = f + 1 n 0 (3.9) where | n 0 k , n ij , n , for n 0 k = 0, 1, n ij ∈ N 0 , are the basis vectors of a Fock space H ′ generated by new fermionic, f + k , f k , k = 0, 1, and bosonic, b + ij , b + , b ij , b, i, j = 1, 2, i ≤ j, creation and annihilation operators with the standard (only nonvanishing) commutation relations we can represent the Verma module as polynomials in the creation operators of the Fock space H ′ . First, we find the action of the negative root operators E ′−α on the basis vectors. After a simple calculation, one obtains Using expressions (3.11)-(3.20) and the mapping (3.9), we reconstruct the action of the operators E ′α , E ′−α , g ′i 0 in the Fock space H ′ , namely, Note that the additional parts E ′α , E −′α do not obey the usual properties if one should use the standard rules of Hermitian conjugation for the new creation and annihilation operators, To restore the proper Hermitian conjugation properties for the additional parts, we change the scalar product in the Fock space H ′ as follows: for any vectors |Ψ 1 , |Ψ 2 with some, yet unknown, operator K ′ . This operator is determined by the condition that all the operators of the algebra must have the proper Hermitian properties with respect to the new scalar product: These relations permit one to determine the operator K ′ , Hermitian with respect to the usual scalar product | , as follows: where ( n lm )! = n 11 !n 12 !n 22 !. One can show by direct calculation that the following relation holds true: V n ′0 k , n ′ lm , n ′ | n 0 k , n lm , n V ∼ δ n 0 1 +2n 11 +n 12 −n n ′0 1 +2n ′ 11 +n ′ 12 −n ′ δ n 0 2 +n 12 +2n 22 +n n ′0 2 +n ′ 12 +2n ′ 22 +n ′ . For low pairs of numbers (n 0 1 + 2n 11 + n 12 − n, n 0 2 + n 12 + 2n 22 + n), with n, n ij being the numbers of "particles" associated with b + , b + ij for i ≤ j (where b + reduces the spin number s 1 by one unit and increases the spin number s 2 by one unit simultaneously) and n 0 k being the number of "particles" associated with f + k , the operator K ′ reads 3.36) This expression for the operator K ′ will be used later in constructing the examples of Section 7. Thus, we have constructed the additional parts o ′ a , (3.21)- (3.30), for the constraints o a . In the next section, we determine the algebra of the extended constraints and find the BRST operator corresponding to this algebra.

The Converted Superalgebra and the BFV-BRST Operator
The superalgebra of the converted operators O I , has the same form as the superalgebra of the initial operators o I , and therefore it is determined by the relations of Table 1 and include H i into the converted first-class constraint system, with respect to which we construct the BRST operator Q ′ . The sum in (4.2) is taken over k i0 , k lm , k, running from 0 to infinity, and over k i , running from 0 to 1 for i = 0, 1, l, m = 1, 2, l ≤ m. Having constructed Q ′ , we extract from it the operators H i , enlarged by means of the ghost variables C, P up to new operators σ i , ), which will be used to describe, by virtue of the equations (σ i +h i )|χ = 0, the direct sum of the Fock subspaces H (n 1 ,n 2 ) of a definite generalized spin s = (n 1 + 1 2 , n 2 + 1 2 ) in the enlarged Hilbert space H tot = H ⊗ H ′ ⊗ H gh for |χ ∈ H tot . In this case, the remaining operator Q, independent of the ghost variables η i , is covariant and nilpotent in each space H (n 1 ,n 2 ) for the converted constraint system O I without H i . Then, substituting instead of the parameters −h i the operators σ i , we obtain a nilpotent BRST operator in the complete space H tot without η i H , P i H , which encodes the superalgebra of the converted constraints {O I } \ {H i } for fermionic HS fields with two rows of the Young tableaux.
The construction of a nilpotent fermionic BRST operator for a Lie superalgebra is based on a principle similar to those developed in [44,45]: see the general analysis of the BFV quantization in the reviews [33,34,35]. Following the prescription of [34], the BRST operator constructed on a basis of the superalgebra presented in Table 1 can be found in an exact form, with the use of the (CP)-ordering of the ghost coordinate C I and momenta P I operators, as follows: with the constants f K IJ written in a compact x-local representation, according to Table 1, Q ′ has the form Here, we imply summation over the repeated index i, and the raising (lowering) of the indices i, j in quantities f ij is made by the two-dimensional Euclidian metric tensor g ij (g ij ), respectively, bosonic and fermionic ghost "coordinates" corresponding to their canonically conjugate ghost "momenta" Following [34], they obey the nonvanishing (anti)commutation relations they also possess the standard ghost number distribution, gh(C i ) = −gh(P i ) = 1, providing the property gh(Q ′ ) = 1, and have the Hermitian conjugation properties of zero-mode pairs, 5 The property of the BRST operator to be Hermitian is defined by the rule and is calculated with respect to the scalar product | in H tot with the measure d d x, which, in its turn, is constructed as the direct product of the scalar products in H, H ′ and H gh . The operator K in (4.7) is the tensor product of the operator K ′ in H ′ and the unit operators in H, Thus, we have constructed a Hermitian BRST operator for the entire superalgebra of O I . In the next section, this operator will be used to construct a Lagrangian action for fermionic HS fields of spin (s 1 , s 2 ) in a flat space.

Construction of Lagrangian Actions
The construction of Lagrangians for fermionic higher-spin fields in a d-dimensional Minkowski space can be developed by partially following the algorithm of [44], which is a particular case of our construction, corresponding to n 2 = 0. As a first step, we extract the dependence of the BRST operator Q ′ (4.4) on the ghosts η i H , P i H , so as to obtain the BRST operator Q only for the system of converted first-class constraints {O I } \ {H i }:

2)
extended by the ghost Wick-pair variables, has the form Second, we choose a representation of the Hilbert space permitting us to find the BRST cohomology spaces for the first-class constraint system, (5.5) and to extract from H tot the Hilbert subspace that does not depend on the η i H operators (since H i are not first-class constraints as the other O I ), The sum in (5.6) is taken over k 1 , k 2i , k 3i , k 12 no , k 13 , k 14 , k 15 , running from 0 to infinity, for i, l, m, n, o = 1, 2, l ≤ m, n ≤ o, and over the other indices, running from 0 to 1. Next, we derive from the equations determining the physical vector, Q ′ |χ = 0, and from the reducible gauge transformations, δ|χ = Q ′ |Λ , δ|Λ = Q ′ |Λ (1) , . . ., δ|Λ (s−1) = Q ′ |Λ (s) , a sequence of relations: The middle set of equations in (5.7)-(5.10) determines the possible values of the parameters h i and the eigenvectors of the operators σ i . Solving these equations, we obtain a set of eigenvectors, |χ (n 1 ,n 2 ) , |Λ (n 1 ,n 2 ) , . . ., |Λ (s) (n 1 ,n 2 ) , n 1 ≥ n 2 ≥ 0, and a set of eigenvalues, (5.11) with (n 1 , n 2 ) related to spin, s = (n 1 , n 2 ) + (1/2, 1/2). The values of n 1 , n 2 are related to the spin components s 1 , s 2 of the field, because the proper vector |χ (n 1 ,n 2 ) corresponding to (h 1 , h 2 ) has the leading term χ 0···0 (µ)n 1 ,(ν)n 2 (x), independent of the auxiliary and ghost operators, which corresponds to the field Φ (µ)n 1 ,(ν)n 2 (x) with the initial value of spin s = (s 1 , s 2 ) in the decomposition (5.6), (5.12) where the values of (n 1 , n 2 ) can be composed of the set of coefficients {k r } \ {k 1 , k 4 } in (5.6) by the formulae Therefore, relations (5.7)-(5.10) guarantee both the extraction of vectors with the required value of spin and the nilpotency of Q in the corresponding Hilbert subspace. If one fixes the value of spin, then the parameters h i are also fixed by (5.11). Having fixed the value of h i , we should substitute it into each of the expressions (5.7)-(5.10). Third, we should extract the zero-mode ghosts from the operator Q as follows: whereT Here,T 0 , ∆Q are independent of q 0 , p 0 , η 0 , P 0 . We also expand the state vector and gauge parameters in powers of the zero-mode ghosts: Following the procedure described in [24,44], we get rid of all the fields except two, |χ 0 0 , |χ 1 0 . Namely, after the extraction of zero-mode ghosts from the BRST operator Q (5.14), as well as from the state vector and the gauge parameter (5.17), (5.18), the gauge transformation for the fields |χ k 0 , k ≥ 2 has the form (5.19) implying, by induction, that we can make all the fields |χ k 0 , k ≥ 2 equal to zero by using the gauge parameters |Λ k 1 . Then, considering the equations of motion for the powers q k 0 , k ≥ 3 and taking into account that |χ k 0 = 0, k ≥ 2, we can see that these equations contain the subsystem |χ k−2 20) which permits us to find, by induction, that all the fields |χ k 1 , k ≥ 1 are equal to zero. Finally, we examine the equations of motion for the power q 2 0 : (5.21) in order to express the vector |χ 0 1 in terms of |χ 1 0 . Thus, as in the totally symmetric case, there remain only two independent fields: |χ 0 0 , |χ 1 0 . The first equation in (5.7), (5.14), the decomposition (5.17), and the above analysis then imply that the independent equations of motion for these vectors have the form where {F, G} = F G + GF for any quantities F, G.
The additional parts o ′ I coincide with those of the massless case, whereas the converted set of constraints has the form O m I =ǒ I + o ′ I , no longer having a central chargeγm for massive HS fields = δ ij acting in the Fock space H m ), determined by adding the terms induced by dimensional reduction The set of O m I satisfies the same HS symmetry superalgebra as the one for massless half-integer irreducible representations of the Poincare group). The generalized spin and BRST operators,  (5.40) Note that the b + l -independent vectors |χ n ′ l , for l = 0, 1, 2, have a decomposition in oscillator powers, given by (5.6). At the same time, one may follow the dimensional reduction [52,53] of a massless HS field theory of the same type in a (d + 1)-dimensional flat space R 1,d . To this end, we make a projection R 1,d → R 1,d−1 over the sphere S 1 with a simple decomposition, To make a projection from R 1,2N −1 to R 1,2N −2 , one may use the following representation of γ M =γ Mγ -matrices, which can be realized inductively, according to (5.43), as an outer product of the unity 1 2 and Pauli σ i -matrices, i = 1, 2, 3, for any even d: where the gamma-matrices of a d = 2N-dimensional space are used for d = 2N + 1, and the matrix γ d 2 N is introduced (5.47). By the Pauli theorem, any set of γ µ 2 [ d 2 ] -matrices for a fixed d can be obtained from the given set by using a similarity transformation. In this case, a respective massless spinor possesses 2 N components and contains two Weyl spinors in a transposed form Φ L|a(M )n 1 ,(N )n 2 , Φ R|b(M )n 1 ,(N )n 2 T (x, x d ), being an element of an ISO(1, 2N −1) reducible massless representation for a half-integer spin, and having the property of an eigenvector for γ d 2 N , Φ R,L = (1/2)(1 ∓ ıγ d )Φ. Any element of an irreducible massless representation has to be chiral, e.g., right-handed, 0, Φ R|b(M )n 1 ,(N )n 2 T (x, x d ), which transforms by the projection (5.44) into the set Φ R|b(µ) n 1 −r 1 ,(ν) n 2 −r 2 (x). According to (5.45)-(5.47) Left-handed massless spin-tensors can be subject to a similar treatment and thereby represent after projection independent massive fermions. For the doubled massive components Φ A(µ)n 1 ,(ν)n 2 = (Φ R|a(µ)n 1 ,(ν)n 2 , Φ R|a(µ)n 1 ,(ν)n 2 ) T , the representation (5.34) with 2 N -dimensionalγ µ -matrices is also valid.
Therefore, the quantitiesγT 0γ , identical with T 0 for massless HS fields, transform for massive fields asγT 0γ = T * 0 , T * 0 = −ıγ µ ∂ µ −γm. Hence, for the coefficient functions entering (along with the matrixγ) into the decomposition of any vector composed of |χ j 0 (n 1 ,n 2 ) , |Λ (s)j 0 (n 1 ,n 2 ) , the part of a vector |Ξ (n 1 ,n 2 ) homogeneous with respect toγ is subject to in comparison with the massless case. 8 A simultaneous construction of Lagrangian actions describing the propagation of all massless (massive) fermionic fields with two rows of the Young tableaux in Minkowski space is similar to the case of totally symmetric spin-tensors in flat spaces [44], and we only note that a necessary condition for solving this problem is to replace in Q ′ , Q, K the parameters −h i by the operators σ i in an appropriate way, and to discard the condition (5.26) for the fields and gauge parameters. Amongst other things, this completes both the conversion procedure for the initial constraint system {o I } \ {g i 0 } and the construction of a nilpotent BRST operator in the entire Hilbert space for the set of converted constraints {O I } \ {H i }.
In the section to follow, we outline a proof of the fact that the action, in fact, reproduces the correct equations of motion (2.2)-(2.5).

Reduction to the Initial Irreducible Relations
Let us briefly show the fact that it is only the solutions of the equations of motion (2.2)-(2.5) that determine the space of BRST cohomologies of the operator Q (5.2) with a vanishing ghost number in the Fock space H for the basic fermionic field with spin s = (n 1 + 1/2, n 2 + 1/2). To solve this problem, we can follow two ways: the first one is realized, for instance, in [26] for massless totallysymmetric bosonic fields in a flat space-time, and the second one, for totally-symmetric fermionic fields [44,45]. We will use the technics of [44,45], taking into account the fact that the spectrum of component fields for an arbitrary vector |χ (n 1 ,n 2 ) in (5.6) for k 1 = k 4 = 0 is essentially larger than the spectrum for a totally-symmetric fermionic vector |χ (n 1 +n 2 ,0) , for whose description one should not use the operators q + 2 , p + 2 , f + 2 , η + 2 , P + 2 , η + 12 , P + 12 , η + 22 , P + 22 , η + , P + , b + , b + 12 , b + 22 , a +ν 2 and the corresponding conjugations. 9 As a consequence, the character of proof is more involved even in comparison with the case of the AdS space [45].
In the standard manner, the proof consists of two steps. First, in order to simplify the spectrum of the gauge parameters |Λ (s) j 0 and the fields |χ j 0 , j = 0, 1, we apply to them a gauge-fixing based on the structure of gauge transformations (5.28)-(5.31) and extract the physical field |Φ (n 1 ,n 2 ) alone, by using (2.7) for s = (n 1 + 1/2, n 2 + 1/2): Second, we use a part of the Lagrangian equations of motion (5.24), (5.25) in order to select from them only the equations of motion for |Φ (n 1 ,n 2 ) , and to remove all of the remaining auxiliary fields of lower spins.
Proceeding by induction, we may use the algorithm which has been applied to the treatment of the vectors (6.13), (6.14) in order to eliminate the dependence on P + 11 related to all the vectors down to |Λ (0) j 0 , whereas the P + 22 -independent terms in |Λ (l) j 0 for l ≥ n 2 are restricted by relations of the form (6.17).
Let us now turn to the gauge-fixing of the fields. We expand the fields in the powers of the ghosts P + ij by analogy with the gauge parameters: Further, we need to restrict the vectors by the gauge conditions (6.17), which follow from the gauge transformations, and then we eliminate the terms coupled to P + 11 . Having completed the above procedure, we briefly mention that the remaining gauge ambiguity is sufficient to eliminate the auxiliary oscillators b + ij , b + , f + i from the field |χ 00000 00 , |χ k 00 = 6.19) and therefore, in view of gh(|χ 00000 00 ) = 0, this field has no dependence on the ghost "coordinates", so that, after the gauge-fixing, we conclude |χ 00000 00 = |Φ . (6.20) The second step of establishing an equivalence of equations (2.2)-(2.5) with the Lagrangian equations (5.24), (5.25) is more involved and is based on a detailed expansion of equations (5.24), (5.25) in the powers of p + i , q + i , η + i , η + ij , η + and then in the powers of b + ij , f + i . We only state the result that after gauge-fixing |χ k 00 and |χ k 012 |χ k 022 , |χ k 04 , expanded by analogy with (6.19), the only independent equations among (5.24), (5.25) have the form t 0 |Φ = t i |Φ = t|Φ = 0 , (6.21) and all of the auxiliary fields can be made equal to zero.
In what follows, we consider some examples of the Lagrangian formulation procedure.

Examples
Here, we shall realize the general prescriptions of our Lagrangian formulation in the case of fermionic fields of lowest spins.

Spin-(3/2,1/2) Field
In the case of a field of spin (3/2,1/2), we have (n 1 , Since s max = 0, the corresponding Lagrangian formulation is an irreducible gauge theory and describes a totally symmetric fermionic field of spin s = 3/2. The nonvanishing fields |χ i 0 (1,1) and gauge parameters |Λ 0 0 (1,0) , (for |Λ 1 0 (1,0) ≡ 0, due to gh(|Λ 1 0 (1,0) ) = −2), have the following Grassmann grading and ghost number distribution: These conditions determine the dependence of the fields and gauge parameters on the oscillator variables in a unique form, with the help of the operators corresponding only to the first row of the Young tableaux, (7.4) Substituting (7.2), (7.3) into (5.27), we find the action (up to an overall factor) for a free massless field of spin (3/2, 1/2) on a flat background: (7.5) In deriving the action (7.5), we have used the expressions (3.36), (4.8) for the operators K (1,0) 10 . A substitution of (7.2)-(7.4) into (5.28), (5.29) permits one to find the gauge transformations (5.28), (5.29) in the form (7.6) 10 For n 2 = 0, we have the case of totally symmetric spin-tensors in a d-dimensional flat space [44], so that the total Hilbert space H tot and all of the operators acting on it can be factorized from q + 2 , η + 2 , η + 12 , η + 22 , η + , q 2 , η 2 , η 12 , η 22 , η, f + 2 , b + 12 , b + 12 , b + and their canonically conjugate operators. In the expressions for the action (5.27) and the sequence of gauge transformations (5.28)-(5.31), we must set n 2 = 0 and use the above restrictions for H tot . In particular, the operator K ′ (n1,0) has an exact form [44], Let us present the action in terms of the physical field ψ µ alone. To this end, we get rid of the field ψ by using its gauge transformation and the gauge parameter ξ 2 . Having expressed the field χ by using the equation of motion χ = iγ µ ψ µ , we can see that the terms with the Lagrangian multiplier χ 1 turn to zero. As a result, we obtain the action which is invariant with respect to the residual gauge transformation δψ µ = ∂ µ ξ 1 .
To obtain a Lagrangian description for a massive fermionic field, we may take two ways, either following the dimensional reduction procedure (5.41)-(5.48), starting directly from the action (7.7) presented for a (d + 1)-dimensional Minkowski space, or following the prescription (5.36)- (5.40), starting from the expansion (7.2), (7.3), where it is only |χ 0 0 that changes to |χ 0 0m , (7.8) In the latter case, the Lagrangian action and the gauge transformations for h 1 m = (1 − d)/2 read as follows: Then one gets rid of the fields ψ, ϕ by using the respective gauge transformations and gauge parameters ξ 2 , ξ 1 . Once again, using the corresponding equation of motion, we express the field χ as χ = iγ µ ψ µ and obtain the Rarita-Schwinger Lagrangian in a d-dimensional flat space [44], The same result follows from dimensional reduction applied to a massless theory in R 1,d (e.g., with an odd d+1), for a spin-3/2 massive fermionic field ψ M = (ψ µ , ϕ), which contains the Stueckelberg field ϕ, (5.49) and due to the relation χ = iγ N ψ N before Eq. (7.7), with allowance for the structure of |χ 1 0 (1,0) in (7.2) and the resolution of the respective gauge transformations δ(ψ µ , ϕ) = (∂ µ , m)ξ.
We have realized the Verma module construction [47] in order to obtain an auxiliary representation in Fock space for the above superalgebra with second-class constraints. As a consequence, the converted Lie superalgebra of HS symmetry has the same algebraic relations as the initial superalgebra; however, these relations are realized in an enlarged Fock space. The generators of the converted Cartan subalgebra contain linearly two auxiliary independent number parameters, whose choice provides the vanishing of these generators in the corresponding subspaces of the total Hilbert space extended by the ghost operators in accordance with the minimal BFV-BRST construction for the converted HS symmetry superalgebra. Therefore, the above generators, enlarged by the ghost contributions up to the "particle number" operators in the total Hilbert space, covariantly determine Hilbert subspaces in each of which the converted symmetry superalgebra consists of the first-class constraints alone, labeled by the values of the above parameters, and constructed from the initial irreducible Poincare-group relations.
It is shown that the Lagrangian description corresponding to the BRST operator, which encodes the converted HS symmetry superalgebra, yields a consistent Lagrangian dynamics for fermionic fields of any generalized spin. The resulting Lagrangian description, realized concisely in terms of the total Fock space, presents a set of generating relations for the action and the sequence of gauge transformations for given fermionic HS fields with a sufficient set of auxiliary fields, and proves to be a reducible gauge theory with a finite number of reducibility stages, increasing with the value of generalized spin. We elaborate a dimensional reduction procedure used to obtain a gauge-invariant Lagrangian description for a massive fermionic HS field in a ddimensional Minkowski space R 1,d starting from a Lagrangian description for a massless fermionic HS field of the same generalized spin, albeit in R 1,d−1 .
We have outlined a proof of the fact that the solutions of the Lagrangian equations of motion (5.24), (5.25), after a partial gauge-fixing, correspond to the BRST cohomology space with a vanishing ghost number, which is determined only by the relations that extract the fields of an irreducible Poincare-group representation with a given value of generalized spin.
As examples demonstrating the applicability of the general scheme, we have derived gaugeinvariant Lagrangian formulations for the field of spin (3/2, 1/2) and for the rank-2 antisymmetric spin-tensor in a manifest form in both massless and massive cases 11 . In principle, the suggested algorithm permits one to derive manifest actions for any other half-integer spin fields characterized by two rows of the corresponding Young tableaux.
The basic results of the present work are given by relations (5.27), where the action for a field with an arbitrary generalized half-integer spin is constructed, as well as by relations (5.28)-(5.31), where the gauge transformations for the fields are presented, along with the sequence of reducible gauge transformations and gauge parameters.
Concluding, we would like to discuss a number of additional points. First, the gauge-invariant description of massless and massive HS field theories with a mixed symmetry is an interesting starting point for a systematic construction of a Lagrangian formulation for HS interacting vertices with mixed-symmetry fermionic HS fields, including the case of the AdS space, in order to provide a description of the high-energy limit for open superstrings; see the arguments in favor of this suggestion in [39]. Second, the role of fermionic HS fields in the above limit of superstring theory in connection with the AdS/CFT correspondence signals the importance of extending the obtained results to the case of fermionic HS fields with a mixed symmetry in the AdS space. Thus, the present Lagrangian description takes a first step towards an interacting theory with mixed-symmetry fermionic HS fields, including the case of curved backgrounds, and then towards a covariant construction (following, e.g., the BV formalism) of the generating functionals of Green's functions, including the quantum effective action; examples of such calculations can be found, e.g., in [49]. Third, we estimate an extension of the obtained results to the case of arbitrary fermionic HS fields with any number of rows in the corresponding Young tableaux as a challenging technical problem. One of the possible approaches to this problem may rely on creating a computer algorithm which would permit one to obtain the HS symmetry superalgebra and calculate the action with the sequence of gauge transformations in an analytic component form for fermionic fields of a given generalized spin. Finally, the example of a field of spin (3/2, 3/2) in Section 7.2 has demonstrated a possibility of extracting a large number of auxiliary fields until the point when the component form of the action and gauge transformations can be derived in a manifest form. In our forthcoming work [54], we plan to realize this possibility, which should permit one to significantly reduce the spectrum of fields and gauge parameters in order to simplify the component structure of the basic results of the present work, however, with a possible appearance of additional off-shell constraints for the fields and gauge parameters.