Gauge-Invariant Lagrangian Formulations for Mixed-Symmetry Higher-Spin Bosonic Fields in AdS Spaces

Abstract

We deduce a non-linear commutator higher-spin (HS) symmetry algebra which encodes unitary irreducible representations of the AdS group—subject to a Young tableaux $Y(s_1,\dots ,s_k)$ with $k\ge 2$ rows—in a d-dimensional anti-de Sitter space. Auxiliary representations for a deformed non-linear HS symmetry algebra in terms of a generalized Verma module, as applied to additively convert a subsystem of second-class constraints in the HS symmetry algebra into one with first-class constraints, are found explicitly in the case of a $k=2$ Young tableaux. An oscillator realization over the Heisenberg algebra for the Verma module is constructed. The results generalize the method of constructing auxiliary representations for the symplectic $sp\left(2k\right)$ algebra used for mixed-symmetry HS fields in flat spaces [Buchbinder, I.L.; et al. Nucl. Phys. B 2012, 862, 270–326]. Polynomial deformations of the $su(1,1)$ algebra related to the Bethe ansatz are studied as a byproduct. A nilpotent BRST operator for a non-linear HS symmetry algebra of the converted constraints for $Y(s_1,s_2)$ is found, with non-vanishing terms (resolving the Jacobi identities) of the third order in powers of ghost coordinates. A gauge-invariant unconstrained reducible Lagrangian formulation for a free bosonic HS field of generalized spin $(s_1,s_2)$ is deduced. Following the results of [Buchbinder, I.L.; et al. Phys. Lett. B 2021, 820, 136470.; Buchbinder, I.L.; et al. arXiv 2022, arXiv:2212.07097], we develop a BRST approach to constructing general off-shell local cubic interaction vertices for irreducible massive higher-spin fields (being candidates for massive particles in the Dark Matter problem). A new reducible gauge-invariant Lagrangian formulation for an antisymmetric massive tensor field of spin $(1,1)$ is obtained.

1. Introduction

The great interest in higher-spin field theory is mainly explained by a hope to re-examine the problems of a unified description for the variety of elementary particles and all of the known interactions beyond the Standard Model, as well as by a possible insight into the origin of Dark Matter [1,2] beyond the scope of models with sterile neutrinos [3] or vector massive fields [4]. Massive higher-spin fields in constant-curvature spaces are legitimate candidates for Dark Matter, providing a contribution to the total energy density at the level of about $27\%$ in a universe with the geometry of a 4D de Sitter space; see [5,6] for a review. Such a model calls for a study involving the compactification of extra dimensions after the Big Bang. In the case of a Big Bang singularity, this indicates that the physics is no longer described by classical gravity but rather by a quantum version of gravity involving higher-spin fields [7], or, perhaps, a superstring field theory model is required. The problems of Dark Matter and Dark Energy provide insight into the inflation theory, which consistently describes the post-Planck classical early universe and plays a crucial role in explaining the late-time cosmic acceleration; for a review, see, for example, ref. [8] and the references therein. Note that the simplest candidate for Dark Energy is given by a positive-valued Einstein cosmological constant $\Lambda$, constructed along with Cold Dark Matter, which has proved to be quite successful in describing a large span of observational data related to polarization anisotropies in Cosmic Microwave Background Radiation; see, for example, refs. [9,10,11].

Should these expectations prove to be viable, it appears that the development of higher-spin field theory will be of high relevance in view of its close relation to superstring theory in constant-curvature spaces, which operates an infinite set of massive and massless bosonic and fermionic HS fields subject to a multi-row Young tableaux (YT) $Y(s_1,\dots ,s_k)$, $k\ge 1$; for a review, see, for example, refs. [12,13,14,15,16,17,18] and the references therein.

A description of such theories, aimed at the final purpose of Lagrangian formulation, demands some advanced and sophisticated group-theoretical techniques, related to constructing different representations of the (super)algebras underlying the theories in question. Whereas for a Lie (super)algebra—being the case relevant to HS fields in flat spaces—the deduction and the structure of such ingredients as Verma modules and generalized Verma modules [19] are rather well established, the similar case of non-linear algebraic and superalgebraic structures—which corresponds to HS fields in AdS spaces—has not been classified until now, with the exception of totally symmetric bosonic [20,21] and fermionic [22,23] HS fields.

There are two well-established approaches to constructing gauge-invariant Lagrangians for higher-spin fields by using the respective metric and frame-like formulations. In the first approach, the fields remain unconstrained, i.e., all the conditions (d’Alembert equation, Young symmetry, absence of trace and divergence) that select an irreducible Poincaré [24,25,26] or (A)dS [27,28,29] group representation of a given mass and spin are implemented on equal footing to obtain a Lagrangian. In the second approach, some of the conditions (normally, those related to the trace) for the fields and gauge parameters are implemented in a consistent manner which is imposed by hand, outside the construction of a Lagrangian. At the level of free fields, there exist two efficient approaches to the above objectives, known as BRST methods (initially developed to quantize gauge field theories [30,31,32], constrained dynamical systems using the BFV–BRST procedure [33,34,35]), with respective complete (e.g., [36,37,38]) and incomplete [39] BRST operators (implied by String Field Theory [40,41]), whose Lagrangian descriptions for one and the same higher-spin field in a d-dimensional Minkowski space-time are shown to be equivalent [42]. For HS fields in AdS spaces, descriptions of irreducible representations (in the metric-like formalism) for the (A)dS group with both integer and half-integer spins are rather different from those for the Poincaré group in a flat space-time, even for free theories. In all of the known cases, Lagrangian descriptions for one and the same higher-spin field deduced using the constrained (incomplete) BRST approach with additional holonomic constraints and the approach with a complete BRST operator do not coincide. Successful Lagrangian descriptions for integer and half-integer spins in AdS spaces using the approach of a complete BRST operator are suggested for massless and massive particles of integer spins in [21,37], and for massive particles of half-integer spins in [22]. Some problems of this approach have not been solved (in view of the yet unknown form of holonomic constraints consistent with an incomplete BRST operator), even for free fields of a given higher spin.1

The present article develops the research line initiated by [44], and its objectives are achieved using a regular method of constructing a Verma module and a Fock space for quadratic algebras, whose negative/positive-root vectors in a Cartan-like triangular decomposition are entangled due to the presence of a special parameter r, being the inverse square of the AdS radius. The resulting ingredients allow one to obtain Lagrangian formulations (LFs) for free mixed-symmetric integer HS fields in a d-dimensional AdS space with $Y(s_1,\dots ,s_k)$ by using the Fronsdal metric-like formalism [45,46,47] in the BRST approach. Such an LF is the starting point for an interacting HS field theory in the scope of conventional Quantum Field Theory. An application of the BRST construction with a complete BRST operator to a free HS field theory in AdS spaces consists of several stages and solves a problem being inverse to that of the method [33,34,35], thus reflecting the concept of BV–BFV duality [48,49,50] as a particular case of the AKSZ model2 [52]. First of all, the conditions that determine representations of a given mass and spin are regarded as a topological (having no Hamiltonian) gauge system of mixed-class operator constraints, $o_I$, $I=1,2,\dots$, in an auxiliary Fock space $\mathcal{H}$. Second, the entire system of $o_I$, which forms a quadratic commutator algebra, is additively converted (see [53,54] for the conversion methods) within a deformed (in powers of the parameter r) algebra of $O_I$, $O_I=o_I+o_I^{\prime }$, defined in a larger Fock space, $\mathcal{H}⨂{\mathcal{H}}^{\prime }$, with first-class constraints $O_{\alpha }$, $O_{\alpha }\subset O_I$. Third, one needs to find a Hermitian and nilpotent BRST operator $Q^{\prime }$ for a non-linear algebra of converted operators $O_I$, which contains a BRST operator Q for the subsystem $O_{\alpha }$. Fourth, a Lagrangian action $\mathcal{S}$ is constructed for a given HS field through an inner product $\langle \left|K\right|\rangle$ in a total Hilbert space ${\mathcal{H}}_{tot}$, $\mathcal{S}\sim \langle \chi \left|KQ\right|\chi \rangle$ so that $\mathcal{S}$ obeys reducible gauge transformations $\delta |\chi \rangle =Q|\Lambda \rangle$, with $|\chi \rangle$ containing the initial and auxiliary fields. As a result, the corresponding equations of motion are to reproduce the initial AdS group conditions.

The above algorithm, as applied to bosonic [55,56] and fermionic [57,58] HS fields in flat spaces, does not encounter any problems in view of a linear Lie structure of the initial constraint algebra $[o_I,o_J\}=f_{IJ}^K{o}_K$ with structure constants $f_{IJ}^K$. Indeed, for the same algebra of additional parts $o_I^{\prime }$, it is sufficient to use the construction of a Verma module (VM) for the integer HS symmetry algebra $sp\left(2k\right)$ [59], which is in one-to-one correspondence with the Lorentz $so(1,d-1)$ algebra of irreducible unitary representations subject to $Y(s_1,\dots ,s_k)$, $k\le \left[\frac{d-2}{2}\right]$, for massless fields, due to the Howe duality [60]. Then, an oscillator realization of the symplectic algebra $sp\left(2k\right)$ in a Fock space ${\mathcal{H}}^{\prime }$ [59] (orthosymplectic algebra $osp\left(1\right|2k)$ for half-integer massless field representations [61]) has a polynomial form as compared to totally symmetric HS fields in AdS spaces [22,62], where the (super)algebras of $o_I$ are non-linear and different from those of $o_I^{\prime }$ [20,21,23]. A problem of equal complexity arises in constructing a BRST operator $Q^{\prime }$ for the algebra of converted constraints $O_I$, which, in the transition to AdS spaces, does not have a form, being quadratic in ghost coordinates $\mathcal{C}$, and requires a complete analysis of the relations, starting from a resolution of the Jacobi identities; see [63] for the details of finding a relevant operator $Q^{\prime }$ and [64] for classical quadratic algebras.

The details of a Lagrangian description for mixed-symmetry massless HS tensors on (A)dS backgrounds have been studied in a “frame-like” formulation [65,66,67,68,69], whereas such descriptions for mixed-symmetry massive bosonic fields with off-shell traceless constraints in the (A)dS case are known for a Young tableaux with two rows [70,71,72], the main result for Lagrangian formulation being [72]. In a metric-like formalism, (non-Lagrangian) equations of motion from the viewpoint of a BRST description with an incomplete BRST operator for arbitrary massless HS fields is discussed in [73,74], whereas in the BRST–BV approach with an incomplete BRST operator and also in the light-cone approach, mixed-symmetric HS fields are examined in [75,76,77,78]. Diverse aspects of mixed-symmetry HS field Lagrangian dynamics in Minkowski spaces are discussed in [79] and the references therein, and the case of interacting mixed-symmetry HS fields in AdS spaces is considered in [80,81]. In the case of cubic interaction (see [82,83,84,85,86,87,88,89,90,91,92] for the study of cubic vertices in different approaches), irreducible higher-spin fields in flat spaces are classified by Metsaev [93] using the light-cone formalism. We intend to follow the application of the BRST approach to Lagrangian cubic interaction vertices for irreducible totally symmetric bosonic fields in flat spaces [38,94,95,96], however, as applied to HS fields (including mixed-symmetric ones) in AdS spaces.

The article is devoted to solving the following problems. First, we deduce an HS symmetry algebra for bosonic HS fields in a d-dimensional AdS space subject to an arbitrary $Y(s_1,\dots ,s_k)$; second, we develop the Verma module construction for an HS symmetry algebra, with a two-row YT $Y(s_1,s_2)$ and an oscillator realization for a given non-linear algebra as a formal power series in the oscillators of the related Heisenberg algebra; third, we construct a BRST operator for a converted HS symmetry algebra and an unconstrained LF for free bosonic HS fields in AdS spaces subject to $Y(s_1,s_2)$; and fourth, we develop a procedure for constructing cubic interaction vertices by using three copies of massive HS fields in AdS spaces.

The article is organized as follows. In Section 2, we examine bosonic HS fields, which includes, in Section 2.1, the construction of an HS symmetry algebra $\mathcal{A}(Y\left(k\right),Ad{S}_d)$ for HS fields subject to a Young tableaux with k rows; then an auxiliary theorem on a deformation of polynomial general commutator algebras under an additive conversion procedure is presented in Section 2.2. In Section 3, we deduce a manifest form of an HS symmetry algebra of the additional parts for arbitrary HS fields, formulate the problem of Verma module construction for the algebra ${\mathcal{A}}^{\prime }(Y\left(k\right),Ad{S}_d)$ in Section 3.1 and solve it explicitly for a quadratic algebra ${\mathcal{A}}^{\prime }(Y\left(2\right),Ad{S}_d)$ of the additional parts $o_I^{\prime }$ in Section 3.2. We find a Fock space realization for ${\mathcal{A}}^{\prime }(Y\left(2\right),Ad{S}_d)$ in Section 3.3. In Section 4, we manifestly deduce a non-linear algebra ${\mathcal{A}}_c(Y\left(k\right),Ad{S}_d)$ of converted operators $O_I$ in Section 4.1; we also present the form of a BRST operator for this algebra with $k=2$ in Section 4.2 and develop an unconstrained Lagrangian formulation for massive bosonic HS fields with a two-row Young tableaux $Y(s_1,s_2)$ in Section 4.3. The general concept of finding cubic interaction vertices for massive HS fields in AdS spaces, using the approach with a complete BRST operator, is examined in Section 5. Some examples of fields with lower spins are presented in Section 6. In the Conclusion section, we summarize our results and discuss some open problems. In Appendix A, we prove a certain proposition, and then in Appendix B, we consider its application to a polynomial algebra related to the Bethe ansatz. In respective Appendices Appendix C.1 and Appendix D, we present calculations for the Verma module construction with the algebra ${\mathcal{A}}^{\prime }(Y\left(2\right),Ad{S}_d)$ in terms of a series of lemmas and also give its oscillator representation. Finally, in Appendix E, we prove that the resulting Lagrangian does indeed reproduce appropriate conditions for a field to determine an irreducible representation of the AdS group.

As a rule, we apply the notation of conventions of [38], and also the respective notation $ϵ\left(F\right)$, $gh\left(F\right)$, $[H,G\}$, $\left[x\right]$, ${\overrightarrow{s}}_k$, ${\left(\overrightarrow{s}\right)}^3$ for the Grassmann parity and ghost number of a homogeneous quantity F, as well as for a supercommutator, the integer part of a real-valued x, an integer-valued vector $(s_1,s_2,\dots ,s_k)$, and a triple $(\overrightarrow{s}{}_{k_1}^1,\overrightarrow{s}{}_{k_2}^2,\overrightarrow{s}{}_{k_3}^3)$.

2. HS Fields of Integer Spin in AdS Spaces

In this section, we obtain a number of special HS symmetry non-linear algebras which encode mixed-symmetry tensor fields as elements of irreducible AdS group representations with a generalized spin $\mathbf{s}=(s_1,\dots ,s_k)$ and mass m in AdS${}_d$ space-times. We examine the problem of a Verma module construction for one of the algebras and solve it explicitly for a non-linear algebra with a two-row Young tableaux, as well as for a polynomial algebra of order $(n-1)$. The construction of a Fock space representation for a non-linear algebra with a resulting Verma module finalizes the solution of the problem under consideration.

2.1. HS Symmetry Algebra $\mathcal{A}(Y\left(k\right),Ad{S}_d)$ for Mixed-Symmetry Tensor Fields with $Y(s_1,\dots ,s_k)$

A massive generalized integer spin $\mathbf{s}=(s_1,\dots ,s_k)$ ($s_1\ge s_2\ge \dots \ge s_k>0$, $k\le [d/2]$) AdS group irreducible representation in an AdS${}_d$ space is realized on mixed-symmetry tensors ${\Phi }_{{\mu }^1\left(s_1\right),{\mu }^2\left(s_2\right),\dots ,{\mu }^k\left(s_k\right)}\equiv {\Phi }_{{\mu }_1^1\dots {\mu }_{s_1}^1,{\mu }_1^2\dots {\mu }_{s_2}^2,\dots ,{\mu }_1^k\dots {\mu }_{s_k}^k}\left(x\right)$, which correspond to the Young tableaux

(1)

subject to the Klein–Gordon (2), divergence-free (3), tracelessness (4), and mixed-symmetry (5) equations for $\beta =(2;3;\dots ;k+1)⟺(s_1>s_2;s_1=s_2>s_3;\dots ;s_1=s_2=\dots =s_k)$ [28], namely,

$$\begin{array}{ccc}\hfill & & \{{\nabla }^2+r[(s_1-\beta -1+d)(s_1-\beta )-\sum _{i=1}^k{s}_i]+m^2\}{\Phi }_{{\mu }^1\left(s_1\right),{\mu }^2\left(s_2\right),\dots ,{\mu }^k\left(s_k\right)}=0,\hfill \end{array}$$

(2)

$$\begin{array}{ccc}& & {\nabla }^{{\mu }_{l_i}^i}{\Phi }_{{\mu }^1\left(s_1\right),{\mu }^2\left(s_2\right),\dots ,{\mu }^k\left(s_k\right)}=0,i,j=1,\dots ,k;l_i,m_i=1,\dots ,s_i,\hfill \end{array}$$

(3)

$$\begin{array}{ccc}& & g^{{\mu }_{l_i}^i{\mu }_{m_i}^i}{\Phi }_{{\mu }^1\left(s_1\right),{\mu }^2\left(s_2\right),\dots ,{\mu }^k\left(s_k\right)}=g^{{\mu }_{l_i}^i{\mu }_{m_j}^j}{\Phi }_{{\mu }^1\left(s_1\right),{\mu }^2\left(s_2\right),\dots ,{\mu }^k\left(s_k\right)}=0,l_i<m_i,\hfill \end{array}$$

(4)

$$\begin{array}{ccc}& & {\Phi }_{{\mu }^1\left(s_1\right),\dots ,\{{\mu }^i\left(s_i\right)\underbrace{,\dots ,{\mu }_1^j\dots }{\mu }_{l_j}^j\}\dots {\mu }_{s_j}^j,\dots {\mu }^k\left(s_k\right)}=0,i<j,1\le l_j\le s_j,\hfill \end{array}$$

(5)

where the underlined brackets denote the fact that the indices inside do not take part in symmetrization, i.e., the present symmetrization involves only the indices ${\mu }^i\left(s_i\right),{\mu }_{l_j}^j$ in $\{{\mu }^i\left(s_i\right)\underbrace{,\dots ,{\mu }_1^j\dots }{\mu }_{l_j}^j\}$.

To obtain the HS symmetry algebra of $o_I$ for a description of all integer HS fields, we introduce, in a standard manner, a Fock space $\mathcal{H}$ generated by k pairs of bosonic creation $a_{{\mu }^i}^i\left(x\right)$ and annihilation $a_{{\nu }^j}^{j+}\left(x\right)$ operators,3 $i,j=1,\dots ,k,{\mu }^i,{\nu }^j=0,1\dots ,d-1$,

$$\begin{array}{c}\hfill [a_{{\mu }^i}^i,a_{{\nu }^j}^{j+}]=-g_{{\mu }^i{\nu }^j}{\delta }^{ij},{\delta }^{ij}=diag(1,1,\dots 1),\end{array}$$

(6)

and a set of constraints for an arbitrary string-like vector $|\Phi \rangle \in \mathcal{H}$, which we call a basic vector,

$$\begin{array}{ccc}\hfill & & |\Phi \rangle =\sum _{s_1=0}^{\infty }\sum _{s_2=0}^{s_1}\cdots \sum _{s_k=0}^{s_{k-1}}\frac{{ı}^{s_1+\dots +s_k}}{s_1!\dots s_k!}{\Phi }_{{\mu }^1\left(s_1\right),{\mu }^2\left(s_2\right),\dots ,{\mu }^k\left(s_k\right)}\left(x\right)\prod _{i=1}^k\prod _{l_i=1}^{s_i}{a}_i^{+{\mu }_{l_i}^i}|0\rangle ,\hfill \end{array}$$

(7)

$$\begin{array}{ccc}\hfill & & {\tilde{l}}_0|\Phi \rangle =\left(l_0+{\tilde{m}}_b^2+r\left((g_0^1-2\beta -2)g_0^1-{\displaystyle \sum _{i=2}^k}{g}_0^i\right)\right)|\Phi \rangle =0,l_0=[D^2-r\frac{d(d-2(k+1\left)\right)}{4}],\hfill \end{array}$$

(8)

$$\begin{array}{ccc}\hfill & & \left(l^i,l^{ij},t^{i_1{j}_1}\right)|\Phi \rangle =\left(-i{a}_{\mu }^i{D}^{\mu },\frac{1}{2}{a}_{\mu }^i{a}^{j\mu },a_{\mu }^{i_1+}{a}^{j_1\mu }\right)|\Phi \rangle =0,i\le j;i_1<j_1,\hfill \end{array}$$

(9)

with particle number operators, a central charge, and a covariant derivative in $\mathcal{H}$, respectively,4

$$\begin{array}{ccc}\hfill & & g_0^i=-\frac{1}{2}\{a_{\mu }^{i+},a^{\mu i}\},{\tilde{m}}_b^2=m^2+r\beta (\beta +1),\hfill \end{array}$$

(10)

$$\begin{array}{ccc}& & D_{\mu }={\partial }_{\mu }-{\omega }_{\mu }^{ab}\left(x\right)\left(\sum _i{a}_{ia}^{+}{a}_{ib}\right),a_i^{\mu (+)}\left(x\right)=e_a^{\mu }\left(x\right)a_i^{a(+)},\hfill \end{array}$$

(11)

with a vielbein $e_a^{\mu }$, a spin connection ${\omega }_{\mu }^{ab}$, and tangent indices $a,b$, $a=0,1\dots ,d-1$. The operator $D_{\mu }$ is equivalent (as applied in $\mathcal{H}$) to the covariant derivative ${\nabla }_{\mu }$ with the d’Alembertian $D^2=(D_a+{\omega }^b{}_{ba})D^a$. The set of $k(k+1)$ primary constraints (8) and (9) with $\left\{o_{\alpha }\right\}$ = $\left\{{\tilde{l}}_0,l^i,l^{ij},t^{i_1{j}_1}\right\}$ is equivalent to Equations (2)–(5) for all admissible values of spin, and for the field ${\Phi }_{{\mu }^1\left(s_1\right),{\mu }^2\left(s_2\right),\dots ,{\mu }^k\left(s_k\right)}$ with a fixed spin $\mathbf{s}=(s_1,s_2,\dots ,s_k)$, once, in addition to (8), (9), we have to add k more constraints with $g_0^i$,

$$\begin{array}{c}\hfill g_0^i|\Phi \rangle =(s_i+\frac{d}{2})|\Phi \rangle .\end{array}$$

(12)

The condition for the algebra of $o_{\alpha }$ to be closed under the $[,]$-multiplication leads to an enlargement of $o_{\alpha }$ by adding the operators $g_0^i$ and the Hermitian conjugates $o_{\alpha }^{+}$,

$$\left(l^{i+},l^{ij+},t^{i_1{j}_1+}\right)=\left(-i{a}_{\mu }^{i+}{D}^{\mu },\frac{1}{2}{a}_{\mu }^{i+}{a}^{j\mu +},a_{\mu }^{i_1}{a}^{j_1\mu +}\right),i\le j;i_1<j_1,$$

(13)

with respect to the inner product in $\mathcal{H}$,

$$\begin{array}{ccc}\hfill \langle \Psi |\Phi \rangle & =& \int d^{d}x\sqrt{\left|g\right|}\sum _{i=1}^k\sum _{s_i=0}^{s_{i-1}}\frac{{(-1)}^{s_1+\dots +s_k}}{s_1!\dots s_k!}{\Psi }_{{\mu }^1\left(s_1\right),{\mu }^2\left(s_2\right),\dots ,{\mu }^k\left(s_k\right)}^{*}\left(x\right){\Phi }^{{\mu }^1\left(s_1\right),{\mu }^2\left(s_2\right),\dots ,{\mu }^k\left(s_k\right)}\left(x\right).\hfill \end{array}$$

(14)

This fact guarantees the Hermiticity of a corresponding BFV–BRST operator, with the self-conjugate operators $(l_0^{+},{g_0^i}^{+})=(l_0,g_0^i)$ taken into account (hence a real value of the Lagrangian $\mathcal{L}$) for the system of all the operators $\{o_{\alpha },o_{\alpha }^{+},g_0^i\}$. We call the algebra of these operators an integer higher-spin (integer HS) symmetry algebra in an AdS space, with a Young tableaux with k rows5 and denote it as $\mathcal{A}(Y\left(k\right),Ad{S}_d)$.

The maximal Lie subalgebra of the operators $l^{ij},t^{i_1{j}_1},g_0^i,l^{ij+},t^{i_1{j}_1+}$ is isomorphic to the symplectic algebra $sp\left(2k\right)$ (see [59] for details; later on, we refer to it as $sp\left(2k\right)$, whereas the only nontrivial quadratic commutators in $\mathcal{A}(Y\left(k\right),Ad{S}_d)$ are due to the operators containing $D_{\mu }$: $l^i,{\tilde{l}}_0,l^{i+}$. For the purpose of an LF construction, it is sufficient to have a simplified (having no central charge ${\tilde{m}}_b^2$), so-called modified, HS symmetry algebra ${\mathcal{A}}_m(Y\left(k\right),Ad{S}_d)$ with the operator $l_0$ (8) instead of ${\tilde{l}}_0$ so that the AdS mass term ${\tilde{m}}_b^2+r\left((g_0^1-2\beta -2)g_0^1-{\sum }_{i=2}^k{g}_0^i\right)$ is restored later on, as usual, within a conversion procedure and an appropriate LF construction.

The algebra ${\mathcal{A}}_m(Y\left(k\right),Ad{S}_d)$ of the operators $o_I$, from the viewpoint of a Hamiltonian analysis for dynamical systems, contains the first-class constraint $l_0$, as well as the $2k$ differential $l_i,l_i^{+}$ and $2k^2$ algebraic $t^{i_1{j}_1},t_{i_1{j}_1}^{+},l^{ij},l_{ij}^{+}$ second-class constraints $o_a$, and also the operators $g_0^i$, composing an invertible matrix ${\Delta }_{ab}\left(g_0^i\right)$ for a topological (having a zero Hamiltonian) gauge system due to

$$\begin{array}{c}\hfill [o_a,o_b]=f_{ab}^c{o}_c+f_{ab}^{cd}{o}_c{o}_d+{\Delta }_{ab}\left(g_0^i\right),[o_a,l_o]=f_{a\left[l_0\right]}^c{o}_c+f_{a\left[l_0\right]}^{cd}{o}_c{o}_d.\end{array}$$

(15)

Here, $f_{ab}^c,f_{ab}^{cd},f_{a\left[l_0\right]}^c,f_{a\left[l_0\right]}^{cd},{\Delta }_{ab}$ are antisymmetric with respect to the permutations of the lower indices; the constant quantities and the quantities ${\Delta }_{ab}\left(g_0^i\right)$ form a non-vanishing $2k(k+1)\times 2k(k+1)$ matrix $\parallel {\Delta }_{ab}\parallel$ in the Fock space $\mathcal{H}$ on the surface $\Sigma \subset \mathcal{H}$, $\parallel {\Delta }_{ab}{\parallel }_{|\Sigma }\ne 0$, which is determined by the equation $o_{\alpha }|\Phi \rangle =0$. The set of $o_I$ satisfies non-linear relations, being additional to those for $sp\left(2k\right)$ and given by the multiplication

Table 1. The HS symmetry non-linear algebra ${\mathcal{A}}_m(Y\left(k\right),Ad{S}_d)$.

$[\downarrow ,\to \}$	${\mathit{t}}^{{\mathit{i}}_1{\mathit{j}}_1}$	${\mathit{t}}_{{\mathit{i}}_1{\mathit{j}}_1}^{+}$	${\mathit{l}}_0$	${\mathit{l}}^{\mathit{i}}$	${\mathit{l}}^{\mathit{i}+}$	${\mathit{l}}^{{\mathit{i}}_1{\mathit{j}}_1}$	${\mathit{l}}^{{\mathit{i}}_1{\mathit{j}}_1+}$
$t^{i_2{j}_2}$	$A^{i_2{j}_2,i_1{j}_1}$	$B^{i_2{j}_2}{}_{i_1{j}_1}$	0	$l^{j_2}{\delta }^{i_{2}i}$	$-l^{i_2+}{\delta }^{j_{2}i}$	$l^{\{j_1{j}_2}{\delta }^{i_1\}{i}_2}$	$-l^{i_2\{i_1+}{\delta }^{j_1\}{j}_2}$
$t_{i_2{j}_2}^{+}$	$-B^{i_1{j}_1}{}_{i_2{j}_2}$	$A_{i_1{j}_1,i_2{j}_2}^{+}$	0	$l_{i_2}{\delta }_{j_2}^i$	$-l_{j_2}^{+}{\delta }_{i_2}^i$	$l_{i_2}{}^{\{j_1}{\delta }_{j_2}^{i_1\}}$	$-l_{j_2}{}^{\{j_1+}{\delta }_{i_2}^{i_1\}}$
$l_0$	0	0	0	$-r{\mathcal{K}}_1^{i+}$	$r{\mathcal{K}}_1^i$	0	0
$l^j$	$-l^{j_1}{\delta }^{i_{1}j}$	$-l^{i_1}{\delta }^{j_{1}j}$	$r{\mathcal{K}}_1^{j+}$	$W_b^{ji}$	$X_b^{ji}$	0	$-\frac{1}{2}{l}^{\{i_1+}{\delta }^{j_1\}j}$
$l^{j+}$	$l^{i_1+}{\delta }^{j_{1}j}$	$l^{j_1+}{\delta }^{i_{1}j}$	$-r{\mathcal{K}}_1^j$	$-X_b^{ij}$	$-W_b^{ji+}$	$\frac{1}{2}{l}^{\{i_1}{\delta }^{j_1\}j}$	0
$l^{i_2{j}_2}$	$-l^{j_1\{j_2}{\delta }^{i_2\}{i}_1}$	$-l^{i_1\{i_2}{\delta }^{j_2\}{j}_1}$	0	0	$-\frac{1}{2}{l}^{\{i_2}{\delta }^{j_2\}i}$	0	$L^{i_2{j}_2,i_1{j}_1}$
$l_{i_2{j}_2}^{+}$	$l_{\{i_2}^{i_1+}{\delta }_{j_2\}}^{j_1}$	$l_{j_1\{j_2}^{+}{\delta }_{i_2\}{i}_1}$	0	$\frac{1}{2}{l}_{\{i_2}^{+}{\delta }_{j_2\}}^i$	0	$-L^{i_1{j}_1,i_2{j}_2}$	0
$g_0^j$	$-F^{i_1{j}_1,j}$	$F^{i_1{j}_1,j+}$	0	$-l^i{\delta }^{ij}$	$l^{i+}{\delta }^{ij}$	$-l^{j\{i_1}{\delta }^{j_1\}j}$	$l^{j\{i_1+}{\delta }^{j_1\}j}$

.

First of all, note that Table 1 does not include any columns with $[,\}$-products of all $o_I$ with $g_0^j$, which can be observed from the rows containing $g_0^j$, as follows, ${[b,o_I\}}^{+}$ = $-[o_I^{+},b\}$, with allowance for the closure of the HS algebra with respect to the Hermitian conjugation. Second, the operators $t^{i_2{j}_2},t_{i_2{j}_2}^{+}$ are defined so as to satisfy the following properties:

$$(t^{i_2{j}_2},t_{i_2{j}_2}^{+})\equiv (t^{i_2{j}_2},t_{i_2{j}_2}^{+}){\theta }^{j_2{i}_2},{\theta }^{j_2{i}_2}=(1,0)\mathtt{f}\mathtt{o}\mathtt{r}(j_2>i_2,j_2\le i_2),$$

(16)

with the Heaviside $\theta$-symbol6 ${\theta }^{ji}$. Third, the products $B_{i_1{j}_1}^{i_2{j}_2},A^{i_2{j}_2,i_1{j}_1},F^{i_1{j}_1,i},L^{i_2{j}_2,i_1{j}_1}$ are determined by the explicit relations

$$\begin{array}{ccc}\hfill B^{i_2{j}_2}{}_{i_1{j}_1}& =& (g_0^{i_2}-g_0^{j_2}){\delta }_{i_1}^{i_2}{\delta }_{j_1}^{j_2}+(t_{j_1}{}^{j_2}{\theta }^{j_2}{}_{j_1}+t^{j_2}{}_{j_1}^{+}{\theta }_{j_1}{}^{j_2}){\delta }_{i_1}^{i_2}-(t_{i_1}^{+}{}^{i_2}{\theta }^{i_2}{}_{i_1}+t^{i_2}{}_{i_1}{\theta }_{i_1}{}^{i_2}){\delta }_{j_1}^{j_2},\hfill \end{array}$$

(17)

$$\begin{array}{ccc}\hfill A^{i_2{j}_2,i_1{j}_1}& =& t^{i_1{j}_2}{\delta }^{i_2{j}_1}-t^{i_2{j}_1}{\delta }^{i_1{j}_2},F^{i_2{j}_2,i}=t^{i_2{j}_2}({\delta }^{j_{2}i}-{\delta }^{i_{2}i}),\hfill \end{array}$$

(18)

$$\begin{array}{ccc}\hfill L^{i_2{j}_2,i_1{j}_1}& =& \frac{1}{4}\{{\delta }^{i_2{i}_1}{\delta }^{j_2{j}_1}\left[2g_0^{i_2}{\delta }^{i_2{j}_2}+g_0^{i_2}+g_0^{j_2}\right]-{\delta }^{j_2\{i_1}\left[t^{j_1\}{i}_2}{\theta }^{i_2{j}_1\}}+t^{i_2{j}_1\}+}{\theta }^{j_1\}{i}_2}\right]\hfill \\ & & -{\delta }^{i_2\{i_1}\left[t^{j_1\}{j}_2}{\theta }^{j_2{j}_1\}}+t^{j_2{j}_1\}+}{\theta }^{j_1\}{j}_2}\right]\}.\hfill \end{array}$$

(19)

They satisfy the obvious additional properties of antisymmetry and Hermitian conjugation

$$\begin{array}{ccc}& & A^{i_2{j}_2,i_1{j}_1}=-A^{i_1{j}_1,i_2{j}_2},A_{i_1{j}_1,i_2{j}_2}^{+}={\left(A_{i_1{j}_1,i_2{j}_2}\right)}^{+}=t_{i_2{j}_1}^{+}{\delta }_{j_2{i}_1}-t_{i_1{j}_2}^{+}{\delta }_{i_2{j}_1},\hfill \end{array}$$

(20)

$$\begin{array}{ccc}& & {\left(L^{i_2{j}_2,i_1{j}_1}\right)}^{+}=L^{i_1{j}_1,i_2{j}_2},F^{i_2{j}_2,i+}={\left(F^{i_2{j}_2,i}\right)}^{+}=t^{i_2{j}_2+}({\delta }^{j_{2}i}-{\delta }^{i_{2}i}),\hfill \end{array}$$

(21)

$$\begin{array}{ccc}\hfill & & {B^{i_2{j}_2}{}_{i_1{j}_1}}^{+}=(g_0^{i_2}-g_0^{j_2}){\delta }_{i_1}^{i_2}{\delta }_{j_1}^{j_2}+(t_{j_1}^{+}{}^{j_2}{\theta }^{j_2}{}_{j_1}+t^{j_2}{}_{j_1}{\theta }_{j_1}{}^{j_2}){\delta }_{i_1}^{i_2}-(t_{i_1}{}^{i_2}{\theta }^{i_2}{}_{i_1}+t^{i_2+}{}_{i_1}{\theta }_{i_1}{}^{i_2}){\delta }_{j_1}^{j_2}.\hfill \end{array}$$

(22)

Fourth, the independent quantities ${\mathcal{K}}_1^k,W_b^{ki},X_b^{ki}$ in Table 1 are quadratic in $o_I$:

$$\begin{array}{ccc}\hfill W_b^{ij}& =& [l^i,l^j]=2r\left[(g_0^j-g_0^i)l^{ij}-(\sum _m{t}^{m[j}{\theta }^{[jm}+t^{[jm+}{\theta }^{m[j})l^{i]m}\right],\hfill \end{array}$$

(23)

$$\begin{array}{ccc}\hfill {\mathcal{K}}_1^k& =& r^{-1}[l_0,l^{k+}]=\left(4\sum _i{l}^{ki+}{l}^i+l^{k+}(2g_0^k-1)-2(\sum _i{l}^{i+}{t}^{+ik}{\theta }^{ki}+l^{i+}{t}^{ki}{\theta }^{ik})\right),\hfill \end{array}$$

(24)

$$\begin{array}{ccc}\hfill X_b^{ij}& =& \left\{l_0+r\left(K_0^{0i}+\sum _{l=i+1}^k{\mathcal{K}}_0^{il}+\sum _{l=1}^{i-1}{\mathcal{K}}_0^{li}\right)\right\}{\delta }^{ij}\hfill \\ & & -r\left[4{\sum }_l{l}^{jl+}{l}^{li}-{\displaystyle \sum _{l=1}^{j-1}}{t}^{lj+}{t}^{li}-{\displaystyle \sum _{l=i+1}^k}{t}^{il+}{t}^{jl}-{\displaystyle \sum _{l=j+1}^{i-1}}{t}^{li}{t}^{jl}+(g_0^j+g_0^i-j-1)t_{ji}\right]{\theta }^{ij}\hfill \\ & & -r\left[4{\sum }_l{l}^{jl+}{l}^{li}-{\displaystyle \sum _{l=1}^{i-1}}{t}^{lj+}{t}^{li}-{\displaystyle \sum _{l=j+1}^k}{t}^{il+}{t}^{jl}-{\displaystyle \sum _{l=i+1}^{j-1}}{t}^{il+}{t}^{lj+}+t_{ij}^{+}(g_0^j+g_0^i-i-1)\right]{\theta }^{ji}.\hfill \end{array}$$

(25)

In (25), we used the quantities $K_0^{0i},i,j=1,\dots ,k$, ${\mathcal{K}}_0^{ij}$, composing a Casimir operator ${\mathcal{K}}_0\left(k\right)$ for the $sp\left(2k\right)$ algebra:

$${\mathcal{K}}_0\left(k\right)=\sum _i{K}_0^{0i}+2\sum _{i,j}{\mathcal{K}}_0^{ij}{\theta }^{ji}=\sum _i\left({\left(g_0^i\right)}^2-2g_0^i-4l^{ii+}{l}^{ii}\right)+2\left(\sum _{i=1}^k\sum _{j=i+1}^k(t_{ij}^{+}{t}^{ij}-4l_{ij}^{+}{l}^{ij}-g_0^j)\right).$$

(26)

The algebra ${\mathcal{A}}_m(Y\left(k\right),Ad{S}_d)$ may be regarded as a non-linear deformation (in powers of r) of the integer HS symmetry algebra $\mathcal{A}(Y\left(k\right),R^{1,d-1})$ [59] in Minkowski space, namely,

$${\mathcal{A}}_m(Y\left(k\right),Ad{S}_d)=\mathcal{A}(Y\left(k\right),R^{1,d-1})\left(r\right)=\left(T^k\oplus T^{k*}\oplus l_0\right)\left(r\right)+\supset sp\left(2k\right),$$

(27)

for a k-dimensional commutative (in $R^{1,d-1}$) algebra $T^k=\left\{l_i\right\}$ and its dual $T^{k*}=\left\{l^{i+}\right\}$, which represents a semidirect sum of the symplectic algebra $sp\left(2k\right)$, being an algebra of internal derivations of $(T^k\oplus T^{k*})$. For HS fields with a single spin $s_1$, $k=1$, and a two-component spin $(s_1,s_2)$, $k=2$, the algebra ${\mathcal{A}}_m(Y\left(k\right),Ad{S}_d)$ coincides with the respective familiar HS symmetry algebras in AdS spaces given by [20,63].

Now, we utilize the results of a special theorem, and then proceed to a conversion of the algebra of $o_I$ so as to obtain the algebra of $O_I$ with first-class constraints alone.

2.2. On Additive Conversion for Polynomial Algebras

In this subsection, having in mind the problem of an additive conversion for non-linear algebras (superalgebras) with a subset of second-class constraints, we need to use an important statement based on the following (see [23] for a detailed description).

Definition 1.

A non-linear commutator superalgebra $\mathcal{A}$ of basis elements $o_I$, $I\in \Delta$ (with Δ being a finite or infinite set of indices) is called a polynomial superalgebra of order n, $n\in \mathbf{N}$ if the set $\left\{o_I\right\}$ is subject to the n-th order polynomial supercommutator relations

$$\begin{array}{ccc}\hfill & & [o_I,o_J\}=F_{IJ}^K\left(o\right)o_K,F_{IJ}^K=f_{IJ}^{\left(1\right)K}+\sum _{n=2}^{\infty }{f}_{IJ}^{\left(n\right)K_1\dots K_{n-1}K}\prod _{i=1}^{n-1}{o}_{K_i},\hfill \\ \hfill & & f_{IJ}^{\left(n\right)K_1\dots K_n}\ne 0,\mathrm{and}{f}_{IJ}^{\left(k\right)K_1\dots K_n{K}_{n+1}\dots K_k}=0,k>n,\hfill \end{array}$$

(28)

with structure coefficients $f_{IJ}^{\left(n\right)K_1\dots K_{n-1}K}$ of generalized antisymmetry with respect to the permutations of lower indices, $f_{IJ}^{\left(n\right)K_1\cdots K_n}=-{(-1)}^{\epsilon \left(o_I\right)\epsilon \left(o_J\right)}{f}_{JI}^{\left(n\right)K_1\cdots K_n}$.

Now, we are in a position to formulate our basic statement of Section 2.2, presented as the following.

Proposition 1.

Let $\mathcal{A}$ be a polynomial superalgebra of order n with basis elements $o_I$ defined in a Hilbert space $\mathcal{H}$. Then, for a set ${\mathcal{A}}^{\prime }$ of elements $o_I^{\prime }$ defined in a new Hilbert space ${\mathcal{H}}^{\prime }$ ($\mathcal{H}\bigcap {\mathcal{H}}^{\prime }=\varnothing$) and supercommuting with $o_I$, as well as for a direct sum of sets ${\mathcal{A}}_c=\mathcal{A}+{\mathcal{A}}^{\prime }$ of the operators $O_I$, $O_I=o_I+o_I^{\prime }$, defined in the tensor product $\mathcal{H}\otimes {\mathcal{H}}^{\prime }$, the requirement to be in involution

$$[O_I,O_J\}=F_{IJ}^K(o^{\prime },O)O_K,$$

(29)

implies that the sets $\left\{o_I^{\prime }\right\}$ and $\left\{O_I\right\}$ form the respective polynomial commutator superalgebra ${\mathcal{A}}^{\prime }$ of order n and the non-linear commutator superalgebra ${\mathcal{A}}_c$ with the composition laws

$$\begin{array}{ccc}\hfill & & [o_I^{\prime },o_J^{\prime }\}=f_{IJ}^{\left(1\right)K_1}{o}_{K_1}^{\prime }+\sum _{m=2}^n{(-1)}^{m-1+{\epsilon }_{K_{\left(m\right)}}}{f}_{IJ}^{\left(m\right)K_m\cdots K_1}\prod _{s=1}^m{o}_{K_s}^{\prime },\hfill \end{array}$$

(30)

$$\begin{array}{ccc}\hfill & & [O_I,O_J\}=\left(f_{IJ}^{\left(1\right)K}+\sum _{m=2}^n{F}_{IJ}^{\left(m\right)K}(o^{\prime },O)\right)O_K,\hfill \end{array}$$

(31)

for the values ${\epsilon }_{K_{\left(n\right)}}$, ${\epsilon }_{K_{\left(n\right)}}={\sum }_{s=1}^{n-1}{\epsilon }_{K_s}\left({\sum }_{l=s+1}^n{\epsilon }_{K_l}\right)$. The structure functions $F_{IJ}^{\left(m\right)k}(o^{\prime },O)$ in (30) are constructed with respect to the coefficients $f_{IJ}^{\left(m\right)K_1\dots K_m}\equiv f_{IJ}^{K_1\dots K_m}$ used in (28) as follows:

$$\begin{array}{ccc}\hfill & & F_{IJ}^{\left(m\right)K}=f_{IJ}^{K_1\cdots K_m}\prod _{p=1}^{m-1}{O}_{K_p}+\sum _{s=1}^{m-1}{(-1)}^{s+{\epsilon }_{K_{\left(s\right)}}}{f}_{ij}^{\widehat{K_s\cdots K_1}\widehat{K_{s+1}\cdots K_m}}\prod _{p=1}^s{o}_{K_p}^{\prime }\prod _{l=s+1}^{m-1}{O}_{K_l},\mathtt{w}\mathtt{h}\mathtt{e}\mathtt{r}\mathtt{e}\hfill \end{array}$$

(32)

$$\begin{array}{ccc}\hfill & & f_{ij}^{\widehat{K_s\cdots K_1}\widehat{K_{s+1}\cdots K_m}}=f_{ij}^{K_s\cdots K_1{K}_{s+1}\cdots K_m}+f_{ij}^{K_s\cdots K_{s+1}{K}_1{K}_{s+2}\cdots K_m}{(-1)}^{{\epsilon }_{K_{s+1}}{\epsilon }_{K_1}}+\cdots +\hfill \\ \hfill & & f_{ij}^{K_{s+1}{K}_s\cdots K_1{K}_{s+2}\cdots K_m}{(-1)}^{{\epsilon }_{K_{s+1}}{\sum }_{l=1}^s{\epsilon }_{K_l}}+(f_{ij}^{K_{s+1}{K}_s\cdots K_{s+2}{K}_1{K}_{s+3}\cdots K_m}{(-1)}^{{\epsilon }_{K_{s+2}}{\epsilon }_{K_1}}+\hfill \\ \hfill & & \cdots +f_{ij}^{K_{s+1}{K}_{s+2}{K}_s\cdots K_1{K}_{s+3}\cdots K_m}{(-1)}^{{\epsilon }_{K_{s+2}}{\sum }_{l=1}^s{\epsilon }_{K_l}}){(-1)}^{{\epsilon }_{K_{s+1}}{\sum }_{l=1}^s{\epsilon }_{K_l}}+\cdots +\hfill \\ \hfill & & {(-1)}^{{\sum }_{l=s+1}^m{\epsilon }_{K_l}{\sum }_{p=1}^s{\epsilon }_{K_p}}{f}_{ij}^{K_{s+1}\cdots K_m{K}_s\cdots K_1},\hfill \end{array}$$

(33)

where the sum in (33) contains $\frac{m!}{s!(m-s)!}$ terms, with all the possible ways of arrangement for $(K_{s+1},\dots ,K_m)$ among the indices $(K_s,\dots ,K_1)$ in $f_{ij}^{K_s\cdots K_1{K}_{s+1}\cdots K_m}$, without changing the separate ordering of the indices7 $K_{s+1},\dots ,K_m$ and $K_s,\dots ,K_1.$

The correctness of the proposition is examined in Appendix A. Turning to the structure of the superalgebra ${\mathcal{A}}_c$, we emphasize that, in contrast to $\mathcal{A}$ and ${\mathcal{A}}^{\prime }$, we call it a non-homogeneous polynomial superalgebra of order n, due to the form of relations (32); see Note 15 in Appendix A for remarks. In the case of a Lie superalgebra ($n=1$), the structures of superalgebras $\mathcal{A}$, ${\mathcal{A}}^{\prime }$ and ${\mathcal{A}}_c$ are identical, and used as the integer HS symmetry algebra $\mathcal{A}(Y\left(k\right),R^{1,d-1})$ [59]. For quadratic algebras (n = 2), the algebraic relations for ${\mathcal{A}}^{n-1}$, ${\mathcal{A}}^{\prime n-1}$ and ${\mathcal{A}}_c^{n-1}$ do not coincide with each other, due to the presence of structure functions $f_{IJ}^{\left(2\right)K_1{K}_2}$, which is originally shown for the algebra $\mathcal{A}(Y\left(1\right),Ad{S}_d)$ with totally symmetric HS tensors in an AdS space [21], having the form

$$\begin{array}{ccc}\hfill & & [o_I,o_J]=f_{IJ}^{\left(1\right)K_1}{o}_{K_1}+f_{IJ}^{\left(2\right)K_1{K}_2}{o}_{K_1}{o}_{K_2},[o_I^{\prime },o_J^{\prime }]=f_{IJ}^{\left(1\right)K_1}{o}_{K_1}^{\prime }-f_{IJ}^{\left(2\right)K_2{K}_1}{o}_{K_1}^{\prime }{o}_{K_2}^{\prime },\hfill \end{array}$$

(34)

$$\begin{array}{ccc}\hfill & & [O_I,O_J]=\left(f_{IJ}^{\left(1\right)K}+F_{IJ}^{\left(2\right)K}(o^{\prime },O)\right)O_K,F_{IJ}^{\left(2\right)K}=f_{IJ}^{\left(2\right)K_{1}K}{O}_{K_1}-(f_{IJ}^{\left(2\right)K_{1}K}+f_{IJ}^{\left(2\right)K{K}_1}){o^{\prime }}_{K_1}.\hfill \end{array}$$

(35)

The relations (34) and (35) are sufficient to determine the form of multiplication laws for the algebra of the additional parts $o_I^{\prime }$, ${\mathcal{A}}^{\prime }(Y\left(k\right),Ad{S}_d)$ and for the converted $O_I$, operator algebra ${\mathcal{A}}_c(Y\left(k\right),Ad{S}_d)$.

As a new result, we present an explicit form for the cubic commutator algebras ${\mathcal{A}}^{\prime }$, ${\mathcal{A}}_c$,

$$\begin{array}{ccc}\hfill [o_I^{\prime },o_J^{\prime }\}& =& f_{IJ}^{\left(1\right)K_1}{o}_{K_1}^{\prime }-f_{IJ}^{\left(2\right)K_2{K}_1}{o}_{K_1}^{\prime }{o}_{K_2}^{\prime }+f_{IJ}^{\left(3\right)K_3{K}_2{K}_1}{o}_{K_1}^{\prime }{o}_{K_2}^{\prime }{o}_{K_3}^{\prime },\hfill \end{array}$$

$$\begin{array}{ccc}\hfill [O_I,O_J\}& =& \left(f_{IJ}^{\left(1\right)K}+\sum _{l=2}^n{F}_{IJ}^{\left(l\right)K}(o^{\prime },O)\right)O_K,\hfill \\ \hfill F_{IJ}^{\left(3\right)K}& =& f_{IJ}^{\left(3\right)K_1{K}_{2}K}{O}_{K_1}{O}_{K_2}-\left(f_{IJ}^{\left(3\right)K_1{K}_{2}K}+f_{IJ}^{\left(3\right)K_2{K}_{1}K}\right){o^{\prime }}_{K_1}{O}_{K_2}\hfill \end{array}$$

(36)

$$\begin{array}{ccc}\hfill & & +\left(f_{IJ}^{\left(3\right)K_2{K}_{1}K}+f_{IJ}^{\left(3\right)K_{2}K{K}_1}+f_{IJ}^{\left(3\right)K{K}_2{K}_1}\right){o^{\prime }}_{K_1}{o^{\prime }}_{K_2},\hfill \end{array}$$

(37)

if the commutator relations for the initial algebra $\mathcal{A}$ are given by (28) for $n=3$.

In Appendix B, we consider the example of a polynomial 3-parametric algebra $\mathcal{A}$ of order $n-1$, $n\in \mathbf{N}$, being a polynomial deformation of the $su(1,1)$ algebra for $n>2$.

3. Auxiliary HS Symmetry Algebra ${\mathcal{A}}^{\prime }(\mathit{Y}\left(\mathit{k}\right),{\mathit{AdS}}_{\mathit{d}})$

The procedure of additive conversion for the non-linear HS symmetry algebra $\mathcal{A}(Y\left(k\right),Ad{S}_d)$ of the operators $o_I$ implies establishing, first of all, an explicit form of the algebra ${\mathcal{A}}^{\prime }(Y\left(k\right),Ad{S}_d)$ for the additional parts $o_I^{\prime }$, and second, a representation of ${\mathcal{A}}^{\prime }(Y\left(k\right),Ad{S}_d)$ in terms of appropriate Heisenberg algebra elements acting in a new Fock space ${\mathcal{H}}^{\prime }$. The structure of non-linear commutators of the initial algebra implies a need for converting all the operators $o_I$ in order to construct an unconstrained LF for a given HS field ${\Phi }_{{\left({\mu }^1\right)}_{s_1},{\left({\mu }^2\right)}_{s_2},\dots ,{\left({\mu }^k\right)}_{s_k}}$.

The former step is based on determining a multiplication table ${\mathcal{A}}^{\prime }(Y\left(k\right),Ad{S}_d)$ for the operators $o_I^{\prime }$, following the structure of the algebra $\mathcal{A}(Y\left(k\right),Ad{S}_d)$, given by (34) and Table 1.

As a result, the required composition law for ${\mathcal{A}}^{\prime }(Y\left(k\right),Ad{S}_d)$ is the same as the one for the algebra $\mathcal{A}(Y\left(k\right),Ad{S}_d)$ in its linear Lie part, i.e., for the $sp\left(2k\right)$ subalgebra of elements $(l^{\prime ij},l^{\prime ij+},t^{\prime i_1{j}_1},t_{i_1{j}_1}^{\prime +},g_0^{\prime i})$, and is different in the non-linear part of Table 1, determined by the isometry group elements $l_i^{\prime },l_j^{\prime +},l_0^{\prime }$. The corresponding non-linear submatrix of the multiplication matrix for ${\mathcal{A}}^{\prime }(Y\left(k\right),Ad{S}_d)$ has the form given by

Table 2. The non-linear part of the algebra ${\mathcal{A}}^{\prime }(Y\left(k\right),Ad{S}_d)$.

$[\downarrow ,\to \}$	${\mathit{l}}_0^{\prime }$	${\mathit{l}}^{\prime \mathit{i}}$	${\mathit{l}}^{\prime \mathit{i}+}$
$l_0^{\prime }$	0	$r{\mathcal{K}}_1^{\prime bi+}$	$-r{\mathcal{K}}_1^{\prime i}$
$l^{\prime j}$	$-r{\mathcal{K}}_1^{\prime j+}$	$-W_b^{\prime ji}$	$X_b^{\prime ji}$
$l^{\prime j+}$	$r{\mathcal{K}}_1^{\prime j}$	$-X_b^{\prime ij}$	$W_b^{\prime ji+}$

.

Here, the functions ${\mathcal{K}}_1^{\prime i+}$, ${\mathcal{K}}_1^{\prime i}$, $W_b^{\prime ji}$, $W_b^{\prime ji+}$$(X_b^{\prime ij}-l_0^{\prime })$ have the same definition as those in (23)–(25) for the initial operators $o_I$, albeit with an opposite sign for $(X_b^{\prime ij}-l_0^{\prime })$,

$$\begin{array}{ccc}\hfill & & W_b^{\prime ij}=2r\left[(g_0^{\prime j}-g_0^{\prime i})l^{\prime ij}-(\sum _m{t}^{\prime m[j}{\theta }^{[jm}+t^{\prime [jm+}{\theta }^{m[j})l^{\prime i]m}\right],\hfill \end{array}$$

(38)

$$\begin{array}{ccc}& & {\mathcal{K}}_1^{\prime j}=\left(4\sum _i{l}^{\prime ji+}{l}^{\prime i}+l^{\prime j+}(2g_0^{\prime j}-1)-2(\sum _i{l}^{\prime i+}{t}^{\prime +ij}{\theta }^{ji}+l^{\prime i+}{t}^{\prime ji}{\theta }^{ij})\right),\hfill \end{array}$$

(39)

$$\begin{array}{ccc}\hfill & & X_b^{\prime ij}=\left\{l_0^{\prime }-r\left(K_0^{\prime 0i}+\sum _{l=i+1}^k{\mathcal{K}}_0^{\prime il}+\sum _{l=1}^{i-1}{\mathcal{K}}_0^{\prime li}\right)\right\}{\delta }^{ij}\hfill \\ & & +r\left\{4{\sum }_l{l}^{\prime jl+}{l}^{\prime li}-{\displaystyle \sum _{l=1}^{j-1}}{t}^{\prime lj+}{t}^{\prime li}-{\displaystyle \sum _{l=i+1}^k}{t}^{\prime il+}{t}^{\prime jl}-{\displaystyle \sum _{l=j+1}^{i-1}}{t}^{\prime li}{t}^{\prime jl}+(g_0^{\prime j}+g_0^{\prime i}-j-1)t_{ji}^{\prime }\right\}{\theta }^{ij}\hfill \\ & & +r\left\{4{\sum }_l{l}^{\prime jl+}{l}^{\prime li}-\sum _{l=1}^{i-1}{t}^{\prime lj+}{t}^{\prime li}-\sum _{l=j+1}^k{t}^{\prime il+}{t}^{\prime jl}-\sum _{l=i+1}^{j-1}{t}^{\prime il+}{t}^{\prime lj+}+t_{ij}^{\prime +}(g_0^{\prime j}+g_0^{\prime i}-i-1)\right\}{\theta }^{ji}.\hfill \end{array}$$

(40)

Now, we can outline the points leading to an oscillator representation for the elements of an auxiliary HS symmetry algebra.

3.1. On Representations of ${\mathcal{A}}^{\prime }(Y\left(k\right),Ad{S}_d)$

Here, we assume that a generalization of the Poincaré–Birkhoff–Witt theorem for the second-order algebra ${\mathcal{A}}^{\prime }(Y\left(k\right),Ad{S}_d)$ does indeed hold true (for a generalization of the PBW theorem involving a quadratic algebra, see [19]), and so we begin to construct a Verma module based on a Cartan-like decomposition8 extended from that of $sp\left(2k\right)$ ($i\le j$, $l<m$),

$${\mathcal{A}}^{\prime }(Y\left(k\right),Ad{S}_d)=\{l_{ij}^{\prime +},t_{lm}^{\prime +},l_i^{\prime +}\}\oplus \{g_0^{\prime i},l_0^{\prime }\}\oplus \{l_{ij}^{\prime },t_{lm}^{\prime },l_i^{\prime }\}\equiv {\mathcal{E}}_k^{-}\oplus H_k\oplus {\mathcal{E}}_k^{+}.$$

(41)

Distinct from the case of a Lie algebra, the element $l_0^{\prime }$ does not diagonalize the elements of the upper ${\mathcal{E}}_k^{-}$ (lower, ${\mathcal{E}}_k^{+}$), triangular subalgebra, due to the quadratic relations (39), as in the case of totally symmetric HS fields in AdS spaces [20,21]. Additionally, the negative-root vectors $l_i^{\prime +},l_j^{\prime +}$ do not commute.

Since the Verma module over a semi-simple finite-dimensional Lie algebra g (an induced module $\mathcal{U}\left(g\right){⨂}_{\mathcal{U}\left(b\right)}{|0\rangle }_V$ with a vacuum vector9 ${|0\rangle }_V$) is isomorphic, due to the PBW theorem, as a vector space to a polynomial algebra $\mathcal{U}\left(g^{-}\right){⨂}_C{|0\rangle }_V$, it is clear that g can be realized by first-order inhomogeneous differential operators acting on these polynomials.

We examine a generalization of a Verma module for quadratic algebras: $g\left(r\right)$ presents a $g\left(r\right)$-deformation of a Lie algebra g in such a way that $g(r=0)=g$. Thus, we consider a Verma module for such non-linear algebras, assuming that the PBW theorem is valid for $g\left(r\right)$. The latter will be proved later on by an explicit construction of the Verma module.

Let us consider a quadratic algebra ${\mathcal{A}}^{\prime }(Y\left(k\right),Ad{S}_d)$ as an r-deformation of the Lie algebra ${\mathcal{A}}^{\prime }(Y\left(k\right),$ $R^{1,d-1})$, with (27) taken into account (see [59] for details);

$${\mathcal{A}}^{\prime }(Y\left(k\right),Ad{S}_d)=\left(T^{\prime k}\oplus T^{\prime k*}\oplus l_0^{\prime }\right)\left(r\right)+\supset sp\left(2k\right),T^{\prime k}=\left\{l_i^{\prime }\right\},T^{\prime k*}=\left\{l_i^{\prime +}\right\}.$$

(42)

Therefore, according to (41), the basis vector of the Verma module $|\overrightarrow{N}{\rangle }_V$ has the form $|\overrightarrow{N}{\rangle }_V=\left|{\overrightarrow{n}}_{ij},{(\overrightarrow{n},{\overrightarrow{p}}_m)}_l{\rangle }_V\right.$=$\left|{\overrightarrow{n}}_{ij},n_1,p_{12},\dots ,p_{1k},n_2,p_{23},\dots ,p_{2k},\dots ,p_{k-1k},n_k{\rangle }_V\right.$,

$$|\overrightarrow{N}{\rangle }_V\equiv \prod _{i\le j}^k\left(l_{ij}^{\prime +}\right){}^{n_{ij}}\prod _l^k\left[\prod _{m,m>l}\left(\frac{l_l^{\prime +}}{m_l}\right){}^{n_l}\left(t_{lm}^{\prime +}\right){}^{p_{lm}}\right]{|0\rangle }_V,{\mathcal{E}}_k^{+}{|0\rangle }_V=0,$$

(43)

with the (vacuum) highest-weight vector ${|0\rangle }_V$, non-negative integers $n_{ij},n_l,p_{lm}$, and arbitrary constants $m_l$ with the dimensionality of mass.

Now, we are in a position to construct a Verma module for the algebra ${\mathcal{A}}^{\prime }(Y\left(2\right),Ad{S}_d)$, thus leaving the general case of an arbitrary Young tableaux with $k>2$ rows out of the scope of the paper. The solution of a similar problem for the auxiliary polynomial algebra ${\mathcal{A}}^{\prime }$ is given by Appendix B.

3.2. Verma Module for Quadratic Algebra ${\mathcal{A}}^{\prime }(Y\left(2\right),Ad{S}_d)$ with $k=2$ Rows

Let us first specify a commutator multiplication law for the algebra ${\mathcal{A}}^{\prime }(Y\left(2\right),Ad{S}_d)$, belonging to ${\mathcal{A}}^{\prime }(Y\left(k\right),Ad{S}_d)$. To do so, we can choose ${\theta }^{ij}={\delta }^{i2}{\delta }^{j1}$, and therefore the only surviving operators are $t^{ij}=t^{12}$, $t_{ij}^{+}=t_{12}^{\prime +}$.

As a result, the Lie part of Table 1 and Table 2 for $k=2$ is the same as in the case of a bosonic Lie subalgebra in [58], once the following expressions for the non-vanishing operators $B^{\prime 12}{}_{12}$, $A^{\prime 1212}$, $F^{\prime 12,j}$, $F^{\prime 12,j+}$, $L^{\prime i_1{j}_1,i_2{j}_2}$,

$$\begin{array}{ccc}& & B^{\prime 12}{}_{12}=(g_0^1-g_0^2),A^{\prime 12,12}=0,(F^{\prime 12,j},\equiv F^{\prime j}=t^{\prime 12}({\delta }^{j2}-{\delta }^{j1}),\hfill \end{array}$$

(44)

$$\begin{array}{ccc}& & L^{\prime i_1{j}_1,i_2{j}_2}=\frac{1}{4}\{{\delta }^{i_2{i}_1}{\delta }^{j_2{j}_1}\left[2g_0^{i_2}{\delta }^{i_2{j}_2}+g_0^{i_2}+g_0^{j_2}\right]-{\delta }^{j_2\{i_1}\left[t^{12}{\delta }^{j_1\}1}{\delta }^{i_{2}2}+t^{12+}{\delta }^{j_1\}2}{\delta }^{i_{2}1}\right]\hfill \\ & & -{\delta }^{i_2\{i_1}\left[t^{12}{\delta }^{j_{2}2}{\delta }^{j_1\}1}+t^{12+}{\delta }^{j_1\}2}{\delta }^{j_{2}1}\right]\},\hfill \end{array}$$

(45)

are identical to the same operators $o_I$ from the initial algebra $\mathcal{A}(Y\left(2\right),Ad{S}_d)$. Since the non-linear part of ${\mathcal{A}}^{\prime }(Y\left(2\right),Ad{S}_d)$ is determined by the same functions $W_b^{\prime ji},{\mathcal{K}}_1^{\prime i},W_b^{\prime ji+},{\mathcal{K}}_1^{\prime i+}{X}_b^{\prime ij}$ as those for arbitrary k, its manifest expression is implied directly by (38)–(40) as follows:10

$$\begin{array}{ccc}\hfill W_b^{\prime ij}& =& 2r{ϵ}^{ij}\left[(g_0^{\prime 2}-g_0^{\prime 1})l^{\prime 12}-t^{\prime 12}{l}^{\prime 11}+t^{\prime 12+}{l}^{\prime 22}\right],\hfill \end{array}$$

(46)

$$\begin{array}{ccc}\hfill {\mathcal{K}}_1^{\prime j}& =& \left(4\sum _i{l}^{\prime ji+}{l}^{\prime i}+l^{\prime j+}(2g_0^{\prime j}-1)-2l^{\prime 2+}{t}^{\prime +12}{\delta }^{j1}-2l^{\prime 1+}{t}^{\prime 12}{\delta }^{j2}\right),\hfill \end{array}$$

(47)

$$\begin{array}{ccc}\hfill X_b^{\prime ij}& =& \left\{l_0^{\prime }-r\left(K_0^{\prime 0i}+{\mathcal{K}}_0^{\prime 12}\right)\right\}{\delta }^{ij}+r\{\left[4{\sum }_l{l}^{\prime 1l+}{l}^{\prime l2}+(g_0^{\prime 2}+g_0^{\prime 1}-2)t_{12}^{\prime }\right]{\delta }^{i2}{\delta }^{j1}\hfill \\ & & +r\{\left[4{\sum }_l{l}^{\prime l2+}{l}^{\prime 1l}+t_{12}^{\prime +}(g_0^{\prime 2}+g_0^{\prime 1}-2)\right]{\delta }^{i1}{\delta }^{j2}.\hfill \end{array}$$

(48)

with the totally antisymmetric $sp\left(2\right)$-invariant tensor ${ϵ}^{ij}$, ${ϵ}^{12}=1$, and the operator ${\mathcal{K}}_0^{12}=\left(t_{12}^{+}{t}^{12}-4l_{12}^{+}{l}^{12}-g_0^2\right)$ deduced from a Casimir operator for the $sp\left(4\right)$ algebra in (26) for $k=2$.

All the Lie part quantities of Table 1, except for $o_I^{\prime }$, and those of Table 2 are specified by (44)–(48) used in the HS symmetry algebra ${\mathcal{A}}^{\prime }(Y\left(2\right),Ad{S}_d)$ of the additional part $o_I^{\prime }$.

The Cartan-like decomposition (41) and the Verma module basis vector (43) of the algebra ${\mathcal{A}}^{\prime }(Y\left(k\right),Ad{S}_d)$ are reduced to those for ${\mathcal{A}}^{\prime }(Y\left(2\right),Ad{S}_d)$ in the form

$$\begin{array}{ccc}\hfill & & {\mathcal{A}}^{\prime }(Y\left(2\right),Ad{S}_d)=\{l_{ij}^{\prime +},t_{12}^{\prime +},l_i^{\prime +}\}\oplus \{g_0^{\prime i},l_0^{\prime }\}\oplus \{l_{ij}^{\prime },t_{12}^{\prime },l_i^{\prime }\}\equiv {\mathcal{E}}_2^{-}\oplus H_2\oplus {\mathcal{E}}_2^{+},\hfill \end{array}$$

(49)

$$\begin{array}{ccc}\hfill & & |\overrightarrow{N}{\left(2\right)\rangle }_V=\left|{\overrightarrow{n}}_{ij},n_1,p_{12},n_2{\rangle }_V\right.\equiv {\displaystyle \prod _{i\le j}^2}\left(l_{ij}^{\prime +}\right){}^{n_{ij}}\left(\frac{l_1^{\prime +}}{m_1}\right){}^{n_1}\left(t_{12}^{\prime +}\right){}^{p_{12}}\left(\frac{l_2^{\prime +}}{m_2}\right){}^{n_2}{|0\rangle }_V,{\mathcal{E}}_2^{+}{|0\rangle }_V=0.\hfill \end{array}$$

(50)

Here, in contrast to the case of a Lie algebra with totally symmetric HS fields in AdS spaces [20,22,58], the negative-root vectors $l_1^{\prime +},t^{\prime +},l_2^{\prime +}$ do not commute, which means that $|\overrightarrow{N}{\left(2\right)\rangle }_V$ is not an eigenvector for the operators $t_{12}^{\prime +},l_2^{\prime +}$, thereby featuring substantial peculiarities in constructing a Verma module for ${\mathcal{A}}^{\prime }(Y\left(2\right),Ad{S}_d)$.

By definition, the highest-weight vector ${|0\rangle }_V$ is an eigenvector of the Cartan-like subalgebra $H_2$,

$$(g_0^{\prime i},l_0^{\prime }){|0\rangle }_V=(h^i,m_0^2){|0\rangle }_V,$$

(51)

with certain real numbers $h^1,h^2,m_0$, whose values are to be determined later on, at the end of LF construction, in order for the Lagrangian equations of motion to reproduce the initial AdS group irrep conditions (2)–(5).

Then, we determine the action of negative-root vectors from the subspace ${\mathcal{E}}_2^{-}$ on the basis vector $|\overrightarrow{N}{\left(2\right)\rangle }_V$, which does not present the action of the raising operators in a manifest form and reads as

$$\begin{array}{ccc}\hfill l_{lm}^{\prime +}\left|\overrightarrow{N}{\left(2\right)\rangle }_V\right.& =& \left|{\overrightarrow{n}}_{ij}+{\delta }_{ij,lm},{\overrightarrow{n}}_s{\rangle }_V\right.,\hfill \end{array}$$

(52)

$$\begin{array}{ccc}\hfill l_l^{\prime +}\left|\overrightarrow{N}{\left(2\right)\rangle }_V\right.& =& {\delta }^{l2}{\displaystyle \prod _{i\le j}^2}\left(l_{ij}^{\prime +}\right){}^{n_{ij}}{l}_2^{\prime +}\left|{\overrightarrow{0}}_{ij},{\overrightarrow{n}}_s{\rangle }_V\right.+{\delta }^{l1}{m}_1\left|{\overrightarrow{n}}_{ij},n_1+1,p_{12},n_2{\rangle }_V\right.,l=1,2,\hfill \end{array}$$

(53)

$$\begin{array}{ccc}\hfill t_{12}^{\prime +}\left|\overrightarrow{N}{\left(2\right)\rangle }_V\right.& =& {\displaystyle \prod _{i\le j}^2}\left(l_{ij}^{\prime +}\right){}^{n_{ij}}{t}_{12}^{\prime +}{|{\overrightarrow{0}}_{ij},{\overrightarrow{n}}_s\rangle }_V-2n_{11}\left|n_{11}-1,n_{12}+1,n_{22},{\overrightarrow{n}}_s{\rangle }_V\right.\hfill \\ & & -n_{12}\left|n_{11},n_{12}-1,n_{22}+1,{\overrightarrow{n}}_s{\rangle }_V\right..\hfill \end{array}$$

(54)

Here, we used the following notation: first, ${\overrightarrow{n}}_s\equiv (n_1,p_{12},n_2)$ and ${\overrightarrow{0}}_{ij}\equiv (0,0,0)$ in accordance with the definition of $|\overrightarrow{N}{\left(2\right)\rangle }_V$; second, ${\delta }_{ij,lm}={\delta }_{il}{\delta }_{jm}$ for $i\le j,l\le m$ so that the vector $\left|\overrightarrow{N}+{\delta }_{ij,lm}{\rangle }_V\right.$ in (52) is subject to the definition (50), increasing the coordinate $n_{ij}$ in the vector $|\overrightarrow{N}{\rangle }_V$, for $i=l,j=m$ by a unit while leaving intact the values of the remaining ones.

In turn, the action of Cartan-like generators on the vector $|\overrightarrow{N}{\left(2\right)\rangle }_V$ is given by the relations

$$\begin{array}{ccc}\hfill g_0^{\prime l}\left|\overrightarrow{N}{\left(2\right)\rangle }_V\right.& =& \left(2n_{ll}+n_{12}+n_l+{(-1)}^l{p}_{12}+h^l\right)\left|\overrightarrow{N}{\left(2\right)\rangle }_V\right.,l=1,2,\hfill \end{array}$$

(55)

$$\begin{array}{ccc}\hfill l_0^{\prime }\left|\overrightarrow{N}{\left(2\right)\rangle }_V\right.& =& {\prod }_{i\le j}^2\left(l_{ij}^{\prime +}\right){}^{n_{ij}}{l}_0^{\prime }\left|{\overrightarrow{0}}_{ij},{\overrightarrow{n}}_s{\rangle }_V\right..\hfill \end{array}$$

(56)

To obtain the representations (52)–(56), we used a formula for the product of operators A, B,

$$\begin{array}{ccc}\hfill & & A{B}^n=\sum _{k=0}^n\frac{n!}{k!(n-k)!}{B}^{n-k}{\mathrm{ad}}_B^{k}A,{\mathrm{ad}}_B^{k}A=\left[\right[\dots [A,\stackrel{k\mathrm{times}}{\overbrace{B\}\dots \},B}\}},\mathtt{f}\mathtt{o}\mathtt{r}n\ge 0.\hfill \end{array}$$

(57)

Finally, the action of positive-root vectors from the subspace ${\mathcal{E}}_2^{+}$ on the vector $|\overrightarrow{N}{\left(2\right)\rangle }_V$, according to the rule (57), reads as follows:

$$\begin{array}{ccc}\hfill t_{12}^{\prime }\left|\overrightarrow{N}{\left(2\right)\rangle }_V\right.& =& {\displaystyle \prod _{i\le j}^2}\left(l_{ij}^{\prime +}\right){}^{n_{ij}}\left(\frac{l_1^{\prime +}}{m_1}\right){}^{n_1}\left(t_{12}^{\prime +}\right){}^{p_{12}}{t}_{12}^{\prime }\left|{\overrightarrow{0}}_{ij},0,0,n_2{\rangle }_V\right.-{\displaystyle \sum _l}l{n}_{l2}\left|{\overrightarrow{n}}_{ij}-{\delta }_{ij,l2}+{\delta }_{ij,1l},{\overrightarrow{n}}_s{\rangle }_V\right.\hfill \\ & & +p_{12}(h^1-h^2-n_2-p_{12}+1)\left|{\overrightarrow{n}}_{ij},n_1,p_{12}-1,n_2{\rangle }_V\right.,\hfill \end{array}$$

(58)

$$\begin{array}{ccc}\hfill l^{\prime 1}\left|\overrightarrow{N}{\left(2\right)\rangle }_V\right.& =& -m_1{n}_{11}|{\overrightarrow{n}}_{ij}-{\delta }_{ij,11},n_1+1,p_{12},n_2{\rangle }_V\hfill \\ \hfill & & +\left\{\prod _{i\le j}^2\left(l_{ij}^{\prime +}\right){}^{n_{ij}}{l}^{\prime 1}-\frac{n_{12}}{2}\prod _{i\le j}^2\left(l_{ij}^{\prime +}\right){}^{n_{ij}-{\delta }_{ij,12}}{l}_2^{\prime +}\right\}\left|{\overrightarrow{0}}_{ij},{\overrightarrow{n}}_s{\rangle }_V\right.,\hfill \end{array}$$

(59)

$$\begin{array}{ccc}\hfill l^{\prime 2}\left|\overrightarrow{N}{\left(2\right)\rangle }_V\right.& =& -m_1\frac{n_{12}}{2}|{\overrightarrow{n}}_{ij}-{\delta }_{ij,12},n_1+1,p_{12},n_2{\rangle }_V\hfill \\ \hfill & & +\left\{\prod _{i\le j}^2\left(l_{ij}^{\prime +}\right){}^{n_{ij}}{l}^{\prime 2}-n_{22}\prod _{i\le j}^2\left(l_{ij}^{\prime +}\right){}^{n_{ij}-{\delta }_{ij,22}}{l}_2^{\prime +}\right\}\left|{\overrightarrow{0}}_{ij},{\overrightarrow{n}}_s{\rangle }_V\right.,\hfill \end{array}$$

(60)

$$\begin{array}{ccc}\hfill l^{\prime 11}\left|\overrightarrow{N}{\left(2\right)\rangle }_V\right.& =& n_{11}(n_{11}+n_{12}+n_1-p_{12}-1+h^1)\left|{\overrightarrow{n}}_{ij}-{\delta }_{ij,11},{\overrightarrow{n}}_s{\rangle }_V\right.\hfill \\ & & +\frac{n_{12}(n_{12}-1)}{4}\left|{\overrightarrow{n}}_{ij}-2{\delta }_{ij,12}+{\delta }_{ij,22},{\overrightarrow{n}}_s{\rangle }_V\right.\hfill \\ & & +\left\{\prod _{i\le j}^2\left(l_{ij}^{\prime +}\right){}^{n_{ij}}{l}^{\prime 11}-\frac{n_{12}}{2}\prod _{i\le j}^2\left(l_{ij}^{\prime +}\right){}^{n_{ij}-{\delta }_{ij,12}}{t}_{12}^{\prime +}\right\}\left|{\overrightarrow{0}}_{ij},{\overrightarrow{n}}_s{\rangle }_V\right.,\hfill \end{array}$$

(61)

$$\begin{array}{ccc}\hfill l^{\prime 12}\left|\overrightarrow{N}{\left(2\right)\rangle }_V\right.& =& {\displaystyle \frac{n_{12}}{4}\left(n_{12}+\sum _l(2n_{ll}+n_l+h^l)-1\right)\left|{\overrightarrow{n}}_{ij}-{\delta }_{ij,12},{\overrightarrow{n}}_s{\rangle }_V\right.}\hfill \\ & & +\frac{1}{2}{p}_{12}{n}_{11}(h^2-h^1+n_2+p_{12}-1)\left|{\overrightarrow{n}}_{ij}-{\delta }_{ij,11},n_1,p_{12}-1,n_2{\rangle }_V\right.\hfill \\ & & +n_{11}{n}_{22}\left|{\overrightarrow{n}}_{ij}-{\delta }_{ij,11}-{\delta }_{ij,22}+{\delta }_{ij,12},{\overrightarrow{n}}_s{\rangle }_V\right.\hfill \\ & & {\displaystyle +\left\{{\displaystyle \prod _{i\le j}^2}\left(l_{ij}^{\prime +}\right){}^{n_{ij}}{l}^{\prime 12}-\frac{n_{22}}{2}\prod _{i\le j}^2\left(l_{ij}^{\prime +}\right){}^{n_{ij}-{\delta }_{ij,22}}{t}_{12}^{\prime +}\right\}\left|{\overrightarrow{0}}_{ij},{\overrightarrow{n}}_s{\rangle }_V\right.}\hfill \\ & & -\frac{n_{11}}{2}{\displaystyle \prod _{i\le j}^2}\left(l_{ij}^{\prime +}\right){}^{n_{ij}-{\delta }_{ij,11}}\left(\frac{l_1^{\prime +}}{m_1}\right){}^{n_1}\left(t_{12}^{\prime +}\right){}^{p_{12}}{t}_{12}^{\prime }\left|{\overrightarrow{0}}_{ij},0,0,n_2{\rangle }_V\right.,\hfill \end{array}$$

(62)

$$\begin{array}{ccc}\hfill l^{\prime 22}\left|\overrightarrow{N}{\left(2\right)\rangle }_V\right.& =& n_{22}(n_{12}+n_{22}+p_{12}+n_2-1+h^2)\left|{\overrightarrow{n}}_{ij}-{\delta }_{ij,22},{\overrightarrow{n}}_s{\rangle }_V\right.\hfill \\ & & +\frac{n_{12}{p}_{12}}{2}(p_{12}-1+h^2-h^1+n_2)\left|{\overrightarrow{n}}_{ij}-{\delta }_{ij,12},n_1,p_{12}-1,n_2{\rangle }_V\right.\hfill \\ & & +\frac{n_{12}(n_{12}-1)}{4}\left|{\overrightarrow{n}}_{ij}+{\delta }_{ij,11}-2{\delta }_{ij,12},{\overrightarrow{n}}_s{\rangle }_V\right.\hfill \\ & & +{\displaystyle \prod _{i\le j}^2}\left(l_{ij}^{\prime +}\right){}^{n_{ij}}\left(\frac{l_1^{\prime +}}{m_1}\right){}^{n_1}{l}^{\prime 22}\left|{\overrightarrow{0}}_{ij},0,p_{12},n_2{\rangle }_V\right.\hfill \\ & & -\frac{n_{12}}{2}{\displaystyle \prod _{i\le j}^2}\left(l_{ij}^{\prime +}\right){}^{n_{ij}-{\delta }_{ij,12}}\left(\frac{l_1^{\prime +}}{m_1}\right){}^{n_1}\left(t_{12}^{\prime +}\right){}^{p_{12}}{t}_{12}^{\prime }\left|{\overrightarrow{0}}_{ij},0,0,n_2{\rangle }_V\right..\hfill \end{array}$$

(63)

It is easy to see that, in order to complete the calculation in (53), (54), (59)–(63), we need to find the action of the positive-root vectors $l^{\prime m},l^{\prime 1m}$, $m=1,2$, the Cartan-like vector $l_0^{\prime }$, and the negative-root vectors $l_2^{\prime +},t_{12}^{\prime +}$ on $|{\overrightarrow{0}}_{ij},{\overrightarrow{n}}_s{\rangle }_V$, and also the action of the remaining operators $t_{12}^{\prime },l^{\prime 22}$ on an arbitrary $|{\overrightarrow{0}}_{lm},0,0,n_2{\rangle }_V$ in terms of linear combinations of certain vectors. We solve this rather non-trivial technical problem explicitly in Appendix C, with an introduction of some auxiliary quantities, which we call a primary block operator, ${\widehat{t}}_{12}^{\prime }$, (A37), (A38) and derived block operators, ${\widehat{t}}_{12}^{\prime +},{\widehat{l}}_2^{\prime +},{\widehat{l}}_0^{\prime },{\widehat{l}}_m^{\prime },{\widehat{l}}_{m2}^{\prime },m=1,2$, (A42), (A43), (A47)–(A51), (A56), presenting the result in the following form.

Theorem 1.

A Verma module for the non-linear second-order algebra ${\mathcal{A}}^{\prime }(Y\left(2\right),Ad{S}_d)$ does exist, is determined by the relations (52), (55), (64)–(72), and is expressed using the primary ${\widehat{t}}_{12}^{\prime }$ and derived block operators ${\widehat{t}}_{12}^{\prime +},{\widehat{l}}_2^{\prime +},{\widehat{l}}_0^{\prime },{\widehat{l}}_m^{\prime },{\widehat{l}}_{m2}^{\prime },m=1,2$, (A42), (A43), (A47)–(A51), (A56) in the final form

$$\begin{array}{ccc}\hfill t_{12}^{\prime }\left|\overrightarrow{N}{\left(2\right)\rangle }_V\right.& =& p_{12}(h^1-h^2-n_2-p_{12}+1)\left|{\overrightarrow{n}}_{ij},n_1,p_{12}-1,n_2{\rangle }_V\right.\hfill \\ & & -\sum _{l}l{n}_{l2}\left|{\overrightarrow{n}}_{ij}-{\delta }_{ij,l2}+{\delta }_{ij,1l},{\overrightarrow{n}}_s{\rangle }_V\right.+{\widehat{t}}_{12}^{\prime }\left|\overrightarrow{N}{\left(2\right)\rangle }_V\right.,\hfill \end{array}$$

(64)

$$\begin{array}{ccc}\hfill t_{12}^{\prime +}\left|\overrightarrow{N}{\left(2\right)\rangle }_V\right.& =& -\sum _l(3-l)n_{1l}\left|{\overrightarrow{n}}_{ij}-{\delta }_{ij,1l}+{\delta }_{ij,l2},{\overrightarrow{n}}_s{\rangle }_V\right.+{\widehat{t}}_{12}^{\prime +}\left|\overrightarrow{N}{\left(2\right)\rangle }_V\right.,\hfill \end{array}$$

(65)

$$\begin{array}{ccc}\hfill l_l^{\prime +}\left|\overrightarrow{N}{\left(2\right)\rangle }_V\right.& =& {\delta }^{l1}{m}_1\left|\overrightarrow{N}\left(2\right)+{\delta }_{s,1}{\rangle }_V\right.+{\delta }^{l2}{\widehat{l}}_2^{\prime +}\left|\overrightarrow{N}{\left(2\right)\rangle }_V\right.,\hfill \end{array}$$

(66)

$$\begin{array}{ccc}\hfill l_0^{\prime }\left|\overrightarrow{N}{\left(2\right)\rangle }_V\right.& =& l_0^{\prime }\left|{\overrightarrow{0}}_{ij},{\overrightarrow{n}}_s{\rangle }_V\right.{|}_{[{\overrightarrow{0}}_{ij}\to {\overrightarrow{n}}_{ij}]},\hfill \end{array}$$

(67)

$$\begin{array}{ccc}\hfill l_1^{\prime }\left|\overrightarrow{N}{\left(2\right)\rangle }_V\right.& =& l_1^{\prime }\left|{\overrightarrow{0}}_{ij},{\overrightarrow{n}}_s{\rangle }_V\right.{|}_{[{\overrightarrow{0}}_{ij}\to {\overrightarrow{n}}_{ij}]}-m_1{n}_{11}\left|{\overrightarrow{n}}_{ij}-{\delta }_{ij,11},{\overrightarrow{n}}_s+{\delta }_{s,1}{\rangle }_V\right.\hfill \\ \hfill & & -\frac{n_{12}}{2}{\widehat{l}}_2^{\prime +}\left|{\overrightarrow{n}}_{ij}-{\delta }_{ij,12},{\overrightarrow{n}}_s{\rangle }_V\right.,\hfill \end{array}$$

(68)

$$\begin{array}{ccc}\hfill l_2^{\prime }\left|\overrightarrow{N}{\left(2\right)\rangle }_V\right.& =& l_2^{\prime }\left|{\overrightarrow{0}}_{ij},{\overrightarrow{n}}_s{\rangle }_V\right.{|}_{[{\overrightarrow{0}}_{ij}\to {\overrightarrow{n}}_{ij}]}-m_1\frac{n_{12}}{2}\left|{\overrightarrow{n}}_{ij}-{\delta }_{ij,12},{\overrightarrow{n}}_s+{\delta }_{s,1}{\rangle }_V\right.\hfill \\ \hfill & & -n_{22}{\widehat{l}}_2^{\prime +}\left|{\overrightarrow{n}}_{ij}-{\delta }_{ij,22},{\overrightarrow{n}}_s{\rangle }_V\right.,\hfill \end{array}$$

(69)

$$\begin{array}{ccc}\hfill l_{11}^{\prime }\left|\overrightarrow{N}{\left(2\right)\rangle }_V\right.& =& n_{11}(n_{11}+n_{12}+n_1-p_{12}-1+h^1)\left|{\overrightarrow{n}}_{ij}-{\delta }_{ij,11},{\overrightarrow{n}}_s{\rangle }_V\right.\hfill \\ & & +\frac{n_{12}(n_{12}-1)}{4}\left|{\overrightarrow{n}}_{ij}-2{\delta }_{ij,12}+{\delta }_{ij,22},{\overrightarrow{n}}_s{\rangle }_V\right.+l_{11}^{\prime }\left|{\overrightarrow{0}}_{ij},{\overrightarrow{n}}_s{\rangle }_V\right.{|}_{[{\overrightarrow{0}}_{ij}\to {\overrightarrow{n}}_{ij}]}\hfill \\ & & -\frac{n_{12}}{2}{\widehat{t}}_{12}^{\prime +}\left|{\overrightarrow{n}}_{ij}-{\delta }_{ij,12},{\overrightarrow{n}}_s{\rangle }_V\right.,\hfill \end{array}$$

(70)

$$\begin{array}{ccc}\hfill l_{12}^{\prime }\left|\overrightarrow{N}{\left(2\right)\rangle }_V\right.& =& {\displaystyle \frac{n_{12}}{4}\left(n_{12}+\sum _l(2n_{ll}+n_l+h^l)-1\right)\left|{\overrightarrow{n}}_{ij}-{\delta }_{ij,12},{\overrightarrow{n}}_s{\rangle }_V\right.}\hfill \\ & & +\frac{1}{2}{p}_{12}{n}_{11}(h^2-h^1+n_2+p_{12}-1)\left|{\overrightarrow{n}}_{ij}-{\delta }_{ij,11},n_1,p_{12}-1,n_2{\rangle }_V\right.\hfill \\ & & +n_{11}{n}_{22}\left|{\overrightarrow{n}}_{ij}-{\delta }_{ij,11}-{\delta }_{ij,22}+{\delta }_{ij,12},{\overrightarrow{n}}_s{\rangle }_V\right.+l_{12}^{\prime }\left|{\overrightarrow{0}}_{ij},{\overrightarrow{n}}_s{\rangle }_V\right.{|}_{[{\overrightarrow{0}}_{ij}\to {\overrightarrow{n}}_{ij}]}\hfill \\ & & -\frac{n_{22}}{2}{\widehat{t}}_{12}^{\prime +}\left|{\overrightarrow{n}}_{ij}-{\delta }_{ij,22},{\overrightarrow{n}}_s{\rangle }_V\right.-\frac{n_{11}}{2}{\widehat{t}}_{12}^{\prime }\left|{\overrightarrow{n}}_{ij}-{\delta }_{ij,11},{\overrightarrow{n}}_s{\rangle }_V\right.,\hfill \end{array}$$

(71)

$$\begin{array}{ccc}\hfill l_{22}^{\prime }\left|\overrightarrow{N}{\left(2\right)\rangle }_V\right.& =& n_{22}(n_{12}+p_{12}+n_2+n_{22}-1+h^2)\left|{\overrightarrow{n}}_{ij}-{\delta }_{ij,22},{\overrightarrow{n}}_s{\rangle }_V\right.\hfill \\ & & +\frac{n_{12}{p}_{12}}{2}(p_{12}-1+h^2-h^1+n_2)\left|{\overrightarrow{n}}_{ij}-{\delta }_{ij,12},{\overrightarrow{n}}_s-{\delta }_{s,12}{\rangle }_V\right.\hfill \\ & & +\frac{n_{12}(n_{12}-1)}{4}\left|{\overrightarrow{n}}_{ij}+{\delta }_{ij,11}-2{\delta }_{ij,12},{\overrightarrow{n}}_s{\rangle }_V\right.+{\widehat{l}}_{22}^{\prime }\left|\overrightarrow{N}{\left(2\right)\rangle }_V\right.\hfill \\ & & -\frac{n_{12}}{2}{\widehat{t}}_{12}^{\prime }\left|{\overrightarrow{n}}_{ij}-{\delta }_{ij,12},{\overrightarrow{n}}_s{\rangle }_V\right..\hfill \end{array}$$

(72)

To obtain these relations, we applied the obvious rule

$$\prod _{i\le j}^2\left(l_{ij}^{\prime +}\right){}^{n_{ij}}(l_0^{\prime },l_n^{\prime },l_{lm}^{\prime })\left|{\overrightarrow{0}}_{ij},{\overrightarrow{n}}_s{\rangle }_V\right.\equiv (l_0^{\prime },l_n^{\prime },l_{lm}^{\prime })\left|{\overrightarrow{0}}_{ij},{\overrightarrow{n}}_s{\rangle }_V\right.{|}_{[{\overrightarrow{0}}_{ij}\to {\overrightarrow{n}}_{ij}]},l,m,n=1,2,l\le m,$$

(73)

when the multipliers $\left(l_{ij}^{\prime +}\right){}^{n_{ij}}$ act as the only raising operators for the vectors $(l_0^{\prime },l_k^{\prime },l_{km}^{\prime })\left|{\overrightarrow{0}}_{ij},{\overrightarrow{n}}_s{\rangle }_V\right.$.

The above result has obvious corollaries: first of all, in the case of ${\mathcal{A}}^{\prime }(Y\left(2\right),Ad{S}_d)$ reduced to the quadratic algebra ${\mathcal{A}}^{\prime }(Y\left(1\right),Ad{S}_d)$, given in [20] for the vanishing components $n_{l2}=n_2=p_{12}=0$, $l=1,2$, of an arbitrary VM vector $\left|\overrightarrow{N}{\left(1\right)\rangle }_V\right.\equiv \left|n_{11},0,0,n_1{,0,0\rangle }_V\right.$, and, secondly, in the case of the AdS space reduced to the Minkowski space $R^{1,d-1}$, when the non-linear algebra ${\mathcal{A}}^{\prime }(Y\left(2\right),Ad{S}_d)$ at $r=0$ turns to the Lie algebra $\left(T^{\prime 2}\oplus T^{\prime 2*}\oplus l_0^{\prime }\right)\left(0\right)+\supset sp\left(4\right)$ for $k=2$ in (42). In the former case, we obtain the Verma module [20], familiar from the results of Theorem 1, whereas in the latter case, we obtain a new Verma module (see Appendix C), with the above Lie algebra being different from that for $sp\left(4\right)$ in [55], and also in [59] for $k=2$.

3.3. Fock Space Realization of ${\mathcal{A}}^{\prime }(Y\left(2\right),Ad{S}_d)$

In this section, we obtain, on a basis of the constructed Verma module, a realization of ${\mathcal{A}}^{\prime }(Y\left(2\right),Ad{S}_d)$ as a formal power series in the creation and annihilation operators $(B_a,B_a^{+})=((b_i,b_{ij},d_{12}),(b_i^{+},b_{ij}^{+},d_{12}^{+}))$ in ${\mathcal{H}}^{\prime }$, whose number coincides with that of second-class constraints among $o_I^{\prime }$, i.e., with $dim({\mathcal{E}}_2^{-}\oplus {\mathcal{E}}_2^{+})$. This task is solved by following the results of [97] and the algorithms suggested in [98,99], initially elaborated for a simple Lie algebra, and then enlarged to a non-linear quadratic algebra11 $\mathcal{A}(Y\left(1\right),Ad{S}_d)$ [20]. To this end, we make use of the following mapping for an arbitrary basis vector in the Verma module and the vector $\left|{\overrightarrow{n}}_{ij},{\overrightarrow{n}}_s\rangle \right.$ in a new Fock space ${\mathcal{H}}^{\prime }$:

$$\begin{array}{ccc}\hfill & \left|{\overrightarrow{n}}_{ij},{\overrightarrow{n}}_s{\rangle }_V\right.⟷\left|{\overrightarrow{n}}_{ij},{\overrightarrow{n}}_s\rangle \right.={\prod }_{i\le j}^2\left(b_{ij}^{+}\right){}^{n_{ij}}{\prod }_{l=1}^2\left(b_l^{+}\right){}^{n_l}\left(d_{12}^{+}\right){}^{p_{12}}|0\rangle ,& \\ & \mathrm{for}{b}_{ij}|0\rangle =b_l|0\rangle =d_{12}|0\rangle =0,|\overrightarrow{N}\left(2\right)\rangle {}_{[\overrightarrow{N}=\overrightarrow{0}]}\equiv |0\rangle .& \hfill \end{array}$$

(74)

Here, the vectors $\left|{\overrightarrow{n}}_{ij},{\overrightarrow{n}}_s\rangle \right.$ for non-negative integers $n_{ij},n_l,p_{12}$ are the basis vectors of the Fock space ${\mathcal{H}}^{\prime }$, generated by six pairs of bosonic operators $(B,B^{+})$, being the basis elements of the Heisenberg algebra $A_6$ with the standard (non-vanishing) commutation relations

$$[B_a,B_b^{+}]={\delta }_{ab}⟺[b_{ij},b_{lm}^{+}]={\delta }_{il,jm},[b_i,b_m^{+}]={\delta }_{im},[d_{12},d_{12}^{+}]=1.$$

(75)

To present the elements of the Verma module as a formal power series in the generators of the algebra $A_6$, we use an additive correspondence between the special Verma module vector $\left|A_{{\overrightarrow{0}}_{ij},0,0,n_2}{\rangle }_V\right.$ (A30) and the Fock space ${\mathcal{H}}^{\prime }$ vector $\left|{\overrightarrow{0}}_{ij},0,0,n_2\rangle \right.$; see (A87) in Appendix D. Summarizing, we present our basic result in the form of Theorem A1, given in the same Appendix D as polynomials for the trivial negative-root vectors $l_1^{\prime +}$ as well as for the particle number operators $g_0^{\prime i}$ (A90), the remaining root vectors, and the Cartan-like vector $l_0^{\prime }$ from ${\mathcal{A}}^{\prime }(Y\left(2\right),Ad{S}_d)$, as power series in oscillators $B_a,B_b^{+}$. The latter ones are given by (A91)–(A101). To obtain the above representations, we use the primary block operator ${\widehat{t}}_{12}^{\prime }$ (A92) and the derived block operators ${\widehat{t}}_{12}^{\prime +},{\widehat{l}}_0^{\prime },{\widehat{l}}_m^{\prime },{\widehat{l}}_{m2}^{\prime },m=1,2$.

The additional parts $o_I^{\prime }(B,B^{+})$, as formal power series in the oscillators $(B,B^{+})$, do not obey the usual properties,

$${\left(l_{lm}^{\prime }\right)}^{+}\ne l_{lm}^{\prime +},{\left(t_{12}^{\prime }\right)}^{+}\ne t_{12}^{\prime +},{\left(l_0^{\prime }\right)}^{+}\ne l_0^{\prime },{\left(l_m^{\prime }\right)}^{+}\ne l_m^{\prime +},l\le m,$$

(76)

if one uses the standard rules of Hermitian conjugation for the new creation and annihilation operators, ${\left(B_a\right)}^{+}=B_a$. The standard Hermitian conjugation properties for $o_I^{\prime }$ are restored by changing the inner product in ${\mathcal{H}}^{\prime }$ as follows:

$$\begin{array}{c}\hfill {\langle {\Phi }_1|{\Phi }_2\rangle }_{\mathrm{new}}=\langle {\Phi }_1|K^{\prime }|{\Phi }_2\rangle ,\end{array}$$

(77)

for any vectors $|{\Phi }_1\rangle ,|{\Phi }_2\rangle$ with a certain non-degenerate operator $K^{\prime }$. This operator is determined by the condition that all the operators of the algebra should have the standard Hermitian properties with respect to the new inner product:

$$\langle {\Phi }_1|K^{\prime }{E}^{-\prime \alpha }|{\Phi }_2\rangle =\langle {\Phi }_2|K^{\prime }{E}^{\prime \alpha }|{\Phi }_1{\rangle }^{*},\langle {\Phi }_1|K^{\prime }{G}^{\prime }|{\Phi }_2\rangle =\langle {\Phi }_2|K^{\prime }{G}^{\prime }{|{\Phi }_1\rangle }^{*},$$

(78)

for $(E^{\prime \alpha };E^{-\prime \alpha })=(l_{lm}^{\prime },t_{12}^{\prime },l_m^{\prime };l_{lm}^{\prime +},t_{12}^{\prime +},l_m^{\prime +})$, $G^{\prime }=(g_0^{\prime i},l_0^{\prime })$. The relations (78) determine an operator $K^{\prime }$ which is Hermitian with respect to the standard inner product $\langle |\rangle$,

$$\begin{array}{c}\hfill K^{\prime }=Z^{+}Z,Z=\sum _{({\overrightarrow{n}}_{lm},{\overrightarrow{n}}_s)=(\overrightarrow{0},\overrightarrow{0})}^{\infty }\left|\overrightarrow{N}{\left(2\right)\rangle }_V\right.\frac{1}{\left({\overrightarrow{n}}_{lm}\right)!\left({\overrightarrow{n}}_s\right)!}\langle 0|\prod _{r=1}^2{b}_r^{n_r}{d}_{12}^{p_{12}}\prod _{l,m\ge l}{b}_{lm}^{n_{lm}},\end{array}$$

(79)

where $\left({\overrightarrow{n}}_{lm}\right)!=n_{11}!n_{12}!n_{22}!$, $\left({\overrightarrow{n}}_s\right)!=n_1!n_2!p_{12}!$, and the normalization ${}_V{\langle 0|0\rangle }_V=1$ is assumed.

Theorem A1 has the same consequences as those of Theorem 1, which concerns, first of all, a flat-space limit of the algebra ${\mathcal{A}}^{\prime }(Y\left(2\right),Ad{S}_d)$, and therefore a new representation for the Lie algebra $\left(T^{\prime 2}\oplus T^{\prime 2*}\oplus l_0^{\prime }\right)\left(0\right)+\supset sp\left(4\right)$. Second, modulo the oscillator pairs $b_{m2},b_{m2}^{+},b_2,b_2^{+},d_{12},d_{12}^{+},m=1,2$, the deduced representation coincides with the one for a totally symmetric HS field quadratic algebra ${\mathcal{A}}^{\prime }(Y\left(1\right),Ad{S}_d)$ in [20,[21] for the Weyl ordering of quadratic combinations of $O_I$.

The set of equations that constitute the results of Theorems 1 and A1 presents a general solution of the second problem (mentioned in Introduction) of an LF construction for mixed-symmetry HS tensors in AdS spaces with a given mass and spin $\mathbf{s}=(s_1,s_2)$.

4. Construction of Lagrangian Actions

In order to construct a general Lagrangian formulation for an HS tensor field of fixed generalized spin $\mathbf{s}=(s_1,\dots ,s_k)$, we should explicitly determine a composition law for the deformed algebra ${\mathcal{A}}_c(Y\left(k\right),Ad{S}_d)$, then we have to specify this to the case of a YT with $k=2$ rows, find a BRST operator for the non-linear algebra ${\mathcal{A}}_c(Y\left(2\right),Ad{S}_d)$, and, finally, reproduce a correct gauge-invariant Lagrangian formulation for the basic bosonic field ${\Phi }_{{\left(\mu \right)}_{s_1},{\left(\nu \right)}_{s_2}}$.

4.1. Explicit Form of ${\mathcal{A}}_c(Y\left(k\right),Ad{S}_d)$

As in the case of the algebra ${\mathcal{A}}^{\prime }(Y\left(k\right),Ad{S}_d)$ of $o_I^{\prime }$, it is only the multiplication law for the quadratic part of the initial algebra $\mathcal{A}(Y\left(k\right),Ad{S}_d)$ that is modified, whereas the linear part is given by the same $sp\left(2k\right)$ algebra as for the maximal Lie subalgebra, $\mathcal{A}(Y\left(k\right),Ad{S}_d)$ and ${\mathcal{A}}^{\prime }(Y\left(k\right),Ad{S}_d)$, with the same form of commutators $[O_a,O_I]$, $O_a\in sp\left(2k\right)$. From (34), (35) and Table 1, the non-linear part of the algebra ${\mathcal{A}}_c(Y\left(k\right),Ad{S}_d)$ can be recovered using

Table 3. The non-linear part of the converted algebra ${\mathcal{A}}_c(Y\left(k\right),Ad{S}_d)$.

$[\downarrow ,\to ]$	${\mathit{L}}_0$	${\mathit{L}}^{\mathit{i}}$	${\mathit{L}}^{\mathit{i}+}$
$L_0$	0	$-r{\widehat{\mathcal{K}}}_1^{i+}$	$r{\widehat{\mathcal{K}}}_1^i$
$L^j$	$r{\widehat{\mathcal{K}}}_1^{j+}$	${\widehat{W}}_b^{ji}$	${\widehat{X}}_b^{ji}$
$L^{j+}$	$-r{\widehat{\mathcal{K}}}_1^j$	$-{\widehat{X}}_b^{ij}$	$-{\widehat{W}}_b^{ji+}$

.

Here, the functions ${\widehat{\mathcal{K}}}_1^i$, ${\widehat{W}}_b^{ji}$, ${\widehat{X}}_b^{ij}$ (hence, the Hermitian conjugate quantities ${\widehat{\mathcal{K}}}_1^{i+}$, ${\widehat{W}}_b^{ji+}$) are given by

$$\begin{array}{ccc}\hfill {\widehat{W}}_b^{ij}& =& 2r\left\{\left[G_0^j-G_0^i-(g_0^{\prime j}-g_0^{\prime i})\right]L^{ij}-l^{\prime ij}(G_0^j-G_0^i)-{\sum }_m[(T^{m[j}-t^{\prime m[j}){\theta }^{[jm}{L}^{i]m}\right.\hfill \\ \hfill & & \left.-l^{\prime [im}{T}^{mj]}{\theta }^{j]m}+(T^{[jm+}-t^{\prime [jm+}){\theta }^{m[j}{L}^{i]m}-l^{\prime [im}{T}^{j]m+}{\theta }^{mj]}]\right\},\hfill \end{array}$$

(80)

$$\begin{array}{ccc}\hfill {\widehat{\mathcal{K}}}_1^j& =& 4{\sum }_{i=1}^k\left\{\right(L^{ji+}-l^{\prime ji+}\left)L^i-l^{\prime i}{L}^{ji+}\right\}+2\left(L^{j+}-l^{\prime j+}\right)G_0^j-2\left(g_0^{\prime j}+\frac{1}{2}\right)L^{j+}\hfill \\ \hfill & & -2{\sum }_{i=1}^k\left\{\left[(L^{i+}-l^{\prime i+})T^{ij+}-t^{\prime ij+}{L}^{i+}\right]{\theta }^{ji}+\left[(L^{i+}-l^{\prime i+})T^{ji}-t^{\prime ji}{L}^{i+}\right]{\theta }^{ij}\right\},\hfill \end{array}$$

(81)

$$\begin{array}{ccc}\hfill {\widehat{X}}_b^{ij}& =& \{L_0+r[{\widehat{K}}_0^{0i}-2g_0^{\prime i}{G}_0^i+4l^{\prime ii+}{L}^{ii}+4l^{\prime ii}{L}^{ii+}+\sum _{l=i+1}^k({\widehat{\mathcal{K}}}_0^{il}-t^{\prime il+}{T}^{il}-t^{\prime il}{T}^{il+}\hfill \\ \hfill & & +4l^{\prime il+}{L}^{il}+4l^{\prime il}{L}^{il+})+\sum _{l=1}^{i-1}\left({\widehat{\mathcal{K}}}_0^{li}-t^{\prime li+}{T}^{li}-t^{\prime li}{T}^{li+}+4l^{\prime li+}{L}^{li}+4l^{\prime li}{L}^{li+}\right)]\}{\delta }^{ij}\hfill \\ & & -r\{4{\sum }_l\left[(L^{jl+}-l^{\prime jl+})L^{li}-l^{\prime li}{L}^{jl+}\right]-\sum _{l=1}^{j-1}\left[(T^{lj+}-t^{\prime lj+})T^{li}-t^{\prime li}{T}^{lj+}\right]\hfill \\ \hfill & & -\sum _{l=i+1}^k\left[(T^{il+}-t^{\prime il+})T^{jl}-t^{\prime jl}{T}^{il+}\right]-\sum _{l=j+1}^{i-1}\left[(T^{li}-t^{\prime li})T^{jl}-t^{\prime jl}{T}^{li}\right]\hfill \\ \hfill & & +\left(G_0^j+G_0^i-(g_0^{\prime j}+g_0^{\prime i})-j-1\right)T^{ji}-t^{\prime ji}\left(G_0^j+G_0^i\right)\}{\theta }^{ij}\hfill \\ & & -r\{4{\sum }_l\left[(L^{jl+}-l^{\prime jl+})L^{li}-l^{\prime li}{L}^{jl+}\right]-\sum _{l=1}^{i-1}\left[(T^{lj+}-t^{\prime lj+})T^{li}-t^{\prime li}{T}^{lj+}\right]\hfill \\ \hfill & & -\sum _{l=j+1}^k\left[(T^{il+}-t^{\prime il+})T^{jl}-t^{\prime jl}{T}^{il+}\right]-\sum _{l=i+1}^{j-1}\left[(T^{il+}-t^{\prime il+})T^{lj+}-t^{\prime lj+}{T}^{il+}\right]\hfill \\ \hfill & & +\left(T_{ij}^{+}-t_{ij}^{\prime +}\right)\left(G_0^j+G_0^i\right)-\left(g_0^{\prime i}+g_0^{\prime j}+i+1\right)T_{ij}^{+}\}{\theta }^{ji}.\hfill \end{array}$$

(82)

Here, the quantities ${\widehat{K}}_0^{0i}$, ${\widehat{\mathcal{K}}}_0^{ij}$, ${\widehat{K}}_0^{1i}$ are the same as those of (26), albeit expressed in terms of $O_I$.

Let us follow our experience in the study of the (super)algebra ${\mathcal{A}}_c(Y\left(1\right),Ad{S}_d)$ [20,21,22], when, in order to find an exact BRST operator, we choose the Weyl (symmetric) ordering for the quadratic combinations of $O_I$ in the r.h.s. of (80)–(82) as $O_I{O}_J=\frac{1}{2}(O_I{O}_J+O_J{O}_I)+\frac{1}{2}[O_I,O_J]$. As a result, for this kind of ordering, Table 3 must contain certain quantities ${\widehat{W}}_{bW}^{ij}$, ${\widehat{\mathcal{K}}}_{1W}^j$, ${\widehat{X}}_{bW}^{ij}$ (and ${\widehat{W}}_{bW}^{ij+}$, ${\widehat{\mathcal{K}}}_{1W}^{j+}$), which have the form

$$\begin{array}{ccc}\hfill {\widehat{W}}_{bW}^{ij}& =& r\left\{\left[{\mathcal{G}}_0^j-{\mathcal{G}}_0^i\right]L^{ij}+{\mathcal{L}}^{ij}\left[G_0^j-G_0^i\right]-{\sum }_{m=1}^k[{\mathcal{T}}^{m[j}{\theta }^{[jm}{L}^{i]m}+{\mathcal{L}}^{[im}{T}^{mj]}{\theta }^{j]m}\right.\hfill \\ \hfill & & \left.+{\mathcal{T}}^{[jm+}{\theta }^{m[j}{L}^{i]m}+{\mathcal{L}}^{[im}{T}^{j]m+}{\theta }^{mj]}]\right\},\hfill \end{array}$$

(83)

$$\begin{array}{ccc}\hfill {\widehat{\mathcal{K}}}_{1W}^j& =& 2{\sum }_{i=1}^k\left\{{\mathcal{L}}^{ji+}{L}^i+{\mathcal{L}}^i{L}^{ji+}\right\}+{\mathcal{L}}^{j+}{G}_0^j+{\mathcal{G}}_0^j{L}^{j+}-{\sum }_{i=1}^k\left\{\right[{\mathcal{L}}^{i+}{T}^{ij+}\hfill \\ \hfill & & +{\mathcal{T}}^{ij+}{L}^{i+}]{\theta }^{ji}+\left[{\mathcal{L}}^{i+}{T}^{ji}+{\mathcal{T}}^{ji}{L}^{i+}\right]{\theta }^{ij}\},\hfill \end{array}$$

(84)

$$\begin{array}{ccc}\hfill {\widehat{X}}_{bW}^{ij}& =& \{L_0+r[{\mathcal{G}}_0^{\prime i}{G}_0^i-2{\mathcal{L}}^{ii+}{L}^{ii}-2{\mathcal{L}}^{ii}{L}^{ii+}+\frac{1}{2}{\sum }_{l=i+1}^k({\mathcal{T}}^{il}{T}^{il+}+{\mathcal{T}}^{il+}{T}^{il}\hfill \\ \hfill & & -4{\mathcal{L}}^{il+}{L}^{il}-4{\mathcal{L}}^{il}{L}^{il+})+\frac{1}{2}{\sum }_{l=1}^{i-1}\left({\mathcal{T}}^{li}{T}^{li+}+{\mathcal{T}}^{li+}{T}^{li}-4{\mathcal{L}}^{li+}{L}^{li}-4{\mathcal{L}}^{li}{L}^{li+}\right)]\}{\delta }^{ij}\hfill \\ & & -r\{2{\sum }_l\left[{\mathcal{L}}^{jl+}{L}^{li}+{\mathcal{L}}^{li}{L}^{jl+}\right]-\frac{1}{2}\sum _{l=1}^{j-1}\left[{\mathcal{T}}^{lj+}{T}^{li}+{\mathcal{T}}^{li}{T}^{lj+}\right]-\frac{1}{2}\sum _{l=i+1}^k[{\mathcal{T}}^{il+}{T}^{jl}\hfill \\ \hfill & & +{\mathcal{T}}^{jl}{T}^{il+}]-\frac{1}{2}\sum _{l=j+1}^{i-1}\left[{\mathcal{T}}^{li}{T}^{jl}+{\mathcal{T}}^{jl}{T}^{li}\right]+\frac{1}{2}\left({\mathcal{G}}_0^j+{\mathcal{G}}_0^i\right)T^{ji}+\frac{1}{2}{\mathcal{T}}^{ji}\left(G_0^j+G_0^i\right)\}{\theta }^{ij}\hfill \\ & & -r\{2{\sum }_l\{{\mathcal{L}}^{jl+}{L}^{li}+{\mathcal{L}}^{li}{L}^{jl+}\}-\frac{1}{2}\sum _{l=1}^{i-1}\left[{\mathcal{T}}^{lj+}{T}^{li}+{\mathcal{T}}^{li}{T}^{lj+}\right]-\frac{1}{2}\sum _{l=j+1}^k[{\mathcal{T}}^{il+}{T}^{jl}\hfill \\ \hfill & & +{\mathcal{T}}^{jl}{T}^{il+}]-\frac{1}{2}\sum _{l=i+1}^{j-1}\left[{\mathcal{T}}_{il}^{+}{T}_{lj}^{+}+{\mathcal{T}}_{lj}^{\prime +}{T}_{il}^{+}\right]+\frac{1}{2}\left[{\mathcal{T}}_{ij}^{+}\left(G_0^j+G_0^i\right)+\left({\mathcal{G}}_0^i+{\mathcal{G}}_0^j\right)T_{ij}^{+}\right]\}{\theta }^{ji},\hfill \end{array}$$

(85)

with the notation ${\mathcal{O}}_I$ for the quantity ${\mathcal{O}}_I=(O_I-2o_I^{\prime })$. Note that the ordering in (83)–(85) does not contain any linear terms (except for $L_0$ in the r.h.s. of the last relation) as compared to (80)–(82).

Thus, we deduce the algebra of converted operators $O_I$ underlying an HS field subject to an arbitrary unitary irreducible AdS group representation with spin $\mathbf{s}=(s_1,\dots ,s_k)$ in an AdS space so that the problem is to find a BRST operator for ${\mathcal{A}}_c(Y\left(k\right),Ad{S}_d)$.

4.2. BRST Operator for Converted Algebra ${\mathcal{A}}_c(Y\left(2\right),Ad{S}_d)$

In this section, we turn to HS fields subject to a YT with two rows. The non-linear algebra no longer has a closed form, due to the operator functions $F_{IJ}^{\left(2\right)K}(o^{\prime },O)$ in (35), and, as shown in [63], this leads to an emergence of higher-order structure functions, due to the quadratic algebraic relations (83)–(85) and their Hermitian conjugation, corresponding to the quantities $F_{IJ}^{\left(2\right)K}(o^{\prime },O)$ for $i,j=1,2$. In [63], new structure functions are found, $F_{IJK}^{RS}\left(O\right)$, being of the third order [35] and implied by a resolution of the Jacobi identities $[[O_I,O_J],O_K]+cycl.perm.(I,J,K)=0$ as follows (33):

$$\begin{array}{c}\hfill \left\{F_{IJ}^M{F}_{MK}^P+[F_{IJ}^{\left(2\right)P},O_K]+cycl.perm.(I,J,K)\right\}=F_{IJK}^{RS}\left(O_R{\delta }_S^P-\frac{1}{2}{F}_{RS}^P\right),\end{array}$$

(86)

for $F_{IJ}^M\equiv \left(f_{IJ}^M+F_{IJ}^{\left(2\right)M}\right)$. The structure functions $F_{IJK}^{RS}(o^{\prime },O)$ are antisymmetric with respect to a permutation of any two of the lower indices $(I,J,K)$ and the upper ones $R,S$, and do exist because of non-trivial Jacobi identities for the $k(2k-1)$ triples $(L_i,L_j,L_0)$, $(L_i^{+},L_j^{+},L_0)$, $(L_i,L_j^{+},L_0)$.

The construction of a BFV–BRST operator $Q^{\prime }$ for ${\mathcal{A}}_c(Y\left(2\right),Ad{S}_d)$ is studied in [63] and has the general form

$$\begin{array}{c}\hfill Q^{\prime }={\mathcal{C}}^I\left[O_I+\frac{1}{2}{\mathcal{C}}^J(f_{JI}^P+F_{JI}^{\left(2\right)P}){\mathcal{P}}_P+\frac{1}{12}{\mathcal{C}}^J{\mathcal{C}}^K{F}_{KJI}^{RP}{\mathcal{P}}_R{\mathcal{P}}_P\right],\end{array}$$

(87)

with the $\left(\mathcal{C}\mathcal{P}\right)$-ordering for the ghost coordinates ${\mathcal{C}}^I$ = $\{{\eta }_0$, ${\eta }_G^i$${\eta }_i^{+}$, ${\eta }_i$, ${\eta }_{ij}^{+}$, ${\eta }_{ij}$, ${\vartheta }_{12}$, ${\vartheta }_{12}^{+}\}$, and their conjugate momenta12 ${\mathcal{P}}_I$ = $\{{\mathcal{P}}_0$, ${\mathcal{P}}_G^i$, ${\mathcal{P}}_i$, ${\mathcal{P}}_i^{+}$, ${\mathcal{P}}_{ij}$, ${\mathcal{P}}_{ij}^{+}$, ${\lambda }_{12}^{+}$, ${\lambda }_{12}\}$. Explicitly, $Q^{\prime }$ is given by

$$\begin{array}{ccc}\hfill Q^{\prime }& =& Q_1^{\prime }+Q_2^{\prime }+r^2\left\{{\eta }_0{\sum }_{i,j}{\eta }_i{\eta }_j{\epsilon }^{ij}\left[\frac{1}{2}{\sum }_m\right(G_0^m\left[{\lambda }_{12}{\mathcal{P}}_{22}^{+}-{\lambda }_{12}^{+}{\mathcal{P}}_{11}^{+}+i{\mathcal{P}}_{12}^{+}{\sum }_l{(-1)}^l{\mathcal{P}}_G^l\right]\right.\hfill \\ & & -i(L_{11}^{+}{\lambda }_{12}^{+}-L_{22}^{+}{\lambda }_{12}){\mathcal{P}}_G^m+4L^{mm}{\mathcal{P}}_{m2}^{+}{\mathcal{P}}_{1m}^{+})-L_{12}^{+}{\mathcal{P}}_G^1{\mathcal{P}}_G^2+2L^{12}{\mathcal{P}}_{22}^{+}{\mathcal{P}}_{11}^{+}]\hfill \\ & & +{\eta }_0{\sum }_{i,j}{\eta }_i^{+}{\eta }_j[{\sum }_m\left({(-1)}^m\frac{i}{2}{G}_0^m{\sum }_l{\mathcal{P}}_G^l+2(L_{22}^{+}{\mathcal{P}}^{22}-L^{11}{\mathcal{P}}_{11}^{+})\right){\lambda }_{12}{\delta }^{1j}{\delta }^{2i}\hfill \\ & & +{\epsilon }^{\{1j}{\delta }^{2\}i}(ı{\sum }_m\left[\frac{1}{2}{T}^{12}{\lambda }_{12}^{+}-2L^{12}{\mathcal{P}}_{12}^{+}{(-1)}^m\right]{\mathcal{P}}_G^m+2\left[L^{12}{\lambda }_{12}-T^{12}{\mathcal{P}}^{12}\right]{\mathcal{P}}_{22}^{+}\hfill \\ & & +2\left[T_{12}^{+}{\mathcal{P}}^{12}-L^{12}{\lambda }_{12}^{+}\right]{\mathcal{P}}_{11}^{+})-T^{12}\left[{\mathcal{P}}_G^1{\mathcal{P}}_G^2{\delta }^{2i}{\delta }^{1j}+2{\mathcal{P}}^{11}{\mathcal{P}}_{22}^{+}{\delta }^{1i}{\delta }^{2j}\right]\hfill \\ & & \left.-2{\sum }_m{(-1)}^m\left[(G_0^m{\mathcal{P}}^{11}+ıL^{11}{\mathcal{P}}_G^m){\delta }^{1i}{\delta }^{2j}-(G_0^m{\mathcal{P}}^{22}+ıL^{22}{\mathcal{P}}_G^m){\delta }^{2i}{\delta }^{1j}\right]{\mathcal{P}}_{12}^{+}]+h.c.\right\},\hfill \end{array}$$

(88)

for ${\epsilon }^{ij}=-{\epsilon }^{ji},{\epsilon }^{12}=1$, with the standard form of the linear $Q_1^{\prime }$ and quadratic $Q_2^{\prime }$ terms in the ghosts ${\mathcal{C}}^I$; for details, see [63]. The Hermiticity of the nilpotent operator $Q^{\prime }$ in the total Hilbert space ${\mathcal{H}}_{tot}$ = $\mathcal{H}⨂{\mathcal{H}}^{\prime }⨂{\mathcal{H}}_{gh}$ is defined by the rule

$$Q^{\prime +}K=K{Q}^{\prime },\mathrm{for}K=\widehat{1}\otimes K^{\prime }\otimes {\widehat{1}}_{gh},$$

(89)

with the operator $K^{\prime }$ given by (79), which provides the Hermiticity of $o_I^{\prime }$ in ${\mathcal{H}}^{\prime }$.

4.3. Lagrangian Formulation

The construction of Lagrangians for bosonic HS fields in AdS${}_d$ spaces can be achieved by partially following the algorithm of [21,37] (see also [59]), which is a particular case of our construction, corresponding to $s_2=0$. As a first step, we extract the dependence of the BRST operator $Q^{\prime }$ (88) on the ghosts ${\eta }_G^i,{\mathcal{P}}_G^i$,

$$\begin{array}{ccc}\hfill Q^{\prime }& =& Q+{\eta }_G^i({\sigma }^i+h^i)+{\mathcal{B}}^i{\mathcal{P}}_G^i,\hfill \end{array}$$

(90)

with certain (non-essential later on) operators ${\mathcal{B}}^i$ and a BRST operator13 Q, which corresponds only to the converted first-class constraints $\left\{O_I\right\}\setminus \left\{G_0^i\right\}$,

$$\begin{array}{cc}\hfill Q=& \frac{1}{2}{\eta }_0{L}_0+{\sum }_i{\eta }_i^{+}{L}_i+{\sum }_{l\le m}{\eta }_{lm}^{+}{L}_{lm}+{\vartheta }_{12}^{+}{T}_{12}+\frac{ı}{2}{\sum }_l{\eta }_l^{+}{\eta }_l{\mathcal{P}}_0\hfill \\ & -{\vartheta }_{12}^{+}{\sum }_n(1+{\delta }_{1n}){\eta }_{1n}^{+}{\mathcal{P}}_{n2}+{\vartheta }_{12}^{+}{\sum }_n(1+{\delta }_{n2}){\eta }_{n2}{\mathcal{P}}_{1n}^{+}+\frac{1}{2}{\sum }_n{\eta }_{n2}^{+}{\eta }_{1n}{\lambda }_{12}\hfill \\ & -\frac{1}{2}{\displaystyle \sum _{l\le m}}(1+{\delta }_{lm}){\eta }_m{\eta }_{lm}^{+}{\mathcal{P}}_l-[{\vartheta }_{12}{\eta }_2^{+}{\mathcal{P}}_1+{\vartheta }_{12}^{+}{\eta }_1^{+}{\mathcal{P}}_2]\hfill \\ & +r\left\{{\eta }_0{\sum }_i{\eta }_i^{+}[2{\mathcal{L}}_{ii}{\mathcal{P}}_i^{+}+2{\mathcal{L}}_i^{+}{\mathcal{P}}_{ii}+{\mathcal{G}}_0^i{\mathcal{P}}_i+2\left({\mathcal{L}}_{12}{\mathcal{P}}_{\{1}^{+}+{\mathcal{L}}_{\{1}^{+}{\mathcal{P}}_{12}\right){\delta }_{2\}i}\right.\hfill \\ & -{\delta }^{1i}\left({\mathcal{L}}_2{\lambda }_{12}^{+}+{\mathcal{T}}_{12}^{+}{\mathcal{P}}_2\right)-{\delta }^{2i}\left({\mathcal{L}}_1{\lambda }_{12}+{\mathcal{T}}_{12}{\mathcal{P}}_1\right)]\hfill \\ & -\frac{1}{2}{\sum }_{i,j}{\eta }_i^{+}{\eta }_j^{+}{\epsilon }^{ij}\left[{\sum }_m{(-1)}^m{\mathcal{G}}_0^m{\mathcal{P}}_{12}-\left({\mathcal{T}}_{12}{\mathcal{P}}_{11}+{\mathcal{L}}_{11}{\lambda }_{12}\right)+{\mathcal{T}}_{12}^{+}{\mathcal{P}}_{22}+{\mathcal{L}}_{22}{\lambda }_{12}^{+}\right]\hfill \\ & \left.+2{\eta }_i^{+}{\eta }_j\left[{\sum }_m{\mathcal{L}}_{jm}^{+}{\mathcal{P}}_{im}-\frac{1}{8}\left({\mathcal{T}}_{12}^{+}{\lambda }_{12}+{\mathcal{T}}_{12}{\lambda }_{12}^{+}\right){\delta }^{ij}+\frac{1}{4}{\sum }_m{\mathcal{G}}_0^m{\lambda }_{12}{\delta }^{j1}{\delta }^{i2}\right]\right\}\hfill \\ & +r^2\left\{{\eta }_0{\sum }_{i,j}{\eta }_i{\eta }_j{\epsilon }^{ij}\left[\frac{1}{2}{\sum }_m\left(G_0^m\left[{\lambda }_{12}{\mathcal{P}}_{22}^{+}-{\lambda }_{12}^{+}{\mathcal{P}}_{11}^{+}\right]+4L_{mm}{\mathcal{P}}_{m2}^{+}{\mathcal{P}}_{1m}^{+}\right)+2L_{12}{\mathcal{P}}_{22}^{+}{\mathcal{P}}_{11}^{+}\right]\right.\hfill \\ & +{\eta }_0{\sum }_{i,j}{\eta }_i^{+}{\eta }_j[2{\sum }_m\left(L_{22}^{+}{\mathcal{P}}_{22}-L_{11}{\mathcal{P}}_{11}^{+})\right){\lambda }_{12}{\delta }^{1j}{\delta }^{2i}-2T_{12}{\mathcal{P}}_{11}{\mathcal{P}}_{22}^{+}{\delta }^{1i}{\delta }^{2j}\hfill \\ & +2{\epsilon }^{\{1j}{\delta }^{2\}i}\left(\left(L_{12}{\lambda }_{12}-T_{12}{\mathcal{P}}_{12}\right){\mathcal{P}}_{22}^{+}+\left(T_{12}^{+}{\mathcal{P}}_{12}-L_{12}{\lambda }_{12}^{+}\right){\mathcal{P}}_{11}^{+}\right)\hfill \\ & \left.-2{\sum }_m{(-1)}^m\left(G_0^m{\mathcal{P}}_{11}{\delta }^{1i}{\delta }^{2j}-G_0^m{\mathcal{P}}_{22}{\delta }^{2i}{\delta }^{1j}\right){\mathcal{P}}_{12}^{+}]\right\}+h.c.\hfill \end{array}$$

(91)

The generalized spin operator ${\overrightarrow{\sigma }}_2=({\sigma }^1,{\sigma }^2)$, extended by ghost Wick-pair variables,

$$\begin{array}{c}\hfill {\sigma }^i=G_0^i-h^i-{\eta }_i{\mathcal{P}}_i^{+}+{\eta }_i^{+}{\mathcal{P}}_i+\sum _m(1+{\delta }_{im})({\eta }_{im}^{+}{\mathcal{P}}_{im}-{\eta }_{im}{\mathcal{P}}_{im}^{+})+({\vartheta }_{12}^{+}{\lambda }_{12}-{\vartheta }_{12}{\lambda }_{12}^{+}){(-1)}^i,\end{array}$$

(92)

supercommutes with Q. The nilpotency $Q^{\prime }{}^2=0$ entails the system of equations

$$\begin{array}{cccc}\hfill & Q^2=2ı\sum _i{\mathcal{B}}^i{\sigma }^i,\hfill & \hfill & [Q,{\sigma }^i\}=0;\hfill \end{array}$$

(93)

$$\begin{array}{cccc}\hfill & Q^{+}K=KQ,\hfill & \hfill & {\left({\sigma }^i\right)}^{+}K=K{\sigma }^i.\hfill \end{array}$$

(94)

We choose a representation for the Hilbert space ${\mathcal{H}}_{tot}$ to be coordinated with the decomposition (90) so that the operators $({\eta }_i,{\eta }_{ij},{\vartheta }_{12},{\mathcal{P}}_0,{\mathcal{P}}_i,{\mathcal{P}}_{ij},{\lambda }_{12},{\mathcal{P}}_G^i)$ annihilate the vacuum vector $|0\rangle$, and suppose that the field vectors $|\chi \rangle$, as well as the gauge parameters $|\Lambda \rangle$, do not depend on the ghosts ${\eta }_G^i$,

$$\begin{array}{ccc}\hfill |\chi \rangle & =& \sum _n{\prod }_l^2{\left(b_l^{+}\right)}^{n_l}{\prod }_{i\le j}^2{\left(b_{ij}^{+}\right)}^{n_{ij}}{\left(d_{12}^{+}\right)}^{p_{12}}{\left({\eta }_0^{+}\right)}^{n_{f0}}\prod _{i,j,l\le m,n\le o}^2{\left({\eta }_i^{+}\right)}^{n_{fi}}{\left({\mathcal{P}}_j^{+}\right)}^{n_{pj}}{\left({\eta }_{lm}^{+}\right)}^{n_{flm}}{\left({\mathcal{P}}_{no}^{+}\right)}^{n_{pno}}\hfill \\ & & \times {\left({\vartheta }_{12}^{+}\right)}^{n_{f12}}{\left({\lambda }_{12}^{+}\right)}^{n_{\lambda 12}}|\Phi {\left(a_i^{+}\right)}_{{\left(n\right)}_l{\left(n\right)}_{ij}{p}_{12}}^{n_{f0}{\left(n\right)}_{fi}{\left(n\right)}_{pj}{\left(n\right)}_{flm}{\left(n\right)}_{pno}{\left(n\right)}_{f12}{\left(n\right)}_{\lambda 12}}\rangle .\hfill \end{array}$$

(95)

The brackets ${\left(n\right)}_{fi},{\left(n\right)}_{pj},{\left(n\right)}_{pno}$ in (95) imply, for example, for ${\left(n\right)}_{pno}$, the set of indices $(n_{p11},n_{p12},n_{p22})$. The above sum runs over $n_l$, $n_{ij}$, $p_{12}$, in the range from 0 to infinity, as well as over the remaining n from 0 to 1. The Hilbert space ${\mathcal{H}}_{tot}$ decomposes into a direct sum of Hilbert subspaces with definite ghost numbers: ${\mathcal{H}}_{tot}={⨁}_{k=-6}^6{\mathcal{H}}_k$. Denote by $|{\chi }^k\rangle \in {\mathcal{H}}_{-k}$ the state (95) with the ghost number $-k$, i.e., $gh\left(\right|{\chi }^k\rangle )=-k$. Thus, the physical state with the ghost number zero is $|{\chi }^0\rangle$, and the gauge parameters $|\Lambda \rangle$ with the ghost number $-1$ are $|{\chi }^1\rangle$, and so on. For the vanishing of all the auxiliary creation operators $B^{+}$ and ghost variables ${\eta }_0,{\eta }_i^{+},{\mathcal{P}}_i^{+},\dots$, the vector $|{\chi }^0\rangle$ must contain only the physical string-like vector $|\Phi \rangle =|\Phi {\left(a_i^{+}\right)}_{{\left(0\right)}_l{\left(0\right)}_{ij}{0}_{12}}^{{\left(0\right)}_{fo}{\left(0\right)}_{fi}{\left(0\right)}_{pj}{\left(0\right)}_{flm}{\left(0\right)}_{pno}{\left(0\right)}_{f12}{\left(0\right)}_{\lambda 12}}\rangle$,

$$\begin{array}{ccc}\hfill |{\chi }^0\rangle & =& |\Phi \rangle +|{\Phi }_A\rangle ,|{\Phi }_A\rangle |{}_{[B^{+}={\eta }_0={\eta }_i^{+}={\mathcal{P}}_i^{+}={\eta }_{lm}^{+}={\mathcal{P}}_{no}^{+}={\vartheta }_{12}^{+}={\lambda }_{12}^{+}=0]}=0.\hfill \end{array}$$

(96)

One can show, using some of the equations of motion and gauge transformations, that the vector $|{\Phi }_A\rangle$ can be completely removed.

The equation for the physical state $Q^{\prime }|{\chi }^0\rangle =0$ and the tower of reducible gauge transformations $\delta |\chi \rangle$ = $Q^{\prime }|{\chi }^1\rangle$, $\delta |{\chi }^1\rangle =Q^{\prime }|{\chi }^2\rangle$, …, $\delta |{\chi }^{(s-1)}\rangle =Q^{\prime }|{\chi }^{\left(s\right)}\rangle$ lead to the following relations:

$$\begin{array}{ccc}\hfill Q|\chi \rangle & =& 0,({\sigma }^i+h^i)|\chi \rangle =0,\left(ϵ,gh\right)\left(\right|\chi \rangle )=(0,0),\hfill \end{array}$$

(97)

$$\begin{array}{ccc}\hfill \delta |\chi \rangle & =& Q|{\chi }^1\rangle ,({\sigma }^i+h^i)|{\chi }^1\rangle =0,\left(ϵ,gh\right)\left(\right|{\chi }^1\rangle )=(1,-1),\hfill \\ \hfill \dots & & \dots \dots \hfill \end{array}$$

(98)

$$\begin{array}{ccc}\hfill \delta |{\chi }^{s-1}\rangle & =& Q|{\chi }^s\rangle ,({\sigma }^i+h^i)|{\chi }^s\rangle =0,\left(ϵ,gh\right)\left(\right|{\chi }^s\rangle )=(smod2,-s).\hfill \end{array}$$

(99)

Here, $s=5$ is the maximal stage of reducibility for a massive bosonic HS field, due to the subspaces ${\mathcal{H}}_k=\varnothing$, for all integer $k\le -7$. The middle set of Equations (97)–(99) determines the possible values of the parameters $h^i$ and the eigenvectors of the operators ${\sigma }^i$. Solving the spectral problem, we obtain a set of eigenvectors, $|{\chi }^0{\rangle }_{{\overrightarrow{n}}_2}$, $|{\chi }^1{\rangle }_{{\overrightarrow{n}}_2}$, …, $|{\chi }^s{\rangle }_{{\overrightarrow{n}}_2}$, $n_1\ge n_2\ge 0$, and a set of eigenvalues,

$$\begin{array}{c}\hfill {\sigma }_i{|\chi \rangle }_{{\overrightarrow{n}}_2}=\left(n^i+\frac{d-1-4i}{2}\right){|\chi \rangle }_{{\overrightarrow{n}}_2},-h^i=n_i+\frac{d-1-4i}{2}i=1,2,n_1\in \mathbf{Z},n_2\in {\mathbf{N}}_0.\end{array}$$

(100)

It is easy to see that, in order to construct a Lagrangian for a field corresponding to a certain Young tableaux (1), the numbers $n_i$ must be equal to the numbers of boxes in the i-th row of the corresponding Young tableaux, i.e., $n_i=s_i$. Thus, the state ${|\chi \rangle }_{{\overrightarrow{s}}_2}$ contains the physical field (7) and all of its auxiliary fields. Here, we fix some values of $n_i=s_i$, and after the substitution $h^i\to h^i\left(s_i\right)$, the operator $Q_{{\overrightarrow{s}}_2}\equiv Q_{|h^i\to h^i\left(s_i\right)}$ becomes nilpotent in each subspace ${\mathcal{H}}_{tot{\overrightarrow{s}}_2}$ whose vectors satisfy (97) for (100). Having in mind the Lagrangian equations of motion (those corresponding to (2)–(5) for $k=2$), the sequence of reducible gauge transformations acquires the form

$$\begin{array}{ccc}\hfill & & Q_{{\overrightarrow{s}}_2}|{\chi }^0{\rangle }_{{\overrightarrow{s}}_2}=0,\delta |{\chi }^s{\rangle }_{{\overrightarrow{s}}_2}=Q_{{\overrightarrow{s}}_2}{|{\chi }^{s+1}\rangle }_{{\overrightarrow{s}}_2},s=0,\dots ,5.\hfill \end{array}$$

(101)

By analogy with the totally symmetric bosonic HS fields [21,37], one can show that the Lagrangian action for a fixed spin ${\overrightarrow{n}}_2={\overrightarrow{s}}_2$ is defined up to an overall factor, as follows:

$$\begin{array}{ccc}\hfill & & {\mathcal{S}}_{0|{\overrightarrow{s}}_2}^m=\int d{\eta }_0{}_{{\overrightarrow{s}}_2}\langle {\chi }^0|K_{{\overrightarrow{s}}_2}{Q}_{{\overrightarrow{s}}_2}|{\chi }^0{\rangle }_{{\overrightarrow{s}}_2},\mathrm{for}|{\chi }^0\rangle \equiv |\chi \rangle ,\hfill \end{array}$$

(102)

where the usual inner product for the creation and annihilation operators is assumed, with the measure $d^{d}x\sqrt{\left|g\right|}$ over the AdS space. The vector $|{\chi }^0{\rangle }_{{\left(s\right)}_2}$ and the operator $K_{{\overrightarrow{s}}_2}$ in (102) are, respectively, the vector $|\chi \rangle$ (95), subject to the spin distribution relations (100) for the HS tensor field ${\Phi }_{{\left({\mu }^1\right)}_{s_1},{\left({\mu }^2\right)}_{s_2}}\left(x\right)$, and the operator K (89), where the substitution $h_i\to -(n_i+\frac{d-1-4i}{2})$ has been made. The corresponding LF for the bosonic field of spin $\mathbf{s}$, subject to $Y(s_1,s_2)$, is a reducible gauge theory of $L=6$ stage of reducibility at the most.

One can prove that the equations of motion (101) reproduce only the basic conditions (2)–(5) for HS fields of a given spin ${\overrightarrow{s}}_2$ and mass m. Indeed, the corresponding analysis is presented in Appendix E and repeats a similar proof of the paper [59] for arbitrary bosonic HS fields in flat spaces (with the AdS space being the only specific one), and allows one to gauge away all the auxiliary fields so that the remaining physical vector satisfies only the AdS group irreducible conditions (2)–(5). Therefore, the resulting equations of motion, due to the representation (96), acquire the form

$$L_0{|\Phi \rangle }_{{\overrightarrow{s}}_2}=(l_0+m_0^2){|\Phi \rangle }_{{\overrightarrow{s}}_2},(l_i,l_{ij},t_{i_1{j}_1}){|\Phi \rangle }_{{\overrightarrow{s}}_2}=(0,0,0),i\le j,i_1<j_1.$$

(103)

The above relations allow one, in a unique way, to determine the parameter $m_0$ in terms of $h^i\left(s_i\right)$,

$$m_0^2=m^2+r\left\{\beta (\beta +1)+\frac{d(d-6)}{4}+\left(h^1-\frac{1}{2}+2\beta \right)\left(h^1-\frac{5}{2}\right)+\left(h^2-\frac{9}{2}\right)\right\},$$

(104)

whereas the values of parameters $m_1,m_2$ remain arbitrary, and can be used to provide the characteristic properties of a Lagrangian for a given HS field.

The general action (102) presents a direct recipe to obtain a Lagrangian for any component field ${\Phi }_{{\left({\mu }^1\right)}_{s_1},{\left({\mu }^2\right)}_{s_2}}\left(x\right)$, starting from a general vector $|{\chi }^0{\rangle }_{{\overrightarrow{s}}_2}$ since we only need to calculate the vacuum expectation values of products for certain creation and annihilation operators. Examples of applying the general action are given below in Section 6. We emphasize that Lagrangian formulations for massive bosonic HS fields described by a Young tableaux with two rows subject to traceless and Young constraints in (A)dS spaces have only been known in a frame-like form: see [72].

5. BRST Approach to Cubic Interaction Vertices

Here, we follow our results on constructing general cubic vertices for massless [38,94] and massive [96] totally symmetric fields of integer higher spin in Minkowski spaces by using two approaches. The first one is based on an ambient formalism of embedding an AdS${}_d$ space in a $(d+1)$-dimensional Minkowski space [100]; see also [43] and the references therein. Now, we use the second of the above approaches to a description of AdS${}_d$ spaces: with no reference to flat space-times.

For completeness, we draw the reader’s attention to the results of Lagrangian cubic vertices construction in AdS${}_d$ spaces [101,102], using various methods, basically for reducible interacting totally symmetric HS fields, Maxwell-like Lagrangians, and the $d=3$ case [103], including interaction with gravity for a partially massless spin-2 field [104]. Also, cubic vertices deduced from the Fradkin–Vasiliev recipe [105,106] are shown to have a smooth flat limit for AdS vertices [107,108].

To construct a cubic vertex, we consider three copies of vectors $|{\chi }^{\left(i\right)}{\rangle }_{\overrightarrow{s}{}_2^i}$ and gauge parameters $|{\chi }^{\left(i\right)1}{\rangle }_{\overrightarrow{s}{}_2^i}$, …, $|{\chi }^{\left(i\right)6}{\rangle }_{\overrightarrow{s}{}_2^i}$ from a corresponding Hilbert space ${\mathcal{H}}_{tot}^{\left(i\right)}$, with the respective vacuum vectors ${|0\rangle }^i$ and oscillators for $i=1,2,3$. This allows one to obtain a deformed action and a sequence of deformed reducible gauge transformations for a massive triple ${\left(m\right)}_3\equiv (m_1,m_2,m_3)$, with accuracy up to the first order in the parameter g:

$$\begin{array}{c}{S}_{\left[1\right]|{\left(\overrightarrow{s}\right)}^3}^{{\left(m\right)}_3}[{\chi }^{\left(1\right)},{\chi }^{\left(2\right)},{\chi }^{\left(3\right)}]=\sum _{i=1}^3{\mathcal{S}}_{0|\overrightarrow{s}{}_2^i}^{m_i}+g\int \prod _{e=1}^{3}d{\eta }_0^{\left(e\right)}\left({}_{\overrightarrow{s}{}_2^e}{\langle {\chi }^{\left(e\right)}{K}^{\left(e\right)}|V^{\left(3\right)}\rangle }_{{\left(\overrightarrow{s}\right)}_2^3}^{{\left(m\right)}_3}+h.c.\right),\hfill \end{array}$$

(105)

$$\begin{array}{ccc}& & {\delta }_{\left[1\right]}|{\chi }^{\left(i\right)}{\rangle }_{\overrightarrow{s}{}_2^i}=Q^{\left(i\right)}|{\chi }^{\left(i\right)1}{\rangle }_{\overrightarrow{s}{}_2^i}-g\int \prod _{e=1}^{2}d{\eta }_0^{(i+e)}\left({}_{\overrightarrow{s}{}_2^{i+1}}\langle {\chi }^{\left(i+11\right)}{K}^{(i+1)}|{}_{\overrightarrow{s}{}_2^{i+2}}\langle {\chi }^{\left(i+2\right)}{K}^{(i+1)}\right|\hfill \\ & & \phantom{{\delta }_{\left[1\right]}|{\chi }^{\left(i\right)}{\rangle }_{s_i}}+(i+1\leftrightarrow i+2)\left)\right|\tilde{V}{}_0^{\left(3\right)}{\rangle }_{\left(s\right){}_2^3}^{{\left(m\right)}_3},\hfill \end{array}$$

(106)

$$\begin{array}{ccc}& & {\delta }_{\left[1\right]}|{\chi }^{\left(i\right)1}{\rangle }_{\overrightarrow{s}{}_2^i}=Q^{\left(i\right)}|{\chi }^{\left(i\right)2}{\rangle }_{\overrightarrow{s}{}_2^i}-g\int \prod _{e=1}^{2}d{\eta }_0^{(i+e)}\left({}_{\overrightarrow{s}{}_2^{i+1}}\langle {\chi }^{(i+1)2}{K}^{(i+1)}|{}_{\overrightarrow{s}{}_2^{i+2}}\langle {\chi }^{\left(i+2\right)}{K}^{(i+1)}\right|\hfill \\ & & \phantom{{\delta }_{\left[1\right]}|{\chi }^{\left(i\right)}{\rangle }_{s_i}}+(i+1\leftrightarrow i+2)\left)\right|\tilde{V}{}_1^{\left(3\right)}{\rangle }_{\left(\overrightarrow{s}\right){}_2^3}^{{\left(m\right)}_3},\hfill \\ & & \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \hfill \end{array}$$

(107)

$$\begin{array}{ccc}& & {\delta }_{\left[1\right]}|{\chi }^{\left(i\right)5}{\rangle }_{\overrightarrow{s}{}_2^i}=Q^{\left(i\right)}|{\chi }^{\left(i\right)6}{\rangle }_{\overrightarrow{s}{}_2^i}-g\int \prod _{e=1}^{2}d{\eta }_0^{(i+e)}\left({}_{\overrightarrow{s}{}_2^{i+1}}\langle {\chi }^{(i+1)6}{K}^{(i+1)}|{}_{\overrightarrow{s}{}_2^{i+2}}\langle {\chi }^{\left(i+2\right)}{K}^{(i+1)}\right|\hfill \\ & & \phantom{{\delta }_{\left[1\right]}|{\chi }^{\left(i\right)}{\rangle }_{s_i}}+(i+1\leftrightarrow i+2)\left)\right|\tilde{V}{}_5^{\left(3\right)}{\rangle }_{\left(\overrightarrow{s}\right){}_2^3}^{{\left(m\right)}_3},\hfill \end{array}$$

(108)

with some unknown three-vectors $|V^{\left(3\right)}{{\rangle }_{\left(\overrightarrow{s}\right){}_2^3}^{{\left(m\right)}_3},|\tilde{V}{}_l^{\left(3\right)}\rangle }_{\left(\overrightarrow{s}\right){}_2^3}^{{\left(m\right)}_3}$, $l=0,1,\dots ,5$, defined in a total Hilbert space ${⨂}_{i=1}^3{\mathcal{H}}_{tot}^{\left(i\right)}$. Here, ${\mathcal{S}}_{0|\overrightarrow{s}{}_2^i}^{m_i}$ is the free action (102) for the field $|{\chi }^{\left(i\right)}{\rangle }_{\overrightarrow{s}{}_2^i}$; $Q^{\left(i\right)}$ is the BRST charge (91) corresponding to the spin $\overrightarrow{s}{}_2^i,i=1,2,3$; $K^{\left(i\right)}$ is the operator K (89) related to the spin $\overrightarrow{s}{}_2^i,i=1,2,3$ for a massive field; and g is a deformation parameter, usually called a coupling constant. We also use the convention $[i+3\simeq i]$.

The construction of a cubic interaction involves finding some 3-vectors $|V^{\left(3\right)}{\rangle }_{\left(\overrightarrow{s}\right){}_2^3}^{{\left(m\right)}_3}$, $|\tilde{V}{}_l^{\left(3\right)}{\rangle }_{\left(\overrightarrow{s}\right){}_2^3}^{{\left(m\right)}_3}$. For this purpose, we can apply the set of fields, constraints, and ghost operators related to the spins ${\overrightarrow{s}}_2^1,{\overrightarrow{s}}_2^2,{\overrightarrow{s}}_2^3$, as well as the respective conditions of gauge invariance for the deformed action under the deformed gauge transformations, and also the preservation of the form of gauge transformations for the fields $|{\chi }^{\left(i\right)}{\rangle }_{{\overrightarrow{s}}_2^i}$ under the first-level gauge transformations ${\delta }_{\left[1\right]}|{\chi }^{\left(i\right)1}{\rangle }_{{\overrightarrow{s}}_2^i}$, having the form of l-th level gauge transformations for the gauge parameters $|{\chi }^{\left(i\right)l}{\rangle }_{{\overrightarrow{s}}_2^i}$ under the $(l+1)$-th level gauge transformations ${\delta }_{\left[1\right]}|{\chi }^{\left(i\right)l+1}{\rangle }_{{\overrightarrow{s}}_2^i}$ for $l=1,\dots ,5$ at the first power in g:

$$\begin{array}{ccc}& & g\int \prod _{e=1}^{3}d{\eta }_0^{\left(e\right)}{}_{{\overrightarrow{s}}_2^j}\langle {\chi }^{\left(j\right)1}{K}^{\left(j\right)}\left|{}_{{\overrightarrow{s}}_2^{j+1}}\langle {\chi }^{(j+1)}{K}^{(j+1)}|{}_{{\overrightarrow{s}}_2^{j+2}}\langle {\chi }^{(j+2)}{K}^{(j+2)}\right|\mathcal{Q}(V^3,{\tilde{V}}_0^3)=0,\hfill \end{array}$$

(109)

$$\begin{array}{ccc}& & g\int \prod _{e=1}^{2}d{\eta }_0^{\left(e\right)}{}_{{\overrightarrow{s}}_2^{j+1}}\langle {\chi }^{(j+1)2}{K}^{(j+1)}|{}_{{\overrightarrow{s}}_2^{j+2}}\langle {\chi }^{(j+2)}{K}^{(j+2)}|\left(\mathcal{Q}({\tilde{V}}_0^3,{\tilde{V}}_1^3)-Q^{(j+2)}|\tilde{V}{}_1^{\left(3\right)}\rangle \right)=0,\hfill \\ & & \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \hfill \end{array}$$

(110)

$$\begin{array}{ccc}& & g\int \prod _{e=1}^{2}d{\eta }_0^{\left(e\right)}{}_{{\overrightarrow{s}}_2^{j+1}}\langle {\chi }^{(j+1)6}{K}^{(j+1)}|{}_{{\overrightarrow{s}}_2^{j+2}}\langle {\chi }^{(j+2)}{K}^{(j+2)}|\left(\mathcal{Q}({\tilde{V}}_4^3,{\tilde{V}}_5^3)-Q^{(j+2)}|\tilde{V}{}_5^{\left(3\right)}\rangle \right)=0,\hfill \end{array}$$

(111)

where

$$\begin{array}{ccc}& & \mathcal{Q}({\tilde{V}}_l^3,{\tilde{V}}_{l+1}^3)=\sum _{k=1}^3{Q}^{\left(k\right)}|\tilde{V}{}_{l+1}^{\left(3\right)}{\rangle }_{\left(\overrightarrow{s}\right){}_2^3}^{{\left(m\right)}_3}+Q^{\left(j\right)}\left(|\tilde{V}{}_l^{\left(3\right)}{{\rangle }_{\left(\overrightarrow{s}\right){}_2^3}^{{\left(m\right)}_3}-|\tilde{V}{}_{l+1}^{\left(3\right)}\rangle }_{\left(\overrightarrow{s}\right){}_2^3}^{{\left(m\right)}_3}\right),j=1,2,3,\hfill \end{array}$$

(112)

for $l=-1,\dots ,4$ and ${\tilde{V}}_{-1}^3\equiv V^3$.

Following our results [38,94], we assume coincidence for the vertices $|V{}^{\left(3\right)}\rangle$ = $|\tilde{V}{}_0^{\left(3\right)}\rangle$ = $|\tilde{V}{}_e^{\left(3\right)}\rangle$, $e=1,\dots ,5$, which provides the validity of the operator generating equations at the first order in g (the highest orders being necessary to find the quartic and higher vertices),

$$Q^{tot}|V{}^{\left(3\right)}{{\rangle }_{\left(\overrightarrow{s}\right){}_2^3}^{{\left(m\right)}_3}=0,\overrightarrow{\sigma }{}_2^{\left(i\right)}|V{}^{\left(3\right)}\rangle }_{\left(\overrightarrow{s}\right){}_2^3}^{{\left(m\right)}_3}=0,$$

(113)

along with the spin conditions as a consequence of the generalized spin Equation (100) for each sample (with $|{\chi }^{\left(i\right)}{\rangle }_{{\overrightarrow{s}}_2^i},|{\chi }^{\left(i\right)1}{\rangle }_{{\overrightarrow{s}}_2^i},\dots ,{|{\chi }^{\left(i\right)6}\rangle }_{{\overrightarrow{s}}_2^i}$), providing the nilpotency of a total BRST operator $Q^{tot}\equiv {\sum }_i{Q}^{\left(i\right)}$ when evaluated on the vertex, due to Equation (93) and $\{Q^{\left(i\right)},Q^{\left(j\right)}\}=0$ for $i\ne j$.

A local dependence on the space-time coordinates in the vertices (given by Figure 1) $|V^{\left(3\right)}\rangle$, $|\tilde{V}{}_e^{\left(3\right)}\rangle$ implies

$$|V^{\left(3\right)}{\rangle }_{\left(\overrightarrow{s}\right){}_2^3}^{{\left(m\right)}_3}=\int d^{d}x\sqrt{\left|g\right|}\prod _{i=1}^3{\delta }^{\left(d\right)}\left(x-x_i\right)V^{\left(3\right)|}{}_{\left(\overrightarrow{s}\right){}_2^3}^{{\left(m\right)}_3}\prod _{j=1}^3{\eta }_0^{\left(j\right)}|0\rangle ,|0\rangle \equiv {\otimes }_{e=1}^3{|0\rangle }^e,$$

(114)

for $(\epsilon ,gh)V^{\left(3\right)|}{}_{\left(\overrightarrow{s}\right){}_2^3}^{{\left(m\right)}_3}=(0,0)$.

Here, we have a conservation law ${\sum }_{i=1}^3{p}_{\mu }^{\left(i\right)}=0$ for the momenta associated with all the vertices. Once again, as in the case of flat space-times [38,96], the deformed gauge transformations form a closed algebra, which implies, after a simple calculation,

$$\begin{array}{ccc}& & \left[{\delta }_{\left[1\right]}^{{\chi }_1^1},{\delta }_{\left[1\right]}^{{\chi }_2^1}\right]|{\chi }^{\left(i\right)}\rangle =-g{\delta }_{\left[1\right]}^{{\chi }_3^1}|{\chi }^{\left(i\right)}\rangle ,\hfill \\ & & |{\chi }_3^{\left(i\right)1}\rangle \sim \int \prod _{e=1}^{2}d{\eta }_0^{(i+e)}\left(\langle {\chi }_2^{(i+1)1}K|\langle {\chi }_1^{(i+2)1}|K+\left(i+1\leftrightarrow i+2\right)\right)-\left({\chi }_1^1\leftrightarrow {\chi }_2^1\right)\left\}\right|V{}^{\left(3\right)}\rangle ,\hfill \end{array}$$

(115)

with the Grassmann-odd gauge parameter ${\chi }_3^1$ being a function of the parameters ${\chi }_1^1,{\chi }_2^1:$${\chi }_3^1={\chi }_3^1({\chi }_1^1,{\chi }_2^1)$.

Equation (113) for identical vertices, $V{}^{\left(3\right)}=\tilde{V}{}_0^{\left(3\right)}=\dots =\tilde{V}{}_5^{\left(3\right)}$, along with the commutator of gauge transformations (115), determine the cubic interaction vertices for irreducible massive mixed-symmetric, $Y^{\left(i\right)}\left(2\right)\equiv Y(s_1^{\left(i\right)},s_2^{\left(i\right)})$, higher-spin fields in an AdS${}_d$ space. To solve the generating Equation (114), we should find BRST-closed forms, e.g., having, in the flat-space limit $r\to 0$, the following expressions, in a so-called minimal derivative scheme for different masses $m_i\ne m_j$, $i\ne j$,

$$\begin{array}{ccc}\hfill & & L_t^{\left(i\right)}=\widehat{p}{}_{\mu }^{\left(i\right)}{a}_t^{\left(i\right)\mu +}+\frac{m_{i+1}^2-m_{i+2}^2}{m_i}{b}_t^{\left(i\right)+}-ı\widehat{\mathcal{P}}{}_{0|t}^{\left(i\right)}{\eta }_t^{\left(i\right)+}+\mathcal{O}\left(r\right),i=1,2,3,t=1,2;\hfill \end{array}$$

(116)

$$\begin{array}{ccc}& & L_{st}^{(ii+1)+}=a_s^{\left(i\right)\mu +}{a}_{\mu |t}^{(i+1)+}-\frac{1}{2}{\mathcal{P}}_s^{\left(i\right)+}{\eta }_t^{(i+1)+}-\frac{1}{2}{\mathcal{P}}_t^{(i+1)+}{\eta }_s^{\left(i\right)+}+\frac{b_s^{\left(i\right)+}}{2m_i}{L}_t^{(i+1)}-\frac{b_t^{(i+1)+}}{2m_{i+1}}{L}_s^{\left(i\right)}+\mathcal{O}\left(r\right);\hfill \end{array}$$

(117)

$$\begin{array}{ccc}& & L_{st}^{\left(ii\right)+}=\frac{1}{2}{a}_s^{\left(i\right)\mu +}{a}_{\mu |t}^{\left(i\right)+}-(1/2)b_s^{\left(i\right)+}{b}_t^{\left(i\right)+}+b_{st}^{\left(i\right)+}+{\mathcal{P}}_s^{\left(i\right)+}{\eta }_t^{\left(i\right)+}+{\mathcal{P}}_t^{\left(i\right)+}{\eta }_s^{\left(i\right)+}+\mathcal{O}\left(r\right),\hfill \end{array}$$

(118)

for $s,t=1,2$, $\widehat{p}{}_{\mu }^{\left(i\right)}=p{}_{\mu }^{(i+1)}-p{}_{\mu }^{(i+2)}$, $p{}_{\mu }^{\left(i\right)}=-ıD_{\mu }^{\left(i\right)}$ and $\widehat{\mathcal{P}}{}_{0|t}^{\left(i\right)}=\mathcal{P}{}_{0|t}^{\left(i\right)}-\mathcal{P}{}_{0|t}^{(i+2)}$. The above operators are such as those suggested in the case of a Minkowski space-time for reducible mixed-symmetry integer spins [89], with an incomplete BRST operator and without the vanishing of non-dynamical algebraic (trace and mixed symmetry) constraints, as the operator is evaluated on the vertices (see [96] for details), being, at the same time, augmented by the trace and mixed-symmetry operators, according to our presentation [96] of cubic vertices for irreducible totally symmetric higher-spin fields in ${\mathbb{R}}^{1,d-1}$ with a complete BRST operator.

A solution for the cubic vertex (114) satisfying the generating Equation (113) requires extending the operators (116)–(118) by means of mixed-symmetric oscillators, and adding some new BRST-closed trace and mixed-symmetric operators, which poses a separate problem outside the scope of the present article.

6. Examples

Here, we apply the general prescriptions of our Lagrangian formulation to the case of free mixed-symmetry bosonic fields at the lowest values of rows and spins.

6.1. Spin-$(s_1,0)$ Totally Symmetric Tensor Field

Let us consider a totally symmetric field ${\Phi }_{{\mu }^1\left(s_1\right)}$ corresponding to spin $(s_1,0)$, i.e., to the Young tableaux $Y\left(s_1\right)$. We suppose that our result should reduce to the one examined in [21] for totally symmetric massive bosonic fields in an AdS space. According to our procedure, we have $n_1=s_1$, $n_2=0$. It can be shown that in the case $n_2=0$ (95), all the components related to the second row in the Young tableaux are equal to zero, namely,

$$\begin{array}{c}\hfill n_{a2}=n_{b2}=n_{b22}=n_{b21}=n_b=n_{f2}=n_{p2}=n_{f22}=n_{p22}=n_{f12}=n_{p12}=n_{fg12}=n_{pg12}=0.\end{array}$$

(119)

Therefore, the state vector reduces to

$$\begin{array}{ccc}\hfill |\chi \rangle & =& \sum _n(a_1^{{\mu }_1+}\cdots a_n^{{\mu }_n+}){\left(b_1^{+}\right)}^{n_{b1}}{\left(b_{11}^{+}\right)}^{n_{b11}}{\left({\eta }_0^{+}\right)}^{n_{f0}}{\left({\eta }_1^{+}\right)}^{n_{f1}}{\left({\mathcal{P}}_1^{+}\right)}^{n_{p1}}{\left({\eta }_{11}^{+}\right)}^{n_{f11}}{\left({\mathcal{P}}_{11}^{+}\right)}^{n_{p11}}\hfill \\ & & \times \chi {\left(x\right)}_{{\mu }_1\cdots {\mu }_{n_{a1}}0n_{p1}0n_{p11}000}^{n_{b_1}0n_{b11}000n_{f0}{n}_{f1}0n_{f11}000}|0\rangle ,\hfill \end{array}$$

(120)

which corresponds to the one of [21], with the Weyl ordering in the right-hand sides of non-linear supercommutators for the HS symmetry algebra ${\mathcal{A}}_c(Y\left(1\right),Ad{S}_d)$. Then the BRST and spin operators (supercommuting with each other) are reduced to $\tilde{Q}$ and $\tilde{\sigma }$:

$$\begin{array}{ccc}\hfill \tilde{Q}& =& \frac{1}{2}{\eta }_0{L}_0+{\sum }_i{\eta }_i^{+}{L}^i+{\eta }_{11}^{+}\left(L_{11}+{\eta }_1{\mathcal{P}}_1\right)+\frac{ı}{2}{\eta }_1^{+}{\eta }_1{\mathcal{P}}_0\hfill \\ & & +r\left\{{\eta }_0{\eta }_1^{+}\left[2{\mathcal{L}}_{11}{\mathcal{P}}_1^{+}+2{\mathcal{L}}_1^{+}{\mathcal{P}}_{11}+{\mathcal{G}}_0^1{\mathcal{P}}_1\right]+2{\eta }_1^{+}{\eta }_1{\mathcal{L}}_{11}^{+}{\mathcal{P}}_{11}\right\}+h.c.;\hfill \\ \hfill \tilde{\sigma }& =& G_0^1-h^1-{\eta }_1{\mathcal{P}}_1^{+}+{\eta }_1^{+}{\mathcal{P}}_1+2({\eta }_{11}^{+}{\mathcal{P}}_{11}-{\eta }_{11}{\mathcal{P}}_{11}^{+}).\hfill \end{array}$$

(121)

The corresponding Lagrangian formulation for a free massive field of higher integer spin is given by the action

$$\begin{array}{c}\hfill {\mathcal{S}}_{0|s_1}^m[\varphi ,{\varphi }_1,\dots ]={\mathcal{S}}_{0|s_1}^m{[|\chi \rangle }_{s_1}]=\int d{\eta }_0{}_{s_1}\langle \chi |K\tilde{Q}{|\chi \rangle }_{s_1},\end{array}$$

(122)

which is invariant under the reducible gauge transformations

$$\begin{array}{c}\hfill {\delta |\chi \rangle }_{s_1}=\tilde{Q}|{\chi }^1{\rangle }_{s_1},\delta |{\chi }^1{\rangle }_{s_1}=\tilde{Q}|{\chi }^2{\rangle }_{s_1},\delta {|{\chi }^2\rangle }_{s_1}=0,\end{array}$$

(123)

and thus determines, at most, a first-stage reducible abelian gauge theory.

A cubic vertex for three copies of massive interacting totally symmetric fields $(m_i;s_i)$, for $i=1,2,3$, being determined as a solution of the generating equations

$$\tilde{Q}{}^{tot}|V{}^{\left(3\right)}{{\rangle }_{\left(s\right){}^3}^{{\left(m\right)}_3}=0,\tilde{\sigma }{}^{\left(i\right)}|V{}^{\left(3\right)}\rangle }_{\left(s\right){}^3}^{{\left(m\right)}_3}=0,$$

(124)

for $\tilde{Q}{}^{tot}={\sum }_i\tilde{Q}{}^{\left(i\right)}$, may have various structures, due to various relations among the masses, and thus poses a separate problem. For non-coincident masses, however, the vertex may have a structure generated by the monomials (116)–(118), for $t=s=1$.

We emphasize that for massless totally symmetric fields of integer helicities ${\lambda }^{\left(i\right)}$, the corresponding BRST and spin operators are simplified to the form presented in [37] so that, in addition to the above restrictions (119), there are no oscillators $b^{\left(i\right)},b^{\left(i\right)+}$, and thus the additional parts $o_I^{\prime }$ become finite polynomials. The cubic vertices—for three massless fields, for one massless and two massive totally symmetric HS fields, as well as for two massless and one massive totally symmetric HS fields in an AdS space-time—can be found according to our approach [38,94,96].

6.2. Spin-$(1,1)$ Antisymmetric Tensor Field

Our next example is the simplest mixed-symmetry case, namely, a spin-$(1,1)$ rank-2 totally antisymmetric tensor field. Below, we denote all the gauge parameters by primed characters, and all of the gauge parameters of the first level for the gauge transformations, by double-primed characters. The values of the parameters $h_i$ are equal to $h_i=-\left(\frac{d+1-4i}{2}\right)$.

6.2.1. Action

Let us decompose the state vector (95), having a zero ghost number14 and obeying (100) at $n_1=n_2=1$, in the ghost oscillators, as well as in the auxiliary creation operators $b_{11}^{+}$, $b_{22}^{+}$, $b_{12}^{+}$, $d_{12}^{+}$,

$$\begin{array}{ccc}\hfill |{\chi }^0{\rangle }_{1,1}& =& |\tilde{\Phi }{\rangle }_{1,1}+{\mathcal{P}}_1^{+}{\eta }_1^{+}|{\tilde{\varphi }}_3{\rangle }_{-1,1}+{\mathcal{P}}_1^{+}{\eta }_2^{+}|{\varphi }_4{\rangle }_{0,0}+{\mathcal{P}}_2^{+}{\eta }_1^{+}{|{\varphi }_5\rangle }_{0,0}\hfill \\ & & +{\vartheta }_{12}^{+}{\mathcal{P}}_1^{+}|A_1{\rangle }_{1,0}+{\vartheta }_{12}^{+}{\mathcal{P}}_{11}^{+}|{\varphi }_6{\rangle }_{0,0}+{\lambda }_{12}^{+}{\eta }_1^{+}|A_2{\rangle }_{1,0}+{\lambda }_{12}^{+}{\eta }_{11}^{+}{|{\varphi }_7\rangle }_{0,0}\hfill \\ & & +{\eta }_0({\mathcal{P}}_1^{+}|\tilde{H}{\rangle }_{0,1}+{\mathcal{P}}_2^{+}|A_4{\rangle }_{1,0}+{\mathcal{P}}_{11}^{+}|{\tilde{\varphi }}_8{\rangle }_{-1,1}+{\mathcal{P}}_{12}^{+}{|{\varphi }_{10}\rangle }_{0,0}\hfill \\ & & +{\lambda }_{12}^{+}|{\tilde{T}}_2{\rangle }_{2,0}+{\lambda }_{12}^{+}{\mathcal{P}}_1^{+}{\eta }_1^{+}{|{\varphi }_{11}\rangle }_{0,0}),\hfill \end{array}$$

(125)

where the states $|\dots \rangle$ in the right-hand sides depend on $a_{i,\mu }^{+}$, $b_i^{+}$, $d_{12}^{+}$, $b_{1i}^{+}$, and are expanded according to

$$\begin{array}{ccc}\hfill |\tilde{\Phi }{\rangle }_{1,1}& =& {|\Phi \rangle }_{1,1}+d_{12}^{+}|T_1{\rangle }_{2,0}+b_{12}^{+}|{\varphi }_1{\rangle }_{0,0}+b_{11}^{+}{d}_{12}^{+}{|{\varphi }_2\rangle }_{0,0},\hfill \end{array}$$

(126)

$$\begin{array}{ccc}\hfill |\tilde{H}{\rangle }_{0,1}& =& {|H\rangle }_{0,1}+d_{12}^{+}|A_3{\rangle }_{1,0},|{\tilde{T}}_2{\rangle }_{2,0}=|T_2{\rangle }_{2,0}+b_{11}^{+}{|{\varphi }_9\rangle }_{0,0},\hfill \end{array}$$

(127)

$$\begin{array}{ccc}\hfill {|\Phi \rangle }_{1,1}& =& \left(-a_1^{+\mu }{a}_2^{+\nu }{\Phi }_{\mu \nu }\left(x\right)-i{b}_1^{+}{a}_2^{+\mu }{H}_{\left(2\right)\mu }\left(x\right)-i{b}_2^{+}{a}_1^{+\mu }{A}_{\left(7\right)\mu }\left(x\right)+b_1^{+}{b}_2^{+}{\varphi }_{\left(19\right)}\left(x\right)\right)|0\rangle ,\hfill \end{array}$$

(128)

$$\begin{array}{ccc}\hfill |T_j{\rangle }_{2,0}& =& \left(-a_1^{+\mu }{a}_1^{+\nu }{T}_{\left(j\right)\mu \nu }\left(x\right)-i{b}_1^{+}{a}_1^{+\mu }{A}_{(4+j)\mu }\left(x\right)+{\left(b_1^{+}\right)}^2{\varphi }_{(16+j)}\left(x\right)\right)|0\rangle ,\hfill \end{array}$$

(129)

$$\begin{array}{ccc}\hfill |\tilde{H}{\rangle }_{0,1}& =& \left(-i{a}_2^{+\mu }{H}_{\mu }\left(x\right)+b_2^{+}{\varphi }_{\left(16\right)}\left(x\right)\right)|0\rangle ,\hfill \end{array}$$

(130)

$$\begin{array}{ccc}\hfill |A_j{\rangle }_{1,0}& =& \left(-i{a}_1^{+\mu }|0\rangle A_{\left(j\right)\mu }\left(x\right)+b_1^{+}{\varphi }_{(11+j)}\left(x\right)\right)|0\rangle ,j=1,2,3,4,\hfill \end{array}$$

(131)

$$\begin{array}{ccc}\hfill |{\tilde{\varphi }}_k{\rangle }_{-1,1}& =& d_{12}^{+}{|{\varphi }_k\rangle }_{0,0},k=3,8,\hfill \end{array}$$

(132)

$$\begin{array}{ccc}\hfill |{\varphi }_i{\rangle }_{0,0}& =& {\varphi }_{\left(i\right)}\left(x\right)|0\rangle .\hfill \end{array}$$

(133)

Then the relation (102) gives the Lagrangian action in a ghost-independent form

$$\begin{array}{ccc}& & {\mathcal{S}}_{(1,1)}^m\left[|{\chi }^0{\rangle }_{1,1}\right]={}_{1,1}\langle \tilde{\Phi }|K_{1.1}\{\frac{1}{2}{L}_0|\tilde{\Phi }{\rangle }_{1,1}-L_1^{+}|\tilde{H}{\rangle }_{0,1}-L_2^{+}|A_4{\rangle }_{1,0}-L_{11}^{+}|{\tilde{\varphi }}_8{\rangle }_{-1,1}-L_{12}^{+}{|{\varphi }_{10}\rangle }_{0,0}\hfill \\ & & \phantom{{\mathcal{S}}_{(1,1)}^m}-T_{12}^{+}|{\tilde{T}}_2{\rangle }_{2,0}+2r[(l_{11}^{+}-l_{11}^{\prime +})|{\tilde{\varphi }}_3{\rangle }_{-1,1}+(l_{12}^{+}-l_{12}^{\prime +}){\left(\right|{\varphi }_4\rangle }_{0,0}+|{\varphi }_5{\rangle }_{0,0})\hfill \\ & & \phantom{{\mathcal{S}}_{(1,1)}^m}+\frac{1}{2}(l_2^{+}-l_2^{\prime +})|A_1{\rangle }_{1,0}\left]\right\}+{}_{-1,1}\langle {\tilde{\varphi }}_3|K_{1.1}\{\frac{1}{2}(L_0-2r[g_0^1-g_0^{\prime 1}])|{\tilde{\varphi }}_3{{\rangle }_{-1,1}-L_1\tilde{H}\rangle }_{0,1}\hfill \\ & & \phantom{{\mathcal{S}}_{(1,1)}^m}+T_{12}^{+}|{\varphi }_{11}{\rangle }_{0,0}-|{\tilde{\varphi }}_8{\rangle }_{-1,1}+r(t_{12}^{+}-t_{12}^{\prime +}){\left(\right|{\varphi }_4\rangle }_{0,0}+|{\varphi }_5{\rangle }_{0,0}\left)\right\}+{}_{0,0}\langle {\varphi }_4|K_{1.1}\{(L_0-r\sum _i(g_0^i\hfill \\ & & \phantom{{\mathcal{S}}_{(1,1)}^m}-g_0^{\prime i}\left)\right)|{\varphi }_5{\rangle }_{0,0}-L_1|A_4{\rangle }_{1,0}-|{\varphi }_{11}{\rangle }_{0,0}+\frac{1}{2}|{\varphi }_{10}{\rangle }_{0,0}\}+{}_{0,0}\langle {\varphi }_5|K_{1.1}\{-L_2{|\tilde{H}\rangle }_{0,1}\hfill \\ & & \phantom{{\mathcal{S}}_{(1,1)}^m}-r(l_1-l_1^{\prime })|A_1{\rangle }_{1,0}+\frac{1}{2}|{\varphi }_{10}{\rangle }_{0,0}-|{\varphi }_{11}{\rangle }_{0,0}\}+{}_{1,0}\langle A_1|K_{1.1}\{(L_0+r[g_0^1-g_0^{\prime 1}])|A_2{\rangle }_{1,0}+2r(l_1^{+}\hfill \\ & & \phantom{{\mathcal{S}}_{(1,1)}^m}-l_1^{\prime +}\left)\right|{\varphi }_7{\rangle }_{0,0}+L_1|\tilde{T}{\rangle }_{2,0}+L_1^{+}|{\varphi }_{11}{\rangle }_{0,0}+\frac{r}{2}(t_{12}-t_{12}^{\prime })|\tilde{H}{\rangle }_{0,1}-\frac{r}{2}\sum _m[g_0^m-g_0^{\prime m}]\left)\right|A_4{\rangle }_{1,0}\}\hfill \\ & & \phantom{{\mathcal{S}}_{(1,1)}^m}+{}_{0,0}\langle {\varphi }_6|K_{1.1}\left\{L_0|{\varphi }_7{\rangle }_{0,0}+L_{11}^{+}|\tilde{T}{\rangle }_{2,0}-|{\varphi }_{11}{\rangle }_{0,0}-\frac{1}{2}{|{\varphi }_{10}\rangle }_{0,0}\right\}\hfill \\ & & \phantom{{\mathcal{S}}_{(1,1)}^m}-{}_{1,0}\langle A_2|K_{1.1}{T}_{12}|\tilde{H}{\rangle }_{0,1}-{}_{0,0}\langle {\varphi }_7|K_{1.1}{|{\varphi }_{10}\rangle }_{0,0}\}\hfill \\ & & \phantom{{\mathcal{S}}_{(1,1)}^m}-\frac{1}{2}\left({}_{0,1}\langle \tilde{H}|K_{1.1}|\tilde{H}{\rangle }_{0,1}+{}_{1,0}\langle A_4|K_{1.1}{|A_4\rangle }_{1,0}\right)+h.c.,\hfill \end{array}$$

(134)

and a tensor form (for the leading field term),

$$\begin{array}{ccc}\hfill {\mathcal{S}}_{(1,1)}^m& =& \int d^{d}x\sqrt{\left|g\right|}\{{\Phi }_{\mu \nu }({\nabla }^2-\frac{d(d-6)}{4}+m_0^2){\Phi }^{\mu \nu }-4T_{\left(2\right)\left(\mu \nu \right)}{\Phi }^{\mu \nu }\hfill \\ & & +2A_{\left(4\right)\mu }{\nabla }_{\nu }{\Phi }^{\mu \nu }+2H_{\nu }{\nabla }_{\mu }{\Phi }^{\mu \nu }+{\varphi }_{\left(10\right)}{\Phi }_{\mu }^{\mu }+\mathrm{more}\}.\hfill \end{array}$$

(135)

Here, the mass parameter $m_0^2$ (104) equals to

$$m_0^2=m^2+r\left\{4+\frac{d(d-10)}{2}\right\}.$$

(136)

Let us now consider the set of gauge transformations.

6.2.2. Zero-Level Gauge Transformations

We decompose the vector (95), satisfying (100), for $n_1=n_2=1$, and having the ghost number $-1$, as follows:

$$\begin{array}{ccc}\hfill |{\chi }^1{\rangle }_{1,1}& =& {\mathcal{P}}_1^{+}\left(|H^{\prime }{\rangle }_{0,1}+d_{12}^{+}{|A_1^{\prime }\rangle }_{1,0}\right)+{\mathcal{P}}_2^{+}|A_2^{\prime }{\rangle }_{1,0}+{\mathcal{P}}_{11}^{+}{d}_{12}^{+}|{\varphi }_1^{\prime }{\rangle }_{0,0}+{\mathcal{P}}_{12}^{+}{|{\varphi }_2^{\prime }\rangle }_{0,0}\hfill \\ & & +{\lambda }_{12}^{+}\left(|T_1^{\prime }{\rangle }_{2,0}+b_{11}^{+}{|{\varphi }_3^{\prime }\rangle }_{0,0}\right)+{\lambda }_{12}^{+}{\mathcal{P}}_1^{+}{\eta }_1^{+}{|{\varphi }_4^{\prime }\rangle }_{0,0}\hfill \\ & & +{\eta }_0\left({\mathcal{P}}_1^{+}{\mathcal{P}}_2^{+}|{\varphi }_5^{\prime }{\rangle }_{0,0}+{\mathcal{P}}_1^{+}{\lambda }_{12}^{+}|A_3^{\prime }{\rangle }_{1,0}+{\mathcal{P}}_{11}^{+}{\lambda }_{12}^{+}{|{\varphi }_6^{\prime }\rangle }_{0,0}\right),\hfill \end{array}$$

(137)

where

$$\begin{array}{ccc}\hfill |T^{\prime }{\rangle }_{2,0}& =& \left(-a_1^{+\mu }{a}_1^{+\nu }{T}_{\mu \nu }^{\prime }\left(x\right)-i{b}_1^{+}{a}_1^{+\mu }{A}_{\left(4\right)\mu }^{\prime }\left(x\right)+{\left(b_1^{+}\right)}^2{\varphi }_{\left(11\right)}^{\prime }\left(x\right)\right)|0\rangle ,\hfill \end{array}$$

(138)

$$\begin{array}{ccc}\hfill |H^{\prime }{\rangle }_{0,1}& =& \left(-i{a}_2^{+\mu }{H}_{\mu }^{\prime }\left(x\right)+b_2^{+}{\varphi }_{\left(10\right)}^{\prime }\left(x\right)\right)|0\rangle ,\hfill \end{array}$$

(139)

$$\begin{array}{ccc}\hfill |A_j^{\prime }{\rangle }_{1,0}& =& \left(-i{a}_1^{+\mu }{A}_{\left(j\right)\mu }^{\prime }\left(x\right)+b_1^{+}{\varphi }_{(6+j)}^{\prime }\left(x\right)\right)|0\rangle j=1,2,3,\hfill \end{array}$$

(140)

$$\begin{array}{ccc}\hfill |{\varphi }_i^{\prime }{\rangle }_{0,0}& =& {\varphi }_{\left(i\right)}^{\prime }\left(x\right)|0\rangle .\hfill \end{array}$$

(141)

Substituting (125), (137) into the relations (98) on the left, we obtain the following gauge transformations for the fields:

$$\begin{array}{ccc}\hfill \delta {\Phi }_{\mu \nu }& =& -2T_{\mu \nu }^{\prime }+{\nabla }_{\mu }{H}_{\nu }^{\prime }+{\nabla }_{\nu }{A}_{\left(2\right)\mu }^{\prime }-\frac{{\eta }_{\mu \nu }}{2}{{\varphi }^{\prime }}_{\left(2\right)},\hfill \end{array}$$

(142)

$$\begin{array}{ccc}\hfill \delta T_{\left(1\right)\mu \nu }& =& T_{\mu \nu }^{\prime }+{\nabla }_{\{\mu }{A}_{\left(1\right)\nu \}}^{\prime }-\frac{{\eta }_{\mu \nu }}{2}{{\varphi }^{\prime }}_{\left(1\right)},\hfill \end{array}$$

(143)

$$\begin{array}{ccc}\hfill \delta T_{\left(2\right)\mu \nu }& =& \left({\nabla }^2-r\frac{d(d-6)}{4}+m_0^2\right)T_{\mu \nu }^{\prime }-{\nabla }_{\{\mu }{A}_{\left(3\right)\nu \}}^{\prime }\hfill \\ & & +\frac{{\eta }_{\mu \nu }}{2}\left({{\varphi }^{\prime }}_{\left(6\right)}+2r{{\varphi }^{\prime }}_{\left(4\right)}+\frac{r}{2}{{\varphi }^{\prime }}_{\left(5\right)}\right)-r{\nabla }_{\{\mu }{A}_{\left(2\right)\nu \}}^{\prime },\hfill \end{array}$$

(144)

$$\begin{array}{ccc}\hfill \delta A_{\left(1\right)\mu }& =& -2A_{\left(1\right)\mu }^{\prime }-A_{\left(2\right)\mu }^{\prime }-H_{\mu }^{\prime },\hfill \end{array}$$

(145)

$$\begin{array}{ccc}\hfill \delta A_{\left(2\right)\mu }& =& -2{\nabla }^{\nu }{T}_{\mu \nu }^{\prime }-\frac{1}{m_1}\left(m_0^2-r\left[\frac{d^2}{4}-\frac{17}{4}\right]\right)A_{\left(4\right)\mu }^{\prime }-{\nabla }_{\mu }{{\varphi }^{\prime }}_{\left(4\right)}+A_{\left(3\right)\mu }^{\prime }\hfill \\ & & +\frac{r}{2}\left(H_{\mu }^{\prime }-2A_{\left(1\right)\mu }^{\prime }+A_{\left(2\right)\mu }^{\prime }(1+2d)\right),\hfill \end{array}$$

(146)

$$\begin{array}{ccc}\hfill \delta A_{\left(3\right)\mu }& =& \left({\nabla }^2-r\frac{(d+4)(d-6)}{4}+r(d-\frac{3}{2})+m_0^2\right)A_{\left(1\right)\mu }^{\prime }+A_{\left(3\right)\mu }^{\prime }+r{A}_{\left(2\right)\mu }^{\prime },\hfill \end{array}$$

(147)

$$\begin{array}{ccc}\hfill \delta A_{\left(4\right)\mu }& =& -{\nabla }_{\mu }{{\varphi }^{\prime }}_{\left(5\right)}+\left({\nabla }^2-r\frac{d(d-6)}{4}+m_0^2+r(d-\frac{7}{2})\right)A_{\left(2\right)\mu }^{\prime }-A_{\left(3\right)\mu }^{\prime }+r(H_{\mu }^{\prime }-2A_{\left(1\right)\mu }^{\prime }),\hfill \end{array}$$

(148)

$$\begin{array}{ccc}\hfill \delta A_{\left(5\right)\mu }& =& m_1{A}_{\left(1\right)\mu }^{\prime }+{\nabla }_{\mu }{{\varphi }^{\prime }}_{\left(7\right)}+A_{\left(4\right)\mu }^{\prime },\hfill \end{array}$$

(149)

$$\begin{array}{ccc}\hfill \delta A_{\left(6\right)\mu }& =& -m{A}_{\left(3\right)\mu }^{\prime }-{\nabla }_{\mu }{{\varphi }^{\prime }}_{\left(9\right)}+\left({\nabla }^2-r\frac{d(d-6)}{4}+m_0^2\right)A_{\left(4\right)\mu }^{\prime }\hfill \\ & & -r({\nabla }_{\mu }{{\varphi }^{\prime }}_{\left(8\right)}-A_{\left(2\right)\mu }^{\prime }+(3-d)A_{\left(4\right)\mu }^{\prime }),\hfill \end{array}$$

(150)

$$\begin{array}{ccc}\hfill \delta A_{\left(7\right)\mu }& =& m_2{A}_{\left(2\right)\mu }^{\prime }+{\nabla }_{\mu }{{\varphi }^{\prime }}_{\left(10\right)}-\frac{m_2}{m_1}{A}_{\left(4\right)\mu }^{\prime },\hfill \end{array}$$

(151)

$$\begin{array}{ccc}\hfill \delta H_{\mu }& =& {\nabla }_{\mu }{{\varphi }^{\prime }}_{\left(5\right)}+\left({\nabla }^2-r\frac{d(d-6)}{4}+m_0^2+r(d-\frac{3}{2})\right)H_{\mu }^{\prime }-A_{\left(3\right)\mu }^{\prime }+r{A}_{\left(2\right)\mu }^{\prime },\hfill \end{array}$$

(152)

$$\begin{array}{ccc}\hfill \delta H_{\left(2\right)\mu }& =& m_1{H}_{\mu }^{\prime }+{\nabla }_{\mu }{{\varphi }^{\prime }}_{\left(8\right)}-A_{\left(4\right)\mu }^{\prime },\hfill \end{array}$$

(153)

$$\begin{array}{ccc}\hfill \delta {\varphi }_{\left(1\right)}& =& {{\varphi }^{\prime }}_{\left(2\right)}-2{{\varphi }^{\prime }}_{\left(3\right)},\hfill \end{array}$$

(154)

$$\begin{array}{ccc}\hfill \delta {\varphi }_{\left(2\right)}& =& {{\varphi }^{\prime }}_{\left(1\right)}+{{\varphi }^{\prime }}_{\left(3\right)},\hfill \end{array}$$

(155)

$$\begin{array}{ccc}\hfill \delta {\varphi }_{\left(3\right)}& =& -{\nabla }^{\mu }{A}_{\left(1\right)\mu }^{\prime }-\frac{1}{m_1}\left(m_0^2-r\left[\frac{d(d+4)}{4}-\frac{29}{4}\right]\right){{\varphi }^{\prime }}_{\left(7\right)}+{{\varphi }^{\prime }}_{\left(1\right)}+{{\varphi }^{\prime }}_{\left(4\right)},\hfill \end{array}$$

(156)

$$\begin{array}{ccc}\hfill \delta {\varphi }_{\left(4\right)}& =& -{\nabla }^{\mu }{H}_{\mu }^{\prime }+\frac{1}{m_2}\left(m_0^2-r\left[\frac{d(d-8}{4}+\frac{14}{4}\right]\right){{\varphi }^{\prime }}_{\left(10\right)}+2r(3-d){{\varphi }^{\prime }}_{\left(7\right)},\hfill \\ & & +\frac{1}{2}{{\varphi }^{\prime }}_{\left(2\right)}-{{\varphi }^{\prime }}_{\left(4\right)}-{{\varphi }^{\prime }}_{\left(5\right)},\hfill \end{array}$$

(157)

$$\begin{array}{ccc}\hfill \delta {\varphi }_{\left(5\right)}& =& -{\nabla }^{\mu }{A}_{\left(2\right)\mu }^{\prime }-\frac{1}{m_1}\left(m_0^2-r\left[\frac{d^2}{4}-5\right]\right){{\varphi }^{\prime }}_{\left(8\right)}+\frac{1}{2}{{\varphi }^{\prime }}_{\left(2\right)}-{{\varphi }^{\prime }}_{\left(4\right)}+{{\varphi }^{\prime }}_{\left(5\right)},\hfill \end{array}$$

(158)

$$\begin{array}{ccc}\hfill \delta {\varphi }_{\left(6\right)}& =& -2{{\varphi }^{\prime }}_{\left(1\right)}-{{\varphi }^{\prime }}_{\left(2\right)},\hfill \end{array}$$

(159)

$$\begin{array}{ccc}\hfill \delta {\varphi }_{\left(7\right)}& =& \frac{d-3}{2}{\varphi }_{\left(3\right)}^{\prime }+T_{\mu }^{\prime }{}^{\mu }+{\varphi }_{\left(4\right)}^{\prime }-\frac{1}{m_1^2}\left(m_0^2-r\left[\frac{d^2-17}{4}\right]\right){\varphi }_{\left(11\right)}^{\prime }+\frac{1}{2}{\varphi }_{\left(2\right)}^{\prime },\hfill \end{array}$$

(160)

$$\begin{array}{ccc}\hfill \delta {\varphi }_{\left(8\right)}& =& \left({\nabla }^2-r\frac{d(d-6)}{4}+m_0^2\right){{\varphi }^{\prime }}_{\left(1\right)}+{{\varphi }^{\prime }}_{\left(6\right)}+r{{\varphi }^{\prime }}_{\left(5\right)}-2r{\nabla }^{\mu }{A}_{\left(1\right)\mu }^{\prime }+\frac{2r^2}{m_2}(3-d){{\varphi }^{\prime }}_{\left(10\right)}\hfill \end{array}$$

(161)

$$\begin{array}{ccc}\hfill \phantom{\delta {\varphi }_{\left(8\right)}}& & -2r\left(m_0^2-r\left[\frac{d(d+4)-29}{4}\right])\right){{\varphi }^{\prime }}_{\left(7\right)},\hfill \end{array}$$

(162)

$$\begin{array}{ccc}\hfill \delta {\varphi }_{\left(9\right)}& =& \left({\nabla }^2-r\frac{d(d-6)}{4}+m_0^2\right){{\varphi }^{\prime }}_{\left(3\right)}-{{\varphi }^{\prime }}_{\left(6\right)},-2r({{\varphi }^{\prime }}_{\left(4\right)}\hfill \\ & & +r({{\varphi }^{\prime }}_{\left(8\right)}+\frac{r}{2}{{\varphi }^{\prime }}_{\left(5\right)}-2(4-d){{\varphi }^{\prime }}_{\left(11\right)-}),\hfill \end{array}$$

(163)

$$\begin{array}{ccc}\hfill \delta {\varphi }_{\left(10\right)}& =& \left({\nabla }^2-r\frac{d(d-6)}{4}+m_0^2\right){{\varphi }^{\prime }}_{\left(2\right)}-2{{\varphi }^{\prime }}_{\left(6\right)}+2r{{\varphi }^{\prime }}_{\left(5\right)}+2r\{{\nabla }^{\mu }{A}_{\left(2\right)\mu }^{\prime }-\frac{2r}{m_1}(3-d){{\varphi }^{\prime }}_{\left(7\right)}\hfill \\ \hfill \phantom{\delta {\varphi }_{\left(8\right)}}& & -\frac{1}{m_1}\left(m_0^2-r\left[\frac{d^2-17}{4}\right]\right){{\varphi }^{\prime }}_{\left(8\right)}+{\nabla }^{\mu }{H}_{\mu }^{\prime }+\left(m_0^2-r\left[\frac{d(d+8)+7}{4}\right]\right){{\varphi }^{\prime }}_{\left(10\right)}\},\hfill \end{array}$$

(164)

$$\begin{array}{ccc}\hfill \delta {\varphi }_{\left(11\right)}& =& {\nabla }^{\mu }{A}_{\left(3\right)\mu }^{\prime }+\frac{1}{m_1}\left(m_0^2-r\left[\frac{d^2-17}{4}\right]\right){{\varphi }^{\prime }}_{\left(9\right)}+\left({\nabla }^2-r\frac{d(d-6)}{4}+m_0^2\right){{\varphi }^{\prime }}_{\left(4\right)}-{{\varphi }^{\prime }}_{\left(6\right)}\hfill \\ \hfill \phantom{\delta {\varphi }_{\left(8\right)}}& & +r(2d-3){{\varphi }^{\prime }}_{\left(4\right)}+r\left(d-\frac{5}{2}\right){{\varphi }^{\prime }}_{\left(5\right)}+2r\left(T^{\prime \mu }{}_{\mu }+\frac{1}{m_1^2}\left(m_0^2-r\left[\frac{d^2-17}{4}\right]\right){{\varphi }^{\prime }}_{\left(11\right)}\right.\hfill \\ & & \left.+\frac{3-d}{2}{{\varphi }^{\prime }}_{\left(3\right)}+\frac{1}{2}{\nabla }^{\mu }{H}_{}^{\prime }\mu +\frac{1}{2m_2}\left(m_0^2-r\left[\frac{d(d-8)+7}{4}\right]\right){\varphi }_{\left(10\right)}^{\prime }+\frac{r}{m_1}(3-d){\varphi }_{\left(7\right)}^{\prime }\right),\hfill \end{array}$$

(165)

$$\begin{array}{ccc}\hfill \delta {\varphi }_{\left(12\right)}& =& -2{\varphi }_{\left(7\right)}^{\prime }-{\varphi }_{\left(8\right)}^{\prime }-\frac{m_1}{m_2}{\varphi }_{\left(10\right)}^{\prime },\hfill \end{array}$$

(166)

$$\begin{array}{ccc}\hfill \delta {\varphi }_{\left(13\right)}& =& -{\nabla }^{\mu }{A}_{\left(4\right)\mu }^{\prime }+m_1({\varphi }_{\left(3\right)}^{\prime }-{\varphi }_{\left(4\right)}^{\prime })+\frac{2}{m_1}\left(m_0^2-r\left[\frac{d(d-4)}{4}-\frac{13}{4}\right]\right){\varphi }_{\left(11\right)}^{\prime }+{\varphi }_{\left(9\right)}^{\prime }\hfill \\ & & +\frac{r}{2}\left(-\frac{m_2}{m_1}{\varphi }_{\left(10\right)}^{\prime }+2{\varphi }_{\left(7\right)}^{\prime }+{\varphi }_{\left(8\right)}^{\prime }(2d-1)\right),\hfill \end{array}$$

(167)

$$\begin{array}{ccc}\hfill \delta {\varphi }_{\left(14\right)}& =& \left({\nabla }^2-r\frac{d(d-10)-8}{4}+r(d-\frac{3}{2})+m_0^2\right){\varphi }_{\left(7\right)}^{\prime }+{\varphi }_{\left(9\right)}^{\prime }+r{\varphi }_{\left(7\right)}^{\prime }+2r\frac{m_1}{m_2}{\varphi }_{\left(10\right)}^{\prime },\hfill \end{array}$$

(168)

$$\begin{array}{ccc}\hfill \delta {\varphi }_{\left(15\right)}& =& \left({\nabla }^2-r\frac{d(d-6)}{4}+m_0^2+r(2d-\frac{9}{2})\right){\varphi }_{\left(8\right)}^{\prime }-{\varphi }_{\left(9\right)}^{\prime }-r\left(2{\varphi }_{\left(7\right)}^{\prime }+\frac{m_1}{m_2}{\varphi }_{\left(10\right)}^{\prime }\right),\hfill \end{array}$$

(169)

$$\begin{array}{ccc}\hfill \delta {\varphi }_{\left(16\right)}& =& m_2{{\varphi }^{\prime }}_{\left(5\right)}+\left({\nabla }^2-r\frac{d(d-6)}{4}+r(d-\frac{3}{2})+m_0^2\right){{\varphi }^{\prime }}_{\left(10\right)}-\frac{m_2}{m_1}({{\varphi }^{\prime }}_{\left(9\right)}\hfill \\ & & +r{{\varphi }^{\prime }}_{\left(8\right)})+r\frac{m_2}{2m_1}{{\varphi }^{\prime }}_{\left(7\right)},\hfill \end{array}$$

(170)

$$\begin{array}{ccc}\hfill \delta {\varphi }_{\left(17\right)}& =& m_1{\varphi }_{\left(7\right)}^{\prime }+{\varphi }_{\left(11\right)}^{\prime },\hfill \end{array}$$

(171)

$$\begin{array}{ccc}\hfill \delta {\varphi }_{\left(18\right)}& =& -m{\varphi }_{\left(9\right)}^{\prime }+\left({\nabla }^2-r\frac{d(d-6)}{4}+m_0^2\right){\varphi }_{\left(11\right)}^{\prime }+\frac{1}{2}{\varphi }_{\left(6\right)}^{\prime },\hfill \end{array}$$

(172)

$$\begin{array}{ccc}\hfill \delta {\varphi }_{\left(19\right)}& =& m_2{{\varphi }^{\prime }}_{\left(8\right)}+m_1{{\varphi }^{\prime }}_{\left(10\right)}-2\frac{m_2}{m_1}{{\varphi }^{\prime }}_{\left(11\right)}.\hfill \end{array}$$

(173)

Let us now turn to the gauge transformation for the gauge parameters.

6.2.3. First-Level Gauge Transformations

We decompose the vector (95), satisfying (100) at $n_1=n_2=1$ and having the ghost number $-2$, according to

$$\begin{array}{ccc}\hfill |{\chi }^2{\rangle }_{1,1}& =& {\mathcal{P}}_1^{+}{\mathcal{P}}_2^{+}|{\varphi }_1^{″}{\rangle }_{0,0}+{\mathcal{P}}_1^{+}{\lambda }_{12}^{+}|A^{″}{\rangle }_{1,0}+{\mathcal{P}}_{11}^{+}{\lambda }_{12}^{+}{|{\varphi }_2^{″}\rangle }_{0,0},\hfill \end{array}$$

(174)

where

$$\begin{array}{ccc}\hfill |A^{″}{\rangle }_{1,0}=\left(-i{a}_1^{+\mu }{A}_{\mu }^{″}\left(x\right)+b_1^{+}{\varphi }_{\left(3\right)}^{″}\left(x\right)\right)|0\rangle ,& & |{\varphi }_i^{″}{\rangle }_{0,0}={\varphi }_{\left(i\right)}^{″}\left(x\right)|0\rangle .\hfill \end{array}$$

(175)

Substituting (137) and (174) into the relations (99) on the right, we find the gauge transformations for the gauge parameters:

$$\begin{array}{cccc}\hfill & \delta T_{\mu \nu }^{\prime }={\nabla }_{\{\mu }{A}_{\nu \}}^{″}-\frac{1}{2}{\eta }_{\mu \nu }({\varphi }_{\left(2\right)}^{″}+r{\varphi }_{\left(1\right)}^{″}),\hfill & \hfill & \delta A_{\left(1\right)\mu }^{\prime }=-A_{\mu }^{″},\hfill \end{array}$$

(176)

$$\begin{array}{cccc}\hfill & \delta A_{\left(2\right)\mu }^{\prime }={\nabla }_{\mu }{\varphi }_{\left(1\right)}^{″}-A_{\mu }^{″},\hfill & \hfill & \delta A_{\left(3\right)\mu }^{\prime }=({\nabla }^2-r\frac{d(d-6)-34}{4}+m_0^2)A_{\mu }^{″},\hfill \end{array}$$

(177)

$$\begin{array}{cccc}\hfill & \delta A_{\left(4\right)\mu }^{\prime }=m_1{A}_{\mu }^{″}+{\nabla }_{\mu }{\varphi }_{\left(3\right)}^{″},\hfill & \hfill & \delta H_{\mu }^{\prime }=-{\nabla }_{\mu }{\varphi }_{\left(1\right)}^{″}+A_{\mu }^{″},\hfill \end{array}$$

(178)

$$\begin{array}{cccc}\hfill & \delta {\varphi }_{\left(1\right)}^{\prime }=-{\varphi }_{\left(2\right)}^{″}+r{\varphi }_{\left(1\right)}^{″},\hfill & \hfill & \delta {\varphi }_{\left(2\right)}^{\prime }=2{\varphi }_{\left(2\right)}^{″}-2r{\varphi }_{\left(1\right)}^{″},\hfill \end{array}$$

(179)

$$\begin{array}{cccc}\hfill & \delta {\varphi }_{\left(3\right)}^{\prime }={\varphi }_{\left(2\right)}^{″}-r{\varphi }_{\left(3\right)}^{″},\hfill & \hfill & \delta {\varphi }_{\left(5\right)}^{\prime }=({\nabla }^2-r\frac{d(d-6)-40}{4}+m_0^2){\varphi }_{\left(1\right)}^{″},\hfill \end{array}$$

(180)

$$\begin{array}{cccc}\hfill & \delta {\varphi }_{\left(4\right)}^{\prime }=-{\nabla }^{\mu }{A}_{\mu }^{″}-\frac{1}{m_1}\left(m_0^2-r\frac{d^2-14}{4}\right){\varphi }_{\left(3\right)}^{″},\hfill & \hfill & \delta {\varphi }_{\left(6\right)}^{\prime }=({\nabla }^2-r\frac{d(d-6)-20}{4}+m_0^2){\varphi }_{\left(2\right)}^{″}+2r{\nabla }^{\mu }{A}_{\mu }^{″}\hfill \\ \hfill & \phantom{\delta {\varphi }_{\left(5\right)}^{\prime }}+{\varphi }_{\left(2\right)}^{″}+\frac{5}{2}r{\varphi }_{\left(1\right)}^{″},\hfill & \hfill & \phantom{\delta {\varphi }_{\left(5\right)}^{\prime }=}-\frac{1}{m_1}\left(m_0^2-r\frac{d(d+2)-23}{2}\right){\varphi }_{\left(3\right)}^{″}-10r^2{\varphi }_{\left(1\right)}^{″},\hfill \end{array}$$

(181)

$$\begin{array}{cccc}\hfill & \delta {\varphi }_{\left(7\right)}^{\prime }=-{\varphi }_{\left(3\right)}^{″},\hfill & \hfill & \delta {\varphi }_{\left(8\right)}^{\prime }=m_1{\varphi }_{\left(1\right)}^{″}+{\varphi }_{\left(3\right)}^{″},\hfill \end{array}$$

(182)

$$\begin{array}{cccc}\hfill & \delta {\varphi }_{\left(9\right)}^{\prime }=({\nabla }^2-r\frac{d(d-6)-34}{4}+m_0^2){\varphi }_{\left(3\right)}^{″},\hfill & \hfill & \delta {\varphi }_{\left(10\right)}^{\prime }=-m_2{\varphi }_{\left(1\right)}^{″}-\frac{m_2}{m_1}{\varphi }_{\left(3\right)}^{″},\hfill \end{array}$$

(183)

$$\begin{array}{cc}\hfill & \delta {\varphi }_{\left(11\right)}^{\prime }=m_1{\varphi }_{\left(3\right)}^{″}-\frac{1}{2}{\varphi }_{\left(2\right)}^{″}.\hfill \end{array}$$

(184)

6.2.4. Gauge-Fixing and Partial Use of Equations of Motion

We now fix the gauge symmetry completely, using the gauge-fixing conditions (A129) obtained in our general consideration. It is easy to see that the following fields are gauged away:

$${\varphi }_{\left(1\right)},{\varphi }_{\left(2\right)},{\varphi }_{\left(9\right)},{\varphi }_{\left(13\right)},{\varphi }_{\left(15\right)},{\varphi }_{\left(17\right)},{\varphi }_{\left(18\right)},{\varphi }_{\left(19\right)},A_{\left(5\right)},A_{\left(6\right)},A_{\left(7\right)},H_{\left(2\right)},T_{\left(1\right)}⟶0.$$

(185)

Then, using the equations of motion for all the fields, except the antisymmetric part of the basic field ${\Phi }_{\left[\mu \nu \right]}$, we find that we are left only with ${\Phi }_{\left[\mu \nu \right]}$ so that the corresponding Lagrangian action (134), up to a total derivative of a certain quantity $f^{\mu }$, has the form

$$\begin{array}{ccc}\hfill {\mathcal{S}}^m\left[{\Phi }_{\left[\mu \nu \right]}\right]& =& \int d^{d}x\sqrt{\left|g\right|}\{{\Phi }_{\left[\mu \nu \right]}({\nabla }^2+m^2){\Phi }^{\left[\mu \nu \right]}+2\left({\nabla }_{\mu }{\Phi }^{\left[\mu \lambda \right]}\right)\left({\nabla }^{\nu }{\Phi }_{\left[\nu \lambda \right]}\right)\hfill \\ & =& \int d^{d}x\sqrt{\left|g\right|}\left\{-\frac{1}{3}{F}_{\mu \nu \lambda }{F}^{\mu \nu \lambda }+m^2{\Phi }_{\left[\mu \nu \right]}{\Phi }^{\left[\mu \nu \right]}+{\nabla }_{\mu }{f}^{\mu }\right\},\hfill \end{array}$$

(186)

where $F_{\mu \nu \lambda }$ stands for the field strength of ${\Phi }_{\left[\mu \nu \right]}$, namely,

$$\begin{array}{c}\hfill F_{\mu \nu \lambda }={\nabla }_{\lambda }{\Phi }_{\left[\mu \nu \right]}+{\nabla }_{\mu }{\Phi }_{\left[\nu \lambda \right]}+{\nabla }_{\nu }{\Phi }_{\left[\lambda \mu \right]}.\end{array}$$

(187)

As a result, we obtained the gauge-invariant action (134) for a massive rank-2 antisymmetric higher integer spin field interacting with an AdS background field $g_{\mu \nu }$, containing a complete set of auxiliary fields and gauge parameters, as well as the action of a non-gauge formulation (186). Note that an equivalent form of the Lagrangian formulation for the field ${\Phi }_{\left[\mu \nu \right]}$ is constructed in [62], using an antisymmetric basis of the initial oscillators.

7. Conclusions

In this paper, we obtained quadratic non-linear HS symmetry algebras for a description of arbitrary integer HS fields defined in d-dimensional AdS space and subject to a k-row Young tableaux $Y(s_1,\dots ,s_k)$. We showed that the difference of the resulting algebras $\mathcal{A}(Y\left(k\right),Ad{S}_d)$, ${\mathcal{A}}^{\prime }(Y\left(k\right),Ad{S}_d)$ and ${\mathcal{A}}_c(Y\left(k\right),Ad{S}_d)$, corresponding, respectively, to the initial set of operators, their additional parts, and the converted set of operators within an additive conversion procedure, is due to purely non-linear parts, related to the AdS radius ${\left(\sqrt{r}\right)}^{-1}$, in the set of AdS space isometry operators.

To obtain the above algebras, we start by embedding the bosonic HS fields into the vectors of an auxiliary Fock space, regarding the fields as coordinates of Fock space vectors, and reformulate the theory in these terms. We realize the conditions that determine an irreducible AdS group representation of a given mass and generalized spin in terms of differential operator constraints imposed on the Fock space vectors. These constraints generate a closed non-linear algebra of HS symmetry, which contains, with the exception of k basis generators in the Cartan subalgebra, a system of first- and second-class constraints. The above algebra coincides, modulo the isometry group generators, with its Howe-dual $sp\left(2k\right)$ symplectic algebra. The construction of a correct Lagrangian description requires the initial symmetry algebra to be deformed to the algebra ${\mathcal{A}}_c(Y\left(k\right),Ad{S}_d)$, introducing the algebra ${\mathcal{A}}^{\prime }(Y\left(k\right),Ad{S}_d)$, different only due to the respective quadratic parts given by (83)–(85) and Table 1 and Table 2, with the use of Weyl ordering for the right-hand sides.

We generalized the method of constructing a Verma module [19], starting from Lie (super)algebras [59,97,98,99] and the quadratic algebra ${\mathcal{A}}^{\prime }(Y\left(1\right),Ad{S}_d)$ for totally symmetric HS fields [20,21], to the case of a non-linear algebra underlying the mixed-symmetric HS bosonic fields in an AdS space with a two-row Young tableaux. Theorem 1 presents our basic result in this respect. We show, as a by-product of the Verma module construction, that the Poincaré–Birkhoff–Witt theorem is valid in the case of the algebra in question, thereby providing a lifting of the Verma module for the Lie algebra $\mathcal{A}(Y\left(2\right),R^{1,d-1})$—isomorphic to $\left(T^2\oplus T^{2*}\right)+\supset sp\left(4\right)$—to the Verma module for an algebra quadratic in the deformation parameter r. The same is certainly expected to hold true in the case of the general algebra ${\mathcal{A}}^{\prime }(Y\left(k\right),Ad{S}_d)$, for which, we believe, it is pure machinery to obtain an explicit form of the Verma module in a recursive manner by means of some new primary and derived block operators, such as ${\widehat{t}}_{12}^{\prime }$ (A92) and ${\widehat{t}}_{12}^{\prime +}$ (A94).

We obtained a representation for 15 generators of the algebra ${\mathcal{A}}^{\prime }(Y\left(2\right),Ad{S}_d)$ as a formal power series over the Heisenberg–Weyl algebra with six pairs of Grassmann-even oscillators, which, in the flat-space limit $r=0$, takes a polynomial form identical with familiar results, at least for $m=0$ [55], and appears to be novel in the massive case [56] for $k=2$ [59]. Theorem A1 finalizes our second basic result, solving the Fock space realization problem for ${\mathcal{A}}^{\prime }(Y\left(2\right),Ad{S}_d)$ by using the approach (A90)–(A101) of a generalized Verma module construction.

On the basis of an exact BRST operator $Q^{\prime }$ constructed for the nonlinear algebra ${\mathcal{A}}_c(Y\left(2\right),Ad{S}_d)$ of 15 converted constraints $O_I$ (having a third order in the ghost coordinates), by analyzing the corresponding structure of Jacobi identities, we consistently develop a construction of gauge-invariant Lagrangian descriptions for bosonic HS fields of a given spin $\mathbf{s}=(s_1,s_2)$ and mass in an AdS${}_d$ space. To this end, in the standard manner, we extract from $Q^{\prime }$ (88) the related BRST operator Q (91), which is associated with the converted first-class constraints alone. The latter operator is nilpotent only in those proper subspaces of the total Hilbert space which have vanishing eigenvalues of the spin operators ${\overrightarrow{\sigma }}_2=({\sigma }^1,{\sigma }^2)$ (92). The corresponding Lagrangian formulation is, at most, a 5-th-stage-reducible abelian gauge theory, and is given by (101) and (102), with a specific mass $m_0$ (104). These last relations may be regarded as our main result in solving the general problem of constructing a Lagrangian formulation for non-Lagrangian initial AdS group irreducible representation relations, which describe a bosonic HS field with two rows in the Young tableaux (2)–(5). It should be noted that unconstrained Lagrangians for free mixed-symmetry HS fields with two rows in the Young tableaux on AdS backgrounds, in both the metric-like and frame-like formalisms, have not been obtained until now. We emphasize that a Lagrangian description of massive bosonic HS fields described by a two-row Young tableaux in (A)dS spaces has been known only in the frame-like form [72], with off-shell traceless and Young constraints.

We used these results to study a deformation procedure in the approach with a complete BRST operator, along the lines of [38,94,96], so as to obtain an interacting theory, in particular, a system of generating Equation (112) for the cubic vertices of three copies of initial Lagrangian formulations, with a respective set of constraints and oscillators. The above system is shown to be specified by three Equation (113), in the particular case of coincident cubic vertices $V^3={\tilde{V}}_l^3={\tilde{V}}_{l+1}^3$ for $l=0,\dots ,5$, so that a nontrivial solution should be a BRST (Q)-closed three-vector with vanishing values of spin ${\overrightarrow{\sigma }}_2$. By construction, all the holonomic constraints (tracelessness and Young symmetry) are included on equal footing with the remaining differential constraints in the total BRST operator. This guarantees an equal number of physical degrees of freedom in interacting and free (undeformed) Lagrangian formulations, thus allowing the interacting model to describe an irreducible interacting triple of massive HS fields (see [109] for the recent progress in the necessary and sufficient conditions for an interacting model, obtained with the use of the constrained BRST approach so as to preserve irreducibility for interacting HS fields in flat spaces). Unfortunately, there is no familiar application of Lagrangian dynamics in an AdS space, even for a free HS field of a given mass and spin, in the approach with an incomplete BRST operator, due to the absence of off-shell BRST extended constraints.

In addition, we consider a number of simple examples of the proposed Lagrangian descriptions: we deduce one for a totally symmetric HS field obtained earlier in [21] and another one for an antisymmetric massive tensor field of spin $(1,1)$, whose first-stage reducible unconstrained Lagrangian formulation with a complete set of auxiliary fields and gauge parameters was obtained in Section 6.2. The final ungauged Lagrangian is given by (186) after an application of partial gauge-fixing and a resolution of some of the equations of motion and is shown to coincide with a Lagrangian given by a different approach [62].

As a first by-product, we obtain polynomial deformations of the $su(1,1)$ algebra related to the conversion analyzed in Appendix B. Second, we demonstrate that a BRST Lagrangian formulation for an HS field of mass m and spin $(s_1,s_2)$ leads to equations of motions, which are equivalent to the initial irreducibility conditions (2)–(5). Third, we obtain a new Lagrangian formulation for the HS field of mass m and spin $(s_1,s_2)$ in a d-dimensional flat space-time starting from a general description for the same field in an AdS space in the limit $r\to 0$, due to new representations for the Verma and Fock modules given, respectively, by (A63)–(A71) and (A107)–(A112), with the gauge-invariant Lagrangian action ${\mathcal{S}}_{0|{\overrightarrow{s}}_2}^m=\int d{\eta }_0{}_{{\overrightarrow{s}}_2}\langle \chi |K_{{\overrightarrow{s}}_2}^0{Q}_{{\overrightarrow{s}}_2}^0{|\chi \rangle }_{{\overrightarrow{s}}_2}$ for the BRST operator $Q_{{\overrightarrow{s}}_2}^0=Q_{{\overrightarrow{s}}_2}{|}_{r=0}$ and $K_{{\overrightarrow{s}}_2}^0=K_{{\overrightarrow{s}}_2}{|}_{r=0}$. This Lagrangian is different from the one obtained in [56] in view of the dependence on two additional mass-like parameters $m_1,m_2$.

From the mathematical standpoint, the construction of a Verma module for the algebra ${\mathcal{A}}^{\prime }(Y\left(2\right)$, $Ad{S}_d)$ opens a possibility of analyzing the module structure and searching for singular and subsingular vectors in it so that, in principle, this could then allow one to construct new (non-scalar) infinite-dimensional representations of the given algebra. The above results allow one, certainly, to understand the problem of (generalized) Verma module construction for the HS symmetry algebras and superalgebras underlying HS bosonic and (respectively) fermionic fields in AdS spaces subject to a multi-row Young tableaux.

We expect, first of all, to use the deformation procedure for irreducible totally symmetric massless HS fields elaborated for free fields in [37] as applied to AdS spaces, and also anticipate the concept of BRST–BV quantization as applied to deduce interacting models in the BRST approach, while adapting the algorithm developed in flat spaces [110], albeit with a complete BRST operator, as in the case of a minimal BRST–BV action [111]. As a subsquent problem to be solved, we intend to apply our results to an explicit construction of a generalized Verma module, a BRST operator, and an LF for irreducible massive half-integer HS fields of spin $(n_1+1/2,n_2+1/2)$ subject to $Y(n_1,n_2)$ in AdS spaces.

Among other directions of application and development of the suggested approach, such as finding LFs and implementing free theories and cubic vertices for irreducible massive half-integer higher-spin fields on AdS backgrounds, having in mind the possibility of taking a flat-space limit [107] for numerous cubic vertices [105,106], one should also mention the problems of constructing quartic and higher vertices, as well as the related problems of locality (see the discussions in [112,113,114,115,116,117,118]), which, as we believe, may be addressed using the BRST method. We also expect that (ir)reducible HS fields can be employed as composite fields in the recent approach [119,120].