Global and local properties of AdS(2) higher spin gravity

Two-dimensional BF theory with infinitely many higher spin fields is proposed. It is interpreted as the AdS(2) higher spin gravity model describing a consistent interaction between local fields in AdS(2) space including gravitational field, higher spin partially-massless fields, and dilaton fields. We carry out analysis of the frame-like and the metric-like formulation of the theory. Infinite-dimensional higher spin global algebras and their finite-dimensional truncations are realized in terms of o(2,1) - sp(2) Howe dual auxiliary variables.


Introduction
In the recent years, higher spin gauge theories in three, four and higher dimensions have attracted considerable interest (e.g., see reviews [1,2,3,4,5] and references therein), while comparatively little attention has been paid to two-dimensional higher spin theories [6,7,8,9,10,11]. One of the reasons for this is that higher spin gravity in two dimensions does not necessarily share some of characteristic features of its higher dimensional cousins such as (A)dS background geometry or infinitely many propagating massless modes of all spins. So for example conventional 2d Fronsdal-type equations of motion both for massless or massive fields of higher spins s ≥ 1 do not propagate local degrees of freedom. For that matter the two-dimensional case is somewhat analogous to that in three dimensions, where higher spin Chern-Simons theory also describes no local degrees of freedom [12,13,14,15].
It follows that in two dimensions the notion of higher spin gauge fields should be clearly defined. We can, at least formally, introduce gauge fields of higher ranks and impose one or another set of gauge invariant equations and/or constraints. Then some of the resulting gauge systems have no local degrees of freedom, while others describe matter modes as particular components of higher rank gauge fields. In the former case the respective gauge fields often result from higher dimensional gauge systems by taking d = 2. In particular, both global and gauge transformations remain intact, while local degrees of freedom disappear.
In view of the above we propose to consider a particular 2d topological field theory as higher spin gravity with the cosmological constant. The theory is formulated as two-dimensional BF model with A-valued 0-form and 1-form fields, where A is some finite-dimensional or infinite-dimensional higher spin Lie algebra. 1 In [10] we explicitly considered the finite-dimensional case of A = sl(N, R) for N ≥ 2. The point is that the gauge algebra can be represented in the higher spin basis where generators are arranged as subalgebra sl(2, R) rank-s irreps so that the respective connections are identified with two-dimensional spin-(s + 1) fields. The case of N = 2 corresponds to the Jackiw-Teitelboim dilaton gravity [17,18], while taking N ≥ 3 gives rise to particular higher spin extensions. The N = 3 theory was also discussed in [11] in the framework of Poisson sigma-models, mainly form the holographic perspective.
It is remarkable that a ground state of the model under consideration is given by the AdS 2 spacetime. It follows that the gauge sector of the sl(N, R) higher spin gravity model comprises gauge fields in AdS 2 space with spins s = 2, 3, ..., N and masses m 2 s = s(s − 1)Λ, where Λ is the cosmological constant. Using their global symmetry properties one finds that the fields are to be treated as "topological partially-massless" fields of maximal depth [10]. Recall that the system does not have local degrees of freedom. It follows that the AdS 2 higher spin gravity can be interpreted as a consistent theory of topological yet interacting partially-massless higher spin fields given in a closed form. It is worth noting that partially-massless fields in higher dimensions do have local degrees of freedom [19,20,21,22], while their interactions at the action level are known only in the cubic approximation [23].
In this paper we formulate AdS 2 higher spin gravity with (in)finitely many fields as BF theory for the infinite-dimensional higher spin gauge algebra A = hs[ν] and its finite-dimensional truncations [24,25]. Note that similar models with an infinite higher spin algebra were partly discussed in [7,9]. Here we focus on the following issues.
• Local tensor fields in the AdS 2 higher spin gravity: frame-like versus the metriclike formulation. We study in detail the interplay between the BF formulation of the higher spin gravity which is actually the frame-like formulation and its metriclike formulation which extends the original Jackiw-Teitelboim dilaton gravity.
• Global higher spin symmetry algebras: 2 a formulation using the Howe duality o(2, 1) − sp(2) between AdS 2 global symmetry algebra and auxiliary symplectic algebra. We explicitly describe previously unknown realization of higher spin algebras A = hs [ν] in terms of o(2, 1) − sp(2) vector doublet variables. 3 Gauging algebra A defines local invariance of the BF theory under consideration.
• BF action for A-valued gauge fields: introducing particular trace operation on the infinite-dimensional gauge algebra A we define various (in)finite-dimensional truncations directly at the action level. We study a perturbative expansion of the action around the AdS 2 background.
The outline of the paper is as follows.
Section 2: The linearized AdS 2 higher spin gravity is formulated via the BF action functional. The action, the equations of motion, and the gauge symmetry transformations are given explicitly. The BF formulation under consideration is treated as a particular frame-like formulation which is known to be a generalization of the zweibein description of 2d gravitational systems. As a by-product, we propose a higher spin generalization of 2d Maxwell theory obtained as higher spin BF theory extended by a particular quadratic potential. Section 3: BF systems are treated in the framework of the unfolded formulation that pursues the cohomological understanding of both lower spin and higher spin systems (see the review [2] for details). The section contains a detailed discussion of various mathematical structures underlying the cohomological interpretation of the dynamics. The main objects here are the so-termed σ + and σ − nilpotent operators acting on the field space of the model. Elements of the space are differential p-forms taking values in any rank o(2, 1) finite-dimensional irreps. Using the σ ± -cohomology we perform a cohomological reduction of the initial field space to a certain subspace: a transition from the frame-like formulation of the model to its metric-like form. We compute σ ± -cohomology groups that completely identify the local structure of the (linearized) metric-like theory: gauge symmetry, independent metric-like fields, equations of motion and their Bianchi identities.
Sections 4 and 5: Nilpotent operators σ + and σ − correspond to two different cohomological reductions of the initial field space. So, in the one-form sector of the BF higher spin model we find that the system is equivalent either to massive scalar theory with a mass proportional to the cosmological constant and dependent on the spin, or to higher rank current conservation conditions. The scalar/current equations are invariant with respect to particular type of trivial on-shell symmetries/improvements that eliminate all local degrees of freedom. We suggest that these two forms of a single system are analogous to the well-known classical duality phenomenon occurring in the WZNW theory when second-order equations can be represented as the first-order conservation condition [28]. The same analysis is done in the zero-form sector of the model. Section 6: It summarizes the metric-like formulation developed in the previous sections. We list the metric-like equations of motions in the zero-form and one-form sectors of the BF higher spin gravity model in both cases of the σ ± cohomological reductions. Finally, the model is interpreted as the higher spin gauge-dilaton theory extending the Jackiw-Teitelboim dilaton gravity. Also, we consider two metric-like action functionals which give rise to dual metric-like equations of motion. We find out that the BF action is a "parent" action for the two dual metric-like formulations.
Section 7: Using manifestly covariant o(2, 1) − sp(2) vector notation we elaborate a realization of the one-parametric higher spin algebra hs[ν] introduced in Refs. [24,25]. Our realization is derived from the general d-dimensional oscillator description of the Eastwood-Vasiliev higher spin algebra for d ≥ 3 [29,30]. The approach is based on the Howe dual pair o(2, d − 1) − sp(2) realization in the bimodule of formal power series in auxiliary doublet variables [30,31]. Specifying to d = 2 we find out that hs[ν] is to be identified as quotient algebra obtained by singling out a particular ideal. The Howe duality o(2, 1) − sp(2) used to describe quotient higher spin algebras may be useful in many respects, in particular, for considering general non-linear two-dimensional higher spin models not necessarily of BF type. Indeed, the Howe duality is known to be crucial to built a consistent interacting higher spin theory in d ≥ 4 dimensions [30]. Section 8: It defines the full non-linear BF formulation of the AdS 2 higher spin gravity. Since the gauge algebras are realized as quotient algebras, the corresponding BF actions are formulated using particular projecting technique that allows to factor out elements of ideals directly inside the action. Quadratic higher spin actions studied in section 2 result from a linearization around the AdS 2 background solution.
Section 9: It summarizes our results and discusses future research directions. Details of the σ ± -cohomology computation are given in Appendix A. Details of the projecting technique are given in Appendix B.

Quadratic higher spin BF action
Let G s be a linear space of differential p-forms on a two-dimensional manifold taking values in finite-dimensional o(2, 1) totally symmetric and traceless representations of arbitrary rank 4 where p = 0, 1, 2 is a rank of a differential form (at p ≥ 3 differential forms are identically zero). Using o(2, 1) Levi-Civita tensor one shows that all non-symmetric finitedimensional o(2, 1) irreducible representations either vanish identically, or are described by hook-type traceless tensors .
(2.2) Two-dimensional higher spin fields are defined to be elements of G s . In two spacetime dimensions both massless and massive Wigner groups trivialize and whence it follows that only scalar and spinor modes may propagate. However, by a slight abuse of notation, we identify parameter s as a spin.
When considering gravitational systems parameterized by the negative cosmological constant Λ, it is convenient to represent gravitational fields as o(2, 1) connection 1-forms basis elements (see, e.g., [18]). Using antisymmetric basis one represents the connection as W AB m = −W BA m which is dual to the original connection via W mAB = ǫ ABC W C m . Flat connections satisfy the zerocurvature condition, which component form is given by The frame field and Lorentz spin connection are introduced in a standard fashion using the compensator V A normalized such that In what follows, we use V A in the form V A = (0, 0, L). The o(2, 1) covariant decomposition of W A m is given by It is well-known that AdS 2 spacetime solves constraint (2.3). The corresponding connection will be denoted W 0 = h a m , −1/ √ −Λ w m . The zero-curvature constraint expresses Lorentz spin connection w m via the frame h a m , while the latter defines AdS 2 4 A spacetime M 2 is a general two-dimensional manifold with local coordinates x m , Lorentz world indices run m, n = 0, 1, Lorentz fiber indices run a, b = 0, 1, o(2, 1) fiber indices run A, B, C = 0, 1, 2, o(2, 1) invariant metric is η AB = (+ − −). The spacetime derivative is denoted as ∂ m = ∂/∂x m , the de Rham differential is d = dx m ∂ m . The Levi-Civita tensor ǫ ABC is normalized as ǫ 012 = +1. Two-dimensional anti-de Sitter spacetime AdS 2 has a radius L and a signature (+−), so that the cosmological constant is Λ = −1/L 2 . The Levi-Civita tensor ǫ mn is normalized as ǫ 01 = +1. Symmetrization of indices has a unit weight and is labelled by parentheses. spacetime metric g mn through the standard identification g mn = η ab h a m h b n , where η ab = (+−) is the fiber Minkowski metric.
Let us consider particular elements of the space G s which are 0-form field Φ A 1 ...A s−1 , 1-form field Ω A 1 ...A s−1 along with 2-form field strength From now on, we systematically omit the wedge product symbol ∧. Representing the zero-curvature condition (2.3) evaluated on the background connection W 0 as R(W 0 ) ≡ D 0 D 0 = 0 one observes that higher spin field strengths are invariant with respect to the following gauge transformations where gauge parameters ξ A 1 ... A s−1 are 0-forms taking values in the same finite-dimensional representations. Note that the Bianchi identities D 0 R in two dimensions are trivial since any 3-form vanishes identically. The 0-form fields are assumed to be gauge invariant, Fields (2.5) are referred to as frame-like fields as these generalize the gravitational connection W A m to any number of fiber indices and any rank of differential form. Let us consider now the BF action for a single rank-s system, (2.9) The equations of motion obtained by varying with respect to Φ A 1 ...A s−1 and Both the action and the equations of motion are invariant with respect to gauge transformations (2.7) and (2.8). In Section 8 the action (2.9) will be obtained from a full non-linear BF higher spin action by a linearization around AdS 2 background W 0 .
The original BF theory (2.9) can be deformed in various ways. For instance, augmenting its action by a quadratic term where V 2 = ǫ ab h a ∧ h b is the volume 2-form built of AdS 2 background frame fields, one obtains the following equations Eliminating the auxiliary field Φ A 1 ...A s−1 by using its own equation of motion one arrives at the action of the form where R ⋆ 1 is the Hodge dual field strength. Note that now the action explicitly depends on the background AdS 2 metric. The rank-s equations of motion following from (2.13) generalize the Maxwell equations and describe no local degrees of freedom (see also our comments in the end of Section 4.3). For the simplest case s = 1 the action (2.11) is the well-known action for the Maxwell field A m on the background metric g mn with the auxiliary scalar variable f : Representing the Maxwell action in this form is useful in the analysis of 2d Maxwell-dilaton theories of gravity, since the dynamical field enters the action linearly (see, e.g., [32]).

Cohomological view of BF equations
In order to analyze the dynamical content of the BF action (2.9) we employ homological tools developed within the unfolded formulation (see the review [2] for details). Indeed, one observes that the BF equations of motion are explicitly formulated as the zerocurvature and the covariant constancy conditions imposed on the frame-like fields which are differential forms taking values in certain o(2, 1) irreps, see (2.10). Fortunately, such a geometrical setting naturally fits the unfolded formulation.
Most importantly, using the unfolded machinery helps to obtain the metric-like formulation of the BF theory. For instance, in order to obtain the Jackiw-Teitelboim dilaton gravity theory from o(2, 1) BF theory one should carefully identify the metric and scalar fields along with auxiliary fields, use local Lorentz symmetry to set an antisymmetric part of the zweibein equal to zero, split all the equations into dynamical and constraint ones [18]. It is remarkable that all these operations can be done in a systematic manner using cohomology groups of certain nilpotent operators called σ ± acting on the field space G s (2.1). In other words, using the σ ± -cohomology provides precise guidelines how to pass from a frame-like (i.e., BF) formulation to a metric-like formulation where the higher spin fields are higher rank Lorentz tensor fields.
In order to make using the cohomological methods more manifest it is convenient to reformulate given BF equations as off-shell system. It means that the right-hand-sides of BF equations are not zero but some arbitrary sources. Sending the sources to zero implies going on-shell. Indeed, put equations (2.10) off-shell as follows (3.1) , where the right-hand-sides of the equations are now arbitrary differential 1-form and 2-form, respectively, taking values in rank-(s − 1) irreducible o(2, 1) representation, (2.1). By definition, sources B (1) are C (2) are invariant with respect to gauge symmetry transformations (2.7) and (2.8), and therefore the off-shell system (3.1)-(3.2) retains the same gauge symmetry as the on-shell one (2.10).

σ ± operators
Most conveniently, the cohomological analysis of off-shell o(2, 1) covariant equation system (3.1) -(3.2) is performed in terms of Lorentz o(1, 1) ⊂ o(2, 1) algebra component fields. To this end, we rewrite elements of the field space G s (2.1) in Lorentz basis, where Lorentz fields are totally symmetric and traceless, Therefore, in Lorentz basis space G s is given by a direct sum of subspaces spanned by differential p-forms T a 1 ...a k (p) with fixed value of k = 0, ... , s − 1. Such elements will be denoted as T (p) (k). It is worth recalling that a o(1, 1) totally symmetric and traceless tensor T a 1 ...a k has just two independent components. This is most obvious in the light-cone parametrization T a 1 ...a k ∼ (T ++···+ , T −−···− ), where a number of ± equals k. However, keeping o(1, 1) symmetry manifest is convenient when analyzing the dynamical content of the theory.
The space G s incorporates all tensor fields of the theory, including zero-forms, oneforms and associated two-forms (2.5), along with their 0-form gauge symmetry parameters (2.7). For a given spin s there are two natural gradings in the space G s : by a rank of differential forms and by a number of Lorentz indices. On the other hand, there exist two nilpotent algebraic operators acting on G s that shift the gradings by one.
Let us define operators σ ± acting on G s as follows σ ∓ : T (p) (k ± 1) → T (p+1) (k). Their component action is given by 5 where h a m is the AdS 2 background frame, while exact expressions for coefficients α (k) , β (k) and γ (k) are given below, see (3.14) and (3.12). The operators satisfy where covariant derivative ∇ m = ∂ m +w m is evaluated with respect to AdS 2 background Lorentz spin connection w m . It is worth noting that conditions (3.6) can be understood as realization of the zero-curvature condition D 2 0 = 0 (2.3) in the Lorentz component basis [33], It is convenient to define the Euler operator N counting a number of Lorentz indices, NT (p) (k) = kT (p) (k). Then, [N, σ ± ] = ±σ ± and [N, ∇] = 0. Operator N provides the space G s with a finite grading, where a subspace G (k) s is spanned by homogeneous elements of degree k. By definition, operator σ − decreases a degree by one, operator σ + increases a degree by one.
The space G s can be endowed with an inner product given by Modulo an overall coefficient, operators σ − and σ + are mutually conjugated with respect to the above inner product. The following properties are elementary: Exact expressions for the coefficients. Coefficients γ (k) in (3.5) are fixed by the algebraic symmetry conditions (3.4) as Coefficients α (k) and β (k) are defined by conditions (3.6). Namely, one arrives at the equation system, for k = 1, ... , s − 1. The explicit solution reads (3.14) Using proper field redefinitions one can set either β (k) = 1 or α (k) = 1 for k = 1, ... , s −1 so that the solution is unique. Here, we choose the former case indicating that the dynamical systems under consideration are extended from Minkowski to AdS space.

Cohomological analysis
Below we shortly describe the general idea of the cohomological reduction of the off-shell BF system (3.1)-(3.2) using σ ± nilpotent operators (see Ref. [2] for more details).
The unfolded equations (3.15) are invariant with respect to gauge transformations given by where (p − 1)-forms ε (p−1) (k) are gauge parameters. In fact, the gauge symmetry transformation appears at p = 1 only. Indeed, in the case p = 0 the gauge fields have no associated gauge parameters, while in the case p = 2 the corresponding equations of motion vanish identically.
Quantities Z (p+1) (k) on the right-hand-side of (3.15) are not completely arbitrary and are restricted by the Bianchi identity which is a differential (p + 2)-form. It is obtained by using conditions (3.6). For p = 1 the Bianchi identity is a 3-form that vanishes identically.
Note that the unfolded equations, gauge transformations and identities are decomposed according to the grade degree (3.8). On the other hand, operators σ ± enter all equations algebraically. It suggests that the gauge system (3.15) -(3.17) can be analyzed recurrently, starting either from the minimal grade degree k = 0 equations or, from the maximal grade degree k = s − 1 equations. In both cases, one arrives at the linear systems of the type for some X, Y ∈ G s built of the sources, fields, parameters, and their derivatives. It follows that one is inevitably led to compute Im σ ± and Ker σ ± , and, moreover, the cohomology group H(σ ± ) = Ker σ ± /Im σ ± as the operators σ ± are nilpotent.
By way of example, let us identify independent equations of motion contained in the gauge system (3.15)- (3.17). Consider the equations of motion (3.15) parameterized by the sources Z (p+1) (k). 6 Those o(1, 1) irreducible components of the sources Z (p+1) (k) that belong to Im σ ± can be shifted to zero by appropriate shift redefinitions of fields in terms σ ± π (p) (k ∓ 1) in (3.15). Representing now the Bianchi identity as (3.18) one finds that non-vanishing irreducible components of Z (p+1) (k) not belonging to Ker σ ± are auxiliary. That is to say these components are expressed through the derivatives of components belonging to the cohomology H (p+1) (σ ± ) = Ker σ ± /Im σ ± p+1 , where the slash denotes restriction to (p + 1)-forms. Note that cohomology elements of H (p+1) (σ ± ) represent independent equations of motion and these nonetheless are not arbitrary being restricted by the residual Bianchi identity.
A number of independent identities between independent equations of motion is equal to a number of independent elements of the next cohomology group H (p+2) (σ ± ). Note that for p = 1 the Bianchi identities are identically vanishing 3-forms and therefore any 2-from always belongs to Ker σ ± . Consequently, there are no differential constraints in this case and only field redefinitions associated with Im σ ± are possible. These field redefinitions allow one to shift all non-zero tensors on the right-hand-side of the unfolded equations (3.15) to zero except for the cohomology elements.
Independent fields and gauge parameters can be considered similarly. So, the independent dynamical fields are particular o(1, 1) irreducible components of π (p) identified with elements of H (p) (σ ± ), while other irreducible o(1, 1) components are either auxiliary fields expressed via dynamical ones, or Stueckelberg fields that can be shifted to zero by appropriate gauge transformation. Residual gauge parameters are given by o(1, 1) irreducible components identified with elements of H (p−1) (σ ± ).
In this way, for a given p = 0, 1, 2 we come to the well-known dynamical interpretation of different cohomology groups [34,2,35] specified to two spacetime dimensions: All higher cohomology groups are empty, H (p) (σ ± ) = ∅ for p ≥ 3, because in d = 2 dimensions differential p-forms with p ≥ 3 vanish identically. As a corollary, there are no reducible gauge parameters and identities for identities.
It is important to note that the above interpretation of the cohomology elements (3.19) gives rise to different forms of one dynamical system reduced via either σ + or σ − operators. Generally, this happens because the respective cohomology groups are non-isomorphic (see below).
Theorem. The cohomology groups of operators σ ± in G s are given by The proof is straightforward and relegated to Appendix A. 7 A few comments are in order.
• The cohomology groups establish a cross-duality relation: It underlines dual interpretations of the BF higher spin theory that we develop in the following sections.
• Scalar elements of H (1) (σ ± ) are two different scalar components of grade k = 1 element of G s , while tensor components are given by the same maximally symmetric traceless component of maximal grade k = s − 1 element of G s (see Appendix A for more details).
• Each of the second cohomology groups H (2) (σ ± ) contains a single non-vanishing element. It is worth noting that in d ≥ 4 dimensions H (2) (σ − ) contains two non-vanishing elements called the Einstein cohomology elements and the Weyl cohomology elements [33]. 8 These cohomology elements are given by differential gauge-invariant combinations of d-dimensional Fronsdal fields and have an elegant interpretation. Indeed, in order to obtain Fronsdal equations of motion one equates the Einstein cohomology element to zero, while the Weyl cohomology element remains arbitrary modulo the Bianchi identities. It follows that the Weyl elements parameterize on-shell nontrivial gauge invariant combinations of dynamical fields, i.e., the physical degrees of freedom. In the d = 2 case H (2) (σ ± ) is spanned by a single element. 9 Equating this element to zero inevitably makes the theory topological. We refer elements of H (2) (σ ± ) to as the Weyl tensors/scalars. 7 Our results on H(σ + ) cohomology (see a comment in footnote 5) can be obtained from ddimensional consideration of [35] by taking d = 2. However, the case of d = 2 is strongly degenerate so that making a direct substitution of d = 2 should not be taken for granted. Also, H(σ − ) has not been discussed before. In particular, an explicit computation of the cohomology has technical features specific to two dimensions that are crucial when analyzing the reduced unfolded equations. 8 See footnote 5. In higher spacetime dimensions one considers the σ − cohomology only because its elements are interpreted as fields, parameters, and equations of the Fronsdal theory of massless fields. A dynamical interpretation of the higher spacetime dimensional σ + cohomology has not been elaborated yet. 9 Along with the second item above this may imply that Fronsdal action in two dimensions at s > 1 is a total derivative. E.g., in the s = 2 case, the Einstein tensor does vanish identically. On the other hand, the 2d Maxwell action is not a total derivative: the respective variational equation is ∂ m F = 0, where F stands for dualized Maxwell tensor. Nonetheless, the theory is topological because the general solution reads F = const allowing for linear potentials only.
4 Off-shell unfolded equations for one-form fields Component form of fields. Lorentz components of 0-form gauge parameters (2.7), 1-form gauge fields (2.5), and 2-form field strengths (2.5) will be denoted as all of them satisfy the irreducibility conditions (3.4).
Using general formulas (3.15), along with (3.5) and (3.12), (3.14), we find that the component form of the field strength is given by [ .

(4.2)
Analogously, the component form of the gauge symmetry transformations (2.7) is given by

(4.3)
Off-shell equations of motion. Consider now the unfolded equations in the oneform sector (3.2) written in the Lorentz component form as follows where o(1, 1) totally symmetric and traceless tensors C a 1 ...a k (s) are the Lorentz components of the 2-form source C parameterizing the right-hand-side of (3.2). The expres- In the case s = 1, the cohomology groups are isomorphic, Therefore, the only equation of motion in (4.4) says that the Maxwell tensor admits a dual representation, i.e., R mn ≡ F mn = ǫ mn C (1) . Whence it follows that there are no restrictions imposed on F mn , and the theory is off-shell. By some means, going on-shell constrains C (1) . For instance, by taking C (1) = 0 one obtains the BF topological theory; other possible constraints are discussed in Section 4. 3. In what follows we always assume s ≥ 2.
For s ≥ 2 and p = 1 the cohomology groups H (p) (σ ± ) are not isomorphic. This implies that the cohomological reduction of the equation system (4.4) could be done in two different ways giving rise to two different but dynamically equivalent theories.
Following the general discussion of Section 3.2, the unfolded equations (4.4) can be represented in two forms depending on particular operator σ ± : within the σ + cohomological reduction, and, within the σ − cohomological reduction. In (4.6) the ellipsis refers to proper symmetrizations of derivatives and trace terms. The proof is analogous to that of the theorem of Section 3.2. The representations (4.5) and (4.6) are convenient in practice because all field redefinitions have been done that remove all right-hand-side tensors / ∈ H (2) (σ ± ). In both cases, we see that field redefinitions produce derivative transformations setting all the source components to zero except for those corresponding to the cohomology elements.
The existence of two operators σ ± used for the corresponding cohomological reductions implies two dual descriptions of the same system (4.4). 10 We show that the σ + cohomological reduction yields the massive scalar Klein-Gordon equation on the hyperboloid with non-vanishing right-hand-side given by scalar Weyl tensor. The σ − cohomological reduction yields the current conservation condition with non-vanishing right-hand-side given by the higher rank Weyl tensor. In both cases, we impose partial gauge conditions setting a part of dynamical fields to zero.
Recall that the Bianchi identity (3.17) for the equation system (4.4) is trivial thereby implying that the cohomology elements are arbitrary. Imposing algebraic and/or differential constraint on Weyl scalars/tensors is discussed in Section 4.3. For instance, equating all the cohomology elements to zero one obtains the BF higher spin theory with the action (2.9).
The cohomological approach says that the field space G s in the sector of 1-form fields ω a 1 ... a k m decomposes into Stueckelberg fields, auxiliary fields, and dynamical fields given by the cohomology H (1) (σ + ). The above three types of fields appear as particular irreducible Lorentz components of ω a 1 ... a k m , cf. (A.2).
In the case s > 1, the vanishing higher rank field strengths at k = 0 are constraints allowing to express auxiliary fields via derivatives of independent dynamical fields given by a scalar and a rank-s traceless tensor ϕ, ϕ a 1 ...as ∈ H (1) (σ + ) (3.20). Other Lorentz components of ω a 1 ... a k m are Stueckelberg ones shifted to zero by algebraic parts of the gauge transformations (4.3).
The minimal grade degree equation ǫ mn R mn = C (s) is the only off-shell equation of motion for dynamical fields. Gauge fixing all Stueckelberg fields to zero and expressing all auxiliary fields via the dynamical fields, one shows that the minimal grade equation is reduced to the following order-s differential equation where AdS 2 = ∇ a ∇ a is the wave operator on the AdS 2 background, and coefficient ρ 0 is given by (3.14). Non-vanishing spin-dependent coefficients κ s , ρ s are fixed by gauge symmetry transformations with an independent gauge parameter ξ a 1 ...a s−1 ∈ H (0) (σ + ), see (3.20). Lower grade degree k = 0, 1, ..., s − 2 gauge parameters ξ a 1 ...a k are Stueckelberg ones used to shift some Lorentz components in ω a 1 ... a k m to zero.
The dynamical equation (4.8) can be simplified. To this end, a field ϕ a 1 ...as is completely gauged away by imposing the higher-spin gauge ϕ a 1 ... as = 0 . (4.12) Indeed, a traceless rank-k tensor in d = 2 dimensions has two independent components for any k ≥ 2. It follows that a number of independent components of a rank-(s − 1) gauge parameter equals a number of equations in (4.12). The higher spin gauge can be viewed as an extension of the standard conformal gauge in 2d gravity which makes the metric proportional to Minkowski tensor. Then, the only dynamical field is given by a scalar component of the cohomology group, ϕ ∈ H (1) (σ + ).
Imposing the higher spin gauge (4.12) and solving the constraints in (4.7) one finds that the leftover equation reduces to the massive scalar equation with particular value of the mass-like term [10] where we redefined the right-hand-side as ρ −1 s C (s) → C (s) .
The massive scalar equation (4.13) is invariant with respect to residual gauge transformations (4.11) provided that the gauge parameter ξ a 1 ...a s−1 ∈ H (0) (σ + ), satisfies the generalized Killing equation on the hyperboloid, The above constraint is clearly explained as the stability transformation of the higher spin gauge condition (4.12) for transformations (4.10).
A few comments are in order.
• In the spin-2 case the above equation reproduces the gauge-fixed linearized equation of motion of the Jackiw-Teitelboim model in the one-form sector [18,36,10]. We see that the higher spin extension is described by the scalar field as well, but with a different spin-dependent mass term (4.9) and higher derivative leftover gauge symmetry (4.14).
• Mass m 2 s (4.9) differs from the conformal value of mass • Mass m 2 s coincides with the value of the Casimir operator of o(2, 1) global symmetry algebra of AdS 2 space realized on tensor fields .
• Since the theory propagates no local degrees of freedom, the scalar field equation (4.13) at C (s) = 0 becomes a constraint equation for auxiliary field ϕ that can be solved by defining the respective Green's function:
In the case s > 1, the vanishing higher rank field strengths at k = 0, ..., s − 2 are constraints allowing to express auxiliary fields via derivatives of independent dynamical fields given by a scalar and a rank-s traceless tensor φ, φ a 1 ...as ∈ H (1)  Solving the constraints (4.15) yields the following expression where τ s is some non-vanishing spin-dependent coefficient, the parenthesis contain terms that depend on field φ only, while the ellipsis refers to appropriate symmetrizations of derivatives and trace terms. Independent gauge transformations are given by δφ a 1 ...as = 1 Λ ∇ a 1 · · · ∇ as ξ + ... , (4.18) where the ellipsis refers to proper symmetrizations and trace terms, while a scalar gauge parameter ξ ∈ H (0) (σ − ) (3.20). The mass coefficient m 2 s is given by (4.9). The maximal grade degree equation R a 1 ...a s−1 = V 2 C a 1 ...a s−1 is the only off-shell equation of motion for dynamical fields. Gauge fixing all Stueckelberg fields to zero and expressing all auxiliary fields via the dynamical fields using (4.16), one shows that the maximal grade equation is reduced to the following order-(s−1) differential equation where the ellipsis refers to proper symmetrizations and trace terms.
Higher order equation (4.19) can be simplified by imposing a gauge condition. Indeed, using the scalar field transformations (4.17) one introduces the scalar gauge condition along with the residual gauge parameter equation which are dual cousins of higher spin gauge condition (4.12) and generalized Killing equations (4.14). It follows that dynamical equation (4.19) takes the form For equation (4.21) with the vanishing right-hand-side C (s) a 1 ...a s−1 = 0 one identifies φ a 1 ...as with spin-s conserved current on the hyperboloid. 11 Higher order derivative transformations (4.18) with the scalar gauge parameter satisfying (4.20) are treated now as "improvement" transformations for conserved currents. Indeed, "improvements" are higher order derivative transformations with an antisymmetric tensor parameter which in d = 2 dimensions is dualized to a scalar via the Levi-Civita tensor.

Off-shell field spaces
In the framework of the unfolded formulation one may introduce the so-called Weyl module as a linear space which elements parameterize all possible gauge-invariant differential combinations of dynamical fields ∈ H (1) (σ ± ) that remain arbitrary on-shell.
In d ≥ 4 dimensions, the Weyl module is derived by solving the Bianchi identities: one "unfolds" the original higher spin Weyl tensor, i.e. introduces new variables (infinite of them) that parameterize independent combinations of derivatives of the Weyl tensor [2].
In d = 2 dimensions the Bianchi identities in the one-form sector trivialize due to H (3) (σ ± ) = ∅, see (3.17) and (3.19). Whence, the Weyl tensor ∈ H (2) (σ ± ) remains completely arbitrary function of spacetime variables. However, it does not yield local degrees of freedom in the theory. Indeed, recall that contrary to the higher-dimensional case, the cohomology H (2) (σ ± ) contains the only element, cf. (3.20). In other words, the Einstein cohomology (higher spin equations of motion) and the Weyl cohomology (higher spin Weyl tensors) coincide in two dimensions. It follows that keeping the Weyl element arbitrary implies the theory is off-shell. On the other hand, choosing the Weyl element to be a particular function can be treated as "going on-shell". E.g., setting all Weyl tensors to zero results in the zero-curvature equations of motion (2.10). There are various ways of how to put our topological system on-shell. We discuss some of them in Section 4.3.2.

Unfolding Weyl tensors
Despite the lack of 2d Bianchi identities, one can still associate to Weyl tensors infinite sets of components which comprise their all possible derivative combinations. Namely, by off-shell field space for the Weyl scalar C (s) ∈ H (2) (σ + ) we call the following set of components where elements are totally symmetric and traceful, η mn W where the ellipses in (4.23) refers to proper symmetrizations and all possible trace terms. For a given k the projector P b 1 ... b k contains a finitely many arbitrary coefficients not fixed by the above definition of W 0 . Note that in d = 2 dimensions only symmetric combinations of covariant derivatives are possible because any non-symmetric ∇ a 1 ...∇ a k C can be reduced to a collection of symmetrized combinations by using the Levi-Civita tensor and commutator [∇, ∇] ∼ Λ.
Quite analogously, by off-shell field space for the Weyl tensor C (s) Generally, off-shell field space elements are not related to each other. A natural option suggested in [34] is to consider particular constraints for elements of the off-shell field space relating components with different values of k as while element W (s) remains arbitrary. It follows that the form of relations (4.23) is not changed, while arbitrary coefficients in projectors P b 1 ... b k are uniquely fixed modulo a single free coefficient to be identified with the mass parameter. We refer the off-shell field space W 0 supplemented with constraints (4.26) to as the off-shell Weyl module W 0 . The same consideration can be applied to off-shell module W s−1 .

Going on-shell
Recall now that dynamical fields propagated by the unfolded equations (4.4) are considered as auxiliary, see our comments in the end of Section 4.1. Indeed, these are completely expressed via the Weyl tensors which parameterize the right-hand-sides of the dynamical equations. Such a phenomenon is characteristic of topological field theories coupled to external dynamical systems with or without local degrees of freedom (see a recent discussion in [37]). In particular, this is the way one couples matter fields to 3d topological Chern-Simons theory. In this case, Chern-Simons strength tensor turns out to be proportional to a matter current so that respective gauge fields are auxiliary carrying no physical degrees of freedom. However, added topological modes may have a profound impact on dynamics of the matter system, giving rise, for instance, to anyonic statistics.
In our case, the problem of coupling a field theory with an (in)finite number of degrees of freedom to the topological unfolded theory given by equations (4.4) reduces to the equivalent problem of specifying Weyl tensors via imposing appropriate constraints on elements of the off-shell field spaces. Note that choosing particular Weyl tensors actually puts the topological system (4.4) on-shell. Other way round, going on-shell in the topological theory (4.4) is nicely interpreted as coupling to external field theory.
By way of example, specify the off-shell field space W 0 to the off-shell Weyl module W 0 given by (4.26), and impose the tracelessness condition (4.27) The above constraint yields the massive Klein-Gordon equation of motion on AdS 2 spacetime imposed on the Weyl scalar C (s) [8,34]. It follows that an external field theory is identified here as the scalar field theory coupled to (linearized) topological spin-s BF theory. The dynamical field ϕ in equation (4.13) is auxiliary and expresses now in terms of the Klein-Gordon field C (s) .
As another possible option let us mention a truncation of the off-shell Weyl W 0 by imposing the following constraint = 0 is equivalent to constraint ∇ b F = 0 which is the dualized Maxwell equation. Recall here that dualized Maxwell tensor F mn = ǫ mn F is identified with scalar C (1) and two off-shell field spaces considered above coincide, being actually a single space W 0 . Also, one may truncate all elements of W 0 by imposing constraint W (1) ≡ F = 0 that appears as the equation of motion in the Abelian BF theory.
Another example of a theory with no local degrees of freedom identified with an external field theory is given by equations (3.1)-(3.2) with the right-hand-sides given by (2.12). In this case, the right-hand-side of unfolded equation (4.4) is parameterized by 0-form field subjected to another unfolded equation which describes no local degrees of freedom as well (see the next section).

Off-shell unfolded equations for zero-form fields
Consider now the unfolded equations in the zero-form sector (3.15). By analogy with where ≡ 0 .

(5.4)
According to the general consideration of Section 3.2, the system (5.2) can be algebraically reduced using one or another type of nilpotent operators σ ± . In both cases, the cohomological theorem (3.20) guarantees that the true dynamical fields in the system are either φ ∈ H (0) (σ − ), or φ a 1 ...a s−1 ∈ H (0) (σ + ). It is worth noting that the right-hand-sides of the independent equations of motion obtained through the cohomological reduction are parameterized by two independent elements of H (1) (σ ± ). In this respect, the situation is different from that in the gauge sector, where the reduced equations of motion are parameterized by a single Weyl scalar/tensor. It is similar to the higher dimensional picture, where the right-handsides of the equations also contain two independent cohomology elements, the Einstein part and the Weyl part, see the discussion in the end of Section 3.2. 12

Explicit σ + -reduction: zero-form sector
The σ + cohomological reduction of the unfolded equations (5.2) gives rise to the following independent equations of motion where ϕ a 1 ··· a s−1 ∈ H (0) (σ + ) and B +(s) , B +(s) a 1 ....as ∈ H (1) (σ + ), and indices are symmetrized with a unit weight. The tensors on the right-hand-sides of (5.5) are not arbitrary and are subjected to the Bianchi identities (5.4). Following (3.19) and (3.20), we find that there is a single identity between independent equations (5.5) corresponding to a scalar element I (s) ∈ H (2) (σ + ), where κ s , ρ s are some non-vanishing spin-dependent coefficients, cf. (4.8), while mass parameter m 2 s is given by (4.9). By way of example, consider the spin-2 case. Here, the unfolded equations of motion where Λ is the cosmological constant, and B and B a are subjected to the Bianchi identities (5.4) Using the σ + cohomological reduction and field redefinitions one finds from the second equation in (5.7) that ϕ = − 1 2 ∇ a ϕ a . Considering the Bianchi identities (5.8) one shows that the first equation in (5.7) is a differential consequence of the second equation. The resulting equations that follow from the second equation in (5.7) for the independent field ϕ a ∈ H (0) (σ + ) read where B + , B + (ab) ∈ H (1) (σ + ); cf. equations (5.5). Note that redefining fields by a dualization via ǫ ab -tensor yields the following system 13 . This form is useful when analyzing Killing symmetries of the gauge dynamical field, see Section 5.3. The Bianchi identities (5.6) take the form 10) or, equivalently, ǫ ab ∇ a ∇ c B b c + ΛB ab = 0. We see that there is a single identity corresponding to a single element of the second cohomology I ∈ H (2) (σ + ).

Explicit σ − -reduction: zero-form sector
The σ − cohomological reduction of the unfolded equations (5.2) gives rise to the following independent equations of motion Following (3.19) and (3.20), we find that there is a tensor identity between independent equations (5.11) corresponding to a tensor element I (s) where τ s is some non-vanishing spin-dependent coefficients, cf. (4.8); the ellipses refers to proper symmetrizations and trace terms.
By way of example, consider the spin-2 case. Here, the equations of motion and the Bianchi identities are the same as in the previous section, see (5.7) and (5.8). The cohomological analysis goes along the same lines. So, using the σ − cohomological reduction one finds from the first equation in (5.7) that φ a = −∇ a φ. It follows that the resulting equation for the independent field φ ∈ The trace and traceless parts of the above equation are cf. equations (5.11). Equations (5.13) reproduce the Jackiw-Teitelboim linearized equations in the zero-form sector [18]. The Bianchi identities (5.12) take the form 14) or, equivalently, ǫ bc ∇ b B c a = 0. We see that there is an o(1, 1) vector identity corresponding to independent elements of the second cohomology I a ∈ H (2) (σ − ).

Background symmetries
The unfolded equations in the zero-form sector (3.1) can be considered from a different perspective. Provided the right-hand-side is vanishing, the equations (3.1) are interpreted as stability transformations for a particular 1-form background gauge field Ω 0 . From (2.7) it follows that the stability transformation equation reads Taking into account the analysis of the unfolded equations in the zero-form sector, the system (5.16) can be treated in two different ways, using either σ − or σ + cohomological reduction. Whence it follows that there are two possible interpretations of the stability transformations.
Using the σ + cohomological reduction one finds out that (5.16) reduces to equations (4.10)-(4.11) or (5.5) on tensor parameters ξ a 1 ...a s−1 at s = 1, 2, ..., ∞ subjected to the Bianchi identity (5.6). For a given s, the solution to the stability equations depends on a finitely many integration constants interpreted as constant o(1, 1) tensors parameterizing higher spin global symmetry transformations of the AdS 2 background spacetime. 14 For instance, in the spin-2 case stability transformation equations can be rewritten in the form ∇ a ξ b + ∇ b ξ a = 0 (see our comments below (5.9)) and their explicit solution reproduces the well-known o(2, 1) Killing vector parameterized by three integration constants representing three o(2, 1) generators.
On the other hand, using the σ − cohomological reduction one finds out that (5.16) reduces to equations (4.17)-(4.18) or (5.11) on a scalar parameter ξ (s) at s = 1, 2, ..., ∞ subjected to the Bianchi identities (5.12). In this case the stability transformations describe trivial "improvement" transformations of the respective spin-s conserved currents. Contrary to the general "improvement" transformations that are invariance transformations of the conservation condition, the trivial "improvements" do not change the conserved current itself. It seems that there is no any "background conserved current" similar to the background spacetime, so that an interpretation of trivial "improvements" remains unclear.
6 Summary of the metric-like formulation 6

.1 Metric-like equations of motion
Below we list the metric-like equations following from the σ ± cohomological reductions of the original spin s > 1 unfolded equation system (3.1) and (3.2) analyzed in Sections 4 and 5.

1-form sector:
AdS 2 − s(s − 1)Λ ϕ = C , ϕ a 1 ... as = 0 (6.1) 0-form sector: Recall that the metric-like equations in the one-form sector have been obtained using the higher spin gauge (4.12) in the σ + case, and the scalar gauge (4.20) in the σ − case. In particular, the above equations are supplemented with the leftover gauge transformations and the Bianchi identities in the one-form and the zero-form sectors, respectively. Note also that the metric-like equations of motion are of order 1, 2, s − 1, s in derivatives.

Dual metric-like higher spin actions
Let us consider linearized frame-like action (2.9) in the metric-like form. To this end, we represent the action in Lorentz basis The idea is to fix Stueckelberg (shift) gauge symmetries and eliminate auxiliary fields using their own equations of motion substituting then the independent metriclike fields and the field strengths back to the frame-like action (6.5). In particular, this is the way one shows the equivalence of the frame-like o(2, 1) BF theory with the original metric-like Jackiw-Teitelboim model [18].
As we have already seen, a reduction to the independent dynamical sector can be done in two equivalent ways associated either to σ + or σ − cohomology. Moreover, when considering both one-form and zero-form sectors simultaneously one has four equivalent reductions which we denote as (σ ± , σ ± ) reduction, where the first and second sigmas refer to corresponding reduction in the one-form and zero-form sector, respectively. However, at the action level one finds out that there are only two possible ways to perform a reduction to the metric-like form. Equations obtained via (σ − , σ − ) or (σ + , σ + ) reductions cannot be derived as variational equations since the number of the independent fields do not coincide with the number of the equations of motion.
Equations obtained via the (σ + , σ − ) reduction can be derived as the Euler-Lagrange equations of motion following from the action where R(ϕ, ϕ a 1 ...as ) is the 2-from field strength of grade degree k = 0 (4.7) expressed in terms of the dynamical fields. The equations of motion of the theory (6.6) take the form (6.1) and (6.4) (using the higher spin gauge). In particular, the linearized action and equations of the Jackiw-Teitelboim model follow from (6.6) at s = 2.
Analogously, equations obtained via the (σ − , σ + ) reduction follow from the other action where R a 1 ...a s−1 (φ, φ a 1 ...as ) is the 2-form field strength of grade degree k = s − 1 (4.15) expressed in terms of the dynamical fields. The equations of motion of the theory (6.7) take the form (6.2) and (6.3) (using the scalar gauge).
On the other hand, both types of the cohomological reductions describe the same dynamical system. It suggests there exists a duality mapping between two linear theories given by (6.6) and (6.7). It would be interesting to provide an exact definition of such a mapping originated from the cohomology cross-duality and to study its properties and implications beyond the linear approximation.

The model interpretation
The equations of motion in the one-form sector have been previously interpreted as describing topological maximal depth partially-massless higher spin fields on the AdS 2 background [10]. It should be noted that such an interpretation follows from (σ + , σ − ) -reduction described by action (6.6).
In this case, the equations of motion in both zero-form and one-form sectors (in the gauge fixed form) are given by the same Klein-Gordon equation AdS 2 −s(s−1)Λ ϕ = 0 and AdS 2 − s(s − 1)Λ φ = 0 for two scalars ϕ and φ. The general solution depends on two arbitrary functions of spacetime coordinates so that it can be interpreted as left and right waves. However, there are gauge symmetry in the one-form sector and additional tensor constraint along with the Bianchi identities in the zero-form sector that eventually eliminate the functional freedom leaving no local modes (only a finitely many integration constants). The absence of propagating degrees of freedom leaves enough room for interpretation of the equations of motion under consideration. We set that fields in the one-form sector are gauge fields, while those in the zero-form sector are dilaton fields, both topological.
The spectrum of the model can be interpreted as follows. The BF higher spin theory given by action (6.6) describes: (one-form sector) topological s = 1 massless Maxwell field and s = 2 graviton field along with increasing spin s = 3, 4, ... partiallymassless gauge fields of the maximal depth; (zero-form sector) topological dilaton fields with increasing masses m 2 s = −s(s − 1)Λ. In this form action (6.6) can be treated as a higher spin gauge-dilaton extension of the original (linearized) Jackiw-Teitelboim dilaton gravity model.

7
The higher spin algebras in two dimensions To formulate a non-linear BF higher spin theory the fields should be represented as connections of some (in)finite Lie algebra. In the case of finitely many fields a higher spin algebra can be identified with sl(N, R) algebra provided that its basis elements are represented as where generators T A 1 ...A k are rank-k totally symmetric and traceless sl(2, R) algebra tensors [41,14,15]. Gauging algebra (7.1) yields a finite collection of 0-form and 1form fields of the type (2.5). A natural infinite-dimensional generalization of (7.1) should have the following structure ∞ s=1 ls T (ls) where the numbers l s are multiplicities of spin-s basis elements. Note that (7.2) contains also infinitely many copies of gl(1, R) generator T corresponding to the spin-1 Abelian connection.
A convenient way to realize higher spin algebras with generators T A 1 ...A s−1 (7.2) is to represent them as homogeneous polynomials of degree-(s − 1) in auxiliary vector variables. It is remarkable that such a vector realization can be obtained using ddimensional oscillator approach based on the o(2, d − 1) − sp(2) Howe duality proposed by Vasiliev [30,31]. In what follows, we use the o(2, 1) − sp(2) Howe duality to describe the one-parametric family of 2d higher spin algebras hs[ν] originally introduced by Feigin as quotients of the universal enveloping algebra U(sl(2)) [24], and by Vasiliev as the enveloping algebra of the Wigner deformed oscillator algebra [25].

Oscillator approach
Following the original papers [30,31], we consider auxiliary doublet variables Y A α , with sp(2) vector index α and o(2, M) vector index A, 15 and consider polynomials expanded in the auxiliary variables as follows where expansion coefficients are totally symmetric in both groups of indices.
Define now the Weyl star-product It follows that the auxiliary variables satisfy the following commutation relations Y A α , Y B β * = ǫ αβ η AB . A space of polynomials (7.3) endowed with the star-product (7.4) is the Weyl algebra A M +2 .
The algebra A M +2 is a bi-module over o(2, M) and sp(2) algebras. Their basis elements are realized as bilinear combinations of the auxiliary variables In general, associative algebra S M +2 (as well as Lie algebra hc(1|2:[M, 2])) contains various two-sided ideals I. For instance, there exists the maximal ideal spanned by elements where g αβ (Y ) is an arbitrary polynomial transforming as an sp (2) [31]. It is singled out by reality conditions where the involution † of the complex algebra S M +2 is defined as where a ∈ C, and the bar stands for the complex conjugation.
In what follows, we explicitly consider the case of M = 1 and study quotient higher spin algebras corresponding to different ideals, including the maximal one. We show that hc(1|2: [1,2])/I 1 is a finite-dimensional algebra. Therefore, in order to produce an infinite-dimensional higher spin algebra one should use non-maximal ideals. We identify two infinite families of ideals that yield both finite-and infinite-dimensional quotient higher spin algebras. Our analysis also applies to the case of M = 2, where the AdS 3 global symmetry algebra o(2, 2) ≈ o(2, 1) ⊕ o(2, 1), and each factor can be considered by analogy with the case of M = 1.  (2) invariance condition (7.7) says that these tensors are of particular index symmetry type. It follows that the resulting expansion coefficients of (7.3) are o(2, M) traceful tensors with index symmetry described by rectangular two-row Young diagrams In the M = 1 case, any o(2, 1) traceful two-row rectangular tensor (7.10) can be decomposed into one-row tensors because any traceless o(2, 1) tensor with indices described by two-row Young diagram with more than one cell in the second row vanishes identically, while those with a single cell in the second row are dualized using the Levi-Civita tensor, see (2.2).
It follows that a linear space of the algebra S 3 spanned by sp(2) singlets (7.7) can be represented as an infinite collection of one-row traceless Young diagrams. Indeed, let T m denote a spin-m o(2, 1) irrep given by a totally symmetric traceless o(2, 1) tensor. Then, one can show that a linear space of S 3 as o(2,1) module is decomposed in a direct sum where a superscript l stands for multiplicity, cf. (7.2). Elements of linear space (7.11) can be depicted on the following plot: • · · · • · · · • · · · • · · · (7.12) Here, irreps T k are depicted as length-k Young diagrams, dots • correspond to scalar components T 0 . Irreps T k resulted from decomposing a traceful two-row rectangle of length m − 1 are disposed vertically, k = 0, ..., m. Note that an each line on the plot successively depicts all basis elements of gl(N) algebra, where N = 1, 2, 3, ....
The other way around, traceless symmetric tensors can be rearranged as traces of a given totally symmetric traceful tensor. It suggests that the linear space can be described by traceful symmetric tensors of all ranks from zero to infinity, each in a single copy. It can be equivalently seen by dualizing traceful rectangular o(2, 1) diagrams (7.10). It follows that the linear space of S 3 can be represented as where G k denotes a rank-k symmetric traceful o(2, 1) tensor; it follows that G k = T k ⊕ T k−2 ⊕ · · · . On the plot (7.12) a tensor G k corresponds to the k-th vertical column.
Let us now notice that when indices A, B, ... run just three values it is possible to introduce new variables (7.14) which are in fact Hodge dualized o(2, 1) basis elements (7.5), and hence satisfy the commutation relations T A , T B * = ǫ ABC T C . One can show that any sp(2) singlet F (Y ) can be equivalently rewritten as an arbitrary polynomial F (T ). Indeed, the sp(2) invariance condition (7.19) says that expansion coefficients of any F ∈ S 3 (7.3) have even numbers of sp(2) and o(2, 1) vector indices, and can be represented as where each group of two vector indices |A i A i+1 | is antisymmetric (see [43] for more details). Using the definition (7.14) along with (7.15) one finds that (7.3) can be completely rewritten as polynomials of o(2, 1) bilinears T A with totally symmetric expansion coefficients. Note that T A are sp(2) singlets. It follows that the space S 3 of sp(2) singlets is now naturally realized as functions of sp(2) invariant variables. The action of Howe dual algebra sp(2) becomes implicit.
In this way, we establish that the associative algebra S 3 of sp(2) singlets and the universal enveloping algebra U(o(2, 1)) are isomorphic, o(2, 1)) . can be collectively represented as three independent sp(2) generators. In particular, any multiple trace of F ∈ S 3 is to be proportional to the following combination [43] t αβ · · · t γρ c 2 · · · c 2 . (7.17) Here, sp (2) indices are assumed to be symmetrized. Totally antisymmetric combinations of t αβ produces powers of the sp(2) Casimir element c 2 .
By way of example consider particular polynomial subjected to the sp(2) invariance condition (7.7). It follows that an expansion coefficient F AB, CD is described by a "window" Young diagram . On the other hand, the expansion coefficient is traceful so that a decomposition into traceless parts yields a linear combination where the ellipsis denote proper symmetrization of indices, while F 0 AB, CD , F 1 AB , and F 2 are traceless components. Substituting the above decomposition into F (Y ) one finds that the second term is proportional to t αβ , while the third term is proportional to For the case of M = 1 the first term in decomposition (7.18) identically vanishes, F 0 AB, CD = 0. The second and the third terms correspond to T 2 and T 0 elements depicted in the third vertical column on the plot (7.12).  (7.19) are given by totally symmetric o(2, 1) traceless tensors. It is worth noting that analogous decomposition for elements of S M +2 algebra is 3-parametric, while taking M = 1 leaves only 2 parameters. The absent branch corresponds to traceless two-row rectangular o(2, M) Young diagrams. In the case M = 1 this branch reduces to the two first terms.

It follows that a trace decomposition of any F (Y ) ∈ S 3 reads [43]
One concludes that the first line in (7.12) contains T k for k ≥ 2 that appear as coefficients in front of symmetrized combinations t (α 1 α 2 * ... * t α 2k−1 α 2k ) , while subsequent lines necessarily contain powers of c 2 . Any tensor on the plot (7.12) is proportional to particular combination (7.17) except for the first two scalar T 0 and vector T 1 representations.

Quotient higher spin algebras
Algebra S 3 is not simple. In what follows, we consider two types of ideals I ⊂ S 3 along with respective quotient algebras S 3 /I which we call vertical and horizontal ones according to their graphical interpretation (7.12) and trace decomposition (7.19).
For instance, factoring out the maximal ideal I 1 spanned by elements (7.8) yields the quotient H 1 = S 3 /I 1 spanned by a finitely many basis elements (7.20) corresponding to gl(2, R) ≈ gl(1, R) ⊕ sl(2, R) algebra. Indeed, using the trace decomposition (7.19) one notes that all elements in (7.12) save for T 0 and T 1 are proportional to sp(2) generators t αβ . It follows that all such elements belong to the ideal I 1 and therefore are to be factored out.

Horizontal factorization
The maximal ideal is the first element in a family of two-sided ideals where and g α 1 ...α 2k (Y ) is a rank-2k symmetric sp(2) tensor: t γρ , g α 1 α 2 ... * = δ α 1 ρ g γ α 2 + . . . , where the ellipses denotes all possible symmetrizations. Using the associativity of the * -product, the sp(2)-invariance condition (7.7), and the following elementary properties where F (Y ) ∈ S 3 , one shows that I k ⊂ S 3 is a two-sided ideal. Note that ideals (7.21) form an infinite flag sequence I 1 ⊃ I 2 ⊃ · · · ⊃ I k ⊃ · · · . (7.24) A quotient algebra H k = S 3 /I k is given by cf. (7.13). It is finite-dimensional and isomorphic to a direct sum of general linear algebras To prove (7.26) one notes that factoring out elements proportional to (7.22) for a given k is equivalent to truncating the plot (7.12) starting from (2k + 1)-th column. The remaining elements form (7.25).

Vertical factorization
Another type of ideals is given by a family where I t (c 2 ) is a t-th order * -product polynomial in the sp(2) Casimir element c 2 . Using the sp(2) invariance condition (7.7) one shows that I t ⊂ S 3 are two-sided ideals. From (7.19) and (7.12) it follows that the resulting quotient algebra H t = S 3 /I t is given by Any polynomial I t (c 2 ) can be decomposed into elementary monomials, so that an ideal corresponding to I 1 = c 2 + ν, where ν is a constant parameter, is special. Taking t = 1 in (7.28) one arrives at the quotient algebra H 1 ν = S 3 /I 1 ν given by Recalling that S 3 ≈ U(o(2, 1)) (7.16) and using the relation c 2 = C 2 + 3 4 obtained by taking M = 1 in formula (7.6), one finds that the above factorization is equivalent to factoring out elements proportional to C 2 + 3 4 from the universal enveloping algebra U(o(2, 1)). In this way, we obtain that H 1 ν = U(o(2, 1))/I C 2 + 3 4 +ν , and, therefore, H 1 ν is isomorphic to the higher spin algebra hs[ν] [24,25,44]. On the other hand, the algebra hs[ν] is spanned by polynomials of two spinor variables q α and an idempotent element K with commutation relations [q α , q β ] = 2iǫ αβ (1 + νK), {q α , K} = 0 [25].
Note that the two types of factorizations can be visualized on the plot (7.12). The horizontal factorization corresponds to truncating the plot horizontally starting from (2k + 1)-th column. The vertical factorization corresponds to truncating the plot vertically starting from t-th row.
For a given ν, all other ideals I 1 µ for µ = ν and ideals I k (7.21) for any k in the quotient S 3 /I 1 ν become the trivial ideal which is the entire quotient itself. Indeed, factoring out I 1 ν one obtains that in the quotient algebra H 1 ν the sp(2) Casimir element takes a particular value c 2 = −ν. Consider now ideal I µ ⊂ S 3 with parameter µ = ν. Using definition (7.29) one shows that elements of I µ restricted to quotient H 1 ν are of the form (µ − ν)g, where g ∈ H 1 ν . As a result, I 1 µ ≈ H 1 ν for µ = ν, and I 1 µ ≈ ∅ for µ = ν, so that the ideal becomes trivial. The same reasoning applies to another type of ideals I k restricted to the quotient algebra H 1 ν . To this end, taking in (7.21) elements g α 1 ...α 2k (Y ) = T α 1 ...α 2k (Y ) * g(Y ), where ∀g(Y ) ∈ S 3 , and using the formula where τ k is some non-vanishing normalization coefficient, one shows that I k contains elements g(Y ) * k−1 m=0 * (c 2 + α m ), where α m = m(2m + 1). Substituting the quotient value c 2 = −ν one finds that I k contains elements of the form g( For general values ν the appearance of these elements implies that the ideal I k is trivial, i.e., I k ≈ H ν .

However, for particular integer values
one finds that the ideal I k restricted to H 1 ν 0 is non-trivial, and, therefore, can be factored out. Indeed, ideal I k restricted to H 1 ν 0 does not contain any powers of the sp(2) Casimir element since c 2 = −ν 0 . On the other hand, it contains combinations T α 1 α 2 ...α 2l for l ≥ k only, cf. (7.22) and (7.24). Since the horizontal factorization yields a finite-dimensional quotient, we conclude that the result of such a double factorization is finite-dimensional as well: examining the plot (7.12) one finds out that basis elements of the double factorization span a general linear algebra, Note that the rank of the algebra (7.33) is even. In Conclusions 9 we discuss how to take account of odd values.
Finally, one can use a combination of the two types of ideals in a single factorization. For instance, consider a composite two-sided ideal I p 1 = t αβ * I p (c 2 ) * g αβ (Y ) provided that a sp(2) symmetric tensor g αβ is not proportional to t αβ , and I(c 2 ) is some p-th order polynomial in c 2 . The resulting quotient algebra is given by

Factorization via (quasi-)projectors
To describe quotients of algebra S 3 explicitly one employs the projecting technique elaborated in [33,31]. 17 Given a quotient H of algebra S 3 with respect to some ideal I one introduces a quasi-projector ∆ satisfying the basic property Then, it follows that elements of quotient H = S 3 /I can be parameterized as follows An educated guess is to consider the following ansatz Note that z = 2c 2 − 9/2, where c 2 is sp(2) Casimir operator. Variable z is invariant with respect to both sp(2) − o(2, 1) Howe dual algebras, [t αβ , z] * = 0 and [T A , z] * = 0. In particular, In Appendix B we explicitly analyze the projecting conditions (7.35) imposed on ∆(z) (7.37). We show that the horizontal projecting condition is given by an ordinary 2k-th order differential equation for function ∆ k (z). The vertical projecting condition is an ordinary 4-th order differential equation for function ∆ ν (z). In both cases the searched-for solutions have the form of the series ∆(z) = κ 0 z α + κ 1 z α+1 + κ 2 z α+2 + · · · , for some degree α ≥ 0 and fixed coefficients κ i depending on either k or ν. Also, we analyze solutions with parameter ν taking particular values (7.32).

Non-linear higher spin BF action
As a starting point, we formulate a non-linear higher spin theory in two dimensions as BF theory with gauge fields taking values in the adjoint representation of the infinitedimensional Lie algebra hc(1|2 : [1,2]) explicitly discussed in Section 7.2. After that, using the factorization procedure of Section 7.4 we describe reduced theories with fields taking values in the quotient higher spin Lie algebras.
The fields of the theory are 0-forms and 1-forms taking values in hc(1|2 : [1,2] From (7.11) it follows that the expansion coefficients in the auxiliary variables of (8.1) are 0-form and 1-form fields taking values in totally symmetric traceless o(2, 1) representations of any rank. Each independent field enters in infinitely many copies, cf. (7.2). We assume that fields (8.1) satisfy the reality conditions where the conjugation † is defined by (7.9).
The higher spin curvature associated to 1-form gauge fields (8.1) is defined as while the infinitesimal gauge transformations are where ε = ε(Y |x) is 0-form gauge parameter taking values in the algebra hc(1|2: [1,2]), and is the gauge covariant derivative.
Consider now an invariant bilinear form on the higher spin algebra needed to build a BF action. To this end, define a trace of any element F (Y ) ∈ hc(1|2: [1,2]) as follows [50] Tr(F (Y )) = F (0) .
The trace satisfies the cyclic property that can be directly shown using the definition (7.4) and the property that F is even function, F (Y ) = F (−Y ). It follows that the algebra hc(1|2 : [1,2]) can be endowed with the following invariant bilinear form where the gauge covariant derivative D m is given by (8.5). The equation (8.10) is the covariance constancy condition involving both fields Ψ and W m , while equation the (8.11) is the zero-curvature condition involving fields W m only. It follows that the gauge sector of the theory can be analyzed independently. Adding invariant potentials to the action results in that the curvature acquires non-vanishing right-hand-side. For instance, additional terms proportional to the second-order invariant operator I 2 = Tr(Ψ * Ψ) yields the deformation (2.12) discussed earlier within the linearized theory.
By construction, the higher spin BF action is invariant under the gauge symmetry transformations (8.4). On the other hand, the theory is manifestly diffeomorphism invariant as it is formulated via differential forms, while containing no metric tensor. The diffeomorphism transformations of fields (8.1) are given by the respective Lie derivatives that can be represented as follows The terms proportional to the field equations represent the trivial invariance transformations vanishing on the mass-shell. Indeed, given any action S[φ i ] depending on fields φ i , i = 1, 2, 3, ... one has a trivial invariance transformation δφ i = M ij δS/δφ j , where the parameter matrix is antisymmetric M ij = −M ij . Symmetries which differ by these trivial terms are equivalent. In our case, 0-form Ψ and 1-form W are identified with φ 1 and φ 2 . It follows that modulo the trivial transformations the diffeomorphisms are just a particular gauge transformation with a field-dependent gauge parameter, and, therefore, can be disregarded as independent symmetries. 18

Linearization around AdS 2 background
The higher spin theory (8.9) contains the gravitational subsector since the higher spin algebras under consideration always contain o(2, 1) subalgebra. Moreover, the ground state of the model is identified with the AdS 2 spacetime. It seems natural to have AdS 2 spacetime as the background, because in this way higher dimensional higher spin gauge theories extend to the 2d case while keeping their main characteristic features intact: higher spin gauge fields and the AdS background geometry. One should note, however, that contrary to d ≥ 4 higher spin theories the AdS 2 background is not necessarily required to have a consistent interacting theory. 19 Recall that switching on the cosmological constant Λ = 0 is indispensable to guarantee consistent gravitational interactions of gauge massless higher spin fields. In two and three dimensions it seems that taking Λ = 0 does not prevent having a consistent theory with higher spin symmetries because higher spin fields carry no local degrees of freedom.
Fixing the background connection W 0 we treat dynamical fields Ω as fluctuations, where W 0 satisfies the o(2, 1) zero-curvature condition (2.3) and describes AdS 2 spacetime. A background value of Ψ is discussed below, while perturbations over Ψ 0 are defined as where Φ are dynamical fields. Up to the second order in the fields the non-linear curvature (8.3) decomposes as 18 In particular, for the spin s = 1 two components of the diffeomorphism parameter ξ n (x) combine into a single scalar gauge parameter ε(x). For the spin s = 2 case one shows that the gauge transformation of the frame with o(1, 1) vector parameter ε a (x) and the diffeomorphism with parameter ξ n (x) are identified [51]. For the higher spins s > 2 diffeomorphism parameters form a subspace in the gauge parameter space. 19 See, e.g., Refs. [52], where 3d flat higher spin theory was discussed. where Substituting the perturbative expansions (8.14), (8.15) into the equations of motion (8.10), (8.11) one finds that the background fields satisfy the following equations The first equation above is the zero curvature-condition (2.3), while the background field Ψ 0 remains unknown. Next, the first-order equations are given by Suppose now that Ψ 0 is x-independent, that is dΨ 0 = 0. Then, the second equation in (8.18) It follows that o(2, 1)-invariant non-vanishing vacuum value of the 0-form field is a function of the sp(2) basis elements only are some (Y, x)-independent (constant) sp(2) symmetric tensor parameters, T α 1 α 2 ...α 2k is given by (7.22) and c 2 is sp(2) Casimir operator. 20 Recall that these properties guarantee the sp(2) invariance of Ψ 0 , cf. (7.19). The fluctuation field Ω is also sp(2) invariant, and therefore it commutes with any combination of t αβ . As a result, [Ω, Ψ 0 ] * = 0.
It follows that the linearized equations of motion (8.19) take the form The Abelian part of the gauge transformation (8.4) for fluctuations has the form where the linearized derivative D 0 reproduces the definition (2.6), while the above transformations themselves reproduce (2.7) and (2.8).
Now, the trace decomposition (7.19) that brings the higher spin algebra hc(1|2: [1,2]) into the basis where all basis elements are given by traceless o(2, 1) tensors (7.12) is expressed via the sp(2) generators. It follows that field Ω m decomposes into irreducible components as where components Ω On the other hand, field equations (8.22) can be represented via the background covariant derivative as D 0 Ω = 0 and D 0 Φ = 0, cf. (3.1), (3.2). Therefore, using D 0 t αβ = 0 one finds out that the field equations (8.22) can be decomposed into o(2, 1) irreducible components as well. In each irreducible spin-s sector equations of motion take the form (2.10); each pair of equations (2.10) comes in infinitely many copies. Whence, the spectrum of the model contains infinitely many copies of all integer spin-s subsystems,

Reduced BF higher spin models
The spectrum of the AdS 2 higher spin gravity model (8.9) is infinite and degenerate. It can be truncated in two possible ways.
• Horizontally reduced model: finitely many fields with spins bounded from above, each field appears in several copies.
• Vertically reduced model: infinitely many fields of all spins from zero to infinity, each field appears in a single copy.
It is clear that such reduced models are governed by respectively horizontal and vertical quotient higher spin algebras of Section 7.3.
We propose to describe reduced models with fields taking values in the quotient higher spin algebras by the BF action (8.9) modified by the projecting operator ∆ in the following manner 21 Tr ∆ * Ψ * R , (8.26) where, according to particular factorization, one chooses either the horizontal projector ∆ k or the vertical projector ∆ u of Section 7.4. By inserting ∆ we reduce the original spectrum of fields to a smaller subset of fields identified with representatives of the quotient algebra. Indeed, ∆ is defined to send all elements of the corresponding ideals in hc(1|2: [1,2]) to zero (7.35).
Reduced action (8.26) is invariant with respect to the gauge transformations (8.4). Additionally, it acquires a new type of invariance due to a degeneracy of the form (8.27), Let us consider a perturbative expansion of the reduced model (8.26). Both zerothorder and first-order equations are again equations (8.18) and (8.19) but now multiplied by ∆. A natural choice for the background is to take the AdS 2 connection W 0 as the vacuum 1-form field because it solves the equation of motion (8.29). As the background 0-form field we take an x-independent Ψ 0 (Y ). From (8.30) it follows that ∆ * [W 0 , Ψ 0 ] * = 0 which means that Ψ 0 can be chosen to be an element of the ideal, Ψ 0 ∈ I. However, using the shift symmetry (8.28) one observes that it can be equivalently set to zero. Therefore, from the very outset one can choose W = W 0 and Ψ 0 = 0 as representatives of the zeroth equivalence class in the quotient higher spin algebra.

Conclusions and outlooks
In this paper, we proposed a new class of two-dimensional higher spin models interpreted as the AdS 2 higher spin gravity and explored some of its global and local properties. The model is formulated by virtue of topological BF action for fields taking values in particular higher spin symmetry algebra containing o(2, 1) ≈ sl(2, R) subalgebra. Our analysis follows methods used within the unfolded approach to higher spin dynamics. In particular, we developed a two-dimensional version of the unfolded formulation resulting in a cohomological understanding of the BF dynamics. Using two different nilpotent operators acting on the field space of BF model we elaborate two metric-like formulations of the model. Our analysis of the linearized BF equations of motion both for 0-forms and 1-forms accomplishes the analysis of the 1-form sector performed earlier in [10]. We also discuss a new type of duality between two metric-like formulations obtained from a single BF frame-like theory.
We suggested a particular formulation of two-dimensional higher spin algebra hs[ν] employing the o(2, 1) −sp(2) Howe duality. In this way we extend the Vasiliev oscillator construction of d ≥ 4 higher spin Eastwood-Vasiliev algebras to the d = 2 case. Infinitedimensional higher spin algebras and their finite-dimensional truncations are realized as particular quotient algebras for which reason we classified relevant cases of ideals and corresponding factorizations. We explicitly described the projecting technique used to define the BF actions for fields taking values in the quotient algebras.
The d = 2 classification of ideals and factorizations extends to any d case. Obviously, using the ideals generated by the sp(2) Casimir operator and its powers one arrives at some quotient algebra with connections identified with higher spin partially-massless fields of any depth (e.g., see discussion in [43]). It should be realized as the symmetry algebra of higher order singleton representations of o(2, d) algebra [53]. 22 One can also discuss reduced models based on double factorizations of the form (7.34).
It is important to note that a given BF theory with a finite-dimensional algebra is necessarily topological one. The situation is more intricate in the case of an infinitedimensional algebra. For instance, the BF action for higher spin algebras considered in this paper is topological. On the other hand, a particular BF theory proposed in Ref. [8] describes self-interactions of matter fields via higher spin currents built of these matter fields. Nonetheless, the model is not topological because BF fields take values in a peculiar infinite-dimensional algebra containing hs[ν] as a subalgebra. The rationale behind this observation is that a BF action formulated on an infinite-dimensional field space may leave a room for local degrees of freedom.
In particular, it follows that BF actions may contain current interactions of matters fields, and, therefore, it is tempting to speculate that higher spin BF action has to do somehow both with currents and matter fields on equal footing. This idea conforms with the duality between the metric-like formulations described in this paper. Indeed, we find out that BF equations of motion can be simultaneously treated as matter field equations and conservation conditions. Below we list some interesting issues left beyond the scope of the paper.
• The form and properties of the mapping between two metric-like descriptions of the free field higher spin theory discussed in Section 6.2. The original linearized BF higher spin action functional can be treated as a parent action for the two dual formulations. • It is interesting to realize the universal enveloping algebra U(o(2, 1)) in terms of extended o(2, 1) − osp(n, 2) Howe dual pairs with arbitrary n ≥ 2.
• The role of parameter ν in the vertical reduced model is to be clarified. We have seen that the linearized equations of motion are independent on ν. It appears that ν comes out in the next orders. 23 • The flat space limit Λ → 0 in the BF higher spin models. The resulting theory should be a higher spin extension of the two-dimensional Poincare gravity suggested in [55] and further discussed in [36,56]. It should be governed by a non-semisimple higher spin algebra extending the (1 + 1) Poincare algebra.
Among other things, the AdS 2 higher spin gravity is interesting because the respective action functional is given in a closed form that makes possible to analyze many conventional questions like higher spin black hole solutions, supersymmetric higher spin extensions, quantization, etc. In particular, it is interesting to consider matter fermions interacting via higher spin fields and, therefore, to formulate a higher spin extension of the Schwinger model in AdS 2 spacetime. 24 Further, topological field theories are known to induce local degrees of freedom at the boundary. This is also the case for two-dimensional higher spin theories of the type considered in the present paper. The problem has been already partly discussed in the literature [9,11].
Acknowledgements. I am grateful to M.A. Grigoriev, S.E. Konstein, R.R. Metsaev, and E.D. Skvortsov for many useful discussions and comments. This research was supported by Russian Science Foundation grant 14-42-00047.

A Computation of the cohomology groups
In what follows, we compute the cohomology of the nilpotent σ ± operators acting on the space G s . To this end, one recalls some relevant group-theoretical facts on o(1, 1) Lorentz algebra representations and their tensor products.
Introducing a collective notation for symmetrized indices (a 1 ...a k ) ≡ a(k), one finds that a frame-like tensor T m a(k) being a tensor product of totally symmetric and traceless tensor with a vector decomposes into two o(1, 1) irreps of spins k − 1 and k + 1.
Recalling that a dimension of any integer spin o(1, 1) (non-scalar) irrep equals 2, the above statement can be simply understood as 2 2 = 2 + 2. On the other hand, any totally symmetric and traceful frame-like tensor A m a(k) decomposes into k n=0 T m a(n) , where T m a(n) are traceless with respect to fiber o(1, 1) tensors. The decompositions clarify the formula dim A m a(k) = 2(2k + 1).
To summarize, the following decompositions are useful in practice where A and T are two different scalar components. Their appearance is due to the relation A a|b = 1 2 A (a|b) + 1 2 A [a|b] = 1 2 A (a|b) + 1 2 ǫ ab A = 1 2 T (ab) + 1 4 η ab T + 1 2 ǫ ab A, where η mn T (mn) = 0 and ǫ ab is 2d Levi-Civita tensor. Vertical slash denotes independent groups of indices.
Consider operators σ ± given by (3.5) that act on the module G s of differential pforms which take values in o(1, 1) finite-dimensional irreps, T a 1 ...a k (p) , where p = 0, 1, 2 24 E.g., see a discussion of a particle moving in lineal gravitational fields [56]. and k = 0, 1, ..., s − 1, see Section 3.1. For the case s = 1 the cohomology computation is trivial so we give detailed consideration of the spin s ≥ 2 case only. (1) also vanish since it is exact, except for the case k = s − 1. Equation (A.5) at k = 1 should be analyzed separately because in this case decomposition into irreducible components is different, see (A.3). It follows that the closer condition sets to zero the antisymmetric part, while symmetric one is arbitrary. For s > 2 symmetric and traceless component cancels due to the exactness condition, while for s = 2 it remains intact. One concludes that cohomology is given by rank-s totally symmetric component and a scalar component T which comes as a trace part of T a (1) . Therefore, H (1) (σ − ) = {T, T a 1 ...as }, see (3.20).
Then, consider cohomology group H (2) (σ − ) defined by the following chain of conditions , δT a(k) where T a(k) , and T a(k+2) , 0 ≤ k ≤ s − 1, are respectively 2-forms, 1-forms, and 0-forms. Being a 3-from the first equation in (A.6) is identically satisfied. On the other hand, analysis of the exactness conditions in (A.6) is similar to previously done computation of H (0) (σ − ) and H (1) (σ − ). Repeating the reasoning we find that H (2) (σ − ) = {T a 1 ...a s−1 }, see (3.20).  where D (k) stands for k-th degree of the second-order differential operator 2) The ordinary differential equation D (k) ∆ k = 0 has 2k independent solutions. Among them we single out only those that have the form of the series ∆ = κ 0 z α + κ 1 z α+1 + κ 2 z α+2 + · · · , for some α ≥ 0. It turns out that α = 0 and there are k independent solutions of this type, ∆ i , i = 1, ..., k. Since equation D (k) ∆ = 0 comes as differential consequences of equation D (k−1) ∆ = 0, one concludes that k − 1 solutions ∆ i , where i = 1, ..., (k − 1) solve equation of lower rank and therefore can be found by induction, while the highest rank solution ∆ k does describe factorization (B.1). From the algebraic perspective, a set of analytical solutions to the horizontal projecting equation is clearly explained by the flag sequence of ideals (7.24).
An explicit form of solutions can be found straightforwardly provided that differential operator (B.2) is represented as D = 2(N z + 1) d dz + 1, where N z = z d dz is the Euler operator, so that searching for a solution in the form of power series yields a recurrent equation system. It is worth noting that using the horizontal factorization via projector (B.1) yields finite-dimensional quotient algebras (7.26) with basis elements realized as infinite formal power series of auxiliary variables Y A α , and not as bilinear combinations as one might expect from (7.5).
On the other hand, we know that the k = 1 horizontal projection yields the quotient H k ≈ gl(2, R) (7.26), while the double factorization in the case ν 0 = 0 yields H 1 ν 0 /I 1 ≈ gl(2, R) (7.33). The resulting quotients obviously coincide. Note, however, that for k > 1 the horizontal quotient algebra H k and the double quotient algebra H 1 ν 0 /I k are not isomorphic anymore, while the respective projectors do not coincide as well, see (B.8).