Higher Spin Gravities and Presymplectic AKSZ Models

As a step towards quantization of Higher Spin Gravities we construct the presymplectic AKSZ sigma-model for $4d$ Higher Spin Gravity which is AdS/CFT dual of Chern-Simons vector models. It is shown that the presymplectic structure leads to the correct quantum commutator of higher spin fields and to the correct algebra of the global higher spin symmetry currents. The presymplectic AKSZ model is proved to be unique, it depends on two coupling constants in accordance with the AdS/CFT duality, and it passes some simple checks of interactions.


Quantization of Higher Spin Gravities (HSGRA) is an important open problem together with
the problem of constructing more viable HSGRA. Concrete quantum checks of HSGRA require concrete models. Topological theories in 3d, which are higher spin extensions of Chern-Simons formulation of 3d gravity [1][2][3][4] or 3d conformal gravity [5][6][7], are well-behaved perturbatively due to the lack of any propagating degrees of freedom, but their non-perturbative definition requires further study, see e.g. [8]. Amplitudes of 4d conformal HSGRA [9][10][11] were studied in [12][13][14]. The most elaborate checks have been performed for Chiral HSGRA [15][16][17][18][19], which was shown to be one-loop finite [18][19][20]. In addition, there is a number of interesting computations of vacuum one-loop corrections [21][22][23][24][25][26][27][28][29][30]. Finally, the one-loop correction to the four-point function in the bulk dual of the free vector model can be reduced to a CFT computation and seems to be consistent with the vacuum one-loop results [31]. 2 The only class of perturbatively local HSGRAs with propagating massless fields and an action seems to be given by Chiral HSGRA and its various truncations. This was proved for flat space in [17] and it is likely to be the case for AdS 4 as well. However, Chiral Theory does not have a covariant action at the moment and this hampers the study of quantum corrections.
Nevertheless, the bulk dual of vector models [46,47] can be approached from the opposite end: abandoning perturbative locality one can get a very explicit description of this HSGRA at the level of formal algebraic structures; hence the name "Formal HSGRA". In practice, this means that one can construct an L ∞ -algebra that extends a given higher spin algebra.
Equivalently, one can think of the corresponding Q-manifold, where Q -an odd, nilpotent are only formally consistent and should not be taken at face value as PDE's or as field's theory equations, we can continue studying the formal structures restricted by the higher spin symmetries with the hope that we either find a 'canonical' Q with certain prescriptions that would lead to a reasonable field theory or the formal approach itself will allow one to define the theory as to be able to compute various physical observables.
The classical equations of motion ( * ) are non-Lagrangian as they stand. There is no theorem prohibiting the action principle for HSGRA extended with suitable auxiliary fields; and yet no one has succeeded in formulating it. This prevents applying the standard quantization methods (canonical or path-integral) to this class of field-theoretical models. Therefore, it seems reasonable to try other approaches to the quantization problem that are not so closely related to the Lagrangian or Hamiltonian form of classical dynamics. Let us stress that all holographic HSGRAs are Lagrangian theories, more or less by definition. The actions, however non-local they may be, can be reconstructed [42] from the CFT correlators. At the free level such actions reduce to the Fronsdal action [49] for an appropriate set of fields.
A possible alternative to the conventional quantization methods has been proposed long ago in [50]. It is based on the concept of a Lagrange anchor, which can be regarded as a covariant counterpart of (degenerate) Poisson brackets. In the non-degenerate case, the Lagrange anchor is just inverse to the integrating factor in the inverse problem of the calculus of variations. In the HSGRA context, the quantization method above was first considered in [51]. However, the Lagrange anchor behind the topological model of [51] does not reproduce the standard propagators for free higher spin fields in Fronsdal's theory. Therefore, the relation between the two theories is unclear beyond the classical equations of motion. In [52], the canonical Lagrange anchor was constructed for the subsystem of ( * ) that describes a free scalar field.
Contrary to the Lagrange anchor of the work [51], it involves an infinite number of spacetime derivatives, and it seems unlikely that one can remove them all by the inclusion of higher spin fields and interactions. As shown in [53], the Lagrange anchors for higher spin fields described by the Bargmann-Wigner equations also contain higher derivatives, the number of which increases with spin. In the two most recent papers [54], [55] a similar quantization method was implemented to reproduce propagators for free higher spin fields.
In the present paper, we consider another approach to quantizing (non-)Lagrangian theories. It employs the concept of a covariant presymplectic structure introduced in [56], [57]. The notion of presymplectic structure is, in a sense, dual to that of Lagrange anchor; both coincide for Lagrangian theories, being non-degenerate and inverse to each other. The degeneracy of a presymplectic structure reduces the algebra of physical observables admitting quantization.
More precisely, the kernel distribution of the presymplectic structure must annihilate quantizable observables. Hence, the bigger the kernel, the smaller the algebra of quantum observables.
Since all the gauge symmetry generators of the classical equations of motion belong to the kernel distribution, the quantizable observables are automatically gauge-invariant. As with the Lagrange anchor, classical equations of motion do not specify a compatible presymplectic structure uniquely, and different choices may lead to different algebras of quantum observables.
While the Lagrange anchor aims at the path-integral quantization of (non-)Lagrangian dynamics, the concept of presymplectic structure is more adapted to the deformation quantization.
As was proved in [116], each covariant presymplectic structure defines and is defined by some Lagrangian. The corresponding Euler-Lagrange equations, however, are weaker than the original ones whereby admitting more solutions. That is why we call such Lagrangians 'weak'. One may regard a weak Lagrangian as a solution to the inverse problem of the calculus of variations where the integrating factor is not necessarily invertible. Finding a covariant presymplectic structure is thus fully equivalent to constructing a weak Lagrangian.
We will show that for 4d HSGRA the corresponding weak Lagrangian has the form of an AKSZ sigma-model 3 [63], or more precisely, its presymplectic counterpart [61], [62]. For free higher spin fields such a weak Lagrangian was proposed in [64]. There is a number of immediate advantages of the presymplectic approach. Among these are (i) minimality: one does not have to introduce any auxiliary fields on top of what are already present in Eq. ( * ); (ii) background independence and gauge invariance: we do not have to pick any particular vacuum, like AdS 4 and the gauge symmetry is fully taken into account to every order in the weak curvature expansion; (iii) relation to the canonical quantization and to the Lagrangian formulation that 3 Historically, the importance of the AKSZ construction for the development of action principles for HSGRA was pointed out in [51], see also [59,60], which have been an important reference frame for our work even though we deviate significantly from these papers by considering the presymplectic case, where the groundwork has been laid in [61,62].
HSGRAs must have in principle; (iv) a complete classification and an explicit description of admissible presymplectic structures in terms of the Chevalley-Eilenberg cohomology of the underlying higher spin algebra, which can be reduced to a much simpler Hochschild cohomology and, finally, computed with the help of the techniques of [65].
The main results of this paper can be summarized as follows: • we prove the existence of the weak action -presymplectic AKSZ sigma-model -for 4d formal HSGRA and construct the first few terms in the weak-field expansion. Higher order corrections are proved to be unobstructed and can be found with the help of general techniques from [66][67][68][69]. The action depends on one additional coupling constant in accordance with conjectured duality to Chern-Simons vector models [35]; • In arbitrary dimension d the presymplectic AKSZ action for HSGRA begins with where ω and C are a connection and a matter field valued in a given higher spin algebra, V(ω, . . . , ω, C) is a cocycle of this higher spin algebra and − is the invariant trace; • for free fields we show that the corresponding presymplectic structure gives the correct quantization for fields with s ≥ 1 (the scalar field's presymplectic structure starts to contribute only at the next order). It also gives the correct commutation relations for the global higher spin symmetry currents. The action correctly reproduces the free field dynamics, i.e., on AdS 4 it turns to be a genuine action, rather than a weak action; • it is shown that the presymplectic structure obtained from the Fradkin-Vasiliev part [70] of the cubic action 4 agrees with the one resulting from the presymplectic AKSZ model, as well as the cubic vertex does. Therefore, the perturbative analysis performed up to the cubic level over the anti-de Sitter background is consistent with the background independent considerations of this paper. We also clarify the origin of the action.
The paper is organized as follows. In the first sections we, to a large extent, review all the required material: formal dynamical systems are defined in Sec. 2; the details specific to HSGRA in four dimensions are introduced in Sec. 3; we discuss presymplectic AKSZ sigmamodels in Sec. 4 with an example of gravity elaborated on. The covariant phase space of presymplectic models is discussed in Sec. 5. The central section, to which a skilled reader can immediately scroll down, is Sec. 6, where we present the presymplectic AKSZ action for 4d HSGRA and discuss its properties and relations to other results in the literature. Sec. 7 elaborates on the general properties of the presymplectic AKSZ models over the maximally symmetric higher spin backgrounds. The main statements of Sec. 6 are supported by technical Appendices A -D, where all necessary cohomology groups are computed.

Formal dynamical systems
The formal dynamical systems we are going to discuss are defined in terms of Q-manifolds, see e.g. [72][73][74]. By a Q-manifold we understand a Z-graded manifold endowed with an integrable vector field Q of grade 5 one. Since Q is odd, the integrability condition is a non-trivial restriction on Q. In mathematics, such vector fields Q are usually called homological or Q-structure [63,75]. Let us present some typical constructions of Q-manifolds that we will need later for reference.
Here ξ A are global coordinates on G [1] relative to the basis {e A }. By definition, It is easy to see that the condition Q 2 = 0 is exactly equivalent to the Jacobi identity in G.
Endowed with the action of Q, the space of smooth functions C ∞ By definition, |v i | = |x i | + n. We will refer to the Q-manifold (M, Q) as the first (tangent) prolongation of (M, Q). One can obviously iterate this construction producing bigger and bigger Q-manifolds. Moreover, the shifted tangent bundle may well be replaced by an arbitrary tensor bundle of M. The resulting Q-manifolds are particular examples of Q-vector bundles [76], [77].
In order to define a formal dynamical system we need a pair of Q-manifolds: the source and the target. As a source manifold M we always take the total space of the shifted tangent bundle T [1]M of a space-time manifold M equipped with the canonical homological vector field (2.3). Let N denote a target manifold with homological vector field Q. We will treat N in the sense of formal differential geometry [78], identifying 'smooth functions' on N with formal power series in globally defined coordinates w A : In particular, the homological vector field Q is given by the series for some structure constants Q A A 1 ···An . Besides, two more assumptions about the structure of the target Q-manifold N will be made.
1. There are no coordinates of negative grade among {w A }, so that the manifold N is actually N-graded.
2. The expansion (2.6) starts with quadratic terms in coordinates, that is, Notice that the structure constants Q A A 1 A 2 of a minimal homological vector field (2.6) satisfy the Jacobi identity of a graded Lie algebra as in Example 2.2.
Given the pair of Q-manifolds above, we identify the classical fields with the smooth maps w : T [1]M → N of degree zero. In terms of local coordinates, each map w is given by a set of relations w A = w A (x, θ), where w A (x, θ) are smooth functions of x's and θ's with |w A (x, θ)| = |w A |. The true field configurations are, by definition, those relating the homological vector fields, i.e., w * (d) = Q. Upon identification (2.2), the last condition takes the form of a system of differential equations, namely, The l.h.s. is given here by the differentials of the forms w A ∈ Λ • (M), while the r.h.s. involves exterior products of the same forms. Equations (2.7) are thus adopted as field equations for a collection of form fields on M. In what follows we will systematically omit the wedge product sign and write Eq. (2.7) simply as where the infinitesimal gauge parameters ε A are differential forms of appropriate degrees. Actually, the general coordinate transformations of the form fields w A are specifications of (2.9). If ξ is a vector field generating a one-parameter group of diffeomorphisms of M, then by Cartan's formula Hereinafter, the sign ≈ means "equal when the equations of motion hold", i.e., on-shell. Therefore we can set ε A = i ξ w A to reproduce the action of an infinitesimal diffeomorphism ξ on the solution space to the field equations (2.8).
Remark 2.4. The gauge symmetry (2.9) is known to be strong enough to gauge away all local degrees of freedom for any finite collection of fields {w A } provided that dim M > 1. The last fact can be seen as follows. Notice that each form field w A of degree > 0 comes 'with its own' gauge parameter ε A such that deg w A = deg ε A + 1. In a topologically trivial situation, the corresponding gauge transformation δ ε w A = dε A + . . . allows one to remove all physical modes of the field w A subject to the first-order differential equation dw A = . . . This suggests that all physical degrees of freedom are accommodated in the zero-form fields, let us enumerate them w a . Without a detailed analysis it is clear that the solution space of the equations dw a = . . . is parameterized by constants c a whose number is equal to the number of zero-forms w a 's. 6 Therefore if we are interested in non-topological field theories, i.e., models with propagating degrees of freedom, then we have to consider infinite multiplets of zero-form fields.
Remark 2.5. It is known that any system of (gauge) partial differential equations can be brought into the form (2.7) at the expense of introducing an infinite number of auxiliary fields [79,80] and references therein. In the context of HSGRA this is known as an unfolded representation [48,81]. Of course, care must be exercised in treating (2.7) as a system of partial differential equations whenever an infinite number of fields and interaction vertices are involved into the game. That is why we refer to (2.7) as a formal dynamical system. In this paper, we leave aside all subtle analytical issues related to the field equations (2.7) focusing upon their formal consistency. 6 In general, the higher-degree forms w A may also add up to the physical sector a finite number of global degrees of freedom associated with the de Rham cohomology of the space-time manifold M .
Remark 2.6. Notice that the system remains consistent if we omit all the vertices in the r.h.s. of (2.7) except the quadratic ones. The resulting system is defined solely in terms of the underlying Lie algebra G. Although the truncated equations of motion are still non-linear, one may think of them as describing free field dynamics; the 'genuine' interaction vertices start form the cubic order. From this perspective switching on a consistent interaction amounts to deforming a quadratic homological vector field. Hence, all non-trivial interactions admitted by a free gauge system are fully controlled by the cohomology of the graded Lie algebra L as discussed in Example 2.2. This leads to a refined version of the Nöether procedure (see e.g. [82]) and may be regarded as a main technical advantage of the formal dynamical system approach. We will illustrate this point in the next section.

Higher spin gravity in four dimensions
The gauge theory of interacting higher spin fields delivers the major class of formal dynamical systems with propagating degrees of freedom. Being general covariant and involving a massless field of spin two, they are usually referred to as Higher Spin Gravities (HSGRA). As explained in the previous section, the structure of the corresponding equations of motion is largely controlled by the underlying graded Lie algebra G.
Below we specify the algebra G for the case of 4d HSGRA.
(1) G is concentrated in degrees zero and one, so that G = G 0 ⊕ G 1 as a vector space and This allows us to regard the odd commutative ideal G 1 as a module over the even subalgebra G 0 .
(2) The G 0 -module G 1 is given by the adjoint representation of G 0 . Therefore G 0 ≃ G 1 as vector spaces.
(3) G 0 = gl n (A), that is, G 0 is the Lie algebra of n × n-matrices with entries in an associative algebra A.
where the complex associative algebra A is given by the smash product A = A 1 ⋊ Z 2 of the first Weyl algebra A 1 and the cyclic group Z 2 generated by an involutive automorphism of A 1 . More precisely, A is the unital C-algebra on the three generators y 1 , y 2 , and κ obeying the relations Hereinafter, α = 1, 2 and the brackets (braces) stand for the commutator (anti-commutator). The first relation in (3.1) is just Heisenberg's commutation relation for the 'canonical variables' y 1 and y 2 . Adding the generator κ makes the outer automorphism In what follows we will denote the generators of the left and right tensor factors in A = A ⊗ A by (y α , κ) and (ȳα,κ), so that the general element of A can be written as f , g, h, and v being some ordered complex polynomials in y's andȳ's. The complex conjugation of the ground field C extends to the semi-linear anti-involution of A that takes y α and κ to (y α ) * =ȳα and κ * =κ.
Remark 3.1. The Lie algebra G 0 contains sp 4 (R) ≃ so(3, 2) as a real subalgebra. The latter is spanned by the unit n × n-matrix 1 1 multiplied by real quadratic polynomials in y's andȳ's; in so doing, the complex conjugate polynomials {y α , y β } and {ȳα,ȳβ} generate the complexified Lorentz subalgebra so(3, 1), while y αȳβ correspond to the AdS 4 transvections. The algebra so (3,2), being the Lie algebra of isometries of 4d anti-de Sitter space, suggests that the empty AdS 4 may appear as a natural vacuum solution of 4d HSGRA.
Remark 3.2. The full associative algebra generated by y's andȳ's is clearly isomorphic to the second Weyl algebra A 2 = A 1 ⊗ A 1 . Let hs ⊂ A 2 denote the subalgebra of all elements commuting with κκ. It is spanned by the even polynomials f (y,ȳ) = f (−y, −ȳ) and is called the higher spin algebra [83,84], hence the notation. Since the element κκ generates a Z 2 subgroup in the Klein four-group Z 2 × Z 2 = {1, κ,κ, κκ}, one can also characterize hs as a Z 2 -invariant subalgebra of A 2 . Extending hs with κ andκ, we get the smash product algebra hs ⋊ (Z 2 × Z 2 ) called usually the extended higher spin algebra. Given the graded Lie algebra G = G 0 ⊕ G 1 above, we can define 4d HSGRA as a formal dynamical system with a one-form field ω and a zero-form field C, both taking values in gl(A).
Eqs. (2.7) assume now the form Here ⋆ combines the wedge product of differential forms with the matrix product in gl(A). To look for the dynamical equations for interacting massless fields of all spins in this form was first proposed in [48]. Geometrically, the target space of fields C and ω is given by an infinitedimensional Q-manifold coordinatized by variables of degree zero and one. In the smooth setting such Q-manifolds are known to be equivalent to Lie algebroids [85]. This allows one to Omitting the interaction vertices V k , we are left with the system describing a free HSGRA.
Indeed, in that case one can view Eq. In a slightly different language this is the content of the Flato-Fronsdal theorem [86] stating that the tensor product of free massless 3d scalar and fermion with themselves decomposes into a direct sum of massless fields with all spins. As was shown by Dirac [87] many years before that, the free massless 3d scalar and fermion, as representations of so(3, 2), can be realized as even and odd vectors in the Fock space wherein A 2 acts naturally. 7 Together with the Flato-Fronsdal theorem, this equips the higher spin multiplet with the action of A 2 (understood as a Lie algebra). The appearance of Z 2 is a bit harder to explain. The group Z 2 realizes 7 Let us choose creation and annihilation operators, [a α , a †β ] = δ α β , instead of y α andȳα. Then, sp 4 (R) is realized by the bilinears in a and a † . The scalar/fermion states |v correspond to the span of f (a † )|0 for even/odd f (a † ) that act on the vacuum |0 .
an isomorphism that allows one to treat elements from the tensor product |v |w (states) as elements |v w| from the higher spin algebra (operators). As a result one can embed the states into the higher spin algebra. This automorphism is realized by κ,κ. The smash product algebra A, which extends the higher spin algebra hs with κ,κ, is useful: it is the deformation of A that allows one to reconstruct all vertices V k in (3.3), see below and [69] for explicit formulas.
Remark 3.4. It should be noted that the spectrum of fields generated by the graded Lie algebra G above is superfluous, containing more fields than actually needed to describe 4d HSGRA.
The true physical degrees of freedom are accommodated in the graded subalgebra G ′ ⊂ G, where G ′ 0 = gl(hs)Π, Π = (1 + κκ)/2 and G ′ 1 = gl(hs)K, K = (κ +κ)/2, K 2 = Π, see Remark 3.2. The corresponding field configurations are given by ω = ω(y,ȳ)Π and C = C(y,ȳ)K such that ω(−y, −ȳ) = ω(y,ȳ) and C(−y, −ȳ) = C(y,ȳ). The use of the 'extended' algebra G offers considerable technical advantages over G ′ . In particular, it makes possible applying the Künneth formula to the tensor product A = A ⊗ A. The redundant fields do not interfere the dynamics of the physical ones and can easily be excluded at the end of all calculations. This final projection onto the subspace of physical fields will always be implied in the sequel.
A closer inspection of these equations shows that they are equivalent to the Bargmann-Wigner equations for the (matrix-valued) massless fields of all integer spins, see e.g. [48,88].  ν being a complex parameter. The resulting one-parameter family of algebras A(ν) is known under the name of deformed oscillator algebra. Implicitly, it was first introduced by E. Wigner in his 1950 paper [89] on foundations of quantum mechanics. See [90][91][92] for subsequent discussions. The general element of A(ν,ν) = A(ν) ⊗ A(ν) is still given by an ordered polynomial in y's,ȳ's, κ, andκ. 8 There are explicit, albeit somewhat cumbersome, formulas for the product of such polynomials [93][94][95][96][97][98]. Choosing, for example, the symmetric (or Weyl) ordering for y's andȳ's, while keeping the generators κ andκ in the rightmost position, one can write the following expansion for the •-product of two polynomials a(y,ȳ) and b(y,ȳ): Here * stands for the usual Weyl-Moyal product in A 2 and the collection of bilinear operators {φ nm } defines a two-parameter deformation of the full algebra A 'in the directions of κ and κ'. A nice integral representation for the first-order deformations φ 10 and φ 01 can be found in The surprising thing is that knowledge of the interaction vertices for a particular vacuum solution C = µ1 1 permits reconstruction of the r.h.s. of Eqs. (3.3a, 3.3b) for arbitrary C! As was shown in [69], it is possible to express all the V's through compositions of the bilinear maps φ nm entering the expansion (3.8). For example, (3.9) Here we wrote down explicitly the gl-indices i's and j's. As is seen, they are all contracted in chain. Of course, all the vertices are defined modulo field redefinitions. The modulus of the complex parameter ν can obviously be absorbed by rescaling C (or µ for C = µ1 1, which is why the value of µ played no role). Therefore, setting ν = e iθ we are left with the only free parameter θ, which allows one to interpolate between the HSGRAs of type A (θ = 0) and type B (θ = π/2) [47]. In the context of AdS/CFT correspondence, these two theories 9 should be dual [46,47,99], respectively, to the free boson and fermion vector models on the boundary of AdS 4 . For general θ, this family of HSGRAs is expected to be dual to Chern-Simons matter theories [35].
Historically, explicit expressions for the first two vertices V 3 and V 4 were found in the works [48], [100]. In the subsequent paper [101], a systematic method was developed for generating all the formal interaction vertices in 4d HSGRA. A direct relation of the interaction problem with that of deformations of extended higher spin algebras was established in our papers [68,69]; this solves the problem of formal HSGRA for any given higher spin algebra and allows one to construct classical integrable systems out of any family of associative algebras. There is a number of formal HSGRA models available in the literature [60,69,[101][102][103][104][105][106][107][108], which include original models, variations and different realizations.

Presymplectic AKSZ models
Like the previous two sections this one is mostly expository. In order to formulate a class of field-theoretic models in the title we need to equip the target space N with one more geometric structure discussed below.
First of all, we note that the N-grading on the target space N can be conveniently described by means of the Euler vector field A presymplectic structure of grade n is, by definition, a closed two-form Ω on N such that The pair (N , Ω) is called a presymplectic manifold. If the matrix (Ω AB ) of the two-form Ω happens to be non-degenerate in some (and hence any) coordinate system, then one speaks of a symplectic structure and a symplectic manifold. In that case the inverse tensor Π = Ω −1 defines a Poisson structure on N of grade (−n). Denote by ker Ω the space of all vector fields V on N such that i V Ω = 0. Since dΩ = 0, the vector fields of ker Ω span an integrable distribution on M. We will refer to it as the kernel distribution of Ω.
A vector field X (a function H) is said to be Hamiltonian if there exists a function H (a vector field X) such that One can easily verify that the bracket is well-defined and satisfies all the required properties: bi-linearity, graded anti-symmetry, the Leibniz rule, and the Jacobi identity. In the case of symplectic manifolds any function is a Hamiltonian and {H, F } = Π(dH, dF ). 3.
The first two statements follow immediately from Cartan's homotopy formula To prove the last statement, write the presymplectic form in terms of local coordinates as In case deg N > |Ω| some of the differentials dw A do not enter Ω and the two-form is necessarily degenerate. If now |Ω| > 2 deg N , then |Ω AB | > 0 and the supermatrix (Ω AB ) is again noninvertible being composed of elements that are at least linear in w's.
The one-form Θ of item (1) is called a presymplectic potential for Ω. It is clear that 11 |Ω| = |Θ| and the equation defines Θ up to an exact one-form dΦ, e.g. one can always take Θ = 1 n i N Ω. In the following we are interested in presymplectic Q-manifolds (N , Ω, Q) that satisfy the additional condition L Q Ω = 0. According to item (2) of the proposition this means that the homological vector field Q is Hamiltonian, i.e., H being the Hamiltonian. Since (L Q ) 2 = L Q 2 = 0, one may also say that Ω is a cocycle of the differential L Q : Λ 2 (N ) → Λ 2 (N ) increasing the grade of a two-form by one unit. As one might suspect, only non-trivial Q-cocycles will be of interest to us below.
Consider now the N-graded presymplectic Q-manifold (N , Q, Ω) as the target space of a formal dynamical system (2.8), whose source manifold M fulfills the only condition 10 It follows from the definition that Ω AB = Ω BA (−1) |w A ||w B |+|Ω|(|w A |+|w B |)+1 . 11 To avoid confusion we note that the grade is defined by the Euler vector field (4.1); it counts the total degree of w's irrespective of whether they appear as variables w A or differentials dw A . In other words, the Euler vector field does not take the exterior differential d into account.
Given these data, the AKSZ Lagrangian for the form fields w A reads 12 where Θ A (w) and H(w) are exterior polynomials in w's defined by Eqs. (4.5) and (4.6). By definition, L is a form of top degree on M. With the help of (4.6) one can readily bring the variation of L into the form As usual the first term defines the Euler-Lagrange (EL) equations while the second term specifies possible boundary conditions.
If the two-form Ω happens to be symplectic, then its matrix (Ω AB ) is invertible and the EL equations (4.10) are fully equivalent to the field equations (2.8). In this case, (Ω AB ) plays the role of the so-called integrating multiplier in the inverse problem of variational calculus.
Moreover, passing to global Darboux coordinates on (N , Ω) considerably simplifies the 'kinetic term' in (4.8), bringing the Lagrangian into the form Remark 4.2. The Lagrangians of the form (4.11) are called AKSZ (sigma-)models after Alexandrov, Kontsevich, Schwartz, and Zaboronsky [63], who proposed them in the mid-1990s. A good deal of topological field theories can be formulated or re-formulated within the AKSZ approach for an appropriate quartet (M, N , Q, Ω), see [72,[109][110][111][112] for reviews and developments. There is also a deep relationship between the AKSZ construction of topological field theories and the Batalin-Vilkovisky (BV) formalism of gauge theories [113,Ch. 17]. This manifests itself in a simple and elegant form of the BV master action associated with the gauge invariant Lagrangian (4.11). In order to construct the corresponding BV action one simply allows the fields w : T [1]M → N to be the maps of arbitrary Z-degree and this yields automatically the right spectrum of auxiliary (anti-)fields of the BV formalism. When evaluated on such promoted form fields w and integrated over M, the Lagrangian (4.11) gives the desired BV master action. 12 We emphasize that w A ≡ w A (x, dx) are fields, while w A are just target space coordinates. Correspondingly, dw A is a space-time form of degree |w A | + 1 and should not be confused with the 1-form dw A of grade |w A | on the target space. The grade (as opposite to degree) of all fields dw A is 1. Since dw A and dw A have different grade/degree, the position of various variables is important in the formulas below.
Remark 4.3. Contrary to the symplectic case, the general presymplectic AKSZ models have received much less attention in the literature. Here we should mention the three papers [61,62,114], where the presymplectic AKSZ approach to gauge theories was put forward together with the analysis of some models. In the first paper, the authors discuss the presymplectic AKSZ models from the perspective of frame-like formulations known for many gauge theories, including higher spin fields. The second paper introduces the AKSZ formalism into the geometric theory of PDEs through the notion of an intrinsic Lagrangian. It was also demonstrated that a large class of Lagrangian gauge theories admits a presymplectic ASKZ action.
The papers [62,114] focus on the presymplectic AKSZ formulation for Einstein's gravity and [114] adds a BV interpretation. We also review the last model and the scalar field in Examples 4.4 and 4.5 below.
Turning back to the general case, we need to examine the equality relating Eqs. (2.8) and (4.10). Rather than discuss this in full generality, let us only highlight some key points. First of all, any solution to the equations E B = 0 obviously satisfies E A = 0 whatever the matrix (Ω AB ). Furthermore, if the matrix is degenerate it seems reasonable to declare the EL equations to be 'weaker' than the original ones. This, however, is not always the case. The crux of the matter is hidden integrability conditions that must be allowed for.
In order to make them explicit, suppose that all null-vectors of the matrix Ω AB (w) come from the kernel distribution of the presymplectic structure Ω. Let us further assume that the distribution ker Ω is spanned by a set of vector fields K a on N . The integrability of ker Ω implies the commutation relations for some structure functions f 's. Besides, it follows from the identity L Q i Xa Ω = 0 that the for some functions U's. Now Eq. (4.10) says that the forms E A determining the l.h.s. of Eq. (2.8) constitute a null-vector of the matrix Ω AB (w) . Under the assumptions above, this amounts to the equality Here λ a are new form fields of appropriate degrees and identification (2.2) is implied. Clearly, the last equations are completely equivalent to the EL equations (4.10). Although the new fields λ a enter the equations in a pure algebraic way, their dynamics are not entirely arbitrary.
Indeed, applying the de Rham differential to both sides of (4.15) and using Rels. (4.13) and (4.14), we find If all the vector fields K a , being linearly independent, are contained in the r.h.s. of Eq. (4.15), then the last condition amounts to The analysis of the general case is complicated by two points. First, it may happen that every generating set {K a } for ker Ω consists of linearly dependent vectors. This implies the existence of left null-vectors Z α for the rectangular matrix {K A a }, so that Z a α K A a = 0. If the system of null-vectors {Z α } is complete, then we can still divide the l.h.s. of Eq. (4.16) by the K a 's at the expense of introducing new form fields ξ α and adding the term ξ α Z a α to the r.h.s. of Eq. (4.17). The verification of formal integrability gives further equations for ξ α and so on. 13 The second point concerns the case where not all the null-vectors K a are actually present in Eq. (4.15). Here, besides the differential equations, one can find some algebraic constraints on λ a associated with those K a 's that dropped out of the r.h.s. of Eq. (4.15) on account of degree.
(Both the points are exemplified below.) Under reasonable assumptions one may repeat this construction once and again. The result is a formal dynamical system equivalent to the EL equations (4.10). All the above comments on equivalence to the original equations E A = 0 are still valid for the fully extended system. We are going to detail this construction elsewhere. 13 The problem we have to deal with here is quite similar to that of reducible gauge symmetries, see e.g. [113,Ch. 10]. From the algebraic standpoint, each extension defines and is defined by a certain free resolution of the C ∞ (N )-module ker Ω.
The last comment concerns gauge symmetries. One should realize that not every gauge transformation (2.9) of the equations E A = 0 can be promoted to a symmetry of the corresponding AKSZ model (4.8), unless Ω is non-degenerate. Some necessary and sufficient condition for this to happen are discussed in [61]. In particular, the general coordinate transformations (2.10) may no longer be part of (2.9). On the other hand, the null-vectors of the presymplectic structure Ω give rise to extra gauge symmetries of the form Example 4.4 (Pure gravity). In order to construct a first-order formulation of 4d gravity with negative cosmological constant, we can start with the Lie algebra G = so(3, 2). As explained in Example 2.2, it gives rise to the homological vector field  Notice that it does not depend on λ. The 2-form Ω GR , being obviously closed, turns out to be degenerate. In degree one and two, the kernel distribution is spanned by the family of null-vectors where the constant parameters C ab,cd and H abc,d form traceless Lorentz tensors.
In accordance with the general philosophy, the target space coordinates e a and ω ab are promoted to the one-form fields e a = e a µ dx µ and ω ab = ω ab µ dx µ associated, respectively, with the vierbein and the spin-connection on M. Applying now the general formulas of Proposition 4.1, we can write the following AKSZ Lagrangian: Varying it w.r.t. the spin connection and vierbein, we learn that Here De a = de a − ω a b e b is the Lorentz-covariant differential of the vierbein and R ab = dω ab − ω a c ω cb is the curvature tensor of D. As the presymplectic form (4.19) has no null-vectors of degree 1 'in the direction of e', the first equation (4.22a) is actually equivalent to It says that the spin connection ω ab is torsion free and can be uniquely expressed via e a and de a . As to the second equation ( where C ab,cd is now a collection of zero-form fields. Considered as a Lorentz tensor, C ab,cd is anti-symmetric in the first and second pairs of indices and has zero trace. Checking the formal integrability of the full system (4.23), (4.25), as explained above, we get both algebraic and differential constraints on C's. The former are given by e a e b e c C ab,cd = 0 and force the tensor C ab,cd to have the symmetry of 'window' Young diagram. Then Eq. (4.25) identifies C ab,cd as the Weyl tensor. As for the differential constraints, they have the standard form dC ab,cd = . . ., where the r.h.s. involves new zero-form fields C ab,cd,e with the symmetry of a two-row Young diagram. These new fields owe their existence to the over-completeness of the null-vector system (4.20): as C ab,cd,e cannot have three anti-symmetric indices, the shift C ab,cd → C ab,cd + C ab,cd,n e n does not affect K 1 C . This gives the null-vectors Z's of the second generation and so on. The above extension procedure never stops generating an infinite number of zero-form fields C ab,cd,··· .
This results in a formal dynamical system of the Einstein gravity (with infinitely many zeroforms, as promised for a field theory), whose explicit form is not known, but can, in principle, be obtained with the help of [79,80].
Example 4.5. (Gravity + scalar field). The above AKSZ formulation of pure gravity can easily be upgraded to include interaction with a scalar field. To illustrate the idea of how one can extend the construction based on an algebra with some of its modules we consider the simplest finite-dimensional module. Specifically, we extend the underlying Lie algebra so (3,2) to the inhomogeneous anti-de Sitter algebra iso(3, 2) and prescribe the abelian ideal of the latter degree one. 15 This results in the graded Lie algebra G = G 0 ⊕ G 1 , where G 0 = so(3, 2) and G 1 ≃ R 3,2 (as vector spaces). The associated homological vector field on G[1] takes the form Here (ϕ, π a ) are new coordinates of grade zero associated with the ideal G 1 . In order to be The kernel of the form Ω is generated by the vector fields (4.20) together with (4.28) Here the constant parameters H ab and C ab form, respectively, anti-symmetric and traceless Lorentz tensors. The corresponding AKSZ Lagrangian now reads Eliminating the auxiliary fields π a with the help of their equations of motion, one obtains the standard second-order formulation for the scalar ϕ coupled to Einstein's gravity with cosmological constant. The mass of ϕ, however, is below the unitarity bound unless λ = 0. By definition, the space G 1 corresponds to the vector representation of so (3,2). Instead one can take any finite-dimensional module of so(3, 2) associated with the totally-symmetric traceless tensors of definite rank and get other discrete values of masses below unitarity. In order to let the mass be arbitrary one has to start with an infinite-dimensional so(3, 2)-module, see e.g.

Covariant phase space of a presymplectic AKSZ model
A glance at the AKSZ Lagrangian (4.8) is enough to observe its similarity with the least action principle in Hamiltonian mechanics and this is more than just an analogy. Actually, the targetspace presymplectic form Ω induces a presymplectic structure on the covariant phase space of fields, which then endows the algebra of physical observables with a Poisson bracket. We will not dwell here on the covariant Hamiltonian formalism in field theory referring the reader to the papers [56,57,64,116] for general discussions and examples. The basic idea is to treat the boundary term in (4.9) as defining a 'functional one-form' on the space of fields: Here Σ ⊂ M is an arbitrary Cauchy surface of initial data for the equations of motion. Treating now the variation symbol δ as an exterior differential on the functional space of fields and applying it to (5.1), we get the two-form Our convention here is that It is fit to and motivated by the concept of variational bi-complex, see e.g. [117][118][119].
Following the general recipe, the space of Hamiltonian function(al)sĤ = Σ H(w), w ∈ Sol(E A ), is then endowed with the covariant Poisson bracket (4.4), which can be viewed as a precursor for the canonical quantization of the theory.
Another important remark is that the correspondence between the presymplectic structure Ω on the target space N and its functional counterpartΩ restricted to either solution space is far from being one-to-one. Indeed, shifting the integrand in (5.2) by an on-shell exact form is Q-exact whenever λ = 0: This is no surprise as the Lie algebra G underlying the homological vector field (4.18) is simple.
It is easy to realize that Q-invariant presymplectic structures Ω of grade 3 correspond to one-cocycles of G with coefficients in S 2 G * , the symmetrized tensor square of the coadjoint form Ω is found, the corresponding presymplectic potential Θ = 1 3 i N Ω defines immediately the 'kinetic' term of an AKSZ Lagrangian a , e 0 a ) in the graded Lie algebra G = G −1 ⊕ G 0 underlying the free theory. To control formal powers series C a we also introduce the vector field where 3) The leading term Q 0 , being a homological vector field by itself, determines the graded Lie algebra G as explained in Example 2.2. A similar expansion for the thought-for presymplectic structure can be written as Ω = Ω m + Ω m+1 + · · · , L N C Ω n = nΩ n , (6.4) a 1 ···anabc ω a ω b dω c dC an + C a 1 · · · C a n−2 Ω (2) a 1 ···anabc ω a ω b ω c dC a n−1 dC a n−2 , where we suppose that Ω m = 0. The defining conditions of a presymplectic structure impose an infinite set of linear relations on the structure constants Ω (0) , Ω (1) , and Ω (2) . In particular, on substituting expansions (6.2) and (6.4) into the condition of Q-invariance (6.6), we obtain the chain of equations Chern-Simons matter theories, which we discuss below.
Equations recap. While the general theorems of Appendix D guarantee that the presymplectic structure to be discussed below is unobstructed, we would like to work out a few terms explicitly as they are the most important ones. To this end let us write down Eq. (3.3) up to the first order in C as The simplest vertex has the factorized form [69,121], see [48] for the explicit expression, where Φ ν represents the first deformation, cf. (3.8), of the extended higher spin algebra A: 16 For manipulations below, it is convenient to rewrite (6.9) in a more compact form as Here and we do not write the gl-indices of fields explicitly; it is assumed that all the matrix indices are contracted in chain as in (3.9). The main property of the space-time 2-form Φ ν (ω, ω), which provides the formal integrability of the differential equations (6.12) modulo C 2 , is DΦ ν (ω, ω) ≈ 0 mod C . The formulas above will suffice for all formal manipulations. In order to make contact with the field theory approach we have to adjust the cubic vertex (6.10). 17 We will only need the fact that there exists a better representative V can ν , one may call it 'canonical', such that • ω, C) = νH αα ∂ α ∂ β C(y,ȳ = 0) +νHαα∂α∂βC(y = 0,ȳ) . More generally, one can perform a field redefinition in equations (6.9) that maps the vertex (6.10) to another representative, e.g. V can ν , which always obeys (6.16).
Presymplectic structure and AKSZ to LO. With the conventions and notation above the leading order term of the thought-for family (6.4) of presymplectic structures on N = G [1] is given by Here µ is an arbitrary complex parameter and the brackets − stand for the matrix trace and the trace in the ⋆-product algebra A. It is important that µ differs from ν in the vertex (6.10).
For the case of HSGRA in d > 4 the doubling of φ's in (6.10) does not take place and there is a clear distinction between the cocycle participating in Θ and the cocycle that deforms the equations, which we will make a few remarks about at the end.
As is clear from the explicit form of Θ 1 , one can instead choose for any representative V µ in (6.9) as long as µ = ν. The last inequality ensures non-triviality of the presymplectic structure to the leading order. Explicitly, 19a) (6.19b) 17 As it stands the vertex (6.10) does not lead to the desired field equations even at the free level [48]. What happens is that the free equations mix fields of different spins. Nevertheless, one can diagonalize the equations by performing a linear change of variables; in so doing, the non-commutativity of covariant derivatives, [∇ a , ∇ b ] ∼ λ, plays a crucial role. This example reveals a general fact that a formally integrable Q, i.e., Q 2 = 0, has to be adjusted as to make contact with field theory.
Notice that the component Ω (0) 1 vanishes on the maximally symmetric background C = 0. With the help of (6.15) we find for free fields in AdS 4 : 18 This is an admissible presymplectic structure [64]. For free fields, the parameter ν in (6.15) does not play any role and can be eliminated by rotating (anti-)holomorphic Weyl tensors (Cα (2s) ) C α(2s) by the phase of ν. It is also clear that for µ = ν the presymplectic structure (6.20), being proportional to the r.h.s. of (6.15), is trivial.
It is especially interesting to see how the presymplectic AKSZ action look like to the leading order. The equations to this order describe free fields C propagating over a maximally symmetric higher spin background ω. not come from an action. Therefore, the main function of Θ 1 is to provide a presymplectic treatment of (6.21). The corresponding presymplectic AKSZ Lagrangian reads It acquires a particularly nice form for the canonical vertex V can µ . Surprisingly, in that case it describes the right propagating degrees of freedom on AdS 4 [122]. (In general, one might expect to obtain only a Lagrangian whose EL equations are weaker than the ones we need).
The presymplectic structure (6.20) or, equivalently, the action (6.22) can be checked against the well-known Fradkin-Vasiliev action [70] and its precursor [123]. First, we need to decompose the curvature R = dω − ω ⋆ ω as follows: where R − (R + ) takes the m < n (m > n) part of the sum and R 0 corresponds to the terms with m = n. Note that the sum is over even m + n since we consider bosonic fields only. The two-form R 0 contains the (higher spin) torsions. As with the conventional gravity, one can solve the zero-torsion equation for some auxiliary fields in a purely algebraic way. With account of this equation, we can write the following integral for the Pontryagin topological invariant: Here − combines the matrix trace and the trace on the Weyl algebra A 2 . The Fradkin-Vasiliev action now reads where σ = sgn(N y − Nȳ) and N y = y ν ∂ ν , Nȳ =ȳν∂ν are the number operators for y andȳ. The purpose of σ-map is to project out the torsion and to flip the sign of R − terms. For example, the usual Pontryagin's invariant and the MacDowell-Mansouri action [124] have the form 19 As is well-known in the case of Yang-Mills theory and gravity one can add up the action and the topological invariant to get where τ ± ∈ C are combinations of the coupling constant g 2 and the theta-angle θ, τ ± = (a/g 2 ± iθ/b) (here, a, b are some numerical constants). This introduces one more coupling θ, which is invisible in the equations of motion. This trick makes the action reminiscent that of 3d (higher spin) gravity, which is a non-degenerate sum of two Chern-Simons actions. With the help of τ the SL(2, Z)-action on the space of 3d conformal field theories [125] extends to the higher spin fields in bulk [126,127]. 20 Any non-degenerate combination (τ − = τ + ) gives 19 We put i in front of MacDowell-Mansouri action assuming the usual reality conditions, (R αα ) * = Rαα, but will ignore all such factors below. Note also that the RR-action does not contain the Maxwell/Yang-Mills action. Nevertheless, it can be added by hand as C αα C αα − CααCαα. The Pontryagin term does include the s = 1 component. 20 It would be interesting to check if the normalization via a, b that can be read off from the spin-two part of the action is consistent for higher spin fields as well.
an action which is consistent with the required gauge symmetries up to the cubic order since (6.26) has this property [70]. The cubic part of the action can be evaluated on-shell to where we indicated that V can consists of two terms and only one of them contributes to the first (second) part of the action. (Note that (6.15) involves the AdS 4 vierbein h αα rather than the full SO(3, 2)-connection • ω.) It is worth noting that each action of the family (6.28) fixes the relative coefficient between cubic vertices V s 1 ,s 2 ,s 3 for various spins s 1,2,3 that contribute to the action, which is usually not the case for cubic interactions (any linear combination of consistent cubic vertices V s 1 ,s 2 ,s 3 is consistent again). This rigidity/uniqueness of the action is due to taking the higher spin algebra into account. Moreover, the relative normalization of free kinetic terms is also fixed in the free limit. The latter point is subtle: one could rescale all ω α(n),α(m) , n + m = 2s − 2, by some spin-dependent constants a s and change this normalization, but then one would have to change the star-product accordingly. Therefore, the relative normalization makes sense as long as we insist on using the Moyal-Weyl star-product for R and for the gauge symmetries After this detour into the Fradkin-Vasiliev action and Pontryagin invariant, we can extract the presymplectic structure and compare it with (6.20). The free part of the action (6.28) differs from the usual first order (AKSZ-type) actions by a total derivative. Varying the action (6.28) and extracting the boundary term, we find the following presymplectic potential: On restricting to the solution space, we can replace the curvature R with its on-shell value (6.15) and get an equivalent presymplectic potential Θ on the target space that yields This is exactly (6.20) with µ's replaced by 2τ 's. Furthermore, (6.31), when decomposed into fields with definite spin, is equivalent to the presymplectic form which comes from the free first-order action of Ref. [123]. It coincides in form with the presymplectic structure of the linearized gravity, cf. (4.19), and is one of the admissible presymplectic structures found in [64].
As another small test of interactions, the presymplectic AKSZ Lagrangian (6.22) does reproduce the correct cubic on-shell vertex (6.29) upon appropriate identification between τ and µ.
However, since beyond the free level it is only a weak action (unless otherwise shown) we do not suggest to literally compare the actions in the literature, e.g. [34,42,71] with (6.21). In general, in physics terms, weak actions can suffer from defects of two kinds: (1) some interactions can disappear because the presymplectic structure is degenerate; (2) some extra interactions can be found due to the action being invariant under a subset of symmetries only. We expect that the presymplectic AKSZ action proposed in the paper does not suffer from either defect, the only real issue being locality, see also discussion in Section 8.
Presymplectic AKSZ Lagrangian (6.21) can also clarify 21 the very form of the original action (6.26) or its refinement (6.28), i.e., the fact that it is very close to a topological one. Free parameters τ ± of (6.28) get expressed in terms of a single coupling constant by requiring the action to be Hermitian and parity invariant. On relaxing these conditions (or at least the parity in view of the importance of the θ-term) we get a two-parameter family of actions, which is closely related to the fact that there are two independent deformations of the equations of motion and of the extended higher spin algebra, cf. (6.11). Therefore, the two-term structure of action (6.28) and the appearance of 'almost' the trace is explained by (6.21) without any reference to AdS 4 or to cubic approximation.
Let us summarize, presymplectic AKSZ action (6.22) does reproduce the correct free dynamic over AdS 4 ; it does reproduce the correct presymplectic structure, which leads to the canonical quantization of free higher spin fields; it does reproduce the presymplectic structure coming from cubic action (6.28) together with the normalization that is fixed by the higher spin symmetry. The last point should not be too surprising in view of the fact that both (6.22) and (6.28) are fixed by the higher spin symmetry. However, (6.28) is fixed up to the cubic terms over AdS 4 , while (6.22) is fixed up to C 2 -terms, which is much more powerful. Nevertheless, it provides an additional check of the structure of interactions that goes beyond the classification [128][129][130] of cubic vertices V s 1 ,s 2 ,s 3 within the Noether procedure.
Let us make a few comments on the presymplectic structure of higher spin fields and its relevance for quantization. In the simplest, non-gauge, case of the scalar field C(x), Ω directly corresponds to the canonical commutation relations: which allows one to work out the commutators of physical observables. In particular, for the components of magnetic field B i and electric field E j we get Similar commutation relations can be found for the curvatures of higher spin fields. Indeed, let φ α(s),α(s) be the traceless part of the Fronsdal field, which can be associated with the totally symmetric part of the higher spin vierbein e α(s−1),α(s−1) . The expressions for the Weyl tensors are very simple in the spinorial language: C α(2s) = ∇ αα · · · ∇ αα φ α(s),α(s) , Cα (2s) = ∇ αα · · · ∇ αα φ α(s),α(s) .
where the dots complete the r.h.s. to a traceless, symmetric and transverse bi-tensor. Note that (6.32) and (6.31) are equivalent in AdS 4 , but not in flat space-time.
The correct quantization of free higher spin fields is guaranteed by the fact that Ω 1 coincides with the one obtained from the standard actions, all of which are equivalent to the Fronsdal action. In particular, the frame-like actions are known [123] to reduce to the Fronsdal action. Note that the presymplectic structure should be used to compute the commutator of observables. The complete classification of observables in 4d HSGRA was obtained in [65] with the important results found first in [131,132]. For the free theory one set of gauge invariant observables is given by the components of C, e.g. by the electric and magnetic fields (6.35) and by the higher spin generalizations thereof. Another set of observables, currents of the global higher spin symmetry, is considered in Sec. 7.
Presymplectic structure and AKSZ to NLO. For the formal manipulations below, we will use the presymplectic potential Θ 1 in the form (6.17). Applying the homological perturbation theory (6.7, 6.8), one can extend (6.17) to higher orders in C's to make it invariant w.r.t.
the full homological vector field (6.2) underlying 4d HSGRA. In particular, the sub-leading term of expansion (6.4, 6.5) has the following structure: By definition, the coefficients of the two-form Ω (0) 2 vanish modulo C 2 ; this allows us to ignore the contribution of Ω (0) 2 in the first-order approximation (6.12) to the field equations. As for the rest two terms, we find One can check that All this allows us to reconstruct the corresponding AKSZ Lagrangian up to the second order in the zero-form field C. A straightforward calculation gives Let us check directly that the Lagrangian above leads to the equations of the form (4.10). We It remains to observe that the last term vanishes due to the identity that follows immediately form (6.14) and (6.42).
We see that the EL equations for the Lagrangian (6.44) are satisfied by all solutions to the equations (6.12) of 4d HSGRA. As expected, the total derivatives in (6.44) exactly reproduce the presymplectic potential Θ = Θ 1 + Θ 2 + . . . defined by Rels. (6.17, 6.39), cf. (4.9). We claim that the corresponding on-shell presymplectic structurê and ω, C obey equations (6.12), becomes trivial modulo C 2 whenever µ = ν. Indeed, we proceed as Hence, Θ degenerates into the sum of on-shell d-and δ-exact terms, so thatΩ = 0 modulo C 2 . The current (6.48) is conserved up to the first order in C, that is, dJ ≈ 0 (mod C 2 ); its conservation, however, inevitably breaks down at the next order, see [65].
The general conclusion is that for µ = ν, the presymplectic structure (6.46) is equivalent to that of the form (6.4) with m > 2. In other words, even for µ = ν the presymplectic structure remains non-trivial, but starts at NNLO. However, this does not seem to be satisfactory from the physical point of view as long as we want to reproduce the correct quantization of free higher spin fields on maximally symmetry backgrounds, C = 0, e.g. on AdS 4 . Therefore, we Another important comment is that the scalar field of the higher spin multiplet is somewhat separated from s > 0 fields: its presymplectic structure does not appear at LO and enters at NLO. Of course, the higher spin symmetry relates it to those of higher spin fields, which were found to be the correct ones. In order to extract the correct presymplectic structure for the free scalar field one has to choose Θ 1 of the form (6.18) and perform the corresponding redefinitions of Θ 2 (6.39), which is a problem closely related to the issue of (non-)locality of HSGRA. We hope that many useful statements can be proved first at the formal level, while the non-locality problem awaits its satisfactory resolution.
Lastly, the presymplectic structure is unobstructed, see Appendix D for the proof. Therefore, it can be extended to any order and the general expression can be written with the help of the techniques introduced in [69] for equations. The presymplectic structure and AKSZ action depend on two coupling constants, as required by the AdS/CFT duality with Chern- Simons matter theories (one should set µ = ν =Ñ −1 e iθ andν = ν * ,μ = −µ * , where θ = π 2 λ, N = 2N sin πλ πλ are expressed in terms of the number of fields N and t'Hooft coupling λ = N/k for Chern-Simons level k).

Higher spin waves and currents
As has been mentioned above, the most symmetric vacuum in HSGRA corresponds to C = 0.
In this case, the highly non-linear equations of motion (3.3) are greatly simplified taking the form of zero-curvature condition for the gauge connection ω associated with the Lie algebra gl(A). In other words, the solutions to describe the most symmetric geometric backgrounds against which the free higher spin fields can consistently propagate. A particular solution to these equations is provided by the anti-de Sitter space, in which case ε being an infinitesimal gauge parameter. Besides, system (7.2) enjoys the global symmetries of the form where the infinitesimal parameter ξ obeys the condition • D ξ = 0. Notice the term involving Ψ ν , which survives even for the anti-de Sitter background. In that case the κ-independent part of ξ accommodates the whole set of Killing's tensors of AdS 4 . Noteworthy also is the fact that the global symmetries form an associative algebra, Mat(A), w.r.t. the ⋆-product of ξ's, and not just the Lie algebra gl(A) w.r.t. their ⋆-commutators.
A nice property of (7.1, 7.2) is that this is a fully consistent system on its own. It requires only the Hochschild cocycle Φ ν and does not suffer from non-localities. It describes propagation of higher spin fields on backgrounds more complicated than just the anti-de Sitter space, providing thus a way to overcome the Aragone-Deser no-go [133]. In d = 3 there is no r.h.s. in (7.2) since higher spin fields do not have propagating degrees of freedom. For d > 3, the Hochschild cocycle is required. The simplest solutions beyond empty AdS d could be topological black holes [134] and higher spin generalizations thereof [135] along the 3d lines, see e.g. [136]. 22 The linearization of (6.43) over the vacuum C = 0 and ω = • ω gives the following quadratic Lagrangian: 23 The Lagrangian is obviously invariant under the gauge transformations (7.3), but not under the action of global symmetries (7.4) as one can easily check. Nonetheless, we can still use the underlying presymplectic structure to assign conserved currents to the global symmetry transformations. The linearized presymplectic form Ω • = dΘ • on the target space induces that on the configuration space of fieldsC andω. The latter is given by 22 We are grateful to Per Sundell for this suggestion. 23 Note that to this order it is easy to write the Lagrangian in a more suitable form as • ω,C) , see footnote 17. One just needs to perform the field redefinitions that lead from the factorized vertex to V can , inducing a certain change of Λ µν intoΛ µν . It is the Λ-term that gives the presymplectic structure for the free scalar field.

(7.5)
In the case of AdS 4 background, this presymplectic structure and the Lagrangian L • were first found in [64].
Let X ξ denote the variational vector field defined by the r.h.s. of equations (7.4). We leave it to the reader to check that the vector field X ξ , being tangent to the solution space, is Hamiltonian relative toΩ • , i.e., • ω, • ω, ξ) ⋆C .

(7.7)
By construction, the Hamiltonian J ξ defines a conserved current for the higher-spin wave equations (7.2). The last fact is easy to verify directly: Here we used identities (6.14), (6.42), and (6.45) together with the cyclicity of the trace. As with the presymplectic structure (7.5), the currents (7.7) become trivial for µ = ν.
Notice that the conserved currents J ξ are not invariant under the gauge transformations (7.3) in the sense that The violation of gauge invariance occurs due to the explicit dependence ofω and cannot be removed by adding d-exact terms to the currents J ξ . Of course, this non-invariance does not affect the integrated conserved charges. It is significant that the currents (7.7) have non-zero projections onto the physical sector if one takes ξ ∈ gl(hs), see Remark 3.4.
Gauge invariant and non-invariant conserved currents for the free higher spin fields on the AdS 4 background have been intensively studied in the literature, see [137][138][139] and references therein. In [139], an infinite family of gauge non-invariant currents of the formω ×ω have been explicitly constructed. Our gauge non-invariant currents (7.7) have a different structure, schematicallyC ×ω +C ×C. The gauge invariant currents were completely classified in Ref. [137,138]. Being gauge invariant, they may only depend on the zero-form field asC ×C. It goes without saying that none of the currents above can be extended to or come from the nonlinear theory. Furthermore, a straightforward cohomological analysis exposed in the Appendices shows that the aforementioned currentsω ×ω andC ×C cannot be even extended from AdS 4 to the general higher spin background (7.1).
The Poisson bracket associated with the presymplectic structure (7.5) makes the conserved currents (7.7) into the Lie algebra: In general, the currents J ξ being quadratic in fields, one may expect a non-trivial central charge to appear in the r.h.s. of (7.10) upon quantization. The vanishing of the second cohomology group H 2 (gl(A)) (see [140,Sec. B.4]), however, precludes such a possibility, so that the commutation relations (7.10) have to survive quantization as they are.

Final comments and discussion
We have constructed a presymplectic AKSZ sigma-model for 4d HSGRA at the formal level, that is, we showed that it does exist and depends on the right number of coupling constants. We also worked out the first two orders explicitly and proved that the higher orders are unobstructed.
To a great surprise the free action turns out to be a genuine action for higher spin fields rather than just a weak action. The action leads to the right quantum commutators for various observables: the gauge-invariant field strengths and the currents of the global higher spin symmetry (leftover of the gauge symmetry at the free level). The AKSZ action reproduces some of the cubic vertices, while the rest should come from higher orders in the C-expansion. It is particularly remarkable that one can define and classify all suitable presymplectic structures independently of Lagrangians; the only input one needs for that are classical equations of motion. In this paper, we solved this classification problem for 4d HSGRA. Applying cohomological analysis, we found that all physically relevant presymplectic structures form a two-parameter family that can be read off or encoded in an AKSZ-type sigma-model (6.43).
Moreover, we were able to find the explicit expressions (6.46) for (some representatives of) these presymplectic structures up to the second-order in C. As an immediate application of these presymplectic structures, we computed the Poisson brackets of the conserved currents (7.7) in free theory. We also argued that the classical commutation relations of the currents (7.10) should survive quantization.
The deformation quantization of finite-dimensional presymplectic manifolds has been considered in the works [141,142]. Appropriate adaptation of this method to the context of field theory (see e.g. [143] and references therein) will hopefully work for the model at hand. As an alternative, one can try to apply the path-integral quantization to the presymplectic AKSZ model of Sec. 6. It is well to bear in mind that the weak Lagrangian (6.43) may not be a Lagrangian in the ordinary sense and its relevance to the path-integral quantization of HSGRA calls for further investigation.
Quantization II. More abstractly one could check what are the possible counterterms in a HSGRA, which does not require actual quantization. The beauty of the counterterm argument is that one might be able to prove renormalizability or finiteness even without having to quantize anything. However, the naive argument -the more symmetries, the less counterterms -does not seem to apply without further fuss: there exists infinitely many invariants of the higher spin symmetry that can serve as potential counterterms [65]. Nevertheless, additional physical restrictions might potentially rule most of them out. Therefore, the direct quantization of HSGRA is an important open problem too.
Vacuum corrections. The on-shell AKSZ action is proportional to the corresponding Hamiltonian; the latter is an example of on-shell observables which were classified in [65]. The on-shell action starts with ω 4 C 2 -type terms and appears to vanish on any maximally symmetric background (C = 0).
However, the value of the classical action on AdS 4 vacuum should not be zero. Moreover, it is known to contain a rather peculiar numerical factor 1 16 log(4)− 3ζ (3) π 2 for the ∆ = 1 boundary conditions [46,47], the free energy of the free scalar field on the three-sphere. Some contribution might come from the boundary terms (a higher spin analog of the Gibbons-Hawking term) that the action should be supplemented with. All possible on-shell non-trivial three-forms can be found in [65] and again there is no candidate among ω 3 -type observables. A more rigorous approach to the problem would be to carefully take into account the near boundary behaviour of fields and gauge parameters, which may lead us to reconsider the problem of boundary terms.
It should also be borne in mind that the on-shell action is actually ill-defined and requires regularization. Formally, it is the indeterminate form 0 · ∞, where 0 corresponds to the onshell value of the integrand, while ∞ comes from integration over an infinite volume. To evaluate the integral properly, one could choose a family of solutions approaching the AdS 4 vacuum; in so doing, each C of the family must satisfy an appropriate fall-off condition to ensure convergence. The limit of the integral as C → 0 may then be identified with the value of the presymplectic AKSZ action (6.43) on the AdS 4 vacuum. Unfortunately, not much exact solutions to HSGRA are available in the literature to implement the above regularization procedure. It seems reasonable to use a family of solutions that preserves as much symmetries as possible, e.g. as in [131,144].
Note that in the ordinary (non-higher-spin) cases the on-shell value is proportional to the volume of (Euclidian) AdS d , which is divergent and needs to be regularized. One can also regard the space-time integral as the volume of the quotient SO(d + 1, 1)/SO(d, 1), which relates it to the 'gauge' group. In the higher spin case the group is not yet well-defined, see e.g. [145].
Moreover, higher spin transformations mix the metric g µν with all higher spin fields and the on-shell value of the action should not be thought of as the volume of AdS d , rather it should be related to the (regularized) volume of the higher spin group.
How degenerate the presymplectic AKSZ action is? In general, as is discussed in Sec. 4, presymplectic actions cannot reproduce all of the equations of motion. Nevertheless, in some cases, e.g. gravity, the hidden integrability conditions allow one to get equations that are completely equivalent to the one we started with. In order to have such a miracle one needs the right balance between fields and equations (e.g. in gravity the torsion constrain has the same number of components as spin-connection ω a,b and the Einstein equations can, in principle, be reproduced from the variation of vielbein e a , which is what happens). However, in any HSGRA in d > 2 this balance is broken: the equations for C and ω are one-and two-forms valued in a higher spin algebra. This argument is not precise enough since we do not have to reproduce all of the equations, only the ones that lead to the differential equations for the dynamical fields (for example, torsion-like constraints can be imposed by hand).
Nevertheless, the following observation can save the day: variation with respect to ω can, in principle, reproduce all of the C-equations, (C.5). Suppose this is true. Then we can show, see Appendix E, that the ω-equations resulting from the integrability of the C-equations have to be exactly (3.3a) up to a two-form B The information about the kernel is also needed for the identification of physical observables that admit consistent quantization using the presymplectic structure. We are going to address all these issues in future publications.
Higher dimensions. It is instructive to have a look at the presymplectic AKSZ model for the d-dimensional formal HSGRA. As argued in [65], the basis of the Hochschild cohomology of the (extended) higher spin algebra is given by a two-cocycle φ and a (d − 2)-cocycle ψ. The two-cocycle is the one that drives the deformation of the free equations. The (d − 2)-cocycle contributes to higher form observables. We see that at d = 4 both of these, rather different in nature, cocycles happen to be two-cocycles. Algebraically the doubling of two-cocycles is due to the fact that the extended higher spin algebra A = A ⊗ A is the tensor square of A = A 1 ⋊ Z 2 .
With the cocycle ψ we can write the following natural expression for the presymplectic potential on fields: It is also plausible that there exists a field redefinition that brings this Θ 1 into the form such that for the anti-de Sitter vacuum • ω= h a P a + 1 2 w a,b L ab we have Here, h a is a vielbein, w a,b is a spin-connection, P a and L ab are the generators of transvections and Lorentz transformations, andω is the fluctuation about • ω. This gives the canonical presymplectic form for free fieldsω,C on C = 0 background. The non-linear presymplectic AKSZ action should have then the form We are going to detail this construction elsewhere, but it is tempting to compare it with the Einstein-Hilbert action in the frame-like formulation The first factor is nothing else but the Chevalley-Eilenberg cocycle of the Poincaré algebra.
In 3d, one should not be surprised that to leading order the only presymplectic structure is the one of Chern-Simons theory and the presymplectic AKSZ action is just the Chern-Simons action. It would be interesting to explore the 3d case further.
As a step towards the complete AKSZ sigma-model it is worth noting that the non-linear equations of any HSGRA are integrable in the sense of being equivalent to the following 'free system' [69]: Here * is the product in the deformed (extended) higher spin algebra A , A Hochschild, cyclic, and Lie algebra cohomology In this appendix, we collect some basic definitions and constructions related to the cohomology of associative and Lie algebras. For a more coherent exposition of the material we refer the reader to [146], [147], [148]. A word about notation: all unadorned tensor products ⊗ and Hom's are taken over k, a ground field of characteristic zero. We systematically follow the Koszul sign convention. As in the main text, the grade of a homogeneous element a is denoted by |a|. Many formulas below are considerably simplified if one uses the shifted gradeā = |a|−1.  f (a 1 , . . . , a p )a p+1 (A.1) 25 We are grateful to Karapet Mkrtchyan for suggesting this peaceful resolution. (a 1 , . . . , a k a k+1 , . . . , a p+1 ) .
In the special case M = A * it is convenient to identify the spaces C p (A, A * ) with Hom(A ⊗(p+1) , k).
Considering, for example, the ground field k as a one-dimensional algebra over itself one readily concludes that C 2n cyc (k) ≃ k and C 2n+1 cyc (k) = 0. Hence, HC 2n cyc (k) ≃ k and HC 2n+1 cyc (k) = 0. It follows from the definition that the cyclic cohomology groups are contravarinat functors of the algebra, so that any algebra homomorphism h : A → B induces a homomorphism h * : HC p (B) → HC p (A) in cohomology.
Depending on coefficients some standard operations can be defined on Hochschild and cyclic cohomology. Below we discuss two such operations: the cup product and the action of derivations. The former relates the cohomology of algebras A and B with that of their tensor product A ⊗ B and is defined as follows.

As usual the hats indicate
In other words, the cohomology group H p (gl n (A)) does not depend on the size of matrices provided it is large enough.

B Cohomology of Weyl algebras and their smash products
The polynomial Weyl algebra A n over C is a unital algebra on 2n generators q i and p j subject to Heisenberg's commutation relations There is no need to antisymmetrise all p + 1 arguments due to cyclicity of g and tr.
It is known to be a simple Noetherian domain with a k-basis consisting of the ordered monomials in q's and p's, see e.g. [154].
The Hochschild cohomology groups of Weyl algebras are known for various coefficients. For instance, applying the Koszul resolution (see e.g. [155], [156]) yields An explicit formula for a non-trivial 2n-cocycle τ 2n generating the group HH 2n (A n , A * n ) was found in the 2005 paper [157] by Feigin, Felder, and Shoikhet. It was derived as a consequence of Shoikhet's proof [158] of Tsygan's formality conjecture. It should be noted that fifteen years earlier Vasiliev had found an explicit expression for τ 2 in the context of 4d HSGRA [48]. In order to present the cocycle τ 2 explicitly it is convenient to identify the elements of A 1 with the polynomials a(q, p) in (commuting) indeterminates q and p endowed with the Weyl-Moyal star-product and m(a ⊗ b) = ab. We also introduce the maps α 01 , α 12 , α 02 : and similarly for α 12 and α 02 . Finally, we define the homomorphism µ : Here a(0) is the constant term of the polynomial a(q, p). Now the expression for the 2-cocycle reads τ 2 (a 0 , a 1 , a 2 ) = µ • F (α 01 , α 12 , α 02 )(a 0 ⊗ a 1 ⊗ a 2 ) , (B.9) where the operator F (α 01 , α 12 , α 02 ) ∈ End(A 1 ⊗ A 1 ⊗ A 1 ) is determined by the following entire analytic function of three variables: Clearly, the action of the operator F is well defined on polynomials. Unlike the Hochschild 2-cocycles of Refs. [157] and [48], the cocycle (B.9) enjoys cyclic invariance, τ 2 (a 0 , a 1 , a 2 ) = τ 2 (a 2 , a 0 , a 1 ) , (B.11) thereby generating the cyclic cohomology group HC 2 (A 1 ) ≃ C.
In the context of HSGRA, Weyl algebras usually appear in smash products with finite groups of their automorphisms. Recall that, given an associative k-algebra A and a finite group G ⊂ Aut(A), the skew group algebra A ⋊ G (aka smash product algebra) is defined to be Here a g denotes the action of g ∈ G on a ∈ A.
Notice that the automorphism group of A n contains a subgroup Sp 2n (C) acting by linear transformations on the 2n-dimensional complex space V spanned by the generators q's and p's. If G is a finite subgroup of Sp 2n (k), then g |G| = e for any g ∈ G and the action of g is diagonalizable in V . Denote by 2µ g the multiplicity of the eigenvalue 1 of the operator g : V → V . Notice that µ g = µ hgh −1 and the set of all element g ∈ G with 2µ g = p is invariant under conjugation. The next theorem is due to Alev, Farinati, Lambre, and Solotar [155] (see also [156]).
Theorem B.1. Let n p (G) denote the number of conjugacy classes of elements g ∈ G with 2µ g = p, then As an example, consider the most simple skew group algebra A = A 1 ⋊ Z 2 . Here the group Z 2 = {e, κ} acts on A 1 by the involution Since 2µ e = 2 and 2µ κ = 0, all non-trivial groups of Hochschild cohomology mentioned in the theorem above are A) . (B.14) In particular, we see that the algebra A admits a unique non-trivial deformation.
Since HH n (A, A * ) = 0 for n > 2 one can easily see from Connes' exact sequence (A. 19) that see [140,Appendix B.4] for details. Furthermore, the groups HC • (A) form a free HC • (C)module generated (via S) by the pair of elements φ 0 ∈ HC 0 (A) and φ 2 ∈ HC 2 (A). The first one is the trace φ 0 = Tr : A → C defined by the projection onto the one-dimensional subspace C(1⊗κ) ⊂ A. We can normalize it by setting Tr(1⊗κ) = 1. The trace is known to give rise to a non-degenerate inner product on A defined by (a, b) = Tr(ab). This inner product allows one to identify the A-bimodules A and A * . An explicit expression for a 2-cocycle representing the class φ 2 is given in [140,Eq. (4.15)]. We refer to φ 0 and φ 2 as primary classes of cyclic cohomology.
All the other classes in (B.15) are obtained from these two by successive application of the periodicity operator: S n φ 0 ∈ HC 2n (A), S n φ 2 ∈ HC 2n+2 (A).
As discussed in Sec. 3, the skew group algebra A = A 1 ⋊ Z 2 is a building block for the extended higher spin algebra A underlying 4d HSGRA. The latter is given by the tensor square One may also regard it as the smash product of the Weyl algebra A 2 and the Klein four-group These are the primary cocycles of HC • (A).

C Cohomology of Grassmann algebras
Let Λ n denote the Grassmann algebra over C on n generators θ 1 , . . . , θ n subject to the relations As the algebra Λ n is supercommutative one concludes immediately that HH 0 (Λ n , Λ n ) = Z(Λ n ) = Λ n and HH 1 (Λ n , Λ n ) = Der(Λ n ) .
Geometrically, one can think of Λ n as the algebra of smooth functions on a supermanifold G with odd coordinates θ's.
the r.h.s. being the space of polyvector fields on G. In terms of the odd coordinates θ i , each p-vector φ is given by where the coefficients φ i 1 ···ip ∈ Λ n are totally symmetric in permutation of indices.
The cyclic cohomology of Λ n is also well known. As was shown by Kassel where HC 2k (C) ≃ C, HC 2k+1 (C) = 0, and the complex dimensions of the vector spaces V p are encoded by the Poincaré series In particular, dim HC 0 (Λ n ) = dim Λ n = 2 n . Actually, Λ n is a bialgebra with the standard This enables us to equip the complex vector space HC • (Λ n ) with the structure of a gradedcommutative algebra. Indeed, composing the k-algebra homomorphism ∆ : Λ n → Λ n ⊗ Λ n with the cup product (A.8), we get a new product 29 that makes HC • (Λ n ) into a graded associative algebra. The algebra HC • (Λ n ) contains HC • (C) as a subalgebra and the action of the periodicity map S on HC • (Λ n ) is induced by the inclusion The map S generates the first summand in (C.3) and acts trivially on the V p 's.
As was first shown in [159] cyclic cocycles representing the spaces V p admit a nice interpretation in terms of the polyvector fields (C.2). See [160] for subsequent discussions. Specifically, to each p-vector φ one first associates a Hochschild p-cocycle by the rule φ(a 0 , a 1 , . . . , a p ) = a 0 φ i 1 ···ip ∂ i 1 a 1 · · · ∂ ip a p ∈ C p (Λ n , Λ * n ) .
Here a k ∈ Λ n and the integral sign stands for the Berezin integral on the Grassmann algebra.
One can easily check that the property ∂φ = 0 is automatically satisfied for any φ. The cyclicity condition φ(a 0 , a 1 , . . . , a p ) = (−1) (|a 0 |+1)(|a 1 |+···+|ap|+p) φ(a 1 , . . . , a n , a 0 ) , (C.7) however, requires the p-vector φ to be divergence-free, i.e., In order to analyse the last condition, it is convenient to identify the polyvector fields with the elements of the Berezin algebra B n on n anti-commuting generators θ i and the same number of commuting generators y i : θ i θ j = −θ j θ i , y i y j = y j y i , θ i y j = y j θ i . 29 Recall that cyclic cohomology is a contravariant functor of algebra.
Clearly, there is the one-to-one correspondence Upon this identification the operator of divergence (C.8) passes to the odd Laplace operator Since ∆ 2 = 0, we may say that the divergence-free polyvectors correspond to the cocycles (perhaps trivial) of the odd Laplacian: The non-trivial cocycle can easily be computed with the help of the homotopy operator h = θ i y i ·. Clearly, In view of this relation there is the only (up to a multiplicative constant) non-trivial ∆-cocyle All other cocycles are of the form φ = ∆ψ.
Since ∆ is a second-order differential operator, the product of two ∆-cocyles is not a cocycle in general. Instead, the ∆-cocycles form a graded Lie algebra w.r.t. the bracket This is just the Schouten bracket on polyvector fields. It follows from the definition that the bracket is differentiated by ∆; and hence, it maps ∆-cocycles to ∆-cocycles. Taken together the odd Laplacian and the bracket endow B n with the structure of Batalin-Vilkovisky algebra.
Let us now specify the above constructions to the case of Λ 2 . The corresponding Berezin algebra B 2 is generated by θ,θ, y, andȳ. As a complex vector space, B 2 is spanned by the monomials y nȳm , θy nȳm ,θy nȳm , θθy nȳm .

D Presymplectic structures in 4d HSGRA
We begin with an algebraic reformulation of the problem. At the free level the homological vector field underlying 4d HSGRA comes from the graded Lie algebra G described in items C A being linear coordinates in the tangent spaces. By definition, |C A | = 0. Since Q is quadratic, it defines and is defined by some graded Lie algebra G = G −1 ⊕ G 0 , so that N = G [1]. One can regard G as the trivial extension of the Lie algebra G 0 with its adjoint module put in degree −1 (adjoint extension). It is the homological vector field (D.2) that determines the free equations of 4d HSGRA. One can iterate this construction to produce higher prolongations of the homological vector field (D.1). Of particular interest to us is the second prolongation of It is easy to see that the commutation relations in gl(A ⊗ Λ 2 ) coincide exactly with those in G = G −1 ⊕ G 0 ⊕ G 0 ⊕ G 1 if we set f 0 ,f 0 ∈ G 0 , f −1 ∈ G −1 , and f 1 ∈ G 1 . On passing to field theory, the element (D.7) is promoted to the 'superfield' f (θ,θ) = ω + θC +θδω +θθδC , (D. 8) which accommodates the zero-and one-form fields C and ω together with their variational differentials δC and δω. defined by the cup product (A.8). The problem thus reduces to identifying those cohomology classes on the right that correspond to free presymplectic structures and their obstructions to deformation. We proceed with a closer examination of the right tensor factor in (D.10).

By Theorem
Keeping the notation of Appendix C, we endow the Berezin algebra B 2 with an auxiliary Z-grading by setting w(θ) = w(y) = 0 , w(θ) = −1 , w(ȳ) = 1 . (D.11) We will refer to this grading as the weight, lest one confuse it with many other degrees. Besides, we introduce the differential δa = (a,ȳ) = ∂a ∂θ , ∀a ∈ B 2 . (D.12) Clearly, w(δ) = 1, δ 2 = 0, and [∆, δ] = 0. Taken together with the Schouten bracket (C.13) this differential makes the space B 2 into a differential graded Lie algebra. When restricted to Λ 2 ⊂ B 2 , δ becomes a derivation of the Grassmann algebra Λ 2 . By formula (A.17) it induces a homomorphism L * δ : HC p (Λ 2 ) → HC p (Λ 2 ) in cyclic cohomology, which then trivially extends to the tensor product (D.10). As should be evident from (D.8) the differential (D.12) just mimics the action of the de Rham differential on Λ(N ). The non-trivial presymplectic structures come from those elements of B 2 that are both ∆-and δ-closed and have weight one. 31 It follows from 31 It is well to bear in mind that the Berezin integral (C.6) implies one more differentiation byθ, so that the resulting cyclic cocycles have weight two and correspond to 2-forms on N .
Turning now to the left tensor factor in (D.10), we recall that the HC • (C)-module HC • (A) is freely generated by the four primary classes (B.21). Of these, only Φ 2 andΦ 2 can generate presymplectic structures on N of degree 3. In other words, all 'free' presymplectic structures of 4d HSGRA come from the two infinite series of cyclic cohomology classes ̟ n = Φ 2 ⊔ β n ,̟ n =Φ 2 ⊔ β n . (D.20) The direct computation of the ⊔-products shows that the resulting presymplectic structures are of the form Ω (1) n , see Eq. (6.5). The corresponding presymplectic potentials originate from the classes ϑ n = Φ 2 ⊔ γ n ,θ n =Φ 2 ⊔ γ n . (D.21) Notice that the remaining class (D. 19) gives no presymplectic structure in degree 3.
We claim that all free presymplectic structures associated with the series (D.20) survive upon switching on interaction. More precisely, by means of homological perturbation theory of Sec. 6, they can always be deformed so as to become compatible with non-linear field equations (3.3a, 3.3b), whatever the interaction vertices. The existence of such a deformation is ensured by the absence of obstructing cocycles. The last fact can be seen in two equivalent ways. First, one can try to deform a free presymplectic 2-form by itself. For reasons of degree, all the obstructing cohomology classes, if any, must belong to the linear span of Φ 2 ⊔ α n and Φ 2 ⊔ α n . However, Eq. (D.14) says that the corresponding 2-forms are not closed for n > 1; and hence, they cannot appear as obstructions to deformation. The remaining case n = 1 is also excluded as the corresponding 2-forms do not depend on C, while the interaction vertices do.
An alternative possibility is to deform the free presymplectic structure through the deformation of its presymplectic potential. Again, by degree considerations, all potential obstructions are spanned by the classes Φ 4 ⊔ B n0 = L * δ (Φ 4 ⊔ B n1 ), so that the corresponding 1-forms turn out to be exact. The exact 1-forms represent natural ambiguity in the choice of a presymplectic potential and can thus be disregarded. The details are left to the reader. All in all, we see that the classes (D.20) span the space of all presymplectic structures in 4d HSGRA.
Finally, let us note that the map I : HC 2 (A) → HH 2 (A, A * ) of the long exact sequence (A.19) is actually an isomorphism, so that each Hochschild 2-cocycle is cohomologous to a cyclic one. The existence of a non-degenerate trace on A implies further isomorphisms A ≃ A * and HH 2 (A, A * ) ≃ HH 2 (A, A). In view of these isomorphisms it is little wonder that the same pair of the Hochschild 2-cocycles of HH 2 (A, A) define the cubic vertices (6.10) and the presymplectic structure (6.17) of 4d HSGRA gravity.

E Integrability of zero-form equations
Since equations (3.3) for ω and C do not seem to be Lagrangian, it is interesting to to ask the following question: to which extent do the equations of motion for C's control those for ω's?
The equations in question have the form Here the indices α and i, labelling the fields, are essentially equivalent and originate from the same algebra gl(A). At this point, however, it is useful to consider them as independent. We also assume (and this is indeed the case for HSGRA) that the right sides of equations (E.1) define a Lie algebroid with anchor V . In other words, the vector fields V α = V i α ∂/∂C i form an integrable distribution on a manifold coordinatized by C i : Notice that we do not assume the vector fields V α to be linearly independent in general position.

(E.3)
Let us now examine the integrability conditions for the second equation (3.3b). Applying the de Rham differential d to both sides, we find If the V α 's are linearly independent, then the last condition implies the first equation (E.1a).
In that case Eq. (E.1b) 'knows' about Eq. (E.1a), and we may omit the later without any consequences for the system. In the general case, there may be some null-vectors Z α A = Z α A (C) spanning the kernel of the anchor V , that is, Then Eq. (E.4) implies that for some collection of 2-forms B A . These 2-forms should be regarded as new independent variables describing arbitrariness in the dynamics of ω's whenever equations (E.1a) are omitted.
The dynamics of B's are not completely arbitrary. Checking the integrability condition and assuming the null-vectors Z A to be linearly independent, one can find for some structure functions U's. In such a way we arrive at the natural extension of the original system (E.1) by the 2-form fields B A subject to (E.7).
When applied to 4d HSGRA, these ideas instruct us to look for null-vectors (E.5). To leading order in C, one gets then the equation Z ⋆ C − C ⋆Z = 0, whereZ(y,ȳ) = Z(y, −ȳ) (we deal with the physical fields only). It is easy to see that the last equation has only constant solutions, Z = const · 1 1, which also satisfy the entire equation (E.5).