An Action for Matter Coupled Higher Spin Gravity in Three Dimensions

We propose a covariant Hamiltonian action for the Prokushkin and Vasiliev's matter coupled higher spin gravity in three dimensions. The action is formulated on ${\cal X}_4 \times {\cal Z}_2$ where ${\cal X}_4$ is an open manifold whose boundary contains spacetime and ${\cal Z}_2$ is a noncommutative twistor space. We examine various consistent truncations to models of BF type in ${\cal X}_4$ and ${\cal Z}_2$ with B$^2$ terms and central elements. They are obtained by integrating out the matter fields in the presence of a vacuum expectation value $\nu\in \mathbb{R}$ for the zero-form master field. For $\nu=0$, we obtain a model on ${\cal X}_4$ containing Blencowe's action and a model on ${\cal Z}_2$ containing the Prokushkin--Segal--Vasiliev action. For generic $\nu$ (including $\nu=0$), we propose an alternative model on ${\cal X}_4$ with gauge fields in the Weyl algebra of Wigner's deformed oscillator algebra and Lagrange multipliers in the algebra of operators acting in the Fock representation space of the deformed oscillators.


Introduction
Three-dimensional gravity with negative cosmological constant, as defined by the sl(2, R)⊕sl(2, R) Chern-Simons (CS) action of [1,2], provides a rich framework for testing various aspects of quantum gravity in a setting that is simpler than in higher dimensions, yet nontrivial. Although AdS 3 gravities are topological, they admit black holes [3] and possess moduli spaces at conformal infinity governed by infinite-dimensional conformal symmetry algebras [4,5]; see [6] for a review.
As for higher spin gravities, these are simpler as well in three dimensions, where the massless higher spin fields are topological, and hence the spectrum requirements on the gauge algebras simplify considerably. Topological higher spin gravities based on the principal embedding of sl (2, R) into sl(N, R) were shown in [7,8] to have asymptotic W N symmetries. In [9] holographic correspondences were conjectured for higher spin gravities with infinite-dimensional gauge algebras hs(2) ⊕ hs (2) and their deformation hs(λ) ⊕ hs(λ) coupled to (complex) bulk scalars.
In the above works, the (classical) gauge sector is assumed to be described by various CS generalisations [10,11,12] of the Achucarro-Townsend supergravity Lagrangian [1]. On the other hand, one class of matter coupled higher spin gravities is described on-shell by the Prokushkin-Vasiliev (PV) equations [13], and off-shell by the Prokushkin-Segal-Vasiliev (PSV) action principle in twistor space [14]. However, as spacetime is absent in the latter, its relation to the CS formulations has remained unclear. Moreover, there exists a second class of matter coupled higher spin gravities based on action principles in three dimensions with an extra dynamical two-form [12], whose relation to the PV system is unclear.
In this paper, we provide the PV system with an action principle of covariant Hamiltonian type on a six-manifold given by the direct product of a closed twistor space Z 2 and an open four-manifold X 4 whose boundary X 3 contains spacetime M 3 . When subjected to the variational principle combined with natural boundary conditions, the action yields the PV equations on its five-dimensional boundary X 3 × Z 2 . The action is constructed such that upon integrating out the matter fields in the presence of an expectation value ν for the PV zero-form, the effective action for the gauge fields can be consistently truncated to models of BF type on X 4 . These model contain B squared terms 1 containing the standard symplectic structure of three-dimensional Fronsdal fields.
For ν = 0, we obtain a model on X 4 containing Blencowe's action, as well as a model on Z 2 containing the Prokushkin-Segal-Vasiliev action. In these models, the dual spaces that contain the gauge fields and Lagrange multipliers are isomorphic. For generic ν (including ν = 0), we shall also consider a model on X 4 in which the aforementioned two spaces are not isomorphic.
However, its existence depends on the finiteness of the trace of the vacuum-to-vacuum projector of a deformed oscillator induced from six dimensions. If existing, such a model would provide 1 The B field of the BF-like models originate from the even Lagrange multiplier in the covariant Hamiltonian action, denoted by T in (3.27).
an alternative to the BF-like Blencowe model based on Vasiliev's supertrace [15], whose sixdimensional origin remains unclear.
The master action to be constructed here is analog of the one for the four-dimensional Vasiliev system found in [16]. In particular, it does not extend the closed and central two-form of the PV system into a dynamical field off-shell. The inclusion of a dynamical two-form in an action that is an analog of that for four-dimensional models given recently in [17] and which makes contact with both the PV system as well as the action proposed in [12] will be treated elsewhere.
The paper is structured as follows: In Section 2, we cast the PV system as a differential algebra on the direct product of twistor space and spacetime. After this preparation, we propose a covariant Hamiltonian action on X 4 × Z 2 , with a term quadratic in Lagrange multipliers, in Section 3. In Section 4, we examine the consistent truncations to the BF-like version of the Blencowe model on X 4 and the PSV action on Z 2 . We summarize our results and provide an outlook in the Conclusions in Section 5. In Appendix A, we review the mass deformation. In Appendix B, we present the proposed ν-deformed BF-like model on X 4 .

Prokushkin-Vasiliev models
In this section, we rewrite the PV equations [13] as a differential algebra generated by master fields on a noncommutative manifold valued in an associative algebra, or equivalently, as an associative bundle with fusion rules. In particular, we shall identify the minimal bosonic model and its massive deformation. For a recent, in-depth treatment of the weak-field perturbative analysis of PV systems to first nontrivial order in interactions, see [18].

Differential algebra
The master fields are defined locally on the direct product M 3 × Z 2 of a commutative three-dimensional real manifold M 3 with coordinates x µ , µ = 1, 2, 3, and a non-commutative two-dimensional real manifold Z 2 with coordinates z α , α = 1, 2. The fields are valued in an associative algebra generated by a real oscillator y α , α = 1, 2, coordinatizing an internal noncommutative manifold Y 2 , and a set of thus coordinatizing the Clifford algebra Cℓ N , that we shall denote by C N for brevity. The dependence of the master fields on (y α , z α ) is treated using symbol calculus, whereby they belong to classes of functions (or distributions) on Y 2 × Z 2 that can be composed using two associative products: the standard commutative product rule, denoted by juxtaposition, and an additional noncommutative product rule, denoted by a ⋆. In what follows, we shall use the normal ordered basis in which the star product rule is defined formally by whereas a more rigorous definition requires a set of fusion rules (see below). In particular, the above composition rule rigorously defines the associative Weyl algebra Aq(4). This algebra consists of arbitrary polynomials in y α and z α , modulo whose symmetric and anti-symmetric parts, respectively, define the normal order and the (ordering independent) commutation rules, viz. 2 The basis one-forms (dx µ , dz α ) obey where the graded star commutator 3 of differential forms is given by with deg denoting the total form degree on M 3 × Z 2 . To describe bosonic models, we impose The doublet variables y α and z α form Majorana spinors once the equations are cast into a manifestly Lorentz covariant form. 3 We will explicitly use commutators and anticommutators, in Section 3.
where π is the automorphism of the differential star product algebra defined by The hermitian conjugation is defined by 11) and the reality conditions on the master fields read where the differential obeys the PV field equations can be written as The element J is closed and central in the space of π-invariant forms, viz.
as can be seen from the fact that κ, which is referred to as the inner Klein operator, obeys

Lorentz covariance
Introducing the equations can be rewritten as (2.21) In view of {S α , κ} ⋆ = 0, which follows from the bosonic projection, the above equations define a deformed oscillator algebra, fibered over M 3 , for which B plays the role of deformation parameter.
The equations can be cast into manifestly Lorentz covariant form [19,20] by introducing a bona fide Lorentz connection ω αβ = dx µ ω αβ µ on M 3 and defining in terms of which the master field equations on M 3 take the form and 26) which are related by r = 1 4i r αβ (y α ⋆ y β − z α ⋆ z β ) = d X ω + ω ⋆ ω. The deformed Lorentz generators obey the algebra where the induced transformations act on the component fields of M γδ . The above commutation rules are an example of a more general construction wherein a Lie algebra L acts on a space M via Lie derivatives and is a representation of L in an associative algebra with product ⋆ obeying which can be seen to obey the Jacobi identity using [L X , L X ] = L [X,Y ] and the Leibniz' rule

Original PV model and its truncations
By taking N = 4 and identifying we recover the original PV system, in which ψ 1 is used to define the AdS 3 translation operators.
By imposing the following conditions on the master fields, conditions that will be justified later on from the existence of an action principle, i.e. by taking them to be valued in the subalgebra using the anti-automorphism defined by yields a model with one propagating scalar and one topological master field. This model is identical to the original PV model based on the algebra ho + 01 (1, 0|4) 4 . In all of the above models, the component along Γ plays the role of a real mass parameter, denoted by ν; see Appendix A.

Associative bundle
The master fields equations define an associative algebra bundle A over M 3 [21,22]. The fiber algebra A| p at a generic point p ∈ M 3 is related by a similarity transformation to that at a reference point p 0 ∈ M 3 . To describe the latter, it is convenient to separate the variables in and Ω(Z 2 ). One can then use the factorization formula to solve the deformed oscillator algebra at p 0 formally in terms of auxiliary integrals facilitated by analytical continuation methods in Ω(Z 2 ) [23]. Thus, by the left and right action of the Weyl algebra Aq(2). We write indicating the properties of Aq (2)[Σ] as a left (λ) and right (ρ) module of Aq(2). The associative structure of A requires a fusion rule such that if N ΣΣ ′ Σ ′′ = 1 then the left-hand side is to be computed using (2.3) with z α = 0 and expanded into the basis of Aq(2)[Σ ′′ ] such that all nontrivial products are finite and the resulting multiplication table is associative. For example, massless particles and various types of algebraically special exact solution spaces arise within Gaussian sectors. The Weyl algebra , with reference state being the identity operator, is also included (as the sector corresponding to twistor space plane waves), typically with 5 N 1Σ Σ = N Σ1 Σ = 1.

Covariant Hamiltonian action
In this section we begin by discussing some generalities on covariant Hamiltonian actions on We then determine the constraints on the Hamiltonian such that it leads to a master action in which the master field content, including the Lagrange multipliers, are extended to consist of sum of even and odd forms of appropriate degree, and central elements. This action yields a generalized version of the PV field equations.

Generalities
In order to formulate the theory within the AKSZ framework [26] using its adaptation to noncommutative higher spin geometries proposed in [27], we assume a formulation of the PV system that treats Z 2 as being closed and introduce an open six-manifold M 6 with boundary where X 3 is a closed manifold containing M 3 as an open submanifold. On M 6 , we introduce a two-fold duality extended [28,16,29] 6 set of differential forms given by

3)
5 An exception is fractional spin gravity [25] whose fractional spin sector Ψ and Lorentz singlet sector U have Starting from a universally Cartan integrable system and replacing each p-form by a sum of forms of degrees p, p + 2, . . . , p + 2N , and each structure constant by a function of off-shell closed and central terms, i.e. elements in the de Rham cohomology valued in the center of the fiber algebra, with a decomposition into degrees 0, 2, . . . , 2N , yields a new universally Cartan integrable system, referred to as the N -fold duality extension of the original system. More generally, one may consider on-shell duality extensions by including on-shell closed complex-valued functionals into the extension of the structure constants [30,29]. valued in A ⊗ C 4 and where the subscript denotes the form degree. We let {J I } denote the generators of the ring of off-shell closed and central terms, i.e. elements in the de Rham cohomology of M 6 valued in the center of A ⊗ C 4 , which hence obey (off-shell) for any differential form f on M 6 valued in A ⊗ C 4 . Following the approach of [16], we consider actions of the form where Tr A⊗C 4 denotes a cyclic trace operation on A ⊗ C 4 . We assume a structure group gauged by A and that S, T and B belong to sections, and (3.6) makes explicit the covariant Hamiltonian Thus, the coordinate and momentum master fields, defined by (X α ; P α ) := (A, B; T, S) , (3.8) lie in subspaces of A that are dually paired using Tr A , which leads to distinct models depending on whether these subspaces are isomorphic or not. In the reductions that follow, we shall consider the first type of models, while a model with coordinates and momenta in non-isomorphic spaces is treated in Appendix B. Moreover, for definiteness, we shall assume that and the associative bundle A defined in (2.40) is chosen such thať is finite (and globally defined on X 4 ). The action can then be written as We shall furthermore assume that Tr A⊗C 4 f , (3.12) and Tr Tr Tr A⊗C 4 g ⋆ f , (3.14) from which it follows that H is a graded cyclic ⋆-function.

Constraints on H
The Hamiltonian is constrained by gauge invariance, or equivalently, by universal on-shell Cartan integrability 7 . In addition, it is constrained by the requirement that the equations of motion on M 6 reduce to a desired set of equations of motion on ∂M 6 upon assuming natural boundary conditions. To examine the above, we let and consider the total variation where Ω ij is a graded anti-symmetric constant matrix 8 and the Cartan curvatures are given by where ∂ i denotes the graded cyclic derivative defined by for any graded cyclic ⋆-function U. We find integrable systems. 8 Adopting the conventions of [27], we take Ω ik Ω kj = −δ i j .
Requiring A and B to be free to fluctuate on ∂M 6 , the variational principle implies The Cartan integrability requires Moreover, imposing the set of boundary equations is a two-fold duality extension of the PV system, viz.
for a set of complex constants F n,I 1 ...I k .

The master action
In order to obtain a model that admits consistent truncations to three-dimensional CS higher spin gravities, we need to assume that V contains a term that is quadratic in T . The simplest possible such action is given by The total variation where the Cartan curvatures read The generalized Bianchi identities are The gauge transformations We take (ǫ B ; η T , η S ) to belong to sections of the structure group and impose 10 We have also assumed that (A, B) fluctuate on ∂M 6 , which implies 11 The resulting boundary equations of motion thus provide a duality extended version of the Prokushkin-Vasiliev equations, that is free from any interaction ambiguity, following a variational principle.
In the action (3.27), the relative coefficient of the BSS and T T terms is fixed uniquely by

Consistent truncations
In this section we perform consistent truncations of the covariant Hamiltonian master action in six dimensions down to various models on X 4 and Z 2 . The truncations consist of integrating out the fluctuations in B around its vacuum expectation value νΓ followed by reductions on Z 2 and X 4 . On X 4 , we reach BF -like models with Lagrangian forms containing Blencowe's action for ν = 0 and a ν-deformed version thereof that we present in Appendix B. For ν = 0, the reduction to Z 2 yields the Prokushkin-Segal-Vasiliev (PSV) action.
A consistent truncation a system with action S[ϕ] and equations of motion E(ϕ) = 0 amounts to an Ansatz ϕ = ϕ(ϕ ′ ) off-shell such that E(ϕ(ϕ ′ )) = 0 are equivalent to a set of equations 10 Following the AKSZ approach, the Batalin-Vilkovisky classical master equation requires that the ghosts corresponding to (η T , η S ) vanish at ∂M 6 off-shell. 11 Following the AKSZ approach, the Batalin-Vilkovisky classical master equation requires that (3.43) holds off-shell.

Reduction to BF-like extension of Blencowe's action
Starting from the equations of motion (3.32) and setting B = 0 yields and which together form a Cartan integrable system containing (4.1) as a subsystem, i.e. the free differential algebra generated by (A, T, S) contains a subalgebra generated by (A, T ). Assuming ∂M 6 to consist of a single component, it follows from S| ∂M 6 = 0 that S can be reconstructed from (A, T ) on-shell 12 from (4.2). Therefore, the system (4.1) is a consistent truncation of the original system (3.32) on-shell. Tr
There are two independent embeddings of Blencowe's model into the above master action.
They can be obtained by choosing the fiber algebras and equipping A ⊗ C 4 with trace operations as follows 13 : The factorization formula (2.39) then yields 14 We truncate the models further as follows: where W ± and T ± are independent of Γ i and κ y , and (4.28) Inserting (4.26) and (4.27) into (4.20) and using m = 0 : 13 The (graded cyclic) supertrace obeys STr Aq(2) f ⋆ g = STr Aq(2) g ⋆ π(f ) and STr Aq(2) f = f (0) provided that f (y) is the symbol of f defined in Weyl order. 14 We use the normalizations dz α dz α = −2dz 1 dz 2 = −2d 2 z and d 2 yd 2 zκ ⋆ f (y) = 4π 2 f (0) . m = 1 : for f independent of Γ i and κ y , yields the following four-dimensional Hamiltonian extension of Blencowe's action: which is thus reached for both m = 0 and m = 1.
Assuming that X 4 = X 3 ×[0, ∞[ and that all fields fall off at X 3 ×∞, and assuming furthermore that X 3 has a simple topology such thatǧ the elimination of the Lagrange multipliers yields where now d denotes the exterior derivative on X 3 . Equivalently, from which we identify using the conventions of [31]. Relaxing the assumption onǧ +ǧ ′ by taking it to be a nontrivial element in the de Rham cohomology of X 3 , Blencowe's action is accompanied by the extra term

Reduction to PSV action
Instead of reducing (3.27) on Z 2 one may consider a reduction on X 4 under the assumption that as well as ∂Z 2 = ∅ . The absence of any boundary condition on T implies that its integration constant in form degree zero contains local degrees of freedom. To exhibit the model on Z 2 , we first perform a consistent truncation by setting B = νΓ , (4.40) leading to the reduced action We then proceed by introducing a volume formJ [4] on X 4 and background potentialsW (0) [3] and V (0) [1] on X 4 and Z 2 , respectively, defined by d XW (0) [3] =J [4] , F (0) In particular, we takeW (0) [3] to be independent of the internal coordinates y α . We next perform a further truncation by taking h = J + iJ [4] , µ = µ 0 , g = g ′ [2] , (4.43) and considering the Ansatz [3] , (4.44) with α, β ∈ R and fluctuating fields where Ω(Z 2 ) ⊗ A ′ consists of an algebra of π-invariant master fields. Defining 4.47) and suppressing the subscripts denoting form degrees, one has where we used that J ⋆ D ′ C ′ ≡ 0 being a 3-form on Z 2 and The equations of motion of (4.41) on the above Ansatz read while plugging the Ansatz back into the action (4.41) yields Tr from which it follows that the Ansatz leads to a nontrivial and consistent truncation provided In order to define the combined integration over Z 2 and trace operation A, we may take The background connection V (0) thereby is flat. The simplest choice amounts to take V (0) = 0 .
We then choose Tr C 4 (4.53) and make the redefinition 15 so that and the reduced action now reads where Vol(X 4 ) = X 4J [4] . The above action is identified with the original action given in [14] upon taking µ 0 = −i and Vol(X 4 ) = 2 α .

Conclusions
We have presented an action principle for the bosonic sector of Prokushkin and Vasiliev's threedimensional matter coupled higher spin gravity. By integrating out the matter fields, in a fashion that amounts to a consistent truncation in the classical limit, followed by consistent dimensional reductions, we have found that the action contains various higher spin CS models as well as the action [14]  Turning to applications, it would be interesting to see to what extent the action, possibly supplemented by boundary terms, can be used to compute the free energy and entropy of exact solutions of the PV system, such as the recent nontrivial examples found in [32]; for related proposals for on-shell actions, see [12] and [33]. The action could also have a bearing on the one-loop corrections from matter fields to the higher spin CS gauge sector. The above implementations may be useful in solidifying the Gaberdiel-Gopakumar (GG) conjecture [9]. In particular, radiative corrections may be of importance in matching symmetry algebras [34] beyond the realm of CS actions; for reviews of the CS approximation, see [35], and for existing works beyond the CS approximation, see [36,37,18].
As for alternatives to the PV system, an interesting action for matter coupled higher spin gravity has been presented in [12]. Its four-dimensional covariant Hamiltonian reformulation is given by the BF-like action where (A, A, B, B; T, T , S, S) are forms of degrees (1, 1, 0, 2; 2, 2, 3, 1) valued in Aq + (2) ∼ = hs( 1 2 ) and The action, with its dynamical two-formB, cannot be obtained from the six-dimensional master action (3.27), as there is an obstruction due to the presence of the central term h. Instead, it is natural to seek a connection between (5.1) and the PV system via a six-dimensional model on X 4 × Z 2 built along the same lines as the nine-dimensional Frobenius-Chern-Simons model in [17]. The construction of such a model will be presented elsewhere.
Comparing higher spin gravities in three and four dimensions, the latter admit covariant Hamiltonian actions in nine dimensions [16,17], though it remains unclear whether they contain the standard symplectic structure for Fronsdal fields [38] 16 . Essentially, this is due to the presence of extra auxiliary fields in the unfolded description of Fronsdal fields on-shell, whose inclusion into a strictly four-dimensional off-shell formulation remains problematic. However, the on-shell actions receive contributions as well from boundary terms [21,27] given by topological invariants that reduce on-shell to higher spin invariants [21]. These invariance, which are inserted on the eight-dimensional boundaries, are given by integrals over closed p-cycles in spacetime of on-shell (de Rham) closed p-forms in their turn given by integrals over twistor space of constructs built from spacetime curvatures. In terms of these observables, spacetime emerges in limits where physical states labelled by spacetime points become separated from each other [39]; see also the Conclusions of [23]. In particular, for p = 0, the resulting zero-form charges [40] of the (minimal bosonic) Type A and Type B models [41,20] were shown in [42,43] to provide free theory correlation functions at the leading classical order, in accordance with the proposal made in [44,45,46,47]. For boundary conditions corresponding to free fields, this proposal requires that correlation functions with separated points do not receive any radiative corrections, in agreement with the covariant Hamiltonian approach [16] 17 ; for a similar usage of zero-form charges in 3D, see [12].
To conclude, we view higher spin gravity as a useful laboratory for exploring the treatment of quantum field theory with local degrees of freedom by combining the AKSZ approach [26] to topological field theories on manifolds with boundaries and Cartan and Vasiliev's formulation of nonlinear partial differential equations as free differential algebras with infinite zero-form towers.
To question its universality, it would be desirable to treat models in which nontrivial radiative corrections arise in an as simple context as possible. To this end, three-dimensional models might prove to be fruitful and we hope that our action will be helpful in this endeavour.

A Massive vacuum of PV system
In this Appendix, we rewrite the massive vacuum of the PV system [13] using the Clifford algebra variables. The vacuum solution reads α . The constraints along M 3 can be solved using a gauge function that commutes to Γ, viz.
The integrability on Z 2 implies the existence of a doubleỹ α obeying for some matrix A ∈ C 4 . Thus one can take Remarkably, as found in [13], the solution obeys At the level of the complexified algebra, a solution is given bỹ where X, Y , A and B are built out of gamma matrices Γ i and the building blocks dt t e ityz (y α + z α ) , τ α := ν 1 0 dt (t − 1) e ityz (y α + z α ) , (A. 10) with yz := y α z α , obey For the above Ansatz, Eqs. (A.5), (A.6) and (A.8), respectively, are equivalent to A solution forz α that commutes to C + 4 is obtained by taking [13] 18 , and the relation between the PV generators and ours is given in (2.31), which yields This solution, however, does not satisfy the required reality conditions, i.e. ((ỹ C α ) † , (z C α ) † ) form a set of deformed oscillators that is linearly independent from (ỹ C α ,z C α ) This is remedied by a highly nontrivial modification found in [13] given bỹ where Φ(a, b; z) is the confluent hypergeometric function. 18 Using the gauge function, the first order fluctuation B (1) can be written as To this end, we observe that the equations of motion (3.32) admit a consistent truncation given by as can be seen from the fact that the resulting field equations, form a Cartan integrable system. Inserting (B.1) into the master action (3.27), the resulting consistently truncated action is given by Its gauge symmetries take the form 4) and the equations of motion and boundary conditions are given by To reach a Blencowe type action, we assume Eq. (4.11) and make the reduction where the subscripts indicating the form degree will be suppressed henceforth, and V (0) is a twistor space background connection obeying To obtain the alternative model, we make the choice corresponding to m = 1 in Section 4.1. We also take W ∈ Ω [1] (X 4 ) ⊗ Aq + (2; ν) ⊗ C + 4 , (B.14) T ∈ Ω [2] (X 4 ) ⊗ ρ(End + (F σ ν )) ⊗ C + 4 , (B. 15) where F σ ν is the Fock representation space of Aq(2; ν) with ground state having eigenvalue σ of the Klein operator −Γ, and ρ : End(F σ ν ) → Aq(2; ν) (B.16) is a monomorphism given by the deformed oscillator realization of End(F σ ν ). The space End + (F σ ν ) consists of the endomorphisms that commute to Γ. Its oscillator realization where w = {a + , a − } ⋆ , a ± = u ± α y α , u −α u + α = i 2 , (B.20) is the symbol of the oscillator realization of the ground state projector in End(F σ ν ) given in Weyl order; for details, see [25,31]. The reduced action readš Tr C 4 ΓJ ⋆ T ⋆ ( F + g + g ′ + 1 2 T ) , (B.21) where T ⋆ ( F + g + g ′ ) and T ⋆ T lie in ρ(End + (F ν )). The Lagrangian is finite provided that is finite. If so, the dual pairing displayed in (B.14) and (B.15) is non-degenerate, and the equations of motion and boundary conditions read where the first equation is valued in Aq + (2; ν) and the second equation in ρ(End + (F ν )). Elimi-natingŤ via the first equation (by inverting the monomorphism), yieldš which is formally divergent for gauge fields given by finite polynomials, unless It is possible to construct a deformation of Blencowe's action by making use of Vasilievs (graded cyclic) supertrace operation STr ν on the Weyl algebra Aq(2; ν) of the deformed oscillator algebra (B.10) [15], which is uniquely characterized by STr ν 1 = 1 and STr ν Γ = ν (and hence differs from the trace operation proposed above). Using this operation, it straightforward to deform Blencowe's action in X 3 and uplift it to BF-type model in X 4 with actioň where T and W are valued in Aq + (2; ν) ⊗ C + 4 . Whether there exists a modification of (B.13) that yields the STr ν operation starting from the master action in six dimensions, possibly by using the trace operation (4.23) for m = 0, remains to be seen 19 . Thus, provided that N σ ν is finite, we have found a covariant Hamiltonian action in which the gauge fields and the Lagrange multipliers belong non-isomorphic dual spaces that is an alternative to ν-deformed Blencowe's action. Whether this action admits a coupling to matter is unknown, while the action presented above appears to be amenable to such couplings on M.