Canonical Deformation of $N=2$ $AdS_{4}$ SUGRA

It is known that one can define a consistent theory of extended, $N=2$ anti-de Sitter (AdS) Supergravity (SUGRA) in $D=4$. Besides the standard gravitational part, this theory involves a single $U(1)$ gauge field and a pair of Majorana vector-spinors that can be mixed into a pair of charged spin-$3/2$ gravitini. The action for $N=2$ $AdS_{4}$ SUGRA is invariant under $SO(1,3)\times U(1)$ gauge transformations, and under local SUSY. We present a geometric action that involves two"inhomogeneous"parts: an orthosymplectic $OSp(4\vert 2)$ gauge-invariant action of the Yang-Mills type, and a supplementary action invariant under purely bosonic $SO(2,3)\times U(1)\sim Sp(4)\times SO(2)$ sector of $OSp(4\vert 2)$, that needs to be added for consistency. This action reduces to $N=2$ $AdS_{4}$ SUGRA after gauge fixing, for which we use a constrained auxiliary field in the manner of Stelle and West. Canonical deformation is performed by using the Seiberg-Witten approach to noncommutative (NC) gauge field theory with the Moyal product. The NC-deformed action is expanded in powers of the deformation parameter $\theta^{\mu\nu}$ up to the first order. We show that $N=2$ $AdS_{4}$ SUGRA has non-vanishing linear NC correction in the physical gauge, originating from the additional, purely bosonic action. For comparison, simple $N=1$ Poinacar\'{e} SUGRA can be obtained in the same manner, directly from an $OSp(4\vert 1)$ gauge-invariant action. The first non-vanishing NC correction is quadratic in $\theta^{\mu\nu}$ and therefore exceedingly difficult to calculate. Under Wigner-In\"{o}n\"{u} (WI) contraction, $N=2$ AdS superalgebra reduces to $N=2$ Poincar\'{e} superalgebra, and it is not clear whether this relation holds after canonical deformation. We present the linear NC correction to $N=2$ $AdS_{4}$ SUGRA explicitly, discuss its low-energy limit, and what remains of it after WI contraction.


Introduction
In our quest for the theory of "Quantum Gravity", we must be prepared to go beyond some deeply rooted assumptions on which we are accustomed, in particular, at very short distances (very high energies) we might have to abandon the notion of a continuous spacetime and the associated mathematical concept of a smooth manifold that describes it. One distinguished approach to the problem is Noncommutative (NC) Field Theory -a theory of relativistic fields on noncommutative space-time, based on the method of deformation quantization by NC ⋆-product [1][2][3]. One speaks of a deformation of an object/structure whenever there is a family of similar objects/structures whose "distortion" from the original, "undeformed" one, can be somehow parametrized. In physics, this so-called deformation parameter appears as some fundamental constant of nature that measures the deviation from the classical (i.e. undeformed) theory. This way of "quantizing" space-time is essentially different from the standard QFT quantization procedure for matter fields. Different space-time dimensions (the usual 3 + 1) are regarded as being mutually "incompatible", in a sense that there exist a lower bound for the product of uncertainties ∆x µ ∆x ν for a pair of two different coordinates. To capture this "pointlessness" of space-time, one introduces an abstract algebra of NC coordinates as a deformation of the classical structure. These NC coordinates, denoted byx µ , satisfy some non-trivial commutation relations, and so, it is no longer the case thatx µxν =x νxµ . Abandoning this basic property of space-time leads to various new physical effects that were not present in theories based on classical spacetime. The simplest case of noncommutativity is the so-called canonical (or θ-constant) noncommutativity, where θ µν are components of a constant antisymmetric matrix, and Λ N C is the length scale at which NC effects become relevant. Deformation parameter is a fundamental constant, like the Planck length or the speed of light.
Instead of deforming abstract algebra of coordinates one can take an alternative, but equivalent approach in which noncommutativity appears in the form of NC products of functions (fields) of commutative variables (coordinates). These products are called star products (⋆-products). In particular, to establish canonical noncommutativity, we use the Moyal ⋆-product, The leading term in the expansion of the exponential is the ordinary commutative product of functions, and the higher order terms represent NC (non-classical) corrections.
Up to date, we still lack direct physical evidence of Supersymmetry (SUSY), at least in its simplest form. Nevertheless, its beneficial influence on high-energy physics (improved renormalizability in QFT and a natural resolution of the hierarchy problem), along with its mathematical consistency and unification power (especially unification of gravity and the Standard Model within Supergravity (SUGRA), and ultimate unification scheme such as Superstring theory), motivate us to seriously consider SUSY as an integral part of our description of nature. Since the pioneering work of Freedman, van Nieuwenhuizen and Ferrara [4,5], and Deser and Zumino [6], the theory of supergravity has become a welldeveloped field of research. SUGRA provides a natural unification of gravity with other fields by imposing gauge principle on SUSY, the associated gauge field being the spin-3/2 gravitino field described by a Majorana vector-spinor. It was demonstrated in [7,8] that one can have a consistent theory of extended N = 2 AdS 4 SUGRA with complex (U (1)charged) gravitino field. In this paper, we propose a geometric way of obtaining N = 2 AdS 4 SUGRA action and perform its NC deformation. The obtained NC correction can be regarded as a low-energy signature of the underlying theory of quantum gravity. We calculate the correction explicitly and discuss some of its properties.
Similarly, one can establish NC SUGRA by gauging an appropriate supergroup [42][43][44][45][46][47][48][49][50] and subsequently performing canonical deformation. Since pure gravity can be obtained by gauging AdS group SO(2, 3), orthosymplectic supergroup OSp(4|1) appears as a natural choice for pure N = 1 Poincaré SUGRA. Bosonic sector of osp(4|1) superalgebrasymplectic algebra sp(4) -is isomorphic to AdS algebra so(2, 3) that reduces to Poincaré algebra under Wigner-Inönü contraction [51]. The subject of NC SUGRA has been treated in [52,53]. Classical action for OSp(4|1) SUGRA presented in [53] is manifestly invariant under OSp(4|1) gauge transformations, and we will use it as a motivation. However, to obtain explicit NC deformation of this action is exceedingly difficult, because the first nonvanishing NC correction is quadratic in θ µν . Taking a lesson from [54][55][56] that inclusion of Dirac spinors coupled to U (1) gauge field produces (much simpler) linear NC correction, we will make a transition to OSp(4|2) SUGRA that involves a pair of Majorana spinors that can be mixed into a pair of charged spin-3/2 gravitini coupled to U (1) gauge field. We present a geometric action that consists of two "inhomogeneous" parts: an OSp(4|2) gauge-invariant action of the Yang-Mills type, and a supplementary action, invariant under purely bosonic SO(2, 3) × U (1) sector of OSp(4|2), that has to be included in order to obtain complete N = 2 AdS 4 SUGRA at the classical level; this additional bosonic term produces a non-trivial linear NC correction to N = 2 AdS 4 SUGRA, after deformation.
In Section 2, we introduce undeformed geometric action for OSp(4|2) SUGRA and make comparison with the similar action for OSp(4|1) SUGRA. In Section 3, we perform NC deformation by using the Seiberg-Witten approach, and study the first order NC correction to N = 2 AdS 4 SUGRA. Section 4 contains discussion and proposals for further investigation. Appendices A and B contain supplementary material.

Classical Orthosymplectic SUGRA
We consider two classical (i.e. undeformed) SUGRA models based on orthosymplectic OSp(4|N ) gauge group: the simple N = 1 AdS 4 SUGRA, describing pure supergravity with the negative cosmological constant, and the extended N = 2 AdS 4 SUGRA that involves also a pair of charged gravitini fields coupled to U (1) gauge field. We focus our attentions on the latter (N = 2), since the former (N = 1) has been treated extensively in [53], including its NC deformation, and we discuss it just for comparison. Some significant differences of the two models in question are manifested already at the level of their classical actions, and this reflects drastically on the structure of their NC corrections after deformation.

Classical OSp(4|2) SUGRA
Orthosymplectic group OSp(4|2) has 19 generators, and they are of two kinds -bosonic and fermionic. Ten bosonic generatorsM AB = −M BA (A, B = 0, 1, 2, 3, 5) span AdS Lie algebra so(2, 3) (symmetry algebra of AdS 4 ), where η AB is flat 5D metric with signature (+, −, −, −, +). By splitting this set of generators into sixM ab AdS rotation generators (a, b = 0, 1, 2, 3) and four AdS translation generatorŝ M a5 , we can recast so(2, 3) algebra in a more explicit form: If we introduce a new set of generators (M ab ,P a ) defined byM ab :=M ab andP a := l −1M a5 = αM a5 , where l is a length scale related to AdS radius and α = l −1 (we will use both parameters in the following formulae), the algebra (2.2) transforms into: In the limit α → 0 (or l → ∞), AdS algebra reduces to Poincaré algebra, in particular, we obtain [P a ,P b ] = 0 with all other commutators left unchanged. This is a famous example of the Wigner-Inönü (WI) contraction, the contraction parameter being α (or l). This Liealgebra contraction (or deformation) can be extended to AdS superalgebra, and we will be interested, later on, in its effect on the NC correction of N = 2 AdS 4 SUGRA.
A representation of the AdS sector of osp(4|1) superalgebra can be obtained by using 5D gamma matrices Γ A satisfying Clifford algebra {Γ A , Γ B } = 2η AB ; the AdS generatorsM AB are represented by 6 × 6 super-matrices, that reduce to 4 × 4 matrices M AB = i 4 [Γ A , Γ B ] in the AdS subspace, see Appendix A. One choice of Γ-matrices is Γ A = (iγ a γ 5 , γ 5 ), where γ a are the usual 4D γ-matrices. In this particular representation, the components of M AB are given by M ab = i 4 [γ a , γ b ] = 1 2 σ ab and M a5 = − 1 2 γ a .
The ten AdS bosonic generators M AB are accompanied by eight independent fermionic generatorsQ I α , with spinor index α = 1, 2, 3, 4 and SO(2) index I = 1, 2, comprising a pair of Majorana spinors, and one additional bosonic generatorT related to SO(2) ∼ U (1) extension. Together, they satisfy osp(4|2) superalgebra (consistency requires that fermionic generatorsQ I α transform as components of an AdS Majorana spinor): with antisymmetric tensor ε IJ , ε 12 = 1. Matrix C −1 is the inverse of the charge-conjugation matrix (spinor metric) for which we use the following representation given in terms of Pauli matrices: C = −σ 3 ⊗ iσ 2 , and C αβ = −C βα . Numerically we have C −1 = −C, but the index structure of the two is different since C αγ (C −1 ) γβ = δ β α . More visually, An explicit matrix representation of osp(4|2) superalgebra is given in Appendix A.
Orthosymplectic supergroup OSp(2n|m) (symplectic sector is always even-dimensional) consists of those super-matrices U that preserve the graded metric with some real 2n × 2n matrix Σ αβ = −Σ βα , and some real m × m matrix ∆ ij = ∆ ji . Considering only infinitesimal transformations U = 1 + ǫM , generated by some osp(2n|m)valued supermatrix (bosonic blocks A 2n×2n and D m×m have ordinary commuting entries, and fermionic blocks B 2n×m and C m×2n have Grassmann-valued entries), the defining relation becomes Super-transpose, super-hermitian adjoint and super-trace are defined by imposing the stan- Now, the key observation is that a pair of Majorana fields χ I µ (describing a pair of neutral spin-3/2 gravitini) constitute the fermionic sector of the osp(4|2) connection supermatrix Ω µ . We can expand this super-connection over the basis {M ab ,M a5 ,Q I α ,T } with the corresponding gauge fields {ω ab µ , ω a5 µ ,χ I µ , A µ }, as . Equivalently, we can expand Ω µ over the re-scaled basis {M ab ,P a ,Q I α ,T }, but with a different set of gauge fields {ω ab µ , e a µ : where we again have so(2, 3) gauge field ω µ = 1 4 ω ab µ σ ab − α 2 e a µ γ a , two independent Majorana spinors ψ I µ , and (dimensionless) U (1) vector potential A µ . We will use this particular representation because it makes WI contraction more transparent.
The two Majorana spinors, ψ 1 µ and ψ 2 µ , can be combined into an SO(2) doublet, It can be readily confirmed that gauge supermatrix (2.12) satisfies the defining relation for the elements of osp(4|2) superalgebra (C is the charge-conjugation matrix (2.5)), (2.14) Field strength associated with AdS gauge field ω µ is It was shown in the seventies that one can relate AdS gauge field theory to gravity (GR) by identifying ω ab µ with the Lorentz spin-connection, ω a5 µ with the re-scaled vierbein field αe a µ ; vierbein is related to the metric tensor by η ab e a µ e b ν = g µν and e = det(e a µ ) = √ −g. Consequently, R ab µν can be identified with the curvature tensor, and F a5 µν with re-scaled torsion αT a µν . Therefore, in the AdS setting, we have a natural unification of vierbein and spin-connection as components of a general SO(2, 3) gauge field; each transforms as a gauge field and stands on equal footing. In order to establish this identification, one has to break the original AdS gauge symmetry to the Lorentz SO(1, 3) gauge symmetry by introducing an auxiliary field φ = φ A Γ A [30]. This field transforms in the adjoint representation of SO(2, 3) and it is constrained by η AB φ A φ B = l 2 . We can now start with an action of the Yang-Mills type, originally suggested by McDowell and Mansouri [28], invariant under SO(2, 3) gauge transformations: where we have AdS covariant derivative in the adjoint representation, We choose the physical gauge by setting φ a = 0 and φ 5 = l, and thus obtain: which is the standard GR action (written in the first order formalism) involving the Einstein-Hilbert term, negative cosmological constant Λ = −3/l 2 = −3α 2 , and the topological Gauss-Bonet term that can be omitted.
Therefore, we can write the SO(2, 3) field strength as and we see that the vierbein and torsion terms vanish under WI contraction.
By generalization, we introduce Osp(4|2) field strength F µν associated with the superconnection Ω µ , with extended AdS field strength F µν (summation over I = 1, 2 is implied) involving extended curvature tensor R mn µν and extended torsion T m µν , given by Electromagnetic field strength is also modified by a bilinear current term J (e) , Note that Pauli matrix iσ 2 mixes the two Majorana components in J (e) .
In the fermionic sector of F µν we introduced where D µ stands for SO(2, 3) covariant derivative. The fact that Majorana spinors ψ 1 µ and ψ 2 ν are not charged is reflected in the manner in which they couple to the gauge field A µ . Using them, we can define two charged Dirac vector-spinors ψ ± µ = ψ 1 µ ± iψ 2 µ , related to each other by C-conjugation, ψ − µ = ψ c+ µ = Cψ +T µ , that do couple to A µ in the right way. Using the Pauli matrix iσ 2 we can unify (2.27) and (2.28) as Now consider an action, similar to the one defined in (2.18) for pure gravity, but now appropriately generalized to be invariant under extended OSp(4|2) gauge transformations, The action is real and we introduced a pair of free parameters, a and b, that will by fixed later. The first part of (2.30) is the quadratic Yang-Mills type of action, and the second part (b-term) is necessary for having local SUSY after the symmetry braking.
After the gauge fixing, field Φ 2 /l 2 that appears in the second term of (2.30) becomes a projector that reduces any osp(4|2) supermatrix to its so(2, 3) sector, and the classical OSp(4|2) gauge-invariant action (2.30) reduces to The term that is quadratic in the Lorentz SO(1, 3) covariant derivative D L µ can be transformed by partial integration, where we invoked the commutator of two Lorentz covariant derivatives Term of the same type appears in the first part of the action (2.35). These two contributions have to cancel each other in order to have SUSY, and this implies the constraint b = −a/2. Moreover, to obtain the correct normalization of the Einstein-Hilbert term, we set a = il/32πG N = il/4κ 2 , yielding However, this is not the full N = 2 AdS 4 SUGRA action. The gravity part is correct (we can omit the topological Gauss-Bonet term) and we also get the correct kinetic term for the gravitino doublet. There are also two bilinear source terms, electric and magnetic, But we are missing the contribution from the SO(2) part of the bosonic sector, in particular, the kinetic term for U (1) gauge field A µ . The reason for this defect can be traced back to the specific form that the auxiliary field assumes in the physical gauge Φ| g.f. (2.32); it completely annihilates the SO(2) sector of any osp(4|2) supermatrix. To restore the missing terms, we must introduce an additional action, supplementing (2.30). In [55], following the approach of [57], we defined a classical action invariant under SO(2, 3) × U (1) gauge transformations (∼bosonic sector of OSp(4|2)) that involves an additional auxiliary field f = 1 2 f AB M AB . Its role is to produce the canonical kinetic term for U (1) gauge field in the absence of Hodge dual operator (this is of course the crucial point, we are trying to construct a purely geometrical action that does not involve the metric tensor g µν explicitly). This auxiliary field f is a U (1)-neutral 0-form that takes values in so(2, 3) algebra, and it transforms in the adjoint representation of SO (2, 3).
The way to proceed is to employ this auxiliary field method to include the modified U (1) field strength F µν defined in (2.26). However, there seems to be no way to construct an OSp(4|2) gauge invariant action that is compatible with this procedure. Therefore, we will use an action, analogues to the one in [55], invariant under purely bosonic SO(2, 3) × U (1) sector of OSp(4|2), involving the bosonic field strength f µν : . The action is given by (2.40) Note that, by doing this, we lose the complete OSp(4|2) gauge invariance of the undeformed action before the symmetry breaking. Nevertheless, we will obtain the correct action for N = 2 SdS 4 SUGRA in the physical gauge, and this is the only requirement that has to be satisfied in order to perform NC deformation.
After calculating traces (see Appendix B) we obtain We conclude that parameter c must be real, otherwise the second term, involving F µν , would be purely imaginary and would not contribute (and this term is the one that we need to include). Therefore, assuming real c, the first term (involving gravitational quantities like curvature tensor and torsion) becomes purely imaginary and vanishes after adding its complex conjugate (c.c.). Also, d must be purely imaginary for the procedure to work.
Gauge fixing yields By varying this gauge fixed action over f ab and f a5 independently, we obtain algebraic equations of motion (EoMs) for the components f ab and f a5 of the auxiliary field f , respectfully, and they are given by Inserting them back into the action (2.42), we obtain (2.44) To get the consistent normalization we set the prefactor to (8κ 2 ) −1 , yielding another constraint 16ilc 2 = 3d for the parameters c and d. To make the connection with the results of [55], we take c = 1/32l and d = i/192l, implying (2.45) Therefore, after imposing the physical gauge, the original bosonic action (2.40), invariant under SO(2, 3) × U (1) gauge transformations, reduces to SO(1, 3) × U (1) gauge-invariant action containing the canonical kinetic term for U (1) gauge field A µ in curved space-time, and two additional terms involving gravitino current J (e)µν =Ψ µ iσ 2 Ψ ν , .

(2.46)
This is exactly the piece that was missing in (2.38). With this result in hand, we have the complete classical N = 2 AdS 4 SUGRA action [43,44], (2.47) The most important characteristics of this SUGRA model are the negative cosmological constant Λ = −3α 2 = −3/l 2 , and the fact that U (1) coupling strength is equal to the WI contraction parameter α. Under WI contraction (α → 0) the N = 2 AdS 4 SUGRA action consistently reduces to the N = 2 Poincaré SUGRA action.
In terms of charged Dirac vector-spinors ψ ± µ = ψ 1 µ ± iψ 2 µ (actually, we can use only one of them since they are related to each other by C-conjugation) the action becomes . For later purpose, we note that action (2.48) contains a mass-like term for charged gravitino (we absorb the parameter κ −1 into ψ + µ to obtain the canonical dimensions), with mass-like parameter equal to the WI contraction parameter.

OSp(4|1) SUGRA
The OSp(4|1) supergroup has 14 generators; ten bosonic AdS generatorsM AB , and four fermionic generators,Q α , comprising a single Majorana spinor (describing a single neutral gravitino). Supermatrix for the OSp(4|1) gauge field Ω µ is given by Consider the following action invariant under OSp(4|1) gauge transformations [52], In the physical gauge, the OSp(4|1) gauge-invariant action (2.51) exactly reduces to N = 1 AdS 4 SUGRA action [43,44,53], It contains Einstein-Hilbert term with the negative cosmological constant Λ = −3/l 2 , Rarita-Schwinger kinetic term for neutral gravitino, and a mass-like gravitino term that is needed in the presence of the cosmological constant to insure the invariance under local SUSY (gravitino actually remains massless). Topological Gauss-Bonet term is omitted. Cosmological constant and the mass-like term vanish under WI contraction, yielding minimal N = 1 Poincaré SUGRA. Note that we do not need additional action terms in (2.51) to obtain a consistent classical theory.
It is shown in [53] that linear (in θ µν ) NC correction to (2.51) vanishes, and that one has to calculate the second order NC correction in order to see NC effects, which is exceedingly difficult. In the following section, we use the Seiberg-Witten approach to NC gauge field theories, to calculate linear NC correction to N = 2 AdS 4 SUGRA, and conclude that it is not equal to zero. The non-vanishing part comes from the additional bosonic action, S A .

NC deformation
Canonical deformation of the orthoymplectic action (2.30) is obtained by replacing ordinary commutative field multiplication with Moyal ⋆-product, yielding an NC action (denoted by "⋆") manifestly invariant under NC-deformed OSp(4|2) ⋆ gauge transformations, We denote NC fields by a "hat" symbol.
In the Seiberg-Witten approach [3,[34][35][36][37], NC gauge field theory is completely defined by its commutative (classical) counterpart. For some non-Abelian gauge group G with generators T A satisfying Lie algebra relations [T A , T B ] = if C A B T C , commutator of two infinitesimal gauge transformations δ ǫ 1 and δ ǫ 2 closes in the algebra, There is, however, a difficulty, in general, concerning the closure axiom for NC gauge transformations. Namely, for a given pair of NC gauge parametersΛ 1 andΛ 2 we would like to find a third one,Λ 3 , such that Now, if NC gauge parameterΛ is supposed to be Lie algebra-valued,Λ(x) =Λ A (x)T A , then, for some generic NC fieldΨ that transforms in the fundamental representation of the gauge group (although the argument holds in any representation), we have The NC closure rule consistently generalizes its commutative counterpart.
However, (3.5) implies that commutator of two NC gauge transformations does not generally close in the Lie algebra, because anti-commutator {T A , T B } does not in general belong to this algebra (except for U (N ) gauge group). To overcome this difficulty, we will apply the universal enveloping algebra (UEA) approach. Enveloping algebra is "large enough" to ensure that closure property of NC gauge transformations holds, provided that NC gauge parameterΛ is UEA-valued. NC covariant derivative (for a generic gauge group G) in the fundamental representation is defined by whereV µ stands for the corresponding NC gauge field, and it transforms as Therefore, NC gauge field must also be UEA-valued and it can be represented in its basis. But, UEA has an infinite basis, and it seems that by invoking it we actually introduced an infinite number of new degrees of freedom (new fields) in the NC theory, rendering it unrealistic. This problem is resolved by the Seiberg-Witten (SW) map [34][35][36][37]. Essentially, we assume that classical gauge transformations induce the corresponding NC gauge transformations, This allows us to represent every NC fields as a perturbation series in powers of the deformation parameter θ µν with expansion coefficients built out of commutative fields, e.g. Λ ǫ = ǫ+Λ (1) +Λ (2) +.... At zeroth order, NC fields reduce to their undeformed counterparts. For example, NC gauge parameter and potential can be represented aŝ After these general considerations, we return to the NC action (3.1). Field strengthF µν appearing in (3.1) is defined in terms of OSp(4, 2) ⋆ gauge potentialΩ µ aŝ (3.13) It transforms in the adjoint representation of OSp(4, 2) ⋆ supergroup as well as the NC auxiliary fieldΦ, (3.14) At this point it would be tempting to proceed by directly imposing the physical gauge. However, this operation would not yield an action with an appropriate symmetry because gauge fixing does not commute with NC deformation. A bypass is provided by the SW map. Representing NC fields in terms of commutative ones, as prescribed by the SW map, we obtain a perturbative expansion of OSp(4|2) * gauge-invariant NC action (3.1) in powers of the deformation parameter θ µν . By construction, SW map ensures invariance of the expansion under ordinary OSp(4|2) gauge transformations, order-by-order.
Now we present some relevant steps in the expansion procedure of the NC action (3.1). Our goal is to calculate and analyze linear NC correction to the classical action (2.30). According to the SW map, the first order NC corrections of the auxiliary field Φ and the OSp(4|2) field strength F µν are given bŷ where D µ stands for the OSp(4|2) covariant derivative (associated to Ω µ ).
Generally, for a pair of NC fieldsÂ andB, the linear NC correction to their product is Â ⋆B In particular, if both fields transform in the adjoint representation of OSp(4|2) ⋆ , we have Â ⋆B where cov(Â (1) ) is the covariant part of A ′ s first order NC correction, and cov(B (1) ), the covariant part of B ′ s first order NC correction. Successive application of this rule gives us the first order NC correction to the classical action (2.30): This linear NC correction is real and invariant under OSp(4, 2) gauge transformations.
However, a careful examination shows that after the gauge fixing it vanishes completely, But we still have the additional NC action S ⋆ A invariant under purely bosonic NC-deformed SO(2, 3) ⋆ × U (1) ⋆ gauge transformations. The only additional SW expansion we need is that off , namelyf The first order NC correction to (3.2) before gauge fixing is given by where we can distinguish the linear f -part and the quadratic f 2 -part, and all terms are manifestly SO(2, 3) × U (1) invariant by the virtue of SW map.
After calculating traces and evaluating the gauge-fixed action S (1) A | g.f. on the EoMs of the components of the auxiliary field f (as it turns out, to obtain the first order NC correction, we only need to insert zeroth order (classical) EoMs (2.43) in the gauge-fixed first order NC action S (1) A,EoM f f | g.f. , (3.23) with the individual terms: e µ a e ν m e ρ b e σ n + e ρ a e σ m e µ b e ν n + 2e ρ n e σ m (e µ a e ν b − e ν a e µ b ) And finally, the f 2 -term, Action (3.23) represents the first order NC correction to N = 2 AdS 4 SUGRA. It involves various new couplings between U (1) gauge field, gravity and gravitini fields that appear due to space-time noncommutativity. As it stands, this action seems too complicated to be analyzed in its entirety. However, we can restrict ourselves to some particular domain of parameters and work with an approximated NC action. In particular, we will derive a lowenergy approximation of (3.23), by taking into account terms at most quadratic in partial derivative. Therefore, we include only terms linear in curvature, and linear and quadratic in torsion. Additionally, we assume that spin connection ω ab µ and the first order derivatives of vierbeins are of the same order. Note also that the torsion constraint T a µν = 0 (2.25) gives us T a µν = −iΨ µ γ a Ψ ν . These assumptions yield a very simple action, This mass-like term for charged gravitino ψ + µ , minimally coupled to gravity, appears due to space-time noncommutativity, and "renormalizes" the corresponding term (2.49) in the classical SUGRA action (2.48). If we again absorb κ −1 in ψ + µ to obtain the canonical dimensions, the mass-like parameter is ∼ l P Λ 2 N C /l 4 , and it vanishes under WI contraction.
After WI contraction, the action (3.23) reduces to (3.31) At this point we are confronted with an interesting question. The fact that N = 2 AdS 4 superalgebra contracts to N = 2 Poinacaré superalgebra when l → ∞ is consistently reflected on the level of classical (undeformed) action (

Discussion and Outlook
Let us emphasize the main points of this paper and propose some further paths of investigation. At this stage, our primary goal was to obtain explicit NC correction to N = 2 AdS SUGRA in D = 4. We stared from a classical (undeformed) action (2.30) of the Yang-Mills type (already advocated in the literature), invariant under orthosymplectic OSp(4|2) gauge transformations. However, this action alone is not enough to obtain N = 2 AdS 4 SUGRA after fixing the gauge (for which we use a constrained auxiliary field). In particular, one has to add a supplementary action (2.40) endowed with SO(2, 3) × U (1) gauge symmetry (bosonic sector of OSp(4|2)) that provides the missing terms in the classical SUGRA action (e.g. the kinetic term for U (1) gauge field). Therefore, we have the following schema:

N=2 AdS SUGRA in D=4
This situation seems curious considering that a similar OSp(4|1) gauge-invariant action (2.51) in the same gauge reduces to the complete N = 1 AdS 4 SUGRA action. We may conclude that extended N > 1 AdS 4 SUGRA cannot be obtained simply by gauging the corresponding orthosymplectic group OSp(4|N ) and subsequently fixing the gauge. For N > 1 we would have to include an additional term similar to (2.40) that involves non-Abelian Yang-Mills gauge field. NC deformation is performed following the Seiberg-Witten approach to NC gauge field theory that involves universal enveloping algebra-valued gauge field and perturbative expansion of the NC-deformed action in powers of the deformation parameter θ µν . The expanded action possesses gauge symmetry of the corresponding classical action, order-by-order, and we focus only on the linear NC correction that remains after the gauge fixing. For the OSp(4|2) gauge-invariant part, linear NC correction vanishes. The reason why this result strikes us as curious is related to some previously establish facts about canonical NC deformation of the similar models. Namely, canonical deformation of pure gravity, regarded as a gauge theory of SO(2, 3) group, leads to quadratic NC correction [25,26,38]. However, after including charged matter (Dirac spinors) coupled to U (1) gauge field, linear NC correction appears [54][55][56]. Since we can take a pair of Majorana vector-spinors of OSp(4|2) SUGRA and form a pair of U (1)-charged Dirac vector-spinors, related to each other by C-conjugation, we expected to obtain a non-vanishing first order NC correction from the OSp(4|2) action (3.1), as well.
However, the supplementary bosonic action provides a non-trivial linear NC correction that is calculated explicitly (3.23). It involves various new interaction terms that are present due to space-time noncommutativity. The full action is difficult to analyze, but we can restrict ourselves to the low-energy sector of the theory by taking into account only terms that are at most quadratic in partial derivatives. This leaves us with a single mass-like term for charged gravitino minimally coupled to gravity.
WI contraction eliminates many of these new interaction terms, but not all of them (3.31). The ones remaining may help us understand the relation between the canonical NC deformation and WI contraction, at least in this particular case. N = 2 AdS superalgebra reduces to N = 2 Poincaré superalgebra under WI contraction and the same holds for their classical actions. Therefore, it may be the case that the same relation pertains even after canonical NC deformation. To confirm this assumption directly, we have to calculate linear NC correction to N = 2 Poincaré SUGRA, and make the comparison.
Let us just mention that there are additional two terms with OSp(4|2) gauge symmetry that we could include. We denote them by S ′ and S ′′ and they are given by S ′ = a ′ 128πG N l STr d 4 x ε µνρσ F µν D ρ Φ D σ ΦΦ + c.c. , with free dimensionless parameters a ′ , a ′′ , and OSp(4|2) covariant derivative D µ . Their SO(2, 3) gauge-invariant counterparts were analyzed in [26]. After gauge fixing, they modify the coefficients in the classical action but do not introduce new terms. In particular, they give us a freedom to eliminate the cosmological constant in the classical action. NC deformation of S ′ and S ′′ will certainly change our final result, but their importance is not yet clear. Analysis of these additional NC corrections remains to be done.