A general relativity formulation for the Doplicher-Fredenhagen-Roberts noncommutative space-time
aa r X i v : . [ h e p - t h ] A p r A general relativity formulation for the Doplicher-Fredenhagen-Robertsnoncommutative space-time
M. J. Neves
1, 2, ∗ and Everton M. C. Abreu
2, 3, 4, † Department of Physics and Astronomy, University of Alabama, Tuscaloosa, AL 35487, USA Departamento de F´ısica, Universidade Federal Rural do Rio de Janeiro, BR 465-07, 23890-971, Serop´edica, RJ, Brazil Departamento de F´ısica, Universidade Federal de Juiz de Fora, 36036-330, Juiz de Fora, MG, Brazil Programa de P´os-Gradua¸c˜ao Interdisciplinar em F´ısica Aplicada, Instituto de F´ısica,Universidade Federal do Rio de Janeiro, 21941-972, Rio de Janeiro, RJ, Brazil (Dated: April 16, 2020)The Doplicher, Fredenhagen and Roberts (DFR) noncommutative (NC) formalism is propose in acurved space-time. In DFR approach, the NC parameter is promoted to the set of coordinates ofthe space-time. As consequence, the field theory defined on this space contains extra dimensions.We propose a metric containing the new coordinates to explain the length measurements of thisextended space. We promote the Minkowski part of this extended metric to a metric defined in acurved manifold. Thus, we can build up a NC gravitation model with extra-dimensions. Based onthe curved metric, we propose an Einstein-Hilbert action to describe the NC gravitation model onthe DFR space. We present the NC version for the gravitational field equations. The weak fieldapproximation is considered in the field equations to obtain the wave equations for the gravitonpropagation on the DFR space.
PACS numbers: 11.15.-q; 11.10.Ef; 11.10.NxKeywords: Noncommutative DFR space-time, Noncommutative Gravitation, Gravitation model with extra-dimensions.
I. INTRODUCTION
The need to control both the ultraviolet and infrared(UV/IR) divergences that eventually emerge in severalcalculations in QFT, motivated theoretical physicists tosuggest the modification of the structure of the stan-dard space-time. One of the possible solutions was topromote the continuum space-time of QFT to a discretespace-time with a noncommutative (NC) algebra of co-ordinates. In this formalism, the space-time coordinates x µ = (cid:0) x = ct , x , x , x (cid:1) are promoted to operatorsˆ X µ = (cid:16) ˆ X , ˆ X , ˆ X , ˆ X (cid:17) . They satisfy non-trivialcommutation relations in which the result is an antisym-metric constant matrix, called θ µν , that play a funda-mental rˆole in the theory and its dimension dwells atPlanck scale, which classifies it as a semi-classical quan-tity, at least. Thereby, we have a kind of fuzzy space-timewhere the position coordinates uncertainty can introducea fundamental length scale in the theory. The space-timedefined on these coordinates is also called NC space-time.The paper by H. S. Snyder was the first published workthat considered the space-time as a NC one [1]. However,shortly after, C. N. Yang [2] showed that Snyder’s NCformalism does not heal the infinities of QFT, and thisresult doomed Snyder’s NC theory to years of ostracism.After the interesting result that the algebra obtainedfrom string theory embedded in a magnetic field back- ∗ Electronic address: [email protected] † Electronic address: [email protected] ground is a NC plane of coordinates, the noncommuta-tivity (NCY) concept of space-time was rekindle [4]. Oneof the ways, the most famous at least, of introducingNCY is through the Moyal-Weyl (MW) product wherethe NC parameter, θ µν , is an antisymmetric constant ma-trix, similar to Snyder’s formalism parameter. However,for calculations at higher orders in the perturbative se-ries, the MW product turns out to be highly nonlocal,and it compels us to work with low orders in θ µν pa-rameter. Besides, since the θ -parameter is a Planck scalequantity, we can eliminate its higher-orders without lossof a definite result. It can also be demonstrated that thetime-space components ( θ i , space-like NCY) can causeunitarity problems [3]. Although it keeps the transla-tional invariance, the Lorentz symmetry is broken [5].As an example, for the hydrogen atom problem, sincea constant parameter in the algebra means a fixed axis,it breaks the rotational symmetry of the model, and re-moves the degeneracy of the energy levels [6]. One wayto recover the Lorentz symmetry is to use the so-calledDoplicher, Fredenhagen and Roberts (DFR) formalismthat introduced another kind NC space-time. In thisformalism, the NC parameter is promoted to ordinarycoordinate of space-time, such that, θ µν ˆΘ µν , whereˆΘ µν keeps both the square length dimension and Planckscale features [7, 8]. As a consequence, the DFR space-time is extended to ten dimensions: four relative to theMinkowski space-time and six relative to the eigenval-ues of ˆΘ µν coordinates, namely, it has the same numberof dimensions as SUGRA. Posteriorly, many authors ex-tended the DFR formalism to include a canonical mo-mentum associated with θ µν [9–18]. Besides the Lorentzinvariance, the causality properties in this space must bepreserved [19–21]. Although the Lorentz symmetry is re-covered in DFR, the field theories defined on this spaceare unitary if we impose ˆΘ i = 0, as we said before, forthe commutators between time and spatial coordinates[27, 28]. To sum up, we guarantee the unitarity of themodel but the Lorentz symmetry is broken. Thus, fromnow on, we will be dealing with a model that preservesjust the translation symmetry. Hence, we have a NCYassociated with the spatial coordinates, the DFR spaceis reduced for 4 + 3 dimensions.One of the challenges of NC formalism and its struc-tures is to construct a NC gravity [22], which meansthat a Planck scale parameter will be part of a gravi-tation structure which suggest a semi-classical, at least,quantum-type formulation of gravity. In this letter, wewill propose a gravitation model defined on the DFRspace of 4 + 3 dimensions [17, 23–26]. This gravitationmodel is motivated by many extensions of gravity the-ories in higher dimensions, like SUGRA [29–32]. Theintroduction of NC effects in gravity is also a subjectthat has a massive literature [33–46]. The main motiva-tion of DFR formalism is to analyze gravity effects in aNC space-time. However, a formal structure, with thegeneral relativity main ingredients was not provided sofar.In this paper we have proposed a Einstein-Hilbert ac-tion in 4 + 3 dimensions and the correspondent gravi-tational field equations in DFR framework. As a par-ticular case, the NC gravity in four dimensions must becontained in the equations for this extended space-time.The commutative limit is recovered when the length scaleof the theory goes to zero, and the usual Einstein-Hilbertaction and the field equations from the general relativityare reobtained. Using the linear approximation of thegravitational field, we obtain the wave equation for thegraviton on the DFR space-time.We have considered the following organization of theideas here. In section II, we provided a very brief re-view of DFR main points. In section III, we introducethe main ingredients of DFR general relativity structure.In section IV, the conclusions and perspective were de-scribed. II. THE NC DFR FRAMEWORK
The original DFR formalism has the compo-nents of the NC antisymmetric parameter θ µν = (cid:0) θ , θ , θ , θ , θ , θ (cid:1) promoted to operators ˆΘ µν = (cid:16) ˆΘ , ˆΘ , ˆΘ , ˆΘ , ˆΘ , ˆΘ (cid:17) . In this letter, as we saidbefore, to avoid unitarity problems, we will consider thatthe time-space components are null, i.e. , ˆΘ i = 0, whichmeans that our approach is NC purely spatial. Thisstatement solves any problem of unitarity that couldemerge in the corresponding effective gravitation theory,which is no more dimensionally analogous to SUGRA.Thereby, we have a NC space-time of D = 4 + 3 dimen-sions. The extended observable DFR algebra is repre-sented by the following commutators h ˆ X i , ˆ X j i = i ε ijk ˆΘ k , h ˆ X i , ˆΘ j i = 0 , h ˆΘ i , ˆΘ j i = 0 , h ˆ X µ , ˆ P ν i = i η µν ˆ1l , h ˆ P µ , ˆ P ν i = 0 , h ˆΘ i , ˆ P ρ i = 0 , h ˆ P µ , ˆ K i i = 0 , h ˆΘ i , ˆ K j i = i δ ij ˆ1l , h ˆ X i , ˆ K j i = i ε ijk ˆ P k , (1)where we have adopted the Minkowski metric η µν =diag(+1 , − , − , − ij and ˆ K ij as our NCoperators, namely, ˆΘ ij = ε ijk ˆΘ k and ˆ K ij = ε ijk ˆ K k , re-spectively. The ˆ K i operator is the conjugated canonicalmomentum associated with the coordinate operator ˆΘ i .Here we have adopted natural units c = ~ = 1, where the θ i -coordinates has squared length dimension, and the k i momentum has inverse of squared length.The components of rotation and boost generatorsˆΣ µν = (cid:16) ˆΣ i , ˆΣ ij (cid:17) in the Lorentz group are defined byˆΣ i = ˆ X ˆ P i − ˆ ξ i ˆ P , ˆΣ ij = ˆ ξ i ˆ P j − ˆ ξ j ˆ P i − ˆΘ i ˆ K j + ˆΘ j ˆ K i , (2)where ˆ ξ i = ˆ X i + i ε ijk ˆΘ j ˆ P k / X i [47–51].With the help of Eq. (1), we can construct the com-mutation relations h ˆΣ µν , ˆ P ρ i = i (cid:0) η µρ ˆ P ν − η νρ ˆ P µ (cid:1) , h ˆΣ i , ˆ K j i = 0 , h ˆΣ ij , ˆ K k i = i (cid:0) δ jk ˆ K i − δ ik ˆ K j (cid:1) , h ˆΣ µν , ˆΣ ρσ i = i (cid:16) η µσ ˆΣ ρν − η νσ ˆΣ ρµ − η µρ ˆΣ σν + η νρ ˆΣ σµ (cid:17) . (3)Consequently, the operators ˆ P µ , ˆΣ µν and ˆ K i close the DFR Poincar´e algebra of generators of translations, ro-tations and Lorentz boosts. The element of the alge-bra in Eq. (3), that commutes with all the generators (cid:16) ˆ P µ , ˆΣ µν , ˆ K i (cid:17) is defined by the first Casimir opera-tor. Using the Schur’s Lemma, the first Casimir opera-tor is proportional to the squared mass m of the par-ticle : ˆ P µ ˆ P µ − λ ˆ K i ˆ K i = m P µ and ˆ K i can be writ-ten in terms of the derivatives ˆ P µ i ∂ µ and ˆ K i i ∂ θ . Thereby, the first Casimir operator on-shell condi-tion leads to the field equations of scalar and fermionsfields [52]. From now on, the notation θ means the θ i -components θ = (cid:0) θ , θ , θ (cid:1) and ∂ θ = ( ∂ θ , ∂ θ , ∂ θ )is the derivative operator in relation to the coordinates (cid:0) θ , θ , θ (cid:1) . The length scale ( λ ) was introduced in thedispersion relation to keep the squared mass dimension ofthe operator ˆ K i ˆ K i . However, there is a reason to intro-duce this parameter connected to the extra dimensionsof the coordinate ˆΘ i which we will see in a moment.Since we have all the symmetries described by the alge-bra in Eq. (3) and the dispersion relation of the particlesare well defined, we can construct the DFR action. Theaction to describe an arbitrary field φ defined in the DFRspace is given by the action S ( φ ) = Z d x d θ W ( θ ) L ( φ⋆, ∂ µ φ⋆, ∂ θ φ⋆ ) , (4)where the volume element contains the θ -integrationmeasure W as a function of the three coordinates θ .The function W is introduced in the context of NCfield theory to smooth the divergences of the integra-tion in the extra θ -space. It has two basic properties: (i)It should be a even function of coordinates θ , namely, W ( − θ ) = W ( θ ), which implies that the integration inthe θ -space is isotropic; (ii) For large coordinates θ , itfalls to zero quickly so that all θ -integrals are well de-fined, and so, a normalization condition is assumed whenintegrated in the θ -space. All the properties involving thefunction W [9, 10, 53–55]. Following these properties, thesimplest function W has the Gaussian form W ( θ ) = (cid:18) πλ (cid:19) exp (cid:18) − θ λ (cid:19) . (5)The length parameter λ can be interpreted as the NCscale energy (Λ NC ) in terms of the expected value ofthe operator ˆΘ = ˆΘ i ˆΘ i : Λ NC = (12 / h ˆΘ i ) / ≡ λ − .There are many phenomenological papers in the litera-ture to set bounds on the NC scale Λ NC . In NC quan-tum electrodynamics, some processes yield the range ofΛ NC & . − . NC &
160 GeV with 95 % C.L. [54, 56].The light-by-light scattering yields a lower bound ofΛ NC & . NC &
10 TeV [58].An important point in the DFR scenario is that theproduct between two fields as function of NC variables keeps the usual form of the MW product. The Weylsymbol provides a map from the operator algebra to thefunctions algebra equipped with a star-product ⋆ via theMW correspondence in which the star-product ⋆ is de-fined by f ( x, θ ) ⋆ g ( x, θ ) = e i θ · ( ∇×∇ ′ ) f ( x, θ ) g ( x ′ , θ ) (cid:12)(cid:12)(cid:12) x ′ = x , (6)for any arbitrary functions f and g of the coordinates( x µ , θ ), with ∇ and ∇ ′ being gradients operators in re-lation to x and x ′ , respectively. In both sides of Eq.(6) we have that f and g are NC functions since theydepend on the coordinate x µ . If f depends on ( x , θ ),and g depends only on θ , the Moyal product betweenthese functions is reduced to the usual product, i.e. , f ( x, θ ) ⋆ g ( θ ) = f ( x, θ ) g ( θ ), because θ commutes with x µ , and with itself.The integration measure in Eq. (4) suggests to con-struct a general metric that encompasses the Minkowskispace-time and the extra space associated with the θ -coordinates. Thereby, we propose the line element ofthis extended space-time into the form ds = Ξ AB ( θ ) dX A ⋆ dX B , (7)where the coordinates X A are redefined by X A = (cid:8) x µ , θ / (2 πλ ) (cid:9) . Note that the components of θ in X A are defined as being dimensionless due to the length scale λ . The components of the metric Ξ AB ( θ ) are elementsof the 7 × AB ( θ ) = diag (cid:18) η µν , e − θ λ , e − θ λ , e − θ λ (cid:19) . (8)The inverse of the metric in Eq. (8) is defined byΞ AB ( θ ) Ξ BC ( θ ) = δ CA which can be represented by theidentity matrix 7 ×
7. With this definition, the function W can be written in terms of the determinant of Ξ( θ )matrix: d x d θ W ( θ ) = d X √− Ξ. The commutativelimit in Eq. (8) can be obtained when λ → λ → e − θ λ (2 πλ ) = δ (3) ( θ ) , (9)where the θ -part in Eq. (7) is zero due to δ (3) -Dirac prop-erty, i.e. , δ (3) ( θ ) d θ = 0. Thus, it is direct to check thatthe line element in Eq. (7) recovers the usual Minkowskispace-time, and the action in Eq. (4) is reduced to the4D QFT commutative case. III. THE DFR GRAVITATION FRAMEWORK
Following the definition of the extended metricΞ AB ( θ ), we generalize the line element in Eq. (7) suchthat its Minkowski part can be modified for a curvedspace-time given by ds = G AB ( x, θ ) ⋆ dX A ⋆ dX B , (10)where η µν is promoted to g µν ( x ) and G AB ( x, θ ) describesthe space of a curved manifold attached to the extra θ space G AB ( x, θ ) = diag (cid:18) g µν ( x ) , e − θ λ , e − θ λ , e − θ λ (cid:19) . (11)The inverse of this metric is extended to the Moyal-product if exists a matrix G AB ( x, θ ), such that G AB ( x, θ ) ⋆ G BC ( x, θ ) = δ CA . (12)Using the extension from the metric in Eq. (11), we pro-pose the Einstein-Hilbert action in the DFR frameworksuch as S EH ( g µν ) = − κ Z d X √− G ⋆ R , (13)where κ = √ πG ≃ . × − m, is the couplingconstant written in terms of the gravitational constant G = 6 . × − m . The Ricci scalar is defined by R = G AB ( x, θ ) ⋆ R AB , in which the Ricci tensor R AB can be written as R AB = ∂ A Γ C CB − ∂ C Γ CAB ++ 12 (cid:0) Γ CAD ⋆ Γ DCB + Γ
DCB ⋆ Γ CAD (cid:1) − Γ CAB ⋆ Γ DDC , (14)and Γ C AB is the connectionΓ
CAB = 12 G CD ⋆ ( ∂ A G BD + ∂ B G AD − ∂ D G AB ) . (15)The derivative operators with capital index in Eqs.(14) and (15) means the derivative with relation to the X A coordinates defined previously: ∂ A = ∂/∂X A =( ∂ µ , λ ∂ θ i ). Since we have all the tensor structure inthis extended space, we propose an Einstein’s field equa-tion for DFR as being R AB − G AB ( x, θ ) ⋆ R = − πG T AB , (16)where T AB is the energy-momentum tensor for matterfields (scalars, spinors, vectors and other ones) with thecomponents T AB = (cid:0) T µν , T µθ i , T θ i θ j (cid:1) . Note that T µθ i and T θ i θ j are the new components due to the extra θ -dimension. As an example, the energy-momentum tensorof a NC scalar field φ on the DFR space is given by T AB = 12 ( ∂ A φ ⋆ ∂ B φ + ∂ B φ ⋆ ∂ A φ ) − G AB ( x, θ ) ⋆ L φ , (17)where L φ is the Lagrangian of a scalar field φ with mass m L φ = 12 G AB ( x, θ ) ⋆ (cid:0) ∂ A φ ⋆ ∂ B φ (cid:1) − m φ ⋆ φ . (18)Note that the connection, the Ricci tensor and theenergy-momentum tensor are symmetric by exchanging the indexes A ↔ B . This symmetry is kept at thecommutative limit. More explicitly, the components of T AB = (cid:0) T µν , T µθ i , T θ i θ j (cid:1) can be written as T µν = 12 ( ∂ µ φ ⋆ ∂ ν φ + ∂ ν φ ⋆ ∂ µ φ ) − g µν ( x ) ⋆ L φ ,T µθ i = λ ∂ µ φ ⋆ ∂ θ i φ + ∂ θ i φ ⋆ ∂ µ φ ) , (19) T θ i θ j = λ (cid:0) ∂ θ i φ ⋆ ∂ θ j φ + ∂ θ j φ ⋆ ∂ θ i φ (cid:1) − δ ij e − θ i λ L φ . The energy-momentum tensor T AB is covariantly con-served in DFR space, i.e. , it satisfies the continuity equa-tion with the covariant derivative operator: ∇ A ⋆ T AB =0. The conserved components of energy-momentum ten-sor are T B = (cid:0) T , T i , T θ i (cid:1) , where T is the energydensity, T i is the spatial momentum density of the Eu-clidean space and the component T θ i can be understoodas the spatial momentum of the scalar field in the extra θ -dimension. Using the result in Eq. (17), the conservedmomentum component T θ i is given by T θ i = λ (cid:16) ˙ φ ⋆ ∂ θ i φ + ∂ θ i φ ⋆ ˙ φ (cid:17) , (20)which is a component associated with the DFR, and itgoes to zero at the commutative limit λ →
0. Using thesymmetry of the tensors in Eq. (16), we can verify thatit has 25 non-linear differential equations with correc-tions in the θ ij parameter by the Moyal product between G AB ( x, θ ) and R . The field equation in Eq. (16) con-tains the NC Einstein’s tensor in 4D dimensions if we set A = µ and B = ν .The action in Eq. (13) and the field equation in Eq.(16) are non-linear and both also contain the Moyal prod-uct series that makes it difficult to deal with calculations.Hence, we make the weak field approximation wherethe metric G AB ( x, θ ) can be written as G AB ( x, θ ) ≃ Ξ AB ( θ ) + κ h AB ( x, θ ) , where Ξ AB ( θ ) is the metric inEq. (8), and the symmetric field h AB has the compo-nents h AB ( x, θ ) = { h µν ( x, θ ) , h µθ i ( x, θ ) , h θ i θ j ( x, θ ) } .We are concerned with the linear terms in the field equa-tion, where the Moyal product becomes the usual productof functions. Thus, we obtain the Ricci tensor and thescalar one in the first order of h AB ( x, θ ), R AB ≃ κ (cid:0) ∂ A ∂ B h − ∂ A ∂ C h BC − ∂ B ∂ C h AC + (cid:3) θ h AB (cid:1) ,R ≃ κ (cid:0) (cid:3) θ h − ∂ A ∂ B h AB (cid:1) , (21)where (cid:3) θ := ∂ A ∂ A = (cid:3) + λ ∂ θ is the extendedD’alembertian operator, and h ( x, θ ) = h AA ( x, θ ). Asusual in effective gravity, we choose the harmonic gauge,in which the h AB field satisfies the Donder gauge condi-tion ∂ A h = 2 ∂ C h AC . In this gauge condition, we obtainthe wave equation for h AB ( x, θ ) for the case where wehave no matter fields ( T AB = 0), namely, (cid:16) (cid:3) + λ ∂ θ (cid:17) h AB ( x, θ ) = 0 . (22)This is the wave equation concerning the components of h AB ( x, θ ) with the propagation in the θ -space. Note thatthe component h µν ( x, θ ) that represents the spin-2 gravi-ton propagates in the extra dimension of the NC space.It would be interesting to investigate the contributionsof this NC space to the gravitational potential. This isan ongoing research. IV. CONCLUSIONS
One of the main motivations to investigate NC fieldmodels is that field theories with space and time NCYgive an interesting opportunity to check the possiblebreakdown of the standard concept of time and the wellknown structure of QM at the Planck scale. We proposedin this letter a gravitation model for the Doplicher, Fre-denhagen and Roberts NC framework. The DFR modelis a NC approach includes the θ µν parameter as a coor-dinate of the system, and consequently, this formalismincludes extra dimensions on the space-time. To keepan unitary QFT in DFR formalism, we defined θ i = 0.Thus, we have a space-like NCY with three extra coor-dinates, i.e. , ( θ , θ , θ ), and the DFR space-time hasseven dimensions: four from Minkowski space-time plusthree spatial coordinates θ . The product between fieldsdefined in this space can be constructed via MW product,which is similar to the usual NC formalism where θ µν isa constant parameter. We presented a metric for this extended space as the Minkowski components completedwith Gaussian components associated with the weightfunction W ( θ ) defined in the NC action. The commu-tative limit is recovered using the Dirac delta identityin Eq. (9). After that, we substituted the Minkowskiterm by the metric g µν ( x ), defined on a smoothed man-ifold that represents a NC curved space-time. Thereby,we constructed a gravitation model with extra spatialdimensions in the DFR approach. The action and thecorresponding field equations are given by Eqs. (13) and(16), respectively. We used the linear approximation inthe NC Einstein’s equation where the wave equation withthe propagation in the θ -space is obtained for the com-ponents of weak field h AB ( x, θ ). As a perspective, theanalysis of the action in Eq. (13) as an effective gravita-tion model in terms of the field h AB ( x, θ ) is an ongoingresearch that will be published elsewhere. Acknowledgments
The authors thank CNPq (Conselho Nacional de De-senvolvimento Cient´ıfico e Tecnol´ogico), Brazilian sci-entific support federal agency, for partial financialsupport, Grants numbers 313467/2018-8 (M.J.N.) and406894/2018-3 (E.M.C.A.). M. J. N. also thanks the De-partment of Physics and Astronomy at the University ofAlabama for the kind and warm hospitality. [1] H. S. Snyder, Phys. Rev. (1947) 38.[2] C. N. Yang, Phys. Rev. (1947) 874.[3] J. Gomis and T. Mehen, Nucl. Phys. B 591 (2000) 265;M. Chaichian, M. Chaichian, A.Demichev, D. Preˇsnajderand A. Tureanu, Phys. Lett. B 515 (2001) 426.[4] N. Seiberg and E. Witten, JHEP 9909 (1999) 032.[5] R. Szabo, Phys. Rep. (2003) 207.[6] M. Chaichian, M. M. Sheikh-Jabbari and A. Tureanu,Phys. Rev. Lett. (2001), 2716-2719.[7] S. Doplicher, K. Fredenhagen and J. E. Roberts, Phys.Lett. B (1994) 29.[8] S. Doplicher, K. Fredenhagen and J. E. Roberts, Com-mun. Math. Phys. (1995) 187.[9] H. Kase, K. Morita, Y. Okumura and E. Umezawa, Prog.Theor. Phys. (2003) 663.[10] K. Imai, K. Morita and Y. Okumura, Prog. Theor. Phys. (2003) 989.[11] A. Deriglazov, Phys. Lett. B (2003) 83.[12] E. M. C. Abreu, M. V. Marcial, A. C. R. Mendes and W.Oliveira and G. Oliveira-Neto JHEP 1205 (2012) 144.[13] S. Saxell, Phys. Lett. B (2008) 486.[14] R. Amorim, Phys. Rev. D (2008) 105003.[15] R. Amorim, J. Math. Phys. (2009) 022303.[16] R. Amorim, J. Math. Phys. (2009) 052103.[17] R. Amorim, Phys. Rev. Lett. (2008) 081602.[18] E. M. C. Abreu, A. C. R. Mendes, W. Oliveira and A.Zagirolamim, SIGMA 6 (2010) 083, and the referencestherein. [19] E. M. C. Abreu and M. J. Neves, Int. J. Mod. Phys. A (2012) 1250109.[20] E. M. C. Abreu and M. J. Neves, Int. J. Mod. Phys. A (2013) 1350017.[21] E. M. C. Abreu and M. J. Neves, Nucl. Phys. B (2014) 741.[22] V. Rivelles, Nucl. Phys. Proc. Suppl. 127 (2004) 63; M.Kober, Int. J. Mod. Phys. A (2015) 1550085; K. S. Gupta,E. Harikumar, T. Juric, S. Meljanac and A. Sansarov,Adv. High Energy Phys. 2014 (2014) 139172; M. Kober,Class. Q. Grav. 28 (2011) 225021;Y-G. Miao, Z. Xue andS-J. Zhang, Gen. Relat. Grav. 44 (2012) 555; P. Nicol-ini, Phys. Rev. D 82 (2010) 044030; O. Bertolami and C.A. D. Zarro, Phys. Lett. B 673 (2009) 83; M. Chaichian,M. Oksanen, A. Tureanu and G. Zet, Phys. Rev. D 79(2009) 044016; G. Fucci and I. G. Avranidi, Class. Q.Grav. 25 (2008) 025005; R. Banerjee, P. Mukherjee andS. Samantha, Phys. Rev. D 75 (2007) 125020; S. Mar-culescu, Phys. Rev. D 74 (2006) 105004; P. Das and S.Ghosh, Phys. Rev. D 98 (2018) 084047; A. K. Mitra, R.Banerjee and S. Ghosh, JCAP 1810(2018) 057.[23] R. Amorim, Phys. Rev. D (2010) 105005.[24] R. Amorim, Phys. Rev. D (2008) 105003.[25] R. Amorim, J. Math. Phys. (2009) 022303.[26] R. Amorim, J. Math. Phys. (2009) 052103.[27] A. Basseto, F. Vian, L. Griguolo and G. Nardelli JHEP Nuclear
Physics B (2012) 219-239.[29] B. de Wit,
Supergravity , Lectures notes Les HouchesSummerschool (2001) 147 pages, hep-th/0212245.[30] P. Nath,
Supersymmetry, Supergravity and Unification ,Cambridge University Press, Cambridge, (2016).[31] F. Brandt, Lectures on Supergravity (2002),
An intro-duction to 4-dimensional N = 1 supergravity .[32] S. P. Martin,
A Supersymmetry Primer (2016),hep-ph/9709356.[33] M. R. Douglas and N. A. Nekrasov, Rev. Mod. Phys. (2001) 977.[34] R. J. Szabo, Class. Quant. Grav. 23 (2006) R199.[35] R. J. Szabo, General Relativity and Gravitation (2010) 1-29.[36] L. Alvarez-Gaum´e, F. Meyer and M. A. Vazquez-Mozo,Nucl. Phys. B (2006) 92.[37] X. Calmet and A. Kobakhidze, Phys. Rev. D (2005)045010.[38] E. Harikumar and V. Rivelles, Class. Quant. Grav. 23(2006) 7551.[39] V. Rivelles, Phys. Lett. B (2003) 191.[40] H. Steinacker, JHEP (2007) 049.[41] H. Steinacker, Nucl. Phys. B (2009) 1.[42] R. Banerjee, H. S. Yang, Nucl. Phys. B (2005) 434.[43] M. R. Douglas and C. Hull, JHEP 9802 (1998) 008.[44] P. A. Horvathy and M. S. Plyushchay, JHEP 0206 (2002)033.[45] P. A. Horvathy and M. S. Plyushchay, Phys. Lett. B (2004) 547.[46] P. A. Horvathy and M. S. Plyushchay, Nucl. Phys. B (2005) 269.[47] J. Gamboa, M. Loewe and J. C. Rojas, Phys. Rev. D (2001) 067901.[48] A. Kokado, T. Okamura and T. Saito, Phys. Rev. D (2004) 125007.[49] A. Kijanka and P. Kosinski, Phys. Rev. D (2004),127702.[50] X. Calmet, Phys. Rev. D (2005), 085012.[51] X. Calmet and M. Selvaggi, Phys. Rev. D (2006)037901.[52] E. M. C. Abreu and M. J. Neves, Int. J. Mod. Phys. A (2017) 1750099.[53] C. E. Carlson, C.D. Carone and N. Zobin, Phys. Rev. D (2002) 075001.[54] J. M. Conroy, H. J. Kwee and V. Nazarayan, Phys. Rev.D , 034017 (2004).[55] S. Saxell, Phys. Lett. B (2008) 486.[56] C. D. Carone and H. J. Kwee, Phys. Rev. D (2006)096005.[57] R. Horvat, D. Latas, J. Trampetic and J. You, Light-by-Light Scattering and Spacetime Noncommutativity , arXiv: 2002.01829v1.[58] E. Akofor, A. P. Balachandran, A. Joseph, L. Pekowskyand B. A. Qureshi, Phys. Rev. D79