Four-dimensional Gravity on a Covariant Noncommutative Space

We formulate a model of noncommutative four-dimensional gravity on a covariant fuzzy space based on SO(1,4), that is the fuzzy version of the $\text{dS}_4$. The latter requires the employment of a wider symmetry group, the SO(1,5), for reasons of covariance. Addressing along the lines of formulating four-dimensional gravity as a gauge theory of the Poincar\'e group, spontaneously broken to the Lorentz, we attempt to construct a four-dimensional gravitational model on the fuzzy de Sitter spacetime. In turn, first we consider the SO(1,4) subgroup of the SO(1,5) algebra, in which we were led to, as we want to gauge the isometry part of the full symmetry. Then, the construction of a gauge theory on such a noncommutative space directs us to use an extension of the gauge group, the SO(1,5)$\times$U(1), and fix its representation. Moreover, a 2-form dynamic gauge field is included in the theory for reasons of covariance of the transformation of the field strength tensor. Finally, the gauge theory is considered to be spontaneously broken to the Lorentz group with an extension of a U(1), i.e. SO(1,3)$\times$U(1). The latter defines the four-dimensional noncommutative gravity action which can lead to equations of motion, whereas the breaking induces the imposition of constraints that will lead to expressions relating the gauge fields. It should be noted that we use the euclidean signature for the formulation of the above programme.


Introduction
The aim to go beyond the notion of classical spacetime has been with us since decades. In particular, there is no obvious physical reason to expect that the commutativity of coordinates and the corresponding description of spacetime to hold up to arbitrary small length scales. Spacetime noncommutativity, inspired by the successes of quantum mechanics and quantum field theory, was suggested as their logical extension providing an interesting alternative framework to describe physics close to the Planck scale. As a bonus it was expected that the natural appearance of a minimal length scale could act as an effective ultraviolet cutoff in field theories defined on the new spacetime structure . Moreover, on the gravity side, noncommutativity could be used to resolve the singularities of general relativity [31,32]. On the other hand a diffeomorphism-invariant gravity theory is obviously invariant with respect to transformations whose parameters are functions of spacetime, just as in the local gauge theories. Then, naturally, it has been long believed that gravity theory can be formulated as a gauge theory [33][34][35][36][37][38]. Then merging the above two approaches by describing gravity as gauge theory on noncommutative spaces is a logical consequence. In addition, the prospect of regularizing quantum field theories, or even better, building finite ones are the features that render this approach as a promising framework.
The construction of quantum field theories on noncommutative spaces is a difficult task though and, furthermore, problematic ultraviolet features have emerged [7][8][9]. Still noncommutative geometry has been proposed as an appropriate framework to accommodate particle models with noncommutative gauge theories [10] (see also [21][22][23]39,40]). It is worth-noting that a very interesting development in the framework of the noncommutative geometry is the programme in which extra dimensions of higher-dimensional theories are considered to be noncommutative (fuzzy) [24][25][26][27][28][29]. This programme overcomes the ultraviolet/infrared problematic behaviours of theories defined on noncommutative spaces. A very welcome feature of such theories is that they are renormalizable, contrary to all known higher-dimensional theories. Even more desirable is that this programme results to phenomenologically promising four-dimensional unified theories.
We recall that a fuzzy two-sphere [5] (see also [41,42]) is constructed from finite dimensional matrices and the size of matrices represents the number of quanta on the noncommutative manifold. The fuzzy sphere, S 2 F , at fuzziness level N-1, is the noncommutative manifold whose coordinate functions are N×N matrices proportional to the generators of the N-dimensional representation of SU (2). Introducing a cutoff parameter N-1 for angular momentum in a two-sphere, the number of independent functions is N 2 . Then one can replace the functions defined on this noncommutative manifold by N×N matrices, and therefore algebras on the sphere become noncommutative. A generalization to a higher dimensional sphere is, however, not straightforward. In particular, in the case of a four-dimensional sphere, the same procedure leads to a number of independent functions which is not a square of an integer. Therefore, one cannot construct a map from functions to matrices. One can restate this difficulty algebraically. Algebras of a fuzzy four-sphere have been constructed in [43] and the difference from the fuzzy two-sphere case is that the commutators of the coordinates do not close in the fuzzy four-sphere case. This is the source of the difficulties to analyse field theories on the fuzzy four-sphere (see [43] and references therein) 1 .
In [46] (see also [47,48]), we started a programme realizing gravity as noncommutative gauge theory in three dimensions. Specifically, we considered three-dimensional noncommutative spaces based on SU(2) and SU(1,1), as foliations of fuzzy two-spheres [49,50] and fuzzy two-hyperboloids [51], respectively. This onion-like construction led to a matrix model, which was analysed in a straightforward way. In the present work we would like to follow a different strategy, originally suggested by Yang [2], as an extension of the first noncommutative model of Snyder [1], which preserves the spacetime isometries 2 . The latter is a property of the fuzzy sphere, but cannot be extended in other cases in a straightforward manner. The suggestion is that in order to formulate a fuzzy covariant noncommutative space, one has to employ a wider symmetry containing all generators of the first one. Then, in practice, in the case of the fuzzy de Sitter four-dimensional spacetime dS 4 , that we would like to construct here, instead of using its algebra of generators, SO(1,4) we consider a larger one in which this can be embedded 3 . Such algebras are the SO (1,5) or the conformal SO (2,4). The larger algebra, say SO (1,5) contains all the generators of the subalgebra SO (1,4) and the commutation relations of the latter are covariant as seen in the larger group.
The next important point is that the coordinates and other operators, too, can be represented by N×N matrices in some higher-dimensional representation of the enlarged algebra. Of course in this procedure one has to identify the various generators of the algebras with operators corresponding to various observables such as coordinates, momenta, angular momenta etc. To realize the above picture of a fuzzy covariant noncommutative spacetime in the present work we consider the four-dimensional fuzzy de Sitter space of a given radius embedded in a five-dimensional spacetime. However in our construction we describe four-dimensional gravity by gauging the SO(1,4) subgroup of SO(1,5) 4 . To facilitate the algebra we construct the whole model in the Euclidean signature. Therefore, we start with a space with isometries given by the SO(5) group, extend, for reasons of covariance, to the SO(6) symmetry and then we gauge its maximal SO(5) subgroup to formulate gravity as a gauge theory on the above space. The results we obtain could be reformulated to the Lorentzian signature.
The outline of the paper is as follows. In sections 2, 3 we briefly present the fourdimensional Einstein and conformal gravity theories for reasons of completeness but also, for reference on some of their important features in the next sections. In section 4, we present the construction of the four-dimensional fuzzy space on which we formulate gravity as a (noncommutative) gauge theory and describe the construction of the gauge theory of the isometry group of the above space. In section 5, we comment on the constraints employed in order to break the initial symmetry to the desirable one -the Lorentz. In section 6, we propose an action and comment on the obtaining of the equations of motion. Eventually, in section 7, we write our conclusions and stress out the main features of our model. In Appendix A we give some details on how to deal with the general problem of the (non-) covariance transformation property of the field strength tensor on such covariant spaces and how this can be relieved, while in Appendix B, we give the calculations and results of the transformations of the fields and the component tensors. 3 The SO(5)∼SO (1,4) in the present formalism, would have the same problem in its description by N×N matrices, as in the fuzzy four-sphere case. 4 For an alternative approach (based on the star product) for the gauging of the same group in the noncommutative regime to formulate gravity, see [55,56] 2 Four-dimensional Einstein gravity as a gauge theory An alternative way to recover the celebrated results of the theory of general relativity is to approach the whole concept from a gauge-theory point of view. This undertaking began in the middle 50's [33] and kept the interest of the physicists for the next three decades [34][35][36][37][38]. Here we recall the whole idea since it is of central importance in our present project. Therefore, in this section, we briefly present the whole construction of the four-dimensional Einstein gravity as a gauge theory, in order to make this project complete and self-contained.
In the gauge-theory approach of the four-dimensional gravity, at first, one has to consider the vielbein formulation of general relativity. The gauge theory is constructed on the four-dimensional Minkowski spacetime, M 4 , and the gauge group is the Poincaré group, ISO (1,3). The choice of the Poincaré group as the symmetry group is very reasonable, as it is the isometry group of the Minkowski spacetime. The generators of the group satisfy the following commutation relations: where η ab is the (mostly positive) Minkowski metric, M ab are the Lorentz group generators and P a are the generators of the translations. According to the gauging procedure, one has to introduce the gauge potential, A µ , which can be decomposed on the generators of the algebra, accompanied by functions that are identified as the gauge fields. Specifically in this case, these functions are the vierbein, e a µ and the spin connection, ω ab µ , which correspond to the translations, P a , and Lorentz generators, M ab , respectively. The gauge connection is decomposed as follows: The above gauge connection is assigned in the adjoint representation, which means that it transforms according to the following rule: where ǫ = ǫ(x) is the gauge transformation parameter that is also expanded on the ten generators of the algebra: The transformations of the component gauge fields are given after combining the expressions (2) and (4) with (3). The resulting expressions for the transformations of the vierbein and the spin connection are: The corresponding field strength tensor (the curvature) of the gauge theory is defined as: Since the field strength tensor takes also values in the algebra that is gauged, it can be also expanded on its generators: where R µν a (e) and R µν ab (ω) are the component curvatures of the gauge fields, which are identified as the torsion and curvature, respectively. Their expressions are obtained by replacing equations (2) and (8) into the (7): As for the dynamic part of the theory, one would consider the obvious choice of the Yang-Mills action of the Poincaré group. However, since the intention is to end up with the action of the Einstein gravitational theory, the Poincaré symmetry has to be broken to the Lorentz. Spontaneous symmetry breaking is the indicated way to realize it and can be induced by the inclusion of a scalar field which belongs to the fundamental representation of the SO(1,4) [36,37]. The latter (de Sitter group) can be equally chosen instead of the Poincaré, and for the present purpose, i.e. the breaking of the initial symmetry to the Lorentz, it is preferred, since it is a semisimple group and all generators can be considered on equal footing, i.e. use a single gauge index to describe all ten generators. The spontaneous symmetry breaking breaks the four generators that are related to the translations, resulting to a constrained theory with vanishing torsion and the SO(1,4) Yang -Mills action considered at first, reduced to an action including the Ricci scalar and preserving only the Lorentz symmetry. This action is indeed the desired Einstein-Hilbert action.
Summing up, Einstein four-dimensional gravity can be considered as a gauge theory of the Poincaré algebra up to retrieving the results of the transformations of the fields and the expressions of the curvature tensors. For the dynamic part, in order to obtain the correct action, one has to begin with a Yang -Mills action obeying the de Sitter symmetry, instead of the Poincaré, and include a scalar field for inducing a spontaneous symmetry breaking which eventually leads to the Einstein-Hilbert action.
An equivalent argument leading to the desired, Lorentz symmetry, is to demand the gauge symmetry of the vacuum to be the SO(1,3), Lorentz group. Therefore, the field strength tensor that corresponds to the generators that break, i.e. the translations, has to be vanishing, recovering in this way the torsionless condition. This condition plays the role of the constraint that is necessary in order to result with the desired symmetry. From this constraint, one obtains an expression of the spin connection as a function of the vielbein: Nevertheless, if one considers the Yang-Mills action for the remnant Lorentz symmetry, one ends up with an R(M) 2 action, which is not the desired one, when aiming to reproduce the results of general relativity, because this would impose the gravitational coupling constant to be dimensionless -which is not. For the expected Einstein-Hilbert action, retrieving the correct dimensionality of the coupling constant as well, one has to consider an action in a non-straightforward way, that is building Lorentz invariants out of the quantities of the theory. The one that includes the Ricci scalar is the right choice, leading to the desired Einstein-Hilbert action.

Four-dimensional conformal gravity as a gauge theory
In this section we briefly discuss the case of the four-dimensional conformal gravity (for detailed analysis see [57][58][59], also [60,61]). The gauge group parametrizing the conformal symmetry in four-dimensional Minkowski spacetime is SO(2,4), but instead, for simplicity, we consider the corresponding euclidean signature, therefore the symmetry group is the SO (6). The generators of the conformal algebra are the translations, (P a ), the Lorentz rotations, (M ab ), the conformal boosts, (K a ) and the dilatations, (D), satisfying the following commutation relations 5 : where a, b, c, d = 1 . . . 4. To proceed with the gauging procedure, one has to define the gauge potential: in which for every generator, a gauge field has been corresponded. The above connection obeys the following infinitesimal transformation rule: where ǫ is a parameter taking values in the Lie algebra of the symmetry group and therefore can be expanded on its generators: Starting from (14) and using (13) and (15), calculations lead to the following transformations of the various gauge fields after comparison of the terms: The well-known formula for the curvature: gives the expressions of the component curvatures of the gauge fields of the theory: The action is taken to be of Yang-Mills form, being SO(6) gauge invariant. According to [57][58][59], there exist some convincing arguments that give specific constraints which induce the symmetry breaking of the initial symmetry. In particular, these constraints are the torsionless condition, R(P ) = 0 along with an additional constraint on R(M), together leading to expressions of ω ab µ and f a µ in terms of e a µ and b µ and finally, as a third constraint, the fact that b µ gauge field can be gauged away. The resulting action is the well-known Weyl action.
Besides the above arguments, we suggest another way of breaking the initial symmetry, this time to the Lorentz. This breaking, along with the transformations of the fields and component tensor expressions we presented above, serves our purpose for the present paper, because we want to use these results as the commutative limit of the noncommutative gravity theory we develop later. This could occur with the inclusion of two scalars in the 6 representation of the SO(6) gauge group [62]. This comes in a very natural way, taking one step beyond the way the de Sitter group (and not the Poincaré for reasons explained in the previous section) is broken to the Lorentz by a scalar in the fundamental of SO(1,4), as described in the previous section for the case of the Einstein gravity. These two scalars could induce a spontaneous symmetry breaking in a complete theory that includes matter fields, giving rise to the constraints that lead us to a resulting four-dimensional action that respects Lorentz symmetry.
Moreover, we can also extend the argument we used in the previous section, in the fourdimensional Poincaré gravity, as an alternative way to break the initial symmetry. Since we want to preserve the Lorentz symmetry out of the initial SO(6), we consider directly that the vacuum of the theory is SO(4) invariant, which in turn means that every other tensor, except for the R(M), has to vanish. This vanishing will produce the constraints of the theory leading to specific expressions that relate the gauge fields. In particular, in [55], it is described that if both tensors R(P ) and R(K) are set simultaneously equal to zero, then the constraints of the theory yield that their corresponding gauge fields, f a µ , e a µ are equal -up to a rescaling factor -and also b µ = 0.
4 Four-dimensional noncommutative gravity on a fuzzy covariant space

Construction of the fuzzy covariant space
In this subsection, we present the construction of the four-dimensional fuzzy space, on which we elaborate gravity as a gauge theory.
We want to construct a gravitational model on a four-dimensional fuzzy space, specifically the fuzzy version of the dS 4 . The de Sitter space is defined as a submanifold of the five-dimensional Minkowski spacetime in the same way the four-sphere is defined as an embedding in the five-dimensional euclidean space. Specifically, the embedding equation that defines dS 4 is: where A, B = 0, . . . , 4 and η AB is the mostly positive metric of the five-dimensional Minkowski spacetime, that is η AB = diag(−1, +1, +1, +1, +1).
However, the coordinates of the fuzzy de Sitter space, X a , must satisfy the following commutation relation, which describes the noncommutativity of the space: Recalling the case of the fuzzy two-sphere with coordinates the three rescaled SU(2) generators in an (large) N-dimensional representation and radius r, the right hand side of (20) is also a generator of the SU(2) algebra, ensuring covariance, that is θ ab = C abc X c , where C abc is a rescaled Levi-Civita symbol. In the fuzzy de Sitter space we want to construct, the problem is that when trying to identify the coordinates with some generators of SO(1,4), covariance is broken, because the algebra is not closing, i.e. θ ab cannot be assigned to generators into the algebra [52] 6 . The requirement of the preservation of covariance, imposes us to use a group with a larger symmetry, in which we will be able to incorporate all generators and the noncommutativity in it. The minimum extension of the symmetry leads to the adoption of the SO(1,5) group. Therefore, in order to end up with a fuzzy dS 4 , with coordinates represented by N-dimensional matrices (like in fuzzy two-sphere case), respecting covariance as well, we are led to the enlargement of the symmetry to the SO(1,5) group. From now on, for convenience, we will use the euclidean signature, which means that the isometry group and the extended symmetry group will be SO(5) and SO (6), respectively, instead of SO(1,4) and SO(1,5) 7 . Now, in order to formulate the above four-dimensional fuzzy space, we consider the extended algebra of SO (6). We denote its generators by J AB = −J BA , with A, B = 1, . . . , 6 and they satisfy the following commutation relations: Now, we decompose the above generators in an SO(4) notation, identifying the component generators as: with m = 1, . . . , 4. For dimensional reasons we have introduced an elementary length λ. The coordinates, momenta and non-commutativity tensors are X m , P m and Θ mn , respectively. Both coordinates and momenta satisfy the following algebra: where m, n = 1, . . . , 4. The extended kinematical algebra i.e. the algebra of spacetime transformations is: [h, Θ mn ] = 0 The above algebra, in contrast to the Heisenberg algebra (see [63]), admits finite dimensional representations for X m , P m and Θ mn , thus we have obtained a model of space-time which is a finite quantum system. In analogy with the fuzzy two-sphere, spaces like the one above are called fuzzy covariant non-commutative spaces [52,53,64]. In the following section we will formulate a gauge theory of gravity on the above four-dimensional fuzzy space.

Noncommutative gauge theory of four-dimensional gravity
Here, we are going to present the formulation of gravity as a gauge theory on the fourdimensional space that was constructed in the previous section 8 . The whole procedure is developed as a noncommutative analogue of the work presented in sections 2 and 3, using upon them the framework of noncommutative gauge theories [66].
Determination of the gauge group and representation by 4 × 4 matrices Recalling the previous section, we argued that for the sake of preserving gauge covariance, we were led to enlarge the symmetry group of the space to the SO(6). Thinking along the lines of the commutative four-dimensional gravity, described in section 2, where the isometry group, (the Poincaré group), was gauged in order to retrieve the desired results, here we have to gauge the isometry group of the covariant space, that is the SO(5), as seen as a subgroup of the SO (6), in which we ended up with.
However, it is known that in noncommutative gauge theories, the use of the anticommutators of the generators of the algebra is inevitable, as we have explained in detail in our previous works [46][47][48] (see also [67]). Of course, the anticommutators of the generators (not in a specific representation) of an algebra do not necessarily yield operators that belong to the algebra and this is exactly the case for the generators of SO(5). The indicated treatment is to specify the representation in which the generators belong and include the operators produced by the anticommutators into the algebra, considering them as generators, too. This will result in the extension of the initial gauge group to one with larger symmetry. In our case, application of this recipe leads to the extension of SO(5) to the SO(6)×U(1) (∼U(4)) group 9 with generators being represented by 4x4 matrices, i.e. the representation should be fixed to the 4 of SO(6) i.e. to the fundamental representation of U(4).
We start with the four γ-matrices (in the Euclidean signature) satisfying: where m, n = 1, . . . 4 and also defining Γ 5 as Γ 5 = Γ 1 Γ 2 Γ 3 Γ 4 . The generators of the SO(6) group are: a) Six Lorentz rotation generators: The construction of Γ-matrices starts with Pauli matrices: and γ−matrices are built as tensor products of them as: In particular, we obtain: where i, j, k = 1, 2, 3 and: The generators defined above, obey the following algebra: Noncommutative gauge theory of gravity Previously, we decomposed the SO(6) generators of the SO(6)×U(1) in the SO(4) notation, in order to identify the generators in the appropriate language and then we calculated their commutation relations. Therefore, we are ready to advance with the (noncommutative) gauging procedure.
First, one has to define the covariant coordinate of the theory, which is given by the following relation:X The coordinateX m is covariant by construction, which means that it transforms covariantly under a gauge transformation: where ǫ = ǫ(X) is the gauge parameter, which is a function of the coordinates, but also takes values in the algebra, (34), given in the previous section. Therefore, it can be written as an expansion on the generators of the algebra. Consequently, the gauge parameter is written explicitly as: Considering that the gauge transformation does not affect the coordinate X m , i.e. δX m = 0, one can find the transformation property of the A m included in the (36). In analogy with the commutative case, the way A m transforms indicates that it can be considered as the potential, that is the gauge connection of the theory. In our case, A m is a function of matrices-coordinates X a of the fuzzy dS 4 . The A m (X) takes values in the U(4) algebra, meaning that it can be spanned on its generators: From the above equation, one can read that we have introduced one gauge field for each generator. The component gauge fields depend on the coordinates of the space, X a , meaning that they are N×N matrices, where N is the dimension of the representation in which the coordinates are described. Multiplication between every gauge field and its corresponding generator is not the usual one, but the tensor product is used, since the product consists of matrices of different dimensions, recalling that the generators are described by 4×4 matrices. Therefore, every term of the connection is a 4N×4N matrix.
Having determined the gauge connection, the covariant coordinate is written explicitly as:X Furthermore, for the U(4) gauge theory we are developing, what is left to determine is the field strength tensor 10 , which, for the noncommutative case, is defined as: which, since it is valued in the algebra, it is expanded in terms of the component curvatures as: R mn (X) = R ab mn (X)⊗M ab +R a mn (X)⊗P a +R a mn (X)⊗K a +R mn (X)⊗D +R mn (X)⊗1l .

The constraints for breaking the symmetry to SO(4)×U(1)
The gauge symmetry we would like to end up with is the one described by the Lorentz group, in the euclidean signature we use, the SO(4). A straightforward way to do this is by considering a constrained theory in which the rest of the component tensors are vanishing, in order to break the initial SO(6)×U(1) symmetry to the desired SO(4)×U(1), which will be the remnant gauge symmetry of the vacuum of the field theory. However, this approach leads to an over-constrained theory, that is evident after counting the degrees of freedom that survive after the breaking. Therefore, it is rather wise to impose at first the constraint:R a mn (P ) = 0 , that is the torsionless condition, which is more or less expected, as it is considered in previous sections as well, in the cases of Einstein and conformal gravity. Furthermore, the possible interpretation of b a m as a second vielbein 11 would give a theory with two metrics or two vielbeins, which is not desirable in our case. Therefore, we are led to adopt as a solution of the constraints, the e a m = b a m . The consideration of this solution leads to the expression of the spin connection ω ab m as a function of the rest of the fields, e a m , a m ,ã m . Taking into consideration the torsionless condition with the specific solution e a m = b a m , we result with the correct number of degrees of freedom, leading to the noncommutative analogue of the four-dimensional gravity with reduced symmetry SO(4)×U(1).
In order to go on with obtaining the expression relating the fields, we employ the following two identities: The constraint of the torsionless condition takes the following form: where D m = X m + a m , i.e. the covariant coordinate of an abelian gauge theory. The above equation leads to the following two: Using the identities (43), the above equations lead to the desired expression of the spin connection in terms of the rest of the fields: According to [68], the vanishing of the field strength tensors in a gauge theory considered on a simply connected space means that, locally, the corresponding gauge fields vanish as well. This argument would be very welcome in our case because it would certainly simplify the expressions of the tensors. Nevertheless, the above argument cannot be applied in our case in general, because after identifying the vielbein with gauge fields of our spacetime, we mixed gauge theory and geometry. It holds that the vielbein is considered to be invertible in every point, therefore, adopting the above argument, that is setting the vielbein to zero, would lead to degenerate vielbein matrices, inducing degeneracy to the metric tensor of the space [69]. However, we could setã m = 0. This will also modify the (46), producing an even simpler expression of the spin connection in terms of the vielbein, which reads: At this point we need to punctuate that the U(1) field strength tensor, R mn (1l) is not set to zero, which means that this U(1), strongly related to the noncommutativity, remains unbroken in the resulting theory after the breaking, since we still have a theory on a noncommutative space. Of course, the corresponding field, a m , would vanish if we assumed the commutative limit of the broken theory, in which noncommutativity is lifted and a m decouples being super-heavy. In this limit, the gauge theory would be just SO(4).
Another way of getting the desired SO(4) symmetry as a remnant symmetry after the breaking of SO (6), is to extrapolate the argument that we developed for the conformal gravity in the commutative case (section 3) of the inclusion of two scalar fields in the fundamental representation of SO (6), to the noncommutative case. We are confident that the spontaneous symmetry breaking will lead to constraints equivalent to the one in (42).

The action
Since we have employed a gauge theory formulation for constructing the gravitational model, the most reasonable choice for the action is of Yang-Mills form 12 : where the Tr is the trace over the matrices representing the coordinates (takes the role of the integration of the commutative case) whereas the tr is the trace over the generators of the algebra. The first term of the above action includes the field strength (curvature) tensor of the gauge theory, while the second one is the (non-topological) kinetic term of the 2-form field. Considering the vacuum state of the theory, i.e. applying all the constraints considered in the previous section, the surviving terms consisting the action will be: S = 2Tr(R ab mn R cd rs ǫ abcd ǫ mnrs +4R mn R rs ǫ mnrs + where in order to result with the above expression of the action with indices in the SO(4) notation (a, b, c, d = 1, . . . 4), we began from the action expressed in the SO(5) notation (A, B, C, D = 1, . . . , 5) for the gauge indices, calculated the trace of the generators filtering out most of the terms and then decomposed the remaining terms to the SO(4) notation. The commutation and anticommutation relations for the generators of the algebra are given in (72).
Inserting the expressions of the component tensors given in (93)-(97) and expressing the ω gauge field in terms of the rest of the surviving gauge fields, as given in (47), then variation with respect to the (surviving) gauge fields would lead to the equations of motion.

Summary and Conclusions
In the present work we have presented a four-dimensional gravity model on a covariant noncommutative space. The chosen space is a generalization of the celebrated fuzzy twosphere in certain basic respects. It is a noncommutative version of dS 4 that keeps the fundamental property of covariance of the two-sphere, which preserves all isometries of the fuzzy space and in particular the Lorentz invariance, which is of particular interest in the present four-dimensional case. Another property that shares with the fuzzy two-sphere is that its coordinates can be represented by finite matrices. Then gravity is built by gauging the isometries of the constructed noncommutative space. It should be noted though that the requirement of covariance led us to enlargement of the isometries of the fuzzy dS 4 , while, furthermore, the construction of a gauge theory on such a noncommutative space led us to an enlargement of the gauge group and in fixing its representation. Also, the definition of the field strength tensor imposed the inclusion of a 2-form dynamic gauge field for reasons of covariance. The introduction of such a gauge field in gauge theories constructed on covariant spaces like the one we use, is a treatment that can be applied in general. That is why we described in detail this subject (Appendix A), for bookkeeping and use as a future reference.
Then, straightforward calculations led to the transformations of the gauge fields and the expressions of the component tensors. However, the part of the isometry group we decided to gauge, even more after the enlargement noncommutativity imposed, was larger than the one we wanted to result with. For this reason, we went on with a symmetry breaking which gave rise to a constrained gauge theory, with the preferred symmetry. Eventually, we proposed an action of Yang-Mills type, including the kinetic term of the dynamic 2-form field. Imposition of the constraints give an expression of the action which can be varied in order to obtain the equations of motion.
Finally, it should be stressed that the constructed gravity is a matrix model giving promises for improved UV properties as compared to ordinary gravity. Clearly, the latter, as well the inclusion of matter fields is going to be a subject of further study.

A Gauge covariant field strength tensor and gauge invariance
In this appendix, we briefly present the gauge covariance of the field strength tensor and the gauge invariance of the action we propose.
In general, the field strength tensor of a noncommutative gauge theory can be written in terms of the covariant coordinate plus an extra term which contributes in such a way that the whole expression of the field strength tensor transforms covariantly under a gauge transformation. The two most widely-known types of spaces on which gauge theories are built on are of constant and Lie-type noncommutativity, specifically defined by the relations: where θ mn is a constant antisymmetric tensor and C mnp is a rescaled Levi-Civita symbol. LetX m be the covariant coordinate in both cases, then expressions of the field strength tensors are given by: On the one hand, the fuzzy space on which we are building the gauge theory of gravity is covariant, pointing at a relation with the fuzzy sphere case, but, on the other hand, the right hand side of the relation defining non-commutativity does not depend on the coordinates, pointing at the case of constant noncommutativity. The truth is that it cannot be classified into any of these cases, therefore it has to be examined explicitly.
The commutation relation of the fuzzy space in our case is: Due to the fact that the right hand side does not depend on the coordinates X m , one would be led to a definition of the corresponding field strength tensor as: Nevertheless, if we assume a gauge transformation 13 of the field strength tensor, straightforward calculations lead to a result according to which, the field strength tensor does not transform covariantly, specifically: where ǫ is a gauge parameter. In order to ameliorate the above problematic result, one has to modify the definition of the field strength tensor to: whereΘ mn = Θ mn ⊗1l+B mn , where B mn is a non-Abelian two-form gauge field, which takes values in the U(4) algebra 14 , as the rest of the gauge fields and transforms covariantly as: which gives δΘ mn = i[ǫ,Θ mn ]. From the above equation we can find the transformations of the component fields of B mn , following exactly the same procedure we describe in Appendix B for the rest of the gauge fields, specifically: Therefore, calculations lead to the following expression of the (infinitesimal) transformation of the field strength tensor: that is a desired covariant transformation.
Since, both terms in the expression of the field strength tensor transform covariantly, it is straightforward to show that the corresponding Yang-Mills action is gauge invariant: Therefore, gauge covariant transformation of the field strength tensor and gauge invariance of the action are ensured.
At this point, it is also important to define the field strength tensor,Ĥ mnp , of the 2-form gauge field:Ĥ In order to show that the above field strength tensor transforms covariantly under a gauge transformation, we start from the expression: and using the transformation properties ofX m andΘ mn , given in equations (36) and (56), respectively, and the Jacobi identity, we find: Next, we expand theĤ on the generators of the algebra: and calculate each component using the definition of the field strength tensorĤ mnp : The inclusion of this 2-form gauge field has to be prominent in the action of the theory, too. Among other terms that will be present, one has to add a kinetic term for this field: S B = Tr trĤ mnpĤ mnp (68)

B Calculations of the field transformations and curvatures
In this appendix, we present our results for the transformations of the gauge fields and the component curvatures. In the end, we check whether our results are valid, after the consideration of the commutative limit.
For calculative convenience, we are going to employ the SO(5) notation and indices and then return to the SO(4) ones. As we showed, the generators of SO(6)×U(1) are encountered in a specific representation given by 4×4 matrices: Turning to the SO(5) notation, we introduce the matrices Γ A with A, B = 1 . . . 5 , satisfying: Taking this into consideration, the above generators can be written in the following more compact form: The generators in this SO(5) notation satisfy the following commutation and anticommutation relations [70]: In turn, the covariant coordinate is written as: and the gauge parameter as: The definition of the field strength tensor is: and is also decomposed on the generators: After the above rewriting of the algebra and the gauge-theory related expressions, we continue with the calculations.
The transformation rule of the covariant coordinate is: Replacing the expressions (73) and (74) into the above rule, we obtain the transformations of the component gauge fields in the SO(5) notation: From the field strength tensor definition, (75), along with the definition of the covariant coordinate in the SO (5) Next, in order to return to the SO(4) notation and give the above results in the SO(4) desirable language, we make the following decompositions of the SO(5) generators: