Intertwining noncommutativity with gravity and particle physics

Eur. Phys. J. Spec. Top. https://doi.org/10.1140/epjs/s11734-023-00830-8 THE EUROPEAN PHYSICAL JOURNAL SPECIAL TOPICS Review Intertwining noncommutativity with gravity and particle physics G. Manolakos1,a , P. Manousselis2,b , D. Roumelioti2,c , S. Stefas2,d , and G. Zoupanos2,3,4,5,e 1 2 3 4 5 Institute of Theoretical Physics, Faculty of Physics, University of Warsaw, ul. Pasteura 5, 02-093 Warsaw, Poland Physics Department, National Technical University, Athens, Greece Theory Department, CERN, Geneva, Switzerland Max-Planck Institut für Physik, Munich, Germany Institut für Theoretische Physik der Universität Heidelberg, Heidelberg, Germany Received 20 October 2022 / Accepted 28 March 2023 © The Author(s) 2023 Abstract Here we present an overview on the various works, in which many collaborators have contributed, regarding the interesting dipole of noncommutativity and physics. In brief, we present the features that noncommutativity triggers both in the fields of gravity and particle physics, from a matrix-realized perspective, with the notion of noncommutative gauge theories to play the most central role in the whole picture. Also, under the framework of noncommutativity, we examine the possibility of unifying the two fields (gravity-particle physics) in a single configuration. 1 Introduction An ultimate anticipation of many theoretical physicists is the existence of a unification picture in which all fundamental interactions are involved. To this end, a huge amount of serious research activity has been carried out, including works that elaborate the very interesting notion of extra dimensions. Superstring theories [1] consist a solid framework, with the heterotic string theory [2] (defined in ten dimensions) being the most promising, in the sense that it potentially admits experimental compatibility, due to the fact that the Standard Model (SM) gauge group can be accommodated into the gauge groups of the grand unified theories (GUTs) that emerge after the dimensional reduction of the initial E8 × E8 . Besides the superstring theories, a few years before their formulation, an alternative approach of generalized dimensional reduction, as compared to the simple one of higher-dimensional gauge theories was formulated. This insightful and significant project which shared common goals with one of the superstring theories, was initially explored by Forgacs–Manton and then a e-mail: [email protected] (correspondingauthor) b e-mail: [email protected] c e-mail: [email protected] d e-mail: [email protected] e e-mail: [email protected] 0123456789().: V,-vol by Scherk–Schwartz studying the coset space dimensional reduction (CSDR) [3–6] and the group manifold reduction [7], respectively. Besides the above, a very interesting framework which could be a competitive candidate in accommodating a gravitational theory but also particle physics models at high energy scale (Planck scale) is that of noncommutative geometry [8–50], in which the commutativity of the coordinates is not an inherent property of spacetime. An interesting virtue of the above framework is the regularization of quantum field theories, or, even better, the construction of finite ones. Of course, such an undertaking is rather complicated and, furthermore, it has presented unwelcome issues regarding its ultraviolet behavior [12–15] (see also [16–18, 90]). Despite that, noncommutative geometry is considered as a solid framework regarding the accommodation of particle physics models, formulated as in the familiar way, that is as gauge theories on noncommutative spaces [19–21] (see also [22–27]). It is remarkable that superstring theories and noncommutative geometry share a common ground, since, in M-theory and open string theory, the effective physics on D-branes can be expressed as a noncommutative gauge theory, in the presence of a nowherevanishing background antisymmetric field [28, 29]. In addition, the type IIB superstring theory (and those related to it through dualities), in its non-perturbative 123 Eur. Phys. J. Spec. Top. formulation as a matrix model [30], consists a noncommutative theory. A major contribution in the framework of noncommutativity was that of Seiberg and Witten [29], who devised a mapping mechanism between commutative and noncommutative gauge theories, correlating the noncommutative degrees of freedom of a gauge theory to their commutative counterparts. Based on the above mapping, important developments have been done [31–35], for instance the construction of a noncommutative version of the SM [36–38]. Unfortunately, the main problem of this approach was not solved, that is the free parameters of the model could not get further reduced by extensions of this type (contrary to supersymmetric theories [51]). Delving a little deeper in the notion of noncommutative geometry, since coordinates are not commutative quantities, it can be linked to a potential quantum structure, which can be supposed that it occurs at very small distances (Planck length), since the behavior of the spacetime fabric at these scales is effectively unknown. According to the above line of thoughts, it is a rather natural step to examine the noncommutative version of general theory of relativity (GR), with aspirations that the latter would provide new insights particularly in regions that a spacetime singularity is encountered in the conventional GR framework. This noncommutative gravitational theory would consist a generalization of GR, ideal for examining higher curvature scales (than a critical one), in which, localization of a point would be impossible to occur. Therefore, in case of high-scales phenomena, the conventional notion of coordinates breaks down and should be substituted by elements of a noncommutative algebra. On the contrary, at less extreme energy scales (e.g. LHC) the rest of the interactions are successfully formulated using gauge theories, while at higher scales (but not of Planck level) a very attractive unification picture of these three interactions is provided by the GUTs. The gravitational interaction does not join this picture, since, in principle, it is formulated geometrically according to GR. Nevertheless, besides the geometric one, there exists an alternative, gauge-theoretic approach to gravity [52–64], which started with Utiyama’s pioneering study [52] and was subsequently evolved as a gauge theory of the de Sitter SO(1,4) group, spontaneously broken by a scalar field to the Lorentz SO(1,3) group [54]. Apart from GR, Weyl gravity has also been formulated as a gauge theory of the conformal group in four dimensions [59, 60]. In this case, part of the gauge fields spectrum is identified as the vielbein and the spin connection, which guarantees the interplay between the gauge-theoretic and geometric approaches through the (equivalent to the regular—second order—) first order formulation of GR. Now, taking into consideration the aforementioned gauge-theoretic formulation of gravity and integrating it to the noncommutative framework in which gauge theories are well-formulated has led to the construction of models of noncommutative gravity [66–75]. In these works, the Moyal–Weyl type of noncommutativity is used, the star-product approach is followed (in which the noncommutative quantities are still 123 ordinary functions but with an upgraded product) and, last, the Seiberg–Witten map is used [29]. Furthermore, there exists an alternative approach to the construction of noncommutative gravitational models, which makes use of the matrix-realization of the noncommutative quantities [30, 76–87] (see also [11, 88, 89]). Here, due to length restrictions of our article, instead of outlining our approach in the introduction, we have chosen to develop it step-by-step in the next chapters, accompanied by more detailed references on the subject. 2 Gauge theories on noncommutative space As explained in the introduction, in order to accommodate the gravitational interaction into the framework of noncommutative geometry, following [65], it is meaningful to recall how gauge theories are rephrased in it. Let a scalar field, φ(X), where X consists the coordinate system of the noncommutative space. The scalar field transforms non-trivially under a local (infinitesimal) gauge transformation as: δφ(X) = ε(X)φ(X), (1) where ε(X) is the coordinate-dependent parameter of the transformation. Contrary to the scalar field, the coordinates transform trivially under gauge transformation (as expected), therefore the product Xµ φ(X) transforms as: δ(Xµ φ(X)) = Xµ ε(X)φ(X). (2) By observation and taking into consideration that the underlying framework is the noncommutative one, it is understood that the above transformation is not a covariant one. In order to covariantize it, drawing lessons from the conventional gauge theories, the covariant coordinate is utilized (in analogy to the covariant derivative), the definition of which is given through its transformation: δXµ = [ε(X), Xµ ], (3) which is obtained by the requirement of the quantity δ(Xµ φ(X)) to transform covariantly, that is δ(Xµ φ(X)) ≡ ε(X)Xµ φ(X). (4) From the above configuration, in order to relate the noncommutative coordinate, Xµ , to the noncommutative covariant coordinate, Xµ , a field Aµ with transformation δAµ (X) = −[Xµ , ε(X)] + [ε(X), Aµ (X)] (5) has to be introduced, specifically according to the relation Xµ = Xµ + Aµ . It is clear from the latter that the Eur. Phys. J. Spec. Top. introduced field Aµ admits the interpretation of the noncommutative gauge connection. In turn, due to the above interpretation of the Aµ , a noncommutative version of a field strength tensor is corresponded in accordance to the ordinary gauge theories (including an extra term to its definition for reasons of covariance), which is dependent on the kind of noncommutative space on which the gauge theory is constructed. A rather important feature of the noncommutative gauge theories, in discordance to the conventional gauge theories, is that the anticommutators of the various operators related to the gauge algebra become relevant. This feature is designated when one considers the commutator of two elements which belong to the gauge algebra, ε(X) = εa (X)Ta and φ(X) = φa (X)Ta : [ε, φ] =  1  a b 1 ε , φ [Ta , Tb ] + εa , φb {Ta , Tb }, 2 2 (6) where Ta denote the algebra generators. The quantities εa and φb are functions of the spacetime coordinates, which means that, in the commutative case, their corresponding commutator vanishes, and subsequently so does the product of the last term in the above relation. Nevertheless, in the noncommutative case, by definition, the coordinates do not commute with each other and thus so do functions that depend on them. Therefore, the aforementioned last term of the above relation becomes non-vanishing giving the anticommutators an essential role in the construction of gauge theories in the noncommutative setting, contrary to the conventional ones. This discrepancy gives rise to terms that originate from the anticommutators, which, in principle, are not members of the gauge algebra. Consequently the closure property of the initial algebra holds no more. One way out is to extend the algebra perpetually including all operators that pop from the anticommutators, with this extension leading to the universal enveloping algebra, which, although useful in other contexts (e.g., in [31, 70, 91]), in ours it is not the appropriate way to proceed. Another way out is to restrict the number of the newly-added operators to a finite (minimum) number by choosing a specific representation, which is the one preferred. Now, let us briefly emphasize on the category of the covariant noncommutative spaces on which we focus [10, 92–97], which have the property that Lorentz covariance is preserved [98–101]. Moreover, another property of the noncommutative spaces is the preservation of the isometries of the corresponding commutative space. The noncommutative spaces forming this special subclass are called fuzzy and the most typical example is the fuzzy 2-sphere [10] (see also [102, 103]) which shares the same isometry group, SO(3), with its commutative analogue, i.e. the ordinary sphere. The fuzzy sphere admits a construction through finitedimensional matrices, the size of which is interpreted as the number of the quanta of the space. The coordinates of the fuzzy sphere with N − 1 level of fuzziness are N × N matrices which are multiples of the SU(2) generators in the N −dimensional representations. The aforementioned construction is deployed through the truncation of the angular momentum by the introduction of a cut-off parameter N − 1 and therefore leaving N 2 independent functions. Thus, the above functions may be represented by N × N matrices leading to a noncommutative algebra of the sphere. Attempting to generalize the above argument to the four-dimensional case, i.e. the one that interests us, it is seen that such a mapping of functions to matrices cannot be achieved, as the number of independent functions does not coincide with the square of some integer number. Therefore, the construction of the fuzzy 4sphere and field theories on it is a more subtle task. For alternative constructions see [99] and referenced studies in it (see also [104, 105]). Overall, it has already become clear that, according to our (and others’) perspective and approach, gauge theories play an important role in the construction of both particle physics models as well as gravitational ones in the framework of noncommutativity. In the ensuing, we will present constructions that are related to both, starting with the particle physics side followed by the gravitational one. 3 Fuzzy particle physics model In this section we describe a fuzzy particle physics model in which fuzziness is introduced through the notion of extra dimensions. The latter are not supposed to be in an arbitrary form but they rather consist specific (extra-dimensional) manifolds, particularly noncommutative analogues (fuzzy) of the well-defined and well-studied coset spaces. Before we move on with the discussion of the specific model, it is rather considerate first to write down some information of the backbone of the whole construction, that is the dimensional reduction of a higher-dimensional gauge theory with fuzzy extra dimensions, in the most general case, in which the extra dimensions consist an arbitrary fuzzy coset space. 3.1 Dimensional reduction of a higher-dimensional theory with fuzzy extra dimensions Suppose a higher-dimensional theory defined on the spacetime M 4 × (S/R)F of D = d + 4 dimensions, where (S/R)F is a fuzzy coset space. Let this theory be gauge invariant under the transformations of the group G = U (P ), with generators T I and be described by the following action of Yang–Mills type: SYM 1 = 2 4g  d4 x kTr trG FM N F M N , (7) where trG is the trace related to the generators of the algebra of the gauge group G and kTr. Also, 123 Eur. Phys. J. Spec. Top. although the theory is higher-dimensional, the integration is written in such a way as it is performed on the four-dimensional component. The correct dimensions regarding the integration are restored by the presence of the Tr which is effectively the integration operator referring to the coordinates of the extra dimensions, since they now are represented by matrices. Moreover, k is a parameter related to the volume of the fuzzy coset space (see [106] for the case of the fuzzy sphere) and FM N is the field strength tensor of the gauge theory with M , N = 0, . . . , D − 1. That means that the tensor FM N consists of components which lie only on the four-dimensional space, components that lie only on the extra-dimensional one, as well as mixed components of the two spaces, that is FM N = (Fµν , Fµb , Faν , Fab ), where μ, ν = 0, . . . 3 and a, b = 4, . . . D − 1. Specifically, the mixed and fuzzy parts of the tensor are: Fµa =∂µ φa + [Aµ , φa ] = Dµ φa Fab =[Xa , Ab ] − [Xb , Aa ] + [Aa , Ab ] − C c ba Aac , where φa stands for the covariant coordinate, that is the noncommutative analogue of the covariant derivative (see Sect. 4 for a more detailed discussion). The initial action (7), after taking into consideration the above two expressions of the components of the field strength tensor, takes the following form: SYM =  d4 xTrtrG   k 2 k 2 F + (D φ ) − V (φ) , µ a 4g 2 µν 2g 2 (8) where V (φ) is a function that involves the terms that 2 originate from the Fab one:  k Fab Fab TrtrG 4g 2 ab   k = − 2 TrtrG [φa , φb ][φa , φb ]−4Cabc φa φb φc +2R−2 φ2 . 4g V (φ) = − (9) Due to its expression, the above is identified as the potential of the theory. The form in which the action arrives after the above substitutions of the field strength tensor according to the splitting we considered in its components, (8), manifestly lets us naturally interpret it as a four-dimensional action. The above interpretation is also possible at the level of the gauge transformation; let λ(xµ , X a ) be an infinitesimal parameter of a local transformation of the initial gauge group, G = U (P ). It admits the interpretation of a local transformation of another group on a gauge theory exclusively on M 4 , if treated in the following way: λ(xµ , Xa ) = λI (xµ , X a )T I = λh, I (xµ )T h T I , (10) where, as mentioned already, T I are the (Hermitian) generators of the gauge group U (P ) of the initial 123 higher-dimensional theory and λI (xµ , X a ) are N × N anti-Hermitian matrices, i.e. functions of the matrixrealized coordinates. Since we are now dealing with N × N antisymmetric matrices, λI (xµ , X a ), they can be thought as an element of the U (N ) group and, as such, they can be written down as decompositions on the corresponding generators, T h , as λI (xµ , X a ) = λh, I (xµ )T h , where λh, I (xµ ) being merely functions of the four-dimensional coordinates identified as the Kaluza–Klein modes of λI (xµ , X a ) and, they can be further understood as a field that takes values in the Lie algebra of the U (N ) × U (P ) group, which is equivalently the algebra of the U (NP ) group. Also, under same treatment, the gauge connection of the initial gauge theory can be viewed from a four-dimensional perspective as a four-dimensional one of the U (NP ) group. Last, the same applies on the scalars as well. The above reduction, although straightforward and rather simple, gives rise to a very important and antiintuitional feature (which can also be encountered in more complicated ones), that is the gauge symmetry of the four-dimensional theory is larger than that of the initial, higher-dimensional one. In other words, one may start with a higher-dimensional Abelian gauge theory and end up with a non-Abelian in four dimensions. Again from the above reduction, it is also understood that the scalars belong to the adjoint representation of the four-dimensional gauge group, as relics of the initial gauge fields which are also in the corresponding adjoint, fact that leads to the understanding that they cannot trigger the electroweak symmetry breaking. This is a crucial reason why the above simple model cannot be evolved into a more complicated and promising one and consequently, another more elaborate one has to be sought. Following the above consideration, an improved fourdimensional model may be obtained by employing a fuzzy version [40, 41, 107] of the CSDR [108]. Due to the fuzziness, the fuzzy CSDR will enjoy the above welcome feature that was noted above, that of the enlargement of the gauge symmetry as the number of dimensions drops to four, contrary to the conventional CSDR in which this feature is absent. An important observation is that in the fuzzy case choosing an Abelian gauge theory in high dimensions one is led to non-Abelian gauge symmetry in four dimensions. Another virtue the fuzzy CSDR inherits from the fuzziness is that both the higher-dimensional and the resulting four-dimensional theories are renormalizable. The matter of renormalizability was initially argued in [40, 107] but an even more convincing argument came after the whole problem was examined from another perspective. In a few words, instead of beginning with a higher-dimensional theory and performing a dimensional reduction to approach a four-dimensional theory, the starting point was overturned, in the sense that one may begin with a renormalizable four-dimensional gauge theory of SU(N ) with scalars populating a multiplet in such a way that fuzzy extra dimensions forming a fuzzy sphere can be developed dynamically [42]. Eur. Phys. J. Spec. Top. Such a theory can be interpreted as a six-dimensional gauge theory since it develops non-trivial vacua, with the geometrical and gauge symmetry determined by the parameters of the initial Lagrangian. Also, the tower of (massive) Kaluza–Klein modes that emerges is finite, in consistency with the view of a compactified gauge theory in higher dimensions. Briefly, the virtues of the above model are, first, that, avoiding supersymmetry or fine tuning, extra dimensions emerge dynamically due to the fact that fuzzy spaces consist solutions of matrix models. Second, the four-dimensional gauge group is SU(n1 ) × SU(n2 ) × U (1) or SU(n), while gauge groups consisting of more than two simple groups are not observed in this kind of models. Third, the induction of a magnetic flux occurs in a rather natural manner in the case of vacua having non-simple gauge symmetry. Subsequently, these nice features of the above mechanism suggest that it is worth attempting to accommodate particle physics models with phenomenological orientation. To this end, chiral fermions are included and relevant studies have shown that, without imposing any additional constraints, best case scenario is that of obtaining four-dimensional theories populating bi-fundamental representations (mirror fermions) [44, 45]. Although having mirror fermions is not a killing result for phenomenological compatibility [109], chiral fermions are significantly preferred. Such an outcome occurs after an additional mechanism is applied in the above, specifically that of a Z3 orbifold projection of an N = 4 supersymmetric SU(3N ) gauge theory which eventually leads to an N = 1 SU(N )3 gauge theory [46]. The case of N = 3 (trinification group) is of particular interest (see [110–112]). Let us see in some detail the above concepts through a particle physics model in which fuzzy extra dimensions are dynamically generated to form fuzzy spheres and chiral fermions are involved due to the orbifold projection mechanism [46]. As mentioned earlier, let us consider an N = 4 Supersymmetric Yang–Mills (SYM) theory in four dimensions with gauge symmetry parametrized by SU(3N ). The particle spectrum consists of the gauge field, Aµ , three complex scalars, φi , and four Majorana fermions ψ p all in the adjoint representation of the gauge group. The orbifold projection is parametrized by the action of a Z3 discrete group and is realized through its embedding into the R-symmetry, that is SU(4)R . The choice of the embedding is not unique and this choice determines the amount of the resulting sypersymmetry [113]. Here, the discrete group is considered to get embedded into the SU(3) subgroup of the R-symmetry, a choice which leads to N = 1 remnant supersymmetry, breaks the gauge symmetry to SU(N ) × SU(N ) × SU(N ) and the only fields that make it to the resulting SYM theory are the ones that are Z3 invariant. It should be noted that the fermions reside in chiral representation of the resulting group and come in three identical copies, that is three chiral families. At the level of the initial N SYM theory the F-part of the scalar potential is: VF (φ) = 1 Tr [φi , φj ]† [φi , φj ] 4 (11) and its form remains the same after the orbifold projection filters the spectrum. The D-part of the potential is VD = 12 D2 = 12 DI DI , where DI = φ†i T I φi with T I the generators of the gauge group in chiral representations. Minimization of the total potential gives [φi , φj ] = 0. However, the introduction of soft supersymmetric breaking terms with scalar part which respects the orbifold symmetry: VSSB = 1 2 † m2i φi φi + i 1 2 hijk φi φj φk + h.c., i, j, k (12) contributes in such a way to the minimization of the potential that leads to different kind of vacua. The potential of the combination of all three contributions is given in the following form: V = 1 ij † ij (F ) F + VD , 4 (13) for suitable parameters, where: † F ij = [φi , φj ] − iεijk (φk ) . (14) The first term of the above potential is positive definite, therefore its global minimum is obtained if the following equations hold: † [φi , φj ] = iεijk (φk ) , † † [(φi ) , (φj ) ] = iεijk φk , † φi (φi ) = R2 , (15) † where (φi ) is the hermitian conjugate of φi and it holds that [R2 , φi ] = 0. The above relations point towards the fuzzy sphere defining relation, which becomes even more transparent by considering the (untwisted) complex scalar fields, φ̃i , which are defined as φi = Ωφi , for † Ω = 1, Ω3 = 1, [Ω, φi ] = 0, Ω† = Ω−1 and (φ̃i ) = φ̃i , † i.e. (φi ) = Ωφi . To conclude, fuzzy extra dimensions equipped with orbifold projection consist a valuable asset when it comes to particle physics models attaching importance to them in the aspects of chirality, renormalizability and phenomenological viability. 123 Eur. Phys. J. Spec. Top. 4 Noncommutative gravity 4.1.2 Gauge theory of three-dimensional gravity on R3λ As mentioned in the introduction, gauge theories in the noncommutative framework are employed for constructing both particle physics and gravitational models. Having reviewed the first part, from now on we will examine the second one in two cases, that of three and four dimensions. Now, having determined the background threedimensional space, a gravitational model can be realized by considering the gauge theory of a suitable group on it [120, 121]. The above methodology is a loan from the formulation of three-dimensional GR as a Chern–Simons gauge theory on the three-dimensional Minkowski spacetime background. Specifically, the gauge group that successfully fits in the above configuration is the isometry group of the background space, that is the threedimensional Poincaré group, ISO(1, 2) [64]. As noted in the introduction, the above translation of GR to the gauge-theoretic setting is achieved by considering the first order formulation of gravity, that is the formulation in which dynamics is described by the vielbein and the spin connection instead of the metric. A very important consideration is that the latter two dynamical quantities are integrated in the gauge-theoretic setting as gauge fields related to the translational and Lorentz part of the Poincaré group, respectively. Back in the noncommutative framework, the first thing that has to be determined, as understood by the above information, is the gauge group. Along the lines of the commutative case, the one that is employed is again the isometry group of the background space R3λ , that is SO(4), which leads to the construction of a non-Abelian noncommutative gauge theory. As pointed out in Sect. 2, such theories complicate things as anticommutators become important and the way out is to fix the representation and extend the group by those operators that the anticommutators give. The resulting gauge theory is that of the U (2) × U (2) and the corresponding generators are represented by 4 × 4 matrices (for details but also for the manipulation of the Lorentzian case see the original publication [120]). The set of the eight generators of the resulting gauge group is: {Pa , Ma , I, γ5 }, where a = 1, 2, 3 and their specific form is given after employing the Pauli matrices of which the commutation and anticommutation relations are known and therefore the commutation and anti-commutation relations of the generators are calculated straightforwardly to be: 4.1 Fuzzy gravity in three dimensions on the R3λ space First we review a matrix-realized description of a gauge-theoretic construction of noncommutative gravity in three dimensions. For that to occur, a threedimensional noncommutative space is required as a background in order to accommodate the whole construction. Therefore, we start our discussion with the description of that three-dimensional space. 4.1.1 The R3λ space The path of the description of the noncommutative space that is used in this case passes through the fuzzy sphere, which is a very fundamental and well-defined fuzzy space that is also equipped with the property of covariance [10, 114]. A glimpse at its properties has already been given in the setup of the particle models of the previous section as it had a very important role in the corresponding construction as well. Moreover, the definition of the fuzzy sphere is given by the commutation relation the coordinates satisfy and the radius constraint, that is: 3 Xi Xi = λ2 j(j + 1) := r2 . [Xi , Xj ] = iλǫijk Xk , i=1 (16) The first relation is actually a rescaled version of the algebra the angular momentum operators satisfy due to the identification Xi = λJi , where λ fixes the dimensions and Ji operators are in a unitary irreducible (high) representation of SU(2). The second relation is a rescaled version of the Casimir operator which is manifestly interpreted as the radius constraint of the fuzzy sphere. The above definition of the fuzzy sphere leads smoothly to the definition of the space that is used as background for the construction of the gauge theory, namely R3λ [115–118], if the radius constraint is switched off by considering unitary reducible representations for the generators. As known, a reducible representation may be written in a block diagonal form of irreducible representations, or to rephrase, a block diagonal form of fuzzy spheres. This property of R3λ gives rise to a very illustrative view, that is the visualization of the above space as a foliation of the threedimensional Euclidean space by multiple fuzzy spheres of all possible radii, according to the number of the fuzziness level [119]. 123 [Pa , Pb ]=iǫabc Mc , [Pa , Mb ]=iǫabc Pc , [Ma , Mb ]=iǫabc Mc , 1 1 1 {Pa , Pb }= δab I, {Pa , Mb }= δab γ5 , {Ma , Mb } = δab I, 2 2 2 [γ5 , Pa ] = [γ5 , Ma ] = 0, {γ5 , Pa } = 2Ma , {γ5 , Ma } = 2Pa . (17) In turn, having at hand the above relations, one may proceed with the construction of the noncommutative gauge theory by considering the covariant coordinate1 : X µ = δµ a Xa + A µ , 1 (18) In the particle models setting it was denoted by φa while now Xa is used. Eur. Phys. J. Spec. Top. where Aµ is the gauge connection which expands on the various generators as Aµ = AIµ (X) ⊗ T I , where T I denotes an arbitrary generator, therefore I = 1, . . . , 8. Also, AIµ are the gauge fields taking values in the algebra of U (2) × U (2) but also they depend on the matrix-represented coordinates of the underlying space, that is why in the following decomposition of the gauge connection: Aµ (X) = eµ a (X) ⊗ Pa + ωµ a (X) ⊗ Ma + Aµ (X) ⊗ iI + õ (X) ⊗ γ5 (19) and therefore of the covariant coordinate: Xµ = Xµ ⊗ iI + eµ a (X) ⊗ Pa + ωµ a (X) ⊗ Ma + Aµ (X) ⊗ iI + õ (X) ⊗ γ5 , (20) the tensor product is present. In a similar way, the parameter of the gauge parameter, ε(X), expands on the generators as an element of the gauge algebra: a + ε0 (X) ⊗ iI + ε̃0 (X) ⊗ γ5 . (21) Having the necessary information at hand, the transformation rules of the various gauge fields are calculated in a straightforward way, taking also into consideration Eqs. (5) and (6): δeµ a (23) δωµ a = −∂µ λa − ǫabc (−iλb ωµc − iξb eµc ). (24) The above results are identical to the ones obtained in the conventional gauge-theoretic approach of threedimensional GR up to some rescalings of generators, gauge fields and parameters in the case that a cosmological constant is present, therefore, switching off noncommutativity leads to recovering the results of the corresponding gauge-theoretic approach in the commutative case. Getting back in track, in order to propose an action, the corresponding field strength (curvature) tensor of the theory has to be written down. This is achieved by considering the anticipated formula of the commutator of the covariant coordinates but augmented by an extra term, that is linear as the right hand side of the commutation relation of the coordinates (16), the presence of which is necessary in order that it transforms covariantly: Rµν (X) = [Xµ , Xν ] − iλǫµνρ X ρ . a ε(X) = ξ (X) ⊗ Pa + λ (X) ⊗ Ma δeµ a = −∂µ ξ a − ǫabc (−iξb ωµc − iλb eµc ) i = − i[Xµ + Aµ , ξ a ] + {ξb , ωµc }ǫabc 2 i abc + {λb , eµc }ǫ 2 (25) The above tensor expands on the various generators of the algebra as: Rµν (X) =Tµν a (X) ⊗ Pa + Rµν a (X) ⊗ Ma + Fµν (X) ⊗ iI + F̃µν (X) ⊗ γ5 . (26) Relating the various relations and definitions, (20), (25) and (26), the components of the curvature tensor are obtained and the commutative limit leads to the corresponding relations of the conventional gauge-theoretic approach (see the detailed expressions and discussion in Ref. [120]). + i[ε0 , eµ a ] + λa , õ + [ε̃0 , ωµ a ], δωµ a = − i[Xµ + Aµ , λa ] i i + {ξb , eµc }ǫabc + {λb , ωµc }ǫabc 2 2 + i[ε0 , ωµ a ] + ξ a , õ + [ε̃0 , eµ a ], δAµ = − i[Xµ + Aµ , ε0 ] i i − [ξ a , eµa ] − [λa , ωµa ] − i ε̃0 , õ , 4 4 1 δ õ = − i[Xµ + Aµ , ε̃0 ] + [ξ a , ωµa ] 4 1 (22) + [λa , eµa ] + i ε0 , õ . 4 Before we move on to the dynamics of the theory, it is meaningful to make a small detour commenting on the above transformations and what happens when a commutative limit is considered. In this condition, the two Abelian gauge fields introduced for reasons of noncommutativity are taken out of the picture, the conventional derivation is recovered [Xµ , f ] → −i∂µ f and the commutators vanish leading to: 4.1.3 The action for a three-dimensional fuzzy gravity To conclude the study of the three-dimensional case, the action that is proposed is aligned to the ordinary case, in which a Chern–Simons type is employed, that is:   i 1 S0 = 2 Tr ǫµνρ Xµ Xν Xρ − m2 Xµ X µ , (27) g 3 which, following its variation, leads to the field equation: [Xµ , Xν ] + 2im2 ǫµνρ X ρ = 0. (28) Variation gives the field equations which admit the background space as a solution for 2m2 = −λ, as expected. Introducing the gauge fields into the above picture by replacing the coordinates with their covariant counterparts, one ends up with:   i µνρ λ 1 µ S = 2 Tr tr ǫ Xµ Xν Xρ + Xµ X , g 3 2 (29) 123 Eur. Phys. J. Spec. Top. in which Tr is effectively the integration operator coordinates and tr acts on the generators. Taking into consideration the non-vanishing traces tr(Pa Pb ) = δab , tr(Ma Mb ) = δab and, finally, performing the variation with respect to the various gauge fields, the field equations are obtained: Adopting an SO(4) notation, after the introduction of a length parameter, λ, the generators become: Jµ6 Tµν a = 0, Rµν a = 0, Fµν = 0, F̃µν = 0. (30) 5 Fuzzy gravity in four dimensions In this section a four-dimensional noncommutative gravitational model as a (noncommutative) gauge theory is constructed. Besides the fact that the fourdimensional case is rather more interesting than the three-dimensional one, the general setting and the methodology that are encountered in the former will be essentially an extension of those of the latter. In this case too, the starting point is the determination of the background space. 5.1 An approach to the fuzzy four-sphere The background noncommutative space that has been chosen to accommodate the construction of the gauge theory is the four-dimensional version of the fuzzy sphere, that is the fuzzy four-sphere, SF4 .2 In order to find the definition of the space in this four-dimensional case, it would be tempting to attempt a straightforward translation of the fuzzy two-sphere defining relations (commutation and radius) to the four dimensions. Quickly recalling, the coordinates in the fuzzy sphere case came from rescaling the angular momentum operators, i.e. the generators of the isometry group SO(3). If we tried, in the four-dimensional case, to identify four out of the ten generators of the corresponding isometry group SO(6) to the coordinates, we would fall into a dead end because there is no subalgebra that would be nicely closing, or, in other words, the virtue the fuzzy two-sphere had but the four-sphere does not is that of covariance. However, covariance is an essential property of the background fuzzy space, therefore either it should be abandoned or find a way and restore covariance. Selecting the second option, a way to achieve the restoration of the covariance is to consider the coordinates as a subset of a larger group in which the corresponding subalgebra will close [95]. The minimal cost one would pay to achieve that is to go to SO(6) [122, 123] with its generators obeying the following algebra: [JAB , JCD ] = i(δAC JBD + δBD JAC − δBC JAD − δAD JBC ). (31) 2 In this case too, the signature of the space is Euclidean but a construction for the Lorentzian case in which the fuzzy de Sitter space, dSF4 , is employed is possible. 123 1 Θµν ,  λ Pµ , = 2 Jµν = Jµ5 = J56 1 X5 , λ 1 = h, 2 (32) where μ, ν = 1, . . . , 4, Xµ , Pµ and Θµν denote the coordinates, momenta and noncommutativity tensor, respectively, and h is an operator bearing information of the radius constraint [123]. Taking these redefinitions into consideration, the above commutation relation consisting the algebra of SO(6) becomes: [Xμ , Xν ]=i λ2 Θμν ,  [Xμ , Pν ] = iδμν h,  Θμν λ2 2 λ [Xμ , h] = i Pμ .  [Pμ , Pν ]=4i [Pμ , h]=4i  Xμ , λ2 (33) The first commutation relation in the above set of relations corresponds to the defining one of the background fuzzy space, obviously closing into an SO(4) subalgebra of the total SO(6), as aimed. The rest of the commutation relations that come from the decomposition of the initial commutation relation of the SO(6) algebra are the spacetime transformations: [Θµν , Θρσ ] = i(δµρ Θνσ + δνσ Θµρ − δνρ Θµσ − δµσ Θνρ ), (34) [h, Θμν ] = 0, = i(δμρ Xν − δμν Xρ ), [Pμ , Θνρ ] = i(δμρ Pν − δμν Pρ ), (35) where the first one is the SO(4) subalgebra (fourdimensional rotations) and the second one shows how the coordinate vector transforms under these rotations, that is as vectors, validating the covariance property in a rather pronounced way. 5.2 On the gauge group As noted throughout the text so far, when constructing a noncommutative gauge theory, the anticommutators become important and, therefore, the initial algebra of the gauge group, here the isometry group SO(5), expands. For the sake of a finite enlargement of the algebra, the representation of the generators gets fixed and, furthermore, for a minimal one, that representation is chosen to be the four-dimensional one and, consequently, the algebra of the gauge group extends to SO(6)×U (1). The four-dimensional matrices representing the sixteen generators are given in terms of combinations of the the (Euclidean) 4 × 4 gamma matrices as: i Mab = − [Γa , Γb ], 4 Ka = 1 Γa , 2 Eur. Phys. J. Spec. Top. i P a = − Γ a Γ5 , 2 1 D = − Γ5 , 2 I4 , (36) which, satisfy the well-known anticommutation relation {Γa , Γb } = 2δab I4 , where a, b = 1, ..., 4 and Γ5 = Γ1 Γ2 Γ3 Γ4 . Therefore, one can now write down a complete list of commutation and anticommutation relations of all the generators: [Ka , Kb ] = iMab , [Pa , Pb ] = iMab , [Pa , D] = iKa , [Ka , Pb ] = iδab D, [Ka , D] = −iPa , [D, Mab ] = 0, [Ka , Mbc ] = i(δac Kb − δab Kc ), [Pa , Mbc ] = i(δac Pb − δab Pc ), √ 2 {Pa , Kb } = {Mab , D} = − ǫabcd Mcd , 2 [Mab , Mcd ] = i(δac Mbd + δbd Mac − δbc Mad − δad Mbc ), √ 2 1 ǫabcd D, {Mab , Mcd } = (δac δbd − δbc δad )I4 − 8 4√ √ 2 ǫabcd Kd , {Mab , Kc } = 2ǫabcd Pd , {Mab , Pc } = − 4 1 1 {Ka , Kb } = δab I4 , {Pa , Pb } = δab I4 , 2 8 {Ka , D} = {Pa , D} = 0. (37) Having determined the gauge group, the representation and, therefore, the above relations, it is now possible to move on with the construction of the corresponding gauge theory. 5.3 Action and equations of motion · [Xρ + Aρ , Xσ + Aσ ] − κ2 (Θρσ + Bρσ ) ,   ǫµνρσ Xν , [Xρ , Xσ ] − κ2 Θρσ = 0, ǫµνρσ [Xρ , Xσ ] − κ2 Θρσ = 0, (40) where tr denotes the trace over the generators of the algebra of the gauge group. The above expression of the action can resemble formally the (ordinary) fourdimensional Chern–Simons action, after the following identifications: • The covariant coordinate: Xµ ≡ Xµ + Aµ , where Aµ is identified as the gauge connection: Aµ = eµa ⊗Pa +ωµab ⊗Mab +bµa ⊗Ka +ãµ ⊗D+aµ ⊗I4 • The covariant noncommutative tensor Θ̂µν ≡ Θµν + Bµν , where Bµν is a 2-form field • The (covariant) field strength tensor of the theory: Rµν ≡ [Xµ , Xν ] − κ2 Θ̂µν , which, as an element of the gauge algebra, can be decomposed in various component tensors4 : Rµν (X) = R̃µν a ⊗ Pa + Rµν ab ⊗ Mab + Rµν a ⊗ Ka + R̃µν ⊗ D + Rµν ⊗ I4 . (41) In turn, using κ2 = iλ , the expression of the action becomes:   iλ2 S = Trtr [Xµ , Xν ] − Θ̂µν    2 iλ Θ̂ρσ ǫµνρσ := TrtrRµν Rρσ ǫµνρσ [Xρ , Xσ ] −  (42) and variations lead once again to two kinds of field equations: [Xρ , Xσ ] − κ2 Θρσ ǫµνρσ , (38) ǫµνρσ Rρσ = 0, ǫµνρσ [Xν , Rρσ ] = 0, (43) which are understood as the vanishing of the field strength tensor and a noncommutative analogue of the Bianchi identity, respectively. with the following field equations: (39) which are obtained after varying with respect to X and 2 Θ. From the second equation, in case κ2 = iλ , the 3 S =Trtrǫµνρσ [Xµ + Aµ , Xν + Aν ] − κ2 (Θµν + Bµν ) 2 From here on, there are two equivalent ways to proceed, the first one is to follow the straightforward methodology, that is the introduction of the covariant coordinate and the gauge fields, the calculation of the field strength tensor and then the determination of an action. Here, we choose the alternative route, in which an initial topological action is proposed and the gauge fields, covariant coordinate and field strength tensor are involved as a consequence of the introduction of dynamics. According to the above, the starting action is3 : S = Tr [Xµ , Xν ] − κ2 Θµν defining relation of noncommutativity of the space is recovered and, therefore, the first equation is automatically satisfied. Dynamics to the above action are introduced after the consideration of gauge fields as fluctuations of X and Θ: Despite the first term of this action, Eq. (38), Tr[Xμ , Xν ][Xρ , Xσ ]ǫμνρσ vanishes identically, it remains present for later use. 5.4 Symmetry breaking of the action The above action of Chern–Simons type (42), is invariant under SO(6) × U (1) gauge symmetry which is a rather big amount of symmetry as a result of the 4 The detailed expressions of the components of the field strength tensors as well as the transformations of the gauge fields can be found in the original papers [122, 123]. 123 Eur. Phys. J. Spec. Top. enlargement of the group. For reasons of compatibility of the two frameworks (commutative-noncommutative), especially when the commutative limit is considered, the above gauge group can break down to a smaller one either by imposing certain constraints [122] or by introducing an auxiliary gauge field and letting it acquire a vev [123]. According to the second recipe, the action (42) becomes: S = TrtrG λΦ(X)Rµν Rρσ ǫµνρσ + η Φ(X)2 − λ−2 IN ⊗ I4 , (44) (45) which means that the initial action remains effectively untouched. Now, let the scalar field, Φ, consist exclusively of the symmetric component, therefore it is expressed as: Φ(X) = φ̃a (X) ⊗ Pa + φa (X) ⊗ Ka + φ(X) ⊗ I4 + φ̃(X) ⊗ D. The symmetry breaking takes place when the scalar field is gauge-fixed along the direction of the D generator, that is: Φ(X) = φ̃(X) ⊗ D|φ̃=−2λ−1 = −2λ−1 IN ⊗ D. Putting the above value of the scalar field into the action (44) and perform the trace over the generators, the action becomes: Sbr = Tr √  2 ab cd ǫabcd Rµν Rρσ − 4Rµν R̃ρσ ǫµνρσ . 4 (46) The consideration that the scalar field consists only of the symmetric part of the algebra of SO(6) and implying at the same time that it is not charged under the U (1), the gauge symmetry of the resulting action is the SO(4) × U (1). Specifically, the generators that remain unbroken in the above mechanism are: a) Pa which leads to the torsionless condition which, in turn results to relation between the gauge fields ω, e and ã, b) Ka which lead to the condition Rµν a = 0, which in turn implies a relation of proportionality between e, b gauge fields and c) D which requires ãµ = 0 [124]. Therefore, the remnant symmetry of the spontaneously broken theory is SO(4) × U (1) and the independent fields are e and a. Last, the resulting expression of the component tensor Rµν ab , after taking into consideration the following results of the conditions from the breaking ãµ = 0 and bµ a = 2i eµ a , is: 123 +  iλ2 3i  a Bµν ab . eµ , eν b − 8  (48) 6 A quick glimpse of the metric where η is a Lagrange multiplier and λ is a parameter of dimension of length. On-shell it holds that: Φ2 (X) = λ−2 IN ⊗ I4 ,     Rµν ab = Xµ + aµ , ων ab − Xν + aν , ωµ ab     (47) + i ωµ ac , ωνb c − i ωµ bc , ωνc a Working with the (noncommutative) gauge-theoretic approach to gravity, the notion of the metric is not explicitly present. Instead, the quantity that relates the above with the geometric approach is the vierbein which has been introduced in the construction as a gauge field related to the generators of the translations. In the conventional framework, the above two are related through the well-known “square root of the metric” equation which, according to Ref. [124], in the noncommutative regime and specifically in the star-product realization, it reads: gµν = eµa ⋆ eνa , where, in the same source, it is argued that the above relation leads to theories of complex gravity. In order to translate the above to the matrix realization of noncommutativity which is of our interest, we consider the following version of the above relation: gµν =(eµa ⊗ Pa )(eνb ⊗ Pb ) = 1 a b e e ⊗ iMab 2 µ ν 1 1 + eµa eνb ⊗ δab I4 ≡ Gµν + iBµν , 2 8 (49) where the commutation and anticommutation relations of the P generators as found in Eq. (37) have been used. By inspection, both real and imaginary parts are encountered, therefore it can be deduced that the metric is complex in this case as well. Also, considering the gµν to be hermitian leads to the understanding that Gµν is symmetric while the Bµν is antisymmetric under the exchange of the spacetime indices μ, ν. Therefore, comparing to the above expression of the metric, (49), the following identification is achieved: 1 a 1 a eµ eνa ⊗ I4 = {e , eνa } ⊗ I4 , 16 32 µ 1 1 = eµa eνb ⊗ Mab = [eµa , eνb ] ⊗ Mab . 2 4 Gµν = (50) Bµν (51) It has been routed for future projects to delve more into the above observations regarding the metric. Eur. Phys. J. Spec. Top. 7 Noncommutative gravity on fuzzy 3-sphere Also, along the lines of the fuzzy 2-sphere, the quadratic Casimir element is related to the radius constraint: Although the three-dimensional case was examined earlier, here is the right place to examine the case of a noncommutative gravitational model in three-dimensions in case the background space in not R3λ like in the previous three-dimensional case but the fuzzy 3-sphere, SF3 . It is examined now because the methodology of building this version of the fuzzy 3-sphere is a lowerdimensional analogue of the construction of the fuzzy four-sphere as it got realized in the previous section. Like the four-dimensional fuzzy sphere, this space is also both of Lie and Moyal type since the noncommutative tensor is both constant and at the same time a generator of the isometry group. So, following the same methodology as in [123], starting from the isometry group SO(4) an extension is required for reasons of covariance, therefore the SO(5) is eventually used. Then, a 2-step procedure is followed in which, in the first step, the space is manifested as an embedding in the four-dimensional Euclidean spacetime, while in the second step in the three-dimensional one. The resulting space will be a realization of the fuzzy 3-sphere and will finally be employed for the construction of a gravity model on it. 1 1 1 = − TrJ 2 = JM N J M N = Jmn J mn + Jm4 J m4 ⇒ 2 2 2   SO (5) SO (4) X m Xm = λ2 C2 ≡ r2 , − C2 7.1 First step: SO(5) ⊃ SO(4) (54) where r2 ≡ λ2 L, and where it has been assumed that the SO(5) and SO(4)generators are of the same spin representation, L. 7.2 Second step: SO(4) ⊃ SO(3) From the first equation of (53), if μ, ν, ρ, σ = 0, . . . , 2: • For m = μ, n = ν, r = ρ, s = σ, where μ, ν, ρ, σ = 0, . . . , 2: [Θμν , Θρσ ] = i(δμρ Θνσ + δνσ Θμρ − δνρ Θμσ − δμσ Θνρ ). • For m = μ, n = ν, r = ρ, s = 3: [Θµν , Θρ3 ] = i(δµρ Θν3 − δνρ Θµ3 ). • For m = μ, n = 3, r = ρ, s = 3: [Θµ3 , Θρ3 ] = iΘµρ . From the second equation of (53): We start with the SO(5) group which has 10 hermitian generators, JM N , where M , N = 0, . . . , 4 which obey: [JM N , JP Σ ] = i (δM P JN Σ + δN Σ JM P − δN P JM Σ − δM Σ JN P ) , (52) where δM N is the 5-dim Kronecker delta. The SO(5) ⊃ SO(4) decomposition yields: • For M = m, N = m, P = r, Σ = s: [Jmn , Jrs ] = i(δmr Jns + δns Jmr − δnr Jms − δms Jnr ). • For M = m, N = 4, P = r, Σ = 4: [Jm4 , Jn4 ] = iJmn . • For M = m, N = n, P = r, Σ = 4: [Jmn , Jr4 ] = i(δmr Jn4 − δnr Jm4 ), where m, n, r, s = 0, . . . , 3. Setting Θmn ≡ Jmn , Xm ≡ λJm4 , where λ is a parameter of dimension of length, the above commutation relations become: [Θmn , Θrs ] = i(δmr Θns + δns Θmr − δnr Θms − δms Θnr ), [Θmn , Xr ] = i(δmr Xn − δnr Xm ), SO (5) C2 λ2 [Xm , Xn ] = i Θmn .  • For m = μ, n = ν, r = ρ: [Θµν , Xρ ] = i(δµρ Xν − δνρ Xµ ). • For m = μ, n = ν, r = 3: [Θµν , X3 ] = 0. • For m = μ, n = 3, r = 3: [Θµ3 , X3 ] = −iXµ . • For m = μ, n = 3, r = ρ: [Θµ3 , Xρ ] = iδµρ X3 . From the third equation of (53): 2 • For m = μ, n = ν: [Xµ , Xν ] = i λ Θµν . 2 • For m = μ, n = 3: [Xµ , X3 ] = i λ Θµ3 . Setting Pµ ≡ λ1 Θµ3 , h ≡ λ1 X3 , the commutation relations regarding all the operators Θµν , Xµ , Pµ , h are: [Θμν , Θρσ ] = i(δμρ Θνσ + δνσ Θμρ − δνρ Θμσ − δμσ Θνρ ), λ2  Θμν , [Xμ , Xν ] = i Θμν , 2 λ   λ2 [Pμ , h] = −i 2 Xμ , [Xμ , h] = i Pμ , λ  [Θμν , Pρ ] = i(δμρ Pν − δνρ Pμ ), [Pμ , Pν ] = i [Θμν , Xρ ] = i(δμρ Xν − δνρ Xμ ), [Pμ , Xν ] = iδμν h, [Θμν , h] = 0. (55) (53) The embedding relation, (54), in SO(3)notation, becomes: 123 Eur. Phys. J. Spec. Top. X m Xm =r2 ⇒ X µ Xµ + X3 X 3 = r2 ⇒ h  1 =± (Xµ X µ − r2 ), λ2 (56) where just like in the case of the fuzzy 4-sphere, h is an operator related to the radius constraint. Also, it is worth noting that in the limit where h → 0, a new realization of the fuzzy 2-sphere, SF2 emerges. 7.3 Determination of the action This noncommutative space, as a fuzzy version of the 3-sphere, has its isometries parametrized by the SO(4) group, therefore the construction of the corresponding noncommutative gauge theory will coincide to the one presented above in the R3λ case. So, according to the results of Sect. 4, the minimum expansion of this group, in order that the resulting operators of the anticommutators to be contained, is again U (2) × U (2) which consists of the same set of generators satisfying the same algebra as it is written in (17). The difference between the two cases of SF3 and R3λ emerges when considering the action. Although, in this case too, an action of three-dimensional Chern–Simons type is used, compared to (27), the second term that is related to the kind of noncommutativity is different:    2 µνρ i Xµ Xν Xρ + κ {Xµ , Θνρ } (57) S = Tr ǫ 3 Variation of the above action will lead to the corresponding field equations. It is expected that our background space should satisfy the derived equations. We consider that, in principle, X and Θ are independent fields. Variation with respect to Θ and X respectively gives: ǫµνρ Xµ = 0, ǫµνρ [Xµ , Xν ] − 2iκ2 Θµν = 0. (58) Despite the second field equation is satisfied by the SF3 2 when κ2 = λ4 , meaning that the background space is indeed a solution, the first one, Xµ = 0, is effectively a trivialization. If we move on to examine the case in which gauge fields are introduced as fluctuations, set2 ting κ2 = λ4 , introducing a trace over the gauge algebra and making use of similar definitions as the ones above Eq. (41), the action given in Eq. (57) will become:    i λ2 Xµ Rνρ + {Xµ , Θ̂νρ } S = Trtr ǫµνρσ 6 4 (59) and, in turn, will give the following field equations: ǫµνρ Xµ = 0, 123 ǫµνρ Rµν = 0. (60) Again, the second equation is the vanishing of the field 2 strength tensor, Rµν = [Xµ , Xν ]− iλ 2 Θ̂µν , which would be an acceptable field equation, but the first one trivializes the model. Summing up, the above construction of a threedimensional noncommutative gravity on a fuzzy 3sphere has a lot in common with the one constructed in Sect. 3 on the R3λ space. Nevertheless, there are some differences in the two gauge theories that become manifest when the kind of space manifestly enters in the calculations. The action is again of threedimensional Chern–Simons type, but involves the Θ operator instead of only the X , (57). This consideration led to the field equations of Eq. (60), which, due to the first one, imply that no interesting physical conclusions are achieved by this model. Maybe the employment of an alternative action would produce more interesting results, leaving this task for a future study. 8 Unification of conventional and fuzzy four-dimensional gravity with gauge interactions As it has been already discussed in the introduction there exists a gauge theoretic construction of GR [52–64] in addition to the geometric one, namely the description of GR as a gauge theory of the Lorentz group with the spin connection as the corresponding gauge field which would enter in the action through the corresponding field strength. Usually the dimension of the tangent space is taken to be equal to the dimension of the curved manifold. However the dimension of the tangent group is not necessarily the same as the dimension of the manifold [125]. In Ref. [126] (see also [127]) the authors have considered higher-dimensional tangent spaces in four dimensional space-time and managed in this way to achieve unification of gauge interactions with gravity. The geometric unification of gravity and gauge interactions is realized by writing the action of the full theory in terms only of the curvature invariants of the tangent group, which contain the Yang–Mills actions corresponding to the gauge groups describing in this way together the GR and the internal GUT in a unified manner. The best model found so far that unifies gravity and a chiral GUT is based on SO(1, 13) in a 14-dimensional tangent space. In order to make clear the specific model let us consider the decomposition of SO(14) under the maximal subgroup: SO(14) ⊃ SU(2) × SU(2) × SO(10), where the compact isomorphic image of SO(1, 3), SO(4) = SU(2) × SU(2) is used for convenience. The decomposition of the adjoint representation under the above subgroups is given by: Eur. Phys. J. Spec. Top. 91 = (3, 1, 1) = (1, 3, 1) + (1, 1, 45) + (2, 2, 10) 32′ =(1, 1, 4) + (3, 3, 2) + (5, 1, 2) and of the spinor by: ¯ 64 = (2, 1, 16) + (1, 2, 16). Noting that the (2, 1), (1, 2) under SU(2) × SU(2) are the L, R handed fermions respectively, one finds that the spinor of SO(1, 13) describes two chiral 16L ¯ R = 16L ). Without going fermionic families (since the 16 further in the analysis the result is that it is indeed possible to achieve unification of GR with internal symmetries expressed as an SO(10) GUT in this case and the number of families will be even, if more 64 are added. In Ref. [126] in their analysis preferred to make heavy one of the families by suitable spontaneous symmetry breaking. Let us first note that the above analysis can be extended using the SO(18) and then consider the decompositions under the following maximal subgroups: SO(18) ⊃ SO(8) × SO(10) and then SO(8) ⊃ SU(2) × SP (4) 32 =(1, 4, 1) + (3, 2, 3) + (5, 2, 1) and SP(4) ⊃ SU(2) × SU(2). In turn choosing one SU(2) from the SO(8) decomposition and another from the Sp(4) decomposition to form the gauging of the Lorentz group that produces the GR, one can obtain again SO(10) unification with 4 families from the spinor, 256 of SO(18). A similar procedure can be applied starting with SO(22) and the decompositions under the following maximal subgroups: SO(22) ⊃ SO(12) × SO(10) with the decomposition of the adjoint: 231 = (1, 45) + (66, 1) + (12, 10) one can find that from the spinor of of SO(22), 1024 chiral 16L are obtained but with a multiplicity 10, which is far too high to become realistic and can be excluded on this basis. With the above procedure considered as a possible way to unify gravity with gauge internal interactions, we naturally tried to extend it in our four-dimensional construction of fuzzy gravity in which we had to enlarge the tangent space to SO(6) with the SO(4) taken as the maximal subgroup of SO(5), which in turn was the maximal subgroup of SO(6), i.e. the SO(4) was taken from the following chain of maximal subgroups: SO(6) ⊃ SO(5) ⊃ SO(4). (61) In turn the gauging procedure to construct fuzzy gravity led us to consider the SO(6) × U (1) as the appropriate gauge group. A possible way to proceed in the unification of fuzzy gravity with internal gauge symmetries, as in the previous examples, would be to consider the SO(16) as the unifying group. Then the taking the the maximal subgroups SO(16) ⊃ SO(6) × SO(10) under which the adjoint is decomposed as: 120 = (15, 1) + (1, 45) + (6, 10) and the spinor as: ¯ 128 = (4, 16) + (4̄, 16) we could consider in the gauging of SO(6) × U (1) to identify e.g. the U (1) with the one resulting from the maximal decomposition of SO(10) ⊃ SU(5) × U (1) with decomposition of the adjoint as: and then: SO(12) ⊃ SU(2) × SU(2) × SU(2), with decomposition of the adjoint: 66 = (3, 1, 1) + (1, 3, 1) + (1, 1, 3) + (3, 3, 3) + (5, 3, 1) + (5, 1, 3) and of the spinor: ¯ 45 = 1(0) + 10(4) + 10(−4) + 24(0). The problem arises from the fact that according to the decomposition chain (61) we have that the spinor (antispinor) of SO(6) decomposes as follows: SO(6) ⊃ SO(5) ⊃ SO(4) = SU(2) × SU(2) 4 = 4 = (2, 1) + (1, 2), ¯ 1024 = (32, 16) + (32′ , 16). which means that from the decomposition of the spinor 128 of SO(16) we obtain: Then choosing the last two SU(2)s of the above decomposition to be gauged and produce GR, under which the decomposition of the spinors is: SO(16) ⊃ SO(6) × SO(10), ¯ 128 = (4, 16) + (4̄, 16), 123 Eur. Phys. J. Spec. Top. which in turn under SO(4) = SU(2) × SU(2) becomes: SO(16) ⊃ SU(2) × SU(2) × SO(10) ¯ 128 = ([(2, 1) + (1, 2)], 16) + ([(2, 1) + (1, 2)], 16), which unfortunately is not a chiral theory. It seems that this is a general feature of our construction, which originates from the chosen chain (3) in our construction. Therefore, this very interesting way of unifying fourdimensional gravity and internal interactions of particle physics does not work in the case of the constructed four-dimensional fuzzy gravity. in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecomm ons.org/licenses/by/4.0/. Data availability statement No data associated in the manuscript. 9 Conclusions In this review, we presented a panoramic view of the studies on the physical perspective of ours on noncommutative geometry, which have been carried out by various collaborators over the years, updated by meaningful comments and extensions. First, we presented the general approach on how gauge theories are formulated in the noncommutative setting, then, a particle physics model in which noncommutativity becomes manifest through fuzzy extra dimensions and, last, gravitational models in three and four dimensions constructed on background fuzzy spaces as gauge theories. At the end, an attempt of unification between the gravity and GUTs was made, which although successful in the commutative case, it failed to result to a fruitful outcome in the noncommutative (fuzzy) one. Overall, intertwining noncommutativity with physics consists a very promising initiative leading to interesting features and models, as in the quantum gravity realm, as in particle physics and extra dimensions. Acknowledgements We would like to thank all our collaborators in the projects that are covered in this review, Paolo Aschieri, Danijel Jurman, Athanasios Chatzistavrakidis, Lara Jonke, John Madore and Harold Steinacker. We appreciate the useful discussions with Patricia Vitale, Ali Chamseddine, Denjoe O’Connor, Dimitra Karabali, V. Parameswaran Nair, Maja Burić, Dumitru Ghilencea, Ichiro Oda, Emmanuel Saridakis, and Viatcheslav Mukhanov. Finally, we are grateful to Dieter Lüst and Harald Grosse for their constant encouragement. Affiliated authors of NTUA have been supported by the Basic Research Programme, PEVE2020 of National Technical University of Athens, Greece. One of us (GZ) would like to thank the DFG Exzellenzcluster 2181:STRUCTURES of Heidelberg University, MPP-Munich, A.v.Humboldt Foundation and CERN-TH for support. This work (GM) has been supported by the Polish National Science Center Grant no. 2017/27/B/ST2/02531. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction 123 References 1. M.B. Green, J.H. Schwarz, E. Witten, Superstring Theory. Vol. 1, 2: Introduction (Cambridge University Press, 1987), p. 469 2. D.J. Gross, J.A. Harvey, E.J. Martinec, R. Rohm, Heterotic string theory 1. The free heterotic string. Nucl. Phys. B 256, 253–284 (1985). https://doi.org/10.1016/ 0550-3213(85)90394-3 3. P. Forgacs, N.S. Manton, Space-time symmetries in gauge theories. Commun. Math. Phys. 72, 15–35 (1980). https://doi.org/10.1007/BF01200108 4. D. Kapetanakis, G. Zoupanos, Coset space dimensional reduction of gauge theories. Phys. 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