PoS(CORFU2019)236 https://pos.sissa.it/ ∗ [email protected] George.Zoupanos@cern.ch [email protected] Speaker. We review the construction of aa four-dimensional gauge-theoretic noncommutative gravity way. model formulated Fordimensions in (General consistency, Relativity, we Weyl Gravity), expressed include as a gaugereminder theories. brief of Also, review we the give of a formulation brief gravity ofinformation theories of gauge in our theories four recent on work noncommutativetheory is on spaces. included, a specifically four-dimensional Last, covariant a noncommutative gravity the space. model essential constructed as a gauge ∗ Copyright owned by the author(s) under the terms of the Creative Commons c Attribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). Corfu Summer Institute 2019 "School and(CORFU2019) Workshops on Elementary Particle Physics and31 Gravity" August - 25 September 2019 Corfù, Greece E-mail: G. Zoupanos National Technical University of Athens E-mail: G. Manolakos P. Manousselis National Technical University of Athens E-mail: Gauge theory of gravity on acovariant noncommutative four-dimensional space PoS(CORFU2019)236 . , ] 4 42 33 dS , , 41 32 ], which , , . In this ) 13 40 31 3 , , , G. Manolakos 1 ( 12 39 , , SO 11 38 , 10 , ] gauging the de Sitter 9 , 13 8 , , 7 12 , , 6 , 11 5 ]. Besides the above approaches, , ], who were based on Yang’s early , 4 23 10 space with the property of preserving 37 , , 3 ]. Also, for alternatives see [ 9 , ] which preserve the Lorentz invariance, dS 2 , 30 , 8 , 37 1 , , ] as a gauge theory on a fuzzy space, the 29 7 , , 36 45 6 28 1 , , 5 ]). , 27 52 4 , , ] the above gauge-theoretic approach of gravity has been 26 51 , , 22 , 25 . Next, on the above (covariant) noncommutative space, the 50 , ) , 21 5 , , 24 49 1 , ( 20 , 48 SO 19 , to ) 18 4 , , -product and the Seiberg-Witten map [ 1 17 ? ( , ] (see also [ SO 16 , spontaneously broken by a scalar field to the Lorentz group, ], we have constructed a covariant fuzzy 47 ) , , 4 36 ]). In general, it is understood that the formulation of noncommutative gravity is ac- , 15 1 46 ]. In the noncommutative framework in order to result with models of noncommutative , from ( 34 , 9 4 , 14 45 SO 8 dS , ] and was extended by other authors [ Four-dimensional gravity is described, in a geometric way, by the successful Theory of Gen- The purpose of this article is to present and highlight the features of a four-dimensional non- In the above treatments the authors make use of the concepts of constant noncommutativity If we consider that, at the regime of small distances (Planck length), the spatial coordinates 44 2 , , 1 gauging of a subgroup ofgroup the in total a isometry fixed representation. tookthe place, covariance During of leading the the to development field the ofour strength enlargement the proposition tensor of of gauge led the theory, the to action the gauge the isfield requirement addition of added of of Yang-Mills earlier. type, a including 2-form the gauge kinetic field. term of Eventually, the 2-form gauge 2. Approaches of gravity as gauge theory 2.1 Gauge-theoretic approach of 4-d Einstein’s Gravity construction of noncommutative gravitational modelsrealization can of be matrix achieved geometries using [ the(see noncommutative also [ Drawing motivation from the workpublication of [ Heckman-Verlinde [ Lorentz invariance. Requiring thisfuzzy property led to the enlargement of the isometry group of the gravity [ 43 commutative gravity model, constructed recently [ is based on gauging the Lorentz[ and Poincaré groups in the pioneer works of Utiyama andcontext Kibble the Weyl gravity has been constructedgroup as [ a gauge theory of the four-dimensional conformal companied by the breaking of Lorentzexistence invariance of by “covariant noncommutative deformations. noncommutative However, spaces” the allows [ the construction of field theories from noncommutative deformations. [ (Moyal-Weyl), the taken into consideration and translated inestablished the formulation noncommutative of framework gauge making theories use on of noncommutative the spaces. well- exhibit a noncommutative behaviour, thenparametrized the by structure the of framework the of(LHC noncommutative spacetime scale) geometry. at the Referring this Strong, now regime Weak and toand could Electromagnetic higher are be interactions distances described are formulated togetherPlanck by in distances) gauge the the theories three Standard interactions Model. areUnified studied Theories. in At the Although interesting much none frameworkthere smaller of of does Grand distances the exist (Gauge) a above (but gauge-theoretic two larger approach pictures of then include it, see the [ gravitational interaction, Noncommutative Gravity 1. Introduction group, PoS(CORFU2019)236 ]. ab 7 , M (2.8) (2.4) (2.7) (2.1) (2.2) (2.6) (2.3) (2.5) 6 , 5 is given , µ 4 , de Sitter, , A ) 2 3 G. Manolakos , , 1 1 ( , ISO 0 , are identified as the ab ] = µ b P ω , a , P [ and , , ab , b a . M ] ab . µ µ c ) e ] e P ab M b ω . ) produces the transformations of the ν b ) [ a ( ] M x a A , ) ( λ , ab cb ] η 2.3 x µ µ ε 2 ab ( − µν , , i.e. A µ ω b ab a µ R c ξ ω a λ ] = A 1 2 M [ + [ 1 2 ab ] 2 1 bc λ ν µ + + [ 2 + M A + a ω 2 ε , a and µ a P − a [ µ P ) P + a ) P ∂ ∂ e ) [ ab a P x 2 ( x ) along with ( ( a ξ λ ( = , a a , which are based on the Poincaré, µ µ ] µ µ 2.4 4 µν ∂ ∂ ξ b ) = e ] A R d A = = δ ( , gauge groups, respectively. The above isometry groups of M AdS ) a ) = ) = c ab [ µν 3 x ) = µ x a , ( µ ) and ( [ e ( R A 2 ε ( µ η δ ( 4 2.2 A δω µν SO R ] = cd are the component tensors, i.e. the curvatures corresponding to the com- M , ab ab , is introduced which, by definition, takes values within the Poincaré algebra: M µν is the parameter of the gauge transformation, which is an element of the algebra and anti-de-Sitter [ µ R ) 4 A x ( dS ε and = a is the mostly positive metric tensor of the Minkowski spacetime and the generators ε , and Anti-de Sitter, µν ab ) correspond to the Lorentz transformations and the local translations, respectively. Then the R η 4 a , P 1 , de Sitter ( 4 ponent gauge fields, identified as the torsion and curvature, respectively. Their explicit forms are where and gauge potential, Combination of the equations ( by: gauge fields: Being also valued in the algebra of generators it is written as an expansion on them: In order to approach four-dimensional gravityvierbein in formulation a of gauge-theoretic General way, Relativity. first Then one ifits has a sign, to cosmological employ the constant the is corresponding present, spaces accordingM on to which the gauge theories are constructed are the Minkowski where where the gauge fieldsvierbein associated and spin with connection, the respectively. The gauge transformation of the connection The standard procedure when buildingsponding gauge field strength theories tensor: continues with the definition of the corre- the spaces are the indicatedmological gauge constant groups is in absent, eachdetermined that case. by is the Let the following us commutation case first relations: of recall the the Poincaré case group. in which In cos- this case, the algebra is Noncommutative Gravity eral Relativity. On theprinciple, other at hand, a possible the unification gauge-theoretic of the alternative gravitational description interaction with of the gravity rest ones, aims, [ in SO in which and therefore takes values in it: PoS(CORFU2019)236 ), a P (2.9) (2.11) (2.12) (2.10) ]. 6 , ] (see also 4 10 G. Manolakos , , of the theory ). The algebra 9 µ c D ] , A b 8 δ a [ , K ) 2 ab M ] = c − K , , D . a ab ab b K c δ , a ] M ( µ ν [ b f 2 ] ): ω ν ) and the dilatations ( e + , ac a ] = c 2.7 µ ab ] D [ K b b µ µ [ ω K δ b , 2 a ω [ a 2 P + P − [ 2 − ab ab a ] ] M , ν 3 ] = ) into the ( ν a c ω e ab P µ K µ µ , [ [ 2.8 ω − ∂ ∂ ab 1 2 2 2 M [ + ] = a ) = ) = ) and ( D P e , , ω a ( a ] µ ] ( a 2.2 c b e K [ ab δ ), the conformal boosts ( µν d = [ a R µν [ ab , µ R M a M A P 4 ] = ] = D cd , 4. The gauging procedure dictates that the gauge potential, a M P , ... [ 1 ab M = [ d , c , b , ]). a The generators of the conformal algebra of SO(2,4) comprises of the local translations ( Regarding the action of the theory, the indicated choice according to the standard procedure of For this specific purpose, the alternative choice of the four-dimensional de Sitter group is In a nutshell, as far as the transformations of the fields and the calculation of the tensors is Another successful translation of a gravity theory into the gauge-theoretic context is that of 12 , 11 is written down as expansion on the algebra generators: where the Lorentz transformations ( they satisfy is determined by the following commutation relations: [ building gauge theories is an action oftions. Yang-Mills type, However, gauge this invariant is under not ISO(1,3) the transforma- caseEinstein-Hilbert, since, which such we a already choice, would knowward lead that route to is is an the necessary action to not desired be identical one. followed.more to the Along For convenient these this to lines, reason, instead gauge less of themechanism the straightfor- induced Poincaré SO(1,4) by group, group it the and is presence rather of incorporate a a scalar spontaneous field in symmetry the breaking fundamental representation [ preferred to the onetwo of cases, the in Poincaré the because, de Sitterspontaneous although case symmetry the breaking the number due generators to are of the distributedto the scalar on the generators field equal translations leads footing. and is to results equal In the toGauss-Bonnet this in breaking a term context, of more with the the constrained vanishing part theory torsion. associated involvingand The the is resulting Ricci actually action scalar the of along Einstein-Hilbert with the action. a theory is Lorentz invariant concerned, Einstein’s gravity can bethe described dynamical as part, that a is gauge theinstead theory action along of of the with the theory, Poincaré an is group.breaking extra obtained which by scalar In leads making field, to turn, us the the of correct the presence Einstein-Hilbert de action. of Sitter group which2.2 induces Gauge-theoretic appropriate approach spontaneous of Weyl gravity Weyl gravity theory. In this case,that the group is to be SO(2,4). gauged isfield the The strength four-dimensional tensor) transformations conformal are group, of obtained straightforwardly.is the The of fields gauge Yang-Mills and invariant type action the and thatImposition is curvature is of considered these tensors broken constraints by (derived recover imposing from the particular desired the scale conditions invariant on Weyl action the [ curvature tensors. Noncommutative Gravity obtained after replacing equations ( PoS(CORFU2019)236 ) ], ], ), ab M 46 µ 10 ( ω , R 2.13 (2.17) (2.14) (2.18) (2.13) (2.15) (2.16) 9 , 8 can be fixed G. Manolakos µ b . , b ] a , ν M b f D ε a ε b µ a µ , [ . µ f b f i f a K . ε + K + a a + b a ] µ a a . K µν ν P , K a i f e , ε ε R K bc bc a ] µ ε µ µ δ ] + δ [ ε + b ν , e ib ac + ac ] b ] µ 4 A . Furthermore, the field ν − ν µν M c + , D A µ ε ω − µ R ω D b Pb b cd b ac ε A ε µ µ δ + [ µ [ + [ [ Kb i , f e ab ε + ε , ω db ] µ ab and ab µ − i 4 ν ab + + ab δ ] Pa ω ∂ µ a M ] ] i b ] ε ν ω ) lead to the following expressions of the various µ ν ν 4 M + ω a ν ab A = e i b b ca µ − b 0 along with an additional constraint on the e µν ab f µ a a µ P a [ [ ε µ µ − R [ ε M µ ab [ 2.17 ∂ µ [ ω M a f e ε + f 2 1 µ D 2 ε ) = D 2 1 ε b + + ie + a P Ka + + = µ µ 4 a ( = ε a ] ] ] + a ab ] a ie ie ν ν ν R P a µ µ ν + ) and ( f µν 2 4 e b a e P A ω µ µ µ R a µν ab [ [ [ ε + + P µ ˜ M − ∂ ∂ ∂ [ R δ ε a a ε 2.16 2 2 ∂ 2 P D K µ ε ε ε = = ∂ µ µ µ ε 1 2 ∂ ∂ ∂ ) = ) = ) = ) = µν P K D R = 0. Therefore, taking all the constraints into account, the initial action M = = = = ( ( ( ), is transformed under the following rule: ( a a a a µ µ µ µ ab ab µν b µν b f µ e µν R µν R δ δ R 2.12 δ R δω . ): 1 2.14 in terms of the rest independent fields a µ ) and ( f Besides the above breaking of the conformal symmetry, there exists another breaking route via constraints [ 1 2.12 tensor. If these constraints are solvedand algebraically, they yield expressions of the gauge fields Then putting together equations ( specifically the torsionless condition, to the specific gauge of which leads to a Lorentz invariant action, more explicitly the Einstein-Hilbert action. component curvature tensors: takes the form of the well-knowntransformations Weyl action, which is invariant under diffeomorphisms and scale ( Noncommutative Gravity in which it is evident that eachare generator identified admits a as gauge gauge field. fields TheThe vierbein of gauge and potential, the the ( spin theory, connection as they were in the previous case of Einstein gravity. The transformations of the various gauge fields is obtained by the combination of equations ( The choice of theSO(2,4) action symmetry. of This the symmetry theory breaks is after the of imposition Yang-Mills of type specific that constraints is [ gauge invariant under the in which the inclusionvalued of in a the gauge Lie algebra transformationform: of parameter the has SO(2,4), taken therefore place. it is The written parameter down in is the following expansion In turn, the field strength tensor is determined by the relation: and is also expanded on the various generators: PoS(CORFU2019)236 (3.5) (3.6) (3.1) (3.2) (3.3) (3.7) (3.4) 0. In = a , while in X , then it is ) a 1 δ ( G. Manolakos X , therefore an , is introduced U is given by: a G φ ) = a X G , that is ( λ G matrices. It is worth-noting , is obtained: a P A × P , ] . a . , φ ) A φ a , , . X ) ] X , ( λ φ ) a X a a φ on a fuzzy space, depending on the noncom- ( φ X ) A , ) ( λ ] + [ λφ X a ). a λ + λ ( λ X X a 5 , ( λ 6= 2.1 X a ) = φ X ) = [ φ φ [ ) = ≡ a ) a is simply a function of the coordinates, ) = φ φ a − φ ], that is the noncommutative analogue of the covariant X ) a ( X as the gauge connection of the theory. Putting together ( φ ( ( X δ = X δ 53 a ( λ ( a a A with gauge transformation parameter δ δφ λ A X δ δφ , i.e. the group including all hermitian . The field belongs to a representation of a gauge group ) matrix, then it can be viewed as a gauge transformation of the non-Abelian a P X ( ), the gauge transformation of the connection, P U × 3.6 = P ), ( G ]. The spontaneous symmetry breaking could be viewed as a generalization of the case is a 3.5 ) 47 X ( λ In this section we include the basics regarding the construction ofIn gauge theories noncommutative on geometry, noncom- gauge fields arise in a very natural way and are intertwined From our point of view, such an outcome could be produced after employing an alternative Let us now begin with considering a field equations ( Eventually, the covariant coordinate is defined as: where it is straightforward to identify In case the transformation parameter that the coordinates are invariant under transformations of the gauge group, which is satisfied in case that: gauge group infinitesimal gauge transformation considered as an infinitesimal Abeliancase transformation and the gauge group is turn, let us perform a gauge transformation on the product of a coordinate and the field: by its transformation rule: mutative spaces, since it is fundamental for our purposes. with the notion of covariant coordinate [ 3. Gauge theories on noncommutative spaces muting coordinates symmetry breaking mechanism, induced bySO(2,4) two [ scalar fields in theof fundamental representation the of de Sitter grouprepresentation breaking of SO(1,4) down (see to previous the section Lorentz group by a single scalar in the fundamental Noncommutative Gravity derivative as we will stress later. The above transformation is not a covariant one since, in general, it holds: Drawing ideas from the methodologydefined of for ordinary similar gauge reasons, in theories, the in noncommutative which case, the covariant covariant derivative coordinate, is PoS(CORFU2019)236 . 3 = ] (3.8) (3.9) (4.1) (4.2) MN 37 η G. Manolakos , as it is the one we . Accordingly, the , 4 2 c φ dS c ab C − ] b φ , a φ can be be defined as an embedding as a gauge field 4 = [ 4 a S c dS A A , , . c ab ] 2 and C ) was fixed to an antisymmetric tensor, then R mn 4 ab θ − F i = ] , 4.2 dS b being a rescaled Levi-Civita symbol. Identified λ N A 6 x ] = , n a M = [ mnr X x A 4. Drawing lessons from the well-known fuzzy sphere , C ab ], where the same problem is encountered in their construction of m F MN X 50 ] + [ δ [ η , a ,..., , is defined as follows: A 1 49 , , with ab b r = F X X [ n r , − mn m ] C b A = assigned in an N-dimensional representation, the right hand side in , a . The construction of its fuzzy analogue is achieved if the coordinates, ) X is the metric tensor of the five-dimensional Minkowski spacetime, mn ) 2 [ ] 1 θ ( 34 ≡ + MN , SU η 1 ab F + , 1 4 and + , ], in which the corresponding coordinates-operators are corresponded to the (rescaled) 1 25 + ,..., , , 0 1 ] is to make use of a group of larger symmetry, in which all generators and the noncommu- ), has to be identified with a generator of the underlying algebra. Such an identification 24 = − We proceed with the construction of a model of four-dimensional noncommutative gravity as Here we construct the fuzzy version of four-dimensional de Sitter space, For more details see [ For more information on this issue, see [ 37 ( 2 3 , 4.2 N , are considered, instead of numbers, to be operators that fail to commute with each other: , m 36 a gauge theory on aspace covariant we fuzzy defined in space. order to First, build we the need theory on to4.1 it. present Four-dimensional the fuzzy four-dimensional covariant spaces fuzzy of which is easily proven to be covariant under a gauge transformation, The above scheme will begauge used theory in on fuzzy the covariant following space. sections in the construction of4. gravity model Four-dimensional as noncommutative gravity X use as the background space inin our the model. five-dimensional The Minkowski continuous space: M diag where the spacetime indices are case [ otherwise, if for instance the right hand side in eq( the fuzzy four-sphere. eq( tativity would be included in it. Targeting minimal extension of the symmetry, we are led to adopt ensures covariance, i.e the Lorentz invariance would beinterested, violated. such an identification However, cannot in be theSince achieved, covariance since fuzzy is the de an algebra indispensable Sitter would property not case[ of be in the closing space which [ we we are are building, a promising suggestion three generators of corresponding field strength tensor, Noncommutative Gravity giving an a posteriori explanation of the interpretation of PoS(CORFU2019)236 , ) 5 ( (4.6) (4.3) (4.5) (4.8) (4.9) (4.4) (4.7) (anti- (4.10) (4.11) SO ) 6 ( , the sym- , has been ), in which ) λ 5 SO G. Manolakos , 2.1 1 ( SO space, i.e. the ) 4 . , np ) dS h Θ 1 2 BC , J mq , the various component gen- mn = δ 4 ]. AD , Θ ) δ and therefore the spacetime ob- − 56 m ) 2 p 54 J p P ¯ h , − X λ P 2 mq mn i , ¯ h λ 51 4 Θ mn m Θ mn AD i , δ J P δ notation np ¯ , respectively, satisfying the following h ) λ δ 2 − 37 ] = BC , − 4 ] = and n space, respecting covariance, is obtained n mn δ n ( m − h P = P X , . 4 m Θ , X − ) 6 P m 2 m mp SO 4 mp mp ¯ m h ( dS , P X λ AC δ δ J [ [ Θ ]. To facilitate the whole construction and the m i J ( ( . and SO 4 ¯ ¯ nq X h h , 45 7 0 i i δ . To describe this space an approach similar to the BD [ m ⊃ m 4 δ P 6, which obey the following commutation relation: ) X ) + S ] = , , 5 , 1 5 ] = ] = ] = λ ( + h h , m nq , mn np np 1 mn X SO mn m = ,..., Θ ( BD Θ . Thus we have resulted with the Euclidean version of the Θ Θ δ P Θ J 1 5 2 ) [ ⊃ , , ¯ , h ¯ mp h m 6 i ) SO λ m m h AC J = δ ( 6 [ i P ( X ( δ [ [ ( B ¯ , h i ] = i , SO SO n ] = mn A P n , Θ ] = ] = X ¯ m 1 h , pq X m CD [ J = X Θ , with , [ , AB mn AB mn J J [ Θ [ 4. For reasons of correct dimensionality, an elementary length, , that is a covariant fuzzy 4 dS ,..., 1 group. Therefore, a fuzzy version of = ) 5 n , , 1 ( m The embedding path we consider is In the procedure of the construction of Einstein gravity as gauge theory (section Explicit formulation of the above fuzzy space is performed by considering the SO 4 Writing down these generators as decompositionserators in are identified as follows: commutation relations: 4.2 Gravity as gauge theory on the four-dimensional fuzzy covariant space the isometry group (the Poincaré group)gauge was group the is one the that isometry was group gauged, of accordingly the in Euclidian this version case of the the fuzzy symmetric) generators, J introduced in the abovenoncommutativity identifications. tensor are The denoted coordinates as and momenta operators as well as the It is notable to referdimensional matrices that representing the the above operators algebra, in contrast to the Heisenberg algebra, admits finite- calculations we adopt the Euclidean signature, which means that instead ofcovariant the fuzzy Noncommutative Gravity the after the symmetry is enlarged to the metry group we consider is that of where tained above is considered to be a finite quantum system [ fuzzy sphere is possible, specifically wematrices. will make use of the approximation by finite-dimensional PoS(CORFU2019)236 (4.12) (4.13) 4 matrices , × G. Manolakos b < a , b Γ a a Γ ) i 2 iP bc 3 − − M σ ad and ⊗ ] = ), 4. δ ] = ) 5 b 1 D . Therefore, it is deduced that the 4 − Γ Γ , σ 4 ( , 2 1 a Γ . a ad = 3 K − [ Γ 1 SU M [ Γ 3 i 2 4 Γ = ⊗ bc , Γ , , 3 δ − 1 ), with the generators being 4 D 5 1 a ab , σ ) Γ − 2 Γ ab = ab 4 iP a , , produce operators that, in principle, are not iM σ ( = δ δ ac = a ) i Γ ab 2 ) 5 5 U i Γ 5 2 ⊗ ) c M ] = ( Γ c 2 1 ] = 1 Γ = K ∼ − P ] = D bd b 8 σ ( , b } δ P , ab SO = ab a ) b P , = 1 δ = δ a , a 1 X + Γ a [ a ( P 2 , ⊗ − [ − a K bd Γ 2 b U [ b ]). Specifically in the current case, the anticommutation , Γ σ K M P , { × D , , 17 ) ac ac ac ab 1 = a ab 6 δ δ δ (or the fundamental of . σ 4 ( δ ( ( ( iM i iK i i 0 i ) group. In general, in noncommutative gauge theories, anticom- Γ ⊗ 6 ) ( SO 1 6 ] = ] = ] = ] = ] = ] = ] = σ ( b b to SO D bc bc cd ab P K matrix is defined as ) = , , , SO a M M 5 M M a 5 1 a , , ] (see also [ ( , , P X [ Γ Γ a a K [ U(1) gauge group are identified as the following matrices: D ab [ P 39 [ K [ . SO [ × , M 1 [ 38 4. The -matrices is determined by tensor products of the Pauli matrices: Γ ,..., 1 = b , a Aiming at the specific expressions of the matrices representing the generators, the Euclidean -matrices are employed, satisfying the following well-known anticommutation relation: Γ in the spinor representation of where The form of the e) one U(1) generator: mutators of the generators of theour algebra previous play works a [ key role, as we have already explained in detail in c) four generators of the local translations: P d) one generator for special conformal transformations: D Noncommutative Gravity viewed as a subgroup of the relations of the generators of theelements gauge of group, the algebra.generators The are treatment assigned to and then this include pathology allerators operators is into produced to the by fix the algebra, anticommutators the extending ofeventually representation the the led in gen- to initial extend which Lie the algebra. Following the above reasoning, we are generators of the SO(6) a) Six generators of the Lorentz transformations: M b) four generators of the conformal boosts: K It is straightforward to calculate the commutation relations of the various operators: After the determination ofa the noncommutative commutation gauge relations theory of can the be generators, performed in the a process straightforward of way. building First, the covariant PoS(CORFU2019)236 ) m × × 1 A ( ) ) 6 6 U ( ( (4.20) (4.19) (4.18) (4.14) (4.17) (4.15) (4.16) , since × m SO SO ) A 6 ( . This leads 4 G. Manolakos . SO a dS . P 1 ⊗ ⊗ is the dimension of ) ) m X X a N ( ( a a + ˜ ξ K ]. D + ⊗ 45 ) ⊗ ab X m Σ ( ˜ a a m ⊗ of the fuzzy space + b ) , a m , X . 1 K , that is taking values in the X ( . ) + ε mn ⊗ ) ab ⊗ matrices, where X ˆ Θ ) ( λ a mnp X m , 2 ( ˆ N X matrices. ] ab b ¯ h + ( m λ ) yields the transformation rule of m H i Σ × m ˆ A + N D X a 4 , 4 matrices. Concluding the reasoning, all terms ⊗ − N + ab mnp ⊗ ] ε ) ˆ 4.16 + [ × × n Σ 1 ) i 9 X ˆ H X D ( X N is a 2-form gauge field, valued in the algebra of ⊗ ⊗ , ( = ) ab 0 m ⊗ m m ˜ ˆ m ε ) X X Trtr mn X ˆ ) and ( ω X , but, at the same time, it is valued in the ( X + m B δ ( = = took place in order to make the field strength tensor co- ab = [ + a m X m a m K ˜ 4.15 B a ω ˆ P mn mn X ⊗ S + + ), ( ⊗ R B ) a ) . The X P X ( ( a mn 4.14 ⊗ field affects the form of the action of the theory, with an addition of a m ξ ) matrices), B e mn X + ( N + 1 a B m 1 ) = follows the same pattern with the × e ⊗ X m ⊗ ) N ( + A 0. In accordance with the commutative case, the transformation rule of X m 1 ( mn is covariant by construction and this property is expressed as: A 0 = ⊗ Θ ε m m ˆ m X = = X X ε δ = is the transformation parameter of the gauge theory, which is a function of the noncom- mn , meaning that they have the form of ) ˆ m m Θ ˆ X X X U(1). The introduction of ( . The presence of 5 ε × algebra, besides that it is a function of the coordinates . The component gauge fields that were introduced above are functions of the space coor- More information on this issue is given in Appendix A of the original paper of [ In turn, the covariant coordinate is also given as: The connection ) ) 5 1 1 ( ( variant a kinetic term of the form: in which SO(6) The next step is tocase, calculate the the field field strength tensor strength is tensor defined of as: the gauge theory we are building. In our Noncommutative Gravity coordinate of the theory is defined as: it holds that implies that it can be interpreted as the connection of the gauge theory. to the following relation: the representation of thegenerators, coordinates. the tensor product In is turn,dimensions, used, since among because generators of are the represented the by gauge fact 4 that fields the and factors their are matrices corresponding of different dinates, of the expression of the gauge connection are 4 The coordinate in which various gauge fields have been introduced for every generator of the algebra of where algebra. Therefore, it can be written as a decomposition on the sixteen generators of the algebra: Combination of the equations ( mutative coordinates ( U U PoS(CORFU2019)236 ), ]. In when 4.19 (4.22) (4.23) (4.21) (4.24) (4.25) 45 a m b in terms of = G. Manolakos a ab m m e ω a , K } . ] m ⊗ ˜ a ) , f gh a [ X m a ( e . a { 1 i mn = matrices and stands in the place R ⊗ − N can be viewed as a second vielbein ] f gh ) + , a a a m × m a X tensor. This way, a breaking of the e ( b P , N ) abc f gh mnrs m mn 1 ⊗ δ ( ε ) R D } [ 1 , 3! U X + rs ( − 0 is achieved. However, simply by counting the a R D mn ) = , ˜ 1 R ⊗ ) = } ( mn ) 10 P + nb U ( X R e space is gauge invariant. For details see Appendix A of the first and ( , ab { a × 4 mn Σ ˜ ) ab mn R m dS ˜ 4 R ⊗ Trtr ( ω ) { + = i X SO ( f ghd − : ε S ] ab 6 0 leads to: . The explicit expression of the spin connection in terms of the mn cd n m abcd R ˜ a ω ε , ) = , b m P m = ) = ( a e ˜ , [ X R a ( abc f gh m ]. e δ mn abcd ε 45 symmetry to the R ) 1 ( U × ) field strengthis a covariant tensor, therefore the above action is invariant under a gauge 6 ]. ( 45 mnp ˆ SO . A brute force way to achieve this is to render every tensor not related to the Lorentz ) Nevertheless, the desired action of the theory is that of Lorentz, in Euclidean signature, the A Yang-Mills action of a theory on the fuzzy Let us now focus on the dynamics of the theory determining the action. In our case it is natural Belonging in the underlying algebra, the field strength tensor of the gauge connection, ( H 4 6 ( the independent fields, other fields is obtained after employing the following two identities: solving the constraint. This choice leads to an expression of the spin connection degrees of freedom, adoption ofthis the reason, above we breaking perform would the lead symmetry to breaking an imposing overconstrained less theory. straightforward For constraints [ of the theory, would renderTherefore, the in theory order to as avoid bimetric, the which bimetric is interpretation, it not is what preferable we to desire consider in the this case. SO subgroup as vanishing, with minor exception of the where Tr denotes the traceof over integration. the coordinates The tr which denotes are the trace over the generators of the algebra. initial which is also encounteredscribed in as the gauge theories. cases Now, of the fact the that Einstein the and gauge field conformal gravity theories when de- paper of [ turn, the first condition is the torsionless condition: can be expanded in terms of the component curvature tensors: 4.3 The action and symmetry breaking by constraints to consider one of Yang-Mills type Noncommutative Gravity The transformation. Having defined all necessary elements, thecurvature transformations tensors of can the now gauge be fieldsthe obtained. and first The the paper explicit of component expressions ref.[ and calculations can be found in Solution of the constraint PoS(CORFU2019)236 , 4 dS gauge (4.26) (4.29) (4.27) (4.28) would ) ω 4 ( G. Manolakos SO . } n ˜ a , a . m e ) . { ) } mnp = m H } ˜ a , nb ] mnrs a e 0, because this choice would lead mnp n ε , ˜ e H rs = { ab . 3 4 m R ] a ], translating the argument developed bc µ [ ω e nd + mn 0, will, in turn, modify the expression of { δ 47 e ˜ R [ , = 4 m ) ] + abcd ) and then varying with respect to the inde- 6 m ε D + ( [ nd a e ). , 4.28 SO and mnrs m abcd mnpcd ε 11 ε D gauge symmetry down to the desired 4.23 [ b H m ]. On the other hand, the field that can be fixed in a ) e ab 6 abcd ] 13 3 4 ( ε abcd a m mnp ε cd e = SO , H rs − . This fixing, ˜ ( m 1 3 R ac b m n decouples being superheavy and also the gauge theory is just m D ab a [ e + ω m mn 3 4 tensor, related to the noncommutativity of the space, is not con- − a R ) part of the initial theory remains unbroken in the resulting theory − ( 1 ) ( ] = = 1 2Tr cd ( mn n ): ac R n U ), producing an even more simplified expression of the spin connection = ω , ω b m 4.24 S e 4.27 [ , that is the covariant coordinate of an Abelian noncommutative gauge theory. m 0 we do not apply the above argument, a abcd ], one could make use of the argument that the vanishing of the field strength ε + ) = 55 m P X ( ˜ R = m , will vanish in case the commutative limit of the broken theory is considered. In this limit, m D . An alternative way to break the a ) Now, after the symmetry breaking, i.e. including the constraints, the action will be: In this review we presented a four-dimensional gravity model as a gauge theory on a fuzzy ver- 4 ( sion of the (Euclidian version) four-dimensional de Sitter space. The constructed fuzzy space, in the case of conformalspontaneous symmetry gravity breaking to induced the by noncommutative thethe scalars above framework. by would the also Our imposition lead expectation of to is the a constraints that constrained ( theory the as pendent gauge fields would produce the equations of motion. 5. Conclusions consists a four-dimensional covariant noncommutative space. In turn, although the initial gauge in terms of the vielbein: The expression of the spinaccount connection the in identities, terms ( of the independent fields is given after taking into to a degenerate metric tensor of the space [ Eventually, writing down the explicit expressions of the component tensors as well as the tensor in a gauge theorythe can case lead of to the vanishing of the corresponding gaugegauge field. in which However, it in is set tothe zero spin is connection, the ( ˜ Noncommutative Gravity where According to [ SO be achieved with thetwo induction extra of scalar a fields spontaneous in symmetry the breaking 6 by representation including of in the theory field in terms of the non-vanishing gauge fields, ( The above equation leads to the following two: sidered as vanishing. The after the breaking as itfield, is still a theorythe on noncommutativity a is noncommutative relieved, space. However, the corresponding It should be noted that the PoS(CORFU2019)236 ) 1 ( U × ) 5 , 1 G. Manolakos ( SO (1961) 212. 2 (1956) 1597. 101 (1982) 976. 25 (1982) 988. (1976) 393. 25 (1977) 1376]. 48 , we were led to extend it to 38 ) 4 , 1 ( 12 SO (1980) 1466. doi:10.1103/PhysRevD.21.1466 , that is 4 21 dS “Lorentz" symmetry, we were led to impose certain conditions (constraints) ) 3 , 1 ( (1977) 739 Erratum: [Phys. Rev. Lett. 38 SO doi:10.1103/PhysRevD.25.976; E. A. Ivanov and J.Theories. Niederle, 2. “Gauge The Formulation Special of Conformal Gravitation Case,” Phys. Rev. D Of Gravitation Theories.,” E. A. Ivanov1. and The J. Poincare, Niederle, De “Gauge Sitter Formulation and of Conformal Gravitation Cases,” Theories. Phys. Rev. D doi:10.1103/PhysRevLett.38.1376, 10.1103/PhysRevLett.38.739 doi:10.1103/PhysRevD.25.988 Rev. Lett. Holonomy Group,” Phys. Rev. D doi:10.1103/PhysRev.101.1597 doi:10.1063/1.1703702 Torsion: Foundations and Prospects,” Rev. Mod. 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West, “Spontaneously Broken De Sitter Symmetry and the Gravitational in order to break thetions symmetry of of the the constraints, initial gauge theare group. action obtained. takes The After its latter the final will breaking, be formthe including included and application the in with of solu- future variation work. the the It symmetrywe equations should breaking, constructed be of reduce the noted to motion expressions the that, ones at of ofbe the the the point also conformal nnoncommutative before gravity highlighted gauge in that the theory the commutative limit.teraction above Last, in is it a the should matrix high-energy modelpromises regime which for where gives improved noncommutativity insight UV into can properties theinclusion be as of gravitational considered compared matter in- and fields to is also ordinary a gravity. giving subject of The further latter, study. asAcknowledgements: well as the We would like to thankIvanov, Larisa Ali Jonke, Chamseddine, Danijel Paolo Jurman,manuel Aschieri, Alexander Saridakis, Thanassis Harold Kehagias, Steinacker, Chatzistavrakidis, Dieter Kelly Evgeny Stelle, Lust, Patriziadiscussions. Denjoe Vitale and O’Connor, Christof One Em- Wetterich of for useful usGerman (GZ) and has States been Governments, at supportedand from the within the Institute the Excellent for Excellence Grant TheoreticalLAPTh Enigmass Initiative - Physics, of funded Annecy LAPTh. Heidelberg by and MPI University GZ the - would Munich like for to their thank hospitality. theReferences ITP - Heidelberg, for the sake of theThen, inclusion in of accordance the to anticommutators theof of standard the the fields procedure, generators and we in the movedrespecting a the component on fixed curvature by respresentation. tensors. calculating the Since transformations we were aiming to result with a theory Noncommutative Gravity group was the isometry group of PoS(CORFU2019)236 (2009) (2009) (1988) 46. 0906 G. Manolakos 0906 311 (1999) 032 gravity: ? ) 9909 3 , 2 (1985) 233. 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