PoS(CORFU2018)096 https://pos.sissa.it/ ∗ [email protected] [email protected] [email protected] Speaker. First, we briefly review thedimensions. description Specifically, of we gravity recall theoriesthree the as and procedure gauge four in theories dimensions which in are theapplied three recovered results for in and of the a four General case gauge-theoretic Relativity of approach.gauge in the theories Also, Weyl on gravity, the noncommutative too. procedure spaces,models is Then, we are constructed after review as reminding our gauge briefly theories most on the recent noncommutative formulation works spaces. of in which gravity ∗ 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 2018 "School and(CORFU2018) Workshops on Elementary Particle Physics and31 Gravity" August - 28 September, 2018 Corfu, Greece G. Zoupanos Physics Department, National Technical University, GR-15780Institute Athens, of Greece Theoretical Physics, D-69120 Heidelberg,Max-Planck Germany Institut für Physik, Fohringer RingLaboratoire 6, d’ D-80805 Annecy Munchen, de Germany Physique Theorique,E-mail: Annecy, France P. Manousselis Physics Department, National Technical University, GR-15780E-mail: Athens, Greece G. Manolakos Physics Department, National Technical University, GR-15780E-mail: Athens, Greece Noncommutative Gauge Theories and Gravity PoS(CORFU2018)096 ]- 1 G. Manolakos ]. However, the 1 ]). The gauge fixing 6 , 5 . Proceeding in the same , 1 4 ] 8 , 7 ] (see also [ 3 ] who modified the above consideration, adopt- 2 1 ]. A very improtant feature of this framework is the potential 35 ]-[ 13 ]. The first part of the construction, that is the calculation of the transformation ]. 9 12 See also [ Another contribution in this aspect is related with the gauge-theoretic approach of Weyl gravity Moreover, in the absence of cosmological constant, 3-d Einstein gravity can be also described An appropriate framework for the construction of physical theories at the high-energy regime Three out of four interactions of nature are grouped together under a common description by 1 ]. Pioneer in this field was Utiyama, whose work was focused on describing 4-d gravity of 12 regularization of quantum field theories and theing construction quantum of field finite theories theories. on Nevertheless, noncommutative build- spaces is a tedious task and, moreover, problematic (and supergravity) as a gauge theory of the 4-d conformal group [ of the gauge fields (dreibeincase. and However, the spin dynamic connection) part and isaction less the is tedious curvature recovered than tensors after that the is ofidentical consideration the similar to 4-d of to the case. a the 3-d Chern-Simons The 4-d Einstein-Hilbert’s 3-d actionan action. Einstein-Hilbert functional, ISO(1,2) Thus, which Chern-Simons 3-d gauge is, theory. Einstein in gravity fact, is precisely equivalent to as a gauge theorySitter of groups, the SO(1,3) 3-d and Poincaréis SO(2,2), group, present respectively are ISO(1,2). [ employed, in In case turn, a the cosmological 3-d constant de Sitter and Anti de (Planck scale), in which commutativitynoncommutative of geometry the [ coordinates cannot be naturally assumed, is that of [ of this scalar field led toTherefore, a the spontaneous 4-d symmetry gravitational breaking, theory recovering of thetheory General Einstein-Hilbert with Relativity action. the was successfully presence described of as a a scalar gauge field. spirit as in the previousvarious cases, curvature the tensors transformations are ofof obtained. the Yang-Mills gauge type, The fields as action it and isalong is the with expected. determined expressions gauge of to Then, fixing the be constraints ofunderstood are SO(2,4) the that imposed gauge fields, Weyl on gravity invariant the admits the final a curvature gauge-theoretic action tensors interpretation is and of actually the the conformal group. Weyl action. Therefore, it is General Relativity as a gauge theory, localizing the Lorentz symmetry, SO(1,3) [ ing the inhomogeneous Lorentz groupalong with (Poincaré the group), spin ISO(1,3), connection, the asertheless, vierbein the were the also gauge dynamics identified group as of gauge in Generalgauge-theoretic fields which, origin Relativity of of the remained the theory. unretrieved, Nev- Poincaré since gaugeaction. group there that was Solution would no be to action identified this asYang-Mills of problem action the was (instead Einstein-Hilbert of given the with Poincaré thefundamental one) representation consideration along of of with the an the gauge involvement SO(1,4) of group, gauge a SO(1,4) invariant scalar [ field in the the Standard Model in whichteraction they are is described not by part gauge of theories.of this picture, General However, the admitting relativity. gravitational a in- In separate,an order geometric undertaking formulation, to in that which make is gravity contact admits the among a theory gauge-theoretic the approach, two besides the different geometric pictures, one there [ has been Noncommutatve gravity 1. Introduction results were not considered to beconvincing successful, way. since A the few inclusion years of later, the it vielbein was did Kibble not [ happen in a PoS(CORFU2018)096 , ], ]. 3 λ ]). R 76 59 23 U(1), , ]-[ × 22 49 , ) in a fixed 21 ]. Motivated C G. Manolakos 78 ] allows one to ]. 36 ]. Such approaches, 70 48 ]-[ ]. In general, formula- , ] (see also [ 67 63 47 20 ]). Despite that, the frame- ], we considered a noncom- ]. The authors of the above U(2) gauge group, in a fixed 19 × 65 45 ]-[ 42 ] and, specifically, for 3-d models, ], but also [ ] and [ ]. 41 62 18 46 , ]-[ 61 37 , 60 2 ] (see also [ 17 ] for field theories on this space), which is actually the 75 ]), in which the corresponding background space is the ], a noncommutative deformation of a general conformal field 73 , 66 72 , that is the 3-d Minkowski spacetime foliated by fuzzy hyperboloids -product formulation, noncommutative gravitational models can be 2 , ? 1 λ ] who were based on Yang’s early work [ R 66 ] (see also ref. [ ] (see also [ -product and the Seiberg-Witten map [ 74 ? 71 ]. Along these lines, in ref.[ Second, a 4-d gravity model as a noncommutative gauge theory is constructed [ However, besides the Also, taking into account the above correspondence between gravity and (ordinary) gauge In this proceedings contribution, our recent contributions in the above field of noncommutative 65 ]. In this case too, the initial gauge group, SO(1,3), is eventually extended to GL(2, , 77 64 3-d Euclidean space foliated by multiplethe fuzzy above fuzzy spheres space of admits different an radii.Noncommutativity SO(4) As implies symmetry, the explained which enlargement in is of ref.[ in the factrepresentation, SO(4) the to in gauge the group order U(2) we that considered. theLorentz anticommutators analogue of of the the generators abovecommutative space close. construction is was the In also the explored,[ same in spirit, which the the correspondingrepresentation, non- for the same reasonsposed as is in a functional the of Euclidean Chern-Simonsaddition, case. the type commutative and In limit its is both considered, variation signatures, retrieving produces the the the expressions equations action of the of pro- 3-d motion. Einstein In by gravity. Heckman-Verlinde [ mutative version of the 4-dLorentz de invariance. Sitter space, Some which ofspace, SO(1,5), is the are in generators identified fact of as afinal its the covariant noncommutative gauge algebra coordinates. fuzzy group of As space, is in the preserving for a the final the minimal previous symmetry 3-d extension same case, group of reasons. the ofdetermined the and this According initial, their to SO(1,4), gauge the specifically transformationsare and standard the obtained. their SO(1,5) procedure, Eventually, corresponding an the component action curvature corresponding ofbreaks tensors Yang-Mills gauge imposing type certain fields is conditions employed are (constraints) and on its the initial curvature gauge tensors symmetry and the gauge fields. The work of noncommutative geometry isparticle considered physics to models, be formulated a as suitable noncommutative gauge background theories for [ accommodating use it as methodology for thehave construction been of models considered of before, noncommutative gravity.employing see Such the approaches for Chern-Simons example gauge refs. theoryworks formulation, to [ which see we [ refered makeformulation use of of the constant noncommutativity (Moyal-Weyl) and also use the constructed using the noncommutative realization of matrixspecifically geometries for Yang-Mills [ matrix models, were proposedAlso, in for the alternative past approaches few on years, the seetion refs. subject of see [ noncommutative [ gravity impliesinvariance. that However, there the exist noncommutative specific deformations noncommutative deformations break whichinvariance the preserve and the Lorentz Lorentz the corresponding background[ spaces are called covariant noncommutativetheory spaces defined on 4-d dS or AdS spacetime has been employed, see also [ introduced in ref. [ theories, the well-established formulation of noncommutative gauge theories [ gravity are included. First, wemutative gravity briefly [ review our proposition for a matrix model of 3-d noncom- Noncommutatve gravity ultraviolet features have been encountered [ PoS(CORFU2018)096 ]- 1 (2.6) (2.7) (2.4) (2.2) (2.3) (2.5) (2.1) and the a P G. Manolakos . 0 , b c , ] = µ b a , ω P , , ) M ac b a a ν ω µ ac P , µ [ λ ( e ω , b ω bc µ ab + , 3 and determine the algebra through ] c + b ν e , a ε c b b νρ c a 2 λ ω , P P ν , ]. Specifically for the dynamic part, the R a µ λ a 1 − + ω ξ A 12 µ abc b − ac bc ε e = ν µ µ b = e + [ ξ a ω ω ε abc µ ab c a ] = ab µ ε ∂ µ − b , λ ω 3 ω a M = ab + µνρ µ , − M + , with ε a ε a ω a ab P a µ µ x bc [ ν µ e 3 ξ λ D ∂ ν M ω d µ µ , ∂ ∂ ∂ − = c + Z abc − µ a ab ε = = 2 1 M a ν P A ν a a µ ω e ab = δ abc = µ µ e µ µ a ε e δ ∂ ∂ M δω = EH3 = = ] = S µ b a ab A µν M µν , T R a M [ Let us begin with the 3-d case, in which Einstein gravity is precisely described by a Chern- The outline of the present contribution is as follows: First we briefly recall the gauge-theoretic In this section we briefly review the gauge-theoretic approach of various gravity theories [ The ISO(1,2) algebra is generated by six operators, i.e. the three local translations, ], which consists the basis of our works in which the whole scheme is translated to the frame- 12 and the component curvature tensors: lead to the calculation of the gauge transformations of the component fields: According to the gauging procedure, theas: gauge connection and gauge parameter are written down where the vielbein andabove the expressions spin along connection with the are transformation identified rule of as the the gauge gauge connection: fields of the theory. The [ three Lorentz transformations, the following commutation relations: work of noncommutativity. 2.1 3-d Einstein gravity Simons gauge theory of the ISO(1,2), Poincaré group [ 3-d Einstein-Hilbert action is written down as: approaches of gravitational theories mentioned above.about Then, the we include construction the of necessary gauge information suggestions theories for in noncommutative gravity the models noncommutative in framework. threedown and Next, our four conclusions we dimensions. and review Eventually, we comment our write on the results. 2. Gauge-theoretic approach of gravity Noncommutatve gravity commutative limit of the modelfields reduces and the the obtained ones expressions of of the gauge tensors transformations to those of of the the conformal gravity. which, as it will be explained, is identical to a Chern-Simons functional of ISO(1,2). PoS(CORFU2018)096 ). 2.1 (2.9) (2.8) (2.12) (2.10) (2.11) are the ab M G. Manolakos , 0 ] = , and Lorentz generators, b a P P , ] a ν P are the generators of the local [ A . , a : , µ P ] ab c A , is defined and it is expressed as a µν . P µ M R b ) [ A + [ ab x a ( µ η M A . 2 ab ab ν µ µν ∂ ab ω ]. Here we mention briefly the main features R ] = M − 6 1 2 2 1 bc ν λ ]-[ + M A + 4 1 , a µ a a P ] = ∂ P ) P b a [ x P ( , µν , a ] = a T ] ν µ P [ b e ] = D d , µ M µν ) = c [ D R x a [ ( µ η , which correspond to the translations, = [ 4 A is the metric tensor of the 4-d Minkowski spacetime, ab µ µν ) ω R ] = 1 , cd 1 , M 1 , , 1 ab − M [ ( diag = ab η First, for this gauge-theoretic approach of 4-d gravity, the vierbein formulation of General Now, gravitational interaction in four dimensions is described by General Relativity, which , respectively. The consideration of the vierbein as a gauge field implies the mixing between and the spin connection, ab a µ Also, consideration of a Chern-Simons action functional leads to the Einstein-Hilbert action, ( generators of the Lorentz grouptranslations. (Lorentz transformations) According and tocase the described standard in gauging the previous procedure,decomposition section, on that the the is gauge generators potential, of the the one Poincaré algebra, followed as: in the 3-d The functions accompanying the generators of the algebraas in the the above gauge decomposition fields are identified ofe the theory and, specificallyM in this case, they are identified as the vierbein, Relativity has to be considered. Thegroup), gauge group since is reasonably it chosen to is becorresponding the the algebra ISO(1,3) isometry (Poincaré satisfy group the ofalgebra: following the commutation 4-d relations, Minkowski which spacetime. in fact The determine generators the of the of this gauge-theoretic approach of the 4-d Einstein’s gravity. consists a solid and successful theory, havingis passed many geometric, tests since differentiating its it early days. fromAiming Its the at formulation rest a interactions, connectiongauge-theoretic which between approach are gravitational of described and gravity as took the place gauge other [ theories. interactions, the undertaking of the In addition, 3-d gravitationaltheoretic approach. theory In this with case, the cosmological gaugeand groups SO(2,2), constant considered respectively, are depending is the on 3-d the also dS signof or of described AdS the the groups, constant. in gauge SO(1,3) The theory a procedure is ofbecause gauge- the the of construction the same, difference obtaining in results thenon-zero: right that hand generalize side the of the above commutator of of the the translations, ISO(1,2) which case, is now 2.2 4-d Einstein gravity Noncommutatve gravity starting from the defining relation of the curvature two-form, and expanding it on the generators of the algebra as: where PoS(CORFU2018)096 (2.18) (2.14) (2.19) (2.20) (2.16) (2.17) (2.13) (2.15) ) in the 2.18 G. Manolakos ) and ( 2.12 , . , ab b c , b ] . M ν µ b ) ] ] e ω ab ν ν b ω . e a ( ac ] A M , , ) µ λ ab ) lead to the expressions of the transfor- ab cb [ ] x µ µ µ ε ( [ ω − A µν , 2 ω b ω ab µ R c 2.13 ξ 2 a λ + [ A − 2 1 [ ] ab 2 1 − λ ν ab µ + a + [ ] 2 A ] + ν a ω 5 ε ν µ a P [ − ω e µ ) ) with ( P + ∂ µ µ ∂ e ) [ [ ab 2 a ( x ∂ ∂ a ξ λ ( = 2 2 a 2.14 µ µ µ µν ) = ∂ ∂ ξ A R A ) = ) = ( = = δ e ω a ) = ( µν ( ab ) and ( a x ) = µ ( µ R e A ab ε µν ( δ R µν 2.12 δω µν R are the curvatures associated to the component gauge fields, identified R ) ω ( ab µν R ], that is also included in the theory. The choice of the 4-d de Sitter group is an is the gauge transformation parameter which is also expanded on the generators of 5 and , ) ) 3 x e ( ( ε a = µν ε R ) leads to their explicit expressions: An alternative way to obtain an action with Lorentz symmetry, is to impose that the action is 2.17 where as the torsion and curvature, respectively. Replacement of the equations ( Combination of the equations ( ( Moving on with the dynamicMills part type, of the being theory, invariant the underthe most the Einstein-Hilbert reasonable action, ISO(1,3) choice is gauge which anthe is group. action initial Lorentz of However, action invariant Yang- the has and, aim to therefore,symmetry be is breaking, the broken to induced Poincaré to by result symmetry the a with of the Lorentz. scalar SO(1,4) field This [ which can belongs be to carried the out fundamental through representation a of spontaneous invariant only under the Lorentz symmetry and not under the total Poincaré symmetry with which alternative and preferred choice tocan that be of considered the on equal Poincaréthe footing. group, translational The since generators, spontaneous all symmetry resulting generators breaking to ofinvariant leads action a the to involving theory algebra the the with Ricci breaking vanishing scalar of Hilbert (and torsion action. a constraint Concluding, topological and Einstein’s Gauss-Bonnet 4-d a term), gravity Lorentz that theorytheory is is but not Einstein- to equivalent to an a SO(1,4) pureappropriate Poincaré gauge potential gauge term theory in with the the Lagrangian, which inclusion leads of to a a spontaneous scalar symmetry field breaking. and the addition of an mations of the gauge fields: Noncommutatve gravity the internal symmetry andordinary spacetime gauge making theories. this The gauge kind connection of transforms construction according to special, the compared following rule: to the The corresponding field strength tensor of the gauge theory is defined as: and is expanded on the generators as it is valued in the algebra: the algebra: where PoS(CORFU2018)096 (2.22) (2.25) (2.21) (2.23) (2.24) . G. Manolakos c µ e , ) ), the conformal c ] c b ab ρ δ e a M [ σ , ∂ K ) 2 − ab c M σ ] = e c ]). − ρ K , ∂ . 11 , D a ( , a b ab ab K K σ a δ 10 a M µ e ( , [ K f a ] 2 ε ρ ε e + , , + µ 1 2 ] = c D ] b D A b µ − D K δ b , ε ) a + [ [ a a µ + P P + ε ] (see also [ e [ 2 µ 9 ν ab ab ∂ , ∂ M 8 , 6 M ] = term, which is not the correct one, since the tar- = − , a c ab ), the Lorentz transformations, ( ab µ 7 a P ε K 2 a ν M , ) ω µ e P ε − µ ab 1 2 D M 2 1 ∂ ( M ( + [ = R ] = + b a ν a D µ P e , P , a A µ a a 2 1 ). The algebra of SO(2,4) is determined by the commutation ] ] ε P c e b K δ ε D [ δ − = d ) [ = a [ b µ µ , ε e M a A ν P 4 ∂ − ] = ] = b ν 4. According to the gauging procedure, the gauge potential of the theory is D cd e , a µ M ... P , ∂ [ 1 ( ), is: ab a ν = M e [ d 1 2 2.23 , ) and the dilatations, ( c = a , b K is a gauge transformation parameter valued in the Lie algebra of the SO(2,4) group and for , ab µ a ε ω Besides the above, also Weyl gravity admits a gauge-theoretic formulation, specifically of the Let us start with the identification of the generators of the conformal algebra of SO(2,4). The 4-d conformal group, SO(2,4). In this case,of too, the the curvature transformations tensors of are the obtained fields in andis a the straightforward an expressions way. SO(2,4) The invariant action action that of isthe Yang-Mills considered type curvature initially which tensors. breaks After by the theis imposition consideration identical of of to constraints the the on constraints, scale invariant the Weyl resulting action [ action of the theory get is the Einstein-Hilbert(zero) action. of the coupling Also, constant such ofincludes gravity. an In a order action dimensionful to coupling would result constant, with implystraightforward the the way, the Einstein-Hilbert action that action, wrong has is which to dimensionality the betensor) construction considered of of in Lorentz the an invariants theory. alternative, out non- correct of The one, the one ensuring quantities that the (curvature is correctRicci built dimensionality scalar by of and the certain the corresponding coupling contractions action constant, of is and eventually the the is curvature Einstein-Hilbert2.3 identified action. tensor as 4-d is the Conformal the gravity leading to Weyl or Einstein gravity fifteen generators are the local translations, ( where defined and is expanded on the various generators as: However, straightforward consideration of anwould action lead of to Yang-Mills type an with action Lorentz involving symmetry, the boosts, ( relations of the above generators: in which a gauge fieldthe has spin been connection associated are identified with as each gaugeconnection, generator. fields ( of In the this theory. case, The transformation too, rule the of the vierbein gauge and where this reason it can be written as: Noncommutatve gravity one begins. This means that theimposition curvature tensor of related the to the torsionless translations condition, hasaction to that with be Lorentz is zero, symmetry. that a Solution is constraint of the with that this the constraint is vielbein: leads necessary to for a resulting relation of with the an spin connection PoS(CORFU2018)096 (2.28) (2.29) (2.26) (2.27) ], that is the G. Manolakos 9 , 8 , . The inclusion of 7 . , 2.2 b ] a , ν M b f D ε a ε b µ a µ , [ in terms of the independent . . The two constraints admit µ f b ) f a i f a K µ ], being a generalization of the ε f + K M + a + ( b a ] µ a 80 a . µν ν R P , K i f e ε ε R bc bc and a µ µ µ δ ] + δ [ + b ν e ib ab ac ac ] b µ ] 4 A ν − ν µν M c + , ω ε ω − µ R ω Pb b cd b ac A ε µ µ δ µ [ + [ [ ), leads to the expressions of the transforma- Kb i , f e ab ε , ω db ] µ ab ab − i 4 ν ab + + δ ] Pa ω µ M ] ] i 2.25 b ] ε ν ω ν ν 7 + ω a ν ab ]. From our prespective, the latter can be achieved A i b b ca µ − b 0 can be employed and, inclusion of all constraints e µν f µ a a µ P a [ [ µ µ 81 − R [ ε µ ab [ ∂ = [ ω M a f e + f 2 1 µ D 2 ε ) and ( µ ε b + + ie + a b ), the expressions of the component curvature tensors are Ka + + = µ µ 4 a ε a ] ] ] a ab ] a ie ie ν ν ν 2.23 P µ ν + f µν 2 4 e b a e 2.28 ω µ µ µ R µν ab [ [ [ ), ( + + µ ˜ M − ∂ ∂ ∂ [ R a a ε 2 2 ∂ 2 P D K µ ε ε ε = ∂ 2.24 µ µ µ 0 and an additional constraint on ) and ( 1 2 ∂ ∂ ∂ ) = ) = ) = ) = µν P K D R M = = = = ( ( ( ) = ( 2.27 a a a a µ P µ µ ab ab µν ( µν b f µ e µν R R µν R δ δ R δ R δω . Also, the gauge fixing µ b and a µ e Besides the above breaking of the conformal symmetry which led to the Weyl action, it is Moreover, the argument used in the previous section in the four-dimensional Poincaré gravity an algebraic solution leading tofields expressions of the fields through an alternative symmetry breaking mechanism,fields specifically in with the the inclusion fundamental of representation two ofcase scalar the of conformal the group [ breaking ofscalar in the the 4-d fundamental de representation Sitter of group SO(1,4), down as explained to in the section Lorentz group by the inclusion of a possible to employ an alternative breakingmetry, explicitly route, the this Einstein-Hilbert time action leading [ to an action with Lorentz sym- in the initial action lead to the well-known Weyl action. case, that is anfor alternative this way case to of break conformal the gravity. initial Since symmetry it to is the desired Lorentz, to can result be with generalized the Lorentz symmetry out of Combining the equation ( obtained: torsionless condition, two scalars could trigger aresulting spontaneous action symmetry would breaking be the in Einstein-Hilbert adetails one, theory on respecting with this Lorentz matter issue symmetry. will Calculations fields be and and included the in a future work. tions of the gauge fields of the theory: Noncommutatve gravity Combination of the equations ( Regarding the action, at firsttial, it SO(2,4), symmetry is gets taken broken to by be the an imposition SO(2,4) of invariant certain of constraints, Yang-Mills [ type. The ini- The field strength tensor is defined by the following relation: and is expanded on the generators as: PoS(CORFU2018)096 , a µ a e X , a (3.5) (3.6) (3.1) (3.4) (3.2) (3.3) (3.7) µ f 0. Now, = P Hermitian a G. Manolakos X × δ is a P ], and a gauge group, ) 36 X [ ( a λ X , is obtained: a are simultaneously set to zero, A ) , is introduced by its transforma- K a ( φ R is a function of the coordinates, . ] ) . a , , X φ ) and A ( φ a , , . ) X ) λ ] X , ( λ φ P ) a X a a ( φ ( φ X ) A , R ( , has to be vanishing. Setting these tensors to λ is: ] + [ λφ X ) a λ + λ ( ) λ X a M a 8 , λ 6= ( X a X ) = ( R 0. X ) = [ φ φ [ ) = ≡ a ) φ a ) = φ = φ a − φ X a ( X ( φ ( µ ( X δ = δ b ( λ a a δ δφ A X δ ]. Obviously, the results can be easily generalized in the cases 15 of the fuzzy sphere, written in terms of powers of ) ), the gauge transformation of the connection, ], it is argued that if both tensors a X 3.6 81 ( φ ), ( is the gauge transformation parameter. If ) 3.5 is identified as the gauge connection of the noncommutative gauge theory. Combining X ( a λ A Let a field Let us now briefy recall the basic concepts of the formulation of gauge theories on noncommu- Therefore, the covariant coordinate is defined as: where equations ( zero will produce the constraintsIn of particular, the in [ theory leading tothen from expressions the that constraints relate of the the gauge theory fields. it is understood that the corresponding gauge fields, of other noncommutative spaces, too. G. An infinitesimal gauge transformation of where then it is an infinitesimalmatrix, Abelian then transformation the transformation and is G=U(1), non-Abelian and whileinvariant the under if gauge an group infinitesimal is G=U(P). transformation The of coordinates the are the gauge group, G, namely which is satisfied when: tative spaces, in order to use themFor later convenience, for the the construction following of the methodology(fuzzy) noncommutative space, is gravity the models. performed fuzzy on sphere the [ most typical noncommutative 3. Noncommutative gauge theories Noncommutatve gravity the initial SO(2,4), the vacuummeans of that the every other theory tensor, is except considered for to the be directly SO(4) invariant, which are equal - up to a rescaling factor - and the gauge transformation of the product of the field and a coordinate is not covariant: since, in general, it holds: Drawing lessons by the construction ofdefined, ordinary in the gauge noncommutative theories, case, in the whichtion covariant a property: coordinate, covariant derivative is PoS(CORFU2018)096 (3.8) (3.9) (4.2) (4.1) , gives ]). There- 2.1 74 G. Manolakos ]). The coordi- 75 , c φ c ]. During the gauging ab C 76 − ] b ] (see also [ φ , 74 a φ = [ c ,.... 1 A , , k . c ab 2 ] X / C ]. 1 ab i jk , − F 77 0 ] , b λε λ i = A 9 , a = [ ] = A j , as it is explained in ref.[ , ` ab 3 λ ] X F ` , , first considered in ref.[ R i [ ] + [ δ 3 λ a X ⊕ [ R A , = b as a gauge field is demonstrated. Next, the field strength tensor, X [ a H − A ] , are the operators which satisfy the following commutation relation: b i X A , , a 3 λ X [ R , are allowed to be set in a reducible representation and this is equivalent i X ≡ ab F ]). 82 , satisfy the commutation relation of the SU(2) algebra, just like the coordinates of a 3 λ (cf. [ R 2 The solid framework for constructing noncommutative gauge theories, as described in the pre- The gauge group that is adopted for the construction of the theory is the SO(4), that is the In the Lorentzian signature, an analogous construction is encountered, specifically the foliation of the 3-d 2 , is defined on the fuzzy sphere as: ab vious section, combined with the description of 3-d gravity as a gauge theory, section which is covariant under a gauge transformation: In the next sections,gravity the models as above gauge methodology theories. is applied on the construction4. of A noncommutative 3-d noncommutative gravity model parametrization of the symmetry of the Minkowski spacetime by fuzzy hyperboloids of different radii [ fore, the coordinates, to a sum of fuzzy 2-spheresdiscrete of foliation different of radii. 3-d Thus, Euclidean the spacefoliation noncommutative by space fuzzy can 2-spheres, be each viewed as fuzzy a sphere being a leaf of the procedure, the typical problem ofalgebra the is non-closure encountered. of The indicated the solution anticommutators forthe of this group problem the is to generators to accommodate pick of the a the specific generators representation and of enlarge the algebra to the minimal extend, including rise to the construction ofappropriate noncommutative a space, 3-d on which model the ofA noncommutative noncommutative suitable gauge theory gravity. 3-d is fuzzy First, constructed space on. fuzzy one is spheres has constructed of to by different identify the radii, called an foliation of the 3-d Euclidean space by multiple Noncommutatve gravity In the above relation, the role of nates of and are described by matrices in reducible representations of the algebra of SU(2) (cf. [ F fuzzy sphere do. However, unlikecoordinate the operators case are of related, are the not fuzzyresentation, accommodated sphere, in but the an in irreducible generators a (higher-dimensional) of reducible rep- SU(2), one.that to The the which employment coordinates of can a be reducible expressedsome representation as irreducible of matrices representation, SU(2) in corresponded means a to block-diagonalHilbert a form, space with fuzzy would each be: sphere block of being certain radius. Therefore, the The three coordinates of PoS(CORFU2018)096 , , ] ] a a (4.9) (4.5) (4.6) (4.8) (4.3) (4.7) (4.4) µ µ (4.10) (4.11) (4.12) (4.13) e e , , 0 0 ˜ ε ε [ [ i , i 2 ) G. Manolakos ρ ] + a A ] + µ a . µ , + . ω ρ ω , c ρ ˜ 1l A , 0
X 0 ˜ ρ M ε . ab } ( [ ε µν c i δ [ ρ i
5 ν 2 1 2 abc µν γ } ε λε ω c i i , = ] + ν ⊗ ] + λε b e i µ − } µ µ µ . , ˜ b ] = ˜ A
e b A A − 5 ω ] b , , µ ] γ M a { a a + ν , M ν ω 2 ξ λ ˜ a , ⊗ e 1l [ A { [ , i a , i , 0 i 2 M a ρ 2 ε e . µ + µ M } − { ⊗ [ ˜ , ] c A } + X ω + + µ [ ν ] µ c i . ρ 1l e ˜ ν A µ , A i , , abc ˜ , abc − µν A ω 5 b νρ c ] + [ + ε 0 ] , , ε ⊗ γ µ a P ε a } a 0 b } R 0 e [ λε ν c ˜ ν ε ab µ c i i ε µ { [ M ξ e δ ω i abc ω λ 1 2 , , 1 2 ε { − , + , b X i ⊗ a ] + a ] − b µ a µ , µ a ] ν µ µ e = e , µ a [ 10 a M ω e abc ρ µνρ { ω abc [ e µ ] = } X a { i ε e , i ε b ε ρ b 1 2 , ⊗ i 2 ω 4 a ρ + 2 2 µ U(2) in a fixed representation, due to the inclusion of , ω a M M a µν λ − a Tr + ] + , ρ , [ λ P − × X i a a a ] + λ µν ] + ] µ [ a λε 2 P P = a + i abc µ ⊗ i [ abc µ ν ω { ˜ 4 = [ ε a S λε e e ε A µ , i − , , , } P ] + } e ν c a , a c ν ν
µ , µν − A ] ] + λ ⊗ µ ξ + c A e A a , µ
, 1l [ a R 1l ] i ω + µ b ˜ b M i A + + , ξ ab µ e µ µ ν , a , − ˜ δ ν ν a A ⊗ X ω abc ω a ] ν ξ = 1 2 [ , X X [ ε { i to the SO(4) set of generators. { µ ν [ [ ξ a i i ω i i ε ν [ 2 2 [ X A 5 i 2 = e − γ [ − − ] } − + − = ] ] ] = a ] + ] + b ] ] ] + − a ν b ν ν a a ν µ P ν 0 0 ] ˜ P ω e X A , ˜ ˜ ξ λ ε ε ν , A , , , , a , , , , ˜ X a , A P µ µ µ µ µ µ µ µ a P , µ { [ A A A A a A A A A µ ω e [ + + + + [ + + + + µ µ µ µ i µ µ µ µ i 2 2 X X X X X X X X [ [ [ [ [ [ [ [ i i i i i i i i + + − − − − = = = = a a = = = = µν µν µν µν ˜ a a After the determination of the fuzzy space and the gauge group, the procedure that is followed µ µ F F µ T R µ ˜ e A A δ δ δ δω Also, a gauge transformation parameteralgebra is as: introduced and it is expanded on the generators of the The above relations combined with the transformationfollowing rule transformations of of the the covariant gauge coordinate, fields: produce the the corresponding component curvature tensors are obtained: Also, making use of the definitionthe of field the strength covariant tensor: coordinate in the following defining relation of Next, in order to write downduced the for expression each of generator. the Therefore, gauge the connection, covariant coordinate a is gauge defined field as: has to be intro- is a modification of the onegeometry. of At the first, continuous the case, commutation adjustedgroup and in are anticommutation the obtained: relations framework of of the noncommutative generators of the gauge Noncommutatve gravity the operators that are produced byinterest, the the anticommutators. final Accordingly, in gauge the group specifictwo is SO(4) more case the operators of 1land U(2) Finally, the action that is proposed is of Chern-Simons type, specifically: PoS(CORFU2018)096 . 4 = is a (5.1) (5.2) MN (4.14) η mnr C cannot be space, with mn G. Manolakos 4 θ , where r X r mn C = . ]. In order to facilitate the ), should be identified as a 0 78 mn 5.2 θ = µν ˜ F is: 4 0 , , = 2 R mn µν θ i = F N x ] = 11 . , n 4. Analogy to the fuzzy sphere case, in which the M 0 X x . However, preservation of covariance is necessary for 2.1 , 3 = m ] MN ,..., X a [ η 1 ], where the same problem emerges in the construction of the fuzzy µν 66 R 86 = , n , 68 , is the metric tensor of the 5-d Minkowski spacetime, 0 m = MN a η µν T . In order to obtain the fuzzy analogue of this space, one has to consider ) 1 + 4 and , is defined as a submanifold of the 5-d Minkowski spacetime and can be viewed 1 , to be operators that do not commute with each other: 4 + m , ,..., X 1 0 + = ]. , 1 N 78 + , , 1 M − For more details on this issue, see [ In this section, the construction of a 4-d gravity model as a noncommutative gauge theory is The 4-d background space that is employed in this case is the fuzzy 4-d de Sitter space, dS ( 3 four-sphere. generator of the underlying algebra, ensuring covariance, that is construction we make use of the Euclideansymmetry signature, therefore, group instead is of considered the to SO(1,5), the be resulting that of SO(6). our purpose, therefore a group with larger symmetry,but in also which all the operators noncommutativity identified tensor as coordinates cansymmetry be leads included to in the it, is consideration considered. of The the enlargement SO(1,5) of the group. Therefore, a fuzzy dS rescaled Levi-Civita symbol. However, inbe this achieved, in fuzzy the de sense Sitter thatSO(1,4) case, such would such break an Lorentz an identification invariance, identification of since cannot the theidentified algebra as coordinate would generators operators not into with the be generators algebra closing, [ of i.e. its coordinates being operators represented byis N-dimensional matrices, obtained respecting after covariance, the too, enlargement of the symmetry to the SO(1,5) [ reviewed. First, the constructionis of constructed, an is appropriate presented 4-d andexplored fuzzy then [ the space, features on of which the the gravity gravity model model on5.1 this 4-d Fuzzy fuzzy de space Sitter are space 5. A 4-d noncommutative gravity model Noncommutatve gravity The equations of motion are obtainedgauge after fields: variation of the above action with respect to the various The continuous dS as the Lorentzian analogue of the definitionspace. of the Specifically, four-sphere the as defining, an embedding embedding in equation the of 5-d dS Euclidean where its coordinates, It is worth-noting that the resultsto of the ones the of above the construction continuous of case noncommutative in 3-d section gravity reduce diag where the spacetime indices are corresponding coordinates are identified asN-dimensional the representation, three implies (rescaled) generators that of the SU(2) right in hand an (large) side, ( PoS(CORFU2018)096 (5.3) (5.5) (5.6) (5.7) (5.8) (5.4) (5.9) (5.10) (5.11) G. Manolakos , the same pattern is 4 ) . , np ) h Θ 1 2 BC , has been introduced in the J mq λ = δ AD mn . δ − 56 m Θ , in section J P 2 − ) 2.3 3 λ ¯ h ) mq 2 , p λ ¯ h p R i λ Θ m AD X P i 4 . In analogy to the 3-d translation of J P np 4 ¯ h mn mn λ δ 2 BC . δ δ ] = ] = δ fall into the general class of fuzzy spaces, 5.1 m − n h − = . − , 4 P X − n n 6 , m 0 2 mp P X ¯ m h m X AC λ J Θ [ P i J [ mp mp ] = nq 4 , δ ]. δ , δ BD 12 ( ( m , mn h ¯ 85 δ ¯ h h X i , therefore the kind of spacetime obtained above is a i + ] = Θ ]), the above algebra admits finite-dimensional matri- mn 1 mn λ + , ]. h mn nq δ , Θ h 83 [ ¯ 6, which satisfy the following commutation relation: h ] = ] = = 84 m 2 Θ Θ i BD ¯ h P , J 5 λ np np [ i mp m Θ 67 Θ AC J δ ] = ,..., and , , ( δ , n 1 ( ¯ m m , h P ] = m i i P , n 66 X P [ = [ mn m X , , B X Θ ] = m ] = [ , m ¯ 1 h X pq A X CD [ J = Θ , , space of the previous section, mn AB mn J J 4 , with [ Θ [ BA J , respectively. The coordinate and momentum operators satisfy the following 4. For dimensional reasons, an elementary length, − mn Θ = ,..., AB 1 and = m n P , , m For a string theory approach on such a model, see [ In this section we review the construction of a noncommutative 4-d gravity model as a gauge For the explicit formulation of the above 4-d fuzzy space, let us consider the SO(6) generators, m 4 X The above generators can be writtengenerators as being a identified decomposition as in various an operators, SO(4) including notation, the with coordinates: the component that is the fuzzy covariant spaces [ followed, this time translating the 4-d case presented in section theory on the fuzzy dS finite quantum system. Spaces like the above fuzzy dS 5.2 A noncommutative gauge theory of 4-d gravity above identifications, in which the coordinates, momentaas and noncommutativity tensor are denoted The algebra of spacetime transformations is: In contrast to the Heisenberg algebraces (see [ to represent the operators the gauge-theoretic description of gravity on the fuzzy space commutation relations: denoted as J Noncommutatve gravity with PoS(CORFU2018)096 U(1) (5.13) (5.14) (5.12) × , G. Manolakos b space, namely 4 < a , b Γ 4 matrices: a × Γ i 2 3 matrices − σ . Therefore, the generators of 4 4 and ⊗ ] = × Γ , is encountered in this case, too 5 b 1 3 4 4 Γ Γ σ . Γ , 2 1 2 k . a , = Γ σ − Γ 1 k 1 [ 3 i σ 4 ⊗ Γ Γ = ⊗ 3 , , 3 ⊗ − = 5 1l σ , σ 1l 5 2 Γ 2 1 = ab 1 2 a , Γ σ = δ a − Γ ab 2 5 i = Γ 2 ⊗ 4 matrices (in the 4 representation of SO(6)). Γ j 2 1 = 1 = × − k Γ σ 13 i } , = Γ b = 1 Γ 4 = a space was constructed and its symmetry group was Γ i a 2 Γ 2 , ⊗ 4 i a 2 Γ 2 − Γ σ − { = , 1 = = i j σ 4 k M 4 Γ ⊗ M 1 matrix is defined as σ 5 Γ = 1 Γ ]). Specifically, the anticommutation relations of the generators of the 39 3 and: 4. Also the , 2 , 1 ,... 1 = -matrices are employed, satisfying the following anticommutation relation: U(1) gauge group are identified as: ] (see also [ , in which the isometry group (the Poincaré group) was chosen to be the gauged, in this = Γ k × , b 73 j , 2.2 , , i a -matrices are determined as tensor products of the Pauli matrices, specifically: 72 However, the same problem related to the anticommutators of the generators of the algebra In order to obtain the specific expressions of the matrices representing the generators, the four In the previous section, the fuzzy dS Γ , 71 where that emerged in the 3-d case of noncommutative gravity, section the SO(5), viewed as a subgroup of the SO(6) group. group with the generators being represented by 4 where e) one U(1) generator: 1l. The Euclidean found to be the SO(6). Recallingsection the case of the construction ofcase Einstein the gravity role as of gauge the theory gauge in group will be given to the isometry group of the fuzzy dS gauge group, SO(5), produce operators that,troubleshoot in this general, problem, do the not representation of belong theduced to generators by the has the algebra. to anticommutators Therefore, be of fixed to the andfying (fixed) all generators them operators have as pro- to generators, be too. included Thus, into the the initial algebra, gauge identi- group, SO(5) is extended to the SO(6) c) four generators of the local translations: P d) one generator for special conformal transformations: D Noncommutatve gravity 5.3 Determination of the gauge group and representation by [ the SO(6) a) Six generators of the Lorentz transformations: M b) four generators of the conformal boosts: K Therefore, the generators of the algebra are represented by the following 4 PoS(CORFU2018)096 = m ) and X (5.15) (5.16) (5.19) (5.17) (5.18) N ma- δ × , 5.16 1l of the fuzzy G. Manolakos ⊗ transforms in . m ) m a X X , namely P A ( m m ⊗ X a ) X + ( ) a D , therefore they have the bc ˜ ξ m a ⊗ M X ) + iP ad X ab δ − ( Σ m − ˜ a ⊗ ] = ad ) D , 4N matrix. ) + M X a ( . X × bc K ( ) ab δ [ a λ X , K − ( 4 matrices. Therefore, it is concluded that , ] + ab m a ac m × ⊗ D ˆ A D ) X iP iM M ab ) , X + c δ ⊗ ) ε bd ( i [ c , is a function of the coordinates 1l ) K i a δ ] = ] = P , is expressed as: m 14 m X b U(1) algebra, which means that it can be expanded ab ⊗ b m D + A = ( ab ] = P ˆ δ , X 0 m × b δ , a ˜ m U(1) algebra. Therefore, it can be decomposed on the ε a bd P X − ˆ ) + X X , P − [ × b [ M + a X b δ , the = ( K a is obtained by the combination of the equations ( K P ac ε , [ , m K δ ac ab m ˆ ac ( D X δ ab i Σ δ A , ⊗ . ( ( ab i a i ) 0 ⊗ δ iM i ) X ] = iK ( ] = X ] = cd a ] = ] = ( ] = bc ξ bc b ] = M ab b ab , K M P m M D + M , , , , , ab , ω 1l a a a a a M D K X K P P + ⊗ [ [ [ [ [ [ [ a ) P X ( ⊗ 0 ) ε X = ( a m ε is the gauge transformation parameter. It is a function of the coordinates (N e N matrices, where N is the dimension of the representation in which the coordinates ) , but also is valued in the SO(6) × , but also takes values in the SO(6) X m 4 ) = ( X ε X ( ). In accordance to the corresponding procedure in the commutative case, the Having determined the commutation relations of the generators of the algebra, the noncom- m A 5.17 The property of covariance of the coordinate where Taking into account that a gauge transformation acts trivially on the coordinate 0, the transformation property of the such a way admitting the interpretationof of the the connection gauge of transformation the parameter, gauge theory. Similarly to the case mutative gauging procedure can be initiated. First, the covariant coordinate is defined as: 5.4 Noncommutative gauge theory of gravity ( space dS on its sixteen generators as follows: where it becomes manifestcomponent that gauge one fields gauge are functions field of has the been coordinates of corresponded the to space, each generator. The Noncommutatve gravity Straightforward calculations lead to the followingisfy: commutation relations, which the operators sat- form of N are accommodated. Therefore, instead ofcorresponding the ordinary generators, product, the between tensor thedimensions, gauge product since fields is the and generators used, their are since represented the by 4 factors are matrices of different each term in the expression of the gauge connection is a 4N trices), sixteen generators of the algebra: PoS(CORFU2018)096 U(1) × U(1). (5.24) (5.23) (5.20) (5.21) (5.22) × . 1l . G. Manolakos ⊗ 1l ) X ⊗ ( m a mn R + + D D ⊗ ⊗ m U(1) noncommutative gauge ˜ ) a × X + ]. ( would admit an interpretation of a 78 mn a K m ˜ R ), can be expanded in terms of the U(1) symmetry to the SO(4) b , ⊗ . × + a m mn a b 5.21 ˆ Θ ]. Accordingly, the first constraint is the field will contribute in the total action of K mnp 2 + ˆ ¯ 78 h ⊗ , λ mn H i ab ) 0 Σ B X − ( mnp ] ˆ ⊗ a ) = n ) mn ˆ H 15 X P R X , ( ( . The m a + 5 ˆ mn ab X Trtr ˜ a m R P ω = = [ ⊗ + is a 2-form gauge field, which takes values in the SO(6) B ) mn a X S P mn ( R a ⊗ B mn ) ˜ R X ( + a m . The e ab mn Σ + 1l B ⊗ ) + ⊗ X 1l m ( field was introduced in order to make the field strength tensor covariant, since in X ⊗ ]. ab mn = mn 78 mn R Θ m B ˆ X = field strength tensor transforms covariantly under a gauge transformation, therefore the ) = mn X ( ˆ mnp Θ ˆ mn Details on this generic issue on such spaces are given in Appendix A of [ The gauge symmetry of the resulting theory, with which we would like to end up, is the one The field strength tensor of the gauge connection, ( After the decomposition of the gauge connection and the introduction of the gauge fields, the H 5 R which is also encountered indescribed as the (ordinary) cases gauge in theories. whicha Next, the second the vielbein Einstein gauge of and field the conformal theory, gravity which theories would are lead to a bimetric theory, which is not our case of All necessary information for theexpressions determination of of the the component transformations curvature of tensorstions the is lie gauge in obtained. fields ref.[ and The the explicit expressions and calcula- torsionless condition: described by the Lorentz group, inconsider the directly Euclidean a signature, constrained the SO(4). theorynot in In be this which imposed direction, the to one only vanish could tors would component of be curvature the the tensors algebra, ones that achieving that would a corresponds breaking to of the the Lorentz initial and SO(6) the U(1) genera- 5.5 The constraints for the symmetry breaking and the action The next step is totheory, calculate which, the for field the strength fuzzy tensor de Sitter for space, this is SO(6) defined as: where algebra. The its absence it does not transform covariantly The above action is gauge invariant. component curvature tensors, since it is valued in the algebra: However, counting the degrees of freedom,constrained adoption of theory. the above Therefore, breakingthe would it lead symmetry is to breaking an more in over- a efficient less to straightforward follow way a [ different procedure and perform covariant coordinate is now written as: Noncommutatve gravity the theory with a kinetic term of the following form: PoS(CORFU2018)096 ]. 78 (5.30) (5.27) (5.28) (5.29) (5.25) (5.26) : G. Manolakos 6 . This fixing, m a . } n ˜ a , , a } m . e ] m . ˜ { a ) , f gh } a [ = m m a e } ˜ a { ), leading to a more simplified , i = nb ] a e n − , e ] f gh 5.28 { , a ab . a m , which is associated to the noncom- m ] e ) bc [ ω , 1l nd abc f gh { δ ( mnrs m e δ , ε D mn } [ m 1 ] + 3! in terms of the rest of the independent fields, R rs D − [ nd ab R e m = , , and ω m } abcd mn 16 ε D nb space is gauge invariant, for details see Appendix A of [ [ b R m e and ), the above equations lead to the desired expression 4 , { e ] 3 4 ab abcd a m m ε 5.25 Trtr e = ω , 0, because such a choice would lead to degeneracy of the − { ( = m i ac , decouples in the commutative limit of the broken theory. b = n in the solution of the constraint should be considered. Also, m f ghd D m − [ e a ε ω a S a ] µ m 4 3 − e b cd n − abcd = ω ε 0, one obtains: ]. The field that can be gauge-fixed to zero is the ˜ ] = , = a b m cd m = 12 n e ac e ) = n [ ], the vanishing of the field strength tensor in a gauge theory could lead defined on the fuzzy dS ω P , abc f gh ω 2 ( b 87 m δ F ˜ abcd R e being the covariant coordinate of an Abelian noncommutative gauge theory. [ ε m a abcd ). ε + 0, does not imply m X 5.24 ) = = . In order to obtain the explicit expression of the spin connection in terms of the other P ( m m ˜ ˜ R a D , 0, will modify the expression of the spin connection, ( m Alternatively, another way to break the SO(6) gauge symmetry to the desired SO(4) is to in- A Yang-Mills action tr Next, for the action of the theory, it is natural to consider one of Yang-Mills type 6 a = , a m m ˜ duce a spontaneous symmetry breakingSO(6), extrapolating by the including argument two developed for scalarmutative the framework. fields case It of in is the expected the that conformal 6would the gravity spontaneous to representation lead symmetry the of to breaking noncom- induced a by constrainedconstraints the ( scalars theory as the one that was obtained above by the imposition of the Therefore, in the commutative limit, the gauge symmetry would be just SO(4). Taking into consideration the identities, ( expression: mutativity, is not consideredpart to of be the vanishing. symmetry since TheHowever, the the breaking symmetry takes corresponding breaking place field, does in the not (high-energy) affect noncommutative regime. the U(1) 5.6 The action and equations of motion a metric tensor of the space [ for the spin connection: to the vanishing of thetensor, associated gauge field. However, the vanishing of the torsion component Noncommutatve gravity interest. Thus, the relation Next, according to ref.[ It should be noted that the U(1) field strength tensor, this leads to the expression of thee spin connection The above equation leads to the following two: fields, the following two identities are employed: Solving the constraint where PoS(CORFU2018)096 (5.31) U(2) and . × gauge field ) G. Manolakos ω mnp which is a foli- H 3 λ R mnp ˜ H 4 3 1 conformal supergravity + = abcd ε N mnpcd H ab mnp H N matrices (it replaces the integration of the 3 1 × + 17 mnrs ε ), then variation with respect to the independent gauge rs R 5.29 mn ˜ R 4 + ]. Gauging of the latter leads to mnrs 88 ε abcd ε cd rs R ab ]. In the present noncommutative context it seems more natural pursuing the gaug- mn R 91 ( , 90 2Tr , U(1). Then, we went on following the standard procedure for the calculation of the trans- = × 89 In this review the construction of noncommutative 3-d and 4-d gravity models as gauge theo- In the 3-d case, the noncommutative background space we employed is the In the 4-d case, the noncommutative background space we employed is the fuzzy version of the Another possible direction of further investigation is to consider our construction embedded Replacing with the explicit expressions of the component tensors and writing the , 7 S , 8 ing of the supergroup of SU(2,2/1) algebra. This possibility appeared to be fruitful in relating the [ ries were revisited. Although for boththe cases corresponding the works main in procedure the isthe continuous similar, two since regime, cases. they let are us based summarize on and conclude separately for fields would lead to the equations of motion. 6. Conclusions 4-d de Sitter space. Itrespecting is Lorentz worth-mentioning that invariance, it which consiststhe is a gauge of 4-d group, covariant major SO(5), noncommutative importance space, whichSO(6) in was our enlarged for case. the same Next, reasons we as determined in the previous case, to the in the Lie superalgebraspacetime of symmetry SU(2,2/1) group which [ is isomorphic to the algebra of the superconformal Noncommutatve gravity where Tr denotes the trace over the coordinates-N ation of the 3-d Euclidean spaceSO(4) by symmetry multiple which fuzzy is spheres. the one This we onion-likeof chose construction the as admits gauge gauge an group group. would The not anticommutatorsfixed close, of its the that representation. generators is why Then, we followingnate promoted the and the calculated standard symmetry the procedure, transformations to we of the the defined U(2) tensors. fields the and Naturally, the covariant the expressions coordi- of action the we componenttion curvature determined were was obtained of after Chern-Simons its typeones variation. and of The the the results equations commutative obtained case. of in mo- the above construction reduce to the formations of the fields and theto expressions result of with the a component theory curvatureto tensors. respecting break Since the the we Lorentz initial desired symmetry, symmetry.variation we After will imposed the lead certain symmetry to constraints breaking, the the in equationsbe action of order noted takes motion. that, its The before final latter form theones will and of be symmetry its part the breaking, of conformal the our gravity results future inabove is work. the of a commutative It the limit. matrix should above model Finally, construction giving itand insight reduce also should into giving be to promises the also the for gravitational stressed improved interaction thatlatter, UV in as properties the the well as the high-energy compared to inclusion regime ordinary of gravity. matter Clearly, fields the is going to be a subject of further study. continuous case) and tr denotesbreaking, the that is trace including over the the constraints, generators the of surviving terms the of algebra. the action After will the be: symmetry in terms of the surviving gauge fields, ( PoS(CORFU2018)096 38 (1988) 46. ] (see also G. 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B Acknowledgements We would like to thank AliIvanov, Larisa Chamseddine, Jonke, Paolo Danijel Aschieri, Jurman, Thanassis Alexander Chatzistavrakidis,Emmanuel Kehagias, Evgeny Saridakis, Dieter Harold Lüst, Steinacker, Denjoe Kelly O’Connor, Stelle,useful Patrizia discussions. Vitale The and work Christof of WetterichAction two for MP1405, of while us both (GM would andparticipation like GZ) in to was the thank partially Workshop ESI supported “Matrix - by ModelsTheory”, Vienna the for Jul for COST Noncommutative 09 the Geometry - hospitality and 13, during String funded 2018. their by One the of German us and (GZ)Heidelberg States has University Governments, been and at supported from the within the Institute the Excellentthe for Excellence Grant ITP Theoretical Initiative Enigmass - Physics, of Heidelberg, LAPTh LAPTh. - GZ Annecy would and like MPI to - thank Munich for theirReferences hospitality. Noncommutatve gravity Connes-Lott model [ PoS(CORFU2018)096 ˇ co G. 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