JHEP08(2020)001 Springer July 7, 2020 May 29, 2020

JHEP08(2020)001 Springer July 7, 2020 May 29, 2020 : . The latter August 4, 2020 : March 10, 2019 4 : : Revised Accepted Received Published , U(1). The latter deﬁnes the × a,b,c,d Published for SISSA by https://doi.org/10.1007/JHEP08(2020)001 [email protected] , and G. Zoupanos a . 3 U(1), and ﬁx its representation. Moreover, a 2-form dynamic gauge × 1902.10922 P. Manousselis The Authors. a c Non-Commutative Geometry, Gauge Symmetry, Models of Quantum Gravity

We formulate a model of noncommutative four-dimensional gravity on a co- , [email protected] Fohringer Ring 6, D-80805 Munich,Laboratoire d’ Germany Annecy de PhysiqueAnnecy, Theorique, France E-mail: George.Zoupanos@cern.ch Physics Department, National Technical University, Zografou Campus 9, Iroon PolytechniouInstitute str, of GR-15780 Theoretical Athens, Physics, Greece D-69120 Heidelberg, Germany Max-Planck Institut fürPhysik, b c d a Open Access Article funded by SCOAP relating the gauge fields.formulation of It the should above be programme. noted that weKeywords: use the Euclidean signature for the ArXiv ePrint: field is included instrength the tensor. theory for Finally, reasonsthe the of Lorentz gauge covariance group of with theory thefour-dimensional an is transformation extension noncommutative considered of of gravity to a action thewhereas U(1), be field which the i.e. can spontaneously SO(1,3) breaking lead broken induces to to the equations imposition of of motion, constraints that will lead to expressions a four-dimensional gravitational modelwe consider on the the fuzzy SO(1,4)we want de subgroup to Sitter of gauge spacetime. thegauge the theory SO(1,5) on isometry In algebra, such part turn, agroup, in of noncommutative first the which space the SO(1,5) directs we full us were to symmetry. led use an Then, to, extension the as of construction the gauge of a Abstract: variant fuzzy space basedrequires on the SO(1,4), employment of thatance. a is Addressing wider along the symmetry the fuzzy linesof group, of version the the formulating of four-dimensional SO(1,5), Poincarégroup, gravity the for as spontaneously dS reasons a gauge broken of theory covari- to the Lorentz, we attempt to construct G. Manolakos, Four-dimensional gravity on anoncommutative covariant space JHEP08(2020)001 9 ] the action 57 , 56 13 17 ]. Moreover, on the 19 51 10 – U(1) 7 1 × 4 9 ]. In particular in refs [ 61 – 54 – 1 – 16 1 15 ]. On the other hand a diffeomorphism-invariant gravity theory is obviously 53 , 52 4.1 Construction of4.2 the fuzzy covariant Noncommutative space gauge theory of four-dimensional gravity obtained is one with cosmologicalspacetime constant, to which which when we positive referdescribing defines gravity in the as the de gauge following. Sitter theory (dS) In on Then noncommutative addition, merging spaces the the is above prospect afinite two reasonable of ones approaches consequence. regularizing are by the quantum features field that theories, render or this even approach better, as a building promising framework. cutoff in field theoriesgravity defined side, on noncommutativity could thetivity be new [ used spacetime to structure resolveinvariant [ the with singularities respect of tojust general as transformations rela- in whose the parameters localtheory can are gauge be theories. functions formulated as Then, of a naturally, spacetime, gauge it theory has [ been long believed that gravity nates and the corresponding descriptionscales. of Spacetime spacetime noncommutativity, inspired to by holdquantum field the up theory, successes was to suggested of arbitrary as quantum small theirnative mechanics logical length framework extension and to providing describe an interesting physics alter- that close the to the natural Planck appearance scale. of As a a minimal bonus length it scale was could expected act as an effective ultraviolet 1 Introduction The aim to go beyondparticular, the there notion is of no classical obvious spacetime physical has reason been to with expect us that since the decades. commutativity of In coordi- A Gauge covariant field strength tensorB and gauge Calculations invariance of the field transformations and curvatures 5 The constraints for breaking6 the symmetry to The SO(4) action 7 Summary and conclusions 3 Four-dimensional conformal gravity as a4 gauge theory Four-dimensional noncommutative gravity on a fuzzy covariant space Contents 1 Introduction 2 Four-dimensional Einstein gravity as a gauge theory JHEP08(2020)001 . ] ], 2 ]). 2 74 , , 64 1 ] and [ – N ma- 73 ]. Still 67 × 62 14 , – 2 8 38 – 31 ] was the first to asso- 1 ] (see also [ 17 ], as an extension of the first – ]) is constructed from finite- 2 N matrices proportional to the ] in connection to their research pro- 15 66 × 78 , – 65 76 ]. This programme overcomes the ul- , at fuzziness level N-1, is the non- 2 F 50 – ]. 39 covariant noncommutative spacetimes 69 , – 2 – 68 ] (see also [ 5 ], which preserves the spacetime isometries. 1 ], respectively. This onion-like construction led to a matrix ]), we started a programme realizing gravity as a noncommu- ] and the difference from the fuzzy two-sphere case is that 75 72 67 , 71 1 ] (see also [ 70 Formulating gravity in the noncommutative framework is a rather difficult task, since In ref. [ We recall that a fuzzy two-sphere [ The construction of quantum field theories on noncommutative spaces is a difficult For more details aboutThis fuzzy idea four-sphere see has [ been also revived recently by other authors [ 1 2 ciate the position operators withinvariant discrete elements of specetime. a Lie Snyder’s algebra position in operators order to belong construct to a the Lorentz four-dimensional grammes. to follow a different strategy,noncommutative originally model suggested of by Snyder Yang [ [ common noncommutative deformations break Lorentzcial invariance. types However, of there deformations are thatwhich spe- constitute preserve the the spacetime isometries. More specifically, Snyder [ tative gauge theory in three dimensions.commutative spaces Specifically, based we on considered SU(2) three-dimensionaland and non- SU(1,1), fuzzy as two-hyperboloids foliations [ ofmodel, fuzzy two-spheres which [ was analysed in a straightforward way. In the present work, we would like to matrices. One canhave restate been this constructed difficulty inthe algebraically. [ commutators Algebras of of a thethe fuzzy source coordinates four-sphere of do the not difficultiesreferences to close therein). analyse in field theories the on fuzzy the four-sphere fuzzy four-sphere case. (see [ This is Then one can replace thetrices, functions and defined on therefore this algebras noncommutativea on manifold higher the by N dimensional sphere sphere becomea is, noncommutative. four-dimensional however, A sphere, not generalization the straightforward. to samewhich In procedure is particular, leads not in to a a the square number case of of of an independent integer. functions Therefore, one cannot construct a map from functions dimensional matrices andnoncommutative the manifold. size ofcommutative matrices The manifold represents fuzzy whose the coordinate sphere,generators functions number S of are of the N quanta N-dimensionalN-1 on representation for of the angular SU(2). momentum Introducing in a a cutoff two-sphere, parameter the number of independent functions is N are considered to betraviolet/infrared noncommutative (fuzzy) problematic [ behaviours ofA theories very defined welcome on featureknown noncommutative of higher-dimensional spaces. such theories. theories Evento is phenomenologically more that promising appealing they four-dimensional is unified are that theories. renormalizable, this contrary programme to results all task though and, furthermore,noncommutative problematic geometry ultraviolet has features been havedate emerged proposed particle [ as models an with noncommutative appropriateIt gauge framework theories is [ to worth-noting accommo- thattive a geometry very is interesting the development programme in in the which framework extra of dimensions noncommuta- of higher-dimensional theories JHEP08(2020)001 N N × × To facilitate 4 N matrices. In addition, using × ] considered the four-dimensional 79 , that we would like to construct here, 4 ]. 81 , 80 – 3 – ] introduced SO(1,5) in order to allow continuous 2 ]. 21 Such algebras are the SO(1,5) or the conformal, SO(2,4). The ], where the authors build a general conformal field theory defined 3 76 ], where the quantized version of Connes-Lott approach to standard N matrices in some higher-dimensional representation of the enlarged 82 × SO(1,4) in the present formalism, would have the same problem in its description by N ∼ N matrices, see e.g. [ × Finally, we should stress that field theories on noncommutative spaces are not quantum The next important point is that the coordinates and other operators, too, can be The SO(5) For an alternative approach (based on the star product) for the gauging of the same group in the 3 4 model is discussed. Similarly, a gravity theory on noncommutative spacetime ismatrices, not as in a the the- fuzzy four-sphere case. noncommutative regime to formulate gravity, see [ Lorentzian signature. It shouldmatrices. be We noted approximate fields that defined wethe on do manifolds tools by not of N approximate noncommutative geometry, manifoldsspace we by of formulate N N differential geometry on a particular theories, see e.g. [ scribe four-dimensional gravity by gaugingthe the SO(1,4) algebra subgroup we of construct SO(1,5). thewith a whole space model with in isometries given thethe by Euclidean SO(6) the symmetry signature. SO(5) and group, Therefore, extend, thenas we for we a start reasons gauge gauge of its theory covariance, to maximal on SO(5) the subgroup above to space. formulate The gravity results we obtain could be reformulated to the algebra. Of course ingebras with this operators procedure corresponding one toangular has various momenta observables to such etc. identify as the To coordinates,spacetime various realize momenta, in generators the the of above present the work picturegiven radius we al- of embedded consider in a the a fuzzy five-dimensional four-dimensional spacetime. covariant fuzzy noncommutative However de in our Sitter construction space we of de- a latter can be embedded. larger algebra, say SO(1,5)commutation contains relations all of the the generators latter of are the covariant as subalgebra seen SO(1,4)represented in and by the the larger N group. tended in other cases inis a that straightforward in manner, order as towider we formulate symmetry explained a containing above. fuzzy all covariant The noncommutative generatorsthe suggestion space, of fuzzy one the de has first to Sitter one. employ four-dimensionalinstead Then, a spacetime, of in dS using practice, in its the algebra case of of generators, SO(1,4) we consider a larger one in which the variant noncommutative spacetimes appear toinvariance, which overcome is the usually encountered problem in ofinvariance noncommutative breaking spaces. and, Lorentz They moreover, preserve they Lorentz a provide natural a further short-distance step,spacetime cutoff. we to promote In local the the symmetrya Lorentz present space. and invariance work, write of The as down covariant simplest noncommutative the covariant space corresponding is gravity the theory fuzzy on sphere, such which however cannot be ex- translations in Snyder’s framework and finallyconformal Kastrup [ algebra SO(2,4). Thework natural of framework noncommutative geometry, of with the possible above applicationsby in early Heckman-Verlinde [ string works theory, in was explored theon frame- covariant noncommutative versions of four-dimensional (A)dS spacetime. Therefore, co- de Sitter algebra, SO(1,4). Then, Yang [ JHEP08(2020)001 ]. ]. In 61 , 88 57 ] where 85 , ]. Therefore, 84 83 ]). Accordingly, four- 89 ] for the problem of emergent gravity. 87 , 86 – 4 – ] and kept the interest of the physicists for the next three 54 ]. Gauging the Poincarégroup leads to the Einstein-Cartan theory [ 61 – ), which can be interpreted as a new and non-trivial matrix model. To formulate ], a similar approach is followed gauging the de Sitter group and considering 55 56 6.1 The outline of the paper is as follows. In sections 2, 3 we briefly present the four- de Sitter group, SO(1,4) and introducelatter, a with scalar field the in its appropriateleads fundamental representation. potential, to The the induces Einstein-Hilbert aIn action, spontaneous ref. starting symmetry [ from breakingconstraints an which on action the of curvature Yang-Mills tensors type for the [ final action (see also [ An alternative way toto recover the approach celebrated the results wholebegan of concept in the from the theory a middle ofdecades gauge-theoretic [ general 50’s point relativity [ of is order view. to recover the This dynamics undertaking of Einstein’s gravity, one has to employ the four-dimensional mation property of the fieldrelieved, strength while tensor in on appendix suchof B, covariant the spaces we fields and give and how the the this calculations component can and tensors. be results of the2 transformations Four-dimensional Einstein gravity as a gauge theory the above space. Inthe section initial 5, symmetry we to the comment desirableand on one the comment — constraints the on employed Lorentz. the inwrite In obtaining our order section conclusions 6, to of and we break the stress proposesome out an equations the action details of main on features motion. of how our to Eventually, model. in deal In section with appendix A 7, the we we general give problem of the (non-) covariance transfor- note that in ourmutative present geometry. work For we thisespecially do in purpose, not the we aim beginning referof at of the to this a four-dimensional review introduction. review fuzzy articles space Ingauge of on throughout section theory the which 4, the and we framework we describe paper, formulate of present gravity the noncom- the as construction a construction of (noncommutative) the gauge theory of the isometry group of and probably therematrix are models connections are to studied. Matrix Also, see theory refs. approach, [ dimensional see Einstein e.g. and [ conformalfor gravity reference theories on for some reasons of of their completeness important but features also, in the next sections. However, we should the fact that gauge theory onit noncommutative is spaces natural is well to formulated. usethe Therefore, the noncommutative we gauge space, feel approach fields to areYang noncommutative space fluctuating gravity. as around In can the our be Euclideanaction formulation ( read version on by of the Snyder- covariantquantum coordinate. theory Using of gauge gravity theory on we noncommutative derive spaces the is an open and interesting project related to the finitedefined degrees or of even freedom) finite. andframework Eventually, hope of our that noncommutative aim the geometry is quantizedmannian but to version geometry the formulate is will generalization a not be in obvious gravity well andin theory this is the in framework under present the of paper, research by general we Rie- mathematicians try [ to give a more down to earth approach via gauge theory, given ory of quantum gravity. We have reasons to expect (in the present Euclidean approach JHEP08(2020)001 (2.7) (2.2) (2.3) (2.4) (2.5) (2.6) (2.1) , which ab µ ω , ] = 0 b ,P a , respectively. The gauge P [ ab . M , , . ] ab . b c µ ] are the Lorentz group generators P ab M e ν b ) b . [ a M x a ab ] , ) ( ,A η λ ] cb µ x M ab µ ( and the spin connection, , − A µ ω µ ab b a c ω ξ ] = 2 µ λ a A 1 2 + [ [ e 1 2 bc ab ] λ , and the gauge group is the Poincarégroup, ν µ + + [ 2 4 + ω A a ,M − µ a µ a [ P – 5 – + ∂ ) P P ∂ , which can be decomposed on the generators of ab [ ) a x µ ξ λ ( x = ( a µ µ A , a µ µ ∂ ∂ ] ξ ) = 2 e b ] = = d δA A , and Lorentz generators, ( a a M ) = ab ) = µ P c µν µ x [ x ). The resulting expressions for the transformations of the ( a ( R δe [ µ δω η 2.3 A ] = 4 cd ) with ( ,M ab 2.4 M ) is the gauge transformation parameter that is also expanded on the ten [ x ( is the (mostly positive) Minkowski metric, ) and ( = are the generators of the translations. According to the gauging procedure, one ab 2.2 η a P In the gauge-theory approach of the four-dimensional gravity, at first, one has to The corresponding field strength tensor (the curvature) of the gauge theory is defined as: The transformations of thesions component ( gauge fields arevierbein given and after the combining spin the connection expres- are: where generators of the algebra: The above gauge connection istransforms assigned according in to the the adjoint following representation, rule: which means that it the algebra, accompanied byin functions that this are case, identifiedcorrespond these as to the functions the gauge are translations, fields.connection the Specifically is vierbein, decomposed as follows: where and has to introduce the gauge potential, consider the vielbein formulation ofthe general four-dimensional Minkowski relativity. spacetime, The M gaugeISO(1,3). theory is The constructed choice on ofit the is Poincarégroup as the the isometry symmetrythe group following group of commutation is the relations: very Minkowski reasonable, spacetime. as The generators of the group satisfy ISO(1,3) gauge theory. Hereour we present recall project. theof whole Therefore, the idea in four-dimensional since this Einstein it section,complete gravity is we and as of briefly self-contained. a central present gauge importance the theory, whole in in construction order to make this project dimensional Einstein’s gravity can be formulated as a gauge theory, but not as a pure JHEP08(2020)001 (2.8) (2.9) (2.11) (2.10) . c µ e ) ρc e σ ∂ − σc e ρ , ∂ ( . ab b σb c M , e ] ) ν b ] ρa ω ω ν e ( e ac 1 2 ab ab µ [ µ − µν [ ω ) ω 2 R a 2 µ 1 2 − e ): − ν + ab ∂ a ] ] a 2.7 ν ν P − e ω ) – 6 – µ µ a e [ [ ν ( ∂ ∂ e a µ ∂ µν ( R ) = 2 ) = 2 νb e e ) into the ( ω ( ( 1 2 a ) = ab 2.8 A µν − action, which is not the desired one, when aiming to repro- ( µν ) R ) are the component curvatures of the gauge fields, which are b 2 µν R µ ) ω e ( R ν M ) and ( ab ∂ ( ]. The latter (de Sitter group) can be equally chosen instead of R µν − 2.2 60 b R ν – e µ 57 ∂ ( ) and νa e e ( a 1 2 µν = R ab µ Nevertheless, if one considers the Yang-Mills action for the remnant Lorentz symmetry, An equivalent argument leading to the desired, Lorentz symmetry, is to demand the Summing up, Einstein four-dimensional gravity can be considered as a gauge theory of ω one ends up withduce an the results of generalconstant relativity, to because this be would dimensionless impose — the gravitational which coupling is not. For the expected Einstein-Hilbert action, strength tensor that corresponds tobe the vanishing, recovering generators in that this break, wayof the i.e. the torsionless the constraint condition. that translations, This is hasconstraint, condition necessary to one plays in obtains the order role an to expression result of with the the spin desired connection symmetry. as From this a function of the vielbein: action, one has to beginof with the a Poincaré,and Yang-Mills include action awhich obeying scalar eventually the leads field de to for Sitter the inducing symmetry, Einstein-Hilbert instead a action. spontaneous symmetrygauge breaking symmetry of the vacuum to be the SO(1,3), Lorentz group. Therefore, the field at first, reduced tosymmetry. an This action action is including indeed the the Ricci desired scalar Einstein-Hilbert and action. the Poincaréalgebra up preserving to only retrieving the the resultsexpressions Lorentz of of the the transformations curvature of tensors. the fields For and the the dynamic part, in order to obtain the correct the Poincaré,and for the presentLorentz, purpose, it is i.e. preferred, the since breaking it isequal of a footing, the semisimple i.e. group initial and use symmetry all a generatorssymmetry to can single breaking the be breaks gauge considered the index on four to generatorsto describe that a all are constrained theory related ten with to generators. vanishing the torsion translations, The and resulting the spontaneous SO(1,4) Yang-Mills action considered As for the dynamic partMills of action the of theory, one theaction would consider Poincarégroup. of the the obvious However, Einstein choicethe since of Lorentz. gravitational the the Spontaneous theory, Yang- intention symmetry the is breakinginduced Poincarésymmetry is to by has the the end indicated to inclusion up way of beof to with a realize broken the scalar the it to field SO(1,4) and which can [ belongs be to the fundamental representation replacing equations ( expanded on its generators: where identified as the torsion and curvature, respectively. Their expressions are obtained by Since the field strength tensor takes also values in the algebra that is gauged, it can be also JHEP08(2020)001 ) a ]). ). K (3.1) (3.2) (3.3) (3.4) 94 , 4.16 93 , ) ab M ], also [ , − c ] 5 92 b – D δ a [ ab 90 δ K , . a ] = 2 ] = 2( a b c K ), the conformal boosts, ( K a a µ ab K ,K ,K , f a ] M ab P + , [ + M µ [ D D A µ b D , + [ + c + , also Weyl’s gravity is successfully formu- ] b , µ ab 2 δ ∂ ab a a [ M K M = P ab – 7 – − µ ab µ ω M 1 2 D ] = 2 ] = 1 2 c ]. In the present section we briefly discuss the case + = + ,D 92 ,P a µ a – a P ab A a K P ), the Lorentz rotations, ( [ 90 µ a δ M a e P [ P = = µ , ] ] A c b ), satisfying the following commutation relations: δ 4. To proceed with the gauging procedure, one has to define the d [ D a [ , ... a M P = 1 ] = 4 ] = cd ,D a ,M P is a parameter taking values in the Lie algebra of the symmetry group and therefore [ a, b, c, d ab M [ Although these are the commutation relations found in literature, we have used a different but equivalent 5 set of commutation relations for the following results, that will be given in section 4.2, equations ( where can be expanded on its generators: gauge potential: in which for everyobeys generator, the a following gauge infinitesimal field transformation has rule: been corresponded. The above connection where spacetime is SO(2,4), butsignature, instead, for therefore simplicity, the we symmetry consideralgebra are group the the corresponding is translations, euclidean ( theand SO(6). the dilatations, The ( generators of the conformal sors, the action withtype. which one Then, starts the initial isthe gauge an curvature symmetry SO(2,4) tensors, is gauge broken which invariantthe by action originate theory imposition of from is of Yang-Mills the physical specificof Weyl arguments, constraints the action on and four-dimensional [ conformal theThe gravity final (for gauge action detailed group analysis of parametrizing see the [ conformal symmetry in four-dimensional Minkowski 3 Four-dimensional conformal gravity as aBesides gauge Einstein’s gravity, theory discussed inlated section as a gaugeof theory the of transformations the of conformal the group, fields SO(2,4). and Again, calculating after the the expressions determination of the curvature ten- an action in a non-straightforward way,tities that of is building the Lorentz theory. invariantsthe out The of desired the one Einstein-Hilbert quan- that action. includes the Ricci scalar is the right choice, leading to retrieving the correct dimensionality of the coupling constant as well, one has to consider JHEP08(2020)001 ), M (3.5) (3.7) ( R , . b ] ν a f , and finally, as a third M b a µ D µ b [ b µ , f a b µ if K + f and + b a ] . + µ , ν a a P µ e bc a bc if K e a δ δ ] (3.6) µ µ [ + ν ac µ ac ] ib e ] b ν 4 ν b ,A ω + − ω µ M c b − b µ µ cd A [ [ P b [ δ ac e f , i , Kb µ db ab ] ab ω + + − in terms of µ ν δ ]. This comes in a very natural way, taking ab P a i ] ] 4 ] b µ a ν ν ] ω ν ), calculations lead to the following transfor- iω a 95 µ b b ν + µ iω ca A a a f e – 8 – f µ − b µ µ a 3.4 µ [ [ [ P [ − µ e ∂ ω [ f + ab a f M D and µ + + + a + Ka = 2 ie a a ab µ µb ] ] ] ab ) = 0 along with an additional constraint on µ ] ν a ν ) and ( ν ie ie µ ν µν ω b f e P + 4 e ( ω µ µ µ R [ [ [ 3.2 µ ab R [ + 4 + 2 ∂ ∂ ∂ − M ∂ a a gauge field can be gauged away. The resulting action is the D K P µ ), has to vanish. This vanishing will produce the constraints of ∂ µ µ µ µ ) = ) = 2 ) = 2 ) = 2 b 1 2 ∂ ∂ ∂ M ( P D K M ( ( = = = = ( ( R a a a a µ µν ab ab µ µ µν µ µν R δb ) and using ( µν δe R δf R δω R 3.3 ], there exist some convincing arguments that give specific constraints which 92 – Moreover, we can also extend the argument we used in the previous section, in the four- Besides the above arguments, we suggest another way of breaking the initial symmetry, 90 dimensional Poincarégravity, as an alternativewe way want to to break preserve thethat the initial the Lorentz symmetry. vacuum symmetry of Since outtensor, the of except theory the for is the initial SO(4) SO(6), invariant, we which consider in directly turn means that every other one step beyond thein way the the previous de section) Sitter isas group broken described to (and the in not Lorentz the the by previouscould a Poincaréfor reasons section induce scalar for in a explained the the spontaneous fundamental case symmetryfields, of of breaking giving SO(1,4), the in rise to Einstein a the complete gravity.respects constraints These theory Lorentz that two that lead symmetry. scalars includes us matter to a resulting four-dimensional action that this time to thecomponent Lorentz. tensor expressions This we presented breaking, above, alongbecause serves our with we purpose the for want the transformations present togravity of paper, theory use the we fields these develop and later. results6 representation as This of could the the occur commutative SO(6) with limit gauge the group of inclusion [ of the two noncommutative scalars in the are the torsionlesstogether condition, leading to expressions of constraint, the fact that well-known Weyl action. The action is takento to [ be ofinduce Yang-Mills the form, symmetry being breaking SO(6) of gauge the invariant. initial symmetry. According In particular, these constraints gives the expressions of the component curvatures of the gauge fields of the theory: The well-known formula for the curvature: mations of the various gauge fields after comparison of the terms: Starting from ( JHEP08(2020)001 ) ], are 80 4.2 (4.1) (4.2) a µ , where , e c a , with co- µ X 4 f abc C = ab θ cannot be assigned to gen- , must satisfy the following a , the right hand side of ( ab r θ X +1). , +1 , , . 2 ) are set simultaneously equal to zero, ab +1 R K , ( iθ = 0. ], where the authors encounter the same problem = R µ +1 b 78 , B , ] = 1 b x 67 – 9 – A − x ,X ) and a P AB X ( [ η R is the mostly positive metric of the five-dimensional = diag( . The de Sitter space is defined as a submanifold of the 4 AB AB The requirement of the preservation of covariance, imposes η η 6 ]. 76 4 and ,..., is: 4 = 0 A, B is a rescaled Levi-Civita symbol. In the fuzzy de Sitter space we want to construct, the However, the coordinates of the fuzzy de Sitter space, We want to construct a gravitational model on a four-dimensional fuzzy space, specif- For a detailed analysis of this issue, see also [ 6 abc ordinates represented by N-dimensional matrices (likecovariance in as fuzzy well, two-sphere case), we respecting areFrom now led on, to for the convenience, enlargement we of will the use symmetry the to euclideanwhen the trying signature, to SO(1,5) which build group. means the that fuzzy four-sphere. the covariance is broken, because theerators algebra into is the not algebra closing, [ us i.e. to use a grouperators with and the a noncommutativity larger in symmetry,the it. in adoption The which of minimum we the extension SO(1,5) will of group. be the Therefore, able symmetry in to leads order incorporate to to all end gen- up with a fuzzy dS Recalling the case of theators fuzzy in two-sphere with an coordinates (large) the N-dimensionalis three representation also rescaled and SU(2) a radius gener- generatorC of the SU(2) algebra,problem ensuring is covariance, that that is when trying to identify the coordinates with some generators of SO(1,4), Minkowski spacetime, that is commutation relation, which describes the noncommutativity of the space: five-dimensional Minkowski spacetime inembedding the in same the five-dimensional waythat euclidean the defines space. four-sphere dS is Specifically, the defined embedding as equation an where approach one has to predicate theof background spacetime the in order isometry to extractdeveloped group the on information which this is space. eventually gauged. Theically the corresponding fuzzy gauge version of theory the is dS 4.1 Construction of theIn fuzzy this covariant subsection, space which we we present elaborate the construction gravitythe spacetime as of is a the obtained gauge four-dimensional as theory. a fuzzy solution space, Although, of on the generally equations in of gravity motion, theories, in the gauge-theoretic it is described thatthen if the both constraints tensors ofequal the — theory up yield to that a their rescaling corresponding factor gauge — fields, and also 4 Four-dimensional noncommutative gravity on a fuzzy covariant space the theory leading to specific expressions that relate the gauge fields. In particular, in [ JHEP08(2020)001 , 6 mn (4.3) (4.4) (4.5) (4.6) (4.7) ,..., = 1 and Θ m P A, B , . , ) m ) (4.10) mn h X m np 1 2 Θ BC , with P 2 J Θ ]. In the following sec- ~ 2 = λ ]), admits finite dimen- The whole procedure is ~ i BA λ mq AD ]. 97 i J δ 56 9 96 δ identifying the component , − 99 8 − − 77 ] = 4 ]. ,J , = n h] = mq m 98 , AD 76 P ,P Θ J m AB ~ m J λ np X 2 [ BC δ P [ δ = − − 6 ) (4.8) SO(4). ) (4.9) p m mp p AC ⊃ Θ X P J ,J nq mn , thus we have obtained a model of spacetime δ mn m BD δ δ δ – 10 – SO(5) + X mn − − 1 + λ ⊃ n nq n X P Θ = , BD , 5 , J and Θ mp mp mp m m h δ δ δ mn ( ( ( X m AC 7 Θ ~ ~ ~ 2 δ mn P i i i ~ 2 ( λ ,J δ ~ , i i λ ~ i i m mn ] = ] = 0 (4.11) ] = ] = X ] = Θ pq np np mn ] = ] = 1 ~ Θ Θ Θ n n h] = 4 CD Θ , , , , , = 4. The extended kinematical algebra i.e. the algebra of spacetime ,P ,J m m m ,X mn [h P m P X m [ [ mn [ 4. For dimensional reasons we have introduced an elementary length AB [Θ X J X [ J ,..., [ [ ,..., = 1 = 1 m, n m Now, in order to formulate the above four-dimensional fuzzy space, we consider the This translation to the euclidean signature could be interpreted asThe the embedding case path we ofFor consider the a is noncommutative fuzzy SO(6) gravity four-sphere motivated from string theory see [ 7 8 9 . The coordinates, momenta and non-commutativity tensors are upon them the framework of noncommutative gauge theories [ which bears the sameirreducible isometry representations, group which SO(5), implies along the imposition with of the the restriction radius that constraint. large-N matrices are set into 4.2 Noncommutative gaugeHere, theory we of are four-dimensional gravity goingdimensional to space present that the wasdeveloped constructed formulation as in of a the gravity noncommutative previous as analogue section. of a the gauge work theory presented in on sections the 2 four- and 3, using The above algebra, insional contrast representations to for the Heisenbergwhich is algebra a (see finite [ quantum system.above are In called analogy fuzzy with the covarianttion non-commutative fuzzy we spaces two-sphere, will spaces [ formulate like a the gauge one theory of gravity on the above four-dimensional fuzzy space. where transformations is: λ respectively. Both coordinates and momenta satisfy the following algebra: Now, we decompose the above generatorsgenerators in as: an SO(4) notation, with instead of SO(1,4) and SO(1,5). extended algebra of SO(6). Weand denote its they generators satisfy by the following commutation relations: isometry group and the extended symmetry group will be SO(5) and SO(6), respectively, JHEP08(2020)001 Re- U(1) (4.13) (4.12) × 3 σ , ⊗ 4 matrices. 1 and σ ! × 5 1 , a < b = Γ b − 3 1 2 Γ Γ 1 0 0 a − Γ

i 2 . The generators of the SO(6) = − 4 3 Γ σ 3 ] = Γ b 2 , Γ , Γ a , , 1 5 , 1 Γ a . 2 Γ 2 1 ab [Γ 1 σ ! a δ i = Γ i 4 Γ = ⊗ ⊗ 0 i 5 2 − − a 3 1 = 2 i − 0 σ σ = } b as Γ – 11 –

= = = Γ ab 5 a 5 2 , = a Γ Γ 2 Γ ]). Of course, the anticommutators of the generators σ { 100 ! -matrices (in the Euclidean signature) satisfying: . γ 1 1 0 0 1 , U(1) is a coincidence with the SO(6) symmetry related to our space and should , 1

σ 1 × ] (see also [ = ⊗ ⊗ 1 72 1 2 4 and also defining Γ – σ σ σ with generators being represented by 4x4 matrices, i.e. the representation 70 = = ,... 10 1 4 Γ Γ = 1 matrices are built as tensor products of them as: m, n − one generator for special conformal transformations: D = four generators for conformal boosts: K four generators for translations: P one U(1) generator: Six Lorentz rotation generators: M γ We start with the four However, it is known that in noncommutative gauge theories, the use of the anticom- The extension to SO(6) c) e) U(4)) group a) d) b) 10 ∼ not be confused. The construction of Γ-matrices starts with Pauli matrices: and group are: should be fixed to the 4 of SO(6) i.e. to the fundamental representation of U(4). where treatment is to specify the representation inerators which produced the by generators the belong anticommutators and intotoo. include the the This op- algebra, will considering result them in asIn the generators, extension our of case, the application initial gauge( of group this to one recipe with leads larger symmetry. to the extension of SO(5) to the SO(6) as a subgroup of the SO(6), in which wemutators ended of the up generators with. of theprevious algebra works is [ inevitable, as we(not have in explained a in detail specific inlong representation) our to of the an algebra algebra and do this not is necessarily exactly yield the operators case that for be- the generators of SO(5). The indicated calling the previous section,we we were argued led that tothe for enlarge lines the the of sake symmetry the ofisometry group preserving commutative group, of four-dimensional gauge (the the gravity, Poincarégroup), covariance, was describedhere space gauged we in to in have to section order the gauge to 2, SO(6). the retrieve where isometry the group the Thinking desired of along the results, covariant space, that is the SO(5), as seen Determination of the gauge group and representation by 4 JHEP08(2020)001 , . a 1 m = 0, X iP ⊗ ) (4.20) (4.17) (4.18) (4.19) (4.14) (4.15) (4.16) − m X ( δX ] = . m a a ,D P a + , i.e. ⊗ K is a function of m D [ ) X ). In analogy with ⊗ m X ( ) A a D, ˜ ξ ab X 4.18 ( ab + m iM iδ a ab ] = ] = M ) + ˜ b b ⊗ X k ,P ,P ( ) , σ a a . a k X P ) K ) takes values in the U(4) algebra, [ ⊗ σ K ( [ X 3 X ab ⊗ ( ⊗ , σ ( λ ) included in the ( ) ] Previously, we decomposed the SO(6) m 1 1 2 m bc m 1 2 + X A m A − ˆ ( X M a A + D = m = , ad b [ j 1 δ ⊗ k i Γ ) Γ i ⊗ − 4 ) + . The = – 12 – Γ X 4 Γ m i ( 2 X ad m i 2 0 ( X ˆ − X M ab − δ = transforms indicates that it can be considered as the bc + ˜ = M δ = a m m ij k ⊗ ˆ − K 4 X A ) M ac ⊗ M ) X c ) ) ( M c ), given in the previous section. Therefore, it can be written K ab X P bd m ( of the fuzzy dS δ ab U(1) in the SO(4) notation, in order to identify the generators in ab a ω δ δ ξ 4.16 × m + + − − + X a bd b b 1 is covariant by construction, which means that it transforms covariantly P , 3 and: K P M , , ⊗ ⊗ m ab a ac ac ac 2 ˆ ) ) δ δ δ , . X ( ( ( iM iK i i i ) is the gauge parameter, which is a function of the coordinates, but also takes X X ( ( = 1 X a 0 m ( ] = ] = ] = ] = ] = ] = 0 e b bc bc cd ab = ,D ) = ,K i, j, k a ) = ,M ,M a ,M X P a a [ X ( First, one has to define the covariant coordinate of the theory, which is given by the The generators defined above, obey the following algebra: K ab D,M ( P [ [ K [ [ m M [ meaning that it can be spanned on itsA generators: From the above equation,generator. one can read The that component we have gauge introduced fields one depend gauge field on for the each coordinates of the space, Considering that the gauge transformation doesone not can affect find the the coordinate transformationthe property of commutative the case, thepotential, way that is thematrices-coordinates gauge connection of the theory. In our case, values in the algebra,as ( an expansion onwritten the explicitly generators as: of the algebra. Consequently, the gauge parameter is The coordinate under a gauge transformation: where the appropriate language andwe then are we ready calculated to their advance commutation with relations. the Therefore, (noncommutative) gaugingfollowing procedure. relation: Noncommutative gauge theorygenerators of of gravity. the SO(6) where In particular, we obtain: JHEP08(2020)001 , . . a 1 m m b ˜ a ⊗ , (5.1) ) (4.22) (4.23) (4.21) m = X a ( , a . m a e m 1 mn e 4N matrix. ⊗ R × + m U(1), which will a D × U(1) + ⊗ ) × D X ( ⊗ m mn a ˜ R + ˜ + a a , K K mn ⊗ ⊗ ˆ ) Θ a m 2 would give a theory with two metrics b X ~ , ( iλ a 12 + − mn ab ] R ) = 0 n M ˆ P + X ( ⊗ , a a – 13 – ) . The consideration of this solution leads to the P m a mn ˆ as a function of the rest of the fields, X m ⊗ X ˜ ( b R ) ab ab m = [ X U(1) symmetry to the desired SO(4) m = ( ω ω × a a m mn + e mn R a ˜ R as a second vielbein P N matrices, where N is the dimension of the representation + a m ⊗ × b ab ) M which, for the noncommutative case, is defined as: X ( ⊗ a 11 m ) e X ( + 4 matrices. Therefore, every term of the connection is a 4N ab 1 × mn ⊗ R m X ) = = X ( m Having determined the gauge connection, the covariant coordinate is written explicitly Details on its definitionWe and note gauge that covariance our property symmetry are group given is in not appendix the A. conformal group but is related to it. ˆ X mn 11 12 solution of the constraints,expression the of the spinTaking connection into consideration the torsionless condition with the specific solution that is theprevious torsionless sections condition, as well, which inpossible is the interpretation more cases of of or Einsteinor less and two conformal expected, vielbeins, gravity. Furthermore, as which the is it not is desirable considered in in our case. Therefore, we are led to adopt as a in order to break thebe initial the SO(6) remnant gauge symmetryleads of to the vacuum an of over-constrained thethat theory, field survive that theory. after is the However, this breaking. evident approach after Therefore, it counting is the rather degrees wise to of impose freedom at first the constraint: 5 The constraints forThe breaking gauge the symmetry symmetry we togroup, would in SO(4) the like euclidean to signatureconsidering we end a use, up the constrained with SO(4). theory in is A which straightforward the the way to rest one do of described this the is by component by tensors the are Lorentz vanishing, R At this point, all that isfields necessary and for the the determination expressions ofintermediate of the calculations transformations the of are component the given curvatures gauge in is the obtained. appendix The B. results and some which, since it is valued inas: the algebra, it is expanded in terms of the component curvatures as: Furthermore, for the U(4) gaugefield theory strength we tensor, are developing, what is left to determine is the in which theits coordinates corresponding are generator described.the is product not Multiplication consists the of betweendescribed usual matrices every by one, of 4 gauge different but dimensions, field the recalling and tensor that the product generators is are used, since meaning that they are N JHEP08(2020)001 ). ) is 5.1 (5.6) (5.2) (5.3) (5.4) (5.5) 1 ( mn R . } U(1). n ˜ a , × , } a , would vanish if we m . m ] e . m ˜ a { ) a , } a fgh [ m = m e a } ˜ a { = 0. This will also modify , i = ] nb a m − n , e a e ] . fgh a { ab ] m a m bc [ ω nd , e δ { abc fgh m , e δ D m ] + 1 [ 3! D nd − [ , e = m abcd } b D nb – 14 – [ m and e ] and , e 3 4 a abcd m ab m = , e ω − ( { m ac i b n m fghd D e ω [ ]. However, we could set ˜ − 3 4 ] − cd − abcd 102 n ] = = (the two surviving fields of the theory), which reads: , ω = cd b ), the above equations lead to the desired expression of the spin ac n m m n e ], the vanishing of the field strength tensors in a gauge theory con- , i.e. the covariant coordinate of an abelian gauge theory. The above [ a 5.2 ω abc fgh , ω m b δ m a 101 e abcd [ + m abcd X , and the a = m e m decouples being super-heavy. In this limit, the gauge theory would be just SO(4). ), producing an even simpler expression of the spin connection in terms of the D m 5.5 a Another way of getting the desired SO(4) symmetry as a remnant symmetry after the At this point we need to punctuate that the U(1) field strength tensor, According to [ In order to go on with obtaining the expression relating the fields, we employ the breaking of SO(6), isgravity to in extrapolate the the commutativefundamental representation argument case of that (section SO(6), we to 3)the developed the spontaneous of noncommutative symmetry for the case. breaking the will inclusion We conformal lead are of to confident constraints that two equivalent scalar to the fields one in in ( the not set to zero,remains which unbroken in means the thaton resulting this a theory U(1), noncommutative after space. strongly theassumed related breaking, the Of to commutative since course, limit the we the ofand still noncommutativity, the corresponding have broken field, a theory, in theory which noncommutativity is lifted the ( vielbein, applied in our case in general,spacetime, because we after mixed identifying gauge theto vielbein theory with be and gauge invertible geometry. fields in of every Itvielbein our point, holds to therefore, that adopting zero, the the would vielbeinmetric above lead argument, is tensor that to considered of is degenerate the setting vielbein the space matrices, [ inducing degeneracy to the sidered on a simplyvanish connected as space well. means Thistainly that, simplify argument the locally, would expressions the be of corresponding the very tensors. gauge welcome in Nevertheless, fields the our above case argument cannot because be it would cer- Using the identities ( connection in terms of the rest of the fields: The constraint of the torsionless condition takes the following form: where equation leads to the following two: analogue of the four-dimensional gravity with reduced symmetry SO(4) following two identities: we result with the correct number of degrees of freedom, leading to the noncommutative JHEP08(2020)001 (6.1) (6.2) ) (modified B.19 } )–( 13 b , n ] (6.3) ,e ! a B.15 , nd m · · } e d mnp , e { s i m mnrs H ab mnrs ,e mnrs mnp c D r [ ]]+ ˆ mn ! e H ! mnp { pf B rs ˜ i cd H abcd 2 mn ,e R 4 mnp ~ ˜ c rs n iλ B ˆ n ]]+ H − 2 mn e B D [ ~ ˜ 2 4 3 pf R + iλ }− ~ 4 ] iλ ,e abcd = − r abef qg − ]] D e ab ,e mnrs [ }− p m ] qh n ,e abcd rs qg ,e D mnpcd n [ cdef n – 15 – R ,e cd H , ω D e s [ D p c rs [ ab b fg mn 3 4 D g [ ,e R q s = 0 f R − q ab mnp ,e ] D h ]] m [ d fg ,e H 5 mn ] ˜ a pf 3 4 pf 2 f R pe 4 q ,e ,e − √ 2 , ,e 4 ,e m m ]] ] a √

m m D D pf pe b [ [

D c is gauge invariant, as it is explained in appendix A. [ p ,e ,e = Tr tr Γ = s e 2 r + Tr [ abef a S F ) for the generators of the algebra have been used, as well as the D D m acde [ [ = Tr ) and applying the tr on the generators, we end up with the following d e p d efgh S p ,e e cdef chje B.13 2 { m A.13 e j i p √ p 16 8 9 D ), ( e 9 [ ,e ), ( { 3 4 r + − i 8 9 D 4.16

[ 4.23 3 4 + Tr 2 4Tr 4 √ − × Considering the vacuum state of the theory, i.e. applying all the constraints considered A Yang-Mills action tr = 13 S and inserting the expressionsaccording of to the the above component constraints)the tensors in action given the after in above the action, ( symmetry will breaking: give the explicit expression of in the previous section: where in order to resultrelations with the ( above expression, thecyclicity commutation property and of anticommutation the trace. tensors, ( expression of the action: where the Tr is thethe trace integration over of the the matricesof representing commutative the the case) algebra. coordinates whereas (takes The thetensor the of first tr role the term is gauge of theory, of the while2-form the trace the field. second above over one Writing action the down is includes the the generators (non-topological) the decompositions kinetic field of term the strength of tensors the (curvature) in terms of their component Since we havemodel, employed the a most gauge reasonable choice theory for formulation the action for is constructing of Yang-Mills the form: gravitational 6 The action JHEP08(2020)001 ) (6.4) rs B , while, 4 + rs (Θ 2 ~ iλ − ]] c p ,e s , keeping the fundamental D 4 [ pd e − ] d p ,e s . D [ ! pc ,e ] mnp qd H – 16 – ,e r mnp D [ ˜ H c q 4 e [ − 9 32 abcd ]+ sa Last, variation with respect to the (surviving) gauge fields would ,e 14 a mnpcd r e H [ ab 5 16 mnp ]+ H s 2 4 ,D √ r U(1) action of Yang-Mills type, including the kinetic term of the dynamic 2-form

× D [ ) in appendix B. +Tr × Then, straightforward calculations led to the transformations of the gauge fields and the Explicit calculations are quite cumbersome to prove the gauge invariance of the action in a straightfor- B.20 14 ward way. Nevertheless, based onsymmetry the breaking, corresponding we are commutativefor cases quite which our confident inspired that future our gauge workspontaneously. choices invariance to should for This include the be approach, scalar among secured. fieldsthe other Also, produced in welcome it action the features, is after theory will postponed the and directly symmetry realize ensure breaking. the the gauge breaking invariance of of the initial symmetry led to a constrained gaugean theory, SO(6) with the preferred symmetry.field. In this Imposition spirit, of we the proposed constraintsin leads order to to an obtain expression the ofthe equations the conformal of action gravitational motion. which theory, can not Note be that the varied in one the of commutative Einstein’s limit general we relativity. recover described in detail this subject (appendix A), for bookkeeping and useexpressions as of a the future reference. component tensors.to However, gauge, the even part more of afterone the the isometry enlargement we noncommutativity group wanted imposed, we to was decided result larger with. than the For this reason, we considered a symmetry breaking which furthermore, the construction of aan gauge enlargement theory of on the such gauge athe field noncommutative group strength space and tensor led imposed in us the fixingcovariance. to inclusion its of The a representation. introduction 2-form of Also, dynamic such gauge theant field a spaces definition for like gauge reasons of the field of one in we gauge use, is theories a constructed treatment on that covari- can be applied in general. That is why we property of covariance of theand, two-sphere, in which preserves particular, all thefour-dimensional isometries case. Lorentz of the Another invariance, fuzzy property whichcoordinates space that is can shares of with be particular the representedisometries fuzzy interest by of two-sphere in finite the is the constructed matrices. thatrequirement noncommutative present its of space. Then covariance led gravity It us to is should an be built enlargement of noted by the though isometries gauging of that the the the fuzzy dS 7 Summary and conclusions In the present work we havecommutative presented space. a four-dimensional The gravity chosen model space onin is a a certain covariant non- generalization basic of aspects the celebrated and fuzzy is two-sphere a noncommutative version of dS breaking) is expected toexplicit expressions leave of it the gauge invariant. transformationsin of ( This the component could tensors, which belead are to proved given the by equations making of use motion. of the It should be noted that a gauge transformation on the above action (after the symmetry JHEP08(2020)001 ]. In addition, our 79 ]. 76 – 17 – ]. Also, it should be noted that our present noncommutative 102 In general, the field strength tensor of a noncommutative gauge theory can be written Finally, it should be stressed that the constructed gravity model is a matrix model In addition, let us make a couple of comments on our approach. First, we should The noncommutative gravity model we propose hopefully provides a certain insight the gauge invariance of the action we propose. in terms of the covariant coordinatethe plus whole an expression extra of term the which field contributesformation. strength in tensor such The a transforms way covariantly two under that most a gauge widely-known trans- types of spaces on which gauge theories are built their hospitality. A Gauge covariant field strengthIn tensor this and appendix, gauge we invariance briefly present the gauge covariance of the field strength tensor and their participation in theString Workshop Theory”, “Matrix Jul 09–13, Models 2018. forInitiative One Noncommutative funded of Geometry by us the and (GZ) German hasPhysics, been and Heidelberg supported States within Governments, University the at and Excellence would the from like Institute for the to Theoretical thank Excellent the Grant ITP Enigmass — of Heidelberg, LAPTh. LAPTh GZ — Annecy and MPI — Munich for We feel that weStelle. have We benefited would a alsoIvanov, lot Dieter like Lüst,Denjoe O’Connor, from to Harold thank discussions Steinacker anddiscussions. Paolo with Christof Aschieri, Wetterich Ali The for Thanassis useful Chamseddine work Chatzistavrakidis, ofAction and Evgeny two MP1405, Kelly of while us both (GM would like and to GZ) thank was ESI partially — supported Vienna by for the the COST hospitality during giving promises for improved UVlatter, properties as as well compared the to inclusion ordinary of gravity. matter Clearly, fields the is going toAcknowledgments be a subject of further study. we proposed. Forpoint this of reason the the space vielbeinautomatically and, field avoided [ under was considered thisapproach to consideration, is not be singularities related invertible related to at to Ashtekar’s formalism every this or issue loop were quantum gravity in general. the conformal group. Theleading low-energy limit to can the be usualwork obtained commutation consists by a relations Wigner-Inönücontraction, formulation of of a quantumrealization four-dimensional of mechanics covariant fuzzy the [ space, starting providing suggestions a gravity of Heckman-Verlinde [ note that we did not aim at a quantization (in the usual sense) of the gravity theory concerning the gravitational interactionis at naturally the introduced. Planck scale,leads In to in an addition, which already existing the noncommutativity the theory fact (conformal picture gravity) that is that a the emerges veryare commutative welcome from considered result. limit our to Therefore, of construction be our is noncommutative model and that are at also high part energies of the the coordinates commutation relations of JHEP08(2020)001 D. ⊗ (A.4) (A.5) (A.6) (A.7) (A.1) (A.2) (A.3) mn ˜ B + , one would a m K X ⊗ . a p = 0 and consequently mn ˆ X B m + δX mnp , ab p iC M X ⊗ − , ] ] . ab mnp n 1 1 ˆ , mn X ⊗ is a rescaled Levi-Civita symbol. iC . , B ⊗ 1 mn m + mn ˆ ˆ ] = Θ a , of the field strength tensor, straight- mn X ⊗ mnp n ] P 2 Θ Θ C ~ 15 ⊗ , 2 iλ = [ mn [ mn ,X a ~ 2 iλ ˆ Θ m Θ − ~ mn mn 2 ] iλ ˜ ~ X , B − λ n [ [ ] i i ˆ − + X n ] , 1 ˆ = – 18 – X m ] = is a non-Abelian two-form gauge field, which takes ⊗ , mn , ,F n ˆ X m mn mn ˆ mn B X ,X mn mn , F B δ B = [ m iθ iθ = = [ = [ X − [ mn ˆ ] mn ] = F mn n mn B n as the rest of the gauge fields and transforms covariantly as: F ˆ ]. From the above equation we can find the transformations of X , where , following exactly the same procedure we describe in appendix δF , ,X 16 mn m mn m mn ˆ ˆ Θ B X B X [ , + [ i 1 = [ ⊗ = mn mn F mn ˆ Θ δ = Θ is a constant antisymmetric tensor and be the covariant coordinate in both cases, then expressions of the field strength mn is a gauge parameter. In order to ameliorate the above problematic result, one has mn ˆ θ Θ m = 0. ˆ The commutation relation of the fuzzy space in our case is: On the one hand, the fuzzy space on which we are building the gauge theory of gravity X A gauge transformation is considered not to affectTherefore, the it coordinates, can that be is written as mn 15 16 Θ which gives the component fields of δ where values in the U(4) algebra, where to modify the definition of the field strength tensor to: Nevertheless, if we assume aforward calculations gauge lead transformation to atransform result covariantly, specifically: according to which, the field strength tensor does not Due to the fact that thebe right led hand to side a does not definition depend of on the the corresponding coordinates field strength tensor as: coordinates, pointing at the case ofbe constant classified noncommutativity. into The truth any is of that these it cases, cannot therefore it has to be examined explicitly. is covariant, pointing atthe a right relation hand with side the of fuzzy the sphere relation case, defining but, non-commutativity on does the not other depend hand, on the where Let tensors are given by: are those of constant and Lie-type noncommutativity, specifically defined by the relations: JHEP08(2020)001 ) ] 4.18 mn (B.1) (A.8) (A.9) (A.11) (A.12) (A.13) (A.14) (A.10) ˆ Θ , of the ,δ p ˆ mnp X D, ˆ H . ⊗ ]+[ 5 Γ mn 1 2 mnp . ˆ Θ ˜ H − , . p + ˆ ] X , a δ 5 ]) = 0 K Γ mn ˆ F a ˆ Θ ]+[ ⊗ , given in equations ( Γ , , [ i a p 2 pm ˆ mn ˆ F X − i ˆ Θ ˆ Θ mnp + ,δ H , n ] + [ ˆ . a , F ˆ ] ] + mnp X ] Γ and pm ˆ ˆ m F 1 2 H ab ˆ ˆ Θ mn H m , ]+[ , ˆ [ F M ˆ i n , , X mnp ( [ b pm ˆ , ⊗ i X ˆ [ H Γ U(1) are encountered in a specific representa- ˆ i Θ a T r , ab = × Γ n = – 19 – ] + [ i ˆ 2 X mnp ) = np δ mnp − mn ˆ H F ˆ ˆ ˆ Θ = Tr tr F H , δ + δ ]+[ ˆ B ] = F δ m a b S np ˆ X Γ P + [ ˆ , Θ ˆ a ⊗ F ,δ ˆ 3 1 a [Γ F m δ i 4 ˆ = X mnp − ˜ on the generators of the algebra: H = ]+[ ˆ mnp = Tr( : + H ˆ np ab H 1 ˆ Θ δS , mnp ⊗ 4 matrices (SO(4) notation): ˆ m H × ˆ ,M X 1 mnp δ [ H 1 3 = ), respectively, and the Jacobi identity, we find: = A.7 mnp As we showed, the generators of SO(6) the inclusion of this 2-form gauge field has to be prominent in the action of the theory, At this point, it is also important to define the field strength tensor, Since, both terms in the expression of the field strength tensor transform covariantly, ˆ mnp H ˆ H δ are valid, after the consideration of the commutative limit. tion given by 4 B Calculations of theIn field this transformations appendix, and wecomponent curvatures curvatures present and our their results transformations. for In the the end, transformations we check of whether the our results gauge fields, the and therefore it isstrength possible tensor to calculate each component usingtoo. the definition Among other of terms the that field will be present, one has to add a kinetic term for this field: Next, we expand the transformation, we start from the expression: and using theand ( transformation properties of 2-form gauge field: In order to show that the above field strength tensor transforms covariantly under a gauge it is straightforward to show that the corresponding Yang-Mills action is gauge invariant: Therefore, gauge covariant transformation of theof field the strength action tensor are and gauge ensured. invariance of the (infinitesimal) transformation of the field strength tensor: that is a desired covariant transformation. B for the rest of the gauge fields. Therefore, calculations lead to the following expression JHEP08(2020)001 , 5 . 5 (B.6) (B.7) (B.8) (B.9) (B.2) (B.3) (B.4) (B.5) ... (B.11) (B.10) mn , ,B = 1 a m , . a 1 mn , AB . B ) AB → A, B M M m → BC AB ⊗ D, ⊗ A 2 ) M M . ) with − X mn E ,A ⊗ X ( AD ) Γ ( ) δ = A m 5 AB − ˜ m a AB AB Γ ,B m 5 A M ≡ A a AD ( , + 5 . ABCDE m + 1 mn a ] M B AB 2 A B K ⊗ Γ ,A 8 ,B Γ Γ BC √ a mn , δ m ab ⊗ . F b ⊗ mn A = 2 ) 1 − ) ˆ mn Θ a + [Γ ≡ X 2 )1 + B Γ ~ X i , ( 4 AB a A i ( λ AC m ) δ B Γ − A m BC → P A m M , − δ ( , A Γ ⊗ ] = a A = 2 ) n P BD AB + AD } → ˆ A + δ DE δ X MN – 20 – AB B 1 = mn , δ A 1 (Γ + m M Γ − 5 m ⊗ , A − ⊗ ˆ Γ ) A X ) a BD N BD mn Γ ,M ,B X Γ δ X ,A 5 { Γ F ( A M i = [ 2 ( a ) Γ m ABCDE a + m AC − m AC MP δ A mn e ,M 2 A 1 , δ δ ( ˆ = ( ( F 1 1 8 + ab i i √ ≡ + ⊗ 5 1 a ) 5 M 1 = = a 1 m ⊗ ( ] = ] = ⊗ } } → ) m C ,A mn CD NP ,M Γ CD X X F ] ab AB , ( ]: b m 0 ,M ,M Γ ω = ,M AB , ) = M 103 a ≡ AB M AB X mn [Γ [Γ ) = { ,M ( ˆ M ab F 5 i M [ 4 m m X { Γ ( ˆ A − , X ( a = Γ → ab → M AB A m Γ A Accordingly, we would decompose the gauge fields to the SO(4) notation: decompositions in order todecompositions recover of the the SO(5) results generators in are: the SO(4) desired language. First, the and identification of the SO(5) component generators with the SO(4) ones is: and is also decomposed on the generators: After the abovecould rewriting continue of with the the algebra calculations and in the this gauge-theory SO(5) related notation expressions, and we then use the following The definition of the field strength tensor is: In turn, the covariant coordinate is written as: and the gauge parameter as: The generators in this SO(5)tation notation relations [ satisfy the following commutation and anticommu- satisfying the well-known anticommutation relation: Taking this into consideration, thecompact above (SO(5) generators notation) can form: be written in the following more We start from an SO(5) notation and introduce the matrices Γ JHEP08(2020)001 } } m ab abcd d . (B.16) (B.14) (B.15) (B.12) (B.13) m ˜ a ] , K d a , ω = 0 m ˜ b ξ abcd ˜ ξ { } , b } abcd ] c b 1 2 m ˜ cd ξ ] 2 [ ,D m ] a ), the transfor- 4 i a m , e √ } − { 2 a , ω P } − 4 b nab ˜ , e ξ m b − { . √ a m ab { 4.16 ˜ ξ ,ω 0 , e λ 2 1 [ = = , b [ b − ab b a i 2 } } a m 16 λ 8 c → λ ω √ { } − [ { abcd 0 ,D ,P 1 8 bc } a + ] + ] m + a ab m } m } − K ab } ]+ a m a , ω ) ˜ a { M a m ,e m c n 0 , m { a ˜ ˜ a , ω , b na , b cd λ , , e 0 b a ab { λ 0 ≡ ˜ ˜ m ξ { [ ˜ 2 λ { a i { 5 i [ 2 { [˜ 1 2 2 i 1 2 , 8 4 1 2 1 4 , ξ √ 1 a } − + }− b ξ + D m ab a ]+ ] + ( } − − } n δ } a m m , b na 1 8 m ,e ab a ˜ ˜ a a → abcd abcd abcd . m ): ξ , , ,b , e abcd = ] 0 ] a a A { ma a 2 cd m ] , ω d ξ b ξ 4 2 1 [˜ } cd b m b { { √ [ { b m i – 21 – cd 4.23 ξ M 4 i m 1 2 2 , b 1 4 1 2 − , ξ − , ω ,P ] bc ) b , ω a − { abcd − ] + mn 1 ˜ λ a ]+ ab ξ ]+ 0 ] + ) in the SO(4) notation, taking into account the above [ ] P ] [ a ˜ m ˜ ξ ) n m a B i a m [˜ i { 2 m m m 2 2 i 2 ˜ a ) and ( 2 ad 2 , ω ,a ≡ 8 , b ˜ a √ , 2 B.4 ~ δ , b , e 0 4 a , √ √ iλ n 5 m 0 0 0 ξ bc − √ [ a a a [ mn i δ [ [ 4.22 − [ − + 1 4 i i , i ] = + B − , λ − d 2 n ) and taking into consideration the above anticommutation ]+[ cd } ] + ab P n ~ ] + ] + abcd bd abcd ] + ] + λ m X iλ ab , [ a a δ abcd 0 0 ( ] ,D ] ,ω ˜ ˜ ξ ,a 1 4.21 ] − d abcd ac − , ξ , , , λ , cd m ab n ab ] d ] δ → m ab m m a m m m m m ( 2 n δ n )–( X , e M ω a a a a a [ ˜ a [ , e , ω 1 8 1 2 { √ , bc ,e b c AB − + + + + + ξ λ ξ m λ ] [ [ [ = = = = 4.17 ma abcd a n i i i m m m m m e } } } } 2 2 2 [ ) along with the corresponding commutation relations ( 2 8 4 b b c + ,a 2 X X X X X cd [ [ [ [ [ √ √ √ 8 1 i i i i i m m 16 √ ,K ,K ,K − − − − − − + + X X a a ,M B.13 − + ab P K ab = = = = = of are obtained using ( { = [ = [ M { a a { M m m ab m m m { ˜ a mn mn mn δb δe δ δa ˜ R R δω R The same holds forof the component curvatures. The expressions of the component tensors relations ( mations of the component gauge fields are obtained: Making use of ( in the SO(4) notation are obtained: Instead, in order tochoose avoid the rewriting alternative and (but calculatingthe equivalent) everything anticommutation and relations in more of the straightforward ( SO(5) route,decompositions notation, that and we is identifications. to Then rewrite the anticommutation relations of the generators as well as the components of the SO(5) gauge parameter: JHEP08(2020)001 . ] ] cd ] mn (B.20) (B.17) (B.18) (B.19) mn ] mn d ,R ,R ab 0 abcd ,R mn a ab ] 0 [˜ ] mn λ ]) a n [˜ [ d mn ,R d i i B m m ˜ abcd bc B mn 2 mn ˜ + + R ,e 2 ~ λ ˜ B i [ c , R iλ ~ } } n 2 2 0 , iλ 4 b ~ [ b [˜ bc √ iλ abcd i }− 4 λ mna − [ b mn c }− ] i − ˜ n R d 2 , n 2 } − ]) mb ,e ,R ] + a d abcd } √ a } cd ,e ξ m ,e ac a m c } ab i { e mn m λ m ab 2 mna { b m 1 2 8 { mn n ,ω } − i ˜ 2 a b 2 ([ √ ,R ,ω ˜ ω } n , R c ,R − a 2 { e , a + nb a ξ − ] i [ 4 n + 0 mnb b ξ { ] √ } b ˜ mn [ cd { ˜ } + − d { 1 2 R { i i 4 ] b + } , i 2 n c 1 2 mn ,R mn cd mn } − 0 ab nb n b ˜ + ] ˜ }− ,ω n R ] + λ ,R b ,R { ,e } , ] + { ,ω ab ac c ,b a n ab 1 2 m b m ˜ ξ ab mn a ˜ m ξ ˜ [ a λ m m ω mn ˜ [ e , { R mnab b ,ω ] + { ω a , ([ 1 2 i { m { } − a ,R abcd i i b mb 2 ,R mn abcd ξ a b { [ − ξ i +2 − – 22 – mn ab { abcd i [ ] 2 i 2 i i ,R ]+ ˜ 4 λ 2 R ), respectively -up to some tuning of the numerical [ cd a 2 abcd √ − 16 , ab ˜ i ξ + 8 8 √ abcd abcd ] ]+ a m [ √ mn i a i 3.7 a ξ } m m 2 ]) ]) { ,ω 2 + ab } − + ] + ,R n ,e ,b 1 2 √ } − b cd cd } b n a m } m n ˜ a mn ξ [ a m ), which permits any use, distribution and reproduction in a mn ab + mna ,ω ,ω ] + ˜ ] + mna ) and ( ,e R + b ,R + ˜ n n R n b mn , n cd n abcd n a , b ξ a X [˜ mn ˜ a ,R [ 3.5 [ a ˜ ξ X i X a mn { ˜ ˜ ,R ξ [ R [ − ˜ { − ξ [ 2 b i − 2 , ] 8 ] ˜ ξ { ] 1 2 − − ,R √ 1 cd cd ] ] } − { b 2 16 1 cd + ab n a n a λ ξ − n } n n [ [ − { CC-BY 4.0 ] a ,ω ] + ] ,ω n mnb ,e ,ω ,b ] + ] + b a a ab m m This article is distributed under the terms of the Creative Commons abcd abcd ,e m m m b a ,R mn mn mn mn mn a a a m i i ([ ([˜ ˜ ˜ ab ˜ a R R 2 2 + + + 2 2 4 2 , ,R , ,R ,R { λ 2 4 0 0 0 0 0 √ √ m m m { i √ √ 2 [ [ [ [ [ X X X i i i i i − + − − − − ), which demonstrates that the transformation of the field strength tensor of the ======[ = [ = [ a a a a ab ab mn mn A.8 Eventually, we give the transformations of the component tensors, starting from mn mn mn mn ˜ mn mn R ˜ ˜ R R R δ δR R δ δR δR Open Access. Attribution License ( any medium, provided the original author(s) and source are credited. eq. ( theory is covariant.action) are: Their explicit expressions (related to the gauge invariance of the It is very welcomecurvatures give that the results the ( abovecoefficients- results after of considering the thethat transformations commutative was limit. of introduced the duethe In fields commutative to this and limit noncommutativity limit, is their decouples, thedescribed the therefore U(1) in SO(6), section gauge i.e. the 3. the field gauge conformal theory gravity, in in euclidean signature, we JHEP08(2020)001 , , 6 to all (2005) 4 ] R ]. 707 model in (2000) 360 (1996) 53 3 ]. φ SPIRE 567 ]. IN Nucl. Phys. B Proc. 376 ]. , (2007) 191601 ][ ]. SPIRE (1998) 363 IN ]. 99 hep-th/9804001 SPIRE Commun. Math. Phys. ][ SPIRE [ Nucl. Phys. B IN , , Gravity and the structure of IN SPIRE 294 [ 2 [ IN SPIRE [ IN CP Nucl. Phys. B Phys. Lett. B ][ , , theory on noncommutative hep-th/0607235 (1999) 1305 , Cambridge University Press, 4 (1992) 69 (1947) 874 [ φ The Standard Model as a Phys. Rept. ]. (1947) 38 9 Phys. Rev. Lett. Noncommutative perturbative dynamics 14 ]. ]. , 72 , 257 hep-th/0603044 Quantum field theory on noncommutative Gauge bosons in a noncommutative [ 71 SPIRE ]. SPIRE SPIRE IN (2008) 605 IN IN ][ hep-th/0403232 [ WKB approximation in noncommutative gravity – 23 – ][ [ , Academic Press inc., San Diego, CA, U.S.A. (1994). 12 On the ultraviolet behavior of quantum fields over Phys. Rev. SPIRE (2006) 054 , The spectral action principle Conceptual explanation for the algebra in the Phys. Rev. IN , ][ 04 ]. Renormalization of Exact renormalization of a noncommutative Finite gauge theory on fuzzy Class. Quant. Grav. (1989) 485 (2005) 13 ]. , Int. J. Mod. Phys. A ]. ]. , JHEP 71 SPIRE , 217 Particle models and noncommutative geometry IN SPIRE arXiv:0712.4024 hep-th/9912072 ][ SPIRE SPIRE [ [ IN . IN IN ][ ][ ][ hep-th/9606001 [ Quantized space-time London Math. Soc. Lect. Note Ser. Adv. Theor. Math. Phys. On quantized space-time Noncommutative geometry , An introduction to noncommutative differential geometry and its physical The fuzzy sphere , (2007) 125 (2000) 020 Phys. Lett. B (1991) 29 , ]. ]. 3 Divergencies in a field theory on quantum space Lett. Math. Phys. 02 18 , hep-th/0407089 (1997) 731 [ SPIRE SPIRE hep-th/9605001 arXiv:0706.3690 IN hep-th/9812180 IN noncommutative geometry: the low-energy[ regime geometry 186 noncommutative approach to the Standard[ Model dimensions 145 Suppl. JHEP orders noncommutative manifolds [ space-times and the persistence of[ ultraviolet divergences SIGMA [ applications Cambridge, U.K. (1999). noncommutative algebras C.P. Martin, J.M. Gracia-Bondia and J.C. Varilly, M. Dubois-Violette, J. Madore and R. Kerner, A.H. Chamseddine and A. Connes, A.H. Chamseddine and A. Connes, H. Grosse and H. Steinacker, A. Connes and J. Lott, S. Minwalla, M. Van Raamsdonk and N. Seiberg, H. Grosse and R. Wulkenhaar, H. Grosse and H. Steinacker, M. Chaichian, A. Demichev and P. Preˇsnajder, M. Burić,J. Madore and G. Zoupanos, T. Filk, J.C. Varilly and J.M. Gracia-Bondia, J. Madore, M. Burić,T. Grammatikopoulos, J. Madore and G. Zoupanos, H.S. Snyder, C.N. Yang, A. Connes, J. Madore, [7] [8] [9] [5] [6] [1] [2] [3] [4] [18] [19] [16] [17] [14] [15] [11] [12] [13] [10] References JHEP08(2020)001 ]. 52 ] ] ] ] (1999) ] SPIRE IN 09 ][ Prog. Theor. ]. , ]. Eur. Phys. J. C , JHEP , ]. hep-th/0611093 SPIRE hep-th/0206003 [ ]. ]. Noncommutative hep-th/0104153 [ IN SPIRE ]. [ [ hep-th/0102129 IN reduced model as [ hep-th/0107055 ]. SPIRE Fundam. Theor. Phys. ][ [ The Standard Model on N , IN SPIRE SPIRE SPIRE hep-ph/0111115 IN IN ][ (2007) 82 [ IN ][ ][ SPIRE [ (2002) 023 (1993) 84 IN Construction of non-Abelian gauge (2001) 383 A large 767 (2001) 148 ][ M theory as a matrix model: a 08 ]. Space-time structures from IIB matrix (2003) 413 21 305 ]. Field equations of massless fields in the 604 29 (2002) 363 SPIRE (1990) 323 JHEP hep-th/9711162 Classical bosons in a noncommutative Noncommutative differential geometry and IN Noncommutative GUTs, Standard Model and Enveloping algebra valued gauge , [ 23 Noncommutative geometry and matrix theory: [ SPIRE 31 hep-th/9612115 IN [ ]. hep-th/9802085 hep-th/9610043 Seiberg-Witten maps and noncommutative ][ [ [ Nucl. Phys. B – 24 – Describing curved spaces by matrices ]. , Phys. Lett. B hep-th/0205214 , [ Eur. Phys. J. C SPIRE (1998) 003 Nucl. Phys. B , IN Non-Abelian noncommutative gauge theory via , (1989) 1709 Eur. Phys. J. C SPIRE ][ 02 (1997) 467 ]. , 6 IN (1998) 713 (1997) 5112 Eur. Phys. J. C ][ J. Math. Phys. , , (2003) 45 498 String theory and noncommutative geometry 99 55 SPIRE JHEP hep-th/0508211 , IN [ 651 ][ hep-th/0006246 [ Nucl. Phys. B hep-ph/9209226 On a quark-lepton duality On a noncommutative extension of electrodynamics Phys. Rev. D (2006) 1295 , Class. Quant. Grav. , , ]. ]. ]. ]. ]. Nucl. Phys. B Prog. Theor. Phys. 114 , , hep-th/9908142 [ (2000) 521 SPIRE SPIRE SPIRE SPIRE SPIRE IN IN IN IN IN [ noncommutative space-time C,P,T Yang-Mills theories for arbitrary gauge[ groups Standard Model: model building noncommutative extra dimensions [ theories on noncommutative spaces [ new interpretation of the[ matrix model transformations for non-Abelian gauge groups17 on noncommutative spaces model Phys. conjecture superstring (1993) 285 [ compactification on tori 032 geometry new models of gauge theory X. Calmet, B. Jurˇco,P. Schupp, J. Wess and M. Wohlgenannt, P. Aschieri, B. Jurˇco,P. Schupp and J. Wess, G. Barnich, F. Brandt and M. Grigoriev, M. Chaichian, P. M.M. Preˇsnajder, Sheikh-Jabbari and A. Tureanu, B. Jurˇco,P. Schupp and J. Wess, B. Jurˇco,L. Möller,S. Schraml, P. Schupp and J. Wess, B. Jurˇco,S. Schraml, P. Schupp and J. Wess, H. Aoki, S. Iso, H. Kawai, Y. Kitazawa and T. Tada, M. Hanada, H. Kawai and Y. Kimura, K. Furuta, M. Hanada, H. Kawai and Y. Kimura, T. Banks, W. Fischler, S.H. Shenker and L. Susskind, N. Ishibashi, H. Kawai, Y. Kitazawa and A. Tsuchiya, A. Connes, M.R. Douglas and A.S. Schwarz, N. Seiberg and E. Witten, M. Dubois-Violette, R. Kerner and J. Madore, J. Madore, J. Madore, M. Dubois-Violette, J. Madore and R. Kerner, [36] [37] [34] [35] [32] [33] [31] [28] [29] [30] [26] [27] [24] [25] [21] [22] [23] [20] JHEP08(2020)001 1 ]. (2006) 03 SPIRE ]. 09 The SFIN A (2003) 441 (1961) 212 IN [ (1956) 1597 , , (2005), (2015) 442 (2010) 537 2 29 JCAP 63 58 SPIRE 101 , JHEP IN Fuzzy extra , (2016) 203 ][ Dynamical generation 191 from orbifolds for fuzzy Phys. Rev. ]. ]. 3 J. Math. Phys. ]. hep-th/0109162 , , ]. , Eur. Phys. J. C Fortsch. Phys. Fortsch. Phys. , , , Higher-dimensional unification Can noncommutativity resolve the Dimensional reduction over fuzzy Unified theories from fuzzy extra Renormalizable theories from SU(3) SPIRE SPIRE SPIRE IN IN Orbifolds, fuzzy spheres and chiral On the fermion spectrum of IN SPIRE hep-th/0306136 ][ [ ][ IN ][ ][ . (2004) 529 ]. Higher-dimensional unified theories with fuzzy extra – 25 – Springer Proc. Math. Stat. Resolving cosmological singularities ]. 36 ]. , ]. (2017) 322 hep-th/0310072 SPIRE arXiv:1008.2049 arXiv:0706.0398 [ [ [ IN The trinification model Higher-dimensional unified theories with continuous and Fermions on spontaneously generated spherical extra SPIRE SPIRE arXiv:1002.2606 14 ]. summer school in modern mathematical physics ][ [ IN IN ]. ]. [ SPIRE rd ]. ][ ]. IN SPIRE ][ , in 3 SPIRE SPIRE IN (2004) 034 (2010) 063 (2007) 017 SPIRE Eur. Phys. J. C IN IN ][ SPIRE 6 IN , (2010) 100 04 ][ ][ 09 IN ][ ][ 05 Lorentz invariance and the gravitational field JHEP Invariant theoretical interpretation of interaction decays in the noncommutative Standard Model SIGMA JHEP hep-th/0401200 Quantum field theory on noncommutative spaces arXiv:1612.05860 , , , , arXiv:0704.2880 [ JHEP gg ]. ]. , hep-th/0503039 Phys. Part. Nucl. Lett. , , γγ hep-th/0606021 [ SPIRE SPIRE → IN IN arXiv:1602.03673 arXiv:1504.07276 arXiv:0909.5559 hep-ph/0202121 (2017) 009 [ [ [ big bang singularity? [ spheres fuzzy coset spaces as extra dimensions fermions dimensions with continuous and fuzzy coset spaces as extra dimensions dimensions spontaneously generated fuzzy extra dimensions[ with fluxes 026 dimensions: dimensional reduction, dynamical(2007) generation 25 and [ renormalizability fuzzy higher dimensions pg. 135 [ of fuzzy extra dimensions, dimensional reduction and symmetry breaking Z [ coset spaces dimensions R. Utiyama, T.W.B. Kibble, R.J. Szabo, M. Maceda, J. Madore, P. Manousselis and G. Zoupanos, A.H. Chamseddine and V. Mukhanov, G. Manolakos and G. Zoupanos, G. Manolakos and G. Zoupanos, A. Chatzistavrakidis and G. Zoupanos, D. Gavriil, G. Manolakos, G. Orfanidis and G. Zoupanos, A. Chatzistavrakidis, H. Steinacker and G. Zoupanos, A. Chatzistavrakidis, H. Steinacker and G. Zoupanos, P. Aschieri, H. Steinacker, J. Madore, P. Manousselis and G. Zoupanos, H. Steinacker and G. Zoupanos, P. Aschieri, J. Madore, P. Manousselis and G. Zoupanos, P. Aschieri, T. Grammatikopoulos, H. Steinacker and G. Zoupanos, P. Aschieri, J. Madore, P. Manousselis and G. Zoupanos, P. Aschieri, J. Madore, P. Manousselis and G. Zoupanos, W. Behr, N.G. Deshpande, G. Duplancic, P. Schupp, J. Trampetic and J. Wess, [54] [55] [51] [52] [53] [49] [50] [47] [48] [45] [46] [43] [44] [41] [42] [39] [40] [38] JHEP08(2020)001 ] ]. , ]. SPIRE , Nucl. IN , ][ SPIRE ]. Proceedings, Phys. Lett. B Progress in IN Fortsch. Phys. , , ][ , in ]. , in ]. . The Poincaré, . The special SPIRE 1 2 arXiv:1809.03879 ]. (2003) 051 IN ]. 11 SPIRE ]. SPIRE IN SPIRE IN ]. ][ (2018) [ IN ][ arXiv:1208.0348 SPIRE Gravity as a gauge theory on JHEP SPIRE [ IN , IN [ 4 SPIRE ]. ]. arXiv:1312.6145 ][ S IN [ ]. models on the fuzzy sphere and their ][ 4 ]. φ ]. SPIRE SPIRE (2012) 070 Scalar and spinor field actions on fuzzy IN (1977) 1376] [ IN SPIRE ]. (1982) 976 [ ][ IN SPIRE hep-ph/0205040 08 38 [ SPIRE [ IN Matrix 25 Bounding noncommutative QCD IN [ arXiv:1809.02954 ]. SPIRE PoS(CORFU2017)162 [ (2014) 045019 ][ , IN hep-th/0109084 JHEP Review of the phenomenology of noncommutative – 26 – [ ][ , 90 (1993) 239 hep-th/0307292 4 F Anomaly freedom in Seiberg-Witten noncommutative SPIRE Quantum gauge theories on noncommutative [ S , Cocoyoc, Mexico (1980), pg. 545. IN Unified geometric theory of gravity and supergravity (2004) 179 (1980) 1466 28 (1982) 988 ][ Non-commutativity in unified theories and gravity Erratum ibid. (2017) 177 Gauge theories of gravity and supergravity 19 21 Phys. Rev. D [ 25 hep-lat/0606013 The construction on noncommutative manifolds using Scalar field theory on fuzzy , On gauge formulations of gravitation theories Gauge formulation of gravitation theories. Gauge formulation of gravitation theories. (2002) 013 [ ]. 263 Spontaneously broken de Sitter symmetry and the gravitational Monte Carlo simulation of a NC gauge theory on the fuzzy (2003) 068 hep-th/0204256 03 Phys. Rev. D [ bundle over , 07 2 F SPIRE , H. Ezawa and S. Kamefuchi eds., (1985), pg. 57 [ (1977) 739 S arXiv:1802.07550 IN [ JHEP (2006) 016 ][ Phys. Rev. D , 38 hep-ph/0107291 Phys. Rev. D Lett. Math. Phys. , JHEP as a [ , , 11 Noncommutative gauge theory and gravity in three dimensions , (2002) 177 3 Noncommutative gauge theory on fuzzy four sphere and matrix model Int. J. Mod. Phys. A CP , 637 ]. JHEP , (2001) 201 (2018) 1800047 : fuzzy SPIRE 4 IN hep-th/0212170 [ coherent states three-dimensional space 66 Springer Proc. Math. Stat. three-dimensional noncommutative spaces S [ G. Zoupanos, sphere Phys. B geometry continuum limits quantum field theory gauge theories 518 de Sitter and conformal cases conformal case Phys. Rev. Lett. holonomy group group theoretical methods in physics H. Grosse and P. Preˇsnajder, VitaleA. and Géré,P. J.-C. Wallet, G. Manolakos and G. Zoupanos, D. Jurman, G. Manolakos, P. Manousselis and G. Zoupanos, J. Medina and D. O’Connor, A. Chatzistavrakidis, L. Jonke, D. Jurman, G. Manolakos, P. Manousselis and D. O’Connor and B. Ydri, Y. Kimura, J. Medina, I. Huet, D. O’Connor and B.P. Dolan, I. Hinchliffe, N. Kersting and Y.L. Ma, B.P. Dolan, D. O’Connor and P. Preˇsnajder, F. Brandt, C.P. Martin and F. Ruiz, C.E. Carlson, C.D. Carone and R.F. Lebed, E.A. Ivanov and J. Niederle, E.A. Ivanov and J. Niederle, T.W.B. Kibble and K.S. Stelle, K.S. Stelle and P.C. West, E.A. Ivanov and J. Niederle, S.W. MacDowell and F. Mansouri, [73] [74] [71] [72] [69] [70] [66] [67] [68] [64] [65] [62] [63] [59] [60] [61] [57] [58] [56] JHEP08(2020)001 ]. , 01 75 (2003) Nucl. 894 SPIRE , Phys. , , IN 44 JHEP ]. ][ , , Cambridge, SPIRE (1985) 233 , IN (2019) 205005 ]. [ ]. Eur. Phys. J. C Nucl. Phys. B 119 36 , (2001) 33 , J. Math. Phys. SPIRE , 504 SPIRE IN : international conference IN General relativity with spin ][ arXiv:1901.03522 (1976) 393 ][ ]. [ 2001 , Ph.D. thesis, Department of 48 Phys. Rept. Grand unification in the spectral , SPIRE Gauge theory of the conformal and IN [ Strings Phys. Lett. B , Cambridge University Press, Cambridge, (2019) 010 , Class. Quant. Grav. Cambridge University Press , , , in 07 -dimensional fuzzy spheres, matrix models and 4 arXiv:1704.02863 ]. arXiv:1507.08161 [ (1977) 304 Rev. Mod. Phys. ]. [ – 27 – JHEP ]. , , Supergravity 69 Supergravity as a gauge theory of supersymmetry SPIRE ]. fuzzy anti-de Sitter space from matrix models Covariant cosmological quantum space-time, higher-spin Covariant SPIRE Curvature in noncommutative geometry IN ]. Conformal supergravity SPIRE ]. IN D ][ ]. IN ]. ][ Covariant non-commutative space-time (2015) 011 ]. ]. ][ SPIRE ]. (2017) 375202 SPIRE IN SPIRE 11 Noncommutative de Sitter and FRW spaces IN SPIRE ][ IN 50 SPIRE [ IN ][ Phys. Lett. B SPIRE SPIRE IN [ ]. SPIRE , IN IN Supersymmetry and higher spin fields Deforming Einstein’s gravity Invariant actions for noncommutative gravity [ [ ][ IN JHEP , ][ Scalar modes and the linearized Schwarzschild solution on a quantized Introduction to supersymmetry SPIRE Position operators, gauge transformations, and the conformal group (1977) 39 IN J. Phys. A arXiv:1508.06058 arXiv:1309.1598 Quantum gravity in de Sitter space , arXiv:1401.1810 [ [ (1966) 1021 [ Review of M(atrix)-theory, type IIB matrix model and matrix string theory 129 ]. hep-th/0106109 hep-th/0202137 143 [ SPIRE IN arXiv:1905.07255 hep-th/0009153 U.K. (2012). Theoretical Physics, Imperial College of Science and Technology, London, U.K. (1976). Phys. B superconformal group [ and gravity in the IKKT matrix model and torsion: foundations and prospects U.K. (2012) [ arXiv:1708.00734 FLRW space-time in Yang-Mills matrix[ models arXiv:1901.07438 (2001) [ 2534 [ Pati-Salam model higher spin Rev. (2014) 100 (2015) 58 (2015) 502 A.H. Chamseddine, A.H. Chamseddine and P.C. West, M. Kaku, P.K. Townsend and P. van Nieuwenhuizen, E.S. Fradkin and A.A. Tseytlin, D.Z. Freedman and A. Van Proeyen, F.W. Hehl, P. Von Der Heyde, G.D. Kerlick and J.M. Nester, P.G.O. Freund, H.C. Steinacker, M. Sperling and H.C. Steinacker, F. Fathizadeh and M. Khalkhali, E. Witten, B. Ydri, A.H. Chamseddine, A.H. Chamseddine, A. Connes and W.D. van Suijlekom, M. Sperling and H.C. Steinacker, H.A. Kastrup, A.H. Chamseddine, J. Heckman and H. Verlinde, M. Burićand J. Madore, D. Jurman and H. Steinacker, 2 [93] [94] [90] [91] [92] [88] [89] [86] [87] [83] [84] [85] [81] [82] [78] [79] [80] [76] [77] [75] JHEP08(2020)001 , 9 24 311 JHEP , : loop amplitudes, 2 Phys. Rev. D Conformal groups and , Nucl. Phys. B ]. , ]. , in , A. Barut and Int. J. Mod. Phys. A , SPIRE IN ]. SPIRE [ IN Comments on noncommutative ][ gravity coupled to fermions SPIRE IN = 4 ][ Gauge theory on noncommutative spaces D ]. Superstring theory. Volume ]. – 28 – arXiv:1806.10134 hep-th/0605113 , SPIRE [ ]. IN SPIRE ][ General relativity with a topological phase: an action hep-th/0001203 IN [ , Cambridge University Press, Cambridge, U.K. (1987). Noncommutative SPIRE ][ IN (2006) 92 Modeling position and momentum in finite-dimensional Hilbert [ , Berlin, Heidelberg, Germany (1985), pg. 3. 753 ]. (2000) 161 ]. -dimensional gravity as an exactly soluble system Springer 16 arXiv:0902.3817 SPIRE [ hep-th/0611174 IN SPIRE [ [ Emergent gravity from noncommutative spacetime IN From Heisenberg algebra to conformal dynamical group hep-th/0311163 [ Group theory of the spontaneously broken gauge symmetries , Nucl. Phys. B , (2009) 086 ´ Alvarez-Gaumé,F. Meyer and M.A. Vazquez-Mozo, (1988) 46 principle (2009) 4473 06 anomalies and phenomenology gravity Eur. Phys. J. C (1974) 1723 spaces via generalized Pauli operators related symmetries.physical results and mathematicalH.D. background Doener eds., L. Smolin and A. Starodubtsev, H.S. Yang, P. Aschieri and L. Castellani, M.B. Green, J.H. Schwarz and E. Witten, E. Witten, (2 + 1) L. J. Madore, S. Schraml, P. Schupp and J. Wess, A. Singh and S.M. Carroll, A. Barut, L.-F. Li, [98] [99] [96] [97] [95] [103] [104] [100] [101] [102]