Special Geometries Emerging from Yang-Mills Type Matrix Models 1 1 0 2 Daniel N. Blaschke∗ n UniversityofVienna,FacultyofPhysics a J Boltzmanngasse5,A-1090Vienna(Austria) 7 E-mail: [email protected] 2 ] Ireviewsomerecentresultswhichdemonstratehowvariousgeometries,suchasSchwarzschild h t andReissner-Nordström,canemergefromYang-Millstypematrixmodelswithbranes. Further- - p more, explicit embeddings of these branes as well as appropriate Poisson structures and star- e h products which determine the non-commutativity of space-time are provided. These structures [ aremotivatedbyhigherordertermsintheeffectivematrixmodelactionwhichsemi-classically 1 leadtoanEinstein-Hilberttypeaction. v 5 1 3 5 . 1 0 1 1 : v i X r a CorfuSummerInstituteonElementaryParticlesandPhysics-WorkshoponNonCommutativeField TheoryandGravity, September8-12,2010 CorfuGreece ∗Speaker. (cid:13)c Copyrightownedbytheauthor(s)underthetermsoftheCreativeCommonsAttribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/ GeometriesEmergingfromMatrixModels DanielN.Blaschke 1. Background In past years, various approaches to quantum space-time have been pursued. One possibility istoreplaceclassicalspace-timebyanon-commutativeonewherethecoordinatefunctionsxµ are promoted to Hermitian operators Xµ on a Hilbert space H . These “coordinate” operators satisfy certainnon-trivialcommutationrelations [Xµ,Xν]=iθµν, (1.1) whichinthesimplestcasereducetoaHeisenbergalgebra,i.e. withconstantθµν. Forareviewof suchnon-commutativefieldtheoriesseee.g.[1–4]. Inordertoincorporategravityinthiscontext, however,adynamicalnon-constantcommutatorθµν isrequired,whichsemi-classicallydetermines aPoissonstructureonspace-time. Incidentally,matrixmodelsofYang-Millstype1naturallyrealize thisidea—forareview,see[7]and[9–11]. Ourstartingpointishencethematrixmodelaction S =−Tr[Xa,Xb][Xc,Xd]η η , (1.2) YM ac bd whereη denotestheflatmetricofaD-dimensionalembeddingspacewitharbitrarysignatureand ab Xa are Hermitian matrices on H which in the semi-classical limit are interpreted as coordinate functions. If one considers some of the coordinates to be functions of the remaining ones [12] such that Xa ∼xa =(xµ,φi(xµ)) in the semi-classical limit, one can interpret the xa as defining theembeddingofa2n-dimensionalsubmanifoldM2n (cid:44)→RD equippedwithanon-trivialinduced metric g (x)=∂ xa∂ xbη =η +∂φi(x)∂ φj(x), (1.3) µν µ ν ab µν i j via pull-back of η , and where µ,ν ∈1,...,2n and i,j ∈2n+1,...,D. Here we consider this ab submanifoldtobeafourdimensionalspace-timeM4,andfollowing[12]wecaninterpret −i[Xµ,Xν]∼{xµ,xν} =iθµν(x) (1.4) pb as a Poisson structure on M4. Furthermore, we assume that θµν is non-degenerate, so that its inversematrixθ−1 definesasymplecticform µν 1 Θ = θ−1dxµ∧dxν (1.5) 2 µν onM4. However, it is not the induced metric which is “seen” by scalar fields, gauge fields, etc., but theeffectivemetric[9] (cid:113) detθ−1 µν Gµν =e−σθµρθνσg , e−σ ≡ . (1.6) ρσ (cid:112) detG ρσ 1In fact, a supersymmetric version, the 10-dimensional IKKT model [5], is expected to be UV finite and hence mightrepresentacandidateforsomeformofquantumgravitycoupledtomatter[6–8]. 2 GeometriesEmergingfromMatrixModels DanielN.Blaschke Therefore, an interesting special case where G = g may be considered. In fact, this corre- µν µν sponds to having a (anti-) self-dual symplectic form, i.e. (cid:63)Θ =±iΘ. This case, however is re- strictedto4-dimensionalsubmanifoldsM4,asinfourdimensionsonealwayshas|G|=|g|which makestheassumptionsabovepossible. (Fordetails,see[7].) Let us consider the following example in order to make the effective geometry clearer: The gaugeinvariantkinetictermofatestparticlemodelledbyascalarfieldφ hastheform S[φ]=−Tr[Xa,Φ][Xb,Φ]η ab (cid:90) √ ∼ d4x detθ−1θµν∂ xa∂ φθρσ∂ xb∂ φη µ ν ρ σ ab (cid:90) √ = d4x detθ−1Gνσ∂ φ∂ φ. (1.7) ν σ 2. Curvature g (x) M2n µν Figure1: Embeddingandinducedmetric ThebarematrixmodelEqn.(1.2)withoutmatterleadstothefollowinge.o.m. forXc: [Xa,[Xb,Xc]]η =0. (2.1) ab Furthermore,onecanderivethematrixenergy-momentumtensorTab whichreads[13,14] (cid:18) (cid:19) 1 (Gg) Tab=Hab− ηabH ∼ eσ ηab−Gµν∂ xa∂ xb , µ ν 4 4 1 Hab= [[Xa,Xc],[Xb,X ]] ∼ −eσGµν∂ xa∂ xb, c + µ ν 2 H =Habη , (2.2) ab andwhoseconservationfollowsdirectlyfromthematrixequationsofmotion(2.1)above: [Xa,Ta(cid:48)b]η =0. (2.3) aa(cid:48) Interestingly, there is a close connection between the matrix energy-momentum tensor and the projectorsonthetangential/normalbundleofM ∈RD Pab=gµν∂ xa∂ xb, Pab=ηab−Pab. (2.4) T µ ν N T 3 GeometriesEmergingfromMatrixModels DanielN.Blaschke Namely,inthespecialcasewherebothmetricscoincide,i.e. theself-dualcasewhereG =g , µν µν onehasTab∼eσPab andHab∼−eσPab inthesemi-classicallimit. N T Furthermore, one easily derives the relation ∇g∇gxa =Pab∇G∇Gx , where ∇g/∇G are the σ ν N σ ν b covariant derivatives defined with standard Christoffel symbols with respect to g/G, respectively. Hencethecurvaturetensorwithrespecttotheinducedmetricgcanbewrittenas R [g]=∇g∇gxa∇g∇gx −∇g∇gxa∇g∇gx ρσνµ σ µ ρ ν a σ ν µ ρ a =Pab∇G∇Gx ∇G∇Gx −Pab∇G∇Gx ∇G∇Gx , (2.5) N σ µ a ρ ν b N σ ν a µ ρ b wherethefirstlineissimplytheGauss-Codazzitheorem,andLatinindiceswerepulleddownwith the embedding metric η . Using the tensorC :=∂ xa∇G∂ x allows to relate the curvature ab α;µν α µ ν a tensorsassociatedwithG/g: R [g]=(Gg)ηR [G]+∇GC −∇GC −C C gαβ+C C gαβ. (2.6) ρσνµ µ ρσνη σ µ;ρν ρ µ;σν α;σµ β;ρν α;σν β;µρ It was previously shown in [13, 14], that the Einstein-Hilbert action emerges in the effective matrixmodelaction. Inparticular,acertaincombinationoforder10matrixtermssemi-classically leadsto √ (cid:90) g SO(X10)∼ d4x(2π)2e2σ(R[g]−3Rµν[g]hµν)+O(∂h2), whereG =g +h isalmostself-dual. Intheself-dualcase(i.e. h=0),thisreducesto µν µν µν (cid:16) (cid:17) (cid:90) √ S =Tr 2Tab(cid:3)X (cid:3)X −Tab(cid:3)H ∼−2 d4x ge2σR, (2.7) O(X10) a b ab where(cid:3)Y ≡[Xa,[X ,Y]],andadditionallyonefindstheorder6matrixterms a (cid:18) (cid:19) 1 S =Tr [Xc,[Xa,Xb]][X ,[X ,X ]]−(cid:3)Xa(cid:3)X O(X6) 2 c a b a (cid:90) √ (cid:18)1 (cid:19) ∼ d4x g θµρθηαR −2eσR+2eσ∂µσ∂ σ . (2.8) µρηα µ 2 Ingeneral, however, thedegreesoffreedomaregivenbytheembeddingφi andthedeviation fromtheself-dualPoissonstructureA ,i.e.: µ θ−1=θ¯−1+F =θ¯−1+∂ A −∂ A , µν µν µν µν µ ν ν µ δ g =δφiφjη +φiδφjη , φ µν ij ij δ F =∂ δA −∂ δA , A µν µ ν ν µ 1 h =−eσ¯(θ¯−1gF) −eσ¯(Fgθ¯−1) − g (θ¯F)+O(F2), (2.9) µν µν µν µν 2 where θ¯−1 denotes a self-dual Poisson structure with respect to a given metric g (φi). It was in µν µν factarguedin[14],thatthetreelevelactionEqn.(1.2)shouldsingleoutalmostself-dualgeometries andthatcertainpotentialtermssetthenon-commutativityscaleeσ ≈const. In the following section, we will consider two examples of geometries which are expected to solvethee.o.m. oftheeffectivematrixmodel(i.e. includinghigherordercontributions)tooagood approximation[15],atleastatsomedistancefromthehorizons. 4 GeometriesEmergingfromMatrixModels DanielN.Blaschke 3. SpecialGeometries 3.1 SchwarzschildGeometry We now continue with the special example of Schwarzschild geometry, and our construction involvestwosteps[15,8]: First, the choice of a suitable embedding M4 ⊂ RD must be made such that the induced geometryonM4givenbyg istheSchwarzschildmetric,andthenonneedstofindasuitablenon- µν degenerate Poisson structure on M4 which solves the e.o.m. ∇µθ−1 =0 for self-dual symplectic µν formΘ. Bothstepsarefarfromuniqueapriori. However,thefreedomisconsiderablyreducedby requiringthatthesolutionshouldbea“localperturbation”ofanasymptoticallyflat(ornearlyflat) “cosmological”background. Thisisclearonphysicalgrounds, havinginmindthegeometrynear a star in some larger cosmological context: It must be possible to approximately “superimpose” our solution, allowing e.g. for systems of stars and galaxies in a natural way. This eliminates the well-knownembeddingsoftheSchwarzschildgeometryintheliterature[16–18],whicharehighly non-trivialforlargerandcannotbesuperimposedinanyobviousway. Furthermore,theembeddingshouldbeasymptoticallyharmonic(cid:3)xa →0forr→∞,inview of the fact that there may be terms in the matrix model which depend on the extrinsic geometry, and which typically single out such harmonic embeddings2. Additionally, θµν should be non- degenerate,andθµν →const.(cid:54)=0asr→∞. WestartbyconsideringEddington-Finkelsteincoordinatesanddefine: (cid:12) (cid:12) t =tS+(r∗−r), r∗=r+rcln(cid:12)(cid:12) r −1(cid:12)(cid:12), (3.1) (cid:12)r (cid:12) c wheret denotes the usual Schwarzschild time, r is the horizon of the Schwarzschild black hole S c and r∗ is the well-known tortoise coordinate. The metric in Eddington-Finkelstein coordinates {t,r,ϑ,ϕ}isgivenby (cid:16) r (cid:17) 2r (cid:16) r (cid:17) ds2=− 1− c dt2+ cdtdr+ 1+ c dr2+r2dΩ2, (3.2) r r r whichisasymptoticallyflatforlarger,andmanifestlyregularatthehorizonr andthusallowsus c tofindanembeddingwhichfulfillstherequirementslistedabove. Inparticular,weneedatleast3 extradimensions: (cid:114) 1 r φ +iφ =φ eiω(t+r), φ = c, (3.3) 1 2 3 3 ω r where φ is time-like and ω is some parameter which does not enter the metric (3.2). Hence, our 3 2Thiscanholdonlyasymptotically,sinceRicci-flatgeometriescaningeneralnotbeembeddedharmonically[19]. 5 GeometriesEmergingfromMatrixModels DanielN.Blaschke Figure2: EmbeddedSchwarzschildblackhole. 7-dimensionalembeddingisgivenby   t  rcosϕsinϑ       rsinϕsinϑ    xa= rcosϑ  (3.4)  1(cid:112)rccos(ω(t+r))  ω r   1(cid:112)rcsin(ω(t+r))  ω r  1(cid:112)rc ω r withbackgroundmetricη =diag(−,+,+,+,+,+,−). ab On the top of Fig. 2, a schematic view of the outer region of the Schwarzschild black hole is shown. Afterpassingthroughthehorizonr=r ,theextradimensionsφ “blowup”inacone-like c i manner. As indicated in the lower half of this figure, every point of the cone is in fact a sphere √ whose radius r becomes smaller towards the bottom of the cone (i.e. T ∝ 1/ r). The twisted verticallinesdrawnintheconearelinesofequaltimet. Forthesymplecticform,werequire(cid:63)Θ =iΘ,sothattheeffectiveandtheinducedmetricco- incide,i.e. Gµν =eσθµρθνσg =gµν,and lime−σ =const.(cid:54)=0. Onethenfindsthesolution[15] ρσ r→∞ Θ =iE∧dt +B∧dϕ, S E =c (cosϑdr−rγsinϑdϑ)=d(f(r)cosϑ), 1 B=c (cid:0)r2sinϑcosϑdϑ+rsin2ϑdr(cid:1)= c1d(r2sin2ϑ), 1 2 (cid:16) r (cid:17) γ = 1− c , f(r)=c rγ, f(cid:48)=c =const., (3.5) 1 1 r 6 GeometriesEmergingfromMatrixModels DanielN.Blaschke fromwhichfollows (cid:16) r (cid:17) e−σ =c2 1− csin2ϑ ≡c2e−σ¯ , (3.6) 1 r 1 where c is an arbitrary constant. Clearly, near the horizon e−σ ≈const. is not fulfilled, meaning 1 our approximations break down in that region. Asymptotically, however, this seems to be a valid solutionwhichapproximatelyfulfillsallrequirementslistedatthebeginningofthissection. Furthermore,Eqn.(3.5)suggeststoworkinDarbouxcoordinatesxµ ={H ,t ,H ,ϕ}corre- D ts S ϕ spondingtoKillingvectorfieldsV =∂ ,V =∂ wherethesymplecticformΘ isconstant: ts ts ϕ ϕ Θ =ic dH ∧dt +c dH ∧dϕ, 1 ts S 1 ϕ (cid:0) (cid:1) =c d iH dt +H dϕ , 1 ts S ϕ 1 H =rγcosϑ, H = r2sin2ϑ. (3.7) ts ϕ 2 TherelationstotheKillingvectorfieldsaregivenby E =c dH =c E dxµ =i Θ, E =Vνθ−1, 1 ts 1 µ Vts µ ts νµ B=c dH =c B dxµ =i Θ, B =Vνθ−1, (3.8) 1 ϕ 1 µ Vϕ µ ϕ νµ andthemetricinDarbouxcoordinatesreads eσ¯ eσ¯ ds2 =−γdt2+ dH2+r2sin2ϑdϕ2+ dH2. (3.9) D S γ ts r2sin2ϑ ϕ Hence,aMoyaltypestarproductcaneasilybedefinedas   0 i 0 0 (g(cid:63)h)(x )=g(x )e−2i(cid:16)←∂−µθDµν→−∂ν(cid:17)h(x ), with θµν =ε−i 0 0 0, (3.10) D D D D  0 0 0 1   0 0 −1 0 where ε =1/c (cid:28)1 denotes the expansion parameter. Transforming back to embedding coordi- 1 nates,thestarproductreads (cid:34) (cid:32) iε (cid:18)←− ir zeσ¯ ←− (cid:19) →− (g(cid:63)h)(x)=g(x)exp ∂ c + ∂ ieσ¯ ∧ ∂ 2 t r2γ z t (cid:33)(cid:35) (cid:18)(cid:16)←− ←− z(cid:17)r eσ¯ (cid:16)←− ←− (cid:17) 1 (cid:19) (cid:16) →− →− (cid:17) c + ∂ −∂ + ∂ x+ ∂ y ∧ x∂ −y∂ h(x) (3.11) t zr r2 x y x2+y2 y x wherethewedgestandsfor“antisymmetrized”,andwhenconsideringtheexpansioncaremustbe takenwiththesequenceofoperatorsandthesidetheyacton. Toleadingorderonehencefindsthe 7 GeometriesEmergingfromMatrixModels DanielN.Blaschke starcommutators  0 −rcy rcx −i izf1+2(1) izf2−1(1) izφ3  r2 r2 r r 2r2  −rrcr2ycx −e0−σ¯ e−0σ¯ −rcrxrczy3z −xyf1f+2r1+2((γγ)) −xyf2f−1r2−1((γγ)) −xγy2φγr3φ23 (cid:104) (cid:105)  r2 r3 r r 2r2  −i xa(cid:63),xb =εeσ¯  i rcyz −rcxz 0 −iωφ iωφ 0   r3 r3 2 1  −izf1+2(1) yf1+2(γ) −xf1+2(γ) iωφ 0 −iωzφ32 −iωzφ3φ2  r r r 2 2r2 2r2  −izfr2−1(1) yf2−1r(γ) −xfr2−1(γ) −iωφ1 iω2zrφ232 0 iω2zφr23φ1 −izφ3 yγφ3 −xγφ3 0 iωzφ3φ2 −iωzφ3φ1 0 2r2 2r2 2r2 2r2 2r2 +O(ε3), (3.12) where (cid:18) (cid:19) Y f±(Y)= φ ±ωφ . (3.13) ij 2r i j This defines a Poisson structure on M4, but it could also be viewed as a Poisson structure on the 6-dimensionalspacedefinedbyφ2+φ2=φ2 whichadmitsM4 assymplecticleaf. 1 2 3 Higher orders in this star product, however, lead to non-commutative corrections to the em- beddinggeometry,suchasφ (cid:63)φ +φ (cid:63)φ (cid:54)=φ (cid:63)φ . 1 1 2 2 3 3 3.2 Reissner-NordströmGeometry SimilartotheSchwarzschildcase,onecanfindanembeddingwithself-dualsymplecticform also for the Reissner-Nordström metric. In spherical coordinates xµ ={t,r,ϑ,ϕ} the according lineelementreads (cid:18) 2m q2(cid:19) (cid:18) 2m q2(cid:19)−1 ds2=− 1− + dt˜2+ 1− + dr2+r2dΩ. (3.14) r r2 r r2 Thegeometryhastwoconcentrichorizonsat (cid:16) (cid:112) (cid:17) r = m± m2−q2 . (3.15) h Shiftingthetime-coordinateaccordingto (cid:16) (cid:17)−1 t =t˜+(r∗−r), withdr∗≡ 1−2m+q2 dr, (3.16) r r2 onearrivesat (cid:18) 2m q2(cid:19) (cid:18)2m q2(cid:19) (cid:18) 2m q2(cid:19) ds2=− 1− + dt2+2 − dtdr+ 1+ − dr2+r2dΩ. (3.17) r r2 r r2 r r2 Wechoosea10-dimensionalembeddingM1,3 (cid:44)→R4,6 whichhastheadvantageofhavingsimilar propertiescomparedtotheSchwarzschildcase3. Theadditionalcoordinatesφ aregivenby i (cid:114) 1 2m φ +iφ =φ eiω(t+r), φ = , 1 2 3 3 ω r q φ +iφ =φ eiω(t+r), φ = , (3.18) 4 5 6 6 ωr 3Thischoice,ofcourse,isfarfromunique.Alternatively,wecouldhaveuseda7-dimensionalembedding,butwhich wouldhavebeenvalidonlyuptotheinnerhorizon. Infact,allphysicallyrelevantgeometriesshouldbeembeddablein 10-dimensions,atleastlocally[20]. 8 GeometriesEmergingfromMatrixModels DanielN.Blaschke where φ , φ and φ are time-like coordinates. Like in the previous case, ω does not enter the 3 4 5 inducedmetric(3.17),butishiddenintheextradimensionsφ. Forr→∞,theφ becomeinfinites- i i imally small and hence asymptotically, the four dimensional subspace becomes flat Minkowski space-time. An according self-dual symplectic form can be derived which in metric compatible Darboux coordinatesreads 1(cid:0) (cid:1) Θ = idH ∧dt˜+dH ∧dϕ , t˜ ϕ ε r2(cid:18) q2(cid:19) H =γrcosϑ, H = 1− sin2ϑ, t˜ ϕ 2 r2 (cid:18) 2m q2(cid:19) γ = 1− + . (3.19) r r2 Thenon-commutativityscaleintheouterregion(i.e. atsomedistancetothehorizon)isgivenby (cid:18) q2(cid:19)2 e−σ¯ =γsin2ϑ+ 1− cos2ϑ, (3.20) r2 andtheReissner-NordströmlineelementinDarbouxcoordinatesreads eσ¯ eσ¯ ds2 =−γdt˜2+ dH2+r2sin2ϑdϕ2+ dH2, (3.21) D γ t˜ r2sin2ϑ ϕ a form similar to the according Schwarzschild metric (3.9). In the limit q→0 these expressions reduce to those in the Schwarzschild case4. Once more, a Moyal type star product can be defined as (cid:16)←− →− (cid:17) (g(cid:63)h)(x )=g(x )e−2i ∂µθDµν∂ν h(x ), (3.22) D D D with the same block-diagonal θµν as before. Higher orders in this star product lead to non-com- mutative corrections to the embedding geometry, such as φ (cid:63)φ +φ (cid:63)φ (cid:54)=φ (cid:63)φ and φ (cid:63)φ + 1 1 2 2 3 3 4 4 φ (cid:63)φ (cid:54)=φ (cid:63)φ (see[15]fordetails). 5 5 6 6 4. Outlook Inthisshortproceedingnote, explicitembeddingsofSchwarzschildandReissner-Nordström geometriesincludingself-dualsymplecticformshavebeendiscussedinthecontextofapproxima- tivesolutionstothee.o.m. ofaneffectivematrixmodelofYang-Millstype. Itwaspointedout,that in a future effort the embeddings should be modified near the horizons to account for nearly con- stanteσ. Openquestions,amongothers,concerndeviationsfromG=gandhigherorderquantum effects. Acknowledgements Manythanksgototheorganizersofthe2010workshoponnon-commutativefieldtheoryand gravityinCorfu,whichwasawonderfulandstimulatingconference. 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