Special Geometries Emerging from Yang-Mills Type Matrix Models

0.3 MB·English

by  Daniel N. Blaschke

Special Geometries Emerging from Yang-Mills Type
Matrix Models
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Daniel N. Blaschke∗
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UniversityofVienna,FacultyofPhysics
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J Boltzmanngasse5,A-1090Vienna(Austria)

7 E-mail: [email protected]
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Ireviewsomerecentresultswhichdemonstratehowvariousgeometries,suchasSchwarzschild
h
t andReissner-Nordström,canemergefromYang-Millstypematrixmodelswithbranes. Further-
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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

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leadtoanEinstein-Hilberttypeaction.
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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].
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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
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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.
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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].
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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
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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
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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].
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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. Thisworkwassupportedby
theAustrianScienceFund(FWF)undercontractP21610-N16.
4Note,that3oftheextradimensions,namelyφ4−6reducetoapointinthislimitsince limφ6(q)=0.
q→0
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GeometriesEmergingfromMatrixModels DanielN.Blaschke
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