Perturba)vefeaturesofthe
wavefunc)onoftheuniverseforpuregravity
JuanMaldacenaIns)tuteforAdvancedStudy
Pascos2011,Cambridge
Outline
• Thewavefunc)onoftheuniverseinEAdSanddS
• Thewavefunc)onfor5ddeSiGer
• 4ddeSiGerorEAdSandconformalgravity.
DeSiGerspace
• Expandinguniverse(Poincarepatch)
!
ds2
= "dt2
+ e2tdx
2
Asympto)cfuture
!
ds2
="d#2 +dx
2
#2
η=0
past
Proper)me
Conformal)me
ComovingvsPhysicaldistances.• xis``comovingposi)on’’.Physicaldistanceisexponen)allygrowing.(x=constant,geodesicofapar)cle‘’atrest’’.)
• Transla)onsymmetrymomentumisconserved.
!
eikx
!
ds2
="d#2 +dx
2
#2
Horizon
Crossedatη=x,orkη=1
FollowafixedkmodeEarly)mes,largephysicalmomentum,likeplanewavesinflatspaceBunchDaviesvacuum.Lookingatafixedkmodeatlate)meslookingatsuperhorizondistances.
!
ds2
="d#2 + dx
2+ hijdx
idx
j
#2
PuregravityLookatmetricfluctua)ons.
Gravitywavefluctua)onsbecomeconstantatlate)mesWavefunc)onbecomes``scaleindependent’’forlargescalefactors:
Forsuperhorizondistances! ! e!M2
H2 k3h2
Gaussianforeachmomentummode
SimplestThreepointfunc)on.CanbecomputeddirectlybyexpandingtheEinsteinac)ontocubicorder.ConformalsymmetryrestrictsitsformOnly3possibleshapesEinsteingravityproducesonlyoneoftheseshapes.Convenienttouseaspinorhelicityformalism,almostiden)caltotheoneusedtodescribescaGeringamplitudesinflatspace.
JM&Pimentel
NonGaussiancorrec)ons
FlatspaceamplitudesfromdScorrelators
• Computecorrela)onfunc)onsofstresstensors
• Wedonothave``energy’’conserva)on• SingularityoftheAdS(ordS)treeamplitudeistheflatspacetreeamplitude.
JM,Pimentel,Raju,…
(BothdeltaFunc)onstripped)
Inprogress
PolchinskiGiddingsPenedones….
!T (1) · · ·T (n)" #!n
i=1 |!ki|(|k1|+ · · ·+ |kn|)n!1
An,Flat
A
Imaginecompu)ngalltreediagramsLeadingcontribu)ontohigherpointfunc)ons.
Containedinaclassicalsolu)onofEinstein’sequa)onwithfixedfuture(andpastBD)boundarycondi)ons.
Fixtheboundarycondi)onsforthemetricinthefuturetoanarbitraryshape.Impose(interac)ng)BunchDaviesboundarycondi)onsinthepast.Solu)ondecayswhenη‐(1+iε)∞.Feynmanboundarycondi)onsinflatspace.Thisprescrip)onworkstoanyorderinperturba)ontheory.
!
"# eiS = eiM
2
H2
g (R +12)$
Evaluatetheclassicalac)ononaclassicalsolu)on
Ψ[]
Focusononeoftheoscilla)ngfactorsintheHartleHawkingpicture(asweusuallydowhenlookingattheKleinGordonequa)on).
Divergent“counterterms”Purephases,dropoutfrom|Ψ|2Interes)ngpart
Late)mebehavior
!R("2gb) = !R(gb)
! ! eiS = eic!d4x
!g(R+12) = eic
!d3x
!gb+ic
!d3x
!gbRb!R(gb)
c =M2
pl
H2
EAdSvs.dS• Thecomputa)onofthedSwavefunc)onisverysimilartothecomputa)onoftheEAdSwavefunc)on.
• InEAdS:Alsoevaluatethe``wavefunc)on’’,asinHartle‐Hawking.Wefocusontheexponen)allyincreasingwavefunc)oninthiscase.
• Inperturba)ontheory,theyarerelatedinaverysimpleway.
(Genera)ngfunc)onofcorrela)onfunc)ons)
EAdSdSanaly)ccon)nua)on
z ! "i! , RAdS = "iRdS ,
ds2 = R2AdS
dz2 + dx2
z2, ! ds2 = R2
dS"d!2 + dx2
!2
g ! e!wz , z " # " g ! eiw! , ! " $#
Inflatspacecon)nua)onfromEuclideanspaceIndeSiGercon)nua)onfromEAdS.
Theboundarycondi)onsalsotransformproperly:
Decayingoscilla)ngwithonefrequency
• Thisworksalsoatlooplevel.
• Expecta)onvaluesvs.Wavefunc)ons:
!BD|!!|BD" Analy)ccon)nua)onfromSphere
![gb] = !gb|BD" Analy)ccon)nua)onfromEAdS.
JMHarlow,Stanford
TurningEAdScomputa)onsintodSones
• Wecouldconsidertheac)onforanS3boundaryinEAdS.(TheCFTpar))onfunc)ononS3)GivesusualHartle‐HawkingfactorforS3
• BlackholefreeenergiesinEAdSGiveHartle‐HawkingfactorsforS2xS1β.Metricsarecomplex!.
|!|2 ! eSdS
SimilartoHartle,Hawking,Hertog
JM
DeSiGterentropy
dS/CFT• Thewavefunc)onΨ[gb]=Z[gb]CFT
• Atoneloopweexpecttostartgetngexponen)alsuppressions
• Suppressionoffluctua)onsatshortdistances.• Likeanexclusiveamplitudeinamasslessgaugetheory.• Objec)onstodS/CFTgoaway.(Bubbledecaysfieldtheorieswithboundaries,etc..)
! ! eiM2
H21!3c
! !gb" 1
!3c
! !g+···
StromingerWiGen(JM)
EAdS4ordS4gravitywavefunc)on
• Canbeevaluatedattreelevelusingtheclassicalsolu)on.
• Wewillshowthatthewholecomputa)oncouldalsobeviewedasaprobleminconformalgravity.
ConformalGravity
• Gravitythatinvolvesonlythe“conformalclass”ofthemetric.
• Overallrescalingsofthemetric(orWeyltransforma)onsofthemetric)donotmaGer.
• Ac)ondependsonlyontheWeyltensor
!
gµ" #$2gµ"
!
S = W2"
• Equa)onsofmo)on4thorderinderiva)ves
• Leadstoghosts.• Aroundflatspace,thesolu)onsgolike
!
eiEt,te
iEt
andcomplexconjugates.
Flatspacehamiltoniannondiagonalizable
Twoproper)es:
• Solu)onsofpuregravityEinstein’sequa)onswithacosmologicalconstantarealsosolu)onsoftheequa)onsofmo)onofconformalgravity.
• Renormalizedac)onondSSameasac)onofconformalgravityonasolu)onofEinstein’sequa)ons.
AndersonMiskovicOleaArosContrerasOleaTroncoso,Zanelli
!
Euler = W2"" + 2 Ricci
2" #1
3R2
Equa)onsofmo)onofWeylgravityInvolvesRiccitensor.ForEinsteinspaces:
Evalua)ngtheEinsteinac)ononanEinsteinspaceSameasevalua)ngthe4volume.
SE,Renormalized =
!d4x
!g " Boundary =
!d4x
!gW 2 " (Euler Number)
SE !!
"g !
!(W 2 # E)
Usefuliden)ty:
Conformalgravitylagrangian~(Einsteinequa)ons)^2
Rµ! ! gµ!
!Sconformal
!gµ!! gµ! " 0
• IfwecanselecttheEinsteinsolu)onsfromthemorenumeroussolu)onsofconformalgravitywecanforgetabouttheEinsteinac)onandcomputeeverythingintermsoftheconformalgravityac)on.
• WegetanexplicitlyIRfinitecomputa)on.
• Asimpleboundarycondi)ononthefieldsofconformalgravityselectstheEinsteingravitysolu)ons.
• Conformalgravityequa)ons:4thorder.2boundarycondi)onsinthepastfromBunchDavies(orEAdScondi)ons).Twointhefuture:
gij(! = 0) = gbij , "!gij(! = 0) = 0
ds2 =!d!2 + (g0 + !2g2 + !3g3 + · · · )dxdx
!2Einsteinsolu)ons.
No)mederiva)ve
StarobinskiFeffermanGraham
!
"Conformal[h,h'= 0] ="Einstein,Renormalized [h]
‐ Wegetthe``right’’signfortheconformalgravityac)onfordSandthe``wrong’’oneforEAdS
‐ Theoverallconstantissimplythe``central’’charge,orthedeSiGerentropy,whichisgivenbyM2/H2
‐ ThisisalsotheonlydimensionlesscouplingconstantforpuregravityindS(orAdS)…(attreelevel).
!
c =M
2
H2ec
!W 2!E = eSE,Renormalized
WecanusethepropagatorsofconformalgravitywithaNeumanncondi)on+thever)cesofconformalgravityOrTheusualpropagatorsofEinsteingravity
!
h = (1" ik#)eik#
Ordinaryde‐SiGerwavefunc)ons:
Canbeviewedasthecombina)onofconformalgravitywavefunc)onsobeyingtheNeumannboundarycondi)on.
Ghosts?
• Withaboundarycondi)on,conformalgravitygavethesameresultsasordinarygravity.Thuswegotridoftheghosts.
• Allwedid,wastoevaluatetheghostwavefunc)onsatzerovaluesfortheghostfields.
• Thisisrathertrivialattreelevel.
QuantumQues)ons
• SomeversionsofN=4conformalsugraappeartobefinite.
• (oneoftheseappearsfromthetwistorstringtheory)
BerkovitsWiGen
FradkinTseytlin
Quantumques)ons:
• CanthequantumtheorywithaNeumannboundarycondi)onbeinterpretedastheresultofaUnitarybulktheory?
‐Onlysuperhorizonwavefunc)on,onlyonesnapshot.
‐Weexpectproblemswithunitarityhowdotheseappear?
‐Gravity+Pauli‐VillarsghostfieldMakingmasscomparabletoAdS(ordS)scalegivesconformalgravity.
Conclusions• ConformalgravitywithNeumannboundarycondi)onsisequivalent(attreelevel)toordinarygravityonsuperhorizondistances.
• InAdS:Thepar))onfunc)onofconformalgravitywithNeumannboundarycondi)onsisthesameasthatofordinarygravity
• Thisisnon‐linear,butclassical(orsemiclassical)rela)on
• Itwouldbeinteres)ngtoseewhathappensinthequantumcase.OneprobablyneedstodoitforN=4conformalsugra,whichisfinite.
Anotherapplica)onofconformalgravity
Solu)onofthetreelevel5dmeasureforpure5dgravity.
Findingtheprobabilityfordifferentshapesforthespa)alsec)ons.
5dpuregravityindeSiGer
• Gravitywithposi)vecosmologicalconstant• ConsidertheBDvacuumintheweaklycoupledregime,
R3
GN! 1
ds2 =!d!2 + gijdxidxj
!2gij = !ij + hij
!(gij) Wavefunc)onoftheuniverse
• Similarto4dcase.• UsetheEAdSdSanaly)ccon)nua)on.• Onecrucialdifference:
!(1
!2gij) = ecAdS[ 1
!4
! !g+ 1
!2
! !gR+log !
!W 2"E+Finite(g)]
InEuclideanspace,wehavearealanswer:
cAdS =R3
AdS
GN! i
R3dS
GN
Alltermsbecomepurelyimaginary,includingthefiniteterm.Theonlyrealpartarisesvia
log ! ! log |"0|+ i#
2
Ac)onofconformalgravity Givesatopologicalterm,theEulernumber.
Starobinski,Fefferman‐GrahamHeningson‐Skenderis
|!|2 = e!cdS!!d4x
"g(W 2!E)
(Dependsonthemetricofthefourdimensionalslice)
ItiscompletelylocalItwasnon‐localinevenbulkdimensions.Inthreebulkdimensions,ordS3gravity,wegetonlytheEulernumberonlythetopologyofthespacemaGers.
Conclusions,5d
• Infivedimensionalde‐SiGerthereisahugesimplifica)onifwecomputethewavefunc)on.
• Wesimplygettheac)onofconformalgravityin4d.Thisisthe4dspa)alsliceofthe5dgeometryatsuperhorizondistances.
StringsandRigidstrings
Susual =
!!g
Srigid =
!!gK̂i
abK̂i,ab
Inducedmetric
Theproblemofcompu)ngaWilsonloopinAdSisequivalenttocompu)ngaWilsonloopinflatspacewiththerigidstringac)on,withanextraNeumanboundarycondi)ononthefields.ValueoftheWilsonloop‐counterterm=Valueoftherigidstringac)on.
Polyakov
Alexakis
Weylinvariantintargetspace.
(PointedoutbyPolyakov)
Xµ(! = 0) = fµ(") , #!Xµ(! = 0) = 0
MembranesindSandrigidstrings
Membrane(domainwallin4d)iscreatedintheprobeapproxima)on.(Orconnec)ngsameenergyvacua).ItsdSboundaryisatwodimensionalsurface.ThetreelevelprobabilitythatthissurfacehasagivenshapeGivenbytherigidstringac)on.
|!(X)|2 = e!R3T!(Srigid!Euler)
Sameargumentusingtheconformalanomalyforthemembraneac)onBerenstein,Corrado,JMFischlerGraham,WiGen
dSΣ3
(!!)2
TheEnd