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