New Compactifications of Supergravities and Large N QCD

We construct supergravity backgrounds representing non-homogeneous compactifications of d=10,11 supergravities to four dimensions, which cannot be written as a direct product. The geometries are regular and approach $AdS_7\times S^4$ or $AdS_5\times S^5$ a

9

991 anJ 91 3v711808/9ht-pe:hviXraImperial/TP/97-98/68hep-th/9808117

NewCompacti cationsofSupergravities

andLargeNQCD

JorgeG.Russo

TheoreticalPhysicsGroup,BlackettLaboratory,ImperialCollege,LondonSW72BZ,U.K.

Abstract

Weconstructsupergravitybackgroundsrepresentingnon-homogeneouscompacti ca-tionsofd=10,11supergravitiestofourdimensions,whichcannotbewrittenasadirectproduct.ThegeometriesareregularandapproachAdS7theyaregenericallynon-supersymmetric,exceptinacertain×S4orAdS5“extremal”×limit,S5atin nity;whereaBogomol’nyiboundissaturatedandanakedsingularityappears.Byusingthesespaces,onecanconstructamodelofQCDthatgeneralizesbyone(ortwo)extraparametersarecentlyproposedmodelofQCDbasedonthenon-extremalD4brane.Thisallowsforsomeextrafreedomtocircumventsome(butnotall)limitationsofthesimplestversion.

August1998

We construct supergravity backgrounds representing non-homogeneous compactifications of d=10,11 supergravities to four dimensions, which cannot be written as a direct product. The geometries are regular and approach $AdS_7\times S^4$ or $AdS_5\times S^5$ a

1.Introduction

InthelastfewmonthstherehavebeenanumberofsuggestionsgeneralizingthedualitiesproposedbyMaldacena[1]tothecaseoffour-dimensionalgaugetheorieswithlesssupersymmetries(seee.g.refs.[2-10]).Inmostoftheexamplessofarconsidered,thesupergravitybackgroundsareoftheformAdSn×X,whereXissomeEinsteinmanifold,andtheyareexpectedtoberelatedtoaconformal eldtheory“attheboundary”[1,11,2],but,becauseofconformalinvariance,nottoacon ninggaugetheory(forareviewofsupergravitycompacti cationsonanti-deSitterbackgrounds,see[12]).TheinterestingmodelintroducedbyWitten[5]andfurtherinvestigatedin[13-16]exhibitscon nement,but,aspointedoutintheseworks,itsapplicationsto3+1dimensionalQCDaresomewhatlimitedbythefactthatthereisasinglescaleinthegeometry.Themodelarisesasdimensionalreductionof4+1dimensionalSU(N)superYang-Millstheory,whichasquantum eldtheoryisnon-renormalizable;inordertodecoupleKaluza-Kleinstates,atleasttwoscalesseemnecessary:onerepresentingΛQCD,andanotheroneforthemassespropertiesofthespectrumseemstorequireanunderstandingofstringtheoryinRamond-ofKaluza-KleinparticlesMKK,withΛQCD MKK.Inaddition,understandingcertain

Ramondbackgrounds.Unfortunately,suchstringmodelsarenotsolvable;tobeabletouncoverthepropertiesoflargeNQCD,onewouldliketohavearegimewherethereisasupergravitydescription.

Thepresentworkisanattemptinthisdirection.WewillassumethatasupergravitydescriptionofpureYang-Millstheoryatlarge’tHooftcouplingexistsandstartwiththemostgeneralregulargeometrythatcanplausiblydescribeacon ningtheory,inthesensethatwillbeexplainedbelow.Thenon-supersymmetricbackgroundofsection3general-izestheWittenQCDmodelinmuchthesamewaytheKerrblackholegeneralizestheSchwarzschildblackholesolution.Itis,however,astaticgeometry–whatis“time”intheKerrsolutionhereplaystheroleofanangularcoordinate–,whichisregulareverywhere,andcontainsanextraparameterawithrespecttotheWittenQCDmodel.Whenthisparameterissmall,themodelreducestotheWittenmodel,withΛQCD～=MKK.Whenthisparameterislarge,oneobtainsthedesireddecouplingofunwantedKaluza-Kleinpar-(ratherthan4+1)dimensionalone.

ticles,ΛQCD MKK,whichguaranteesthatthegaugetheoryattheQCDscaleisa3+1

1

We construct supergravity backgrounds representing non-homogeneous compactifications of d=10,11 supergravities to four dimensions, which cannot be written as a direct product. The geometries are regular and approach $AdS_7\times S^4$ or $AdS_5\times S^5$ a

http://doc.guandang.netrgeNQCDfromsupergravity

Inordertointroduceournotation,inthissectionwereviewtheapproachtonon-supersymmetricYang-Millstheoryof[5].

2.1.ConstructionofthemetricfromtheblackM5-brane

Westartwiththemetricforthenon-extremalM5-braneofeleven-dimensionalsuper-gravity,whichisgivenby

ds2 1/3

11

=f

hdt2

+

dx21

+...+dx52

+f

2/3

dr2

,

h=1

2m

r3

α′).Thereisaneventhorizonatr=rH,rH=(2m)1/3.The

Hawkingtemperatureisgivenby

T3r2

H=

H

h

+r2d 24e2(φ(r) φ∞)=f 1/2

,

(2.5)(r).

(2.6)

Thesupergravitysolutionthatisrelatedtoa eldtheoryisobtainedbytakingthe(“decou-pling”)low-energylimitlP→0in(2.1).Thisisdonebyrescalingvariablesasfollows[1,18]:

r=U2l3

P,

m=

1

(πN)1/3

(1

U6U2(1

U60

We construct supergravity backgrounds representing non-homogeneous compactifications of d=10,11 supergravities to four dimensions, which cannot be written as a direct product. The geometries are regular and approach $AdS_7\times S^4$ or $AdS_5\times S^5$ a

2.2.WittenQCDmodel

Thelow-energytheoryonthenon-extremalD4braneis4+1dimensionalSU(N)Yang-Millstheoryat nitetemperatureTH.Inthepathintegralformulation,thegaugetheoryat nitetemperatureisdescribedbygoingtoEuclideantimet→ iτ,andidentifyingτperiodicallywithperiod1/TH.Zero-temperature3+1Yang-MillstheorycanbedescribedR0=(2πTH) 1,where–asinthethermalensemble–fermionsobeyantiperiodicboundaryconditions.Atenergiesmuchlowerthan1/R0,thetheoryise ectively3+1dimensional.Becauseoftheboundaryconditions,fermionsandscalarparticlesgetmassesproportionalThelow-energytheoryisthuspureYang-Millstheory.

totheinverseradius,sothat,asR0→0,theyshoulddecouplefromthelow-energyphysics.

2

Thegaugecouplingg4inthe3+1dimensionalYang-Millstheoryisgivenbytheratio

bymakingx4→ ix0,andviewingτasparametrizingaspace-likecirclewithradius

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