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arXiv:hep-th/9909047v25Oct1999Bicocca-FT/99/27NYU-TH/99/09/02hep-th/9909047TheSupergravityDualofN=1SuperYang-MillsTheoryL.Girardelloa,M.Petrinib,M.PorraticandA.Zaffaronia1(a)DipartimentodiFisica,Universit`adiMilano-BicoccaandINFN,SezionediMilano,Italy(b)TheoreticalPhysicsGroup,BlackettLaboratory,ImperialCollege,LondonSW72BZ,U.K.(c)DepartmentofPhysics,NYU,4WashingtonPl.,NewYork,NY10003,USAABSTRACTWefindanexact,N=1supersymmetrickinksolutionof5dgaugedsupergravity.WeassociatethissolutionwiththeRGflowfromN=4superYang-Millstheory,deformedbyarelevantoperator,topureN=1superYang-MillsintheIR.WetestthisidentificationbycomputingvariousQFTquantitiesusingthesupergravitydual:thetensionofelectricandmagneticstringsandthegauginocondensate.Asdemandedbyouridentification,ourkinksolutionisatruedeformationofN=4,thatexhibitsconfinementofquarks,magneticscreening,andspontaneouschiralsymmetrybreaking.1e-mail:[e-mail],[e-mail],[e-mail],alberto.zaff[e-mail]1IntroductionInpreviouspapers[1,2]weexploredseveralpropertiesofrelevantdeformationsofstrongly-coupledN=4superYang-Millsusingitsdual[3,4,5]descriptionintermsof5dgaugedsupergravity.InthispaperwepresentaholographicRGflowfromN=4SYMtopureN=1SYMintheIR.Wefindagreementwithfieldtheoryexpectations,namely,quarksconfine,monopolesarescreened,andthereisagauginocondensate.Thispaperisthenaturalcontinuationof[1,2],wherewestudiedthesupergravitydescriptionofRGflows.Inparticular,in[1]weprovedtheexistenceofakinksolutionof5dsupergravitythatwasinterpretedasthedualofamass-deformedN=4superYang-Mills,flowingintheIRtoanon-supersymmetric,interacting,localconformalfieldtheory(CFT)(seealso[6]).Wealsointroducedac-functionandweprovedac-theoremvalidforallsupergravityflowsinvolvingthemetricandanarbitrarynumberofscalars.Suchasetup,namelycoupledgravity-scalarequationsisremarkablyversatile,anditcanalsodescribedeformationsofN=4flowingtonon-conformal,interactingtheories.MostofthesolutionsasymptotictotheN=4UVfixedpointgotoinfinityinthespaceofparametersdescribedbythe5dsupergravity.ItisnaturaltointerpretthesesolutionsasRGflowstonon-conformalIRtheories.Exampleswereconsideredin[2,7].Thegenericsolutionsdiscussedin[2]exhibitdifferentbehavioursintheIR,fromconfinementtoscreening,associatedwithanIRsingularitythatiscommontomany10dsolutions[8,9,10,11,12,13].In[7]theCoulombbranchofN=4YMwasstudied.InviewofthedoubleroleofsupergravitysolutionswithanasymptoticAdSregion(de-formationsofanUVfixedpointversusthesametheoryinadifferentvacuum[14,15]),wemayaskwhetherthesolutionsin[2]aretruedeformationsofN=4YMorsimplydifferentstates(e.g.spontaneouslybrokenvacua)ofthesametheory.Thecaseof[8,9]turnedouttocorrespondtoadifferentvacuumofN=4YM;thesameiscertainlytruealsofor[7].Unlikethesolutionsin[7,8,9],theonesin[2]areexpectedtodescribedeformations.However,uptonow,noexplicit,analyticallysolvableexampleofdeformationswasgiveninliterature.Oneofthepurposesofthispaperistoprovidesuchanexample,describingthepureN=1YMcase.Anotherimportantquestionisthereliabilityofsupergravitysolutions.Problemswerepointedoutin[7],afteranexplicitcomparisonofsupergravitypredictionstogaugetheoryresults.Inthescreeningcasediscussedin[7],thetypeIIBstringgoesnearthesingularity,andanyactualcomputationrequirestheformofthesolutionintheIR,wherelargecorrectionstothesupergravityresultareexpected.Itisthenimportant,tobetterunderstandtheseriousnessofthisproblem,toconsiderothersolutionswherecomparisonwithquantumfieldtheoryexpec-tationcanbemade.Thesimplestsuchcase,whichcorrespondstoadeformationoftheN=4theory,istheflowtopureN=1YM.Inthispaper,weexhibitasolutionwithallqualitative1featuresofpureN=1superYMtheory,inparticular,quarkconfinement.Eveninourcasetheconfiningstringprobesahigh-curvaturesingularity;itisthereforeencouragingtofindthatatleastinsomecasesthisproblemdoesnotchangethequalitativepicture(confinement,magneticscreening,chiralsymmetrybraking)emergingfromfieldtheory.Tobettercontroloursolutionsweareledtoconsidersupersymmetricexamples.TheequallyinterestingN=0solutionsarecertainlylessreliable,eventhoughtheyareeasiertostudy[2].Thispaperisorganisedasfollows:inSection2,wegiveabriefreviewofthebasicresultsintheinterpretationof5dsupergravitysolutionsasRGflows,following[1,2].WealsodiscussthecasesofflowstoIRconformalandnon-conformaltheories,withparticularattentiontoopenproblems.Section3describesindetailstheflowtoN=1YManditsproperties.Thecomputationofelectricandmagneticstrings,andtheidentificationofthesupergravityfielddescribingthegauginocondensatearedescribed.ThatSectionalsocontainsabriefdiscussionofthestabilityofchirally-symmetricversuschiral-symmetrybreakingvacua.TheAppendixcontainsvarioususefulformulaeusedthroughoutthepaper.2RGFlowfrom5dSupergravity:aBriefReviewTofixonceforallournotations,wewillusecoordinateswheretheAnti-de-Sitterspacereadsds2=dy2+e2y/Rdxμdxμ,μ=0,1,2,3.(1)RistheAdSradiusanditwillbehenceforthsetto1,unlessexplicitlystated.ThefifthcoordinateyofAdS5hasanaturalinterpretationasanenergyscale[3,16,17,18].Asin[1],weassociatelargerenergieswithincreasingy.Welookforthesupergravitydes