On the Geometry of the String Landscape and the Sw

arXiv:hep-th/0605264v126May2006CALT-68-2600,HUTP-06/A017OntheGeometryoftheStringLandscapeandtheSwamplandHirosiOoguri1andCumrunVafa21CaliforniaInstituteofTechnologyPasadena,CA91125,USA2JeffersonPhysicalLaboratory,HarvardUniversityCambridge,MA02138,USAAbstractWemakeanumberofconjecturesaboutthegeometryofcontinuousmoduliparam-eterizingthestringlandscape.Inparticularweconjecturethatsuchmoduliarealwaysgivenbyexpectationvalueofscalarfieldsandthatmodulispaceswithfinitenon-zerodiameterbelongtotheswampland.Wealsoconjecturethatpointsatinfinityinamodulispacecorrespondtopointswhereaninfinitetowerofmasslessstatesappear,andthatneartheseregionsthemodulispaceisnegativelycurved.Wealsoproposethatthereisnonon-trivial1-cycleofminimumlengthinthemodulispace.Thisleadsinparticulartothepredictionoftheexistenceofaradiallymassivepartnertotheaxion.Theseconjecturesputstrongconstraintsoninflatonpotentialsthatcanappearinaconsistentquantumthe-oryofgravity.Ourconjecturesaresupportedbyanumberofhighlynon-trivialexamplesfromstringtheory.Moreoveritisshownthattheseconditionscanbeviolatedifgravityisdecoupled.May20061.IntroductionThefactthatstringtheoryseemstoofferadiverserangeofpossibilitiesforvacuahasbeenviewedasadrawbackforthetheory:Wecannotconvergeonaprecisepredictionforthetheory.Howeverdespitethisdiversityofoptionsforthestringlandscape,ithasbeenpointedoutin[1]thattherearealsoanumberofpatternsthatseemtoemerge.Noteveryeffectivefieldtheorythatappearsconsistentseemstoariseinstringtheory.Itisnaturaltoconjecturethatthesetheoriesarenotfullyconsistentasquantumgravitationaltheories.Suchtheoriesbelongtotheswampland.Theconditiontobeontheswamplandcanbeliftedifwedecouplegravity.Inotherwordsafullyconsistentquantumfieldtheorycannotalwaysbecoupledtogravity.Itisnaturaltoconjecturethatpointsontheswamplandare“anomalousquantumgravitationaltheories,”whichareanomalousinamoresubtleway,thanhasbeendiscoveredinthecontextofquantumfieldtheories.Itwouldbeimportanttoimprovetherestrictionsfortheoriestoariseinstringtheory.Bytabulatingsuchrestrictionsandfindingthecriteriathatdistinguishtheswamplandfromthelandscapeonecanhopetohaveadeeperunderstandingintotheuniversalityclassofquantumgravitationaltheories.Themainaimofthispaperistotakesomemodeststepsinthisdirection.Forsomerelatedconjecturesdistinguishingswamplandfromthelandscapesee[2,3,4,5].Oneconjecturalcriteriontobeonthestringlandscapeisthatthevolumeofthemodulispaceseemstypicallyfinite.Thereare,however,wellknowncounter-examplestothisseeminglygeneralphenomenon.Considercompactificationonacircleofradiusr.Themodulispaceforthecirclehasametricds2=drr2,(1.1)andthevolumeintegralisdivergent.In[1],itwaspointedoutthatthisvolumedivergencecorrelateswiththecutoffinmass.Namely,letǫbeafixedscaleforthelowenergyeffectivetheoryandthatweinsistthatallthehighermassivescaleshavemassgreaterthanǫ.Forlargerandsmallǫ,theregionofthemoduliofthemodulispacesatisfyingthisconstraintisspecifiedbyr1/ǫ.Clearlythevolumeofthisregionofthemodulispaceisfinite,Z1/ǫdrr=−logǫ.(Thelowerboundfortherintegralshouldalsoberegularizedinasimilarfashion,aswillbeclearinourexamples.)Wefindthelogarithmicvolumedivergenceaswetakethelimit1ǫ→0.Thus,inthiscase,thevolumedivergenceisrelatedtotheemergenceofinfinitelymanyextralightparticles.Eventhoughthisdivergencemightseemspecialto1-dimensionalmodulispaces,herewewishtoformulateconjecturesapplicabletoeverymodulispaceMencounteredinstringtheory.Ourconjecturesapplybothtothemodulispaceofscalars,aswellasthesubspacesparameterizingminimumlociforthepotentialsdefinedonsuchspaces.Inthiswayourconjecturescanbeviewedasverypowerfulconstraintsonwhatpotentialscanappearinaconsistenttheoryofquantumgravity.Ourconjecturessuggestthefollowingpictureforthemodulispaceofaconsistentquantumgravitationaltheory:Themodulispaceisparameterizedbyexpectationvalueofscalarfields.Weconjecturethattherearepointsinfinitelyfarawayfromoneanotheronthemodulispaceandthatthepointsnearinfinityarepointswhereatowerofmasslessmodesappear(withexponentiallysmallmassasafunctionofthedistancetosuchpoints).Infinitedistancesingularitiescombinedwithfinitevolumetypicallyimplynegativecurvatures.Thus,weconjecturethatthecurvaturebecomesnegativenearpointsatinfinity.Wealsoproposethattherearenonon-trivialloopsofminimumlengthinthemodulispaceofscalars.Aswewillelaboratelaterthisisinlinewiththeintuitionthatthedualitygroupsaregeneratedbydiscretegaugesymmetriesrealizedatdifferentpointsonthemodulispace.Theorganizationofthepaperisasfollows:Insection2wepresentthepreciseformforourconjectures.Insection3wepresentexamplesofhowstringtheorysupportsourconjecturesandhowifwedecouplegravitywecanviolatethem.Insection4wediscusssomefieldtheoreticconsiderationsrelatedtoourconjectures.Insection5weendwithsomeconcludingthoughts.2.TheConjecturesInthissectionwepresentourconjectures.WeclaimthatourconjecturesapplytoconsistentquantumtheoriesofgravitywithfinitePlanckmassin4andhigherspacetimedimensions.Wedonotconsider3orlowerdimensionsasgravitydoesnotcontainprop-agatingdegreesoffreedominthesedimensions,thoughsomeofourconjecturesmaybeapplicableto3dimensionalcasesasw