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Type IIB Killing spinors and calibrations

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002epS423v890504/0ht-pe:hviXraDCPT-04/17hep-th/0405098
TypeIIBKillingspinorsandcalibrations
EmilyJ.Hackett-Jones1andDouglasJ.Smith2
CentreforParticleTheory
DepartmentofMathematicalSciences
UniversityofDurham,DurhamDH13LE,U.K.
ABSTRACT
Inthispaperwederivethefullsetofdifferentialequationsandsomealgebraicrelationsforp-formsconstructedfromtypeIIBKillingspinors.TheseequationsarevalidforthemostgeneraltypeIIBsupersymmetricbackgroundswhichhaveanon-zeroNS-NS3-formfieldstrength,H,andnon-zeroR-Rfieldstrengths,G(1,G(3andG(5.Ourmotivationistousetheseequationstoobtaingeneralisedcalibrationsforbranesinsupersymmetricbackgrounds.Inparticular,weconsidergiantgravitonsinAdS5×S5.Thesenon-staticbraneshaveaninterestingconstructionviaholomorphicsurfacesinC1,2correspondingtothesebranesandshowthattheysatisfythe×correctC3.Weconstructthep-formsdifferentialequations.Moreover,weinterprettheequationsascalibrationconditionsandderivethecalibrationbound.Wefindthatgiantgravitonsminimise“energyminusmomentum”.

1Introduction
Recentlytherehasbeenmuchinterestinclassifyingsupersymmetricsolutionsofsuper-gravitytheoriesinvariousdimensions[1–16].OnetechniquewhichhasprovedparticularlyeffectiveistousetheKillingspinorsofthebackgroundtoconstructformsofdifferentde-grees.Forexample,theauthorsofRefs.[2,3]usedp-forms,φ,intheclassificationofgeneralsupersymmetricsolutionsof11-dimensionalsupergravity.Thecomponentsofφaregivenby
¯ΓM1...MpǫφM1...Mp=ǫwhereǫisaKillingspinorof11-dimensionalsupergravityandΓMareDiracmatrices.
Thesep-formsobeyalgebraicanddifferentialrelationsdescendedfromtheFierzidentitiesandtheKillingspinorequation,respectively.Moreover,theformsdefineamathematicalstructureknownasaG-structure,whichisthereductionoftheSpin(10,1framebundletoaG-sub-bundle.ThetypeofG-structurethatarisescanthenbeusedtoclassifythesupersymmetricsolutionsof11-dimensionalsupergravity[2,3].Similartechniqueshavebeenused[4,6,9–16]to(partiallyclassifysupersymmetricsolutionsinvariouslower-dimensionalsupergravitytheories.
Aswellastheiruseinclassifyingsupersymmetricbackgrounds,theformsconstructedfromKillingspinorsarerelatedtogeneralisedcalibrationsforbranes.Forexample,inRef.[17]itwasshownthatgeneralisedcalibrationsforM-branesnaturallyemergefromthediffer-entialequationssatisfiedbytheforms.HerewewillbeinterestedincalibrationsforbranesintypeIIBbackgrounds.SomeexamplesofgeneralisedcalibrationsinparticulartypeIIBbackgroundshavebeenfound[18].However,herewewillbeinterestedinfindingcalibra-tionsfornon-staticprobebranes,whichhasnotbeeninvestigatedpreviously.WebeginbyconsideringthemostgeneralsupersymmetricbackgroundsoftypeIIBsupergravity.Thatis,weconsiderbackgroundswhichadmitatleastoneKillingspinorandhavebackgroundfieldstrengths,H,G(1,G(3andG(5non-zero.Weconstructp-formsfromtheKillingspinorsandderivethefullsetofdifferentialequationsfortheseforms.Somealgebraicrelationsbetweentheformsandthefieldstrengthsarealsoderived.ThesedifferentialandalgebraicequationscouldthenbeusedintheclassificationoftypeIIBsupersymmetricbackgrounds,asdemonstratedinRefs.[14–16]forsomespecialclassesof10-dimensionalbackgrounds.However,ourfocuswillbeonusingtheforms,andtheircorrespondingdifferentialequations,toconstructgeneralisedcalibrationsfornon-staticD3-branesinIIBbackgrounds.Inparticular,wewillconsidergiantgravitonsinAdS5×S5.
Giantgravitonsarenon-staticsphericalbranesinAdS5×S5.Thefactthattheyarenon-staticmakesthemaninterestingexampletoconsiderfromthepointofviewofcalibrations,asmostpreviousworkoncalibrationshasinvolvedstaticprobebranes.AninterestingconstructionofgiantgravitonshasbeenproposedbyMikhailov[19].InthisconstructionthespaceAdS5×S5isembeddedinC1,2×C3.Thegiantgravitonworld-volumethenarisesfromtheintersectionofaholomorphicsurfaceinC3withtheembeddedS5.Oneofthebenefitsofconstructinggiantgravitonsinthiswayisthatthesupersymmetryprojectionconditionsbecomeverysimple.ThisisessentiallybecauseKillingspinorsinAdS5×S5
1

lifttocovariantlyconstantspinorsinC1,2×C3,andconsequentlyeverythingsimplifiesinthehigherdimensionalspace.
Theplanofthispaperisasfollows.In§2weconsiderthegravitinoKillingspinorequa-tionfortypeIIBsupergravityandweuseittoderivedifferentialequationsfortheforms.Thenin§3wederivesomealgebraicidentitiesfortheformsusingFierzidentitiesandthealgebraicKillingspinorequation.In§4.1-4.3wediscusstheMikhailovconstructionofgiantgravitonsinsomedetail.Thenin§4.4theformscorrespondingtotheseholo-morphicgiantgravitonsareshowntoobeythecorrectdifferentialequations.In§5weconsidertherelationshipbetweenthedifferentialequationsderivedin§2andgeneralisedcalibrations.Inparticular,weareinterestedinprobeD3-branesinbackgroundswherethefieldstrengthsH,G(1andG(3aresettozero,whichisthecaseforAdS5acalibrationboundforthesebranesandthenshowthattheholomorphicgiant×S5.Wefindgravi-tonssaturatethisboundin§5.2.Thesecalibratedgiantgravitonshaveminimal“energyminusmomentum”intheirhomologyclass.Moreover,in§5.3weshowthatdualgiantsalsosaturatethecalibrationboundandtheyminimisethesamequantityastheordinarygiants.Ourconclusionsaregivenin§6.
2Differentialequationsforthep-forms
WebeginbyconsideringtheKillingspinorequationsfortypeIIBsupergravity.Partialresults[14–16]havebeenobtainedforbackgroundswhichpreserve4-dimensionalPoincar´einvariance.AlsoinRef.[18]somedifferentialconditionswerederivedasgeneralisedcali-brationsfor5-braneswrappingspecialLagrangian3-cycles.However,thefullsetofequa-tionsforcompletelygeneraltypeIIBbackgroundshasnotbeengivenuntilnow.TypeIIBsupergravityhastwoKillingspinorequations.Oneisalgebraic,andarisesfromrequiringthatthevariationoftheaxino-dilatinovanishes.Thesecondequationisdifferential,andarisesfromvaryingthegravitino.Inthissectionwewillbeinterestedinthesecond(grav-itinoequation,andwewilluseittocomputederivativesofformsconstructedfromKillingspinors.In§3wewilldiscussalgebraicrelationsbetweentheforms,someofwhichcanbeobtainedfromthealgebraicKillingspinorequation.WeexpectthatboththedifferentialandalgebraicrelationswederivewillplayanimportantroleinthefullclassificationofsupersymmetrictypeIIBbackgrounds.
FollowingRef.[20],thegravitinoKillingspinorequationinthestringframeisDMǫ=0,whereǫisa32-dimensionalchiralspinor,withtwo16-dimensionalcomponents1,i.e.
ǫ=

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