D-branes, Orientifolds and K-theory

The complete D-brane spectrum in $\Zop_2$ orientifolds is computed. Stable non-BPS D-branes with both integral and torsion charges are found. The relation to K-theory is discussed and a new K-theory relevant to orientifolds is suggested.

hep-th/0206174

AEI-2002-003

D-branes,OrientifoldsandK-theory

B.Stefa´nski,jr.1

1

arXiv:hep-th/0206174v1 19 Jun 2002Max-Planck-Institutf¨urGravitationsphysik,Albert-Einstein-InstitutAmM¨uhlenberg1,D-14476Golm,Germanybogdan@aei.mpg.deAbstract:ThecompleteD-branespectruminZZ2orientifoldsiscomputed.Stablenon-BPSD-braneswithbothintegralandtorsionchargesarefound.TherelationtoK-theoryisdiscussedandanewK-theoryrelevanttoorientifoldsissuggested.1IntroductionD-branes[2]playanimportantroleinstringtheory.Theyclassifyalargenumberofstringvacua,provideacanvasforgaugetheoriesandareakeyingredentofmanydualities.UnderstandingthespectrumofD-branesinagivenon-shellclosedstringbackgroundisanimportantandoftendi cultsubject.Forclosedstringbackgroundswithaconformal eldtheorydescription,D-branesaremostsuccinctlyrepresentedasboundarystates[8]-coherentstatesconstructedoutofclosedstringoscillators.AnothertoolusedinthestudyofD-branespectraisK-theory[9,5].SinceD-branescoupletoR-Rpotentialsitwasinitiallythoughtthattheymaybeclassi edby(possibly

rational)cohomology.Withthediscoveryofbraneannihilationthroughtachyoncon-densation[3]thoughtheK-theorydescriptionofD-branesseemstobethemorenaturaldescription.Thetwodescriptionsdi erintheclassi cationoftorsion-chargedbranes.Hence,torsionchargedD-branesprovideagoodtestoftheK-theorydescription.

Inthisnotewereviewtheboundaryconformal eldtheory(BCFT)constructionofD-branesinthe ×I4orientifolds[10,11,12]oftypeIIBtheory[1].WeshallfocusonD-branesintheGPmodelandreferthereaderto[1]fordetailsoftheBZDPmodel.TheGPmodelhasanO9-planeandsixteenO5-planes,andinordertocanceltheresultingtadpolesoneintroduces16D9-branesand16D5-branes.PlacingoneD5-braneateachoftheO5-planesoneobtainsalocallychargecancelingcon guration.TheD5-branescanonlymoveoftheO5-planesinpairs,muchasfractionalbranesinorbifolds.Unlikefractionalbranes,theydonotcoupletotwistedsectorsandhavethusbeendubbedstuck[1].

WhenclassifyingD-branes,wewill nditconvenienttofollowthenotationof[7]:wewillrefertoaDp-braneasaD(r,s)-branewithr+s=pandthebranehavingr+1Neumanndirectioninthenon-compactsix-dimensionalspacetimeandsNeumanndirectionsonT4.ForexamplethetadpolecancelingD5-andD9-braneswillbelabeledasD(5,0)-andD(5,4)-branes,respectively.

The complete D-brane spectrum in $\Zop_2$ orientifolds is computed. Stable non-BPS D-branes with both integral and torsion charges are found. The relation to K-theory is discussed and a new K-theory relevant to orientifolds is suggested.

Therestofthisnoteisorganisedasfollows.Insection2wediscussintegrallycharged

non-BPS

D-branes

and

their

decay

channels,whileinsection3wepresentsomeexamplesoftorsionchargednon-BPSD-branes.Insection4wecomparetheBCFTconstructiontoK-theorypredictions;weshowthattheBZDPmodelisdescribedbyorthogonalequiv-ariantK-theory.InparticularthereisgoodagreementbetweenK-theoryandBCFTfortorsionchargedbranes.WealsosuggestanewtwistedversionoftheequivariantorthogonalK-theorywhichshoulddescribeD-branesintheGPorientifold.

2Integrallychargednon-BPSD-branes

Integrallychargednon-BPSD-branescarrytwistedR-RchargesbutareneutralundertheuntwistedR-Rcharge.IntheGPmodeltwotypesofsuchtruncatedbraneswereencountered.Firstly -invarianttruncatedbranesfromtheI4orbifold[7]arepresent¯forr= 1,3ands=1,3.Secondly,DDpairs,inwhich mapstheD-branetothe¯D-brane,occurforr= 1,3ands=0,4.Intheorbifoldthelatterareboundstatesoftwofundamentalobjects;intheorientifoldhowever,theyformoneobject.

Thedecaychannelsandstabilityregionsofthe -invarianttruncatedbranesaremod-i edbythe projection.Intheorbifoldthesearestablefor1

R⊥≥

R||≤1,2√(1)

takeα′=1throughout.2Thisisconsistentsincetherearer= 1,3,s=2BPSfractionalbranesintheGPmodel.3ThisisfortunateasintheGPorientifoldtherearenoBPSfractionalr= 1,3s=0,4branesintowhichthetruncatedbranescoulddecay.1We

The complete D-brane spectrum in $\Zop_2$ orientifolds is computed. Stable non-BPS D-branes with both integral and torsion charges are found. The relation to K-theory is discussed and a new K-theory relevant to orientifolds is suggested.

Intheuncompacti edtheorytherearetwo(5,2)-braneswithuntwistedandtwistedNS-NScouplings:h+andh ,bothofwhicharestablefor4

1

2

Further,itwasshownin[1]that

|h+,(5,2) +|h ,(5,2) =|g(5,2) .

|g +|g =|0 ,(6).(5)gcanbethoughtofasaZ2D7-braneoftypeIanditsimageunderI4andsosatis es

(7)

with|0 representingtheclosedstringvacuum.Sinceh+andh aredistinct,the(5,2)-branesinthenon-compacttheorycarryZZ2⊕ZZ2charge.Inthecompacttheorytheh-braneswillcoupletofourtwistedNS-NSsectors,andbyturningonsuitableWilsonlinesand uxesinthecompactdirectionsone ndsmanymoreh-branes[1].

TheabovetorsionchargedD7-branesareinfactunstablesincetheopenstringstretch-ingbetweenthemandtadpole-cancelingD-braneshasatachyonicgroundstate.TheD7-braneswillasaresultdecay;theabovediscussiondoeshoweverindicatesthepresenceofatorsionchargeinthetheory.IntheBZDPorientifoldwefound[1]torsion-chargedD-branes(forexampleaD-particle)whichhavenosuchtachyonicinstabilities.ItisinterestingtonotethatsuchaD-particleisstableforallvaluesofcompacti cationradii.ThegD7-braneofequation(4),hasnotachyonicstatesfromopenstringswithbothendpointsonitsworldvolumeforradiisatisfyingequations(1)and(2).ForradiioutsidetheseregionstheD7-branebecomesunstableanddecays.Thedecaychannelsaremosteasilyfoundbyconsideringthebraneasan -invariantsuperpositionoffractional(5,2)-branesintheI4orbifoldasdepictedin gure1.Thedecays√intheorbifoldareknownandturnouttobe -invariant.ForexampleforR||≥

¯2decaysintoastuckD(5,0)-braneatone xedpointandstuckD(5,0)-braneat¯another.TheD(5,0)-bran …… 此处隐藏：10202字，全部文档内容请下载后查看。喜欢就下载吧 ……