Type IIB orientifolds with discrete torsion

We consider compact four-dimensional ${\bf Z_N}\times {\bf Z_M}$ type IIB orientifolds, for certain values of $N$ and $M$. We allow the additional feature of discrete torsion and discuss the modification of the consistency conditions arising from tadpole c

1

002 ebF 61 3v502101/0ht-pe:hviXraInternationalc

fJournalofModernPhysicsA,WorldScienti cPublishingCompanyTYPEIIBORIENTIFOLDSWITHDISCRETETORSION

ROBERTL.KARP ,F.PAULESPOSITO,LOUISWITTEN

PhysicsDepartment,UniversityofCincinnati

Cincinnati,Ohio45221-0011,USA

Weconsidercompactfour-dimensionalZN×ZMtypeIIBorientifolds,forcertainvaluesofNandM.Weallowtheadditionalfeatureofdiscretetorsionanddiscussthemodi -cationoftheconsistencyconditionsarisingfromtadpolecancellation.Wepointoutthedi erencesbetweenthecaseswithandwithoutdiscretetorsion.

Orientifoldcompacti cations1ofthetypeIIBsuperstringcircumventtheprob-lemthatthetypeItheorydoesnotproduceachiralspectrumwhencompacti edonaCalabi-Yauthreefoldwithstandardembeddingofthegaugedegreesoffree-dom.Independentlyofthisdiscretetorsion(DT)wasintroducedasaphasefactorrelatedtotheB- eld,allowedbymodularinvariance2inorbifoldcompacti cationsoftheclosedstringtheories.IntheopenstringtheoriestheanalogousnotionofDTwasdiscoveredrelativelyrecently,onlyafterD-braneswerebetterunderstood3.InadditiontherelationshipbetweenclosedandopenDThasbeenfurtherclari ed4.ThepioneeringworkforZ2orientifoldswasquicklygeneralizedtoZnfordif-ferentvaluesofn’s5.ThecaseZ2×Z2wasinvestigated6andgeneralized7.ThequestionofnoncompactorientifoldswithDTwasaddressedaswell8.ThegeometricaspectsofDTwaspartlydescribed9andtherehasrecentlybeenarevivalofinterestinthesubject10.

ThecompleteorientifoldgroupweconsiderhereisG1+ G2with h h′∈G1forh,h′∈G2.WerestrictourattentiontoG1=G2=ZN×ZM.Thegeneratorofeitherofthefactorswillhavetheformθ=exp(2iπ(v1J45+v2J67+v3J89)),withJmntheSO(6)Cartangenerators,actingonthecompactT6(complexi ed)coordinatesZ1=X4+iX5,Z2=X6+iX7andZ3=X8+iX9asθZi=e2iπviZi.IfwechosethetwistvectorsoftheZNandZMgeneratorsθandωtobeoftheformvθ=v=1M(0,1, 1),weendupwithN=1d=4supersymmetry.Undoubtedly,therearemanyequallyinterestingchoicesthatdonothavethisform.

Toderivethemasslessspectraweworkinlight-conegauge.TheGSOpro-jecteduntwistedmasslessRamondstates|s0s1s2s3 transformasθ|s0s1s2s3 =e2iπv·s|s0s1s2s3 .

InthispaperwewillbemainlyinterestedintheKleinbottlevacuumtovacuumamplitude;theMobiusstripandthecylinderinfacthavesimilarexpression.The

We consider compact four-dimensional ${\bf Z_N}\times {\bf Z_M}$ type IIB orientifolds, for certain values of $N$ and $M$. We allow the additional feature of discrete torsion and discuss the modification of the consistency conditions arising from tadpole c

2TypeIIBorientifoldswithdiscretetorsion

Kleinbottleamplitudeisgivenby

K=

V4

(4παt)

2′

2

Trh{

1+( 1)F

2t[α2

,xayb

)=

χ(a,b)β]

(N

α,β

2

η α,β

=0

2

,xayb

)=(1

1)χ(a,b)

2

]

(N

2

β+2ui

]

[α

β+2u3]

2

2

+2ui]

( 2sin2πu3)

1

2

1

2+2u3

]

[1

2

1

We consider compact four-dimensional ${\bf Z_N}\times {\bf Z_M}$ type IIB orientifolds, for certain values of $N$ and $M$. We allow the additional feature of discrete torsion and discuss the modification of the consistency conditions arising from tadpole c

TypeIIBorientifoldswithdiscretetorsion3

nonzeroone.InwhatfollowswefocusontheK(y3,xya)=0contributions,whichwillbeproportionalto1/V1,asopposedtotheK(y3,ya)contributionsthatareinfactproportionaltoV1.Wealsohave (y3,xayb)=( 1)a.ItturnsoutthatK(y3,xya)=0fora=0,3,whileforothervaluesofatheyallequalacommon

11

valueproportionalto [] []/ [ ].Inthelimitt→0thetwistedKlein1bottleamplitudeswillgivethecontribution(2t)(64π2α′)/V1withoutDT,andthenegativeofthiswithDT.Similarly,theuntwistedKleinbottlecontributionthatcontributeswithafactorof1/V1turnsouttobeK(1,xya),fora=1,3,4,5.Inthet→0limittheseaddupthecontribution3(2t)(64π2α′)/V1.ThusinthecasewithoutDTwehaveatadpolecontributionproportionalto(2t)(256π2α′)/V1,whichturnsouttorequire32D-branestobecanceled.Thisagreeswiththealreadyknownresult7.Ontheotherhand,forthecasewithDTthetadpolecannotbecanceled,renderingthemodelperturbativelyinconsistent,inthesenseof11.

ThenextinterestingcaseisZ3×Z6.MoregenerallyfornoddtheZn×Z2nDTis (yn,xayb)=e(2πi/n)n( b)=1,andonceagainDThasnoe ect.

TheZ4×Z4modelisinterestingtoanalyzeaswell.Itiswasknown7thatwithoutDTthismodelwasperturbativelyinconsistent.OurhopewasthatDTwouldchangethetadpolecancellationconditions,andallowforaconsistentsolution.Itiselementarytoshowthat (x2,xayb)=1i b=1,3; (x2y2,xayb)=1i a b= 3, 1,1,3,and (y2,xayb)=1i a=1,3.UnfortunatelyitturnsoutthatwiththeseconstraintsK(x2, )=K(x2y2, )=0,andK(y2, )=0,implyingthatevenbyturningonDTwecannotperturbativelysavethemodel.

Acknowledgments

R.L.K.wouldliketothankR.G.Leighforusefulconversations.ThisworkwassupportedinpartbytheDepartmentofEnergyunderthecontractnumberDOE-FGO2-84ER40153.RLKwasalsosupportedinpartbytheNationalScienceFoun-dationgrantDMS-9983320.

Note

Afterthistalkwasgivenanexhaustivetreatmentofthesubjectappeared12thatoverlapspartlywithourresults.

6

6

6

We consider compact four-dimensional ${\bf Z_N}\times {\bf Z_M}$ type IIB orientifolds, for certain values of $N$ and $M$. We allow the additional feature of discrete torsion and discuss the modification of the consistency conditions arising from tadpole c

4TypeIIBorientifoldswithdiscretetorsion

References

1.A.Sagnotti,TalkpresentedattheCargeseSummerInstituteonNon-PerturbativeMethodsinFieldTheory,1987.P.Horava,Nucl.Phys.B327,461(1989).Phys.Lett.B231,251(1989 …… 此处隐藏：4346字，全部文档内容请下载后查看。喜欢就下载吧 ……