BPS Amplitudes, Helicity Supertraces and Membranes in M-Theory

THU-99/35hep-th/9912225

BPSAmplitudes,MembranesHelicityinM-Theory

Supertracesand

BernarddeWit1,2aandDieterL¨ust1,3b

1

InstituteforTheoreticalPhysics,UniversityofCalifornia,

SantaBarbara,CA93106,USA

2

InstituteforTheoreticalPhysics,UtrechtUniversity,Princetonplein5,3508TAUtrecht,TheNetherlands3

Institutf¨urPhysik,HumboldtUniversit¨atzuBerlin,

Invalidenstrasse,D-10115Berlin,Germany

Abstract

WestudyBPSdominatedloopamplitudesinM-theoryonT2.Forthispurposewegeneralizetheconceptofhelicitysupertracestoninespacetimedimensions.Thesetracesdistinguishbetweenvariousmassivesupermulti-pletsandappearascoefficientsintheirone-loopcontributionston-gravitonscatteringamplitudes.ThiscanbeusedtoshowthatonlyultrashortBPSmultipletscontributetotheR4termintheeffectiveaction,whichwasfirstcomputedbyGreen,GutperleandVanhove.Therearetwoinequivalentultra-shortBPSmultipletswhichdescribetheKaluza-Kleinstatesandthewrappedmembranesthatcoverthetorusanumberoftimes.Fromtheperspectiveofthetype-IIstringstheycorrespondtomomentumandwindingstatesandD0orD1branes.

December1999

M-theoryisbelievedtoprovideaunifyingframeworkofallknownsuperstringtheories.Itslow-energylimitisdescribedbyeleven-dimensionalsupergravity[1].Thelattertheoryexhibitsnonrenormalizableultravioletbehaviorwhichwillpresum-ablybecuredonceoneincludestheadditionalM-theorydegreesoffreedom.Uponcompactificationtolowerdimensions,allthedualitysymmetriesofstringtheory,suchasS-,T-andU-duality,shouldbecomemanifest.Howeverthefundamental,microscopicformulationofM-theoryissofarunknown.Matrixtheory[2,3]isoneattemptinthisdirection.Closelyrelatedtothisistheideathatsupermembranes[4]constitutethefundamentaldegreesoffreedomofM-theory.Supermembranetheorymaynotsufferfromtheincompletenessofperturbativestringtheory.Unlikestringtheory,whichhasbothastringtensionaswellasacouplingconstant,ithasnoconventionalperturbativeexpansionasitsonlyparameteristhemembranetensionTm.

InthispaperweconsiderBPS-dominatedamplitudescinM-theorycompact-ifiedonatwo-dimensionaltorusT2,orequivalentlytype-IIA/Bsuperstringthe-orycompactifiedonS1.BPSstatesplayanimportantroleinthecomputationofloopamplitudesintheorieswithextendedsupersymmetry.ThemassesoftheBPSstatesareoftensupposedtobeexactfunctionsofthemodulisothatamplitudesthatreceivetheircontributionsexclusivelyfromBPSstatesarealsoexact,evenbeyondperturbationtheory.Herewewillfocusontheone-loopfour-gravitonscat-teringamplitudewhichprovidesthecoefficientofthegravitationalR4terminthecorrespondinglow-energyeffectiveaction.Computingthisone-loopamplitudeineleven-dimensionalsupergravitycompactifiedonT2toninespacetimedimensions[5,17,18,19,20],thenonvanishingcontributionisassociatedwiththesumoverthetowerofdoubly-chargedKalazu-Kleinstateswhichcirculateinthenine-dimensionalloop.

Fromanine-dimensionalperspectiveonecanstudytheone-loopcontributionstotheR4termcomingfromavarietyofsupermultiplets.Inordertoanalyzewhatkindofmassivestatescancontributetotheloopamplitude,weadopttheconceptofhelicitysupertraces[13,14,15]butnowextendedtoninedimensions.WewillthenshowthattheR4termsareexclusivelygeneratedbytheultrashortBPSmultiplets.Aswasrecentlydicussed[16]therearetwoinequivalentultrashortmultiplets,onecorrespondingtotheKaluza-Kleinstatesfromeleven-dimensionalsupergravityandtheotheronecorrespondingtotheKaluza-KleinstatesfromIIBsupergravity.ThelatterstatescanbeinterpretedasthewrappedmembranestatesofM-theory.Sum-mingoverbothtypesofstatesthusyieldstheexactanswerforM-theory,asothermassivesupermultipletscannotcontribute.Wethenshowthatincludingthesum

overthewrappedmembranestates(usingthesamesubtractionmethodasusedin[5]fortheKaluza-KleinstatesonT2)yieldsthecorrectamplitudeconsistentwithT-duality.TheR6termsreceivecontributionsfromintermediateBPSmultipletsbutnotfromlong(non-BPS)multiplets,whileR8andhigher-ordertermsreceivecontributionsfromallsupermultiplets

Letusnowfirstgeneralizetheconceptofhelicity(super)tracesformassivestatestoninespacetimedimensionswherethegroupofrest-framerotationsisgivenbySO(8).Following[14]wedefineageneratingfunctionforthehelicitytracesby

󰀮·H󰀮],Zr(y)=TrrgSO(8)=Trrexp[iφ

󰀃

󰀄

󰀃

󰀄

(1)

wherethetraceandtheSO(8)groupelementgSO(8)aredefinedinarepresentation

r.ObviouslythegeneratingfunctiondependsonlyonthegroupconjugacyclassesparametrizedbyfouranglesφiassociatedwiththetorusoftheCartansubalgebra.TheHidenotethegeneratorsoftheCartansubalgebraintherepresentationrandthevariablesyiaredefinedbyyi=exp[iφi/2].Thegeneratingfunctionssatisfiesthefollowingproperties,

Zr⊕r′(y)=Zr(y)+Zr′(y),Zr⊗r′(y)=Zr(y)Zr′(y),Zr(y)=Zr(y−1),Zr(y=1)=dim(r),

(2)

whichareeasytoprove.Alltheaboveresultscanreadilybegeneralizedtoasupertraceforwhichwewillnotintroduceanynewnotation.Note,however,thatthesupertraceforyi=1isequaltothegradedtraceoftheidentityandwillthusvanishforsupermultiplets.

Thehelicitytracesarenown-thranksymmetrictensorsdefinedby(nosumma-tionoverrepeatedindices)

Bn(r)=

1

∂yi1

···yin

∂

24

i=j=k=l

Z8c(y)=

1

󰀍

󰀃

yiyjykyl+

−1−1

6yiyjykyl

+

−1−1−1−1

yiyjykyl

󰀄

,

Z280(y)Z35′0(y)Z35′′

0

(y)Z56v(y)=

1

2

󰀍󰀃

y2iy2j

+

2y2iy−j

2

+

y−i2y−j

2

i=j

󰀄

+3,

=

1

2

y2iy2j

+

2y2iy−j

2

+

y−i2y−j

2

i=j

󰀃

󰀄+3,

=1

󰀍2

y2iy2

j

+

2y2iy−j

2

+

y−i2y−j

2

+3,

i=j

󰀃

󰀄

=

1

󰀍6

i=j󰀍=k=l

󰀃3y3

iyjykyl−1

+

y3iy−j1

y−k1y−l

1

+

yiyjykyl−3

+

3yiy−j1y−k1

yl−3

󰀄

+1

6

󰀍

󰀃

y3

iyj

ykyl+

3y3

iyjy−k1y−l

1

+

3yiyjy−k1y−l

3

+

y−i1y−j1y−k1y−l

3

i=j=k=l

󰀄

+1

+[8s+8c+56s+56c]fermion.(5)

ThisisthemultipletthatcomprisestheKaluza-KleinstatesofIIAsupergravitycompactifiedonS1,whicharethemomentumstatesofthecompactifiedIIAstring.ThereforethisparticularmultipletisknownastheKKAmultiplet.AlsotheD0-branesoftheIIAsuperstringtransformaccordingtothismultiplet.ThesecondultrashortmultipletistheKKBmultiplet.Nowonlythenoninvariantcentralchargeisdifferentfromzeroandequalinmagnitudetotherestmass.Themultipletdecomposesaccordingto

(8v+8c)×(8v+8c)=[10+10+280+280+350+35′′0]boson

+[8s+8s+56s+56s]fermion.

(6)

ThissupermultipletcomprisestheKaluza-KleinstatesofIIBsupergravitycompact-ifiedonS1.Clearly,theBPSstatesassociatedwithamembranewrappedaround

T2inelevendimensionswillalsoconstituteKKBmultiplets.

Intermediatemultipletsexistofmassive1/4BPSstatesannihilatedby8super-charges.Thismultipletcarriesbothtypesofcentralcharges.Thesmallestmultipletcontains212=211+211states.Theydoappearinstringtheoryasmixedstatesthatcarrybothwindingandmomentumandhaveanonzerooscillatornumberinordertosatisfythemass-shellcondition.Hencetheycarrymassesoftheorderofthestringscale.Thesmallestmultipletassociatedwiththelowestspinsdecomposesas

(8v+8s)×(8v+8c)×(8v+8c)

(7)

(oritsconjugate).

Finallytherearethelong(non-BPS)multipletswhereallthesuperchargesactnontrivially.Thesmallestonecomprises216statesanddecomposesinto

(8v+8s)×(8v+8s)×(8v+8c)×(8v+8c).

(8)

Beforeproceedingtothedeterminationofthegeneratingfunctionsforthefourclassesofsupermultiplets,letusdefinethesefunctionsfortwoN=1supermultipletsconsistingof(8v+8s)and(8v+8c).ObservethatthesearepreciselytheBPSmultipletsthatoneobtainsfromcompactifyingtheten-dimensionalsupersymmetricgaugetheoryonS1.WedenotethecorrespondinggeneratingfunctionsbyZ[s]andZ[c].Theyaregivenbyd

Z[s](y)=Z8v(y)−Z8s(y),Z[c](y)=Z8v(y)−Z8c(y),

(9)

andcanbecalculatedfromtheexpressionsgivenin(4).Itthenfollowsstraightfor-wardlythatthehelicitytracesgeneratedbyZ[s]andZ[c]vanishforn<4.Fromtheaboveresultitfollowsthat

ZKKA(y)=Zr(y)Z[s](y)Z[c](y),ZKKB(y)=Zr(y)Z[2c](y),ZINTERM(y)=Zr(y)Z[s](y)Z[2c](y),

2ZLONG(y)=Zr(y)Z[2s](y)Z[c](y),

(10)

whererdenotesthespinrepresentationoftheCliffordvacuum.Thissufficesto

showthatthehelicitysupertracesBnarevanishingforKKAandKKBmultipletswhenevern<8,forintermediatemultipletswhenevern<12andforlongmultipletswhenevern<16.Thisresulthasimportantimplicationsfortheone-loopgravitonamplitudesintype-IIstringtheories,aswewillshowbelow(obviouslythereisacorrespondingresultforN=1theoriespertainingtotheone-loopgaugefieldamplitudes).

SubsequentlywereviewthespectrumoftheBPSstatesofM-theorycompactifiedtoninespacetimedimensionsonT2[23,24,16].TheBPSmassformulabasedonthesupersymmetryalgebrawithamembranewindingcharge,takestheform

Mq1,q2,p=

1

|q1−τq2|+TmA|p|.Aτ2

(11)

TheultrashortBPSmultipletsareeithergivenbyKalazu-Kleinstateswithmomen-tumnumbers(q1,q2)alongthetwoperiodsofT2orbywrappedmembraneswhichcoverthetorusptimes.TheKaluza-Kleincharges(q1,q2)transformasdoubletsundertheSO(2)automorphismgroupofthesupersymmetryalgebra;thechargeassociatedwiththewrappedmembranestatesisinvariantunderthisgroup.In(11)AdenotesthevolumeofT2,measuredwithrespecttothe11-dimensionalmetric,τ=τ1+iτ2denotesthecomplexstructureofT2andTmdenotesthemembranetension.

TheBPSmassformula(11)canbeinterpretedinthecontextoftype-IIsuper-stringscompactifiedonS1.Forthispurposeletusrecallthewell-knownrelationsbe-tweenthestringandM-theoryparameters[25,26].ForsimplicitywesetTm=1and

22

assumethattheM-theorymetricisdiagonalandhastheformGij=diag(r9,r10),sothatA=r9r10,τ1=0andτ2=r9/r10.Theten-dimensionalIIAstringcouplingandthenine-dimensionalradiusoftype-IIAonS1(inthestringframe)arerelatedtotheM-theoryparametersby

√3/2AA

g10=r10,R9=r9

thestringmetricinunitsofthestringscale1/

AR9

√

1

r9

.

(13)

=

1r10

,

Bg10=

SubsequentlywecanexpresstheBPSmassformula(11)inthestringframeinterms

ofIIAandIIBstringtheoryvariables,respectively,

Mq1,q2,p=

󰀋󰀋󰀋

q1

Ag10

ThereforefromtheperspectiveoftheIIAstringtheory,q1istheIIAKaluza-Kleinmomentumnumber,whileq2istheD0-branecharge.TheM-theorymembranewrappingnumberpbecomestheperturbativewindingnumberintheIIAstring.Ontheotherhand,fromtheIIBperspective,q1andq2arethewindingnumbersoftheelemenarystringandofthesolitonicD1string,andthemembranewrappingnumberpistheIIBKaluza-Kleinmomentum.TheIIBstrong-weakcouplingS-dualityinterchangestheelementarystringswiththeD1strings:q1↔q2.UndertheIIA/Bduality,q1↔p,theIIA/IIBmassformulasareinterchangedprovidedoneinterchangestheD0withtheD1states.Clearlythepresenceofthewrappedmembranesisessential,astheycorrespondeithertotheIIAwindingstatesortotheIIBmomentumstates,respectively.

In(perturbative)type-IIstringtheorythevariousBPSstatesemergeasfollows.UltrashortBPSstates,whichpreserve16ofthetotal32supersymmetries,mustbe1/2BPSstatesbothwithrespecttotheleft-movingN=1andalsowithrespecttotheright-movingN=1supersymmetryalgebra.Thereforethesestatesarenotallowedtocarryanyoscillatorexcitations,i.e.NL=NR=0.HencetheirSO(8)helicitiesareentirelydeterminedbythebosonicandfermionicgroundstates,(8v+8s(c))L×(8v+8c)R,leadingtothedecompositionin(5)and(6)forthetwo

2

ultrashortmultiplets.Usingthestringlevelmatchingcondition,p2L=pR,where

AA

±pR9forIIA,andsimilarlyforIIB,itfollowsthattheultrashortBPSpL,R=q1/R9

statescancarryeithernonvanishingKKAquantumnumbersqiornon-vanishingKKBquantumnumbersp,butnotboth.

Theintermediate,1/4BPSstatesareshort1/2BPSmultipletswithrespecttotheleft-movingN=1supersymmetryalgebrabutarelongmultipletswithrespecttotheright-movingN=1supersymmetryalgebra(orviceversa).Thereforethey

2

requireNL=0andNRarbitrary.Thelevelmatchingconditionp2L=pR+2NRnowtellsusthatintermediatemultipletsmusthavebothwindingandmomentumandtuscarrybothtypeofcharges.ForNR=1wehavepreciselythedecomposition(7),whileforhigherNRwehaveintermediateBPSmultipletsofhigherspin.FinallythelongmultipletshaveNL,NR=0.ThesmallestlongmultipletwithNL=NR=1haspreciselythedecomposition(8)andotherswillhavehigherspins.

6

󰀋󰀋A󰀋+|p|R9

=

󰀋󰀋B󰀋q1R9

+q2

BR9

BR9

.(14)

Afterthisperusalofthevarioussupermultipletsfromcomplementaryviewpoints,webrieflydiscussthehelicitygeneratingpartitionfunctionofM-theoryonT2.Forstringtheoryithasbeenargued[15]thatthispartitionfunctionisrelatedtothestringamplitudesformulti-gravitonscatteringwhichwewillturntoshortly.Hencewestartfromthecorrespondingexpressionfortheperturbativetype-IIstringinninedimensions,

Z(φ,φ¯;q,q¯)=Str󰀆qL0q¯L¯0eiφiHi+iφ¯iH¯i󰀇

.

(15)

Hereqandq¯arerelatedtotheworld-sheetmodularparameterintheusualway.

TheexpressionforZ(φ,φ¯;q,q¯)canbewrittenas

Z(φ,φ¯;q,q¯)∼

1

√√

ThentheM-theoryBPSsumtakesthefollowingform:

Γ1,1(q,q¯)=

󰀍

q

τq2

+ATmp||q1−Aτ

2

2

τq2

−ATmp||q1−Aτ2

q¯.

2

(19)

q1,q2,p

However,Γ1,1doesnotdependontheanglesandthereforeplaysonlyaminor

roleonwhatfollows.Clearly,apartfromΓ1,1,theexpression(17)factorizesinaholomorphicandananti-holomorphicpart.Theappearanceofthesumoverthe

󰀅

powersoftheta-functions,4i=1θa(φi),isofcrucialimportanceforwhatfollows.UsingtheRiemannidentitytheθ-functions,wecanrewrite(17)inthefollowingway:

¯;q,qZ(φ,φ¯)=

1

∂φi1

···

∂

Hencethegravitonamplitudecanbeexpandedintermsofpowersoftheopera-¯iψ¯−iwhicharetheCartansubalgebracurrentsofSO(8)inthetorsψiψ−iandψ

left-andright-movingsectorgeneratedbytheworld-sheetfermions.Theircharges

󰀈

¯iψ¯−iaretheCartanalgebrageneratorsof¯i=(2πi)−1󰀈ψHi=(2πi)−1ψiψ−iandH

SO(8).Furthermoretheamplitudefactorizesintocontributionsfromtheleft-andtheright-movingsector.Eachcontributionthusdecomposesintoalinearcombi-nationofthehelicitysupertracesthataregeneratedbythefunctions(9).BecausethenumberofgravitonsisequaltothehighestpowerofofhelicityoperatorsHi

¯i,then-gravitonamplitudeisproportionaltoB2nandlowerhelicitysuper-andH

traces.Thisprovesthatthefour-gravitonamplitudemustbeproportionaltothehelicitysupertraceB8andwillthusreceiveconstributionsfromonlyultrashortBPSmultiplets.Likewisethesix-gravitonamplitudewillreceivecontributionsfrombothultrashortandintermediateBPSmultiplets.Thecontributionfromthelattercon-tainsatermproportionaltoB12.Foreightandmoregravitonsallsupermultipletswillinprinciplecontribute.

Havingestablishedtheseresults,webrieflyconsidertheR4termsinmoredetail.Herewecastthecalculationintermsofnine-dimensionalfieldtheory,wheretherelevantamplitudehasthestructureofaboxdiagraminmassiveϕ3theoryinninespacetimedimensions.InviewofourpreviousresultsitsufficestorestrictourselvestocontributionsfromtheKKAandKKBstates.SinceweevaluatethecoefficientoftheR4termatzeromomentum,suchanintegraltakestheform

1

(q2+M2)4

=1(4π)9/2

󰀂

∞

0

dtt−3/2e−tM.

2

(24)

Observethatthisintegralhasalinearultravioletdivergence,whichreflectsitselfinthesingularbehaviouroftheintegrandatt=0.Theintegralhasthedimensionofamass,whichisappropriateforacouplingconstantofR4inninespacetimedimensions.Wewillnowuse(24)andtheBPSmassformula(11)anddeterminethecontributionsfromboththeKKAandtheKKBstates.Fortheformerweobtain

AKKA4

===

1(4π)9/21(4π)52(4π)6

󰀂󰀂

∞0

dtt

−3/2

q1,q2

∞

0

dtt1/2

′,q′q12

󰀍

󰀍

e−t(Aτ2)

−1

−1|q

1−τq2|

2

e−πtτ2

′+τq′|2|q12

f(τ,τ¯),(25)

wherethemodularfunctionf(τ,τ¯)isdefinedby[5]

f(τ,τ¯)=

󰀍

τ2

3/2

′,q′)=(0,0)(q12

Inthesecondlineof(25)weperformedaPoissonresummationandchangedtheintegrationvariablet→πA/t.Observethattheultravioletdivergenceisnowassociatedwiththeupperintegrationboundaryandisonlypresentfortheterm

′′

withq1=q2=0.Thiscontributionwasdroppedinthethirdline.Likewise,weobtainfortheKKBstates,

AKKB4

===

1(4π)9/21(4π)54(4π)6

󰀂󰀂

∞0

dtt

−3/2

∞

0

dt1

p′=0

󰀍

󰀍

p′

󰀍

p

e−tTmA

22

p2

e−πtp

′2

13

3

πTmA.

2

󰀇

(28)

ThisresultisinvariantundertheIIBS-dualitysymmetryτ→(aτ+b)/(cτ+d).

ThecontributionfromtheKKBstatesissuchthattheresultiscompatiblewithT-dualityoftype-IIstringtheory.

Letusconsiderthetwopossibledecompactificationlimitstotheten-dimensionalIIA/Btheories.Hereweneedtheresultthat,forτ1=0andτ2large,thefunctionf(τ,τ¯)hastheform

f(τ,τ¯)=2ζ(3)τ2

3/2

+

2

√3

(4π)6󰀌󰀉

ζ(3)

3

π2+···+

󰀊

1

B2

(R9)

󰀎

.(30)

B

ForlargeR9thelasttermvanisheswhilethefirsttwotermsyieldthetreeandtheone-loopcontributionstotheR4term,andtheellipsesdenoteexponentially

B−2

suppressedtermsin(g10),whichcorrespondtononperturbativeIIBD-instanton

√

contributions.(Thefactor

√3(4π)6󰀌󰀉

ζ(3)

2π3

1

2

.π3

󰀎

(31)

10

AA2

Nowtheellipsesdenotetermsthatareexponentiallysuppressedin(R9/g10).Drop-pingthesetermsaswellasthesecondtermtermwhichallvanishinthedecom-pactificationlimit,weareleftwiththeIIAstringtreeandone-loopcontributiontoR4(thelatteroriginatedfromtheKKBmultiplets).HenceperturbativeT-dualityismanifestismanifestintheresults

ItisanintriguingquestionwhythesubtractionmethodbasedonPoissonre-′′

summationandthesubsequentsubtractionoftheq1=q2=0andp′=0termsleadstothecorrectresult.ApuzzlingfeatureofthecalculationisthatthesumovertheKKAandthesumovertheKKBstatesbothincludethemasslessstates,whichseemstomakenosense.CorrectingforthisrequirestosubtractaninfinitetermwhichcouldinprinciplebecancelledagainstthepositiveinfinitecontributionsofthemassiveBPSstates.However,whythecut-offshouldbefine-tunedsuchthatthiscancellationtakesplaceremainsamystery.Withinthecontextofnine-dimensionalsupersymmetrytherearenootherstatesthatcouldpossiblycanceltheinfiniteterms.ThecoefficientoftheR4termisnotgenericallyfinite,whichreflectsitselfinthefactthattheKKAcontributionsdisappearinthedecompactificationlimittoelevendimensions,A→∞,whichissomewhatcounterintuitive.Westressthatsupersymmetrydoesnotseemtoberelevantforobtainingfiniteresults,inviewofthefactthatthesupertracesfortheKKAandKKBmultipletsdonotvanish,andnocancellationseemstoarisebetweentheinifinitecontributionsfromthetwotypesofmultiplets.

Acknowledgements

WethankMohabAbou-Zeid,EliasKiritsis,WolfgangLerche,HermannNicolaiandKostasSkenderisforclarifyingdiscussions.ThisresearchwassupportedinpartbytheNationalScienceFoundationunderGrantNo.PHY94-07194throughtheInstituteforTheoreticalPhysicsinSantaBarbara.Wethanktheinstituteforthehospitalityextendedtousduringthiswork.TheresearchwasalsosupportedbytheEuropeanCommissionTMRProgramundercontractERBFMRX-CT96-0045,inwhichHumboldtUniversityatBerlinandUtrechtUniversityareassociated.Appendix

FortheconvenienceofthereaderwepresenttheweightvectorsforanumberofSO(8)representationswhichcanbeusedintheconstructionofthegeneratingfunctions(4).Toeachoftheweightvectorsonemustaddallpossiblepermutations.Incertaincasesthenumberofminussignsmustbeevenorodd.Thisisindicatedby,e.g.[+20;even],implyingthat20independentpermutationsshouldbeaddedwithanevennumberofminussigns.Thesubscripts0,v,sandcrefertothedifferent

11

conjugacyclassesofweightvectors.

8v:8s:8c:280:350:35′0:35′′0:56v:56s:56c:

󰀮λ=(±1,0,0,0)[+6];

󰀮λ=(±12,±12)[even];󰀮,±1)[odd];λ=(±1

2

2

󰀮λ=(±1,±1,0,0)[+20],󰀮λ′1,2,3,4=(0,0,0,0);󰀮λ=(±2,0,0,0)[+6],󰀮λ′=(±1,±1,0,0)[+20],󰀮λ′′=(0,0,0,0);

1,2,3

󰀮λ=(±1,±1,±1,±1)[even],󰀮λ′=(±1,±1,0,0)[+20],󰀮λ′′󰀮λ=(±1,±1,±1,±1)[odd],󰀮λ′=(±1,±1,0,0)[+20],󰀮λ′′

1,2,3

=(0,0,0,0);=(0,0,0,0);

1,2,3

󰀮λ=(±1,±1,±1,0)[+24],󰀮λ′1,2,3=(±1,0,0,0)[+6];󰀮λ=(±32,±12)[+3;odd],󰀮,±1)[even];λ′=(±1

1,2,3

󰀮)[+3;even],λ=(±32,±2󰀮λ′1,2,3=(±12,±12)[odd].

2

1

2

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