Branes, Fluxes and Duality in M(atrix)-Theory

We use the T-duality transformation which relates M-theory on T^3 to M-theory on a second T^3 with inverse volume to test the Banks-Fischler-Shenker-Susskind suggestion for the matrix model description of M-theory. We find evidence that T-duality is realiz

hep-th/9611202,PUPT-1668

Branes,FluxesandDualityinM(atrix)-Theory

arXiv:hep-th/9611202v3 10 Dec 1996OriJ.Ganor,SanjayeRamgoolamandWashingtonTaylorIVDepartmentofPhysics,JadwinHallPrincetonUniversityPrinceton,NJ08544,USAoriga,ramgoola,wati@puhep1.princeton.eduWeusetheT-dualitytransformationwhichrelatesM-theoryonT3toM-theoryonasecondT3withinversevolumetotesttheBanks-Fischler-Shenker-SusskindsuggestionforthematrixmodeldescriptionofM-theory.We ndevidencethatT-dualityisrealizedasS-dualityforU(∞)N=4Super-Yang-Millsin3+1D.WearguethatKaluzaKleinstatesofgravitonscorrespondtoelectric uxes,wrappedmembranesbecomemagnetic uxesandinstantonicmembranesarerelatedtoYang-Millsinstantons.TheT-dualitytransformationofgravitonsintowrappedmembranesisinterpretedasthedualitybetweenelectricandmagnetic uxes.Theidenti cationofM-theoryT-dualityasSYMS-duality

ingtheequivalencebetweencompacti edM(atrix)theoryandSYM,we ndanaturalcandidateforadescriptionofthelight-cone5-braneofM-theorydirectlyintermsofmatrixvariables,analogoustotheknowndescriptionoftheM(atrix)theorymembrane.

November,1996

We use the T-duality transformation which relates M-theory on T^3 to M-theory on a second T^3 with inverse volume to test the Banks-Fischler-Shenker-Susskind suggestion for the matrix model description of M-theory. We find evidence that T-duality is realiz

1.Introduction

Duringthepasttwoyears,evidencehasbeenaccumulatingwhichindicatesthataconsistentquantumtheory(M-theory)underlies11Dsupergravity[1,2].Recently,anex-citingconjectureforamicroscopicdescriptionofM-theoryhasbeenputforwardbyBanks,Fischler,ShenkerandSusskind[3].TheBFSSmodelincorporatesinanaturalwaythenon-commutativenatureofmicroscopicspacetime[4]andthequantizationofthemem-brane[5].

Theauthorsof[3]haveshownthattheirmodelcontainsmanyofthefeaturesofM-theory:thesupermembrane,correctgravitonscatteringamplitudes,toroidalcompacti- cationandpartial11DLorentzinvariance.FurtherevidencefortheBFSSconjecturewassuppliedin[6]wherethebehaviorofamembraneina5-branebackgroundwasstudied.Questionswhichremainopenincludeageneraldescriptionofcompacti cation,anintrinsicdescriptionofa5-braneandacompleteproofof11DLorentzinvariance(asuggestioninthisdirectionhasbeenmadein[7]).

ThepurposeofthepresentpaperistopassM(atrix)-theorythroughonemoretestbyconsideringitsbehaviorunderT-duality.T-dualityrelatescompacti edtypeIIAtocompacti edtypeIIBsoinordertogetan“automorphism”ofM-theoryweneedtocompactifythetypeIIAtheoryonT2andapplyT-dualitytwice.ThisgivesaconnectionbetweenM-theoryonT3withvolumeVandM-theoryonT3withvolume1/V.Underthisduality,wrappedmembranestatesareexchangedwithKaluza-Kleinstatesofthegravitonandunwrappedmembranesbecomewrapped5-branes[8,9].

OurdiscussionbeginswithareviewoftheM(atrix)-theorydescriptionoftoroidalcompacti cationgivenin[3,10].TheresultingmodelisalargeNlimitofU(N)Super-Yang-Millstheory.Werecastthederivationinaslightlydi erentlanguageandexplainhowtwistedsectorsoftheU(N)bundleappear.WethenarguethatT-dualityisrealizedasS-dualityintheSYMtheoryandthatgraviton membranedualityisrelatedtoelectric magneticduality.Finally,wediscusssomeissuesrelatedto ndinganexplicitconstructionofthe5-braneinM(atrix)theory.

Thepaperisorganizedasfollows:Section2isareviewoftoroidalcompacti cationoftheM(atrix)-model.InSection3werelateS-dualityofN=4SYMtoT-duality.InSection4,thetransformationsofgravitonsintomembranesinM-theoryarediscussed.Werelatemembranestomagnetic uxesandgravitonswithKKmomentumtoelectric uxes.InSection5weusetheinterpretationofthemembranesintermsofmagnetic uxtoobtaintheenergyandcountingofmembranestates.WealsorelatetheYang-MillsinstantontoaEuclideanmembrane.InSection6wesuggestaformulationofa5-branewrappedaroundthelight-conedirectionsintermsofBFSSmatrixvariablessatisfyingacertainrelation.WealsodiscusstheimplicationsoftheT-duality/S-dualityequivalenceforthesearchfora5-braneinM(atrix)theorywhichextendsalong5transversedirections.

We use the T-duality transformation which relates M-theory on T^3 to M-theory on a second T^3 with inverse volume to test the Banks-Fischler-Shenker-Susskind suggestion for the matrix model description of M-theory. We find evidence that T-duality is realiz

2.ReviewofCompacti cation

Onewayofunderstandingtoroidalcompacti cationofM(atrix)-theoryisbyconsid-eringasectoroftheN→∞0-branetheoryinwhichtheXmatricessatisfycertainsymmetryconditions.Thisdescriptionforcompacti cationonthed-dimensionaltorusTdcanbegivenasfollows[3,10]:In niteunitarymatricesUiarechosenfori=1...dthatcommutewitheachother,

UiUj=UjUi,(2.1)

andgenerateasubgroupofU(∞)isomorphictoZd.Thecompacti edtheoryisgivenbyrestrictingtothesubspaceofX’swhichareinvariantundertheZdaction:

Ui:Xµ →Ui 1XµUi+eiµ(2.2)

wheretheei’sformabasisofthelatticewhoseunitcellisTd.Thecountablyin nitedimensionalvectorspaceonwhichtheX’sactcanbewrittenasatensorproduct

V=VN Hd(2.3)

whereVNisanN-dimensionalspaceandHiscountablyin nitedimensional.Onecanthentake

Ui=I I1 ··· Ii 1 Si Ii+1 ··· Id(2.4)

whereIjistheidentityonthejthHandSiisashiftoperator

(Si)k,l=δk+1,l(2.5)

(theindicesk,lrunoverallintegersZ).ByrestrictingtheLagrangiantothesubspaceofX’sinvariantunder(2.2)oneobtainstheLagrangianfor(d+1)dimensionalSYMtheory.Thegauge eldsare

A(µd

i=1xi ei)=l1,...,ld∈Z e2πi lixiµX(0,l1)···(0,ld)(2.6)

d.Whend=3,theinversesquaredcouplingwhere eiformabasisofthedualtorusT

constantisthevolumeofT3(in11-dimensionalPlanckunits)andthe(3+1)-dimensional

3.theoryisde nedonthedualtorusT

Forlateruse,wewillrecastthederivationinadi erentlanguage.WenotethattheresultingmatrixXµinvariantunder(2.2)isjustthematrixoftheoperator

µ=i µ+Aµ

(2.7)

We use the T-duality transformation which relates M-theory on T^3 to M-theory on a second T^3 with inverse volume to test the Banks-Fischler-Shenker-Susskind suggestion for the matrix model description of M-theory. We find evidence that T-duality is realiz

actingon eldsinthefundamentalrepresentation

φk(d i=1xi ei),k=1...N(2.8)

d.TherequisiteformoftheXmatricesiswhicharesectionsofthetrivialbundleoverT

k(n1...nd):obtainedbywriting µintheFourierbasisofφ

φk( xi ei)= k(n1...nd)eφ2πi

{ni} nixi.(2.9)

TheoperatorsUjcanbetakentoactonsectionsby

Ujφk( xi ei)=e2πixjφk(

From(2.7)itisclearthatifwealsotakeXµforµ=d+1,...,9tobethematricesoftheoperators

φ →Φµφ(2.11) xi ei)(2.10)

whereΦµarethescalar eldsofSYM,thentheBFSSLagrangianwillreducetotheSYMLagrangian.

3.CheckingT-duality

Anon-trivialdualityofM-theoryisobtainedbycompactifyingonT3,regardingthetheoryastypeIIAonT2andT-dualizingtwice(oncealongeachdirectionoftheT2).

3.1.ReviewofT-dualityforM-theory

WewillbeusingelevendimensionalPlanckunitslp=1.T-dualityonM-theoryisobtainedfromtherelations(hereGµνisthemetricincoordinates0,...,6,11.Notethatourspace-timecoordinatesarex0...x9,x11):

M theory

1/21/2R9Gµν;R9R7,R9R8;λst3/2R9=(3.1)

togetherwithtypeIIAT-dualitytwice(hereliarelengthsinstringunits):

TypeIIA

1 1 1 1gµν;l8,l9;l8l9λ(3.2)

We use the T-duality transformation which relates M-theory on T^3 to M-theory on a second T^3 with inverse volume to test the Banks-Fischler-Shenker-Susskind suggestion for the matrix model description of M-theory. We find evidence that T-duality is realiz

Toobtain

(3.3)V2/3Gµν;V 2/3R7,V 2/3R8,V 2/3R9

whereV=R7R8R9(theRHSactuallycomesoutwithR7andR8switchedbutin(3.3)wehavecombinedare ection).ThiscanbegeneralizedtoslantedT3’sandtoincludethe3-formCµνρ[8]:1τ=iV7,8,9+C7,8,9 → M theory

We use the T-duality transformation which relates M-theory on T^3 to M-theory on a second T^3 with inverse volume to test the Banks-Fischler-Shenker-Susskind suggestion for the matrix model description of M-theory. We find evidence that T-duality is realiz

3.2.T-dualityintheMatrixmodel

WehaveseenthatM-theoryonT3isequivalenttoN=4U(N)SYMonthedual 3:torusT

H=1

g

g26 2+θB21

A=1|DiΦA|2+

g2+θ

τ(3.9)

3arerescaledby(Imτ)2/3withoutagreeswith(3.4).Accordingto(3.3),thesidesofT

changingitsshape.ThankstotheexactconformalinvarianceofN=4Yang-Millswecan

3andde nethematrixmodelonaT 3ofvolume1.ThishastobeaccompaniedrescaleT

byrescalingofthesixscalarsXµbyV 1/3.TheS-dualXµshouldthusberescaledbyV1/3.SinceS-dualityofSYMmultipliestheHiggsVEVsbyV=1/g2we ndthataltogetherXµisrescaledbyV1/3.ThisisinaccordwiththeWeylrescalingofGµνin(3.3).

4.T-dualityactiononbranes

InthissectionweexaminehowtheactionofT-dualityonstatesismanifestedinthematrixmodel.SinceS-dualityexchangeselectricandmagnetic uxes,anaturalguessisthatelectric(magnetic)statescorrespondtograviton(membrane)states.

AsdescribedinSection3,boththewindingnumberofamembraneandtheKK

3.Moreover,momentumareparameterizedbyvectorsintheduallatticewithunitcellT

T-dualityexchangesthewindingandthemomentum.Electricandmagnetic uxesarealsoparameterizedbyvectorsintheduallattice.TheirexchangeunderS-dualityisinaccordwiththeiridenti cationwithbranes.Weshallnowexaminethecorrespondenceinmoredetail.

We use the T-duality transformation which relates M-theory on T^3 to M-theory on a second T^3 with inverse volume to test the Banks-Fischler-Shenker-Susskind suggestion for the matrix model description of M-theory. We find evidence that T-duality is realiz

4.1.Reviewofelectricandmagnetic uxes

WerecallthatU(1)isanormalsubgroupofU(N)and

U(N)=(U(1)×SU(N))/ZN(4.1)

3.OnT 2 T 3wecanhavenon-trivialWewillconsiderU(N)Super-Yang-MillsonT

U(N)-bundles.StateswithoneunitofU(1)magnetic uxsatisfy:

tr{B}=2π.(4.2) 2T

Tobuildthecorrespondingbundle,wepick[11]:U 1V 1UV=e 2πi

NU 1φ(x+1,y)=V 1φ(x,y+1).(4.4)

Astatewithoneunitofelectric uxinthedirectionofS1 T3(whereS1issomecycleofthetorus)satis es: tr{E}=2π.(4.5)

S1

4.2.Fluxesandcompacti edM(atrix)-theory

Inthissectionweshowhowthemembraneof[3]naturallybecomesastatewithmagnetic uxaftercompacti cation.Toseethis,webeginbyshowinghowtwistedU(N)bundlesareobtainedfromtheconstruction(2.2).Theidenti cationofXµas(i µ+Aµ)attheendofSection2suggeststhatwewritethesameoperatorasamatrixactingonsectionsofthetwistedbundle.Giventhisidenti cation,wehave

[X7,X8]=[ 7, 8]=iB78

whereinthesectorwithoneunitof ux,B78isthescalarmatrix2π(4.6)

NyUφ(x,y),

φ(x,y+1)=Vφ(x,y),

(4.7)

We use the T-duality transformation which relates M-theory on T^3 to M-theory on a second T^3 with inverse volume to test the Banks-Fischler-Shenker-Susskind suggestion for the matrix model description of M-theory. We find evidence that T-duality is realiz

where

U= 1

e2πi

N

U 1V 1UV=e , 2πi V= 111 .. . (4.8)

N(y+k+Np)x.(4.11)

|2<∞.So,Hereφissomearbitrarycontinuousfunctionde nedon( ∞,∞)with|φ

∈L2(IR)wecan ndthespaceonwhichXµisde nedisnowcontinuous(though,sinceφ

acountablebasisasbefore).

(w)we ndWritingXµinthebasisofφ

(X7)kl(w,w′)=2π

N)w′7ap+k l

N,

(X8)kl(w,w′)=iδ′(w w′)δkl s δ(w′+Np+k l w)e2πi(q+

p,q∈Z,s∈ZNN,q+s

N,q+

8sN)x2πi(q+es

Nx+

p,q∈Z,s∈ZN ap+k lNe2πi(p+k lN)(y+l)(4.13).

From(4.10)itisobviousthattheBFSSactiongoesovertotheSYMaction.TheU1andU2matriceswhichde nethesectorarestillgivenbymultiplicationbyeixandeiy.AsN×Nmatricestheyaredi erentfromthosefortheuntwistedsectorbecausetheyactonsections(4.11)ratherthanfunctions.ThusthereareseveralwaystoembedZdsubgroupswhicharenotmutuallyconjugate.

We use the T-duality transformation which relates M-theory on T^3 to M-theory on a second T^3 with inverse volume to test the Banks-Fischler-Shenker-Susskind suggestion for the matrix model description of M-theory. We find evidence that T-duality is realiz

WealsonotethatthereisanotherwaytorealizethetwistedXµ’s.Forcompacti -cationsonT2,wetakeavectorspace(onwhichXµwillact)thatisaproductofa nitedimensionalvectorspaceVNofdimensionNandsingleHilbertspaceHwithacountablyin nitebasis

V=VN H.(4.14)

Notethedi erencebetween(4.14)and(2.3)–hereonlyonecopyofH(ratherthantwo)isusedforcompacti cationonT2.Thisdi erencemayberelatedtotheformoftheexpansion(4.11).WerealizeU1andU2as

U1=U eiP/√N,(4.15)

whereQ,ParecanonicaloperatorsactingonW([Q,P]=2πi)andU,Vareasin(4.8).U1andU2commuteandonecancheckthattheexpansionof(2.2)agreeswith(4.12).

Nextweshowtherelationbetweenelectric uxandgravitonstates.Theoperator

3isgivenbythatmeasuresthetotalelectric uxinsideT

Using

Xµ=i µ+Aµ,

wecanwritetheelectric uxas

tr{Ei}= ˙i=Tr{X˙i}=Tr{Πi}A(4.18)(4.17)tr{Ei}.(4.16) 3T 3T 3T

whereTristhetraceinthein nitebasisofthematrixmodel.Thus,electric uxinSYMnaturallycorrespondstomomentumalongthei-thdirectionoftheT3inM-theory.

5.Moreonmembranes

Inthissectionwedescribefurtherevidenceinfavoroftheidenti cationsbetweenmagnetic uxandLorentzianmembranes.WealsoproposeaconnectionbetweenYang-MillsinstantonsandEuclideanmembranes.

We use the T-duality transformation which relates M-theory on T^3 to M-theory on a second T^3 with inverse volume to test the Banks-Fischler-Shenker-Susskind suggestion for the matrix model description of M-theory. We find evidence that T-duality is realiz

5.1.Energyandcountingofwrappedmembranestates.

ConsiderM-theoryonT2.Wewouldliketocalculatetheenergyofthemembraneusingthefactthatthemembranestatescorrespondtostatesinthesectorwithnon-zeromagnetic uxintheU(1)subgroup.In“mathematical”conventionswherethereisan

2overallgYMinfrontoftheaction,theunitof uxis

trB=2π.(5.1)

TakingBoftheformB0timestheidentitymatrix,we ndthat

B0=1

Ri.In

thissubsectionwerestorethe11DPlancklength

lp=g1/3√

α′

2 2gYM trB=2(2π) 4(R1R2)2

p2+M2= g√

g√g√α′T2(R1R2)2

α′duetothemomentum

alongR11tothekineticenergy.Thisexplainswhythesquareoftheareaappearsinthenumerator,andgivesthetensionofthemembrane,

T2=1

6(2π)4lp(5.7)

We use the T-duality transformation which relates M-theory on T^3 to M-theory on a second T^3 with inverse volume to test the Banks-Fischler-Shenker-Susskind suggestion for the matrix model description of M-theory. We find evidence that T-duality is realiz

inagreementwith[3].

M-theoryonR9×T2shouldhave,for xedmomentumonR9,onenormalizableBPSmultiplet(annihilatedbyhalfthesupersymmetries)withthequantumnumbersofann-wrappedmembrane.Thisisknown[12]byusingtherelationbetweenM-theoryandtypeIIstrings.InthecontextoftherelationbetweenM-theoryandlargeNU(N)YangMills,andtheabovedescriptionofmembranewindingnumberasthemagnetic uxintheU(1)wecanarguethatthisistherightcounting.SoweneedtoknowthenumberofstatescomingfromlargeNU(N)YangMillswhichsitina28dimensionalrepresentationofsupersymmetryandcarrynunitsofmagnetic ux.Followingargumentsin[4]andin[13],andinagreementwiththediscussionattheendofSection4,preciselythisYangMillsquestionarisesifwewanttocountthenumberofnormalizableboundstatescarryingNunitsof2-branechargeandnunitsof0-branechargeintypeIIAtheory.Weknowthattheuniqueboundstate[14]ofNtypeIIA2-branescanbegivennunitsofmomentumalongtheeleventhdimension.Weconcludethatthereshouldbeaunique1/2BPSsaturatedstatecomingfromthesectorwithnunitsofmagnetic uxinU(N)YangMills.ThisstatementinthelargeNlimitshowsthatM-theoryonT2hasexactlyoneBPSmultipletcarryingthechargeofamembranewrappedntimesontheT2.

5.2.Euclideanmembranes

SinceLorentzinvarianceisnotmanifestintheBFSSformulation,itisinterestingtocheckwhataninstantonicmembranelookslike.SincetheMinkowskimetriciscrucialtotheIMFformulation,wewillthinkofaEuclideanmembraneasatransitionbetweentwodi erentvacuawhichwewillsoonidentify.Let’stakeaEuclideanmembranethatwrapsallofT3.Sinceitislocalizedindirections1...6,thecorrespondingSYMsolutionmusthaveallsixadjointscalarssettoascalarmatrix,say:

Φ1=···=Φ6=0.(5.8)

ThisallowsforanordinaryYang-Millsinstantonsolutioninthegauge elds.Itisthustemptingtoguessthatafullywrappedmembranecorrespondstoatransitionbetweentwovacuawhichdi erbyalargegaugetransformationrelatedtoπ3(U(N)).Infavorofthisclaimwerecallfrom(3.4)thattheθ-angleisrelatedtoC7,8,9.AmembraneinstantonthatwrapsT3isatransitioninwhich (inanappropriategauge):

dxd1 6dx11dtθ(5.9)

changesbyoneunit.Ifwereplaced

g2+iθ

We use the T-duality transformation which relates M-theory on T^3 to M-theory on a second T^3 with inverse volume to test the Banks-Fischler-Shenker-Susskind suggestion for the matrix model description of M-theory. We find evidence that T-duality is realiz

6.5-branesinM(atrix)theory

Wenowdiscussseveralissuesrelatedtotheconstructionofa5-braneinM(atrix)theory.A5-branewhichextendsalongthelight-conedirectionsx±(andfourmoredi-rections)hasbeendiscussedin[6].However,the5-branedescribedbytheseauthorswasessentiallygivenasabackgroundforthe0-branetheory.Byusingtherelationbetweenthe0-brane eldsonatorusandthecovariantderivativeoperatoronthedualtorus,we ndanaturaldescriptionofthelight-cone5-branewhichisintrinsictothe0-branevariablesofM(atrix)theory.WealsodiscussthepossibilityofusingT-dualitytodescribea5-branethatoccupies vetransversedirectionsandisboostedalongtheBFSSpreferreddirection.

6.1.Thelight-cone5-brane

ItwasshownbyWitten[15],andinamoregeneralformbyDouglas[16],thataninstantonona(p+4)-branecarriesp-branecharge.Thisresultisessentiallyduetothefactthattheworldvolumetheoryonthe(p+4)-braneincludesaChern-Simonsterm

C∧eF.(6.1)

Σp+5

whereCisasumoverRR elds.Theterm

C(p+1)∧F∧F(6.2)

inparticularcouplesaninstantontothe eldC(p+1),underwhichp-branesareelectricallycharged.Asoneapplicationofthisresult,YangMillsinstantonsembeddedinathree-braneworld-volumetheoryappearasD-instantons.AfteraT-dualitywhichconvertsthreebranestozerobranes,theseD-instantonsareconvertedtomembraneinstantons.Relatingthistothetransformationfromthe0-branevariablesofM-theoryonT3totheYangMillsvariables,thisgivesanotherpieceofevidenceinfavouroftheidenti cationproposedinsection5.2.

Wecanusethissametypeofrelationtoshowhowa5-braneofM-theorycanbeconstructeddirectlyfromtheXmatrix eldsofBFSS.Wrappingthe5-branearoundthelight-conedirections,weshould nda4-braneintheresultingIIAtheory.Letuscompactifyindimensions6-9.Sincea4-braneonatorusT4goestoa0-braneonthedual

4,theconnection(2.7)indicatesthatacon gurationoftheXmatricessatisfyingtorusT

Tr ijklXiXjXkXl=8π

(6.3)

We use the T-duality transformation which relates M-theory on T^3 to M-theory on a second T^3 with inverse volume to test the Banks-Fischler-Shenker-Susskind suggestion for the matrix model description of M-theory. We find evidence that T-duality is realiz

willhaveaunitof4-branecharge.(Theindicesi laresummedoverthecompacti edcoordinates6-9.Theantisymmetricproductof4X’sisjust1

We use the T-duality transformation which relates M-theory on T^3 to M-theory on a second T^3 with inverse volume to test the Banks-Fischler-Shenker-Susskind suggestion for the matrix model description of M-theory. We find evidence that T-duality is realiz

7.Conclusions

WehaveseenthatT-dualityisrealizedinanaturalwayintheM(atrix)-theoryofBFSSasS-dualityoflargeNU(N)N=4Yang-Millstheory.Thelarge/smallvolumedualityofM-theoryismappedtotheweak/strongcouplingdualityofSYMandgraviton(0-brane)/membranedualityismappedtoelectric/magneticduality.Wehavealsoseenthattherearedi erentinequivalentembeddingsoftheZdsymmetrygroupoftranslationsintheBFSSmodel.Thedi erentembeddingsgiverisetodi erentU(N)bundlesintheSYMtheory.ThewrappedmembraneofM-theoryisidenti edwithstatescarryingmagnetic ux.

Theconnectionsdevelopedinthispaperprovideanaturalframeworkinwhichtotrytounderstandthe5-braneinM(atrix)theory.Theunwrapped5-braneisnaturallyrelatedthroughS-dualityof3+1dimensionalSuper-Yang-Millstheorytoa2-branecon gurationwhichcanbeunderstoodinmatrixvariablesX.Furthermore,the5-branewrappedaroundthelight-conedirectionshasanaturaldescriptionasan“instanton”of4+1dimensionalSuper-Yang-Mills,whichallowsustodescribeitintermsofasetofmatrixvariablessatisfyingtherelationTr ijklXiXjXkXl=8π.InthelanguageoftypeIIAstringtheory,thefactthatitispossibletodescribethe4-braneintermsoffundamental0-brane eldsisessentiallytheT-dualoftheresultthatinstantonsona4-branecarry0-branecharge.Remarksalongtheselineswerealsomadein[17].Itwouldseemthattheabilityof0-branestoformthehigherdimensionalbranesofM-theoryisastrongargumentinfavoroftheconjectureofBFSSthatinfact0-branesformacompletedescriptionofallthedegreesoffreedominM-theory,atleastintheIMFframe.

Acknowledgments

WewishtothankA.Hashimoto,R.Gopakumar,D.J.Gross,I.Klebanov,G.Lif-schytz,S.MathurandE.Wittenforhelpfuldiscussions.OJGisalsogratefultoT.BanksforadiscussionduringtheRutgersUniversitytheorygroupmeeting.TheresearchofSRandWTissupportedbyNSFGrantPHY96-00258andtheresearchofOJGissupportedbyaRobertH.DickefellowshipandbyDOEgrantDE-FG02-91ER40671.

Noteadded

Asthisworkwascompleted,apaperbyL.Susskind[18]appearedwhichdiscussesthesameissue.

We use the T-duality transformation which relates M-theory on T^3 to M-theory on a second T^3 with inverse volume to test the Banks-Fischler-Shenker-Susskind suggestion for the matrix model description of M-theory. We find evidence that T-duality is realiz

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