On the Supersymmetry and Gauge Structure of Matrix

arXiv:hep-th/0003161v118Mar2000UT-Komaba00-05hep-th/0003161March,2000OntheSupersymmetryandGaugeStructureofMatrixTheoryY.Kazama†andT.Muramatsu‡InstituteofPhysics,UniversityofTokyo,Komaba,Meguro-ku,Tokyo153-8902JapanAbstractSupersymmetricWardidentityforthelowenergyeffectiveactioninthestandardbackgroundgaugeisderivedforarbitrarytrajectoriesofsupergravitonsinMatrixThe-ory.Inourformalism,thequantum-correctedsupersymmetrytransformationlawsofthesupergravitonsaredirectlyidentifiedinclosedform,whichexhibitanintricateinterplaybetweensupersymmetryandgauge(BRST)symmetry.Asanapplication,weexplicitlycomputethetransformationlawsforthesource-probeconfigurationat1-loopandconfirmthatsupersymmetryfixestheformoftheactioncompletely,includingthenormalization,tothelowestorderinthederivativeexpansion.†[e-mail]‡[e-mail]1IntroductionBynowaconsiderableamountofevidencehasbeenaccumulatedfortheMatrixtheoryforM-theory,originallyproposedbyBanks,Fischler,ShenkerandSusskind[1]andlaterre-interpretedbySusskind[2]intheframeworkofdiscretelight-conequantization.Inparticular,justtomentiononlythedirectcomparisonwithelevendimensionalsuper-gravity,completeagreementforthemulti-gravitonscattering(includingtherecoileffects)at2-loop[3,4]andthatforthetwo-bodypotentialbetweenarbitraryfermionicaswellasbosonicobjectsat1-loop[5]canbecitedashighlynon-trivialandremarkable.Despitesuchimpressivepiecesofevidenceaswellasgeneralsupportivearguments[6,7],thedeepreasonandthemechanismoftheagreementareyettobefullyunderstood.Evidently,oneofthekeysshouldbetheunderstandingofthestructureandtheroleofthesymmetries.Thewell-knownsymmetriespresentbothintheMatrixtheoryandinthesupergravitytheoryaretheglobalSpin(9)invariance,theCPTinvarianceandtheinvarianceunder16supersymmetries.Perhapslessfamiliaristhegeneralizedconformalinvariance[8,9,10],thegeneralizationoftheconformalsymmetrythatplaysthemajorroleintheAdS/CFTcorrespondenceofMaldacena[11].Besidesthesesymmetries,thesupergravitypossessesthegeneralcoordinateinvariancewhiletheMatrixtheoryhastheYang-Millstypegaugesymmetry,whichmustbedeeplyconnected.Finally,althoughnotyetclearlyidentified,theagreementofthemulti-bodyscatteringamplitudesstronglysuggeststhattheelevendimensionalLorentzinvarianceispresentinahighlynon-trivialmannerintheMatrixtheory.ExceptfortheelevendimensionalLorentzinvariance,theabove-mentionedsymme-triesoftheMatrixtheoryareeasilyrecognizedintheoriginalaction.However,sincethesupergravityinteractionsbetweenvariousobjectsariseonlyafterintroducingthecorre-spondingbackgroundsandintegratingoutthequantumfluctuationsaroundthem,therealizationsofthesesymmetriesareingeneralmodifiedfortheeffectiveactionofinterest.Besides,beinganoff-shellquantity,theformoftheeffectiveactiondependsonthechoiceofthegaugeaswellasonthedefinitionofthefields.Further,thereisalwaysavastdegreeofambiguitiesasonecanaddtotalderivatives.Forthesereasons,thestudyofhowthesymmetriesgovernthestructureoftheeffectiveactionbecomesquitenon-trivial.Inthisarticle,weshallfocusonthesupersymmetry(SUSY),consideredtobethemostpowerfulamongtheoneslistedabove.InfactithasbeenclaimedthatalargedegreeofSUSY,N=16,presentinthetheoryimposesstrongrestrictionsontheformoftheeffectiveactionandleadstoanumberof“non-renormalizationtheorems”[12]–[17].However,uponcloseexaminationsonefindsthatsuchassertionsintheexistingliteraturecanstillbechallenged.Thisisessentiallyduetothelackofcompletelyoff-shellconsideration.AsweshallelaborateinsomedetailinSec.3,tounderstandprecisely2towhatextentSUSYisresponsibleindeterminingtheeffectiveaction,onemustallowthebackgroundfieldstohavearbitrarytime-dependence.Thisinturninevitablyleadstothenecessityofexaminingthegauge(BRST)symmetry,anotherimportantingredientofMatrixtheory,asSUSYandgaugesymmetryareknowntobeintimatelyintertwinedoff-shell.Suchananalysishasnotbeenperformedinthepast.Whatmakestheoff-shellanalysisdifficultisthatwedonotasyethavetheoff-shellunconstrainedsuperfieldformulationinthecaseofN=16supersymmetry.Thispromptsustoresorttotheconventionalmeans,namelytheWardidentityforthesymmetriesinquestion.Inordertobeabletocheckagainsttheavailableexplicitresultfortheeffectiveaction,onewouldliketoobtaintheWardidentityinso-calledthe“backgroundgauge”,inwhichallthecalculationshavebeenperformed.AlthoughthederivationofsuchaWardidentityisexpectedtobeatext-bookmatter,thiswasnottobe:Onemustcarefullydis-entanglethedependenceonthebackgroundfieldoftheeffectiveactionbymakinguseofBRSTWardidentities.TheresultisasomewhatcomplicatedWardidentity,inwhichthesupersymmetryandtheBRSTsymmetryareintertwined.AnotablefeatureofourWardidentityisthatonecanreadofftheeffectivequantum-correctedSUSYtransformationsinclosedformandthisshouldserveasastartingpointofvarioustrulyoff-shellinvestiga-tions.Asasimpleapplication,wecomputethetransformationlawstothelowestorderinthederivativeexpansionat1-loopandanalyzetherestrictionimposedbySUSYontheeffectiveactionforabackgroundwitharbitrarytime-dependence,tothecorrespondingorder.Wefindthatatthisordertheeffectiveactionisindeedfullydeterminedbytherequirementofsupersymmetr