N=2 and N=4 supersymmetric Born-Infeld theories from nonline

Starting from nonlinear realizations of the partially broken central-charge extended N=4 and N=8 Poincar\'e supersymmetries in D=4, we derive the superfield equations of N=2 and N=4 Born-Infeld theories. The basic objects are the bosonic Goldstone N=2 and

LNF-00/040(P)

JINR-E2-2000-311

hep-th/0012236

December2000

N=2andN=4supersymmetricBorn-Infeld

theoriesfromnonlinearrealizations

arXiv:hep-th/0012236v1 25 Dec 2000S.Belluccia,1,E.Ivanovb,2,S.Krivonosb,3aINFN-LaboratoriNazionalidiFrascati,P.O.Box13,I-00044Frascati,ItalybBogoliubovLaboratoryofTheoreticalPhysics,JINR,141980Dubna,Moscowregion,RussianFederationAbstractStartingfromnonlinearrealizationsofthepartiallybrokencentral-chargeextendedN=4andN=8Poincar´esupersymmetriesinD=4,wederivethesuper eldequationsofN=2andN=4Born-Infeldtheories.ThebasicobjectsarethebosonicGoldstoneN=2andN=4super eldsassociatedwiththecentralchargegenerators.Byconstruction,theequationsaremanifestlyN=2andN=4supersymmetricandenjoycovarianceunderanothernonlinearlyrealizedhalfoftheoriginalsupersymmetries.Theyprovideamanifestlyworldvolumesupersymmetricstatic-gaugedescriptionofD3-branesinD=6andD=10.FortheN=2casewe nd,tolowestorders,theequivalencetransformationtothestandardN=2Maxwellsuper eldstrengthandrestore,uptothesixthorder,theo -shellN=2Born-InfeldactionwiththesecondhiddenN=2supersymmetry.

E-Mail:

1)bellucci@lnf.infn.it

2)eivanov@thsun1.jinr.ru

3)krivonos@thsun1.jinr.ru

Starting from nonlinear realizations of the partially broken central-charge extended N=4 and N=8 Poincar\'e supersymmetries in D=4, we derive the superfield equations of N=2 and N=4 Born-Infeld theories. The basic objects are the bosonic Goldstone N=2 and

1Introduction

SupersymmetricextensionsoftheBorn-Infeld(BI)theoryinpersedimensionsarecurrentlyunderintensivestudy.ThisvastinterestismainlyduetothefactthatthecorrespondingactionsnaturallyariseinstringtheoryastheworldvolumeactionsofDp-branes(see[1]andrefs.therein).Itisofimportancetohavesuper eldformulationsofsupersymmetricBItheories,withalllinearlyrealizedsupersymmetriesbeingmanifest.

TheN=1,D=4superBIactionconstructedin[2]isnotentirelydeterminedbyN=1supersymmetry.Aswasshownin[3],thisfreedomcanbe xedbyrequiringtheactiontopossessanextrahiddennonlinearlyrealizedN=1supersymmetrycompletingthemanifestonetoN=2,D=4supersymmetry.SuchaformoftheN=1BIactionnaturallycomesoutwithinthenonlinearrealizationofN=2supersymmetrypartiallybrokentoN=1,withthevectorgaugeN=1multipletastheGoldstoneone.Incomponents,theactionisreducedtothestatic-gaugeformoftheworldvolumeactionofthe“space- lling”D3-brane[3,4].

Untilnow,theactionofrefs.[3,4]remainstheonlyexampleofasuper eldBIactioninD≥4havingbothmanifestandhiddensupersymmetries,andsoadmittinganinterpretationastheGoldstonesuper eldaction(oraworldvolume-supersymmetricformofsomeD3-braneaction).TheN=2BIactionconstructedin[5]possessesnohiddensupersymmetry[6].SoitcanbeviewedmerelyasapartoftheasyetunknownactionoftheGoldstoneN=2vectormultipletsupportingthespontaneouspartialbreakingN=4→N=2aswassuggestedin

[7,8].AsforN=4BIactionwhichisacandidatefortheGoldstonemultipletactionassociatedwiththebreakingN=8→N=4inD=4(thestatic-gaugeactionofD3-braneinD=10

[9]),onlythe rst(quartic)nonlinearcorrectiontotheN=4Maxwellactionisknown[1].BecauseoflackingtheGoldstonesuper eldBIactionsforthesecases,itisnaturaltotrytodeducetherelevantsuper eldequationsofmotionasanon-polynomialgeneralizationoftheN=2andN=4superMaxwellequations.In[10,11]wederivedcovariantsuper eldequationsforafewexamplesofsuperbranesinD=3andD=4byapplyingtheformalismofnonlinearrealizationstotheappropriatepartiallybrokensupersymmetries.Inparticular,werecoveredtheN=2→N=1BIsysteminthisnewsetting[10].

Hereweapplythesamemethodtoderivethesuper eldequationsforthepartial-breakingpatternsN=4→N=2andN=8→N=4inD=4,withtheN=2andN=4vectormultipletsastheGoldstoneones.Inbothcases,thecovariantgauge eldstrengthsatis esadisguisedformoftheBIequationwhichtakesthefamiliarformaftera eldrede nition.Weelaborateinmoredetailonthe rstcase.Inparticular,weshowthattheequationforthecomplexscalar eldcorrespondstothestatic-gaugeNambu-Gotoactionof3-braneinD=6.To rstordersintheGoldstonesuper eld,we ndtheequivalencetransformationtotheordinaryN=2vectormultipletsuper eldstrengthandrestoretheinvarianto -shellactiontosixthorder.The rstcorrectiontotheactionof[5]arisesinsixthorderanditcoincideswiththeonefoundin[6]fromadi erentreasoning.Wespeculateonapossibilitytogeneratenon-abelianversionsofsuperBItheoriesfromthenonlinear-realizationsapproach.

2VectorGoldstonemultipletforN=4→N=2

WewishtoderivetheN=2supersymmetricBItheoryasatheoryofthepartialbreakingofN=4,D=4supersymmetrydowntoN=2supersymmetry,withthevectorN=2multiplet

Starting from nonlinear realizations of the partially broken central-charge extended N=4 and N=8 Poincar\'e supersymmetries in D=4, we derive the superfield equations of N=2 and N=4 Born-Infeld theories. The basic objects are the bosonic Goldstone N=2 and

astheGoldstoneone.Toapplythenonlinearrealizationstechniques[10],we rstlyneedtospecifyN=4,D=4supersymmetrytostartwith.InthecaseofN=2→N=1breakingthebasicobjectwastheGoldstoneN=1spinorsuper eldassociatedwiththespontaneouslybrokenhalfoftheN=2fermionicgenerators.Thissuper eldisrelatedbya eldrede nitiontothestandardchiralspinorN=1Maxwellsuper eldstrength.IntheN=2Maxwelltheory,thebasicobjectisacomplexscalarN=2o -shellsuper eldstrengthWwhichischiralandsatis esoneadditionalBianchiidentity:

α¯i¯α(a)D˙W=0,DiW=0,¯ikW¯.(b)DikW=D

(2.1)Here,

iDα=

˙¯αi θ

˙j¯ij≡D¯iDαD.α˙ αiθi αα˙,i¯Dα,Dαj˙i= 2iδj αα˙,(2.2)(2.3)jDij≡DαiDα,

Thesuper eldequationofmotionforWreads

¯ikW¯=0,DikW+D

and,togetherwith(2.1b),amountsto

¯ikW¯=0.DikW=D(2.5)(2.4)

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