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Light-Cone Superspace Bps Equations, and Osp(2,2|16)

Permanent Link: http://ufdc.ufl.edu/UFE0043717/00001

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Title: Light-Cone Superspace Bps Equations, and Osp(2,2|16)
Physical Description: 1 online resource (91 p.)
Language: english
Creator: Hearin, Patrick W
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2011

Subjects

Subjects / Keywords: supersymmetry
Physics -- Dissertations, Academic -- UF
Genre: Physics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: The BPS bound is formulated in light-cone superspace for the N = 4 superYang-Mills theory. As a consequence of the superalgebra all momenta are shown to be expressed as quadratic forms in the relevant supertransformations, and these forms are used to derive the light-cone superspace BPS equations. Finally, the superfield expressions are expanded out to component form, and the Wu-Yang Monopole boosted to the infinite momentum frame is shown to be a solution. A possible non-linear realization of the OSp(2, 2|16) superconformal algebra is prescribed with methods used to derive the OSp(2, 2|8) theory in light-cone superspace. This theory is shown to have an inconsistency in the dynamical constraints which cannot be resolved, and no non-linear field theory can be consistently realized. This inconsistency is also derived in the covariant OSp(2, 2|16) theory.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Patrick W Hearin.
Thesis: Thesis (Ph.D.)--University of Florida, 2011.
Local: Adviser: Ramond, Pierre.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2013-06-30

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Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2011
System ID: UFE0043717:00001

Permanent Link: http://ufdc.ufl.edu/UFE0043717/00001

Material Information

Title: Light-Cone Superspace Bps Equations, and Osp(2,2|16)
Physical Description: 1 online resource (91 p.)
Language: english
Creator: Hearin, Patrick W
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2011

Subjects

Subjects / Keywords: supersymmetry
Physics -- Dissertations, Academic -- UF
Genre: Physics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: The BPS bound is formulated in light-cone superspace for the N = 4 superYang-Mills theory. As a consequence of the superalgebra all momenta are shown to be expressed as quadratic forms in the relevant supertransformations, and these forms are used to derive the light-cone superspace BPS equations. Finally, the superfield expressions are expanded out to component form, and the Wu-Yang Monopole boosted to the infinite momentum frame is shown to be a solution. A possible non-linear realization of the OSp(2, 2|16) superconformal algebra is prescribed with methods used to derive the OSp(2, 2|8) theory in light-cone superspace. This theory is shown to have an inconsistency in the dynamical constraints which cannot be resolved, and no non-linear field theory can be consistently realized. This inconsistency is also derived in the covariant OSp(2, 2|16) theory.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Patrick W Hearin.
Thesis: Thesis (Ph.D.)--University of Florida, 2011.
Local: Adviser: Ramond, Pierre.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2013-06-30

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2011
System ID: UFE0043717:00001


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LIGHT-CONESUPERSPACEBPSEQUATIONS,ANDOSP(2,2j16)ByPATRICKHEARINADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2011

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c2011PatrickHearin 2

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ToMaryT.Hearin 3

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ACKNOWLEDGMENTS IwouldliketoacknowledgemymotherMaryT.Hearinwhopassedawaytwoyearsagofromcancer.Mymotherencouragedmetopursuemyeducation,andIwouldnothavebeenabletoachieveanythingwithoutherhelp.Iwillalwaysrememberherhelpingmewithfractionsasachild,andherunconditionalloveforme.Eventhoughshewasdiagnosedwithterminalcancer,shenevergaveuphope,neverstoppedbeinghappy,andalwaystriedtoenjoyallthatlifehadtogive.R.I.P.motherIwillalwaysloveyou,andmissyou.Iwouldalsoliketothank,myotherfamilymembersWayneA.Hearin,SusanBergmanwhohelpedmethroughmymothersdeath,myadvisorPierreRamondwhoguidedmethroughmyworkintoughtimes,andtherestoftheUniversityofFloridaPhysicsDepartmentwhichhasalwaysbeenverykindtome. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 ABSTRACT ......................................... 7 CHAPTER 1INTRODUCTION ................................... 8 2INTRODUCTIONTOSUPERSYMMETRY ..................... 10 2.1SupersymmetryAlgebra ............................ 10 2.2MasslessRepresentations ........................... 10 2.3MassiveRepresentations ........................... 11 2.4DerivingTheBPSBoundForTheN=2Symmetry .............. 13 2.5DerivingTheBPSBoundForTheN=4Symmetry .............. 18 2.6FactorizingTheHamiltonian .......................... 21 2.7ExactBPSSolutions .............................. 22 3LIGHT-CONESUPERSPACEBPSTHEORY ................... 24 3.1DimensionalReduction ............................ 24 3.2N=4Light-ConeBPSFreeTheory ..................... 30 3.3ConservedSuperchargeForTheN=4Yang-Mills .............. 35 3.4Non-LinearLight-ConeBPS .......................... 39 3.5ComponentBPSEquationSolution ...................... 42 4THREEDIMENSIONALSUPERCONFORMALTHEORIES ........... 52 4.1BLGTheoryAndSO(8)Triality ........................ 52 4.2SO(16)Generalization ............................. 54 4.3Light-ConeSuperspaceOSp(2,2j16)FreeTheory ............. 59 4.4KinematicalConstraints ............................ 60 4.5DynamicalConstraints ............................. 67 4.5.1CalculatingP)]TJ /F5 11.955 Tf 6.76 -.3 Td[(a,J)]TJ /F5 11.955 Tf 6.75 -.3 Td[(a ........................ 67 4.5.2Derivationof[P)]TJ /F3 11.955 Tf 6.75 -.3 Td[(,J)]TJ /F3 11.955 Tf 6.76 -.3 Td[(](1)a ...................... 69 4.5.3Dynamicalinconsistency ........................ 72 5CONCLUSION .................................... 74 APPENDIX ACONVENTIONS ................................... 76 BCOMPONENTEXPANSION ............................ 80 COSp(2,2j16)FREETHEORY ............................ 85 5

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REFERENCES ....................................... 89 BIOGRAPHICALSKETCH ................................ 91 6

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyLIGHT-CONESUPERSPACEBPSEQUATIONS,ANDOSP(2,2j16)ByPatrickHearinDecember2011Chair:PierreRamondMajor:Physics TheBPSboundisformulatedinlight-conesuperspacefortheN=4superYang-Millstheory.Asaconsequenceofthesuperalgebraallmomentaareshowntobeexpressedasquadraticformsintherelevantsupertransformations,andtheseformsareusedtoderivethelight-conesuperspaceBPSequations.Finally,thesupereldexpressionsareexpandedouttocomponentform,andtheWu-YangMonopoleboostedtotheinnitemomentumframeisshowntobeasolution. Apossiblenon-linearrealizationoftheOSp(2,2j16)superconformalalgebraisprescribedwithmethodsusedtoderivetheOSp(2,2j8)theoryinlight-conesuperspace.Thistheoryisshowntohaveaninconsistencyinthedynamicalconstraintswhichcannotberesolved,andnonon-lineareldtheorycanbeconsistentlyrealized.ThisinconsistencyisderivedinthecovariantOSp(2,2j16)theory. 7

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CHAPTER1INTRODUCTION Supersymmetryrequiresanequalnumberofbosons,andfermions.Thepurposeofusingsuchasymmetryistounderstandtheconsequencesofitinthequantummechanicsofelds.Themostinterestingconsequenceoccursinthequantumtheoryofthemaximallysymmetricgaugetheoryinfourdimensions,sinceithasanUVnitequantumtheory[ 1 ],[ 2 ].Nowtheconsequencesofthissymmetryingravityhavebecomealargeportionofmodernresearch[ 3 ],[ 4 ],[ 5 ].Inthisthesismassivesupersymmetryinfour-dimensionalsuperspacewillbethersttopicderived,andthensuperconformaltheoriesinthreedimensionswillbediscussed. ThenumberofsupersymmetriesisdenotedbyN,andforN=4Yang-Millstheorytheeldcontentisonevectoreld,sixscalars,andfourMajoranafermions.ThistheoryisobtainedbydimensionalreductionofaN=1tendimensionalgaugetheorytofourdimensions,yieldingaprojectiveunitaryPSU(2,2j4)symmetry.ThenotationforthesuperalgebradenotesthebosonicsuperconformalalgebraSO(4,2)SU(2,2),andR-symmetrySU(4)[ 7 ].Inthisthesiscovariantdimensionalreductionwillbereviewed[ 8 ],andthenthelight-conemethodofdimensionalreduction[ 9 ]isusedtoderivenewresults.Finally,afterthemasslesstheoriesarerealized,theirsymmetriesarebrokenwiththeHiggsMechanism.ThenafterthemasslesssymmetriesarebrokentheBogomol'nyi,Prasad,Sommereld(BPS)boundcanbederived[ 8 ],[ 10 ],[ 11 ],[ 12 ]. Thelight-coneformalism[ 13 ]willbeutilizedfortheN=4Yang-Millstheoryinlight-conesuperspace[ 14 ]toderivetheBPSbound[ 15 ].TheBPSboundisderivedinlight-conesuperspacebydimensionallyreducingthenon-linearkinematicalanddynamicalconservedcharges,andthenvaryingthesuperspaceexpressions.ThisderivationusesthepositivedenitenatureofthequadraticformtoderiveaBogomol'nyiequationinlight-conesuperspace.Theseequationsareexpandedouttocomponentformandcomparedtocovarianttheories.Finally,asimplesolutiontothelight-coneBPS 8

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equationsisderivedfromboostingtheWu-Yangmonopoletotheinnitemomentumframe[ 16 ]. Afullyinteractingsuperconformaltheoryinthreespace-timedimensionsistheBLGtheory:theBagger,Lambert,Gustavsson(BLG)covarianttheorywasformulatedin[ 17 ],[ 18 ].Thistheoryhasaeldcontentofeightbosons,andafermionwitheightphysicaldegreesoffreedom.TheseeldstransformgloballyundertheR-symmetrySO(8):thescalarsunderthevectorrepresentation,andthefermionsunderaspinorrepresentation.DuetothetrialityofSO(8)anothertheorywiththesameeldcontentcanbeconsistentlyrealizedwiththescalarsnowtransformingunderaspinorrepresentation[ 19 ].Thistrialitytheoryisthebasisforthenaltopicofthisthesis:generalizingtheBLGtheorytoatheorywithonehundredtwentyeightscalars,andonehundredtwentyeightfermions.ThislargertheoryhasanR-symmetrySO(16).Atthenon-linearlevelthistheorybecomesinconsistentduetotheamountSO(16)Fierzidentities.Afterthecovarianttheoryisshowntobeinconsistentthelight-coneformalismisusedtoshowthisinconsistencyfromalgebraicrstprinciples. Thelight-conesuperpaceBLGtheorywasderivedin[ 20 ].ThistheoryisathreedimensionalsuperconformaltheorywithanSO(8)R-symmetrycontainedinitsfullsymmetryOSp(2,2j8).ThenotationforthesuperalgebradenotesthebosonicsuperconformalalgebraSO(2,2)Sp(2,2),andR-symmetrySO(8).AgeneralizationofthistheorywillbediscussedusingtheN=8light-conesupereld.ThisgeneralizedtheoryhasaSO(16)R-symmetrycontainedinthesuperconformalalgebraOSp(2,2j16).Thenon-lineardynamicalalgebrabecomesinconsistent,andthedynamicalconstraintsarenotsatised.Thisconclusionagreeswiththecovarianttheory:thereisnoknownconsistentinteractingtheorywiththissuperconformalalgebra. 9

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CHAPTER2INTRODUCTIONTOSUPERSYMMETRY 2.1SupersymmetryAlgebra ForanarbitraryextendedsupersymmetrythealgebrainfourdimensionsconsistsofthePoincarealgebraP,M,where=0,1,2,3isthefourdimensionalspacetimeindex,andthesuperchargesQm,Q_m,wherem=1,..,N,=1,2,_=1,2.TheLorentzspinorindices=1,2,_=1,2transformundertheLorentzspingroupSU(2)SU(2).ThereisalsothecentralchargesZmn,whichcommuteswithallotheroperatorsinthesupersymmetryalgebra.Theseoperatorssatisfythealgebra fQm,Q_ng=2mn()_P, (2)fQm,Qng=2Zmn, (2)[Qm,M]=1 2()Qm, (2)[P,M]=gP)]TJ /F2 11.955 Tf 11.96 0 Td[(gP, (2)[M,M]=gM)]TJ /F2 11.955 Tf 11.96 0 Td[(gM)]TJ /F2 11.955 Tf 11.95 0 Td[(gM+gM, (2) where=[,],therepresentationforthegammamatricescanbereviewedinAppendix A .Thisisthemostgeneralsupersymmetryalgebra,andallnon-lineareldtheoriesmustclosethisalgebraconsistently.Nowthedifferentrepresentationsforthisalgebrawillbecalculated. 2.2MasslessRepresentations Representationscanbefoundbyplacingparticlesintheirrelevantreferenceframes,andusingthesuperalgebratoinferwhichsuperchargeshavenon-zeroanticommutationrelations.Thesenon-zeroanticommutationsdenethedifferentmultipletsbyactingwithconjugatedsuperchargeoperatorsontheCliffordvacuumj>.TheCliffordvacuumisdenedbythefactthatnon-conjugatesuperchargesannihilateit.Since,theCliffordvacuumisanirreduciblerepresentationofthesuperPoincare 10

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algebraithashelicityorspindependingontheparticle'smass.ForanalgebrawithNsupersymmetryoperatorsinfourdimensionsamasslessCliffordvacuumhashelicityU(1),andeigenvalueh.ChoosingthemasslessparticlestobeineitherreferenceframeP=(E,0,0,E)forceshalftheconjugatesuperchargestoannihilatetheCliffordvacuum.TheremainingconjugatesuperchargesgeneratethemasslessmultipletbyactingontheCliffordvacuumstatewiththegreatestorleasthelicity,yieldingh2Nstates. InsertingamasslessframewheretheparticleisalignedalongthethirdspaceaxisP=(E,0,0,E)intothesupersymmetryalgebraEq.( 2 )forceshalfofthesuperchargestoanti-commute fQm,Qng=20B@2E0001CAmn.(2) Therefore,onlythe=1spinorcomponentshaveaCliffordalgebra am=1 2p EQm1, (2)am=1 2p EQm1, (2)fam,ang=mn. (2) ThenumberofwaysthecreationoperatorscanbeappliedntimestotheCliffordvacuumj>withhstatesishPNn=10B@Nn1CA.Thebinomialtheoremyields2Nstatesforeachhelicitystate,orh2Ntotal. 2.3MassiveRepresentations Massivemultipletscanhavemorestatesthenthemassless,becauseplacingtheparticlesintherestframedoesnotforceanyoftheconjugatesuperchargestoannihilatetheCliffordvacuumthathas(2s+1)statesundertheSU(2)spingroupwiththirdcomponentofspineigenvalues;furthermore,sincenoneoftheconjugate 11

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superchargesannihilatetheCliffordvacuumtheblockdiagonalizedcentralchargeeigenvaluesdonotvanish.Thus,fors=0,thelargestmassiverepresentationhasnocentralchargescreatingsaturation,andiscalledthelongmultiplet,ithas22Nstates.Thesmallermultipletsarecalledshortmultiplets,andaredenedbyhowmanyoftheN 2centralchargeeigenvaluessaturatetheBPSbound.Whennofthecentralchargeeigenvaluessaturatethebound,thisgenerates22(N)]TJ /F8 7.97 Tf 14.7 0 Td[(n)states.Thus,fortheN=4theorythenumberofstatesforthedifferentmultipletsis:massless16,halfmultiplet16,quartermultiplet64,andthelongmultiplet256. Particleswithmassalwayshaveareferenceframeatrest:P=(M,0,0,0).Thesupersymmetryalgebrais fQm,Qng=2mnM, (2)fQm,Qng=2Zmn. (2) PuttingZpqindiagonalformZmn=Ump~ZmnUnq,yieldsadiagonalmatrixZ=i2D,whereDisadiagonalmatrixN 2N 2witheigenvaluesthatarethecentralcharge.Thus,theindexm,ncanbesplitintotwosetsofindicesa,b,R,S=1,2 fQaR,QbSg=2abRSM, (2)fQaR,QbSg=2RSabZb, (2) wheretheindexbisnotsummedover,andZaaretheeigenvaluesofthecentralcharge.Finally,theoperators aR=1 2(Q1R+Q2R), (2)bR=1 2(Q1R)]TJ /F5 11.955 Tf 11.96 0 Td[(Q2R), (2) 12

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satisfythealgebra faR,aSg=RS(2M+Z), (2)fbR,bSg=RS(2M)-222(Z), (2)faR,aSg=fbR,bSg=faR,bSg=0. (2) Since,theanticommutatorisanabsolutevalueaftercontractingtheR,SindextheBPSboundis 2MZa.(2) Takingdifferentamountsoftheb'stobezeroyieldsdifferentmassiverepresentationsasdiscussedatthebeginningofthissection. 2.4DerivingTheBPSBoundForTheN=2Symmetry InthissectionwewillreviewcovariantdimensionalreductionofaN=1,D=6theorytoaN=2,D=4theory,andthenderivetheBPSbound.Thisderivationisinstructive,andwillpreparethereaderforthenextsectionwherethedimensionalreductionoftheN=4D=4theoryfromaN=1,D=10theoryisreviewed.Thenthelight-coneformalismfordimensionalreductionwillbereviewed,andusedtoderivenewresultsinchapterthree. ThesixdimensionalN=1theoryiscomposedofamasslessvectorparticleAaAandaeightcomponentspinor a.Theseeldstransformundersupersymmetry,thePoincaregroupR5,1nSO(5,1),andtheadjointrepresentationofthegaugegroupG,withstructureconstantfabc.Thelowercaselettersa,b,c,d,edenotetheadjointrepresentation'sindices,andA,Barethesixdimensionalspace-timeindices.The 13

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Poincare,gaugeinvariantLagrangianis L=)]TJ /F3 11.955 Tf 10.49 8.09 Td[(1 4FaABFaAB+i1 2 a~)]TJ /F8 7.97 Tf 6.78 -1.79 Td[(ADabA b,(2) whereDabA=@Aab)]TJ /F2 11.955 Tf 11.96 0 Td[(gfabcAcA, (2)FaAB=@AAaB)]TJ /F5 11.955 Tf 11.95 0 Td[(@BAaA+gfabcAbAAcB, (2) andgisthecouplingconstant.TheLagrangianwasshownin[ 7 ]tobesupersymmetricwhentheWeylconstraintisimposedunderthetransformationsAaA=i~)]TJ /F8 7.97 Tf 6.77 -1.79 Td[(A a, (2) a=1 4~)]TJ /F8 7.97 Tf 6.78 4.93 Td[(ABFaAB, (2) whereisa8componentconstantspinorthatparameterizestheglobalsupersymmetrictransformations,and~)]TJ /F8 7.97 Tf 6.78 4.34 Td[(AB=[~)]TJ /F8 7.97 Tf 6.77 4.34 Td[(A,~)]TJ /F8 7.97 Tf 6.78 4.34 Td[(B].Theequationsofmotionare DabAFbAB=)]TJ /F2 11.955 Tf 11.69 8.09 Td[(i 2fabc b~)]TJ /F8 7.97 Tf 6.78 4.94 Td[(B c, (2)~)]TJ /F8 7.97 Tf 6.77 -1.8 Td[(ADabA b=0. (2) TakingthesupersymmetricvariationoftheLagrangianyieldsafewtermsthatarenotatotalderivative gfabc a~)]TJ /F8 7.97 Tf 6.77 -1.79 Td[(A b~)]TJ /F8 7.97 Tf 6.77 4.94 Td[(A c+ a~)]TJ /F8 7.97 Tf 6.77 -1.79 Td[(A b c~)]TJ /F8 7.97 Tf 6.77 4.94 Td[(A.(2) 14

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UsingtheFierzexpansionthesetermsgotozero,becauseoftheWeylcondition =)]TJ /F5 11.955 Tf 9.3 0 Td[(6 [ 7 ].Thisprocessyieldsaconservedcurrent JA=~)]TJ /F8 7.97 Tf 6.77 4.93 Td[(BC~)]TJ /F8 7.97 Tf 6.77 4.93 Td[(A aFaBC. (2) TheN=2symmetryinfourdimensionsisobtainedbydimensionalreductionoftheN=1,D=6theory,andhasthesymmetries:localgaugeinvariance,Poincareinvariance,aglobalSU(2)R-Symmetryfromdimensionalreduction,supersymmetry.Toobtaintheon-shellmasslessN=2,D=4LagrangianwecancompactifytwoofthespacedimensionsintheN=1,D=6theory.Thevectoreldcomponentsinthecompactieddimensionsyieldthe2N=2scalarsPa,Sa.Theeightcomponentfermionsreducetotheform a=a0B@1)]TJ /F2 11.955 Tf 9.3 0 Td[(i1CAleavingoneDiracfermion,ortwoMajoranafermionsthattransformundertheR-SymmetrySU(2),2.Allspinorindicesareleftimplicitinthissection.DimensionallyreducingtheN=1,D=6theorywiththegammamatrixrepresentations ~)]TJ /F11 7.97 Tf 6.78 4.93 Td[(=3, (2)~)]TJ /F9 7.97 Tf 6.78 4.94 Td[(4=53, (2)~)]TJ /F9 7.97 Tf 6.78 4.94 Td[(5=11, (2)~)]TJ /F9 7.97 Tf 6.78 4.93 Td[(6=12, (2) yieldstheN=2,D=4Lagrangian LD=4N=2=)]TJ /F3 11.955 Tf 10.49 8.09 Td[(1 4FaFa+1 2DabSbDacSc+1 2DabPbDacPc++iaD)]TJ /F3 11.955 Tf 13.15 8.09 Td[(1 2fabcfadeSbPcSdSe)]TJ /F2 11.955 Tf 11.96 0 Td[(fabc(a5Pbc)]TJ /F2 11.955 Tf 11.95 0 Td[(iaSbc), (2) 15

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whichisinvariantunderthedimensionallyreducedtransformations Aa=i(a)]TJ /F3 11.955 Tf 13.14 0 Td[(a), (2)Pa=a5)]TJ /F3 11.955 Tf 11.83 0 Td[(5a, (2)Sa=i(a)]TJ /F3 11.955 Tf 11.84 0 Td[(a), (2)a=(Fa+iDabPb5)]TJ /F2 11.955 Tf 11.95 0 Td[(DabSb+igfabcPbSc5), (2) wheretheparameterreduceslike=0B@1)]TJ /F2 11.955 Tf 9.3 0 Td[(i1CAtothefourdimensionalparameter.Usingthefollowingrelations ~)]TJ /F11 7.97 Tf 6.77 4.94 Td[(~)]TJ /F11 7.97 Tf 6.77 4.94 Td[(=3, (2)~)]TJ /F11 7.97 Tf 6.77 4.94 Td[(4~)]TJ /F11 7.97 Tf 6.78 4.94 Td[(=253, (2)~)]TJ /F11 7.97 Tf 6.77 4.93 Td[(5~)]TJ /F11 7.97 Tf 6.78 4.93 Td[(=)]TJ /F3 11.955 Tf 9.3 0 Td[(21, (2)~)]TJ /F9 7.97 Tf 6.77 4.94 Td[(45~)]TJ /F11 7.97 Tf 6.77 4.94 Td[(=)]TJ /F3 11.955 Tf 9.3 0 Td[(2(51), (2) theN=2currentbecomes JD=6=JD=40B@1i0)==iFaAB)]TJ /F8 7.97 Tf 6.77 4.93 Td[(AB)]TJ /F11 7.97 Tf 6.77 4.93 Td[(a==(Fbab+iDabPb5)]TJ /F2 11.955 Tf 11.95 0 Td[(DabSb)]TJ /F2 11.955 Tf 11.96 0 Td[(gfabcSbPc5) a0B@1i1CA. (2) 16

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Nowwewillcalculatethesupersymmetricvariationoftheconservedcurrentusingthegammamatrixidentities =+)]TJ /F5 11.955 Tf 11.95 0 Td[()]TJ /F2 11.955 Tf 11.95 0 Td[(i5,==(+)]TJ /F5 11.955 Tf 11.96 0 Td[()]TJ /F2 11.955 Tf 11.96 0 Td[(i5)+(+)]TJ /F5 11.955 Tf 11.96 0 Td[()]TJ /F2 11.955 Tf 11.96 0 Td[(i5))]TJ -448.72 -26.89 Td[()]TJ /F5 11.955 Tf 11.95 0 Td[((+)]TJ /F5 11.955 Tf 11.95 0 Td[()]TJ /F2 11.955 Tf 11.95 0 Td[(i5))]TJ /F2 11.955 Tf 11.96 0 Td[(i(+)]TJ /F5 11.955 Tf 11.96 0 Td[()]TJ /F2 11.955 Tf 11.95 0 Td[(i5)5, (2) thevariationoftheconservedcurrentreducestoJ=2iT+@(Faay)+@(Faay)5, (2) whereTisthestressenergytensor,anda=Sa+i5Pa.NowwhenthescalareldbecomesconstantontheboundaryorasphereingaugespaceSaSa=2,PaPa=2,whereisthescalarvacuumexpectationvalue.ThisboundaryconditionbreaksthemasslesstheoryduetotheHiggsMechanism.Takingthezerocomponentofthesupercurrentandintegratingoverthreedimensionalspacetimeyieldsthemassivesupersymmetryalgebra fQ,Qg=2iP+QE+QB5,(2) where QB=Zd3x@iBi=ZS2!1d2xBi,QE=Zd3x@iEi=ZS2!1d2xEi. (2) 17

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Nowtheconstantscalareld'svacuumexpectationvaluebreaksthemasslessgaugesymmetrytoatorusU(1)r,whereristherankofthegaugegroupGwiththeHiggsMechanism,andbreakstheR-symmetrySU(2)!U(1);therefore,intherestframeP=(0,00,M),andthegaugeischosentobeAa0=0,thentheBPSboundcanbederivedduetothepositivedenitenatureofthesuperchargemakingitseigenvaluespositive.SubstitutingtherestframeintothealgebraandcomputingtheeigenvaluesyieldstheBPSbound MQB.(2) So,themassisboundedbelowwiththemagneticcharge.Thisboundiswhatthisthesiswillderiveinlight-conesuperspacefortheN=4,D=4theory.But,rstthecovariantN=4,D=4theory'sBPSboundwillbederivedsimilartothissection'scalculation. 2.5DerivingTheBPSBoundForTheN=4Symmetry InthissectionwewillfollowthesamecovariantcalculationforthemaximalN=4,D=4gaugetheorythatwascalculatedinthelastsection.ThiswasoriginallydonebyOsbornin[ 8 ]. ThetendimensionalN=1theoryiscomposedofamasslessvectorparticleAaMandathirty-twocomponentspinora.Theseeldstransformundersupersymmetry,thePoincaregroupR9,1nSO(9,1),andtheadjointrepresentationofthegaugegroupG.TheformoftheLagrangianisthesameasEq.( 2 ),excepttheindicesrunoverthetendimensionalspacetimeindicesM,N.ThetransformationsarealsothesameasEq.( 2 ),Eq.( 2 )withtheexceptionofthespacetimeindex.ThecurrentisconservedwhenthefermionsareMajorana,andWeyl[ 8 ],andissimilartoEq.( 2 ). TheN=4LagrangianinD=4canbeobtainedbycompacticationofaN=1,D=10gaugetheorywithgammamatrices 18

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)]TJ /F11 7.97 Tf 6.77 4.94 Td[(=13, (2))]TJ /F9 7.97 Tf 6.77 4.93 Td[(3+i=1i1, (2))]TJ /F9 7.97 Tf 6.77 4.93 Td[(6+i=5i1, (2))]TJ /F9 7.97 Tf 6.77 -1.79 Td[(11=112, (2) wherei,iarefourbyfourmatricesthatobey:[i,j]=2ijkk,[i,j]=2ijkk,fi,jg=)]TJ /F3 11.955 Tf 9.3 0 Td[(2ij,fi,jg=)]TJ /F3 11.955 Tf 9.3 0 Td[(2ij. DimensionalreductionofsixdimensionsyieldsthefourdimensionalLagrangianwithafourdimensionalvectoreldAa,sixscalarsSi,Ci,i=1,2,3.Thefermionsreduceto=0B@1)]TJ /F2 11.955 Tf 9.3 0 Td[(i1CA,whereisasetoffourMajoranaspinnorsinD=4thattransformunderthe4,4oftheSU(4)R-symmetry.Allspinorindicesareleftimplicit.Theresultofdimensionalreductionis Aa=i()]TJ /F3 11.955 Tf 14.14 2.66 Td[( ), (2)Cai=i a+ i, (2)Sai=i5 a+ i5, (2) a=(Fa)]TJ /F2 11.955 Tf 11.95 0 Td[(iDab(iCbi+iiSbi5)++i 2gfabcijk(CaiCbjk+SaiSbjk)+gfabcijCbiScj5). (2) TheglobalsupersymmetrycurrentreducestoJD=10=JD=40B@1i1CA,where 19

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JD=4=iFaa)]TJ /F5 11.955 Tf 11.95 0 Td[(iDabCbi a)]TJ /F2 11.955 Tf 11.96 0 Td[(ii5DabSbia++gfabc[iijCaiSbjc)]TJ /F3 11.955 Tf 13.15 8.08 Td[(1 2ijk(CaiCbjk+SaiSbjk)c]. (2) Derivingthealgebrafollowingtheprevioussectionyields J=2iT)]TJ /F3 11.955 Tf 11.95 0 Td[(2Aii)]TJ /F3 11.955 Tf 11.95 0 Td[(2iBi5i+i(F,DCi)5i)]TJ /F5 11.955 Tf 11.96 0 Td[((F,DSi)i, (2) whereTisthestressenergytensorandAi,Biareextradimensionalcomponentsofthestressenergytensor T=FF+DCiDCi+DSiDSi+i 2D)]TJ /F2 11.955 Tf 11.95 0 Td[(gL, (2)Ai=)]TJ /F5 11.955 Tf 9.3 0 Td[(@(FCi)+(DF)]TJ /F2 11.955 Tf 11.96 0 Td[(J)Ci, (2)Bi=)]TJ /F5 11.955 Tf 9.3 0 Td[(@(FSi)+(DF)]TJ /F2 11.955 Tf 11.96 0 Td[(J)Si. (2) Nowwhenthescalareldbecomesconstantontheboundary,orasphereingaugespaceCaiCai=2,SaiSai=2,themasslesssymmetriesbreak.And,takingthezerocomponentofthecurrentandintegratingoverthreedimensionalspacetimeyieldsthemassivesupersymmetryalgebra fQ,Qg=2P)]TJ /F5 11.955 Tf 11.96 0 Td[([(2ii+25i)QEi+(25i)]TJ /F3 11.955 Tf 11.96 0 Td[(2ii)QBi.(2) AgaintheconstantscalareldsbreakthemasslessgaugesymmetrytoatorusU(1)r,andbreakstheR-symmetrySU(4)!Sp(4);therefore,thesameargumentastheN=2casederivestheBPSbound.Intherestframe,andgaugeAa0=0,theeigenvaluesofthesupersymmetricalgebrayield 20

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MQB.(2) 2.6FactorizingTheHamiltonian Nowthatthemassiveboundshavebeenderivedusingthealgebratheseboundscanbereproducedintheclassicalenergyfunctional.Theclassicalenergyfunctionaltakesthefermionstobezero.MassivetheoriesarerealizedbybreakingthemasslesssymmetrieswiththeHiggsMechanism,andthentheBPSboundcanbederivedbyminimizingtheenergy.ForsimplicityaSU(2)gaugesymmetryischosentobreaktoU(1),andtheR-symmetriesbreaktoN=2SO(3)!SO(2),N=4:SO(6)!SO(5).ToderivetheBPSboundfromtheeldtheorythegaugechoiceisA0=0,andtheenergymustbeminimizedbysettingsupersymmetricpotentialstozeroN=2:[P,S]=0,N=4:[Ci,Cj]=[Si,Sj]=[Ci,Si]=0,leavingtheenergies HN=2=1 2Zd3xf~Ba~Ba+~DabPb~DacPc+~DabSb~DacScg, (2)HN=4=1 2Zd3xf~Ba~Ba+~DCai~DCai+~DSai~DSaig, (2) wherei=1,2,3.ToSaturatetheBPSboundfortheN=2theorythescalarscanbewrittenasSa=a+^Sa,Pai=ab+^Pa,andforN=4theorythescalarsareCai=aai+^Cai,Sai=abi+^Sai,whereai,biareconstantandobeyaiai=bibi=1 2.Thenewscalarelds^Sa,^Pa,^Cai,^SaiareeldcongurationsofthevacuumDabi^Sa=Dabi^Pa=Dabi^Caj=Dabi^Saj=0,andaa=2.SubstitutingtheseconditionsintotheN=2,N=4energiesyieldsaHamiltonianthatistheexactonefactorizedbyBogomol'nyiin[ 11 ] H=1 2Zd3xf(~Ba)]TJ /F5 11.955 Tf 13.89 3.02 Td[(~Da)(~Ba)]TJ /F5 11.955 Tf 13.88 3.02 Td[(~Da)+2~Ba~Dag,(2) 21

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SincethescalarshaveanontrivialexpectationvaluethevacuummaybeplacedattheboundaryCaiCai+SaiSai=2.Therefore,ontheboundarythescalarsbecometheirvacuumexpectationvalue,yieldingthemagneticcharge H=1 2Zd3xf(Bia)]TJ /F2 11.955 Tf 11.96 0 Td[(Dabib)(Bai)]TJ /F5 11.955 Tf 13.89 3.03 Td[(~Db)g+QB,(2) Thus,theBPSboundMQBissaturatedwhentheeldcongurationssatisfyBogomol'nyiequations Bai=Dabib.(2) TheBogomol'nyiequationsdescribemagneticmonopolesthathavetopologicalsymmetries,andwillbereviewedinthenextsection. 2.7ExactBPSSolutions PrasadandSommereld[ 12 ]showedthattheexactsolutionstotheBPSequationsEq.( 2 )aremonopoles.Theymadeananzatzthatthesolutionbesphericallysymmetric a=ya yH(y), (2)Aai=)]TJ /F5 11.955 Tf 9.3 0 Td[(aijyj y(1)]TJ /F2 11.955 Tf 11.96 0 Td[(K(y)), (2) wherey=gr.SubstitutingthisintotheBogomolnyiequationEq.( 2 )yieldsacoupledordinarynonlinearequation y_K=)]TJ /F2 11.955 Tf 9.3 0 Td[(KH, (2)y_H=H)]TJ /F3 11.955 Tf 11.95 0 Td[((K2)]TJ /F3 11.955 Tf 11.95 0 Td[(1), (2) 22

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Thesolutiontotheseequationsare H(y)=ycoth(y))]TJ /F3 11.955 Tf 11.96 0 Td[(1, (2)K(y)=y sinh(y). (2) Thesesolutionsdescribeamagneticmonopoledenedbyatopologicalchargequantization.In[ 21 ]theyshowthatthemostgeneralsolutiontoEq.( 2 )is Aa=1 2eabcb@c+1 aB,(2) whereBisanarbitraryvector.Substitutingintothechargeyields m=)]TJ /F3 11.955 Tf 10.62 8.09 Td[(1 Zd3x@i(Baia)=)]TJ /F3 11.955 Tf 10.49 8.09 Td[(4 eT,(2) whereTisthezerocomponentofthetopologicalcurrent T=1 83abc@a@b@c.(2) ThisisnotaNoethercurrentsinceitsconservationdoesnotrelyontheequationsofmotion.ItcanbeshownthatitisanintegersincetheSU(2)scalareldscreateasphericalsystemna=a ,resultinginchargequantization QEQB=4n,(2) wherenZ.TopologicallythisisthewindingnumberofthemapsattheboundaryS2!S2,whichareclassiedby2(S2)=Z. 23

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CHAPTER3LIGHT-CONESUPERSPACEBPSTHEORY 3.1DimensionalReduction Inthissectionlight-conedimensionalreductiontechniqueswillbebrieyreviewed,andthefourdimensionalconservedlight-conesuperchargeswillbederived.Toderivethelight-conetheoryfromtendimensionsthephysicaldegreesoffreedommustbeeliminated.Intendimensionsthevectorparticlehaseightdegreesoffreedom;consequently,twenty-fourfermionicdegreesoffreedommustbeeliminatedoutoftheoriginalthirty-twocomponentstoachievesupersymmetry.TheWeylconstraintcanbeimposedinanyevendimension,whiletheMajoranaconstraintwhichcanbeimposedin2mod8,4mod8dimensions,andbothcanbesimultaneouslyimposedin2mod8;therefore,intendimensionsbothoftheseconstraintscanbeimposedonthespinnora=aTC, (3))]TJ /F9 7.97 Tf 6.78 -1.79 Td[(11a=a, (3) wherea=ay)]TJ /F9 7.97 Tf 6.78 4.33 Td[(0.Theseconstraintseliminatesixteenfermionicdegreesoffreedom,andwiththeequationsofmotiontheymakethenumberoffermionsequaltothenumberofbosons.1 ThePoincare,gaugeinvariantLagrangianisthesameonedimensionallyreducedinchaptertwo L=)]TJ /F3 11.955 Tf 10.5 8.08 Td[(1 4FaMNFaMN+i1 2a)]TJ /F8 7.97 Tf 6.78 -1.8 Td[(MDabMb,(3) 1TherepresentationusedtodenetheconstraintsEq.( 3 ),Eq.( 3 ),andisreviewedinAppendix A 24

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whereDabM=@Mab)]TJ /F2 11.955 Tf 11.96 0 Td[(gfabcAcM, (3)FaMN=@MAaN)]TJ /F5 11.955 Tf 11.95 0 Td[(@NAaM+gfabcAbMAcN, (3) andissupersymmetricwhentheMajoranaandWeylconstraintsareimposedunderthetransformationsAaM=i)]TJ /F8 7.97 Tf 6.78 -1.79 Td[(Ma+DabM!b, (3)a=1 4)]TJ /F8 7.97 Tf 6.77 4.94 Td[(MNFaMN+fabcb!c, (3) whereisa32componentconstantspinorthatparameterizestheglobalsupersymmetrictransformations,!aisthelocalgaugeparameter,and)]TJ /F8 7.97 Tf 6.77 4.34 Td[(MN=[)]TJ /F8 7.97 Tf 24.67 4.34 Td[(M,)]TJ /F8 7.97 Tf 12.25 4.34 Td[(N].TheprevioussupersymmetrictransformationsEq.( 3 ),Eq.( 3 )havethegaugesymmetryaddedtothemtomakethegaugeandsupersymmetrycompatible,andisdiscussedin[ 22 ].Theconservedcurrentforthesupersymmetrictransformationsisthesumoftheoriginalsupersymmetriccurrentplusthegaugecurrent.BothcurrentsareconservedwhentheequationsofmotionDabMFbMN=)]TJ /F2 11.955 Tf 11.69 8.09 Td[(i 2fabcb)]TJ /F8 7.97 Tf 6.77 4.93 Td[(Nc, (3))]TJ /F8 7.97 Tf 6.78 -1.79 Td[(MDabMb=0, (3) areused,sotheconstantsinfrontofthecurrentsarearbitrary;therefore,thesupercurrent JM=)]TJ /F3 11.955 Tf 10.5 8.09 Td[(1 4FaPQ)]TJ /F8 7.97 Tf 6.77 4.94 Td[(PQ)]TJ /F8 7.97 Tf 6.77 -1.79 Td[(Ma)]TJ /F2 11.955 Tf 11.96 0 Td[(iFaMNDabN!b+i1 2gfabca)]TJ /F8 7.97 Tf 6.77 -1.79 Td[(Mb!c,(3) 25

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isconservedwhentheequationsofmotionareused.Thepreviousexpressionisthesupersymmetriccurrentforthemodiedsupersymmetrictransformations. NowallthepertinentN=1tendimensionaltheory'squantitieshavebeencalculated,andtheN=4fourdimensionaltheorycanbederivedusingdimensionalreductiononthesixextradimensions.ThetendimensionalPoincarealgebraSO(9,1)isdimensionallyreducedtoSO(3,1)SO(6),wheretheSO(3,1)isthefour-dimensionalPoincaregroupandSO(6)SU(4)istheR-symmetry.Inthelight-coneformalismthetendimensionalN=1superalgebra fQ,Qg=()]TJ /F8 7.97 Tf 24.27 -1.8 Td[(M)PM,(3) isprojectedintotwosets,thekinematicaldenotedbyplusandthedynamicaldenotedbyminus,withthematrices=)]TJ /F3 11.955 Tf 10.5 8.09 Td[(1 2)]TJ /F6 7.97 Tf 6.77 4.94 Td[()]TJ /F6 7.97 Tf 6.77 4.94 Td[(, (3)Q=Q,Q=Q. (3) AfterprojectingandapplyingtheMajorana,WeylconstraintsthespinnorsbecomeQ+=)]TJ /F2 11.955 Tf 9.3 0 Td[(i)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(0,0,0,q1,...,0,0,0,q4,q1,0,0,0,...,q4,0,0,0, (3)Q)]TJ /F3 11.955 Tf 10.4 1.8 Td[(=i)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(0,0,Q1,...,0,0,Q4,0,0,Q1,0,0,0,...,Q4,0,0. (3) ThisreductionleavesfourGrassmannvariablesforeachprojectionthattransformgloballyunderthefundamentalrepresentationofSU(4)(4,4),withspinorindicesm,n=1,2,3,4.TheplusprojectionEq.( 3 )containsthekinematicalchargesqmanditscomplexconjugate.While,theminusprojectionEq.( 3 )containsthedynamicalchargesQmanditsconjugate,whereinfourdimensionsoperatorsthataredynamicalaredenotedbycapitalcalligraphicletters.Whenthesesuperchargesaresubstitutedinto 26

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Eq.( 3 )thisyieldsthealgebrafqm,qng=p 2mnP+, (3)fQm,Qng=p 2mnP)]TJ /F3 11.955 Tf 7.08 -4.93 Td[(, (3)fqm,Qng=p 2mnP, (3)fqm,Qng=p 2Zmn, (3) whereP+=1 p 2(P0+P3),P)]TJ /F3 11.955 Tf 10.41 -4.34 Td[(=1 p 2(P0)]TJ /F2 11.955 Tf 11.96 0 Td[(P3),P=1 p 2(P1+iP2). Dimensionalreductionmakesthesixextradimensionalmomentumconstantrealnumbers,orthecentralcharge.ThecentralchargesEq.( 3 )commutewiththesuperPoincarealgebraafterreductiontofourdimensions.Thesechargestransformundertheanti-symmetricrepresentationofSU(4),6,andaredenedbythematrixZmn=1 p 2mnIPI,whereI=4,5,6,7,8,9andthematrixmnIisdenedinAppendix A :thisoperatorhassixrealcomponents,sinceitobeysthedualitycondition Zmn=1 2mnpqZpq.(3) Nowtheeldtheoryisreducedtofourdimensions.First,theunphysicaldegreesoffreedom:Aa+,Aa)]TJ /F3 11.955 Tf 7.08 -4.34 Td[(,a)]TJ /F1 11.955 Tf 10.4 2.96 Td[(areeliminatedfromtheLagrangian;thus,thetemporallight-conecoordinateeldAa+ischosentobeequaltozero,orthelight-conegaugecondition,andtheequationsofmotionareusedtosolveforAa)]TJ /F1 11.955 Tf 7.08 -4.34 Td[(,a)]TJ /F1 11.955 Tf 7.08 2.95 Td[(.2Whenthelight-conegaugeconditionischosenAa+=0thisconstrainsthegaugeparametertobeindependentof 2Thelight-conemethodforeliminatingtheunphysicaldegreesoffreedomcanbereviewedin[ 22 ]. 27

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thex)]TJ /F1 11.955 Tf 10.4 -4.34 Td[(coordinateAa+=@+!a=0,andduetothesupersymmetricvariationEq.( 3 )thegaugeparametercanbesolvedfor3 !a=)]TJ /F2 11.955 Tf 9.3 0 Td[(i+)]TJ /F9 7.97 Tf 6.77 4.94 Td[(+1 @+a+.(3) NowalloftheunphysicaldegreesoffreedomhavebeeneliminatedleavingthemasslesslittlegroupintendimensionssymmetrySO(8)SO(9,1)oftheLagrangian;consequently,thedimensionalreductionSO(8)SO(2)SO(6)reducesthebosoniceldAaK,whereK=1,2,4,5,6,7,8,9,transformingunderSO(8)into:afour-dimensionalvectoreldtransformingunderU(1)helicity,Aa=Aa1+iAa2,andscalarsdenedbythematrixCamn=1 p 2mnIAaI.Thescalarstransformunderthe6ofSU(4),andhavethedualityCamn=1 2mnpqCapqthatleavessixrealcomponents.Theleftoverspinor+=)]TJ /F2 11.955 Tf 9.3 0 Td[(i)]TJ /F3 11.955 Tf 6.65 -9.69 Td[(a1,0,0,0,...,a4,0,0,0,0,0,0,a1,...,0,0,0,a4, (3) isMajorana,Weylandtheamtransformunder:U(1)helicity,thefundamentalrepresentationofSU(4),4,andtheamtransformunder4.TheresultsfortheLagrangian,andthesupertransformationreductioncanbefoundin:[ 9 ],[ 22 ]respectively. Theonlyquantityneededforthisthesisisthefourdimensionallight-conesupercharges,andtheyarenotfoundinthepreviousliterature.Toreducethesuperchargethesupersymmetryparameter'sprojections=++)]TJ /F17 7.97 Tf 10.41 6.13 Td[(4dividethepluscomponentofthecurrentEq.( 3 )into)]TJ /F1 11.955 Tf 10.41 1.79 Td[(multiplyingthekinematicalcharge,and+multiplying 3ThegaugeparameterEq.( 3 )wasoriginallyderivedin[ 22 ].4Thesupersymmetricparametersprojections+,)]TJ /F1 11.955 Tf 10.41 1.8 Td[(yieldtwodifferentparametersuponreduction,calledm,min[ 22 ].Forthepurposesofthisarticle,theseparametersaresetequaltoeachotherm=m,withoutlossofgenerality;furthermore,thesupersymmetricparameterisrescaledbyafactorof)]TJ 9.3 9.96 Td[(p 2whenreduced,orthefourfourdimensionalparametersinsideofare)]TJ /F9 7.97 Tf 14.02 4.71 Td[(1 p 2mand)]TJ /F9 7.97 Tf 14.02 4.71 Td[(1 p 2m.Thisisthesamerescalingthatisusedin[ 22 ]tomatchsuperspaceexpressions. 28

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thedynamical;5thus,thekinematicalchargesare6 qm=2iZd3xf@+Camnan)]TJ /F5 11.955 Tf 11.95 0 Td[(@+Aaamg,(3) whered3x=dx)]TJ /F2 11.955 Tf 7.09 -4.34 Td[(dxdx.Usingthematrixidentities)]TJ 7.03 2.66 Td[(\000+=)]TJ /F2 11.955 Tf 9.3 0 Td[(i+18=2i0B@0+001CA18, (3)\000)]TJ /F9 7.97 Tf 20.33 4.94 Td[(+=)]TJ /F2 11.955 Tf 9.3 0 Td[(i+18=2i0B@00)]TJ /F3 11.955 Tf 17.05 -4.33 Td[(01CA18, (3)\000I)]TJ /F9 7.97 Tf 6.78 4.94 Td[(+=)]TJ 9.3 10.54 Td[(p 2i0B@0001CA0B@0ImnImn01CA, (3)\000I)]TJ /F9 7.97 Tf 6.78 4.94 Td[(+=)]TJ 9.3 10.54 Td[(p 2i0B@0001CA0B@0ImnImn01CA, (3))]TJ /F8 7.97 Tf 6.78 4.94 Td[(I)]TJ /F8 7.97 Tf 6.77 4.94 Td[(J)]TJ /F9 7.97 Tf 6.78 4.94 Td[(+=)]TJ /F2 11.955 Tf 9.3 0 Td[(i0B@0)]TJ /F5 11.955 Tf -31.22 -28.24 Td[(+01CA0B@ImpJpn00ImpJpn1CA, (3) 5Thetechniquethatdividesthekinematicalfromthedynamicalsymmetrieswasoriginallyusedin[ 22 ].6Fieldtheoryexpressionsaredenotedbyq,Q,P. 29

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where=1 p 2(123),=1 p 2(1+i2),andtheidentityCamn@+Cbnp1 @+ap=)]TJ /F9 7.97 Tf 10.49 4.7 Td[(1 2(CamnCbnpcp+1 2Canp@+Cbnp1 @+am)thedynamicalchargesreducetoQm=2iZd3xf@Camnan)]TJ /F5 11.955 Tf 11.95 0 Td[(@Aaam)]TJ /F2 11.955 Tf 11.96 0 Td[(gfabc(@+AaAb1 @+cm)]TJ -267.19 -31.01 Td[()]TJ /F3 11.955 Tf 12.85 2.66 Td[(AaCbmncn+Camn@+Cbnp1 @+cp+i1 p 2amb1 @+cs)g. (3) Thesechargesobeynon-lineartransformationrulesunderthePoncairealgebra,mainlythetransformationunderthehelicityoperatorj=x@)]TJ /F3 11.955 Tf 13.06 0 Td[(x@;although,theserulesaremuchsimplerinsuperspace,andwillbediscussedinthenextsection.Thesuperchargereductiontofourdimensionsistheonlynewresultinthissection.Nowthatallfour-dimensionalquantitieshavebeenderivedthesuperspaceexpressionscanbeformulated. 3.2N=4Light-ConeBPSFreeTheory ThefourdimensionalN=4superconformaltheorycanbeformulatedinlight-conesuperspacewithGrassmannvariablesm,m,wherem=1,2,3,4,andtransformunder4,4ofSU(4).Thesecoordinateshaveananti-commutingderivative@m=@ @mthathasthecomplexconjugation(@m)=)]TJ /F3 11.955 Tf 10.55 2.66 Td[(@m,andsatises f@m,ng=mn.(3) Thenewsuperspacecoordinateisdenedby y=(x,x,x+,y)]TJ /F3 11.955 Tf 10.4 -4.94 Td[(=x)]TJ /F4 11.955 Tf 9.75 -4.94 Td[()]TJ /F2 11.955 Tf 19.32 8.09 Td[(i p 2mm).(3) 30

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WiththisredenitionofspacethetwovectoreldsAa,Aa,sixscalarsCamn,andeightfermionsm,mareplacedintoasuperelda(y)=1 @+Aa(y)+i p 2mnCamn(y)+i @+mam(y)++p 2 6mnpqmnpq(y)+1 12mnpqmnpq@+Aa(y), (3) wherethescalarsareconstrainedbythedualityconditionCamn=1 2mnpqCapq. Thetransformationsonthesupereldcanbeexpressedintermsofthechargeoperatorsinsuperspaceforthefreetheoryqm=)]TJ /F5 11.955 Tf 9.29 0 Td[(@m+i1 p 2m@+, (3)Qm=@ @+qm. (3) Takingadifferentlinearcombinationoftheanti-commutingderivativeandGrassmannvariableyieldsthechiralderivative dm=)]TJ /F5 11.955 Tf 9.3 0 Td[(@m)]TJ /F2 11.955 Tf 11.96 0 Td[(i1 p 2m@+.(3) Foramasslessfreetheorythesuperchargeandchiralderivativesatisfyfqm,qng=p 2imn@+,fQm,Qng=p 2imn@@ @+,fdm,dng=)]TJ 9.3 10.54 Td[(p 2imn@+,fdm,qng=0, (3) andthechiralderivativeannihilatesthechiralsupereld dma=0;(3) 31

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thus,asupersymmetrictransformationpreserveschiralityforthelight-coneformalism.Finally,thereistheinside-outconstraint dmdna=1 2mnpqdpdqa,(3) anditallowsthesupereldtobeconjugatedwithcovariantderivatives a=1 48d41 @+2a,(3) whered4=mnpqdmdndpdqandd4=mnpqdmdndpdq.Thetechniqueforreducingtothecomponenttheorycanbereviewedin[ 22 ],andAppendix B .7 Todenemassivetheorieswemustconsiderthecentralcharge.AfterapplyingSchur'slemmatheblockdiagonalcentralchargeiscomposedoftwocomplexeigenvaluesZn,where:Z1=Z2,Z3=Z4;thus,thealgebratakestheform:fqm,qng=p 2mnP+, (3)fQm,Qng=p 2mnP)]TJ /F3 11.955 Tf 7.08 -4.93 Td[(, (3)fqm,Qng=p 2mnZn, (3) wheretheninEq.( 3 )isnotsummedover,andthematrixmnisatwoindexanti-symmetricmatrix mn=0BBBBBBB@0100)]TJ /F3 11.955 Tf 9.3 0 Td[(1000000100)]TJ /F3 11.955 Tf 9.29 0 Td[(101CCCCCCCA.(3) 7Foracompletereviewoflight-conesuperspacethereaderisalsoreferredto[ 9 ],[ 14 ]. 32

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Dividingtheidentity14=1++1)]TJ /F1 11.955 Tf 10.41 -4.34 Td[(andtheanti-symmetricmatrix=++)]TJ /F17 7.97 Tf 10.41 0 Td[(81+=1 p 2+12,1)]TJ /F3 11.955 Tf 10.41 -4.94 Td[(=1 p 2)]TJ /F4 11.955 Tf 9.74 -4.94 Td[(12, (3)+=1 p 2i+2,)]TJ /F3 11.955 Tf 10.41 -4.93 Td[(=1 p 2i)]TJ /F4 11.955 Tf 9.74 -4.93 Td[(2, (3) allowsthekinematicalanddynamicalchargestobearrangedintotwooperatorsam=1 p 2[(1+)mnQn)]TJ /F3 11.955 Tf 11.96 0 Td[((+)mnqn+(1)]TJ /F3 11.955 Tf 7.09 -4.94 Td[()mnqn)]TJ /F3 11.955 Tf 11.95 0 Td[(()]TJ /F3 11.955 Tf 7.09 -4.94 Td[()mnQn], (3)bm=1 p 2[(1+)mnQn+(+)mnqn+(1)]TJ /F3 11.955 Tf 7.08 -4.93 Td[()mnqn+()]TJ /F3 11.955 Tf 7.08 -4.93 Td[()mnQn]. (3) Inthelight-conecoordinaterest-frame (P+=1 p 2M,P)]TJ /F3 11.955 Tf 10.4 -4.94 Td[(=1 p 2M,P=0,P=0),(3) theseoperatorssatisfythealgebrafam,ang=fbm,bng=fam,byng=0, (3)fam,ayng=mn(M+p 2Re(Zn)), (3)fbm,byng=mn(M)]TJ 11.95 10.54 Td[(p 2Re(Zn)). (3) Theanti-commutationsfam,ayng,fbm,byngarepositivedeniteform=n;thus,themassisboundedbelowbythecentralchargeeigenvaluesforeachm Mp 2Re(Zm).(3) 8Thematrices=1 p 2(03)and=1 p 2(1+i2)arediscussedinAppendix A 33

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ThedifferentmultipletsaregeneratedwhensomeamountoftheoperatorsbymannihilatetheCliffordvacuum,withspins,thatisannihilatedbythenon-conjugateoperatorsam,bm.Since,thereareonlytwocentralcharges,annihilationofthevacuumbyby1orby2setsthersteigenvaluetothemass,andannihilationbyby3orby4setsthesecondeigenvaluetothemass.ThehalfmultipletisdenedbyallthebymannihilatingtheCliffordvacuumstatewiths=0 bymjs=0>=0,(3) andallthecentralchargesareequaltothemass M=p 2Re(Zm).(3) Togeneratethehalfmultiplettheoperatorisdividedintotwosetsdependingonwhethertheyraiseorlowerthethirdcomponentofspina+y1=ay1,a+y2=ay2, (3)ay1=ay3,ay2=ay4, (3) wheretheplusandminusdenotethisraisingandloweringofthethirdcomponentofspin:[a+i,j]=)]TJ /F9 7.97 Tf 10.5 4.71 Td[(1 2a+i,and[ayi,j]=1 2ayi,wherei=1,2.Thus,theoperators i=1 p 2Mai,(3) satisfytwoCliffordalgebrasfi,yjg=ij,fi,yjg=0,fi,jg=0,fi,jg=0, (3) 34

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andactingontheCliffordvacuumwithiyields:js=0>ones=0,yijs=0>fours=1 2,y1y2js=0>twos=1,yiyjjs=0>fours=0,yiyjykjs=0>fours=1 2,+y1y2+y2y1js=0>ones=0. (3) Thismultiplethasthesameeldcontentasamasslessmultiplet;thus,intheeldtheorythismutipletcanbeobtainedbyusingtheHiggsmechanismonthemasslessN=4eldtheory.ThealgebraicformulationofthehalfmultipletwillbeusedtoderivetheBPSboundintheeldtheorybasedpurelyonalgebraicrstprinciplesandnosuperuousassumptions,becausetheboundcanbederivedbasedonpreservinghalfthesupersymmetries. 3.3ConservedSuperchargeForTheN=4Yang-Mills Originally,in[ 14 ]thelight-coneHamiltonianwasshowntobeaquadraticform P)]TJ /F3 11.955 Tf 10.4 -4.94 Td[(=i1 p 2Zd3xd4d4fma1 @+mag,(3) andthenon-linearinnitesimaldynamicaltransformation ma=1 @+(@ab)]TJ /F2 11.955 Tf 11.95 0 Td[(fabc@+c)qmb,(3) 35

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wasderivedusingalgebraicrstprinciples.9 Inthisarticletheequations:Eq.( 3 ),Eq.( 3 )superspaceexpressions10qm=2Zd4xd4d4fa1 @+qmag, (3)Qm=2Zd4xd4d4fa@ @+2qma)]TJ /F3 11.955 Tf 13.15 8.09 Td[(2 3gfabca1 @+2(qmb@+c)g, (3) willbeusedtoshowthatallmomentaarequadraticforms,becauseforaunitarytransformationtheinnitesimalchangeoftheconservedchargeFduetoanothertransformationGistheiralgebraiccommutator[G,F] iGF=[G,F].(3) Usingthepreviousequationthesuperchargescanbeshowntoobeythehelicityproperties[qm,j]=)]TJ /F9 7.97 Tf 10.5 4.71 Td[(1 2qm,[Qm,j]=1 2Qm ijqm=1 2qm,ijQm=)]TJ /F3 11.955 Tf 10.5 8.09 Td[(1 2Qm,(3) wherefrom[ 14 ] j=x@)]TJ /F3 11.955 Tf 12.14 0 Td[(x@+1 2(m@m)]TJ /F3 11.955 Tf 12.68 2.65 Td[(m@m))]TJ /F2 11.955 Tf 22.46 8.09 Td[(i 4p 2[dm,dm]1 @+.(3) Furthermore,usingEq.( 3 )allmomentacomefromtheinnitesimalvariationoftheassociatedchargesduetothesupersymmetricalgebraEq.( 3 ),Eq.( 3 ), 9ThedynamicaltransformationEq.( 3 )isacovariantderivativefortheresidualgaugesymmetrytransformation,thisphenomenonisdiscussedin[ 22 ].10Theequations:Eq.( 3 ),Eq.( 3 )superspaceformswherederivedfortheN=2theorycoupledtoaWess-Zuminomultiplet[ 23 ]. 36

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Eq.( 3 ),Eq.( 3 )respectively:P+mm=1 p 2mnfqm,qng=1 p 2m[nqn,qm]=1 p 2imnqnqm, (3)P)]TJ /F5 11.955 Tf 7.09 -4.94 Td[(mm=1 p 2mnfQm,Qng=1 p 2m[nQn,Qm]=1 p 2imnnQm, (3)Pmm=1 p 2mnfqm,Qn,g=1 p 2m[nQn,qm]=1 p 2imnnqm, (3)Zmnmn=1 p 2mnfqm,Qng=1 p 2m[nQn,qm]=1 p 2imnnqm, (3) wherethenotationnn,nqndenotestheinnitesimalvariationwithrespecttotheoperatorinthesubscript.Forexample,thecentralchargequadraticformcanbederivedfromthealgebrabyvaryingthekinematicalchargeEq.( 3 )withthedynamicaltransformationEq.( 3 )Zmnmn=1 p 2imnnqm==p 2imnZd4xd4d4fna1 @+qma)]TJ /F3 11.955 Tf 13.26 2.66 Td[(a1 @+nqmag. (3) Integratingbypartsandusingtheinside-outconstraintyieldsthequadraticform Zmn=2p 2iZd3xd4d4fqma1 @+nag.(3) VaryingthedynamicalchargeEq.( 3 )withthekinematicaltransformationEq.( 3 )alsoyieldsthepreviousresult.Theothermomenta'squadraticformsalsofollowfromthelight-conesupersymmetricalgebra,andtheirderivationsaresimilartothecentralcharge: 37

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mnP)]TJ /F3 11.955 Tf 10.4 -4.94 Td[(=2p 2iZd3xd4d4fma1 @+nag, (3)mnP+=2p 2iZd3xd4d4fma1 @+nag, (3)mnP=2p 2iZd3xd4d4fma1 @+nag, (3)Zmn=2p 2iZd3xd4d4fma1 @+nag, (3) wheretheinnitesimalkinematicaltransformations(withoutthesupersymmetricparameter)aredenotedbyma=qma,ma=qma.Calculatingthesequadraticformsbyvaryingtheconservedchargesprovesthatthelongitudinalquadraticformsarepositivedenite,becausetheyareequaltotheanti-commutatorsthatarecomplexconjugatesofeachother.TheoffdiagonaltermsinEq.( 3 ),Eq.( 3 ),Eq.( 3 )aretriviallyzerointhisconstruction,becausetheanti-commutatorsEq.( 3 ),Eq.( 3 ),Eq.( 3 )arezeroforoffdiagonalcharges.ThecentralchargequadraticformisalsoantisymmetricZmn=)]TJ /F2 11.955 Tf 9.3 0 Td[(Znm,becausethekinematicalchargesqm,qninthequadraticformcanbeintegratedbypartstoswitchtheindicesm,n.ThisantisymmetrycanalsobecomprehendedbyobservingthatthecomponentformofEq.( 3 )indicesarejustthescalareld'sindicesCamnwhichisdiscussedlater. ThemethodforderivingEq.( 3 ),Eq.( 3 ),Eq.( 3 ),Eq.( 3 )isnew,andthequadraticformexpressionsforthecentralcharge,pluscomponent,andtransversemomentaarealsoallnewresults.Thesetechniquesyieldasimplewaytocalculatequadraticformsfromthedynamicaltransformationbyderivingtheconservedcharge.Makingasimpleanzatzthattheconserveddynamicalchargeisaconstantmultipleofthedynamicaltransformations Qm=Qm=KZd3xfa@ @+2qma)]TJ /F2 11.955 Tf 11.95 0 Td[(Cgfabca1 @+(qmb@c)g=K(Q0m+CQ1m),(3) 38

PAGE 39

whereC,Karerealnumbers,theconstantCcanbecalculated C=1P)]TJ /F2 11.955 Tf 6.76 1.39 Td[(Q0m 0P)]TJ /F2 11.955 Tf 6.76 1.62 Td[(Q1m=2 3.(3) ThisresultyieldsthatthequadraticformsEq.( 3 ),Eq.( 3 ),Eq.( 3 ),Eq.( 3 )cannowbedetermineduptoanoverallconstantK.Thistechniqueisusefulforndingquadraticformsfornewconformaleldtheoriesderivedinthelight-coneformalism.Nowthatthequadraticformshavebeenformulated,theBPStheorywillbederivedusingthem.AllrelevantsuperspaceexpressionsareexpandedtocomponentforminAppendix B 3.4Non-LinearLight-ConeBPS InthissectiontheBPSboundwillbederivedfortheN=4eldtheoryinsuperspace.TheBPSboundfoundfromthealgebraEq.( 3 )holdsforalltheparticlesinthetheory;consequently,theeldtheorymusthaveafactorizationofthemassthatholdsforallparticles.Toachievethisfactorizationthemasslesssymmetriesarebrokenbyconstantscalarvacuumcongurationsthatminimizethepotential1 16g2fabcfade[CbmnCcpqCdmnCepq+1 @+(Cbmn@+Ccmn)1 @+(Cdpq@+Cepq)]. (3) Thispotentialhasnon-localtermsthatdonotchangetheminimumfromzero,becausethesetermsaresquared.Foranyconstantscalareldthesenon-localtermsvanish;therefore,themodulispaceforthistheoryisdescribedbytheconditionsfabcCbmnCcpq=0,fabcCbmn@+Ccmn=0, (3) whichisthenormalN=4orbifold R6r=W,(3) whereristherankofthegaugegroupGandWisitsWeylgroup[ 24 ].Atanon-singularpointinthepreviousorbifoldthegaugegroupisbrokentotheCartantorusU(1)rand 39

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theR-symmetryisbrokentoSO(5)Sp(4)bytheconstantHiggseld C=H,(3) whereHistheCartansub-algebra,isavectorofrealnumberswithlengthr,andtherootbasislischosensuchthatl6=0.Thisschemeyieldsrmasslessmultipletsandasmanymassivemultipletsasthenumberofrootgenerators[ 25 ]. TocalculatetheBPSboundthespontaneoussymmetrybreakingisachievedbyaspatialboundaryconditionimposedonthescalarelds.Thisboundaryconditionrequirestheenergybeniteontheboundary,orthetheorymustmaptothevacuum;accordingly,theeldtheoryontheboundarymustreducetothegaugevacuummanifoldparameterizedbytheHiggseldEq.( 3 ).Foranequal-timetheorythisboundaryisS2,andtheinvariantdeformationsofthegaugevacuummanifoldtotheboundaryyieldtopologicalconservationlawscharacterizedbythesecondhomotopygroupofthegaugevacuummanifold2(G=U(1)r)=1(U(1)r)=Zr. Forthelight-coneformalismtheseniteenergysolutionsandtopologicalconservationlawsareinthelight-conecoordinatesystem,oronthespatialboundaryx+=)]TJ /F2 11.955 Tf 9.3 0 Td[(x)]TJ /F1 11.955 Tf 10.41 -4.34 Td[(whichisanequal-timefoliationofspace.Since,thelight-conecoordinatesystemisonthelight-front,orconstantx+,thisequal-timefoliationneedstobeboostedtotheinnitemomentumframe.Toboosttotheinnitemomentumframethestaticequal-timeboundaryisboostedalongthethirdaxis,andthentherapiditygoestopositiveinnity.Thisprocessyieldsaniteboundaryintermsofx1,x2,x)]TJ /F1 11.955 Tf 10.41 -4.33 Td[(thatspontaneouslybreakslight-conetheoriesinthesamemannerasequal-timetheoriesdiscussedinthepreviousparagraph.Aswewillseelaterabrokenlight-conetheory'stopologicalsolitonsaretheinnitemomentumframeboostoftheequal-timesolutions. 40

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ToderivetheBPSboundtheblockdiagonalcentralchargeanti-commutationsEq.( 3 )areusedinsteadoftheotherbasisEq.( 3 )todiagonalizethequadraticform mnZn=2p 2iZd4xd4d4fma1 @+nag.(3) JustlikethealgebraEq.( 3 ),Eq.( 3 )thekinematicalanddynamicalvariationsaredivided3m=1 p 2[(1+)mnn)]TJ /F3 11.955 Tf 11.95 0 Td[((+)mnn+(1)]TJ /F3 11.955 Tf 7.08 -4.94 Td[()mnn)]TJ /F3 11.955 Tf 11.95 0 Td[(()]TJ /F3 11.955 Tf 7.08 -4.94 Td[()mnn], (3)rm=1 p 2[(1+)mnn+(+)mnn+(1)]TJ /F3 11.955 Tf 7.08 -4.94 Td[()mnn+()]TJ /F3 11.955 Tf 7.08 -4.94 Td[()mnn], (3) wherethesearetheinnitesimalversionoftheoperatorsa,brespectively.Theseinnitesimaltransformationshaveanassociatedconservedcharge bm=1 p 2[(1+)mnQn+(+)mnqn+(1)]TJ /F3 11.955 Tf 7.08 -4.94 Td[()mnqn+()]TJ /F3 11.955 Tf 7.09 -4.94 Td[()mnQn].(3) VaryingthisconservedchargewithEq.( 3 ) n[mbm,byn]=in(mrmbyn),(3) yieldstheeldtheoryequivalentoftheoperatorEq.( 3 ),apositivesemi-denitequantity.Afterintegratingbypartsandusingtheinsideoutconstraintthenalexpressionintherestframeisfbm,byng=4iZd43d4d4frma1 @+rynag=mn(M)]TJ 11.95 10.54 Td[(p 2Re(Zn)). (3) Thesurfacetermsthatcomefromintegrationbypartsinthepreviousequationarezero,becausetheydependinverselyontheradius.Thenon-zerotermsarechargesthatonlydependonangularvariablesontheboundary,andallthesetermsarecontainedinthecentralchargequadraticform;therefore,thesupereldequivalentoftheBogomol'nyi 41

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techniqueis Mmn=4iZd3xd4d4frma1 @+rynag+mnp 2Re(Zn),(3) andwhenm=nthequadraticformispositivedenitemakingtheBPSbound Mp 2Re(Zm),(3) valideldtheoreticallyforallelds.Saturationisachievedwhentheequation ryma=0,(3) issatised.Theotherinnitesimaltransformationsyieldthemass M=p 2Re(Zm)=1 2iZd3xd4d4f3na1 @+3ynag.(3) Thesesuperspaceequationshaveasimpleinterpretationintermsofthecomponenttheorywhichisderivedinthenextsection. 3.5ComponentBPSEquationSolution InthissectionMassivethecomponentdifferentialequationswillbederivedfromEq.( 3 ),andthenshowntobetheBolgomo'nyiequationsinlight-conegauge.Usingtheinside-outconstraintonEq.( 3 ),andmanipulatingthesystemofequationsyields(m+mnn)a==qma+mn1 @+(@ab)]TJ /F2 11.955 Tf 11.95 0 Td[(fabc@+c)qnb=0. (3) ThesolutiontothisequationhasmassequaltothecentralchargeeigenvaluesM=1 p 2Re(Z1+Z3). (3) TheeigenvaluescanbederivedfromthecomponentformofEq.( 3 )withtheidentityCamp@+CbnqCcpq=1 4Camn@+CbpqCcpq,andthefermionsequaltozero 42

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Zmn=2[@+AaDabCbmn)]TJ /F5 11.955 Tf 11.95 0 Td[(@+Camn1 @+(Dab@+Ab)++1 4gfabc@+Camn1 @+(@+CbpqCcpq))]. (3) ContractingtheSU(4)indiceswithmnyieldsthecentralchargeeigenvalues M=1 p 2Re(Z1+Z3)=1 2p 2mnRe(Zmn)==1 2p 2[@+AaDabb+@+AaDabb)]TJ /F5 11.955 Tf 11.95 0 Td[(@+a@+Aa)]TJ /F3 11.955 Tf 7.09 -4.94 Td[(], (3) wherea=mnCamn.IntegratingbypartsyieldstheU(1)light-coneelectricchargeplustheequationofmotion Aa)]TJ /F3 11.955 Tf 10.4 -4.93 Td[(=1 @+2[Dab@+Ab+Dab@+Ab)]TJ /F3 11.955 Tf 13.15 8.09 Td[(1 2gfabc(@+CbmnCcmn)],(3) withthepotentialminimizedbytheconditionsEq.( 3 ) M=1 p 2Zd3xf(@(@+A3)+@(@+A3))]TJ /F5 11.955 Tf 11.96 0 Td[(@+(@+A3)]TJ /F3 11.955 Tf 7.08 -4.94 Td[()g==1 p 2Zd3xf@F3+g=1 p 2QLCE, (3) whereontheboundarya=2a3asdiscussedinthelastsection. 43

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ExpandingoutEq.( 3 )withthefermionsequaltozeroyields@+Aa=1 4mnDabCbmn, (3))]TJ /F3 11.955 Tf 13.15 8.09 Td[(1 4mn@+Camn=1 @+(Dab@+Ab)+1 4gfabc1 @+(Cbmn@+Ccmn). (3) Theseequationscanbederivedfromthestressenergytenser.Themassintherestframeandfermions/potentialsettozerocanbefactorizedwithanarbitraryfourbyfouranti-symmetriccomplexmatrixXmnM=1 p 2(T+)]TJ /F3 11.955 Tf 9.74 -4.94 Td[(+T++)==1 p 2[1 2(Fa+)]TJ /F2 11.955 Tf 7.08 -4.93 Td[(Fa+)]TJ /F4 11.955 Tf 9.74 -4.93 Td[()]TJ /F2 11.955 Tf 11.96 0 Td[(FaFa)+Fa+iFa+i+1 4DabiCbmnDaciCcmn+1 2Dab+CbmnDac+Ccmn++1 16g2fabcfade[CbmnCcpqCdmnCepq]==1 p 2[1 2j(Xmn(Fa+)]TJ /F3 11.955 Tf 9.74 -4.94 Td[(+Fa)+D+abCbmn)j2+j(XmnFa+i)]TJ /F3 11.955 Tf 13.15 8.09 Td[(1 2DiabCbmn)j2]++1 16g2fabcfadeCbmnCcpqCdmnCepq)]TJ /F3 11.955 Tf 13.15 8.09 Td[(1 2(XmnFa+)]TJ /F2 11.955 Tf 7.08 -4.94 Td[(D+abCbmn+XmnFa+)]TJ /F2 11.955 Tf 7.08 -4.94 Td[(D+abCbmn)++1 2(XmnFa+iDiabCbmn+XmnFa+iDiabCbmn)], (3) wherethematrixXmnsatisesXmnXmn=1,andXmnCamnhastobereal.Theseconditionsdonotspecifytheanti-symmetricmatrixmnforXmn,butXmn=1 2mnsatisestherequiredidentities.Thus,settingXmn=1 2mnthelasttermsinEq.( 3 )becometheelectricchargebyusingtheequationofmotionEq.( 3 ),andthebound 44

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M1 p 2QLCE,(3) agreeswithEq.( 3 ),andissaturatedbytheequations: Fa+i=1 4Diabb, (3)Fa+)]TJ /F3 11.955 Tf 9.75 -4.94 Td[(+Fa=)]TJ /F3 11.955 Tf 10.49 8.09 Td[(1 2D+abb, (3)fabcCbmnCcpq=0, (3)fabcCbmn@+Ccmn=0. (3) Theequations:Eq.( 3 ),Eq.( 3 )agreewithEq.( 3 ),Eq.( 3 )afterchoosingthelight-conegaugeandsubstitutingthedependentvectoreld;furthermore,theequations:Eq.( 3 ),Eq.( 3 )arethepotentialminimizationconditionsEq.( 3 ). Theequations:Eq.( 3 ),Eq.( 3 ),Eq.( 3 )havesolutionsintermsofthe'tHooftPolyakovmonopole[ 26 ],[ 27 ],becausetheyarejustlinearcombinationsoftheequaltimeBPSequations.Tocomprehendhowtheseequationshave'tHooftPolyakovmonopolesolutionthetendimensionalmassintherestframewithfermionsequaltozeroisreduced,andfactorizedintheequal-timecoordinatesystemyielding 45

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M=Zd3xf1 2(BaxBax+EaxEax)+1 4(DabxCbmnDacxCcmn+Dab0CbmnDac0Ccmn)++1 4g2fabcfadeCbmnCcpqCdmnCepqg==Zd3xfjXmnBax)]TJ /F3 11.955 Tf 18.14 8.09 Td[(1 p 2aeiDabxCamnj2+jXmnEax)]TJ /F3 11.955 Tf 18.13 8.09 Td[(1 p 2iaeiDabxCamnj2g++asin()QE+bcos()QB, (3) where,arephases,anda,barearbitraryrealnumberssuchthata2+b2=1.Thus,saturationofthebound Masin()QE+bcos()QB,(3) occurswhen Bax=1 2p 2aeiDabxa, (3)Eax=1 2p 2iaeiDabxa, (3)Dab0b=0, (3)fabcCbmnCcpq=0, (3) andthesolutionsarefoliatedintheequal-timecoordinateorstatic 46

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a=ra r2(coth())]TJ /F3 11.955 Tf 11.95 0 Td[(1), (3)Aa0=aieira r2(coth())]TJ /F3 11.955 Tf 11.95 0 Td[(1), (3)Aax=beiaxyry r2(1)]TJ /F5 11.955 Tf 11.96 0 Td[(csch()), (3) where=gr,andristhethreedimensionalradius.11Therefore,sincetheeldstrengthsatises1 2i(aei)]TJ /F2 11.955 Tf 11.96 0 Td[(bei)Dabb=1 p 2(Ea)]TJ /F2 11.955 Tf 11.96 0 Td[(iBa)=1 p 2(Fa+1+iFa+2), (3))]TJ /F2 11.955 Tf 11.95 0 Td[(i(aei)]TJ /F2 11.955 Tf 11.95 0 Td[(bei)D3abb=)]TJ /F2 11.955 Tf 9.3 0 Td[(Ea3+iBa3=Fa+)]TJ /F3 11.955 Tf 9.74 -4.93 Td[(+Fa, (3) whereEa=Ea1+Ea2,Ba=Ba1+iBa2,thesolutiontoEq.( 3 ),Eq.( 3 ),isEq.( 3 ),Eq.( 3 ),Eq.( 3 ),witha=b=1 p 2,=)]TJ /F5 11.955 Tf 9.3 0 Td[(=)]TJ /F5 11.955 Tf 10.5 8.09 Td[( 4. (3) The'tHooftPolyakovsolutiondoesnotsatisfyEq.( 3 ),buttheequationsofmotionchangeoncetheeldsareassumedstatic,andthischangesthesaturationequationstoEq.( 3 ),Eq.( 3 ),Eq.( 3 ). Theequations:Eq.( 3 ),Eq.( 3 )havebeenchosentobeinlight-conegaugeandthe'tHooftPolyakovsolutionsdonotsatisfythem,becausethe'tHooftPolyakovsolutionsarefoliatedinequal-time.Tondthelight-conegaugesolutionsthe 11Massfactorizationsinequal-timetheoriescanbereviewedin[ 28 ],[ 29 ]. 47

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static'tHooftmonopolecanbeboostedtothespatiallight-conecoordinatesx,x,x)]TJ /F1 11.955 Tf -435.58 -28.24 Td[(thatarefoliatedinx+,orthelight-front.Thisboostiscomplicatedbythehyperbolicfunctions,andthelimitoftherapiditygoingto 4isunknown.Asimplerwaytondasolutionfoliatedinlight-coneistoboosttheboundarybehaviorforthe'tHooftPolyakovmonopole,ortheWu-YangmonopoleA1=A2=A33=A30=0, (3)ANi=1 2(1)]TJ /F3 11.955 Tf 11.95 0 Td[(cos())ijxj, (3)ASi=1 2(1+cos())ijxj, (3) whereisthetransverseradius2=x21+x22.BoostingtheWu-Yangmonopolealongthethirdaxisandtakingthelimitoftherapiditytoinnityyieldstheultrarelativisticmonopole[ 16 ]12 Ai=ijxj 2(x)]TJ /F3 11.955 Tf 7.09 -4.93 Td[().(3) Thescalarsareinvariantundertheboost;thus,thescalarsolutionmustbederivedfromtheultra-relativisticvectoreld.Ifahasonecomponent 1=0,2=0,3=,(3) 12Equation( 3 )isrelatedto(4),(5)in[ 16 ]bytheredenitionofthestepfunctionfoundinAppendix A 48

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thisreducesEq.( 3 ),Eq.( 3 )to@+A=1 4@, (3)@A=)]TJ /F3 11.955 Tf 10.5 8.09 Td[(1 8@+, (3) whereA=A1+iA2.Sincebaredistributionfunctionswithoutintegralsarenotwelldened,thesolutionsareintegratedoveradistanceL;therefore,thesolutionstoEq.( 3 ),Eq.( 3 )are13A=)]TJ /F2 11.955 Tf 9.3 0 Td[(i1 4x 2ZL 2)]TJ /F16 5.978 Tf 7.79 3.26 Td[(L 2dy)]TJ /F4 11.955 Tf 7.08 -4.94 Td[(f(y)]TJ /F4 11.955 Tf 9.74 -4.94 Td[()]TJ /F2 11.955 Tf 11.95 0 Td[(x)]TJ /F3 11.955 Tf 7.08 -4.94 Td[()g, (3)=)]TJ /F2 11.955 Tf 9.3 0 Td[(iln( L)ZL 2)]TJ /F16 5.978 Tf 7.78 3.25 Td[(L 2dy)]TJ /F4 11.955 Tf 7.09 -4.93 Td[(f(y)]TJ /F4 11.955 Tf 9.74 -4.93 Td[()]TJ /F2 11.955 Tf 11.95 0 Td[(x)]TJ /F3 11.955 Tf 7.08 -4.93 Td[()g. (3) SubstitutingtheprevioussolutionsintothechargeEq.( 3 )yieldszero Zd3xf@F+g=LZdSfIm(x 2)g=0,(3) wherethesurfaceisthetransverseS1.Thisisatraitsharedbyallultra-relativisticmonopolesbecauseoftheirpuregaugenatureeverywhereoutsideofthex)]TJ /F1 11.955 Tf 10.41 -4.34 Td[(interval()]TJ /F8 7.97 Tf 10.5 4.71 Td[(L 2,L 2). TheWu-Yangmonopolesingularitiesarenotpresentinthe'tHooftPolyakovmonopoleinanequal-timetheory.Itisunknownhowtoformulatea'tHooftPolyakovequivalentsolutioninthelight-conetheory,buthopefullyitwillsolvesomeoftheproblemswiththesingularitiesattheorigininthesolutionsEq.( 3 ),Eq.( 3 ), 13Thex)]TJ /F1 11.955 Tf 10.4 -4.34 Td[(dependenceinEq.( 3 ),Eq.( 3 )isjustaniteversionoftheGreensfunction1 @+.Thereasonforusinganon-zeronitedistanceLisforawelldenedbehaviorattheboundary.ThereismoresolutionsthenEq.( 3 ),Eq.( 3 ).Theothersolutionshaveadifferentdependanceon:A/1 n,/1 n)]TJ /F9 7.97 Tf 6.59 0 Td[(11 n)]TJ /F15 5.978 Tf 5.76 0 Td[(1. 49

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liketheequal-timetheory.ThesolutionsEq.( 3 ),Eq.( 3 )aretheonlyknownsolutionstothelight-coneBPStheory.Thesesolitonscomefromthestaticmonopoleboundaryconditionsthatareboostedtotheinnitemomentumframediscussedforarbitrarygaugegroupinchapterthreesectionfour.Inthelight-frontframetheseultra-relativisticparticleshavemassequaltotheelectricchargethatiszeroforthesolutionsEq.( 3 ),Eq.( 3 ). Theanalysisinthissectionsofarhasbeenbasedonsolutionswiththefermionssettozero.ThesupereldformulationyieldsequationsthatgeneralizetheBPSequationstoincludefermions.ThesearethesupersymmetricversionoftheBPSequations(SBPS)mnDab1 @+bn=0, (3)@+Aa=mn1 4(DabCbmn)]TJ /F3 11.955 Tf 18.13 8.09 Td[(1 p 2igfabcbm1 @+cn), (3)@+am=1 3()]TJ /F5 11.955 Tf 9.3 0 Td[(mnpqnpDabbq)]TJ /F2 11.955 Tf 11.95 0 Td[(gfabcnp(Cbnpcm)]TJ /F3 11.955 Tf 13.53 2.65 Td[(Cbmncp)]TJ /F5 11.955 Tf 11.96 0 Td[(@+Ccmn1 @+cp)), (3))]TJ /F3 11.955 Tf 13.16 8.09 Td[(1 4mn@+Camn==1 @+(Dab@+Ab)+1 4gfabc(1 @+(Cbmn@+Ccmn)+1 p 2i(1 @+bm@+cm)]TJ /F3 11.955 Tf 11.96 0 Td[(3bmcm)), (3)am=gfabcmn[@+Ab1 @+cn+Ccnp1 @+bp]. (3) ThesegeneralizedBPSequationsdescribeanon-lineareldtheoryofparticleswithmassequaltotheelectricchargewhichisfoliatedinthelight-coneframe,andarethesupersymmetricgeneralizationoftheultra-relativisticsolutionEq.( 3 ),Eq.( 3 ).Insuperspacethesolutionstothepreviousequationsmakeupasupersoliton~athat 50

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satisesEq.( 3 ).Inthisthesis,therewillnotbeanattempttondthesolutionstotheSBPSequations.Theseequationsarepresentedtonotethatthebosonictheoryisasubsetofthearbitrarytheorygivenbythepreviousequations,becauseequations:Eq.( 3 ),Eq.( 3 )canbeobtainedbysettingthefermionstozerointheSBPSequations. 51

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CHAPTER4THREEDIMENSIONALSUPERCONFORMALTHEORIES 4.1BLGTheoryAndSO(8)Triality InthischaptertheworkdonebyDmitryBelyaev,PatrickHearin,andPierreRamonddidtogeneralizetheBLGtheorywillbederived.ThissectionwillstartwithaBLGtheoryreview.DuetothetrialityoftheSO(8)theory,theBLGcanbeformulatedwherethescalareldsXaAtransformunderthespinorrepresentation8S,A=1,...,8,andthefermionsa_A_A=1,...,8transformunder8C[ 31 ].Thenon-linearcommutator[1,2]XaAwillbederivedtoderivetheSO(16).Thisisanewcalculation,since[ 20 ]onlyderivedthefreetheory.ThemostgeneraltransformationsforthistheoryXaA=a(I)]TJ /F8 7.97 Tf 6.77 4.93 Td[(Ia)A, (4)a_A=b[(I)()]TJ /F8 7.97 Tf 11.65 4.94 Td[(JDabXb)_A++gfabcd(I)[c1()]TJ /F8 7.97 Tf 11.66 4.94 Td[(JXb)_A(Xc)]TJ /F8 7.97 Tf 6.78 4.94 Td[(IJXd)+c2()]TJ /F8 7.97 Tf 11.66 4.94 Td[(IJKXb)_A(Xc)]TJ /F8 7.97 Tf 6.78 4.94 Td[(JKXd)], (4)~Aab=digfabcdIc)]TJ /F8 7.97 Tf 6.77 4.93 Td[(IXd, (4) where()]TJ /F8 7.97 Tf 11.65 4.33 Td[(I)_ABareSO(8)gammamatrices,I=1,...8,andarethethreedimensionalgammamatrices=0,1,2.~AabistheChern-Simonseld,andalleldstransformunder3-Liealgebrawithconstantfabcd.Thecovariantderivativeis Dab=ab@+Aab. (4) Thecoefcientsa,b,c1,c2,darearbitraryrealnumbers. CalculatingthecommutatorinvolvingXaAyieldsthenalexpression 52

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[1,2]XaA=DabXbA+abXbA, (4) where=AI2I1,ab=BgIJfabcd(Xc)]TJ /F8 7.97 Tf 6.77 4.34 Td[(IJXd),IJ=I1J2)]TJ /F5 11.955 Tf 11.73 0 Td[(J1I2.Theordergtermsinthebosoniccommutatorare ([1,2]XaA)g=gfabcdIJ[c1[(Xb)_A(Xc)]TJ /F8 7.97 Tf 6.77 4.94 Td[(IJXd)+()]TJ /F8 7.97 Tf 31.15 4.94 Td[(IKXb)_A(Xc)]TJ /F8 7.97 Tf 6.78 4.94 Td[(JKXd)]++c2[2()]TJ /F8 7.97 Tf 21.45 4.94 Td[(IKXb)_A(Xc)]TJ /F8 7.97 Tf 6.77 4.94 Td[(JKXd)+()]TJ /F8 7.97 Tf 31.15 4.94 Td[(IJKLXb)_A(Xc)]TJ /F8 7.97 Tf 6.78 4.94 Td[(KLXd)]. (4) NowthesetermscanbeFierzed,forexampletheidentityIJfabcd()]TJ /F8 7.97 Tf 11.66 4.93 Td[(IKXb)A(Xc)]TJ /F8 7.97 Tf 6.77 4.93 Td[(JKXd)==gIJfabcd[C0XbA(Xc)]TJ /F8 7.97 Tf 6.77 4.94 Td[(IK)]TJ /F8 7.97 Tf 6.77 4.94 Td[(JKXd)+C1()]TJ /F8 7.97 Tf 11.66 4.94 Td[(MNXb)A(Xc)]TJ /F8 7.97 Tf 6.78 4.94 Td[(IK)]TJ /F8 7.97 Tf 6.78 4.94 Td[(MN)]TJ /F8 7.97 Tf 6.78 4.94 Td[(JKXd)++C2()]TJ /F8 7.97 Tf 11.66 4.94 Td[(MNPQXb)A(Xc)]TJ /F8 7.97 Tf 6.77 4.94 Td[(IK)]TJ /F8 7.97 Tf 6.77 4.94 Td[(MNPQ)]TJ /F8 7.97 Tf 6.77 4.94 Td[(JKXd)], (4) hasthegaugeterminthersttermintheexpansion,andthetermXc)]TJ /F8 7.97 Tf 6.77 4.34 Td[(IK)]TJ /F8 7.97 Tf 6.77 4.34 Td[(MN)]TJ /F8 7.97 Tf 6.77 4.34 Td[(JKXd=0becauseitissymmetric,andtheconstantsC1,...,C4canbefoundbytracing.Thelasttermintheexpansionneedstoberelatedtothegaugeparameter.ThistermcanbeshowntobenumericallyrelatedduetotheFierzidentity0=IJfabcd()]TJ /F8 7.97 Tf 11.66 4.94 Td[(IKLSXb)A(Xc)]TJ /F8 7.97 Tf 6.78 4.94 Td[(JKLSXd)==IJfabcd[C0XbA(Xc)]TJ /F8 7.97 Tf 6.77 4.93 Td[(IKLS)]TJ /F8 7.97 Tf 6.77 4.93 Td[(JKLSXd)+C1()]TJ /F8 7.97 Tf 11.66 4.93 Td[(MNXb)A(Xc)]TJ /F8 7.97 Tf 6.78 4.93 Td[(IKLS)]TJ /F8 7.97 Tf 6.78 4.93 Td[(MN)]TJ /F8 7.97 Tf 6.78 4.93 Td[(JKLSXd)++C2()]TJ /F8 7.97 Tf 11.66 4.93 Td[(MNPQXb)A(Xc)]TJ /F8 7.97 Tf 6.77 4.93 Td[(IKLS)]TJ /F8 7.97 Tf 6.78 4.93 Td[(MNPQ)]TJ /F8 7.97 Tf 6.77 4.93 Td[(JKLSXd)], (4) 53

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beingzerobecause)]TJ /F8 7.97 Tf 6.77 4.34 Td[(JKLSissymmetric.AllthetermsintheFierzidentitiescanbereduced )]TJ /F8 7.97 Tf 6.77 4.93 Td[(IKi)]TJ /F8 7.97 Tf 6.78 4.93 Td[(JK=K1UIJi+K2VIJi, (4))]TJ /F8 7.97 Tf 6.77 4.94 Td[(IJKLi)]TJ /F8 7.97 Tf 6.78 4.94 Td[(KL=K3UIJi+K4VIJi, (4))]TJ /F8 7.97 Tf 6.77 4.94 Td[(KLi)]TJ /F8 7.97 Tf 6.78 4.94 Td[(IJKL=K5UIJi+K6VIJi, (4) wheretheconstantsK1,...K6arearbitraryrealnumbers,UIJi=)]TJ /F8 7.97 Tf 19.4 4.93 Td[(Ii)]TJ /F8 7.97 Tf 6.78 4.93 Td[(J)]TJ /F3 11.955 Tf 11.96 0 Td[()]TJ /F8 7.97 Tf 6.77 4.93 Td[(Ji)]TJ /F8 7.97 Tf 6.78 4.93 Td[(I, (4)VIJi=)]TJ /F8 7.97 Tf 19.39 4.94 Td[(IJi+Xii)]TJ /F8 7.97 Tf 6.78 4.94 Td[(IJ, (4) andirunsovertheset0=1, (4)4=)]TJ /F9 7.97 Tf 19.4 4.94 Td[(4. (4) Since,thei=4termscanbenumericallyrelatedtothegaugeparameterusingtheFierzidentityEq.( 4 ).AfternormalizingthegaugeandsuperparametersyieldsEq.( 4 ). 4.2SO(16)Generalization Thissectinwillderivethecommutator[1,2]XaAforasuperconformaltheorywithR-symmetrySO(16).InthistheorytheindicesA,_B=1,...,128transformunderthetwoinequivalentspinorrepresentationsofSO(16).Themostgeneraltransformationsforthis 54

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theoryareXaA=a(I)]TJ /F8 7.97 Tf 6.77 4.94 Td[(Ia)A, (4)a_A=b[(I)()]TJ /F8 7.97 Tf 11.66 4.93 Td[(JDabXb)_A++gfabcd(I)[c1()]TJ /F8 7.97 Tf 11.66 4.93 Td[(JXb)_A(Xc)]TJ /F8 7.97 Tf 6.77 4.93 Td[(IJXd)+c2()]TJ /F8 7.97 Tf 11.66 4.93 Td[(IJKXb)_A(Xc)]TJ /F8 7.97 Tf 6.77 4.93 Td[(JKXd)++c3()]TJ /F8 7.97 Tf 11.65 4.94 Td[(JKLMNXb)_A(Xc)]TJ /F8 7.97 Tf 6.77 4.94 Td[(IJKLMNXd)+c4()]TJ /F8 7.97 Tf 11.65 4.94 Td[(IJKLMNPXb)_A(Xc)]TJ /F8 7.97 Tf 6.78 4.94 Td[(JKLMNPXd)], (4)~Aab=digfabcdIc)]TJ /F8 7.97 Tf 6.78 4.94 Td[(IXd, (4) where()]TJ /F8 7.97 Tf 11.65 4.33 Td[(I)_IJareSO(16)gammamatricesI=1,...,16,andarethethreedimensionalgammamatrices.Thecoefcientsa,b,c1,c2,c3,c4,darearbitraryrealnumbers. CalculatingthecommutatorinvolvingXaAforthistheoryyieldsnalexpression [1,2]XaA=DabXbA+abXbA, (4) where=AI2I1,ab=BgIJfabcd(Xc)]TJ /F8 7.97 Tf 6.77 4.33 Td[(IJXd),IJ=I1J2)]TJ /F5 11.955 Tf 12.2 0 Td[(J1I2,andA,B.Theordergtermsinthebosoniccommutatorare 55

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([1,2]XaA)g=gfabcdIJ[c1[(Xb)_A(Xc)]TJ /F8 7.97 Tf 6.78 4.94 Td[(IJXd)+()]TJ /F8 7.97 Tf 31.15 4.94 Td[(IKXb)_A(Xc)]TJ /F8 7.97 Tf 6.77 4.94 Td[(JKXd)]++c2[2()]TJ /F8 7.97 Tf 21.45 4.93 Td[(IKXb)_A(Xc)]TJ /F8 7.97 Tf 6.78 4.93 Td[(JKXd)+()]TJ /F8 7.97 Tf 31.15 4.93 Td[(IJKLXb)_A(Xc)]TJ /F8 7.97 Tf 6.77 4.93 Td[(KLXd)]++c3[4()]TJ /F8 7.97 Tf 21.45 4.93 Td[(KLMNXb)_A(Xc)]TJ /F8 7.97 Tf 6.77 4.93 Td[(IJKLMNXd)+()]TJ /F8 7.97 Tf 31.15 4.93 Td[(IKLMNPXb)_A(Xc)]TJ /F8 7.97 Tf 6.77 4.93 Td[(JKLMNPXd)]++c4[6()]TJ /F8 7.97 Tf 21.45 4.94 Td[(IKLMNPXb)_A(Xc)]TJ /F8 7.97 Tf 6.77 4.94 Td[(JKLMNPXd)+()]TJ /F8 7.97 Tf 31.15 4.94 Td[(IJKLMNPQXb)_A(Xc)]TJ /F8 7.97 Tf 6.78 4.94 Td[(KLMNPQXd)]]. (4) NowtheremainingtermsthatarenotthegaugeparametercanbeFiertzed.Forexample,whentheFiertztransformationisusedonthetermIJfabcd()]TJ /F8 7.97 Tf 11.66 4.93 Td[(IJKLXb)_A(Xc)]TJ /F8 7.97 Tf 6.77 4.93 Td[(KLXd)==gIJfabcd[C0XbA(Xc)]TJ /F8 7.97 Tf 6.77 4.93 Td[(IJKL)]TJ /F8 7.97 Tf 6.78 4.93 Td[(KLXd)+C1()]TJ /F8 7.97 Tf 11.66 4.93 Td[(MNXb)A(Xc)]TJ /F8 7.97 Tf 6.78 4.93 Td[(IJKL)]TJ /F8 7.97 Tf 6.77 4.93 Td[(MN)]TJ /F8 7.97 Tf 6.77 4.93 Td[(KLXd)++C2()]TJ /F8 7.97 Tf 11.65 4.94 Td[(MNPQXb)A(Xc)]TJ /F8 7.97 Tf 6.77 4.94 Td[(IJKL)]TJ /F8 7.97 Tf 6.78 4.94 Td[(MNPQ)]TJ /F8 7.97 Tf 6.77 4.94 Td[(KLXd)]++C3()]TJ /F8 7.97 Tf 11.65 4.93 Td[(MNPQRSXb)A(Xc)]TJ /F8 7.97 Tf 6.78 4.93 Td[(IJKL)]TJ /F8 7.97 Tf 6.77 4.93 Td[(MNPQRS)]TJ /F8 7.97 Tf 6.77 4.93 Td[(KLXd)]++C4()]TJ /F8 7.97 Tf 11.65 4.93 Td[(MNPQRSTUXb)A(Xc)]TJ /F8 7.97 Tf 6.77 4.93 Td[(IJKL)]TJ /F8 7.97 Tf 6.77 4.93 Td[(MNPQRSTU)]TJ /F8 7.97 Tf 6.77 4.93 Td[(KLXd)], (4) manytermscanbecanceledwhenweswitchthetwoanti-symmetricforms,andFiertzagain 56

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0=IJfabcd()]TJ /F8 7.97 Tf 11.65 4.94 Td[(KLXb)_A(Xc)]TJ /F8 7.97 Tf 6.77 4.94 Td[(IJKLXd)==gIJfabcd[C0XbA(Xc)]TJ /F8 7.97 Tf 6.77 4.93 Td[(KL)]TJ /F8 7.97 Tf 6.77 4.93 Td[(IJKLXd)+C1()]TJ /F8 7.97 Tf 11.66 4.93 Td[(MNXb)A(Xc)]TJ /F8 7.97 Tf 6.78 4.93 Td[(KL)]TJ /F8 7.97 Tf 6.78 4.93 Td[(MN)]TJ /F8 7.97 Tf 6.78 4.93 Td[(IJKLXd)++C2()]TJ /F8 7.97 Tf 11.65 4.93 Td[(MNPQXb)A(Xc)]TJ /F8 7.97 Tf 6.77 4.93 Td[(KL)]TJ /F8 7.97 Tf 6.77 4.93 Td[(MNPQ)]TJ /F8 7.97 Tf 6.77 4.93 Td[(IJKLXd)]++C3()]TJ /F8 7.97 Tf 11.65 4.94 Td[(MNPQRSXb)A(Xc)]TJ /F8 7.97 Tf 6.78 4.94 Td[(KL)]TJ /F8 7.97 Tf 6.78 4.94 Td[(MNPQRS)]TJ /F8 7.97 Tf 6.77 4.94 Td[(IJKLXd)]++C4()]TJ /F8 7.97 Tf 11.65 4.94 Td[(MNPQRSTUXb)A(Xc)]TJ /F8 7.97 Tf 6.77 4.94 Td[(KL)]TJ /F8 7.97 Tf 6.77 4.94 Td[(MNPQRSTU)]TJ /F8 7.97 Tf 6.77 4.94 Td[(IJKLXd)]. (4) AddingthelastexpressiontotheoriginalFierzedtermleaves IJfabcd()]TJ /F8 7.97 Tf 11.66 4.93 Td[(IJKLXb)_A(Xc)]TJ /F8 7.97 Tf 6.78 4.93 Td[(KLXd)==gIJfabcd[C0XbAXc()]TJ /F8 7.97 Tf 11.65 4.94 Td[(IJKL)]TJ /F8 7.97 Tf 6.77 4.94 Td[(KL+)]TJ /F8 7.97 Tf 18.73 4.94 Td[(KL)]TJ /F8 7.97 Tf 6.77 4.94 Td[(IJKL)Xd++C2()]TJ /F8 7.97 Tf 11.65 4.93 Td[(MNPQXb)AXc()]TJ /F8 7.97 Tf 11.65 4.93 Td[(IJKL)]TJ /F8 7.97 Tf 6.77 4.93 Td[(MNPQ)]TJ /F8 7.97 Tf 6.77 4.93 Td[(KL+)]TJ /F8 7.97 Tf 18.73 4.93 Td[(KL)]TJ /F8 7.97 Tf 6.77 4.93 Td[(MNPQ)]TJ /F8 7.97 Tf 6.77 4.93 Td[(IJKL)Xd]++C4()]TJ /F8 7.97 Tf 11.65 4.93 Td[(MNPQRSTUXb)AXc()]TJ /F8 7.97 Tf 11.65 4.93 Td[(IJKL)]TJ /F8 7.97 Tf 6.77 4.93 Td[(MNPQRSTU)]TJ /F8 7.97 Tf 6.77 4.93 Td[(KL+)]TJ /F8 7.97 Tf 18.73 4.93 Td[(KL)]TJ /F8 7.97 Tf 6.77 4.93 Td[(MNPQRSTU)]TJ /F8 7.97 Tf 6.77 4.93 Td[(IJKL)Xd], (4) becausethematrices)]TJ /F8 7.97 Tf 6.78 4.34 Td[(KL)]TJ /F8 7.97 Tf 6.78 4.34 Td[(MN)]TJ /F8 7.97 Tf 6.78 4.34 Td[(IJKL+)]TJ /F8 7.97 Tf 19.13 4.34 Td[(IJKL)]TJ /F8 7.97 Tf 6.77 4.34 Td[(MN)]TJ /F8 7.97 Tf 6.77 4.34 Td[(KL,)]TJ /F8 7.97 Tf 6.78 4.34 Td[(KL)]TJ /F8 7.97 Tf 6.78 4.34 Td[(MNPQRS)]TJ /F8 7.97 Tf 6.77 4.34 Td[(IJKL+)]TJ /F8 7.97 Tf 19.14 4.34 Td[(IJKL)]TJ /F8 7.97 Tf 6.78 4.34 Td[(MN)]TJ /F8 7.97 Tf 6.78 4.34 Td[(KLaresymmetric. AllthetermsintheFierzidentitiescanbereducedto 57

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)]TJ /F8 7.97 Tf 6.77 4.94 Td[(IKi)]TJ /F8 7.97 Tf 6.78 4.94 Td[(JK=K1UIJi+K2VIJi,)]TJ /F8 7.97 Tf 6.77 4.93 Td[(IJKLi)]TJ /F8 7.97 Tf 6.77 4.93 Td[(KL=K3UIJi+K4VIJi,)]TJ /F8 7.97 Tf 6.77 4.93 Td[(KLi)]TJ /F8 7.97 Tf 6.78 4.93 Td[(IJKL=K5UIJi+K6VIJi,)]TJ /F8 7.97 Tf 6.77 4.94 Td[(KLMNi)]TJ /F8 7.97 Tf 6.77 4.94 Td[(IJKLMN=K7UIJi+K8VIJi,)]TJ /F8 7.97 Tf 6.77 4.93 Td[(IJKLMNi)]TJ /F8 7.97 Tf 6.78 4.93 Td[(KLMN=K9UIJi+K10VIJi,)]TJ /F8 7.97 Tf 6.77 4.93 Td[(IKLMNPi)]TJ /F8 7.97 Tf 6.77 4.93 Td[(JKLMNP=K11UIJi+K12VIJi,)]TJ /F8 7.97 Tf 6.77 4.94 Td[(IJKLMNPQi)]TJ /F8 7.97 Tf 6.78 4.94 Td[(KLMNPQ=K13UIJi+K14VIJi,)]TJ /F8 7.97 Tf 6.77 4.93 Td[(KLMNPQi)]TJ /F8 7.97 Tf 6.77 4.93 Td[(IJKLMNPQ=K15UIJi+K16VIJi, (4) wheretheconstantsK1,...K16arerealnumbers,UIJi=)]TJ /F8 7.97 Tf 19.4 4.94 Td[(Ii)]TJ /F8 7.97 Tf 6.78 4.94 Td[(J)]TJ /F3 11.955 Tf 11.96 0 Td[()]TJ /F8 7.97 Tf 6.77 4.94 Td[(Ji)]TJ /F8 7.97 Tf 6.78 4.94 Td[(I, (4)VIJi=)]TJ /F8 7.97 Tf 19.39 4.93 Td[(IJi+Xii)]TJ /F8 7.97 Tf 6.78 4.93 Td[(IJ, (4) andirunsovertheset0=1, (4)4=)]TJ /F9 7.97 Tf 19.4 4.93 Td[(4, (4)8=)]TJ /F9 7.97 Tf 19.4 4.94 Td[(8. (4) Since,thei=4,8termscannotbenumbericallyrelatedtothegaugeparameterusingtheFierzidentitiesEq.( 4 ).ThereistomanydifferenttypesofUIJi,VIJiterms,and 58

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notenoughFierzidentitiestoconstrainthemtothegaugeparameter:thisproblemmakesthistheoryinconsistent. 4.3Light-ConeSuperspaceOSp(2,2j16)FreeTheory InthissectiontheOSp(2,2j16)theorywillbeproofedtobeinconsistentfollowingthecalculationsfortheOSp(2,2j8)theoryin[ 20 ],byusingtheN=8light-conesupereld[ 30 ]. =1 @+2h(y)+im1 @+2 m(y)+imn1 @+Bmn(y))]TJ /F5 11.955 Tf 11.96 0 Td[(mnp1 @+mnp(y))]TJ /F5 11.955 Tf 11.96 0 Td[(mnpqDmnpq(y)++i~mnpmnp(y)+i~mn@+Bmn(y)+~m@+ m(y)+4~@+2h(y), (4) wheremareGrassmannvariables,m=1,...,8,y=(x,x+,y)]TJ /F3 11.955 Tf 12.51 -4.33 Td[(=x)]TJ /F4 11.955 Tf 10.58 -4.33 Td[()]TJ /F8 7.97 Tf 18.39 4.7 Td[(i p 2mm),a1...an=1 n!a1...an,~a1...an=a1...anb1...b8)]TJ /F16 5.978 Tf 5.75 0 Td[(nb1...b8)]TJ /F16 5.978 Tf 5.76 0 Td[(n,andthesupereldcontains128bosons,128fermions. Followingthecalculationin[ 19 ]thefreetheoryforOSp(2,2j16)willbederived,thenthekinematical,anddynamicalconstraintsarecalculated.TheconformalgroupSO(3,2)operatorshavethesameformastheOSp(2,2j8)theoryP+=)]TJ /F2 11.955 Tf 9.3 0 Td[(i@+,P=)]TJ /F2 11.955 Tf 9.3 0 Td[(i@,P)]TJ /F3 11.955 Tf 10.4 -4.94 Td[(=)]TJ /F2 11.955 Tf 11.69 8.09 Td[(i 2@2 @+,J+=)]TJ /F2 11.955 Tf 9.3 0 Td[(x@+,J+)]TJ /F3 11.955 Tf 10.4 -4.94 Td[(=i(A+1 2(x@+1)),J)]TJ /F3 11.955 Tf 10.41 -4.94 Td[(=)]TJ /F2 11.955 Tf 9.3 0 Td[(i@ @+A,D=i(A)]TJ /F3 11.955 Tf 25.36 8.09 Td[(1 2x@),K+=ix2@+,K=2ixA,K)]TJ /F3 11.955 Tf 7.09 -4.94 Td[(2i1 @+A(A)]TJ /F3 11.955 Tf 25.35 8.09 Td[(1 2),Qm=1 p 2@ @+qm,sm=ixqm,Sm=)]TJ /F2 11.955 Tf 9.3 0 Td[(iqm1 @+A,Qm=1 p 2@ @+qm,sm=)]TJ /F2 11.955 Tf 9.3 0 Td[(ixqm,Sm=iqm1 @+A, (4) whereA=x)]TJ /F5 11.955 Tf 7.09 -4.34 Td[(@+)]TJ /F9 7.97 Tf 13.27 4.71 Td[(1 2(x@+N)+3 2,N=m@m+m@m.Thegeneratorsoftheconformalgrouparechosentobehermitionwithrespecttothefollowinghermitianform <1,2>=iZd3xd8d8f11 @+32g,(4) 59

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since(x)]TJ /F5 11.955 Tf 7.09 -4.34 Td[(@+)y=)]TJ /F2 11.955 Tf 9.29 0 Td[(x)]TJ /F5 11.955 Tf 7.08 -4.34 Td[(@++4,and Ay=A)]TJ /F3 11.955 Tf 34.65 8.09 Td[(1 2.(4) ThehermiticitypropertiesareOy=OforO=(P+,P,P)]TJ /F3 11.955 Tf 7.09 -4.94 Td[(,J+,J+)]TJ /F3 11.955 Tf 7.09 -4.94 Td[(,J)]TJ /F3 11.955 Tf 7.08 -4.94 Td[(,D,K+,K,K)]TJ /F3 11.955 Tf 7.08 -4.94 Td[(),(Om)y=OmforOm=(qm,Qm,sm,Sm)(Tmn)y=Tnm,(T)y=T,(Tmn)y=)]TJ /F3 11.955 Tf 11.29 2.66 Td[(Tmn. (4) TheR-symmetryisSO(16)whichwasoriginallyderivedasthemaximallycompactsubgroupofE8(8)inthreedimensionallight-conesupergravity[ 31 ] Tmn=1 2p 2(qmqn)]TJ /F3 11.955 Tf 13.15 8.09 Td[(1 8mnqpqp)1 @+, (4)T=1 4p 2[qm,qm]1 @+, (4)Tmn=qmqn1 @+,Tmn=qmqn1 @+. (4) AlloperatorsOactlinearlyOa=iOa.TheOSp(2,2j16)freealgebraislistedinAppendix C 4.4KinematicalConstraints InthissectionthekinematicalconstraintsfortheOSp(2,2j16)theorywillbederivedfollowing[ 19 ].Thefollowingconstraintsmustbesatisedbytheanzatz 60

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dmintQa=0, (4)[P+,intQ]a=0, (4)[P,intQ]a=0, (4)[J+,intQ]a=i p 2qa, (4)[q,intQ]a=0, (4)[J+)]TJ /F3 11.955 Tf 6.76 -.3 Td[(,intQ]a=i 2intQa, (4)[U(1),intQ]a=)]TJ /F3 11.955 Tf 10.5 8.08 Td[(1 2intQa, (4)[coset,intQ]a=0, (4)[coset,intQ]a=i 2int!mnmQna, (4) andtheinsideoutconstraint intQa=d8 4@+4(intQa),(4) whered8=1 8!mnpqijkldmnpqijkl. TheseconstraintsleaveananzatzthatisthesameastheOsp(2,2j8)theory a(1)a=XI(,,)a==Xfabcd1 @+A(EE@+BaE)]TJ /F11 7.97 Tf 6.58 0 Td[(E)]TJ /F11 7.97 Tf 6.58 0 Td[(1 @+M(E@+CbE)]TJ /F11 7.97 Tf 6.59 0 Td[(@+Dc)), (4) 61

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excepttheoperatorsE=em^qm,E=em^dm,E=em^dmindexrunsovereightfortheOsp(2,2j16)theory.1Theconstantfabcdisleftarbitrary.Themasterformulacanbeusedtoderiveallconstraints2 [O,X]a=(4Xi=1[Xi,O]X)]TJ /F9 7.97 Tf 6.59 0 Td[(1Ui+fOg12+fOg34)Xa,(4) whereOisanyfreeoperator,Uiinserttheoperatorinthegivenpositioni=1,2,3,4inthenon-linearterm Xa=fabcdX1bX2(X3cX4d), (4) andthetwotripletsare fO12g=O)-223(OU1)-222(OU2, (4)fO34g=OU2)-222(OU3)-222(OU4. (4) NowthemasterformulacanbeusedtoderiveallthekinematicalconstraintswithX=mQm.TherstapplicationofthemasterformulaistotheU(1)operatorrewrittenas 1Thehatdenotes^qm=1 @+qm.2Themasterformulacanbefoundin[ 19 ]. 62

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T=i 4p 2@+[qm,qm]==i 4p 2@+[dm+ip 2m@+,dm)]TJ /F2 11.955 Tf 11.96 0 Td[(ip 2m@+]==i 4p 2@+([dm,dm])]TJ /F2 11.955 Tf 11.95 0 Td[(ip 2([dm,m@+]+[dm,m@+]+2[m,m]@+)==2)]TJ /F2 11.955 Tf 29.42 8.08 Td[(i 2p 2@+dmdm)]TJ /F3 11.955 Tf 13.15 8.08 Td[(1 2(mdm+mdm)+1 p 2mm@+, (4) tosimplifythetripletsto fTg12+fTg34=4I(,,)a,(4) becausethelastthreetermsintheU(1)operatorarederivativeoperatorsandthetermdmdmgoestozerobecauseofchiralityand em^qmUie)]TJ /F11 7.97 Tf 6.59 0 Td[(m^qmUj=e^dUie)]TJ /F11 7.97 Tf 6.59 0 Td[(^dUj.(4) Addinginthecommutatorpartofthemasterformulayields (m@ @m+m@ @m)]TJ /F3 11.955 Tf 11.95 0 Td[(8)I(,,)a=0,(4) oronlytermsoctalin,needtobeconsidered 8,7,26,35,44,53,62,7,8.(4) NexttheJ+)]TJ /F1 11.955 Tf 10.41 -4.33 Td[(constraint[J+)]TJ /F3 11.955 Tf 6.75 -.29 Td[(,mQm]a=i 2mQmaiscalculatedusingtheidentity [A,Q]a=[3+1 2(m@ @m+m@ @m+m@ @m))]TJ /F3 11.955 Tf 11.95 0 Td[(()]TJ /F2 11.955 Tf 9.29 0 Td[(A+B)]TJ /F2 11.955 Tf 11.95 0 Td[(M+C+D)],(4) andtheresultrestrictsthenumberof@+to )]TJ /F2 11.955 Tf 11.96 0 Td[(A+B)]TJ /F2 11.955 Tf 11.95 0 Td[(M+C+D=8.(4) NextthemasterformulaisappliedtoTmn.Thetripletsarecalculatedusing 63

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fqmqn1 @+gij=@ @m@ @n1 @+(@+EUi)(@+E)]TJ /F11 7.97 Tf 6.58 0 Td[(Uj)j=0,(4) and1 @+qmqn=1 @+(dm+ip 2m@+)(dn+ip 2n@+)= (4)=1 @+dmdn+ip 2(mdn)]TJ /F5 11.955 Tf 11.95 0 Td[(ndm))]TJ /F3 11.955 Tf 11.96 0 Td[(2mn@+, (4) yielding fTmngij=0,(4) duetochiralityandEq.( 4 ).Thecommutatorpartyields4Xi=1[Xi,Tmn]X)]TJ /F9 7.97 Tf 6.58 0 Td[(1UiQa==)]TJ /F2 11.955 Tf 9.3 0 Td[(ip@ @p(mn(^U1+^U2)+mn(^U3+^U4))I(0,,)a, (4) sincetheanzatzislinearin.Givingthenalexpression [!mnTmn,Q]a=)]TJ /F2 11.955 Tf 9.3 0 Td[(i!mnp@ @p(mn(^U1+^U2)+mn(^U3+^U4))I(0,,)a.(4) SimilarlytheothercosetoperatorTmnyields [!mnTmn,Q]a=2!mn(@ @m@ @nS+@ @m@ @nT)I(,,)a, (4) whereS=1 @+(@+,@+(1,1)),T=(1,1 @+(@+,@+)),becausethetripletsyieldthesamethingasthecommutatorpartduetoEq.( 4 ). ThecosetalgebraEq.( 4 )issatisedwhentheanzatzissplitintotheevenandoddansatze ((1)Qa)even=X=0,1,2DI(,,)a,(4) 64

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((1)Qa)odd=X=1 2,3 2DI(,,)a,(4) whereD=@ @4)]TJ /F15 5.978 Tf 5.76 0 Td[(2@ @4+2=1 (4+2)!1 (4)]TJ /F9 7.97 Tf 6.59 0 Td[(2)!i1...i8@ @i1...i4)]TJ /F15 5.978 Tf 5.75 0 Td[(2@ @i5)]TJ /F15 5.978 Tf 5.76 0 Td[(2...i4+2,andtherecursionrelation I(,,)a+1=S)]TJ /F9 7.97 Tf 6.59 0 Td[(1TI(,,)a,(4) issatised.ToderivetherecursionrelationinEq.( 4 )theidentity!mn@ @4)]TJ /F9 7.97 Tf 6.59 0 Td[(2@ @4+2@ @m@ @n=!mn@ @6)]TJ /F9 7.97 Tf 6.58 0 Td[(2@ @2+2@ @m@ @n, (4) (4) isused. TheothercosetcommutatorEq.( 4 )yields (1)Qa=imXmS)]TJ /F9 7.97 Tf 6.59 0 Td[(1I(0,,)a,(4) whentherecursionrelationisimposed,andusingtheidentities !mn@ @4)]TJ /F9 7.97 Tf 6.59 0 Td[(2@ @4+2mn=![2]@ @2)]TJ /F9 7.97 Tf 6.58 0 Td[(2@ @4+2, (4)!mn@ @4)]TJ /F9 7.97 Tf 6.59 0 Td[(2@ @4+2mn=![2]@ @4)]TJ /F9 7.97 Tf 6.59 0 Td[(2@ @2+2. (4) Thenotationofthepreviousequationisdenedby A[m]B[n]C[8)]TJ /F8 7.97 Tf 6.59 0 Td[(m)]TJ /F8 7.97 Tf 6.59 0 Td[(n]=1 m!n!(8)]TJ /F2 11.955 Tf 11.95 0 Td[(m)]TJ /F2 11.955 Tf 11.95 0 Td[(n)!i1...i8Ai1...imBim+1...im+nCm+n+1...i8,(4) 65

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whereitisonlytruewhen,aresettozero,andtherighthandsideiszerowhenthepoweroforderivativesisnegative. Finallytheinsideoutconstraintcanbeprovedbythefollowingidentity (I(0,,))=OI(0,,), (4) O=1 28!mnpqijklorstwxyz(1 (8!)2mnpqijkl@ @orstwxyzU2+1 (8!)2mnpqijkl@ @orstwxyzU)]TJ /F9 7.97 Tf 6.59 0 Td[(2++1 (7!)2morstwxy@ @npqijkl@ @zU3 2+1 (7!)2mnpqijko@ @l@ @rstwxyzU)]TJ /F15 5.978 Tf 7.78 3.26 Td[(3 2++1 (2!)2(6!)2mnorstwx@ @pqijkl@ @yzU1+1 (2!)2(6!)2mnpqijor@ @kl@ @stwxyzU)]TJ /F9 7.97 Tf 6.59 0 Td[(1++1 (3!)2(5!)2mnpqiors@ @jkl@ @twxyzU1 2+1 (3!)2(5!)2mnporstw@ @qijkl@ @xyzU)]TJ /F15 5.978 Tf 7.78 3.25 Td[(1 2++1 (4!)2(4!)2mnpqorst@ @ijkl@ @wxyzU0)d8 @+4, (4) where U=()]TJ /F3 11.955 Tf 9.3 0 Td[(1)21 @+2(@+2,1 @+()]TJ /F9 7.97 Tf 6.59 0 Td[(2)(@+()]TJ /F9 7.97 Tf 6.58 0 Td[(2),@+()]TJ /F9 7.97 Tf 6.58 0 Td[(2))).(4) Differentiatingthepreviousidentityyields 1 28!d8 @+4(@ @4)]TJ /F9 7.97 Tf 6.59 0 Td[(2@ 4+2Ia(0,,))=@ @4+2@ 4)]TJ /F9 7.97 Tf 6.59 0 Td[(2UIa(0,,),(4) whichprovestheinside-outconstraint 66

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1 28!d8 @+4(XevenoroddIa(,,))=XevenoroddIa(,,). (4) Thiscalculationyieldstherecursionrelation A=A)]TJ /F11 7.97 Tf 6.59 0 Td[()]TJ /F3 11.955 Tf 11.96 0 Td[(2,B=B)]TJ /F11 7.97 Tf 6.59 0 Td[()]TJ /F3 11.955 Tf 11.95 0 Td[(2,M=M)]TJ /F11 7.97 Tf 6.58 0 Td[(+4,C=C)]TJ /F11 7.97 Tf 6.59 0 Td[(+2,D=D)]TJ /F11 7.97 Tf 6.59 0 Td[(+2. (4) 4.5DynamicalConstraints 4.5.1CalculatingP)]TJ /F5 11.955 Tf 6.75 -.3 Td[(a,J)]TJ /F5 11.955 Tf 6.76 -.3 Td[(a Inthissectiontheinnitesimaltransformationsforthelight-coneHamiltonianandtheboostJ)]TJ /F1 11.955 Tf 10.41 -4.34 Td[(willbederived.Thenthesetransformationsareusedtoderivetherstorderincouplingconstantcommutator[P)]TJ /F3 11.955 Tf 6.76 -.3 Td[(,J)]TJ /F3 11.955 Tf 6.75 -.3 Td[(](1)a.Thiscommutatorhastobezerotosatisfythealgebra,andthisconstraintxesallofthearbitraryconstantsintheanzatzjustasin[ 19 ]. Toderivethelight-coneHamiltoniantotherstorder,thecommutatorsofthedynamicalsupercharges(0)Qa=1 p 2mqm@ @+a,(1)Qa=imXmS)]TJ /F9 7.97 Tf 6.59 0 Td[(1I(0,,)a, (4)(0)Qa=1 p 2mqm@ @+a,(1)Qa=imX@ @mI(0,,)a, (4) iscalculated.Usingthemasterformulawiththepreviousexpressionsyieldsnotriplets,andtheexpressionsforthecommutatorpartare 67

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[0Q,1Q]a=i p 2mnXoddm@ @r(@ @nI[r,1]+@ @nI[1,r]+1), (4)[0Q,1Q]a=i p 2mnXodd@ @m@ @r(nI[r,1]+nI[1,r]+1), (4) whereI[r,1]=ErU1E)]TJ /F8 7.97 Tf 6.58 0 Td[(rU2I(0,,)a,I[1,r]=ErU3E)]TJ /F8 7.97 Tf 6.59 0 Td[(rU4I(0,,)a,andEr=er^@.Afterusingtheidentities Xeven@ @mI=Xodd@ @mI+1 2,XevenmI=XoddmI)]TJ /F15 5.978 Tf 7.79 3.26 Td[(1 2, (4)Xodd@ @mI=Xeven@ @mI+1 2,XevenmI=XoddmI)]TJ /F15 5.978 Tf 7.79 3.26 Td[(1 2, (4)Xeven(m@ @nI+m@ @nI+1 2)=mnXoddI, (4)Xodd(m@ @nI+m@ @nI+1 2)=)]TJ /F5 11.955 Tf 9.3 0 Td[(mnXevenI, (4) thenalresultfortheinnitesimallight-coneHamiltoniantransformationis P)]TJ /F5 11.955 Tf 6.76 -.3 Td[(a=([0Q,1Q])]TJ /F3 11.955 Tf 11.96 0 Td[([0Q,1Q])a=i p 2mm@ @r(XoddIa[r,1]+XoddIa[1,r]+1 2).(4) Nowtheinnitesimalboosttransformationcanbederivedusingthecommutator [K,(1)P)]TJ /F3 11.955 Tf 6.75 1.61 Td[(]a=2i(1)J)]TJ /F5 11.955 Tf 6.75 1.61 Td[(a.(4) ThetripletsofKactingonthegeneralformI(r1,r2)=Er1U1E)]TJ /F8 7.97 Tf 6.59 0 Td[(r1U2Er2U3E)]TJ /F8 7.97 Tf 6.59 0 Td[(r2U4I(0,,)aare fxAg12+fxAg34=()]TJ /F3 11.955 Tf 9.3 0 Td[(3xU0+r13 2U2))I(r1,r2).(4) Usingtheidentities[Er,xA]=(x)]TJ /F5 11.955 Tf 7.08 -4.94 Td[(@+)]TJ /F3 11.955 Tf 13.15 8.09 Td[(1 2N+5 2)Er, (4) [Er@+k,xA]=x[1 2m@ @m)]TJ /F2 11.955 Tf 11.96 0 Td[(k]+r[x)]TJ /F5 11.955 Tf 7.09 -4.93 Td[(@++1 2(m@ @m)-222(N)+5 2)]TJ /F2 11.955 Tf 11.95 0 Td[(k]1 @+,(4) 68

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thenalexpressionis (1)J)]TJ /F5 11.955 Tf 6.75 1.61 Td[(a=)]TJ /F2 11.955 Tf 9.3 0 Td[(x(1)P)]TJ /F5 11.955 Tf 6.76 1.61 Td[(a==1 2[Xeven(F^U1)]TJ /F2 11.955 Tf 11.96 0 Td[(F^U2)I(r,1))]TJ /F12 11.955 Tf 11.95 11.35 Td[(Xodd(F^U3)]TJ /F2 11.955 Tf 11.96 0 Td[(F^U4)I(1,r)+1 2])]TJ -275.71 -36.02 Td[()]TJ /F12 11.955 Tf 11.96 11.36 Td[(Xeven(B^U1+M^U2)I(r,1)+Xodd(C+1 2^U3)]TJ /F2 11.955 Tf 11.95 0 Td[(D+1 2^U4)I(1,r)+1 2, (4) whereF=m@ @m)-222(N,andF=m@ @m)-222(Nand B=B)]TJ /F3 11.955 Tf 13.15 8.08 Td[(5 2,M=M)]TJ /F3 11.955 Tf 11.95 0 Td[((C+D)+4+(2+), (4)C=C)]TJ /F3 11.955 Tf 13.15 8.08 Td[(5 2,D=D)]TJ /F3 11.955 Tf 13.15 8.08 Td[(5 2. (4) 4.5.2Derivationof[P)]TJ /F3 11.955 Tf 6.75 -.3 Td[(,J)]TJ /F3 11.955 Tf 6.76 -.3 Td[(](1)a Inthissectiontheexpressionfortheconstraint[P)]TJ /F3 11.955 Tf 6.76 -.3 Td[(,J)]TJ /F3 11.955 Tf 6.75 -.3 Td[(](1)a=0willbecalculated.Theidentity[0P)]TJ /F3 11.955 Tf 6.76 .79 Td[(,Ka]=)]TJ /F2 11.955 Tf 11.69 8.09 Td[(i 2S@2 @r@R(Ia(r+R,1)+Ia(1,r+R)+1), (4) canbeusedtoderive[(0)P)]TJ /F3 11.955 Tf 6.75 1.62 Td[(,(1)J)]TJ /F3 11.955 Tf 6.76 1.62 Td[(]a=)]TJ /F2 11.955 Tf 9.3 0 Td[(x[(0)P)]TJ /F3 11.955 Tf 6.75 1.62 Td[(,(1)P)]TJ /F3 11.955 Tf 6.75 1.62 Td[(]a)]TJ /F2 11.955 Tf 11.96 0 Td[(i@ @+(1)oddP)]TJ /F5 11.955 Tf 19.62 1.62 Td[()]TJ -225.98 -29.85 Td[()-222(S@2 @r@Rf1 2@ @u[XevenKfu,1g)]TJ /F12 11.955 Tf 11.96 11.36 Td[(XoddKf1,ug+1 2])]TJ -201.25 -36.02 Td[()]TJ /F12 11.955 Tf 11.96 11.36 Td[(Xeven(B^U1+M^U2)(Ka(r+R,1)+Ka(1,r+R)+1)++Xodd(C+1 2^U3)]TJ /F2 11.955 Tf 11.95 0 Td[(D+1 2^U4)(Ka(r+R,1)+1 2+Ka(1,r+R)+3 2)g. (4) Thenextcommutator[0J)]TJ /F3 11.955 Tf 6.75 .8 Td[(,1P)]TJ /F3 11.955 Tf 6.75 .8 Td[(]=[0J)]TJ /F3 11.955 Tf 6.75 .8 Td[(,I[r,1]]+[0J)]TJ /F3 11.955 Tf 6.75 .8 Td[(,I[1,r]], (4) 69

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requirestheidentity[0J)]TJ /F3 11.955 Tf 6.76 .79 Td[(,I[rr,r2]]=2X[@ @+A,Xi]X)]TJ /F9 7.97 Tf 6.59 0 Td[(1iUi+f@ @+Ag12+f@ @+Ag34. (4) Usingtheidentity[EEr@+k,@ @+A]=1 2(m@ @m+r@ @r)]TJ /F3 11.955 Tf 11.96 0 Td[(2k), (4) thecommutatorpartiscalculatedX[@ @+A,Xi]X)]TJ /F9 7.97 Tf 6.59 0 Td[(1iUi=2A^@U0++()]TJ /F3 11.955 Tf 9.3 0 Td[(2B+m@ @m+r1@ @r1)^@U1+(2M+m@ @m+r1@ @r1)^@U2++()]TJ /F3 11.955 Tf 9.3 0 Td[(2C+m@ @m+r1@ @r2)^@U3+()]TJ /F3 11.955 Tf 9.3 0 Td[(2B+m@ @m+r1@ @r2)^@U4. (4) Thetriplets fJ)]TJ /F4 11.955 Tf 7.09 -4.94 Td[(gij=fx@2 @+gij+4f^@gij)-221(fN^@gij,(4) arecalculatedwiththeidentities fx@2 @+g12+fx@2 @+g34=x[0P)]TJ /F3 11.955 Tf 6.75 .8 Td[(,K])]TJ /F2 11.955 Tf 11.96 0 Td[(r1^U2@ @R@ @KK(r1,r2+R+K)+1, (4)f^@g12+f^@g34=)]TJ /F3 11.955 Tf 10.55 2.66 Td[(^@U0+^@U1+^@U3+^@U4, (4)fN^@g12+fN^@g34==m@ @m^@U1+m@ @m^@U2+m@ @m^@U3+m@ @m^@U4)]TJ -280.08 -31.25 Td[()-222(fF^@g12)-222(fF^@g34)]TJ /F3 11.955 Tf 11.96 0 Td[((4)]TJ /F3 11.955 Tf 11.96 0 Td[(2)^@U0)]TJ /F3 11.955 Tf 11.95 0 Td[((4+2)^@U2, (4) 70

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wherefF^@g12=I(t+r,1),fF^@g34=I(1,t+r)+1. (4) Usingtherecombinationidentity(4Xi=1i^@Ui)Ia==(4Xi=1i)^@Ia+S@ @rf(1^U1)]TJ /F3 11.955 Tf 11.96 0 Td[((2+3+4)^U2)Ia[r,1]+(3^U3)]TJ /F5 11.955 Tf 11.96 0 Td[(4^U4)Ia[r,1]+1g, (4) yieldsthenalexpression [0J)]TJ /F3 11.955 Tf 6.75 .79 Td[(,1P)]TJ /F3 11.955 Tf 6.75 .79 Td[(]=)]TJ /F2 11.955 Tf 9.3 0 Td[(x[(0)P)]TJ /F3 11.955 Tf 6.75 1.61 Td[(,(0)P)]TJ /F3 11.955 Tf 6.75 1.61 Td[(])]TJ /F2 11.955 Tf 11.96 0 Td[(i@ @+(1)P)]TJ /F5 11.955 Tf 6.76 1.61 Td[(a)]TJ -194.18 -57.73 Td[()]TJ /F3 11.955 Tf 13.15 8.08 Td[(1 4S@2 @r@RfXodd[(^U2K(1,r+R)+1+(5)]TJ /F3 11.955 Tf 11.95 0 Td[(2B)^U1)]TJ /F3 11.955 Tf 11.95 0 Td[((2(M)]TJ /F2 11.955 Tf 11.95 0 Td[(C)]TJ /F2 11.955 Tf 11.95 0 Td[(D)+13+2)^U2)Ia[r+R,1]+ +((4)]TJ /F3 11.955 Tf 11.95 0 Td[(2)C)^U3)]TJ /F3 11.955 Tf 11.96 0 Td[(((4)]TJ /F3 11.955 Tf 11.96 0 Td[(2)D)^U4)Ka(r,R)+1]++Xeven[(4)]TJ /F3 11.955 Tf 11.96 0 Td[(2B)^U1)]TJ /F3 11.955 Tf 11.96 0 Td[((2(M)]TJ /F2 11.955 Tf 11.96 0 Td[(C)]TJ /F2 11.955 Tf 11.96 0 Td[(D)+14+2)^U2)Ia[r+R,1]++((5)]TJ /F3 11.955 Tf 11.95 0 Td[(2)C)^U3)]TJ /F3 11.955 Tf 11.95 0 Td[(((5)]TJ /F3 11.955 Tf 11.96 0 Td[(2)D)^U4)Ka(r,R)+1]++@ @u[Xodd(Ia[r+R,1]fu,1g+Ia[r,R]f1,ug+1)+Xeven(Ia[r,R]fu,1g+1 2+Ia[1,r+R]f1,ug+3 2)]g. (4) AddingEq.( 4 )toEq.( 4 ),andusingtheidentity 71

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@ @u[Xeven,oddIa[fu,1g+Xodd,evenIaf1,ug+1 2]==4Xeven,odd^U2Ia+Xeven,odd(2)]TJ /F3 11.955 Tf 11.95 0 Td[(2)(^U1)]TJ /F3 11.955 Tf 14.15 2.52 Td[(^U2)Ia+Xodd,even(2+2)(^U3)]TJ /F3 11.955 Tf 14.15 2.52 Td[(^U4)Ia+1 2, (4) yields [oddP)]TJ /F3 11.955 Tf 8.3 .79 Td[(,oddJ)]TJ /F3 11.955 Tf 7.67 .79 Td[(]a=)]TJ /F3 11.955 Tf 10.5 8.09 Td[(1 4S(FOa1)-222(GOa2)+O(f2), (4) whereF=B^U1+M^U2,G=C^U3)]TJ /F2 11.955 Tf 11.96 0 Td[(D^U4,B=B+)]TJ /F3 11.955 Tf 13.15 8.09 Td[(7 2,M=M)]TJ /F3 11.955 Tf 11.96 0 Td[((C+D)+6,C=C)]TJ /F5 11.955 Tf 11.95 0 Td[()]TJ /F3 11.955 Tf 11.95 0 Td[(4,D=D)]TJ /F5 11.955 Tf 11.95 0 Td[()]TJ /F3 11.955 Tf 11.96 0 Td[(4, (4) (OaN=81)odd=@2 @r@RfXodd(Ia[r+R,1])-222(Ia[1,r+R]+1)+2XevenIa[r,R]+1 2gr=R===0, (4) (OaN=82)odd=@2 @r@Rf)]TJ /F12 11.955 Tf 17.26 11.36 Td[(Xeven(Ia[r+R,1]+1 2)-222(Ia[1,r+R]+3 2)+2XoddIa[r,R]+1gr=R===0. (4) 4.5.3Dynamicalinconsistency InthissectionEq.( 4 )isexpandedoutandshownthatitisnotzero.TheexpressionEq.( 4 )reducesto ((B)]TJ /F15 5.978 Tf 7.78 3.26 Td[(3 2)]TJ /F3 11.955 Tf 11.96 0 Td[(5)^U1+(M3 2)]TJ /F3 11.955 Tf 11.95 0 Td[((C)]TJ /F15 5.978 Tf 7.78 3.26 Td[(3 2+D)]TJ /F15 5.978 Tf 7.78 3.26 Td[(3 2))]TJ /F3 11.955 Tf 11.96 0 Td[(5)^U2)(OaN=81)odd++((C3 2)]TJ /F3 11.955 Tf 13.15 8.08 Td[(5 2)^U3+(D3 2)]TJ /F3 11.955 Tf 13.15 8.08 Td[(5 2)^U4)(OaN=81)odd=0, (4) 72

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where(OaN=81)odd=(@2 @r@RfD3 2W3+D1 2W2+D)]TJ /F15 5.978 Tf 7.78 3.26 Td[(1 2W+D)]TJ /F15 5.978 Tf 7.79 3.26 Td[(3 2)Ia[r+R,1])]TJ /F15 5.978 Tf 7.78 3.25 Td[(3 2++(D3 2W4+D1 2W3+D)]TJ /F15 5.978 Tf 7.78 3.26 Td[(1 2W2+D)]TJ /F15 5.978 Tf 7.78 3.26 Td[(3 2W)Ia[1,r+R])]TJ /F15 5.978 Tf 7.78 3.26 Td[(3 2++2(D2W4+D1W3+D0W2+D)]TJ /F9 7.97 Tf 6.59 0 Td[(1W+D)]TJ /F9 7.97 Tf 6.58 0 Td[(2)Ia[r,R])]TJ /F15 5.978 Tf 7.79 3.26 Td[(3 2)gr=R===0, (4)(OaN=82)odd==@2 @r@Rf(D2W4+D1W3+D0W2+D)]TJ /F9 7.97 Tf 6.59 0 Td[(1W+D)]TJ /F9 7.97 Tf 6.59 0 Td[(2)Ia[r+R,1])]TJ /F15 5.978 Tf 7.78 3.26 Td[(3 2)+(D2W5+D1W4+D0W3+D)]TJ /F9 7.97 Tf 6.59 0 Td[(1W2+D)]TJ /F9 7.97 Tf 6.58 0 Td[(2W(Ia[1,r+R])]TJ /F15 5.978 Tf 7.78 3.26 Td[(3 2+2(D3 2W4+D1 2W3+D)]TJ /F15 5.978 Tf 7.79 3.26 Td[(1 2W2+D)]TJ /F15 5.978 Tf 7.79 3.26 Td[(3 2W)Ia[r,R])]TJ /F15 5.978 Tf 7.78 3.25 Td[(3 2gr=R===0, (4) andW=@+(1 @+,1 @+2(@+,@+)). (4) Expandingoutthepreviousexpressionsyieldsdifferenttermswhere@2d8actsonallthreesupereldsplusalltheothercombinations.TryingasolutionthatissimilartotheBBKRcalculationA)]TJ /F15 5.978 Tf 7.78 3.26 Td[(3 2=5,B)]TJ /F15 5.978 Tf 7.78 3.26 Td[(3 2=5,M)]TJ /F15 5.978 Tf 7.78 3.26 Td[(3 2=)]TJ /F3 11.955 Tf 9.3 0 Td[(6,C)]TJ /F15 5.978 Tf 7.79 3.26 Td[(3 2=1,D)]TJ /F15 5.978 Tf 7.78 3.26 Td[(3 2=1,andtheconstantfabcdisanti-symmetricyieldsthatthe@2d8termsdon'tcancel.Duetothecomplexityofthiscalculationasolutiontothiscancelationproblemdoesnotseemtobeattainable.Theevenanzatzwouldnothaveasolutioneitherforthesamereason:therearetoomanytermstocancel. 73

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CHAPTER5CONCLUSION InthisthesistheBPStheoryfortheN=4light-conesuperspacewasderived,andasupereldequivalentoftheBogomolnyiequationsisthemainresult.Thedifferencebetweenthelight-conesuperspaceBPStheoryandthenormalBPScalculationsisthelight-coneincludesfermions,andistherstsuperspaceformulationoftheBPSbound.Thesimplicityofthelight-conechiralsupereldwiththeinside-outconstraintallowsfortheeldtheoriesBPSboundtobeformulatedwiththesameexpressionasthealgebra,theonlydifferencebeingthealgebraicoperatorsareswitchedwithinnitesimaltransformations. Thelight-coneBPSsolutionswerealsofoundinthisthesistobethestaticequal-timeWu-Yangmonopoleboostedtotheinnitemomentumframe.Thissolutionhaszeromassduetoitsnon-localform,andisastartingpointformoregeneralsolutionsthatsatisfythefullnon-Abelianequations,andhavenon-zeromass.Includingthefermionsintothecalculationwouldalsochangethesolutions,butthesearecomplicatedequations,anditisunknownhowtondasolution.Solutionswouldbemucheasiertondwiththesupersymmetricgeneralizationoftheequal-timeBPSequations.Usingthesupersymmetriclight-coneequationsasthestartingpointtheequaltimetheorycouldbederivedbasedontheBosonicBPStheory,butthisprocessisstillcomplicated,anditisunknownhowtoproceed.IfthesolutiontothesupersymmetricBPSequationscouldbeformulated,theseequationswoulddescribeasupermanifoldmodulispace.ThisspacewouldbethesupersymmetricgeneralizationofthehyperkahlerAtiyahandHitchinmanifoldfortheN=4theory. Thelight-coneBPSformalismcanbeextendedtomanydifferenttheories.TherecentlyformulatedBLGtheoryinlight-conesuperspace,superspaceBPSequationsshouldyieldtheNahmequation;although,itisunknownhowformulatethelight-coneBPStheoryintheBLGcase.AsimplereductionoftheN=4supereldtoitsN=2 74

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versionwouldalsobeaninterestingcalculation.ItisunknownwhethertheorieswithoutmaximalsupersymmetrylikeN=2haveaquadraticform,andhowtoformulateitssuperspaceBPStheoryisunknownduetothisfact.Lastly,maximalsupergravitytheoriessuperspaceBPSformalismcanbefoundusingthequadraticform,andmightofferalight-conesuperspacegeneralizationoftheADS/CFTconjecture. Thenon-linearrealizationfortheOSp(2,2j16)algebrainthelight-coneformalismwasattempted,andthenalresultisnoconsistenttheorycouldbederived.ThiswouldmeanthatthereisnoconsistentsuperconformaltheoryinthreedimensionsotherthentheBLGtheory.ThecovarianttheoryforOsp(2,2j16)conrmsthisinconsistencywhentheFierzidentitiesaretakenintoaccount. 75

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APPENDIXACONVENTIONS Infourdimensionsdenotedbyindex=0,1,2,3thelongitudinalandtimecoordinatescanbearrangedintothetwolight-conelines x=1 p 2(x0x3),@=1 p 2(@0@3).(A) Thelight-conemetricisoffdiagonalintheplusminuscoordinatesg+)]TJ /F3 11.955 Tf 10.49 -4.34 Td[(=g)]TJ /F9 7.97 Tf 6.58 0 Td[(+=)]TJ /F3 11.955 Tf 9.3 0 Td[(1.TherestofthecoordinatesareEuclidianandaredenotedbyi=1,2,gij=ij. Thelight-conegaugeconditionmakesthetemporalcomponentofthevectoreldzero:A+=0.TheminuscoordinateA)]TJ /F1 11.955 Tf 10.41 -4.34 Td[(iseliminatedusingtheequationsofmotion.Tosolvefortheminuscomponentofthevectoreldwemustintegrateoverthederivative@+,andintroducenon-localities.Thegreensfunctionfortheminuscoordinatex)]TJ /F1 11.955 Tf 10.41 -4.33 Td[(canbeusedtosolveforthisdependenteld @+G(x)]TJ /F3 11.955 Tf 7.08 -4.94 Td[(,y)]TJ /F3 11.955 Tf 7.08 -4.94 Td[()=(x)]TJ /F4 11.955 Tf 9.74 -4.94 Td[()]TJ /F2 11.955 Tf 11.96 0 Td[(y)]TJ /F3 11.955 Tf 7.09 -4.94 Td[().(A) Thepreviousequationisjusttheequationforastepfunction:G(x)]TJ /F3 11.955 Tf 7.08 -4.34 Td[(,y)]TJ /F3 11.955 Tf 7.08 -4.34 Td[()=1 2(x)]TJ /F4 11.955 Tf 9.74 -4.34 Td[()]TJ /F2 11.955 Tf 11.95 0 Td[(y)]TJ /F3 11.955 Tf 7.08 -4.34 Td[()(x)]TJ /F4 11.955 Tf 9.75 -4.94 Td[()]TJ /F2 11.955 Tf 11.95 0 Td[(y)]TJ /F3 11.955 Tf 7.09 -4.94 Td[()=1forx)]TJ /F5 11.955 Tf 10.41 -4.94 Td[(>y)]TJ /F3 11.955 Tf 7.08 -4.94 Td[(,(x)]TJ /F4 11.955 Tf 9.74 -4.94 Td[()]TJ /F2 11.955 Tf 11.96 0 Td[(y)]TJ /F3 11.955 Tf 7.09 -4.94 Td[()=)]TJ /F3 11.955 Tf 9.3 0 Td[(1forx)]TJ /F5 11.955 Tf 10.4 -4.94 Td[(
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1=0B@01101CA,2=0B@0)]TJ /F2 11.955 Tf 9.3 0 Td[(ii01CA,3=0B@100)]TJ /F3 11.955 Tf 9.3 0 Td[(11CA,(A) thelight-conematricesare+=1 p 2(I2+3)=p 20B@10001CA, (A))]TJ /F3 11.955 Tf 10.41 -4.94 Td[(=1 p 2(I2)]TJ /F5 11.955 Tf 11.95 0 Td[(3)=p 20B@00011CA, (A)=1 p 2(1+i2)=p 20B@01001CA. (A) Thesematriceshavetheproperties:()2=p 2,=0,=p 2+,=p 2)]TJ /F1 11.955 Tf 7.09 -4.34 Td[(. ThefourdimensionalCliffordalgebrasatisesf,g=)]TJ /F3 11.955 Tf 9.3 0 Td[(2g,andhastheequal-timerepresentation 0=0B@0I2I201CA,x=0B@0x)]TJ /F5 11.955 Tf 9.3 0 Td[(x01CA, (A)5=i0123=0B@)]TJ /F2 11.955 Tf 9.3 0 Td[(I200I21CA,C4=0BBBBBBB@0100)]TJ /F3 11.955 Tf 9.3 0 Td[(1000000)]TJ /F3 11.955 Tf 9.29 0 Td[(100101CCCCCCCA, (A) wherex=1,2,3.Thematrices5areusedforthefourdimensionalWeylconstraint =5 ,andC4isusedfortheMajoranaconstraint = TC4,foranyfourdimensional 77

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spinor ,where = T0.TheminussignintheWeylconditionistheanti-Weylspinorconstraint.ThematricesEq.( A )canbeusedtoconstructaCliffordalgebraforthelight-conemetric =0B@001CA,i=0B@0i)]TJ /F5 11.955 Tf 9.29 0 Td[(i01CA, (A)=1 p 2(1+i2)=0B@0)]TJ /F5 11.955 Tf 9.3 0 Td[(01CA. (A) Fromfourdimensionsthehigherdimensionalgammamatricescanbedened.Onecanconstructthetendimensionalmatricesdenotedby)]TJ /F8 7.97 Tf 6.78 4.34 Td[(M,whereM=0,1,2,...,9,bystartingwiththerstfourdimensions )]TJ /F6 7.97 Tf 6.77 4.94 Td[(=iI8,)-278(=iI8.(A) TheothersixdimensionsaredenotedbyI=4,5,6,7,8,9.ThesesixextratendimensionalgammamatricescanbefoundfromasixdimensionalCliffordalgebraconstructedwiththe'tHooftsymbols ymn=ymn4+ymn4)]TJ /F5 11.955 Tf 11.96 0 Td[(ynm4,(A) ~ymn=ymn4)]TJ /F5 11.955 Tf 11.96 0 Td[(ymn4+ynm4,(A) wherem,n=1,2,3,4andthemnpqsymbolisaLevi-Civitatenserdenedbythepermutationgroupoffourobjects;furthermore,theobjects 78

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Imn=ymnIy+i~ymnIy+3, (A)Imn=ymnIy)]TJ /F2 11.955 Tf 11.96 0 Td[(i~ymnIy+3, (A) yieldthematrices ~)]TJ /F8 7.97 Tf 6.78 4.93 Td[(I=0B@0ImnImn01CA,(A) thatsatisfytheCliffordalgebraf)]TJ /F8 7.97 Tf 6.77 4.34 Td[(I,)]TJ /F8 7.97 Tf 12.26 4.34 Td[(Jg=)]TJ /F3 11.955 Tf 9.3 0 Td[(2gIJ,wheregIJisthesixdimensionalmetric.Acompletelistofthepropertiesofthet'Hooftmatricescanbefoundin[ 6 ].Nowthenaltendimensionalgammamatricesare )]TJ /F8 7.97 Tf 6.77 4.94 Td[(I=i5~)]TJ /F8 7.97 Tf 6.77 4.94 Td[(I.(A) Finally,thetendimensionalmatricessatisfytheCliffordalgebra f)]TJ /F8 7.97 Tf 6.77 4.94 Td[(M,)]TJ /F8 7.97 Tf 12.26 4.94 Td[(Ng=2gMN,(A) wheregMNisthetendimensionalmetric,andtheWeyl,Majoranaconstraintsaredenedwiththematrices )]TJ /F9 7.97 Tf 6.77 -1.8 Td[(11=i)]TJ /F9 7.97 Tf 6.77 -1.8 Td[(0...)]TJ /F9 7.97 Tf 17.23 -1.8 Td[(9=50B@I400)]TJ /F2 11.955 Tf 9.3 0 Td[(I41CA, (A)C=iC40B@0I4I401CA, (A) respectively. 79

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APPENDIXBCOMPONENTEXPANSION UsingtheProjectionmethodthecomponentformoftheconservedcharges,andthecentralchargewillbederivedfromsuperspace.Themultinomialtheoremcanbegeneralizedtoincludeoddelds.LettherebeMelds,Nderivativesandnqbezeroforbosonsandoneforfermionsthen dN(MYm=1m)=m1...mNNXfPMi=1ki=NgMYj=1kjYq=10B@Nk1,...,kM1CA()]TJ /F3 11.955 Tf 9.3 0 Td[(1)(PM)]TJ /F15 5.978 Tf 5.76 0 Td[(1l=1Plp=1kl+1np)dmqj.(B) ExpandingthisequationforN=4,M=2,3witheachelddenotedby1=A,...,4=D,andthelowercasedenotesthestatisticalnatureoftheeldnq.Expandingoutthenegativesigncontributionfromthestatisticalnatureoftheelds,yieldsfor2,3: 1Xl=1lXp=1kl+1np=k2n1, (B)2Xl=1lXp=1kl+1np=k2n1+k3(n1+n2). (B) UsingEq.( B ),wendtheexpansionoftwoeldsAandB,thathavestatisticalnaturea,b d4(AB)==d4AB+Ad4B+mnpq(4()]TJ /F3 11.955 Tf 9.3 0 Td[(1)admAdnpqB+6dmnAdpqB+4()]TJ /F3 11.955 Tf 9.3 0 Td[(1)admnpAdqB). (B) ForthreeeldsA,B,Cwithstatisticalnaturea,b,c,weuseEq.( B ),andnd 80

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d4(ABC)=d4ABC+Ad4BC+ABd4C++4mnpq(()]TJ /F3 11.955 Tf 9.3 0 Td[(1)admAdnpqBC+()]TJ /F3 11.955 Tf 9.3 0 Td[(1)a+bdmABdnpqC)+()]TJ /F3 11.955 Tf 9.3 0 Td[(1)admnpAdqBC++(()]TJ /F3 11.955 Tf 9.3 0 Td[(1)a+bAdmBdnpqC)+()]TJ /F3 11.955 Tf 9.3 0 Td[(1)a+bdmnpABdqC+()]TJ /F3 11.955 Tf 9.29 0 Td[(1)bAdmnpBdqC)++(dmnAdpqBC+dmnABdpqC+AdmnBdpqC)++12(()]TJ /F3 11.955 Tf 9.3 0 Td[(1)admAdnBdpqC+()]TJ /F3 11.955 Tf 9.3 0 Td[(1)a+bdmAdnpBdqC+()]TJ /F3 11.955 Tf 9.3 0 Td[(1)bdmnAdpBdqC. (B) NowthecomponentformofthetheorycanbederivedusingEq.( B ),Eq.( B ),andtheprojectionrulesaj=1 @+Aa,dmaj=i1 @+am,dmdnaj=)]TJ /F2 11.955 Tf 9.3 0 Td[(ip 2Camn,dmdndpaj=)]TJ 9.3 10.54 Td[(p 2mnpqaq,dmdndpdqaj=2mnpq@+Aa. (B) 81

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NowwewillexpandouttheconservedchargesEq.( 3 ),Eq.( 3 ),startingwiththefreedynamicaltransformation )]TJ /F3 11.955 Tf 11.96 0 Td[(2Zd3xd4d4fa@ @+qsag=1 48Zd3xfd4(a@@+qsa)g.(B) ThiscanbeexpandedusingEq.( B ) d4(a@@+qma)==a@@+3qsa+1 48mnpq()]TJ /F3 11.955 Tf 9.3 0 Td[(4dmaqsdnpq@ @+a+ (B)+6dmnaqsdpq@ @+a)]TJ /F3 11.955 Tf 11.96 0 Td[(4dmnpaqsdq@ @+a), (B) andthenapplyingtheprojectionrulesEq.( B )yields (Qm)0=)]TJ /F3 11.955 Tf 9.3 0 Td[(2iZd3xf@Aaam+@Camnang,(B) orEq.( 3 )atthefreelevel.Thekinematicalchargeisreducedinthesamemannerasthedynamical.TheresultisobtainedbyreplacingthetransversederivativewiththelongitudinalyieldingEq.( 3 ).NowweexpandEq.( 3 )toorderg Zd3xd4d4f2 3gfabca1 @+2(qmb@+c)g=Zd3xd4f2 3gfabcaqmb@+cg. (B) UsingEq.( B )toexpandthechiralderivative 82

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d4(2 3aqmb@+c)=2 3(@+2aqsb@+c+aqsb@+3c++mnpq[4()]TJ /F3 11.955 Tf 9.83 2.66 Td[(dmaqsdnpqb@+c)]TJ /F3 11.955 Tf 12.48 2.66 Td[(dmaqsbdnpq@+c+dmnpaqsdqb@+c++aqsdmbdnpq@+c)]TJ /F3 11.955 Tf 12.48 2.66 Td[(dmnpaqsbdq@+c+aqsdmnpbdq@+c)++6(dmnaqsdpqb@+c+dmnaqsbdpq@+c+aqsdmnbdpq@+c)++12(dmaqsdnbdpq@+c)]TJ /F3 11.955 Tf 12.48 2.66 Td[(dmaqsdnpbdq@+c+dmnaqsdpbdq@+c)]), (B) thenapplyingtheprojectionrulesEq.( B )yields 2igfabc(AaCbmncn+1 2AaAbcm+1 2CamnCbnpcp++@+AaAb1 @+cm+@+AaAb1 @+cm)g, (B) orEq.( 3 )toorderg. NowwewillreducethecentralchargeEq.( 3 ) Zmn=2ip 2Zd3xd4d4fma1 @+nag,(B) toitscomponentform.UsingEq.( B )thefreetheoryreducesto 83

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(Zst)0=i1 24p 2Zd3xfd4qsaqt@a)]TJ /F5 11.955 Tf 11.96 0 Td[(@+(qsaqt@ @+a)g==)]TJ /F2 11.955 Tf 9.3 0 Td[(i1 8p 2mnpqZd3xfqsdmaqtdnpq@a+qsdmnpaqtdq@a++@+(qsdmaqtdnpq@ @+a+qsdmnpaqtdq@ @+a)g, (B) andusingtheprojectionrulesEq.( B )yields (Zst)0=Zd3xf@(Cast@+Aa)+@+(Cast@Aa)g.(B) UsingEq.( B )theordergtermsreduceto (Zmn)g=i1 24p 2gZd3xd4fqsaqtb@+c+@+(qsa1 @+(qtb@+c))g==)]TJ /F2 11.955 Tf 9.3 0 Td[(i1 24p 2mnpqgZd3xf4dmqsadnpqqtb@+c++4dmnpqsadqqtb@+c+12dmqsadnqtbdpq@+cg, (B) andtheprojectionrulesEq.( B )yield Zd3xf@+(Camn1 @+(@+AbAc))g.(B) Thenalresultis Zmn=Zd3xf@(@+AaCast))]TJ /F5 11.955 Tf 11.95 0 Td[(@+(1 @+(Dab@+Ab)Camn)g.(B) 84

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APPENDIXCOSP(2,2j16)FREETHEORY ThealgebragivenbyEq.( 4 )hasthefollowingnon-zero(anti)commutators:[J+)]TJ /F3 11.955 Tf 7.08 -4.93 Td[(,J+]=iJ+,[J+)]TJ /F3 11.955 Tf 7.09 -4.93 Td[(,J)]TJ /F3 11.955 Tf 7.08 -4.93 Td[(]=)]TJ /F2 11.955 Tf 9.29 0 Td[(iJ)]TJ /F3 11.955 Tf 7.08 -4.93 Td[(,[J+,J)]TJ /F3 11.955 Tf 7.08 -4.93 Td[(]=iJ+)]TJ /F3 11.955 Tf 7.09 -4.93 Td[(,[J+)]TJ /F3 11.955 Tf 7.08 -4.94 Td[(,P+]=iP+,[J+)]TJ /F3 11.955 Tf 7.08 -4.94 Td[(,P)]TJ /F3 11.955 Tf 7.09 -4.94 Td[(]=)]TJ /F2 11.955 Tf 9.3 0 Td[(iP)]TJ /F3 11.955 Tf 7.09 -4.94 Td[(,[J+)]TJ /F3 11.955 Tf 7.08 -4.94 Td[(,K+]=iK+,[J+)]TJ /F3 11.955 Tf 7.08 -4.94 Td[(,K)]TJ /F3 11.955 Tf 7.08 -4.94 Td[(]=)]TJ /F2 11.955 Tf 9.3 0 Td[(iK)]TJ /F3 11.955 Tf 7.09 -4.94 Td[(,[J+,P]=)]TJ /F2 11.955 Tf 9.3 0 Td[(iP+,[J+,P)]TJ /F3 11.955 Tf 7.09 -4.94 Td[(]=)]TJ /F2 11.955 Tf 9.3 0 Td[(iP,[J+,K]=)]TJ /F2 11.955 Tf 9.3 0 Td[(iK+,[J+,K)]TJ /F3 11.955 Tf 7.08 -4.94 Td[(]=)]TJ /F2 11.955 Tf 9.3 0 Td[(iK,[J)]TJ /F3 11.955 Tf 7.08 -4.93 Td[(,P+]=)]TJ /F2 11.955 Tf 9.3 0 Td[(iP,[J)]TJ /F3 11.955 Tf 7.08 -4.93 Td[(,P]=)]TJ /F2 11.955 Tf 9.3 0 Td[(iP)]TJ /F3 11.955 Tf 7.09 -4.93 Td[(,[J)]TJ /F3 11.955 Tf 7.09 -4.93 Td[(,K+]=)]TJ /F2 11.955 Tf 9.3 0 Td[(iK,[J)]TJ /F3 11.955 Tf 7.08 -4.93 Td[(,K]=)]TJ /F2 11.955 Tf 9.29 0 Td[(iK)]TJ /F3 11.955 Tf 7.08 -4.93 Td[(,[K+,P]=2iJ+,[K+,P)]TJ /F3 11.955 Tf 7.09 -4.93 Td[(]=2i(J+)]TJ /F4 11.955 Tf 9.75 -4.93 Td[()]TJ /F2 11.955 Tf 11.95 0 Td[(D),[K)]TJ /F3 11.955 Tf 7.08 -4.94 Td[(,P]=2iJ)]TJ /F3 11.955 Tf 7.08 -4.94 Td[(,[K)]TJ /F3 11.955 Tf 7.09 -4.94 Td[(,P+]=2i(J+)]TJ /F3 11.955 Tf 9.74 -4.94 Td[(+D),[K,P+]=)]TJ /F3 11.955 Tf 9.3 0 Td[(2iJ+,[K,P)]TJ /F3 11.955 Tf 7.08 -4.93 Td[(]=)]TJ /F3 11.955 Tf 9.3 0 Td[(2iJ)]TJ /F3 11.955 Tf 7.08 -4.93 Td[(,[K,P]=2iD,[D,P+]=iP+,[D,P]=iP,[D,P)]TJ /F3 11.955 Tf 7.09 -4.93 Td[(]=iP)]TJ /F3 11.955 Tf 7.08 -4.93 Td[(,[D,K+]=)]TJ /F2 11.955 Tf 9.3 0 Td[(iK+,[D,K]=)]TJ /F2 11.955 Tf 9.3 0 Td[(iK,[D,K)]TJ /F3 11.955 Tf 7.08 -4.94 Td[(]=)]TJ /F2 11.955 Tf 9.29 0 Td[(iK)]TJ /F3 11.955 Tf 7.08 -4.94 Td[(. (C) 85

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SO(16) [Tmn,Tpq]=mpTnq)]TJ /F5 11.955 Tf 11.95 0 Td[(mqTnp,[Tmn,Tpq]=1 4mnTpq)]TJ /F5 11.955 Tf 11.95 0 Td[(pnTmq+qnTmp,[T,Tmn]=)]TJ /F2 11.955 Tf 9.3 0 Td[(Tmn,[Tmn,Tpq]=1 4mnTpq)]TJ /F5 11.955 Tf 11.95 0 Td[(mpTnq+mqTnp,[T,Tmn]=)]TJ /F3 11.955 Tf 11.29 2.66 Td[(Tmn,[Tmn,Tmn]=mpTnq)]TJ /F5 11.955 Tf 11.95 0 Td[(mqTnp+nqTmp)]TJ /F5 11.955 Tf 11.95 0 Td[(npTmq+(mpnq)]TJ /F5 11.955 Tf 11.96 0 Td[(mqnp)T (C) SO(16)withthefermionicgenerators: [Tmn,qp]=1 8mnqp)]TJ /F5 11.955 Tf 11.95 0 Td[(pnqm,[T,qm]=)]TJ /F3 11.955 Tf 10.5 8.09 Td[(1 2qm,[Tmn,qp]=pmqn)]TJ /F5 11.955 Tf 11.96 0 Td[(pnqm,[Tmn,qp]=1 8mnqp)]TJ /F5 11.955 Tf 11.95 0 Td[(mpqn,[T,qm]=1 2qm,[Tmn,qp]=mpqn)]TJ /F5 11.955 Tf 11.95 0 Td[(npqm, (C) andthesameformforthecommutatorswith:sm,sm,Sm,Sm,Qm,Qm. SO(3,2)withthefermionicgenerators: 86

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[J+)]TJ /F3 11.955 Tf 7.09 -4.94 Td[(,qm]=i 2qm,[J+)]TJ /F3 11.955 Tf 7.09 -4.94 Td[(,qm]=i 2qm,[J+)]TJ /F3 11.955 Tf 7.09 -4.94 Td[(,Qm]=i 2Qm,[J+)]TJ /F3 11.955 Tf 7.08 -4.94 Td[(,Qm]=)]TJ /F2 11.955 Tf 11.69 8.09 Td[(i 2Qm,[J+)]TJ /F3 11.955 Tf 7.09 -4.94 Td[(,sm]=i 2sm,[J+)]TJ /F3 11.955 Tf 7.09 -4.94 Td[(,sm]=i 2sm,[J+)]TJ /F3 11.955 Tf 7.09 -4.94 Td[(,Sm]=i 2Sm,[J+)]TJ /F3 11.955 Tf 7.08 -4.94 Td[(,Sm]=)]TJ /F2 11.955 Tf 11.68 8.09 Td[(i 2Sm,[J+,Qm]=)]TJ /F2 11.955 Tf 16.67 8.08 Td[(i p 2qm,[J+,Qm]=)]TJ /F2 11.955 Tf 16.67 8.08 Td[(i p 2qm,[J+,Sm]=i p 2sm,[J+,Sm]=i p 2sm,[J+,Qm]=)]TJ /F2 11.955 Tf 16.67 8.09 Td[(i p 2qm,[J+,Qm]=)]TJ /F2 11.955 Tf 16.67 8.09 Td[(i p 2qm,[J+,Sm]=i p 2sm,[J+,Sm]=i p 2sm,[J)]TJ /F3 11.955 Tf 7.08 -4.93 Td[(,qm]=)]TJ /F2 11.955 Tf 16.67 8.08 Td[(i p 2Qm,[J)]TJ /F3 11.955 Tf 7.09 -4.93 Td[(,qm]=)]TJ /F2 11.955 Tf 16.67 8.08 Td[(i p 2Qm,[J)]TJ /F3 11.955 Tf 7.09 -4.93 Td[(,sm]=iSm,[J)]TJ /F3 11.955 Tf 7.08 -4.93 Td[(,sm]=iSm,[P+,Sm]=qm,[P+,Sm]=)]TJ /F3 11.955 Tf 9.54 0 Td[(qm,[P,sm]=qm,[P,sm]=)]TJ /F3 11.955 Tf 9.54 0 Td[(qm,[P,Sm]=i p 2Qm,[P,Sm]=)]TJ /F2 11.955 Tf 16.67 8.09 Td[(i p 2Qm,[P)]TJ /F3 11.955 Tf 7.09 -4.93 Td[(,sm]=p 2Qm,[P)]TJ /F3 11.955 Tf 7.08 -4.93 Td[(,sm]=)]TJ 9.3 10.54 Td[(p 2Sm,[K+,Qm]=)]TJ 9.3 10.54 Td[(p 2sm,[K+,Qm]=p 2sm,[K,qm]=sm,[K,qm]=)]TJ /F3 11.955 Tf 9.06 0 Td[(sm,[K,Qm]=p 2Sm,[K,Qm]=)]TJ 9.29 10.54 Td[(p 2Sm,[K)]TJ /F3 11.955 Tf 7.09 -4.93 Td[(,qm]=)]TJ /F3 11.955 Tf 9.3 0 Td[(2Sm,[K)]TJ /F3 11.955 Tf 7.08 -4.93 Td[(,qm]=2Qm,[D,qm]=i 2qm,[D,qm]=i 2qm,[D,Qm]=i 2Qm,[D,Qm]=i 2Qm,[D,sm]=)]TJ /F2 11.955 Tf 11.68 8.09 Td[(i 2sm,[D,sm]=)]TJ /F2 11.955 Tf 11.69 8.09 Td[(i 2sm,[D,Sm]=)]TJ /F2 11.955 Tf 11.69 8.09 Td[(i 2Sm,[D,Sm]=)]TJ /F2 11.955 Tf 11.68 8.09 Td[(i 2Sm, (C) 87

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Anti-commutationrelations:fqm,qng=)]TJ 9.3 10.54 Td[(p 2mnP+,fQm,Qng=)]TJ 9.3 10.54 Td[(p 2mnP)]TJ /F3 11.955 Tf 7.08 -4.94 Td[(,fsm,sng=p 2mnK+,fSm,Sng=i1 p 2mnK)]TJ /F3 11.955 Tf 7.09 -4.93 Td[(,fqm,Qng=)]TJ /F5 11.955 Tf 9.29 0 Td[(mnP,fqm,Qng=)]TJ /F5 11.955 Tf 9.3 0 Td[(nmP,fqm,sng=)]TJ /F2 11.955 Tf 9.3 0 Td[(ip 2mnJ+,fqm,sng=ip 2nmJ+,fQm,Sng=)]TJ /F2 11.955 Tf 9.3 0 Td[(imnJ)]TJ /F3 11.955 Tf 7.09 -4.93 Td[(,fQm,Sng=imnJ)]TJ /F4 11.955 Tf -384.45 -58.73 Td[(fqm,Sng=1 p 2Tmn,fqm,Sng=)]TJ /F3 11.955 Tf 15.47 8.09 Td[(1 p 2Tmn,fQm,sng=Tmn,fQm,sng=)]TJ /F3 11.955 Tf 11.29 2.66 Td[(Tmnfqm,Sng=1 p 2[i(J+)]TJ /F3 11.955 Tf 9.74 -4.93 Td[(+D)mn)]TJ /F3 11.955 Tf 11.96 0 Td[((2Tmn+1 4Tmn)],fqm,Sng=1 p 2[)]TJ /F2 11.955 Tf 9.3 0 Td[(i(J+)]TJ /F3 11.955 Tf 9.74 -4.94 Td[(+D)mn)]TJ /F3 11.955 Tf 11.96 0 Td[((2Tmn+1 4Tmn)],fQm,smg=1 p 2[i(J+)]TJ /F4 11.955 Tf 9.74 -4.94 Td[()]TJ /F2 11.955 Tf 11.96 0 Td[(D)mn)]TJ /F3 11.955 Tf 11.95 0 Td[((2Tmn+1 4Tmn)],fQm,smg=1 p 2[)]TJ /F2 11.955 Tf 9.3 0 Td[(i(J+)]TJ /F4 11.955 Tf 9.74 -4.94 Td[()]TJ /F2 11.955 Tf 11.96 0 Td[(D)mn)]TJ /F3 11.955 Tf 11.95 0 Td[((2Tmn+1 4Tmn)]. (C) 88

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REFERENCES [1] S.Mandelstam,LightConeSuperspaceAndTheUltravioletFinitenessOfTheN=4Model,Nucl.Phys.B213,149(1983). [2] L.Brink,O.LindgrenandB.E.W.Nilsson,TheUltravioletFinitenessOfTheN=4Yang-MillsTheory,Phys.Lett.B123,323(1983). [3] L.J.Dixon,UltravioletBehaviorofN=8Supergravity,arXiv:1005.2703. [4] Z.Bern,J.J.M.CarrascoandH.Johansson,ProgressonUltravioletFinitenessofSupergravity,arXiv:0902.3765. [5] Z.Bern,J.J.Carrasco,L.J.Dixon,H.JohanssonandR.Roiban,TheUltravioletBehaviorofN=8SupergravityatFourLoops,Phys.Rev.Lett.103,081301,2009. [6] Z.Bern,J.J.Carrasco,L.J.Dixon,H.Johansson,D.A.KosowerandR.Roiban,Three-LoopSupernitenessofN=8Supergravity,Phys.Rev.Lett.98,161303,2007. [7] LarsBrink,JohnH.SchwarzandJ.Scherk,SupersymmetricYang-MillsTheory,Nucl.Phys.B12177,December1976. [8] H.Osborn,Phys.Lett.83B321,1979. [9] LarsBrink,OlofLindgren,BengtE.W.Nilsson,Nucl.Phys.B212:401,1983. [10] E.WittenandD.Olive,Phys.Lett.78B97,1978. [11] E.B.Bogomolnyi,Sov.J.Nucl.Phys.24,449,1976. [12] M.K.PrasadandC.H.Sommereld,Phys.Rev.Lett.35760,1975. [13] P.A.M.Dirac,FormsOfRelativisticDynamics,Rev.Mod.Phys.21,392,1949. [14] SudarshanAnanth,LarsBrink,Sung-SooKim,PierreRamond,Non-linearRealizationofPSU(2,2j4)ontheLight-Cone,Nucl.Phys.B722(2005)166-190,May2005. [15] PatrickHearin,Light-ConeBPSTheory,May2011,Nucl.Phys.B846:266-249. [16] SauryaDas,ParthasarathiMajumdar,Charge-monopoleversusGravitationalScatteringatPlanckianEnergies,Phys.Rev.Lett.72(1994)2524-2526,July1993. [17] JonathanBaggerandNeilLambert,ModelingmultipleM2s,Phys.Rev.,D75045020,2007;GaugeSymmetryandSupersymmetryofMultipleM2-Branes,Phys.Rev.,D77065008,2008. [18] AndreasGustavsson,AlgebraicstructuresonparallelM2-branes,Nucl.Phys.B81166,2009. 89

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[19] HitoshiNishino,SubhashRajpoot,TrialityandBagger-LambertTheory,Phys.Lett.B671:415,2009. [20] DmitryBelyaev,LarsBrink,Sung-SooKim,PierreRamond,TheBLGTheoryinLight-ConeSuper-Space,JHEP1004:026,2010,January2010. [21] E.Corrigan,D.Olive,D.B.FairlieandJ.Nuyts,Nucl.Phys.B106475,1976. [22] DmitryV.Belyaev,DynamicalSupersymmetryinMaximallySupersymmetricGaugeTheories,October2009. [23] LarsBrink,AnnaTollsten,N=4Yang-MillsTheoryInTermsOfN=3AndN=2LightConeSuperelds,Nucl.Phys.B249:244,1985,May1984. [24] NathanSeiberg,NotesonTheorieswith16Supercharges,RU-97-7,May1997. [25] NicholasDorey,ChristopheFraser,TimothyJ.Hollowood,MarcoA.C.Kneipp,S-dualityinN=4supersymmetricgaugetheorieswitharbitrarygaugegroup,Phys.Lett.B383(1996)422-428,May1996. [26] G.'tHooft,Nucl.Phys.B79(1974)276. [27] A.M.Polyakov,JETPLett.20(1974)194. [28] JeffreyA.Harvey,MagneticMonopoles,Duality,andSupersymmetry,EFI-96-06,March1996. [29] PaoloDiVecchia,DualityinsupersymmetricN=2,4gaugetheories,Nordita98/11-HE,March1998. [30] LarsBrink,Sung-SooKim,andPierreRamond,E8(8)inLightConeSuperspace,JHEP0807,113,2008. [31] S.Mandelstam,Nucl.Phys.B213(1983)149. 90

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BIOGRAPHICALSKETCH PatrickHearinwasbornin1982inSouthPasadena,StPetersburg.Withhismother'shelpheobtainedhisA.A.inEngineeringfromSt.Petersburgcollegein2003;B.S.inPhysics,B.S.inMathematicsfromtheUniversityofFloridain2005;andaPh.DinPhysicsfromtheUniversityofFlorida2011. 91