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Light-Cone Formulation of Maximally Supersymmetric Theories in Superspace

Permanent Link: http://ufdc.ufl.edu/UFE0022321/00001

Material Information

Title: Light-Cone Formulation of Maximally Supersymmetric Theories in Superspace
Physical Description: 1 online resource (82 p.)
Language: english
Creator: Kim, Sung-Soo
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2008

Subjects

Subjects / Keywords: light, super, supergravity, superspace, 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: Maximally supersymmetric theories are discussed in four dimensions in the light-cone superspace formalism. We derive the symmetry generators of N=4 Super Yang-Mills theory which contains maximal spin-one in four dimensions, and show that the on-shell Hamiltonian of the theory can be written as a quadratic form of a fermionic superfield. With maximal spin two, we explore how the non-linear on-shell E7(7) symmetry of N=8 Supergravity in four dimensions can be realized on the light-cone, and find a remarkable result that in this light-cone gauge, all the physical fields, including the graviton, transform under E7(7). We then derive the dynamical supersymmetry transformations to order kappa^2 by requiring the coset E7(7)/SU(8) to commute with the super-Poincare group.
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 Sung-Soo Kim.
Thesis: Thesis (Ph.D.)--University of Florida, 2008.
Local: Adviser: Ramond, Pierre.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2008
System ID: UFE0022321:00001

Permanent Link: http://ufdc.ufl.edu/UFE0022321/00001

Material Information

Title: Light-Cone Formulation of Maximally Supersymmetric Theories in Superspace
Physical Description: 1 online resource (82 p.)
Language: english
Creator: Kim, Sung-Soo
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2008

Subjects

Subjects / Keywords: light, super, supergravity, superspace, 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: Maximally supersymmetric theories are discussed in four dimensions in the light-cone superspace formalism. We derive the symmetry generators of N=4 Super Yang-Mills theory which contains maximal spin-one in four dimensions, and show that the on-shell Hamiltonian of the theory can be written as a quadratic form of a fermionic superfield. With maximal spin two, we explore how the non-linear on-shell E7(7) symmetry of N=8 Supergravity in four dimensions can be realized on the light-cone, and find a remarkable result that in this light-cone gauge, all the physical fields, including the graviton, transform under E7(7). We then derive the dynamical supersymmetry transformations to order kappa^2 by requiring the coset E7(7)/SU(8) to commute with the super-Poincare group.
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 Sung-Soo Kim.
Thesis: Thesis (Ph.D.)--University of Florida, 2008.
Local: Adviser: Ramond, Pierre.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2008
System ID: UFE0022321:00001


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LIGHT-CONEFORMULATIONOFMAXIMALLYSUPERSYMMETRIC THEORIESINSUPERSPACE By SUNG-SOOKIM ADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOL OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT OFTHEREQUIREMENTSFORTHEDEGREEOF DOCTOROFPHILOSOPHY UNIVERSITYOFFLORIDA 2008 1

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c 2008Sung-SooKim 2

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InmemoryofProf.YoungjaiKiem 3

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ACKNOWLEDGMENTS Foremost,Iamdeeplyindebtedtomyadvisor,Prof.PierreRamond,forhispatience, encouragement,andguidance.Ihavebeenfortunatetohaveanadvisorwhotaughtto questionthoughtsandexpressideas,andwhoalsotaughtmethejoyofdoingresearch.I amgratefultoProf.LarsBrinkformanydiscussionsandadvice.Hehasbeenaconstant sourceofinspiration.Workingwithhimhasalwaysbeenfullofexcitement. Iwouldliketothankmysupervisorycommitteemembersfortheirsupport: ProfessorsDavidGroisser,AndreyKorytov,KonstantinMatchevandRichardWoodard.I alsowanttothankCharlesThornforhissupportandvaluablediscussions.Manythanks gotoDr.KyoungchulKong,Dr.Hye-SungLee,Dr.ChristophLuhn,andDr.Aravind Natarajanfortheirinterestandconcern.Ialsoacknowledgemyfellowstudents,Michael Burns,HyoungJeenJeen,ChanghyunKoo,FranciscoRojasfortheirhelpandfriendship, andespeciallyJesusEscobarforreadingandcommentingthismanuscript. IwishtothankEnidCorbinwhohastakencareofmelikehersonandhelpedme throughdiculttimes.Finally,allofmyaccomplishmentswouldnotbepossiblewithout myparents'supportandlove. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS.................................4 ABSTRACT........................................7 CHAPTER 1INTRODUCTION..................................8 2 LC 2 FORMALISM..................................10 2.1Superspace...................................11 2.1.1 N =4Superspace............................12 2.1.2 N =8Superspace............................14 2.2SuperPoincareAlgebra.............................15 2.2.1KinematicalSuperPoincareGenerators................16 2.2.2DynamicalPoincareGenerators....................18 3 N =4SUPERYANG-MILLS............................20 3.1 PSU ; 2 j 4symmetry.............................20 3.1.1KinematicalConformalGenerators..................20 3.1.2DynamicalConformalGenerators...................21 3.2Non-LinearRealization.............................22 3.3HamiltonianasaQuadraticForm.......................28 4 N =8SUPERGRAVITY..............................31 4.1 E 7 OnshellDualitySymmetry........................31 4.1.1Covariant N =8Supergravity.....................31 4.1.2Dualitysymmetries...........................33 4.1.2.1 SU Duality........................34 4.1.2.2 E 7 Duality.........................36 4.2 E 7 InvarianceontheLight-Cone......................41 4.2.1The LC 2 Gauge.............................41 4.2.2DualityTransformationsoftheVectorPotentials...........43 4.2.3TheVectorandScalar LC 2 Hamiltonians...............45 4.2.4HamiltoniantoOrder 2 ........................48 4.2.5 E 7 InvarianceoftheVectorandScalarHamiltonians.......50 4.3NonlinearRealizationof E 7 inSuperspace.................53 4.3.1KinematicalSupersymmetry......................54 4.3.2SUTransformations.........................55 4.3.3 E 7 =SU Transformations......................55 4.4HamiltonianinSuperspace...........................58 5

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5CONCLUSION....................................65 APPENDIX A PSU ; 2 j 4ALGEBRA...............................67 B E 7 BASICS.....................................69 B.1 SU ; 1Analysis................................69 B.2 E 7 Analysis..................................71 CNON-LINEARREALIZATION...........................74 DUSEFULIDENTITIES................................76 REFERENCES.......................................80 BIOGRAPHICALSKETCH................................82 6

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AbstractofDissertationPresentedtotheGraduateSchool oftheUniversityofFloridainPartialFulllmentofthe RequirementsfortheDegreeofDoctorofPhilosophy LIGHT-CONEFORMULATIONOFMAXIMALLYSUPERSYMMETRIC THEORIESINSUPERSPACE By Sung-SooKim August2008 Chair:PierreRamond Major:Physics Maximallysupersymmetrictheoriesarediscussedinfourdimensionsinthelight-cone superspaceformalism.Wederivethesymmetrygeneratorsof N =4SuperYang-Mills theorywhichcontainsmaximalspin-oneinfourdimensions,andshowthattheon-shell Hamiltonianofthetheorycanbewrittenasaquadraticformofafermionicsupereld. Withmaximalspintwo,weexplorehowthenon-linearon-shell E 7 symmetry of N =8Supergravityinfourdimensionscanberealizedonthelight-cone,and ndaremarkableresultthatinthislight-conegauge,allthephysicalelds,including thegraviton,transformunder E 7 .Wethenderivethedynamicalsupersymmetry transformationstoorder 2 byrequiringthecoset E 7 =SU tocommutewiththe super-Poincaregroup. 7

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CHAPTER1 INTRODUCTION Supersymmetryisahypotheticalsymmetrythatinterchangesfermions,whichare thebuildingblocksofmatter,andbosonswhichmediatethefundamentalinteractions. Particles,relatedbysupersymmetry,dierbyhalfaunitofspin,andtheyarecalled superpartners,andtogethertheyformsupermultipletnamedaftertheeldswiththe maximumspin. Maximallysupersymmetrictheoriesinfourdimensionshavemanyremarkable properties:simplicityintheperturbativeexpansionsandcloserelationtohigher dimensionaltheoriessuchasten-dimensionalstringtheoriesandM-theoryineleven dimension.Therearetwomaximallysupersymmetrictheoriesinfourdimensions: N =4 SuperYang-Mills[1]and N =8Supergravity[2]. SuperYang-Millsisanon-Abeliangaugetheorycombinedwithglobalsupersymmetry. Thenumberofthesupersymmetryisdenotedby N .Thus N =4SuperYang-Millsis thesupersymmetricgaugetheorythathasfoursupersymmetrygenerators,whichisthe maximalnumberofthesupersymmetryinfourdimensionforthetheorieswithmaximal spin1.SuperYang-Millshasasuperconformal PSU ; 2 j 4symmetrywhosebosonic symmetryis SO ; 2 SU ,andcanbeobtainedfrom N =1SuperYang-Millsinten dimensionsbydimensionalreduction. Ontheotherhand, N =8Supergravityisalsothemaximallysupersymmetrictheory thatcombinessupersymmetrywithgeneralrelativityincludingthegraviton,andthus supersymmetryislocal.Itcanbeobtainedbydimensionalreductionofeleven-dimensional N =1Supergravity.Italsohasaglobalnon-linearnon-compact E 7 symmetry[3], whichactsasanon-shelldualitysymmetrythattranslatestheequationsofmotiontothe Bianchiidentitiesandviceversa.HoweverthescalareldpartoftheLagrangianisfully invariantunderthenon-linearlyrealized E 7 8

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Expressedinlight-conesuperspace[4{6], N =8Supergravitysharesformalanalogies with N =4SuperYang-Mills:botharedescribedbyachiral constrained supereldwhich capturesallthephysicaldegreesoffreedom.Therearemanyindicationsthatthisformal analogyextendstointeractions,wheretreelevelSupergravityamplitudesarerelatedto thesquareofYang-Millsamplitudes[7,8].Therearehintsbackedbyexplicitcalculations thattheultravioletniteness[4]of N =4SuperYang-Millsmightextend[9,10]to N =8 Supergravity,althoughtheissueisfarfromsettled. Itisthereforehopedthatmanyofthesimplicitiesofthe N =4theorycanbe extendedtoSupergravity.Forexample,thefullyinteractinglight-coneHamiltonianof N =4SuperYang-Millstheorycanbewrittenasa positivedenite quadraticformin superelds[11].Thecubicsupergravityinteractionisencapsulatedby one term,justasin theYang-Millstheory.Thissimplicityofformmightextenttohigherordersin Thereareofcoursedistinctstructuraldierencesbetweenthetwotheories: N =8 Supergravity,unlike N =4SuperYang-Mills,isnotSuperconformalinvariant;however itdoeshavethenon-linearCremmer-Julia E 7 symmetry.Itisthereforenaturaltoask ifthissymmetrycanbeexploitedtobringsimplicitytothequarticandhigher-order interactionsof N =8Supergravity.Thisissueisaddressedinchapter4,basedonthe recentpublication[12]. Thisthesisnotonlysummarizesthepapers[11,12]donealongthisdirection,but alsoprovidestechnicaldetails.Inchapter2,weintroducethelight-coneformulation,and showhowthephysicaleldsofmaximallysupersymmetrictheoriesinfourdimensions,are representedinthelight-coneformalisminsuperspace.Inchapter3,weexplicitlyexpress thelight-coneformofthesuperconformalgroup PSU ; 2 j 4generators,andusethe symmetrytoyieldtheHamiltonianasaquadraticform.Inchapter4,theon-shell E 7 symmetryisdiscussedinconnectionwiththequarticinteractionsof N =8Supergravity. Conclusionthenfollows. 9

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CHAPTER2 LC 2 FORMALISM Withthespace-timemetric )]TJ/F21 11.9552 Tf 9.299 0 Td [(; + ; + ; ; +,thelight-conecoordinatesandtheir derivativesaregivenby x = 1 p 2 x 0 x 3 ; @ = 1 p 2 )]TJ/F21 11.9552 Tf 11.291 0 Td [(@ 0 @ 3 ;{1 x = 1 p 2 x 1 + ix 2 ; @ = 1 p 2 @ 1 )]TJ/F21 11.9552 Tf 13.948 0 Td [(i@ 2 ;{2 x = 1 p 2 x 1 )]TJ/F21 11.9552 Tf 13.947 0 Td [(ix 2 ; @ = 1 p 2 @ 1 + i@ 2 ; {3 sothat @ + x )]TJ/F15 11.9552 Tf 10.406 -4.936 Td [(= @ )]TJ/F21 11.9552 Tf 9.077 -4.936 Td [(x + = )]TJ/F15 11.9552 Tf 13.948 0 Td [(1; @x = @ x =1 ; {4 where x + istakentobealight-conetimeevolutionparameterand @ )]TJ/F15 11.9552 Tf 10.987 -4.339 Td [(isthelight-cone timederivative. The LC 2 formalismreferstothetwo-componentlight-coneformalismwiththe light-conegauge.Infourdimensions,themasslessgaugeelds A x haveonlytwo physicaldegreesoffreedom.Onecanchoosethelight-conegauge A a + x =0 ; andthenusetheequationsofmotiontoeliminatetheunphysicaldegreesoffreedom A a )]TJ/F15 11.9552 Tf 7.084 -4.339 Td [( x .Thusonehasonlytwocomponentsofthegaugeeldswhichareallphysical. Foraspinor x ,onecansplitintoitstwopartsbyintroducingtheprojectionoperator whichismadeofthegammamatrices = P + + P )]TJ/F15 11.9552 Tf 9.077 -4.936 Td [( + + )]TJ/F21 11.9552 Tf 10.987 -4.936 Td [(; 10

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where P + = 1 2 )]TJ/F21 11.9552 Tf 7.085 -4.936 Td [( + ; P + = 1 2 + )]TJ/F21 11.9552 Tf 10.986 -4.936 Td [(; satisfying P + + P )]TJ/F15 11.9552 Tf 12.398 -4.339 Td [(=1since f ; g = )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 ; wherethelight-conemetric+ )]TJ/F15 11.9552 Tf 13.201 0 Td [(componentisgivenby + )]TJ/F15 11.9552 Tf 14.669 -4.339 Td [(= )]TJ/F15 11.9552 Tf 12.1 0 Td [(1.Thentheunphysical degreesoffreedom )]TJ/F15 11.9552 Tf 10.987 -4.339 Td [(iseliminatedbytheequationsofmotion.Therefore,onlythe physicaldegreesoffreedomarepresentedinthe LC 2 formalism.Itisalsoworthnoting thatinthisformalism,oneuseseldredenitionsothattheinteractionhasnotermson whichthelight-conetimederivativesact. 2.1Superspace ThetheoriesexhibitingsupersymmetryarewelldescribedinSuperspace,anextension ofordinaryspacetimetothecoordinatespacewhichincludesanticommutingdimensions whosecoordinatesaregivenbytheGrassmannvariables m andtheircomplexconjugates f m ; n g = f m ; n g = f m ; n g =0 : {5 ThederivativesassociatedwiththeseGrassmannvariablesaredenedas @ @ m ; @ @ m ; {6 andsatisfy f @ @ m ; n g = m n ; f @ @ m ; n g = m n : {7 ThenumberoftheGrassmannvariablesisthesameasthatofsupersymmetryofthe theory.Forexample, m runform1to4forthe N =4SuperYang-Millstheoryandthe Grassmannvariables m transformasthe 4 and m asthe 4 under SU .Forthe N =8 11

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Supergravityinfourdimension,theytransformas m 8 and m 8 under SU m =1 ; 8. InSuperspace,onecandenefunctionsthatcontainthephysicaleldstogether withtheGrassmannvariables,whicharecalled superelds .Thesupersymmetric transformationsonsupereldsaremanifestbecausetheyarerepresentedasrotations andtranslationsinSuperspace.Inordertotthephysicaleldsintosuperelds,onemay constrainthesupereldswiththechiralderivatives d m )]TJ/F21 11.9552 Tf 39.051 8.088 Td [(@ @ m )]TJ/F21 11.9552 Tf 19.062 8.088 Td [(i p 2 m @ + ; d m @ @ m + i p 2 m @ + ; {8 whichsatisfycanonicalanticommutationrelations d m ; d n = )]TJ/F21 11.9552 Tf 11.956 0 Td [(i p 2 m n @ + : {9 Theformoftheconstraintsdependsonthenumberofsupersymmetriesandthedetailis discussedinthefollowingsubsections. 2.1.1 N =4 Superspace The N =4SuperspaceistheSuperspacethatdescribesthemaximallysupersymmetric theorieswith16=2 N physicaldegreesoffreedominvariousdimeions.Inthisthesis,we restrict N =4Superspacetotheoneinfourdimensions,andthusitdescribesthe maximallysupersymmetrictheorywith16physicaldegreesoffreedom: N =4Super Yang-Mills. Thephysicaldegreesoffreedomof N =4SuperYang-Millstheory,thetwogauge eldsdenedas A a = 1 p 2 A a 1 + iA a 2 ; A a = 1 p 2 A a 1 )]TJ/F21 11.9552 Tf 11.955 0 Td [(iA a 2 ; {10 thesixrealscalareldssatisfying 12

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C a mn = 1 2 mnpq C apq ; with 1234 =1 andnallyfourcomplexspin1 2 fermions am and a m m =1 ;:::; 4,arecapturedina complexsupereld[5] a y = 1 @ + A a y + i @ + m a m y + i p 2 am n C a mn y + p 2 6 m n p mnpq aq y + 1 12 m n p q mnpq @ + A a y {11 wherethelight-conecoordinate y aregivenby y = x; x;x + ;y )]TJ/F19 11.9552 Tf 10.405 -4.936 Td [( x )]TJ/F19 11.9552 Tf 9.741 -4.936 Td [()]TJ/F21 11.9552 Tf 21.054 8.088 Td [(i p 2 m m ; {12 where 1 @ + isinterpreted[13]as 1 @ + x )]TJ/F15 11.9552 Tf 7.084 -4.936 Td [(= 1 2 Z d )]TJ/F21 11.9552 Tf 11.955 0 Td [(x )]TJ/F15 11.9552 Tf 7.085 -4.936 Td [( ; with z being1for z> 0and )]TJ/F15 11.9552 Tf 9.298 0 Td [(1for z< 0.Thesupereldalsosatisesthechiral constraint d m a =0; d m a =0 ; {13 aswellasthe inside-out constraints d m d n a = 1 2 mnpq d p d q a ; {14 d m d n a = 1 2 mnpq d p d q a : {15 InthisSuperspace,theactionofSuperYang-Millsisthen 13

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Z d 4 x Z d 4 d 4 L ; {16 where L = )]TJ/F15 11.9552 Tf 10.812 3.154 Td [( a 2 @ +2 a + 4 g 3 f abc 1 @ + a b @ c + 1 @ + a b @ c )]TJ/F21 11.9552 Tf 9.299 0 Td [(g 2 f abc f ade 1 @ + b @ + c 1 @ + d @ + e + 1 2 b c d e ; {17 where g isthecoupling,and f abc arethestructurefunctionsoftheLiealgebra,and Grassmannintegrationisnormalizedsothat R d 4 1 2 3 4 =1. 2.1.2 N =8 Superspace The N =8Superspaceisthesuperspaceforthemaximallysupersymmetrictheories containingthe256=2 N physicaldegreesoffreedom,invariousdimensions,whose supermulipletcontainsthegraviton.Itdescribes N =1eleven-dimensionalSupergravity [19]and N =8Supergravityinfourdimensionsand N =16Supergravityinthree dimensions.Thesevarioustheoriesareobtainedthroughdimensionreductionfrom d =11to d =4,orto d =3.Or,onecanoxdizethefour-dimensionaltheorytothe eleven-dimensionalone[20].Inthisthesis,wefocuson N =8Supergravityinfour dimensions. Inordertodescribeallthephysicaldegreesoffreedomofsuchtheories,asdonein N =4Superspace,oneintroducestheconstrainedchiralsupereld anditscomplex conjugate ,thatsatisfythechiralconstraints d m =0 ; d m =0 ; where m =1 ;:::; 8,andarerelatedbythe inside-outconstraint = 1 4 @ +4 d 1 d 2 d 8 ': {18 14

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Asageneralexpansioninpowersof m ,thechiralsupereldisgivenby y = 1 @ + 2 h y + i m 1 @ + 2 m y + i mn 1 @ + B mn y )]TJ/F21 11.9552 Tf 11.291 0 Td [( mnp 1 @ + mnp y )]TJ/F21 11.9552 Tf 13.947 0 Td [( mnpq D mnpq y + i e mnp mnp y + i e mn @ + B mn y + e m @ + m y +4 e @ + 2 h y ; {19 where y isthesamelight-conecoordinatedenedinEq.2{12and a 1 a 2 :::a n = 1 n a 1 a 2 a n ; e a 1 a 2 :::a n = a 1 a 2 :::a n b 1 b 2 :::b )]TJ/F23 5.9776 Tf 5.756 0 Td [(n b 1 b 2 b )]TJ/F23 5.9776 Tf 5.756 0 Td [(n : Thecoecientscanbeviewedasphysicaleldsinfourdimensions:thesecoecients describeallphysicaldegreesoffreedomof N =8Supergravity:thespin-2graviton h and itscomplexconjugate h ,eightspin3 2 gravitinos m and m ,twentyeightvectorelds B mn and B mn ,ftysixgauginos mnp and mnp ,andseventyrealscalars D mnpq N =8SupergravityactioninthisSuperspacewasrstderivedin[6,24]toorder linearincoupling ,usingalgebraicconsistencyandsimpliedfurtherin[20].Itis remarkablysimple: S = )]TJ/F15 11.9552 Tf 13.42 8.088 Td [(1 64 Z d 4 x Z d 8 d 8 )]TJETq1 0 0 1 243.758 264.738 cm[]0 d 0 J 0.478 w 0 0 m 7.668 0 l SQBT/F21 11.9552 Tf 243.758 257.917 Td [(' 2 @ +4 )]TJ/F15 11.9552 Tf 13.948 0 Td [(2 1 @ +2 @' @' + c:c: + O 2 ; {20 where 2 2 @ @ )]TJ/F21 11.9552 Tf 14.575 0 Td [(@ + @ )]TJ/F15 11.9552 Tf 7.084 -4.338 Td [(.Recently,theorder2 interactiontermswasfoundin[14] usingthequadraticformoftheHamiltonian. 2.2SuperPoincareAlgebra Therearetwotypesofsymmetrygenerators:kinematicalanddynamicalones.While thekinematicalgeneratorsdonotinvolvethetimederivativesandarerealizedlinearlyon elds,thedynamicalgeneratorsdoinvolvetimederivativesand,ingeneral,arerealized 15

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non-linearly onelds.Inorderforthedynamicalgeneratorstoberealizedlinearly,one needstointroduceauxiliaryelds.Unfortunately,ndingsuchauxiliaryeldsisnot aneasytaskandthereisnoguidingprinciplefortheirexistence[6].In LC 2 formalism, thereforeallthedynamicalgeneratorsarenon-linearlyrealizedbecausethereareno auxiliaryelds. 2.2.1KinematicalSuperPoincareGenerators Inthissection,wereviewhowthekinematicalgeneratorsrepresentedonlight-cone, whichactlinearlyonthechiralsupereld = i!K'; where K isakinematicalSuperPoincaregeneratorand isthetransformationparameter. Allthetransformationsensurethechirality d m =0 : Therearethreekinematicalmomenta p + = )]TJ/F21 11.9552 Tf 11.955 0 Td [(i@ + ;p = )]TJ/F21 11.9552 Tf 11.956 0 Td [(i@; p = )]TJ/F21 11.9552 Tf 11.955 0 Td [(i @; {21 ThekinematicalLorentzgeneratorsincludesthetransverse SO rotation j = x @ )]TJ/F15 11.9552 Tf 12.68 0 Td [( x@ + 1 2 m @ @ m )]TJ/F15 11.9552 Tf 14.887 3.155 Td [( m @ @ m )]TJ/F21 11.9552 Tf 11.955 0 Td [(; {22 where = i 4 p 2 @ + d p d p )]TJ/F15 11.9552 Tf 14.021 3.155 Td [( d p d p ; {23 measuresthehelicityofthesupereld.Forexample, =+1forachiralsupereldand )]TJ/F15 11.9552 Tf 11.291 0 Td [(1forananti-chiralsupereld.Theactionofthistransverserotationgeneratoronthe 16

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chiralsupereldisthenexpressedas = i!j .Itisworthnotingthatunderthis SO rotation,thetranslationgenerator p and p transformdierentlyas [ j;p ]= p; [ j; p ]= )]TJ/F15 11.9552 Tf 12.941 0 Td [( p; andthusonecanalsoassignhelicity,inasensethatthehelicityindicatehowanoperator transformunder SO ,onthesegeneratorsfor p and )]TJ/F15 11.9552 Tf 9.299 0 Td [(1for p Theremainingkinematicalgeneratorsare j + = ix@ + ; j + = i x@ + ;j + )]TJ/F15 11.9552 Tf 14.307 -4.937 Td [(= ix )]TJ/F21 11.9552 Tf 9.077 -4.937 Td [(@ + )]TJ/F21 11.9552 Tf 14.081 8.088 Td [(i 2 p @ p + p @ p + i: {24 Wemustnotethatthe j + )]TJ/F15 11.9552 Tf 10.986 -4.339 Td [(generatoris,infact,adynamicalgeneratorsinceitcontains thelight-conetimederivative;however,itbecomeskinematicalifoneworksat x + =0 plane,andthushereitisassumedthatweworkat x + =0. Finally,thekinematicalsupersymmetrygenerators q m + and q + m 1 q m + = )]TJ/F21 11.9552 Tf 19.873 8.088 Td [(@ @ m + i p 2 m @ + ; q + m = @ @ m )]TJ/F21 11.9552 Tf 19.061 8.088 Td [(i p 2 m @ + ; {25 whichsatisfyanticommutationrelation f q m + ; q + n g = i p 2 m n @ + = )]TJ 13.948 10.473 Td [(p 2 m n p + ; {26 andanticommutewiththechiralderivatives f q m + ; d n g = f d m ; q + n g =0 : {27 1 Wemayneglect+"ifitisobvious,anduse q m and q m insteadof q m + and q + m 17

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2.2.2DynamicalPoincareGenerators Aswediscussedearlier,allthedynamicalgeneratorsarerealizednon-linearlyonthe chiralsupereld.Therefore,theiractiononthesupereldisnontrivial.However,theform ofthedynamicalgeneratorswithoutinteractionsareassimpleasthekinematicalones.We herelistthosewithoutinteractions.Thelight-coneHamiltonianisgivenby p )]TJ/F15 11.9552 Tf 14.307 -4.936 Td [(= )]TJ/F21 11.9552 Tf 11.955 0 Td [(i @ @ @ + : {28 ThedynamicalLorentzgeneratorsaretheboosts, j )]TJ/F15 11.9552 Tf 17.047 -4.936 Td [(= ix @ @ @ + )]TJ/F21 11.9552 Tf 15.857 0 Td [(ix )]TJ/F21 11.9552 Tf 9.078 -4.936 Td [(@ + i p @ p )]TJ/F21 11.9552 Tf 11.955 0 Td [( )]TJ/F15 11.9552 Tf 11.956 0 Td [(1 @ @ + ; j )]TJ/F15 11.9552 Tf 17.047 -4.936 Td [(= i x @ @ @ + )]TJ/F21 11.9552 Tf 15.857 0 Td [(ix )]TJ/F15 11.9552 Tf 10.542 -1.781 Td [( @ + i p @ p + )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 @ @ + : {29 Theyalsopreservechiralitybecause [ j )]TJ/F21 11.9552 Tf 9.077 -4.936 Td [(;d m ]= i 2 d m @ @ + ; [ j )]TJ/F21 11.9552 Tf 9.077 -4.936 Td [(; d m ]= i 2 d m @ @ + ; {30 andsatisfythePoincarealgebra.Inparticular [ j )]TJ/F21 11.9552 Tf 9.077 -4.937 Td [(; j + ]= )]TJ/F21 11.9552 Tf 11.955 0 Td [(ij + )]TJ/F19 11.9552 Tf 9.741 -4.937 Td [()]TJ/F21 11.9552 Tf 11.955 0 Td [(j; [ j )]TJ/F21 11.9552 Tf 9.078 -4.937 Td [(;j + )]TJ/F15 11.9552 Tf 9.077 -4.937 Td [(]= ij )]TJ/F21 11.9552 Tf 10.986 -4.937 Td [(: {31 Thedynamicalsupersymmetrygeneratorsarethenobtainedbyboostingthekinematical supersymmetries q m )]TJ/F19 11.9552 Tf 14.307 2.955 Td [( i [ j )]TJ/F21 11.9552 Tf 9.077 -4.936 Td [(;q m + ]= @ @ + q m + ; q )]TJ/F22 7.9701 Tf 7.998 0 Td [(m i [ j )]TJ/F21 11.9552 Tf 9.077 -4.936 Td [(; q + m ]= @ @ + q + m : {32 Noticethatthesearethesquare-roots"ofthelight-coneHamiltonian,inthesensethat f q m )]TJ/F21 11.9552 Tf 9.077 2.956 Td [(; q )]TJ/F22 7.9701 Tf 7.998 0 Td [(n g = i p 2 m n @ @ @ + = )]TJ 13.948 10.473 Td [(p 2 m n p )]TJ/F21 11.9552 Tf 10.987 -4.936 Td [(; {33 18

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implyingthatthelight-coneHamiltonianisaderivablequantityfromthedynamical supersymmetries. 19

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CHAPTER3 N =4SUPERYANG-MILLS 3.1 PSU ; 2 j 4 symmetry N =4SuperYang-Millstheoryisinvariantunderthesupergroup PSU ; 2 j 4,which consistsofthe30bosonicgeneratorsdescribingtheconformalgroup SO ; 2andan internalsymmetry SU SO andthe32fermionicgeneratorsrepresentingthefour supersymmetriesandthesuperconformalsymmetriesandtheircomplexconjugates. The PSU ; 2 j 4includesthesuper-Poincaresymmetry.Unlikethesuper-Poincare symmetry,itisa simple Liesuperalgebra.Therefore,itsucestoknowonebosonic kinematicalconformaltransformationandthentherestofthegeneratorsareobtained fromcommutationrelationswiththesuperPoincaregenerators.ReferAppendixAforthe detailformof PSU ; 2 j 4. 3.1.1KinematicalConformalGenerators Forcomputationalease,wecontinueto x + =0.Westartwiththesimplest kinematicalconformalgenerator K + K + =2 ix x@ + : {1 Thecommutationsrelation [ K + ;p )]TJ/F15 11.9552 Tf 9.078 -4.936 Td [(]= )]TJ/F15 11.9552 Tf 9.299 0 Td [(2 iD +2 ij + )]TJ/F21 11.9552 Tf 10.987 -4.936 Td [(; {2 leadstothedilatationgenerator, D = i x )]TJ/F21 11.9552 Tf 7.085 -4.936 Td [(@ + )]TJ/F21 11.9552 Tf 13.948 0 Td [(x @ )]TJ/F15 11.9552 Tf 14.672 0 Td [( x@ )]TJ/F15 11.9552 Tf 15.143 8.088 Td [(1 2 @ @ )]TJ/F15 11.9552 Tf 15.144 8.088 Td [(1 2 @ @ ; {3 whichsatises 20

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[ x;D ]= ix ;[ d m ;D ]= )]TJ/F21 11.9552 Tf 9.298 0 Td [(i 1 2 d m : {4 Therestofthekinematicalconformalgeneratorsarethenobtainedbyboosting K + K = i [ j )]TJ/F21 11.9552 Tf 7.085 -4.936 Td [(;K + ]=2 ix x )]TJ/F21 11.9552 Tf 7.085 -4.936 Td [(@ + )]TJ/F21 11.9552 Tf 11.955 0 Td [(x @ )]TJ/F21 11.9552 Tf 11.955 0 Td [( @ @ + ; {5 K = i [ j )]TJ/F21 11.9552 Tf 7.085 -4.936 Td [(;K + ]=2 i x x )]TJ/F21 11.9552 Tf 7.085 -4.936 Td [(@ + )]TJ/F15 11.9552 Tf 12.68 0 Td [( x@ )]TJ/F15 11.9552 Tf 12.894 3.155 Td [( @ @ )]TJ/F21 11.9552 Tf 11.955 0 Td [( ; {6 where isthehelicitycounteroftheprevioussection,whichpreservethechiralityofthe supereld,since [ K;d m ]=0;[ K;d m ]=2 i xd m : {7 Thenewfermionicgeneratorscalledconformalsupersymmetries"areobtainedina similarfashion, [ K + ;q m )]TJ/F15 11.9552 Tf 10.41 2.956 Td [(]= )]TJ 11.291 10.473 Td [(p 2 i p 2 xq m + = )]TJ 11.291 10.473 Td [(p 2 s m + ; {8 [ K + ; q )]TJ/F22 7.9701 Tf 7.998 0 Td [(n ]= p 2 )]TJ/F21 11.9552 Tf 11.291 0 Td [(i p 2 x q + n = p 2 s + n ; {9 whicharekinematical.Wenotethattheseconformalsupersymmetriesand K + ,infact, containdynamicaltermswhichareproportionalto x + ,butasweset x + tozerowithout lossofgenerality,andthustheybecomekinematical. 3.1.2DynamicalConformalGenerators Thedynamicalgeneratorsareallobtainedbyboostingthekinematicalones.The dynamicalconformalsupersymmetriesare 21

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s m )]TJ/F15 11.9552 Tf 17.951 2.956 Td [(= i [ j )]TJ/F21 11.9552 Tf 9.077 -4.936 Td [(;s m + ]= i p 2 x )]TJ/F21 11.9552 Tf 9.077 -4.936 Td [(@ + )]TJ/F21 11.9552 Tf 13.948 0 Td [(x @ )]TJ/F21 11.9552 Tf 13.947 0 Td [( @ @ + +1 1 @ + q m + ; {10 s )]TJ/F22 7.9701 Tf 7.998 0 Td [(n = i [ j )]TJ/F21 11.9552 Tf 9.077 -4.936 Td [(; s + n ]= )]TJ/F21 11.9552 Tf 13.947 0 Td [(i p 2 x )]TJ/F21 11.9552 Tf 9.077 -4.936 Td [(@ + )]TJ/F15 11.9552 Tf 14.672 0 Td [( x@ )]TJ/F15 11.9552 Tf 14.887 3.155 Td [( @ @ )]TJ/F21 11.9552 Tf 11.956 0 Td [( +1 1 @ + q + n ; {11 andthedynamicalconformalgenerator K )]TJ/F15 11.9552 Tf 10.986 -4.338 Td [(is K )]TJ/F15 11.9552 Tf 17.048 -4.936 Td [(= i [ j )]TJ/F21 11.9552 Tf 9.077 -4.936 Td [(;K ] =2 i x )]TJ/F21 11.9552 Tf 9.078 -4.936 Td [(@ + )]TJ/F15 11.9552 Tf 14.673 0 Td [( x@ )]TJ/F15 11.9552 Tf 14.887 3.154 Td [( @ @ )]TJ/F21 11.9552 Tf 11.955 0 Td [( +1 x )]TJ/F21 11.9552 Tf 9.077 -4.936 Td [(@ + )]TJ/F21 11.9552 Tf 13.947 0 Td [(x @ )]TJ/F21 11.9552 Tf 13.948 0 Td [( @ @ + +1 1 @ + : {12 Itisworthnotingthatasthedynamicalsupersymmetrygeneratorscanbethoughtofas squarerootoftheHamiltonian,theseconformalsupersymmetriessatisfyasimilarrelation p 2 m n K )]TJ/F15 11.9552 Tf 12.398 -4.936 Td [(= f s m )]TJ/F21 11.9552 Tf 9.982 2.955 Td [(; s )]TJ/F22 7.9701 Tf 7.998 0 Td [(n g : {13 Finally,theinternal SO generators, J m n areobtainedfromtheanticommutatorsof theconformalandnormalsupersymmetries f q m + ; s )]TJ/F22 7.9701 Tf 7.085 0 Td [(n g = )]TJ/F21 11.9552 Tf 9.299 0 Td [(i m n D + j + )]TJ/F15 11.9552 Tf 9.741 -4.936 Td [(+ ij +2 J m n ; {14 whichcommutatewiththeconformalgroup SO ; 2generators.Therefore,thesegenerate SO ; 2 SO SU ; 2 SU ; andtogetherwiththefermionicgenerators,theyformtheentire PSU ; 2 j 4algebra. 3.2Non-LinearRealization Asshownintheprevioussection,dynamicaltransformationsofthesuperPoincare algebraaregeneratedbythelight-coneHamiltonian p )]TJ/F15 11.9552 Tf 7.085 -4.338 Td [(,bytheboosts j )]TJ/F15 11.9552 Tf 10.987 -4.338 Td [(and j )]TJ/F15 11.9552 Tf 7.085 -4.338 Td [(,and 22

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bythesupersymmetries q )]TJ/F15 11.9552 Tf 10.987 1.793 Td [(and q )]TJ/F15 11.9552 Tf 7.085 1.793 Td [(.Thesetransformationsactnon-linearlyonthe supereldsintheinteractingtheory.They,ofcourse,preservethecommutationrelations oftheSuper-Poincarealgebra.Findingthesedynamicaltransformationstoallorders isimportantbecausetheformofthesedynamicaltransformationsdeterminesthefully interactingsuper-Poincareinvariantaction. WereviewanearlierworkalongthisdirectiondonebyBengtsson etal. [6]The authorsdevelopedasystematicprocedureforndingnon-lineardynamicaltransformations orderbyorderinthecouplingconstant g ,byexpandingthedynamicaltransformationsof theeldsasapowerseriesin g p )]TJ/F21 11.9552 Tf 6.753 0.796 Td [( = 0 p )]TJ/F21 11.9552 Tf 6.753 0.796 Td [( + g p )]TJ/F21 11.9552 Tf 6.753 1.614 Td [( + g 2 p )]TJ/F21 11.9552 Tf 6.753 1.614 Td [( + ; q )]TJ/F21 11.9552 Tf 6.753 4.029 Td [( = 0 q )]TJ/F21 11.9552 Tf 6.752 4.029 Td [( + g q )]TJ/F15 11.9552 Tf 9.41 4.029 Td [(+ g 2 q )]TJ/F21 11.9552 Tf 6.753 4.029 Td [( + ; j )]TJ/F21 11.9552 Tf 6.753 0.795 Td [( = 0 j )]TJ/F21 11.9552 Tf 6.752 0.795 Td [( + g j )]TJ/F21 11.9552 Tf 6.752 1.613 Td [( + g 2 j )]TJ/F21 11.9552 Tf 6.752 1.613 Td [( + ; {15 wherethesuperscriptdenotestheorderofthevariation.Sincethekinematicaltransformations havenoorder g corrections,muchinformationisgainedfromthecommutationrelations. Forinstance, [ j ; p )]TJ/F15 11.9552 Tf 8.745 0.795 Td [(] =0 ; [ j + )]TJ/F21 11.9552 Tf 8.745 0.795 Td [(; p )]TJ/F15 11.9552 Tf 8.745 0.795 Td [(] = i p )]TJ/F21 11.9552 Tf 6.752 0.795 Td [(; {16 determinethehelicityandthenumberof @ + inthevariations. TheystartedbymakingeducatedguessesfortheHamiltonianvariationtoorder g g p )]TJ/F21 11.9552 Tf 8.745 1.614 Td [( = )]TJ/F21 11.9552 Tf 13.947 0 Td [(ig@ + [ @ a @ + @ b @ + ] ; {17 aswellasfortheboosts g j )]TJ/F21 11.9552 Tf 8.744 1.614 Td [( = )]TJ/F21 11.9552 Tf 13.947 0 Td [(x g p )]TJ/F21 11.9552 Tf 8.745 1.614 Td [( + g spin ; {18 23

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wherethelatteristhespin"change,and ;; denotethepowersof @ + and a;b are thoseofthetransversederivatives,tobexedbythealgebra.Closureofcommutation relation3{16yields a + b =1 ; + + =0 : Therequirementthat [ j )]TJ/F21 11.9552 Tf 8.745 1.116 Td [(; p )]TJ/F15 11.9552 Tf 8.745 0.795 Td [(] =0 ; {19 holdsorderbyorderin g ,afterlengthycalculations,leadstotheHamiltonianvariation g p )]TJ/F21 11.9552 Tf 8.745 1.614 Td [( a = )]TJ/F21 11.9552 Tf 13.948 0 Td [(igf abc 1 @ + @ b @ + c ; {20 andtheboost g j )]TJ/F21 11.9552 Tf 8.745 1.613 Td [( a = x g p )]TJ/F21 11.9552 Tf 8.745 1.613 Td [( a + igf abc 1 @ + @ @ )]TJ/F15 11.9552 Tf 13.948 0 Td [(1 b @ + c : {21 Finally,byboostingthekinematicalsupersymmetries q )]TJ/F21 11.9552 Tf 8.745 4.029 Td [( = i [ q + ; j )]TJ/F15 11.9552 Tf 6.753 0.796 Td [(] ; q )]TJ/F21 11.9552 Tf 8.745 4.029 Td [( = i [ q + ; j )]TJ/F15 11.9552 Tf 6.753 1.116 Td [(] ; {22 theyobtainedthedynamicalones g q )]TJ/F21 11.9552 Tf 8.745 3.441 Td [( a = )]TJ/F21 11.9552 Tf 11.291 0 Td [(gf abc 1 @ + @ @ b @ + c ; {23 g q )]TJ/F21 11.9552 Tf 8.745 4.03 Td [( a = gf abc d 4 48 @ +3 @ @ b @ + c ; {24 where d 4 mnpq d m d n d p d q .Ofcourse,thesedynamicaltransformationsatisfyallthe commutationsofthesuperPoincarealgebra.ThisishowBengtsson etal determinedthese non-lineartransformationstorstorderin g 24

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Theauthorsthenusedthevariationstogeneratethecubicinteractionvertexfrom thekinetictwo-pointfunction,butdidnotextendtheirmethodtohigherorderin g due toalgebraiccomplications.Asaresult,theirprocedurefellshortofshowingthat f abc isa three-formwhichsatisestheJacobiidentity,andofderivingthefour-pointfunctionusing algebraicmeans.Duetoalgebraiccomplications,theydidnotproceedbeyondtherst orderincoupling. Wenotethatthismethodisverygeneric,anddoesnotmakeuseofsupersymmetry. Yetinsupersymmetrictheories,theHamiltonianisaderivedconcept.Furthermore, Bengtsson etal hadtomakeinspiredguessesfor both theHamiltonianandboosts separately. Aswediscussedearlier, N =SuperYang-Millshasalargerinvariancegroupthan thesuperPoincaregroup,namely PSU ; 2 j 4.Sincethisisa simple Liesuperalgebra,it isenoughtoknowonebosonickinematicalconformaltransformationandtheformofthe non-linearlyrealizedsupersymmetrytoreconstructthewholealgebrafortheinteracting case. Forthatreason,weneedtodetermineonlyonedynamicaltransformationtoorder g andhigher.Wechoosethedynamicalsupersymmetrytransformationonthesupereld q a .Takingintoaccountofchirality d m q a =0,dimensionalanalysis,proper helicityandsimplecommutators,onecanstartwiththefollowingform g q )]TJ/F21 11.9552 Tf 8.745 3.44 Td [( a = )]TJ/F21 11.9552 Tf 11.956 0 Td [(gf abc 1 @ + +1 d@ + b @ + +1 c ; {25 where f abc = )]TJ/F21 11.9552 Tf 12.324 0 Td [(f acb and d m and q + m areinterchangeablebecauseoftheantisymmetry ofthestructurefunction,and isafreeparameter.Theconjugatesupersymmetry transformationcanbeobtainedbytakingcomplexconjugateandthenusingthe inside-out"constraint g q )]TJ/F21 11.9552 Tf 8.745 4.03 Td [( = )]TJ/F21 11.9552 Tf 11.955 0 Td [(gf abc d 4 48 @ + +3 d@ + b @ + +1 c : {26 25

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Wenotethatthedynamicalsupersymmetrytransformations,infact,donotextend beyondorder g ,whichisdiscussedin[11].Thecommutator f q m )]TJ/F21 11.9552 Tf 9.813 5.464 Td [(; q )]TJ/F23 5.9776 Tf 7.029 0 Td [(n g g a = f g q m )]TJ/F21 11.9552 Tf 9.813 5.622 Td [(; 0 q )]TJ/F23 5.9776 Tf 7.029 0 Td [(n g a + f 0 q m )]TJ/F21 11.9552 Tf 9.812 5.464 Td [(; g q )]TJ/F23 5.9776 Tf 7.029 0 Td [(n g a = )]TJ 11.956 10.473 Td [(p 2 m n g p )]TJ/F21 11.9552 Tf 8.745 1.614 Td [( a ; {27 yieldstheHamiltonianvariationtoorder g .Useofchiralityandtherelation q + m = d m )]TJ/F21 11.9552 Tf 13.948 0 Td [(i p 2 m @ + ; leadsto p )]TJ/F21 11.9552 Tf 8.745 0.796 Td [( a = )]TJ/F21 11.9552 Tf 11.955 0 Td [(i @ @ @ + a )]TJ/F21 11.9552 Tf 13.948 0 Td [(igf abc 1 @ + +1 @@ + b @ + +1 c + d 4 48 @ + +3 @@ + b @ + +1 c + O g 2 ; {28 where isstillnotxedyet. Inordertodeterminetheparameter ,wenowusethefactthat PSU ; 2 j 4issimple Liesuper-algebra.Wecan derive theformofthenon-linearboostsusingthecommutator ofthekinematicalconformalgenerator K withtheHamiltonianvariation3{28 [ K ; p )]TJ/F15 11.9552 Tf 8.745 0.796 Td [(] a = )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 i j )]TJ/F21 11.9552 Tf 8.745 1.117 Td [( a ; {29 whichyields g j )]TJ/F21 11.9552 Tf 8.745 1.934 Td [( a = )]TJ/F15 11.9552 Tf 14.673 0 Td [( x g p )]TJ/F21 11.9552 Tf 8.745 1.613 Td [( a )]TJ/F21 11.9552 Tf 15.189 8.088 Td [(ig d 4 f abc 48 @ + +3 @ @ + )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 @ + b @ + +1 c ; {30 neglectingtheorder g 2 contributions.Noticethatthedynamicaltransformations, g q g p )]TJ/F21 11.9552 Tf 8.745 1.614 Td [( and g j )]TJ/F21 11.9552 Tf 8.745 1.614 Td [( ,areexpressedasafunctionof ,andthusdetermining yieldstheformof thesedynamicaltransformations. Tothateect,wenowverifythat 26

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[ j )]TJ/F21 11.9552 Tf 8.745 1.116 Td [(; p )]TJ/F15 11.9552 Tf 8.745 0.796 Td [(] a =0 : {31 Toorder g ,thiscommutatoryieldstermsproportionalto [ j )]TJ/F21 11.9552 Tf 8.745 1.117 Td [(; p )]TJ/F15 11.9552 Tf 8.745 0.796 Td [(] g a = )]TJ/F21 11.9552 Tf 12.668 8.087 Td [(gf abc @ + +2 @ + @ + +1 @ 2 b @ + +1 c )]TJ -163.114 -23.908 Td [()]TJ/F15 11.9552 Tf 13.201 0 Td [(2 @ @@ + b @ + +1 c + ; {32 whichrequires =0.Therefore,byusingthesuperconformalalgebraandchirality wehavearrivedattheuniquerealizationofthe PSU ; 2 j 4algebra:thedynamical supersymmetry g q )]TJ/F21 11.9552 Tf 8.745 3.441 Td [( a = )]TJ/F21 11.9552 Tf 11.956 0 Td [(gf abc 1 @ + )]TJ/F15 11.9552 Tf 9.538 -6.529 Td [( d b @ + c ; {33 theHamiltonianvariation g p )]TJ/F21 11.9552 Tf 8.744 1.614 Td [( a = )]TJ/F21 11.9552 Tf 13.947 0 Td [(igf abc 1 @ + @ b @ + c + d 4 48 @ +3 @ b @ + c + O g 2 ; {34 andnallytheboost g j )]TJ/F21 11.9552 Tf 8.745 1.935 Td [( a = )]TJ/F15 11.9552 Tf 14.672 0 Td [( x g p )]TJ/F21 11.9552 Tf 8.744 1.614 Td [( a )]TJ/F21 11.9552 Tf 15.143 8.088 Td [(ig d 4 f abc 48 @ +3 @ @ )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 b @ + c + O g 2 : {35 Theremainingtransformationscanbeobtainedbyusingthe PSU ; 2 j 4algebra.For example,thesuperconformaltransformationsaregivenas g s )]TJ/F21 11.9552 Tf 6.752 3.441 Td [( a = 1 p 2 g q )]TJ/F21 11.9552 Tf 6.753 3.441 Td [(; K a = p 2 i x g q )]TJ/F21 11.9552 Tf 6.752 3.441 Td [( a = p 2 i xgf abc 1 @ + @ @ b @ + c : {36 27

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Weremarkthatthetotalantisymmetryofthe f abc andtheJacobiidentitiesareobtained byrequiringclosureofthealgebra.Inparticular,wecalculatetheconformalgenerator K )]TJ/F15 11.9552 Tf -459.283 -28.246 Td [(intwoindependentways,fromthecommutator K )]TJ/F15 11.9552 Tf 13.975 0.796 Td [(= i [ j )]TJ/F21 11.9552 Tf 8.745 1.117 Td [(; K ] ; {37 orfromtheanticommutator K )]TJ/F15 11.9552 Tf 13.976 0.795 Td [(= 1 4 p 2 f s m )]TJ/F21 11.9552 Tf 9.527 5.464 Td [(; s )]TJ/F23 5.9776 Tf 7.029 0 Td [(m g : {38 MatchingthesetwoequationsyieldstheJacobiidentityforthestructureconstants.In thisgauge,space-timeandinternalsymmetriesarelinkedtogether;conformalinvariance requiresthegaugesymmetryofYang-Millstheories. 3.3HamiltonianasaQuadraticForm Inthissection,weshowthisalgebraicformulationenablesustowritetheHamiltonian of N =4SuperYang-Millsasaquadraticform ; 2 i Z d 4 xd 4 d 4 1 @ + ; {39 where and arechiralsuperelds. TheHamiltonianfor N =4SuperYang-Mills,canbeeasilyobtainedfromEq.2{17 H = Z d 4 xd 4 d 4 a 2 @ @ @ +2 a )]TJ/F15 11.9552 Tf 13.15 8.088 Td [(4 3 gf abc 1 @ + a b @ c + 1 @ + a b @ c + g 2 f abc f ade 1 @ + b @ + c 1 @ + d @ + e + 1 2 b c d e ; {40 H 0 + H g + H g 2 : {41 ToseethatthefreeHamiltonian 28

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H 0 = Z d 4 xd 4 d 4 a 2 @ @ @ +2 a ; {42 weintroduceafermionicsupereld W a m = @ @ + q + m a ; {43 whichisthe free dynamicalsupersymmetryvariationofthesupereld.Letusconsiderthe quadraticformwith W a m 1 2 p 2 W 0 m ; W 0 m = i p 2 Z d 4 xd 4 d 4 @ @ + q m + a @ @ +2 q + m a : {44 Werewritethisastwoterms i 2 p 2 Z d 4 xd 4 d 4 @ @ + q m + a @ @ +2 q + m a + @ @ + q m + a @ @ +2 q + m a : Usingtheinside-outpropertyofthesuperelds2{14,weget i 2 p 2 Z d 4 xd 4 d 4 @ @ + q m + a @ @ +2 q + m a + @ @ +3 q m + a @ q + m a : Aftersomeintegrationbyparts,thiscanbewrittenas )]TJ/F21 11.9552 Tf 21.324 8.088 Td [(i 2 p 2 Z d 4 xd 4 d 4 a @ @ @ +3 f q m + ; q + m g a ; whichisthesameasEq.3{42, H 0 ,since f q m + ; q + m g =4 p 2 i@ + WegeneralizethisquadraticformtothethefullyinteractingHamiltonianby completingthefermionicsupereld W a toorder g W a = @ @ + q + a )]TJ/F21 11.9552 Tf 11.955 0 Td [(gf abc 1 @ + d b @ + c ; {45 29

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whichistheclassicaldynamicalsupersymmetryvariation.TheHamiltonianisthen H = 1 2 p 2 W a ; W a : {46 Wenotethatthethree-pointfunctioncanbeexpressedas 1 2 p 2 W a ; W a g = i p 2 gf abc Z @ @ +2 a q + m 1 @ + d m b @ + c = 4 3 gf abc Z 1 @ + a b @ c ; {47 wherethelastequalityinvolvesalengthycalculationandthedetailislefttoAppendixD. Similarly,thefour-pointinteractioncanberewrittenas 1 2 p 2 W a ; W a g 2 = Z i p 2 g 2 f abc f ade 1 @ + d m b @ + c 1 @ +2 d m d @ + e {48 = Z g 2 f abc f ade n 1 @ + b @ + c 1 @ + d @ + e + 1 2 b c d e o alsoprovedinthesameAppendixD. 30

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CHAPTER4 N =8SUPERGRAVITY 4.1 E 7 OnshellDualitySymmetry 4.1.1Covariant N =8 Supergravity N =8Supergravitycontainsagraviton h andits8gravitinos i interactingwith mattercomposedof28vectors A [ ij ] ,56spinors [ ijk ] ,and70scalars C [ ijkl ] .Thebrackets indicateantisymmetrization.Thetheoryisinvariantunderaglobal SO symmetry,with i;j;k;l =1 ; 2 ;:::; 8. TheLagrangianwaswrittendownuptoorder 2 longagobydeWitandFreedman [16],inaformwherethe SO symmetryismanifest.CremmerandJuliafoundthatit containedamuchlargerhiddenglobalsymmetry, E 7 actingonthescalarsandtheeld strengths.Thissymmetryisnon-linearlyrealizedonthescalarsandisadualitysymmetry thatinterchangesthestructuralBianchiequationswiththeequationsofmotion.Itis asymmetryofthetheoryonlyafterinvokingtheequationsofmotion,althoughitisa symmetryofthescalarpartoftheLagrangian.WebeginwithareviewofthedeWitand Nicolai'sanalysis[17,18],keepingonlythevectorandthescalarpartsoftheLagrangian. L dWF = L S + L V + L others ; wherethescalarLagrangianisgivenby L S = )]TJ/F15 11.9552 Tf 15.413 8.088 Td [(1 96 @ a ijkl 2 + @ b ijkl 2 )]TJ/F21 11.9552 Tf 15.531 8.087 Td [( 2 384 a + b ijkl a )]TJ/F21 11.9552 Tf 11.955 0 Td [(b klmn @ a + b mnpq @ a )]TJ/F21 11.9552 Tf 11.955 0 Td [(b pqij + O 3 ; andthe28vectorsandtheirinteractionwithscalarsare 31

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L V = )]TJ/F15 11.9552 Tf 10.494 8.088 Td [(1 8 F ij F ij )]TJ 13.151 17.977 Td [(p 2 16 a ijkl F ij F kl + b ijkl F ij e F kl {1 )]TJ/F15 11.9552 Tf 15.413 8.088 Td [(1 32 2 )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [(a ijkl a ijmn )]TJ/F21 11.9552 Tf 11.955 0 Td [(b ijkl b ijmn F kl F mn )]TJ/F15 11.9552 Tf 16.077 8.088 Td [(1 16 2 a ijkl b ijmn F kl e F mn + O 3 ; wherethedualeldsaredenedas e F ij = 1 2 F ij : Thevectoreldsinthetheoryappearexclusivelyintermsoftheeldstrengths F ij which transformastherealantisymmetric 28 of SO .The a ijkl and b ijkl denoterealscalar eldstransformingtwodierent 35 of SO whichsatisfydualities a ijkl = 1 4! ijklmnpq a mnpq ; b ijkl = )]TJ/F15 11.9552 Tf 14.777 8.088 Td [(1 4! ijklmnpq b mnpq : Itisconvenienttointroducethecomplexeld C ijkl C ijkl 1 p 2 )]TJ/F21 11.9552 Tf 5.48 -9.683 Td [(a ijkl + ib ijkl ; {2 anditscomplexconjugate C ijkl ,arerelatedbythedualitycondition C ijkl = 1 4! ijklmnpq C mnpq : {3 Intermsof C ijkl ,thescalarLagrangianreads L S = )]TJ/F15 11.9552 Tf 18.069 8.088 Td [(1 48 @ C ijkl @ C ijkl + 2 2 C ijkl C klmn @ C mnpq @ C pqij + O 3 ; {4 whichcanbeunderstoodasthenon-linearrealizationReferAppendixC.Nowwediscuss thedualitysymmetryontheseelds. 32

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4.1.2Dualitysymmetries FollowingdeWitandNicolai[17,18],wewritethevectorLagrangianintheform L V = )]TJ/F15 11.9552 Tf 13.151 8.087 Td [(1 8 F ij G ij ; {5 where G ij )]TJ/F15 11.9552 Tf 33.126 0 Td [(4 L V F ij ; {6 sincetheLagrangianisquadraticintheeldstrengths.Wethenintroducethe complex eldstrengths F ij = 1 2 F ij + i e F ij ; F ij = 1 2 F ij )]TJ/F21 11.9552 Tf 11.955 0 Td [(i e F ij ; {7 and G ij = 1 2 G ij + i e G ij ; G ij = 1 2 G ij )]TJ/F21 11.9552 Tf 11.955 0 Td [(i e G ij ; {8 withthedualdenedintheusualway, e G ij = 1 2 G ij : {9 Itiseasytoseethatthecomplexeldstrengthsareself-dual F ij = i 2 F ij ; G ij = i 2 G ij : {10 ThevectorLagrangiancannowbewrittenas L V = L + + L )]TJ/F21 11.9552 Tf 10.987 -4.936 Td [(; L )]TJ/F15 11.9552 Tf 14.307 -4.936 Td [(= L + ; {11 where 33

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L + = )]TJ/F15 11.9552 Tf 10.494 8.088 Td [(1 8 F ij G ij : {12 Theequationsofmotionaregivenby @ G ij = @ G ij + G ij =0 ; {13 whiletheBianchiidentitiesread @ e F ij = @ F ij )]TJETq1 0 0 1 331.947 520.213 cm[]0 d 0 J 0.478 w 0 0 m 7.306 0 l SQBT/F32 11.9552 Tf 331.947 510.304 Td [(F ij =0 : {14 Dualitytransformations,electric-magneticduality,exchangeequationsofmotionfor Bianchiidentities.Therearetwowaysofgeneratingsuchtransformations,byenlarging thesymmetryto SU orto E 7 .Weexamineeachcaseseparately. 4.1.2.1 SU Duality Theequationswrittensofararemanifestlycovariantwithrespecttoaglobal SO Wecangeneralizethesetransformationstothelarger SU ,byactingoncomplex eight-componentvectors z i as z i = )]TJ/F21 11.9552 Tf 5.48 -9.684 Td [(R ij + iS ij z j ; {15 where R ij arethe28 real antisymmetricrotationtensorswhichgenerate SO ,and S ij are35 real symmetrictracelessmatricesinthecoset SU =SO .Complexconjugation yields z i = )]TJ/F21 11.9552 Tf 5.48 -9.684 Td [(R ij )]TJ/F21 11.9552 Tf 11.955 0 Td [(iS ij z j : {16 Wecanviewthesetobedualities,byassumingthatunder SU ,thecombinations G ij + F ij transformsasthecomplex 28 ,thatis 34

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)]TJ/F32 11.9552 Tf 5.48 -9.684 Td [(G ij + F ij = )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [(R ik + iS ik )]TJ/F32 11.9552 Tf 12.952 -9.684 Td [(G kj + F kj )]TJ/F15 11.9552 Tf 11.956 0 Td [( i $ j ; {17 whiletheothercombinations G ij )]TJ/F32 11.9552 Tf 11.956 0 Td [(F ij transformsasthecomplexconjugate 28 )]TJ/F32 11.9552 Tf 5.479 -9.684 Td [(G ij )]TJ/F32 11.9552 Tf 11.955 0 Td [(F ij = )]TJ/F21 11.9552 Tf 5.48 -9.684 Td [(R ik )]TJ/F21 11.9552 Tf 11.955 0 Td [(iS ik )]TJ/F32 11.9552 Tf 12.952 -9.684 Td [(G kj )]TJ/F32 11.9552 Tf 11.956 0 Td [(F kj )]TJ/F15 11.9552 Tf 11.955 0 Td [( i $ j : {18 The SU =SO cosettransformations prime onthecomplexeldstrengths prime F ij = iS ik G kj )]TJ/F15 11.9552 Tf 11.955 0 Td [( i $ j ; prime G ij = iS ik F kj )]TJ/F15 11.9552 Tf 11.956 0 Td [( i $ j ; {19 arethedualitytransformationswhichmaptheequationsofmotionintotheBianchi identitiesand vice-versa prime n @ G ij + G ij o = iS ik @ F kj )]TJETq1 0 0 1 362.997 406.151 cm[]0 d 0 J 0.478 w 0 0 m 7.306 0 l SQBT/F32 11.9552 Tf 362.997 396.242 Td [(F kj )]TJ/F15 11.9552 Tf 15.857 0 Td [( i $ j : {20 The SU =SO transformationsareclearlysymmetriesoftheequationsofmotionand oftheBianchiidentities,butnotoftheLagrangian. FromtheLagrangian L + ,expressedintermsofthecomplexeldstrengthstoorder 2 L + = )]TJ/F15 11.9552 Tf 13.15 8.087 Td [(1 8 F ij F ij )]TJ/F21 11.9552 Tf 15.143 8.087 Td [( 8 C ijkl F ij F kl )]TJ/F21 11.9552 Tf 15.261 8.087 Td [( 2 16 C ijkl C klmn F ij F mn ; weobtain G ij = F ij + C ijkl F kl + 2 2 C ijkl C klmn F mn : {21 ConsistencyofthecosetvariationofthisexpressionwiththetwovariationsofEq.4{19 requiresthatthescalareldtransformlinearlyunderthefull SU ,thatis 35

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C ijkl = )]TJ/F21 11.9552 Tf 11.291 0 Td [(iS im C mjkl )]TJ/F15 11.9552 Tf 13.948 0 Td [( i $ j )]TJ/F15 11.9552 Tf 13.948 0 Td [( i $ k )]TJ/F15 11.9552 Tf 13.948 0 Td [( i $ l : {22 Thisisanexactequationwithnoorder corrections.ItfollowsthatthescalarLagrangian 4{4is SU invariant. Ontheotherhand,thecomplexeldstrengthshavemorecomplicatednon-linear cosettransformation F ij = iS im F mj + C mjkl F kl + 2 2 C ijkl C klmn F mn )]TJ/F15 11.9552 Tf 15.857 0 Td [( i $ j ; {23 whichextendstoallordersin .Thetermsontheright-hand-sidetransformdierently orderbyorderin : F mj 28 ,while C mjkl F kl 28 ,andtheorder 2 termhasmore complicatedcosettransformations.Yet,onecancheckthatthecommutatoroftwosuch variationscloseson SO transformation,asrequired. Theextensionto SU dualityontheeldstrengthsismeaningfulonlyinthe interactingcasewhen 6 =0,since G ij )]TJ/F32 11.9552 Tf 11.955 0 Td [(F ij = O 4.1.2.2 E 7 Duality AsshownbyCremmerandJulia,thedualitysymmetriescanbeextendedtothe non-compact E 7 .Toseethis,assemblethecomplexeldstrengthsinonecolumnvector with56complexentries Z = 0 B @ G ij + F ij G ij )]TJ/F32 11.9552 Tf 11.955 0 Td [(F ij 1 C A 0 B @ X ab Y ab 1 C A ; {24 where a;b are SU indices,withupperlowerantisymmetricindicesfor 28 28 .Itstwo components 36

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X ab =2 F ij + C ijkl F kl + 2 2 C ijkl C klmn F mn + ; {25 Y ab = C ijkl F kl + 2 2 C ijkl C klmn F mn + ; {26 are notindependent ,butrelatedby Y ab )]TJ/F21 11.9552 Tf 15.143 8.088 Td [( 2 C abcd X cd + O 2 =0 : {27 Theequationsofmotion4{13andBianchiidentities4{14canbewrittenintermsof Z @ Z + e Z =0 ; {28 where e Z 0 B @ 0 1 1 0 1 C A Z = 0 B @ G ij )]TJETq1 0 0 1 337.015 393.336 cm[]0 d 0 J 0.478 w 0 0 m 7.306 0 l SQBT/F32 11.9552 Tf 337.015 383.427 Td [(F ij G ij + F ij 1 C A = 0 B @ Y ab X ab 1 C A : TheuppercomponentofEq.4{28isthesumoftheequationsofmotionandtheBianchi identities,andthelowercomponentthedierence. Transformationswhichactthesamewayonboth Z and e Z leaveequationsof motionandBianchiidentitiesinvariant.Considerthen Z !E Z ; e Z !E e Z ; intermsofmatrices E ,whichsatisfyin 28blockform E = 0 B @ 0 1 1 0 1 C A E 0 B @ 0 1 1 0 1 C A : {29 Thisrestrictsthematrixtobeoftheform 37

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E = 0 B @ ab cd abcd abcd ab cd 1 C A ; wheretheindicesareraisedandloweredbycomplexconjugation.Wespecializeto innitesimaltransformations with ab cd =0,oftheform X ab = abcd Y cd ; {30 Y ab = abcd X cd ; {31 whichtransform 28 into 28 andviceversa.Itcanbecheckedthatsuchtransformations withreal abcd leavebothequationsofmotionandBianchiidentitiesinvariant,whilethose withpureimaginary abcd aredualitytransformationswhichinterchangethetwo. ThetransformationsofinterestarethosewhichrespecttheconstraintEq.4{27 betweentheupperandlowercomponentsof Z Y ab = 2 )]TJETq1 0 0 1 287.022 361.02 cm[]0 d 0 J 0.478 w 0 0 m 9.234 0 l SQBT/F21 11.9552 Tf 287.022 351.177 Td [(C abcd X ab + O 2 : Explicitly, abcd X cd = 2 C abcd X cd + 2 C abef efmn 2 C mncd X cd + O 2 ; whichrequirethatthescalarstransformnon-linearlyas C abcd = 2 abcd )]TJ/F21 11.9552 Tf 13.151 8.088 Td [( 2 C ef [ ab C cd ] mn efmn + O 2 ; {32 wheretheindicesinsidethesquarebracketsareantisymmetrized.Bycomplexconjugation, oneobtains 38

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C abcd = 2 abcd )]TJ/F21 11.9552 Tf 13.151 8.088 Td [( 2 C ef [ ab C cd ] mn efmn + O 2 : {33 Sincethescalarssatisfytheselfdualitycondition4{3,somust abcd abcd = 1 4! abcdefgh efgh ; {34 restricts abcd to70realparameters.ItalsomeansthattheextraterminEq.4{32is self-dual.RepeateduseofEq.4{31yieldsthecommutator [ 1 ; 2 ] X ab = abef efcd )]TJ/F15 11.9552 Tf 13.947 0 Td [( abef efcd X cd : UsingEq.4{34,theright-hand-sidebecomes [ 1 ; 2 ] X ab = 1 4! 2 mnpq xyzw h 2 abxyzw cdmnpq )]TJ/F21 11.9552 Tf 13.947 0 Td [( abef mnpq xyzw efcd i X cd ; ab cd X cd Itiseasytoseethattheright-hand-sideofthecommutatorvanisheswhenthereareno contractionsbetweentheupperindices a b ,andthelowerones c;d .Thisallowsusto express ab cd intheform ab cd = a c b d )]TJ/F15 11.9552 Tf 13.947 0 Td [( b c a d )]TJ/F15 11.9552 Tf 13.947 0 Td [( a d b c + b d a c ; where a b =6 ac bc = 1 4 4! mnpq xyzw )]TJ/F15 11.9552 Tf 11.291 0 Td [(6 axyzw bmnpq + acef mnpq xyzw cefb ; and a a =0 : {35 39

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Thesamereasoningappliedto Y cd yieldstheconjugateequations ab cd = a c b d )]TJ/F15 11.9552 Tf 13.947 0 Td [( b c a d )]TJ/F15 11.9552 Tf 13.948 0 Td [( a d b c + b d a c ; and a b =6 ac bc = 1 4 4! mnpq xyzw )]TJ/F15 11.9552 Tf 11.291 0 Td [(6 bxyzw amnpq + bcef mnpq xyzw cefa : Furtheralgebrainvolvingtheselfdualcondition4{34yields a b = )]TJ/F15 11.9552 Tf 22.02 8.087 Td [(1 4 4! mnpq xyzw )]TJ/F15 11.9552 Tf 11.291 0 Td [(6 bxyzw amnpq + bcef mnpq xyzw cefa = )]TJ/F15 11.9552 Tf 11.955 0 Td [( b a : {36 Itfollowsthatthe a b arerestrictedtosatisfybothEq.4{35andEq.4{36 a b = )]TJ/F15 11.9552 Tf 11.955 0 Td [( b a ; a a =0 ; whichimpliesthat a b canberepresentedbyatracelessanti-hermitianmatrix a b y = b a = )]TJ/F15 11.9552 Tf 11.956 0 Td [( a b ; whichparametrizesthe63realparametersof SU Withthe70-parameterscosettransformations,theenlargedfulldualitytransformations arethen X ab =2 a e X eb +2 b e X ae + abcd Y cd ; Y ab =2 a e Y eb +2 b e Y ae + abcd X cd ; {37 resultingina133-parametergroup a b and70 abcd .Sincethecosettransformations generatedby abcd arenotunitary,wearedealingwith E 7 ,thenon-compactformwith 40

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SU asmaximalcompactsubgroup,and7= )]TJ/F15 11.9552 Tf 12.611 0 Td [(63istheexcessofnon-compact generators. Tondthe E 7 transformationsof F ij ,werstsubstitute X ab and Y ab inthe E 7 =SU transformation4{30withEq.4{25andEq.4{26 2 F ij + C ijkl F kl + 2 2 C ijkl C klmn F mn + O 3 = ijkl C klmn F mn + 2 2 C klmn C mnpq F pq + O 3 : {38 Then,substitutingthescalartransformations4{32anditerating F ij ,wendthe followingnon-lineartransformationofthecomplexeldstrengths F ij = )]TJETq1 0 0 1 226.002 437.104 cm[]0 d 0 J 0.478 w 0 0 m 7.804 0 l SQBT/F15 11.9552 Tf 226.002 427.261 Td [( ijkl F kl + 2 ijkl )]TJETq1 0 0 1 344.157 437.104 cm[]0 d 0 J 0.478 w 0 0 m 7.804 0 l SQBT/F15 11.9552 Tf 344.157 427.261 Td [( ijkl C klmn F mn + 2 4 ijkl )]TJETq1 0 0 1 285.727 407.355 cm[]0 d 0 J 0.478 w 0 0 m 7.804 0 l SQBT/F15 11.9552 Tf 285.727 397.512 Td [( ijkl C klmn C mnpq F pq + O 3 : {39 Aswementionedbefore,thisequationismeaningfulonlywhen 6 =0.Whilethescalar partofLagrangian L S is E 7 -invariant,thevectorLagrangian L V isnot.Invarianceis attainedonlyafterinvokingtheequationsofmotion. Wenotethatrecentlytheactionofthe E 7 generators,toallordersin wasfound in[23]inthecovariantformalism. 4.2 E 7 InvarianceontheLight-Cone Theformulaeoftheprevioussectionactonthecovarianteldstrengths.Inorderto ndtheiractiononthephysicalelds,weneedtoxthegauge.Toeventuallyconnectto thelight-conesuperspace,whereonlythephysicaldegreesoffreedomaredisplayed,we choosetoworkinthe LC 2 gauge. 4.2.1The LC 2 Gauge TheAbelianeldstrengthsarewrittenintermsofthepotentials A ij through 41

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F ij = @ A ij )]TJ/F21 11.9552 Tf 11.956 0 Td [(@ A ij : The LC 2 formalismisonewherewechosethegaugeconditions A + ij = 1 p 2 )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [(A 0 + A 3 ij =0 ; {40 and inverttheequationsofmotiontoexpress A )]TJ/F22 7.9701 Tf 7.998 0 Td [(ij intermsoftheremainingvariablesin thetheory,thephysicaltransversecomponentsofthe complex vectorpotentials A ij = 1 p 2 A 1 + iA 2 ij ; A ij = 1 p 2 A 1 )]TJ/F21 11.9552 Tf 13.948 0 Td [(iA 2 ij : Alengthybutstraightforwardcomputationyields A )]TJ/F22 7.9701 Tf 7.998 0 Td [(ij 1 p 2 )]TJ/F21 11.9552 Tf 5.48 -9.684 Td [(A 0 )]TJ/F21 11.9552 Tf 11.955 0 Td [(A 3 ij = @ @ + A ij + @ @ + A ij )]TJ/F21 11.9552 Tf 11.955 0 Td [( 1 @ + C ijkl @A kl )]TJ/F21 11.9552 Tf 11.955 0 Td [( 1 @ + )]TJ/F21 11.9552 Tf 5.48 -9.684 Td [(C ijkl @ A kl + @ @ +2 C ijkl @ + A kl + @ @ +2 C ijkl @ + A kl + 2 2 1 @ + h C ijkl C klmn @A mn + C ijkl C klmn @ A mn )]TJ/F15 11.9552 Tf 9.298 0 Td [( C ijkl + C ijkl @ @ + C klmn @ + A mn )]TJ/F15 11.9552 Tf 11.955 0 Td [( C ijkl + C ijkl @ @ + C klmn @ + A mn + @ @ + C ijkl C klmn @ + A mn + @ @ + C ijkl C klmn @ + A mn + O 3 : {41 Furtheralgebrayields 42

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X + )]TJ/F22 7.9701 Tf 7.998 0 Td [(ij =2 @A ij )]TJ/F21 11.9552 Tf 11.955 0 Td [(C ijkl @ A kl + @ @ + C ijkl @ + A kl + @ @ + C ijkl @ + A kl + 2 2 C ijkl C klmn @A mn )]TJ/F21 11.9552 Tf 11.291 0 Td [(C ijkl @ @ + C klmn @ + A mn )]TJ/F21 11.9552 Tf 11.956 0 Td [(C ijkl @ @ + C klmn @ + A mn + @ @ + C ijkl C klmn @ + A mn + @ @ + )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [(C ijkl C klmn @ + A mn + O 3 ; Y + )]TJ/F22 7.9701 Tf 7.998 0 Td [(ij = C ijkl @A kl )]TJ/F21 11.9552 Tf 15.144 8.088 Td [( 2 2 C ijkl C klmn @ A mn + 2 2 C ijkl @ @ + C klmn @ + A mn + C ijkl @ @ + C klmn @ + A mn + O 3 : {42 Notethat Y + )]TJ/F22 7.9701 Tf 7.998 0 Td [(ij vanisheswhen =0. 4.2.2DualityTransformationsoftheVectorPotentials InLC 2 ,thecomplexeldstrengths F + )]TJ/F22 7.9701 Tf 6.586 0 Td [(ij arefoundtobe F + )]TJ/F22 7.9701 Tf 6.586 0 Td [(ij = 1 2 )]TJ/F21 11.9552 Tf 5.48 -9.683 Td [(@ + A )]TJ/F22 7.9701 Tf 7.998 0 Td [(ij + @A ij )]TJ/F15 11.9552 Tf 13.419 3.155 Td [( @ A ij = @A ij )]TJ/F21 11.9552 Tf 13.151 8.088 Td [( 2 C ijkl @A kl )]TJ/F21 11.9552 Tf 13.151 8.088 Td [( 2 C ijkl @ A kl + 2 @ @ + C ijkl @ + A kl + 2 @ @ + C ijkl @ + A kl + 2 4 C ijkl C klmn @A mn + C ijkl C klmn @ A mn )]TJ/F15 11.9552 Tf 9.298 0 Td [( C ijkl + C ijkl @ @ + C klmn @ + A mn + @ @ + C klmn @ + A mn + @ @ + C ijkl C klmn @ + A mn + @ @ + )]TJ/F21 11.9552 Tf 5.479 -9.683 Td [(C ijkl C klmn @ + A mn + : {43 ByvaryingthisexpressionandusingEq.4{23,wearriveatthetransformationofthe vectorpotentialsunder SU =SO prime A ij = iS im A mj + 1 @ + C mjkl @ + A kl + O 2 )]TJ/F15 11.9552 Tf 15.857 0 Td [( i $ j ; {44 43

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andbycomplexconjugation, prime A ij = )]TJ/F21 11.9552 Tf 11.955 0 Td [(iS im A mj + 1 @ + )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [(C mjkl @ + A kl + O 2 )]TJ/F15 11.9552 Tf 15.857 0 Td [( i $ j ; {45 showingacomplicatednon-lineartransformationunder SU =SO .Asinthecovariant case,thetermsontheright-hand-sidedonotsharethesamecosettransformations. Similarly,thecoset E 7 =SU transformationsofthevectorpotentialsareobtained bysubstituting F + )]TJ/F22 7.9701 Tf 6.587 0 Td [(ij inEq.4{39withEq.4{43.Rememberingthatthescalars transformnon-linearlyunder E 7 =SU ,wendthenon-linearcosettransformationsof thevectorpotentials A ij = )]TJETq1 0 0 1 179.835 454.899 cm[]0 d 0 J 0.478 w 0 0 m 7.804 0 l SQBT/F15 11.9552 Tf 179.835 445.056 Td [( ijkl A kl + 2 ijkl )]TJETq1 0 0 1 283.69 454.899 cm[]0 d 0 J 0.478 w 0 0 m 7.804 0 l SQBT/F15 11.9552 Tf 283.69 445.056 Td [( ijkl 1 @ + C klmn @ + A mn + O 2 ; {46 and A ij = )]TJ/F15 11.9552 Tf 11.955 0 Td [( ijkl A kl + 2 ijkl )]TJ/F15 11.9552 Tf 11.955 0 Td [( ijkl 1 @ + )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [(C klmn @ + A mn + O 2 : {47 Theypreservethehelicityofthevectorpotentials.Itisworthrepeatingthatthese formulaearevalidaslongas 6 =0.For =0,these E 7 =SU transformationssimply donotexist. Thesetransformationscanbegeneralizedtoallordersin .Weleavethemathematical detailstotheAppendixB.Herewesimplystatetheresults.Introducetheall-orders shorthandnotationfor G ij G ij = 1 2 im jn )]TJ/F21 11.9552 Tf 11.955 0 Td [( jm in + C ijkl + 2 2 C ijkl C klmn + F mn E ijkl F kl : {48 Under SU ,wend 44

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A ij = R ik A kj + iS ik 1 @ + E ikmn @ + A mn )]TJ/F15 11.9552 Tf 13.947 0 Td [( i $ j : Similarly,under E 7 =SU A ij = )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(1 2 ijkl + ijkl A kl + 1 2 ijkl )]TJETq1 0 0 1 358.061 615.781 cm[]0 d 0 J 0.478 w 0 0 m 7.804 0 l SQBT/F15 11.9552 Tf 358.061 605.938 Td [( ijkl 1 @ + E klmn @ + A mn : Finally,theall-orderstransformationofthescalarsisgivenby E ijkl = 1 2 ijmn + ijmn E mnkl + 1 2 E ijmn mnkl + mnkl )]TJ/F15 11.9552 Tf 10.494 8.087 Td [(1 2 E ijmn mnpq )]TJETq1 0 0 1 305.611 497.649 cm[]0 d 0 J 0.478 w 0 0 m 7.804 0 l SQBT/F15 11.9552 Tf 305.611 487.806 Td [( mnpq E pqkl )]TJ/F15 11.9552 Tf 13.151 8.087 Td [(1 2 ijkl )]TJETq1 0 0 1 429.194 497.649 cm[]0 d 0 J 0.478 w 0 0 m 7.804 0 l SQBT/F15 11.9552 Tf 429.194 487.806 Td [( ijkl : 4.2.3TheVectorandScalar LC 2 Hamiltonians Inthissection,wederivetheformofthelightconeHamiltonianinvolvingthevectors andthescalarsinthe LC 2 gauge.WebeginwiththevectorLagrangian4{1inthe LC 2 gauge,thatissetting A + ij =0andreplacing A )]TJ/F22 7.9701 Tf 7.998 0 Td [(ij usingtheequationsofmotion,toget L V = A ij )]TJ/F21 11.9552 Tf 9.299 0 Td [(@ + @ )]TJ/F15 11.9552 Tf 11.733 -4.936 Td [(+ @ @ A ij + 2 @ + A ij C ijmn @ )]TJ/F15 11.9552 Tf 10.172 -1.914 Td [( A mn + @A ij C ijkl @A kl )]TJ/F21 11.9552 Tf 16.693 8.087 Td [(@ @ + C ijkl @ + A kl )]TJ/F15 11.9552 Tf 18.156 11.242 Td [( @ @ + C ijkl @ + A kl + c:c: )]TJ/F21 11.9552 Tf 12.487 8.088 Td [( 2 2 @ @ + @ + A ij C ijkl @ @ + C klmn @ + A mn + 2 2 )]TJ/F15 11.9552 Tf 10.494 8.088 Td [(1 2 @A ij C ijkl C klmn @ A mn + @A ij C ijkl @ @ + C klmn @ + A mn + c:c: + 2 2 @A ij C ijkl @ @ + C klmn @ + A mn )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(1 2 @ @ + @ + A ij C ijkl @ @ + C klmn @ + A mn + c:c: + 2 4 @ )]TJ/F15 11.9552 Tf 10.172 -1.914 Td [( A ij C ijkl C klmn @ + A mn )]TJ/F21 11.9552 Tf 18.686 8.088 Td [(@ @ + @ + A ij C ijkl C klmn )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [(@A mn + @ A mn + c:c: : {49 Similarly,thescalarpartofthedeWit-FreedmanSupergravityLagrangian4{4isgiven by 45

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L S = )]TJ/F15 11.9552 Tf 13.42 8.088 Td [(1 24 C ijkl @ + @ )]TJ/F19 11.9552 Tf 9.741 -4.936 Td [()]TJ/F21 11.9552 Tf 11.955 0 Td [(@ @ C ijkl + 2 96 C ijkl C klmn @ + C mnpq @ )]TJETq1 0 0 1 335.117 652.277 cm[]0 d 0 J 0.478 w 0 0 m 9.234 0 l SQBT/F21 11.9552 Tf 335.117 642.434 Td [(C pqij + @ )]TJ/F21 11.9552 Tf 7.085 -4.936 Td [(C mnpq @ + C pqij )]TJ/F21 11.9552 Tf 9.298 0 Td [(@C mnpq @ C pqij )]TJ/F15 11.9552 Tf 13.419 3.155 Td [( @C mnpq @ C pqij + : {50 Bothcontain @ )]TJ/F15 11.9552 Tf 10.987 -4.338 Td [(thelight-conetimederivativesintheirinteractions.Inordertogeta sensibleHamiltonian,thesemustbeeliminatedbyeldredenitions.Toorder 2 ,these aregivenbysetting C ijkl = D ijkl )]TJ/F21 11.9552 Tf 15.143 8.087 Td [( 2 4 1 @ + )]TJ/F21 11.9552 Tf 5.479 -9.683 Td [(D pq [ ij @ + D kl ] mn D pqmn + + 3 2 @ + @ + B [ ij 1 @ + D kl ] mn @ + B mn + + 3 2 4! @ + ijklrstu @ + B rs 1 @ + D tumn @ + B mn + + ; {51 A ij = B ij )]TJ/F21 11.9552 Tf 13.15 8.088 Td [( 2 1 @ + D ijkl @ + B kl + 2 8 D ijkl 1 @ + @ + B mn D mnkl + : {52 Withtheseredenitions,weverifythatthenewvectoreld, B ij anditsconjugate transform linearly under SU ,thatisunderthe SU =SO cosettransformations, prime B ij = iS ik B kj )]TJ/F15 11.9552 Tf 11.955 0 Td [( i $ j : Thisallowsustolabelthe i;j;::: indicesastrue SU indices;inparticular,their loweringproducesthebarred"representation. The E 7 =SU variationsoftheseequationsyieldthetransformationpropertiesof thenewelds B ij = )]TJ/F21 11.9552 Tf 13.151 8.088 Td [( 4 mnkl D ijkl B mn + 4 ijkl 1 @ + )]TJETq1 0 0 1 379.648 96.515 cm[]0 d 0 J 0.478 w 0 0 m 10.044 0 l SQBT/F21 11.9552 Tf 379.648 86.672 Td [(D mnkl @ + B mn {53 46

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D ijkl = 2 ijkl )]TJ/F21 11.9552 Tf 13.151 8.088 Td [( 2 mnpq 1 @ + )]TJ/F21 11.9552 Tf 5.48 -9.684 Td [(D mn [ kl @ + D ij ] pq + 2 pq [ ij 1 @ + )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [(@ + D kl ] mn D pqmn )]TJ/F15 11.9552 Tf 9.299 0 Td [(3 mn [ kl @ + )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [(@ + B ij ] B mn + ijklrstu tumn 4! @ + )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [(B mn @ + B rs + : {54 Wenotethattheeliminationofthelight-conetimederivativesfromtheinteractionterms hasrequiredthevariationofthescalarstocontaintermsquadraticinthegaugeelds. Thismixingisafeatureofourgauge,anddoesnotoccurinthecovariantformalism. Thesetransformationsareallweneedtoconstructtheactionof E 7 onthescalar andvectorelds.Recallthatthecomponentsofthe 56 X ij and Y ij ,transformas X ij = ijkl Y kl ; Y ij = ijkl X kl : Theupperlowerindicesontheeldstransformastrue SU octetsanti-octets.In ordertoexpressthisactionof E 7 =SU on B ij ,weneedtogetridof ,soastoobtain thecanonicalformofEqs.4{30and4{31.Suchunwantedtermscanbesystematically eliminatedbysettingthe withthewrongindexstructureby 2 D ,whichentailserrorsof higherorderin .WethenuseLeibnitz'sruletorewritethevariationasatotalvariation minusthevariationofthetermitmultiplies.Thisprocedureyieldstolowestorderin B ij + 2 4 D ijkl D klmn B mn = 4 D ijmn B mn + 1 @ + )]TJETq1 0 0 1 398.895 207.313 cm[]0 d 0 J 0.478 w 0 0 m 10.044 0 l SQBT/F21 11.9552 Tf 398.895 197.47 Td [(D ijmn B mn {55 Theright-handsideisidentiedwith Y ij ,andwewriteitsvariationalsoincanonicalform. Atediouscalculationshowsthattheuppercomponentacquiresextratermsoforder 2 whicharelinearandalsocubicinthegaugeelds.Herewesimplystatetheresult 47

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X ij = B ij + 2 4 D ijpq D pqrs B rs )]TJ/F21 11.9552 Tf 13.15 8.088 Td [( 2 8 D ijpq 1 @ + D pqrs @ + B rs )]TJ/F21 11.9552 Tf 13.151 8.088 Td [( 2 8 1 @ + D ijpq @ + D pqrs B rs )]TJ/F21 11.9552 Tf 10.494 8.087 Td [( 2 4 1 @ + B ij @ + B kl B kl + 2 8 1 @ + )]TJ/F21 11.9552 Tf 5.479 -9.683 Td [(B ij @ + B kl B kl )]TJ/F21 11.9552 Tf 10.494 8.088 Td [( 2 4 B ij 1 @ + )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [(@ + B kl B kl + 2 8 1 @ + )]TJ/F21 11.9552 Tf 5.48 -9.684 Td [(B ij @ + B kl B kl ++ O 3 ; Y ij = 2 D ijmn B mn )]TJ/F21 11.9552 Tf 13.151 8.088 Td [( 4 1 @ + )]TJ/F21 11.9552 Tf 5.48 -9.684 Td [(@ + D ijmn B mn + O 3 ; {56 Thevariationofthethirdtermin X ij doesnotcontributetoorder1 = sincethe transformationisglobal. Wecanalsochecktolowestorderin thatthecommutatoroftwosuchtransformations onthegaugeeldsisindeedan SU transformation. [ 1 ; 2 ] B ij = ijmn 2 mnpq 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [( ijmn 1 mnpq 2 B pq : {57 4.2.4HamiltoniantoOrder 2 TheseeldredenitionsallowustoexpresstheLagrangians4{49and4{50in termsofthenewelds 48

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L V + L S = B ij )]TJ/F21 11.9552 Tf 9.299 0 Td [(@ + @ )]TJ/F15 11.9552 Tf 11.734 -4.936 Td [(+ @ @ B ij + 1 24 D ijkl )]TJ/F21 11.9552 Tf 9.299 0 Td [(@ + @ )]TJ/F15 11.9552 Tf 11.733 -4.936 Td [(+ @ @ D ijkl )]TJ/F21 11.9552 Tf 10.494 8.088 Td [( 2 @B ij @ @ + D ijkl @ + B kl )]TJ/F21 11.9552 Tf 11.955 0 Td [(@B ij D ijkl @B kl + c:c: )]TJ/F21 11.9552 Tf 12.486 8.088 Td [( 2 2 1 2 @B ij D ijkl D klmn @ B mn )]TJ/F21 11.9552 Tf 13.948 0 Td [(@B ij D ijkl @ @ + D klmn @ + B mn + c:c: )]TJ/F21 11.9552 Tf 12.486 8.088 Td [( 2 4 @ @ + @ + B ij D ijkl @ @ + D klmn @ + B mn )]TJ/F21 11.9552 Tf 10.494 8.088 Td [( 2 8 @ @ @ + B ij D ijkl D klmn @ + B mn + @ + D ijkl 1 @ + D klmn @ + B mn + c:c: )]TJ/F21 11.9552 Tf 12.486 8.088 Td [( 2 8 @ + B ij @ @ @ + D ijkl 1 @ + D klmn @ + B mn + c:c: )]TJ/F21 11.9552 Tf 12.604 8.088 Td [( 2 96 D ijkl D klmn @D mnpq @ D pqij )]TJ/F21 11.9552 Tf 13.947 0 Td [(@ + D mnpq @ @ @ + D pqij + c:c: + O 3 ; fromwhichweread-othevectorandscalarHamiltoniantoorder 2 H V + H S = @ B ij @B ij + 1 24 @ D ijkl @D ijkl + 2 @B ij @ @ + D ijkl @ + B kl )]TJ/F21 11.9552 Tf 13.948 0 Td [(@B ij D ijkl @B kl + c:c: + 2 2 1 2 @B ij D ijkl D klmn @ B mn )]TJ/F21 11.9552 Tf 13.948 0 Td [(@B ij D ijkl @ @ + D klmn @ + B mn + c:c: + 2 4 @ @ + @ + B ij D ijkl @ @ + D klmn @ + B mn + 2 8 @ @ @ + B ij D ijkl D klmn @ + B mn + @ + D ijkl 1 @ + D klmn @ + B mn + c:c: + 2 8 @ + B ij @ @ @ + D ijkl 1 @ + D klmn @ + B mn + c:c: + 2 96 D ijkl D klmn @D mnpq @ D pqij )]TJ/F21 11.9552 Tf 13.947 0 Td [(@ + D mnpq @ @ @ + D pqij + c:c: + O 3 : The 2 termscanbefurthersimpliedas 49

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2 4 @B ij D ijkl D klmn @ B mn + 1 2 @ + B ij D ijkl D klmn @ @ @ + B mn + c:c: + 2 8 4 @ @B ij D ijkl )]TJ/F21 11.9552 Tf 13.278 8.088 Td [(@ @ @ + @ + B ij D ijkl + @ @ @ + B ij @ + D ijkl + @ + B ij @ @ @ + D ijkl 1 @ + D klmn @ + B mn + c:c: + 2 96 D ijkl D klmn @D mnpq @ D pqij )]TJ/F21 11.9552 Tf 11.956 0 Td [(@ + D mnpq @ @ @ + D pqij + c:c: : 4.2.5 E 7 InvarianceoftheVectorandScalarHamiltonians Inthissection,wederivethelight-coneHamiltonianinadierentway,throughthe morefamiliarcanonicalLegendretransformationsontheLagrangians L S and L V .As weshow,thisprocedurehastheadvantagethattheHamiltonianthusobtainedis E 7 invariant.Aftertheeldredenitions4{51and4{52,thecanonicalHamiltonian reducestotheonederivedintheprevioussection. Thecanonicalenergy-momentumtensorfor L S isgivenby S = L S )]TJ/F21 11.9552 Tf 28.291 8.088 Td [(@ L S @ @ C ijkl @ C ijkl )]TJ/F21 11.9552 Tf 28.291 8.087 Td [(@ L S @ @ C ijkl @ C ijkl : Thelight-coneHamiltonianisobtainedbysetting =+and = )]TJ/F15 11.9552 Tf 9.298 0 Td [(.Afterabsorbing surfaceterms,oneobtains H can: S = 1 24 @C ijkl @ C ijkl + 2 96 )]TJ/F21 11.9552 Tf 5.48 -9.684 Td [(C ijkl C klmn @C mnpq @ C pqij + c:c: + ; wherewehaveused, C ijkl C klmn @C mnpq @ C pqij = C ijkl C klmn @ C mnpq @C pqij +surfaceterms : 50

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ThisHamiltonianismanifestlyinvariantunder E 7 .Similarly,thevectorcanonical energy-momentumtensorfor L V isgivenby V = 1 4 F G + 1 4 G F + L V ; {58 yielding H can: V = 1 2 F + )]TJ/F22 7.9701 Tf 6.586 0 Td [(ij G + )]TJ/F22 7.9701 Tf 6.587 0 Td [(ij + 1 4 F + mij G )]TJ/F22 7.9701 Tf 6.586 0 Td [(mij + 1 4 G + mij F )]TJ/F22 7.9701 Tf 6.586 0 Td [(mij )-222(L V ; where m;n =1 ; 2arethetransversespacecomponents.Substituting L V = 1 4 F + )]TJ/F22 7.9701 Tf 6.587 0 Td [(ij G + )]TJ/F22 7.9701 Tf 6.587 0 Td [(ij + 1 4 F + mij G )]TJ/F22 7.9701 Tf 6.587 0 Td [(mij + 1 4 F )]TJ/F22 7.9701 Tf 6.587 0 Td [(mij G + mij )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(1 8 F mnij G mnij ; yieldsthesimplerexpression H can: V = 1 4 )]TJ/F21 11.9552 Tf 7.472 -9.683 Td [(F + )]TJ/F22 7.9701 Tf 6.586 0 Td [(ij G + )]TJ/F22 7.9701 Tf 6.586 0 Td [(ij + F 12 ij G 12 ij : {59 Thedualitypropertiesofthecomplexeldstrengthsinlight-conecoordinates F 12 ij = i F + )]TJ/F22 7.9701 Tf 7.998 0 Td [(ij ; G 12 ij = i G + )]TJ/F22 7.9701 Tf 7.998 0 Td [(ij ; {60 allowustorewritethisHamiltonianinthesuggestiveform H can: V = 1 4 n )]TJ/F32 11.9552 Tf 5.479 -9.684 Td [(G + )]TJ/F22 7.9701 Tf 7.998 0 Td [(ij + F + )]TJ/F22 7.9701 Tf 7.998 0 Td [(ij G + )]TJ/F22 7.9701 Tf 7.998 0 Td [(ij + F + )]TJ/F22 7.9701 Tf 7.997 0 Td [(ij )]TJ/F26 11.9552 Tf 11.955 9.684 Td [()]TJ/F32 11.9552 Tf 5.48 -9.684 Td [(G + )]TJ/F22 7.9701 Tf 7.998 0 Td [(ij )]TJ/F32 11.9552 Tf 11.955 0 Td [(F + )]TJ/F22 7.9701 Tf 7.998 0 Td [(ij G + )]TJ/F22 7.9701 Tf 7.997 0 Td [(ij )]TJETq1 0 0 1 479.383 208.41 cm[]0 d 0 J 0.478 w 0 0 m 7.306 0 l SQBT/F32 11.9552 Tf 479.383 198.501 Td [(F + )]TJ/F22 7.9701 Tf 7.998 0 Td [(ij o ; thatis H can: V = 1 4 X + )]TJ/F22 7.9701 Tf 7.998 0 Td [(ab X + )]TJ/F22 7.9701 Tf 6.531 -10.361 Td [(ab )]TJ/F21 11.9552 Tf 11.956 0 Td [(Y + )]TJ/F22 7.9701 Tf 3.93 -8.278 Td [(ab Y + )]TJ/F22 7.9701 Tf 7.998 0 Td [(ab ; {61 = 1 4 e Z + )]TJ/F26 11.9552 Tf 7.084 8.334 Td [( T 2 Z + )]TJ/F21 11.9552 Tf 10.987 -4.937 Td [(; {62 51

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where 2 iswrittenintermsof 28blockmatricesas 2 = 0 B @ 0 )]TJ/F38 11.9552 Tf 9.298 0 Td [(1 1 0 1 C A : AsweshowinAppendixB.2,thisisjustthequadratic E 7 invariantconstructed outoftwo 56 representations Z + )]TJ/F15 11.9552 Tf 10.987 -4.338 Td [(and e Z + )]TJ/F15 11.9552 Tf 7.084 -4.338 Td [(.Hencethevectorlight-coneHamiltonian isinvariantunder E 7 aswell.TheuseofEq.4{42yieldsthevectorlight-cone Hamiltonianin LC 2 gauge H can: V = @A ij @ A ij )]TJ/F26 11.9552 Tf 11.955 16.857 Td [( 2 C ijkl @ A kl )]TJ/F15 11.9552 Tf 18.157 11.243 Td [( @ @ + )]TJ/F21 11.9552 Tf 5.48 -9.684 Td [(C ijkl @ + A kl @ A ij + c:c: + 2 @A ij @ @ + )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [(C ijkl @ + A kl + 2 @ A ij @ @ + C ijkl @ + A kl + O 2 : {63 Thiscanbegeneralizedtoall-orders H can: V = 1 4 n X + )]TJ/F22 7.9701 Tf 7.998 0 Td [(ab X + )]TJ/F22 7.9701 Tf 7.084 -9.199 Td [(ab )]TJ/F21 11.9552 Tf 13.948 0 Td [(Y + )]TJ/F22 7.9701 Tf 7.085 -6.73 Td [(ab Y + )]TJ/F22 7.9701 Tf 7.998 0 Td [(ab o = 1 2 F + )]TJ/F22 7.9701 Tf 7.998 0 Td [(ij E + E ijkl F + )]TJ/F22 7.9701 Tf 7.997 0 Td [(kl = 1 2 n @A ij E + E ijkl @ A kl + @A ij + @ A ij K ij + K ij E + E )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 ijkl K kl o ; where K ij = @ @ + E ijkl @ + A kl )]TJETq1 0 0 1 252.218 242.92 cm[]0 d 0 J 0.478 w 0 0 m 9.366 0 l SQBT/F21 11.9552 Tf 252.218 233.077 Td [(E ijkl @A kl + @ @ + )]TJ/F21 11.9552 Tf 5.48 -9.684 Td [(E ijkl @ + A kl )]TJ/F21 11.9552 Tf 11.955 0 Td [(E ijkl @ A kl : {64 Wenowexpress A ij and C ijkl intermsof B ij and D ijkl byusingEq.4{51andEq. 4{52.ThisyieldstheHamiltonianoftheprevioussection.Thisshowsthatthetwoways ofderivingtheHamiltonian,eitherbyrequiringinthe LC 2 gaugeno @ )]TJ/F15 11.9552 Tf 10.987 -4.339 Td [(termsinthe interactions,orbymakingthefamiliarLegendretransformation,leadtothesameresult. Inthisprocesswenotethatweobtainasecond 56 representationof E 7 : 52

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X + )]TJ/F22 7.9701 Tf 7.998 0 Td [(ij =2 @B ij )]TJ/F21 11.9552 Tf 11.955 0 Td [(D ijkl @ B kl + @ @ + D ijkl @ + B kl + 2 2 D ijkl D klmn @B mn )]TJ/F21 11.9552 Tf 11.956 0 Td [(D ijkl @ @ + D klmn @ + B mn + 1 2 @ D ijkl 1 @ + D klmn @ + B mn + ; Y + )]TJ/F22 7.9701 Tf 7.084 -6.73 Td [(ij = D ijkl @B kl )]TJ/F21 11.9552 Tf 13.151 8.088 Td [( 2 2 D ijkl D klmn @ B mn + 2 2 D ijkl @ @ + D klmn @ + B mn + ; {65 obtainedbyredeningtheeldsinEq.4{42. HavingshowninvarianceofthecomponentHamiltonianof N =8Supergravity,we nowconnecttheseresultstothesupereldformalisminordertogeneratethehigherorder interactionsof N =8Supergravity. 4.3NonlinearRealizationof E 7 inSuperspace Asdescribedinsection2.1.2,allthephysicaldegreesoffreedomarecapturedinthe chiralsupereld denedinEq.2{19 y = 1 @ + 2 h y + i m 1 @ +2 m y + i 2 m n 1 @ + B mn y )]TJ/F15 11.9552 Tf 14.112 8.088 Td [(1 3! m n p 1 @ + mnp y )]TJ/F15 11.9552 Tf 16.769 8.088 Td [(1 4! m n p q D mnpq y + i 5! m n p q r mnpqrstu stu y + i 6! m n p q r s mnpqrstu @ + B tu y + 1 7! m n p q r s t mnpqrstu @ + u y + 4 8! m n p q r s t u mnpqrstu @ + 2 h y : {66 satisfyingtheinside-outconstraint d i d j d k d l = 1 4! ijklmnpq d m d n d p d q '; {67 53

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whichimplies D mnpq satisesthedualitycondition D ijkl = 1 4! ijklmnpq D mnpq : {68 4.3.1KinematicalSupersymmetry Theeightkinematicalsupersymmetrygenerators q m = )]TJ/F21 11.9552 Tf 19.873 8.088 Td [(@ @ m + i p 2 m @ + ; q m = @ @ m )]TJ/F21 11.9552 Tf 19.061 8.088 Td [(i p 2 m @ + ; {69 arelinearlyrepresented q y = m q m y ; {70 whichcanbeexpressedintermsoftheGrassmannvariablesaswell q y = i p 2 m m @ + y ; {71 where m and m aretheanticommutingtransformationparameters.Thiskinematical supersymmetrytransformationsofthephysicaleldsarethengivenby s h =0 ; s h = )]TJ/F21 11.9552 Tf 11.956 0 Td [(i p 2 4 m m ; s m =2 p 2 n @ + B mn ; s m = )]TJ 11.955 10.473 Td [(p 2 m @ + h; s B mn = )]TJ/F15 11.9552 Tf 11.955 0 Td [(3 i p 2 p mnp ; s B mn = )]TJ/F15 11.9552 Tf 11.956 0 Td [(2 i p 2 [ m n ] ; s lmn = )]TJ 13.151 17.977 Td [(p 2 3! k @ + D klmn ; s mnp = )]TJ/F15 11.9552 Tf 11.955 0 Td [(3 p 2 [ p @ + B mn ] ; andnally s D klmn = )]TJ/F15 11.9552 Tf 11.955 0 Td [(4 i p 2 [ n klm ] : 54

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4.3.2SUTransformations Thequadraticoperatorsthataremadeformthekinematicalsupersymmetries T i j = )]TJ/F21 11.9552 Tf 27.015 8.087 Td [(i p 2 @ + q i q j )]TJ/F15 11.9552 Tf 13.15 8.087 Td [(1 8 i j q k q k ; {72 whichsatisfythe SU algebra [ T i j ;T k l ]= k j T i l )]TJ/F21 11.9552 Tf 11.955 0 Td [( i l T k j ; alsoactlinearlyonthechiralsupereld SU 8 y = j i T i j y : 4.3.3 E 7 =SU Transformations Wecannowincludetheothereldsofthetheorybydemandingthatthe E 7 =SU transformationscommutewiththekinematicalsupersymmetries,thatis [ s ; ] y =0 : {73 Webeginbyapplyingthisequationtothevectorpotential.Wespecializetothe componentvector B 12 y [ s ; ] B 12 y =0 ; or s B 12 y = s B 12 y : {74 Sincethisequationholdsforanyparameters m and mnpq ,forsimplicity,weset 3 and 1234 areonlynonzerootherwisezero.Onethenimmediatelyndsthefollowing contradiction:whilethelefthandsideofEq.4{74vanishes 55

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s B 12 y = s )]TJ/F15 11.9552 Tf 11.291 0 Td [(2 i p 2 [1 2] =0 ; wherethe E 7 =SU transformationisobtainedfromEq.4{53and 1 issettobezero, therighthandsideisnonzero s B 12 y = )]TJ/F19 11.9552 Tf 5.479 -9.684 Td [()]TJ/F21 11.9552 Tf 9.299 0 Td [( D 1234 3412 B 12 = )]TJ/F21 11.9552 Tf 11.955 0 Td [( p 2 1234 3 124 B 12 : Thisindicatesthateither s B 12 y or s B 12 y needsmodication.Thekinematical supersymmetrytransformationsthespectrumgeneratingpartofthesymmetryrelate theeldswhosespinsdierby1 = 2,andthus s B 12 y shouldbeunaltered.Hence,the E 7 =SU transformation B 12 y needsmodication! Byaddingthetermsthatmaintainthecommutativity4{73tothevector,wearrive atthegeneralizationofEq.4{53toorder B ij = )]TJ/F21 11.9552 Tf 11.291 0 Td [( klmn 1 4 D ijkl B mn + 1 4! 1 @ + D klmn @ + B ij )]TJ/F15 11.9552 Tf 14.777 8.088 Td [(1 4! ijklmnrs 1 @ + B rs @ + h + i 3! 1 @ + klm ijn )]TJ/F21 11.9552 Tf 17.699 8.088 Td [(i 3! ijklmrst 1 @ + rst n + ijkl 1 @ + 1 4 D klmn @ + B mn )]TJ/F15 11.9552 Tf 19.174 8.088 Td [(1 @ + B kl @ +2 h + i 4! 2 mnp rst klmnprst )]TJ/F15 11.9552 Tf 13.948 0 Td [(3 i 1 @ + kln @ + n + O 2 ; {75 aswellastothe E 7 =SU transformationsofthegravitinossincecommutativity implies s B ij = )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 i p 2 [ i j ] : Theresultis 56

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i = )]TJ/F21 11.9552 Tf 11.291 0 Td [( mnpq 1 4! 1 @ + D mnpq @ + i + 1 3! D mnpi q )]TJ/F15 11.9552 Tf 12.12 8.088 Td [(1 4! mnpqirst 1 @ + rst @ + h + 1 4 imn B pq + 1 3! 1 @ + mnp @ + B iq + O 2 : {76 Applyingcommutativityonthegravitinosyieldsthe E 7 =SU transformationofthe graviton s i = )]TJ 11.955 10.473 Td [(p 2 i @ + h; with h = )]TJ/F21 11.9552 Tf 11.955 0 Td [( ijkl 1 8 B ij B kl + 1 4! 1 @ + D ijkl @ + h + i 6 1 @ + ijk l + O 2 : {77 Allthesetransformationsarenon-linear.Similarequationscanbederivedfortheftysix spinorsandseventyscalars. Theinhomogeneous E 7 =SU transformationsoforder )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 ofthescalareldscan beexpressedinsupereldlanguage,thatis )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 = )]TJ/F15 11.9552 Tf 13.594 8.087 Td [(2 klmn klmn ; whichischiralsince E 7 isaglobalsymmetry: @ + klmn =0.Theorder transformations ofthesuperelditselftakeaparticularlysimpleform.Weneedonlyrequirethatits variationbechiral,withthetensorstructure mnpq mnpq : Assumingthatthelowerindicesarecarriedbytheantichiralderivatives d n leadstothe uniqueformforthehomogeneouspart 57

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' = 4! mnpq 1 @ +2 d mnpq 1 @ + '@ +3 )]TJ/F15 11.9552 Tf 13.948 0 Td [(4 d mnp d q @ +2 +3 d mn @ + d pq @ + + O 2 ; {78 where d k::l isashorthandnotationfor d k d l .Itischiralbyconstruction d n =0, withthepoweroftherstinversederivativesetbycomparingwiththegraviton transformation.Hence, all physicalelds,includingthegravitontransformunder E 7 and canbereadofromthisequation.Wenotethatthese E 7 =SU transformationsdo closeonan SU transformationsonthesupereld [ 1 ; 2 ] = SU '; which,forinstance,leadstoEq.4{57. Wecannowextendthemethodtothedynamicalsupersymmetries,anddeterminethe formoftheinteractionsimpliedbythe E 7 symmetry. 4.4HamiltonianinSuperspace Thelight-conesupereldHamiltoniandensitycanbeobtainedfromthesupereld action2{20 H =2 @ @ @ +4 +2 1 @ +2 @' @' + c:c: + O 2 ; {79 whichcanbederivedfromtheactionofthedynamicalsupersymmetriesonthechiral supereldaswell dyn s = dyn s + dyn s + dyn s + O 3 ; = m @ @ + q m + 1 @ + @ d m '@ +2 )]TJ/F21 11.9552 Tf 13.947 0 Td [(@ + d m '@ + @' + O 2 : Wenowrequirethatthe E 7 =SU commuteswiththedynamicalsupersymmetries 58

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[ ; dyn s ] =0 : {80 Thiscommutativityisvalidonlyonthechiralsupereld.Forexample, [ 1 ; s ] 2 6 =0 ; duetothenon-linearityofthe E 7 transformation.Thishelpsusunderstandhowthe Jacobiidentity [ 1 ; [ 2 ; s ]]+[ 2 ; [ s ; 1 ]]+[ s ; [ 1 ; 2 ]] =0 ; isalgebraicallyconsistent.Inthelasttermthecommutatorofthetwo E 7 =SU transformations,[ 1 ; 2 ],yieldsan SU underwhichthesupersymmetrytransforms. Thisispreciselycompensatedbycontributionsfromthersttwoterms. Althoughthedynamicalsupersymmetrytoorder isalreadyknown,were-derive dyn s fromthecommutativitybetweenthedynamicalsupersymmetriesand E 7 =SU transformations. Theinhomogeneous E 7 transformationslinkinteractiontermswithdierentorder in .Tozerothorder,onends [ )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 ; dyn s ] = )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 dyn s =0 ; {81 since dyn s )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 =0.Tond dyn s thatsatisesboththeaboveequationand theSuperPoincarealgebra,onemaystartwithageneralformthatsatisesallthe commutationrelationswiththekinematicalSuperPoincaregeneratorstheformsofthe kinematicalSuperPoincaregeneratorscanbefoundinAppendixA, dyn s / @ @a @ @b 1 @ + m + n +1 e a ^ @ e b ^ q @ ++ m 'e )]TJ/F22 7.9701 Tf 6.586 0 Td [(a ^ @ e )]TJ/F22 7.9701 Tf 6.587 0 Td [(b ^ q @ ++ n a = b =0 ; 59

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where ^ @ = @ @ + ^ q = m q m @ + .Itisnotdiculttoseethatthisformwithnon-negative m;n satisesEq.4{81.Thenumberofpowersof @ + canbedeterminedbychecking thecommutationrelationbetweentwodynamicalgenerators p )]TJ/F15 11.9552 Tf 6.752 -0.299 Td [(Hamiltonianvariation whichisderivedfromthesupersymmetryalgebraand j )]TJ/F15 11.9552 Tf 6.753 1.393 Td [(theboostwhichcanalsobe obtainedthrough[ j )]TJ/F21 11.9552 Tf 8.745 1.394 Td [(; q ] = dyn s ,yieldingthatthecommutatorbetween j )]TJ/F15 11.9552 Tf 10.654 1.394 Td [(and p )]TJ/F15 11.9552 Tf -459.331 -24.207 Td [(vanishesonlywhen m = n =0,whichleadstothethesameformasEq.4{80writtenin acoherent-likeform dyn s = 2 @ @a @ @b 1 @ + h e a ^ @ e b ^ q @ +2 'e )]TJ/F22 7.9701 Tf 6.587 0 Td [(a ^ @ e )]TJ/F22 7.9701 Tf 6.586 0 Td [(b ^ q @ +2 i a = b =0 : Itisworthnotingthatthisisthesolutionthathastheleastnumberofpowersof @ + in thedenominator,andthustheleastnon-local". Thesamereasoningcanbeappliedtohigherordersin .Toorder ,wendthat commutativity [ )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 ; dyn s ] +[ ; dyn s ] =0 requires )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 dyn s {82 = 4! ijkl 1 @ +3 )]TJ/F15 11.9552 Tf 13.357 3.154 Td [( d ijkl @ @ + '@ +3 q' +4 d ijk @' d l @ +2 q' )]TJ/F15 11.9552 Tf 13.947 0 Td [(3 d ij @@ + d kl @ + q' )]TJ/F15 11.9552 Tf 16.014 3.155 Td [( d ijkl q @ + '@@ +3 +4 d ijk q' d l @@ +2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(3 d ij @ + q' d kl @@ + + d ijkl @ @ +2 q'@ +4 )]TJ/F15 11.9552 Tf 13.947 0 Td [(4 d ijk @ @ + q' d l @ +3 +3 d ij @ q' d kl @ +2 + d ijkl '@@ +2 q' )]TJ/F15 11.9552 Tf 13.948 0 Td [(4 d ijk @ + d l @@ + q' +3 d ij @ +2 d kl @ q' ; where q denotes m q m ,whichcanbewritteninasimplerformbyrewritingitintermsof acoherentstate-likeform: 60

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)]TJ/F20 7.9701 Tf 6.587 0 Td [(1 dyn s {83 = 2 4! ijkl @ @a @ @b @ @ ijkl 1 @ +3 h e a ^ @ e b ^ q e ^ d @ +4 'e )]TJ/F22 7.9701 Tf 6.586 0 Td [(a ^ @ e )]TJ/F22 7.9701 Tf 6.587 0 Td [(b ^ q e )]TJ/F22 7.9701 Tf 6.586 0 Td [( ^ d @ +4 i a = b = =0 : where ^ @ = @ @ + ; ^ q m = q m @ + ; ^ d = m d m @ + ; and @ @ ijkl @ @ i @ @ j @ @ k @ @ l : Wenotethattheabovecoherentstate-likerepresentationwiththeoppositesignsin ^ @ and ^ q isagenericformthatsatisesallcommutationrelationswiththekinematicalgenerators theformsofthekinematicalsuper-PoincaregeneratorscanbefoundinAppendixA, whichimpliesthatsuchstructureshouldappearin dyn s toclosethesuper-Poincare algebratoorder 2 Tond dyn; s thatsatisesEq.4{82,considerthechiralcombination Z ijkl 1 4! ijklmnpq @ @ mnpq e ^ d @ +4 'e )]TJ/F22 7.9701 Tf 6.586 0 Td [( ^ d @ +4 =0 ; {84 = 1 4! ijklmnpq )]TJ/F15 11.9552 Tf 7.545 -6.529 Td [( d mnpq '@ +4 )]TJ/F15 11.9552 Tf 13.947 0 Td [(4 d mnp @ + d q @ +3 +3 d mn @ +2 d pq @ +2 : Theinhomogeneous E 7 transformationof Z ijkl 1 4! ijklmnpq Z mnpq ; hasthesimpleform )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 Z ijkl = 1 4! ijklmnpq d mnpq )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 '@ +4 = 2 ijkl @ +4 '; {85 whichleadstothesolution 61

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dyn s = 2 2 4! @ @a @ @b @ @ ijkl 1 @ +4 e a ^ @ + b ^ q + ^ d @ +5 'e )]TJ/F22 7.9701 Tf 6.587 0 Td [(a ^ @ )]TJ/F22 7.9701 Tf 6.587 0 Td [(b ^ q )]TJ/F22 7.9701 Tf 6.586 0 Td [( ^ d Z ijkl a = b = =0 ; wherewehavexedtheambiguitydiscussedearlierbychoosingtheexpressionwiththe leastnumberof @ + inthedenominator.Thiscoherentstate-likeformisveryecient; Writtenoutexplicitly dyn s consistsof60terms: dyn s = 2 ijkl 48 m )]TJ/F21 11.9552 Tf 5.48 -9.684 Td [( I s + II s + III s + IV s ; where I s = 1 @ +4 h d ijkl @ @ + q m 'Z ijkl +4 d ijk @ q m d l @ + Z ijkl +6 d ij @@ + q m d kl @ +2 Z ijkl +4 d i @@ +2 q m d jkl @ +3 Z ijkl + @@ +3 q m d ijkl @ +4 Z ijkl i ; II s = 1 @ +4 h )]TJ/F15 11.9552 Tf 14.021 3.155 Td [( d ijkl @' q m @ + Z ijkl +4 d ijk @@ + d l @ +2 q m Z ijkl )]TJ/F15 11.9552 Tf 13.947 0 Td [(6 d ij @@ +2 d kl @ +3 q m Z ijkl +4 d i @@ +3 d jkl @ +4 q m Z ijkl )]TJ/F21 11.9552 Tf 13.948 0 Td [(@@ +4 d ijkl @ +5 q m Z ijkl i ; III s = )]TJ/F15 11.9552 Tf 16.642 8.087 Td [(1 @ +4 h d ijkl q m @ @ + Z ijkl +4 d ijk @ + q m d l @ +2 @Z ijkl +6 d ij @ +2 q m d kl @ +3 @Z ijkl +4 d i @ +3 q m d jkl @ +4 @Z ijkl + @ +4 q m d ijkl @ +5 @Z ijkl i ; IV s = 1 @ +4 h d ijkl @ + @ q m @ +2 Z ijkl )]TJ/F15 11.9552 Tf 13.948 0 Td [(4 d ijk @ +2 d l @ +3 @ q m Z ijkl +6 d ij @ +3 d kl @ +4 @ q m Z ijkl )]TJ/F15 11.9552 Tf 13.948 0 Td [(4 d i @ +4 d jkl @ +5 @ q m Z ijkl + @ +5 d ijkl @ +6 @ q m Z ijkl i : Weusethefact,asAnanthetal.[14]haveshown,thatthe N =8supergravity light-coneHamiltoniancanbewrittenasaquadraticformtoorder 2 62

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H = 1 4 p 2 W m ; W m 2 i 4 p 2 Z d 8 d 8 d 4 x W m 1 @ +3 W m ; wherethefermionicsupereld W m isthespecicdynamicalsupersymmetryvariationof dyn s m W m ; with W m = W m + W m + W m + : Uptoorder ,theHamiltonianissimply H = 1 4 p 2 )]TJ/F19 11.9552 Tf 10.461 -9.684 Td [(W m ; W m + )]TJ/F19 11.9552 Tf 5.48 -9.684 Td [(W m ; W m + )]TJ/F19 11.9552 Tf 5.48 -9.684 Td [(W m ; W m ; {86 whiletheHamiltonianoforder 2 consistsofthreeparts: H 2 = 1 4 p 2 )]TJ/F19 11.9552 Tf 10.46 -9.684 Td [(W m ; W m + )]TJ/F19 11.9552 Tf 5.479 -9.684 Td [(W m ; W m + )]TJ/F19 11.9552 Tf 5.479 -9.684 Td [(W m ; W m ; {87 wheretherstpartwascomputedbyAnanthetal.[14] )]TJ/F19 11.9552 Tf 5.48 -9.684 Td [(W m ; W m = i 2 2 @ @a @ @b @ @r @ @s Z d 8 d 8 d 4 x {88 1 @ +5 e a ^ @ + b ^ q m @ +2 'e )]TJ/F22 7.9701 Tf 6.587 0 Td [(a ^ @ )]TJ/F22 7.9701 Tf 6.587 0 Td [(b ^ q m @ +2 e r ^ @ + s ^ q m @ +2 'e )]TJ/F22 7.9701 Tf 6.587 0 Td [(r ^ @ )]TJ/F22 7.9701 Tf 6.586 0 Td [(s ^ q m @ +2 a = b = r = s =0 ; andthesecondandthirdpartsarecomplexconjugateofeachother.Itsucestoconsider )]TJ/F19 11.9552 Tf 5.479 -9.684 Td [(W m ; W m = i 2 4! @ @a @ @b @ @ ijkl Z d 8 d 8 d 4 x {89 @ @ + q m 1 @ +7 e a ^ @ + b ^ q m + ^ d @ +5 'e )]TJ/F22 7.9701 Tf 6.587 0 Td [(a ^ @ )]TJ/F22 7.9701 Tf 6.586 0 Td [(b ^ q m )]TJ/F22 7.9701 Tf 6.586 0 Td [( ^ d Z ijkl a = b = =0 : 63

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Integrationbypartswithrespectto d 'sanduseoftheinside-outconstraint4{67allow foranecientrearrangementoftermstoyieldthenalexpression )]TJ/F19 11.9552 Tf 5.48 -9.684 Td [(W m ; W m {90 = )]TJ/F21 11.9552 Tf 11.955 0 Td [(i 2 4! @ @a @ @b Z d 8 d 8 d 4 x @ @ +4 q m d ijkl e a ^ @ + b ^ q m @ + 'e )]TJ/F22 7.9701 Tf 6.587 0 Td [(a ^ @ )]TJ/F22 7.9701 Tf 7.998 0 Td [(b ^ q m 1 @ +4 Z ijkl a = b =0 : Therefore,theHamiltoniantoorder 2 iswrittenas H 2 = i 2 4 p 2 Z d 8 d 8 d 4 x @ @a @ @b {91 1 2 @ @r @ @s 1 @ +5 e a ^ @ + b ^ q @ +2 'e )]TJ/F22 7.9701 Tf 6.586 0 Td [(a ^ @ )]TJ/F22 7.9701 Tf 6.586 0 Td [(b ^ q @ +2 e r ^ @ + s ^ q @ +2 'e )]TJ/F22 7.9701 Tf 6.586 0 Td [(r ^ @ )]TJ/F22 7.9701 Tf 6.587 0 Td [(s ^ q @ +2 )]TJ/F26 11.9552 Tf 13.948 16.857 Td [( 1 4! @ @ +4 q m d ijkl e a ^ @ + b ^ q m @ + 'e )]TJ/F22 7.9701 Tf 7.998 0 Td [(a ^ @ )]TJ/F22 7.9701 Tf 7.998 0 Td [(b ^ q m 1 @ +4 Z ijkl + c:c: a = b = r = s =0 : 64

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CHAPTER5 CONCLUSION Inthisthesis,wehavediscussedthemaximallysupersymmetrictheoriesinfour dimensionsintermsofthelight-cone LC 2 formalism.Weusedpurelyalgebraic consistenciestoreconstructthedynamicalsupersymmetrytransformationswhichgovern theinteractionsofthemaximallysupersymmetrictheories.Thealgebraictechniqueslead tosomeremarkablenewresults: TherstoneisthatHamiltonianasaquadraticform.TheHamiltonianof N = 4SuperYang-Millstheoryinthe LC 2 formalismcanbewrittenasaquadraticform ofthefermionicsupereld W a thatisspecicformofthedynamicalsupersymmetry transformationonthechiralsupereld H/ W m ; W m : ThesamequadraticstructureoftheHamiltonianof N =8Supergravityisalsofound, whosequartictermsaremorethanninetyterms.Wealsodevelopedanotheralgebraic techniquetocombinesuchmanytermsinacompactandecientway:thecoherent state-likerepresentation. Thesetwomaximallysupersymmetrictheoriesinfourdimensionsshareasalient feature,allthephysicaldegreesoffreedomarecapturedinachiral constraint supereld. Inparticular,bothsatisfytheinside-outconstraintsthatrelateachiralsupereldtoits anti-chiralcomplexconjugate = 1 2 N = 4 d N @ + N = 2 '; where d N = 1 N m 1 m N d m 1 d m N ,whichplayacrucialroleinexpressingtheHamiltonian intoaquadraticform.Thisthereforesuggestthatothermaximallysupersymmetric theoriesthataredescribedbythesamechiral,constraintsupereldmaybeofaquadratic formaswell. 65

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Thesecondresultsisthatallthephysicaleldstransformundertheon-shell E 7 dualitysymmetrygroup.Havingexplicitlyderivedthenon-lineardualitytransformations onthe256physicaleldsof N =8Supergravitywhichiswelldescribedinthe N =8 Superspace,wefoundthatthesetransformationsareelegantlyexpressedintermsthe chiralsupereldatleasttoorder .Surprisingconsequenceisthat,inthisgauge,all elds,includingthegraviton,transformunder E 7 .Thisisunexpectedoutcomewhen itiscomparedwiththeoriginalcovariantformulationsofCremmerandJulia's N =8 Supergravityinwhichthegravitonisinvariant.In LC 2 formalism,thegaugexingprocess andtheeliminationofunphysicaldegreesoffreedomusingtheequationsofmotionwhich yieldsinevitablemixingamongphysicalelds,mayrequirethatalleldstransform. Armedwiththesealgebraictechniques,manychallengingyetintriguingproblems seemtractable.Oneofwhichistoobtaintheinteractiontermsof N =8Supergravityto ordersin using E 7 symmetryinconnectiontonon-linearsigmamodel.Itishopedthat itmightleadtoabetterunderstandingofUVnitenessof N =8Supergravity.Another interestingprojectistond E 8 invariantSupergravityinthreedimensionsintermsof thelight-conedescription.Fairamountofprogresshasbeenmadeandthisworkisclose topublication[15]. 66

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APPENDIXA PSU ; 2 j 4ALGEBRA Thelight-coneformofthe PSU ; 2 j 4algebraisgivenasfollows. [ K;j ]= )]TJ/F21 11.9552 Tf 9.299 0 Td [(K; K;j = K; K; j + = iK + ; K;j + = iK + ; K; j )]TJ/F26 11.9552 Tf 7.085 4.748 Td [( = iK )]TJ/F21 11.9552 Tf 7.084 -4.936 Td [(; K;j )]TJ/F26 11.9552 Tf 7.085 4.748 Td [( = iK )]TJ/F21 11.9552 Tf 7.084 -4.936 Td [(; K + ; j )]TJ/F26 11.9552 Tf 7.085 4.747 Td [( = i K; K )]TJ/F21 11.9552 Tf 7.085 -4.937 Td [(; j + = i K; K + ;j )]TJ/F26 11.9552 Tf 7.085 4.747 Td [( = iK; K )]TJ/F21 11.9552 Tf 7.085 -4.936 Td [(;j + = iK; K + ;j + )]TJ/F26 11.9552 Tf 7.085 4.748 Td [( = )]TJ/F21 11.9552 Tf 9.299 0 Td [(iK + ; K )]TJ/F21 11.9552 Tf 7.084 -4.936 Td [(;j + )]TJ/F26 11.9552 Tf 7.085 4.748 Td [( = iK )]TJ/F21 11.9552 Tf 7.084 -4.936 Td [(; K + ;p )]TJ/F26 11.9552 Tf 7.085 4.747 Td [( = )]TJ/F15 11.9552 Tf 9.299 0 Td [(2 iD +2 ij + )]TJ/F21 11.9552 Tf 7.085 -4.937 Td [(; K )]TJ/F21 11.9552 Tf 7.084 -4.937 Td [(;p + = )]TJ/F15 11.9552 Tf 9.298 0 Td [(2 iD )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 ij + )]TJ/F21 11.9552 Tf 7.085 -4.937 Td [(; K + ;p =2 ij + ; K + ; p =2 i j + ; K )]TJ/F21 11.9552 Tf 7.085 -4.936 Td [(;p =2 ij )]TJ/F21 11.9552 Tf 7.084 -4.936 Td [(; K )]TJ/F21 11.9552 Tf 7.085 -4.936 Td [(; p =2 i j )]TJ/F21 11.9552 Tf 7.084 -4.936 Td [(; K;p + = )]TJ/F15 11.9552 Tf 9.298 0 Td [(2 ij + ; K;p + = )]TJ/F15 11.9552 Tf 9.298 0 Td [(2 i j + ; K;p )]TJ/F26 11.9552 Tf 7.085 4.748 Td [( = )]TJ/F15 11.9552 Tf 9.298 0 Td [(2 ij )]TJ/F21 11.9552 Tf 7.085 -4.936 Td [(; K;p )]TJ/F26 11.9552 Tf 7.085 4.748 Td [( = )]TJ/F15 11.9552 Tf 9.298 0 Td [(2 i j )]TJ/F21 11.9552 Tf 7.085 -4.936 Td [(; [ K; p ]=2 iD +2 j; K;p =2 iD )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 j; j;s m + = )]TJ/F15 11.9552 Tf 10.494 8.087 Td [(1 2 s m + ; [ j; s + n ]= 1 2 s + n j;s m )]TJ/F26 11.9552 Tf 7.989 12.639 Td [( = 1 2 s m )]TJ/F21 11.9552 Tf 7.988 2.956 Td [(; [ j; s )]TJ/F22 7.9701 Tf 7.084 0 Td [(n ]= )]TJ/F15 11.9552 Tf 10.494 8.087 Td [(1 2 s )]TJ/F22 7.9701 Tf 7.085 0 Td [(n j + ;s m )]TJ/F26 11.9552 Tf 7.989 12.639 Td [( = )]TJ/F21 11.9552 Tf 9.299 0 Td [(is m + ; j + ; s )]TJ/F22 7.9701 Tf 7.085 0 Td [(n = )]TJ/F21 11.9552 Tf 9.298 0 Td [(i s + n j )]TJ/F21 11.9552 Tf 7.085 -4.936 Td [(;s m + = )]TJ/F21 11.9552 Tf 9.299 0 Td [(is m )]TJ/F21 11.9552 Tf 7.989 2.956 Td [(; j )]TJ/F21 11.9552 Tf 7.085 -4.936 Td [(; s + n = )]TJ/F21 11.9552 Tf 9.298 0 Td [(i s )]TJ/F22 7.9701 Tf 7.084 0 Td [(n j + )]TJ/F21 11.9552 Tf 7.085 -4.936 Td [(;s m + = 1 2 is m + ; j + )]TJ/F21 11.9552 Tf 7.085 -4.936 Td [(; s + n = 1 2 i s + n j + )]TJ/F21 11.9552 Tf 7.085 -4.936 Td [(;s m )]TJ/F26 11.9552 Tf 7.989 12.639 Td [( = )]TJ/F15 11.9552 Tf 10.494 8.088 Td [(1 2 is m )]TJ/F21 11.9552 Tf 7.989 2.955 Td [(; j + )]TJ/F21 11.9552 Tf 7.085 -4.936 Td [(; s )]TJ/F22 7.9701 Tf 7.085 0 Td [(n = )]TJ/F15 11.9552 Tf 10.494 8.088 Td [(1 2 i s )]TJ/F22 7.9701 Tf 7.084 0 Td [(n 67

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q m )]TJ/F21 11.9552 Tf 8.418 2.956 Td [(;K = p 2 s m )]TJ/F15 11.9552 Tf 172.964 2.956 Td [([ q + n ;K ]= p 2 s + n q m + ; K = )]TJ 9.299 10.473 Td [(p 2 s m + ; q )]TJ/F22 7.9701 Tf 7.085 0 Td [(n ; K = )]TJ 9.298 10.473 Td [(p 2 s )]TJ/F22 7.9701 Tf 7.085 0 Td [(n ; q m )]TJ/F21 11.9552 Tf 8.417 2.955 Td [(;K + = p 2 s m + ; q )]TJ/F22 7.9701 Tf 7.084 0 Td [(n ;K + = )]TJ 9.298 10.474 Td [(p 2 s + n ; q m + ;K )]TJ/F26 11.9552 Tf 7.085 4.748 Td [( = )]TJ 9.299 10.473 Td [(p 2 s m )]TJ/F21 11.9552 Tf 7.989 2.956 Td [(; q + n ;K )]TJ/F26 11.9552 Tf 7.085 4.748 Td [( = p 2 s )]TJ/F22 7.9701 Tf 7.085 0 Td [(n ; s m + ;p = )]TJ 9.299 10.473 Td [(p 2 q m + ; [ s )]TJ/F22 7.9701 Tf 7.084 0 Td [(n ;p ]= )]TJ 9.298 10.473 Td [(p 2 q )]TJ/F22 7.9701 Tf 7.085 0 Td [(n s m )]TJ/F21 11.9552 Tf 7.988 2.955 Td [(; p = p 2 q m )]TJ/F21 11.9552 Tf 8.418 2.956 Td [(; [ s + n ; p ]= p 2 q + n ; s m )]TJ/F21 11.9552 Tf 7.989 2.956 Td [(;p + = p 2 q m + ; s )]TJ/F22 7.9701 Tf 7.085 0 Td [(n ;p + = )]TJ 9.298 10.473 Td [(p 2 q + n ; s m + ;p )]TJ/F26 11.9552 Tf 7.085 4.748 Td [( = )]TJ 9.299 10.473 Td [(p 2 q m )]TJ/F21 11.9552 Tf 8.417 2.955 Td [(; s + n ;p )]TJ/F26 11.9552 Tf 7.084 4.748 Td [( = p 2 q )]TJ/F22 7.9701 Tf 7.084 0 Td [(n ; s m + ; s + n = p 2 m n K + ; s m + ; s )]TJ/F22 7.9701 Tf 7.084 0 Td [(n = p 2 m n K; s m )]TJ/F21 11.9552 Tf 7.989 2.955 Td [(; s )]TJ/F22 7.9701 Tf 7.085 0 Td [(n = p 2 m n K )]TJ/F21 11.9552 Tf 7.085 -4.936 Td [(; s m )]TJ/F21 11.9552 Tf 7.989 2.956 Td [(; s + n = p 2 m n K; q m + ; s )]TJ/F22 7.9701 Tf 7.085 0 Td [(n = )]TJ/F21 11.9552 Tf 9.299 0 Td [(i m n D + j + )]TJ/F15 11.9552 Tf 9.741 -4.937 Td [(+ ij +2 J m n ; q m + ; s + n = )]TJ/F15 11.9552 Tf 9.298 0 Td [(2 i m n j + ; s m )]TJ/F21 11.9552 Tf 7.989 2.956 Td [(; q + n = i m n D + j + )]TJ/F19 11.9552 Tf 9.741 -4.936 Td [()]TJ/F21 11.9552 Tf 11.955 0 Td [(ij +2 J m n ; s m + ; q + n =2 i m n j + ; q m )]TJ/F21 11.9552 Tf 8.418 2.955 Td [(; s + n = i m n D )]TJ/F21 11.9552 Tf 11.955 0 Td [(j + )]TJ/F19 11.9552 Tf 9.741 -4.936 Td [()]TJ/F21 11.9552 Tf 11.955 0 Td [(ij )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 J m n ; q m )]TJ/F21 11.9552 Tf 8.417 2.955 Td [(; s )]TJ/F22 7.9701 Tf 7.084 0 Td [(n =2 i m n j )]TJ/F21 11.9552 Tf 7.085 -4.936 Td [(; s m + ; q )]TJ/F22 7.9701 Tf 7.085 0 Td [(n = )]TJ/F21 11.9552 Tf 9.299 0 Td [(i m n D )]TJ/F21 11.9552 Tf 11.955 0 Td [(j + )]TJ/F15 11.9552 Tf 9.742 -4.937 Td [(+ ij )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 J m n ; s m )]TJ/F21 11.9552 Tf 7.988 2.955 Td [(; q )]TJ/F22 7.9701 Tf 7.084 0 Td [(n = )]TJ/F15 11.9552 Tf 9.298 0 Td [(2 i m n j )]TJ/F21 11.9552 Tf 7.084 -4.936 Td [(; [ J m n ;q r + ]= 1 4 m n q r + )]TJ/F21 11.9552 Tf 11.955 0 Td [( r n q m + ; [ J m n ; q + r ]= )]TJ/F15 11.9552 Tf 10.494 8.088 Td [(1 4 m n q + r + m r q + n ; [ J m n ;q r )]TJ/F15 11.9552 Tf 7.084 2.956 Td [(]= 1 4 m n q r )]TJ/F19 11.9552 Tf 9.742 2.956 Td [()]TJ/F21 11.9552 Tf 11.955 0 Td [( r n q m )]TJ/F21 11.9552 Tf 8.418 2.956 Td [(; [ J m n ; q )]TJ/F22 7.9701 Tf 7.084 0 Td [(r ]= )]TJ/F15 11.9552 Tf 10.494 8.088 Td [(1 4 m n q )]TJ/F22 7.9701 Tf 7.085 0 Td [(r + m r q )]TJ/F22 7.9701 Tf 7.085 0 Td [(n ; [ J m n ;s r + ]= 1 4 m n s r + )]TJ/F21 11.9552 Tf 11.955 0 Td [( r n s m + ; [ J m n ; s + r ]= )]TJ/F15 11.9552 Tf 10.494 8.088 Td [(1 4 m n s + r + m r s + n ; [ J m n ;s r )]TJ/F15 11.9552 Tf 7.084 2.956 Td [(]= 1 4 m n s r )]TJ/F19 11.9552 Tf 9.742 2.956 Td [()]TJ/F21 11.9552 Tf 11.955 0 Td [( r n s m )]TJ/F21 11.9552 Tf 7.988 2.956 Td [(; [ J m n ; s )]TJ/F22 7.9701 Tf 7.084 0 Td [(r ]= )]TJ/F15 11.9552 Tf 10.494 8.087 Td [(1 4 m n s )]TJ/F22 7.9701 Tf 7.085 0 Td [(r + m r s )]TJ/F22 7.9701 Tf 7.085 0 Td [(n ; [ J m n ;J p q ]= m q J p n )]TJ/F21 11.9552 Tf 11.955 0 Td [( p n J m q where J m n = 1 4 @ @ )]TJ/F15 11.9552 Tf 12.894 3.155 Td [( @ @ +2 m n + i p 2 1 @ + q m + q + n : A{1 68

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APPENDIXB E 7 BASICS ThelowestdimensionalrepresentationsofthecompactLiegroups E 7 and SU arepseudorealinthesensethattheyarewrittenintermsofcomplexnumbers,with conjugationbeinganelementofthealgebra.Itisthereforeinstructivetoillustratethe roleofpseudorealityinbuildingquadraticinvariantsinconnectionforthemuchsimpler non-compact SU ; 1group. B.1 SU ; 1 Analysis Inthecompact SU case,considerthe 2 representation Z Z 0 = e i! = 2 Z; writtenintermsoftwocomplexnumbersas Z = 0 B @ x y 1 C A : Foranytwodoublets Z 1 and Z 2 ,wecanformtheinvariantcombination I = Z y 1 Z 2 = x 1 x 2 + y 1 y 2 : Ontheotherhand,wecanconstructanothercombination e Z = 2 Z = i 0 B @ )]TJETq1 0 0 1 344.115 228.251 cm[]0 d 0 J 0.478 w 0 0 m 6.137 0 l SQBT/F21 11.9552 Tf 344.115 221.43 Td [(y x 1 C A ; whichalsotransformsasadoublet,ascanbeseenusingthewell-knownidentity T i = )]TJ/F21 11.9552 Tf 11.956 0 Td [( 2 i 2 : 69

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Thedoubletrepresentationissaidtobe pseudoreal ,thatis,realuptoagrouptransformation 2 isagroupelement.Thisenablesustowriteanotherinvariant I = e Z y 1 Z 2 = Z T 1 2 Z 2 : Thisinvariantvanisheswhen Z 1 = Z 2 :thesingletliesinthe antisymmetric combinationof twodoublets. Ifwemultiplythe 1 ; 2 compactgeneratorsby i ,weobtainthenon-compact generatorsof SU ; 1 i N = i 1 ;i 2 ; 3 : Wenowuse 1 i N 1 = )]TJETq1 0 0 1 332.676 415.021 cm[]0 d 0 J 0.478 w 0 0 m 7.082 0 l SQBT/F21 11.9552 Tf 332.676 408.199 Td [( i N ; tondthatboth Z = 0 B @ x y 1 C A ; e Z = 1 0 B @ x y 1 C A = 0 B @ y x 1 C A ; transforminthesamewayas non-unitary doublets. Wecanreducethenumberofdegreesoffreedombyhalfinthenon-compactcase withoutaectingthetransformationproperties:thechoice e Z = Z isaectedbytaking y = x .Thisisdierentfromthecompactcase.Weconcludethatunlikethecompactcase, wecanimplementthe SU ; 1transformationontwo real variables,afterasuitablechoice ofbasis. Thequadraticinvariantisstillantisymmetric,andgivenby I = Z T 1 2 Z 2 = i x 2 y 1 )]TJ/F21 11.9552 Tf 11.955 0 Td [(y 2 x 1 : 70

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When e Z = Z Z = 0 B @ x x 1 C A ; thequadraticinvariantvanishes. B.2 E 7 Analysis Thefundamental 56 representationofcompact E 7 isalsopseudoreal.Itsquadratic invariantliesinthe antisymmetric productoftwo 56 .Wecanperformaverysimilar analysisbyconsideringtheembedding E 7 SU : 56 = 28 + 28 ; wherethe 28 and 28 arerepresentedbycomplexnumbers x ab = )]TJ/F21 11.9552 Tf 9.299 0 Td [(x ba and y ab = )]TJ/F21 11.9552 Tf 9.299 0 Td [(y ba respectively,where a;b =1 ; 2 ;:::; 8are SU indices.Wewritethe 56 intheform Z = 0 B @ x ab y ab 1 C A ; onwhichthecompact E 7 transformationscanbeunitarilyrealized.Firstthe SU transformationsaregivenby x ab = a c x cb + b c x ac ;y ab = a c y cb + b c y ac ; B{1 wherethesixty-three SU generatorsarewrittenintermsoftracelessantihermitian 8matrices a b = )]TJ/F15 11.9552 Tf 9.298 0 Td [( b a ; a a =0 : B{2 Theremainingseventytransformationsof E 7 areoftheform x ab = abcd y cd ;y ab = abcd x cd ; B{3 71

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parametrizedbythetotallyantisymmetriccomplextensors abcd whichsatisfy abcd = )]TJ/F15 11.9552 Tf 9.299 0 Td [( abcd ; B{4 whichmeansthatthesetransformationsareunitary.Inaddition,thesetensorssatisfythe additionalconstraint abcd = 1 4! abcdefgh efgh ; B{5 andthusrepresent70realdegreesoffreedom. Inthe E 7 non-compactcase,the E 7 =SU transformationsarenotunitaryonly the SU transformationsremainunitary,andwehave abcd =+ abcd : B{6 Ifwewritethenon-compact E 7 nitetransformationsas Z Z 0 = E Z; theysatisfytheconstraints E = 1 E 1 ; E )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 = 3 E y 3 ; B{7 where,in 28blockform, 1 = 0 B @ 0 1 1 0 1 C A ; 2 = i 0 B @ 0 )]TJ/F38 11.9552 Tf 9.299 0 Td [(1 1 0 1 C A ; 3 = 0 B @ 1 0 0 )]TJ/F38 11.9552 Tf 9.298 0 Td [(1 1 C A : B{8 Theseimplythat e Z = 1 Z; B{9 transformsinthesamewayas Z ,andthat 72

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2 = E T 2 E ; B{10 sothatthequadraticinvariantisgivenby I = )]TJ/F21 11.9552 Tf 9.298 0 Td [(iZ T 2 Z 0 = x 0 ab y ab )]TJ/F21 11.9552 Tf 11.955 0 Td [(x ab y 0 ab : B{11 Thismeansthatweneedtwo dierent representationsof E 7 tobuildaquadratic invariant. Wecansustainthe E 7 transformationsonhalfthedegreesoffreedom,byrequiring Z = e Z ,asinlikeinthe SU ; 1case,actingonthe56realcomponents,butthenwe cannotmakeaquadraticinvariant. 73

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APPENDIXC NON-LINEARREALIZATION Non-linearrealizationsofLiesymmetriescanbecastinagenericform,asshown byColeman etal. [21],relyingonlyontheanalyticexpressionofgroupelementsas exponentialinthegenerators,andgroupproperties.Let G beasemi-simpleLiegroup withLiealgebraoftheform f h a gf t g ,withtheschematicstructure [ h;h ] h; [ h;t ] t; [ t;t ] h + t: C{1 Thegenerators h a formasubgroup H ,whichwetaketobecompact.Anynitegroup elementof G ,callit g ,canbewrittenastheproductofthecosetelementandanelement of H as g u;c = e c t e u h ; C{2 intermsofthegroupparameters u a and c u h = P u a h a ;c t = P c t .Thegroup axiomsrequirethattheproductofanytwosuchelementsbeitselfbeagroupelementand thereforewritteninthesameform.Inparticular,foranyelement g u;c of G ,wehave g u;c e t = e 0 t e u 0 h ; C{3 where 0 and u 0 arefunctionsof ;c and u .Thisformulacanbefurthersimpliedwhenwe limitourselvestogroupsforwhich[ t;t ] h .Thisappliestochiralsymmetrygroups, decomposedwithrespecttoavectorsubgroupasconsideredbyColeman etal. ,orto realforms,suchas G = E 7 ,withcompactsubgroup H = SU .Inthiscase,thegroup algebraadmitstheextrasymmetry h a h a ;t !)]TJ/F21 11.9552 Tf 26.567 0 Td [(t ; C{4 correspondingtothegroupelement 74

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g u;c e g = g u; )]TJ/F21 11.9552 Tf 9.298 0 Td [(c : Itfollowsthat e ge )]TJ/F22 7.9701 Tf 6.586 0 Td [( t = e )]TJ/F22 7.9701 Tf 6.586 0 Td [( 0 t e u 0 h : C{5 Wecannoweliminate u 0 ,toarriveat ge 2 A e g )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 = e 2 0 A : C{6 If g isasubgroupelement,thisleadstotheusuallineartransformationlaw, e u h e 2 t e )]TJ/F22 7.9701 Tf 6.586 0 Td [(u h = e 2 0 t ; C{7 whilethecosettransformationsarenon-linearlyrealized e c t e 2 t e c t = e 2 0 t ; C{8 ascanbeseenbycarefullysetting 0 = + ,expandinginpowersof c .Theresultisthe non-linearvariation t = c t + 1 3 [ t; [ t;c t ]]+ : C{9 75

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APPENDIXD USEFULIDENTITIES Thefollowingidentitiesarethesameonesthatappearintheauthor'spaper[11]. iIdentity1 Foranyfunction X ofchiralsupereldsanditsconjugate X Z X 1 @ + X = i 4 p 2 Z d m @ + X d m @ + X: D{1 Provingthisidentityisrathersimple:put @ + @ + onthe X andrewritethe @ + inthe numeratoras i 4 p 2 f d m ;d m g .Thenintegrationbypartswithrespectto d m yieldsthe identity. iiIdentity2 f abc Z 1 @ +2 a b X c =0 : D{2 Usetheinside-outrelationon b followedbyintegrationbypartswithrespectto d 4 Thenswapindices a and b andusetheantisymmetryof f abc toobtainEq.D{2.Asa corollary, f abc Z 1 @ +2 a @ + b X c = )]TJ/F21 11.9552 Tf 11.291 0 Td [(f abc Z 1 @ + a b X c : D{3 iii3-ptfunctionidentity f abc Z a @ @ +3 q + m d m b @ + c = 4 i p 2 3 f abc Z 1 @ + a b @ c : D{4 ThisidentityisessentialtoshowthattheHamiltonianisaquadraticform.Itcanbe veriedbyusingtheexplicitformsof q + and d ,partialintegrations,andtheinside-out 76

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relations.Theprocedureisasfollowstheintegralisomitted.Therststepistoreplace q + and d by and @ @ ,respectively: f abc a @ @ +3 q + m d m b @ + c = i p 2 f abc a @ @ +2 @ @ b @ + c : D{5 Wethenperformthepartialintegrationwithrespectto @ + and @ @ : 4 i p 2 f abc 1 @ +2 a @ b @ + c )]TJ/F21 11.9552 Tf 11.956 0 Td [(i p 2 f abc 1 @ +2 @ @ a @ b @ + c )]TJ/F21 11.9552 Tf 11.955 0 Td [(i p 2 f abc @ @ b 1 @ + a @ c ; D{6 andcalltheresultingterms I II ,and III ,respectively. Thesecondstepistospecializetoeachterm.integrationbypartswithrespectto @ + interm I yields 4 i p 2 f abc 1 @ + a b @ c )]TJ/F15 11.9552 Tf 13.948 0 Td [(4 i p 2 1 @ +2 a c @ + @ b : D{7 ThesecondtermvanishesthankstoEq.D{2.Term I isthen, I =4 i p 2 f abc 1 @ + a b @ c : D{8 Imposingthechiralconditionsonterm II yields d =0 )]TJ/F21 11.9552 Tf 44.517 8.088 Td [(@ @ = i p 2 @ + ; D{9 d =0 )]TJ/F21 11.9552 Tf 44.517 8.087 Td [(@ @ = i p 2 @ + ; D{10 Theseimplythat II = )]TJ/F21 11.9552 Tf 9.299 0 Td [(i p 2 f abc @ @ b 1 @ + a @ b @ + c : D{11 Combining II and III givesus )]TJ/F21 11.9552 Tf 11.955 0 Td [(i p 2 f abc @ @ + @ @ b 1 @ + a @ c : D{12 77

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Now,oneusestheinside-outrelationson c followedbythecommutationrelation [ d m ; @ @ + @ @ ]= q + m ; D{13 toobtain II + III = )]TJ/F15 11.9552 Tf 9.299 0 Td [(2 i p 2 f abc a @ @ +2 @ @ b @ + c ; D{14 Finally,equatingEq.D{5andthesumofterms I II and III provesthe3-ptfunction identity. iv4-ptfunctionidentity Z g 2 f abc f ade n 1 @ + b @ + c 1 @ + d @ + e + 1 2 b c d e o = Z i p 2 g 2 f abc f ade 1 @ + d m b @ + c 1 @ +2 d m d @ + e : D{15 Wehavenotyetsucceededinproducingananalyticproofofthisidentity,butwehave aproofintermsofthecomponentelds.Wecheckedexplicitlythatthefour-scalar interaction,afterintegrationovertheGrassmannvariables, Z )]TJ/F15 11.9552 Tf 10.494 8.088 Td [(1 6 g 2 f abc f ade 1 @ + )]TJ/F15 11.9552 Tf 8.145 -6.662 Td [( C pn b @ + C tnc 1 @ + )]TJ/F21 11.9552 Tf 5.48 -9.684 Td [(C pmd @ + C tm e ; D{16 isfullyreproducedbybothsidesofEq.D{15.Recallthat a;b;c;d;e aregaugeindices. Forcalculationalconvenience,wespecializeto SU .Thenthisexpressionsplitsinto threeoneforeachvalueofthesummedovergaugeindex a ,eachonecontainingdierent elds.Weset C 1 mn D mn ,and C 2 mn E mn ,andtrackdowntermsthatinvolvespecic bi-spinorindicessuchas D 12 D 12 terms.ThenEq.D{16becomes Z 2 3 g 2 1 @ + D 12 @ + E 12 1 @ + E 12 @ + D 12 : D{17 Thistermexactlymatchesthe D 12 D 12 termswhichcomefromthecomponents 78

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Z )]TJ/F15 11.9552 Tf 10.494 8.088 Td [(1 8 g 2 f abc f ade 1 @ + @ + C mn b C mnc 1 @ + @ + C pq d C pqe )]TJ/F15 11.9552 Tf 13.42 8.088 Td [(1 16 g 2 f abc f ade C mnb C pqc C mn d C pq e : D{18 Thealgebraislengthyandnotparticularlyrevealing,althoughtheresultisnon-trivial. Oncewehaveshownitholdsforaparticularcomponent,wecanusethe kinematical supersymmetryvariationstoproducetheotherterms,andthusshowtheveracityofthis claimforallcomponents. 79

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REFERENCES [1]L.Brink,J.H.SchwarzandJ.Scherk,Nucl.Phys.B 121 ,77;F.Gliozzi, J.ScherkandD.I.Olive,Nucl.Phys.B 122 ,253. [2]E.Cremmer,B.JuliaandJ.Scherk,Phys.Lett.B 76 ,409. [3]E.CremmerandB.Julia,Phys.Lett.B 80 ,48;Nucl.Phys.B 159 ,141 [4]L.Brink,O.LindgrenandB.E.W.Nilsson,Phys.Lett.B 123 ,323;S. Mandelstam,Nucl.Phys.B 213 ,149. [5]L.Brink,O.LindgrenandB.E.W.Nilsson,Nucl.Phys.B 212 ,401. [6]A.K.H.Bengtsson,I.BengtssonandL.Brink,Nucl.Phys.B 227 ,31;Nucl. Phys.B 227 ,41. [7]H.Kawai,D.C.LewellenandS.H.H.Tye,Nucl.Phys.B 269 ,1. [8]F.A.Berends,W.T.GieleandH.Kuijf,Phys.Lett.B 211 ,91. [9]Z.Bern,L.J.Dixon,D.C.Dunbar,M.PerelsteinandJ.S.Rozowsky,Nucl. Phys.B 530 ,401;Z.Bern,J.J.Carrasco,L.J.Dixon,H.Johansson, D.A.KosowerandR.Roiban,Phys.Rev.Lett. 98 ,161303 [10]M.B.Green,J.G.RussoandP.Vanhove,JHEP 0702 ,099;Phys.Rev.Lett. 98 ,131602 [11]S.Ananth,L.Brink,S.-S.KimandP.Ramond,Nucl.Phys.B 722 ,166 [12]L.Brink,S.-S.KimandP.Ramond,arXiv:0801.2993[hep-th]. [13]JohnB.KogutandDavisonE.Soper,Phys.Rev.D 1 ,2901 [14]S.Ananth,L.Brink,R.HeiseandH.G.Svendsen,Nucl.Phys.B 753 ,195 [15]L.Brink,S.-S.KimandP.Ramond,arXiv:0804.4300[hep-th]. [16]B.deWitandD.Z.Freedman,Nucl.Phys.B 130 ,105. [17]B.deWit,Nucl.Phys.B 158 ,189. [18]B.deWitandH.Nicolai,Nucl.Phys.B 208 ,323. [19]S.Ananth,L.BrinkandP.Ramond,JHEP 0407 ,082. [20]S.Ananth,L.BrinkandP.Ramond,JHEP 0505 ,003. [21]S.R.Coleman,J.WessandB.Zumino,Phys.Rev. 177 ,2239,C.G..Callan, S.R.Coleman,J.WessandB.Zumino,Phys.Rev. 177 ,2247. 80

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[22]M.K.GaillardandB.Zumino,Nucl.Phys.B 193 ,221. [23]R.KalloshandM.Soroush,arXiv:0802.4106[hep-th]. [24]L.BrinkandP.S.Howe,Phys.Lett.B 88 ,268. 81

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BIOGRAPHICALSKETCH Sung-SooKimwasborninSeoul,Korea.HeearnedhisB.S.andM.S.inphysicsfrom SungkyunkwanUniversityin1996and1998.Afternishinghismilitaryduty,hecameto theUniversityofFloridainthefallof2002andcompletedhisPh.D.programinphysicsin 2008. 82