INTRODUCING SUPERSYMMETRY Martin F. SOHNIUS · Supersymmetry is, by definition, a symmetry between fermions and bosons. A supersymmetric field theoretical model consists of a set

PHYSICSREPORTS(Review Section of PhysicsLetters)128, Nos.2 & 3 (1985)39—204. North-Holland, Amsterdam

INTRODUCING SUPERSYMMETRY

Martin F. SOHNIUSDepartment of Applied Mathematics and TheoreticalPhysics,Universityof Cambridge, CambridgeCB39EW, England

and

The Blackett Laboratory,* Imperial Collegeof Scienceand Technology,LondonSW?2BZ, England

ReceivedFebruary1985

Contents:

Preface 41 4.3. Reducibility andsubmultiplets 85I. Introduction 42 4.4. The productmultiplets 872. Generatorsof supersymmetryandtheir algebra 58 4.5. The “kinetic multiplet” 88

2.1. Generatorsof symmetries 58 4.6. Contragradientmultiplet and invariants 892.2. Canonicalquantisationrules 59 4.7. Dimensions 912.3. Algebraof generators 59 5. The Wess—Zuminomodel 912.4. GradedLie algebras 60 5.1. The Lagrangianandtheequationsof motion 912.5. The Coleman—Mandulatheorem 62 5.2. Auxiliary fields 922.6. The supersymmetryalgebra 63 5.3. Commentson renormalisation 932.7. Summary of algebra 67 5.4. On-shellandoff-shell representations 932.8. Cautionaryremarks 68 5.5. A supercurrent 94

3. Representationsof thesupersymmetryalgebra 69 6. Spontaneoussupersymmetrybreaking(I) 963.1. The “fermions= bosons”rule 69 6.1. Thesuperpotential 963.2. Masslessone-particlestates 71 6.2. Condition for spontaneoussupersymmetrybreaking 973.3. Massiveone-particlestateswithout central charges 74 6.3. The O’Raifeartaighmodel 993.4. One-particlerepresentationswith centralcharges 75 6.4. The massformula 1023.5. Spectrumgeneratinggroups 77 7. N = 1 superspace 1033.6. A representationon fields (theN = 1 chiralmultiplet) 80 7.1. Minkowski space(a toy example) 103

4. N = 1 multiplets andtensorcalculus 83 7.2. Cosetspaces 1054.1. Four-componentnotation 84 7.3. Needfor anticommutingparameters 1064.2. The generalmultiplet 85 7.4. Superspace 107

* Presentaddress.

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INTRODUCING SUPERSYMMETRY

Martin F.SOUNIUS

Departmentof AppliedMathematicsand TheoreticalPhysics,University ofCambridge,CambridgeCB39EW,England

and

TheBlackettLaboratory,Imperial CollegeofScienceand Technology,

LondonSW72BZ, England

INORTH-HOLLAND-AMSTERDMI

Martin F. Sohnius,Introducingsupersymmetry 41

7.5. Representationson superfields 107 12.1. A model with N = 2 supersymmetry 1477.6. Componentfields (generalcase) 109 12.2. The N = 2 mattermultiplet (“hypermultiplet”) 1507.7. Covariantspinor derivatives 110 12.3. The hypermultipletin interaction 1537.8. Chiral superfieldsandchiral parametrization Ill 12.4. Chiral fermions? 1547.9. Submultipletsassuperfields 113 12.5. N = 2 supersymmetryfrom an N = 1 model 1557.10. Superfieldtensorcalculus 114 12.6. SU(2)x SU(2) invarianceand the real form of the

8. Superspaceinvariants 115 hypermultiplet 1568.1. Berezinintegration 115 13. N = 4 supersymmetry 1588.2. Thesuperdeterminant(“Berezinian”) 117 13.1. The N = 4 super-Yang—Millstheory 1588.3. Chiral integrals 118 13.2. Finiteness 1608.4. TheWess—Zuminoactionin superspace 118 14. Supersymmetryin higherdimensions 1638.5. Superfieldequationsof motion 119 14.1. Spinors in higherdimensions 1M8.6. Thenon-renormalisationtheorem 120 14.2. SupersymmetricYang—Mills theory in d = 6 167

9. Supersymmetricgaugetheory (super-QED) 121 14.3. The hypermultipletin d = 6 1699.1. Thegaugecurrentsuperfield 122 14.4. SupersymmetricYang—Mills theory in d = 10 1719.2. Noethercoupling and thegauge superfield 123 15. Thesupercurrent 1759.3. Wess—Zuminogaugeand matterLagrangian 125 15.1. The multiplet of currents 1759.4. The Maxwell Lagrangian 127 15.2. The “multiplet of anomalies” 177

10. Superspacegaugetheories 128 15.3. The charges 17810.1. Ordinarygaugetheories 128 15.4. The supercurrentsuperfield 18110.2. Minimal superspacegauge transformations? 130 16. Conformalsupersymmetry 18210.3. The maximal approach 130 16.1. Deriving theN = I algebrafrom thecurrents 18210.4. Constraints 132 16.2. More about theconformalgroup 18410.5. Prepotentials 134 16.3. Conformal spinors 18510.6. Supersymmetricgaugechoice 134 16.4. The algebraof extendedconformalsupersymmetry 18610.7. Solvingthereality constraint 135 16.5. Multiplets of N = 1 conformalsupersymmetry 18710.8. The field strength superfield and the Yang—Mills 17. Concludingremarks 191

action 135 A. Technicalappendix 19210.9. The matterLagrangian 137 A.!. Minkowski metric 19210.10. SupersymmetricYang—Mills theory(generalcase) 138 A.2. Grouprepresentations 192

11. Spontaneoussupersymmetrybreaking(II) 139 A.3. Representationsof the Lorentzgroup 19411.1. TheFayet—Iliopoulosterm 139 A.4. Two-spinornotation 19511.2. ThesupersymmetricHiggsmodel 141 AS. Four-spinornotation;Majoranaspinors 19711.3. Brokensupersymmetry 143 A.6. Propertiesof bi-spinors; Fierzformula 19811.4. The Fayet—Iliopoulosmodel 144 A.7. Dirac matricesfor arbitraryspace—times 19811.5. Stabilityof massspectra 146 Acknowledgements 203

12. N = 2 supersymmetry 147 References 203

Abstract:A systematicand self-containedintroduction to supersymmetricmodel field theoriesin fiat Minkowskian spaceand to thetechniquesused in

deriving them is given (includingsuperspace).A generaloverviewof supersymmetryandsupergravityis providedin theform of an introductiontothemain body of thereport.

Preface

Modernscienceshareswith both the Greekandearlier philosophiesthe conviction that the observeduniverseis foundedon simple underlyingprincipleswhich can be understoodand elaboratedthroughdisciplined intellectual endeavour.By the Middle Ages, this conviction had, in Christian Europe,becomestratified into a systemof Natural Philosophythat entirely and consciouslyignoredthe realitiesof the physical world and basedall its insights on thought, and Faith, alone. The break with themedievaltradition occurredwhen the scientific revolutionof the 16th and 17th centuriesestablishedanundisputeddominancein the exact sciencesof fact over idea, of observationover conjecture,and of

42 Martin F. Sohnius,Introducingsupersymmetry

practicality over aesthetics.Experimentand observationwere establishedas the ultimate judge oftheory.Modern particlephysics,in seekinga single unified theory of all elementaryparticlesandtheirfundamentalinteractions,appearsto be reachingthe limits of this processandfinds itself forced,in partand often very reluctantly, to revert for guidelines to the “medieval” principles of symmetry andbeauty.

Supersymmetrictheories(the subjectof this report) are highly symmetricand very beautiful.Theyareremarkablein that they unify fermions (matter)with bosons(the carriersof force), either in flatspace (supersymmetry)or in curved space—time (supergravity). Supergravity naturally unifies thegravitationalwith otherinteractions.Noneof the presentmodel theoriesis in any sensecomplete;thehurdleson theway to experimentalprediction— andthus to acceptanceor rejection— havenot yet beencleared.What naive immediatepredictionscan be madeseemto be in disagreementwith nature.Yetthis particularfield of researchappearsto promisesolution of so manyoutstandingproblemsthat it hasexcited great enthusiasmin large parts of the theoretical physics community (and equally largescepticismin others). In a truly philosophicalspirit it has even beensaid of the theory that it is “sobeautiful it mustbe true”.

It is the purposeof this article to makethe concepts,andmanyof thedetails,of thesetheoriesmorewidely accessible.I havebeenfortunate in my friends: many from the “non-particle” domainsof theexactsciences,from e.g. astrophysicsandchemistry,haveexpressedan evermorestridentwish to be letinto thesecretsof the theory as it hasbeendeveloping.They haveurgedme againandagainto exerciseall available means of communication to help them understandwhat it is that we are about.Furthermore,I waspleasantlysurprisedandgreatlyencouragedby the vivid interestin an introductorycourseon theelementsof supersymmetrictheorieswhich I gaveat CERN in the winter of 1981/82.Thepresent article is a widened and, I hope, deepenedversion of those lectures and of the ones Isubsequentlygaveat CambridgeUniversity andat two TriesteSchoolson Supergravity.It is, at present,limited to a thoroughdiscussionof themain aspectsof supersymmetricfield theoriesin flat Minkowsk-ian space—time,but I hopeto completea secondparton supergravityin the not too distantfuture.

Whereas,of necessity,the main body of this report is mathematicallyas rigorous as is compatiblewith clarity andbrevity (I hopeto havemadethis articleof useaswell to my colleaguesin thefield astostudentsor to otherswishing to enterthe field), a feelingof obligationto awider communityhasled meto write, in collaborationwith Dr. J.J.Perry*,aratherlongandqualitativeintroduction.In it we attempttoclarify the aims and the structureof the field of supersymmetryand supergravityin such a way thatsomeonewho is looking for ageneraloverviewmaybe satisfiedby readingonly the introduction.It alsofits the technicalsectionsinto their properplace in the generalframework and tries to clarify theirraisonsd’etre andtheir interconnections.

1. Introduction

The aim of theoreticalphysicsis to describeas manyphenomenaas possibleby a simple andnaturaltheory. In elementaryparticle physics,the hope is that we will eventuallyachievea unified schemewhich combinesall particles and all their interactionsinto one consistenttheory. We wish to makefurtherprogresson the path which startedwith Maxwell’s unification of magnetismandelectrostatics,andwhich hasmorerecentlyled to unified gaugetheoriesof the weakandof the electromagnetic,andperhapsalsoof the stronginteraction.* Institute of Astronomy,Universityof Cambridge.

Martin F. Sohnius, Introducing supersymmetry 43

The purposeof this report is to introducethe readerto a developmentin theoreticalparticlephysicswhich carriesour hopesof being led further alongthat path: supersymmetry.

Beforedescribinganddiscussingthe origins andthe physicalcontentof supersymmetrictheories,wegive a very brief generaldefinition of suchtheories.A lengthierdiscussionof the historicalcontextoutof which theyhavearisenand of their physical and theoreticalassumptionsfollows. Togetherwith acondenseddescriptionof the important featuresof the theoriesthemselves,this forms the main part ofthis introduction, followed by a summaryof wherein the text of the report the detailsareto be found.

Supersymmetryis, by definition, a symmetry betweenfermions andbosons.A supersymmetricfieldtheoreticalmodelconsistsof a set of quantumfields andof a Lagrangianfor them which exhibit suchasymmetry.The Lagrangiandetermines,through the Action Principle, the equationsof motion andhencethe dynamicalbehaviourof the particles. A supersymmetricmodel which is covariantundergeneralcoordinatetransformationsor, equivalently,a model which possesseslocal (“gauged”)super-symmetry is called a supergravitymodel. Supersymmetrictheoriesdescribemodelworlds of particles,createdfrom the vacuumby the fields,andthe interactionsbetweentheseparticles.The supersymmetrymanifestsitself in the particle spectrumand in stringentrelationshipsbetweendifferent interactionprocessesevenif theseinvolve particlesof different spin andof different statistics.

Both supersymmetryandsupergravityaim at a unified descriptionoffermions and bosons,andhenceof matter and interaction.Supergravityis particularly ambitious in its attempt at unification of thegravitational with the other interactions.All supersymmetricmodels succeedto some degreein theseaims,but theyfail in actuallydescribingtheworld as we experienceit andthusaremodels,not theories.We are still striving to find somecontactbetweenoneof the modelsand physicalreality so that thatmodelcould becomean underlyingtheory for natureat its most fundamentallevel.

By “most fundamentallevel” particle physicistsmean at presentthe decompositionof matterintoquarksand leptons(fermions)andthe understandingof all forcesbetweenthem as arisingout of fourtypes of basic interactions, gravitational, weak, electromagneticand strong. These are describedinterms of exchangeparticles (bosons).The framework within which thesebuilding blocks makeup aphysicaltheory is relativistic quantumfield theory.Seenat this level, “unification” ought to include allfour interactions.Thereis, however,a quantitativeandqualitativedifferencebetweenthe gravitationalinteractionandthe otherswhich hashadprofoundconsequencesboth for the structureof the Universeandfor our understandingof it.

The purely attractivenature and the long rangeof the gravitationalforce were responsiblefor itsbeing the first of the forces in nature to be accessibleto the rigorous discipline of experiment andobservation.We now know that it is the only fundamentalforcewhich addsup for macroscopicbodiesand which is independentof the material composition of thesebodies. Gravity as a phenomenonisobvious,relativelyeasyto measure,and— at leastin the Newtonianlimit — rathersimple to understandandto describe.The studyof the effectsof gravitationalforces,first in the solarsystem,then on massivephysicalobjectsin the laboratory,led to the formulation of Newtonianmechanics,the first field in thenatural scienceswhich could be describedin purely mathematicalterms. The initial understandingofgravity andthe subsequentknowledge that grew out of it gaverise to a mathematicaldescriptionofmechanicalprocesses,basedon the conceptof force beingthe causeof acceleratedmotion. This is stillthe current understandingof “force”. In the Industrial Revolution, this scientific knowledge helpedto change the world fundamentally and forever. The “technological forces” of water, steam andmachinery,althoughperhapsnot fully understoodat their ownmost fundamentallevel, could certainlybe exploitedby a technologythat was entirelybasedon classicalmechanics.Technology,in turn, madeexperimentspossible throughwhich physicistslearnedto keepelectric chargesapart.They were thus


able to investigatethe propertiesof the electromagneticforce, the otherlong-rangeforce in nature. Bythe end of the 19th century, this had led to the establishmentof the beautiful edifice of classical

‘electrodynamics,culminatingin Maxwell’s equationsandin yet moreefficient machines.At the turn ofthis century, the observationof processeswhich are basically electromagneticin nature (the constantspeedof light, the spectrumof black-bodyradiation and the photoelectriceffect) brought about aconceptualrevolutionandwith it the breakdownof classicalphysics.At the sametime, the discoveryofradioactivedecaybrought with it the discoveryof the short-rangeweak and strong interactions; theestablishmentof quantum mechanicshad openedthe way for understandingthem as well. Theunderstandingof the gravitationalinteractiontook anotherdirection: GeneralRelativity doesnot seeitas an actual“interaction” but ratheras a global propertyof spaceandtime itself. Interestinglyenough,both developmentsplayed a similarly important role in their impact on the public imagination;theshock of the First World War and of the political revolutions in its wake made the simultaneousbreakdownof theold political order manifest,an orderwhich in its sureview of theworld hadbeensoakin to the determinismof classicalphysics.

Unification of gravity with the otherforces is an elusive goal. Sincegravity is always attractiveandlong ranging, it would make all complex and large physical objectscollapseunder their own weightwere the other interactions not so much stronger at short distances.This difference in strength,necessaryfor the world as we seeit to exist,hasin turn set gravity so far apartfrom therest of physicsthat it is very difficult to think of any experimentwhich could actually test predictionsof a unifiedtheory of all interactions and even less of one that could provide experimentalinput into theconstructionof such a theory.The naturaldomain of Newtoniangravity andof its modernreplacement,Einstein’s theory of GeneralRelativity, is the world of largedistancesandmassiveagglomerationsofmatter;that of the otherforces is the world of atoms,nuclei andelementaryparticles.Many ordersofmagnitudeseparatethe two.

The strong,electromagneticandweakinteractionsarenowfairly well understood.It hasbeenfound thattheir exchangeparticlesarisenaturallyin aquantumfield theory,if thatfield theoryis requiredtobelocallygaugeinvariant.

The conceptof gaugeinvariancegrew out of the observationthat if a “charge” (e.g.electriccharge,total energy,isospin, etc.) is conservedin a dynamicalsystem,then the Lagrangianfor the systemisinvariant under“global gaugetransformations”of the fields. For example,the electricchargeis relatedto invarianceunderphasetransformationss,/,—* ~i~O~/J for all fields i/i which describeparticlesof chargeq.Similarly, the energy is related to time translations~ x)-~çli(t+~t,x).The converseis also true(Noether’stheorem):if the Lagrangianis invariant undersomeinfinitesimal transformation~li—~ ~1i+ ~i,

then there is a conservedcurrent and a conservedchargeassociatedwith this gauge invariance.(“Gauge” is an unfortunatemisnomer,originating in an attempt by H. Weyl in 1918 to relate theelectric chargeto a rescalingtransformation~li~~~eAç1,.)We call the transformations“global” if theirparametersdo not dependon the space—timecoordinates,i.e. if 6 = const.This relationshipbetweenconservedquantumnumbersandglobal symmetriesof the Lagrangianled, in the 1960’s, to a searchforglobally gauge-invariantfield theoriescapableof describingandclassifying all elementaryparticles.The“8-fold way” was a symmetry very much in this vein and it was in this contextthat quarkswere firstpostulatedas building blocks of strongly interactingmatter.But that was not yet the full story.

The requirementof local gauge invariance (also called “gauge invarianceof the 2nd kind”) goesbeyond that which can be inferred from chargeconservation.We now demandinvarianceof theLagrangianundertransformationswith a space—timedependentparameter6 = 6(x). This is in generalonly possibleif an additional field is introduced,the gaugepotential,whosequantawill interactwith all

Martin F Sohnius, Introducing supersymmetry 45

the chargedparticles.This interaction,which resultsin theexchangeof field quanta,will generateforcesbetweenthe particles.“Gauging” the phasetransformationsassociatedwith electriccharge(i.e. makingthem x-dependent)forcesus to introducethe electromagneticfour-vectorpotentialand, as its quanta,the photons.The result is quantumelectrodynamicsor, for classicalfields, Maxwell’s electrodynamics.Requiring other gauge invariances requires additional gauge potentials which give rise to moreexchangeparticles and the other interactions.These exchangeparticlesare the long predictedandrecently discoveredW~andZ°for the weak force andthe gluonsfor the stronginteractions.The latterhaveonly beenindirectly seenin their effects on the distribution of the debrisin high-energyparticlecollisions (jets). To summarise:“gauging” an invarianceof the Lagrangian will always give rise tointeractionandto forces.Note that the name“gaugetheory” is nowadaysusedexclusivelyfor theorieswith local gaugeinvariance.

The gaugetransformationsunderwhich a Lagrangianis invariantfulfill the axiomsfor a group(in themathematicalsense):two subsequentoneswill againbe an invariance,“no transformation”is the unitelement,thereis exactly onetransformationwhich reversesthe effect of eachtransformation,andthreetransformationsare associative.Usingthe standardterminologyfor groups,the respectivegaugegroupsare SU(3) for the strong interactionsand SU(2)x U(1) for the electro-weakinteractions.The SU(3)transformationsacton triplets of quarkswhosepropertiesarevery similar. They are saidto differ onlyin “colour”, hencethe namequantum-chromodynamics(QCD) for theSU(3)gaugetheoryof the stronginteractions.The successof gaugetheoriesin describinga variety of elementaryparticle phenomenainthe past ten years has eclipsed the role played by global invariance, and nowadayssuch globalsymmetriesare thoughtof as more or less accidental— if indeedthey are present.In this context it isalreadyimportantto mention that local (gauged)supersymmetrywill imply gravity.

The theoriesfor both gravitation and elementaryparticle interactionsare well establishedwithintheir respectivedomains. In submicroscopic physics, for the massesand distances involved, thedeviationsintroduced by gravity from the flat Minkowskian metric are so minute that elementaryparticles, or atoms, can safely be treatedas if gravitation did not exist. Any “true”, i.e., generallycovariant,theory should thus be closely approximatedby Lorentz-covariant,non-gravitatingtheories.We must,however,demandof the “true” theory that it be mathematicallyconsistentandthat it predictthecorrectflat limit. Any quantumtheory of gravitationso far fails to do so.

Let usarguebriefly why the gravitationalfield ought to be quantisedat all. Lacking a betterworkinghypothesis,we assumethat the laws of physics extrapolateinto extreme regions. The theory ofgravitation hasso far been testedonly in systemswhereat least one of the massesinvolved is ofastronomicalmagnitude.Yet this same theory is assumedto also describethe gravitational effects,howeverweak,between,say,the protonandthe electronin an hydrogenatom.If the gravitationalfieldwere not quantised,experimentscould be imaginedwhich could, in principle, simultaneouslymeasurethe position and the momentumof an electronto arbitrary precision. If we decide to postulatethevalidity of Heisenberg’suncertaintyrelationeven in this extremecase,we mustassumethe gravitationalfield to be quantised.

Perhapswe should not extrapolateto very small distancesGeneralRelativity’s view of spaceandtime as a continuumwhosestructureis governedby the matterin it. In that case,an entirely differentview of the ultimate high energyphysicswould invalidatemostof our presentattempts.

Once we assumequantisedgravitation, the problem of renormalisability becomesvery important.Renormalisationis requiredin quantumfield theory in order to makesenseof divergentintegralswhichappearin the perturbationexpansionfor physicalprocesses.Suchexpansionsareunfortunatelythe onlycalculationaltools currently availablefor solving the equationsof motion of thetheory; theyareusually

46 Martin F Sohnius, Introducing supersymmetrv

conceptualisedin terms of vaccum polarisation and virtual particle interactions and illustrated byFeynmangraphs.In renormalisabletheoriesthe divergenceswhich appearcan betreatedby redefining,in eachorderof the perturbationexpansion,a finite numberof theoreticalparametersin such a waythat the resultsof “test experiments”arereproduced.Other processescan then be calculateduniquelyto the sameorder.To lowestorder, the parameterswhich must be so renormalisedtypically representvacuumenergies,masses,coupling constantsand factors which multiply wave-functions.Correspond-ingly, onespeaksof “vacuum,mass,couplingconstantandwave-functionrenormalisation”.Oneof thestrongestmotivationsfor gaugetheoriesis their renormalisability.

A theory is called non-renormalisableif infinitely many parametersmust be redefined. Such a“theory” can makeno predictionsandis thereforenot a theoryin the senseof exactscience.In general,and this is rathereasyto see oncethe actualmathematicaldetailsare considered,coupling constantswith negativemassdimensions(for h = c = 1) lead to non-renormalisabletheories.No matterhow weattemptto quantisegravity, we endup with a field theory whosecouplingconstant,Newton’sconstantG, hasdimension[1/mass2lin theseunits and quantumgravity is thereforenon-renormalisable.

The energyat which gravity and quantumeffects becomeof comparablestrengthcan be estimatedfrom the only expressionwith the dimensionof an energythat can be formedfrom the constantsofnatureh, c and G,

E~1=c2V~7~,1019GeV. (1.1)

It is in the regionof this energy,the Planckenergy,whereour presenttheoriesfor the gravitationalandthe otherinteractionsbecomeincompatiblewith eachotherandwherewe expecta possibleunificationof the interactionsto becomemanifest.A point particle with Planckmasswould havea Schwarzschildradiusequal to twice its Comptonwavelength.The veryremotenessof such an energyregion (10’~GeVequalsabout 500 kilowatt-hours)eliminates all hope for direct experiment.Perhaps,if we are verylucky, someisolatedpredictionof such a unified theory could be testableon a systemthat integratesaminute effect over a vast range(proton decay experimentsare of this type, where large numbersofavailableprotonscan makevery smalldecayprobabilitiesmeasurable).We can, however,not expectexperimentalphysicsto give usmuchreliable guidanceinto the “Planck region”.

The SU(3)x SU(2)X U(1) picture of the other forces is not yet a “unified” one. The only propertywhich unites them is that they areeachdescribedby a gaugetheory. The fact that the direct productstructureof SU(2)x U(1) is “skew” (by the weakmixing angle)againstthe naturaldistinctionbetweenthe weakandthe electromagneticforce maysuggestsomeunderlyingunified schemewhich can predictthat angle.A lot of work has,over many years,gone into finding a largergaugegroupwhich woulddescribeall threeinteractionsat somehigh energy. If such a GrandUnification occurredit is knownthat it musthappenat energiesof about 10~GeV, only four ordersof magnitudelessthanE~1.GrandUnified Theories(GUTs) havehad somesuccesses(suchas the predictionof the mixing angle) andsomefailures: at thetime of writing the measuredlower limit for the proton lifetime is growing beyondthepoint to whichonecanstretchthe simplestGUTs. In any case,evena GrandUnified Theorywould atmostunify different kindsof interaction(strongandelectro-weak)with eachotheranddifferent kindsofmatter(quarksandleptons)witheachother.Theunificationof matterwith interactionisnot oneof theaimsof GUTs.

What is it thenthat pointsin the direction of supersymmetrictheoriesfor a solution to the unificationproblem?Already the most obviousdifference betweengravity and, say,electrodynamics,namelytheabsenceof negativegravitationalcharges,can be shownto imply that only a supersymmetrictheory can

Martin F Sohnius,Introducingsupersymmetry 47

unify them.As long as we do not dramaticallydeviatefrom standardquantumfield theory,andwe hopethatthat will not benecessary,apurely attractiveforcemustbe carriedby afield with evenintegerspin.The electromagneticforce, on the otherhand,whichis — of course— not alwaysattractive,is carriedby afield with spin one.A numberof no-gotheorems,aboutwhich we will have to learn morelater, forbidany direct symmetry transformationsbetween fields of different integer spin and actually leavesupersymmetrictheoriesas the only field theoreticalmodels which might achieveunification of allforcesof nature.Supersymmetrytransformationsdo not directly relatefields of different integerspin,rather they relate a graviton (the quantumof gravity, with spin 2) to a photon (spin 1) via a spin ~intermediary,the “gravitino”. A partial unification of matter(fermions)with interaction(bosons)thusarisesnaturally out of the attemptto unitegravity with the otherinteractions.

The no-go theoremsimply that supersymmetryand supergravityare the only possibilities forunification within the framework of quantumfield theory.Failure of thesetheoriesto ultimately giveresultswhicharecompatiblewith the realworld would force us to give up eitherunificationor quantumfield theory. It is our reluctanceto do so which keeps this particular researchalive in spite of theabsence,as it has beensaid, of “even a shredof experimentalevidence”.There is, in principle, onejustification for researchinto theoreticalphysics other than predictingnew data. That is to interpretexistingdata over a wider rangeof parametersthan before.Therefore,if supergravitycould comeupwith a model which is self-consistentand incorporatedGeneralRelativity at the long-distancelow-energyendof its phenomenologyand “standard”particlephysicson the high-energyside, thenit wouldbe a grandsuccessevenwithout any “new” testablephysics.This latter possibility is by no meansruledout.

Apart from the no-go theorems,thereis a further, more technicalpoint that singlesout supersym-metric theories:they may resolvethe non-renormalisabilityproblemof quantisedgravity. In pertur-bativequantumfield theory fermions andbosonstendto contributewith oppositesignsto higher-ordercorrections.Supersymmetryalmost always achievesa fine-tuning betweenthesecontributions whichmakessome a-priori presentand divergentterms vanish.For a long time now there hasbeengreatoptimismthat oneof the supergravitymodelsmaybe entirely free of infinities andthusbe a consistentmodelof quantisedgravitationwith no needfor renormalisation.This hopeis slowly fading,but only tothe extent that no obvious proof can be found for the conjecturedfiniteness.What remainsis aremarkablyfriendly behaviourof all supersymmetrictheorieswhen it comesto quantumdivergences,andthe convictionof manytheoriststhat it would be surprisingif Naturedidn’t makeuseof it. We willlater haveto say more aboutcancellationsof divergencesand the enormousinterestwhich they havearoused,not only for gravity but also for the “hierarchy problem” of GUTs, the thirteen ordersofmagnitudebetweenthe GUT massof 1015GeV/c2andtheW-bosonmass.Normally, agapof this size isnot stablein perturbationtheory becauseof the considerableadmixtureof the largemassto the smallone through exchangeof virtual particles and through vacuum polarisation. The gap can only bemaintainedby repeatedfine-tuningup to highordersin the perturbationexpansion.In supersymmetricversionsof GUTsnew particlesare exchangedandpair-created,and thesenew processescancelsomeof the effectsof the old ones.Massmixing andconsequentfine-tuningcan usually be avoided,andthehierarchy,onceestablished,is stabilised.

Progressin physicsas in all the scienceshasalmost alwaysbeenbasedon an interplay of two quitedifferent approachesto nature: the onestartsby collecting andorderingobservationalor experimentaldata(“Tycho”), thendescribesthesedataby a smallnumberof empiricallaws (“Kepler”), andfinally“explains” theselaws by a theory,basedon a few principles(“Newton”). Theoreticalpredictionsof theoutcomeof further, more refined, observationsandexperimentscan then be made (“discovery of

48 Martin F Sohnius,Introducing supersymmetry

Neptune”).The otherapproachstartsfrom an idea, formulatesit in termsof a theory,andproceedstomakepredictionswhich thenact as testof the theoryandof the original idea. The latter approach— inits pure form — has been most dramatically and singularly successful in Einstein’s developmentofGeneralRelativity. Supersymmetryhas also startedwith just an idea (althoughwithout an Einstein),andat the momentwe areall searchingfor the physicalevidenceto confirm it.

Let usestablishsomeof the rulesof the game.For one, the field is still a game,albeit a seriousone.First andforemostwe postulatethe existenceof asymmetry betweenfermionsandbosonswhich shouldunderly the laws of physics. No experimentalobservationhas yet revealedparticlesor forceswhichmanifestly show such a symmetry.Thereforethe developmentof a theory basedon supersymmetryrequiresan understandingnot only of how the varioussymmetrytransformationsaffect eachother(thealgebra) but also of all possible systems (multiplets of particles or quantumfields) on which thesupersymmetrytransformationscan act. The symmetryoperationswill transformdifferentmembersof amultiplet into eachother.More precisely,the transformationsareto berepresentedby linear operatorsactingon the vector space(the “representationspace”)spannedby the multiplet. Finally, the theorymust predict the time developmentof interactingphysical systems.This is usually achievedby findingappropriateHamiltonians or Lagrangians.The supersymmetrypresent in the physical system willmanifestitself in an invarianceof this Lagrangian— or ratherof its integralover all time, the action— ifall fields undergotheir respectivesupersymmetrytransformations.Becauseof the lack of experimentalinput, a largefraction of the researcheffort of supersymmetrytheoristshas,in fact, beendevotedto thefinding of, andexplorationof, possiblesupersymmetryrespectinginteractions.

The theoretical framework in which to construct supersymmetricmodels in flat space—timeisquantumfield theory,andit mustbe pointed out that

the standardconceptsofquantumfield theory allowfor supersymmetrywithoutanyfurther assump-tions.

The introductionof supersymmetryis not a revolution in the way one viewsphysics.It is an additionalsymmetry that an otherwise“normal” field theoreticalmodel can have. As we shall see,all that isrequiredfor a field theory to be supersymmetricis that it containsspecifiedtypesandnumbersof fieldsin interaction with each other and that the various interaction strengthsand particle masseshaveproperly relatedvalues.As an example,considerthe SU(3)gaugetheory of gluons, which can be madesupersymmetricby including a masslessneutral colour octet of spin ~ particleswhich are their ownantiparticles.Jargonhas it that such spin ~ partnersof the gluons are called “gluinos”. If our modelcontainsnot only gluons but alsoquarks,we mustalsoaddcorrespondingpartnersfor them.Thesehavespin 0 and are commonly called “squarks”. (Procedureslike theseare employedparticularly in theconstructionof supersymmetricGrandUnified Theories— superGUTsor susyGUTs.)

Beforeweproceedto discusstheingredientsof supersymmetricmodels,wemustaddressthequestionoftheFermi—Bose,matter-forcedichotomy.After all, the wave-particledualityof quantummechanicsandthesubsequentconceptof the “exchangeparticle” in perturbativequantumfield theory seemedto haveabolishedthat distinction for good. The recent triumphal progressof gaugetheorieshas, however,reintroducedit: forcesare mediatedby gaugepotentials,i.e., by vector fields with spin one, whereasmatter(including integer-spinmesons)is built from quarksandleptons,i.e., from spin ~fermions.TheHiggs particles, necessaryto mediate the neededspontaneousbreakdownof some of the gaugeinvariances(more about this later), play a somewhatintermediaterole. They must havezero spin andare thusbosons,but they arenot directly relatedto any of theforces.Puristshopeto seethem ariseas


bound statesof the fermions (condensates).Supersymmetrictheories,and particularly supergravitytheories,“unite” fermionsandbosonsinto multiplets andlift the basicdistinction betweenmatterandinteraction.The gluinos, for example, are thought of as carriersof the strong force as much as thegluons,exceptthat as fermionstheyobey an exclusionprinciple andthuswill neverconspireto form acoherent,measurablepotential.The distinctionbetweenforcesand matterbecomesphenomenological:bosons— and particularly masslessones— manifest themselvesas forces becausethey can build upcoherentclassicalfields; fermionsareseenas matterbecauseno two identicalonescan occupythe samepoint in space— an intuitive definition of materialexistence.

For some time it was thought that symmetrieswhich would naturally relate forces and fermionicmatter would be in conflict with field theory. The progressin understandingelementaryparticlesthroughthe SU(3)classificationof the “eight-fold way” (a global symmetry)hadled to attemptsto finda unifying symmetry which would directly relate to eachotherseveralof the SU(3) multiplets (baryonoctet, decuplet,etc.), evenif thesehad different spins. The failure of attemptsto makethose “spinsymmetries”relativistically covariantled to the formulation of a seriesof no-gotheorems,culminatingin 1967 in a paperby Colemanand Mandula [8] which was widely understoodto show that it isimpossible,within the theoreticalframework of relativistic field theory, to unify space—timesymmetrywith internalsymmetries.More precisely,the theoremsaysthat the chargeoperatorswhoseeigenvaluesrepresent“internal” quantumnumberssuch as electric charge, isospin, hypercharge,etc., must betranslationallyandrotationallyinvariant.This meansthat theseoperatorscommutewith theenergy,themomentum and the angular momentum operators.Indeed, the only symmetry generatorswhichtransform at all under both translationsand rotations are those of the Lorentz transformationsthemselves(rotationsand transformationsto coordinatesystemswhich move with constantvelocity).The generatorsof internal symmetriescannotrelate eigenstateswith different eigenvaluesm2 and1(1 + 1)h2 of the mass and spin operators.This meansthat irreduciblemultiplets of symmetry groupscannotcontainparticlesof different massor of different spin. This no-gotheorem,whichis discussedinmuchmoredetailin section2, seemedto rule out exactlythe sort of unity whichwas sought.Oneof theassumptionsmadein ColemanandMandula’sproof did,however,turn out to be unnecessary:theyhadadmittedonly thosesymmetrytransformationswhich form Lie groupswith real parameters.Examplesof such symmetriesare spacerotations with the Euler angles as parametersand the phasetrans-formations of electrodynamicswith a real phaseangle 0 aboutwhich we talked earlier.The chargeoperatorsassociatedwith such Lie groupsof symmetry transformations(their generators)obey welldefined commutationrelations with each other. Perhapsthe best-knownexampleis the set of com-mutators L~L~— L

5L~ [Li, L~]= ihL~, for the angular momentum operators which generatespatial rotations.

Different spins in the same multiplet are allowed if one includes symmetry operationswhosegeneratorsobey anticommutation relations of the form AB + BA = {A, B} = C. This was firstproposedin 1971 by Gol’fand and Likhtman [24], andfollowed up by Volkov and Akulov [70] whoarrived at what we now call a non-linearrealisationof supersymmetry.Their model was not renor-malisable.In 1973,WessandZumino[71] presenteda renormalisablefield theoreticalmodelof a spin ~particle in interaction with two spin 0 particleswhere the particles are relatedby symmetry trans-formations,and therefore“sit” in the samemultiplet. The limitations of the Coleman—Mandulano-gotheoremhad beencircumventedby the introductionof a fermionic symmetryoperatorwhich carriedspin ~, andthuswhenactingon a stateof spinj resultedin a linear combinationof stateswith spinsj + ~

andj — ~. Suchoperatorsmust anddo observeanticommutatorrelationswith eachother. They do notgenerateLie groupsandarethereforenot ruled out by the Coleman—Mandulano-gotheorem.

50 Martin F. Sohnius, Introducingsupersymmetry

In the light of this discovery, Haag,Lopuszañskiand I [34] extendedthe resultsof ColemanandMandulato includesymmetryoperationswhich obeyFermi statistics.We proved that in the contextofrelativistic field theory the only modelswhich can lead to a solution of the unification problemsaresupersymmetrictheories.Theclassof suchtheoriesis, however,ratherrestricted:wefoundthatspace—timeandinternalsymmetriescanonlyberelatedto eachotherby fermionicsymmetryoperators0 of spin~(not~or higher)whosepropertiesareeither exactly thoseof the Wess—Zuminomodel or areat leastcloselyrelatedto them.Only in the presenceof supersymmetrycan multiplets containparticlesof differentspin,suchasthegravitonandthephoton.We classifiedall supersymmetryalgebraswhich canplayarole in fieldtheory.Theseresultsarepresentedin moredetailin section2,andweshallheresketchonlytheessentialsofthe supersymmetryalgebras.

Supersymmetrytransformationsaregeneratedby quantumoperators0 which changefermionicstatesinto bosoniconesandvice versa,

0 fermion)= boson); 0 boson)= fermion). (1.2)

Which particular bosonsandfermionsarerelatedto eachother by the operationof somesuch 0, howmany 0’s thereare, and which propertiesother than the statisticsof the statesare changedby thatoperationdependson thesupersymmetricmodelunderstudy.Thereare,however,anumberof propertieswhich arecommonto the 0’s in all supersymmetricmodels.

By definition, the Q’s changethe statisticsand hencethe spin of the states. Spin is related tobehaviourunderspatialrotations,andthussupersymmetryis — in somesense— a space—timesymmetry.Normally, and particularly so in modelsof “extendedsupersymmetry”(which is definedbelow, N = 8supergravitybeing one example),the 0’s also affect someof the internal quantumnumbersof thestates.It is thisproperty of combininginternal with space—timebehaviourthat makessupersymmetricfield theoriesinterestingin the attemptto unify all fundamentalinteractions.

As a simple illustration of the non-trivial space—timepropertiesof the 0’s considerthe following.Becausefermions andbosonsbehavedifferently underrotations,the 0 cannotbe invariant undersuchrotations. We can, for example, apply the unitary operatorU which, in Hilbert space,representsarotationof configurationspaceby 360°aroundsomeaxis. Thenwe get from eq. (1.2) that

UQ boson)= UQU1U boson)= U fermion)

UQ fermion)= UQU1 U fermion)= U boson).

Sincefermionicstatespick up a minussign whenrotatedthrough 360°and bosonicstatesdo not,

U fermion)= —jfermion); U boson)= boson),

andsince all fermionicand bosonicstates,takentogether,form a basis in the Hilbert space,we easilyseethat we must have

UQU’=—O. (1.3)

The rotatedsupersymmetrygeneratorpicks up a minus sign, just as a fermionic statedoes.One can

Martin F Sohnius,Introducing supersymmetry 51

extendthis analysis and show that the behaviourof the 0’s underany Lorentz transformation— notonly underrotationsof the coordinatesby 360°— is preciselythat of a spinor operator. More technicallyspeaking,the 0’s transformlike tensoroperatorsof spin ~ and,in particular,theydo not commutewithLorentz transformations.The resultof a Lorentz transformationfollowed by a supersymmetrytrans-formation is different from that whenthe orderof the transformationsis reversed.

It is not as easyto illustrate,but it is neverthelesstruethat, on the otherhand,the 0’s are invariantundertranslations: it doesnot matterwhetherwe translatethe coordinatesystem before or after weperform a supersymmetrytransformation.In technicalterms, this meansthat we havea vanishingcommutatorof 0 with the energyand momentumoperatorsE and F, which generatespace—timetranslations:

[Q,E]= [Q,P]=0. (1.4)

The structure of a set of symmetry operationsis determinedby the result of two subsequentoperations.For continuoussymmetrieslike spacerotationsor supersymmetry,this structure is bestdescribedby the commutatorsof the generators,such as the ones given above for the angularmomentumoperators.The commutatorstructureof the 0’s with themselvescan best be examinedifthey areviewedas productsof operatorswhichannihilatefermionsandcreatebosonsin their stead,orvice versa. It can then be shown (seesection2) that the canonicalquantisationrules for creationandannihilation operatorsof particles (and in particular the anticommutatorrules for fermions, whichreflect Pauli’s exclusion principle) lead to the result that it is the anticommutatorof two 0’s, not thecommutator,which is againa symmetrygenerator,albeit oneof bosonicnature.

Let us considerthe anticommutatorof some0 with its HermitianadjointQ~.As spinorcomponents,the 0’s are in generalnot Hermitian, but {Q, Qt} QQ~+ Q~0is a Hermitian operatorwith positivedefinite eigenvalues:

~.. ~ ~ QtoI...)= o~.. ~>J2± 01. ~ (1.5)

This can only be zero for all states ...) if 0 = 0. A more detailed investigationwill show that

{ Q Qt} mustbe a linear combinationof the energyandmomentumoperators:

{Q Qt}= aE+/JP. (1.6)

This relationshipbetweenthe anticommutatorof two generatorsof supersymmetrytransformationsonthe onehand and the generatorsof space—timetranslations (namely energyand momentum)on theother, is central to the entire field of supersymmetryandsupergravity.It meansthat the subsequentoperationof two finite supersymmetrytransformationswill inducea translationin spaceandtime of thestateson which theyoperate.

Thereis a further importantconsequenceof the form of eq. (1.6). Whensummingthisequationoverall supersymmetrygenerators,we find that the fiP termscancelwhile the aB termsaddup, sothat

~ {Q,Qt}ccE. (1.7)all 0

Dependingon the sign of the proportionality factor, the spectrumfor the energywould haveto be

52 Martin F Sohnius, Introducingsupersymmetry

either~0 or ~0 becauseof the inequality (1.5). Fora physicallysensibletheory with energiesboundedfrom belowbut not from above,the proportionality factor will thereforebepositive.

The equations(1.4) to (1.7) arecrucialpropertiesof the supersymmetrygenerators,andmanyof themost important featuresof supersymmetrictheories,whetherin flat or curved space—time,can bederivedfrom them.

Onesuch featureis the positivity ofenergy,as can be seenfrom eq. (1.7) in conjunctionwith (1.5):

the spectrumofthe energyoperatorE (the Hamiltonian) in a theory with supersymmetrycontainsnonegativeeigenvalues.

We denotethe state(or the family of states)with the lowestenergyby 0) andcall it the vacuum.Thevacuumwill havezeroenergy

EI0)=0 (1.8a)

if andonly if

Q~0)= 0 and Qt10)= 0 for all 0. (1 .8b)

Any statewhoseenergy is not zero, e.g. any one-particlestate,cannot be invariant under super-symmetry transformations.This means that theremust be one (or more) superpartnerstate Oil) or0t11) for everyone-particlestate 1). The spin of thesepartnerstateswill be different by ~from that ofii). Thus

each supermultipletmustcontain at leastonebosonand onefermion whosespinsdiffer by ~.

A supermultipletis a setof quantumstates(or, in a different context,of quantumfields) which can betransformedinto oneanotherby oneor moresupersymmetrytransformations.This is exactlyanalogousto the conceptof “multiplet” knownfrom atomic,nuclearandelementaryparticlephysicswhere,e.g.,the proton and the neutronform an iso-spin doublet and can be transformedinto eachotherby aniso-spinrotation.

The translationalinvarianceof 0, expressedby eq. (1.4), implies that 0 doesnot changeenergyandmomentumandthat therefore

all statesin a multipletof unbrokensupersymmetlyhavethe samemass.

Experimentsdo not showelementaryparticles to be accompaniedby superpartnerswith different spinbut identical mass.Thus, if supersymmetryis fundamentalto nature, it can only be realised as aspontaneouslybrokensymmetry.

The term “spontaneouslybrokensymmetry” is usedwhenthe interactionpotentialsin a theory,andthereforethe basic dynamics,are symmetric but the statewith lowest energy, the ground stateorvacuum, is not. If a generatorof such a symmetry acts on the vacuum the result will not be zero.Perhapsthe most familiar example of a spontaneouslybrokensymmetry is the occurrenceof ferro-magnetismin spiteof thesphericalsymmetryof the laws of electrodynamics.(S. Coleman,in a seriesoflecturesin 1973,used insteadthe term “secretsymmetry”. For elementaryparticlephysics,this would

Martin F Sohnius, Introducingsupersymmetry 53

be a betterterm sincethe impressionis avoidedthat the “breaking” is in all casesa physicalprocess— aphase transition— which takes place “spontaneously” at some time and which could perhapsbeobserved.Furthermore,the “secret” emphasisesthe wish of researchersto find such underlyingsymmetry.) Becausethe dynamicsretain the essentialsymmetry of the theory,stateswith very highenergytendto lose the memory of the asymmetryof the ground stateandthe “spontaneouslybrokensymmetry” “gets re-established”.High energymay meanhigh temperature,but we do not yet fullyunderstandthe behaviourof supersymmetryat non-zerotemperatures,essentiallybecausethe statisticsof the occupationof statesis different for fermions andbosons.

If supersymmetryis spontaneouslybroken, the ground state will not be invariant under allsupersymmetryoperations:Q~0)� 0 or

0t10) � 0 for some0. From what was said abovein eqs. (1.8),we concludethat

supersymmetryis spontaneouslybroken if and only if the energy of the lowest lying state (thevacuum)is not exactlyzero.

Whereasspontaneoussupersymmetrybreakingmay lift the massdegeneracyof the supermultipletsbygiving different massesto differentmembersof the multiplets,the multiplet structureitself will remainintact. In particular,we still need “superpartners”for all known elementaryparticles,althoughthesemay now be very heavy or otherwiseexperimentallyunobtainable.The superpartnerscarry a newquantumnumber(calledR-charge).It hasbeenshownthat the highly desirablepropertyof superGUTsmodels,mentionedearlier,namely that they stabilisethe GUT hierarchy, is closely associatedwith astrict conservationlaw for this quantumnumber. If natureworks that way, the lightest particle withnon-zeroR-chargemustbe stable.Whereasthis particlemay beso weaklyinteractingthat it hasnot asyet beenobserved,its presencein the Universecould crucially andmeasurablyinfluencecosmology.

As a matter of convention (and of amusement),fermionic superpartnersof known bosonsaredenotedby the suffix -mo (hence“gravitino, photino, gluino”); the bosonicsuperpartnersof fermionsaredenotedby a prefixeds- (“squark, slepton”).The discoveryof any such bosinosor sfermionswouldconfirm the importantpredictionof superpartnerswhich is commonto all supersymmetricmodels.Itwould be a major breakthroughand would establishsupersymmetryas an importantproperty of thephysicsof natureratherthanjust as an attractivehypothesis.

We havenot yet specified“how much” supersymmetrythereshould be. Do we proposeone spin ~photino as a partnerfor the physicalphoton,or two, or how many?Different supersymmetricmodelsgive different answers,dependingon how manysupersymmetrygenerators0 arepresent,as conservedcharges,in the model. As alreadysaid, the 0 are spinor operators,and a spinor in four space—timedimensionsmust haveat least four real components.The total numberof 0’s must thereforebe amultiple of four. A theory with minimal supersymmetrywould be invariant underthe transformationsgeneratedby just the four independentcomponentsof a single spinor operator0~.with a = 1,. . . ,4.We call this a

theory with N = 1 supersymmetry,

andit would giverise to, e.g., a single unchargedmasslessspin ~photinowhich is its own antiparticle(a“Majorananeutrino”).If thereis moresupersymmetry,thentherewill beseveralspinorgeneratorswithfour componentseach, 0~.~with i = 1,. . - ,N, andwe speakof a

theory with N-extendedsupersymmetry,

54 Martin F Sohnius, Introducingsupersy~nmetry

which will give rise to N photinos. The fundamentalrelationship(1.6) between the generatorsofsupersymmetryis now replacedby

{Q~,(Q~)t}=5.1(aE+flP). (1.9)

Most modelswith extendedsupersymmetryarenaturally invariant underrotationsof their -inos intoeachother. Theserotationsform an internal symmetrygroup(somewhatlike iso-spin). In the caseofsupergravity, this invariance can be “gauged”, i.e., made local, and we arrive at a natural linkbetweenspace—timesymmetries(generalcoordinateinvarianceand supersymmetry)and gauge inter-actions.This exciting possibilityspeaksvery muchin favourof extendedsupergravities.

On the other hand, extendedsupersymmetryis not without serious drawbacks.An importantargumentagainstits physicalrelevanceis that it doesnot allow for chiral fermionsas theyareobservedin nature (neutrinos). The reasonsfor this can only be understoodafter a more detailedstudy ofsupersymmetricgaugetheories,andwe shall comebackto this point in section 12. Here it mustsufficeto say that in order to retain the stabilising influence of N = 1 supersymmetryon the GUT masshierarchyon the onehand,andto permit chiral fermionson the other,we musthavea breakdownchainof the type

extendedsupersymmetry N = 1 supersymmetry no supersymmetry—~ .(at largeE) (at mediumE) (at low E)

with decreasingenergy.The final breakdownof all supersymmetryin the laststepis necessaryto lift themass degeneracyof the supermultiplets.For spontaneoussupersymmetrybreakingat least, thereis aseriousproblemwith this picture: for extendedsupersymmetry,eq. (1.7) remainsvalid evenif the sumruns only overthe space—timeindiceson

{Q~,(Q~)~}~ E for each i. (1.10)

Earlier, we saw that this relationshipmeant that a non-zerovacuumexpectationvalue of E was anecessaryand sufficient condition for spontaneoussupersymmetrybreaking,i.e. for 010) � 0. This isnow thecasefor eachof the individual N supersymmetries,andwe concludethateithernoneof them isspontaneouslybroken,or that all of them are, dependingon whether(E) = 0 or not. Furthermore,theproportionality factor in (1.10) is the same for all values of i and we thereforeexpect the effects ofsupersymmetrybreakingto set in at the sameenergyfor all Q~.This makesa breakdownhierarchysuchas the one requiredseemimpossible.

Argumentssuch as the one just presentedhold strictly only in the absenceof gravity. In thecontextof supergravity,it maythereforebe possibleto overcomethesedifficulties andto arriveat a scenarioofhierarchicalsupersymmetrybreaking after all. This has indeed beenachievedfor oneof the Kaluza—Klein supergravities.Kaluza—Klein theories[12] are characterisedby additional spatial dimensionsinwhich the spaceis very highly curved (radii in the region of the Planck length).Deviationsfrom thephenomenologyof flat spacearethereforeparticularlylargefor suchmodels,andit should not surpriseus to see many “no-go theorems” overcome. It is typical of supersymmetrythat a problemencounteredat one end of the spectrumof physical research,such as the absenceof right-handedneutrinos, should lead us to search for a solution at anotherend, such as in the structure and


dimensionalityof space—timeitself. It is successeslike this which makethe field of supersymmetryandsupergravityso exciting andpotentially so rewarding.

Is therea maximalnumberNmax Of supersymmetriesthat a field theory or asupergravitymodelcanhave?The answeris yes, andthe limits originatein the requirementthat the symmetrytransformationsshould act on multiplets of physical statesand that the underlying theory must be either GeneralRelativity or arenormalisablefield theory in flat space.Fromthe point of view of the algebraalone,assummarisedin eqs. (1.4) and(1.10) andas describedin section2, therewould beno limits on N. Let ustry to understandhow the limits actuallyariseandwhat theyare.

Considera particlesuch as the photonor the graviton. If we apply a supersymmetrytransformationto that particle,we get an -mo:

photon ~ photino.

The samesupertransformationapplied a second time will give us the original particle back, buttranslatedto a differentpoint in spaceandtime — this is the essenceof eq. (1.6).If we haveadditional,different typesof supersymmetrytransformations,as in the caseof N> 1, thenanothertypewill producea different -mo. If appliedtwice, the new supersymmetrytransformationwillgive back the original particlein adisplacedposition, just as the old onedid. But what happensif wefirst apply one type of transformationand then the other?We cannotget back the original photon,becauseof the Kronecker8-symbol in eq. (1.9). Instead,we get yet other superpartnerswith newproperties,

h ~ first photino .~. . . -p oton ,~ . moresuperparners,- secondphotino ~——‘

anda detailedstudy(section3) will show that theynormally havespinsdifferent from both the originalparticle and from the -inos.There is a minimal rangeof spins coveredby multiplets, and this rangeincreaseswith N. More quantitatively,

anymultiplet ofN-extendedsupersymmetrywill containparticles with spinsat leastaslarge as~N.

This meansthat spins� ~mustbepresentfor N >4 andspins�~ for N>8. Renormalisableflat-spacefield theories cannot accommodatespins �~. (The canonical dimensionsof fields which describeparticles increasewith spin, and at a certainthreshold,which happensto be at spin ~, their presencerequires the introduction of coupling constantswith negative mass dimensions.These make fieldtheoriesnon-renormalisable.)In addition, sincegravity cannotconsistentlycoupleto spins�~,we arriveat the famouslimits

Nmax = 4 for flat-spacerenormalisablefield theory,

Nmax = 8 for supergravity.

Both maximallyextendedsupersymmetrictheories,namelyN = 4 super-Yang—Millstheory and N= 8supergravity,areessentiallyunique: thereis only onemultiplet in eachcasewhosespinsfit into the slots0 � s � 1 and 0 � s � 2, respectively.These models have been studiedextensively, and it hasbeen

56 Martin F Sohnius,Introducingsupersymmern’

shown that the N = 4 theory is not only renormalisablebut actuallyfinite. The infinite contributionstoperturbationtheory from the different typesof particlesin the modelcancelexactly.This is discussedinslightly more detail in the sectionon N = 4 supersymmetry.For a model field theory, this propertyishighly remarkableandpossiblyof importancefor therigorousdisciplineof researchinto quantumfieldtheory. For physics, unfortunately, the N = 4 theory seemsirrelevant, since its spectrum(which isuniquelydetermined)in no way resemblesthat of the world of elementaryparticles.

N = 8 supergravity,andparticularly the version which has a local, “gauged”0(8) invariancewhichrotates the gravitinos into each other, hasbeenhailed as the bestcandidateto date for a Theory ofEverything.Like all modelswith extendedsupersymmetry,it can beunderstoodas derivedfrom a fieldtheory in more than four dimensions(more aboutthis in section14). The N = 8 model, at leastin itsnon-gaugedversion, can be derivedfrom a relativelysimple model in elevenspace—timedimensions.Itactuallyprovidesa rathernaturalseparationbetweenthe four dimensionsof the “real world” andtheother seven,which get spontaneouslycompactified.It remainsto be seenwhetherN = 8 supergravityhasother,yet moremiraculouspropertieswhich could pavethe way for the unification of gravity withquantummechanics.Unfortunately, it doesnot seemto sharethe propertyof finitenesswith the N = 4theory.

The presentreport treatsflat-spacesupersymmetry.It is meantto lay a basefor work in the field ofsuperGUTsas well as to be a steppingstonetowardssupergravity,the subjectof asubsequentreport. Iintendto handover to the readera tool kit of techniquesandsoundbasicknowledgeof the field whichshouldenablehim or her to thoroughly enjoy the vast amountof literatureon the subject.In order toachieve this, I havetried to make almost every step of reasoningand of algebraas transparentaspossible,only relying on the “it canbe shownthat. . .“ whenI felt that it canindeedbe shownusingthetechniquespresentedin this report. Wheneverthat is not so, I say it (like in the caseof superspaceperturbationcalculations).

In section 2, a detailedderivation of the supersymmetryalgebrasis presented.Section 3 describespropertiesof their representationson multiplets and includesthe derivation,stepby stepand in fulldetail, of the simplest field multiplet. Further field multiplets, togetherwith the rules of their tensorcalculus, are introduced in section 4. In section 5 the first supersymmetricmodel field theory isdescribed,the Wess—Zumino model, which is used as an example to illustrate the appearanceofauxiliary fields andthe problemof on-shell/off-shellrepresentations.Also in section5, the supercurrentmakesits first appearance.

After a diversioninto a form of spontaneoussupersymmetrybreakingwhich doesnot involve gaugetheories(section 6), the conceptof superspaceis introduced.Sections7 and8 are devotedentirely tosuperspaceand superfield techniques.Nowadays, both the literature on supersymmetryand thelanguageof those working in the field usesa mixed nomenclaturein which things are explainedsometimesin component(multiplet) languageandsometimesin terms of superfields,whicheveris themoreconvenientfor the taskat hand.In thelater sectionsof thisreport, I alsousethe samemixture oflanguage.Therefore,in sections7 and8, I try to clarify this somewhatconfusinghabit by giving carefultreatmentof the correspondencesbetweensuperfieldsandmultiplets.

Sections9 and 10 dealwith the very importantsubjectof supersymmetricgaugetheories.With aneye on supergravity,I chosetwo approachesandpresentedthem both. First, in section 9, I introducesuper-QED as the theory generatedby coupling radiation to the multiplet which contains theelectromagneticcurrent (“Noether coupling”). Using the multiplet of the energy-momentumtensorinstead,one could come to grips with supergravityin a similar way. Subsequently,in section 10, thecomplementary approach is developed where supersymmetric gauge theories are introduced


geometrically,startingfrom superspace.It turnsout that crucial input hasto comefrom elsewhere(“theconstraints”).This input could be the resultsof the Noether procedure,or it could be some otherknowledgesuch as that of the structureof thosemultiplets which aresupposedto exist in the geometricbackgroundcreatedby the gaugetheory.Thesamewill be true for supergravity,which can bederivedusing a similar tandem approach,involving geometryand Noether coupling to the multiplet whichcontainsthe supercurrentandthe energy-momentumtensor.

More aboutspontaneoussupersymmetrybreakingis saidin section11, this time in the presenceofgaugeinteractions.

Sections12 to 14 introduceextendedsupersymmetry.It is shownhow one can arriveat N = 2 andN = 4 by using specialcases of the general N = 1 theory. The finiteness propertiesare discussed.Section 14 dealswith dimensionshigher thanfour. This is meantto provide a basisfor the readerwhowishes to go on to Kaluza—Klein or string theories, or to the many other aspectsof extendedsupersymmetryand supergravitywhich areoften describedin morethan four dimensions.A thoroughintroductioninto the propertiesof spinorsand Dirac matricesfor arbitrary space—timesis provided insection 14 in conjunctionwith appendixA.7.

Section 15 is devotedto the supercurrentandthe propertiesof the multiplet in which it sits. Thissectionprovidesthe natural link to supergravity,sincethe energy-momentumtensormustbe containedin the multiplet of the supercurrent.Since the energydensitygives rise to the gravitational field, thesuper-chargedensitymustgiverise to a particlefield in the samemultiplet with the graviton.This is thegravitino which hasspin ~. In this picture of Noether couplings wherecurrentsare the sourcesforparticle fields, section 15 will be to supergravitywhat subsection9.1 on the electromagneticcurrentsuperfieldwas to super-QED.

Finally, in section 16, conformal supersymmetryis introduced as a natural consequenceof thestructureof the supercurrentmultiplet. Conformalsupergravityandits tensorcalculusareuseful toolsfor the construction of supergravity representations.It may also very well be that the unbrokenconformal invarianceof the N = 4 theory is highly significant for field theory.

The Technical Appendix contains formulas and notation in somewhatmore depth than usual.ParticularlysubsectionsA.2—3 andA.7 on finite dimensionalrepresentationsof groupsandon the Diracmatrices for 0(d+, d_) could almost be independentsections.Having them in the Appendix does,however,makethemeasierto find!

A list of referencesconcludesthe article,andI wish to concludethis introductionwith a few wordsonreferencing.

This report coversthe basicsof supersymmetryand its techniques.Much of the matter presentedherewaswell developedwithin a few yearsof the appearanceof WessandZumino’s first paperandofSalam and Strathdee’spaperon superspace.I havechosenmy presentationwith the hindsightof themanyyearsof researchthat havepassedsince,andwith the knowledgeof the existenceandthe shapeof supergravitytheories.Nevertheless,thematerial itself is ratherold,andmanyof my colleagues,somewith nameswhich have become householdwords among supersymmetrists,will look in vain forreferencesto their later work. I apologisewhere this was an oversight,and I ask for understandingwhereI simply could not, in this context,cover the resultsfor which they are famous. This holds inparticularfor all the work on supergravity.

In previous PhysicsReports [82,83] and books [26,80] on the subject, the authorshave eitherchosento include every referencein existenceat the time, or they haverestrictedreferencesto asymbolic numbersuch as two per chapteror none at all. In eachcaseany argumentaboutproperreferencingcould easilybe defused.Since it would clearly be inappropriateto give no referencesin a

58 Martin F Sohnius,Introducingsupersymntetry

report like this one, andsince the total numberof publishedpaperson the subjecthad exceeded1500already in summer1982 when I compiled a computerlist, neither of thesetwo elegantways out ofcontroversywere availableto me. I choseinsteadto give a relativelysmallnumberof referenceswhereI felt that theywerenecessary,andI hopenot to haveoffendedtoo manyold friends,nor to havemadetoo manynew enemies.

2. Generatorsof supersymmetryand their algebra

In this section,Colemanand Mandula’stheoremaboutthe generatorsof symmetries[8] is described,and thoseadditional possibilitiesarediscussedwhich arise if statistics-changingsymmetries(supersym-metries)are admitted.We shall see that supersymmetrygeneratorsnecessarilymust obey anticom-mutatorrelationswith eachother.This section is basedmainly on the 1975 paperby Haag,1~opuszañskiandmyself [34].

2.1. Generatorsofsymmetries

The generatorof a symmetryis an operatorin Hilbert spacethat replacesone incomingor outgoingmultiparticle statewith anotherand furthermore“leaves the physics unchanged”.Such an operatorshouldactadditively on directproductsof states,andthis implies that it can bewritten asthe productofan annihilationoperatora, which picks a particle out of a multiparticle stateand annihilatesit, andacreationoperatorat, which createsanotherparticlein its steadwith different propertiesand different3-momentum.The mostgeneralsuch productis

G= ~Jd3pd3qa~(p)Kjj(p,q)aj(q). (2.1)

This operatoris determinedcompletely by the integral kernel K11(p,q), a c-numberfunction of themomentap andq andof the otherparticle properties,denotedby the indices i andj. Using the symbol* for the convolutionsin (2.1), we canwrite G symbolically as

G=at*K*a. (2.la)

The operator G will replacesome incoming quantumstate in) by another, G in). G is called ageneratorofa symmetryif, in addition,it commuteswith the S-matrix, i.e., if it doesnot matterwhetherwe “reshuffle” the statebeforeor after an interactionhastakenplace: SG in) = GS in) or

[5, G] = 0. (2.2)

This is a formal expressionfor what was called “leaving the physicsunchanged”.The sumsin (2.1) runover all particlequantumnumbersandthus alsooverall spin values.Thereare

terms which replacebosonsby bosons, and otherswhich replacefermions by fermions,bosonsbyfermions and fermionsby bosons.Any G can be decomposedinto an evenpart B and an oddpart F,

G=B+F, (2.3)


where B containsonly terms which replacebosonsby bosonsand fermions by fermions, while F

replacesbosonsby fermionsandfermionsby bosons.Symbolically,we can writeB= bt*Kbb*b+ft*Kfl*f

(2.4)

F = ft * Kfb * b + bt * Kbf *f,

where /j andf areannihilationoperatorsfor bosonsandfermions,respectively.We assumevalidity ofthe spin-statisticstheorem and thereforethe B’s may changespin by integeramountsor not at all,whereasthe F’s must changethe total spin of a stateby a half-odd amount,and are thusnecessarilysupersymmetrygenerators.

2.2. Canonicalquantisationrules

We assumethat the particleoperatorsobeycanonicalquantisationrules:

[b1(p),b(q)] = ~ Y(p— q); [b, b] [b

t, bt] = 0 (2.5a)

{f, (p), f(q)} = 8.~t53(p — q); {f, f} = {ft, ft} = 0 (2.5b)

[b,f] = [b,ft] = [bt,f] = [bt,ft] = 0. (2.5c)

Theserelationscan be written mostelegantly:weobservethat ô~83(p— q) is the unit element1 of theconvolution product *, and we introducethe “gradedcommutator” symbol [..,..} which denotestheanticommutatorif both operatorsare fermionicand the commutatorin all other cases.The canonicalquantisationrules thenreadsimply

[a, at}=~; [a,a}=[at,at}=0. (2.5)

2.3. Algebraofgenerators

Let usnowtry to concocta third symmetrygeneratorG3 from two knownonesG’ and G2. If bothcommutewith the S-matrix,then so doestheir productG1G2.Thisproduct,however,is not ageneratorof a symmetry in the sensedefinedabovesince it is quadrilinearin particle operators.The canonicalquantisationrulessuggestthat we try the commutator[G1,G2].

We first do this for two bosonicgeneratorsB1 andB2. We can usethe quantisationrules togetherwith the identities

[ab,c] a[b, c] + [a,c]b a{b, c}— {a, c}b, (2.6)

and aftera bit of algebrawe get

[B1,B2] = B3, (2.7a)

wherethe kernelsof B3, which define it, aregiven by


K~b=K~b*K~b—K~b*K~b;K~1=K~t*K~j—K~1*K11. (2.7b)

The quantisationrules have eliminated two of the four particleoperatorsoriginally presentin theproduct, and the commutatorhas turned out to be bilinear in particle operators.It is thus anothergeneratorof a symmetry.

Similarly, we can show that the commutatorof an F with a B gives anotherF:

[F1,B2] = F3 (2.8a)

with

= K~*K~b—K~f*K~b;K~=K~*K~f—K~b*Kj~. (2.8b)

Thereis, however,no way to decomposethe commutator[btf, ftbj which appearsin [F’, F2], in suchaway that only those propercombinations[..,..} appearwhich allow elimination of two of the fourparticle operatorsin the product. We concludethat the commutatorof two odd generatorsis notbilinear in particleoperatorsandhencenot a symmetrygenerator.

On the otherhand,we find that we can evaluatethe anticommutatorof two F’s, usingthe identities

{ab, c} a{b, c} — [a, c]b a[b, c] + {a, c}b (2.9)

as well as the onesof eq. (2.6).The anticommutatorof two F’s turnsout to be the generatorof an evensymmetry:

{F’, F2} = B3 (2.lOa)

with

K~bK~*K~+K~*K)l,; K~f=Kjb*K~f+K~*K),f. (2.lOb)

The B’s and the F’s satisfy the relationscharacteristicof a gradedLie algebra. It is the gradingof thecanonicalquantisationrules, i.e., that the fermions obey anticommutationrelations,which inducesasimilar gradingfor the symmetry generators.The odd generatorsbehavelike fermionic objects,theevenones like bosonicobjects.

2.4. GradedLie algebras

Let us explore some fundamentalpropertiesof graded Lie algebras.They are defined by thealgebraicrelations(2.7—10)which we nowwrite in termsof a basis(B

1, Fa):

[B1,B1] = ic11’~Bk

[Fa,Bj]5ai~F~ (2.11)

{F~,F~}=ya,s’Bi.

The structureconstantscannotbe completelyarbitrary.The c and y havesymmetryproperties,


= _~k ~l (2.12)

and all structureconstantsare subject to consistencyconditionswhich follows from the gradedJacobiidentities

[G1, G2}, G3}+ gradedcyclic 0. (2.13)

The “gradedcyclic sum” is definedjust asthe cyclic sum, exceptthat thereis an additionalminus-signif

two fermionicoperatorsareinterchanged,e.g.:FaF

13B1 + gradedcyclic = FaF13B, + B1FaF13— F8B1Fa. (2.14a)

The full set of gradedJacobiidentitiesis then

[B1,B.], Bk] + [Bk, B1], B1] + [B1, Bk],B1] = 0

[Fa, B1], B1] + [B1, Fa], B1] + [B1, B1], Fa] = 0 (2 14b)

[{Fa, F13}, B1] + {[B1, Pa], F13} {[F13, B1], Fa} = 0 -

[{Fa,F13},Fy]+[{Fv,Fa},Fp]+[{F,9,Fy},Fa]0.

The requirementthat thesebe fulfilled is equivalentto the demandthat certainmatricesconstructedfrom the structureconstantsshouldform a representationr of the algebra, the adjoint representation.Thesematricesare

r(B1)= [~shl r(Fa)= [0 ~a] (2.15a)

with matrix elements

(C)”=ic.”- (5)I3~~ 13

(Fa)~1= ~ (Ia)i~a~. (2.15b)

The requirement that these matrices form a representationis equivalent to the following fourconditions:

(a) the matricesC1 form a representation(the adjoint representation)of the bosonicLie subalgebra

which is spannedby the B’s alone;(b) the matricesS alsoform a representationof the Lie subalgebra(this representationneednot be

irreducible);(c) the Yap’ arenumericalinvariantsunder the Lie subalgebra:

(Sj)a8 y~’ + (51)138Yal — Ya13’~ (CI)k’ = 0 ; (2.16)

(d) there is a cyclic identity involving s and y:

ya13

1s71

8+cyclic(a—*$-+y)=0. (2.17)

62 Martin F Sohnius,Introducingsupersym,netrv

2.5. TheColeman—Mandulatheorem

Colemanand Mandula [8] studied the propertiesof all bosonicgeneratorsof symmetriesin themathematicalframework of relativistic field theory.The assumptionstheymade(non-trivial S-matrix,etc.) canhardly be questionedfrom the point of view of physics,exceptperhapsfor the “strong spectralassumption”that thereshouldbeonly one zero-massstate,the uniquevacuum,and a finite energygapbetweenit and the lowest one-particlestate.This assumption,however,is introducedmainly to avoidinfrared problems and relaxing it — in the hope that infrared problemswill somehowtake care ofthemselves— alters their result only in that the conformal group may then be admitted as symmetrygroup beyond the Poincarégroup if all one-particlestatesare massless[34]. The assumptionof auniquevacuumalso appearsto removetheorieswith spontaneouslybrokensymmetriesfrom their (andour) investigation.It could be interestingto explorethe possibility of further symmetrieswhich showdiscontinuousbehaviourwhen the parametersof the model are changed:as long as the vacuum isunique they are absent,but they appearas soon as spontaneousbreaking of someother symmetryintroducesa degeneratevacuum.

The fact that only Lie groupsof symmetrieswereconsideredexcludedfermionicgenerators,andthussupersymmetryas we know it, from the analysisof ColemanandMandula.Nevertheless,their resultsstill hold for thesubsetof all bosonicgeneratorsand,throughthe conditionsof theprevioussubsection,severely restrict the fermionicgeneratorsas well.

They found that any group of bosonic symmetries of the S-matrix in relativistic field theory is thedirectproductof the Poincarégroupwith an internalsymmetrygroup. Furthermore,the lattermust bethe direct productof a compactsemisimplegroupwith U(1) factors.

The bosonicgeneratorsare thusthe four momentaPM and the six LorentzgeneratorsMMr, plus acertain numberof Hermitian internal symmetry generatorsBr. The algebra is that of the Poincarégroup,

[PM,PV]=0

[PM, M~]= i(nMPPU — ~ (2.18)

[MM~,M,~]= i(nppMM~— — 7JMpM~.+ ,M~0),

togetherwith that of the internalsymmetrygroup,

[Be,B5] = ic~5tB~. (2.19)

The direct-productstructuremanifestsitself in the vanishingof the commutators

[Br, PM] = [Br, MM~,]= 0, (2.20)

in otherwords, the Br mustbe translationallyinvariant Lorentzscalars.The Casimir operatorsof the Poincaré group are the mass-squareoperatorP2 PMPM, and the

generalisedspin operatorW2 W,.WM, whereWM is the Pauli—Lubanskivector:

WM = —~e’°P1M,~. (2.21)


In the rest frameof a massivestatewe have

= (m, 0,0,0) and W2 = —m2L2 (2.22a)

with

L = (M23, M31,M,2). (2.22b)

TheseCasimir operatorscommutewith the entirePoincarégroup andalsowith all internal symmetrygenerators:

[Br, P2] = 0; [Br, 1472] 0. (2.23)

The first of theseequationssaysthat all membersof an irreduciblemultiplet of the internalsymmetrygroup must have the samemass.This is also known as O’Raifeartaigh’s theorem [49]. The secondequationsaysthat theymustalso havethesamespin.

For masslessstateswith discretehelicities we have

WM = APM (A half-integer) (2.24)

andP2 = 1472 = 0. Again, no B,. can changethe helicity since[Br, PM] = [B,.,WM] = 0, andthe statementthat internalsymmetriescannotchangespin still holds.Statedin a positiveway, the Coleman—Mandulano-gotheoremreads:

All generatorsof supersymmetriesmustbefermionic, i.e., theymustchangethe spin by a half-oddamountand changethe statisticsof thestate.

Witten [77] has given a beautiful explanation for the essentialcontent of the Coleman—Mandulatheorem:additional space—timesymmetriesbeyondenergy,momentumand angularmomentumwould,in a relativisticallycovarianttheory,overconstrainthe elasticscatteringamplitudesand allow non-zeroscatteringamplitudesonly for discretescatteringangles.The assumptionof analyticity thereforerulesout such symmetries.

2.6. Thesupersymmetryalgebra

This leavesus to explore the possibilitiesof fermionicgenerators,the odd elementsof the gradedalgebra.

The Jacobiidentities(subsection2.4) link the propertiesof the fermionic sectorof the algebracloselyto thoseof the bosonicsector.The Coleman—Mandulatheoremthereforeprovidesstronglimitationsforthe fermionicgeneratorsaswell. Theresultsof detailedexamination[34] of theselimitations areoftenreferredto as the “Haag—Lopuszañski—Sohniustheorem”.

In ref. [34] a “positive metric assumption”was madeaboutthe Hilbert space.This meansthat theanticommutatorof an operatorwith its adjoint is non-negative,and zero only if the operatoris zeroitself:

(.. I{Q, Qt}I .) = - ~ 2 + 01. )~2>0 if 0 � 0. (2.25)

64 Martin F Sohnius,Introducingsupersymmetry

If thereis no elementof the bosonicsubsetof the generatorsthat possessesall the propertiesrequiredof {Q, Qt}, weconcludethat 0 = 0. This line of argumentis at the nucleusof mostof the statementsinref. [34].

We already know from the general considerationsof subsection2.4 that the supersymmetrygenerators,which from now on are called 0 (rather thanF) in order to follow generalusage,mustcarry a representationof the bosonicsymmetry group. If 0 sits in the representation(j, J’) of theLorentzgroup,then {Q, 0~}will containthe representation(j + J~’J+ j’). Since ~M is the only objectinthe bosonicsectorwhich is in such a representation,namelyin (~,~), we know that all 0’s must be inoneof the two 2-dimensionalrepresentations(~,0) and (0,~) of the algebraof the Lorentzgroup. Thuswe have

[Qaj, MMJ = ~(OM~~)a~~— — (2.26)

[0’ M] = _~QI~(~Mv)~a

with u-matricesas definedin the TechnicalAppendix,eqs. (A.14—17).The index i in Qaj labels all the different 2-spinorsQa that are presentand runs from 1 to some

integerN. We assumethat if 0 is the generatorof a symmetrythen so is Q~.Since (Q,~,)tmustbe inthe complexconjugaterepresentation,andthusbe oneof the 0’s, we numberthosein such a way thatalways

= (0ai)~ (2.27)

The anticommutator{Q, Q} transformsas (~,~) and must thereforebe proportional to the energy-

momentumoperator:

{Qai, Q’13}= 28/(oM)a,3 “/L (2.28)

The factor~ is amatterof properlydefining the 0’s. We would at first assumethat

{Qai, ~ = 2ci1(uM)a~~M

with somecoefficients c’. Since the matrices 0-M are Hermitian (see below), we find, by taking theadjoint, that c1

3 = (c/)*, i.e., that the matrix C with matrix elementsc1’ is Hermitian. Thenthereis a

unitary matrix U which diagonalisesC, andwe redefine

Qai~ U1’ Qai; Q’~-~Q’lS (U~)~’.

If the energyis to bepositive(seebelow), all eigenvaluesc of thematrix C must be positive and we canrescale

0a1~VciQai; Q~-*\/c,Q’6 (nosum)

and get (2.28) for the twice redefined0’s. The redefinitionshave not spoiledthe reality connection(2.27) between0 and Q.

Knowing that the structureconstantson the right-hand side of eq. (2.28) must be numerically


invariant tensorsof the Lorentz group,seeeq. (2.16),wecan concludethat the matriceso.M mustbe the2 X 2 matricesdefinedin the TechnicalAppendix,eq. (A.28), or multiplesthereof.We havenormalisedthe 0’s so that the absolutevalue of the factor in (2.28)is 2. Anotherwidely usedandperhapsmorenatural normalisationwould be 1. The sign on the right-handside of (2.28) is determinedby therequirementthat the energyE P0 shouldbe a positivedefinite operator:weget for eachvalueof theindex i

{Oaj, (Qaj)t} = 2 tr uMPM = 4P

0 (no sumover i), (2.29)

andwe find from (2.25)that the energyis indeedpositivedefinitefor our choiceof sign (thisdependsonthe metric; authorswho use ~ = —1 may havea minus sign, dependingon their other conventions).

The supersymmetrygeneratorscommutewith the momenta,

[QaI,PM][O’a,PM]O. (2.30)

Becauseof its importance,the proof for this equationfollows:The commutatorof a 0 with a ~M could contain the Lorentz representations(1,~) and (0,~). Since

thereare no (1,~)generatorspresent,we get as the most generalpossibility a sumover 0’s (seetheTechnicalAppendix for someof the conventionsused):

[Qal, PM] = cjj(o~)ag ~.

The adjoint of this is

[~alp 1—1 \*( \a$

I ‘ Mi — ~c11,~uM) 1Sf,

so that

[Qa:, PM], Pj = c~1(cjk)* (oMã,.)a13O/3k

From [PM,Pj = 0 and the Jacobi identity we deducethat for the matrix C with matrix elementsC11:

CC*=0,

since the u-matrixfactordoesnot vanish.The mostgeneral{Q, Q} that wecan haveis

{ Qal, Q~~}= 2rta$Z[ijl + a term symmetricin a/3.

The Z0 area linearcombinationof internalsymmetrygeneratorsandhencecommutewith ~M Then

0= rt”13 [{Oaj, ~ PM] = Ea13{Q I [013j,PM]} — (i+*j)

= C 13c•~(o~)$$ {Qai, O~}~(i~-~j)~ c[I,IPM

which meansthat C is symmetric,henceCC= 0 which implies c1 = 0, q.e.d.


The 0’s generallyalsocarry somerepresentationof theinternal symmetry,

[Oai, B,.] = (b,.)/ Qa,. (2.31a)

Since the internal symmetry groupis compact,the representationmatricescan be chosenHermitian,br = b~.For 0 we then have

11)1 B1—_ñi ‘b’1L” a, rJ ‘.i a~.nj -

The largestpossibleinternalsymmetrygroupwhich can actnon-trivially on 0 is thusU(N).The doubleindex structureon the 0’s reflectsthe direct productstructureof the bosonicsymmetry

group: a and a belongto the Poincarégroupand i is the internalsymmetryindex.The last algebraicrelationwhich wehaveto consideris the anticommutatorof 0 with 0 (that of 0

with 0 can then be obtainedby Hermitian conjugation).Lorentzcovariancerequires{Q, Q} to be alinear combinationof bosonicoperatorsin the representations(0,0) and (1,0) of the Lorentz group.The only (1,0) presentin the bosonicsectorof the algebrais the self-dual part of MM,~.Sucha term in{Q, Q} would, however,not commutewith ~M whereas{Q, Q} does,due to eq. (2.30). Thuswe areleftwith the following mostgeneralform for the anticommutator:

{Qaa, Q~~}= 2Ea1SZjj, (2.32)

wherethe Z11 aresomelinear combinationof the internal symmetrygenerators:

Z11=a~,B,.. (2.33)

We nowproceedto showthat

[Z11,anything] = 0, (2.34)

which has given rise to the namecentralchargesfor the Z’s.Fromeqs. (2.31) and(2.32) we get

[Zn 1_Il, \k 7 ~i~I1. ‘, k ~

1), -

1—’rJ — ~Vr)j ~-~kj ‘ ~~Un)j ‘-1k

and then, with (2.33),

~ ZklI = aki (b~)1k

2’k + akl (b,.)1

1’ Zk -

Thesetwo equationssay that the Z11 span an invariant subalgebraof the internal symmetry algebra.

With the help of the Jacobi identity, the commutator[0,4] can be rewritten as a sum of [0, ~M]

commutatorsandthusvanishes.This implies

a11r (b,.)k’ = 0

and therefore [0, Z11] = [Z11,ZkI] = 0, and the invariant subalgebrais Abelian. The direct product

Martin F Sohnius, introducingsupersymmetry 67

structure(semisimple® Abelian) of the internal symmetrygroup now implies that all Z11 lie in theAbelian factor. Thus

andZ1, commuteswith everything.The adjoint of (2.32)is (seeeq. (A.19) for the extraminussign):

~ ~‘~} 2~a,~Z1’ with Z” = (Z

11)t. (2.35)

The Z” are, of course,also centralcharges.The symmetry of {Q, Q} underinterchangeof ai with f3jand the antisymmetryof Ca

13 meanthat

Z1~=—Z,1 i.e. a~,=—a1, (2.36)

which excludescentral chargesfor N = 1. Furthermore,for eachnon-vanishingcentral chargetheremust be a different antisymmetricN x N matrix a~which is a numerical invariant of the internalsymmetrygroup:

a ~, + (b~)11’a~ = 0 (2.37)

(this follows from eq. (2.16)). Centralchargesin the algebrathereforeimposea symplecticstructure onthe semi-simplepart of the internal symmetrygroup. The largestinternal symmetrygroup which cansustaina centralchargeis USp(N), the compactversionof the symplecticgroupSp(N).

2.7. Summaryof algebra

The following equationssummarisethe resultsof the previoussubsection:

[PM,Pj=0 (2.38a)

[PM,M~]= i(nMPP,., — ‘)M~T”P) (2.38b)

[MMP,M,,,,.] = i(~vpMMU— tj~M~~— ThLP1~”V(T+ Th~o~vp) (2.38c)

[B,.,B~]= ic,.5B,. (2.38d)

[Br, PM] = [B,., MMV] = 0 (2.38e)

[Qa,, PM] = ~ PM] = 0 (2.38f)

[Oat, MM,.] = ~(uM,.)a130131 (2.38g)

[Qua MM,,] = ~Q~~(~M,.Ya (2.38h)

68 Martin F Sohnius.Introducingsupersvmmetrv

[Qaj, B,.] = (b,.)1’ Oaj (2.38i)

[Q’ a, B,.] = — ~‘ (b,.) (2.38j)

{Qaa, ~ = 28~(UM)a$PM (2.38k)

{Oa,, Q~~}= 2Ea$Zjj with Z11 = a~,B,. (2.381)

~ ~~‘,3}= 2e~jpZ1’ with Z” = (Z

11)t (2.38m)

[Z,1,anything] = 0. (2.38n)

This concludesthe discussionof possiblesupersymmetryalgebraswhich havethe Poincarégroup asspace—timesymmetry.As alreadymentioned,it is possibleto include the conformal group into thebosonicsector if all massesare zero. This results in what is known as conformal supersymmetry,asubjectcoveredin section16.

If thereis only one 2-spinorsuperchargeQa, i.e., if N = 1, we say that a theory exhibits simple orunextendedsupersymmetry.If N> 1, we speakof extendedsupersymmetry.For simple supersymmetrythe only non-trivially actinginternalsymmetryis asingle U(1), generatedby a chargewhich hasbecomeknown under the name of R:

[Q,R]=Q; [~,R]=—O. (2.39)

Since underparity 0 —* 0 and0 —~ 0, we must haveR —~—R, i.e., the U(1) symmetrygroupis chiral

(this will be muchclearerin 4-spinornotation).

2.8. Cautionary remarks

In the courseof the discussionso far, it hasbecomeincreasinglyclear that supersymmetries,whichwere originally defined rather loosely as any symmetriesthat involve different spins, can appearinrelativistic field theoriesonly in a very specific form: they are generatedby spinorial charges0 whichobservewell-definedanticommutationrelations.It is this schemewhich is generallymeantwhen wespeakof “supersymmetry”or — in the context of curvedspace—time— of “supergravity”.

The restrictionsof the Coleman—Mandulatheoremhaveturnedout to be considerablylessstringentfor fermionicsymmetrygenerators:in particular,we nowhave

[0, W2] � 0.

The O’Raifeartaigh theorem [49], however,

[Q,P2]=0,

still holdsevenfor the spinorialcharges,since [0, PM] = 0. Supersymmetrymultiplets thereforecontaindifferent spinsbut are alwaysdegeneratein massandsupersymmetrymust be brokenin naturewhereelementaryparticlesdo not comein mass-degeneratemultiplets.

Martin F. Sohnius,Introducingsupersymmetry’ 69

I mustpoint out anotherlimitation of the discussionin both refs. [8] and[34]: thesepapersdeal withsymmetriesof the S-matrix, i.e., of the observedscatteringdata. The underlyingfield theory may havemore,or less, symmetry.

A large part of the analysisin [34] is basedon the positivity assumptionof the metric in Hilbertspace, as in eq. (2.22). This and the spin-statisticstheorem, which is used throughout, rule outstatementsabout,say,the ghostsectorof a quantisedgaugetheory andits BRStransformations.

Obviously,sincerefs. [8] and[34]deal with Poincaréinvariant theoriesin flat Minkowski space,theydo not strictly speakingapply to gravity or supergravity.Hencewe arenot surprisedto learn that, forinstance,non-compactinternalsymmetriesappearin someextendedsupergravitytheories.

We see that whateverno-go statementsarecontainedin the presentsectionshouldbe takenwith agrain of salt. Still, we shall now proceedto discusstheorieswhich are supersymmetricin the limitedsenseoutlined so far. In fact, very little is known aboutpossibleinfringementson the no-gotheoremswhich limit supersymmetry algebras.

3. Representationsof the supersymmetryalgebra

Supersymmetrictheoriesdealwith setsof fields or stateswhich carry representationsof oneof thedescribedgradedLie algebras(supersymmetryalgebras).Loosely speaking,arepresentationconsistsoflinear operatorson somevectorspacewhich havethe same“properties”as the objectswhich theyareto represent.In our case,their commutator—anticommutatorrelationsshould be thoseof the super-symmetry algebra.

3.1. The “fermions = bosons” rule

The very nature of the graded algebrasdemandsa grading of the vector space on which arepresentationacts.In less fancy language:since supersymmetrytransformationsrelate fermions andbosons,the representationspacecan be divided into a bosonicanda fermionicsector:

= ~spa~e + (3.1)

The even elementsof the supersymmetryalgebra (the bosonic generators~M’ MMV, B,.) map thesubspacesinto themselves,anda-priori we must allow the mappingto be onto a propersubspaceof theoriginal subspace, o PPcS

bosonic/fermionic bosonic/fermionic


The odd elements(the supersymmetrygenerators)mapthe bosonicsubspaceinto the fermoniconeandvice-versa,i.e.:

obosonic/fermionic fermionic/bosonic

The anticommutatorsare a particular linear combinationof two subsequentmappingswhich can besymbolisedlike this:

first Q second Q

bosonic/fermionic fermionic/bosonic bosonic/fermionic

The relation{Q, Q} = 2o~’P~~,characteristicof supersymmetry,meansthat thereshouldbe a particularway of going from onesubspaceto the otherand back,suchthat the netresult is as if we had operatedwith P1~.on the original subspace.

For alargeclassof representations,the representationr(PM) of the translationoperatorwill map therepresentationspaceonto itself. On aphysical multiparticle state,for instance,we have

r(PM)= (E,p), (3.2)

with E andp thetotal energyand momentumof the state.On quantumfields ~(x), ~M is the generatorof translations~(x)—~./(x + y), andis representedby the partial derivatives

r(PM)=i8M. (3.3)

In both cases,no “degreesof freedom”arelost from the representationspacethrough the actionof ~

This implies that the dimensionof the fermionic subspacemust be the same as that of the bosonicsubspace.This is so because

(a) no picture can havemoredimensionsthan its original, and(b) in the doublemapping00+ 00 no dimensionsget lost.

We thushave the following theoremwhich holdsfor the wide classof representationscharacterisedby(3.2) or (3.3):

Thefermionic andbosonicsubspacesoftherepresentationspaceofa supersymmetryalgebra havethesamedimension.

Whatexactlymakesup the “dimension” of such arepresentationspaceremainsto beseen.Historically, this property of representationsquickly becamean establishedrule of thumb. Since

thereis, of course,no needto ever postulatethe property— it simply follows from the fact that a givenmultiplet carriesa representation— no completeprooffor all possiblecaseshasto my knowledgebeenattempted.Hopefully, the somewhatloose argumentationabovewill makethe readerappreciatethis

Martin F Sohnius.Introducingsupersymmetry 71

“fermions= bosons”rule which hasplayed such an importantrole in the searchfor what is known as“off-shell representations”.

A coupleof caseswherethe rule doesnot hold ought to be mentioned.The adjoint representationofthe superalgebra,see subsection2.4, has r(PM) = 0, andthus is not coveredby the rule. Its represen-tationspacearethegeneratorsthemselves,andusuallythereareindeedadifferentnumberof fermionicand bosonic generators.Another exception are non-linear realisations, where nothing of the aboveargumentsapplies: we don’t havea representationvectorspaceand cannotspeakof “dimension”. Infact, thereareseveralwaysto realisethe N = 1 supersymmetryalgebranon-linearlyon a single chiralspinorandits adjoint [70,78].

3.2. Masslessone-particlestates

We now turn our attentionto representationsof the supersymmetryalgebraon one-particlestates[56]. Sincethe most interestingsupersymmetricmodelsare extensionsof non-Abeliangaugetheoriesand of quantisedgravity, all of which have zero-massexchangeparticles and hencewhole masslesssupermultiplets,the representationson masslessstates are particularly important. They allow tounderstandthe “physical” spectraof themodels.

A masslessone-particlestatecan alwaysbe rotatedinto a standardframewhereits movementis inthe z-direction, i.e. where

= (E, 0, 0, E). (3.4)

The space—timepropertiesof the state are then determinedby its energyE and its helicity A. Thehelicity is the projectionof its spin onto the direction of motion,i.e. the eigenvalueof E1L . P. Fromthe definitions of the Pauli—Lubanskivectorandof the angularmomentum,eqs. (2.21)and (2.22b),wefind that W

0 = L . P. Fora masslesshelicity eigenstateIB, A) we then alwayshaveW0 = AE, or, becauseof Lorentzcovariance,

WMIE,A)=APMIE,A). (3.5)

If we let one of the operators0~,act on the states,energyand momentumremainunchangedsince[0, PM] = 0. The helicity of the resultingstatecan bedeterminedas follows:

W0QaIE,A)OaW0IE,A)+[W0,Qa]IE,A)E(A1J ~~~~3)a13Q

13lE,A).

We haveusedthe particularform (3.4) of themomentumand[L . J~, 0] = . P0,which follows fromeqs.(2.22b)and(2.38g),togetherwith ourparticularrepresentationof O~Mp~Substitutingthe explicit formof u

3, we seethat all Oh lower the helicity by ~ andall 02~raiseit. For Qa the calculation is similar,anddue to the minus sign in eq. (2.38h) the result is the opposite:all Q~raise the helicity by ~ and allQ~lower it.

Using the explicit form (A.28) for the matricesuTM, we find that in the absenceof centralchargesthealgebraof the 0’s on our standardframestatesreducesto

{Q,Q}={~,~}=0; {Q2I,~~2}=0—. (3.6a)

{Q~,,Q’1}=48,’E,


or

{q1, 4’} = 81’; {q3, q1} = {4~,41} = 0 (3.6b)

if we rescale

q (4E)1120

11. (3.6c)

From our positivity assumption,asexpressedby eq. (2.25),we concludethat

021 = 0 (3.7)

on thesestates.The remainingalgebra(3.6b) of the q, is the Clifford algebrafor N fermionicdegreesoffreedom.Any irreducible representationof thisalgebrais characterisedby a Clifford groundstateIE, A0)with

q,JE,Ao)=0, (3.8)

andthe otherstatesaregeneratedby successiveapplicationof the N operators4’:

4’IE, A0) = IE, A0 + ~,i)(3.9)

4’4’IE,A0)= E,Ao+1,ij),

etc.,until we havereachedthe “top” state

4NIE, A0) = IE, A0 + n/2, 1 2~. N) (3.10)

on which any furtherapplicationof a 4 will give zero. In addition to whateverotherlabels (internalsymmetry eigenvalues)the ground statemay have,the other statesalso carry the internal symmetrylabelsof the 4’s which generatedthem.They mustbe totally antisymmetricin theselabels: IE, A, ij) =

—IE, A,ji), etc. The helicity, of course,hasbeenraisedby ~in eachapplicationof a 4’. We thushavethe

following spectrum:

helicity: A0 A0 + ~ A0 + 1 .. A0 + n/2

N N N N (3.11)

numberof states: (0) = 1 (1) = N (2) --- (N) = 1.

The total rangeof helicities is from A0 to A0+ N/2. The total numberof statesand the “fermions=

bosons”rule follow from propertiesof the binomial coefficients:

N N N/2 N N/2 N

(k)2N; (2k)~(2k+1)°~


The first of theseequationsis the binomial expansionof (1 +1)N andthe secondis that of (1 — i)N.

Two particularly important examplesof thesespectra are the N = 4 Yang—Mills multiplet withA0 = —1 andtheN= 8 supergravitymultiplet with A0 = —2:

N = 4 helicity: —1 —~ 0 ~ 1

states: 1 4 6 4 1, (3.12)

N= 8 helicity: —2 —~ —1 —~ 0 ~ 1 ~ 2

states: 1 8 28 56 70 56 28 8 1. (3.13)

Normally, any spectrumof statesthat is derivedfrom a Lorentz-covariantfield theory will exhibitPCT-symmetry[42]. This implies that for everystatewith helicity A thereshouldbe aparity reflectedstatewhich hashelicity —A. Our spectrado not generallyhave this property,e.g. for N = 1, A0 = 0, weget

helicity: 0 ~

states: 1 1.

The spectrumof a Lorentz-covariantfield theory can thereforecontainthesestatesonly in conjunctionwith thoseof the PCT-conjugatemultiplet with A0 =

helicity: —~ 0

states: 1 1

so that the smallest N = 1 multiplet, even for the massless case, is:

N= 1 helicity: —~ 0 ~

states: 1 2 1. (3.14)

We shall see later that this is the spectrumof the masslessWess—Zuminomodel [72], a model fieldtheory which containsa scalar, a pseudoscalaranda Majoranaspin-i field in interaction.

It is a particular property of the N= 4 Yang—Mills theory and of N= 8 supergravitythat theirmultipletsareactuallyPCT-self-conjugate.

Spin ~ doesnot allow renormalisablecoupling and spin ~ does not allow consistentcoupling togravity. Thereforewe see already at this stage that thereare stringent limits on the numberN ofindependentsupersymmetries:

N � 4 for renormalisabletheories;

N� 8 for consistenttheoriesof supergravity.

If these limits were violated, the spectrum of masslessstates would contain helicities ~ and ~,respectively.

74 Martin F Sohnius,Introducingsupersymmetrv

3.3. Massiveone-particlestateswithoutcentralcharges

The space—timepropertiesof massiveone-particlestatesaredescribedby the massm, the totalspin sandthe spin projection53 alongthe z-axis. We assumethe particlesto be in the restframewhere

= (m, 0, 0, 0). (3.15)

O is a tensoroperator ofspin ~,as can be seenfrom [0, L] = ~oQ.Thereforethe resultof the actionof0 on a statewith spin s will be a linear combinationof stateswith spinss + ~and s —

Qlm S53) = ~ c~jms+~s~)+ ~ c~]~jms—~si).

The sameis true for 0 with, of course,different coefficients.In the restframe, the algebrawithout centralchargesreducesto

{Qaj, Q’,i} = 2m6~’8as; {Q, Q} = {Q, ~} = 0 (3.16a)

or

{q~,,4’,~}= 8ap61~ {q1,q1}={4’,4

1}=0 (3.16b)

if we rescale

qai (2m)1”20a1. (3.16c)

This is the Clifford algebra for 2N fermionic degreesof freedom. A representationis thereforecharacterisedby aspin multiplet of groundstatesjmsos3),53 = ....... +s0, which areannihilatedby q~1.The other statesare generatedby successiveapplicationof 4’s to the ground states.A typical statewould be

\_ —Il .. —I,, 1/—qa1 q~.ms0s3 -

The statesaretotally antisymmetricunderinterchangeof index-pairs(&i) *~* (Jij). The maximalspin iscarriedby stateslike

4~q~”-4~”fms0)

and is

Smax s0+N/2. (3.18)

The minimal spin is 0 if N/2 � s0 or s0 — N/2 otherwise.The “top” state is reachedafter all 2Noperators4 have been applied once. It always has spin So, since it can be seen as producedbyapplicationof the operatorpairs q~1421 4~24~2etc.,which all carry spin 0.


The dimensionof a representationwith So = 0 is 22N becausethereare now 2N independentraisingoperators,comparedwith only N in the masslesscase.

Since renormalisability requires massive matter to have spin ~, we deduce from the aboveexpressionfor Smax that we must have

N= 1 for renormalisablecouplingof massivematter.

We seethat, apartfrom supergravitywhererenormalisabilityis not an issueandin theabsenceof centralcharges,the only importantmultipletwith massis that of the massiveWess—Zuminomodel[72],whichhasN = 1 and So = 0 and contains a scalar, a pseudoscalarand the two spin statesof a massiveMajoranaspinor:

N=1 s”: 0~ ~ 0

states: 1 2 1. (3.19)

Since,as we shall see,representationsof the supersymmetryalgebraon fields areformally very similarto thoseon massivestates,the studyof the latteris muchmoreworthwhilethan it would appearat thisstage.

3.4. One-particlerepresentationswith centralcharges

The centralchargesZ,1 on the right-handside of the {Q, Q} anticommutator(2.381) commutewitheverything,seeeq. (2.38n). We can thereforechoosea basisin our representationspacewheretheyarediagonalandrepresentedby complexnumbersz11. Theseform an antisymmetricNXN matrixwhichcanbebroughtinto a standardform with the helpof a unitary matrix U:

51J = U,” U1’ Zk,. (3.20)

The standardform is, for evenN,

2 = [ (3.21)

whereD is a real, diagonal matrix with non-negativeeigenvalues

0�z(,.); r1,...N/2. (3.22)

If N is odd, there is an additional row and column in (3.21) with all zeros.We now use the unitarymatrix U to redefineour 0’s,

U1’ Qaj~Qai; Q’~(U1)

11~ 0’~~,

and introducedouble-indicesi = (a, r) compatiblewith the obvious form of (3.21), i.e. a = 1, 2 andr = 1, . . . ,N/2. Again, for oddN, the last chargeQaN is not touchedby this.

76 Martin F Sohnius, Introducing supersymmetry

The algebra of the 0’s is now

{Qaar, Q~} 28a!~ 8,.~ (uM)ai, ~M (3.23a)

{ Qaar, Q,3b5} =

2Ea1SEab6,.sZ(,.) (3.23b)

{~ar ~~S} = —2e.1jEth8~SZ(r). (3.23c)

For odd N, we also have

{QaN, ~ = 0; {QaN, Qi} = 28N’ (uM)a$PM. (3.23d)

Let us first considerthe masslesscase.As before,we find in the standardframe (3.4) that O2j = 0. Butnow this implies, through (3.23b),that all Z(,.) = 0 and we concludethat

masslessparticle representationsrepresentcentralchargestrivially.

For the massivecase,we introducelinear combinations

At,. ~(Qair ±Qa2n) (3.24)

andtheir Hermitianadjoints.We noticehow dottedandundottedLorentz indices aremixed in suchaway that covarianceundertherotationsubgroupis maintainedsince 0,. and ~a transformin the samewayunderit, seeeqs.(A.15)and (A.17).

In termsof A~,the rest-framealgebra(3.23) now reads

325)

{A~,.,(A~5)t} = 8,.138,.5(m±Z(r)) (

andwe concludeimmediately,from the positivity of the left-handside of the last equation,that

zfr)<m. (3.26)

Let us assumethat this bound is satisfiedfor a numbern0 of eigenvaluesZ(,.) of the centralcharges.

Then the correspondingA arerepresentedtrivially, andafter rescalingthe remaininggeneratorsq~ (m ±Z(r))~

2A~r (3.27)

q,.N m~2QaN (if N odd),

we havethe Clifford algebra for 2(N — no) fermionic degreesof freedom. As far as the spectrumisconcerned,wehavethe samesituationas without centralcharges,exceptthat

N is effectivelyreducedby n0, the numberof centralchargesthat satisfythe boundm = z.


The simplestrepresentationwith centralcharge,theN= 2 hypermultiplet[57, 14, 60] hasone centralchargewhich saturatesthe bound,andthe spectrumis a doubledversionof the massiveWess—Zuminomodel.

3.5. Spectrumgeneratinggroups [21]

We_saw in subsection3.2 that the masslessstateswhich are generatedby applying n operators4’ Q’~to the groundstateIE, A0) carry n antisymmetricindices i1,. .. i,,. Thereare (~‘)of thesestates,

and they all havehelicity A0 + n/2. Thus the stateswith a given helicity form an irreduciblerepresen-tation of the groupU(n), characterisedby the Young-tableaufor n antisymmetricindices,

:: nrowsH

and an R-weight, which can be normalisedto be n. U(n) characterisesthe representationsquiteindependentlyof whetherit is actually a symmetry of the field theory whose spectrumwe may belooking at (the actual internalsymmetrygroupis generatedby the operatorsB,. as discussedin section2, andmaybe only asubgroupof U(N)).

The reasonfor thisis that therepresentationspaceof the algebra(3.6) which wasthestartingpoint ofourdiscussionof masslessparticlemultipletswill automaticallycarryarepresentationof theautomorphismgroup of that algebra.This is the group of all linear transformationsof q, and 4’ into eachothersuchthat the algebraitself remainsinvariant.

In order to constructthe largestsuch groupwe introduce2N realoperators

a1~q1+4 forl=i�N(3.28)

aj~i(q1—41) forl=N+i,

in termsof which the algebra(3.6b) reads

{a1, aj} = 26g. (3.29)

The largest automorphismgroup of this is the orthogonalgroup O(2N) which is generatedby theoperators

= ~i[aj,aj]. (3.30)

Two comments:first, °Lt is clearlynot a “generatorof a symmetry”in the senseof section2, sinceit isnot bilinearin creationand annihilationoperatorsfor particlestates(it is quadrilinear),andsecond,O~really generatesthe coveringgroupspin(2N) of O(2N), just as the matrices~i[yM,y,.] generateSl(2, c)ratherthan 0(1, 3) in the usualDirac theory.

The 2N statesof ourmultiplet carry the oneandonly irreduciblerepresentationof the Diracalgebra(3.29). But, of course, all helicities in the multiplet are containedin this. As a representationof


spin(2N), it is reducibleinto two 2’~~’-dimensionalrepresentations,correspondingto the bosonicandfermionicsubspacesof our multiplet. The fact that spin(2N) respectsstatisticsbut not helicity is easilyunderstood:itsgeneratorsarebilinearin fermionicoperators— andthusarebosonic— but theyarea medleyof 0’s and0’s andthuscannotbe expectedto preservehelicity. The largestautomorphismgroup whichdoesnot mix 0’s and 0’s, andthuswill respecthelicity, is generatedby

H1’=—~[q1,4~] (3.31)

andis U(N). Correspondingly,we found that we could classify thestatesfor eachhelicity in termsof anirreduciblerepresentationof U(N).

We now turn to the spectrumgeneratinggroupfor massiverepresentations.First, we observethatthe algebra (3.16) hasagain the group U(N) as the largest automorphismgroup which leaves theLorentz indicesa and/3 untouched.

Since a state is createdby applying operators4’,. to a ground state, its Lorentz propertiesaredeterminedby the symmetryunderinterchangeof a-indices,given typically by a Youngtableauof theform

The total antisymmetryin index pairs thenrequiresthe U(N) propertiesof this stateto bedeterminedby the mirror-imageYoungtableau

0

and an R-weight which we can normaliseto be the total numberof 4’s applied, i.e. the numberofsquaresin the Youngtableau.

However,eachspin can ariseout of severaltableaux,e.g. spin ~out of

o EF EH~, etc.,

andthus theremaybe severalU(N) representationsfor eachspin. Thereforethestatesof any given spindo not generatean irreduciblerepresentationof this particularautomorphismgroup.The reasonbehindthisis that thereis indeeda largerautomorphismgroupof (3.16) which leavesthe spin untouched.

We notice that “spin” is a propertyof the rotation subgroupratherthan of the full Lorentz groupitself. Under spatial rotations,the operatorsq~.and 4” transformin the sameway and we can treatthem on the samefooting. We defineq,~1with I = 1,. . - 2N to be

~ forI=i~N (3.32)

q~i 4a1 for I = i + N and a = & (numerically).

In termsof these,the algebrabecomes

Martin F Sohnius, Introducingsupersyrnmetry 79

{q~i, q13j} = c,.13 C1~ (3.33)

with C1~the matrix elementsof the antisymmetric2N x 2N matrix

C=[° ~]. (3.34)

The largest automorphismgroup of this algebrawhich leaves,the spin indices untouchedis thesymplecticgroup Sp(2N).The “symplectic reality condition”

(q,.1)t= e,.

13C1~q13~ (3.35)

follows from the definitions (3.32),andin order for it to be invariant,the automorphismgroupmustbeunitary andthusbe the unitary version of Sp(2N), namelyUSp(2N).This can be seenas follows:

The defining condition for Sp(2N) is that its groupelementsS leaveC invariant:

STCS=C. (3.36)

Thecondition that (3.35)be invariant,reads

(Sq)t=eC(Sq) or StTqt=rCSq.

Multiplying with5-ltT we get

qt= e(S1tTCS)q, i.e. C= STCS.

Theunitarity of S follows from comparisonwith (3.36).We concludethat

USp(2N) is the spectrumgeneratinggroupfor massivemultiplets of N-extendedsupersymmetrywithout centralcharges.

The generatorsof thisgroup are

D11 = ~c

13”[q,.

1, q13j]. (3.37)

As an example, let us consider the following tables which give the propertiesof the massiverepresentationswith 5~= 0 for N = 1 andN= 2:

N= 1; s~=0:

spin: 0

SU(1)R: 10+ 12 11 (3.38)

USp(2): 2 1


N= 2; So = 0:

spin: 0 1

SU(2)R: 10+32+ 1~ 21+23 12 (3.39)

USp(4): 5 4 1

N= 2; Sq = 0; z= m:

spin: 0

USp(2): 2 + 2 1 + 1. (3.40)

3.6. A representationon fields (theN = 1 chiral multiplet)

By far the most practically important representationsof supersymmetry,both “on-shell” and“off-shell”, are those on fields. It is they from which supersymmetricfield theoretical models areconstructedandwhich havekept theoreticiansbusyfor manyyears.

In this subsection,the simplestfield representationof the simplestsupersymmetryalgebra,theN = 1chiral multiplet, is constructedexplicitly andin detail.

Elementsof the Hilbert space of a quantum field theory can be generatedby the action offield-valuedoperators~/(x)on a translationallyinvariant vacuum:

x) = ~(x) 0); Jx, x’) = /.(x) 4(x’) JO),

etc., andtranslationsof a statearegeneratedby the energy-momentumoperator:

Jx + y) = e’~”x); Jx + y, x’ + y) = e~~”jx, x’), (3.41)

wherey P yMP~.Thereforethe displacementof a field takesthe form

~(x + y) = e’~”/~(x)e~~” (3.42a)

or, differentially,

[4’ PM] = 9~. (3.42b)

Becausewe know theactionof ~M on fields, thestructureof the supersymmetryalgebrais very similar to

that of a Clifford algebra:{Q, Q} = {~,Q} = 0; {Q, ~} = somethingknown

andwe can proceedto constructarepresentationexplicitly in “seveneasysteps”:Step 1: We choosesomecomplexscalarfield A(x) as “ground state” of our representation.Step2: We imposethe constraint

[A, Q,~]= 0. (3.43)


Although this constraintis somewhatreminiscentof the Clifford ground statecondition(3.8)— with therolesof 0 and0 interchanged— it is ratherarbitraryat this stage.It defineswhat is knownas thechiralmultiplet. Thisnameis appropriatesincethe constraintitself hasa chiral index a.

The following formula,derivedfrom the gradedJacobiidentity, the algebra,andeq. (3.42b),

{[A, 0], Q} + {[A, ~], Q} = [A, {Q, ~}] = 2io~”aMA, (3.44)

implies that A mustbe complex, sinceotherwisewe would have [A, 0] = 0 as well as [A, 0] = 0 andthiswould meanthat A is constant,aMA = 0.Step 3: We define fields ~4r,.(x),F,.13(x) andX.,,3(x) by

[A, 0~] 2i~’,.

{‘/‘a, Q13}~—iF,,13 {~fr,., ~

Step4: We enforcethe algebraon A. Becauseof (3.44) and(3.43) this means

2i(o.M),.~8MA = 2i{~i,.,O~}r2iX,.~

and,from the similar identity for [A, {Q, 0)],

O2~{i/i,., 013}+2i{ ‘/‘~,0~}“2(Fa13+F13,.)

The solution of the latter equationis

F,,,13 =

with F(x) a complexscalarfield.Step5: We define fields A,. and,l~aby

[F,Qa]~Aa; [F,t~a]~a.

Step 6: We enforcethe algebraon ci’. The gradedJacobiidentitiesinvolved are

[{i/i,,,, Q~),~s] + [{çfr,.,O13}, 0~]= [i/i,., {Q,~,Q~}] (3.45)

and a similar onefor [{ci’~Q}, 0]. Theresultingconditionsare

+ 2i(o~”),,,g3Mct’s = 2i(uM)

135 aM~1a, c,,,13A7+ e,.~A13= 0

andhaveonly onesolution:

- —‘~~ 113/ ~ . —Xa MV’ ‘L° )$~~ ~

This is shownby contractingindices a and/3 with the helpof ~

82 Martin F. Sohnius,Introducingsupersvnunetrv

Step 7: We haveto checkthe remainingconditions

[Vi,{Q, 0)] = [F, {Q, 0)1= [F, {O, O}] = 0

[F,{0, On= 2iuM aMF,

which are found to be satisfied.We havethusconstructedarepresentationof theN = 1 supersymmetryalgebraon a multiplet ~ of

fields

= (A; ci’; F). (3.46)

The representationis in termsof the (anti)commutators

[A,Q,.]2icfr,, [A,Q~]O

{~fra,Qis} = —ie~13F {çi’,., 013) = (~M),.~ 8MA (3.47)[FOa]2ôM,/,13(uM)$a.

Usually,one introducesanticommutingspinorparameters~“ and

(3.48)

which are definedto anticommutewith everything fermionic (including themselves)and to commutewith everythingbosonic(including, of course,ordinaryc-numbers).Theyaresomethinglike “fermionicnumbers”. It should be mentioned that there ought to be no mathematicalinconsistencyin this:spinorialquantumfields C(y), e.g., have thesepropertiesif their coordinatesyM are space-liketo eachother andto the region wherewe do our physics [33].

With the help of the spinor parameters,we can define infinitesimal supersymmetryvariations of afield ~ by

—i[4, ~Q+ 0~], (3.49)

in analogywith eq. (3.42b) (seethe TechnicalAppendix for properdefinitions of implied summationover spinor indicesand Hermitianconjugation).

For our multiplet, the transformationlaws are

=

= —~F— i ~ (3.50)

6F= —2i açfrciTM~

andthe whole algebracan bewritten as

[6~,ô2]~= 2i(~io.M~2— ~2°~’ ~1) ôM~~ (3.51)


Let uscountdegreesof freedomfor the multiplet 4. A “degreeof freedom”is an unconstrainedsinglereal field. Therearetotal of four bosonicones:ReA, Im A, ReF andIm F, andfour fermionicones:Re ‘/‘l, Im Vii, Re Vi2 and Im Vi2. The multiplet hasthus 4+4 degreesof freedom;this is the smallestpossiblenumber in four space—timedimensions,sinceany multiplet mustcontain a spinor, andspinorshaveat least two complexor four real components(Weyl and Majoranaspinors,respectively).Themultiplet is thereforeirreducible.

The algebraitself, mostobviously in the form (3.51), prohibits a trivial representationaq5 = 0, sincethiswould imply 4’ = const.Thus,

the only trivial representationofsupersymmetryareconstantfields.

The field multiplet 4, which we havederivedin this subsection,is called by various names: “chiralmultiplet” (which I preferand shall use),“scalarmultiplet” (the namegiven it by WessandZumino intheir first paper[71])and “Wess—Zuminomultiplet”. —

If we had, in step 2, set a constraint [A, 0] = 0 instead of [A, 0] = 0, we would have got ananti-chiral multiplet q~.Such a multiplet can be constructedfrom 4 by Hermitian conjugation,becauseindeed [At, 0] = 0. The multiplet hascomponents

4~(At ci’; Ft) (3.52)

andtransformationlaws

= -Ft~+i~oMI~At (3.53)

= 2i~uMaMciI.

The anti-chiral multiplet containsthe same4+ 4 real field componentsas the chiral one,they are justarrangeddifferently. —

Furtherchiral multiplets can bederivedby giving every field in ~/‘or q~an additional Lorentz index.Such multiplets are not irreducible. Carehasto be takenwhen addinga spinor index becausethisinvertsthe statisticsof the fields. Indeed,eqs.(3.50)and(3.53) canbe usedto obtain the transformationlaws for multiplets with inversestatisticsonly afterall ~havebeencommutedto the right of the fields. Iconcludethis sectionby pointing out that the principle of the “seveneasysteps”can be appliedto muchmorecomplicatedcases,e.g. in supergravitywherethe algebrais morecomplicatedandinvolvesvariousfield dependentgaugetransformations.The stepsare then not soeasy.

4. N = 1 multiplets andtensorcalculus

In this section, the real general multiplet is introduced [71] and more propertiesof the chiralmultiplet of N = 1 supersymmetryarediscussed.We shall seehow wecan combinemultiplets andhowwe can construct“invariant” scalarproductsfrom anytwo of them [731.


4.1. Four-componentnotation

Fromnowon whenevercomponentmultipletsareused,I drop chiral notation (dottedand undottedtwo-spinor indices) and adopt the usual four-componentnotation of relativistic spinor physics.Theadvantagesanddisadvantagesof eithernotationweigh aboutequal(chiral notationis somewhateasiertomanagein calculations,four-spinorsallow a moreconcisepresentationof results),andsinceeachiswidelyusedin articleson supersymmetrictheories,theymustbothbeintroduced.Thebasicformulasrelatingthetwo arecontainedin the TechnicalAppendix, sectionA.5.

For any N-extendedsuperalgebra,we can defineMajoranaspinorsfrom our chiral 0,.,,

(4.1)

In this four-spinornotation,the supersymmetryalgebrawhereit involves 0 becomes

{Q,, 0,}= 2(8IJYMPM +ilm Z,~+iy5ReZ,1) 42

[0,PM]=O; [Q,A~V]=~rM,.0; [Q,R]=i750. (.)

We alsohave,with a Majoranaspinor~’definedfrom i”’:

~ =—i[~,~Qj (43)

[o~,82]~= 2i~~iyM~28M~+2i~’[4, Im .7,,+ y~ReZ,1j~2’.

We notethat whereassomesymplecticsubgroupof U(N) was an automorphismgroupof the algebrainthe chiral notation, eq. (2.37), our definition of the Majorana chargesmixes upper and lowerU(N)-indices,andwe are left with someorthogonalsubgroupof the original automorphismgroup asthe new, smallerautomorphismgroupof (4.2).

A real (Majorana)form of the N = 1 chiral multiplet is obtainedin the following way:We call the real and imaginary parts of the complexfields A andFt in the chiral multiplet by the

namesA, B andF, G andhopethat thereis no confusionover thesymbolsA andF. We thenconstructa Majoranaspinor Vi from the chiral spinor ci’,. and its conjugate~ The transformationlaws for theresultingmultiplet

q5=(A,B;~F,G) (4.4)

can be calculatedfrom the complexversion (3.50)in a straightforwardmanner(J~yMaM):

~A=~Vi; ~B~y5Vi;

= —(F+ y5G)~—iJ(A + y5B) ~‘; (4.5)

bFi~Jçi’; bG—i~ysJVi.

The parameter~is a Majoranaspinorconstructedfrom ~“ and i”. The presenceof ys’5 in (4.5) givesclearparity assignments:A andF are scalars,B and G arepseudo-scalars.


The defining propertyof the chiral multiplet, namely the absenceof ~“ from the variation of thecomplexA-field, manifestsitself nowin that the spinor in 6B is just y~timesthe onein 3A.

We could have derivedthe transformations(4.5) directly in a processsimilar to the “seven easysteps”of section 3, with step1 beingthe choiceof a scalar—pseudoscalarpair (A, B), step2 beingtheconstraintthat 8B is just like 6A with an additionaly~,andwith the algebrato beenforcedbeingthatof eq. (4.2) with N = 1, andthereforewithout centralcharges.

Since~ and4 havethe samereal-fieldcontent, thereis no needto introducea separaterealform ofthe anti-chiralmultiplet.

4.2. Thegeneralmultiplet

Whatwould havehappenedto the constructionof the chiral multipletwithout the chirality constraint[A, 0~~]= 0? Thebasic principle behindthe “seveneasysteps”was that the anticommutator{Q, 0} is“somethingknown” which of coursestill is true. Therefore,we can derive a multiplet from a general“ground state” field C(x). This will takeconsiderablymore than seven steps,and resultsin a largermultiplet

V=(C;x;M,N,AM;A;D), (4.6)

with transformationlaws whichin four-componentnotationread

6C= ~Y5X

= (M+ y5N)~_iyM(AM+ )‘5

3M = ~(A— i /x)&N=~y5(A—i/x) (4.7)3AM i~yMA+~aMx

= —i O~’~8MAJ,— y5~D

6D —iC/ysA.

The scalarM, the pseudoscalarsC, N andD andthe vectorAM area-priori complexandthe spinorsxand A are Dirac spinors.In contrastto the chiral multiplet, however,it is possibleto imposea realitycondition

V= Vt (4.8)

which is defined to meanthat all componentsare real or Majorana.This is known as the real generalmultiplet, “general” becausethereis no constraintimposedon the transformationlaws otherthanthattheymust representthe algebra.This multiplet has8 +8 field components.

4.3. Reducibilityand submultiplets

Closerinspectionof the real generalmultiplet revealsthat it is not an irreduciblerepresentationof

86 Martin F. Sohnius, Introducingsupersymmetry

the supersymmetryalgebra. Instead,we find that the fields A, D and FM,. 3MA,. — ~,.A,. transformentirely among themselves, and thus form a submultipletdV with components

dV=(A;FM,.,D). (4.9)

These are 4+4 real field components,and we call the multiplet by the name curl multiplet. Thissubmultiplet is irreducible,but V itself is not. The transformationlaws

6A =—~ia~~FM,.--ys~D

= —i ~(YM8~— Y,.aM) A (4.10)

6D”—i~/ysA

representthe algebra,providedthat the constraint

(4.11)

holds.This is the casefor F,,,. beinga curl.A different submultiplet of (4.6) is the chiral multiplet

aV=(M,N;A_i/x;aMA,.,D+LIC). (4.12)

Both dV and 8V are submultipletsof V, but their complementsare not: the transformationlaw for x’e.g., containsthe transversepart of AM as well as both M andN This propertyof being reduciblebutnotfully reducibleis very commonfor multiplets of supersymmetry.

Since the dV and av aresubmultiplets,we can constrain V to not contain oneor the other:

dV=O or 8V=O. (4.13a,b)

In thesecasesthe remainingfields now do transformsolely into eachother.For dV = 0, the survivingcomponentsof V can bearrangedinto a chiral multiplet

(4.14)

with the scalarA definedby the solution A,. = ÔMA of 13k,. = 0. For aV= 0, the surviving componentsform a new typeof multiplet whichcontainsC, x and a divergence-freevectorAM:

L=(C;x;A,.). (4.15)

This is called the linear multiplet; it is irreducibleandhastransformationlaws

3C= CYSX

3x = —i yM (AM + 1,~~ (4.16)

6A,.=i~o,.,.8x

whichrepresentthe supersymmetryalgebra,providedthat the constraint

Martin F Sohnius,Introducingsupersymmetrv 87

ÔMA = 0 (4.17)

holds.The structureof a not fully reduciblemultiplet can be summarizedsymbolicallyasZ containsX and Y6X containsonly X, but6Y containsX and Y.

In our cases,we hadZ= VandeitherX= dVand Y= qS, or X= aVandY= L.

4.4. Theproductmultiplets

One can build a chiral multiplet ~/i~ as the productof two othersçl~and &: with [A1,0] = 0 and[A2,0] = 0, we alsohave [A1A2,0] = 0, all for complexA’s. In real componentsthis meansthat thespinor in &(A1B2 + B1A2) is y~times that in 6(A1A2— B1B2). Explicit calculation shows that thecomponentsof the productof two chiral multiplets

43*/)1~4~2 (4.18)

are

A3= A1A2-B1B2

B3 = B,A2+ A1B2

Vi3 = (A1 — ysBi)Vi2+ (A2— ysB2)Vii (4.19)

F3 = F1A2+A1F2+B1G2+ G1B2+ Vi1Vi2

G3= G1A2+A1G2—B1F2—F1B2—Vi1y5Vi2.

The product of three multiplets is associative,and thus the product of any number of them iswell-defined.

Two otherproductsof chiral multiplets can be defined.Oneis denotedby

V = X 4.~ (4.20)

andis neitherchiral nor anti-chiralbut rathera generalrealmultiplet with components

C =A1A2+B1B2

x = (B1 — y5A i)çfr2 + (B2 — y5A2)Vii

M =-G1A2-F1B2-A1G2-B1F2

N = F1A2— G1B2+A1F2—B1G2 (4.21)

A,. = B1 o,,A2+B2 ÔMAl~’ ‘,biy,.y5Vi2

A ~

D =—2FiF2—2GjG2—2a,.Aia~A2—2a,.B1a~B2—i~1/Vi2.

Like 4~~ this is symmetric under interchangeof 4~and ci’2. The third type of product isantisymmetric,is denotedby

V=b1 A ~2, (4.22)


is againa generalreal multiplet and hascomponents

C =A1B2-B1A2

x = (A1+ ysB1)Vi2— (A2+ y5B2)~/ti

M =F,A2-G1B2-A1F2+B1G2

N =F1B2+G1A2—B,F2—A1G2 (4.23)

A,. =A, a,.A2+B1 a,.B2—i~iy,.Vi2

A =(Fi—y5Gi+i/Ai+i/y5Bi)Vi2—(1~-~2)

D = —2G1F2+2F,G2+2a,.B1 3MA

2—2o,.A1 a~B2+iVuiys/Vi2.

Finally, we can also definea productfor two real generalmultiplets

V3= V,~V2 (4.24)

which is againareal generalmultiplet, is symmetricin V, and V2 and hascomponents

C3 = C1C2

X3 =Ci~2+~iC2

M3 = CiM2+MiC2—~kiysx2

N3 =C,N2+NjC2—j~ix2 (4.25)

A~= ~

A3 = C1A2+ ~(N,+ y5M1+ i/C1 — i A’1y5)x2+ (1 ~—*2)

D3 = C,D2+ D1C2 — M1M2 — N1N2—a,.C~ 8TMC

2 — A1,.A~+ Aix2 + ~1A2—

4.5. The“kinetic multiplet”

There are variousways of seeingthat anotherchiral multiplet is containedin */: we can taketheadjoint of (3.47) andobservethat [Ft, 0] = 0; or we can noticethat 6Ft doesnot contain ~“ in (3.53);or we can seethat in (4.5) the spinor in 8Gis just y~timesthe one in 6F In any case,the real fields Fand G arethestart of anotherchiral multiplet, the kinetic multiplet,denotedby T4i. Its componentsare

Tq5 = (F, G; i /,/,; -lilA, -LiB). (4.26)

We can, of course,go on anddo this again;but by inspectionwe seethat

TT~=—LJ4, (4.27)

which is why the multiplet is called kinetic: T is somethinglike the generalisationof the masslessDirac


operator:

T ~ i/.

4.6. Contragradientmultipletand invariants

We nowtry to constructa multiplet which is contragradientto the chiral multiplet. This meansthat

thereshouldbe a bilinear“inner product” of the form

L=Af+Bg+çtl~+Fa+Gb (4.28)

which is invariant.The new fields a, b, x~f andg wouldbe thoseof the contragradientmultiplet.We do,however,alreadyknow that only constantscan be invariant (seethe discussionfollowing eq. (3.51)),sothat we can at most expect to be able to demandthat L be a density,which transformsinto adivergence,

= aTMK,.. (4.29)

If L is a density,then f d~xL is invariant (and constant).Thus we can hope to constructLagrangiansfrom our densities.This behaviouris similar to thatof fields (includingLagrangians)undertranslations—a similarity which makes sense since two successivesupersymmetrytransformationsgenerate atranslation.Using (4.5) we get for 6L

6L A6f+B6g+ Vi6x+F6a + G3b+f~çi’+g~y5Vi— ~(F+ y5G)~

+i~/(A—y5B)x+i ~/Via+i~y5/Vib.

We partially integratethe last threeterms,

~

andwe_demandthat 6L shouldjust bethe divergencein the last line. We now comparecoefficientsforA, B, Vi~F andG andget transformationlaws for the fields of the contragradientmultiplet:

6a=~x; 6b=~y5x3x= —(f+y

5g)~—i/(a+y5b)~

6f~’i~/x; 6g=i~y5/~.

Theseareagainthoseof a chiral multiplet,

çb’(a, b;x;f,g),

which meansthat the chiral multiplet is self-contragradient, andthat we can constructa density fromany two chiral multiplets. We notice that the density (4.28) is just the F-componentof the product

90 Martin F Sohnius,Introducingsupersymrnetry

L = (q5 . ci’)F. (4.30)

This is quite consistentsincethe transformationlaws imply that

anyF-componentof a chiral multiplet is a scalar density.

In a similar fashion, we constructa multiplet which is contragradientto the real generalmultiplet.We start with the ansatz

L=CD’+~A’-MM’-NN’-AMA~+A~’+DC’ (4.31)

and from the known transformationlaws of the fields of the generalmultiplet, we derive thoseof the

new fields by comparingcoefficients.The result is3C’ = ~Y5X’

(M’+ y5N’)~T—iy

M(A,~,+~

= ~y5A’ (4.32)

—(aMA~+ y

5D’) ~— i /(M’ + y5N’) ~

3D’ =—i~/y5A’.

Thesetransformationsare very closely relatedto those of the real generalmultiplet. Indeed,certaincombinationsof thesefields form a real generalmultiplet:

V’ = (C’; x’; M’, N’, A~A’ + i Ix’; D’ — LIC’). (4.33)

Comparisonwith eq. (4.12), or simple inspectionof the transformationlaws (4.32), will show that thisshifted form of the generalmultiplet is particularlysuitedfor detectingthe chiral submultiplet 8V’ andthe linear multiplet L to which V’ reduceswhen av’ = 0. Theotherform of V wasmoresuitedto makethe curl submultiplet dV apparent.

Since

anyD-componentofa generalmultiplet is a scalardensity,

it is not surprisingto seethat the invariant (4.31) can alsobe written as

L = (V. V’)D +4-div. (4.34)

Martin F Sohnius,introducingsupersymmetry 91

4.7. Dimensions

If we definethe (mass)dimension~ of a chiral multiplet as that of its A-component,we find that

dimA=dimB=~1~dimqS

dimVi=J+~ (4.35)

dim F = dim G= ~ + 1

becausethe parameter~ must havedimension—~ so that i~y~~2is a length. The assignments(4.35)then follow from the transformationlaws (4.5). Furthermore,

dim(~1~2)=L11+~2

dimT4 = z~+1(4.36)dim(q~. Tq5)F = 2i + 2

dim(4i”)F = tiLl + 1.

5. The Wess—Zuminomodel

The Wess—Zumino model consists of a single chiral multiplet 4 in renormalisableself-interaction[72].

5.1. TheLagrangianand the equationsof motion

If we want to constructa Lagrangianwhich describesthe motion of fields in time, it mustcontainderivatives.If the maximal numberof derivativesis to be two (for the bosonfields), we must involveexactly one of the T-operationsof subsection4.5. This, together with the demandthat dim L = 4,determinesthe dimensionof themultiplet to be zl = 1. ThemostgeneralrenormalisableLagrangianfora single chiral multiplet (i.e. one with no coupling constantsof negative dimension) is then theWess—ZuminoLagrangian:

L~b.Tcb__~cb~cb)Lo+Lm+Lg (5.1)

(a furtherpossibleterm (Acb)F can be eliminatedby field redefinitions,moredetailsof this canbe foundin section6). Using manyof the tensorcalculus resultsof the previoussection,we can spell this out incomponents:

L0 =~(a,.AaMA+a,.BaMB+i~/Vi+F2+G2)+4-div

Lm = m(AF+ BG+ ~ViVi) (5.2)

Lg = —g[(A2 — B2)F + 2ABG+ i/i(A — y5B)çb].

92 Martin F Sohnius,Introducingsupersymrnetry

The term “4-div” in L0 standsfor the divergencewhich arose out of the partial integration whichreplaced—~(AlilA + B LIB) by the standard~(a,.A)

2+ ~(a,.B)2.Fromthis Lagrangian,the following Euler—Lagrangeequationsof motion can be derived,

F= mA+g(A2—B2)

G= mB+2gAB

i /çi’ = mVi + 2g(A— y5B)Vi (5.3a)

-lilA mF+2g(AF+BG+~fr~)

-LllB”~mG+2g(AG-BF-~çi’y5~),

which can be written most elegantlyin multiplet form

Tq5=mq5+g4i~q~. (5.3b)

5.2. Auxiliaryfields

The Lagrangianand the equationsof motion in their presentform may not be very illuminating asfar as the “physics” of the model is concerned.Notice, however,that the field equationsfor F and Gare purely algebraic.These fields can be eliminatedfrom the Lagrangianand from the equationsofmotion by useof their own field equations.Such fields whose equationsof motion do not describepropagationin time andspace,are calledauxiliary fields. The result of their elimination is in our casethe “on-shell” Lagrangian*

L = ~(a,.AaMA — m2A2)+ ~(aMB 8TMB— m2B2)+ ~i(i/— m)Vi

— mgA(A2 + B2) — g~(A— y5B)Vi— (A

2 + B2)2 (5.4)

andthe “on-shell” equationsof motion

(il + m2)A= —mg(3A2+ B2)—2g2A(A2 + B2)—gçlsçb

(LI + m2)B= —2mgAB— 2g2B(A2+ B2)+ gi,by

5~ (5.5)

(i/— m)çb= 2g(A— y5B)~.

What doesthe Lagrangian(5.4) know of the supersymmetryof the model?Actually, it is not the mostgeneralrenormalisableLagrangianfor the fields involved.All sevenpossibleparity-invariantinteractionterms

A3,AB2,A4, B4,A2B2, AçIJVi and B~fry

5Vi,

‘The somewhat idiosyncraticuseof ‘on-shell” and “off-shell” in supersymmetryis discussedbelow (subsection5.4).


andthe threepossiblemasstermsarepresent,but all massesare the same(O’Raifeartaigh’stheoremagain!)andall sevencouplingsarefixed in termsof themassandasinglecouplingconstantg. This reflectsthe fact that the Lagrangian,evenafter the fields F andG havebeeneliminated,still transformsas adensityunderthe “on-shell” transformations

6A =

3B~y5Vi (5.6)

= —[i/+ m + g(A+ y5B)](A+ y5B)~

which arederivedfrom (4.5) by elimination of the fields F andG, usingtheir own equationsof motion.There is no needto employ the dynamicalequationsof motion (5.5) to show that (5.4) transformsas adensityunder(5.6).

5.3. Commentson renormalisation

Particularlyfrom this last discussion,we seethat it is actuallythe particularrelationshipbetweenthecoupling constantsof the Lagrangian (5.4) that constitutesthe supersymmetryof the model. Thequestionis now whetherthis relationshipis stableunderrenormalisation.This hasbeenfound to be so[72,39], since there actually exists a regularisationwhich preservessupersymmetryfor the model(Pauli—Villars).

Moresurprisingly,theindependentrenormalisationsof m andg arenotneededsinceall contributionstothequadraticallyandthelinearlydivergenttermscancel,dueto fermionicandbosonicloopscontributingwith oppositesigns.Thereis only asingle, logarithmically divergentinfinity which canbe absorbedinto awave-functionrenormalisationwhichiscommonto thewholemultiplet, i.e. thesamefor all thefieldsA,Band Vi.

This absenceof infinities which are not prohibited by the impossibility to write down a supersym-metric counterterm(the counterterm3m (4 4~would, e.g., be perfectly supersymmetric)is called“miraculouscancellations”.They arenow understoodin termsof non-renormalisationtheoremsderivedinsuperspace[30,311.

The “miraculouscancellations”requirethe presence,in a very controlledway, of fields whose loopscontributeconsistentlywith oppositesigns, asbosonsandfermionsdo. A “well-controlled relationshipbetweenfermionsandbosons”is a supersymmetry.

5.4. On-shellandoff-shellrepresentations

Oneof the importantopenproblemsin the field of supersymmetryis to find “off-shell” formulationsfor the higher-Nextendedmodelsor to prove that theydon’t exist.Using the Wess—Zuminomodelasan example,it is straightforwardto explain what is meantby off-shell andon-shell representations.

The transformationlaws (4.5) representthe supersymmetryalgebraon the4 + 4 field componentsofthe chiral multiplet, unconditionallyand independentlyof the dynamics,i.e. of a Lagrangian.Eachofthe threeterms in the Wess—ZuminoLagrangian(5.1) separatelytransformsas a densityunderthosetransformations.This is called off-shellsupersymmetry.

The completeset of field equations(5.2) is againa multiplet, in this caseactually a chiral multiplet.The structureof thesefield equations,however,separatesthem into two classes,algebraicequationsfor


somefields (which arethus auxiliary), andwaveequationsfor the others(which thusdescribedynamicaldegreesof freedom).

The elimination of auxiliary fields takesplaceby enforcing their field equationson the Lagrangianandon the transformationrules.This procedureis not itself supersymmetric:sinceall field equationssitin an irreduciblemultiplet, we can retain the supersymmetryonly by taking all fields “on-shell”, i.e.subject to their equationsof motion. Whatdoesthis meanin practice?

The on-shell Lagrangian (5.4) is still a density under the on-shell transformations (5.6). Thesetransformations,however,havebecomedependenton the model (they dependon m andg), andthereis no part of the Lagrangianwhich separatelytransformsas a densityunderthem.

In gaugetheoriesand supergravitythis leads to an inseparablemarriagebetweenbackgroundandmatterwhich is tolerableat most for themaximally extendedtheories(N = 4 andN = 8) wherethereisno matterandwhereeverythingis a supersymmetricversionof gaugeor gravitationalbackground.

The mostseriousdrawbackof “on-shell formulations” appearsif we calculatethe commutatorof twoof the transformations(5.6). The result is 2i~1/~2 on A and B, as it shouldbe, but

[ô1,~21Vi= 2i~1/~2 Vi— y~(i/—m— 2g(A — y5B))Vi~y,.~2 (5.7)

on ~‘. This meansthat

the on-shellalgebra closesonlyif the equationsof motionhold,

which set the last term in (5.7) to zero. This situationcould in principle be disastrousfor quantumcorrections:therethe fields mustbe takenoff-shell, away from their classicalpathsthrough configura-tion space.What this doesto the on-shell supersymmetrictheoriesis quite unclear.Obviously, if thereexists—evenunknown to us—someunique off-shell version, there should be no problem. But whathappensif the theory were intrinsically only on-shell supersymmetric(asthe N = 4 andN 8 theoriesmay well be) or if therewere severalcompetingoff-shell versions(astherearefor N = I supergravity),is unclear to date.

There are several ways to understandwhy on- and off-shell representationsare different and whyauxiliary fields appearin supersymmetrictheories. At the root of the problem lies the differencebetweenthe vector spacefor the off-shell representations,fields over x

M, and that for the on-shellrepresentations,the Cauchydatafor the field equations.In both casesthe fermions= bosonsrule musthold. For different spins, however,thereis a different relationshipbetweenCauchydata and fields.While, e.g., a real scalarfield A(x) describesone neutralscalarparticle, the Dirac field Vi(x) haseightreal components,but describesonly the four statesof a chargedspin-~particle. Thus,in going from thefields to the states,we havelost somedimensionsof our representationspace,but differently so fordifferent spins. Supersymmetricmodelswith their strict fermions= bosonsrule must somehowwiggleout of this, and they do so by meansof auxiliary fields whose off-shell degreesof freedomdisappearcompletelyon-shell.

55. A supercurrent

As for any continuoussymmetry, there should be a Noether current associatedwith the trans-formations (5.6) and the Lagrangian(5.4). This is a Majoranaspinor-vectorcurrent, given by theformula


ol ÔL 3LJ =—=(6A +6B +&Vi——içl (5.8)M a~\ 3I9TMA oaMB

if 8L = 9MKM. The term KM is determinedonly up to termsof the form t9aM,. with a,,,. = —a,.,,. Thesepossible“improvementterms” havebeenstudiedwidely [19,35], andoneform [72] of the supercurrentis

= /(A — y5B)y,.Vi+imy,,(A—y5B)~+igy~(A—75B)

24’i. (5.9)

This is derivedlike this: for the “off-shell” Lagrangian(5.2) weget

6L= i~ls~’[~~

= 8MK

with

~

+im~(A+y5B)y,.~+ig~y,.(A+y5B)2çb.

Theresultfor the “on-shell” Lagrangian(5.4) is the same,with F and Gexpressedin termsof A andBas given by (5.3). Theothertermsin J,. are

6A + &B 3~”B+ 3Vi ~3MVi= ~3,.(A+ y5B)Vi + ~i~(F+ ysG)y,.Vi+ ~/y,.(A + ysB)Vi.

Adding up, we get (5.9) for .1,., modulo an improvementterm

8aM,. = 2i0[o~MP(A+ y5B)Vi].

Thesupercurrentis conserveddueto the equationsof motion (5.5), asit shouldbe. To verify thisis abittricky andinvolvesan identity which is discussedin the TechnicalAppendix, eq. (A.48).

The existenceof a conservedsupercurrentfor a model with interaction is a highly non-trivialoccurrence.In fact, we could havestartedwith the Lagrangian(5.4), pulled the current (5.9) out of ahat, observedthat it is conservedandthenconsideredthe charges

Q=Jd3xJo. (5.10)

The transformationlaws of the fields could thenhavebeencalculatedas

andif we hadusedthe canonicalequal-timecommutationrelations

96 Martin F Sohnius,Introducingsupersymrnetrv

[A,A}=[B,E]=iTh [A,A]=[B,B]=0; {Vi,Vi}y~8,

the transformationlaws (5.6) would haveemerged.The supersymmetryalgebracould then havebeenderivedfrom thesetransformationlaws. This approachto supersymmetryis quite popularand widelyused.

It is important that the supercurrentshouldactuallyexist for an interactingmodel.Take the free toymodel

L50~= ~8~AÔMA + ~iViJVi.

This Lagrangianhasa supersymmetry

6A=~Vi; 6Vi=-i/A~

with a current

J,. =/Ay,,Vi

which is conserveddue to the free equationsof motion LIlA = /Vi = 0. There is, however,no way tomakethis toy model interactandpreservea conservedsupercurrent:in contrastto the Wess—Zuminomodel, the toy model is not supersymmetricin a non-trivial sense.It cannotbe since it violates the“fermions= bosonsrule”.

6. Spontaneoussupersymmetrybreaking(I)

In this section,we analysepossibleself-interactionsof chiral multipletsandwe seehowit is possibletobreaksupersymmetryspontaneously.

6.1. Thesuperpotential

It is natural to breakup the Lagrangian(5.1) of the Wess—Zuminomodel into a “super”-kinetic anda superpotentialpart:

L=~(4~T~)F—[V(~)]F (6.1)

with

(6.2)

We hereincludethe possibleAj termin the Lagrangianandwill thereforeseeexactlywhy andhow itcan beeliminated.In this notation,the equationsof motion (5.3) take the very compactform

Tq~= V’(~) (6.3)

with V’ thederivativeof the function V with respectto its argument.


Sinceall termsin [V(4)]F areeither linear in F or linear in G or quadraticin Vi (this will betrue aslong aswe stick to renormalisablecouplings),we havea scalingequation

[V(qS)]F= (F8I8F+ Gô/8G+~Vi8I81/J)[V(cb)1F. (6.4)

Let usevaluate(818F)[ V(q5)]F in a generalway: we use a suitableadaptationof the chain rule, withdenotingthe productof two chiral multiplets,andget

V(q5) = ~. ~ V(4)= (0,0;0; 1, 0). V’(q5) = (0,0;0; [V’(~)]A, —[ V’(çb)]B).

This implies the first stepin the following equation

= [V’(cb)]A= F, (6.5a)

the secondbeingthe equationof motion (6.3). Similarly, weget

—~ [V(4)]F = [V’(cb)]B = G, -~= [V(~)]F = [V’(~)]~,. (6.5b)

If we now usethe lettersF andG, not to denoteindependentfieldsbut asa shorthandfor [V’(~)]A and[V’(4)]~, thenthe superpotentialbecomes

= F2+ G2+~i/i[V’(cb)]~,. (6.6)

The “true” potential U must,of course,include the non-kinetictermsin ~ T~)~:

U= —~(F2+ G2)+ [V(4)]~ = ~(F2+ G2)+ ~[ V’(4)]~. (6.7)

We see that for the vacuum, where (Vi) = 0 (otherwiseLorentz invariancewould be spontaneouslybroken), U is nevernegative.This is a consequenceof the positivity of the energyin any supersym-metric theory,spontaneouslybrokenor not.

6.2. Conditionfor spontaneoussupersymmetlybreaking

Supersymmetrywill be spontaneouslybroken if and only if the lowest energy (vacuumenergy)islargerthan zerobecause

(OIEJO) � 0 ~ 0,. 0)� 0, (6.8)

due to

Emin (olEIo) = ~ Jo,.o~J2~ (6.9)


Since,on the otherhand,

Emin = (OIUIO) (6.10)

we find that

supersymmetryis spontaneouslybrokenif and onlyif the minimumofthepotentialis positive.

This canbe summarisedin figs. 6.1 and6.2.

Let us analysethe Wess—Zuminomodel in this respect.The potential is (U) = ~(F)2+~(G)2andF=A+mA+g(A2—B2), G=mB+2gAB.

The minimumof the potentialis at (F) = (G) = 0, i.e. at

(A)=_!(m±Vm2_4gA) form2�4gA2g (6.lla)

(B) =0

U ~Uj ~Uj

supersymmetric non—supersym metric non—supersymmetricground state ground state potential

Fig. 6.1. Various potentialsand their implicationsfor supersymmetry.

U ~ < A>

Fig. 6.2. The potential in thewess—Zuminomodel.

Martin F Sohnius,Introducingsupersvmmetrv 99

m(A) = — —

g for m2�4gA. (6.llb)

(B)= ±~~_V4gA_m2

2gThe field redefinitions*

A~”=A~(A), B~”=B~(B)

will theneliminateA~from the Lagrangianand shift oneof the minimato A = B = 0. Figure6.2 showsthe behaviourof the potentialin theWess—Zuminomodel, assumingA = 0 and mg>0.

6.3. TheO’Raifeartaigh model

A generalisationof the Wess—Zuminomodel is the following renormalisableLagrangianfor severalchiral multiplets tI3a:

L = ~(4~a T43a)F — (V)F (6.12a)

with

V Aacba +~mabcb,.4~+~g,.~çti,.çb~, ~. (6.12b)

The A, m and g arereal and totally symmetric.The equationsof motion arenow

T45a = V = Aa + mab~bb+ gabct~b (6.13)

andthe potential is

UFaFa+~GaGa+~a[~V] . (6.14)t943a it,

We shall see that the condition that therebe no spontaneousbreakingof supersymmetry,namely

(Fa)(Ga)0, (6.15)

can now not always be fulfilled.Let us considerthe examplegiven by O’Raifeartaigh[50], a special caseinvolving threemultiplets

andthe following assignmentsfor the constantswhich definethe model:

* A furtherredefinition t~’~~”= exp(—~iry5)~ris necessaryto recovermanifest parity invarianceif (B) � 0.


A3A; all otherAa=0

= m21 = m; all othermab = 0 (6.16)

g113= g131 = g311 = g; all othergabe = 0.

The equationsof motion for the six auxiliary fields arethen

F1 = mA2+2g(A3A1—B3B1)

F2 = mA1

F3 = A + g(A~— B~)

= mB2+2g(A3B1+B3A1) (6.17)

G2 = mB1

G3 = 2gAtB1

and we cannotset F3, F2 and G2 to zerosimultaneously.Consequently,the bosonicpotential

~ (6.18)

is alwayspositivewith a possibleminimal valueof*

2UminA2 for 2gA~<m2. (6.19)

This value of U is achievedby the choice

(AI) = (B1) = (A2) = (B2) = 0 (6.20)

with (A3) and (B3) arbitrary,say

(A3)=~I2g; (B3)=0 (6.21)

(we set (B3) = 0 for simplicity; this cangenerallybeachievedby suitablefield redefinitions).We plot thepotential as function of (A1) in fig. 6.3. With the assignments(6.20—21) for the vacuumexpectationvaluesof the bosonfields, we find for the fermion massmatrix

M~.=( m 0 0 J. (6.22)

\000/

This hasthreedifferenteigenvaluestm~, — m~ andm~ which are

* if 2gAj > m2, thepotential will havetwo degenerateminima which bothbreaksupersymmetry.

The tilde denoteslinearcombinationsof theoriginal fields.


U

2

Fig. 6.3. The potential in theO’Raifeartaighmodel.

m~12=V~2+m2±p72

m~3=0. (6.23)

The massmatrix for the A-fields, on the otherhand, is

/~2+m2+2gA m,a o\

M~=~( mj~ m2 0 J (6.24)\ 0 00/

with eigenvalues

m = .u2/2+ m2+gA ±V~L2+2gA )2+

= (p]2± + 2gAI~)2 + m2)2—

= m~12+gA(l±~(p2 + 4m

2)112)+o(g2A2) (6.25)

m~3—0.

The situationfor the B-fields is exactlythe same,’exceptthat gA mustbe replacedby —gA everywhere.

In the interestingcaseof very large positiveeu., i.e. for

(6.26)

wehavealargemassscale,~, anda smallone,m2/~.The mass-splittingwithin the multiplets is of order

gA/1a andmay well be almostas largeas the smallmassscaleitself. In this approximation,we get

m~1~a; mAl~m~1+gA//1, mA1~m~1—gAI~a(6.27)

m,~m2I,a; mA

2~m~2+gA/~,mA2~m~2—gA/p..

Figure6.4 illustratesthis spectrum.In the treeapproximation,the “large hierarchylimit”


gA0 gA�O.- - A

1

p A, B, LI), -—— _______

______ A2

A2 B2 W2 ~-ii:

0 A3B34.’3 —~ A, B, 4’3

Fig. 6.4. Massspectrumof theO’Raifeartaighmodel in the“large hierarchylimit”.

(A3) = p/2g~ othermasses

is not at all unnatural.The vacuumdegeneracyin A3 which allowedusto considerthis caseis, however,broken by quantum effects [38,77] and the O’Raifeartaigh model actually does not generateahierarchy: U growslogarithmically for large A3 andthe minimum of the potentialis at relativelysmallp. This is so becausepurespin f—spin 0 modelsare not asymptoticallyfree.

6.4. Themassformula

For gA = 0, i.e. in the absenceof spontaneoussupersymmetrybreaking,the Hermitian matricesM~,M~andM~,commutewith eachotherandcan besimultaneouslydiagonalised.The resultingmultipletsare mass-degenerate,as expected,with massesgiven by eq. (6.23). Accordingly, we find

M~=M~=(M~,)2.

For gA � 0, i.e. in the presenceof supersymmetrybreaking,only a weakercondition

M2A + M~= 2(M~,)2 (6.28)

still holds true and no two of the Hermitian matricesM~,M~and (M~)2can be simultaneouslydiagonalised.The masseigenstatesare thereforedifferentmixturesof multipletsç~and& for A, B andVi and in generalno two of the six particlesA

1, A2, B1, B2, Vii and Vi2 havethe samemass.Fromthetraceof eq. (6.28) for the massmatrices,however,we find the importantrelationship

~ m~= ~ m~ (6.29a)all boson all fermion

states states

Martin F Sohnius, Introducing supersymmetrv 103

or, sincethe numberof statesfor a chargeself-conjugatespin-s particle is (2s+ 1),

~ (~i)~s(2s+ 1)m~= 0, (6.29b)

wherethe index i runsover all chargeself-conjugateparticlesin the model. In the caseof unbrokensupersymmetry,this is trivially fulfilled becausefor each multiplet in the model the massesaredegenerateand becauseof the “fermions= bosons”rule, which can be written as

~ (_l)2s(2s+ 1) = 0. (6.30)

7. N = 1 superspace

A compactandvery usefultechniquefor working out representationsof the supersymmetryalgebraon fields was inventedby A. SalamandJ. Strathdee[55]: the superfieldin superspace.It is particularlyuseful for N = 1 theories;their superfieldstructureis completely known. The techniqueis less welldevelopedfor extendedsupersymmetry,althoughit appearsthat important progresshasbeenmaderecentlyfor N = 2 [23]beyondearlier superspacework [29,46,36].

N = 1 superspacehas coordinatesxM, O~and O’~and servesto representthe algebraof N = 1Poincarésupersymmetryin very muchthe sameway as the Poincaréalgebrais representedon ordinaryspace—time.

7.1. Minkowskispace(a toy example)

In order to get a feeling for what will happen,let usfirst considerordinaryquantumfields4(x) whichdependonly on the four coordinatesxM of Minkowski space.Translationsof thesecoordinatesaregeneratedby the operatorsF,. in the usual way, eq. (3.42), and we can consider4(x) to havebeentranslatedfrom xM = 0:

0(x) = e”” 4(0)e1xP. (7.1)

The transformationlaw (7.1) is compatiblewith themultiplication law

e~~”e1’~~°= e1~~° 7 2

which holdsbecausethe operatorsF,. commutewith eachother.The expressionon the right-handside of eq. (7.1) can be expandedby useof the formula

e~AeB = -~-~[A, B](,.) (7.3a)

,~n.

with

[A, B](O) = A; [A, B](~±j)= [A, B](~),B] , (7.3b)


andthen saysthat 0(x) is completelydeterminedby the propertiesat any given point x0 (herex0 = 0),namelyby the multiple commutatorsof 0(xo) with F,. to arbitraryorder.

For a Lorentz rotationwe have

exp(~A . M) 0(x)exp(_~ A . M) = exp(ix’ . F) exp(~A . M) 0(0)exp(—~A . M) exp(—ix’ . F),

(7.4a)

whereA . M AM~M,.,.,andx’ is given implicitly by

exp(~A . M) exp(ix . F) = exp(ix’ . F) exp(~A . M). (7.4b)

For infinitesimal A, we can easilycalculatex’,

(1 + A . M) eixP = e’~~ (i + A . M) e1xP

= e~x~’(1+~A~M+ [~A •Mjx .F])

= exp{i(x~+x~A,.~)F,.}(1+~A.M)+o(A2),

making useof eqs. (7.3) and the algebraof the Poincarégroup. For finite A,.”, we get the expectedtransformationlaw for the coordinatesunderLorentz transformations:

= x~(eA),.M= (e~)~,,x”. (7.5)

Finally, the action of the Lorentz transformationon 0(0) can at most be somelinear transformation

which actson as yet unwritten indiceswhich 0(0), and hence0(x), mayhave:

exp(~A . M) 0(0) exp(_~ A . M) = exp(—~A 0(0). (7.6)

Here the ~ are some matrix representationsof the algebraof the M,.,.. The minus sign on theright-handside is crucial: a furtherLorentz transformationdoesnot seethe numericalmatrices.~ andwill act “inside”, directly on 0(0)— thisreversesthe factorson the right-handside againstthoseon theleft andwe needthe minus sign to retain the groupmultiplication law. All in all, we get

exp(~A . M) 0(x) exp(-~ A M) = exp(- ~A . 0(e~x). (7.7)

The differentialversion of eq. (7.7) is

[0(x),M,.,.] = i(x,,8,.—x,.8,.)0(x)+.~,,,.0(x). (7.8)


Before going on to repeatthe sameprocedurein superspacefor the superalgebra(2.38), let us try tounderstandin a moreformal mannerwhat happenedhere.

7.2. Cosetspaces

Let g be an arbitraryelementof a group0 (in our particularexampleof theprevioussubsectionthiswas the Poincaré group) which contains a subgroupH (here the Lorentz group). We now defineequivalenceclassesof elementsof G: two elementsg and g’ are consideredequivalentif they can beconnectedby a right multiplicationwith an elementh E H:

g’goh or g1°g’EH. (7.9)

This equivalenceclassis called the (left) cosetof g with respectto H. The set of all cosetsis a manifold,denotedby 0/H.

A set of group elements L(x), labelled by as many parametersas necessary,parametrizesthemanifold if eachcosetcontainsexactly one of the L’s. Once we havechosena parametrizationL(x),eachgroupelementg can be uniquelydecomposedinto a product

g=L(x)oh (7.10)

whereL is the representativememberof thecosetto which g belongsandh connectsL to g within thecoset.A productof g with an arbitrarygroupelement,andin particularwith someL(x), will thus defineanotherL andan h accordingto

goL(x) L(x’)oh. (7.11)

Herex’ andh are in generalfunctionsof both x and g:

x’=x’(x,g), h=h(x,g). (7.12)

The Minkowski spaceof the previous subsectionwas the coset manifold Poincaré/Lorentz.We hadparametrizedit by

L(x)= e~~P (7.13)

anddeterminedx’ and h explicitly as

x’ = x + yh =1 } for g = L(y), seeeq. (7.2)

x’ = e~x (7.14)

h = g } for g = exp(~iA. M) E H, seeeqs. (7.4—5).

That casewasparticularlysimplebecause[F,F] = 0 and[P,M] = F. Thequantumfieldswerewritten as

0(x) = L(x) 0(0)L1(x) (7.15)

106 Martin F Sohnius.Introducingsupersymmetry

andthe action of a groupelementon them was completelydetermined,

g~(x)g~= L(x’) h0(0)h~L1(x’), (7.16)

oncewe knewhow h actedon

h 0(0) h~= exp(—~iA•I)0(0). (7.17)

The caseof the cosetPoincaré/Lorentzis not themostgeneralexampleof acosetspace:dueto thefactthat the translationsform an invariant Abelian subgroup,the “little group” elementh in (7.11) isactuallyindependentof x, see(7.14). This will not be so for all cosetspaces(comparethe representationof the conformal groupon fields over Minkowski spacein section16).

7.3. Needfor anticommutingparameters

We now turn our attentionto the supersymmetryalgebra.Immediately,we stumbleovera problem:it is not the Lie algebraof a groupsince it involves anticommutators.We must, however, be able toexponentiatethe algebrainto a groupin such a way that theproductof two groupelementsis againagroupelement,say

e’°°~ = e(something)

In the general case of non-commutingobjects this required behaviouris describedby the Baker—Campbell—Haussdorffformula,

eA eB = exp{ ~ C~(A,B)], (7.18a)n=t fl.

which can be derivedusingnothingelsebut the associativityof multiplication. The coefficientsC~(A,B)aremulti-commutatorsof A andB. The first five of theseare

C1=A+B

C2 = [A, B]

C3 = ~[A, B], B] + ~[A,[A, B]] (7.18b)

C4 = [A, [A, B]], B]

C5 = —~[A,[A, [A, [A, B]]]] — ~[[A, B], B], B], B] — ~[A, [A, [A, B]]], B]—~[A,[[A, B], B], B]] + [A, [A, [A, B]], B]] + [A, [A, B], B]], B]

Hereandin all otherC~only commutatorsappear,no anticommutators,andwe seemto be in trouble.It is, however,a straightforwardexerciseto show that commutatorslike [00, 00] can be reducedtoexpressionswhich only involve the anticommutatorsof the Q’s if we assumethe parameters0~and 0’~to be anticommutingspinorialquantities,just as the ~‘swere in section5.

Martin F Sohnius,Introducingsupersyinmetry 107

Strictly speaking,such anticommutingparameterscan be avoided(at somecost) in the context ofrigid (flat-space)supersymmetry,but not in supergravitywheretheyappearin transformationlaws likebVi,. = V,.~andtheycertainlycannotbe avoidedin any superspacetreatmentof the subject.

7.4. Superspace

With theseparameters,the supersymmetryalgebracan be integratedto a group G, the super-Foincarégroup, with typical groupelements

g = exp(ix . F + iOQ + ~ ~iA. M). (7.19)

Superspaceis the coset spacesuper-Poincaré/Lorentz.There are infinitely many different ways toparametrizethis manifold, the one most commonly used is “real” or “symmetric” superspace,parametrizedby

L(x, 0, O)= exp(ix ~F+i0~Q,.+i&~). (7.20)

The spaceclearlyhaseightcoordinates,four bosoniconesx~andfour fermionicones0~if we assume= (0”)”.The multiplication laws (7.11) can be evaluatedto give, for the multiplicationwith a secondL,

L(y,~,~)L(x, 0, 0) = L(x’, 0’, 0’) (7.21a)

with transformedcoordinates

= x~+ y~+ i~a~0—i0u~~— — — (7.21b)

0’=0+~ 9’=0+~.

Multiplication with an elementof the Lorentzgroupgives

exp(~iA. M)L(x, 0, 0) = L(x’, 0’, 0’) exp(~iA. M) (7.22a)

with transfonnedcoordinates

x’~= (e~)~~x”

0= 0exp(—~iA~~o,.~);ö’=exp(~iA~~ê,.~)0. (7.22b)

Theevaluationof eq. (7.21a)involvesuseof eq. (7.18)andtheadditional 0-dependenttermsin x’ are aconsequenceof the non-vanishinganticommutator{Q, Q}. They are an essentialfeature of anyrealizationof thesupersymmetryalgebraon a manifold.

Z5. Representationson superfields

A superfieldis definedin completeanalogywith eq. (7.15) as


0(x, 0,0) = L(x, 0,0)0(0,0,0)L1(x, 0,0), (7.23)

and for any group elementthe action on 0 is given by the coordinatetransformations(7.21—22) inconjunctionwith

exp(~iA. M) 0(0,0, 0)exp(—~iA. M) = exp(—~iA~ 0(0, 0, 0). (7.24)

In its infinitesimal form, we write the groupaction on a superfield0 = 0(x, 0, 0) as

= —i[~, ~Q]= —i~r(Q)~

= —i[0, ~] = ii” r(O,~)0 (7.25)= —i[0, y . F] = —iy . r(P) 0

&L0=—21[cb,AM] =—~iAr(M)q5

with the differentialoperatorrepresentationr of the algebragiven by

r(Q,.) = i afar — (V0),.a,.

r(Q,,) = —i i9/30” + (0o~),,9,.(7.26)

r(F,.)=1i9,.

r(M,~)= i (x,.ô~— x~a,.)+ ~0o,,~3/80— ~(0o~,~)”9/80” + ~

The optional chiral transformationswhich aregeneratedby R, cf. eq. (2.39), actas

= —i[0, crR] = —ia r(R) 0 (7.27)

with

r(R) = 0”o/oO” — 0” 9/00” + n. (7.28)

The number n is the chiral weight of the superfield and must be real in order to render theR-transformationscompact.

Theseoperatorsrepresentthe algebra(2.38) for N = 1. Great care hasto be takento get the signsright in theseequations— theyareratherdependenton the conventionsdiscussedin the appendix.If 0doesnot haveoverall spinor indices,wecan write the first two equationsof (7.25) as

[0,0]=r(Q)0; [0,~]=r(~)0.

The differentiations8/80 and 8/80 aredefinedby

~—O~=&~,,; —~~=3~ (7.29)80” 80”

Martin F Sohniu.s,Introducingsupersymmetry 109

andbecauseof the difficult covariancepropertiesof the 8-symbol,ci. eq. (A.24), we should avoidusingthem with any other index positions.In order for thesedefinitions to make sense,we must alwaysanticommute0 or 0 to immediatelybehindthe differentiationoperator.

7.6. Componentfields (generalcase)

The Baker—Campbell—Haussdorffformula (7.3) and the definition (7.23) meanthat a superfieldisactuallydefinedas aTaylor-expansionin 0 and0 with coefficientswhicharethemselveslocal fields overMinkowski space.Since the third powersof 00 and 00 are alreadyzero, dueto the vanishingof thesquareof eachcomponentof 0” and 0”, this expansionwill breakoff rathersoon,andthe following isalreadythe mostgeneralsuperfield(I follow standardconventionsand usethe letter V to denoteit):

V(x, 0, ö~)= C— iOX + i,~’~—~i02(M— iN) + ~i02(M+ iN) — Ou”OA,.

+ i020(A — ~i,8~’)— i020(A~— ~iJx)— ~0202(D+ ~DC). (7.30)

Assuming that it hasno over-all Lorentz indices, this superfield containsas Taylor coefficients fourcomplexscalarfields C(x), M(x), N(x)andD(x),onecomplexvectorA,.(x), two spinors~(x) andA(x)in the (~,0) representationandtwo unrelatedspinors,~‘(x)andA ‘(x) in the (0,~)representationof theLorentzgroup, altogether16 (bosonic)+ 16 (fermionic) field components.

Let usnow considercomplexconjugation.If we define

(Oe)* = 0~ (7.31)

andimpose as a rule that fermionicquantitiesshouldreversetheir orderundercomplexconjugation,

like operatorsdo underHermitianconjugation,thenwe have[L(x, 0, 0)]t = L1(x, 0, i), (7.32)

and a reality conditionon a superfield

[V(x, 0, ö)]~= V(x, 0, i) (7.33a)

is covariantsince V~is a superfieldjust as V was. We can imposesuch a condition on the superfield(7.30) andfind that then

CCI; M=Mt; N=Nt; D=Dt— (7.33b)At=At; ~

so that the total number of componentfields is now only 8 + 8. This is called the real (general)superfield. In order to maintain this reality condition, the supersymmetrytransformationsmust, ofcourse,berestrictedby


(7.34)

andthe chiral weight of a real V must be zero.The transformationlaws under supersymmetrytransformationsfor the componentsof V are

calculatedby comparingcoefficientsin the expansion6V= ~C—iO6~+.~. with the ~V+ ~V, whichis given by eqs. (7.25—26) when acting on (7.30). The result of this calculation, expressedin four-componentnotation,will be the transformationlaws (4.7) of the real generalmultiplet, in otherwords,

the componentfieldsof a generalsuperfieldform a generalmultiplet.

As we know, the generalrealmultiplet is not an irreduciblerepresentation.Herewe encountera majorproblemwith the superspaceapproachto supersymmetry:superfieldsdo in generalnot correspondtoirreduciblerepresentations.The lattermustbe obtainedby imposingconstraintson the superfields.

7.7. Covariantspinorderivatives

Immediatelyafter the conceptof superspaceandsuperfieldswas developed,it was realized[17]thatsomethingelsewas neededto makecontactwith the irreduciblerepresentationsof supersymmetry.Ingeneral,superfieldscontainmore componentsthan acceptablefor irreduciblerepresentations,and theproblemarose— andwassolvedin ref. [17]— of howto imposesupersymmetricconditionson superfieldstoreducethe numberof degreesof freedomwhich theydescribe.

A “supersymmetriccondition” would be, for example,

8,.0(x,0, 0) = 0 (7.35)

—not a very interestingone since it just leavesus a single constantto describethe entire superfield(therecould be a multiplet of constants,but then 0 would be representednon-trivially while P,. wasrepresentedtrivially: 0 would not be an operatoron a positivedefinite Hilbert space).Moreinterestingconditionsinvolve covanantspinorderivativeswhich can beintroducedin a ratherelegantway.

The associativityof groupmultiplication

(g1og2)og3=g1o(g2og3) (7.36)

canbe interpretedto meanthat the “right action” of the groupelementg3 on g2 andthe “left action” of

g1 on g2 commutewith eachother— it doesnot matter in which order we perform the operations.Correspondingly,the left actionof oneL on another,

L(y, ~ ~)L(x, 0, i)=~[1—iy~r(F,.)—i~r(Q)+i~’~r(~a)]L(x, 0,0), (7.37)

which we interpretedas a transformationof the superfield,commuteswith the right action, which we

can write as:

L(x, 0, 0) L(y,~ ~‘) [1+ y~D,.+ ~“D~— ~~(15ia)]L(x, 0, 0). (7.38)

We can explicitly calculatethe D’s andfind


D,. = 8,.

Da = 8/80” — j(O~0)ai9,. (7.39)

= —8/80” +i(0oM)~8,...

The calculation is particularly simple since we need only reverse the roles of parametersandcoordinatesin eqs.(7.21).

Whereasthe left actioninducesa mappingof themanifold GIH on itself whichis a realization of thegroup, the right action inducesan anti-realization (the orderis inverted).Without any calculation,thealgebraof the D’s can begiven: the associativelaw (7.36) implies

{D, r(Q)} = {15, r(Q)} = 0- — — (7.40)

{D,r(Q)}={D, r(0)}= 0,

andthe commutatorsof the D’s with eachother

{D, D} = {15, 15} =0

{D~,i5~} = 2ia~~g8,. (7.41)

[D, 8,,] = [15,8,.]= [a,.,a~] = 0

resemblethe fact that the right actioninducesan anti-realizationon superspace.Having definedthe D’soff by afactor of i from r(Q), this algebrais formally the sameas that for the r(Q).

Theconventionsfor D andD were chosensuch that

(D~ç5)t = D,,çb if (0)~= 41 and0 bosonic. (7.42)

Thereis a differencein the reality conditionsbetweenr(Q) and r(~)on the onehandandD andD onthe other: the former are “Hermitian adjoint” to each other in the same sensein which iO,. isHermitian, in an inner-productspacewith integration as inner product. The latter are “complexconjugate”to eachother in the samesensein which 8,. is real, the gradient of a real field beingreal.Thus,althoughthe D’s seemto havethe samealgebra*asthe r(Q)’s, theydo not correspondto_anothersupersymmetry:as representationsof symmetryoperators,we would haveto takeiD and iD. Thesehavethe oppositealgebraandthe energyPo would not havea definite sign if we interpretedthem asrepresentationsof operatorsin a Hilbert spaceof physicalstates.

The eqs. (7.40) haveas a consequencethat if 0(x, 0, 0) isa superfield,i.e. if it is of the form (7.23)and transforms according to (7.25—26), then so are D41, D41 and 8,.41. The only change is in therepresentationmatrix .X,.,, of the Lorentzgeneratorssincethe derivativescarry Lorentz indices, andinthe chiral weight n which is raisedby one for D41 and lowered by one for D41. Thesederivativesarethereforecovariantundersuper-Poincarétransformationsof the superspacecoordinates.

7.8. Chiral superfieldsand chiral parametrization

Like all covariantderivatives,D and 15 can be used to impose covariantconditions on superfields.* Otherauthorsmay uscdifferent conventions.


The simplest,mostwidely usedandmost importantsuch conditionsare thosefor a chiral superfield0with

= 0 (7.43a)

andan anti-chiral superfield41 with

Da4~= 0. (7.43b)

Note that if 41 is chiral then (41)t is anti-chiral,due to eq. (7.42),and that a superfieldcannotbe both

chiral and anti-chiral exceptif it is a constant.This follows from the algebra(7.41). Also, a superfieldcannotbe both chiral (or anti-chiral) andreal. The conditions(7.43) are first-orderdifferentialequationsandcan easilybesolved:

41(x, 0, 0) = exp(—iOJO)q5(x, 0), ~(x, 0, 0) = exp(i0/0)~(x, 0), (7.44)

with 41(x, 0) and 4(x, 0) not dependingon 0 and 0, respectively.Their Taylor expansionin termsofordinary fields is particularlysimple,

0(x, 0) = A + 20Vi — 02F, q5(x, 0) = At+ 24i0— O2Ft, (7.45)

andgives rise to irreduciblemultiplets which turn out to beexactlythe chiral andanti-chiralmultiplets(3.50)and (3.53).

If we only ever had to deal with chiral superfields, it would have been more convenient toparametrizesuperspacedifferently from (7.20).The choices[17]

L(l)(x, 0, ë) = eix~’e’°°&°°

-- . (7.46)L(

2)(x, 0, 0) = e~” e’°°e’°°

are relatedto our previousparametrizationby the shifts

— i0c.r”O, 0, 0) = L(x, 0, 0) = L(2)(x~+ i0u”O, 0, O~ (7.47)

and,correspondingly,the representationsr(~)andthe covariantderivativesD(1) take a different,shiftedform:

r(l)(Q) = e~’°r(Q) e’°~°= it9/t90

r(l)(Q) = e~°’°r(~)e~°~°= —i9/9~+20,6- - - (7.48)

D(i) = e’°~°De’°’~°= 0/00— 2i,60

15(1)= eDe= —0/00,

and similarly for i = 2. We see that a chiral superfieldis particularly simple in the 1-parametrizationwhere it does not dependon 0, whereasan anti-chiral superfield is simple in the 2-parametrization


where it does not dependon 0. The exponentialsin (7.44) can be seen as the shifts from theseparametrizationsto the real (or “symmetric”) one. Since the “chiral” parametnzationsdo not easilyallow complexconjugation,andthusreality conditions,oneusually rathertoleratesthe shifts in (7.44)than the strangeasymmetryof (7.48) under 0 ~-* 0. We shall mostly ignore chiral parametrizationshenceforth.

7.9. Submultiplets assuperfields

To concludethisinitial introductiontosuperspace,let usrewrite theresultsof subsection4.3in termsofsuperfields.

The curl multiplet dV correspondsto a superfield

Vy~aD2DaV. (7.49)

This can be confinnedby establishingthat the lowest, 0—0-independentcomponentof Wa is the fieldA~(x),which is the lowest-dimensionalmemberof the multiplet dV. We find that indeed

1 j52~] = es” ~ of D~V)= A~

because

(~2componentof D~V)= _~_ [ië2o (A — ~~6i)] — i(J~)~(i~ë)

= ij2(A_.J~)_~ë2(JA~)a = ij2A~.

To concludefrom equality of the lowest componentto the equality of an entiresuperfieldis a verycommontechniquewhich is justified by the definition (7.23) of a superuield.This definitionimplies thatif a superfieldvanishesat the superspaceorigin, 0(0, 0, 0) = 0 in the operatorsense,thenit is identicallyzero.

A D appliedto W~would result in anexpressioninvolving D3 which vanishesdueto {D, i.5} = 0 andthe fact that thereareonly two different Da. ThereforeW~is a chiral superfield

lsawa=o. (7.50)

Since We,, is not of the form for the mostgeneralchiral superfieldwith a spinor index (that would be

D2Va), it must be restrictedby a furthercondition. This isD”Wa+15aW””’O (7.51)

which follows from the identity

DaI52D~= &D2b~~. (7.52)


It is this conditionwhich is responsible,amongotherthings, for theF~,in W,. beinga curl, i.e. for eq.(4.11). The full componentexpansionof Wa ~5

W = exp(—iOJ [A — ~io~”0F,.~+ iOD — i02,6A]. (7.53)

The constrainedmultipletwith dV= 0 correspondsto a superfield

V=~i(4~—41) with 15a~=o. (7.54)

This is the mostgeneralsolutionof Wa = 0.In a similar fashion, it can beshown that the chiral multiplet OV correspondsto a chiral superfieldX

which can be projectedout of V:

x = ~ii.52 V. (7.55)

The linear multiplet with OL = 0 hasasits superfieldequivalenta real L for which

j52~ = D2L = 0. (7.56)

The mostgeneralsolutionof this is

LH(D”Aa+DáA”) with 15aAa ‘O. (7.57)

The chiral weight of W~is —1, that of X is —2, that of 0 mustbe 0 and that of A~mustbe —1.

7.10. Superfieldtensorcalculus

In the superfieldformulation, the crucialproperty which makestensorcalculuspossible,is that theproductoftwo superfieldsis againa superfield:

çbi(x, 0, 0) 02(x, 0, 0) = 413(X, 0, i). (7.58)

This can be seen as a consequenceof the definition (7.23) or, alternatively, of the fact that therepresentation(7.26) of the super-algebrais in termsof first-orderdifferentialoperators.

The superfieldexpressionsfor the four products(4.18—25)are

03=41r02 ~ &“0102

V=~1x4~2~ V=~(0iq52+4s102)

V =411A412 ~ V =~i(41,~2-~1412) (. )

V3=V1~V2 ~ V3=V1V2,

and the fact that 41~is againchiral is dueto the fact that .15 is a first-order differentialoperator,which


obeysthe Leibniz rule,

= (DqS1)02+ OI~02=0. (7.60)

Note that second-orderconditionslike D241 = 0 are not preservedfor a product.

The kinetic superfieldis given by

T41 = (7.61)

We seethat T41 is chiral, D(T0) = 0. The propertyTT = —LI follows from T41 ~D20and

[152 D2] = — 16L11— 8iDJD. (7.62)

8. Superspaceinvariants

The attentivereadermay havenoticed that whereasfor someof the elementsof multiplet calculussimple equivalentsuperspaceformulascould begiven in the previoussection,no mentionwas madeofan equivalent of the “contragradientmultiplets” or of the invariants. Indeed, the construction ofinvariantsfrom superfieldsis the subjectof the presentsection.

8.1. Berezinintegration

The generalmethodby which a translation invariant action is derivedfrom fields is to integrateaLagrangiandensity L(x) over d4x. The result is translationallyinvariant if surfaceterms vanish. Asimilar procedurecan be used to constructsupersymmetryinvariant actions in superspace.As can beseenfrom eqs.(7.26), the actionof 0 is a derivativein 0 plus a term which itself is a total divergenceinx-space(and moreoverparametrizationdependent,cf. eq. (7.48)). Oncewe can define an integral f dOin such a way that it is invariant under 0—* 0+ ~‘, then the integralof anysuperfieldover the whole ofsuperspace(0, 0 andx”) will be invariant.

Suchan integral is known. It is the Berezinintegral [3], definedby

0~Jd0; lucJdOO (8.1)

for eachdifferent 0. Sincea function of anyoneanticommuting0 is alwaysof the form

f(0)=f~+0ff, (8.2)

thesedefinitionsare sufficient to define a generalf dof(0). Assumingthat 0 is not a multiple of ~ the

translationalinvarianceof the integralfollows:

f dof(0+ ~)= JdO(f~+ Ofi + ~f~)= ft = J dof(0). (8.3)


Formally, differentiationandintegrationarethe same,

J dOf(0) = -~f(0), (8.4)

a curious fact for which we can develop an understandingby visualising power seriesin 0 to be“modulo 2” so that raisingthepower(integrating)andlowering thepower (differentiating)are thesamething. This also resultsin such strangeequationsas that for the 8-function:

6(O)=0; 6(—O)———8(0). (8.5)

It should be notedthat anymoresophisticatedmathematicaltreatmentof the Berezinintegrationveryquickly dissolvesthe illusion that it maybe rathertrivial [53].

We define

Jd20~Jd02dO1; Jd2~~JdO~dO2, (8.6)

so that

Jd2OO2=Jd2~ë2=_2. (8.7)

As mentionedbefore,the integralof any superfieldover the whole of superspacewill be an invariant,

o J d4x d20d200(x,0, ~)= 0, (8.8)

providedthat thereis no Jacobiandeterminantto be considered.Actually, the functionalmatrix for thetransformations(7.21b) is

8~ —i(o-”~),. ~~(CO”)aO(x,0 0)— = 0 0 =eX (8.9)

O(x,0,0) 0

with

0 ~io-” ~

x= 0 0 0

0 0 0

andhasunit determinant,no matterhowwe define the determinant.


8.2. Thesuperdeterminant(“Berezinian”)

The superspacetransformations(7.21b)which representsupersymmetrytransformationsin flat-spacefield theory havethe functional matrix (8.9) with unit Jacobiandeterminant.Thereforetherewas noneedto include a determinantin the formula (8.8) for an invariant. In the context of supergravity,however,weshall encountergeneralcoordinatetransformationsof superspaceandthenwe better havesomeconsistentprescriptionfor a Jacobian.

Defining a determinantfor a matrix like (8.9) which has numericalentries (the 5’s) as well as“anticommutingnumber”entries(the i’s), is not quitestraightforward.Sincetwo ~‘sdo not commute,we wouldn’t get the usual determinantmultiplication law if we used the standarddefinition of thedeterminant.We must thereforedefine a new super-determinant(Russianauthorscall it the “Berez-inian”; in the Westit first surfacedin ref. [1]) which doesindeedobservethe law

Sdet(M1M2)= (SdetM1) (SdetM2). (8.10)

To makethingseasier,we startby defining a super-trace.Matriceslike (8.9) areof the form

M= [m1 P21 with ~m1commutingelements (8.11)Ljz1 m2i ~p., anticommutingelements.

Let usconsidertwo possibledefinitions of a super-trace,STrM = tr m1±tr m2. If we nowcalculatethetraceof a commutator,wefind it to vanishfor the definition with the minussign, but not for the usualdefinitionwith the plus sign. We can thereforemaintainthe very importantpropertyof any trace,

STr[M,N]=0, (8.12)

only if we define

STrM~trm1—trm2. (8.13)

Equation(8.12) is very usefulfor our questfor a super-determinant:with

M exp(lnM) (8.14)

we can define

SdetM exp{STr(lnM)} (8.15)

andthen get

Sdet(M1M2)= Sdetexp{ln M1+ inM2+ ~[lnM1, in M2] + ..

= exp{STr(lnM1 + in M2)} = exp{STrIn M1} exp{STrln M2}

= SdetM1 SdetM2

118 Martin F Sohnius.Introducingsupersymmetrv

as required.The first stepfollowed from (7.18), with the dots standingfor further multi-commutators.The second step is a consequenceof (8.15) and of (8.12). Additional terms would appearif thesupertraceof a commutatordidn’t vanish.

Equations(8.15) and (8.13) togetherconstitutea suitable definition for the super-determinantofmatriceswith anticommutingoff-diagonalelements.

8.3. Chiral integrals

The integral (8.8), of whose invariancewe have now convincedourselves,plays an importantrolebecauseit can be usedto constructinvariant actions.Beforewe attemptto do that, we must, however.considerthe specialcaseof an integralovera chiral superfield.In that case,the shift term in (7.44) canbe droppedunderf d~x,andthe rest is independentof 0 so that .1 d~0and hencethe full superspaceintegral givezero. —

On the other hand, the f d~xd20 alone,without the d20, is alreadyan invariant integral for chiralsuperfields. This is so becausefor them the supersymmetryalgebracan be realized as coordinatetransformationsof the chiral subspaceof superspacealone,which has coordinatesx” and O~but not0”. This would be mostobviousin the 1-parametrization.Correspondingly,we have

aJd4xd2041(x,o,o)=0 ifD41=0

(8.16)

~Jd4xd20~(x,0,0)=0 if 1541=0.

8.4. The Wess—Zuminoaction in superspace

Let us now try to write down an invariant action for a chiral superfield41 which, as we know, hasordinary field componentsA, B, Vi~ F and G. First, as in the muitiplet approach,we do somedimensionalanalysis:clearlyeach 0 carriesa (mass)dimensionof —~ (sothat, e.g., the exponentsin eq.(7.44) are dimensionless);if we want the spinorVi to havecanonicaldimension~,we find by looking at(7.45) thatthe wholesuperfieldhasdim 0 = 1. The integrationsd2O andd20 havedimension+ 1 each,asis clear from either (8.1) or (8.4). Thus the chiral measured4x d20 hasdimension—3, andin order toconstructa dimensionlessaction, the most generalterm involving only O’s andno negative-dimensioncouplingconstantsis

A0+~~02+~03.

Clearly, an actionconstructedfrom this cannotdescribedynamics:thereareno time derivatives.Whatwe need for a kinetic term is somethingbilinear in the superfield with either one derivative 0,,(dimension 1) or two derivativesD~or D,. (dimension~).The first alternativecannotbe realizedLorentzcovariantly.In the secondcasewe areluckier: the kinetic superfieldT41 ~D241,can be usedtowrite down the Wess—Zuminoactionfor achiral superfield(asdescribedin section6, the A0 term can beshiftedaway):


I=~Jd4xd20(~0T0_~~02_~413)+h.c.

= ~J d~xd20 (~oj~52~— m 02_g 0~)+ h.c. (8.17)

Workedout into components,this gives the sameLagrangiandensityaswe hadbefore,eqs. (5.2). This

is clear from the relationship

~Jd200 + h.c. = [0]F + 4-div. (8.18a)

for any chiral 41, a formula which follows from eqs. (8.7) and (7.45). A similar relationshipfor theintegralover the wholesuperspaceis

—~ J d~xd20d20 V= [V]D + 4-div. (8.18b)

for a generalsuperfieidV.

8.5. Superfieldequationsof motion

The action of the Wess—Zuminomodelcan berecastin different form. First, what is the “+ b.c.” ineq. (8.17)?Well,

+ h.c. = . . + ~J d~xd20 (~D20 — m ~2 — g ~3)

and we concentrateon the first term of this, which we can rewrite as an integral over the wholesuperspace:

d4x d2~D20= d4x d2OD2(,~41)= d4x d20d2~40.

This follows from the anti-chirality of 41, the equivalenceof integrationover and differentiationwithrespectto anticommutingvariablesandthe fact that the differencebetweenD and — 3/30 is aderivativeterm whichdoesnot contributeunderthe d4x-integral.Just aswe haveconvertedD2 into 2d20,we cannowconvertd20 into ~D2and get

= d4x d2O41152~,

i.e. justthe sameterm againas alreadypresentin (8.17).Basically,what we haveshown is that

120 Martin F Sohnius.Introducingsupersvmmetrv

Jd4xIm(Jd2041152~)

The actionnow reads

1= ~Jd~xd20(~~152~— ¶ 412_ ~ o~)— ~f d~xd20(~~2 + ~ ~3), (8.19)

and while in principle this is a “higher derivative action” (there is a .152 in it), we see that it containsderivativesonly of 0~not of 0. The equationof motion for 41 is thereforesimple to calculate:

0= ~I/b41= ~,(~D2t~ — mçb — g412), (8.20)

i.e. 141 = m41 + g412, the superfieldform of (5.3).Note that we were allowedto simply vary the actionwith respectto 41 becausewe arelooking at an

integral overchiral superspacein which a chiral superfieldis unconstrained.If, for the kinetic term,wehadchosento write

‘kin = ~Jd~xd20 d20410 (8.21)

then variation for 41 would haveproducedthe non-sensicalequationq5 = 0. In order to get the correctresult in this way, we must imposethe chiral constraintby the Lagrangemultiplier:

‘kin = d~xd20d20 (4141 + ISaOAa+ AaDO) (8.22)

This action,whenvariedwith respectto A gives us the chiral constraint,i5~ = 0, andwhen variedwithrespectto 41,

0 “a 3L/ODa0 — ÔL/341

gives

0 =D~A”~

which is the generalsolutionof the free equationof motion~D24i = 0.

8.6. The non-renormalisationtheorem

The fact that the kinetic part of the Wess—Zuminoaction can be written as an integral over thewhole of superspace,eqs. (8.21—22), but the mass and interaction terms cannot, has importantconsequences.Thereis a theorem[30,31] that thosepartsof a Lagrangianwhich can in principle onlybe written as chiral integrals will not receivequantumcorrections.


The observedrenormalisationbehaviour of the Wess—Zumino model [72,39] is a direct andpredictableconsequenceof this: the kinetic term must be renormalised,resulting in a logarithmicallydivergent wave-functionrenormalisation,but there are no independentquadratically and linearlydivergentmassandcouplingconstantrenormalisations,respectively.

Furthermore,since the vacuumenergy is strictly zero in a supersymmetrictheory, we know thattherewill also be no contributionsto the vacuum energy if supersymmetryis unbrokenby renor-malisation,as it is in the Wess—Zuminomodel [79].

9. Supersynimetricgaugetheory (super-QED)

In this section, the simplestsupersymmetricgaugetheory will be introduced,namely super-QED[73].

Quantumelectrodynamics,the gaugetheory of phasetransformationsof the Dirac field,

(9.1)

is a theory of matter,describedby the field 41’ in interactionwith radiation,describedby a gaugevectorfield A,.. Fromwhat we know aboutsupersymmetryrepresentations,seesection2, wethereforeexpectsuper-QEDto describea multiplet of matterwith spins(0,~)in interactionwith a multiplet of radiationwith helicities(±1,±~).Two questionsarise: first, could the radiationmultipletcontainhelicity ~insteadof ~?The answeris no, sincewe want the model to be renormalisable.Second,could matterbe thephotino, the spin-i partner of the photon?The answeris againno, for several reasons.Firstly, thiswould not allow to havemassivematter for unbrokensupersymmetry.Secondly,the photinocan onlyhavetwo physicaldegreesof freedomandmust thereforebe describedby eithera chiral or aMajoranaspinor. In either case,the only possible gaugetransformationhas negative parity— for a Majoranaspinor it takesthe form

Vi -~ exp{y5 a(x)} 41, (9.2)

— and thusrequiresan axial-vectorphoton.Finally, neutralphotonsandchargedmattercannotcoexistin the samemultiplet if the gaugetransformationscommutewith supersymmetrytransformations.Ifthey don’t, we would require a consistentdescriptionof local supersymmetrytransformations,sincethen

[Ssuper,Sgauge(xdependent)]= 5su~r(x-dependent), (9.3)

andconsequentlylocal translations,since

[localsupersymmetry,local supersymmetry]= local translation. (9.4)

“Local translations”x~—* x” + ‘(x), on the otherhand, are generalcoordinatetransformationsand

lead us into the field of supergravity.We conclude:Only in supergravitycantherebe gaugetransformationswhich do not commutewith supersymmetry,

122 Martin F Sohnius.Introducingsupersymmetry

andwhich thereforegive rise to particlesof different chargein the samesupersymmetrymultiplet.We arethusleft, as only startingpoint for super-QED,with havingto couplewholemattermultipiets

(chiral multiplets)to a radiationmultiplet. The spectrumwe expectis that of the massiveWess—Zuminomodel, plus the photon and the photino. The presenceof “gauginos” such as the photino, neutralmasslessspin-i fermions,is an importantpredictionof supersymmetrictheories.

But what is that gaugemultiplet (1,1)? What are its transformationlaws, its auxiliary fields? Thereareseveralwaysto tackle theproblem:

(a) try finding a muitiplet with a vector in it andtry constructingagaugetheory aroundit — this wasthe approachin the original paperby WessandZumino [73];

(b) try couplingto the Noethercurrent of the gaugetransformations— this approach,first suggestedby OgievetskyandSokatchev[48],is didactically moreenlightening,andwill bepursuedin this section;

(c) try applying the generalgeometrictechniquesof gaugetheoriesto superspace— this will be thesubjectof the nextsection.

9.1. Thegaugecurrentsuperfield

The gaugetransformationsof acomplexfour-spinorare thosegiven by eq. (9.1), thoseof a Majoranaspinorcanonly be (9.2). Sincethe latterareunacceptablefor QED, which mustgive a Coulombfield ofpositiveparity, we needat leasttwo chiral multiplets 41~and 02 for a super-QEDmodel.The complexspinor is then Vi = Vii + i412, and the infinitesimal version of the gaugetransformation(9.1) is ~ =

a(x)412 and ~gVi2 = a(X) Vii. Since a supersymmetryvariation must commute with gauge trans-formations,this gaugetransformationlaw will holdfor all fields of the mattermultiplets:

~g01= a(x) 02; &g02 = —a(x)0i. (9.5)

The free Wess—ZuminoLagrangian*for the two multiplets,

L0~[(3,.A1)2+(3~)2+i4I4~+F~+~ (9.6)

is invariant under transformations(9.5) with a = const. The associatedNoether current of suchtransformationsis

j,, =içli1y,,412—A1 3,.A2—B1~9,.B2. (9.7)

As a non-constantfield, j,. must be a memberof a multiplet. In order to find the structureof thatmultiplet, we at first subject the matter fields to their (free) equationsof motion. This gives us theconservationlaw for the Noethercurrent,

(9.8)

and makesit much easierto calculate the supersymmetrytransformationlaws for the gaugecurrent

• Theinteraction term for two multiplets cannot bemadegaugeinvariant.


multiplet. We find that the multiplet contains,besidesj”, the following field combinations,

x = (A2+ 75.82)Vii — (A~+ 75B5)412

C=B1A2-A1B2,

andthat the supersymmetrytransformationsof thesefields areexactlythoseof the linear muitiplet, asgiven in eqs. (4.16).

Let us re-examinethis in termsof superfields.Comparing (9.7) and (9.9) with (4.23) and (7.59), wefind that the gaugecurrentsuperfieldis

J= —~i(0i&— ~1412). (9.10)

The conditions for J to be a linear superfield follow from the chirality of the matter superfields,D41, = 0, andthe equationsof motion ~D

2411= m411:

152j_ —~i(411b2~

2—b2~o)....—2im(411412—0102)=0 (9.lla)

andsimilarly

D2J=0. (9.llb)

9.2. Noethercouplingand the gaugesuperfield

In QED, gaugeinvarianceof the matterLagrangianis achievedby addinga term —j”A,. to it:

L=it/i,6Vi+m4iVi-j~A,.,

wherej~is the Noethercurrent for the phasetransformationon Vi. The gaugetransformationlaw forthe photonfield,

A,.-+A~,,+3,.a, (9.12)

is contragradientto the conservationlaw for the current,so that j”A,. transformsas a total derivative:

J”A,. —~j~A,,+ t9,.(j”a) for t9,,j” = 0. (9.13)

In super-QED,the place of j” is taken by the superfieldJ as given in eq. (9.10) and that of theconservationlaw by the conditions(9.11).The Lagrangianwill be of the form

L(i) = L(o) —j’-’A,, + otherterms (9.14a)

or, in superfields,

If!) = 1(0) — ~Jd4x d20d2OJV (9.14b)


with 1(o) the sum of thefree Wess—Zuminoactionsfor the two superfields411 and02. V is a real generalsuperfieldwhich containsthe gaugefield A,,. The numericalfactor —~ servesto eventuallyreproducethe factor —1 in (9.14a); detailsfollow from eqs. (4.25) and (8.18b).

The specific form of thegaugetransformationson V can be derivedfrom the constraints(9.11) on J:V can transformas

V-* V+D2X+D2Y~V+~gV

without disturbingthe secondterm in (9.14b) if the matterfields are on-shell andJ thereforesatisfies(9.11).D2 Y is the generalform of any chiral superfield,D2Xthat of a (different)anti-chiralone,but ifV is to be real and remain real then one must be the conjugateof the other. Thus the gaugetransformationsof V are

~igV —~i(A—A) with A chiral (9.15a)

(the factor —~iis chosenfor convenience).In components,thisreads

&gCB ~gXVi

6gMF 6gN=~G (9.15b)

~gA,. = 3,.A; SgA 0

6gD =0.

We now proceedto actually check the gauge invarianceof (9.14b) which so far we haveonlyconjecturedin analogy with QED, and we shall find the analogyincomplete.The full set of gaugeparametersis the chiral parametersuperfieldA. The role of the parameterfield a(x) which appearsin(9.5) and in (9.12) is played by the A-componentof A and the full supersymmetricgauge trans-formationsfor the matter superfieldsare thereforenot the eqs. (9.5), which are not supercovariantasa(x) is not a superfield,but rather

~g41iA02 ag02=—A01. (9.16)

For thesegaugetransformations,the massterm in ‘(0) is invariant but the kineticterm transformsas

SgI(O) = ~J d’~Xd20d20~g(0iOi+ 0202)

~Jd4xd2od2o(02q5i_~io2)(A A)~Jd4xd2Od20J~gV.

We see that the Noethercoupling term in (9.14b) would indeed cancel this if J itself were gaugeinvariant as it is in QED. Here,however,J transformsas

~gJ = ~i(4ii01 + ~202) (A — A) = (OiOi + 4~202)~g V

so that the variation of If!) becomes


&gI(i) —~Jd’x d20d20(~‘41’+~

Thingsdon’t seemto work quite like in QED, the first-orderNoethercoupling termis not sufficient torender the action gaugeinvariant. But then, in the presenceof chargedscalar matter (we havethe“siepton” fields A

1, A2, B1 and B2 which are the superpartnersof the “lepton” ~),we expect theLagrangianto at leastcontain “sea-gull” termsA,.A~(A~+ B~)as in scalarelectro-dynamics.Indeed,we can try to cancel the remaining&gI(1) by addingafurther term to the action,

‘(2) I(l)+~Jd4xd20 d20(41i0i+ ~2O2) V2,

but doing this,we now find that

= J d~xd20d2OJV2&~V

whichwantsa term ~JV3to cancelit, andsoon. Clearly, in contrastto evenscalarQED, the action isnon-polynomial.A quick calculationwill showthat the action

‘kin = ~Jd~xd20d2ö~(~1O1+ 4’202)cosh2V— ~Jd~xd

20d2~Jsinh2V (9.17)

is indeed gaugeinvariant. An elegantway of rewriting this is to define a doublet from the mattermultiplets,

1 r411+i412i

(9.18)\/2 L01_i02J

on which the gaugetransformationsarethoseof 0(2),

~ with A = [11 A]~ (9.19)

Then1~kinbecomes

Ikin~Jd4xd20d2~~e_2%/tPwith v= [~ ~ (9.20)

9.3. Wess—Zuminogaugeand matterLagrangian

In supersymmetrizingQED, we were forced to introducetwo complications:first, the single realparameterfield a(x) hadto be replacedwith a whole multiplet (or superfield)of parameters,A, andsecond,the Lagrangianbecamenon-polynomial. In this subsection,we will see how the two com-

126 Martin F Sohnius, Introducingsupersvrnmetrv

plicationscan be played off against each other and how we can fix the additional gaugefreedomsassociatedwith the parameterfields B, 41, F and G in (9.15) in sucha way that the Lagrangianbecomespolynomial. It will still be invariant under the usual gaugetransformationswhich havethe real fieldA(x)as parameter.

To do this, we first observe that the action (9.17) or (9.20) is non-polynomial only in the0-0-independentcomponentof V. This is sobecauseany power seriesin 0 or 0 eventuallyterminates.Thus the Lagrangianwill be polynomial if and only if we choosea gaugewhere C= 0. This gaugechoice is availablesince ~gC= B, which allows to gaugeaway the field C completely.Clearly, such agaugeis not supersymmetric:as remarkedin subsection7.9, for unbrokensupersymmetrya wholesuperfieldwill be zero if its lowestcomponentis zero. On the otherhand, the whole of V cannotbegaugedaway by transformationslike (9.15) andthuswe know that

a polynomialLagrangiancan only be achievedin a non-supersymmetric gauge.

It is mostconvenientactuallyto useup all the superfluousgaugefreedomsand choosea gaugewhere

C=~=M=N=0. (9.21)

Such a gaugeis called a Wess—Zuminogauge, after ref. [73]. To repeat,this gaugeis not supersym-metric; indeedneither6~nor ~iMor ~N vanishin this gauge.The “multiplet” which remainsof V in aWess—ZuminogaugecontainsfieldsA,., A and D with transformationlaws

= i~y,,A

6A = —io”~~3,.A~— y5~D (9.22)

= —iCJy5A.

Thesearenot a representationof the supersymmetryalgebrabut ratheroneof a modified algebra[10]

[8k,82] = 2i~y~~23,.+ ~g (9.23a)

where8~is a field dependentgaugetransformation with parameter

a = —2i~iy~C2A,,. (9.23b)

On the gaugefield A,., the modified algebratakesthe particularly suggestiveform

[8k, 52]A,. = ~ (9.24)

The reasonbehind the modification of the superalgebrais that the Wess—Zuminogauge must bere-establishedafter the first supersymmetrytransformation.This is done by a field dependentgaugetransformationon whoseparameterthe secondsupersymmetrytransformationthenacts.

In the Wess—Zuminogauge,the superfieldV becomes

V= —0o~’0A,.+iO2OA —iO20A—~O202D (9.25)


and the matter Lagrangian (9.17) contains powers of V only up to V2— these are the sea-gullterms— and can be evaluatedas

Lmatter = ~ [v,.A1v”A1+ V,.B1V”B~+ iç~y~V,.ç/~+ F~+ G~—tn(2A1F+ 2B~G1+ 4~ç1~)]

— A~(A1+ y5Bi)t/i2+ A(A2+ y5B2)çfr1 — (A1B2— B1A2)D (9.30)

with

V,.A1 0,.A1 — A~,A2 V,.A2 3,,A2+ A,,A1 (9.31)

(samefor B1 and 41~).The contributionsin the last line of (9.30) arenon-minimalcoupling termsdue tothe supersymmetryof the model.

9.4. TheMaxwellLagrangian

To completethe treatmentof super-QED,we needa supersymmetrisedMaxwell Lagrangianof theform

LMaXWCII = F,,~F1L*~+ otherterms. (9.32)

Sincewe alreadyknow that F,,~sits in the curl-multiplet dV, seeeq. (4.9),which correspondsto achiralsuperfieldWa with an additionalspinor index, seeeq. (7.49), it is ratherstraightforwardto constructaMaxwell action:

‘Maxwell = Tj~Jd4xd20 W2+ h.c. (9.33)

All componentsof the curl multiplet aregaugeinvariant.This can be seenin superfieldform from

~gWa = W2D~(A — A) = 0. (9.34)

The last stepfollowed from DA = 0 and152DA = [b2, D]A = 4iISAA = 0.

We can expand(9.33) into components,using the explicit form (7.53) for W~,andfindLMaxweIl = ~ + ~iAJA + ~D2. (9.35)

As expected,thephysical spectrumcontainsa photonandits superpartner,a masslessneutralphotino.Thefield D(x) is auxiliary. The full Lagrangianof super-QEDis now

L = LMaxweIl + Lmatter. (9.36)

Going back to the form (9.33) of the Maxwell action, we see that it is written as an integral overchiral superspaceonly. Becauseof the non-renormalisationtheorem,the questionariseswhetheror not

128 Martin F Sohnius,Introducingsupersym,netn’

it is possible to write it as an integral over all of superspace.The answer is affirmative: fromWa = ~~D2DaV we find that wecan write

w2= _~152(D~Vb2DaV)

and

J d~xd20 W2= d4x d20d20 VD”D2DO,V

(D” has beenpartially integrated).This is Hermitian becauseof the identity (7.52),andwe get

‘Maxwell = d~xd20 d2OVD’D2DaV. (9.37)

The fact that the action can be expressedas an integral over all of superspacemeansthat we mustexpecta logarithmically divergentwave-functionrenormalisation,commonto the fields of the photonandthe photino.

It is possible,though not altogetherstraightforward,to generalisethe results of this section tonon-Abeiian gauge theories [18,54]. Much more, however,can be learnedfrom a full geometrictreatmentof gaugetheories,the subjectof the nextsection.

10. Superspacegaugetheories

The subjectof this sectionwill be the full treatmentof non-Abeliangaugetheory in superspace.Inapproachandphilosophy,this is basedon papersby Wessand Zumino[74,75] (seealso [59]).

10.1. Ordinary gaugetheories

Ordinarygaugetheoriesin x-spaceare basedon someunitary Lie groupwith algebra

[Tm, T~] jCmn1 1’,; Tm=(Tm)t (10.1)

andgroupelements

g = exp(iamTm) (10.2)

whicharedefinedby realparametersatm = (am)*. The gaugegroupis representedon quantumfields by

unitary transformationsU(g

1) U(g2)= U(g1°g2) (10.3)

whoseeffect, in the caseof gaugecovariantfields, is a reshufflingof field components:


= U(g) 41, U 1(g) — Er i(g)]1J 0~— (e ‘ 0)~, (10.4a)

or, infinitesimally,

= —ia41. (10.4b)

We havewritten the matrix representationr(g) of the groupas

r(g)=e~, (10.5)

wherewe understandthe awithout any further indicesto be a “Lie-algebravalued” parameter,

(10.6)

with .~(T)a matrix representationof the Lie-algebra(10.1).In the caseof theories with local gauge invariance (the interesting ones), the parametersare

space—timedependent:

a = a(x). (10.7)

Thereforethe gradient of a covariantfield is not covariant,

= 3,.(U0U~) = e”’ ~,.0+ (3,.e~~)41,

andin order to mendthis defect,we must introducea gaugeconnection(gauge“potential”) field A,,

which is alsoLie-algebravalued,A,. = A,,

tm.~(Tm), (10.8)

andwhich hasan inhomogeneoustransformationlaw

UA,,U1 = e_i~~(A,.— iD,,) e”’ (10.9a)

or, infinitesimally,

6gA,. = i9,,a + i[A,,, a]. (10.9b)

The inhomogeneousterm in bA,. can cancelthe 3,.a-termin ~ andwe get

U(3,,41+ iA,.41)U~= e_ia (3,,41 + iA,.41) (10.10)

so that

V,.—=3,.+iA,. (10.11)

is a gauge-covariantderivative.From the commutatorof two V’s, we get the gaugecurvature (“field

130 Martin F Sohnius, Introducingsupersvmmetrv

strength”)~

[V,,,Vj0 = [3,.,3~]4+ iF,,~0= iF,,~41, (10.12)

anotherLie-algebravaluedfield. It is given in termsof the A,, by

= 3,.A~— c9aA,, + i[A,., A,,] (10.13)

and transformscovariantly in the adjoint representationof the gaugegroup. In our conventions,thistransformationlaw reads

UF,J.PU1 = ~ e”’; ~gF,,,, = i[F~,,,,a]. (10.14)

Equation(10.12)is sometimescalledthe“Ricci-identity”, in analogywith thecorrespondingcasein generalrelativity. Similarly, the cyclic identity

V1~F,,,,1= 0 (10.15)

is sometimescalled “Bianchi-identity”.It follows from theJacobi-identityfor threeV’s andfrom (10.12).Inthe caseof an Abelian gaugegroup, it reducesto the homogeneousMaxwell equations

~ =0C

10.2. Minimal superspacegaugetransformations?

The original papersby WessandZumino [73]on super-QEDandby FerraraandZumino [18] andSalam and Strathdee[54] on non-Abeliansupersymmetricgaugetheoriessuggestthat it should bepossibleto write a superspacegaugetheory which involvesonly a singlereal superfieldV as connectionanda chiral superfieldA as gaugeparameter.Sucha schemeis minimal: a single field a(x) cannotbe asupersymmetriclocal gaugeparametersinceit cannotbe a superfieldunlessit is a constant.The gaugeparametermust thereforebe a superfield,andthe smallestone availableis the chiral superfield.

As we shall see,the generalisationto superspaceof the geometricapproachoutlined so far in thissectionwill not give us such a minimal scheme(we could, however,havegot the minimal schemebysuitably adaptingthe Noethercoupling method to the non-Abeliancase).For the time being, let usneverthelesspursue.

10.3. Themaximalapproach

For eachof thesuperspacederivatives3,., D,and iSa weintroducea correspondinggaugepotentialsuperfieldA,. (x, 0, 0), A~(x, 0, 0) and A,, (x, 0, 0). As gaugeparameter,we take a general superfieldX(x, 0, 0). We do not as yet makereality assumptionsaboutany of these,becausewe know that theminimal schemeinvolves a chiral superfieidas gaugeparameter,which is intrinsically complex. Inparticular,A,, � (A,.)t, A,. � (A,,)~and X � X~.

To simplify notation, I usecapital Latin indices to denotethe completeset of superspace-indices:A = (is, a, &). Thus:


DA = (3,,,Da, i5~)

AA = (A,., A,,, Ad). (10.16)

The gaugeconnectionstransformin analogywith eq. (10.9) as

AA —~e~~~(AA— iDA) eiX (10.17)

andwe can definecovariantderivativesasbefore

VA~DA+iAA. (10.18)

In contrastto ordinaryspace,where[3,,,3,,] = 0, we havenow

[DA, DB} iTABCDC, (10.19a)

namelythe algebra(7.41) of the spinorderivatives,with

~ = T13a~”= ~ and all other T’s zero. (10.19b)

The “graded commutator” [.., . .} was introduced in subsection2.2. The non-Abelianstructureof(10.19)modifiesthe Ricci identity

[VA,VB} = iTABC V~+ iFAB (10.20)

andthe definition of FAB in termsof AA:

= DAAB DBAA + i[AA, A~}— iTABC A~. (10.21)

With thisdefinition, including the last term,F is covariant,

v -iXi~ iX

~AB-~e ‘ABeandfulfills Bianchi identities.Theseare, however,alsomodified by the presenceof the “torsion” termin (10.19):

~ [VAFBC — iTAB’~FDC] = 0. (10.23)[ABC)

In extendingthe gauge theory set-upto superspace,we had to introduce the complex connectionsuperfieldsAA which betweenthem have16 times as many componentsas we expectfor the minimalversionwith just onereal V. The complexparametersuperfieldX hasfour times as manycomponentsas the chiral A.

Somethingdrasticmusthappento get usbackto the minimal scheme.This cannotbea partial gaugechoicebecauseX hasnot enoughcomponentsto gaugeaway somanysuperfluousonesin AA. Whatweneedare super-andgauge-covariantconstraintson the theory.

132 Martin F Sohnius.Introducingsupersymrnetry

10.4. Constraints

Such constraintsmust set covariantsuperfieldsto zero. Covariant,becausethey should not breakgaugeinvariance; superfields,becausethey should not break supersymmetry.Clearly, the FAB aresuitablecandidatesfor beingconstrained.

For an ordinary gaugetheory this cannot be done. The only F is 13,,,, which containsLorentzrepresentations

F,.,,: (1,0)+(0,1)

which are linked to eachotherby the Bianchi identity. Correspondingly,settingone of them to zero,say

0= F,,,, +

gives the equationof motion 0 = V”F,.V. The lessonwe learnis to be careful not to over-constrainthesystem,as we mayendup with on-shellconditionsfor the fields, or evenwith atrivial theory,13,,, 0.

For superspacegauge theory, the scopeis much wider. Indeed,decomposingFAR into Lorentzrepresentations,we find

FAB: ~ (10.24)

whichleavesplenty of choicefor constraints.Which constraintsto imposeis a highly non-trivial decisionwhich lies at the heartof trying to find

superspaceversionsof supersymmetrictheories.The sizableconfusionwhich originally dominatedthispoint hasbeenconsiderablyreducedby a classificationfor constraintsgiven by Gates,Stelle andWest[26].We will follow their argumentation.

First therearewhat is calledconventionalconstraints*:the transformationlaw for a gaugepotential,eq. (10.17),remainsunchangedif we add to it a covariantfield in the adjoint representation:

AA ~ AA + FA.

We could, e.g., addthe superfield

J~ ..i~$“1~,. 410,, ,,~

to A,, and get a new A,,. For this new, redefined~ we thenget F~W= 0. Thus,

F,,~=0 (10.25)

is a “safe” constraintwhich can always be imposed. It just redefinesA,. suitably. Alternatively, we

could saythat (10.25)can always besolvedto expressA,, in termsof A,, andA~:

A,. = ~ ~ (D,, A~+ IS~A,, + i{A,,, A~}). (10.26)

* A badname— “redefinition constraints”would bebetter,since “conventional”can alsomean“ordinary, unimaginative”— which thesearenot!


Note that all of this only worksbecausethe “torsion” in (10.21)hasan inverse[63]. Such a constraintwhich can be solvedfor oneof the fields in the theory is a conventionalconstraint.

The secondtype of constraintswhich we needarecalled representationpreservingconstraints.Thesecomeabout through a somewhatmore“physical” argumentation:the superfieldswhich describescalarand spinormatter are chiral or anti-chiral.Eventually,we will want to couplethem to gaugefields inorder to haveasupersymmetricextensionof, say,quantumchromodynamics.As we havealreadyseen,derivativesmust be gaugecovariantised,andcorrespondinglywe expecta “gauge-chiral” superfieldtofulfill a constraintof the sort

VaO= 0, (10.27a)

andits conjugate,the “gauge-anti-chiral”superfield,shouldfulfill

Va4~= 0. (10.27b)

Suchdifferentialequationshaveintegrabilityconditions.The equationsareonly consistentfor all 0 if

{V,,, V~}= iF,,~= 0, {V~,V~}= jFa~= 0. (10.28)

The constraints(10.28)are called “representationpreserving”becausetheyare necessaryin order toensurethat the specialsupersymmetryrepresentations“chiral superfield” and “anti-chiral superfield”survive in the presenceof non-zerogaugecouplingwhen thereis a term with a gaugepotentialpresentin the conditions(10.27).

Onemoretypeof constraintwill be necessaryin order to boil downour schemeto the minimal one:a reality constraint. In ordinary gaugetheories,a real gaugeparameternaturally leadsto a real gaugepotentialA,.. An imaginarypart of A,. would just be a covariantfield in the adjoint representationofthe gauge group. Here, where we a-priori allow complex gauge parametersuperfields, we mustexplicitly ensurethat there is only one gauge field, not real and imaginary parts separately.Wethereforeimposethe soft reality constraintthat A,. and(A,.)t be gaugeequivalent,

A,, = (A,.)t + gauge-transformation. (10.29)

We collect our constraints,

= Fa~= Fa~= 0; Im A,. = puregauge (10.30)

andgo aboutsolving them.“Solving a constraint” meansexpressinga field which is subject to some differential condition in

termsof derivativesof otherfields in such a way that the propertiesof the derivativesensurethat theoriginal condition holds. A good exampleof this is provided in elementaryelectro-dynamicslectures,wherethe vectorpotential is usually introducedas a solutionof the constraintswhich the homogeneousMaxwell equationse”~3~F,,,,= 0 impose on the electric and magneticfield strengths.The curl of apotential,F,,,, = 3,.A,,— 3,.A,., is a solutionbecausepartial differentiationsareassumedto commutewitheachother. In our caseof superspacegaugetheory,we havethis alreadybuilt in: we startedout with

134 Martin F Sohnius,Introducingsupersvinmetrv

potentialsrather than field strengths,so that the Bianchi identities (10.23), one of which is thehomogeneousMaxwell equation,are really identities. If we had interpretedthe identities as “con-straints”on the F’s, then eq. (10.21)would havebeenthe solution in termsof potentialsAA. Equations(10.30),however,aretrue constraints:not every set of potentialswill satisfythem.

10.5. Prepotentials

We do, of course,alreadyknow what is necessaryto solve the conventionalconstraint(10.25): A,,cannotbeindependentany more, it mustbeexpressedin termsof differentiationsactingon A,, andA~,see eq. (10.26). The representationpreservingconstraints(10.28) are a bit more difficult to solve: weobservethat, in somevaguesense,the “spaceis spinorially flat”, i.e. that all F’s vanishwhichhavebothlegs in spinorial directions.It is thereforenot surprisingthat the solution is “spinorially pure gauge”.This is jargon for

A,, = —i e2~’D,.e2’~— (10.31)

A~= —i e2~’D~e~21’,

which is a solutionof (10.28) for generalcomplexsuperfieldsU and V.It may look as if we have done too much now: both A,, and A,. are “pure gauge” and A,. is

expressedin termsof them. So can we gaugeaway everything?The answeris, of course,no. Since Uand V aredifferent,andsincewe haveonly onegaugeparametersuperfieldX available,wecan at mostgaugeaway oneof them.Before doingso, let us havea look at the gaugetransformationpropertiesofV and U. In order to satisfy (10.17), the original gaugetransformations(parameterX) must take theform

e_2V—* e~2”~ e_2U-+ e_2L~e”. (10.32)

The solutions(10.31)of the constraintsexpressthe gaugepotentialsin termsof the prepotentialsU andV. Theseare called “prepotentials” becausethey are unconstrainedsuperfieldsin terms of which thepotentialsareexpressedin sucha way that the constraintsaresolved.Justas one can look upon gaugetransformationsof the vectorpotentialas havingarisenout of solving the Bianchi constrainton the fieldstrengthin termsof the potential,sosolving the constraintson AA in termsof prepotentialsintroducesadditional transformationsbeyond(10.32)underwhich the AA areinvariant.Theseare

~e~’~’e~2”, e_2U—* e”~e_2U (10.33)

with A chiral andA’ anti-chiral.

10.6. Supersymmetricgaugechoice

At this stage,oneusually makesa partial gaugechoice. It mayseemsomewhatodd, but the mostuseful oneis to gaugeA,, to zero.This can be doneby choosingiX = 2U in (10.32).We then have

A,. = 0 (10.34a)


Va = (10.34b)

A,. = ~ i5s(e2~~’D,.e251) (10.35)

and the only remainsof the original gauge parameteris a chiral piece, X = A, which leaves theconditionA,. = 0 invariant.

The remaininggaugefreedomon V is now

e2” -+ e~1r’e2~’e~. (10.36)

The gaugechoiceA,, = 0 is supersymmetric,sincea superfieldhasbeengaugedaway and a superfield’sworth of gaugeparametershavebeenfixed. Also, the remaininggaugefreedom(10.36) is expressedintermsof superfields.

10.7. Solvingthe reality constraint

In order to solve (10.29),we first considerthe Abeliancase.There,eq. (10.35) reducesto

A,. =~Do~,DV

andthe constrainthasthe solution

V= Vt+ chiral+ anti-chiral

because“chiral+anti-chiral” is the solutionto D&,LD ~c3,,. We can now makeone last gaugechoice,and chooseA + A’ suchthat strictly

V= Vt (10.37)

which restricts(10.36) to A’ = A. This argumentationcarriesthroughto the non-Abeliancaseas well.Although it is then not possibleto find a supersymmetricgaugewhereA,, = (A,.)~,we can still have(10.37) andgaugetransformations

e2~’ e~e2~’e~ (10.38)

which preserveit. ThenA,. is gauge-equivalentto A~.We havenow succeeded,by imposingthe constraints(10.30)andthe two gaugechoices(10.34) and

(10.37), in reducing everything to the desiredminimal theory, containing one real prepotentialsuperfield V and one chiral gaugeparameterA. In the Abelian case,the theory is identicalto thatdescribedin section8.

10.8. The field strength superfield and the Yang—Mills action

The lowestdimensionalsurviving field strengthsuperfieldsareF,.,, andF,.,,. We can either invokethe Bianchi identities [59], eqs. (10.23),or the explicit solution to showthat the constraintsimply that

136 Martin F Sohnius,Introducingsupersymmetrv

theseareof the form

F,,a~io~,.,,I~Wti;~ 1039

iii — ! f2 A — isv’ \t“‘a — 8~-’~ “‘ a — t, VP’,,)

whereW,, is obviouslychiral andgaugecovariant.Also from the Bianchi identities,onecan show that

VaW,,+VaWa=0. (10.40)

Theseconditionsrestrict W,, to containonechiral spinorA,,,onereal pseudoscalarD,oneantisymmetrictensor13,,,, subjectto V[,.l3,~) = 0,

all in the adjoint representationof the gaugegroup.Since W,. is gauge-covariant(in the adjoint representation)ratherthan gauge-invariant(asit was in

the Abelian case),its explicit form will be gaugedependent.In the Wess—Zuminogaugeit is

W= e’°’°[A — ~io~”0F + i0D — i02u~V,.A] (10.41)

with F,,,, the non-Abelianfield strength(10.13) and the covariantderivativeof the “gaugino” field Athat for the adjoint representationof the gaugegroup:

V,.A = 3,,A + i[A,,, A]. (10.42)

The supersymmetrytransformationswhich one derives from this by naive application of thesuperspaceformulas(7.25—26)are, in 4-spinornotation,

= —~iu”~~F,.,,— y5~D

6F,.,, = —i~(y,,V,,— y,,V,.)A (lO.43a)

W = —i~y~y5V,,A.

Theseinvolve the typical non-lineartermsof non-Abeliangaugetheories,andtheir algebrais that of“supersymmetrymodulogaugetransformations”[10],namelythat of eqs. (9.23).This is of coursedueto having chosenthe Wess—Zuminogauge.In contrastto the Abeliancase,thereis no clear advantagein using W,. insteadof Vitself — both aregaugedependent.Correspondingly,we canreplaceSF,,,, in themultiplet (10.43)by

8A,, = i~y,.A (10.43b)

without changingthe algebra.

The actionfor the gaugefields is nowvery similar to the Abeliancase,

IyM_mfdxd0tr W2+h.c., (10.44)


which is supersymmetricand gauge-invariant.If we work this out into components,we find

LyM = tr(—~F,.,,F~~+ ~iAy”V,.A+ ~D2). (10.45)

Theequationsof motion are

V~13,V= ~{1c,y,.A}(10.46)

iy”V,.A=O, D=0

in componentsand

VaW,,_VaWa=0 (10.47)

in superfieldlanguage(notethe sign changebetweenthe constraint(10.40),which is oneof theBianchiidentities,and the equationof motion).

The action (10.44)can also be written as an integral over the whole of superspace,along the linesdiscussedat the end of the previous section. We therefore expect infinite wave-functionrenor-malisation.

10.9. ThematterLagrangian

In order to gauge covariantisethe kinetic term (8.19) of the Wess—Zuminoaction, it will not besufficient to just makeminimal substitutionsDA -* VA, because4141 by itself is not invariant:

41-*e~1’~41, ~-*q5e1’~

- -.-. - (10.48)4141—~41e”~e”~41�4141.

It is clear,however,that thetransformationsof e~2”,eq. (10.38),aresuch that 41 e_2v0is invariant,andwecan write the kinetic part of the matter action as

‘kin = ~Jd~xd20d2095e

2” 41. (10.49)

The correspondingformula (9.20)for the Abeliancasewas explicitly derivedat the end of subsection9.2. Theterm e~2~’hasageometricmeaning:whereasthechiral superfield0 is alsogauge-chiral,duetothe gaugechoice (10.34), its conjugate0 is anti-chiral but not gauge-anti-chiral.For 41 e21”, however,we have

V,.(41 e2”) = (D,,~ — i~A,,)e2” + ij(D,, e2~’+ i[A,,, e2~’])

= qSD,, e~2”— iq5 e2~”A,,

= qSD,,e~2”— ~ e2~’(e2~’D,.e2”) = 0

and 1 e2” is gauge-anti-chiral,thoughnot anti-chiral.

138 Martin F Sohnius,IntroducingsupersyPnmetry

As in the Abeliancase,we can go to the Wess—Zuminogaugeto evaluate(10.49) in components.Unlessthematteris in a realrepresentationof thegaugegroup, it would be very inconvenientto work ina basis of real matter fields; the following Lagrangianis thereforegiven in two-componentnotation,with complexfields A andF asthey appearin the expansion(7.45) of the chiral multiplet:

Lk,fl = W,,AtV~A+ i~iö~’V,.~+ ~FtF + iAtA,/j — i~’AA+ ~AtDA. (10.50)

A specialcaseis matter in the adjoint representationof the gaugegroup. Then it makessenseto userealfour-componentnotation andthe Lagrangiancan be cast into the form

Lk~,,= tr[W,.AV~A + W,.BV~B+ ~i*y~V,.t/i+ ~F2+ ~G2

— iA[B, D] — i~i[A,A] — i~y5[A,B]]. (10.51)

10.10. Supersymmetric Yang—Millstheory (generalcase)

The most generalN = 1 supersymmetricYang—Mills theory is not just the sum of the Lagrangians(10.45) and (10.50) but can also involve self-interactiontermssimilar to those in the Wess—Zuminomodelas long as thesearegaugeinvariant.The mostgeneralsuperpotentialwould be

V= ba0a+ ~m°~‘0a0b+ ~gabc41a0b0c. (10.52)

The indices are takenfrom the representationof the gauge group under which 0 transforms,thepotentialis gaugeinvariant if the parametersb, m andg arenumericallyinvariantsymmetrictensorsofthat representation.This representationneed,of course,not be irreducible,andthereis wide scopefordifferent such models.

Thefull componentLagrangiannowlooksas follows (in 2-spinornotation):

L = tr[—~F,,,,F~~ + iAu”V,.A + ~D2]+ ~V,.AaV~At” + 1V1aU”~7,,V1”+ ~F,,Fta

+ iAtc~Aat~I/Jb — j~/JA Ab + ~Atc~DabAb

— ~[b”F,. + m” (F,.Ab + t/JaOi,) + gabc(A,.~~~~+ 2cfrat,l’,,Ac)+ h.c.]. (10.53)

This is the most general renormalisablesupersymmetricLagrangian,except for a possible Fayet—Iliopoulos termwhich we shallget to know in the nextsection.Upon elimination of the auxiliary fieldsall termswhich contain eitherFa or Fm or Da” are replacedby

rr— 1r’E’ta 1r~ r~ iiI1~A— ~-‘ — 2Ua1 — 2~—’m-1---’m

with

Fta = mabAb + gabCAbAC

= _~At~~Zat~(Tm)Ab (10.54b)

whereZa”(Tm) arethe representationmatrix elementsof the generatorsof thegaugegroup.

Martin F Sohnius,Introducingsupersymmetry

11. Spontaneoussupersynunetrybreaking(II)

The presenceof the pseudo-scalarauxiliary field D(x) in the gaugemultiplet will haveits effect onthescalarpotentialandthuson the spontaneousbreakingof the symmetriesof the model [13].Since, inparticular, the value of the potential at its minimum determineswhether or not supersymmetryisspontaneouslybroken,we must re-examinethepossibilitiesfor spontaneoussupersymmetrybreakinginthe presenceof gauge interactions. The terms in the Lagrangians(10.45) and (10.50) which cancontributeto the potentialare

L~5= ~D2+ ~DA~’rA+ ~FtF. (11.1)

In this notation,the adjoint representationof the gaugegroupis denotedby vectors(bold italics) andthe matricesi- arethe generatorsof the representationwhich actson the matterfields. The equationsofmotion for the auxiliary fields are

D=—~At’rA; F=0 (11.2)

and putting this backinto U = ~ wefind

U= —L~~=~D2+~jF~2 (11.3)

with D and F now short-handfor the functionsD(otherfields) and F(otherfields) which aregiven bythe equationsof motion,hereby eqs. (11.2). The potentialof a “minimal” gaugemodel,as describedbythe Lagrangians(10.45) and (10.50), has its minimum at (A) = (D) = (F) = 0 and henceat (U) = 0.Neithergaugenor supersymmetryinvarianceare broken,becausethe conditionsfor symmetrybreakingare

(a) gaugeinvarianceis spontaneouslybrokenif the vacuumexpectationvalueis not zero for a fieldwhich is not gauge-invariant;

(b) supersymmetryis spontaneouslybrokenif the vacuumexpectationvalueof theenergy,i.e. of thepotential,is not zero.In the morecomplicatedmodel (10.53)which involvesself-couplingtermsfor mattermultiplets we mayget the O’Raifeartaightype of supersymmetrybreaking [50], provided that one of the multiplets isunchargedso that a bF term is gauge-invariantandcan be presentto triggerthe symmetry breaking.Aswe recall from section6, we get spontaneoussupersymmetrybreakingwhen such a term (linear in anauxiliary field) cannotbe shiftedaway.

11.1. TheFayet—Iliopoulosterm

If the gaugegroup containsan Abelian factor, a similar term (linear in an auxiliary field) can beaddedto the Lagrangian,the “Fayet—Iliopoulosterm”, namedafterref. [13],

LFJ = ~D, (11.4a)

without breakinggaugeinvariance.At the momentwe do not worry aboutpossibleparity-violation bysucha term (D is pseudo-scalarif A,. is a polarvector).Sincethe D’s are in the adjoint representation,

140 Martin F Sohnius,Introducingsupersy~nFnetrv

the Fayet—Iliopoulos term is gauge-invariantonly for an Abelian gauge group. It is, of course,supersymmetricsince D transformsinto a derivative. In superspacenotation,the contribution to theactionis

‘Ft. = —~Jd~xd20d2O~V. (11.4b)

To studythe effectof sucha term,wecan considerthe simplestAbeliangaugemodel[44]whereasinglechiral superfieldundergoesphasetransformations

(11.5)

In contrastto the precedingtwo sections,the gaugecoupling constantg is hereexplicitly written inorder to show its effect on the symmetrybreakingmechanisms.

Sincethereis only oneMajoranaspinorpresentin a single chiral multiplet, the gaugetransformationon it must take the form of y

5-transformationsi/i —~exp{igy5a(x)} cit. This assignsnegativeparityto thechargewhich was why super-QEDhadto involve two chiral multiplets, seesection9.

The total superfieldaction for the model is

i = [i~Jd4xd20 W2+ h.c.]+~Jd~xd20d20(~e_2eVO— 4EV) (11.6)

andthe termsin the Lagrangianwhich arerelevantfor the potentialare

~ = ~D2+ AtDA + ~F~F+ ~D; (11.7)

a massterm is not possiblesince m412 is not invariant under(11.5). The equationsof motion for theauxiliary fields are

D=_~AtA_sc and F=0 (11.8)

andthe potentialis

~ . (11.9)

After elimination of the auxiliary fields, the “on-shell” Lagrangianbecomes

L = ~ +~A/A + W,.AtV,.A+ ~ ~by~V,,çb+ igA1 +1Y5 OAt— igk1 ‘~‘~çbA — U (11.10)

with V,.A = 3,,A+ igAA,. andV,.O = 3,,~i— gy5~A,, and U given in (11.9).

Martin F Sohnius.Introducingsupersymmetry 141

9

Fig. 11.1. The potential for thesimplestgaugemodel.

If we plot (U) against (A)! we find two very different types of potential,dependingon whether~g<0 or ~g>0 (seefig. 11.1).

11.2. ThesupersymmetricHiggs model

Thefirst casewhich weconsider,that of ~g<0, hasthe minimumof the potentialat

(U)=0 and (A)=\/—2~/g_=mIg. (11.11)

We thereforeexpectsupersymmetryto be preservedbut gaugeinvarianceto be spontaneouslybroken.This will turn out to be the supersymmetricversionof the Higgs model [15].

The usual Higgs modelsubstitutionsin the Lagrangian(11.10),with the parameterm as definedby(11.11),

A = (~-+c) exp(igølm), A,. = B,. —--~--3,.@, (11.12a)

togetherwith the moreunusualsubstitution

ci, = exp(—gy5øIm)x, (11.12b)

eliminatethe Goldstoneboson~9from the Lagrangian:

L = ~ + ~(m + gC)2B,.B~+ ~3,.C3~C—~(m+ ~gC)2C2

+~A/A ~ —(m+gC)Ay5~. (11.13)

The fermionmassterm can be diagonalizedby introducingfield combinations


A’~~,~(A+y5x); X’—~7(A—y5X)

in which casethe free part of the fermion Lagrangianis

We see that the spectrumof the model containsan axial-vectorB,., a scalar C and two MajoranaspinorsA and x, all with the same mass m. This mass degeneracy,together with the fact that thepotentialis zeroat its minimum, is a strongindicator for unbrokensupersymmetry.

In order to confirm this, we can study the following action [15] which involves a real generalsuperfieldV andits curl superfieldWa:

I=[~Jd4xd2ow2+h.c.]+~Jd4xd2od2i(e2~~+2gv). (11.14)8g

This is clearlysupersymmetricand can beexpandedinto components

L = ~ + A/A + ~D2- ~ [e2~”]D — ~-D.

2 4g 2g

The exponentialterm is tediousbut straightforwardto evaluate(use the Taylor expansionin 0 and 0ratherthan that for the exponential).The termswhich involve the field componentsM andN are

— 4g2[e 2~~]D= + e~ [(M + ~ ~Y5X)2 + (N + ~ ~x) 2j

and give no contributionat all after the auxiliary fields M and N are eliminated. The part of theLagrangianwhich involvesthe auxiliary field Dgives

2 m4L= ...+~D2+ (e_2~c_1)D_..

2g 8g

andthe total “on-shell” Lagrangianis found to be

L = —~13,,,F~~+ ~~/A— (e2~C— 1)2 — im2g e_2gC,~7y~y5xA,,

+ m2e29~3,,C3~C+~ij/x + ~ — ‘tx] + 4-div.

The field redefinitions


m ~_~C~_t~ Y5X

A,.-*B,.

give exactlythe Lagrangian(11.13)which hasthusbeenshownto besupersymmetric.The supersymmetricHiggs model can be seeneither as a gaugemodelwheregaugeinvariancebut

not supersymmetryis spontaneouslybroken,or as a real generalmultiplet in self-interaction.P. Fayethasobserved[15] that the superfieldaction (11.13) for the latter picturecan be derivedfrom the actionfor the formerone,eq. (11.6), by choosingnot the Wess—Zuminogaugebut what shouldbe called the“Fayet gauge”

41=mlg (11.15)

which is supersymmetric.The existenceof such a gaugewhereapparentlythe matterfields havebeengaugedaway is a consequenceof supersymmetry— the degreesof freedomreappearin C andx whichnowpropagateandin A,. which hasnowacquireda longitudinal part.The gaugechoice(11.15)can beseenasthe supersymmetricversionof the procedureusedin the Higgs model wherea complexfield isreplaced by its modulusand phase,and the latter is then gauged to zero. Here, with a complexparametersuperfieldA, wecan also gaugethe modulusto a constant.Consequently,not only the phasereappearsas the third degreeof freedomof amassivevector,but alsotheothermatter fieldsas a scalaranda Majoranaspinor.

11.3. Brokensupersymmetry

In the secondcaseof fig. 11.1, where~g>0, the minimumof the potentialis at

(U) = ~ and (A) = 0. (11.16)

No shift in fields is required,but afterelimination of the auxiliary fields from theLagrangian,weget thepotential (11.9) which containsa massterm ~g~AtA for the scalarsbut no similar term for one of thespinors.Thus the massdegeneracyis lifted andsupersymmetryis spontaneouslybroken— as expectedfor a modelwherethe lowest energyis not zero but ~ A field which is not the 0-0-independentpartof a superfield, namely the auxiliary field D, hastaken on a vacuum expectationvalue. This fieldappearsin the transformationlaw for thegauginofield A andconsequentlythereis a field independentpiecein the on-shell transformationlaw for A:

= — io”~3,.A,,+ y5~A~A. (11.17a)

This property,which perhapsis betterwritten as

{A,,, Q”) = i(y5),,~~+ (11.17b)

showsA to bethe Goldstonefermion (“goldstino”) of the brokensupersymmetry.


Note that since supersymmetryis not a local gaugeinvarianceof the model, thereis no “super-Higgs” effect and the goldstinois not “eaten up” to provide additional degreesof freedomfor amasslessgaugefield. Such a thing can only happenin the contextof supergravitywherespontaneouslybroken local supersymmetrymayoccur.

11.4. TheFayet—Iliopoulosmodel

In the original paperby FayetandIliopoulos [13],super-QEDwas the theory whosesupersymmetrywas spontaneouslybroken, ratherthan the simpler gaugemodel discussedso far. Taking the super-QED Lagrangian,consistingof the Lmatterof eq. (9.30) andtheLMaXWCII of eq. (9.35)andaddingto it theFayet—Iliopoulosterm (11.4), we get for the scalarpart of the Lagrangian

L~5= ~D2— g(A

1B2—B1A2)D + ~D+ (~j32+ ~GI2 — mA

1F1— mB~G~) (11.18)

which givesthe usualpotentialU = ~(auxiliary fields)2with equationsof motion

D = g(A1B2—B1A2)—~

F1=mA~ G~=mB~. (11.19)

The conditionfor asupersymmetricvacuum,(D) = (F,) = (G,) = 0, cannotbefulfilled for m � 0 � ~,sothat supersymmetryis alwaysspontaneouslybroken for this model. Looking at the term quadraticinfields we find a non-diagonalscalarmassmatrix which is diagonalizedby the transformations

A1=~(A1+B2); A2=~(A2—B1)

1 (11.20)E1=~(A1—B2); A2=~(A2+B1),

resultingin the potential

U = ~(m2— ~g)(A~+A~)+ ~(m2+ ~g)(~ + ~) + ~g2(A~+ M— B~— E~)2+ ~2 (11.21)

Althoughwe canassumethat always~g>0 (otherwisethe rolesof A andE areinterchanged),we mustdistinguishtwo different cases:

(a) If ~g < m2, theminimumof thepotential is at

(AI) = (E~)= 0 and (U) = 1~2 (11.22)

In this casesupersymmetryis spontaneouslybroken, but gaugeinvarianceis not (the gauge trans-formationsfor the new fields are~gAi = aA

2 ~gA2= —aA1 andthe samefor B,). The massspectrumhasbecome


m~=m; m~,=Vm2+~g

rnA,,—mA—O. (11.23)

The masssplitting inside the mattermultiplet establishesfor surethat supersymmetryhasbeenbroken.The only masslessspinor, A, is the goldstino.

(b) If ~g> m2 the minimumof the potentialis at a finite valuefor the matter fields. Onechoice is

(A1) = ~/g with ~2 2(~g— rn

2) (11.24a)

andgaugeinvarianceis spontaneouslybroken.Supersymmetryis as well, since

(U) = ~ (p2+ m2) = ~2_ -~j(~g— rn2)2~~2_ z~U (11.24b)2g 2g

which is largerthan zerobut lessthan~ A Higgs mechanismwill eliminatethe GoldstonescalarA2,

give the vectorA,. a massandmix the threespinorsof the model into masseigenstatesA and ~. Thedetails can be found in the original reference[13] and here it shall suffice to give the final massspectrum:

m,j,,=\/m2+/22; mA=O

mA,~—mA,=/1, rnn,=V2rn. (11.25)

The goldstinois clearly that particularlinearcombinationA of the spinorsA and~‘, which hasremainedmassless. —

Figure 11.2 showsthe behaviourof the potentialfor the two cases,drawn against (A1)!.The spectrumof the Fayet—Iliopoulosmodel for variousvalues of ~g canbe summarisedin fig. 11.3.

A moredetailedanalysiswill show that, in order not to breakparity, the gaugefield A,, must be anaxial vector. This is a consequenceof the fact that the auxiliary field D which appearsin theFayet—Iliopoulosterm wouldbe apseudoscalarwereA,, avector.Thusthe startingpointof this analysiswas not super-QEDafter all but rathera chiral U(1) gaugetheory.

<U>

Ii9

Fig. 11.2. The potential for theFayet—Iliopoulosmodel.


mass

/ _~i,‘. —A~A~

-— ~‘_‘_~/~

______,/~

—- ‘p1 /

,— .-— /—— ‘B .-~ /

I -. /

m A~BjW~---- -Wi---- /

A A~ A ~ Ap A1 A A A

~g-O O<~g<m2 ~g~m2 m2<~g<2m2 ~g>2m2

~— supersymmetrybroken I— gauge invariance broken

Fig. 11.3. The massspectrumof theFayet—Iliopoulosmodel.

The massformula (6.29) is fulfilled in the caseof the Fayet—Iliopoulos model, but not for theminimal modelwith brokensupersymmetryof the previoussubsection.

1.5. Stability of massspectra

A word is in orderon the questionof the behaviourof the massspectrumunderrenormalisation.Oneof the mostattractivefeaturesof supersymmetricfield theoriesfor modelbuilding is the stability ofmasses.This featurepointsa way aroundthe “technicalhierarchyproblem” wherea “spill-over” from alarge massscaletendsto raisea small oneandthus renderslargemassgapslike the thirteenordersofmagnitudebetweenthe GUT-massandthe W-massvery unnatural:the gapmustbe enforcedorderbyorder.This doesnot happenin thosesupersymmetrictheorieswheremassterms do not get separatelyrenormalised.The already mentionednon-renormalisationtheorem for chiral superspaceintegralsprovidessuchstability for masstermslike 5 d4x d20rn412.In the presentsection,however,we haveseenthat the actual“physical” massspectrumis determinednot only by thesetermsbut alsoby the valueofthe Fayet—Iliopoulos parameter~ which multiplies an integral over the whole of superspace,5 d4x d20d20~V, and can thus a-priori receive independentquantum corrections.Answering thequestionof massstability must thereforeinvolve an investigationof the circumstancesunderwhich theFayet—Iliopoulosterm doesindeednot get renormalised[22,41], seealso [31].


12. N = 2 supersymmetry

A largepart of the excitementgeneratedin theoreticalphysics by the conceptof supersymmetryisdue to the non-renormalisationtheoremswhich hold for supersymmetricmodels. The property ofquantumdivergencesto be less singularthan one would ordinarily expectis very commonandindeedpresentalreadyin the simplestof all the models,the N = 1 scalarself-interactionmodelsandthegaugemodelsdiscussedso far.They arethe onesusedfor the modelbuilding of super-GUTs.

The other exciting featureof supersymmetry,however,is the potential for true unification of thegravitationalforce with the otherforcesof nature.Althoughtheseforcesare now understoodto be theconsequencesof a “gaugeprinciple”, a true super-unificationcannotbe madeapparenton the simplegaugemodelsof N = 1 supersymmetry.First of all, an extensionfrom Minkowskiansupersymmetrytosupergravityis clearlycalled for, andsecondly,the gaugegroupmust be much deeperingrainedin thestructureof the theory thanthe “tackedon” gaugegroupswith whichwe havedealtso far. We saw atthe beginningof section9 that thosegaugetransformationswhich do not commutewith supersymmetryarethe oneswhich areintimately relatedto generalcoordinatetransformations,andwe anticipatethatthey will be the oneswhich in the endmayprovide super-unification(“gaugedN = 8 supergravity”isthe catchphrasein this context).

“Not commutingwith supersymmetry”meansmore than just upsettingaparticular non-supersym-metric gaugelike the Wess—Zuminogauge.It meansthat the generatorsof thegaugesymmetrydo notcommutewith thegeneratorsof supersymmetry.We haveseenin section2 that the largestgroupwhosegeneratorsdo not commutewith all the Oat (i = 1, . . . , N) and all the 0,,’ is U(N) or, in the presenceof central charges,USp(N). Thus in order to ever have a non-Abeliangauge group in a gaugedsupergravitytheory, the numberN of independentspinorialsymmetriesmustbe largerthanone.

This makesthe studyof extendedsupersymmetryimperative,andwe shall begin this studyoutsideofthe context of supergravity,by looking at the simplest Minkowskian N = 2 models. Clearly thatpreciousU(2) symmetry will only be a global invariancein thesecases,since supersymmetryitself isungauged.This point was discussedat the beginningof section9.

12.1. A modelwith N = 2 supersymmetry

Since the helicities in a supermultipletof particle stateshaveat least a rangeof ~N, the simplestN = 2 multiplet will contain spins~ and 0. This multiplet must, however,be eithermasslessor haveanon-trivial centralcharge,andwe discardit for the time being(we shall later seethat aself-interactionis not possiblefor it). The nextsimplestmultiplet will alreadycontainspin 1 andthusmustbe a gaugemultiplet and massless.Its spectrum,taking into accountthe doubling necessaryto meet the require-ment of CPT-invariance,is

N = 2 helicity: —1 —~ 0 ~ 1 (12.1)

states: 1 2 1+1 2 1

and comparingthis with the N = 1 case,we seethat it is the superpositionof a gaugemultipletwith achiral multiplet:


helicity: —1 —~ 0 ~ I

N = 1 gaugemultiplet: 1 1 1 1 (12.2)

chiral multiplet: 1 1 + 1 1

As a first try, we thereforelook at theLagrangianfor the non-Abeliangaugetheory of a singlemasslesschiral multiplet in the adjoint representationof the gaugegroup. This is the sum of the Lagrangians(10.45)and (10.51).We eliminatethe auxiliary fields anddefine

A1~A; A2~O. (12.3)

The resulting Lagrangian(to avoid future confusion, the fields A and B of the chiral multiplet arerenamedas M and N)

L = tr[—~F,.,,F~~ + ~iAjy” V,,A, + ~V,,M V”M + ~V,.N V~N

— i,~[A1, M] — iic2y5[A1, N] + ~[M, N]2] (12.4)

has an 0(2)symmetrywhich is moreapparentif we rewrite

A~A ~ Cabc = — ~ A ~ca& (samefor Ay5A) (12.5)

(thisequationfollows from theantisymmetryof the gaugegroup’sstructureconstantsCab,, andfrom eqs.(A.45) in the Appendix).Underthis 0(2) thegaugefield A,. is a singlet,but theold gauginoA is part ofa doublet. Since under the old N= 1 supersymmetrytransformationsBA,,= i~y,,A,supersymmetrycannotcommutewith the 0(2). Its generatorsmust be part of a doublet, the other supersymmetrytransformationbeing WA,. = ~ A more detailedanalysis will indeed show that the Lagrangian(12.4) is a densityunder the following set of transformationswith two MajoranaparametersA1 andA2:

= ~

=

— (12.6)=

= —~io-”~F1~,,,+ ie11y~V,.(M+y5N)~— iy5~[M,N]

which lead to the algebra:

[5(1), 5(2)] = ~ +5gauge+ field equations. (12.7)

Ignoring the gauge transformation(it arisesbecausewe are in a Wess—Zuminogauge)and the fieldequations(they arisebecausewe are in an on-shell formulation without auxiliary fields), we see therelationshipwith the centrepiece


{Q, ~} = 2511YILP,. (12.8)

of the N = 2 superalgebra.The motivationfor studyingextendedsupersymmetrywas its non-Abelianinternalsymmetrygroup.

In the presentexample,however,it seemsthat the internalsymmetry is actually only an 0(2), not theU(2) we hadhopedfor. Rememberingtheremarksmadeafter eq. (4.3), this is not surprising:as longaswe use Majoranaspinorsof the form (4.1), our notation will hide any possiblesymmetry largerthan0(N). One way to overcomethis problem is to revert from Majoranaspinorsto chiral notation with2-componentspinors. However, as said before, presentationof results is much neaterwhen using4-componentspinors.Fortunately,thereis a possibilityof constructingan SU(2)-covariant4-componentspinor from the 2-componentspinorsA,,1 andA ~ (Aaa)t: we define a symplecticMajorana spinorA’ by

r—ie”A •~

A~I —..~‘ I~ A~(A~’iEjji’a) (12.9)l A” i

which satisfiesa symplecticreality condition [56]

A’ = e’1Y

5CAT. (12.10)

The word “symplectic” refers to the antisymmetrictensor e,, which is used as a metric to raiseandlower SU(2) indices. In general,reality conditions of this type are only covariantunder symplecticgroups, like SU(2)= Sp(

2), just as ordinary Majoranaconditionsare only covariantunderorthogonalgroups.

In this SU(2)covariantnotation,the Lagrangian(12.4)becomes

L = tr[—~F,.,,F~~+ ~iA,y~ V,.A’ + W,,M V~M+ ~V,,N V”N

— ~ [At,M] — ~Ay5[At,N] + ~[M, N]

2] (12.lla)

which is clearly invariant underthe SU(2)which rotatesthe spinordoublet,but also underthe chiralU(1) transformations

M+ iN-.e2~~(M+iN)

A’ —*e”~5A’ (12.12)

A,. -~A,..

The transformationlaws (12.6)arein this notation:

BA,. = i~y,.A’

— . (12.13a)oN =i~,y

5A’

= —~io~”~,’F— y~ V,.(M+ y5N)~t— iy

5~’[M, N].


An off-shell version of thesetransformationlaws is known [29]. It involves a real SU(2) triplet ofauxiliary fields D with U(1) weight0. For the off-shell version,the transformationlaws (12.13a)mustbeaugmentedby

8A1=”~—i~1r3•’D

— . . — . S — (12.13b)= r,’ ~y~’V,,A’ + ir,’ ~[A’, M] + i’r/~Tjy5[A’, N]

andthe Lagrangian(12.11)picks up the additional term

L=~”+~trD2. (12.llb)

The inclusion of D will drop the term “+ field equations” from the algebra (12.7), but the term “+

5gauge” is still there:

[5(1) 5(2)] = ~ + ~5gauge, (12.14a)

with the following field dependentparameterfor thegaugetransformation

= —2i ~ + 2i~~’~(M+y5N)~’~

2’. (12.14b)

12.2. TheN = 2 mattermultiplet (“hypermultiplet”)

When a Frenchsuper-marchécarriesnot only food and drink but also car spares,gardenfurnitureand ladies’ underwear,it becomesan hyper-marché.Correspondingly,P. Fayet called N = 2 super-symmetry“hyper-symmetry”[14].Whereasthat namehas not stuckin general,the mattermultiplet ofN = 2 supersymmetryis still called the “hypermultiplet”. The well chosenuseof that word is likely toproducea glimmer in the eyesof the true cognoscenti.*

If N> 1, any massivemultiplet will contain particlesof spin �1, unless thereis a centralchargeZwhich satisfiesthe condition[57,60, 76]

(12.15)

so that the eigenvalueZ of the centralchargeequalsplus or minus the mass,see subsection3.4. Thespectrumof the hypermultipletis given by eq. (3.40). So far no explanationhas beengiven for themysteriousdoubling of statesin that spectrum.Without that doubling,the spectrumwould bethe sameas for the N = 1 chiral multiplet, and the questionthereforearises whetherwe can representN = 2supersymmetryon a single chiral multiplet of N = 1. A secondsupersymmetrygeneratorS,,,differentfrom Q,,, would haveto seethe multiplet as anti-chiral,or S,, would just be a multipleof 0,, again.Theansatz

[A,~]=0; [A,S]=0

[A, 0] = 2iw; [A, S] = acb

* Early referencesto N = 2 modelsare[11]and,particularly, ref. [57]wherethemattermultiplet first appearedin print.


is alreadythe mostgeneralone.At the next level, the requiredproperties{Q, Q} = {S, S}= 2o~’P,,leadto {O~~} = /A and{S, O} = 2ia’/A, andby Hermitian conjugationto

{Q,~}=/At

Thus

[A, {Q, ~ = 2i{41, .~}+a{Q ~c}= (4/a* + a)/At,

an expressionwhich must vanish becausewe require that {Q, S} = 0. This implies the trivial solutionA = const.

A different ansatzis therefore called for which involves two complex fields A,. On them twosupersymmetrygenerators0, can act. Since we only want to double the spectrumof the chiralmultiplet, the commutator[A

1,Q,,~]shouldbe expressiblein termsof no morethantwo different chiralspinors.Since the most generalansatzfor that commutatorwould be E10,, + ci1a(ij), we see that thetriplet Oa(ij) mustbe ruled out. The hypermultipletmust thereforesatisfy a constraintwhich reads[60]

[A1,Q~]+ [A1,0~]= 0 (12.16)

in a notation where0’ is asymplecticMajoranaspinor. In an N = 2 superspacelanguagethisconstraintwould read

Da(j41j) = D&(l0J) = 0.

It definesthe hypermultipletjust as the chiral constraint(3.43) definedthe chiral multiplet. Indeed,justas for the chiral multiplet, we can go through aprocedurelike the “seveneasysteps”of subsection3.6and constructthe entiremultiplet from thisconstraintand knowledgeof the superalgebra.An attemptto do this without a centralchargewill, however,lead to free masslessequationsof motion for thefields. Only if we allow for a central charge, do we get an acceptableresult: a multiplet

(12.17)

of two complex scalarsA.(x), a Dirac spinor O,,(x) and two complex auxiliary scalarsF1(x), with

supersymmetrytransformationsas follows:= 2~41

00 = —i~’F,— i/~’AI (12.18)

OF, = 2~%0.

Thesetransformationssatisfy the algebra

[5(1) ~(2)] = ~ + ~ (12.19)

with the actionof the centralcharge,~ given by


= F,

(12.20)

5~F,=~I1A1.

It is easyto checkthat the centralchargecommuteswith supersymmetry,

[3,3~]= 0 (12.21)

andthat

(12.22)

which is just the condition(12.15)which ensuresthat thereareno higher spinsin the multiplet than ~.

In order to construct an invariant from the hypermultiplet,we use the techniqueof finding acontragradientmultiplet, similar to subsection4.6. The hypermultipletturnsout to be contragradienttoits own Hermitian adjoint andwe find for any two hypermultipletsç~,and

çb~=(a~x;f,)

that the product

(~“ ~,)= iat’F, — iftAI + 2

1~çls (12.23)

is a density. This productdoesnot involve derivatives and can only be used for mass terms in theLagrangian;it is the N = 2 generalisationof (q~. /)F. Since,however,t9~4,is alsoa hypermultiplet,withcomponentswhich do involve derivatives,

= (F, ; /tfr; LJA,) (12.24)

(it plays the role of the kinetic multiplet), it is easyto constructa free Lagrangian from (12.23):

~

= ~8,,At’ô~A

1+ ~Ft~F,+ iciu/0+ m[~iAth13— ~iF~’A1+ 00] + 4-div. (12.25)

The equationsof motion are

= irn4j (12.26)

whenwritten in multiplet form. In components,thesearetheequationsfor free fieldswith massm oncethe auxiliary fields F, havebeeneliminated.

Note that the non-compactZ-transformations(12.20) become compact phase transformationson-shell,with a chargethat is equalto the mass—just the multiplet shorteningcondition! In the absenceof a massterm,the centralchargedisappearsentirely on-shell,as it shouldaccordingto subsection3.4.


12.3. Thehypermultipletin interaction

The Lagrangian(12.25) describesfree fields. In the caseof N = 1, it was possibleto write down atrilinear self-interactionterm for the chiral multiplet. This resultedin the Wess—Zuminomodelwhichwas a supersymmetricgeneralisationof the self-interactingscalarA~4model. A similarconstructionisnot possible for the hypermultiplet.This is quite obvious if we insist on SU(2) invarianceof theLagrangiansincehalf-integerspinsgo togetherwith integer“iso”-spins, andthusthereis no third-orderinvariant. But evenif SU(2) invarianceis sacrificed, no supersymmetricrenormalisableself-couplingtermsexist [47].We can thereforesaythat

the onlyinteractionsof N = 2 matterare gaugeinteractions.

Ideally, we should like to havea full superspacetreatmentof the theory.Unfortunately, however,nosuch treatmentexists for the non-Abeliancase(thereis a solutionof the superspaceconstraintsfor theAbelian case[46],andthe full impactof the recentdevelopmentof “harmonicsuperspace”needsto beawaited[23]).For the time being,we areforced to proceedin a ratherpedestrianway.

The problemweface is that the gaugemultiplet is alreadyin aWess—Zuminogauge,resultingin theadditional 3gauge in the algebra (12.14). If the gauge multiplet is to interact with matter in asupersymmetricfashion, the samealgebramust be representedon all fields involved. This meansthatwe mustextendthealgebraof the transformationsof the gaugefields to include the centralcharge,andthe algebraon the matterfields to includethe gaugetransformations.

The former is easily done, we just assumethe fields of the gaugemultiplet to be invariant underZ-transformations:

3~A,,= = 3~N= SZA’ = = 0. (12.27)

The algebrais now that of eq. (12.14)with the additional S~-termof (12.19). Its genericstructure is

F~(1) ~(2)~ —5) J — 5)traflSJ~tjofl ucentral charge 5)gauge, . a

andon a mattermultiplet it looks like this:

[5(1) 3(2)] = ~ + 2i~’~(3~— iAl — iy5N)~’~

2~. (12.28b)

The fields M and N are Lie-algebravalued and acton the gaugegrouprepresentationindices of thefields of the mattermultiplet. We seethat a non-zerovacuumexpectationvalue for M or N will inducea centralcharge[76].

We can imposethis algebraon a doublet A, of complexscalarfields, just as we could imposethesimpleralgebrawithout gaugetransformations.The constraint(12.16) is to remainunchanged.Thentheresultof aratherlengthy calculationwill bethe following modified transformationlaws for the fields ofthe gaugehypermultiplet,

= 2~çit

80 = —i~’F,—(iy~V,.+M+ y5N)~

1A1 (12.29)

OF, = 2~(y~V,,+iM—iy5N)O—2~A’A1,

154 Martin F Sohnius,Introducingsupersytnmetry

togetherwith modified centralchargetransformations

ô~A1= F,

‘5~O =(x~V,,+iM—iysN)Vi—AtA

1 (12.30)

= (V~V,.+M2+ N2)A

1— 2iA,0— r/DA1 +2iMF,.

The productdensity(12.23)picks up anothertermand becomes

ia~’F,— ift’A

1 + 2,~O+ 2atIMAt. (12.31)

The centralchargetransformationsstill commutewith supersymmetry,therefore&4~is still the kineticmultiplet, andwe can write our Lagrangianas

~

= ~V,.At1V~A,+ ~FtiF, + it/iy~V,.~it+ iA~’A,i/J— itIJA’AI — ci’(M — y

5N)t/i

— ~At~(M2+ N2)A

1 + ~Atir/DA1 + m(~iA~’F,— ~iFt’A

1+ cltifr)+ rnA~’MA1+ 4-div. (12.32)

The equationsof motion remain &/, = im~1but the left-hand side is now given by the morecomplicatedeqs. (12.30)with all their interactionterms.

12.4. Chiral fermions?

We havealreadyseenhow the N = 2 gaugemultiplet (12.13)originatedfrom agaugemultiplet andachiral multiplet of N = 1. Similarly, we now look at the transformationlaws (12.18) for a freehypermultipletandtry to analyseits contentin termsof N = 1 multiplets.Of the two chiral spinorsc,,,from which the symplecticMajoranaparameter~‘ is built, we take ~, call it c,,, breakthe Dirac spinorof the hypermultipletinto chiral partsO~and ,~,,, andfind the following transformationlaws (in chiralnotation)

8A1 =2~0; OA2=—2i~~

8O=~F2—i/~A1 8~=—i~Fl+A4~A2 (12.33)

OF2 = —2i~4O; OF1 = ~

By comparisonwith eqs. (3.50) and (3.53), thesecan be recognisedas those of a chiral multiplet(A1 0; —F2) andan anti-chiralmultiplet (iA2 ,~iF~).This was to be expectedsincethe hypermultipletcontainsa full Dirac spinor whose right-handedand left-handedcomponentsmust belong to a chiraland anti-chiral multiplet, respectively.This has the important consequencethat a matter multiplet ofN = 2 supersymmetrywhich transformsundersomerepresentationof agaugegroupwill alwayscontainboth a right-handedand a left-handedfermion in that representation,and thereforewith the sameinternalquantumnumbers.Thus evenif the massis zero, the representationsof the gaugegrouparenecessarilyvector-like.Thesameis, of course,truefor all higher-N supersymmetricmodels,sincetheir


symmetry groupsalways containN = 2 supersymmetryas a subsymmetry.Realistic modelsthereforerequirethe breakdownof all but the last of the supersymmetriesalreadyat very high energiesin orderto obtainthe V-A structureof the weak interactions.

12.5. N = 2 supersymmetryfrom an N = 1 model

After the considerationsin subsections12.1 and 12.4, we now know that the most generalN = 2gauge theory, describedby the Lagrangians(12.11) and (12.32), can be seen in the light of N = 1supersymmetryas a model with threematter multiplets. Two of theseare possibly massiveand inrepresentationsof the gaugegroupwhich areconjugateto eachother, the third is masslessand in theadjoint representation.

A different question may now be asked: when does an N = 1 supersymmetricgauge theory,describedin generalby the Lagrangian(10.53), havea secondsupersymmetry?This requiresa secondgaugino, so theremust be one matter multiplet ‘I’ in the adjoint representation.Secondly,the theorymust be vector-like,so if thereare further chiral mattermultiplets 4’a in somerepresentationof thegauge group (not necessarilyirreducible) then there must be correspondingmultiplets 4,~in theconjugaterepresentation(4,a (

4)a)t won’t do). The most general N = 1 gauge theory for these(comparesubsection10.10) is describedby a superpotential*

V= b”4~+ b~çb”+bab ~b — mçb ~ +gq5”~P~t’cbb, (12.34)

or rathera slight generalisationof it: the parametersm and g can actually take different values fordifferentirreduciblepartsof Oa. Also, if thesameirreduciblerepresentationappearsmorethanonce,themassterm andthe third-ordercouplingterm maynot be simultaneouslydiagonal.

This N = 1 theory is themostgeneralonewith theright kind of spectrum,andwe can nowconnectitto the N = 2 theory (12.32).After eliminating the auxiliary fields, the spinorfrom ~Pis combinedwiththe gauginoA to form a gauginodoublet At, the spinor from 0~is combinedwith that from (

4~~)ttoform the Diracspinorof the hypermultiplet,the scalarsfrom çba and4,~arecombinedinto the complexdoublet A1, and finally the scalarsfrom !P’ are renamedM and N. After a lengthy calculation, itemergesthat this particularN = 1 model can be recastinto the form of the N = 2 Lagrangian(12.32)(with auxiliary fields eliminated),provided that indeedthe mass and third-ordercoupling termsaresimultaneouslyblock-diagonalandthat

b~= b,’~bab 0 (12.35)

= 1. (12.36)

The latter equation is most important: the essentialdifference between the N = 1 and the N = 2theoriesis that the formerstill hastwo couplingparameters,g andthe gauge-couplingconstantwhichwas set = 1 in both (10.53) and(12.32).The phaseof g can be absorbedinto redefinitionsof fields, andthe correctway of statingthe condition ~I= 1 for N — 2 supersymmetryis that

* If 4,, is in arealrepresentation,theremay be more terms,e.g.m ‘4,,4,, or d~b~4,,4,~~N = 2 supersymmetrywill rule theseout.


thesuper-çb3couplingconstantmustbeequal to thegauge-couplingconstantfor eachirreduciblepairof mattermultipletsçb~and~

It is interesting to note that the higher symmetry is apparentonly in the on-shell version of theLagrangian.The setsof auxiliary fields for N = 1 andN = 2 arequite differentandcannotbe translatedinto eachother.

The constantsb are related to possible F, and D terms in the N = 2 Lagrangianwhich wouldexplicitly break SU(2), but not supersymmetrynor gaugeinvariancefor an Abeliangaugegroup. Thequestionof such N = 2 Fayet—Iliopoulosterms andof the spontaneoussymmetry breakingwhich theymay induce goes beyondthe scopeof this article, except for one point to be made: if one supersym-metry is spontaneouslybroken, then both are. This is a consequenceof eq. (2.29) which saysthat theHamiltonian can be expressedas sum of the squaresof eachof the supersymmetriesseparately.Thusthe condition (E) ~ 0 for the spontaneousbreakingof one supersymmetryis sufficient to ensurethebreakingof the otheroneas well. This situationis different for supergravity,andwe arrive at interestingconclusions:Nature can have extendedsupersymmetryonly at some high energy becauseof theobservedchiral lepton spectrum.Yet one supersymmetryshould survive to relatively low energiesinorder to stabilise the GUT hierarchymassgap. SpontaneousbreakingcannotreduceN � 2 to N = 1supersymmetryin a Poincaréinvariant setting, andwe areleft to concludethat at leastthe high-energybreakingfrom extendedto simple supersymmetrymustbe somehowrelatedto gravity effects.

12.6. SU(2)x SU(2) invarianceandthe realform of the hypermultiplet[41

If the hypermultiplet is in a real representationof the gauge group, then the superpotentialfor thecorrespondingN = 1 theory can bewritten as

V= —mç~çb~+ ~a!1’ab~b (12.37)

with q5~,andç~the two chiral multiplets containedin the hypermultiplet.For a real representationwehave ~‘~ab = ~ and in the absenceof the mass term the superpotentialcan be cast into anSU(2)-invariantform

V~Ei’j’~I~a”V’ab4lb”. (12.38)

Sincewhole multiplets carry the indices of this additional global symmetry group, the new SU(2) willcommutewith supersymmetry.It is thereforedistinctfrom the old SU(2) which, e.g.,rotatedtheA, asadoublet.We distinguishthe two groupsby useof primedandunprimedindices.

Let us now explicitly establishthe SU(2)x SU(2) invarianceof the model.The componentsof A andF, will be combinedinto 2x 2 matricesA

1” andF,” with reality conditionsof the form

(A,l’)t= —e”e,1A/ (12.39)

(samefor F). From the Dirac spinor ~‘ a symplecticMajoranadoublet u” will be formed with reality

conditioncl” e”ysO/J. (12.40)


The explicit expressionsare

A11’ = A

1 A12’ = e

11At1

(12.41)01=0;

andwe see that this is only possible if A andAt~arein the samerepresentationof the gaugegroup(the representationmust thereforebereal). A pseudo-realrepresentationwon’t do: if we write

A12’ = we

11At1

with w somemetric for the representationof the gaugegroup,then

(A1l’)t A~1= —w~e

12e”A12

(12.42)(A

12’)t = w*Ei)A = _W*E

2ie~1Al

andwe can havean SU(2)x SU(2) covariantreality conditionfor A1” only if w~= w’. This is possiblefor orthogonalbut not for symplecticmetrics. Interestinglyenough,pseudo-realrepresentationshaveadifferent property: they allow symplectic reality conditionslike (12.39—40) which involve the gaugegroupindicesratherthan i’ and j’. Consequently,suchgaugetheoriesneedto haveonly half as manymatter fields as one would naively expect [47]. An SU(2)-gaugetheory can, e.g., be formulated on asingle hypermultipletratherthan a doublet of them (thisamountsto gaugingthe “new” SU(2)).

We can rewrite the entirematter part (12.32)of the Lagrangianin termsof the new quantities:

L = —~e”e1’1’(V,,A1” V” A1” + F,” 13”— A1” (M2+ N2)A

1” + i~A1” DAk”)

+ ~i~1.y~V,.0”— i01 A’ A1”— ~~i’ (M— y5N)0”, (12.43)

and the SU(2)x SU(2) invariance becomesmanifest. The transformation laws (12.29—30) remainunchanged,except that an index i’ is tackedonto all matter fields. That this is consistentwith thedefinitions (12.41)is a nice checkon algebraiccorrectness!

The perhapssomewhatawkward reality condition (12.39)canbe solvedin termsof four real fields AandA by

A1” = i3,”A + ~“ A (12.44)

(same for F). This leads to a real form of the hypermultipletwhere the two SU(2)groupshavebeen

identified andonly the diagonalSU(2)subgroupis still manifest.In this notation,the LagrangianisL ~V,,AV~A+W,,A V~A+~iciiy~V,,0+~F

2+~F2

+ ç/1AA - icirTAA — ~ç1i(M—

— ~A(M2+ N2)A— ~A(M2+ N2)A — iADA — ~iA . (D x A), (12.45)

with suppressedSU(2)spinor indiceson ~‘ andA.


13. N = 4 supersymmetry

The largestamountof supersymmetrythat can be representedon a particlemultiplet with spins~1,is N= 4. Higher spinsmeannon-renormalisablecouplings, andthuswe reach,with N = 4, the limit ofsupersymmetricflat-spacetheory.Correspondingly,the N = 4 model [27] is called maximallyextended.Any multiplet (even masslessones)will contain spin-i particles,and thus all N = 4 modelsmust beconstructedsolely from gaugemultiplets of N = 4. This makesthe particles necessarilymasslessandtherewill be no centralchargeson-shell,cf. the discussionafter eq. (3.23).

13.1. TheN = 4 super-Yang—Millstheory

The spectrumof the only N = 4 multiplet without higher spins is given by (3.12). This is a gaugemultiplet becauseit contains spin-i, there is no matter multiplet and we conclude that the onlyinteractioncan bethat of a non-Abeliangaugefield A,, in interactionwith itself andits superpartnersinthe multiplet, the four gauginosandsix scalars,all in the adjoint representationof the gaugegroup.

An N = 4 modelwill by default also be an N = 1 and an N = 2 model. We alreadyknow the mostgeneralN = 1 gaugetheory with the requiredspectrum,it is describedby a gaugemultiplet and threemasslesschiral multiplets, all in the adjoint representationof the gaugegroup. The requirementofN = 2 supersymmetry(seesubsection12.5) leavesonly one possiblesuperpotentialfor these(up to aphase— we choosea convenientone):

V= itr~1[q52,~ . (13.1)

There is no free parameterwhatsoeverleft to play with and the model either hasthe required N = 4supersymmetryor it doesn’t. Fortunately,it does.

To showthis, we start with the Lagrangianfor the N = 1 theory:

L = tr(_~13~F~~+ ~iAy~V,,A + ~D2

+ ~ (~V~AIV~AI+ ~V,.BIV~B1+ ~ V,,~+ ~ + ~

— i[A1, B1 ]D - i~[A,A1] - i~y5[A,B1

~ elfk(ci~[0j, Ak]- ~ys[~, Bk]+ [A1,AJ]Fk -[B1, BJ]Fk + 2[A1, ~]Gk)). (13.2)2 ,jk

We now establishan 0(4) symmetry which does not commutewith supersymmetry.We define fourMajoranaspinorsby

A1=1fr1 fori=1,2,3

A4 = A, (13.3a)

andantisymmetric4x 4 matricesof scalarsandpseudoscalarsby


A11 = — ElIkAk, B11 = Elf kBk

A41 = —A14= A1 B4, = —B14= B1, (13.3b)

and eliminatethe auxiliary fields D, F, and G from the Lagrangian.This now takes the followingmanifestly0(4) invariant form (i, J’ k, I = 1, .. . 4):

L = tr[—~F,,,.F~+ ~iAy~V,.A,+ ~V,,A11V~A11+ ~V,.B,1V~B11

+ ~iA,[A1,A11] + ~iAjy5[A1,B1~]+ ~[A11,Bk!][AlJ, Bk!]

+ ~[AIJ, Ak,] [A11,Ak,] + ~[B11,Bk,] [B11,Bk,]] . (13.4)

The A.1 andB.1 areself-dual and anti-self-dualtensorsof 0(4), respectively:

A11 = 1e11k,Ak,; B,1 = —~E1Jk,Bk,. (13.5)

As in theN = 2 case,we arguethat the old gluino A is now part of a vectorof 0(4)but thegluon is stilla singlet. This meansthat theremust be a vector’s worth of supersymmetrytransformations:N = 4.Indeed,the Lagrangian(13.4) is found to transformas a densityunderthe following transformationswith four Majoranaspinorparameters~,

OA,, = i~,y,,A1

= ~A1— ~A1+ E,jk,AA, (13 6)

= ~y5A1— ~y5A1— EIjk,AY5A,

= ~ + iy~V,,(A,1 + y5B11)~,+ ~i[A11— y5B,,,A1,, + YSBik]A,

which closeinto the on-shell N = 4 algebra

[3(1) 6(2)] = 2i ~ +6gauge+ field equations (13.7a)

with the parameterof the gaugetransformationgiven by

a = ~ + 2~1°(A11+ y5B11)~,(

2. (13.7b)

Just as for N = 2, the internal symmetry group is actually larger than0(N). By defining symplecticMajorana spinors, we could make the somewhatlarger symplectic group USp(4) 0(5) manifest[61,62], but by not insisting on any Majorana or pseudo-Majoranacondition and going to a chiralnotation,we can actuallyshow a yet larger SU(4) symmetry [5]. In order to do this, we break A, intochiral partsA,,

1 andA~. (A,,1)~, and definea complexmatrix of scalars

AA’ _1/,4IV.! if = 2~/~ij~r I

with a reality condition

(M11)t = ~ M”. (13.9)

160 Martin F Sohnius,Introducingsupersyininetry

The N = 4 Lagrangian

L = tr(_~F~~F~+ iA1u”V,,A’ + ~V,.M11V”M”

+ iA1 [A1,Pv.t”] + ik’[k’, It,1~,,]+ ~[J%i~,,T~4’k,][M11,P4k!]) , (13.10)

is then manifestlySU(4) covariant,as are the transformationlaws

= i~u,.A’— iA1o~’

— r~ ~0JV11J — ~,/tf — ~~f/tjr E,1k,1, /‘.

= ~ + 2io~V,,M,1~’+ 2i[M,,, M” ]~,,.

The internalsymmetryis SU(4), not U(4). Therecan beno furtherU(1), sinceany phasetransformationon M,1 is ruled out by the reality condition (13.9), andany phasetransformationon A, would requireoneon M~as well in orderto renderthe Yukawacoupling termsinvariant.

In order to makethe model interactingat all, the gaugegroup must be non-Abelian.There is noknown off-shell formulation for the non-Abeliantheory in terms of unconstrainedfields (but comparerefs. [61,62]).

13.2. Finiteness

The fact that bosonicandfermionic loopstendto contributeto quantuminfinities with oppositesignshasbeenknown since the 1930’s. Only in 1974, however,this was effectively put to work for the firsttime to reduceif not eliminatethe problemof quantumdivergences:if the fermions andbosonsin amodel stand in a well definedcontrolledrelationshipto each other, i.e., if thereis a supersymmetry,certaingroupsof divergentdiagramswill cancelandsomeof the infinities do not appear.Judgingfroma footnote in ref. [72], thispossibility was pointedout to Wessand Zuminoby the late BenjaminLee,andindeedsuch cancellationswere then foundin the Wess—Zuminomodel.

Thesecancellationsare now understoodin termsof a non-renormalisationtheoremin superspace:any invariant which can only be written as an integralover chiral superspacebut not over the wholeofsuperspace,doesnot receiveany renormalisationcorrections[30,31]. Thus,whereasthekinetic termofthe Wess—Zuminomodel does receivesuchcorrections,

J d4x d20d2OqSqS-~ Z

4, J d4x d20d2O/x~, (13.12)

the massandinteraction termsdo not,

d4x d20~2 + h.c.—~-~-Jd4x d2O4Y+ h.c. (13.13)

~Jd4xd2O~3+h.c._~Jd4xd2O~3+h.c., (13.14)

so that massandcouplingconstantdo get renormalised,


m—~Z~1m and g—*Z~213g, (13.15)

but, since.4,, is only logarithmicallydivergent,their renormalisationis only logarithmically divergent.Ina genericfield theory of spin 0 and~ particles,without supersymmetry,we would expect a quadraticdivergencefor the mass and a linear one for the coupling constant(as well as quartic ones for thevacuumenergy— theseare alsoabsentheresincetheywould breaksupersymmetry).

Not having gone into the field of superspaceperturbationmethods(there is an exhaustivebookabout the subject which contains 1001 formulas and no references[26]) this non-renormalisationtheoremfor chiral superspaceintegralsmust remaina pieceof magicwithin this report. A secondpieceof magicwill be the non-renormalisationtheoremfor thefactor e2gV.This follows from the backgroundfield methodfor renormalisinggaugetheories,and has the effect that a super-Yang—Millsmodel withno matter hasonly one renormalisation,the logarithmically divergentZ~,and the coupling constantrenormalisationis relatedto it:

g—~Z~!’2g. (13.16)

A very early investigationwas concernedwith the problem of how many matter multiplets could becoupledto a Yang—Mills multipletbefore asymptoticfreedomwas spoiled,i.e., before

I3(g2)=,1i_~_zg (13.17)

had a positive slopeat g = 0. Quick calculationshowed[18] that up to two chiral multiplets left /3’(O)negative,that for four andmorechiral multiplets it waspositive, andthat for threechiral multiplets thequestioncould not be decidedat the one-looplevel: /3’(O) = 0. Theconditionfor asymptoticfreedomisthat /9 be monotonicallyfalling at g = 0, rather than that /3’ < 0; so a higher order calculationwasrequired. It was found that the result dependson more details than just the number of mattermultiplets: the representationsof the gaugegroupin which theysit andtheir t/i~ self-couplingtermplaya role as well.

Specifically,however,it wasfound that for the N = 4 theory (which is, afterall, an N = 1 theory withthreematter multiplets) the /3-function remainedzero up to threeloops [52, 40, 2, 32, 7] and thatthereforetherewereno divergentgraphsat all up to that order! This naturally led everyoneto suspectthat this mayactuallybe a finite field theoreticalmodel in four space—timedimensions,andargumentsweresoughtto prooffinitenessto all orders.

Theseargumentswere forthcomingfrom threedifferentdirections.The first was followed by Ferraraand Zumino [20] and by West and myself [64] and exploits the conformal invariance of the theory,which will remain a valid symmetry to all ordersof perturbationtheory if the theory is finite. Con-versely, if it is not, there will be a trace-anomaly in the energy-momentumtensor—its trace isproportionalto the /3-functionand breaksconformalinvariance.Supersymmetrylinks this traceto thedivergenceof someaxial current,and if thereis no axial anomalyin N = 4 Yang—Mills theory,therecannotbe a trace-anomalyeither,andthe theory is finite. The only axial currentsin the model are theonesassociatedwith those nine generatorsof SU(4) which are not also generatorsof its real 0(4)subgroup.Thesetransform as an irreduciblerepresentation9 under 0(4). Neither SU(4)nor 0(4) aresymmetry groupsthat are likely to be brokenby anomalies.This is particularly clear for the 0(4). Soevenif oneof theaxial currentconservationlaws weresomehowbroken,all nineof them wouldhaveto


be, being irreducibleunder0(4). It was arguedthat N = 2 supersymmetryassuredthat at leastoneofthe nine would be unbroken (this can probably be shown as well by meansof the USp(4) 0(5)invarianceof the N = 4 theory). Thereforetherewould be no axial current,no trace-anomaly,andnoquantumcorrections.

The secondapproachto the problemconvincedthetheoreticalphysicscommunityof thefinitenessofN = 4. Mandelstam[45] andthen Brink, Lindgrenand Nilsson [6] showedthat finitenesswas obviousonceaparticulargauge,the light-conegauge,was chosenfor A,,.The boson-fermioncount is manifestlyrestoredin a fixed gauge,anda kind of on-shellsuperfieldformalism can beusedto write down possiblecounterterms.Thesewere found to be absent.A detaileddescriptionof this proof cannotbe attemptedhere,sincethe ground-workfor it hasnot beenlaid in this report.

The third approach,by Grisaru, Roéek and Siegel [32] and by Stelle [69], uses the non-renor-malisationtheoremandthe backgroundfield method.It can be presentedherein slightly moredetail, asit leadson from eqs. (13.12—16).We know from subsection12.5 that N = 2 supersymmetrydemandsthat the ~3-couplingconstantand the gauge-couplingconstantmust be the same.If N = 2 supersym-metry is not to be brokenby renormalisation,then this must be true to all orders,andboth g’s mustreceive the samecorrections.The possible renormalisationsof the N = 1 gauge theory with super-potential(12.34—36) are

V2 +ZvV2 ggauge—*Zgggauge

4~ ~Z4,(4~a)

2 g~3 ~ (1318)(~a)2 ~Z~(4,a)2 m -~m+Om

(!pb)2~Z,p(W0”)

2

with relationships

Z~ZV= 1, (Z~)2Z~Z46Z~= 1

(13.19)

(1+8m/m)24Z,~,= 1

betweenthem which follow from the non-renormalisationtheoremsfor e2~’andfor chiral superspaceintegrals.

N = 2 supersymmetryrequiresthat the two coupling constantsbe the sameand that the gauginofrom V andthe spinor from ~Pform a doublet,hencethat

ZgZ~ Z~=Z,~ (13.20)

which in turn implies, through(13.19), that

Z,6Z,,,= 1; Om =0. (13.21)

At this stage(N = 2 supersymmetryassumed)thereare only three renormalisationsleft, namely thewave-functionrenormalisationsfor the three matter multiplets in the N = 1 picture. In the N = 4model,everythingis andmustbecompletelysymmetricin thesethreemattermultiplets (cf. the obvious0(3) symmetryof the Lagrangian(13.2)). Hence

Martin F Sohnius,Introducingsupersytnmetry 163

= = Z,~ (13.22)

andfrom (3.21) it follows that

Z = I for all renormalisations (13.23)

andthe theory is finite.This line of argumenthasbeenimprovedin subsequentwork by Howe, Stelle andTownsend[36].A

weakpoint of the original argumentwas its relianceon N = 2 supersymmetry,which is not manifestin(13.2), in order to guaranteeeqs. (13.20).It is thereforedesirableto write everythingin a fully N = 2covariantway. In order to be able to use the superspacearguments,however, it is necessarytoformulate it in terms of unconstrainedN= 2 multiplets. The hypermultiplet is not suitablesince it issubject to the constraint (12.16). By restricting themselvesto real representationsof the gaugegroup,however,the authorscould use the real form of the hypermultiplet.This still hasa constraint,

[A,Q1]+i[A,T/Q1]=0, (13.24)

but this could berelaxed, replacingthereal hypermultipletwith a muchmorecomplicatedstructure,the“relaxed hypermultiplet”. This involves not one multiplet with a centralchargebut rather threeofthem, without a central chargebut with complicatedconstraint relationshipsbetween them which,however,can be solved in terms of unconstrainedprepotential supermultiplets.The auxiliary fieldstructureis much more complicatedthan for the constrainedhypermultiplet and thereare also many“gauge” degreesof freedom, even for the matter sector which involves only spins 0 and ~. Theadvantagesof having unconstrainedsuperfieldsto work with, and to arguein terms of, are howeverparamount.

With a real representationof the gaugegroup andhencewith the real form of the hypermultiplets,relaxedor not,the “secondSU(2)” is manifest.Sincethis rotates~ ~ (~a)t the equalityof Z,, andZ~is guaranteed,andthus, from eq. (13.21),

Z4, = Z,~= 1 (13.25)

even for a theory with only N = 2 supersymmetry.The backgroundfield formalismwhenappliedto therelaxed hypermultiplet, then showed[37] that indeed the only contributions to the only remainingrenormalisationZ~comefrom theone-loopgraphs.Thus,

If an N = 2 theory with matterin a real representationof thegaugegroup is one-loopfinite, it is finiteto all orders.

According to the resultin ref. [18], this is the casefor that particularN = 2 theory which has masslessmatterin the adjoint representation,i.e., for the N = 4 model. But it is alsotrue for whole families ofN = 2 models,someevenwith masstermsandhencewith no conformal invariance[51].

14. Supersymmetryin higher dimensions

The moreimportantmodelswith extendedsupersymmetryareintimately relatedto Lagrangianfieldtheoriesin morethanfour space—timedimensions.For flat-spacesupersymmetrythis is somewhatof a

164 Martin F Sohnius,Introducingsupersymmefry

mathematicalgimmick, but in the caseof supergravity,two particular avenuesof researchhavestressedthe importanceand possiblephysical significanceof theseadditional dimensions:the “Kaluza—Klein”programme[12] in which the gaugeinteractionsare interpretedas remnantsof gravity in additionalspatial dimensions,and the “string programme” [28] which only works in ten dimensionsin the firstplace.

It is not possible to write about higher dimensions and their role in supersymmetrywithoutacknowledgingthe lateJoel Scherk.It was he whopioneeredthe idea of employing higher dimensionsto get a grip on the complexitiesof extendedsupersymmetry.He co-authoredseveralof the importantpaperswhich laid the foundationsof the field, in particular Gliozzi, Scherkand Olive [27] and Brink,Scherkand Schwarz[5] where, amongotherthings, the N = 4 super-Yang—Millsmodel was derivedfrom a simple model in ten dimensions,and Cremmer,Juliaand Scherk[9] which first employedtheeleven-dimensionalscenarioin the searchfor the N = 8 supergravitywhich was subsequentlyworkedout in full in a monumentalpaperby CremmerandJulia.

As an exercisein getting familiar with higher dimensions,I shall in this sectioncover someof thegroundsof ref. [5], and showhow N = 2 andN = 4 supersymmetricYang—Mills theoriesfit beautifullyinto a world of six andten dimensions,respectively.

14.1. Spinorsin higherdimensions

Before we can talk aboutsupersymmetryin higher dimensions,we must know what spinorsare inhigherdimensions.Supersymmetry,after all, is by definition a symmetrywhich involvesspinorialas wellas tensorialquantities.The finite dimensionalrepresentationsof the Lie algebrasof pseudo-orthogonalgroups0(d±,d_) fall into two classes:somerepresentationsarecontainedin multiple direct productsofthe fundamentalvector representationof the group and others are not. The former are the tensorrepresentations,the latter are the spinorrepresentations.*An easyhandleon the spinorrepresentationsisprovidedby the Dirac matricesandtheir properties.Theseareirreduciblerepresentationsof the Diracalgebra

{Fa,f’b}271ab~1 , (14.1)

from whichit follows that the matrices~Iab with

1ab If’ab 1T~aTb] (14.2)

representthe algebraof the Lorentzgroupin d dimensionswith metric 7)ab;

~Zab = ~(Mab)

(14.3)[Mab, MCd] = ~(7~Mad — 7ibdMac— 7lacMi,ci + ‘qadMbc).

In appendixA.7, which containsmanyimportantdefinitionsandmustbeseenas an integralpart of thepresentsection,we learnthat the (complex)matrix dimensionof this representationis

‘with respectto their global properties,theseare very different: thetensorrepresentationsof the algebraindeedgeneraterepresentationsofthegroup O(d+,d_), thespinor representationsgeneratethoseof thecoveringgroupspin(d).


n = 2~~/2 for evendimensiond(14.4)

n = 2(~~1Y2 for odd dimensiond,

which grows exponentiallywith d. Spinor representationsare thus most certainly not “small”. Theirincreasingsize actuallyputs limits on the largestdimensionswhich can possiblysustainsupersymmetryat all: since the total number of chiral spinor chargesin four dimensionsis limited to N = 4 forrenormalisablemodelsand to N = 8 for supergravity,the total numberof real spinorial chargesmustnot exceed16 or 32, respectively.

It is therefore important to establishthe smallestpossibledimensionof a spinor representation.Equation (14.4) gives the dimension as 2n (a factor of two becausen is the number of complexdimensions).Two waysareavailableto reducethat number,chirality conditionsand reality conditions.

Chirality conditions are possible for even dimensions,where the matrix j’d+I is non-trivial. Itcommuteswith the“ab which implies that the two projections

~Iab 5~ll±VI3Fd+l)~ab (14.5)

(seetable A.2 in the Appendixfor the valuesof /3) arealso representationsof the algebra(14.3). Sincethesechiral representationsinvolve the projectionoperators~(1±\//3 Fd+l), the numberof dimensionsof the representationspacehas beenhalved. This is not possiblefor odd dimensionswhere Td+l 1.

A reality condition (Majorana condition) for a spinorwill havethegeneralform

ci’ = X~L,4 (14.6a)

with X somenon-singularn x n matrix and * the component-wiseconjugate.Since an infinitesimalLorentz transformationactson a spinoras

~i _1~,ab~’ i — 1~abr— 41/5 ‘~‘abY — 4~ 1 ab

andon the complexconjugateas

~,*.1,abr* ,*

— 4/5 Iab’P

the matrix X musthave the property

= X~FabX

in order to makethe Majoranacondition(14.6a)Lorentzcovariant.The only matriceswhich do thisaremultiplesof D andD as definedin eqs. (A.60) and(A.64):

X=D or X=I’,J1D~D. (14.6b)

Not in all cases,however,will this be consistent.Only if at leastoneof the conditions

or ñD*=ô=+i1 (14.7)

166 Martin F Sohriius, Introducing supersymmetry

is fulfilled, can we havea Majoranacondition çtakethe conjugateof eq. (14.6) andinsert).Thus tableA.4 of the Appendix,which gives thesigns of 8 and3, acquiresadditional importancesinceit showsforwhich space—timedimensionsandmetricsMajoranaconditionsarepossible:at leastoneplus signin theentry in the table is required.

A standardway of writing Majoranaconditionsis

0= C~J~ (14.8)

(or 0=F~i0c)with

(14.9)

Finally, the question arisesunder which circumstancesa chiral spinor can also satisfy a Majoranacondition. For this, we musthave

(1 ±V/3Td÷l)0=D(1±V/3*F~+1)0*

or the correspondingequationwith D. We evaluatethis andfind

= ~ = (1 ±V/3*Edfl)D,/,*.

Thereforewe must have\//3 real, i.e. /3 = +1, and3 = +1. This implies that

3 = 6= +1 (14.10)

becauseof eq. (A.66). Startingwith .i5 would haveled to the sameresult.We seethat chiral Majoranaspinorsexist exactlyfor thoseevendimensionalspace—timeswherethe entry in table A.4 is + +.

I summarisethe resultsin table 14.1. Clearly, the maximal dimensionin which N = 4 Yang—Millstheory can be expressedis d = 10, andthe maximal dimensionfor N = 8 supergravityis d = 11. N = 2supersymmetry,with eight spinorial charges,seemsto want to live in six dimensions.We considerthiscasefirst.

Table 14.1

Chirality and realityof spinorsin Minkowski space—timeswith d ~ 12

d 1 2 3 4 5 6 7 8 9 10 11 12

n 1 2 2 4 4 8 8 16 16 32 32 64Chiral spinors — yes — yes — yes — yes — yes — yesReal spinors yes yes yes yes — — — yes yes yes yes yesChiral real spinors — yes — yes — —

Minimal spinor dimension 1 1 2 4 8 8 16 16 16 16 32 64


14.2. Supersymmetric Yang—Millstheory in d = 6

Considerthe following Lagrangian in a six dimensional Minkowski space for a gauge field Aa

(a = 0, . . . , 3, 5, 6) anda chiral spinorA in the adjoint representationof the gaugegroup:

L = tr(—~FabF”+ ~iAF”Vai~t) (14.11)

A = ~(l — r7)A (14.12)

VaA = 3aA + j[Aa, A]. (14.13)

A particularrepresentationof the Dirac matricesis:

F )~/L1

j for~=u,...

r5=[° ~5]; r6=[~ ~] (14.14)

r7=r0...r6=[~ ~]; A=I’0.

First weshowtherelationshipbetweentheLagrangian(14.11)andd = 4, N = Z supersymmetry.ThechiralspinorA can be written as

A = 1~1 (14.15)l-Oj

with x an unconstrained,complex4-spinor.We can then rewrite

VaA ~XY~ V,,XXY5V5X~,~~V6X.

As a nextstep,we assumethat nothingdependson x5 andx6,

85—ô6—0. (14.16)

This is called trivial dimensionalreduction.It gives us a gaugecovarianttransformationlaw for A5 andA6 (becausea5a = 86a = 0) and

V5 6X = i[A5,6, x]

= V,,A5, F,,6 = V,,A6 (14.17)

F56 = i[A5, A6].

The Lagrangian(14.11)now dependsonly on four coordinatesx” andreads


L = tr(—~F,,~F”~+ ~V,,A5V”A5 + ~V,,A6V”A6 + ~ V,,~y

— ,~[x, A6] — ,~ys[X,A5] + ~[A5, A6]2). (14.18)

With the identifications

x = (A1 — iA2)

(14.19)A5=N, A6=M,

this is exactlythe N = 2 Lagrangianin 0(2) notation,eq. (12.4). In the calculation,a numberof termsvanishdueto symmetrypropertiesof bi-spinors.

The N = 2, d = 4 Lagrangiancan thusbe derived from a d = 6 Lagrangianby trivial dimensionalreduction.Since the former is a densityunderthe set of supersymmetrytransformations(12.6),we canusetheseas a startingpoint andtry to constructsupersymmetrytransformationsin six dimensionsunderwhich (14.11) is a density. We first define a spinor parameter~ from the two Majorana spinorparameters~, of the four-dimensionaltransformations.In analogywith (14.15)and (14.19) this is

~= ~ [~i i~~2] (14.20)

andwe evaluate

i~F,,A = ~i~,y,,A1+

i~F5A = ~i~,y5A1+ ~e11~,y5A1 (14.21)

i~f’6A= ~i~A1+

Using the reality propertiesfor the termson the right-handside,we can combinethe BA,,, ~iMand~N

of eqs. (12.6) into a single equation

6Aa = ~~f’aA— ~itj’a~.

The transformationlaw for A can be calculatedfrom ~A,andis

= —~iI”~CF,,,,— iI”5~V,,A

5 — i1”~V,,A6+ ~6~[A

5, A6].

The dimensionallyreducedLagrangian(14.18)is a densityunderthesetransformations.We expect,andindeedverify, that the full Lagrangian(14.11) is a densityunder the obviousgeneralisationof this to195 � 0 � 86:

~Aa = ~~1’aA— jAFa~(14.22)

= _~j~ab~~6


A slightly moreinvolved calculationwill give the commutatorof two supersymmetrytransformationsina form reminiscentof eq. (12.14):

[3(1) 3(2)] = 2i(C°F”~2~— ~2)FaCl))3a + 6gauge+ eq. of motion (14.23a)

with a field-dependentparameterfor thegaugetransformation,

a = —2i(~-°~r”C2~— ~2)raCl))A (14.23b)

All of this is consistentwith the fact that the N = 2 supersymmetryalgebrawith two centralchargesinfour dimensions, I

{Q1, Q1} = 2311y”P,, + 2ie11Z+2ie11y5Z’

[Q1,P,L] = [Q~,Z] = [01, Z’] = 0 (14.24)

can be recastinto a six-dimensionalform

{Q, Q} = (~+ I’7)F°Pa, {Q, Q} = 0

[Q,Pa]0, [Pa,Pb]O (14.25)

with

0 ); Ps=—Z’; P6=—Z. (14.26)V2 01-i02

In this subsection,we have seen that there is a very close relationship indeed between N = 2super-Yang—Millstheory in d = 4 andsimple super-Yang—Millstheory in d = 6. The transitionfrom thelatterto the former is achievedby the trivial dimensionalreduction195 = 86 = 0. The nextsubsectionwillbe devotedto the mattermultiplet in d = 6 andits behaviourunderdimensionalreduction.

14.3. Thehypermultipletin d = 6

The following Lagrangianfor two complexscalarsA andB andan anti-chiral spinor0 = ~(1+ F7)0in six dimensions,

L = 0aAt 19~~4+ 8aBt 19aB + ~i~iFa~0 (14.27)

is a densityundera set of supersymmetrytransformations

&4=~t/i, dB=~çfr(14.28)

= _ira 8a(A~+B~~)


which havea closedalgebraof the form

[3(1) 3(2)] = 2i(~°F”C2~— ~2)ra~(l))19 + eq.of motion. (14.29)

We could “gauge-covariantise”this in order to make the hypermultiplet interact with the gaugemultiplet of the previoussubsection.But since I haven’t given the off-shell formalism for the gaugemultiplet, this would involve changesnot only in the Lagrangian(14.27) and the transformationlaws(14.28) but alsoin (14.22). As this is unnecessarilycomplicatedand not very instructive,it may just besaid that it can be done, without any effect on the moreimportant aspectthat I now wish to discuss,namelynon-trivial dimensionalreductionon a torus.

Clearly, a trivial dimensionalreduction195 = = 0 would reducethe Lagrangian(14.27) to the d = 4Lagrangianfor a free,masslesshypermultiplet.But thereis anotherway of obtaininga sensibletheoryin four dimensionsfrom (14.27):we assumethat all fields areperiodic in x5 andx6 with periods1/m’ and1/m, respectively.At eachpoint in four-dimensionalspace—time,the remainingtwo dimensionsthenhavethe shapeof a donut,

andwecan Fourierdecomposethe fields, e.g.,

A(x~,x5, x6) = ~ exp(—in’m’x5— inmx6)~ (14.30)

The Lagrangianwill becomethe sum of Lagrangiansfor the Fouriercomponents,

I —~ At ~s’A ~ Dt )~P ~1~1 ~‘iJ_,flfl’ — t~~1nn’ ( ,,n’ VpAJ,~n’ nfl’ 41’/’nn’A ~,~‘nn’

— (n’2m’2 + n2m2)~ + ~ — ~-~— 0—nn’ysiIJfln’ +~~ . (14.31)

The fields ~‘=~‘ arethe Diracspinorsin four dimensionswhich form the bottom halvesof the anti-chiralspinors~.=‘ in six dimensions.

We see that our assumptionthat the “world” is a torus in the fifth and sixth dimensionled to aspectrumwith an infinite tower of ever more massive hypermultiplets.Their massesare, after ay5-transformationon 0:

~ \/n2m2+ n’2m’2 with n, n’ integers. (14.32)


Apart from the masslessmode, the lowestmassis M = m’ at n = 0, n’ = 1 if we assumethat m’ ~ m.The standardmassivehypermultiplet of section 12 is the casewhereonly one (massive)mode is

presentin the Fourier decomposition.Having identified the centralchargesas —P5 and —P6 in theprevioussubsection,we seethat the “multiplet shorteningcondition” (12.15),which in generalreads

~ Z~= P,,P”, (14.33)centralcharges

is just the higher dimensionalmasslessequationof motion

8at9a0, (14.34)

andthe equationof motion in four dimensions,

3~4= im/, (14.35)

is just the periodicity condition

= —im~, (14.36)

i.e., the conditionthat the world be a torus in the sixth dimension(section12 assumedm’ = 0 so that thetorus had infinite radiusin thefifth dimension).

14.4. Supersymmetric Yang—Millstheory in d = 10

A quick glance at table 14.1 will show that the maximal dimension which can sustainN = 4supersymmetry(16 real spinorial charges) is d = 10. A 16-componentspinor in ten-dimensionalMinkowskispacemustbe both chiral,

A = ~(1— F11)A, (14.37)

and Majorana,

A = CX’T. (14.38)

Such a spinorfield will describeeight fermionicdegreesof freedom,and we can againhope that a Awhich is minimally coupled to a gaugefield (which has d — 2 degreesof freedom,also 8) may be asupersymmetricsystem,just as it happenedin the caseof d = 6.

Indeed,the Lagrangian

L = tr(—~Fa6F”+ ~iAJ~aV0A) (14.39)


is a densityundersupersymmetrytransformations

SAa = ~~f’aA, ~A= _ijIab~F (14.40)

whoseparameter~ is alsoa chiral Majoranaspinor. The algebraof thesetransformationsis

[3(1) 3(2)] = 2j~l)Fa~(2)19~+ 3gauge+ eq. of motion (14.41a)

with a field-dependentparameterfor the gaugetransformation,

a = ~ (14.41b)

In working out the algebra, it is necessaryto perform a Fierz transformation,take accountof thechirality propertiesof ~ of the symmetry propertiesof FaC, FabcC and F,,bcdeC, and finally of thesecondof eqs. (A.70) for k= 1 andk= 5.

To obtain a theory in four dimensionsfrom this, we haveto perform a trivial dimensionalreduction:

r94+m0 form=i,...6. (14.42)

Reductionon a torus would lead to massivespin-i modesand thus to a non-renormalisabletheory.Indeed, while this particular model is the renormalisable,conformally invariant and finite N = 4super-Yang—Millstheory in four dimensions,it doesn’t haveany of thesenice propertiesin ten.

We expectthe six componentsA5,.. . A10to becomescalarsandthe 16 componentsof A to breakup

into four chiral spinorsA,,,. The Lorentz symmetrygroup 0(1,9) of the d = 10 theory will breakdownto 0(1,3)®0(6)= SL(2,c)® SU(4). Thereforewe can expect to end up with the SU(4)-covariantversion of the N = 4 super-Yang—Millsmodel.

The details of the dimensionalreductionarea rather involved exercisein higher-dimensionalDiracmatrix algebra.Becauseof their similarity to the dimensionalreductionsof supergravitymodels,I shallneverthelesspresentthem here.

We must first considera particularrepresentationof the d = 10 Diracmatrices.We choose

for

F4+,ny5®Fm for m=1,...6 (14.43)

with y,,, y~the standard4 X 4 Dirac matricesin four-dimensionalMinkowski spaceandtm a set of 8 x 8Dirac matrices for six-dimensionalEuclideanspace with ~Jrnn = +3mn. We also havematricesA, C(lo)andf’~~:

A=F0

C(lo) = C(4)® t~6) (14.44)

Fiiyo.~ys(ys)6®tr~f6~ys®t7.

A representationfor thetm could beobtainedfrom theV~for six-dimensionalMinkowski space,seeeqs.


(14.14),by the substitutions

= Fo; t~= iF,~ for m = 1,2,3,5,6. (14.45)

It will, however,be muchmoreconvenientto work with a set of I’m which areobtainedfrom theseby asimilarity transformation

tm = ~ with S= [~y5C]~ (14.46)

The result for the new tm ~5

tm[.~i ~m] (14.47a)0m 0

with

êi=iyiysC; ê2=iy2-y5C; ó3=iy3y5C (14.47b)

= yoy5C; 6~= —iC; = —iy5C.

Here Candthe y’s arethe standard0(1, 3) matriceswhoseexplicit form is given in appendixA.5. Allsix matrices~m areantisymmetric.Anotherpropertyof the ~m will laterbe of interest:the explicit formOf &

001 00 0 0 1

—1 0 0 0010 0

satisfiesthe equations(~~~f)*= ~(5~j1)”= ~EIJkI&lk, SU(4) covariance(or explicit calculation)ensures

this for all ~m andwe have

~ = ~ ~E1~’(ó~m)kt. (14.48)

The explicit forms of the matricesC(6) andF7 are straightforwardto evaluate:

C(6){ ~] (14.49)

[‘7= {~ 0] (14.50)

A general32-componentcomplexspinor would look like


A =(~~)with a,á1,2and i= 1,...4. (14.51)

In this notation,the matricesA, C(10) and [‘11 are nowdeterminedas

/0 0 1 0\ /—1 0 0 0(0001~ (010 0

A=~ 1 0 0 0); ~ 0 0 1 0 (14.52a)

\o 1 0 0/ \o 0 0 —1

/ 0 e,,~31’ 0 0

C(io)= (~_e~~~ô1i 0 0 ~ 6/) (14.52b)

0 0 —rt”~6/ 0

which implies

/0~~\

X’= (~‘,~ k~,~ and AC = (\~). (14.53)

The chirality condition (14.37) is now

(14.54)

andthe Majoranacondition (14.38) is

~ai = kni (As1)tE~~~~ Z~= (x~)~e~~. (14.55)

We seethat the surviving 16 componentsof A arejust four chiral two-spinorsA,,, in the representation4of SU(4).

We proceedto decomposethetermsin the d = 10 Lagrangian(14.39)underthe assumptionof trivialdimensionalreduction,eq. (14.42):

~Fa&F~th = ~ + W,,Mm V~Mm+ ~j[Mm, M~]2 (14.56a)

~iAF” VaA = ~iAF~V,,A — ~AF4±m[A,Mm]

= ~iA1o~V,,A’ +~iA1[A1, ~ m’) Mm] iA’[A”, (óm)jjMm]. (14.56b)

In the last stepwe havemadethe transition from d = 10 covariantnotationto d = 4 chiral two-spinors.


The scalarfields Mm arethe last six componentsof Aa:

Mm~A4+m form=1,...6, (14.57)

andif we define

M,1 ~(êm)ijMm, M” ~(órn1)~’Mm (14.58)

we get the relationship (13.9) becauseof (14.48) and the reality of Mm. Since there is also atrace-orthogonalityrelationshipfor the ~m,

tr ~ = 43 (14.59)

we can write

MmMm M11M

11. (14.60)

Using thison (14.56)givesus, up to a divergence,exactlytheSU(4)-covariantN = 4 Lagrangian(13.10).Clearly, the d = 4, N = 4 transformationlaws (13.11)can be obtainedfrom (14.40) in a similar way.

15. The supercurrent

At the end of section5, I introduceda supercurrentfor the Wess—Zuminomodel. This was a localMajoranaspinor-vectorcurrent J,,,,, constructedfrom the fields of the model and conservedif thosefields weresubjectto their equationsof motion. In the presentsection,I shall presenta moredetailedstudyof the supercurrent,of its propertiesandof the multiplet to which it belongs[19].

15.1. Themultipletof currents

The time componentof the supercurrent,when integratedover d3x, gives rise to the supersymmetrycharges0,,, as alreadystatedin eq. (5.10):

0,, = f d3xJo,,(x). (15.1)

The time-independenceof 0 is, as usual,guaranteedby the conservationlaw for J,, andthe assumedvanishingof surfaceintegrals.As an x-dependentfield, the supercurrentcannotstandalonebut mustbepart of a supermultiplet.The taskat handis to determinethat multiplet.

Fromthe generaldefinition of the supersymmetrytransformations,

= —i[J,,, ~Q],

we getby integrationover d3x


J d3x6J~~= -i[Q, Q~]= -2iy~~P,,.

The momentumfour-vector, on the other hand, is the d3x integral over componentsof the energy-momentumtensor (stresstensor):

P,. =Jd2xOo,~, (15.2)

and we see that the algebra of supersymmetry itself already imposes a very intimate relationshipbetween the supercurrent and the stress tensor. This relationship originates from the intimate relation-ship betweenthe energy-momentumvector and the supersymmetrychargeitself. It meansthat bothcurrentsmust be in the samesupermultipletand is of great importancefor supergravity:Einstein’sequationstatesthat the densityof energyandmomentumis the sourceof thegravitationalfield. As thisdensityis relatedto the super-chargedensityJ,,,,,we expectthegravitationalfield to be relatedto someotherfield whosesourceis J,,,,. This will be the gravitino field ci’,,~,the field of the superpartnerof thegraviton. The gravitino haszeromassfor unbrokensupersymmetryandalwaysspin ~.

In order to find the full supercurrent multiplet, we can start from the old expression (5.9) for J,,,

jokI = J(A — y5B)y,,tfr+ imy,,(A — y5B)t/i + igy,,(A— y5B)

20,

andcalculateits supervariation,usingthe transformationrules(5.6) for the fields A, B and0 as well astheir equationsof motion (5.5). The matter fields are thus taken to be on-shell. The result of thecalculationwill be an utter mess.This is due to the fact that the chosenform of J,, doesnot haveniceconformal properties.This point will becomeclearerlater, for the time being it may suffice to say thatthe particular “improvementterm” of subsection5.5 madefold look ratherpretty, but it doesnot leadto a reasonablysimple current multiplet. A “nice” supercurrentwill havethe property that, at theclassicallevel,

-y~J,,=0 if m=0,

i.e. that the “y-trace” of J,, vanishesfor zero mass.The y-traceis the component(~,0)~(0,~)in thedecomposition of a vector-spinor (~,~)® ((i, 0)~ (0,i)). For the old supercurrentwe get instead

y~Jold = —2J[(A + ysB)0]+ 2im(A —

75B)0,

with the first term violating the nice-nesspostulate.We can, however,improve .1,, by

j = j~d — 19v[cy(A + ysB)0].

As statedin subsection5.5, such improvementsby quantitieswhich are preservedindependentof fieldequations are always possible.

We proceedand calculate the supersymmetryvariationsfirst of the improved J,, and then of thefields which appearin 6)’,, and so on until the multiplet closes.The result of this rather non-trivial


calculationis the following multipletof currents:

6X= ~ 6Y —~i~y5y~J,,

&j~=~ ~

= —2iy~~O,,~+ iy~y5~(8~j~— ~,,~3PJ(~5)) + ~i~°~ ~ — iu,,,8~(X + y~Y)~ (15.3)~

1fr1’ ~ + ~UV,,p4l5t,O~~pU .‘v °vp ~

whereX, Y area scalar—pseudoscalarpair,j~is an axial vector,J,, is aMajoranaspinor-vectorand 0,,.is a symmetrictensor.In our particularcaseof the Wess—Zuminomodel,thesefields are givenin termsof the matterfields by

m 2mX=-~-(A2-B2), Y=—~--AB

j~=~Aa,,B+~iciiy,,ys0

2i (15.4)J,. ‘,19’(A— ysB)y,,tfr+imy,,(A—y

5B)~+igy,,(A—ysB)20—-~-8”[er,,~(A + ysB)0]

= 8,,A 8~A— ~(0,,8~— ~,,~L1)A2+ 8,,B 0,,B — ~ — ~,,,L1)B2+ ~icii(y,,8~+ y~19,,)t/i—

with the L in 0,,,. beingthe Wess—ZuminoLagrangian(5.4).Quite independentlyof the particularrealization(15.4), the fields of (15.3) form a representationof

the supersymmetryalgebra,providedonly that the conservationlaw holdsand that the stresstensorissymmetric:

= 0”O,,,,. = 0 (15.5)

0,,,. = 0,.,,.. (15.6)

Both propertiesare preservedby the transformationlaws and are guaranteedfor the particularrealization(15.4) in termsof on-shellmatterfields.

15.2. The “multiplet of anomalies”

The supercurrentmultiplet (15.3) is not irreducible.The fields X and Y, the y-traceof J,,, the traceof 0,~,,.and the divergenceof j~transforminto eachotherand thusform a submultiplet.Closerstudywill revealthat this is a chiral multiplet with components

4, = (X, Y; —~iy”J,,; ~ 8~j~) (15.7)

in the notation of eqs. (4.4) and(4.5). In the particularrealization(15.4), all componentfields of thismultiplet will disappearwhen m = 0. This is, of course,only a classicalresult: the trace of the stresstensoraswell asthe divergenceof a chiral currentareexpectedto receiveanomalouscontributionsevenfor a masslessmodel. Provided that the supersymmetrycurrent itself has no anomalies, i.e. that


= 0 remainstrue, theseanomaliesmust sit in a multiplet of supersymmetry,a “multiplet ofanomalies”, of which the chiral multiplet (15.7) is an example. Being part of the same irreduciblemultiplet as 0,,~and~ the y-traceof the supercurrentwill get anomalouscontributionsas well. The“nice-ness”condition of the previous subsectionwas nothing but the demandthat the multiplet ofanomalies vanish for zero mass at the classical level.

In the caseof

(15.8)

the multiplet (15.3) reducesto a smaller,irreduciblemultiplet, the conformalcurrentmultiplet

= —~y5J,,

6J,, = —2iy~~0,,,.+ iy~y5~8,.j~+~ y%9~j~ (15.9)

60,,,.= ~ + ~

which representssupersymmetryif all quantities are conserved, if J,. is y-tracelessand if 0,,,, issymmetricandtraceless:

0= 19~j~=8”J,, = a’~0,,,. (15.10)

0 y”J = 0,,,,— 0,.,, = 0,,”. (15.11)

As suggestedby the absenceof atracein thestresstensor,the multiplet (15.9)with conditions(15.10)and (15.11) is closely related to conformalsymmetry. The multiplet of conformal currentsis actuallyunique [68], but thereare many waysof introducinga trace for 0,,,. other than by a chiral anomaliesmultiplet [65,66].

15.3. The charges

The conservationlaws for J,, and 0,,,. allow to definetime-independentchargeoperatorsas integralsof J0 and Oo,, over 3-space,eqs. (15.1) and (15.2). The transformationlaws (15.3) allow to actuallycalculatethe algebrafor thesecharges.Thus we get, as expected,

[0, ~] = i J d3x 6J

0

= 2y~Jd3x0

0,,— ~ J d3x (8,,j~—

eo”~y,,~J d3x8PJ~+ oo,,~Jd3x8~X+ Oo,,Y5~J d3x19” Y

= 2y”~P,,.

The termswith j~,X and Y all vanished because we assumed that surface terms are zero,


Jd3xo’(anything)=0 for i= 1,2,3. (15.12)

Thus the term 0,,j~— ~o,,8~’j~can only contributefor ~ = ~.‘ = 0 when it vanishesanyway.The othertermsinvolve only spatial derivativesin the first place.In a similar fashion, it can be shown from 60,,,,,that [0. P] = 0.

Thesymmetryof 0,,,, allows the constructionof a further conservedcurrent as a first momentof 0,,,.:

~ with 8”m,,,.~=0. (15.13)

If we define

f d~xm,,,.0, (15.14)

we find

[M,,,.,~Q] = if d3x (x,, 60,.

0—x,. 60,,~)

= —~Jd3x [x,,o’,.,,8”J + x,,o-0~19PJ,.— (~*-* p)].

To evaluatethis requiressomealgebra.Using the conservationlaw for J,,, we first write

x,,o,.~I9”J = x,,o~,.030f0 + x,,o,.18~J~= x,, 8’(o~,.1J0— o,.~f~)

x,,a0~8~J,.= x,,o0~19’)’,..

The spatial derivativesnow do contribute since they act on the explicit factorsof x when partiallyintegrated,and we get

[M,,,~~0] = —~!Jd3x m~1(u,.oJj - o,.1J0— ti~~J,.)— (~t **

= —~CJd3x (~,.0J,,— ~,,Jo— uo,,J,.)—~ v)

= —o,,,.Q,

exactlyas expectedfrom eq. (4.2).In the caseof the multiplet of currents (15.3), 0, P,, and M~,,. are all the chargesthat can be

constructed.They arejust the symmetriesof N = 1 Poincarésupersymmetry.In particular,thereis nochiral R-charge.This is readily understandablein terms of the currents, as there is no conservedaxial-vector.For the massiveWess—Zuminomodel itself, we can derive the absenceof R-symmetrydirectly: according to eqs. (7.27), (7.28) and (7.45), the F-componentof a superfieldwill only be


R-invariantif the superfieldhasR-weightn = —2. Thus,the massterm m(4,2)F andtheinteraction termg(4,3)F can neverbe simultaneouslyR-invariantbecause4,2 and4,3 cannotboth haven = —2.

In the absenceof the massterm,however,it is the shortercurrentmultiplet (15.9) that is associatedwith the model, at least at the classical level. This does have a conservedchiral current and acorrespondingcharge

R Jd3xj~ (15.15a)

with

[R,~Q]=ifd3x6J~= —i~y5Q, (15.15b)

as requiredby (4.2). The explicit form of j~gives, with the canonicalquantisationrules for the matterfields,

[A + iB, R] = ~J d3x [A + iB, AE — AB+ ~i

4’iyoy50]= ~(A+ iB)

which correspondsto a chiral weightof n = —~for the mattersuperfield,seeeqs. (7.27) and(7.28). ThesuperfIeld4,3 then hasn = —2, as requiredfor the invarianceof the interactionterm.

Apart from R, the conformal currentmultiplet givesrise to furtherchargesbecausethe tracelessnessconditionsin (15.11) allow the constructionof furtherconservedmoments,namely

d,, = x~0,,,,, with a”d,, = 0

k,,,~=2x,,x~0,.~—x20,,~with a”k,,

0=0 (15.16)

s,, = ix”y,.J,, with t9”s,, = 0.

The new chargesare

D =Jd3xdo

K,, =Jd3xk,~o (15.17)

5,, =Jd3xso~~,

and their algebraicpropertiescan be evaluatedin a mannervery similar to the example of [0, M,,,,,]given above.The resultwill be the algebra ofN = 1 conformalsupersymmetry,to be discussedin moredetail in the next section.


15.4. Thesupercurrentsuperfield

We now turn our attentionto the superspaceformulation of the currentmultiplet. In order to derivea superfieldform for the multiplet (15.3),we first study its representationin termsof matterfields (15.4)and translatethat into superfieldlanguage.We find

X+iY=~m[4,2]9g,,,o

= ~ ~[D,,4,.D,,4]o=e,.,o (15.18)

andthe secondequationsuggeststhe definition of a real superfield

V,,~~D,,4,Dá4,~2~4,J,,~4,, (15.19)

~z (5)... _liij \ —50 LIIaL ULthJ~ — 6~V ,,áJO=6=0.

The componentsX + i Y and j~have the same dimension so that each could be the lowestcomponentof unrestrictedsuperfields.The transformationlaws for j~,X and Y, however,arerelatedby thefact that y”J,, appearsin all of them.This will constrainthe two superfields,andwe now proceedto determinethe exactnatureof that constraint.

A little bit of experienceteachesthat thechiral projectionof y”J,, will be proportionalto the lowestcomponentof the superfieldD” V,,,,. We evaluate, from (15.19) and using the chirality conditionD4, =0,

= _D~D,,4,D~~q5+D,,4,D24,_2i4,j,,~Da4,.

This can be furtherevaluated,againusingthe chirality condition:

-D~D,,qSD~’4,= -{D,,, D~}4,D~çb= -2i I,,,,qSD”çl,

— 2i4,/,,~D”4i= 2i,~’,,~q5D”4,—i4,J,,,.Dr~q5—~D~{D,,,D~}4,.

The last term in thisis

_~4,Da{D,,,Da}4, = ~4,D~D,,D~4,~4,{D~,D,,}D”q5- ~q5D,,D24,

= i4,X,,~D~4,-~4,D,,D24,.

Collectingterms,we find

= D,,4,D24,-~4,D,,D24,.

We nowuse the equationof motion (8.20) andfind very straightforwardly

D” V,,,, = m D,,4,2. (15.20)


If we evaluatethe 0-0-independentcomponentof this, we find agreementwith the propertiesof 6j~,FiX and6 Y

Let us, as abstractlyaspossible,considerthe propertiesof eq. (15.20).In the absenceof amassterm,the right-handsidewill be zero,

Da V,,~ = 0. (15.21)

We can evaluatethis conditionwithout recourseto anyrealizationof V,,,. in termsof matter fields andfind that (15.21) is completely sufficient to define the conformal current superfield with componenttransformationlaws as in eqs. (15.9). In the presenceof a chiral anomaliesmultiplet,

D” V~= DaS with DaS= 0, (15.22)

we get insteadthe Poincarésupercurrentsuperfield with componenttransformationlaws asin eqs. (5.3).

In thecaseof the gaugemultiplet, the supercurrentsuperfieldis given byV,,,. = —l2tr WaWa. (15.23)

The property(15.21)then follows from the constraints0 = V~W,, + VaW’~= Va Wa andthe equationofmotion V~W,.- Va W = 0.

16. Conformalsupersymmetry

In section 15, we saw that the multiplet of currentsis particularlysimple andshort if no massesarepresentin an N = 1 supersymmetricmodel,andif we allow ourselvesto ignore possibleanomalies.Thepropertiesof the currentsaresuch that we can constructconservedchargesR, P,,, M,,,,, D, K,,, 0,, andS,, from them. In the presentsection,we shall explorethe algebraof thesecharges[71],generalisetheresultsto N> 1 [11,34],andgiverepresentationsfor the N = 1 superconformalalgebraon fields [71,58].

16.1. Deriving the N = 1 algebra from thecurrents

Assumethe only conservedcurrent to be a symmetricenergy-momentumtensor 6,,,,. with non-zerotrace. The only chargesare P,. and ~,,., defined by (15.2) and (15.13—14).Assumefurther that theaction of P,. on fields is given by the derivative, eq. (3.42b), and that surfaceterms vanish in thed3x-integrals,eq. (15.12). Thenwe can showthat

[P~,P~]=Jd3x[0o,,,P~]=ijd3x19~6o,, 0

[M,,,.,P~]= i J d3x (x,. i3,,6,,~— x,.a~6,,~)= i~,.,,P,,—

The intermediatestepsare very similar to thosedisplayed after eq. (15.14). The [M, M]-commutatorfollows from the Jacobiidentities,andwe haverederivedthe algebra(2.18) of the Poincarégroup.


Next we assume0,.,. to be traceless.This introducesthe new chargesD and K,,, viz. (15.16) and(15.17).LorentzcovariancerequiresD to be a scalarand K,, to be a vector.By explicit calculationwefind

[P,,,D] = —if d3xx~8,,00,.= iP,,,

againwith intermediatestepsas before.Knowing that no furtherchargescan be derivedfrom 0,,,,, wecan postulatethat the algebramust closeon the 15 chargesP,,, M,,,,, D and K,,. Surprisingly enough,this closure requirementtogether with what we already know about the algebra is sufficient todeterminethe remaining commutators[P, K], [K, D] and [K, K] from the Jacobi identities alone,without further recourseto the currents.The result is the algebra of theconformalgroup

[M,,..,Mpo.] = j(~,,pM,,g— i~,,pMm,— ~ + ii,,wM,.p)

[P,,,~ = ~ —

[K,,,M,,,~]= i(~j,,,,K,7— ~,~K0) (16 1)

[1~.,,D] = iP,,; [K,,, D] = —iK,,

[F,,,K,.] = 2i(~,,,,.D—

Thereis a rescalingfreedomfor K,, which wouldchangethe factorof 2 in the lastequation;as it stands,the normalisationis compatible with our definition of K,, from the current. A vanishing [P,K]commutatorwould also be compatible with the Jacobi identities, in that casethere would be anarbitraryreal factorin [K, D]. With thisexception,the algebraof the conformalgroupfollows from theratherweak assumptionsmadeabove.

Next we include into our analysisthe conserved,y-tracelessMajoranasupercurrentJ,,. We havealreadyshownin subsection15.3 that the charge0, definedby (15.1) from J,,, completesthe algebraofthe Poincarégroupto that of N = 1 Poincarésupersymmetry.The relationshipof supersymmetrywiththe conformal algebra (16.1) is completelydefined by the commutator[0, K,,]. which after somecalculationis found to be

[1Q,K,,]= _iJd3x (2x,,x~60,.o—x280,,o)=~y,,S,

with S the chargedefinedin (15.17) from the first moments,, of J,,. Thecommutator[0, K,,] doesnothavea y-tracelesspart.This propertycan be written as

[0, K,,] = ~y,,y~[Q, K,.] = y,,S, (16.2)

andis actuallysufficient to determineall relationshipsof the algebra ofconformalsupersymmetry,oncethe Poincarésuperalgebraandthe conformalalgebraaregiven. The calculationcan be donesolely bymeansof Jacobi identities, without useof the current multiplet. It leads to the following algebra


relationshipsbeyondthoseof (16.1):

[0, M,,,,] = ~o,,,,.Q; [S,M,,,,] =

[Q,D]=~iQ; [S,D]=—~iS

[Q,P,,]=0; [S,P,,]ry,,Q

[Q,K,,]=y,,S; [S,K,,]=0 (163)

[0, RJ= iy5Q; [S,R] = —iy5S

[R,M,,,.]= [R,P,,]= [R,D]= [R,K,,]=0

{Q, 0) 2y”P,,; {S, ~ = 2y”K,,

{S, Q} = 2iD + ~ + 3iy5R.

The S is defined in (16.2) as a certain multiple of the y-trace of [0, K]. The normalisationhere iscompatiblewith the definition in (15.17). Similarly, the chargeR can be seenas definedby the lastequationin (16.3). Its commutatorswith the otherchargesarefixed by Jacobiidentitiesandwe cannotsetR = 0, sincethe [0, R] commutatoris not zero,but we could arbitrarily rescaleR. The chosenscaleis the onecompatiblewith the alternativedefinition via the chiral current,eq. (15.15).

16.2. More aboutthe conformalgroup

Four-dimensionalMinkowski spaceis a cosetspaceGIH not only of the Poincarégroupbut alsoofthe conformalgroup, with H the subgroupgeneratedby M,,,., D and K,,. We can thereforeuse thetechniquesof section7 to determinethe action of conformaltransformationson Minkowski space[43].The cosetcan be parametrizedas in eq. (7.13) andwe get the samerelizationfor Lorentz rotationsandtranslationsas before, eqs. (7.14). For infinitesimal D-transformations(dilatations) and K-trans-formations(specialconformaltransformations)we get

(1 + irD) ~ = e~~°elx~’(1+ ieD)e~~°= e~”(1 + irD) + 0(e

2)

(1 + ia”K,,) etx~O= exp{i(x~+ 2x ax” — x2a” )P,,} (1 + ia~K,,+ 2ix . aD — 2ix”a~M,,,.)+ o(a2).

The equivalentsof eqs.(3.42) and(7.8), namelythe infinitesimal version of the action of the generatorson fields over Minkowski space,arethus

[4,,P,,] =i8,,4,

[4,,M,,,.] = i(x,,19,.— x,.8,,)4,+ ~ (164)

[çb,D] =ix”19,,çb+izlçb

[4,,K,,] = i(2x,,x~ t9—x219,,)4,+(K,, +2ix,,L1 ~

The matricesi’,,,., ui and #c,, act on indices of 4, and are a finite-dimensionalrepresentationof the


algebraof the “little group” H. Since [~,~�,,,.J= 0, we know from Schur’s lemma that i must be anumberif the I,,,. generatean irreduciblerepresentationof the Lorentz group. This numberis thedimensionof the field 4,. This dimensionhas alreadycomeup on severaloccasions,first in subsection4.7.

Whereasthe coordinatetransformationsassociatedwith the Poincarégroupleavethe line element

ds2 ii,,,, dx” dx~ (16.5)

invariant,the dilatationsandspecialconformal transformationsrescaleit in an x-dependentway,

ds2-+o(x)ds2. (16.6)

Anglesbetweenlines are preserved,hencethe name“conformal”. The 15-parameterconformalgroupwhich we encounterhereis actually themostgeneralconnectedgroupof coordinatetransformationsinfour dimensionswith property(16.6).

The conformal group is the pseudo-orthogonalgroup 0(2, 4). This is easily seen by definingadditional “Lorentz” generators

M,,5~(P,, —K,,); M,,6ns~(P,, + K,,); M56=—D (16.7)

andarrangingthem togetherwith the .M,,. into an antisymmetric6 x 6 matrix M,,6. The commutationrelations(16.1) can then be rewritten in a single equationof the form typical for the algebraof anorthogonalgroup:

[Mab, Med] = ~(7ii,cMad — ?lacMbd— hibdMac + ~ (16.8)

The metric elements1)ab are the usual n,,,., augmentedby the new non-zeroelements

—,~ = = +1, (16.9)

which makethe metric that of 0(2, 4).A spinorrepresentationof this algebrais easilyfound, after all the groundworklaid in appendixA.7

andsection14. The 0(2,4) Dirac matricesarethoseof (14.14)but with [‘6 multiplied with ito accountfor thedifferencebetween0(1, 5), the six-dimensionalLorentzgroup,and0(2,4), the four-dimensionalconformalgroup.We thentake therepresentation~ab, definedin (14.2) and(14.5),andcall it ~Oab. Wefind that the u,,

6 arethe standard4 x 4 matriceso-,,,., augmentedby

u,,5— iy,,,y5, u,,6— y,,; u56— )‘~. (16.10)

Since all of theseare tracelessand havethe property yo(o.,,6)ty~l= O~ab,with two eigenvalues+1 andtwo eigenvalues—1 for Yo, theygenerateSU(2,2), the coveringgroupof 0(2, 4).

16.3. Conformalspinors

This spinorrepresentationof the algebra0(2, 4) is four-dimensionalandcomplex.In the languageofappendixA.7, it is a representationwith a chirality condition. We could also haveMajoranaspinors


with eight real components,but we cannot havespinorsthat areboth Majoranaand chiral since theentry in table A.4 for d = 6 and d_ = 4 is + — and not + +. At least eight real componentsare thusrequiredfor a conformal spinor,which explainsthe necessityfor the additionalchargesS,, in theN = 1superconformalalgebra.

A conformalspinor can be constructedfrom the chiral compo’nentsof the Lorentz spinors0 and Sin the following way [16]:

(16.11)

Like always when a covariant quantity has been properly defined, the algebra (16.3) simplifiesenormouslyand can actuallybewritten in threelines,usingthe i’s:

1 —

[I, Mat,] = ~o,,t,~ [~,Maj,J = —2~~°ab

[I, R] = .~‘; [.~,R] = —~ [Mae,,R] = 0 (16.12)

{2,.~}={~}=0; {Z,~}~O”Mab~3R.

The constraint(16.2)can nowbe rephrasedin termsof representationsof SU(2,2): thereareno chargespresentwhich are in the representations20 or 36 of SU(2,2). Hence the commutatorof .~ with M,,,,which in general could decomposeas 4®15=4020*c~36only contains a 4, namely the 1~’sthemselves. This property connects constraint (16.2) with the argumentsof ref. [34] and of section 2,becausethe anticommutatorof a 20 with the 20* or of a 36 with the 36* would necessarilycontainnon-trivial tensorrepresentationsof SU(2,2) other than the 15 and thus must be zero (this is anextrapolationof theColeman—Mundulatheorem).The positivemetricassumptionfor the Hilbert spacethen meansthat the 20 andthe 36 must be absent.

16.4. Thealgebra of extendedconformalsupersymmetry

Just as extendedPoincarésupersymmetryhasseveral chargesQm, so extendedconformal super-symmetry [11,34,58] hasseveral.~,,i = 1, . . . N. Stretchingthe Coleman—Mandulatheoremsomewhat,we can rule out bosonicchargesin non-trivial representationsof the conformal groupother than theconformal generatorsthemselves.This reducesthe possibleextensionsof (16.12) to

~ lvi,,,,] = 2u,,b.~I, [�‘, MabI = —~~Uab

— — — (16.13)~.�.,,.~}= {.~“,.i’} = 0; ~ 2~’}= 3/ u”~’Mab— 4B/.

The last equation defines chargesB/ as the most general right-hand side compatible with theassumptionsabout the structureof the bosonicsubalgebraand with conformal covariance.The Jacobiidentities fix the remaining commutators

[BI, Mab] = 0

[.L~BJ’I = 3k 2~—~ .~,; ~ By”] = _~i ~k + 13k ~ (16.14)

[B/, Bk’] = 3/Bk1 —3k5B,’,


andfrom the Hermiticity propertyof the last of eqs. (16.13)weget

(B/)t = B1’. (16.15)

Thesefix the tracelesspart of B,’ to be the generatorsof SU(N). The non-zerocommutatorof I withall tracelesscombinationsof B/ meansthat the internalsymmetrygroupmust containthe full SU(N),not just asubgroup.For the tracepart we define

R~ B,1 for N�4 (16.16)

4—N

andfind

[I,, R] = I~ [I’, R] = —1’. (16.17)

The internalsymmetryalgebrais thusthat of U(N) for N� 4.For the specialcaseof N= 4, I alreadymentionedthe absenceof a chiral scalarchargeat the endof

subsection13.1. This now makessense,sinceB,’ commuteswith all l’s andmaythusbeabsentin the{I, I) anticommutator.The algebrawould actually allow such a trace term in {I, I), but it mustcommutewith the .X’s. ElsewhereI haveshown [67] that this trace term is absentif and only if thelowesthelicity in the representationmultiplet is A

0 = —1, i.e. for the Yang—Mills multiplet. ConformalN= 4 supergravitywill havesucha term.The algebrawould also allow an R-chargefor N= 4 which isan outerautomorphism,i.e. which is not generatedby the anticommutators.

16.5. Multiplets of N= 1 conformalsupersymmetry

Already in Wess and Zumino’s first paper [71], the transformationsof the chiral multiplet (4.4)andof the generalmultiplet (4.6) weregiven not only for Poincarésupersymmetrytransformationsbutalsofor conformal ones.In the presentsubsection,thesewill be rederivedin a moregeneralcontext.

Since the commutatorof 0 with K,, gives the conformal spinor chargeS, it should be possibletomake a superconformalmultiplet out of every multiplet of Poincarésupersymmetryby postulatingsuitable transformationlaws of the fields underK-transformations.Thesewill be of the form given ineq. (16.4),andinitially therewill be somefreedomof choicefor the“little group” representationK,,, ii,I,,,.. Most importantly,however,thereis the constraint(16.2) which must be fulfilled by the 0 andtheK-transformations,namelythat a y,. can be “split off” from the commutator[0, K,,]. We shall seethatthis will restrictthe possiblelittle grouprepresentationsfor the fields of a superconformalmultiplet.

We start by imposing as few restrictionsas possible on a general multiplet of superconformalsymmetry.A (complex) field C is taken as the starting point of the multiplet, just as in subsection 4.2.Poincarésupersymmetrytransformationswill generatea multiplet

V=(C;x;M,N,A,,;A;D) (16.18)

out of C, with no reality restrictionson any of the fields. Let us assumethat C transformsunderR-transformations,dilatationsandLorentztransformationsas


[CR] =ñC

[C,D] =ix~19C+iziC (16.19)

[C,Al,,,.] = i(x,,3,. — x,.B,,)C+ ~�,,,.C,

and let us imposethe restriction that the matrix representationI,,,, of the Lorentzgroup (this actsonunwritten indices of C) is irreducible. Because of [I,,,,,LI] = 0 and Schur’s lemma,this meansthat LImust be a number. Since all fields of the multiplet can be derivedfrom (multiple) commutatorsof Cwith 0, andsincewe know that [0, D] = ~iQ,we can actuallycalculatethe representationU of D forthe whole multiplet. The result is a diagonalmatrix which actson a column vector madeout of thecomponentsof V as given in (16.18):

LI =~1+diag(0,~,1,1,1,~,2). (16.20)

The matrix K,,, the representationof the specialconformalchargeK,,, musthave

[K,,, LI] = —K,,. (16.21)

This, togetherwith (16.20), restrictsK,, to the form( 0 0 0 0 0 0 00000000*000000* 0 0 0 0 0 0 (16.22)*0000000*0000000***00

with non-zeroentriesonly wherethereare stars.In particular,

K,,C K,,,x=O. (16.23a)

Dimensionalanalysiswill showthat K,,M and K,,N should havethe samedimensionas C. Sincewe donot allow non-local operatorssuch as ~ thereis no way of constructinga vectorof that dimensionfrom the fieldsof our multiplet, andweconcludethat

K,,M K,,N=0. (16.23b)

The remainingfields A,,,, A andD canandwill haveK-terms. Sincewe now know the LI andthe .1,,,, forall fields of the multiplet (I,,,. is I,,,. plus the vector representationfor A,, or the spinor representationfor x and A), andknow quite a bit aboutK,,, we can plungeinto derivingthe S-transformationsfor themultiplet from (4.7) and(16.4). This is a lengthy calculationwhich cannotpossiblybe repeatedhere.Itconsistsof evaluating

[..,[Q,K,,]}=[.., Q},K,,]—[..,K,,], 0)


for all fields of the multiplet and of imposingthe constraint (16.2) at eachstage.This resultsin theremainingK-terms

K,,A,. = —e,,,.~°C+3,~,,,.C

3i~K,,A = — zIy,,~— Y,,.YsX (16.23c)

= —2iLIa,,C— 3i~A,, — 22~,,,.Ô~C + ~

andthe transformationlaws for the multipletundercombinedtransformations

&V=—i[V,CQ+ëS]. (16.24)

We usethe convenientabbreviations

~— ix”y,,e; X~ LI — y~±~ (16.25)

andfind

= ‘tfl’5X

= (M+ y5N)~— iy”~(A,, + ys19,,C)~+ 2X~y5eC

FiM ~

FiN = ~jy5(A— ilx) — evsXX +2EY5X (16.26)

= iijy,,A + 3,,(~~X)+~= —iu”~~0,,A,.— y

51~D— X~(M—ysN)e+ iy”X~(A,,+ y58,,C)e

FiD = —i~IysA+ 2ëy5X(A— ~iJx).

We can nowtry to imposereality and chirality conditionson this multiplet. If C is to be real,

V= %T’, (16.27)

thenwe must have

i~=0 and ,,.=—(~,,,.)t, (16.28)

i.e., I,,,,. mustbe the generatorof a real representationof the Lorentz group.The transformationlaws(16.26)arethenconsistentwith the usualreality (Majorana)conditionson the fields.

A chirality conditionconsistsof imposinga constraint

(1—iys)~0. (16.29)

Thiswill be consistentonly if


(1—iys)X~=0,

i.e., if

= —~LI and I,,,. = ~ie,,,.,,,,I”. (16.30)

This meansthat a chiral multiplet mustbe in a self-dualrepresentationof the Lorentzgroup(it can onlyhave “undotted indices”) and that its chiral weight must be —~ times its dimension. With theseconditions,(16.29)is consistent,togetherwith

0=M+iN= A,, —iô,,C= A =

andwe areleft with the fieldsof the chiral multiplet:

= (A, B; 1,11; F, G)

C= A+iB; —~i(1+iys)~0; M —i(F—iG). (16.31)

If thereareno externalLorentz indices,wecan write the transformationlaws for the chiral multiplet inthe usualform involving Majoranaspinors:

=

FiB = flY5O

64’ = —(F+ ysG)’q—iJ(A+ y5B)i~—2LI(A— y5B)e (16.32)

FiF =i~J4’—2(LI—1)e4’

FiG = i~jy~J4i+2(LI— l)~y5çli.

The canonicalvalue LI = 1 is clearly singledout: only then can we start a new chiral multiplet with Fand G, the kinetic multiplet, andonly then doesthe chiral weight havethe value —~ which is requiredfor invarianceof the super-4,

3interactionterm,cf. the commentsafter eqs. (15.15).Another interestingcaseis that of

X’O i.e. LI =,~=±,,,,=0, (16.33)

when all K-terms (16.23) vanish and the curl-submultiplet is also a submultiplet under conformaltransformations.

In the caseof

X = 2, i.e. LI = 2, ,~= I,,,,. = 0, (16.34)

we havea chiral submultipletwhich startswith Al andN. With this set to zero,we get the conformallinear multipletwhereA,, musthavedimension3 in order to be conserved.

Let us now turn our attentionbackto the conditions(16.30) for the existenceof a chiral multiplet,and let us try to understandthem in a more fundamentalway than just by seeingthem appearasconsequenceof a lot of algebra.


In the caseof Poincarésupersymmetry,chiral multiplets existedwith arbitrary additional Lorentzindices; in otherwords, the chirality constraintD4, = 0 was covariantfor arbitrary representationsofthe “little group”, which was the Lorentz group.This is not so for conformalsupersymmetry,as couldbe seenin several ways which involve treatment of conformal supersymmetryin superspace.Thesimplestway seemsto be to considerthe shifted superspacesof eq. (7.47). It is clear from (7.48) that achiral superfieldis independentof 0 in a 1-type superspace.Indeed,it would havebeenpossible fromthe beginningto usea superspaceG/H with H generatednot only by M,,,. but alsoby 0~~.This cosetisparametrizedby x” and 0~aloneand is called chiral superspace.For a general superfieldover chiralsuperspace,the action of 0 would be given by

[4,(x),~,,}= 2(Ou”),. ~9,,,4,+ 7~4, (16.35)

anda chiral superfieldis onefor which the matrix representationfor 0 vanishes:

(16.36)

Justas theconformalgroupcould be realizedon Minkowski space,soconformal supersymmetrycan berealizedon Minkowski superspace,with rathercomplicatedx, 0, 0 dependent“little group” elementsh[58].It is alsopossibleto realizeit on chiral superspace.The“little group” H is thengeneratedby Al,,,., K,,,D, S”, S’~andQa, andthequestionwhetherchiral multipletsexist,canbereducedto thequestionwhetherthe constraint(16.36)on the representationis consistent.This is non-trivial, becausethe non-vanishinganti-commutator

~a = (&“~)~M,,,, — 26”,, (~R— iD),

togetherwith the constraint(16.36) implies that the representationof the right-handside mustvanish,i.e., that (16.30)must hold.

Note that it is only requiredthat the Lorentz representationof a chiral superfieldbe self-dual, notthat it vanish. In otherwords, a chiral conformalsuperfieldcan havearbitrarily manyundottedbut nodottedspinorindices.The chiral weight andthe dimensionmustbe related.

We see that these restrictionsfollow quite directly from the structure of the algebra. Similarrestrictionshold for extendedconformal supersymmetry[11,58]. Therethe Lorentz representationof achiral superfieldmustagainbe self-dual,the SU(N) representationmustbetrivial, andthe R-weightisrelated to the dimension by

2N

(16.37)

17. Concluding remarks

This is the point whereone should,in any systematicpresentation,begin to talk aboutsupergravity.The current multiplet of section 15 acts as a sourcefor the fields of supergravityin a supersymmetricextension of Einstein’s equation.The geometricresultsof section 10 can be generalisedto curvedspace—time,andthe knowledgeof conformalsupersymmetryfrom section16 providesan elegantentry


into supergravitytensorcalculus.The technicalaspectsof supergravityare considerablymore com-plicatedthanthoseof flat-spacesupersymmetry,andI hope to presentasecondPhysicsReport in duetime whereI can give them similar in-depth treatment.

A subject related to flat-spacesupersymmetryis superGUTs:supersymmetricversionsof grand-unified theories.However, sincetheir consistenttreatmentshould not be attemptedwithout referenceto the super-Higgseffect, I haveomitted them from this report. The super-Higgseffect, arising out ofspontaneousbreaking of local supersymmetrycan only be treated in the context of supergravity.Indeed,the absenceof masslessGoldstonefermionsfrom nature [80,81] demandsthat theremust besomethinglike asuper-Higgseffect,hencethattheremustbe (broken)local supersymmetry,andhence,by virtue of eq. (9.4), theremust be (broken)supergravity.Conversely,since curvedspacedoesnotpermit constantspinors(thereis no coveringgroup of GL(4) with spinorial representations)we knowthat the only supersymmetrywhich can coexistwith gravity is local, “gauged”supersymmetry.

The way to proceedfrom hereis to cover the basicconceptsof supergravityin a systematicway, andthen to apply thoseto superGUTsandthe super-Higgseffect as well as to the otherexciting frontiersof presentresearch,super-Kaluza—Kleintheoriesandsuperstrings.This holdsas well for the readerwhowishesto learnmoreaboutthesesubjectsasfor the structureof a secondpart of this report.Until then,as far as supergravityis concerned,I wish to refer the readerto van Nieuwenhuizen’sexcellentearlierreport [83]which, aimedat adifferent typeof reader,may not presentthe subjectin as muchdetail as Ihavedonehere,but which insteadcoversa muchwider rangeof developments.I alsowish to refer thereaderto the excellentbooks by Wessand Bagger[84] and by Gates,Grisaru, RoëekandSiegel [26]which the seriousstudentof our subjectwill want to readin parallel with and as complementsto thepresentarticle. (In particular, ref. [26] fills a major gap in my treatment,being entirely devotedtosuperspaceperturbationtechniques.)

A. Technicalappendix

A.1. Minkowskimetric

The flat-spacemetric is takento be

(A.1)

andthe totally antisymmetrictensor is normalisedsuch that

= 1. (A.2)

A.2. Group representations

A matrix representationof agroupis a homomorphismof the groupinto a matrix space:

r:G—*M with r(g,)r(g2)=r(g1og2),

where o denotesthe groupproduct. If r(g) is a homomorphism,then so are r*(g), r_1T(g) and r~1~(g).

We denote the vectors on which theserepresentationsact by u,,, u~,u” and u’~,respectively.The


group transformationson the vectorsare then

u~=r,,’3u~‘_i*\ $ua—~r~,, u~—u,~r,~ A

= (r~T)~sut’ = u” (r1)~”

u” = (r_lt)c~,~,u’3 = u’3 (r_l*)~~.

If the groupis a Lie group,we haveinfinitesimalgroupelementsfor which the representationstaketheform

r,,~ (1 + ieI)j~ generator:I

(r*)a$ ~(/ _iel*)a$; generator: 1”

(r_lT)~~,3 (1 — ieIT)~~s; generator:_1T (A.4)

(r1~)~,,(1 + ieIt)~~,,; generator:I~,

andthe infinitesimal variation of a vector becomes

Fiu,, u,~— u,, = ~ u~, (A.5)

andsimilarly for ut,, u~and u”. The variationsof tensorsaredefinedaccordingly,e.g.:

= ieI,,~t~,

9+ iel8~t,,~

= id,,” ~ — ie(l*)s * t,,,, id,,” t,,~— iet,,y(It)~,~ (A.6)

Fita’9 = id,,” t,j~ — ie(IT)11

7 t,,” = id,,” t,j~— iet,,” I,f~.

Theuseof upperandlower indicesis motivatedby the invarianceof tracesand Kroneckersymbols:

Fit,,” = Fitaa = 0; Fi6”,, = 33$,.= 0. (A.7)

Compactgroupscan haveunitary representations,for which r~t= r andr1~= r*, so that thereis no

needfor dotted indices: u,, = u” and U’~= a,,,.Realgroups havereal representations,for which r =

andr_1t = ri’, andagainthereis no needfor dottedindices: u,, = a,, andu” = a”.TheLevi—Civitatensorsaretotally antisymmetricnumericaltensorsof rank n = dim(r),

c,,1.,,; ~ c”’””; ~

which we normaliseso that

(A.8)

with the sign dependingon otherconventionsused(it is minusin thecaseof eqs.(A.2)) andthe symbol


[S”] denotingtotal antisymmetrisationnormalisedsuchthat

= X[,,~...,,,]. (A.9)

Total symmetrisationis definedaccordinglyanddenotedby (“).

The transformationlaws for the Levi—Civita tensorsare compatiblewith eq. (A.8):

(r(g)e),,~...,,,, = det re,,1.,,,

(r~T(g)e)~1a~= (detr)’e”’”’* ~ —~d \* (A.10)~r (g,e,ai...a, — ~ e r1 ~

= (det r)_l*e~~1~ln

and if the group is unimodular(det r = 1; tr I = 0), the c-tensorsare numerical invariantswhich wenormaliseto plus or minusone (whicheveris the moreconvenient)if all indicesaredifferent.

A.3. Representationsof the Lorentzgroup

The algebra of the Lorentz group, eq. (2.18c), is normally written in terms of six Hermitiangenerators,arrangedinto an antisymmetricsecond-ranktensorAl,,,.. Thealgebracan,however,alsobewritten in termsof six non-HermitiangeneratorsJ±with

J ~(M23±iM°’) andcyclic (1 —*2 —* 3 —* 1) (A. 11)

andthentakesthe form

[J~, J2+] = iJ3± andcyclic,(A.12)

[J±,J~]=0.

Finite dimensional irreducible representationsare classified by pairs of half-integernumbers(j±,J~)which are takenfrom the eigenvaluesj±(j±+ 1) of the two Casimiroperators(.L)2. The algebrais notthat of SU(2)X SU(2) becausethe f’s arenot Hermitian:

(J±)t = L. (A.13)

Correspondingly,whereasthe f’s can be representedby finite dimensionalHermitian matrices, the

M,,,.’s cannot.The representation(~,0), e.g.,hasr(J4=~o; r(J_)=0 (A.14)

(u = Pauli matrices)andhence

r(M°’) ~u°’= —~iu’; r(M~2) ~u12= ~o~3 andcyclic. (A.15)


Thegroupgeneratedby thesematricesis SL(2, c). For the representation(0,~)we have

r(J+)=0; r(J_)=.~u (A.16)

andhence

r(M°1) = ~iu’; r(M12) ~&12 = ~o3 andcyclic. (A.17)

We seethat in eithercasethegeneratorsL of the properrotationsarerepresentedby ~o.Since6” = (u~~)t,it is appropriateto assign“index types” ~ and (&~~)“,,to theserepresen-

tations. We also havethe self-duality relations

= 2ic,,,.~Au; &,~,,.= ~ (A.18)

which reflect the fact that there are only threeratherthan six linearly independenttraceless2x 2matrices.

A.4. Two-spinornotation

The two-spinorindices which correspondto thesefinite dimensionalrepresentationsof the Lorentzgroup can be raised and lowered by meansof the two-dimensionalLevi—Civita tensorswhich wenormalisein the following way:

dl2= E12= Ej~ —c~~+1. (A.19)

If we definethe raisingofspinor indices by

4”‘~e”~ç1i~ 4”~çlt~e”, (A.20)

then the lowering is

4’,,=4/3e~,,; 4ia=ea,~4i”. (A.21)

There is a wide variety of different conventionsin use for (A.19—21), with various advantagesanddisadvantages.The presentnotation hasthe advantagethat reality relationshipsand c-tensorsarecovariant,

(4’~)*...4’d ,~ (4j)*_4, (A.22)

,,$_ ,,7 8~C —E C

(A.23)= ey.

5Ev,,e8$,

andthe disadvantagethat the mixed-indexc-tensorsare antisymmetric,


e,!’=—e”,,=3~ e”$=—e$”=3~, (A.24)

*h ~I \*__an LIIa~~c,,11,,—

As a generalrule, unwrittenspinorindicesarecontractedfrom “ten to four” if theyareundottedandfrom “eight to two” if theyaredotted,

= —~,,4”’; 4’~çba~”= _4Jd~,~• (A.25)

We definecomplexconjugation to includereversalof the orderof spinors,andget

(~4’)*= (~a4’)*= (çt,,,)* (~~)*= 4J.~” = 4i~. (A.26)

Since the c’s are antisymmetric,we must expectand indeedfind that the matricesu” and a” aresymmetricfor every index position:

(O~”’),,p = (u”~)pa; (u~”),,’9= (u”~)”,,,etc. (A.27)

The four matrices(u”),,~with a-°= I ando-’ = Pauli-matrices,

= (1 ,u), (A.28)

are,up to a factor, the only solution of the equation

0 = ~ — ~ — i(u”~~— u~’r~”’) (A.29)

which saysthat a-” ,,~,is a numericallyinvariant tensorif the index ,u is takento transformaccordingtothe vectorrepresentationof 0(1, 3). The a-” are the Clebsch—Gordancoefficientswhichrelate the (~,~)of SL(2,c) to the vectorof 0(1,3). Accordingly,we can associatea mixed bi-spinorwith every vector:

b,,(u”),~. (A.30)

We can alsodefinematriceso~”by

(&~)a13 (a-~~)I~~ (A.31)

andfind that &“ = (1, —o-) and

b,, o,,),”’~jio,,)”~ with ,~“~b,,(ã”)””. (A.32)

Finally, the variousu-matricesarerelatedby the properties

= q”~— iu”~(A.33)

= i,”” — ia” -


A.5. Four-spinornotation;Majorana spinors

Dirac matricescan be constructedfrom the 2x 2 matricesa-” anda” in the following way:

/ 0 u”\ /u”~ 0 \y”~f - and u””—( - ~. (A.34)

\u” 0/ \ 0 u”~/

Thesefulfill

{y”, y} = 2~”” and ~i[y”, y”] = a-”” (A.35)

andactnaturally on four-spinorswhich arecomposedfrom a chiral andan anti-chiral two-spinor:

(A.36)

The adjoint spinor is

(A.37)

andthe charge-conjugatespinor is

(A~) 4’C (4’C)fA = (f, A~). (A.38)

Thesespinorsarerelatedthrough4 x 4 matrices

A (~~); C (~ - 0.,~); C1 = (e~43 ~) (A.39)

in the following way

çl’4’tA 4”=C~/iT. (A.40)

The matricesA and C intertwine the representationy,, of the Dirac algebrawith the equivalentrepresentationsy~,and — y~,respectively:

Ay,,A~= y~ C-’ y,,C= —y~. (A.41)

It follows that the 16 linearly independentmatricesA, Ay,,, Au,,,.,iAy,,y5andAy5 areHermitian,the

six matricesC, y,,y5C and ysCare antisymmetric,and the ten matricesy,,Cand u,,,.Caresymmetric.Definition andpropertiesof y~are

ys=yoyly2y3; (y~)2=—l; Ys=(

0 ~)‘ (A.42)


andthe projectionoperators~(1±iy5) projectout thechiral componentsx~andA” of 4’.

In general, the two chiral componentsof a Dirac spinor are unrelated. If, however, there is aMajorana condition

4’ = (A.43)

i.e. if

~a(x,,)t and 4’=(~, (A.44)\x”J

thenwe havea Majorana spinor with only half as manyindependentcomponents.Thereis a particularrepresentationof the y-matrices,unitarily equivalentto the onedefinedabove,

for which — C= A = ‘Yo. Thenall y,, arepurely imaginary (for our metric)and a Majoranaspinorhasreal components.This representationis called the “Majorana representation”,and should not beconfusedwith the conceptof a Majoranaspinor:eqs.(A.40) andthe condition(A.43) are representationindependent. They imply a relationship between the components of 4’ and 4’ which only in a particularrepresentationmeansthat the componentsare actuallyreal.

Majorana spinors can only exist because4’CC= +4’ for spinors in four-dimensionalMinkowskispace—time.

A.6. Propertiesof bi-spinors; Fierz formula

For two spinorsrand 4’ whosecomponentsanticommute,we havethe following relations:

~4JZ4JC~C; ~yy54’=4JCyy5~C; ~y5çfr=4’Cy5~C

= _4’C~,,,~C; ~o’,,,.4’= _V/co.,,,,~,c. (A.45)

The completenessof the 16 Dirac matricesin 4x 4 spacecan be used to derive the following Fierzrearrangementformulafor two anticommutingspinors:

çli~T = — ~4i) — ky,,(~y”4,)— ~o~,,,,(~o”’4’)— ~y,,y~(~y”y~4’)+ ~y~(~y~4i). (A.46)

Togetherwith propertiesof y-matrices,this gives us

4,4’ — y~i/J9Jys= —~(4’4’)+ ~ys(4iy~4’) — ~o-,,,.(4’u””4i). (A.47)

For a Majoranaspinor 4’, the last term vanishesbecauseof (A.45), and multiplying with 4’ from the

right, wefind that

= y~/i(4iy~ç1’). (A.48)

A. 7. Dirac matricesfor arbitrary space—times

Dirac matricesFa~a = 1, -.. d, aredefinedto be irreduciblerepresentationsof the algebra

Martin F. Sohnius,Introducingsupersymmetry

{F,,,I’t,}r271at,~ (A.49)

for arbitrary dimensionsd and arbitrary signaturesof the metric ~7ab = ±6ab. We denoteby d~the

numberof plus andminussigns in the metric:

d=d++d. (A.50)

TheoremI (without proof):For a given evendimensionanda given signatureof themetric, all irreduciblerepresentationsof the

Dirac algebra(A.49) are equivalentand aren X n matriceswith

n = 2~~/2. (A.51)

This meansthat for any two representations{Fa} and{F~},thereis a non-singularmatrix S such that

F~”SFaS~. (A.52)

TheoremII:For a given odd dimensionand a given signature, thereare two equivalenceclassesof irreducible

representationsin termsof n xn matriceswith

n =

2(,~1V2• (A.51’)

If {F,,} is in oneequivalenceclassthen {1’a} is in the other.I give a proof only for the last statement:clearly, {—f’,,} represents(A.49) if {f’a} does.For all

dimensionswe can definea “generalisedy~”as

Fd+1~[’r . .J’ (A.53)

As a consequenceof eq. (A.49), Fd+1 will anticommutewith all f’,, for evend, but commutefor odddimensionswhere it is thereforea multiple of theunit matrix (Schur’slemma).If therewere an Ssuchthat

SFaS1 = F (A.54)

then

SFd÷1S’= (~1)”f’d÷1, (A.55)

which for odd d is in contradictionwith 1’d+l ~ L For evend we haveindeed

1’d±1Ta[’~1= Fa (even d) (A.56)

as an intertwinerof {f’,,} and {1a}.

200 Marlin F Sahnius,Iniroducingsupersymmetry

Equivalencefor evendimensions:All of the following arerepresentationsof (A.49):

[‘a, 1’a, F~,~ [‘k, ~ F~,~ (A.57)

We introduceintertwiners

Af,,A1 = (A.58)

C’F,,C= —f’~ (A.59)

which togetherwith F~+~allow us to transformany two of the representations(A.57) into eachother.Thus,e.g.,

D’F,,D = —I’~,, (A.60a)

with

D~_CAT. (A.60b)

For any two representations,the matrix S which intertwinesthem is uniqueup to a numerical factor(this is a stronger version of Schur’s lemma). From the Hermitian adjoint of (A.58), the negative of(A,56), the transposedof (A.59) andthe complexconjugateof (A.60) we get thus

A=aAt; Fd÷1=f~J’~I; C=~C~ D=8D~*. (A.61)

Theseequationsareconsistentonly if the numbersa, ~ and 8 satisfy

aa* = = I and 3 = 3* (A.62a)

A free phaseanda free scalefactor in the definition of A can be usedto fix

a = = 1 (A.62b)

(the only redefinitionsstill possiblearethen C—*cC; A—*!cJ~A).The remaining quantities (/3 and thesignsof ~ andof 3) are invariantsthat cannotbe tamperedwith. They aredefinedby the metric, i.e. byd±andd_, and aregiven in tablesA.2 to A.4 below.

Under an arbitrarysimilarity transformation(A.52), the matricesA, C andD transformas

A’ = SItAS~ C’ = SCST D’ = SDS1* (A.63)

(thereis onefree numericalfactorin this,correspondingto therescalingwhich isstill possiblefor C; it hasbeenset to one). Equations(A.63) and(A.55)are consistentwith (A.61).

For the matrix


15=—F~ID (A.64)

we have

L51Ea15 = (A.65)

and

J5 gj5-s* with ~ = /33. (A.66)

Equivalenceclassesfor odddimensions:For odd d, the questionwhich of the eight representations(A.57) fall into which equivalenceclassis

determinedby the behaviourof the productfd±1 underthe respectivetransformations.With

AFaA1 = ±J’~,; C1Ta ±J’~ (A.67)

TableAl Table A.2The equivalenceclasseswhich contain I’, for arbitrary odd The value of $=

dimensions

d = 1,4,5,8,9 mod 4 d = 2,3,6,7, 10, 11 mod4d=1,5,9mod4 d=3,7,lImod4

d_even +1 —1d_even ~ ~ d_odd —l +1dodd J’,,—r~,,,I’i,—fl

Table A.3The sign of ~ (symmetryproperty of C). The tablecontinuesmodulo4 in d

d=1 2 3 4 5 6 7 8 9 10 11 12+ — — — — + + + + —

- Table A.4Thesignsof 8 and 88 (possibleMajoranaconditions,seesubsection14.1). The tablecontinuesmodulo8 in d

and modulo4 in d~

d= 1 2 3 4 5 6 7 8 9 10 11 12

d= 0 + —+ — —— — +— + ++ + —+ — ——

++ + —+ — —— — +— + ++ + —+

2 + ++ + —+ — —— — +— + ++

3 +— + ++ + —+ — —— —

4 — +— + ++ + —+ — ——

5 —— — +— + ++ + —+

6 — —— — +— + ++

7 —+ — —— —

8 + —+ — -—

9 ++ + —+

10 + ++

11 +—


and always AFd+1A1= C1Fd+

1C= Fd±1cc I, we find table A.1 as result. Otherwisethe analysisisexactly parallel to the caseof even d, and I havegiven the tablesA.2, A.3 and A.4, respectively,for /3, ~ and 3. The last two tablesare most easilycalculatedin a specific representationof the Diracmatrices.

Antisymmetric products:For evend, all antisymmetricproducts

r,,~r.

[‘ab

[‘abc albfc] (A.68)

f’aIad

1’[aJ’a

2

are trace orthogonalto eachother andto the unit matrix (i.e. they are traceless).Theirnumberis

so that togetherwith the unit matrix theyarea completebasis in the spaceof Dirac matrices.For all dimensions,the relationship

Falas = ac,,i...a, ~ (A.69a)

holdswith

a = (d — k)! ~ l)k1)/2~1)/2. (A.69b)

For odd dimensions,wheref’~+~is a number,this implies that only the antisymmetricproductsup torank ~(d— 1) arelinearly independent.Their numberis

(d—I)12 d

~ (k)=2~1_1,andtogetherwith the unit matrix they,too, are complete.It is easy,for eachcaseathand,to deriveFlermiticity propertiesof Al’... andsymmetrypropertiesofF...C, as was donefor d = 4 and d. = 3 in the sentencefollowing eq. (A.41).

The following formulas maybe founduseful= (_1)~(~1)/2d!J(d — k)!

f~bFai...akFb= (_1)k(d_2k)Fat...a5

(d — 1)’ (A.70)

= (_1)k(~l~~2(d — 2k) (d — k)!

FCdFabJ~cd — (d2 — 9d + 1 6)Fab.


Acknowledgements

First and foremost I must expressmy gratitude to Dr. J.J. Perry who in effect co-authoredtheIntroduction.But for her insistence,I would haveneverattemptedto placethe subjectinto as generalacontext,andwithout her helpI would hardly havebeenable to do it.

I also thankthoseof my colleaguesandthosestudentswho attendedmy variouslecturesandwhoseenthusiasmhas been a great encouragement.I thank CERN and T. Fabergefor their generoushospitality at a crucialstage.Without the generoussupport of M. Jacob,as headof the CERN theorygroup, and his everlastingpatience,as editor, this report would neverhavebeenfinished. I am verygrateful to him.

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INTRODUCING SUPERSYMMETRY Martin F. SOHNIUS · Supersymmetry is, by definition, a symmetry between fermions and bosons. A supersymmetric field theoretical model consists of a set