KALUZA-KLEIN SUPERGRAVITY

M.J. DUFF and B.E.W. NILSSON

CERN, Geneva,Switzerland

and

C.N. POPE

BlackettLaboratory, Imperial College,LondonSW72BZ, UK.

NORTH-HOLLAND-AMSTERDAM

PHYSICSREPORTS(Review Sectionof PhysicsLetters)130, Nos. 1 & 2 (1986) 1—142. North-Holland,Amsterdam

KALUZA-KLELN SUPERGRAVITY

M.J. DUFF* and B.E.W. NILSSON**CERN, Geneva, Switzerland

and

C.N. POPEBlackettLaboratory, Imperial College,LondonSW72BZ, U.K.

Received4 April 1985

Contents:

1. Why Kaluza—Klein? 3 7.1. SO(8), triality andN = 8 651.1. Introduction 3 7.2. The massspectrum 681.2. Historical developmentof Kaluza—Klein theories 8 8. The squashedseven-sphere 711.3. TheKaluza—Klein recipe 13 8.1. The meaningof squashing 711.4. The d = 5 theory:Kac—Moodysymmetries 17 8.2. N= 1 supersymmetryandG

2 holonomy 762. Why supergravity? 23 8.3. Spontaneoussymmetry breakinginterpretation: the

2.1. Reasonsfor N = 1 supergravityin d = 11 23 ‘spaceinvaders’scenario 792.2. TheLagrangian,itssymmetries,andtheequationsof 8.4. Themassspectrum 82

motion 24 9. Solutionswith otherM7 topologies 843. Spontaneouscompactificationto d = 4 26 9.1. Ricci flat M7 : T

7and K3 x T3 843.1. The Freund—Rubinansatz:M.

4 x M7 groundstate 26 9.2. Othersolutions 863.2. Anti-dc SitterspacetimeandOSp(4/N) 30 10. Non-vanishing~ 91

4. Propertiesof M7 35 10.1. Generalremarks 914.1. Symmetries,groupspacesand cosetspaces 35 10.2. Solutions from spaceswith Killing spinors 934.2. Killing spinors, holonomyand supersymmetry 42 10.3. Solutions from U(1) bundlesover Kähler six-mani-4.3. Propertiesof operatorson M7 46 folds 96

5. The d = 4 massspectrum 54 11. Vacuumstability 995.1. Thed = 4 massoperators 54 11.1. Freund—Rubinsolutions 995.2. The criterion for vacuumstability 63 11.2. Othersolutions 100

6. Classification of known d = 7 Einsteinspaces 63 12. Relation betweend = 11 andd = 4 supergravity 1017. The roundseven-sphere 65 12.1. Thede Wit—Nicolai theory 101

* On leaveof absencefrom theBlackettLaboratory,Imperial College,London5W7 ~BZ,U.K.

** On leaveof absencefrom theInstitute for TheoreticalPhysics,S-41296Göteborg,Sweden.

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MJ. Duff eta!., Kaluza—Kleinsupergravity 3

12.2. Conjectureson theseven-sphere 102 14.3. Fermioncondensatesand thecosmologicalconstant 11612.3. Linearizedmasslessansatzand transformationrules 103 14.4. Towardsa realistic theory? 11712.4. New S

7 solutionsandWarner’sextrema 10713. Consistencyof the Kaluza—Klein ansatz 109 Appendices 118

13.1. GenericKaluza—Klein theories 109 A. Curvaturecalculations 11813.2. d = 11 supergravityandtheseven-sphere iii B. f-algebra 12113.3. Significanceof consistency 113 C. Modifiedmassoperators 123

14. The quantumtheory 114 References 12414.1. Divergencesand anomalies:theChern—Simonsterm 114 Recentdevelopments(addedin proof) 13014.2. Casimir energies 115

1. Why Kaluza—Klein?

1.1. Introduction

Review articlestendto fall into two categories:either they areunbiasedandcomprehensivewith alist of referencesincluding every paperon the subject;or elsethey areselective,focusingattentiononwhatthe authorsthink is important.Thefirst kind is fair but not alwaysveryuseful:oneof the purposesof a reviewshouldbe to sparethe interestedbut busyreaderthetaskof wadingthroughthe irrelevant.In this reviewof Kaluza—Klein supergravitywe havechosenthe latter course,albeit at the risk of bias.So we shall begin by stating our prejudices.

We do not yet know whethersupergravity[1, 2, 3, 4] is a theory of the realworld, nor whetherthereal world hasmorethan four dimensionsas demandedby Kaluza—Klein theories[5, 6, 7]. However,our researchin this areahasconvincedusthat the only way to do supergravityis via Kaluza—Kleinandthat the only viable Kaluza—Klein theory is supergravity. The purposeof this PhysicsReport is topresentthe casefor this point of view and in the processwe shall find someintriguing hints thatKaluza—Klein supergravitymayevenbe realistic.

We begin in section 1.2 with a brief history of the developmentof Kaluza—Klein theoriesand theireventualmergerwith supergravity.Ideason the subjecthavecertainlychangedover the yearsand wepresenta critique of certainconfusionsand misapprehensionswhich pervadethe old Kaluza—Kleinliterature,someof which persistto the presentday. (For example,failure to demand“spontaneous”asopposedto “ad hoc” compactificationof the extradimensions.)The Kaluza—Kleinrecipefor interpret-ing a higher-dimensionaltheory as a four-dimensionaltheory is given in section 1.3. As the classicexampleof theseideaswe discussin section1.4the original five-dimensionalmodelof Kaluza [5] from amodernperspective.It is not arealistictheory,nor is it typicalof the kind of higher-dimensionalmodelsweareinterestedin. Nevertheless,it is of historicalinterestandsufficiently simple that it admitsa fairlycompleteanalysisof its properties,at leastat the classicallevel. In particular,oneis able to write downin closed form the symmetriesof the effective four-dimensionalLagrangian,including the massivestates.The novel featureis the emergenceof infinite parameteralgebrasof the Kac—Moody type [8].

In ouropinion, the most attractiveKaluza—Klein theoriesarethe supersymmetricones.Even here,however,expertopinion is divided betweenthe purists,whose guiding principles arethe mathematicalconsistencyandeleganceof the theory,andthepragmatistswhoareseekingto makeimmediatecontactwith low-energy phenomenology.Two problems spring to mind in this respect: the cosmologicalconstantproblemand the chirality problem[9]. Ideally, we would like to explain the observedchiralrepresentationsof the quarksand leptonsand the fact that the cosmologicalconstantis very small, atleast in the phasein which we live. However, should thesepropertiesappearimmediately at the tree

4 MJ. Duff eta!., Kaluza—Kleinsupergravity

level from the masslesssectorof the higher-dimensionalLagrangianwith which we start?Or, shouldthey ratheremergefrom some effective (perhapsboundstate?)low-energy Lagrangianwhose inter-actionsmay be very different from the fundamentalLagrangian?Before answeringthis questionweshould bearin mind that we are undertakingthe enormouslyambitiousextrapolationof 17 ordersofmagnitudefrom the Planckscaleto the elementaryparticlescaleand a further40 ordersof magnitudeto the cosmologicalscale.

The two theories which have attractedthe most attention are supergravityin d = 11 [10], themaximum dimensionpermittedby supersymmetry[111,andthe d = 10 superstringtheories[12, 13, 14]whose zero-slopelimits yield d = 10 supergravity.Three consistentsuperstringtheoriesare known*:type hA, whosezero-slopelimit yields the non-chiral N = 2 supergravityin d = 10; type JIB, whosezero-slopelimit yields the chiral N = 2 supergravityin d = 10 and which is neverthelessanomaly-free[2471;anda specifictypeI which correspondsto N = 1 supergravityin d = 10 coupledto N = 1 SO(32)Yang—Mills and is also anomaly free [151.The anomaly freedom, one-loop finiteness and possiblefinitenessto all ordersmakesthesetheoriesvery attractive,especiallythe chiral SO(32)version whichseemsclosestto phenomenology.The major drawbackis that, to date, no one hasfound a satisfactorycompactification to d = 4. For the string theoriescompactification is, in any case,a very obscureconcept.For thecorrespondingfield theories,the only knownnon-trivial compactificationsto d = 4occurin the non-chiral typehA but theseare all obtainablefrom thoseof d = 11 supergravity[16,270, 285].No non-trivial (i.e. chirality preserving)compactificationsto d = 4 of typeJIB or typeI areknown*, andin the caseof typeI thereareevenno-gotheorems[171to this effect. By contrast,d = 11 supergravitycompactifiespreferentiallyto d = 4 [18]anda wholebody of literatureon the subjectnow exists.This isalsothe subjectwith which thepresentauthorshavebeenmostclosely involved. As for d < 10, theonlytheories of interest are those which cannot be obtained from reduction of d = 11 or 10. ThesupersymmetricN = 2 Einstein—Maxwell-typetheoriesin d = 6 [19,20, 2841 are particularlyattractive,permitting a monopolecompactificationto d = 4 with zero cosmologicalconstant,chiral fermions andunbroken N = 1. In common with the type I string theories, however, it is necessaryto put theYang—Mills fields in by handalreadyin the higher-dimensionalLagrangian.

In this PhysicsReportwe shall follow mainly the “purist” approach(throughsomeconcessiontopragmatismwill be madeby demandingnon-trivial compactificationto d = 4) andfor the reasonsout-linedin chapter2,we shalldevoteourattentionalmostexclusivelytothe d = 11,N = 1 supergravitytheoryof Cremmer, Juliaand Scherk [101.By requiring no morethan two derivativesthe on-shell theory isunique.Moreover,its field equationsadmit a spontaneouscompactification,i.e., as describedin chapter3 theyadmit solutionsfor which the metric describesaproductof d = 4 anti-deSitter(AdS) spaceandacompact d = 7 Einstein space M7. AdS is the maximally symmetric spacetimewith negative cos-mological constantand its propertiesare discussedin detail along with the correspondingOSp(4/N)supersymmetry.Without specifying M7 we also derive in chapter5 the massspectrumof the resultingd = 4 effective field theory in terms of well-known differential operators.This in turn enablesus todeducethe criterion for vacuumstability, which is treatedin detail in chapter11.

Further propertiesof M7 are discussedin chapter4, beginning with an elementaryintroduction tosymmetries,group spacesand coset spacesG/H. Crucial to the whole programmeof Kaluza—Kleinsupergravityare the notions of Killing spinors [211and holonomy groups [22, 23, 241, and theirrelevanceto supersymmetry.We shall payparticular attentionto the differencebetweenspaceswhichare both homogeneousandsymmetricandthosewhich are homogeneousbut not symmetric.With the

* See,however,thesectionon recentdevelopments.

MI. Duff etaL, Ka!uza—Kleinsupergravity 5

exceptionof the round 57, which admits supersymmetrieswith both orientations,the former spacesbreak all supersymmetries.Of the latter, thosewhich admit N <8 supersymmetryadmit none whentheir orientationis reversed.(This is the “skew-whiffing theorem”[25]). Interestinglyenough,however,both orientationsleadto classicallystablevacua.

Thenextproblemis to specifyM7. Indeed,oneof the majorproblemsin anyKaluza—Klein theory isto selectthe “true vacuum” from amongthe manycandidateground-statesolutions.In particularthereare infinitely many seven-dimensionalEinstein metrics which might serve as vacuum statesin thespontaneouscompactificationof d = 11 supergravitydown to d = 4. Oneobviousway to narrowdownthis choice is to demandstability, andthis will be guaranteedif thereis an unbrokensupersymmetry[26—28].Unfortunately,this doesnot pin down the vacuumuniquely.To complicatemattersfurtherwefind examplesof vacua which are classically stable even when all supersymmetriesare broken. Inchapter6, we list all known d = 7 Einstein spaces,both with and without supersymmetryspecifyingthe stability properties.

In the absenceof any mathematicalcriterion for singling out the “true vacuum”, one must againresortto sometheoreticalprejudice.Severalpointsof view maybefound in the Kaluza—Klein literatureand theydiffer essentiallyin the Kaluza—Kleinorigin of the SU(3)x SU(2)x U(1) gaugebosonsof thestandardmodel.To a traditional model builder,perhapsthe most obviouspossibility is that one looksfor an isometry group G j SU(3)x SU(2)x U(1). Indeed this was the original proposal ofWitten [26]who noted that sevenextradimensionsarenot merelythe maximumnumberpermittedbysupersymmetry[111but alsothe minimumnumberneededfor SU(3)x SU(2) x U(1). Witten classifiedall such spaces,and denotedthem ~ where p, q and r are integers.However, he did not inquirewhetherthey provided solutionsto the d = 11 field equations.Duff and Toms [29] pointed out that

= CP2x S2x S1 did not admitan Einsteinmetric but thatM°~= CP2x S3 andM’°t= XS~did, the

latter admitting a spin structureso that fermions can be globally defined.More recently, Castellani,D’Auria and Fré [30] haveshown that all ~ exceptM~°1,admit an Einstein metric, and that thep-oddsolutionsalwaysadmitof spinstructure.Ofparticularinterest,thep = qspacesadmitanN= 2super-symmetry(seealso [200,272]).The beautyof thesesolutionsis that the SU(3),SU(2)andU(1) couplingconstantswill all be relatedby the geometryeventhough this is not a GUT theory [29]. In particularonemay calculatesin2O~where O~is the weak mixing angle [31, 32], at least at the compactificationscale(~=4O18GeV). Unfortunately,the majorproblemfor thesesolutionslies in the fermion spectrum.Aside from the absenceof chirality alreadyanticipatedby Witten [26], a harmonicanalysisof fermionfields on M”~spacesreveals[33—36]that nowheredo the right quark andleptonrepresentationsappeareither in the zero modesor the non-zeromodes.Onewould then have to arguethat the quarksandleptonsappearas boundstatesbut in this casethe whole ideaof gettingthe right gaugebosonsat theelementarylevel seemsmuch lesscompelling. For this reason,we postponea discussionof M”~Tspacesuntil chapter9, wherewe alsodiscussothercompactificationsnot favoredby our theoreticalprejudicies.

To a supergravityenthusiasta totally differentunificationschemepresentsitself as apossibility.Firstwe recall that sinceits discoveryin 1976, conventional(i.e., four-dimensional)supergravity[1, 2] hasevolvedalong ratherdiverselines: the N = 8 route and the N = 1 route. Thosewho do the maximalN = 8 supersymmetry[37—39]have several motivations.First, it is beautiful.Secondly,one has thefeelingthat if naturecaresat all aboutsupersymmetryit would becrazyto stopat N = 1. Thirdly, N = 8supergravityis the only truly unified theory we have.It is the only knowntheory in whichgravity andallthe other particlesof lower spin appearin one and the samemultiplet. This is not true of N<8supersymmetryand, in particular,it is not true of N = 1. One alsoexpects,therefore,that the N = 8theory will havethe bestbehaviorfrom the point of view of ultraviolet divergences[40]. Thosewho

6 MJ. Duff eta!., Ka!uza—Kleinsupergravity

advocateN = 1 supersymmetry,on the otherhand,do so becauseit is the only supersymmetrylikely tobe relevantfor particlephysicsphenomenologyat presentenergies.For recentreviewsof N = 1 particlephysics see [41,42], and for supersymmetryin general[269].Consequently,the hope has sometimesbeen expressedthat thesetwo approachescould be linked if N = 8 supergravity were to breakspontaneouslyto N = 1 at Elanckianenergies.

We shall see in chapters7 and 8 that this hope is indeed fulfilled: N = 8 supersymmetrybreaksspontaneouslyat the treelevel to N = 1 at ascalegiven by eM~wheree is the couplingconstantof theYang—Mills gauge group SO(8) and M~is the Planck mass. At the same time SO(8) breaks to50(5)x S0(3)but parity remainsunbroken.However,theN = 8 theory possessingthe abovepropertiesis not theungaugedsupergravityof CremmerandJulia [37]nor eventhegaugedSO(8)theory of deWitand Nicolai [38, 39]. Rather it is the N = 8 theory proposedby Duff and Pope [21] obtainedbyspontaneouscompactificationof d = 11 supergravityon the seven-sphere,57• Indeed, the followingfeaturesare uniqueto Kaluza—Klein supergravity:(i) the masslessgravitino of the N = 1 phasecomesfrom a massiveN = 8 supermultiplet(this is the “Space InvadersScenario”of section 8.3), (ii) thescalarsdevelopingnon-zeroVEVs also belongto massiveN = 8 supermultiplets,(iii) parity remainsunbrokenwhenN = 8 breaksto N = 1.

Some time ago [29, 40] we pointed out that the field equationsof N = 1 supergravityin d = 11dimensionsadmit of vacuumsolutionscorrespondingto AdSx S7, and that since S7 admits 8 Killingspinors and since its isometry group is SO(8) this gives rise via a Kaluza—Klein mechanismto aneffective d = 4 theory with N = 8 supersymmetryand local SO(8) invariance.It was this observationwhichfirst arousedour interest in Kaluza—Klein theoriesand providedthe stimulusfor our subsequentresearch.It is to the seven-sphere,therefore,that the bulk of this PhysicsReportis devoted.There isnow a considerableliterature on S7 compactificationof d = 11 supergravity [21, 22, 29, 40, 43—97,194, 195, 198] and in chapters7, 8, 10, 12 and 13 we give an up to date accountpaying particularattentionto the Brout—Englert--Higgs—Kibblespontaneoussymmetrybreakinginterpretation[45,661 ofthe different S7 solutionswhich arepresentlyknown.

In [40,29], we conjecturedthat the masslessN = 8 supermultipletwhich survivesafterdiscardingthemassivemodeswhich ariseon compactificationon the roundS7 coincideswith the gaugedSO(8),N = 8supergravityin d 4 of de Wit and Nicolai [38,39]. This was promptedby the well-known result ofCremmerandJulia that a similarprocedureon the flat seven-torusT7 yields the ungaugedN= 8 theory[37].Chapters12 and 13 deal with the currentstatusof this conjecturewhich, at the time of writing, isstill neitherproved nor disproved owing to the calculationaldifficulty in obtainingall the non-lineartermswhich arisein the d = 4 theory comingfrom d = 11. Nevertheless,we still believeit to betrue andthis suggeststhat the hiddenSU(8)symmetryof N = 8 supergravityin d = 4 [37]mayalsobe sharedbythe Kaluza—Klein theory*. In [45,21], anotherconjecturewasmade.Namely, that compactificationsofthe d = 11 theory with the sameS7 topology as the roundS7 solutionbut with differentgeometrymightcorrespondto non-symmetricextremaof the de Wit—Nicolai effectivepotential.Of course,a necessaryconditionfor thisto work is that the scalarsor pseudoscalarsdevelopingnon-zeroVEVs‘must belongtothe massless35 0~or 35 0 of the de Wit—Nicolai theory.So the squashedS7 [22,53] wherethe Higgsscalarsbelongto themassive300 of 50(8)hasno suchinterpretation.The parallelizedS7 [43]wheretheHiggs belongsto the 35 0-, however,might have this interpretationand indeedan extremumof thefour-dimensionalpotentialwith all theright properties,e.g.,N = 0 and50(7),hassincebeenfound [60,71]. The statusof thisconjectureandthe wholeproblemof discardingmassivemodesin Kaluza—Kleintheoriesis discussedin chapters12 and 13. A particularlyinterestingresult of this equivalenceis the

* See thesectionon recentdevelopments.

MI. Duff etaL, Kaluza—K!ein supergravity 7

recentdiscovery[95] of a new 57 compactificationwhich breaksN= 8 to N = 1 and 50(8) to G2. Inaccordancewith ourpreviousremarks,it alsobreaksparity. It corresponds,in fact, to the G2 extremumof thede Wit—Nicolai effectivepotential.

So far the analysishasbeenmainly classicaland in chapter14 we turn briefly to the problemofquantizationand, in particular to divergencesand anomalies,Casimir energiesand quantumcorn-pactification. We also suggest a possible resolutionof the cosmologicalconstantprobleminvolvingfermion condensates.

Finally, in this chapter,we turn to the all importantquestionof whetherthe 57 theory can providearealisticunification,bearingin mind that 50(8)~ SU(3)x SU(2)x U(1). We shall speculatethat it mightif someof the gaugebosonsandfermionsarecompositeas in the modelof Ellis, Gaillard, Maiani andZumino [98]. It is here,therefore,that we departfrom the naive assumptionthat the standardgaugegroup must be containedin the isometry group of the extradimensions.At this stagethere is oneimportant aspectwe should like to stress.The existenceof two kinds of Yang—Mills gaugeboson,“elementary” and “composite”, is an automaticconsequenceof Kaluza—Klein theoriesand not to beregardedas an extraad hoc ingredient in the theory.This is due to the fact, rarely stressedin theliterature,that Kaluza—Klein theoriesunify gravity andYang—Mills in two different ways.Therearetwogravitationalbosonicsymmetriesin the d-dimensionaltheory: d-dirnensionalgeneralcovariance

zM~~~ziM(z) (1.1.1)

andd-dimensionallocal Lorentz invariance

~ A,’ \... A I \ B,’ \ ~ AB( \.... us AB,’ \

OeM t,z)—a B~Z)CM i,Zj, OWM kz,——L.’Ma ~Z),

which, in our d = 11 case, is a local SO(1, 10). Crudely speaking,the elfbeins CMA(z) are the gaugebosonsof general covarianceand the spin connectionsWMAB(Z) are the gauge bosonsof Lorentzinvariance.However,whereaseMA is an “elementary” field the wM~~Bare “composite” fields. Thespin connectionhasno kinetic term of its own and may, therefore,be expressedin terms of theelfbein and its derivatives.The Kaluza—Klein mechanismthereforegives rise to both elementaryandcompositegaugebosonsin d = 4. Let uswrite ZM = (xi’, y

m)wherexI* arethe coordinatesof spacetimeandym the coordinatesof the extradimensions.Then the elementarygaugefields B~(x)comefrome (x, y) andcorrespondto a gaugegroupgiven by the isometrygroupof the extradimensions;SO(8)in the caseof the round57• The compositegaugefields A~(x)comefrom w,~”(x,y) andcorrespondtothe tangentspacegroupof the extradimensions;S0(7)in the caseof d = 7. Theselatterfields havenokineticenergyterm of their own andmay, therefore,be expressedin termsof the scalarfields comingfrom erna (x, y) andtheir derivatives.In d = 11 supergravity,thereis a third bosonicsymmetryassociatedwith the 3 index gaugefield ~

= ~[MANP],

(1.1.3)ANP(Z) = —APN(z).

Fromthe work of CremmerandJulia [37]we knowthat whenthe ~ field is takeninto account,thehiddensymmetrymay be evenbigger thanS0(7)andmight be enlargeableto SO(8)andthento SU(8).[TheS0(7)generatorsF,~,,aresupplementedby I’a which togethercloseon 50(8)andthen by 751’a& toform the 63 generatorsof a chiral SU(8).] Seealso [275].

8 MI. Duff eta!., Kaluza—K!ein supergrauity

If oneacceptsthe S0(8) [elementary]x SU(8) [composite]picture,then onecan imagine manywaysin which to embedthe SU(3)x SU(2)x U(1) of the standardmodel. Some of them are discussedinsection 14.4.

Apart from the problemsof quantizationtouchedon only briefly in chapter14, thereare, of course,other aspectsof Kaluza—Klein theories not covered by this report. These include Kaluza—Kleinmonopoles[99—1011andtheblossomingsubjectof Kaluza—Kleincosmology[85,102—108]. Nor havewepursuedWeinberg’squasi-Riemannianapproachto higher dimensions[109—111],which might providearesolution to someof the problemsof conventionaltheories.Other reviews on Kaluza—Klein theoriesmay be found in [112—1151and in Proceedingsof the Erice School of Unified Field Theoriesof Morethan Four Dimensions [116],Proceedingsof the Chalk River Workshop on Kaluza—Klein Theories[117],and Proceedingsof theJerusalemWinter Schoolon Physicsin Higher Dimensions[118].

1.2. Historical developmentofKaluza—Kleintheories

Preoccupiedas we are nowadayswith the quest for a unified field theory of the fundamentalinteractions,it is easyto forget that this is not a newactivity. Already in the 1920sKaluza [5] andKlein[6, 7] suggesteda unification of the gravitationaland electromagneticinteractionswhich, at the time,were the only forcesto be well understood.They did so by the ingeniousdeviceof postulatingan extra,fifth, dimensionfor spacetime.In this schemethe electromagneticfield ceasedto haveany fundamentalsignificanceof its own but was seenmerely as componentof gravity, albeit in an extradimension.(Aneven earlier attempt at unification from five dimensionswas madeby Nordstrom [119] prior to thediscoveryof generalrelativity. We aregrateful to P.G.O.Freundfor bringing this to our attention. Ashort history of Kaluza—Klein ideasmay befound in [120].)

Einstein,who was the refereeof Kaluza’s original 1919 paper[5] did not at first like the idea buteventuallywarmedto it, acceptingit for publication in 1921.Indeed,it was Einstein andBergmann[121]who subsequentlyemphasizedKlein’s idea [7] that the compactnessof the gaugegroup(i.e., U(1) ratherthanli), andhencethequantizationof electriccharge,could be accommodatedby postulatingthe extradimensionto be topologically 5’. In suitablecoordinatesall field variablesshould then beperiodic withrespectto the fifth coordinateandcould be expandedin a Fourierseries.When appliedto the metrictensoritself oneobtainedin this way, as explainedin section1.3, an infinite tower of tensor,vectorandscalarmodeswith massesquantizedin units of the inverseradiusof 5’. Much of the earlier literatureaddresseditself to the questionof whetherthe massivestatesshould be discarded,i.e., of whethertoinsist that all fields be independentof the fifth coordinate:the so-called“cylinder condition”. If not,how shouldoneinterpret the fact that with the electricchargeset equalto its experimentallyobservedvalue, the value of m would haveto be enormouslyhigh, i.e., the Planckmass =-105g?In today’sunified field theories,observedparticleslike the electronareassignedzeromassin the field equationsand acquiretheir smallmassvia a spontaneoussymmetrybreakingmechanism.So it is not unreason-able to supposethat the observedparticleslie in the zero-masssectorof the Kaluza—Klein theory,butthis was far from clear in the earlydays.

Evenin themasslesssector, however,anotherproblem was the scalarfield (seesection1.4). Gravityand electromagnetismwere experimentallyindisputablebut what on earthwas the scalar?Again, intoday’s unified theoriesscalarsproliferate and we no longer reject theorieswith elementaryspin 0fields. (At leastnot straightaway.)But in the earlydays,the scalarwas a causeof greatanguish.Manyauthorssimply set it to zero. Apart from problemsof inconsistency,discussedin section 1.4, theresultingtheory would then simply be describedby Einstein’sLagrangianplus Maxwell’s Lagrangian.

Mi. Duff eta!., Ka!uza—K!ein supergravity 9

Thus,with coordinates~M = (xi’, y) and ~ = 1, . . . , 4, a typical “ansatz” for the d = 5 line elementwould be

d~2= [g~(x) + K2A~(x)A~(x)]dx” dx’~+ 2KA~~~(x)dx~dy + dy2. (1.2.1)

Substitutioninto the d = 5 Lagrangianthenyields

K2V_gR = K 2V_gR—~./—gF~F’~”, (1.2.2)

where F~= ~ — ~ But this theory could havebeen (and indeed was) written down withoutrecourseto extradimensionsso what hadbeengained?In our opinion, this ideaof mutilating and/ordoctoring the higher-dimensionaltheory in order to fit one’sfour-dimensionalprejudicescontinuesinthe literature to the presentday and should be avoided.In otherwords, one should take the extradimensionsseriously,both classicallyand quantummechanicallyevenif this gives rise to an infinitetower of massivestatesandevenif thispredictslighter particleswith no currentexperimentalsupport.

Curiously, the retention of the scalar field by Jordan and Thiry [122, 123] in fact stimulatedBrans—Dicke theoriesof gravity [124].For earlier discussionof the scalar,see Veblen and Hoffman[125].Once again,however,the failure of Brans—Dicketheoriesto gain experimentalsupport shouldnot be countedagainstKaluza—Klein theoriessincethe kind of scalarswe find in modernKaluza—Kleintheorieshaveverydifferent interactions.After all, no olle hascomputedthe perihelionshift of Mercuryusing the Lagrangianof N= 8 supergravity,since the Sun and planetsare not describedby N = 8supermultiplets!So onecannotrule out the 70 masslessspin 0 fields in this way. The crucialquestioniswhetherthe low-energyeffectivetheory hasBrans—Dicketypeinteractions.

The arrival of Yang—Mills gaugetheoriesin 1954 presentedan altogetherdifferent challengetohigher-dimensionalgravity theories:could theyalsoaccountfor non-Abelian gaugebosons?Curiouslyenough, Klein discoverednon-Abeliangauge fields in the 1930s while investigating d > 5 gravitytheoriesbut their significancewas neverrealizedor articulated.The first concreteattemptseemsto bethat of DeWitt [126] in 1963, and this was followed up by work of Rayski [127], Kerner [128],Trautmann[129],Cho [130],Choand Freund[131],Cho andJang [132]andothers.The breakthroughwasthe realizationthat the gaugegroupobtainedin d = 4 was connectedto the isometrygroup of theextradimensionswhich in analogywith 5’ were takento be compactin orderto ensurecompactnessofthe gaugegroup.Thus,it wasargued,SU(2)gaugebosonsarosefrom taking threeextradimensionsandassigningto them the geometryof a three-sphereS3 whichwas, afterall, the SU(2)groupmanifold.

A featureof thesetheorieswas the relation betweenthe Yang—Mills gaugecoupling constante,Newton’sconstantG andthe radiusof the extradimensionsm’:

e2-’- Gm2. (1.2.3)

With the wisdom of hindsight, we can now identify several shortcomingsof thesenon-Abeliandevelopments.First, little attention was paid to the question of why the extrak dimensionswerecompactifiedandwhetherthiswas consistentwith the field equationsof the higher-dimensionaltheory.It wasusuallya completelyadhocprocedure.A secondshortcomingwas the failure to realizethat theextra-dimensionalgroundstatemanifold Mk neednot correspondto a groupspaceG in orderto obtainYang-Mills fields with G as their gaugegroup.Now we know that anyMk with G as its isometrygroupwill do, i.e., anymetric admitting the Killing vectorsof G. This could be a homogeneousspace.In this

10 MJ. Duff eta!., Ka!uza—K!ein supergravity

casethe groupG actstransitivelyandwe maywrite the manifold as thecosetspaceMk = G/H whereHis the isotropy subgroupof the isometrygroup G.The useof suchhomogeneousspacesin Kaluza—Kleintheorieswas discussedby Luciani [133].Since k = dim G — dim H, one was no longer obliged to haveonly onegaugebosonfor eachextradimension,ashad previouslybeenassumed.Indeed,as explainedin section 4.1, the isometry group of a group manifold G can be as large as Gx G if we use thebi-invariant metric,so S3 can give SU(2)x SU(2)gaugebosonsandnot merely SU(2).

The aboveconfusionwas not unrelatedto anotherconfusion(andearlyKaluza—Klein literaturewascertainly confusing!) namely, the failure to distinguish between the full metric, the Kaluza—Klein“ansatz” and the ground-statemetric. For example, see section 1.4. Indeed, the word “ansatz” wasoften bandiedaboutbut almost neverdefined! The problemsof the consistencyof the Kaluza—Kleinansatzarediscussedin detailin chapter13.

The history of Kaluza—Klein took a totally different turn with the adventof supersymmetry[3] andsupergravity [1—4].Although no rigorous theorem exists, it is generally believed that in fourdimensionsthere are no consistent masslessinteracting field theorieswith spins greater than two[134—136].In 1978 Nahm[11] observedthat the structureof the supersymmetryalgebras,coupledwiththis spinrestriction,placedan upperlimit on the dimensionof spacetimein which onecan formulateaconsistentsupersymmetricfield theory.

This upper limit takesthe bizarrevalueof d = eleven!In d = 11, wherethe Lorentz groupis S0(1,10) the Diracspinor index runsfrom 1 to 32.Thiscorrespondsin d = 4 to eight four-componentspinors.Accordingly, there are N = 8 supersymmetrygeneratorseach one changing the helicity of physicalstatesby one-halfunit, i.e., the helicitiesin d = 4 takethe values—2, —~, —1, —~, 0, ~, 1, ~, 2. Turningtheargument around, spin �2 means N ~ 8 means d ~ 11. This result is central to the whole ofKaluza—Klein supergravity. If one takes the view that nature is not merely supersymmetricbutmaximally supersymmetric,thenone is forced to d = 11 dimensions.(As explainedin section7.2, N = Iin d = 11 hasmoresupersymmetrythan conventionalN = 8 in d = 4.) In thissenseif Kaluza—Kleinhadnot already existed in 1978, it would havehad to havebeeninvented.Conversely,for the first timeKaluza—Kleinenthusiastshad a guideas to the dimensionalityof spacetimewith which to start.

Yet more was to come.Cremmer,Julia and Scherk [10] were able to constructthe correspondingN = 1, d = 11 supergravityLangrangian,describingthe interactionof the elfbein eMA, the gravitino ~PM

andthe three-indexgaugefield AMNP. As describedin chapter2 the uniquenessof the field equationsmeantthat not only did Kaluza—Klein enthusiastshaveaguide to the dimensionalityof spacetimebutalsoa guide to the correctinteractionsof gravity andmatter.In d < 11, supersymmetrictheoriesarenolonger unique but still very restrictive. It seemsto us that this restrictionon the dimensionand therestrictionson the interactions,which supersymmetryprovides,areessentialto anysuccessfulKaluza—Klein unification. Otherwise,oneis wanderingin thewilderness:the problemof finding theLagrangianof the world in d = 4 is simply replacedby the problem of finding the Lagrangianof the world in d > 4with the extraheadacheof which d to pick. Nothing hasbeengainedby way of economyof thought.Weinberg[110]is fond of recalling the fableof the “stone soup”whendiscussingthis problem.Just asthe promiseof delicioussoupmadeonly from stonesprovedto meanstonesplus meat andvegetables,so the Kaluza—Klein promiseof a unified field theory madeonly from gravity hasproved to meangravity plus a whole variety of matterfields with eachauthorchoosinghis favorite ingredients.But bycombininggravity andmatterinto onesimple superfieldto which no further supermattermaybeadded,d = 11 supergravityhasrealizedEinstein’s old dreamof replacingthe “basewood” of matter by the“pure marble” of geometry.Eleven-dimensionalsupergravityis “marble soup”!

However, we are running aheadof ourselvesbecause,as history would have it, this merger of

Mi. Duff et a!., Ka!uza—K!ein supergravity 11

supergravityandtraditional Kaluza—Kleinideasdid not takeplaceimmediatelyafterthe constructionofthe d = 11 theory. Instead,the existenceof higher-dimensionalsupergravitytheoriesin d = 11 andd < 11 was regardedmerely as a mathematicalcuriousity with no physical significance.To understandthis, we haveto recall the climate of thought in the supersymmetrycommunity aroundthis time. Amajor goal was to construct (in d = 4) the extendedsupergravityLagrangians,especiallythe N = 8theory:a gargantuantask.CremmerandJulia[37]wereable to achievethisby the mathematicaldeviceof “dimensionalreduction”, i.e., theywrote down the d = 11 Lagrangianandthen simply insistedthatall fields be independentof the extra7 dimensions.(From a Kaluza—Kleinpoint of view, “dimensionalreduction”correspondedto compactifyingon the seven-torusT7 andthen discardingthe massivestates.Further duality and other field transformationswere then performed to cast the theory into itsmanifestlyN = 8, E

7 x SU(8) symmetricform.) Dimensionalreduction was subsequentlyapplied to anumberof different supersymmetrictheoriesand proved an invaluableguide to the constructionofd = 4 Lagrangians.A variantof dimensionalreduction introducedby Scherkand Schwarz[137,138]alsoprovideda supersymmetry-breakingmechanism.

Trivial dimensionalreductionwas also applied to the “string” theorieswhich had, as a matter ofinterest, provided anotherway of introducing higher dimensionsinto physics. The original bosonicstring [139]was known to be consistentonly in the critical dimensiond = 26, and the fermonic stringonly in the critical dimensiond = 10 [12—14].Despitethe amazingdiscoverythat the zero-slopelimits ofthesetheoriesyielded gravity and supergravityrespectively [140, 141], the idea of treating the extradimensions as physical and using the equationsof motion to generatea dynamical Kaluza—Kleincompactificationwas neverfully exploited.

By a strangequirk of fate, therefore,the mergerof higher dimensionsand supersymmetrywhichshouldhaveprovided aKaluza—Klein Renaissanceinsteadproducedthe Dark Ages.Extradimensions,though useful, were relegatedto the position of a mathematicalcuriosity to which no physicalsignificance need be attached.Attention was focusedon supersymmetryin d = 4 where interestingthings were happening.In particular,it was realized that the N(N— 1)/2 Abelian vector fields of Nextended supergravitymight be gauged, i.e., they could be converted into non-Abelian 50(N)Yang—Mills fields with coupling constante in a supersymmetricway provided other e-dependentmodificationsof the Lagrangianand supersymmetrytransformationrules weremade.Das and Freed-man[142]first constructedthe N = 2 theoryanddiscovered,amongstother things,that it was necessaryto includea cosmologicalconstantA satisfying

4ITGA = —3e2. (1.2.4)

Subsequently,de Wit and Nicolai [38, 39] constructedthe gaugedN = 8 Lagrangian,with a localSO(8)x local SU(8) symmetry obtainedby gaugingthe S0(8) subgroupof Cremmerand Julia’s E

7.Once again,the cosmologicalrelation (1.2.4)obtained.

In the meantime,back in non-supersymmetricextradimensions,valuableadditionsto the Kaluza—Klein programmewere being made. One vital one was the idea of “spontaneouscompactification”[143—145,283], i.e., to look for stable“ground-state”solutionsof thefield equationsfor whichthe metricdescribesa productmanifold M4 X Mk whereM4 is four-dimensionalspacetimewith the usualsignature(— + + +) and Mk is a compact “internal” spacewith Euclideansignature(+ + +~. .). As shown byCremmer,Horvath,Palla andScherk[146]it was necessaryto augmentpuregravity with matterfieldsin order to achievea satisfactory compactification.Of course, this raised the whole “stone-soup”problem; a problem which was waiting to be cured by supergravity.Another problematical, but

12 Mi Duff eta!., Ka!uza—K!ein supergravity

intriguing, effect was the appearanceof aspacetimecosmologicalconstantrelatedto the radiusm~ofthe compactdimensions

A — —m2. (1.2.5)

Indeed, one of the observationswhich stimulated the presentauthors’ interest in Kaluza—Kleinsupergravitywas that combining (1.2.3) with (1.2.5) yields a formula like (1.2.4). Could the gaugedextendedsupergravitytheoriesbe relatedin someway to supergravityin higherdimensions?

Our growing convictions that the developmentsin Kaluza—Klein and in supergravity should bemergedwere confirmed by threepaperswhich, in our opinion, markedthe beginning of the presentRenaissancein Kaluza—Klein. The first by Freundand Rubin [18]pointedout that the threeindexfieldof d = 11 supergravity,AMNP (M, N, P = 1... 11) provideda dynamicalmechanismwherebysevenofthe eleven dimensionscould compactify spontaneously,the field equationsadmitting solutionsof theform

_13me~= ,a=l...4,FMNPQ — . (1.2.6)

‘. 0 otherwise,

where FMNPO = 43EMANpQ]. As discussedin section 3.1, when substitutedinto the d = 11 Einsteinequationsoneobtains

R,.,.= —12m2g~~, (1.2.7)

Rmn= 6m2gmn, (1.2.8)

where m, n = 5... 11. In otherwords the spacehad factorizedinto a d = 4 Einstein spacetimewithnegativecosmologicalconstantanda d = 7 compactinternalEinstein spacewith positive cosmologicalconstant.The equation

F,,.,~.= ~ (1.2.9)

had already appearedin papersby Duff and van Nieuwenhuizen[147] and by Aurilia, Nicolai andTownsend[148]in the contextof cosmologicalconstantsfrom three-indexfields in four dimensions,butthe miracleof the Freund—Rubintrick was to use it to singleout four from elevensinceFMNPQ hasrankfour. The secondpaper,by Witten [26], consideredthe search for a realistic Kaluza—Klein theory. Itsprimaryaim was to look at compactificationsyielding gaugegroupscontainingtheSU(3)x SU(2)x U(1)of the standardmodel andthe problem of chiral fermion representations.Although it did not concernitself much with solutionsof the d = 11 field equations,it did emphasizethe relation betweentheisometrygroupG of cosetsG/H andthe Yang—Mills groupin d = 4. This relationwasfurtherclarifiedin the third of thesepapersby Salamand Strathdee[149]who laid the foundationsof much of thesubsequentresearch.

All this sparkedoff our observation[21, 29, 40, 45] that the seven extra dimensionsof d = 11supergravity could, via the Freund—Rubin ansatz,yield solutions of (1.2.7) and (1.2.8) of the form(d = 4 AdS) x S7, andthat sinceS7 has isometrygroupSO(8),thiswould give rise to a d = 4 theory withSO(8) invariance.From (1.2.3) and (1.2.7), the gaugecoupling e, the inverseS7 radius m and cosmo-

Mi Duff eta!., Ka!uza—K!ein supergravity 13

logical constantA arerelatedby e2 ‘-= m2andA = —12m2. Hence,we obtain(1.2.4). It was not difficult toprove that S7 also admits8 Killing spinorsandhencegives rise to N = 8 supersymmetryin d = 4, (seesection 4.2). It was natural to conjecturetherefore[21, 29, 40, 45] that the masslesssupermultipletofspins (2, ~, 1, ~, 0~,0) in the SO(8) representations(1, 8, 28, 56, 35, 35) correspondedto the gaugedN = 8 theory of de Wit andNicolai [38,39].

This was the startingpoint of thesubsequentwork set out in this PhysicsReport.

1.3. TheKaluza—Kleinrecipe

Froma modernperspectivea Kaluza—Klein theory works as follows:(i) We start with gravity ~ (M, N = 1... d) plus matter fields (denotedcollectively by ~b)in

dimensiond = 4+ k (signature— + ++ +~. .), describedby the d-dimensionalEinstein—Hubertaction:

S = f ddz~\/-gE ~ (1.3.1)

(ii) We now look for stable“ground-state”solutionsof the field equations(gMN) and (~) whichexhibit “spontaneouscompactification”,i.e., the metric (b.,’) describesa productspaceM

4 x Mk

r~~(x) 0 ](gMN(x,y))— I I’ (1.3.2)

i 0 gmn(y)i

whereM4 is four-dimensionalspacetimewith the usualsignature(— ++ +) and coordinatesx5’ and Mk is

a k-dimensionalcompactspacewith Euclideansignature(++ +~~) andcoordinatesym.Therequirementof maximalsymmetryfor the d = 4 spacetimeM

4 restrictsus to spacesof constantcurvature

~‘ 1,~i° o —o o1~/LP~~U~ ~ ~

andhenceto Einstein spaces~ = AL,, accordingto table 1.Maximal spacetimesymmetrymay still be maintainedby replacingthe direct productansatz(1.3.2)

by the “warped-product”ansatz

rf(y)L,,(x) 0 ](gMN(x, ‘)) = I I• (1.3.4)

L 0 gmn(y)~i

Table IMaximally symmetricspacetimes

A Spacetime Symmetry E>0 theorem? Susy?

>0 de Sitter So(1,4) No No=0 Minkowski Poincaré Yes Yes<0 AdS S0(2,3) Yes Yes

14 Mi Duff etaL, Ka!uza—K!ein supergravity

However, the crucial questionis whethera sensiblefour-dimensionaltheory can be obtainedin thisway. In particular, it is no longer clear whether thereis a canonical split into spacetimeand extradimensions.(To illustrate this problem comparethe direct product metric ds2= d02+d42 with the“warped-product”metric ds2 = do2+ sin2 0 dqY). For certainspecialchoicesof the “warp-factor” f(y),onecan obtaina sensibletheory in d = 4 andthesewill be discussedin chapter12. In what follows, weconfineour attentionto warp-factorone.As far as the extradimensionsareconcerned,we want Mk toyield interestingnon-Abeliangaugegroups,to solve the field equations,andto be compact(in ordertoguaranteea discreted = 4 massspectrum).Typically, this is mosteasilyachievedby taking Mk alsotobe Einstein,Rm~= cgmn.Two relevanttheoremsfrom thedifferentialgeometryof spaceswith Euclideansignatureare (a)compactEinsteinspaceswith c<0 havenocontinuoussymmetries[150],and(b)completeEinstein spaceswith c>0 are always compact [151]. All group spacesadmit at least one Einsteinmetric, as do a large numberof cosetspaces.(In d = 7 all coset spacesadmit at least one Einsteinmetric, excludingthosewith S1 factors[152,206].) Thereare also non-homogeneouscompactEinsteinspaceswith symmetries[153].

In summarythen, we would like A � 0 in order that the ground state admit a positive energytheorem and supersymmetry(table 1); and c> 0. Of course,not all higher-dimensionaltheorieswilladmit such a spontaneouscompactificationand the requirement that they do will place severerestrictions on the sort of theorieswe are interestedin. It should be emphasized,however, thathomogeneousEinstein spacesare not the only candidatesfor Mk and in chapter12 we shall discussinterestingcompactificationsof d = 11 supergravityon S7 which are neitherEinsteinnor homogeneous.

Finally, it has recently been claimed that the Kaluza—Klein programmegoes through with Mknon-compact [154—156].However, thesemanifoldsarenecessarilygeodesicallyincompleteif theyare tohavefinite volume. This problemandthe problemof thediscretenessof themassspectrumhavenot yetbeenfully resolved.

(iii) To determinethe spectrumof the four-dimensionaltheory,we first considersmall fluctuationsofthe d-dimensionalfields abouttheir ground-statevalues

gMN= (gMN)+EMN, (1.3.5)

(1.3.6)

We then substitute (1.3.5) and (1.3.6) into the equationsof motion and retain terms linear in thefluctuations.Following [21]eachfluctuationdenotedgenericallyby ~:(x, y) is thendecomposedas asum of termsof the form

(1.3.7)

whereVtm” (~)areeigenfunctionsof the massoperatorM2:

(1.3.8)

SeeSalamandStrathdee[149]andsection4.3 for adiscussionof harmonicexpansionson homogeneousspaces. In this way, we obtain an effective d = 4 theory with an infinite tower of massivestateswith massesm~quantizedin units of a fundamentalmass m R1 (R is the “size” of Mk)

together with a finite number of masslessstates(including the graviton) coming from the zero-

Mi. Duff etaL, Kaluza—Kleinsupergravity 15

eigenvaluemodesof (1.3.8). All stateshavespins �2. Thus theseextradimensionsneednot conflictwith one’s everyday sensationof inhabiting a four-dimensionalworld (with its inversesquarelaw ofgravitationalattraction)providedR is small. Having found a spontaneouscompactification,of course,one must checkthat the vacuumis stable,i.e., that all the statesin the tower havepositiveenergy.InMinkowski space,thismeansm~� 0. In AdS the problem is slightly moresubtle(seesection3.2). Thecriterion for vacuumstability in d = 11 supergravityis discussedin chapters5 and11.

(iv) If Mk hasa symmetrygroupG, i.e., if it admitsKilling vectorsK’,,, (i = 1 . . . dimG)

V(mK~z)= 0, (1.3.9)

then the masslessstateswill includeYang—Mills gaugefields with gaugegroupG. This is, of course,thewhole beautyof Kaluza—Klein theories,so let usexaminein somedetailwhy thisis true.Considerthed-dimensionalmetric gMN(X, y) and, in particular,the off diagonalcomponentg~,,(x,y). In the Fourierexpansionof step(iii) the lowestterm in the expansionlooks like

,~1,,,(x,y) = A~,’(x)K,,’(y)+.”, (1.3.10)

whereK,,t is a Killing vector. By a Killing vectorK we meanthat the ground-statemetric on Mk has

vanishingLie derivativewith respectto K, i.e.

~tK(gmn)—0. (1.3.11)

The correspondingK definedby

K’ K”1 3/ny” (1.3.12)

obey the Lie algebra

[K’, K’] = f’k K” (1.3.13)

appropriateto G. Now considera generalcoordinatetransformation

—~zM — ~M(z), (1.3.14)

underwhich

= ~~M5~N + ~N5~M. (1.3.15)

andfocusone’sattentionon the very specialtransformation

~M(x,y) = (0, E(x)Kmi(y)) (1.3.16)

with e’ arbitrary.Then from (1.3.15)we maycomputethe transformationrule for ~,,(x, y) andhencefrom (1.3.10)that for A,~1(x).We find

= c9~.s(x) — f’~~A,.,-’(x)e” (x). (1.3.17)

16 Mi Duffel aL, Ka!uza—K!ein supergravily

This is precisely the transformationlaw for a Yang—Mills field with gaugegroup G. Hence G is asubgroupof the d-dimensionalgeneralcoordinategroup.

We havesaid that the gaugegroupG is given by the isometrygroupof Mk. However, this statementrequiresmodification if the matterfields with non-zeroVEV in the ground-statetransformnon-triviallyunderG, i.e., if

11’K(d~)� 0, (1.3.18)

in which casethe unbrokengaugegroup correspondingto masslessgaugebosonsis given by somesubgroupof G.

Substituting the expression(1.3.10) into the d-dimensionalequationsof motion, one obtainstheYang—Mills equationsof motion for A,.,.” (x), with a coupling constante given by

e2=~-G/R2=- m2M2, (1.3.19)

whereG is thefour-dimensionalNewton’sconstantandM~is thePlanckmass~ 10~GeV). HereR isthe “size” of M~,an arbitrary integration constant(related to the constantc ~f Rmn = ~ Forexample in the caseof S7 compactificationof d = 11 supergravityR = m’ is just the S7 radius.However, for more complicatedgeometriesone must be more precise aboutthe meaningof R, andWeinberg[157]has shownhow this is done for an arbitrary geometrywith Killing vectorsin terms ofappropriateroot-mean-squarecircumferences.The precise constantsof proportionality in (1.3.19)dependcrucially on the field contentof thehigher-dimensionaltheory.For d = 11 supergravity,we willreturn to this in chapters12 and 13. Although at the classicallevel, the “size” of Mk is undetermined,Candelasand Weinberg [158] have pointed out that in a certain class of theories admitting acompactificationdue to one-loop radiative corrections,onemay calculateR and hence, in a realistictheory, the fine structureconstanta. Note, however,that oneobtainsa pure numberonly after finetuning the spacetimecosmologicalconstantto A = 0. An alternativeschemefor obtainingA = 0 in amore natural way and henceof calculatingfine structureconstantsas pure numbersis suggestedinchapter14.*

(v) One’sattitudeto the correctmagnitudeof the gaugecouplingconstante appearingin (1.3.19),i.e.the correctsize of the extracompactdimensionsdependson which physicalgaugebosonsonewantstoexplain.Conventionally,oneassumesthat with e of orderunity, R mustbe aboutthe Plancklengthandhencethe fundamentalunit of mass m must be about i0’~GeV. The gauge couplings are thusappropriateto a very high energyscale (and are presumablysubject to renormalization-groupcor-rections). All this meansthat the massivestatesin a Kaluza—Klein theory are very heavy and thus(alwaysassuminge— 1) beyondthe rangeof currentor foreseeableaccelerators.Thephilosophywouldthenbe to look for the observedelementaryparticlesamongstthe masslesssectorof the Kaluza—Kleintheory.Onewould thenhopethat thestatesmasslessattreelevel wouldacquiretheirsmallmasses(smallcomparedto iO’~GeV)via quantumeffects.For low-energyphysics,therefore,onecan ignorethemassivestates.This conventionalwisdomrequiresseveralqualifications,however.

First, including the massivestatesdrasticallychangesthe ultraviolet behaviorof the quantumtheory(seechapter14). Secondly,what appearsto be massivein the “false” vacuummay be masslessin the“true” vacuumandvice-versa(e.g., the “spaceinvaders”phenomenonof chapter8). We cannotdiscard

* See also the section on recent developments.

Mi Duff etaL, Kaluza—K!ein supergravily 17

massivestatesuntil we are sure that (,~MN) and (d3) describethe groundstate,and this is a difficultproblemin practice.Thirdly, someof the massivestates(which Kolb andSlansky[85] havechristened“pyrgons”) maybe stableagainstdecayinto lighter particlesin the Kaluza—Klein theory andhencegiverise to cosmologicalproblems[85].Fourthly, if oneregardsthe gaugebosonsas well as the quarksandleptonsnot aselementaryparticlesbut asboundstatesof theKaluza—Kleinpreonsasin section14.4,thenitis no longerclearthat massivepreonsshouldbediscarded.In thecaseof d = 11 supergravity,for example,preonswhich aremassivein the S0(8)phasemight be masslessin a chiral SU(8) phase.

Finally, although the discarding of the massive sector is often taken for granted in much ofKaluza—Klein literature,recentwork on the S7 compactificationof d = 11 supergravityhasforced ustoreexaminemoreclosely the wholequestionof whetherthis truncationis consistentand, if so, whetherthe resultingtheory is necessarilythe onewhich describesthe low-energysector.This is the subjectofchapter13.

1.4. Thed = 5 theory: Kac—Moodysymmetries

As a concrete,if somewhatunrealistic,exampleof the Kaluza—Klein procedureoutlined in section1.3 let usconsiderpuregravity in five dimensionsdescribedby the action

S=—~——1d~xd0V—gE, (1.4.1)

2iric2i

wherewe areusingcoordinates

= (x”, y) (1.4.2)

with M = 1.. . 5 andj.t = 1. . . 4. We areassumingagroundstateM4 x S

1, i.e.,four-dimensionalMinkowskispacetimesacircle of radiusm’, so that

0= my, 0�O�2ir. (1.4.3)

The action (1.4.1) is invariant under five-dimensional general coordinate transformationswithparameters,~M (z),

— j ~ ,~P.’ L 0ogz,4N — 0M ~ gPN UN c gMP ~ Up gMN.

Next considerthe changeof variables

— ~ + K2çt3A,,.A,, KçbA4,.

g~(z)—4 L . (1.4.5)

The periodicity in the 0 coordinatemeansthat the fields g~,.,.,,,A1. and~ maybeFourierexpandedin theform

~ g1.,,,,(x)et”°, A

1.(x,O)= ~ A1.,,(x)et”°, ~(x,0)= ~

(1.4.6)

18 Mi Duff eta!., Ka!uza—K!ein supergravity

with

g,,n = g1.~-~, (1.4.7)

etc.It is well-known that, after integrating over 0 and retaining only the n = 0 terms in the above

expansiononeobtainsa four-dimensionaltheory of a masslessspin 2, g1.,,0amasslessspin 1, A1.0 andamasslessspin 0, ~, with action [159]

S = Jd~xV-g0 {-~R(go)-~oF~,.,,oF1.~o_ 6K2~ 01. ~ (1.4.8)

where indicesare raisedandlowered by g1.,,0 and whereF~,,= 201,.. A,,1. This action is invariant undergeneralcoordinatetransformationswith parameter~‘ (x),

= ~O” ~1n’O + 3,, ~O’~’g1.~+ ~ 8 g1.,,0,(1.4.9)

~iA1.0=~1. ~ A~0+~0P a~A1.0, ~4o= ~:~P~

local gaugetransformationswith parameterK~~05(x)

= K’8,., ax), (1.4.10)

andglobal scaletransformationwith parameterA,

6A1.0= AA1.0, aq5o = —2A40. (1.4.11)

The symmetryof the vacuum,determinedby the VEVs

(g1.,,0)= ~1.~’ (A1.0)= 0, (4k) = 1, (1.4.12)

is the four-dimensionalPoincarégroupX l~.Thus,the masslessnessof g1.,,0 is due to generalcovarianceand the masslessnessof A1.0 to gaugeinvariancebut 4o is masslessbecauseit is the Goldstonebosonassociatedwith the spontaneousbreakdownof the global scaleinvariance.Note that the gaugegroupisR ratherthan U(1) becausethis truncatedn = 0 theory haslost all memoryof the periodicity in 0.

In this section,however,we wish to analyze the four-dimensionalsymmetriesof the theory whichresultsfrom retainingthe n ~ 0 modesof (1.4.6) andwhich describes,in additionto the abovemasslessstatesan infinite tower of charged,massive,purely spin 2 particles[149]with chargesq,, andmassesm,,

given by

qn = nim, m,, = InIm. (1.4.13)

To this end, and following [8], we also Fourier expandthe general coordinateparameter~M(z) of(1.4.4) in the form

Mi Duff eta!., Kaluza—K!ein supergravity 19

~1. (x, 0) = ~ (x)et”°, (1.4.14)

~5(x, 0) = >~~(x) et”°, (1.4.15)

with ~‘ = ~,. An importantobservationis that the assumedtopology of the ground state,namelyM4 X 5’, restrictsus to generalcoordinatetransformationsperiodicin 0. Whereasthegeneralcovarianceof (1.4.9)andthe local gaugeinvarianceof (1.4.10)simply correspondto then = 0 modesof (1.4.14)and(1.4.15) respectively,the global scaletransformationsof (1.4.11) is no longer a symmetry becauseitcorrespondsto a rescaling

= ~ (1.4.16)

combinedwith a generalcoordinatetransformation

= —A0/m, (1.4.17)

which is now forbiddenby the periodicity requirement.The field 4~ is thereforemerely apseudo-Goldstoneboson.

Just as ordinary generalcovariancemaybe regardedas the local gaugesymmetrycorrespondingtothe global Poincaréalgebraobtainedfrom the restriction

~1.(X)r a1. +w1.,,x”, (1.4.18)

wherea1. and w1.,, = —w,,1. areconstants,so the infinite parameterlocal transformationsof (1.4.14) and

(1.4.15) correspondto an infinite-parameterglobal algebra. To determineits propertieswe makeananalogousrestriction

~1.,,(X) = a,,1. + w1. x~, (1.4.19)

~,,5(x)=C,,, (1.4.20)

wherea,,1., w1.,..,, and C,, areconstants.The correspondinggeneratorsare

p1. = e”‘°O”, (1.4.21)

M,,1.~= emnO(x1.t9~’— xVO1.), (1.4.22)

= ie1”°30, (1.4.23)

andthey definethe following non-compactinfinite-parameterLie algebra[8]

[P~, P~]= 0, (1.4.24)

[M,,”~,Pm°~]= ~~P~+m — ?l1.°F~+m, (1.4.25)

20 Mi Duff eta!., Ka!uza—K!ein supergravity

[M,,”~,Mm’30~I= ?~1.’~M~°+m+ fl~M~m— n”M~~— ~1.0~M~±m, (1.4.26)

[On, Om] = (n — m)Q,,+~, (1.4.27)

[0,,, P~]= mP~+~, (1.4.28)

[0k,, M~fl= mM~im. (1.4.29)

The threeequations(1.4.24—1.4.26)definea Kac—Moodyextensionof the Poincaréalgebra.(Rememberthat n, m = —~, . . . , —1, 0, 1,.. . ,co). Equation (1.4.27) is the Virasoro algebra without a centralextension. Equations (1.4.28) and (1.4.29) describe a mixing between the generalized“internal”generators0,, andthe generalized“spacetime”generatorsP~and~

Although the abovealgebradescribesa symmetryof the four-dimensionalLagrangian,the symmetryof the vacuumdeterminedby the VEVs

(g1.,,)=tj1.,,, (A1.)=0, (4)=1, (1.4.30)

is only Poincaréx U(1). This finite dimensionalsubalgebramay, however,be enlargedto PoincaréxSO(1,2) sinceP~,Ms”, 0-,, Oo, 0, alsoclose.This S0(1,2) symmetrywasalreadyobservedby SalamandStrathdee[149].The relationbetweendiffeomorphismson 5’, theVirasoroalgebraandSO(1,2) waspreviouslydiscussedin [264].

Since the gaugeparameters4~(x)and ~(x) eachcorrespondto spontaneouslybrokengeneratorsexceptfor n = 0, it follows that for n � 0, the fields A1.,, (x) and j,, (x) arethe correspondingGoldstonebosonfields. The correspondinggaugefields g1.,,,,, with two degreesof freedomwill then eachacquiremassesby absorbingthe two degreesof freedomof eachvector GoldstonebosonA1.,, and the onedegreeof freedomof eachscalarGoldstoneboson 43,, to yield a pure spin 2 massiveparticlewith fivedegreesof freedom.This accordswith the observationthat the massivespectrumis purely spin 2 [149].In a somewhatdifferentcontext,the spontaneousbreakdownof generalcovariance,andtheappearanceof massivespin2 particlesvia vector andscalarGoldstonebosonshasbeenobservedbefore [160].

The five-dimensionaltheory outlined in this section is neither realistic nor typical of the Kaluza—Klein theoriesof interest,but it providesa usefultheoretical“laboratory” for illustratingcertainideas.Herewe list someof the lessonsto be learned.

Classicalfeatures

(1) To avoid ghostsandtachyons,extradimensionsmustbespacelike,i.e., + + signature.This is ageneralfeatureof Kaluza—Klein theories.

(2) There are three different metricswhich enterthe discussionin Kaluza—Klein theoriesand theyshouldnot be (but frequentlyare!) confused.

(i) Thefull metric

g~.,’(x,0) = ~-1/3 [~1.~+ K2ç6A

1.A,, K~A1.] (1.4.31)KçbA,,

whereg1.,,, A1. and4 dependon x and0. This describesthegroundstateplus both massless(n = 0) andmassive(n � 0) modes.

Mi Duff eta!., Ka!uza—K!ein supergravity 21

(ii) TheKaluza—Klein“ansatz”As above,but discardthe massivemodes.This describesthe “low energy theory” of graviton g1.,,0,

photonA1.0 andscalar~. In the caseof S’(but not in general)this “ansatz”is the sameas“dimensionalreduction” [161],i.e., taking fields to be independentof the extracoordinates,discardingthe n � 0modesin (1.4.6) so that g1.,,, A,.. and~ dependonly on x.

(iii) Theground-statemetric

(gMN)=[0 ~1. (1.4.32)

This describesthe VEV of gMN(x, 0) andcontainsneithermasslessnor massivefluctuations.It is this“ground-state”metric (iii) which determinesthe unbrokensymmetriesof the vacuum,not the Kaluza—Klein “ansatz” (ii) nor the full metric (i).

(3) It is inconsistentwith the d = 5 field equationsto set 4~’= 1 (asis sometimesdonein theliterature)becausethe *~oequationis

LII (In 4~)= ~K2t~

0F1.,,0F~~, (1.4.33)

andq5o = 1 would imply F

1.,,0F~~= 0. Again oneshouldnot confuse4’ or 4’°with (4’). Scalarfields playacrucial role in Kaluza—Klein theoriesandareespeciallyimportantin the compositeapproachto gaugebosonsof section 14.4.

(4) The M4 x S1 groundstateof the d = 5 theoryyields an AbelianU(1) gaugegroup.As discussedin

section1.3 thisis atypical.(5) Equally atypicalis the fact that the masslessfields are singletsunder the gaugegroup(i.e., that

g1.,,0, A1.0 and 4’°are electrically neutral).Moreover, the appearanceof 4~as a (pseudo-)Goldstone

bosonis alsothe exceptionratherthan the rule. Seethe remarkson d > 5 below.(6) The d = 5 theory hasboth spacetimeandextradimensionsflat. 5’ is consistentwith (thoughnot

implied by) the equationsof motion. d = 4 for spacetimeis chosenby hand.Noneof thesefeaturesistypical. For exampled = 11 supergravitynaturally yields (d = 4 AdS)x M7 as a consequenceof the fieldequations.

(7)TheRicci-flatnessof theextradimensionsimpliesno potential V(4.’) for thescalarfields. This in turnmeansthat (i) the cosmologicalconstantis zeroand(ii) thereis no Higgsmechanism.Both featuresareatypical. Seesection9.1.

Extension to d >5(1) RMN = 0 is consistentwith groundstateM4 X T” whereT” is the k-torus

T” =S’xS1X~~S’ (1.4.34)

k times

and is metrically flat. The gaugegroupwould be G= [U(1)]”.(2) The countof masslessmodesand degreesof freedomis

lspin2 : 2

kspinl : 2k

~k(k+1)spin0 : ~k(k+1)

wherek(k + 1)12 spin 0 comesfrom the k(k+ 1)/2parametersof T”. The connectionbetweenmassless

22 Mi Duff etaL, Ka!uza—Klein supergravily

scalarsand parametersis peculiarto Ricci flat spaces(seesection9.1). Note that the total numberofmasslessdegreesof freedomis

2 + 2k + k(k+ 1)/2 = (4+ k)(1 + k)/2, (1.4.35)

which equalsthe degreesof freedomof the gravitonin 4+ k dimensions.This matchingis not true forgeneralKaluza—Klein theories.

(3) Just as for S1, all masslessmodesare singlets under the gaugegroup for T”, and the scalarsparameterizethe non-linearu-modelGL(k,R)/SO(k),see [37]. SL(k,R) is the subgroupof the d = kgeneralcoordinategroup

y’ —~A’, y’, detA = 1 (1.4.36)

for constantA’,. The GL(k,R) is madeup of this SL(k,R) combinedwith a scalingas in (1.4.11). Again,thesescalarsare only pseudo-Goldstonebosonssince the symmetry is destroyedwhen the massivemodesareincluded.

Inclusionoffermions(1) For fermions we needthe vielbein eM ratherthan the metric (M = general coordinateindex,

A = Lorentz index) e.g.,

(det ~M )t/Ie A~ DM41, (1.4.37)

wheref~ = (y~*y5) is the d = 5 Diracmatrix, (that is how y~got its name!) and

DM = 3M — ~WM[FA, “B]. (1.4.38)

Now the unbrokengauge group is given by the invarianceof (EM”) rather than (gMN), i.e., it ~5

compoundedof generalcovananceand local Lorentz invariance[149].Hencethe y1.A,.. couplingof thephotoncomesfrom both the r~3M andFM~Mterms.Onefinds (a) an infinite tower of massivespin~with m,, = nlm and q,, = n,m, (b) massterms of the y5 type mt/iy5i/i, (c) Pauli terms of the y5 typet/i

75u

1.,,F1.V4,,(d) neitherbreaksP nor T sincethe y5 canbe removedby 4, —~(expiy5ir/4)4t. (This fails if

we startwith ad = 5 massterm.This is not the result of thereductionbut merelyreflectsthefact that,for d odd, masstermsbreakdiscretesymmetries.)

DegreesoffreedomAn importantconceptin Kaluza—Klein theoriesis the counting of on-shell degreesof freedom of

variousfields in d dimensions.In addition to the familiar scalar,vectorandd-bein,we will also requirethe counting for antisymmetric tensor gauge fields. These are given in table 2. These numberscorrespondto thedimensionof irreduciblerepresentationsof the little groupfor masslessparticlesin ddimensions,namely SO(d— 2). Furtherreductionis possiblein d = 2 mod4 for tensorsof rank (d — 2)/2,

whose field strengthsmaybe split into self-dualand antiself-dualparts.The fermiondegreesof freedomin table2 correspondto Diracparticlesin d dimensions.Thesemay,

however,correspondto reduciblerepresentationsof SO(d— 2). In all evendimensionswe maydefineWeyl spinors with half the number of degreesof freedom. For certain dimensionswe can define

Mi Duff etaL, Kaluza—Kleinsupergravity 23

Table 2Degreesof freedomin d dimensions

d-Bein e~ ldd —3)

Gravitino ‘I’M 2”(d —3)Vector B,,, (d—2)Spinor x 2”Scalar 1

For Dirac spinors a = I’2 if d is even anda = (d — 1)/2 if d is odd.

Antisymmetric tensorgaugefields

AMNP ~(d— 4)(d — 3)(d —2)AMN — 3)(d —2)AM (d—2)

A 1

(pseudo-)Majorana[161—163]spinorswith half the numberof degreesof freedom.In d = 2 mod 8 wecan defineMajorana—Weylspinorswith onequarterthe number.For the connectionbetweencasuality,spin andstatisticsin higher dimensions,see [164].

2. Why supergravity?

2.1. ReasonsforN = I supergravityin d = 11

We maintainedin section 1.1 that the only viable Kaluza—Klein theory is supergravityand that theonly way to do supergravityis via Kaluza—Klein. In this sectionwe shall attempt to substantiatethisclaim. Some of the reasonsareas follows.

(i) Themaximum dimensionin which onecan consistentlyformulateasupersymmetricfield theory iseleven [11]. This d = 11 theory of Cremmer,Juliaand Scherk [10]describesthe interactionof gravityeM with the Majoranagravitino ~“M andthree-indexgaugefield ~ Fromtable2, thesefieldshave44, 128 and 84 degreesof freedomrespectively.The Lagrangianandits transformationrulesare givenin section2.2. The continuoussymmetriesare d = 11 generalcovariance,local N = 1 supersymmetry,local SO(1,10) Lorentz invariance,and Abelian gaugeinvarianceof AMNP. There is also a discretesymmetrywhereone performsan odd numberof spaceor time reflectionsandsendsAMNP to ~(Thesignatureis — + + + + + + + + + +). All attemptsto modify the theory in any way, for examplebyreplacingthe three-indexfield by a six-indexfield or by including a cosmologicalconstanthavefailed[165].Moreover, coupling to matter is forbidden by supersymmetry.The theory is almost certainlyuniqueif we demandsecond-orderfield equations,andthereis only one parameter,the gravitationalcoupling [281].Eitherthis is the theory of everythingor it is wrong: thereareno half-measures!In ouropinion, the restrictionof the choiceof dimensionandthe restrictionof thechoiceof matterfields andtheir interactions implied by supergravityare essentialingredientsin any successfulKaluza—Kleinunification.

(ii) The basic idea of Kaluza—Klein is that what we perceive to be internal symmetriesin fourdimensionsare really spacetimesymmetriesin the extradimensions.Carrying this logic to its ultimate

24 Mi Duff eta!., Kaluza—K!ein supergravity

conclusion would demandthat all symmetries in nature are spacetimesymmetries.This extremeKaluza—Klein philosophy is realized by d = 11 supergravityboth in the continuous and discretesymmetries(eachd = 11 “particle” is its own “antiparticle”). This extremecasewould not be true forN> 1 supersymmetrywhich would require d < 11 dimensions,nor for any Kaluza—Klein theory withYang—Mills fields presentalreadyin the extradimensions.A purist might arguethat the Abeliangaugeinvarianceof AMNP is not a spacetimesymmetry.However, BarsandMcDowell [166]haveshownhowthe gM~/A~~~gravity—mattersystemmay be reinterpreted,in first-orderformalism, as a pure gravitytheory with torsion. In this senseeventhe AMNP invariancemayberegardedas a spacetimesymmetryof puregravity! (Such an interpretationis possibleonly for three-indexfields sincethe torsiontensorisrank 3.) Seealso the sectionon recentdevelopments.

(iii) As discussedin section1.2, modernapproachesto higher-dimensionaltheoriesdemandthat theyexhibit spontaneouscompactification.Compactnessensuresthat the d = 4 massspectrumis discreteandalso guaranteescharge quantization. (For remarks on non-compactness,see section 1.3.) Higher-dimensionaltheorieswhere the extradimensionsare compactare indistinguishablefrom four-dimen-sional theorieswith a very specialmassspectrumandthisis why Kaluza—Kleincould berealisticdespitethe science-fictionovertonesof extradimensions.However, not all higher-dimensionaltheoriesexhibitspontaneouscompactification.In d = 11 supergravityspontaneouscompactificationnot only works but,as describedin section3.1, the dimension(d = 4) of spacetimeis now outputratherthan input.

(iv) Supergravity in d = 11 permits the possibility that the extra dimensions are topologicallyseven-spheres.As discussedin chapters7 and 8, S7 enjoys some rather unusual topological andgeometricalproperties.Remarkably, thesesomewhatabstractconcepts from differential geometrytranslateinto somethingconcreteand familiar in the effective four-dimensionaltheory. They cor-respondto a Brout—Englert—Higgs—Kibblemechanismwherebyscalarsand/or pseudoscalarsacquirenon-zeroVEVs, giving rise to a spontaneousbreakdownof gaugesymmetries,supersymmetriesanddiscretesymmetriesof C, P and CP [45]. Of particular interest will be the spontaneousbreaking ofN = 8 supersymmetryto N = 1 [22,53, 95].

Another attractive feature of 57 and of d = 11 supergravity is that it is the only non-trivialcompactificationof any Kaluza—Klein theory to d = 4 knownto avoidthe inconsistenciesof the Kaluza—Klein ansatz[78,96]. The possiblephysicalsignificanceof thisresult is discussedin section 13.3.

2.2. TheLagrangian, its symmetries,and the equations of motion

The Lagrangianof N = 1 supergravityin d = 11 is given with signature(— + + + + + + + + + +) by[10]

I _1 N M AB( \ ~. alt J”MNPr~ 11/ L

—4eeB eA MN ~ 21e1-Mi ~‘N[2I,W W)J ~

2— 4aeFMNPOF + (12)~erM1 - MIIF ... M4FM5 . - . M,A M9. M11

+ 4(12)2e[~MtM~~~I1N + 12 fXY~l,Z](Fwxyz + Ewxyz), (2.2.1)

wheree = deteM, ~P= ~pt fo, DM(w) =

0M — 4WM tAB and

0~MAB= ~(—11MAB + 11ABM — fiRMA) + KMAB, (2.2.2)

Mi. Duff etaL, Kaluza—K!einsupergravily 25

KMAB = ~E_!IIN!MABNP !1’~+ 2(!!’Mtfl ~PA— !PMFA !PB + ~PBFM !PN)], (2.2.3)

(1 A_’)0 A ‘224

— Lu[f.,e,,4]

WMAB = WM~+ ~“NFMAB ~ (2.2.5)

F,,,1~=

4OIMANPQ], (2.2.6)

= FMNPQ - 3~[MCNP~PQI. (2.2.7)

We areusingthe conventionthat M, N~P,... refer to d = 11 world indicesandA, B, C,... refer tod = 11 tangentspaceindices.~MNP is a tensor,ratherthan a tensordensity,and

El

2.. li—C. (2.2.8)

Note that we havedroppedthe hatson d = 11 fields but not on d = 11 C-matrices.

The F-matricessatisfy{CA, t~}= ~ (2.2.9)

where~ is the metric in thetangentspaceandCA,... A~ t[A1.. .tA~.The spinorsappearingin (2.2.1)areanticommutingandsatisfy the Majoranacondition

P= ~1’TC~, (2.2.10)

wherethe chargeconjugationmatrix C is antisymmetricanddefinedby

C’CAC= ~tA. (2.2.11)

Upon variation with respectto eM, ~PM and~ we obtainthe field equations[167]

D I-\ 1 ni -\ — 1r~~ ,~POR 1 i~ I?PQRS1

1’MN~,.W)—2g~rt~ — 3[1 MPQRUN — ggMNrpoRsu j,

= 0, (2.2.13)

r ( \ J~MPQR— — L PORM,. . . M~jM~WJ1 — 5766 1 M1 . . . M4

1 M

5. . . Mg, .

where

DM(w)~!’N = DM(ó~)!1’N+ TMPQRSEPQRS~PN, (2.2.15)

TSMNPQ = ...J(CsMNPO— 8C~~~I’g01S). (2.2.16)

We see that the appearanceof ~(W + ~) and~(F+ C) in the Lagrangianensuresthat only thesupercovariantá’ andF enterthe field equations.

26 Mi Duff eta!., Ka!uza—K!ein supergravity

The action andequationsof motion areinvariant underthe following symmetries.(a) d = 11 generalcovariance,with parameter~M:

aCM = eNA3M~N+ r,9NCMA, (2.2.17)

&~PM= !PNOMSE + ~N3N~pM (2.2.18)

=3A0[MNOP]S’° + ~3

0Aft.,ff.~~ (2.2.19)

(b) local S0(1,10) Lorentz transformationswith parameteraAB = —aBA:

a A B AOeM — ~CM a~,

a!PM = ~laAB!PM, (2.2.21)

aAMNP = 0; (2.2.22)

(c) N = 1 supersymmetrytransfonnationswith anticommutingparametere:

aCM = 1EtI~M, (2.2.23)

6~PM= DM(W)E, (2.2.24)

~iAMNP = ~eFEMN!t’P1; (2.2.25)

(d) Abeliangaugetransformationswith parameterAMN = —ANM:

aCM = 0, (2.2.26)

(2.2.27)

aAMNP = OEMANP]; (2.2.28)

(e) odd numberof spaceor time reflectionstogetherwith

AMNP —* -AMNP. (2.2.29)

This summarizesthe Lagrangianand transformationrules to be used in subsequentchapters.ForpossibleChern—Simonsmodification,seesection 14.1.

3. Spontaneouscompactificationto d = 4

3.1. TheFreund—Rubinansatz:Al4 xM~groundstate

We now justify the claims of chapters 1 and 2 that d = 11 supergravityadmits spontaneouscompactificationto d = 4. We start by looking for solutionsof (2.2.12—2.2.14)which mightbe candidatesfor a grqund state,at least at the tree level.

We are eventuallyinterestedin obtaininga four-dimensionaltheorywhich admitsmaximalspacetime

Mi Duff eta!., Ka!uza—K!ein supergravily 27

symmetry.With signature(— + + +) this means that the vacuum should be invariant under SO(1,4),Poincaré,or S0(2,3) as the cosmologicalconstantis positive, zero, or negative,correspondingto deSitter, Minkowski or anti-de Sitter space(AdS), respectively.Minkowski and AdS havethe virtue ofadmittingapositiveenergytheorem[27,28, 168—171],andpermit the constructionof an S-matrix [172].

The first requirementof maximal symmetryis that the VEV of any fermion field shouldvanishandaccordinglywe set

(3.1.1)

andfocus our attentionon the equationsfor gMN andAMNP which from (2.2.12)and(2.2.14)are

~ 1 i) — lEE’ £‘ POR 1 a’ E’PQRS11~MN— 2gMNI’ — 3[UMPQRJ N — 8gMNVpQRS’ J,

T7 c’1s4NPQ — — . MgNPQa’ a’— 5766 1 M,...M

41 M

5...M8 .

We look for solutionsof the direct productform M4 x M7, as describedin chapter1, compatiblewithmaximalspacetimesymmetry.Thus we set

(g1.,,)= k1.,,(x), (F1.,,,.,,.~)=~ (g~,,)=km,,(y), (Fmnpq) k,,,,~,(y), (3.1.4)

but

(g1.,,)= 0, (F1.,,,~)=(F1.,,pq) (F1.npq) 0, (3.1.5)

wherex1. arethe spacetimecoordinatesand ym the extracoordinates.The y-independenceof k

1.~ andthe x-independenceof ~mn in (3.1.4) are necessary0if gMN is to be a product metric, whereasthey-independenceof F1.,,~and the x-independenceof Fmnpq areconsequencesof the Bianchi identity

O[MFNPQRJO (3.1.6)

and (3.1.5).It is important to note that maximal spacetimesymmetry alonewould not rule out a “warped-

product” ansatzfor which (g1.,,(x, y)) = f(y)~1.,,(x). Note howeverthat sincethe Bianchi identity doesnot involve the metric, the y-independenceof F,,,,,~,is still guaranteed.In what follows we shall confineour attentionto “warp-factorone”, i.e., f(y) = 1. The casef(y) � 1 will be postponeduntil chapter12.

The ansatzof FreundandRubin [18] is to set

= ~ (3.1.7)

~7’mnpq0, (3.1.8)

wherem is a realconstantandthe factorof 3 is chosenfor futureconvenience.The superscriptzeroonthe Levi—Cevita tensor ~1.”$’~~ meansthat ~ = V~. Substituting (3.1.7) and (3.1.8) into the fieldequationswe find that (3.1.3)is trivially satisfiedwhile (3.1.2) yieldsthe productof a four-dimensionalEinsteinspacetime

28 Mi. Duff eta!., Ka!uza—Klein supergravity

= —12m2~1.,,, (3.1.9)

with Minkowski signature(— + + +) and a seven-dimensionalEinstein space

Rmn = ~ (3.1.10)

with Euclideansignature(+ + + + + + +).For futurereferencewe alsorecordtheform takenby thesupercovariantderivativeDM appearingin

(2.2.15), when evaluatedin the Freund—Rubinbackgroundgeometry.First we decomposethe d = 11F matrices

tA(y~®1,yS®Fa), (3.1.11)

where

{ya, Y$} = ~ (3.1.12)

{1’a, Fb} = 2ôa6, (3.1.13)

and where a, /3,... are spacetimeindices for the tangentspace group SO(1,3) and a, b,... areextra-dimensionalindicesfor the tangentspacegroup S0(7).Substituting(3.1.4), (3.1.5) (3.1.7), (3.1.8)and(3.1.11)into (2.2.15)we find that

0

= D,,.. + m’1.y5, (3.1.14)0 0

Dm= Dm ~ (3.1.15)0 0 0 0 a

wherey1.=e1. yaandFm=emFa.

Severalcommentsarenowin order.The constancyof m in the ansatz(3.1.7) is necessaryin order tosatisfy the field equations.However, the ansatz(3.1.8) is not necessaryand indeedwe shall considersolutionswith non-zeroFm,,,,~,in chapter10. The maximally symmetricsolution to (3.1.9) is, in fact,anti-deSitterspacesincethe cosmologicalconstant

A = —12m2 (3.1.16)

is negative.Thereareinfinitely manyseven-dimensionalEinsteinspacesM7 satisfying(3.1.10)and,for

the moment,M7 will be left arbitrary.The importantpoint is that completeEinsteinspacesof positivecurvature and Euclidean signature are automatically compact [151] and hence spontaneouscom-pactificationto d = 4 has indeedbeenachieved.Moreover,as FreundandRubinemphasize,the choiceof dimensionfour for spacetimeis not ad hoc but a consequenceof the field equationssince fornon-zerom in (3.1.7),maximalsymmetrysinglesout thefour-dimensionalLevi—Cevitasymbol.This, inturn, is dictatedby the requirementsof d = 11 supersymmetrythat the F tensorhaverank four. Ofcourse,as FreundandRubinalsopointout, onemight haveset Fmn,,,~proportionalto ~ andF,.,.,,,.~tozero andtherebyobtained(d = 7 AdS) x (compactM4. Is thereany mathematicalway to rule out thisobviouslyunphysicalvacuumstate?There is oneinterestingfeaturewhich differentiatesthe 4+ 7 from

Mi Duff etaL, Ka!uza—K!ein supergravity 29

the 7+4 compactifications.Setting F equal to the volume form on a non-compactspace Mk is

consistentwith the equation

F=dA, (3.1.17)

since k-dimensionalnon-compactspaceshavetrivial cohomologygroupH’~(Mk, R), whereassettingFequalto the volume form on a compactspacemeansthat (3.1.17) cannotbe globally valid, since H”(Mu, R) is non-trivial. In otherwords, if we were to insist that the four-indexfield strengthof d = 11supergravitybe globally derivedfrom a three-indexpotential then spacetimeis uniquely d = 4. We donot know whetherthis argumentcan be strengthenedbut it shouldbe notedin this connectionthat allattemptsto reformulated = 11 supergravityexclusively in termsof a seven-indexfield strengthderivedfrom a six-indexpotentialhavefailed [165].prom now on we shall assumea d = 4 AdS spacetime.Forthe d = 7 case,see[173,277]. Of course,the cosmologicalconstantA = —12m2 appearingin this tree-level approximationto thegroundstatewill prove to bemuchtoo large.We postponeadiscussionof thevalueof A in the true vacuumuntil chapter14.

Next, wenote that (3.1.3)maybecompactlywritten as

d(*F+AAF)=0. (3.1.18)

As noted by Page [57], this meansthat when the d = 11 spacetimehasthe M4 x M7 topology, the

integral

(*F+AAF) (3.1.19)

gives a conservedchargeP which is independentof the point x of spacetime.In the caseof a

Freund—RubincompactificationA A F vanishesand (3.1.7) yields

P = (3m/IT

4) V7, (3.1.20)

where

1/7=Jd~yV~ (3.1.21)

is the volume of M7. Note that P will changesign if in the Freund—Rubinansatzwe replace~ bythe equallyacceptablesolution —3m~1.,,~.It alsochangessign if we reversethe orientationof M7. Thisorientationreversal,sometimesreferredto as “skew-whiffing”, will prove importantfor supersymmetryin section8.2.

Finally, we notethat at the classicallevel the constantm appearing in (3.1.7) is merelyan arbitraryintegrationconstantundeterminedby the theory. One could take m to be zero in which casebothspacetimeandtheextradimensionswill beRicci flat (providedFmnpq = 0) andcompactificationof theextra

30 Mi Duff et a!., Ka!uza—K!ein supergravity

dimensions is no longermandatory.Compact,Ricci flat, manifoldsdo existhoweverandthesearediscussedin chapter9*~Notemoreoverthat, when m = 0, d = 4 is no longer singledout as the the dimension ofspacetimeand that solutionswith compactextra dimensionsare no longer the most symmetric, themaximally symmetricsolution being d = 11 Minkowski space.Solutionswith rn = 0 but Fm,,j~� 0 arediscussedin section10.1.

Now we discusssomeof the propertiesof the four-dimensionalAdS spacetime;the propertiesof theextra-dimensionalcompactspaceM7 will be treatedin chapter4.

3.2. Anti-deSitter spacetimeand OSp(41N).

We summarize in this section some of the useful propertiesof AdS, the maximally symmetricsolutionof (3.1.9).The readeris referredto [174,170—172,279] for details.AdS is the four-dimensionalhyperboloid

h)abZZ = —m2/8 (3.2.1)

embeddedin R5 with Cartesiancoordinatesz’~wherea = (a,4), and flat indefinite metric ~, = diag

(— + + + —). In polar coordinates (t, r, 0, 4’) the line elementmay bewritten

ds2 = —(1 + 4rn2r2) dt2+ (1 + 4m2r2)~dr2 + r2(d02+ sin20d4’2), (3.2.2)

whereOs t < IT/rn, 0 � r < oo, 0 � 0 < IT, Os4’<21r. The topology is 51 X R3 andthe metric hasclosedtime-like curves. However,we maypassto the universalcoveringspaceby insteadallowing t to rangefrom —~ to +c,~.This covering spacehas R4 topology and no closed time-like curves. Problemsassociatedwith the lack of a global Cauchyhypersurfacein the AdS geometrycan be resolved withsuitableboundaryconditions[172].

AdS may be regardedas the coset space SO(2,3)/SO(1,3) and the generatorsof the SO(2,3)isometrygrouparegiven by

L°~’= Za(3/OZb) — Z”(t9/f9Za). (3.2.3)

They satisfythe SO(2,3) algebra

[L~, L~~d]= - r~LM- + i~L”. (3.2.4)

Representationsof SO(2,3) can be denotedD(E0, s) whereE0 is the lowest energyeigenvaluewhich

occursands is the total angularmomentumquantumnumberof the lowestenergystate,analogoustothe massand spin of the Poincarégroup. The representationis unitary provided E0� s + 1 for s = 1,

andE0� s + ~for s = 0, ~. The representationsareall infinite-dimensional.It is important that thenotion of a particlemass be interpretedin the context of AdS. D(~,0) and D(1,~) are the so-called“singletons” [175,176] which have no Poincaré analogue; D(1, O)~D(2, 0) andD(s+ 1, s) aremasslessparticles; particleswith E0> s + 1 are massive. It is known that the masslesswave equationsareconformally invariant for s = 0, ~and 1 andgaugeinvariant for s � 1 [174].

* See also the section on recent developments.

Mi Duff etaL, Kaluza—Kleinsupergravity 31

Abbott and Deser [170]havediscussedthe definition of energyin spacetimeswhose metric tendsasymptoticallyat largeradii r to the metricgiven by (3.2.2) plus terms of order hr. To each generatorLth of (3.2.3) one may associatea surfaceintegralwhich maybe thoughtof asa generalizedmomentumor angular momentum. In particularthe L°4integralcan be interpretedas the total energy.Abbot andDeser[170]show that this energyis non-negativefor sufficiently regularasymptoticallyAdSspacetimes,vanishingif andonly if the spacetimeis exactlyAdS. Note, however,that the notion of total energyistied to a specific value of the cosmologicalconstant. This will prove important in the context ofKaluza—Klein because it meanswe cannot comparethe energiesof two AdS vacua with differentcosmologicalconstants.

Let us assumethat we havea theory of gravity coupledto matter fields. Maximal symmetry meansthat only the metric g

1.,,(x) and the spin 0 fields 41’(x) can be non-vanishingin the backgroundfieldconfiguration,so we focusourattentionon these.The equationsof motion are

R1.,,— ~g1.,,R= 2T1.,,, (3.2.5)

where

= ~1.41~V41+~g1.,,[3,,41I3*~41~— 2V(41)], (3.2.6)

—Li4’1—(OV/34”)=O. (3.2.7)

We look for backgroundsolutionswith g1.,,(x)= ~1.,,(x),the metric on AdS, and 41(x) = = constant

sothat (3.2.7) requiresthat we areat a critical point of the scalareffectivepotentialV(44.Comparisonwith (3.1.9) yields

A =_12m2=2V(~). (3.2.8)

Next, we decomposethe metric andscalarfield into backgroundplus fluctuations

g1.,,(x)= ~‘1.,,(x)+ h1.,,, 41(x) = ~/‘ + S(x). (3.2.9)

Substitution into (3.2.5) yields

— ~g1.,,R+ 12m2h

1.,,= 2(J1.,,+ T1.,,— 6m2~g

1.,,). (3.2.10)

The tilde refersto termslinear in h1.,,. Explicitly

= ~/iLh1.,, + V~V~h,,,,, — ~ (3.2.11)

where all covariant derivatives and contractions are with respect to ~‘1.~ and where ZIL is theLichnerowiczoperator

ZILh1.V = —El h1.,, — 2R1.~.~h~+ 2R~’h,,)~. (3.2.12)

J1.,, is the gravitational pseudotensorcontainingterms of secondand higher order in h1.,,. Since theleft-hand side of (3.2.10) obeysthe Bianchi identity

V1.(R

1.,, — ~g1.,,1~+ 12m2h

1.,,)= 0, (3.2.13)

32 Mi Duff eta!., Ka!uza—Kleinsupergravity

we obtain the conservationlaw

V1.(J1.~ + 7~,.— 6m

2~g1.~)= 0. (3.2.14)

The conservedenergyfunctionalof Abbott andDeseris now given by

E = Jd3x V~(J0V+ D~— 6m2~OV)K~, (3.2.15)

whereK~is the global time-like Killing vectorwhich generatestimetranslationisometry.A static field configuration is stable if the associatedconservedenergyfunctional is positive for

fluctuationsfor which the energyintegralconverges.Abbott and Deser[170]establishedthe positivityof E in (3.2.15)in the casethat the matrix

32V(3.2.16)~

‘I, ~

haspositive eigenvalues,i.e., when 4’ = ~/scorrespondsto a local minimumof the effective potential.However, Breitenlohnerand Freedman[27, 171] were able to show that an AdS backgroundis stablefor fluctuationswhichvanishsufficiently fastat spatialinfinity, not only whenthe critical point is a localminimum but alsowhen it is a maximum or saddlepoint providedthe eigenvaluesof 14, arenot toonegative.Stability can occurevenwhen V(41) is unboundedbelow.This is preciselythe situationwhichobtainsin Kaluza—Klein theories.

Before quotingthe Breitenlohner—Freedmanstability criterion, we first discussapparentmasstermsin AdS for fields of spin 2, ~, 1, ~ and0~.In termsof non-interactingfields propagatingon AdS wedefinethe massmatrix M as follows. For spin 2 fields h

1..,,

LlLh1.~+ 2V(1.V”h~)~— V(1.VV)h~~+24m2h

1.~+ M2h

1.,.= 0. (3.2.17)

For spin ~ fields 411.,

y1.*’PDç/J — My

5y

1.”4i,. = 0, (3.2.18)

where

i5,.. = D1. + my1.7~. (3.2.19)

For spin 1 fields B1.,

LIB1. + V1.V~B~+ M2B,. = 0. (3.2.20)

For spin ~fieldsx~

y1.D1.~— Mys~= 0. (3.2.21)

Mi Duff eta!., Ka!uza—K!ein supergravity 33

For spin O~fields S,

zlS—8m2S+M2S=O, (3.2.22)

with a similar equationfor spin 0 fields P. Here ii is the Hodge—deRham operator(seesection4.3).The ~ in the fermion mass matrix is optional but appearsnaturally in Kaluza—Klein theories.Thedefinition of massis such that Al = 0 correspondsto gaugefields with only two degreesof freedomforspins2, ~ and 1 andconformally invariant waveequationsfor spins 1, ~andO~[27, 171, 174, 177, 178].Conformalinvarianceimpliespropagationon thelight cone;masslessspin2 and~do not propagateon thelight cone[179,180].(With theseassignmentsall membersof themasslessN = 8AdSsupermultiplethaveM = 0. It will turnout, however,thatnot all membersof amassiveAdS supermultiplethavethesameM)

We are now in a position to quote the stability criteria. For spins 2, ~, I and ~ the condition forclassicalstability [170] (i.e. no exponentiallygrowing modes)is that the (mass)2matrix be positivesemi-definite

M2� 0. (3.2.23)

For spins0~,however,exponentiallygrowing modesare avoidedprovided [27,171]

M2�—m2. (3.2.24)

We will makeuseof theseresultswhenwe discussKaluza—Kleinstability in section5.2.Now we turn to AdS supersymmetry.In a supergravitytheory the SO(2,3) invariant vacuummay, in

fact, be invariant under a largersymmetry,namely the global supersymmetryof the supergroupOSp(4/N) where0 s N � 8. By aprocedureanalogousto the constructionof the Killing energyE = L°4, theSO(2,3) generatorsL” are associated with the 10 Killing vectorsK~(x)on AdS satisfying

V(1. K~= 0. (3.2.25)

Moreover,the spinorialgeneratorsQA (A = 1.. . N) of OSp(4/N)areassociatedwith the 4N “Killingspinors” 6A(x) satisfying

~1. e~O, (3.2.26)

where.151. is the samecovariantderivativeappearingin (3.2.19). This will be further justified in section4.2. The remaininggeneratorsof OSp(4/N)are the N(N — 1)/2 generatorsTAB of S0(N).Explicitly thesuperalgebrais given by (3.2.4) togetherwith

{QA ~B} = 2ô/~y,,.,,L’~”+ T~, (3.2.27)

[L~, QA] =2yabQA (3.2.28)

[TAB, TCD] = ,

5BCTAD — 5ACTBD — ÔBDTAC + ,5ADTBC (3.2.29)

[TAB, QC] = oBCQA — 5ACQB (3.2.30)

34 Mi Duff eta!., Ka!uza—K!ein supergravity

where

= [Yab Ya] (3.2.31)

Yb 0

TheindicesA,B,... areS0(N)vectorindices.In the specialcaseof N = 8, however,we areat libertyto replacethem by spinorindicesI, J,... or conjugatespinorindicesI’, J’,... by virtue of triality. Seesection7.1.

In the casethat the backgroundgeometryadmitsan unbrokensupersymmetry,i.e., a Killing spinor,we may use the OSp(4/N) algebra to prove that the Killing energy is positive not only for smallfluctuationsbut for all excitationswhich are asymptoticallyAdS [27,28, 171]. Multiplying (3.2.27)by y°andtracing,we find

L°4=TrQ2�O. (3.2.32)

This quantumoperator relation can be applied at the tree level [28] as was done for Poincarésupersymmetryin [181,182] to prove classicalstability. A moredirect proof remainingat the classicallevel is given in [28].

In the caseN = 1, all linear unitary irreducible representationsof AdS supersymmetryhavebeenclassifiedby Heidenreich[183].They are

Class1: D(~,O)~D(1,~); (3.2.33)

Class2: D(E0,0)+ D(E0+ ~,~)~ D(Eo+ 1, 0), E0 > ~ (3.2.34)

Class 3: D(s+1,s)~D(s+~,s+~), ~ (3.2.35)

Class4: D(E0, s)~ D(E0+ ~,s + ~)~ D(E0+ ~,s— ~)~ D(E0 + 1, s). (3.2.36)

Classhis the singletonsupermultipletwhich was, in fact, first discoveredby Dirac in 1963 [175];it hasno counterpartin Poincarésupersymmetry[176].Class 2 is the Wess—Zumino[171] supermultipletwhichis massless(Al = 0) for E0 = 1 andmassiveotherwise.Class3 is themasslessgaugesupermultipletwith spinss and s + ~with s � ~. Class4 is the massivehigher spin supermultiplet.

The correspondingstudy of OSp(4/N) representationswith N> 1 was neglectedin the literatureuntil their importancein Kaluza—Klein supergravitybecameapparent[44, 53]. For example, corn-pactification on the round S

7 discussedin chapter7 leads to massiveN = 8 supermultipletswithmaximum spin 2. This correspondsto a new type of multiplet shorteninganalogousto the shorteningdue to central chargesin Poincarésupersymmetry,except that in AdS supersymmetrythe S0(N)automorphismgroup is maintained.This is discussedin detail by Freedmanand Nicolai [184].Twofeaturesemerge:(1) OSp(4/N)multiplets maybe decomposedinto the OSp(4/l)multiplets given above;

Mi Duff etaL, Ka!uza—K!ein supergravity 35

(2) In the limit that the radiusof AdS tends to infinity (i.e. as m —p0) and the OSp(4/N) algebracontractsto the N-extendedPoincarésuperalgebra,all short OSp(4/N) rnultiplets becomemassless

Poincarésupermultiplets.Seealso [279].As an illustration of someof the ideasdiscussedin this sectionwe write down the action of the

Wess—Zuminomultiplet consistingof real scalar,pseudoscalarandspinorfields (S,P, x);

= _~V_g{g1.~~a1.so~S+ g1.”3

1.P8,,P+ ~y1.D

1.~

+ (,a2m2— 2,arn2—8m2)S2+ (ji2m2+ 2~m2— 8m2)P2 + ~ (3.2.37)

The parameter /2 in the Lagrangian is related to the mass Al by the formulae

M~=~ —2)rn2, M~= ~2m2, M~=~e + 2)m2. (3.2.38)

Nowwe make useof the Breitenlohner—FreedmanrelationsbetweenE0 and ,~i,

= — i~, Eo,,. = ~±~jpi, E0~= ~±~ji~+ ii, (3.2.39)

to yield

E0~= ~±~V(M/m)2+ 1, Eo~= ~±~IM/mJ, E0~= ~±~\/(M/m)2+ 1, (3.2.40)

The ± sign must always be chosensoas to makeE0 � s+ ~but this still permits a residual ambiguity insome cases. See Breitenlohner and Freedman [27] and Hawking [185].The natural generalizationstohigher spin are [80]

E0

~±~\/(M/m)2+1 0~±~IM/mI~+~~s./(M/m)2+1 1~+~IM/m—2I~+~V(M/m)2+9 2 (3.2.41)

It is interestingto notethat only the + sign is permittedfor s � 1 andhenceN = 4 supersymmetry(butnot N <4) would determinethe signsfor the remainingmembersof thesupermultiplet.

4. Propertiesof M7

4.1. Symmetries, group spaces and coset spaces

So far, we havedemandedonly that the extradimensionsof our d = 11 theorybe a compactd = 7Einstein manifold and here we discuss some further restrictions one might impose. Weshall consideronly orientable manifolds. As discussed in chapter 1 a necessary condition for massless Yang—Mills

36 Mi Duff eta!., Ka!uza—K!ein supergravity

fields in d = 4 is that the metric on M7 admitsa groupG of motions, i.e., that thereexistKilling vectors

V(mKn)’ = 0, (4.1.1)

where i = 1 . . . dimG. G is called the isometry groupof M7. Introducing

K = Kt”9/8y”, (4.1.2)

the K satisfy the Lie algebra

[K’, K’] = f”kK”, (4.1.3)

wheref”~are the structureconstantsof G. The statementthat the isometrygroup of a metric g,,,,,is 0 may equivalentlybe expressedby the statementthat g,~hasvanishingLie derivativewith respectto the Killing vectorfields K’;

~Kgmn = K”t9pgmn + 3m1~gpn+ ônK”gmp = 0. (4.1.4)

Oneexampleof a spacewith symmetrieswould bea groupmanifold. In order to fix our ideasaboutKilling vectorsand isometrygroupsit will be instructiveto examinein detail a d = 3 examplenamelyS3= SU(2). Let U be an arbitrary element of SU(2)which we parametrizein termsof coordinatesym(m = 1, 2, 3) chosenfor convenienceto be the Eulerangles(0, 41, i/i)

— I cos0/2e’~~~2 sin 0/2e~~21 4 1 5)U — I—sin 0/2e~~’2 cos0/2e~~2J’ (

where0� 0 s ~r,0� 41 ~ 2ir, 0< 4, s4IT. Thethreeleft-invariantone-formso~are definedby

U~dU = (4.1.6)

whereTa arethe SU(2) generators

[Ta, Tb] = Ea~Tc. (4.1.7)

Theleft-invarianceof o~isseenbynotingthatit is unchangedunderU —* AUwhereA is aconstantSU(2)matrix. By exterior differentiationof (4.1.6) and using(4.1.7), we obtain the Maurer—Cartanequation

1 bU0 — 26abc° A tT . .

Explicitly, we have

u1=cos4,dO+sin4isinod41, o.2 sin4,dO+cos41sin0d4’, o3=d41+cos0d4’.(4.1.9)

Mi Duff etaL, Ka!uza—K!ein supergravily 37

Wedefine the left-invariant vector fields La = La~(aIt9ym)as the dualsof o~:

(~JL

6) ô~. (4.1.10)

Explicitly

sin4, cos41L1=cos4,a0+—--—3~—sin41cot0a~,L2= —sin41t99+--———3~—cos41cot08~,,L3= a~..

sin0 sin0

(4.1.11)

It follows that theyobey the SU(2)algebra

[La, Lb] = Ea&Lc. (4.1.12)

Similarly, one may define right-invariant one-formsand vector fields, which are unchangedunderU—* UB

1, where B is a constantSU(2)matrix, by replacingU1 dU in (4.1.6) by dUU1. The rightinvariantvectorfields Ra sodefinedsatisfy

ERa,Rb] = ~6ajjcRc. (4.1.13)

Moreover, sincethe left andright multiplicationsareindependent,we have

[La,Rb]0. (4.1.14)

The bi-invariantmetric g,,,, on S3 is now given by

ds2= gmndy dy~= —2 Tr(U~ dUU~dU), (4.1.15)

so-called because it is unchanged under both left (U-+ AU) andright (U—t’ UB~)actionsof the group.It maybe written in termsof the familiar dreibeinse~as

— a begmn—em Cn 0ab,

where

em”dytm =c”, (4.1.17)

However,we could equallywell haveused the right-invariantdreibeins.Hence,the isometrygroupofthis bi-invariant metric on S3 is SU(2)x SU(2) S0(4). In other words, the vector fields ~

A = (0, a), in the adjoint representationof SO(4),given by

K°” = ~ — ~a, Kab = eaI~(_R*~+ LC), (4.1.18)

are Killing vectorssatisfying (4.1.3). Thus, a Kaluza—Klein theory based on S3 with its bi-invariant

38 Mi Duff eta!., Ka!uza—K!einsupergravity

metric (4.1.16) would describe the Yang—Mills gauge bosons of SU(2) x SU(2)andnot merely SU(2) asis sometimes claimed. (See, however, section 13.1.)

Now, however, consider the metric (4.1.16)but wherethe ea = e’~rndytm are rescaledby constantsA1,

A2 andA3,

e1=A

1o~, e2=A

2o-2, e3=A

3o-3. (4.1.19)

We shall refer to this metric as a “squashedthree-sphere”to distinguish it from the “roundthree-sphere”describedby the bi-invariant metric. Although this squashedS3 is still manifestlyinvariant under the left actionof SU(2)it is no longerinvariant underthe right action of the full SU(2)for generalA

1, A2 andA3. Thereis a residualU(1) right action if two of the A’s arechosenequalandwerecoverthe full SU(2) right actionif all threearechosenequal.For all valuesof A1, A2 andA3 the SU(2)left action is transitive and so the squashed53 is still a homogeneousspace, though no longer asymmetricspace.(Transitivity, homogeneity,symmetricspacesandcoset spacesare discussedbelow.)For example, considerthe caseA1 = A2 = 1 and A3 = A then from (4.1.9) and (4.1.19) the line elementtakesthe form

ds2= d02+ sin20 dq52+ A2(d41+ cos0 d41)2, (4.1.20)

andthe SU(2)x U(1) Killing vectorsaregiven by R” and L3. TheRicci tensorhasconstant~igenvalues

12A2 0 flRab=~I 0 2—A2 0 ~, (4.1.21)

L o 0 A2i

andcontrary to one’sintuition, the scalarcurvature

R =~(4—A2) (4.1.22)

can passthroughzeroand becomenegativeas A2 is increased.(As amatterof interest, the spectrumofthe Diracoperatoron sucha squashedS3 hasbeendiscussedby Hitchin [186].Countingthezeromodesturns out to be a non-trivial exercisein numbertheory.)Note also that the round S3 admitsa discreteisometry 41~-~41 which, from (4.1.20), is no longer the case when the sphere is squashed.ThistransformationhasJacobian—1 andhencereversestheorientationof 53~We shall returnto thequestionsof squashingand orientationreversalin chapter8. We shall reservethe word “squashing” for homo-geneousdeformations.Of course, we could always distort the round 53 metric inhomogeneouslybreakingsomeor all of the SU(2)x SU(2).

Theabovediscussionof 53 maybegeneralizedto an arbitrarygroupmanifold. The isometrygroupofthe hi-invariant metric on a non-Abeliangroup manifold G is 0 x 0 correspondingto the left- andright-invariant vectorfields, which reducesto 0 X H, H C 0, if the metric is squashedby rescalingtheleft-invariantone-forms.Onemoral of all this for Kaluza—Klein is that the gaugegroupis determinednot by tbe topology of the space(e.g. S3)but by the metric we put on it. Note howeverfrom (4.1.21)that in our S3 exampleonly theroundmetric is Einstein.(In chapters7 and8 weshall examineS7, whichis not a group space, and find that it admits two Einstein metrics; one round and onesquashed.)

Mi Duff et aL, Kaluza—Kleinsupergravity 39

Moreover, returning to the physically interesting case of d = 7, wenotethat theonly groupmanifoldsofdimension 7 are products of 51~~and 53~~and that the product of a flat space and a curved space cannotbe Einstein. The only relevant example is the seven-torus T7 = [S~]~which is flat (m = 0) and whoseisometry groupis Abelian0 = [U(1)]7.(For Abeliangroups,left andright multiplication is equivalentand the isometry groupof the manifold is only 0 andnot 0 x 0.) For this reason,non-Abeliangroupmanifolds are of no importancefor d = 11 supergravityeven thoughthey always admit an Einsteinmetric. Insteadwe shall concentrateon homogeneousspacesdiscussedbelow correspondingto thecosets0/H whereH is somesubgroupof 0, e.g., 57, S5x S2 54 x 53, CP2x 53,

Note that productsof Einstein spacesarethemselvesEinsteinprovided their Ricci tensors havethesameeigenvalues.However,it shouldbe bornein mind that thereareEinsteinspaceswhich areneithergroup spacesnor cosetspaces,e.g., Page’stwisted S2 bundle over S2 [153]which has isometrygroupU(2). Suchspaceswould still yield non-Abeliangaugebosonsin a Kaluza—Klein theory.

The mostattractiveandeconomicalway to obtainYang—Mills fieldscorrespondsto thecasewhenM7

is a cosetspace.It is to cosetspacesthat we now turn. See,for example[30,113, 149, 187] for adetaileddiscussionof their geometry.We begin by defining a homogeneousspaceas onewhich admits as anisometry the transitiveaction of a group0. (This meansthat 0 mustbe containedin the full isometrygroup but doesnot necessarilycoincide with it (seebelow). A groupacts transitively if any point in thespacecan bereachedfrom anyotherby the group action.A classicexampleof a homogeneousspacewould be the k-sphere,S” with its standardmetric, which admits the transitiveaction of the groupSO(k + 1). For exampleS

2, the unit spherein ii~with Euclideancoordinates(~y, z) given by

x2+y2+z2=1 (4.1.23)

admitsthe transitiveactionof SO(3),the three-dimensionalrotation group,sinceanypoint (x, y, z) onS2 can be reachedfrom anyotherpoint, e.g.,(1,0, 0);

fx\ /1\y ~= R( 0 ~, R CS0(3). (4.1.24)

\z/ \o!

SO(3) is, in fact, the isometrygroup of S2. Therearein fact infinitely manyways of reaching(x, y,z)

from (1,0, 0), sincethereis an S0(2)subgroupof SO(3)which leaves(1,0, 0) fixed;

fh\ 1 of1\i 0 = [~ All 0 J, A C S0(2). (4.1.25)\oI \o/

Hence, if

R’ = R [~~], (4.1.26)

thenboth R’ and R take (1,0, 0) to the samepoint. This is a generalfeatureof homogeneousspacesand the subgroupH of G which leaves a point fixed is called the isotropy subgroup.Thus, the

40 Mi Duff etaL, Ka!uza—Klein supergravity

two-spherewith its standardmetric is the cosetspaceSO(3)/SO(2).This is sometimeswritten

S2= SO(3)/SO(2), (4.1.27)

but it shouldbe bornein mind that thesymbolS2 is hereusedexclusivelyto refer to the standardmetricif, by writing SO(3)/SO(2) it is understoodthat SO(3) acts transitively. If we were to distort thetwo-sphereby writing

x2+y2+A2z2= 1 (4.1.28)

we reducethe isometry group to SO(2), but SO(2) does not act transitively and so this distortedspacewhich is still topologically 52 is no longer homogeneous.(You cannot “squash”a two-sphere.)In whatfollows, the notation0/H will alwaysbe takento meanthat0 actstransitively.There is still a residualambiguity, however,dependingon whether0 is anysubgroupwhich actstransitivelyorwhetherG is thefull isometry group. For example, the round 57with isometrygroupS0(8)mayequivalentlybedescribedbythe cosets

SO(8) Spin(7) SU(4) Sp(2)

SO(7)’ 02 ‘ SU(3)’ Sp(h)’

since SO(8), Spin(7), SU(4)and Sp(2) can all act transitively and SO(7), G2, SU(3)and Sp(l) are the

correspondingisotropy groups.There is a tendencyin the physics literature to reserve0 for theisometry group and H for its isotropy group and indeed, one is always free to parameterizeahomogeneousspacein this way. We shall adopt this conventionunlessotherwisestated.For example,the three-sphereas well as beinga groupmanifold may alsobe regardedas a cosetspace:the roundS

3of (4.1.15) beingSU(2)x SU(2)/SU(2)andthe squashedS3 of (4.1.20) beingSU(2)x U(1)/U(1).

An important aspect of Kaluza—Klein theory is the dimensionof thecosetspace,k,

k = dim 0/H = dim 0—dim H. (4.1.29)

Hence,the most economicalconfigurationoccurswhen H is that maximal subgroupof G with thegreatestdimension.For exampleS7= SO(8)/SO(7)would yield 28 gaugebosonswith only sevenextradimensions.The tangentspacegroup of any k-dimensionalmanifold, i.e., the groupof frame rotations

= A”b(y)em”(y) (4.1.30)

leavingthe metric (4.1.16)invariantis SO(k),for which

= t5cd, (4.1.31)

andit is alwaysthe casethat the isotropy groupis a subgroupof the tangentspacegroup,

HCSO(k). (4.1.32)

Not all cosetspacesG/H are reductiveor symmetric.Let the generatorsof 0 be denotedby TA

Mi Duff etaL, Kaluza—Kleinsupergravity 41

(A = 1 . . . dim 0) satisfying

[TA, TB]=f,.,JTC (4.1.33)

Theymaybe divided into the generatorsT~of H (a = 1 . . . dim H) and Ta of K, the complementof Hin 0 (a = 1 . . . k). Reductivespacesarethosefor which all structureconstantsvanishexcept~ fab”,

fab ~,fabC and those related by antisymmetry, i.e.,

[H, H]= H, [H, K] = K, [K, K]= H+K. (4.1.34)

In fact, all compact coset spaces are reductive. If fab’ also vanishes then the space is said to be symmetric[188],i.e., [K, K] = H The Riemann tensor of a symmetric space is covariantly constant

VtRmnpq= 0. (4.1.35)

All the spaceswe shall be concernedwith arereductivebut not all are symmetric. For example, theround S3 = SU(2)x SU(2)/SU(2)is a symmetricspacebut the squashedS3= SU(2)x U(1)/U(1) is not.Another example,of more relevancefor d = 11 supergravity, is provided by 57~The round S7 =

S0(8)/SO(7) is a symmetric space but the squashed S7 = Sp(2)x Sp(l)/Sp(l)xSp(l) of chapter 8 isnot.

Round spheres 5” arealso maximallysymmetricin that theyadmit the maximumnumberk(k+ 1)12of Killing vectors on a k-dimensional manifold. In this case

Rmnpq= m2(gmpgnq— gmqgnp), (4.1.36)

wherem is the inverse radius.For compactmanifolds, k-spheresare the only maximally symmetricspaces.However, not all symmetricspacesare maximally symmetric, an examplewith k = 2n dimen-sionsbeingCP” = SU(n+ 1)/U(n).

So far, we have started with the action of the group Gon the manifold, equipped with a 0-invariantmetric. This uniquely determinesthe isotropy subgroupH of 0 and hencethe cosetspace0/H. Inpractice,however,we oftenadopta different startingpoint. We pick a group0 with asyet unspecifiedaction on the manifold; indeedthe topology of the manifold is not yet determined.We then examinethe possiblespaceson which 0 can act transitively by determiningthe possiblesubgroupsH of 0 andtheir embeddings.It might be that a given subgroupH can be embeddedin 0 in morethan onewayand these different embeddings can lead to different topologies. For example there are three in-equivalentembeddingsof S0(3)in SO(5) and hencethreedifferent coset spaces SO(5)/SO(3). To seethis, we note that there are two maximal subgroups of S0(5)

S0(5)i SO(3)A X SO(3)B, 50(5)D SO(3)max,

and hencethreeS0(3) candidatesfor H: SO(3)A (or SO(3)B), SO(3)A+B (the diagonal subgroupofSO(3)A x SO(3)B)and SO(3)ma,,(the maximalsubgroup).The resultsaresummarizedin table 3.

Columntwo lists how the fundamentalrepresentationof 0 decomposesunderH and column threelists howan S0(7)tangentspacevectordecomposesunderH. This “H contentof G/H” is obtainedbydecomposing the adjoint representation of Gunder Hand subtracting the adjoint representation of H.

42 Mi Duff et a!., Ka!uza—K!ein supergravity

Table 3Different SO(5)/SO(3)cosetspaces

G/H 5 H-content Topology Parameters

SO(5)/SO(3)A 1 + 2 + 2 1 + I + 1 + 2 + 2 S7 7

SO(5)ISO(3)A+B I + 1 + 3 1 + 3 + 3 V5,2 4

SO(5)/SO(3),,,~. 5 7 1

Thus these three different embeddingshaveled to threedifferent topologies:the seven-sphereS7, the

Stiefel manifold V5,2 and a third onewhich is neitherof these.

Having fixed thetopology of 0/H, theremay still be afreedomin thechoiceof 0-invariantmetric. Ifthe spaceis isotropy irreducible then the metric is fixed, and moreoveris an Einstein metric [188].Isotropy irreduciblesimply meansthat the k-vector of 50(k) is irreducibleunder H, and hencethesymmetric product of two vector representationscontains only one singlet. This means that all0-invariant symmetricrank-two tensorswill be equalup to an overall scale. In particular,the Riccitensoris proportionalto the metric tensorand hencethe spaceis an Einsteinspace.An exampleof anisotropy irreduciblecosetspaceis providedby the third exampleof table3.

When the coset spaceis not isotropy irreducible, as is the casefor 57 and V5,2 in table 3, the0-invariantmetric is no longer unique.In fact thereareas manyparametersas therearesingletsin thesymmetricproductof two k-vectorsof S0(k)decomposedunderH. For example,S

7 has7 parametersand V

5,2 has4. The fact that theseparametersmay be varied while maintaining 0 invarianceandhomogeneityis preciselythe phenomenonof “squashing”referredto earlier.Theimportantquestioninthe contextof Kaluza—Klein is whetherby varying theseparameterswe can find new solutionsof thefield equations.This is difficult to answerin general.For example,as discussedin chapter8, S

7 admitstwo homogeneousEinsteinmetrics,andthisis true for all spheresof dimension4n + 3 with n � 1 [206,

211].

4.2. Killing spinors, holonomy and supersymmetry

Supergravity is atheory of fermions as well as bosonsandit is essentialthereforethat we restrictourattention to spaceson which fermions can be globally defined.Technically this meansthat the spacemustadmit aspin structure,i.e., the secondStiefel—Whitneyclassmustvanish[189,190, 200]. SinceAdSdoes admit a spin structure, the problemdevolvesupon M

7. For example,CP2 doesnot admit a spin

structurewhereasS3doesso an M7 = CP

2X S~wouldbe an unacceptablevacuumfor d = 11 supergravityeven though it admits an Einstein metric. These remarksapply to any Kaluza—Klein theory whichattempts to describefermions[29].Onecandefinefermionson spacesnot admitting a spin structureviathe deviceof a generalizedspin structure[189]but this requiresthat theybe minimally coupledto somesuitablegauge fields as well as to gravity. To date, however,this idea hasnot found applicationinKaluza—Klein theories.It would require,in any event,the presenceof gaugefields alreadyin the higherdimension.In particular,generalizedspin structuresare irrelevant for d = 11 supergravity.They arealso irrelevantfor typehA supergravityin d = 10 (contraryto someclaimsin the literature)in spiteofthe presenceof a U(h) gaugefield becausethefermionsare neutralunderthisU(h). Fermionsdo coupleminimally to a U(h) gauge potential in type IIB supergravityin d = 10. Here the gauge field iscomposite,being built out of elementaryscalars.There is, to date, no known compactificationof thistheory to d = 4. Fromnow on, weshall alwaysassumethat M

7 is a spin manifold.

Mi Duffet aL, Kaluza—Kleinsupergeavily 43

We now turn our attentionto a crucialaspectof Kaluza—Kleinsupergravity,namelythe criterion forunbrokensupersymmetryin the effectived = 4 theory [21—24].Recall from (3.1.1) thatthe VEV of thefermion field ~PM of d = 11 supergravityhasbeenset to zero. Fora supersymmetricvacuum,we requirethat (‘I’M) stayzeroundera supersymmetrytransformation,i.e., that

(~I’I’M) = ‘~I5M~= 0, (4.2.1)

where15M is given in (2.2.15).To solve (4.2.1)we look for solutionsof the form

E(x, y) = s(x)o(y) (4.2.2)

where e(x) is an (anticommuting) four component spinor in d = 4 and ~(y) is a (commuting) eightcomponentspinorin d = 7.

01n the c,~seof Freund—Rubincompactification,the supercovariantderivativeDM decomposes into the D,.~and Drn of (3.1.14)and (3.1.15). Hence,

D1.s(x)= 0, (4.2.3)

Drn~(y) = 0. (4.2.4)

Thus, the problem of counting unbroken supersymmetries is equivalentto the problem of countingKilling spinors.We havealreadyestablishedin section3.2 that AdS admits the maximum number (i.e.four) as far as spacetimeis concerned,andso the-numberN of unbroken generators of AdS supersym-metry is given by the number of Killing spinors on the d = 7 manifold, i.e., the numberof solutionsto

15 _ii_i~ 1 abr 1 ar\rnT1’V~Vm4Wm jab’2mernia)fl,

whereWm~*bandem” arethe spin connectionandsiebenbeinof the ground-statesolutionto (3.1.10)(andwherewe havedroppedthe zerosuperscripts).The generalsolution to (4.2.1) is therefore

r(x, y) = eA(x)i~(y), (4.2.6)

whereA runsover 1 to N. Since~ is an 8-componentspinor,0 � N � 8, consistentwith the well-knownresultfor conventionald = 4 supergravity.

From (4.2.5),Killing spinorsareseento satisfy the integrability condition[22,23]

[‘3m, I5n]~1= ~~Crnn”Fab7l = 0, (4.2.7)

whereCmn*th is the Weyl tensor.The subgroupof Spin (7) (the doublecoverof the tangentspacegroupSO(7)) generated by these linear combinations of the Spin (7) generatorsFab correspondsto the holon-omy group Yt’ of the generalized connection of (4.2.5). Thus, the maximum number of unbroken supersym-metriesNmax is equalto the numberof spinorsleft invariant by ~‘ [22,23]. This in turn is given by thenumberof singletsappearingin thedecompositionof 8of Spin (7)under~ [24].Thesubgroupof ~Wandthebranchingrulesaregivenin table4. We havenot includedU(1) factorsin thistablebut it maybecheckedthat theyyield no newvalues of Nmax.

At this stageit is important to realize that the holonomy argumentbasedon (4.2.7) merelytells us

Mi Duff eta!., Ka!uza—K!einsupergravisy

Table 4

Holonomy and supergravity

8~ NmaxSpin (7) 8 0G

2 1+7 1SU(4) 4+4 0SU(2) xSU(2) x SU(2) (1,2,2)+ (2,2,1) 0SU(3) 1+1+3+~ 2Sp(2) 4+4 0SU(2)xSU(2) (2,2)+(1,3)+(1,1) 1

(2,2)+(1,2)+(1,2) 0(1,2)+(1,2)+(2,1)+(2,1) 0(2,2)+(2,2) 0

SU(2) 1+1+1+1+2+2 41+1+3+3 21+2+2+3 11+7 14+4 02+2+2+2 0

1 1+1+1+1+1+1+I+1 8

the maximumnumberof unbrokensupersymmetries.The actual number, determinedby the Killingspinorequation(4.2.5)may beless. In otherwords, the integrabilitycondition (4.2.7) is a necessarybutnot sufficient conditionfor the existenceof Killing spinors.Thus,althoughthe only permittedvalues ofNmax are0, 1, 2, 4 and8, this doesnot rule out othervalues of N.

The most striking way in which to illustrate the phenomenonof N<Nmax is to consider“skew-whiffing”, i.e., orientation reversalof M7. For eachFreund—Rubinsolutionof d = 11 supergravity(withni � 0), onemayobtain anothersolutionby a reversalof orientationof M7 for exampleby sendingem” to—em” [53].An equivalentway to obtainsuchvacuais to keeptheorientationfixed but insertaminussigninfront of 3m in (3.1.7). Eitherway, as discussedin section3.1, the two vacuaaredistinguich~~by havingoppositesignsfor the Page-charge,P, of (3.1.19). In thiscase,the criterion for unbrokensupersymmetrybecomes

Dm7) + ~mem”1”~i~= 0, (4.2.8)

i.e., the sign of the last term is oppositeto that of (4.2.5). Yet the integrability condition(4.2.7), ~ andhenceNm,,,, remainunchanged.The importantpoint, as we shall now demonstrate,is that with theexceptionof the round57 whereboth orientationsgive N = Nmax = 8, at mostoneorientationcan haveN > 0 [25]. We call this the “skew-whiffing” theorem. (Historically, the first example of thisphenomenonoccurred for the squashedS

7 for which ~° = 02 but for which one orientation gaveN = Nmax = 1 while the othergaveN = 0 [53]).To seethis, supposeonehadboth ~÷ and ~ satisfying

Dmfl± i ~mFm’q±= 0. (4.2.9)

Then the scalarfield

41 ~±~i_ (4.2.10)

Mi Duff etaL, Ka!uza—Kleinsupergravisy 45

satisfies

= 7m24’, (4.2.11)

which follows directly from (4.2.9), which implies

Emn VmVn41 + m2gmn41= 0, (4.2.12)

as maybe seenby integratingIEmnI2 over M7. By a theoremof Yano andNagano[191]this equation,

the equationfor conformalKilling vectorsVm41 on M7, hassolutionsonly for the roundS7. Henceonly

the round S7 can support supersymmetriesfor both orientations.This argumentwould fail if i~+~= 0.Sinceas shownin appendixB,

Fmfl~Fm = 17?~— 1~fl, (4.2.13)

for any spinor ~, it follows that ~+i~- = 0 would imply

Vm ?j+Fm7)_ � 0. (4.2.14)

Fromthis, one finds that

VmVnO, (4.2.15)

and henceusing the Ricci identity of appendixA for covariantderivativesand (3.1.10), that V~= 0.Therefore,i~+,~. cannot be zero,which completesthe proof.

A corollary of the above skew-whiffing theorem is that the round 57 is the only d = 7 symmetricspaceto admit Killing spinors.This is becauseareversalof orientationcorrespondsto sendingK to —Kin (4.1.34) and the algebra is left invariant if and only if the spaceis symmetric. In other words, asymmetricspaceadmits an orientation-reversingisometryand, therefore,the numberof Killing spinors~ mustequalthe numberof ~. As shownabove,this is possibleonly for the round57 or elsea spacewith no supersymmetry.It follows immediately that symmetric spaceslike S

4x S~and ~5 X S~yieldN =0.

Onemust not concludefrom the abovediscussionthat oneorientation necessarilyyields N = Nm,,,,

sincethe spinorsleft invariant by ~Wneednot satisfyeither of the first-orderequations(4.2.9). In otherwords, it is necessaryto go beyondthe second-orderintegrability condition (4.2.7) (see [192]).Asdiscussedin chapter9, for example,therearespaceswith ~‘ = SU(2) but N = 3. In practice,however,itis easierto go back to the first-orderequationsto determineN.

Note that for Ricci flat spaceswith m = 0 the criterion for unbrokensupersymmetryis the sameforboth orientationsand the above discussion no longer applies. In particular for T7, ~C= 1 andN = Nm,,,, = 8 [37]and for K3 x T3, ~° = SU(2) andN = Nm,,,, = 4 [23].

Finally we commenton someglobal, as opposedto local, obstructionsto unbrokensupersymmetry.Strictly speaking,the Weyl tensorcharacterizesthe restrictedholonomygroupof Dm, i.e., it describesthe rotation of a spinorparallel transportedarounda closedloop which is homotopic to zero. If thespaceis not simply connectedtheremaybeglobal obstructionsto the existenceof covariantlyconstantspinors,in addition to any local obstructionimplied by the Weyl tensor.For example,in addition to the

46 Mi Duff eta!., Ka!uza—K!ein supergravitv

ground-statesolutionsT7andS7, therewill alsoexistsolutionsof theform T7/FandSq/F(generalizationsoflensspaces),whereF isadiscretegroup.Thesespaces,like theirT7 or 57coveringspaces,have~ = 0 buttheseglobalconsiderationsimply thattheyadmit fewerthan8covariantlyconstantspinors.SimilarremarksapplytootherM

7 compactifications:globalobstructionsmayrendertheactualnumberof supersymmetriesless thanthat implied by the restrictedholonomy group [231.

4.3. Propertiesofoperatorson Al7

In the massspectrumof table5 of section5.1 we will encountervariousdifferentialoperatorson M7.Without computingthemassspectrumexplicitly, which would requirespecifyingM7, a greatdeal canbelearnedabout the four-dimensionaltheory, e.g., the stability propertiesof section 5.2, merely fromknowledgeof generalpropertiesof thesedifferentialoperators.In thissectionwe shall discusssomeofthesegeneral properties, in particular the existenceof zero-modesand of lower boundson theeigenvaluespectrum.

We beginby discussingthe Hodge—deRhamoperatorzl,., actingon p-forms,definedby

zi=d6+8d, (4.3.1)

whered is the exteriorderivativewhich mapsp-formsinto (p + 1)-forms, and8 is its adjoint,

8 = (_1)P*d*, (4.3.2)

whichmapsp-formsinto (p — 1) forms.A form obeyingdw = 0 is called closed,a form obeying&v = 0 is called coclosed, and a form obeying

= 0 is called harmonic.If w can bewritten as da it is calledexact,while if it can bewritten as 6/3 itis calledcoexact.Explicitly, for a p-form w with componentsWm1.. ~ in the coordinatebasisdy;

WWmmdY’/~”AdYm”, (4.3.3)

p! ‘ ~‘

(dw)mi...mp+i~r(p+1)a1miwm2...mp+i1, (4.3.4)

= ‘V”Wnmi...m,.,i. (4.3.5)

li is a manifestlynon-negativeoperator,asmaybe seenby introducingthe norm of w by

(w,w)_JVgd7ywm,,,,mpwml.mP. (4.3.6)

This norm is non-degenerate,vanishingif andonly if w = 0. By definition, the 8-operator satisfies

(w, dw) = (6w, w), (4.3.7)

andhencefrom (4.3.1)

Mi Duff eta!., Ka!uza—Klein supergravity 47

(w, ~iw)= (dw, dw)+ (6w, 6w), (4.3.8)

proving that

(4.3.9)

with equality holding if and only if zl acts on an w which is closedand coclosed.

Any p-form can be uniquely decomposed into its exact, coexact and harmonic pieces [190],

wda+6/3+wH, (4.3.10)

which are mutually orthogonalwith respectto the norm (4.3.6). The numberof closed but not exactp-formson a manifold is given by the pth Betti numberb~,and hencethereareb~zero-modesof A~,wherethe notation li~meanszl actingon p-forms.

On 0-forms (i.e. scalars), li0 coincides with minus the d’Alembertian,

iiw—LIw, (4.3.11)

while on 1, 2 and3-forms (which is as high aswe needgo in 7 dimensionssincethe Hodge* operator,which commuteswith li, mapsp-formsinto (7— p)-forms) thereareadditionalcurvatureterms:

= LJWm + Rm” w,,, (4.3.12)

Ii2Wmfl = LJWmn—

2Rmpnqw” — 2R~[mWn]p, (4.3.13)

zl3wmnp= Llwmnp — 6R[mn~b’Wp]qr +

3R[mr w,,,,],.. (4.3.14)

Even strongerboundson the eigenvaluespectrumof li~may be obtainedwhen we specializetoEinstein metricsRmn = 6m2gmn. Let us first consider zlo41 = A41. For any connectedM

7 thereis alwaysonezeromodenamely41 = constant.It can alsobe shownthat the first non-zeroeigenvaluesatisfies

A�7m2 (4.3.15)

with equality if and only if M7 is the round 57~To see this, consider the tensor Gmn defined by

Gmn = VmVn41 + c2gmn41, (4.3.16)

whereli041 = A41 and c is someas yet arbitraryconstant.Integrating IGrnnP2 over M7 yields

J d7y \/g~G~~~2= [A2 A(6m2+2c2)+7c4]Jd7yVg412, (4.3.17)

and hence

A2—A(6m2+2c2)+7c4>0, (4.3.18)

48 M.J. Duff etaL, Kaluza—Kleinsupergravily

for anyvalue of c2. Taking c2 = m2, this implies that A lies in the range0� A s m2 or that it satisfiesA � 7m2. Taking c2 = 0 implies that A = 0 or A � 6m2. Combining thesebounds,we see that the firstnon-zeroeigenvaluesatisfies(4.3.15),with equality if andonly if

r ~7~_ 2m ~ mgmn

Suchmodesoccuron the roundS7, but as shownin [191]this is the only M7 whereit can happen.This

correspondsto the existenceof conformalKilling vectorsCm Vm4~on the round S7 satisfying

V(mCn) = —m2gmncb. (4.3.20)

Let usnow considerLI1Vm = AVm, wherefrom (4.3.12)

41=—L1+6m2. (4.3.21)

This immediately gives the bound LI~� 6m2. For p-forms with p � 1, however, we are primarilyinterestedin coclosedforms, i.e., transversevectorsVtm Vm = 0 in thecasep = 1. For suchmodeswe canprove the strongerbound

LI~� 12m2, (4.3.22)

with equality if andonly if Vm is a Killing vector, i.e., a vectorsatisfyingVmVn + VnVm = 0. To prove

this, defineSmn = VmVn + VnVm. (4.3.23)

It follows that

Jd~yVg [ISmnJ2+ 2(12m2— A)I Vm 2] = 0, (4.3.24)

which immediatelyimplies (4.3.22). (Since Vm commuteswith LI, the boundfor longitudinal vectorsisthe sameas that for scalars,i.e., LI~� 7m2.)

Similar boundson 42 and 43 are more difficult to establishowing to the appearanceof theuncontractedRiemanntensorin (4.3.13)and(4.3.14),but thesearenot essentialfor our analysis.Thereis neverthelessoneinterestingpropertyof 43, namely, that its transversethree-formeigenfunctionscanbe put into one-to-onecorrespondencewith the eigenfunctionsof the first-orderoperatorQ definedby

(4.3.25)

Equation(4.3.25) implies that an eigenfunctiona with Qa= ~a satisfies

= ~amnp. (4.3.26)

It is clear from (4.3.1) and (4.3.25) that Qa= ±~aaimplies that 43a = ~a

2a.To prove that the

MJ. Duff etal., Kaluza—Kleinsupergravüy 49

eigenfunctionsof 43 and 0 can be put in one-to-onecorrespondence,consider the set of alleigenfunctionsa satisfying

43a1 = ji2a~ (4.3.27)

for someparticularvalueof p,2 wherethe index i labelsthe independenteigenfunctions.Since * andd

commutewith LI it follows that LI3Qat = p,2Qa’ andhencethat(4.3.28)

whereS’1 areconstants.Su is symmetricsince0 is self-adjoint.Applying 0 to (4.3.28)showsthat

Si1Sik = ,j2ôik (4.3.29)

andsowith suitablechoiceof basisS” canbediagonalizedwitheigenvalues±p,.Thecorrespondenceis thusestablishedfrom (4.3.28).

Now we turn our attention to the Lichnerowicz operator 4L acting on transversetrace-freesymmetrictensors

= LIhmn — 2Rmpnqh’~”+ 2R(m~hn)p. (4.3.30)

Unlike LIpS 4L is not in generalpositivesemi-definite.Indeed,it is difficult to establishlower boundsonthe eigenvaluespectrumevenon Einstein spacesRmn= 6m2gmn,without detailedknowledgeof theRiemanntensor.To seethis,considerthe identity

Jd~yVghmn4Lhmn= Jd~y~ + 24m2hm~zhmn+ 3V(mhr~~)V(mh,,,~)], (4.3.31)

which holdsfor all transverse,trace-freesymmetrictensors.Let usconsiderfirst homogeneousspaces.On such spacesthe 27 eigenvaluesK of the Riemanntensor,definedfor symmetrictrace-free Vmn by

RmpnqV~= KVmn, (4.3.32)

are constants.Denoting the most positive such eigenvalueby Kmax, it follows from (4.3.31) that thelowesteigenvalueAmjn of 4L satisfies

Amin � 24m2— 4Kmax, (4.3.33)

with the inequality being saturatedif andonly if the correspondingeigenmodehmn is a Killing tensor,

V(mhnp)= 0, (4.3.34)

andsatisfies(4.3.32)with K = Kmax. If the spaceis not homogeneous,(4.3.33) still holds if Km~, is nowtakento be the maximum valueattainedby any of the eigenvaluesof (4.3.32) anywherein the space.We will use(4.3.33)in chapter11 in the discussionof stability of compactifications.The Killing tensorin

50 Mi Duff etaL, Kaluza—KleinsupergravOy

(4.3.34), which defines symmetricrank two Killing tensors,is but a special caseof a more general

classof arbitrary rank symmetricKilling tensorshm, m~satisfyingV(mhm

1m)0. (4.3.35)

Anotherclassof geometricalquantitieswhich arisesnaturally in our analysisis YanoKilling tensorsAm,.. . m,~which aretotally antisymmetricand satisfy[193]

VmAmjm = t9[mAm1m]. (4.3.36)

Ordinary Killing vectorsareclearly a specialcaseof both (4.3.35) and(4.3.36).Let us now consider fermions. In commonwith the operator Q, the Dirac operator.0

possessesbothapositiveand negativeeigenvaluespectrum.Acting on spinorscu

= A~, (4.3.37)

onemay show that

AI�~m (4.3.38)

providedR= 42m2. (This holds in particularfor Rmn = 6m2g~~.)The boundin (4.3.38)is saturatedby

Killing spinorscu obeying

Dmçli = ±~mF~cb. (4.3.39)

(Strictly speaking,wereservedthename“Killing spinor” in section4.2 for spinorsobeying (4.3.39)onlywith the + sign which, in our conventions,is the sign appropriateto an unbrokensupersymmetry.)Tosee this, considerthe identity

Jd~yVg~Dmcb- ~mFmV.~2= (A2+ mA — ~m2)J d7y Vgjcu12� 0. (4.3.40)

Hence, either A �—~mor A �~m.Conversely, letting m—~—min (4.3.40), A �~tnor A �—~m.Combining thesetwo bounds,we obtain (4.3.38) which implies a forbiddenregion between—~mand+~mfor R = 42m2.Note, incidentally,that Killing spinorscan exist only on Einsteinspaces,sincefrom(4.3.39)

[Dm, D~]cIi= ~Rmn’thFabcuI = —~m2f~~cb (4.3.41)

andhence,by contractingwith r~andusingthe cyclic identity Rm~npq1 0, we obtain

(R~~— 6m2g~~)F”~1i= 0. (4.3.42)

The Einsteinconditionthen follows by multiplying on the left by cuFF”.

Boundson ~ actingon vector-spinorsçf’m or higherrank tensorspinorsare moredifficult to establish

M.J. Duffel aL, Kaluza—Kleinsupergraviry 51

owing to the appearanceof the full Riemanntensorin .02, but arefacilitated by specialchoiceof ~as for the Lichnerowiczoperator.We may alsodefineKilling tensor-spinorsvia

D(mclImI...m,,)0, (4.3.43)

or

DmlIJmi...m,~= DimV1mi... m,,], (4.3.44)

accordingas ~I1m~.. . m,~is totally symmetricor antisymmetricin tensorindices,whereD is as given in(4.2.5)exceptthat tensor-spinorswith world indicesalsorequireChristoffel symbolsin the definitionoftheir covariantderivativeDm. Similar definitionsapplywith m replacedby —m in (4.2.5).

Finally in this section we presenta discussionof the calculation of the spectrumof the variousdifferential operatorson cosetspaces.We will follow the techniquesof [149].Considera coset spaceG/H,whereH is a subgroupof G. The generators,TA, of G (A = 1 . . . dim G) maybe divided into thegeneratorsT~of H, and Ta of K, the complementof H in G: i.e., a runsover dim H values,andarunsover dim (G/H) = dim G— dimH values.The generatorssatisfythe commutationrelations

r’r rl_t Cp~1A, IBJ JA.B 1C.

All the known coset-spacecompactificationsof d = 11 supergravity involve reductive coset spaces,which as mentionedin section4.1 are characterizedby all structuresconstantsvanishingexcept fa

4cë,

fab lab andfabt~i.e. [H, H] = H, [H, K] = K, [K, K] = H + K. If fabC also vanishes,the spaceissymmetric,as for exampleis the roundsphere.

Our final objectiveis to expressthe eigenvaluesof the variousdifferential operatorson G/H in termsof algebraicgroup-invariantquantities. In order to do this, we need to expressthe action of thestandardRiemanniancovariantderivativeVm on the relevanttensoror spinorharmonicsin terms ofgroupgeneratorsandstructureconstants.The hannonics,Y, transformasirreduciblerepresentationsofthe isometrygroupG. In order to emphasizethe generalityof themethod,andto avoid a profusionofindices,we will suppressnot only the label which characterisesthe irreduciblerepresentation,but alsothe tensoror spinor indices.Thesesuppressedindicesaretangentspaceindices,so in the caseof tensorharmonicsthey areobtainedfrom world indices m, n,..., by meansof the vielbein ema. Acting on Y,the covariantderivativeVm is thereforegivenby

VmY i9mY+ Wm0~)IabY, (4.3.45)

whereWm°’

t’ is the Riemannianspin connectionon GIH andLab arethe generatorsof the tangentspace

groupin the representationappropriateto the tensoror spinornatureof Y

The requiredexpressionfor Vm Y, whosederivationwe discussbelow is,

r v + 1 C SCab ~‘ ~y — a ‘r

~‘~1 2e~J c~ab1 ——em 1a . .

To begin wedenotecoordinateson G/Hby ym. We can nowassociatea uniqueelementL~of Gto eachpoint ym by defining

52 Mi Duffel aL, Kaluza—Kleinsupergravizy

L~= ~ (4.3.47)

wherey’~= emaym. Thus the action of an arbitrary elementg of G definesin a uniquemannera newpoint ym’, anda uniqueelementh of H, via the equation

gL~= L~.h. (4.3.48)

If onenow calculatesL1 dL, which belongsto the Lie algebraof G, onecan define Q~by

L1 dL~= e~zTa + i2~T~, (4.3.49)

whereea = eam dytm is the siebenbein.This can berewritten as

3mL’ = (em’~Ta + ~ T~)L;1, (4.3.50)

where Qã = 11m~idytm. As explained in [149], any given harmonic Y is obtainedfrom L;’ in the

appropriaterepresentationandso3mY= (em” Ta + 11,,, ‘~ T~)Y. (4.3.51)

It nowremainsto re-express12,,~~ Ta in termsof the Lab’S, spin connectionandstructureconstants,andthensubstitute(4.3.51)into (4.3.45).

Taking the exterior derivativeof (4.3.49) gives the Maurer—Cartanequation,which, projecting outthe coefficientsof Ta is

de” = —fi” A edf~~~l— ~e” A edf~c~l. (4.3.52)

Comparingthis with the defining equationfor the Riemannianspinconnection,de~= ._~ab A e’~,gives

b.flet b lc ~b

10ma — ~.&mJ~a

2e mica ,

Recalling that the generatorsT~satisfy the Lie algebraof H, one seesthat one can write, in therepresentationof Y,

T~= —fab L~’c, (4.3.54)

whereL~are the tangentspacegroupgeneratorsintroducedearlier.Substituting(4.3.51), (4.3.53)and(4.3.54)into (4.3.45)weobtainthe requiredresultof eq. (4.3.46).

As an illustration of the useof this formalism,we now applyit to the caseof vectorharmonicson thesquashedseven-sphere,whosegeometryis discussedin detailin chapter8.

The squashedseven-spherecan be regardedas the cosetspaceS0(5)x SU(2)/SU(2)x SU(2) [53].This is demonstratedexplicitly in [58],including aderivationof the structureconstantsf~’.In (4.3.46)werequirethe constantsJab, which can beobtainedfrom [58]by rescalingto ourmetric normalization.Theresult is

fai,c = (11V5)aabc, (4.3.55)

Mi Duff eta!., Kaluza—K!einsupergravity 53

whereaa& are the structureconstantsappearingin the multiplication table of the octonions(aabc=

a[abcl, a011—3q’ a~= ~Eiik, a~

11~= eIJk). Seechapter8.We nowuseeq. (4.3.46)to derive thespectrumof eigenvaluesK

2 of the Hodge—deRham operatoronone-forms(or equivalently,vectors)

4Ya~EYa+RabYb~L~Ya+6m2YaK2Ya, (4.3.56)

wherem2=

We beginby squaring(4.3.46),andthenusing thefact that, actingon vectors,(Lab)’~ = ~ to obtain

L1Y0 + fabcVbYc+ ~Vb(fa&)Yc + ~fabcfdbcYd = (TbTbY)a. (4.3.57)

Beforesubstituting(4.3.55) into (4.3.57),werecall that on the A2 = ~squashedsphere,thereis a Killing

spinor ~ satisfyingVafl ~mTafl, in termsof which aabc may bewritten as

aa& = flFabcfl, (4.3.58)

where i~i~= 1, and Ta are the seven-dimensionalDirac matrices. This leads to the simplification

V°aa& = 0, andso using the standardoctonionicresultthat aa&ad& = 63ad, we obtain—E Ya + (1/V5)aa,,.,VbYc+ j~ijYa= (TbTbY)a. (4.3.59)

On the right-handside, TbT,, can be written as TBTB — T~T1j~,which is simply equalto — (CG — CH),

whereCG and CH are quadraticCasimirsof the groupsG and H. We will postponediscussionof CGuntil later, andconcentratefor now on CH.

For the squashedsphereG= SO(5)x SU(2)~,andH = SU(2)A+cX SU(2)B [53,58], wherethe SU(2)groupsare relatedto the maximal subgroupSU(2)A x SU(2)B in SO(5), i.e., SU(2)A+C is the diagonalsubgroupin SU(2)A X SU(2)~.Thus CH hasthe form aCsu(2)A+C + /

3Csu(2)B, wherea and/3 areuniversalconstants.Using the appropriatelyrescaledstructureconstantsfrom [58],we find

— ‘1,~~, 6ç’— “~SU(2)A+C + ~‘~SU(2)a

wherewe havenormalizedthe Casimirsof SU(2) so that the adjoint representationhasCsu(2)= 2. In

the caseof vectorharmonics,the 7 of S0(7)(the tangentspacegroup)splits as a (1,3) + (2,2) underH.Thus using (4.3.60),we find that CH = ~ for both the (1,3) and(2,2), sousing(4.3.56) we obtainfrom(4.3.59)

aa,,cVbYc= V5(CG—K2)Ya. (4.3.61)

This hasnot yet given a purely algebraicequationfor K2, andit is necessaryto square(4.3.61)oncemore, i.e., to calculateD2Ya, whereDYa aa&VbYc. Using the propertiesof a,,,,,, implied by (4.3.58),andthat VaYa= 0, onefinds

D2Y,,= (6/V5)DY,,+ K2Ya, (4.3.62)

andhenceK2 satisfiesthe quadraticequation

54 Mi Duff eta!., Kaluza—Klein supergravity

5(K2 — C

0)2 = 6(K2 — CG) + K2. (4.3.63)

This result is in agreementwith [92],wherethe vectoreigenvalueson thesquashed57 were derivedby adifferent method.Finally we need to determinethe universalconstantsa and b in the expressionCG = aCSO(

5)+ bCsu(2)~.This could in principle be doneusingthe structureconstantsgiven in [58],butan easierway is to note from (4.3.46)that the eigenvaluesof the scalar Laplacianare simply given by— LIJ Y = CGY, which by comparisonwith [65]immediatelygives

CG= Cso(s)+3Csu(2). (4.3.64)

Thenormalizationof the Casimiroperatorsis suchthat Cso(5)= 6 for theadjoint representation,andasbefore Csu(2)= 2 for its adjoint. Thus the eigenvaluesof the vectorharmonicoperatorare

K2 CG+~±(~CG+~)112, (4.3.65)

with CG given by (4.3.64).To determinewhich 50(5)x SU(2) representationsactually occur,onesimplyhasto decomposethe SO(8)representationsof divergence-freevectorharmonicson the roundS7 (whichhaveDynkin label (n, 1, 0, 0), n >0) underthe SO(5)) SU(2) isometrygroupof thesquashedS7. The ±

signs in (4.3.65) reflect the fact that eachrepresentationmay occur twice. However, somerepresen-tations occur only onceand for theseone hasto determinewhich choice of sign is to be made in(4.3.65).Oneway to do thisis to calculatetheeigenvaluesfor all valuesof the squashingparameter(seechapter 8), so that one can follow them back to the round S7 and hence determinethe signsunambiguously.Thishasbeendonein [92]for vectors.A generalanalysisof harmonicexpansionson cosetspacesis given in [276].

5. The d = 4 mass spectrum

5.1. The d = 4 massoperators

In this sectionwe derive the d = 4 massoperatorsresultingfrom Freund—Rubincompactificationonan arbitraryM

7. As wasdescribedin chapter1 for a generalKaluza—Klein theory,the spacetimefieldscorrespondto the x~-dependentcoefficientsin the harmonicexpansionon M7 of the deviationof thed = 11 fields from their ground-statevalues.The massmatrix of the spacetimefields can be derivedbyexpandingthe d = 11 field equationsto linear order in the fluctuations,andsubstitutinginto thesefieldequationsthe various harmonic expansionsof the d = 11 fields. The massspectrumin spacetimeisfinally obtainedby diagonalizingthe massmatrix, for each typeof field, mode by mode. Sincewe arenot interestedin pure gauge modes, we will perform the calculation in a particular gauge. Thisprocedurealso hasthe advantagethat it eliminatessomeof the mixings presentin the linearizedfieldequations.When carryingout the analysiswe will discoverthat a few specific modesfall outsidethegeneralanalysisandmust thereforebedealtwith separately.Thiswill bedone afterthe generalanalysiswhich we beginby deriving the diagonalizedmassmatrix for bosonicfields and then continuewith theanalogousderivationfor thefermionic fields. Beforedoing so, however,wewill exhibitall the linearizedfield equations.

To thisend,we definethe fluctuationsby

gMN(x,y)=gMN(x,y)+hMN(x,y), (5.1.1)

Mi Duffel aL, Ka!uza—Kleinsupergravity 55

V~M(X,y) = 0 + ~frM(X,y), (5.1.2)

AMNP(X, y)= AMNP(X,y)+ a~p(x,y), (5.1.3)

where~°4MNI’ is apotentialfor theFreund—Rubinbackground,i.e.,J~’~.fl~Q= 4~IMANPol.We nowsubstitutetheseexpansionsinto the d = 11 field equations,andretain termsup to linear orderin the fluctuations.The field equationsfor the bosonic fields are given in (3.1.2) and (3.1.3), which after linearizationyield

W _!~ i L~7 r7Pt. I P— 2’~LnMN ~ V(~V ~

tN)P — 2VMVNflP

— 2~. POR; — 10 ~, :PORS POR ~‘ S ~ ii ~‘ ~‘P0RS3’(M JN)PQR 18gMNEpORsJ 1’(M EN) ORflPS36nMNEPORSV

~O ‘~, ~‘0RSLPT

9gMN’ PORSI T ~ ,

~ fQMNPj1°

1’QMNP~7 ~ R_~7 (~OR °J’MNP\ ~QILSLM°L’ NPIQi 21 V 0”R V 0’.” 1 R .1 -, ‘.,‘~ I OS

— iOMNP010,°~ 51— 2886 ‘ Oi . . . 04,1 05... 08’

whereJMNPQ =48[MaNPO]. Note that the backgroundpotentialAMNP neednot be specifiedsinceonly its

field strengthappearsin theseequations.The covariantderivative *M is defined in the backgroundmetric, and is torsion-free. 4L is the d = 11 Lichnerowicz operatorin the background,which wasdefinedin (4.3.30).The linearizedfermion field equationis

tMNP DM1/P = 0, (5.1.6)

whereall quantitiesaredefinedusingthe backgroundelf bein.The superscript~ is nowsuperfluousatthe linearizedlevel, andwill thereforebe droppedhenceforth.

Wenowproceedwith the derivationof the bosonicmassspectrumby splitting the d = 11 indicesintospacetimeandinternal indices,and so the bosonicequations(5.1.4)and (5.1.5)yield upon substitutingthe Freund—Rubinansatz(3.1.7)and (3.1.8)

LILh~U.+ 2V~,,V”h~,)~— V~,V~h”~+ 2V~Vmh,,)m — V~,Vphmm

= ~ + 24m2g~~h”~— 24m2h,,~, (5.1.7)

+ V~,V’~’h,.,~+ V5,V~~ + V,,V” h~+ V~V~’~ — V5LV~h~~— V~V0h~

= 12m2h/,fl~ (5.1.8)

&‘lmn + 2V(mV”hn)p+ 2V(mV~h~)~— VmVnh~”p— VmVnh”p = 12m2(hmn— gmnh~p)—~(5.1.9)

~ + V,f’~’)+ 9mV,~(h”,,— h~~)+ 18mV,,h”.,~= 0, (5.1.10)

V,~f”m”+ ~ — 3me°”~ V,,hm~= 0, (5 111)

~ + Vqf~m~~= 0, (5.1.12)

56 Mi Duffel aL, Ka!uza—Kleinsupergravisy

V~f~tmflP+ Vqf”~’ = ~ (5.1.13)

wherein (5.1.10)we havemultiplied by ~

The harmonicexpansionsof thefields hMN(x,y) andaMNp(x, y) will takeaparticularlysimple form ifthe following gaugeconditionsareimposed[25]

Vmhm~=0, (5.1.14)

Vtm (hmn—~gmnh~p)=0, (5.1.15)

VmamNp=0 (5.1.16)

(different gaugechoiceswereusedin [80,90, 194, 195]). The expansionsof hMN(x, y) andaMNp(x, y) areseento be

h5,~(x,y) = h~,~(x)Y(y), (5.1.17)

h5,~(x,y) = B,,(x)Y0(y), (5.1.18)

hmn(X,y) S(x)Xmn(y)+~gmn~(x)Y(y), (5.1.19)

a1,,~~(x,y)= a~,~~(x)Y(y), (5.1.20)

a,,~,p(x,y)= a,,..,,(x)Y~,(y), (5.1.21)

aj,,np(x,y)= a,1(x)Y0~(y), (5.1.22)

amnp(x,y) = a(x)Y~~~(y), (5.1.23)

wherethep-forms Ym, .. . mp(Y)are transversemodesof LI,,, andXmn aresymmetric,transversetracelessmodesof LIL. The new quantitiesappearingon the right-handsidesof (5.1.17)to (5.1.23)areall definedthrough theseequationsand are completelyunconstrainedspacetimefields at this stage.In order tomake theseequationsas transparentas possible,we suppressall summationsymbols and the cor-respondingisometryrepresentationindices.We will also drop the explicit coordinatedependence;thiscan bedonewithout introducingambiguitiessince,in the backgroundmetric of the form (3.1.4)—(3.1.5),all operatorssplit into a sum of oneoperatoron AdS and oneon M7. Furthermore,the Lichnerowiczoperatoris from now on denotedLI, and it will be clear from the context whetherLI is the Hodge—deRhamoperatoron p-forms,LI,,, or the Lichnerowiczoperator,

4L of section4.3.It is convenientto start the analysiswith the field equationsfor aMNp. Thus,substituting(5.1.18)and

(5.1.20)—(5.1.23)into (5.1.11)—(5.1.13)andusing the gaugeconditions(5.1.16), wefind

V,,a~=0, V,,a~=0, V,,a”~=0, (5.1.24)

(LIa)Ymnp+ a(LI Ymnp)+ maemnpqr,1V”yrsi = 0, (5.1.25)

(LIa,,)Y0~+a5,(LIY~~)=0, (5.1.26)

Mi Duffel aL, Kaluza—Kleinsupergravity 57

(LIa~,~)~,+~ Y~= 0, (5.1.27)

which arevalid for all modesexceptzero eigenvaluezero-, one- and two-forms, i.e., modessatisfyingLI Y = 0, 4 Ym= 0 or LI Ymn = 0. Althoughwe will return to theseparticularmodesbelow,we mentionherethat therealways exists a solution to LI Y = 0, namely Y= constant,on any internal spaceM7.Furthermore,LI Ymn = 0 hassolutionsonly on spaceswith non-zerosecondBetti numberb2, while nocompactEinsteinspacewith positivecurvaturehasnon-zerofirst Betti numberb1, as is easilyseenfromLI Y~= — E I’m + Rm” Y~,andthereforesuch a spaceadmitsno zeroeigenvalueone-forms.

Turningto (5.1.10), wemakeuseof (5.1.14), (5.1.17),(5.1.19),(5.1.20), (5.1.21)and(5.1.24)to obtain,again,excludingthe specialmodesmentionedabove,

(LI0)Y+ 0(LIY)—9m2(H— ~)Y= 0, (5.1.28)

whereH h~,and 0 is definedby

= ~,,0. (5.1.29)

Next we consider the equationsstemming from the d = 11 Einstein equations.Using the aboveequations,the traceof (5.1.9) yields immediatelyafter elimination of V” hmn by meansof (5.1.15)

(LI~)Y+ (LI Y)+ H(LI Y)= 12m2(~— 7H)Y + ~(40)Y, (5.1.30)

while the tracefreepart of (5.1.9) implies

(LIS)Xmn+ S(LIXmn)= l2m2SXmn, (5.1.31)

7H+5~=0, (5.1.32)

V~B,,.= 0. (5.1.33)

In arriving at (5.1.32) and (5.1.33) it was necessaryto assume LIY� 0, LIY� 7m2Y and4Ym � 12m2Y~,thusadding7m2 modesof LI on zero-formsand12m2 modesof LI on one-formsto theset of specialcasesto bediscussedafter the generalanalysis.

Continuing this,the sameprocedureas abovebut adaptedto (5.1.8) gives

(LIB,,)Y~+B,,(LIY~)=12m2B,,Y0_6me,,~~f~0’V~a,,,,Y,,, (5.1.34)

V~h~= V,,(H+~ + 20). (5.1.35)

Fromthe generaldecompositionof h~,.[196],i.e.,

h,,,. = h~+ Vt,, 1/f)+ (V,,V,. + ~g,,,.LI)4~+ ~g,,,.H, (5.1.36)

whereV~is transverseandh~ is transverseand traceless;we find that (5.1.35)translatesinto

V~=0. (5.1.37)

58 Mi Duffel aL, Ka!uza—K!einsupergravily

Turning to the final field equation,i.e. (5.1.7), we find its traceto be

2(LIH)Y + (zL~)Y+ H(LI Y)+ 2m2GY 72m2HY— ~(LI0)Y, (5.1.38)

where m2G(x) V8’V~h,,,.(X),while the divergenceof (5.1.7) reads

V~’h,,,.(LIY)+ V,.[(LI — 12m2)H+ (LI + 12m2)~+m2G+~LI0]Y = 0. (5.1.39)

Finally, making useof (5.1.36) and(5.1.37)in the trace-freepart of 5.1.7, we find

(LIh~)Y+h~’(LIY) + 24m2h~Y= 0. (5.1.40)

It is now straightforwardto obtain the massoperatorsfor the variousspacetimefields provided, asabove,we excludemodeson M

7 satisfyingany one of the equations

LIY~0=0, LIY=0, LIY=7m2Y, LIYm = l2m2Ym. (5.1.41)

Given the definition of AdS massfor the variousbosonicfields in (3.2.17), (3.2.20)and (3.2.22),we canreadoff the mass operatorsfor h~(x), a,,(x) and S(x) directly from (5.1.40), (5.1.26) and (5.1.31),respectively.The result is presentedin table5, wherethesetowers of modesare denoted2~,1~and0.~2).By diagonalizingthe coupled equationsfor B~and E~ (1/m)&,,,.~,V”a”,i.e. (5.1.27) and(5.1.34),onefinds the ni,assoperatorsfor two towersof vector fields, denotedby 1~’~and1_(2) in table 5where(1) refersto the minussign and(2) to the plus sign.

For the remaining scalar fields, their massesare computed by considering(5.1.28), (5.1.30) and(5.1.32) togetherwith the equationresulting from the elimination of V~h,,,.from (5.1.35) and (5.1.39).Sincesomeof theseequationsare algebraiconly two towersemerge.Their massoperatorsaredenoted0~~1)(the minus sign) andO~(the plus sign) in table5. The last bosonic fields to be discussedare thepseudoscalars.Herethe massoperatorsfollow directly from (5.1.25)and (4.3.26). In table5 0~1)refersto the negativepart of the spectrumof 0 and0_(2) to the positivepart.

To concludethe derivationof thebosonicmassspectrum,we mustnowreturn to the modesthat felloutsidethe analysisabove,namely the modesmentionedin connectionwith (5.1.41). Considerfirst thesituationb

2 � 0, i.e., M7 admitsnon-trivial solutionsto LI Y,,,,, = 0. This hastwo implications. First, theconditionV~’a,,= 0 obtainedpreviouslyis no longerderivable.Secondly,we nowfind a gaugeinvariantfield equationfor at’, which hasto be the casesince,with b2 � 0, the gaugecondition (5.1.16) fails toeliminatethis gaugefreedom.Thus,we can trivially extendthe validity of the massoperatorfor the 1~tower to include thesemasslesspseudovectormodes.An analogousphenomenonoccursin the caseY = constant. Here one finds the masslessgraviton equationand its accompanyingunfixed gaugeinvarianceby discardingthe y-dependencein the field equationsandin the gaugeconditionV

mh~,,= 0.Thus,we can include the masslessmode in the 2~tower. In the scalar sector, however,the modeY = constantmust be excludedfrom the 0~1)tower. This is easily seenby going through the eqs.(5.1.7)—(5.1.13)again,with the result that, for this mode,the spectrumcontainsone scalarfield with amassvalue appropriatefor theO”~tower.

Turning to the 12m2 modesof LI on one-forms,theseare shown in section 4.3 to be in a one-to-one correspondencewith the Killing vectors on M

7. Here the symmetry left over by the gaugeconditions(5.1.14)—(5.1.16)is the Yang—Mills gaugeinvarianceof the d = 4 theory.Thus, we can now

M.i Duffel aL, Ka!uza—Kleinsupergravily 59

add (5.1.33) as an extra gaugecondition, and hencemakethe abovederivationof the massoperators

1”~and 1_(2j valid alsofor modessatisfyingziY,,, = 12m2Y~.

We now considerthelast specialcase,namelymodessatisfyingLI Y = 7m2Y. As was shown in section4.3, thesemodesare relatedto the existenceof conformal Killing vectorsandare, therefore,presentonly if M

7 is the round sevensphere[191].Now, a repetition of the analysisof (5.1.7)—(5.1.13)fortheseparticular scalar modeson M7 will seem to imply the existenceof two scalar fields in fourdimensions.However, also for thesemodesthe gauge conditionsfail to fix the gaugecompletely,leaving the possibilityof addingextragaugeconditions.It is convenientto choose

hm~(x, y) = hmm(X)Y(Y)= 0, (5.1.42)

where Yis a 7m2 modeof LI. By (5.1.19),this means.~(x)= 0, thusleaving only one scalarfield in thetheory.Onemay calculatethat this field hasmass91m2,andhenceit belongsto the Q~3)tower.For the0~1)tower thereforethis LI = 7m2 mode must be omitted. This concludesthe analysisof the bosonicfield equations,and we now continuewith a similar treatmentof the fermionic field equations(5.1.6).

The gaugeconditionsto be usedin the fermionic sectorare[34]

Tmt/i~(x, y) = 0, (5.1.43)

which implies that a split of the d = 11 indices of equations(5.1.6) into spacetimeand internalindicesgives rise to the following equations

— y5y”F” Dpi/i,. — ~my5y~5/’,.+ y”(Dmi/Jm)= 0, (5.1.44)

—y5r”D,,cuim + y5Im(D”clim)+ ~my5Vim + FmT

0Dn(y~V’~)+ Dm(y~Viv)— y”Dvçlim + Fm

75(y” D,.cui,,) = 0. (5.1.45)

In order to arrive at (5.1.44) and (5.1.45) we madeuseof the expressionsfor 15,. and .1.5,,, and thedecompositionof the d = 11 DiracmatricesfA given in (3.1.14), (3.1.15)and(3.1.11),respectively.Wenow proceedto expandV’,,(x, y) and Vim(X, y) in terms of the spinorharmonicsZ(y) andZ~(y),thelatter satisfying

Dm Z~(y) Tm Zm(Y) 0. (5.1.46)

(For moredetailsaboutZ(y) and Z~(y)seesection4.3.) Onefinds, using (5.1.43),

Vi,,(x, y) = çfi,,(x)Z(y), (5.1.47)

çt’m(x, y) = ~(X)Zm(y)+~(X)D~Z(y), (5.1.48)

wherethe J~tmtrace-freeoperatorD~is given by

= Dm + ‘/1’rn~7. (5.1.49)

Mi Duff et aL, Kaluza—K!einsupergravily

Turning to thefermionic field equations,we first takethe y~’traceof (5.1.44)to obtain the followingexpressionfor y’~~D,~Vi,.:

= —~y5(F~D,,+ ~m)y~’Vi1.— 2DmVi~. (5.1.50)

If we usethisequationto eliminatey”~D,,Vi,.from the field equations,andthenuse (5.1.47)and(5.1.48)

we find that (5.1.44)reads

(F-~c”,~)Z = (D,,ç~)Z— Ys(Vi,~— ~y,,~)(Ø + ~m)Z— ~ (5.1.51)

where ç~ yMVi~~while a substitution of (5.1.50) into (5.1.45) resultsin an equationwhich can bedecomposedinto threeseparateequations.By taking the F trace andthe divergenceof the equationresultingfrom (5.1.45)the result is

(I~)Zm= y5,k~m— .0)Zm, (5.1.52)

2ysXDmD~Z+ç~(/~+ ~m)Z = 0, (5.1.53)

çi.Z— (I1~x)Z Y5x(~P— ~m)Z =0. (5.1.54)

Equation(5.1.54)is valid only provided ØZ� ±(7m/2)Z, i.e., from section4.3 providedthat

D~Z=±~mFmZ. (5.1.55)

Theseconstitutethe fermionicanalogueof the bosonicmodesof (5.1.41).We nowcontinuethe analysisassumingthat (5.1.54)holdsandreturn to (5.1.55) later.

Thus,since

D~D~Z= —~(~ + ~m)(~ — ~m)Z, (5.1.56)

(5.1.53)gives an expressionfor ç&Z which whencombinedwith (5.1.54)becomes

(~x)Z=y5x(Ø—~m)Z. (5.1.57)

Finally, (5.1.51)can be seento give

+ y~i/’~(Ø + ~m)L = 0, (5.1.58)

wherethe transverseand ‘y trace-freeVi~Tris definedby the decomposition

= Vi~+D~Vi—~yAi, (5.1.59)

whereD~=D8,~The massoperatorsof the two spin~ andthe four spin ~towerspresentedin table5 arenow readily

seento follow from (5.1.58),(5.1.57)and(5.1.52).The AdSmassusedherefor spins~and~weredefined

Mi Duffel at, Kaluza—K!einsupergravily 61

in (3.2.18)and (3.2.21). Recallingthe caveatunderwhich theseresultswereobtained,we nowreturn tothe modesspecifiedin (5.1.55) and derive the massmatrix for the correspondingAdS fields. Startingwith the minus sign in (5.1.55), it follows from (5.1.53)and (5.1.56)that Vi(x) = 0. By meansof (5.1.51)andthe decompositionof Vi~in (5.1.59)it can thenbe shown that in fact Vi~= ~ with the implicationthat the ~m of .0 on M7 spinors must be excludedfrom the ~1) tower. In the caseof the plussign in (5.1.55), we noticethat the original gaugecondition (5.1.43) can no longer be maintainedsinceFtm1.5mE is identically zero. Insteadwe pick the gauge Vi = 0, which leads to y’~”D4,,~= 0 and(0+ 8my5),~= 0. Thus, thesemodescan be absorbedinto the structureof the towers ~1) and ~4),

respectively.Theresultsof this sectionarecollectedin table5. As in [80],the differenttowersof a given spin are

distinguishedby superscripts.For fermionsandpseudoscalars,the first (second)superscriptrefersto thenegative(positive) part of the spectrumof the relevantfirst-orderoperator,while for the bosonictowerswith superscriptsthe first (second)onecorrespondsto taking the minus(plus) sign in the expressionforthe massoperator.Concerningthesetowers, thereis the following caveat;althoughsome fields areobtainedfrom perfectly respectablemodes of someoperatorson M7, they should neverthelessbeomittedfrom the physicalmassspectrum.Thesefields arein the0~’~tower the 0 and7m

2eigenvaluemodesof 4~and in the ~1) tower the 7m/2 mode of .01/2. Note that for clarity, we havereinstatedsubscriptson the differentialoperatorson M

7 in table5 indicatingwhetherthe operatoractson p-forms,LI,,, second-ranksymmetrictransverseandtracelesstensors,

4L, spinors,01/2, or vectorspinors,03,2. Itremainsonly to computethe spectrumof theseoperatorson the particular M

7 in question.Weemphasizethat table5 is equallyapplicableto both homogeneousandinhomogeneousspaces.(To date,however,the massspectrumhasbeencomputedin full only for theroundS

7 [34,80,90, 194, 195, 279], asdiscussedin section7.2, andfor the N(k, 1) spaces[197]as discussedin chapter9. Partialresultsfor thesquashedS7 of chapter8 maybe found in [65,92, 198] andfor the M(m, n) spacesof chapter9 in [34,35, 36, 199].)

We alsopoint out that by setting m = 0 both in table5 andin the caveataboutexcludedmodes,weobtain the result for a Kaluza—Klein reduction on a Ricci flat space.The masslesssector is thendeterminedsolely by the zero eigenvaluemodesof the relevantoperatorson M

7, information aboutwhich is provided by Betti numbersb,,, etc. It is, however,importantto realizethe only if the internalspaceis Ricci flat is this the case.Thereis, however,a subtlety for spaceswith non-zerob1. Theclassicexampleis T

7 for which the spectrumis completelyknown (seesection9.1).A numberof commentson table 5 arenow in order,recallingthe propertiesof the operatorsgiven in

section4.3.

Table 5Massoperatorsfrom theFreund—Rubinansatz

Spin Massoperator

+

(3/2)~~~’(2) 41,2 + 7m/2

~ (2) + 12m2±6m (41+ 4m2)’12

(1/2)~~~’(1) 41/2 — 9m/2(1/2)~~~(2) 3m/2—

O~1~3) 4o+44m2±12m(40+9m

2)’12

AL—4m2

O_(1)(2) ~2+ 6mQ + 8m2

62 Mi Duffel aL, Kaluza—K!einsupergravity

(i) SinceLI0 hasoneandonly onezeromode,thereis oneandonly onemasslessgravitonin the d = 4

theory.This is reassuring.(ii) Masslessgravitinos, i.e., unbrokensupersymmetries,correspondto —7m/2 modesof ‘~1/2~As

shownin section4.3, thesearealsoin one-to-onecorrespondencewith the Killing spinors.(iii) Masslessspin 1, i.e., unbrokengaugesymmetriescorrespondto 12m

2 modesof LII. As showninsection4.3, thesearealsoin one-to-onecorrespondencewith the Killing vectors,e.g.,the 28 of S0(8) inthe caseof the roundS7 or the 7 x 1 of [U(1)]7in the caseof ‘F7.

(iv) Masslessspin 1÷also appearwhen LI2 haszero modes, i.e., when b2(M7)� 0. Thesemust be

Abelian, however,since zero-modesof LI,, are always singletsunder the isometry group. The Abeliansymmetryis R ratherthan U(1) sinceit is a remnantof the AMNP gaugeinvariancein d = 11. All statesin the spectrumarethusneutralwith respectto thisR ~ symmetry.Thereareno such modeson S

7 butfor exampleT7 has21, K3 x T3 has28 andall M (m, n) have1.

(v) Contraryto assumptionssometimesmadein theKaluza—Klein literature,masslessspin~aregivenneitherby zero modesof ~~1/2 nor by zero modesof ‘~3/2~This maybetracedback to the existenceofnon-minimalfermioncouplingsin the d = 11 theory.Note, however,that thereis no apparentreasontoexpectspin ~fields to be masslessunless,for N> 0, theyaresuperpartnersof gaugefields. For examplethe round S7 with N = 8 yields 56 masslessspin ~, whereasthe N = 0 squashedS7 of chapter8 yieldsnone. Curiously,however,the Q(1, 1, 1) spacewith N = 2 of chapter9 has 16 masslessspin ~ which arenot partnersof gaugefields in addition to thosewhich are.

(vi) Similarly, thereis no apparentreasonto expectmasslessspin 0 unlessthan are demandedby anunbrokensupersymmetry.(Note alsothat for m � 0 the pseudoscalarmassmatrix is not given by 0 andso thereis no reasonto expectmasslessspin 0 for topological reasons.)In the0~’~tower, masslessscalarscorrespondto 16m2 and 40m2 modesof LI

0. Onecan show that in vacuawith N>0, the 16m2

modesbelongto masslesssupermultipletsas describedin section 3.2. For exampleon the round S7,thereare 35 16m2 modesand 294 40m2 modes[194].On the squashedS7, thereare 135 40m2 modes[25]. Moreover, they occur for both the N = 1 and N = 0 cases.The curious occurrenceof masslessscalars for no apparentsymmetry reason is, in fact, a common feature of Freund—Rubin com-pactifications.When masslessspin 0 occur for m � 0, they are not Goldstonebosonsas sometimesoccursfor m = 0.

(vii) Since the skew-whiffing of section4.2 reversesthe spectrumof p1/2, p3/2 and0 but not LI~or4L, the massspectraof spins~, ~and0 aresensitiveto the orientationof M

7 but thoseof spins2, 1 andare insensitive.Finally, it is sometimessuggestedin the literature that the spacetimecosmologicalconstantwhich

arises naturally in spontaneouscompactificationat the tree level should be cancelledby the ad hocaddition of a bare cosmologicalconstantin the original higher-dimensionalLagrangian.We do notadvocatesuch an approach.However, for the sake of interest,we haverepeatedthe bosonicmassspectrumcalculationsof thissectionin the casethat the d = 11 theory is modified by a barecomologicalconstantA11= 54m

2/3 such that /3 = 0 correspondsto the previouscalculation and /3 = 1 to a four-dimensionalspacetimewith A = 0. See appendixC. Although this exercise is somewhatartificial, itteachesus somelessonsaboutKaluza—Klein. In particularwe notethat evenwhen /3 = 1 andspacetimeis flat, that masslessantisymmetrictensorsdo not necessarilycorrespondto zeromodesof LI,, andthatthe dilation modeLI~= 0 in the ~ tower doesnot giverise to a masslessscalar.The only exceptioniswhen m = 0. Theseresultsthusprovide a counterexampleto someof the claims in the Kaluza—Kleinliterature.Note that we havenot includedfermions in this calculation since the addition of a bare Aterm in d = 11 breaksthe supersymmetryexplicitly, andpresumablygives rise to inconsistencies.

Mi Duffel aL, Ka!uza—Kleinsupergravuy 63

5.2. The criterion for vacuum stability

As mentionedin section3.1, thereare infinitely manyseven-dimensionalEinstein spacesM7 whichmight serveas vacuumstatesin the spontaneouscompactificationof d = 11 supergravitydown to d = 4.

This illustrateswhat is a major problemin any Kaluza—Klein theory: how to select the true vacuumfrom amongthemanycandidateground-statesolutions.Oneobviousway to narrowdownthis choiceisto demandthat the vacuumbe stableand, as discussedin section3.2, this will be guaranteedif thereisan unbrokensupersymmetry.The questionnaturally arises,therefore,whetherthe converseis true: areall non-supersymmetricvacuaunstable?This is not without physical interestsincethe experimentallyobservedvacuumhas no supersymmetrybut is presumablystable.This questionwill be answeredin thenegative in chapter 11 but as a preliminary we derive in this section the necessaryand sufficientconditionthat the vacuumsolutionsobtainedby Freund—Rubincompactificationbestableagainstsmallfluctuations,i.e., that theysatisfy (3.2.23)M

2 � 0 for spins>0,and(3.2.24), M2 � —m2 for spin 0.We shall prove that statesof spin 2, ~, 1~,~ and0 alwayshavenon-negativeenergy,that possible

instabilitiescan ariseonly in the 0~2~tower of table 5 and that the criterion for stability can beexpressedby the single inequality [25]

LIL�3m2. (5.2.1)

First we considerthe massoperatorsfor spins>0 in table 5. The M2 � 0 condition will be triviallysatisfiedby the fermionssinceboth ~~1/2 and~~312areself-adjoint andhencehavereal eigenvalues.Spins2~,1’~,1_(2) also satisfyM2� 0 sinceLI~is positivesemi-definite(seesection4.3). This will alsobe truefor i-”~since,as shownin section4.3, LI~� 12m2.

Now weconsiderspinsOt The M2 � —m2conditionwill be automaticallysatisfiedby 0+0) and~sinceLI

0 � 0. Similarly, by writing .M~for 0_(1),(2) in the form (Q + 3m)2— m2 we seethat pseudoscalars

can nevergive rise to instabilities.Interestinglyenough,in both caseswe havereproducedpreciselytheBreitenlohner—Freedmanbound.

Thus the whole problemof stability devolvesupon the 0~2)tower. From (3.2.24), the criterion ofstability is simply given by (5.2.1),namely ‘1L ~ 3m2. Althoughit is difficult to makegeneralstatementsabout boundson the spectrumof the Lichnerowicz operator,the stability propertiesof all knownFreund—Rubinsolutionshavebeenanalyzedandwill be discussedin chapter11.

In appendixC, we haverepeatedthe stability analysisof this section with the barecosmologicaladdition A

11 = 54m2/3.This is valid in the region 0� /3 � 1. It is no longertrue that possibleinstabilities

areconfined to the 0÷(2~tower.They can also arisein the 0~1)and0~”~towers.

6. Classificationof knownd = 7 Einsteinspaces

In chapters3, 4 and 5 we haveanalyzedcompactificationsof d = 11 supergravityto d = 4 of theFreund—Rubinform, taking the d = 4 spacetimeto be AdS but withoutspecifying the internalEinsteinspaceM

7. As it turns out, thereareinfinitely manyd = 7 Einsteinspaces.A priori, eachcould claim tobethe vacuumstate,at leastif it is stable.In table6 we simply list all knownsolutionswithoutprejudiceas to their physicalrelevance.This list is exhaustivefor cosetspaces[200],but presumablythereexistmanyotherinhomogeneoussolutions,in addition to the two known cases*K3 X T

3 andS3 X the twistedS2 bundleover S2.

* Seesectionon recent developments.

64 Mi Duffelat, Kaluza—K!einsupergravisy

Table 6

Solution G N Is2 Stable?

Round57 S0(8) 1 8 0 Yes

Squashed~7 S0(5)x SU(2) G2 1 0 Yes

S5x S2= M(1, 0) SU(4)x SU(2) SO(7) 0 1 No

S4x S0(5)x SU(2)x SU(2) SO(7) 0 0 NoS2x S~x S3= 0(0,0,1) [SU(2)]4 S0(7) 0 2 NoS2x T

1S3 = 0(0, 1, 1) ISU(2)13x U(1) SO(7) 0 2 No

Twisted(S2 x S2) x [SU(2)]’x U(1) SO(7) 0 2 Nocp2x S~= M(0, 1) SU(3)x SU(2)x SU(2) SO(7) 0 1 No

SU(3)x S2 SU(3)x SU(2) SO(7) 0 1 No

S0(5)SO(5) G

2 1 0 YesSO(3),,,,,V52 SO(5)xU(1) SU(3) 2 0 Yes

M(3, 2) SU(3) x SU(2)x U(1) SU(3) 2 1 YesM(m,n) SU(3) x SU(2)x U(1) SO(7) 0 1 Seebelow0(1,1,1) FSU(2)1

3xU(1) SU(3) 2 2 Yes

Q(p, q, r) [sU(2)]3x U(1) S0(7) 0 2 See belowN(1, 1)~ SU(3)x SU(2) SU(2) 3 1 Yes

N(1, l)~~ SU(3) X SU(2) G2 1 1 Yes

N(k,l)~~ SU(3)x U(1) G2 1 1 Yes

T7 [U(1)17 1 8 21 Yes

K3 x ‘I’s [U(1)l’ SU(2) 4 25 Yes

A numberof commentsarenow in order.(1) Therehasbeenatendencyin the literatureto adopta notationfor certainspaceswhich explicitly

includesnon-simply-connectedversionsobtainedby factoring the covering spacesby discretegroups,while for others,suchacharacterizationhasnot beenused.An exampleof the latteris theseven-sphere;therearein fact infinitely manylensspacesL(k; l~,12, 13), wherek is an arbitraryintegerand 15, 12, 13 arearbitraryintegersprimeto k [265].Theseare,of course,all solutionsof thesupergravityfield equations.Examplesof spaceswherenon-simply-connectedversionshavebeenincludedin theirnotationaretheM”spaces[26,30], wherethethird integerparameterr isusedtocharacterizethefundamentalgroupof certaindiscretefactoringsof the coveringspaces.

In order to give a balancedtreatmentof all solutions,we find it convenientto useamoreappropriatenotation for the M” and ~ spaces.Accordingly, for the M”~spaceswe use insteadthe notationM(m, n) of [16, 201], where rn/n = 3p/2q. Similarly for the N”~spaces[202,203] the parameterr isredundantfor this classification,and so we use instead the notation N(k, 1) of [204]. In this case,k/l=(3p+q)/(3p—q).

The M(m, n), Q(p, q, r) [203, 205] and N(k, 1) spacesare all simply connectedif and only if theintegersarerestrictedto be relativelyprime.

(2) For generalvaluesof the integersk and 1, the spacesN(k, 1) admit two distinct Einsteinmetrics[203].Thesearedenotedby N(k, l)~and N(k, ~ However, if I = 0, thesetwo metricsareidentical.

(3) The spaceV5,2 is the Stiefel manifold SO(5)/SO(3)endowedwith its Einsteinmetric [200].

(4) S2x S3 admits a second,non-producthomogeneousEinstein metric [206].This yields a second

~2 X S2x ~ solution,which in fact is the spaceQ°11.(5) With the exceptionof the round S7, all non-Ricci-flat spaceswith Killing spinors admit these

M.J. Duffel a!., Ka!uza—K!einsupergravity 65

spinorsonly for onechoiceof the sign in (4.2.9). This meansthat in all thesecasesthereis a secondnon-supersymmetricsolution obtained by skew-whiffing M7, i.e. reversing its orientation. Thesesolutions are also stable, despite the absenceof supersymmetry[25], and have not been includedexplicitly in the table.

(6) The M(m, n) andQ(p,q, r) solutionsare stableif andonly if theratiosof the integerparameterssatisfy certainconditions.The M(rn, n) solutionsarestableif andonly if [201].

~V~<Im/nI<~V~, (6.1)

which, in termsof the p andq reads

<Ip/qI <rrf~/66. (6.2)

TheQ(p, q, r) solutionsarestablein acertainregioncontainingthepointp = q = r = 1. Theshapeof thisregion is given in [207].

(7) If M7 has secondBetti number b2> 0, the b2 harmonictwo-forms give rise to b2 additionalmasslessvector fields. Since thesedo not couple minimally to any other fields, each one has anon-compactAbelian gauge invarianceR. Thus, the full gauge group is G X R b2 where G is theisometrygroupof M7.

(8) All solutionswith N supersymmetrieshaveisometry group G = S0(N)x K for some group K[30]. This follows on generalgroundsfrom the fact that the bosonicsectorof OSp(4/N)is SO(3,2)xS0(N).

7. The round seven-sphere

7.1. SO(8), triality and N = 8

The maximally symmetric M7 is given by the standardSO(8) invariant seven-sphere.It mayconvenientlybe describedas the surfaceof radiusrn~,

= m2, (7.1.1)

embeddedin R8 with cartesiancoordinatesyA (A = 1, . . . , 8) andflat positivedefinite metric 8AB = diag

(1, 1, 1, 1, 1, 1, 1, 1). The metric inducedon the surface(7.1.1) from the flat metricds2= 8,.~dy’~ dyB (7.1.2)

on R8is manifestlySO(8)invariant,sinceboth (7.1.1)and(7.1.2)arepreservedby S0(8)matricesactingon the column vector

/yl\

y= ( (7.1.3)

66 Mi Duffel aL, Ka!uza—K!einsupergravity

in thestandardmanner.We shall refer to thisgeometryasthe round 57~Later we shall considerspaceswhich are still topologically S7 but which havedifferent geometries.

All pointson the sphereareequivalent,and maybe mappedinto oneanotherunder the transitiveactionof SO(8). This maybe seenby choosinga particularpoint on the sphere,suchas

0

0

0

0

0 ‘ (7.1.4)0

0

m1

and noting that the generalpoint (7.1.3) can be reachedby acting on (7.1.4) with SO(8). The SO(7)subgroup

SO(7)

( ~.) (7.1.5)

leaves(7.1.4)fixed, andsoS7maybedescribedasthecosetspaceSO(8)/SO(7).Asexplainedin section4.1though,theroundS7canequallywellbedescribedasSO(7)/G

2,SU(4)/SU(3)or Sp(2)/Sp(l).It isnotmerelya homogeneousspace,but is also a symmetricspacein the sensedescribedin section4.1.

From (7.1.2), the inducedmetric on the sphere(7.1.1) is

ds2 = (ômn + dytm dy~, (7.1.6)

wherem = 1,. . . , 7. A straightforwardcalculationshowsthat the Riemanntensoris given by

Rmnpq= m2(gmpgnq — gmqgnj,), (7.1.7)

which accordswith thefact that the roundS7 is a spaceof constantcurvature.For practicalcalculations(7.1.6) is not very convenient,andwe shall insteadusethe metric andsiebenbeingiven in chapter8.

The group SO(8) possessesthe unique property of “triality”, i.e., there are three inequivalenteight-dimensionalrepresentationswith Dynkin labels (1, 0, 0, 0), (0, 0, 0, 1) and (0, 0, 1, 0) to which weconventionallyassign the namesvector (8w), spinor (8~)and conjugatespinor (8,,), respectively.(Wefollow the notation of Slansky[208]exceptthat s andv areinterchanged.)This tnality will be crucial inwhat follows.

As a consequenceof (7.1.7), Cmnpq= 0, and from (4.2.7) the holonomy is trivial ~‘ = 1, and henceNmax= 8. Moreover, it is not difficult to show that thereare indeed8 solutionsof the Killing spinorequation(4.2.5) in accordancewith the fact that, with Cm,,pq = 0, all higher-orderintegrabilityconditionsare trivial. Let us denotethesesolutions i~’ (I = 1 . . . 8) wherewe assignthe index I to the 8. of SO(8).Note, however, that by replacingm by —m in the Freund—Rubinansatz(3.1.7), one obtainsa newsolution for which the criterion for unbrokensupersymmetryis obtainedby replacingm by — rn in

Mi Duffel a!., Kaluza—Kleinsupergravily 67

(4.2.5).Therearestill 8 solutionsdenotedby ii” (I’ = 1 . . . 8) wherewe assignthe index I’ to the 8,, ofSO(8). This accordswith our previousobservationsin section4.2 and the fact that 57 is a symmetricspace.Unlessotherwisestated,we shall adoptthe ansatzas given in (3.1.7).

Sincethe round S7 admits 8 Killing spinorsandsinceits isometrygroupis SO(8), the Kaluza—Kleinmechanismwill give rise to an effective d = 4 theory with N = 8 supersymmetryand local SO(8)invariance,describinga masslessN = 8 supermultipletcoupledto an infinite tower of massiveN = 8

supermultipletswith massesquantizedin units of m, the inverseradiusof 57~Combining the internalSO(8) symmetry and N = 8 supersymmetrywith the SO(3,2) of the AdS spacetimebackground,it isreadily seenthat the completesymmetryof the round S7 vacuumstateis OSp(4~8)whosealgebrawasgiven in section 3.2. The masslessand massivesupermultipletswill then correspondto irreduciblerepresentationsof OSp(4 8) but maybereducibleunderSO(8) [44].

It follows without anyfurthercalculationthat the masslesssectorof the d = 4 theory correspondstothe familiar masslessN = 8 supermultipletshown in table 7. The 8, index on Vi,~’correspondsto the 8Killing spinorsi~’ and the 28 index on A,,IIJI to the 28 Killing vectorsK~~’1.We shall denote the 3inequivalent8-dimensionalrepresentationsof S0(8) by the indices I for 8,, I’ for 8,, andA for 8~.Correspondinglythe threeinequivalent35’s aredenoted

s c v35, (If) [I’J’K’L’]÷ [ABCD]35~ [IJKL]_ (I’J’) [ABCD]+35~ [IJKL]+ [I’f’K’L’]_ (AB)

where (If) means symmetric trace-freeand [IfKL]± means totally antisymmetricself-dual (+) oranti-self-dual(—)

÷11IJKL1 — — IJKLMNPQ 7 1 8~ — E [MNPOI±•

Themassiverepresentationsareobtainedby takingthe tensorproductof the masslessmultiplet andthe representation(n,0, 0, 0) which correspondto the eigenmodesof LI

0 andwhich, from table5, yieldthe massivegravitons.This done, we allow eachmasslessspin 2 to eat a spin 1 andspin 0 to becomeamassivespin 2, eachmasslessspin~to eat a spin ~to becomea massivespin ~andeachmasslessspin 1to eat a spin 0 to becomea massivespin 1. The resultsareshownin table8. For example,the lightestsuch multiplet has the S0(8) content {8~, 8,, + 56,,, 56~+ 160w, 160,,+ 224w,,, 112w, 224,,~}and the

Table 7 Table 8The masslessN = 8 supermultiplet MassiveN = 8 supermultiplets

Spin Field S0(8)rep. Dynkin label Spin Dynkin label

2 e,,’ 1 (0,0,0,0) 2 (n, 0,0,0)3/2 ~‘ 8, (0,0,0,1) 3/2 (n,0,0,1)+(n—1,0,1,O)

A,” 28 (0,1,0,0) 1 (n,1,0,0)+(n—1,0, 1, 1)+(n—2, 1,0,0)1/2 XUK 56, (1,0,1,0) 1/2 (n+l,O,1,O)+(n—1,1,1,0)+(n—2,1,0,1)+(n—2,O,O,1)0~ SF~KI.)+ 35, (2,0,0,0) 0~ (n+

2,O,0,0)+(n—2,2,0,0)+(n—2,0,0,0)pI1JKLL 35,, (0,0,2,0) 0 (n,0,2,0)+(n—2,0,0,2)

68 Mi Duffel a!., Ka!uza—K!ein supergravily

next-to-lightesthasthe S0(8)content{35~,56.+ 224w,, 28+350+ 567w, 8, + 160.+ 672~,,+ 840,, 1 + 294w+

300, 35.+ 840~}.Note that the s- andc-typerepresentationsoccuronly for spins~, ~, 0 while spins2, 1and 0~have either no subscriptor a v subscript. This accordswith our previous observationthatskew-whiffing interchangess and c labels,andinterchangesthe positive and negativespectrumof theoperators.03,2, ~~1/2 and0 of table 5,while the spin2, 1 and0~spectraareinsensitive.

Note also that the maximum spin of these massivesupermultipletsis 2 and not 4. This is the“multiplet shortening” phenomenonof section 3.2. We now turn to the calculation of the massspectrumitself.

7.2. Themassspectrurn

We now usethe massoperatorsof table 5 to calculatethemassspectrumon the roundseven-sphere.To obtainthe eigenvaluespectrumof the relevantoperatorsin thiscaseis particularly simple sincewearedealingwith a symmetricspace.Oneway of calculatingthe spectrawould be to embedthe roundseven-spherein EuclideanR8and restrictharmonictensorsin ~ to 57~However, herewe shall usethetechniquefor coset spacesG/H describedby Salam and Strathdee[149] becauseof its wider ap-plicability. This method becomesvery simple in the caseof symmetricspacessince the action of acovariantderivativeon an irreducible tensorharmonic Y is simply

VmY Cern” TaY, (7.2.1)

where T~are the generatorsof G not in H, c is a constantdependingupon the normalization ofgeneratorsTa and erna is thesiebenbeinon 57~In (7.2.1)we havesuppressedany tensoror spinor indicesthat Y may carry as well as any internal indiceslabelingthe isometrygrouprepresentations.Note thatthe derivativein (7.2.1) is covariantwith respectto all tangentspaceindiceson Y. (Seesection4.3 for aderivationof this equation.)

Squaring(7.2.1)we obtain

—LIIY= c2(CG—CH)Y, (7.2.2)

sincein the notationof section4.1

(Ta)2 = (TA)2 — (T~)2= ~Co — CH), (7.2.3)

whereCG andCH arethe quadraticcasimirsof G andH. Thusthe problemhasbeenreducedto the oneof determiningwhich isometry representationsoccurfor each of the relevantoperators.This can bedoneeither by the previouslymentionedmethodof embeddingin R8 or by the techniquediscussedatthe endof section4.3 [149].The resultis given in (7.2.5). Transversalityconditionshavebeenimposedwherevernecessary.

The universalconstantc is most easilydeterminedby consideringa specialcase.For example,onecan usethe 28 Killing vectorsK,. which satisfy —LiK,. — 6m2K,.= 0. By using the normalizationof C

0and CH adoptedby McKay and Patera[209]we find CG = 12 and CH = 6 which implies c

2= m2, i.e.,from (7.2.2)

—LIY= m2(C0—CH)Y. (7.2.4)

M.i Duffelat, Ka!uza—K!einsupergravily 69

We now usethe explicit expressionsfor the operatorswhich appearin table5, as givenin section4.3togetherwith the expressionfor the Riemanntensorin (7.1.7).We obtainthe following eigenvalues:

SO(7)representation Operator S0(8)representation Eigenvalue1 40 (p,O,O,O) p(p+6)m2

7 LI1 (p — 1, 1,0,0) [p(p + 6) + 5]m

221 42 (p—i,0,1,1) [p(p+6)+8]m235 0 (p — 1, 0, 2, 0) — (p + 3)m

(p—i,O,O,2) (p+3)m27 LIL (p—2,2,0,0) [p(p+6)+12]m28 .4~1/2 (p,O,l,O) (p+~)m

(p,O,O,l) —(p+~)m48 11)3/2 (p—i, 1, 1,0) (p+~)rn

(p — 1, 1, 0, 1) — (p + ~)m (7.2.5)

Substituting (7.2.5) into the mass operatorsof table 5 gives the mass spectrumof the roundseven-spherepresentedin table9. This wasfirst calculatedby [80,90, 194].

Severalcommentsarenow in order:(i) As expected,each value of n >0 correspondsto a massiveN = 8 supermultiplet and has

maximum spin 2 in accordancewith the multiplet shorteningphenomenonof section4.2. The lowestenergyvaluesarethosegiven by (3.2.41).The sign ambiguity for spins0 and~is resolvedby noting thatwhen the N = 1 Heidenreichsupermultipletsof section3.2 are combinedinto N = 8 supermultiplets,particlesof the samespin and SO(8)representationmustchoosethe samesign.

(ii) Also as expected,all massessatisfythe positiveenergycriteria (3.2.23)and(3.2.24)in accordancewith the generalresult that supersymmetricvacuaarestable.Curiously,however,all spin 0 exceedtheBreitenlohner—Freedmanbound(3.2.24) except the 112w = (3,0, 0, 0) in n = 1 level of the 0~towerwhich saturatethe bound.

Table9Themassspectrumon theroundseven-sphere

Spin S0(8)rep (Mass)2 Lowest energy

2 (n,0,0,0) (n +3)2—9 (n + 6)/2~(‘) (n, 0,0, 1) (n + 5)/2~(2) (n— 1,0,1,0) (n +6)2 (n +7)12

i~’~ (n,1,0,0) (n+ 1)2—1 (n+4)/2i~ (n—1,0,1,1) (n+3)2—1 (n+6)/21(2) (n—2,1,0,0) (n+5)2—1 (n+8)/2

(n +1,0, 1,0) ~ (n +3)12(n—1,1,1,0) (n+2)2 (n+5)/2(n—2,1,0,1) (n+4)2 (n+7)/2(n —2,0,0,1) (n + 6)2 (n +9)12

0+(1) (n+2,0,0,0) (n—1)2—1 (n+2)/20~” (n,0,2,0) (n + 1)2—1 (n+4)/204~2) (n—2,2,0,0) (n+3)2—1 (n+6)/20(2) (n—2,0,0,2) (n+5)2—1 (n+8)/2~ (n —2,0,0,0) (n + 7)2— I (n + 10)/2

70 Mi Duffelat, Ka!uza—K!einsupergravisy

(iii) As discussedin section3.2, membersof the sameAdS supermultipletdo not in generalhavethesamemass.In particular,masslessspin 0 could belongto either massless(n = 0) or massive (n > 0)supermultiplets.In fact, in addition to the massless35~in n = 0 level of theO~’~tower, onefinds [194]an extramassless294w in the n = 2 level of the 0~1)tower.

(iv) We recall from thecaveatsthat accompaniedtable5 that 7rn2 modesof LI0 (which can occuronly

on the round 57) and7m/2 modesof ~~1/2do not correspondto physical statesin either the 001) ortowersof the massspectrum.Thesemodesbelongto the8~spin 0~andthe 8,, spin ~towers.As pointedout by Sezgin[90], thesestates,if present,would havecorrespondedto N = 8 generalizationsof theN = i singletonrepresentationsdiscussedin section3.2. It is perhapsjust as well that thesestatesareabsentfrom the spectrumsincethereis asyet no adequated = 4 Lagrangianfield theory descriptionofthesestates,oneof the manyproblemsbeingthe fact that the fermionshaveonly half the usualnumberof degreesof freedom.An N = 8 supermultipletwith Smax = ~is certainly bizarre.Note, however,thattheir absencefrom the theory hasto datebeenestablishedonly at the linearizedlevel.

(v) Since 57 is a symmetricspace,its massspectrumshouldbe insensitiveto skew-whiffing. This isindeedthe caseas maybe seenby noting its symmetryunderinterchangeof the s andc labels.

As we remarkedbefore,the major differencebetweenKaluza—Klein theoriesandconventionalfieldtheories is the infinite tower of massive states, as is amply demonstratedby the round S

7 corn-pactification. Let us examine the ramifications of this in more detail. For example,the infinity ofmassivegravitinos is dueto an infinity of spontaneouslybrokensupersymmetries[451as may be seenbyFourierexpandingthe d = 11 supersymmetryparameterin harmonicson 57

E(x, y) = ~ Ek(x)Z”(y), (7.2.6)

where Zk(y) are eigenmodesof (01,2+ 7rn/2) and k denotesthe (n,0,0, 1)+ (n — 1,0, 1,0) represen-tations of S0(8) as described in table 8. The lowest n = 0 mode, which is 8-fold degenerate,correspondsto the unbrokenN = 8 supersymmetryand to the 8 masslessgravitinos, whereastheremaining Ek transformstatesof different mass into eachother. Similar remarksapply to the otherd = ii local symmetriesof generalcovariance,local SO(1,10) andgaugeinvarianceof AMNp andto theFourier expansionof the correspondinggauge parameters.In this way one obtains an infinitedimensionalnon-compactsuperalgebraanalogousto the Kac—MoodyandVirasoro algebrasof section(1.2). It containsa finite-dimensionalsubalgebrawhich like the Salam—StrathdeeS0(1,2) acts as aspectrum generating algebra. Indeed, Salam and Strathdee pointed out that for S” therelevantalgebrais SO(1,k + 1) i.e. SO(1,8) for 57~

This SO(1,8) hasbeenexaminedin somedetail by GUnaydinandWarner[210].That it is a subgroupof the generalcoordinategroupfollows from a constructionanalogousto (1.3.16), wherewe augmentthe 28 Killing vectorsKm48 on S7 by the 8 conformal Killing vectors CmA on S7

~ (x, y) = a(x)~Km~4il(y)+ aA(X)CmA(y). (7.2.7)

Both Km and Cm may beexpressedin termsof the eight7rn2 modesçb’~’of i~

= 7rn2çb” (7.2.8)

by theequations

Mi Duffel at, Ka!uza—Kleinsupergravily 71

Krn~~~B= ~[AV~B] (7.2.9)

Cm’ = Vrn4~’. (7.2.10)

Onemay verify that the generatorsKmAB8m and CmAÔrn do indeedcloseon SO(1,8).

To see that S0(1,8) doesindeedgeneratethe spectrum,considerfor simplicity the scalarLaplacian

4o4)(,,)= A(n)t/~’(n). (7.2.11)

Let ussuppressall SO(8) indices anddenoteby f the 7m2modeof (7.2.8),andrecall from section4.3

that it satisfies

VmVnf —m2gm,,f. (7.2.12)

Definethe scalarfield Vi(n) by

Vi(n) = VmfVm41(,,) + c(,,)f41(,,), (7.2.13)

whereC(n) is a constant.Substitutinginto (7.2.11)anddemandingthat ~ be an eigenfunctionimplies

C(,,) _3m2±V9m2+A(fl). (7.2.14)

Choosingthe + sign implies that Vi(~)= 41(n+1) with eigenvalue

A(,,+l) = A(,,) + m2+ 2m \/9m2+A(,,). (7.2.15)

Solving this recursionrelationwith A~= 0 yields, in agreementwith (7.2.5),

A(n) = n(n + 6)m2. (7.2.16)

Similarly, choosingthe — sign in (7.2.14)implies Vi(~)= 4(~-1)with eigenvalueA(,,l). Having determinedthe massivespin 2 spectrum,the remainingspectrummaybe calculatedeither by repeatingthe aboveprocessfor the otheroperatorsor elseby usingthe N = 8 supersymmetry.

We shall return to the round S7 in chapter12.

8. The squashedseven-sphere

8.1. The meaning of squashing

In section 7.1 we discussedthe round seven-sphere,which can be most convenientlydescribedinterms of its embeddingin 8-dimensionalEuclidean space R8. The round S7 can be said to beisometrically embeddedin R8.The Einstein metric on the round sphere,andits SO(8)isometrygroup,can thenbe understoodby meansof this embedding.However,the SO(8)invariant metric is only oneout of an infinite classof metricswhich onecould put on a spacewith the topology of the seven-sphere.

72 Mi Duffel a!., Ka!uza—K!ein supergravily

One could, for example, deform the surface yAyA = constant in some continuous, but otherwisearbitrarymanner,andobtaina new metric by restricting the metric dyA dyA to this new surface.Anysuchdeformationwould still bevery specialin that the resultingspaceis still isometricallyembeddablein R8. A general deformationof the metric on the round 57 will yield a spacewhich cannot beisometricallyembeddedin R8 i.e., its metriccannotbeobtainedby restrictingdyA dyA to somesurfacein R8. In general,any deformationfrom the round 57 would not be expectedto solve the Einsteinequations;onecan easilyshow for examplethat of the seven-spheremetricsobtainableby embeddingsurfacesin R8, only the round spherehas an Einstein metric. Remarkably,theredoes in fact exist asecondEinsteinmetric on 57 [211],for which the metric canbe obtainedby an isometricembeddingnotin R8, but in HP2, the quaternionicprojectiveplane[22,212]. We, therefore,beginby discussingHP2,andderiving its standardFubini—StudyEinstein metric.

The quaternionicprojectiveplane can be defined by introducing 3 quaternioniccoordinatesQ~.(a = 1, 2, 3) on the flat three-dimensionalquaternionicspaceO-ii~. Since eachquaternioniccoordinatecorrespondsto 4 real coordinates,i-is is isomorphicto flat R12 I-il P2 is thendefinedby identifying pointsin i-is underthe equivalencerelation

(Qi, 02, Q3) = (Q1/.L, Q2~.t,Q~u), (8.1.1)

where~ is any non-zeroquaternion.The two quaternioniccoordinatesq1 and q2, definedby

q1=Q1Q3~, q2=Q2Q31 (8.1.2)

coverall pointsin HP2 for which 03 � 0. We canobtainthe standardmetric on HP2 by first definiug the(round!)eleven-spherein H3 by

QaQa= 1, (8.1.3)

andconsideringthe metric inducedon it from the flat metric

ds2=dQ,. dO,,, (8.1.4)

on H3. The constraint(8.1.3)andmetric (8.1.4)arepreservedunderthe action of the mapping~ givenby

Qa~Qaa, (8.1.5)

wherea is a unit quaternion;i.e., a is an elementof SU(2).The orbitsof ~ arethereforethree-spheres,and the spaceof orbits is the quaternionicprojectiveplane,HP2. Put anotherway, S11 is an SU(2)bundleover lIP2. Writing the S11 metric in termsof q

1 andq2, we obtain

ds2= 4~dq

1 Q303+ dQ3Q3~~2+ (1+ 4kqkY1 d4~dq

1 — (1 + 4kqk)241dq

1 d41q~, (8.1.6)

andso projectingorthogonally to the orbits of ~ yields the metric

ds2= (1 + q~q~)dci, dq

1 —(1 + q~q~)qj dq, d~1q1 (8.1.7)

Mi Duffelat, Kaluza—Kleinsupergravily 73

on HP2. This is the standardFubini—Study metric; it is real, and sincethere are two quaternioniccoordinatesthe spacehas8 real dimensions.

It is convenientto introducea real parametrizationof qi andq2, which wedo by defining

q1=Utanxcosii~, q2=Vtanxsin~,u, (8.1.8)

whereUU = VV= 1; i.e., U, V E SU(2)and0 � x ~ ir/2, 0 � ~t � ir. ParametrizingU by Eulerangles04’, andVi (0�O�ii, 0�4’s2ir, 0�Vi�4ir)

U = ek~

2eio/2et<~~~2, (8.1.9)

andsimilarly V in termsof 0’, 4” andVi’~onecan easilyshowthat

2U1dUkr1+jo2+k~3, 2V~dV—i~1+j22+k1~3, (8.1.10)

wherethe 3 real one-formso~are given by

o-i=cosVid0+sinVisined4’, o-2=—sinVid0+cosVisin0d4’, u3”dVi+cos0d4’,(8.1.11)

with similarexpressionsfor £, in termsof 0’, 4” andVi’. Theseone-formsare left-invariant, i.e. invariantunderU -+ aU or V -* bV wherea, b E SU(2), andsatisfythe SU(2)algebra

dcr~—~e1JKo).A17K, d.~I=—~EI/K.~JA £~,,. (8.1.12)

With thesedefinitions, themetric (7.1.7) on HP2 becomes[22]

ds2= d,y2+ ~sin2xEd~a~+ ~sin2~ + ~cos2X(~i+ cos,,cu~)2], (8.1.13)

wherewe havedefined

vjuj+.~j, wjo,—.~. (8.1.14)

The squashedseven-spheremay be describedas the distance-spherein HP2, which is definedasfollows. First onepicks apoint p in HP2, whichmaybe chosenanywheresinceHP2 is homogeneousandthusall pointsare equivalent(it is in fact thecosetspaceSp(3)/Sp(2)x Sp(l)). The distancesphereis theclosedsurfacesurroundingp which is tracedout by the set of all geodesicsof some given length remanatingfrom p. It is clear that at least for sufficiently small r this surfacemustbe topologically S7,sincein the neighbourhoodof p HP2 is homeomorphicto R~. Themetric of the squashedseven-sphereis the one which this surfaceinherits from HP2. In the metric (8.1.13) the most convenientpoint tochoosefor p is x = 0, which correspondsto q

1 = q2 = 0. Onecan show that geodesicsfrom x = 0 haveaffine parameterx and correspondto curvesfor which all coordinatesexceptx remainconstant,andthusthe squashedseven-spheremetric is obtainedby settingx = constantift (8.1.13).After extractinganoverall constantconformalfactor~sin

2x, anddefining the constant“squashingparameter”A = cosxwe thereforeobtainthe metric [22]

74 Mi Duffelat, Ka!uza—Kleinsupergravigy

ds2= dj~2+~sin2~ + ~A2(v~+ cos,uu~)2. (8.1.15)

We now introduce a notation which will prove very useful for subsequentcalculations.Indicesa, b,..., running from Oto6will be split as a=(0,i,I), where i= 1,2,3 and 1=4,5,6=1,2,3. Aconvenientorthonormalbasisfor (8.1.15) is

e°=d~, e1=~sin~w~,e’=~A(i.’1+cos~~w1). (8.1.16)

The connectionone-formWab, definedby dea= Wb A e”, W(ab)= 0, is thengiven by

= —cot,ue’ +~Ae1, w

01~Ae1,.. wq =[cot~e”+(A/2— 1/A)e~]EIJk,

= _(2A)1EjJke~, w~= —~Aô11e°— ~AE,Jke”. (8.1.17)

Thecurvaturetwo-form0ab = dWab+ (.Vac A Wcb~ ~Rabcae~ A e”) is given by

@01 = (1—~A2)e°A e~+ ~(1— A2)e~Jke’A e~, @01 = ~A2e°A et—~(1— A2)E~ke~A

0,, = (1 —~A2)e1n e +~(1—A2)e’ A e~, @fj= (4A2y1ef A e’+~(1— A2)[BlJke° A e” + e’ A e’l,

= ~A2e’A e3—~(1— A2)e’ A e’— ~(1— A2)eIJke°A ek + ~(1— A2)8ije” A e~. (8.1.18)

Fromthisit follows that the Ricci tensoris diagonal,andgiven by Rat, = diag (a,a, a, a, /3, /3, /3), where

a3—3A2/2, $~rA2+(2A2)’. (8.1.19)

The Einstein condition R,.,, ~ gab, i.e. a = /3, is thussatisfiedby two values of A2 A2 = 1, which is theroundsphere,andA2 = ~,which is the non-standardEinstein metric on the squashed57~

Theisometrygroupof the squashedspheremay bedeterminedby returningto themetric (8.1.13)onHP2. We beginby writing the quaternionsq

1 and q2 as a column vector

w=~ ~i,\q21

andso (8.1.7)becomes

ds2= (1 + wtw)l dwtdw —(1 + wtw)2wtdwdwtw. (8.1.20)

The distancesphereA = cosx = constantcorrespondsto the condition

w~w= (1— A2)/A2. (8.1.21)

The group of transformationswhich preserves(8.1.20) and (8.1.21) thereforeconsistsof left multi-

plicationsby quaternionicunitary 2 X 2 matricesA (i.e. A~A= 1);w-+Aw, (8.1.22)

MJ. Duffel a!., Ka!uza—K!einsupergravily 75

andalso of right multiplicationsby unit quaternionsa(ãa = 1);

(qi) (~1a) (8.1.23)

q2 q2a

Thus the isometry group of the squashedseven-spherefor any value of A2 other than A2= 1 is

Sp(2)x Sp(l), whereSp(n) meansthe group of quaternionicunitary n X n matrices. Sp(2) is locallyisomorphicto SO(5), andSp(l) SU(2). (WhenA2 = 1 the distancespherein (8.1.13) degeneratesto apoint, but the rescaledmetric (8.1.15)is non-degenerateandhasa bigger symmetry,namelySO(8)).

The squashedsphereis a homogeneousspace,on which Sp(2)x Sp(l) actstransitively; i.e. any pointmaybereachedfrom any otherpoint by actingwith Sp(2)x Sp(l). In fact Sp(2)by itself actstransitively,as may be seenby consideringits action (8.1.22) on a typical point, such as w = (~),c2 = (1— A2)/A2.Writing the Sp(2)matrix A as

A=(° ~), (8.1.24)

wherea, /3, y and 6 arequaternions,the unitary conditionA~A= 1 implies

äa+~y”1, /3/3+851, ã/3+~ô0. (8.1.25)

Thus the point (~)may be reachedfrom (~)by choosinga = q1/c, y = q2/c. The conditions (8.1.25)

imply /3/3 = (1 + 1q112/Pq

2l2Y1, 3 = —(q

2q111q212)13, so the Sp(2) matrix A which takes (~)into (~)is

undeterminedup totheSp(i) (~SU(2))subgroupwhichactson /3 andpreserves/3/3 = (1+ q~2/Iq2~

2)~.Thisshowsthat pointson the squashedS7 areparameterizedby the cosetSp(2)/Sp(l).

For many purposesit is more convenientto describethe squashedsphereas the cosetspaceG/HwhereG is the isometrygroupSp(2)X Sp(l) and H is the isotropygroup, i.e. the subgroupof G whichleavesa point fixed. Taking w = (~)again as a representativepoint, the condition for it to be left

invariant by (8.1.22)and (8.1.23)is

(a :)(~)=(~)~ (8.1.26)

whereagaina, /3, y and S arequaternionssatisfying(8.1.25) and a is a unit quaternion,ãa = 1. Thus/3 = y = 0, a = a and86 1. If wedefinethe SP(l)A X SP(l)B subgroupof Sp(2)by matricesof the form

A = (~ ~), (8.1.27)

whereãa = 66 = 1, a E SP(l)A, 8 E SP(l)B,andlabel theSp(l) generatedby a asSp(l)c,thenthisshowsthat

H = Sp(l)A+c X SP(i)B, (8.1.28)

whereSp(l)A+cdenotesthe diagonalSp(i) subgroupof SP(l)A andSp(1)~.

76 Mi Duff eta!., Kaluza—K!ein supergravity

There is a different, but of courseentirely equivalent,way to describethe squashedseven-sphere,basedupon the fact that 57 is an S

3 bundleover 54~In this descriptionthe squashingcorrespondstochangingthe “size” of the S3 fibers relativeto the S4 basespace.We can rederivethe squashedspheremetric by constructingthe standardmetricon thebundlespaceof the k = 1 Yang—Mills instantonon 54~We beginwith the standardEinsteinmetric on 54, which may bewritten as

ds2= d,u2 + ~ ~ ,~2 (8.1.29)

where0 � � ir, and.~, (i = 1, 2, 3) area set of left invariant one-formssatisfyingthe SU(2)algebraof(8.1.12). A suitablegaugepotential for the k = 1 self-dual SU(2)instantonis

A’ = cos2~St, (8.1.30)

from which onecan constructthe following metric on the SU(2)bundleover 54

dS2= d~2+ ~sin2p .~ + A2(o~— A’)2, (8.1.31)

whereA is a constantandtheó~areanotherset of left invariant one-formssatisfyingthe SU(2)algebra(8.1.12). This is really just an “inverse Kaluza—Klein” construction, in which we re-interpret afour-dimensionalmetric plus SU(2)gaugefield as a seven-dimensionalmetric. If A’ had beenzero in(8.1.31)the metric would just describe54 x 53, with A scalingthe size of the 53, andit is becauseA’ infact correspondsto the topologically non-trivial instantonthat the 54 x 53 is “twisted” to give 57~Themetric (8.1.31) looksvery similar to the squashedspheremetric of (8.1.15), and in fact one can easilyestablishby setting

= u,, i~1+j.~2+k.~3V(iwi+jw2+koi3)V

1, (8.1.32)

whereo~,w, and V aredefinedin (8.1.10)and (8.1.14),that ö~and£ indeedsatisfythe SU(2)algebra(8.1.12)andthat the two metricsareequivalent.

8.2. N = 1 supersymmetryand G2 holonomy

We now focus our attention on the non-standardEinstein metric on the squashedseven-sphere,whichcorrespondsto A

2 = ~in (8.1.15)andinvestigatetheexistenceof Killing spinors.From(8.1.19),weseethat it satisfiesthe Einstein equation

Rab~gab. (8.2.1)

When we eventuallyusethis solution in the spontaneouscompactificationof d = ii supergravity,wewill rescale(8.1.15)by the constantconformal factor9(20m2)’ in order that the rescaledmetric willsatisfyR,.,,= 6m2g,.,,.For thepresent,however,it is moreconvenientto work with (8.1.15)directly, andsoin the Killing spinorequation(4.2.5)we must set

rn2=~a. (8.2.2)

Mi Duffelat, Ka!uza—Kleinsupergraviy 77

The Weyl tensor,which appearsin the integrability condition (4.2.7), is therefore related to theRiemanntensorby

C,.bCd = Rabcd — ~j(g,.,,g~— g,.,j,g~). (8.2.3)

Thus,from (8.1.18)with A2= ~,and(4.2.7), the linear combinationsof T,.b which mustpreservea Killingspinorare

C’01 = ~[F01+ ~SUkFfl~] , = ~[FIJ+ I’~j] , C~= ~[—I’~~— ~F11+ ~SlJFk,~— ~ijkF0d

(8.2.4)Co~= —~[F01+ ~,kFj~], C11 = ~[2Ft;+ F1, + s,~kFok],

where Cm,. Cm,.” Fab. The 21 C,.,, in (8.2.4) are not all independent;one can show that thereare 7

redundantequations,since

C,t= 0; C01= e,JkC,~ C~= C~,+ EJkCok. (8.2.5)

The holonomygroupc~Cfor the connection(4.2.5)is thereforegeneratedby the 14 linearly independentcombinationsof Spin(7) generators,which may convenientlybe chosento be C01, C~and C,1 (withC,1= 0).

The fact that ~Cis 14-dimensionalimmediatelysuggeststhat it is theexceptionalgroupG2, whichhasrank 2. A discussionof the classificationof Lie groupsmaybe found in manytextbooks,for example[213].Thealgebraof a Lie groupis completelycharacterizedby giving the simple roots (the numberofsimplerootsis equalto the rank of the group),andit is a straightforwardexerciseto show,by taking thetwo simple roots e,,,, and e,,,2 of G2 to beproportionalto

C1~+ C21+ i(C11 — C2~), (8.2.6)

and

C~j—iC15, (8.2.7)

respectively,that the C,.,., do indeedgenerateG2.The numberof solutionsto the integrability condition for Killing spinors,C,.,,q= 0, is thusequalto

the numberof spinorsin 7 dimensionswhich areleft invariant by the action of G2, andas discussedinsection4.2, the answeris 1. It will prove useful to havean explicit solutionfor this spinor,whichwe doby first decomposingthe d = 7 Dirac matricesF,. as

Fo=yo®1, Fj=yj®1, F

1~iy5®r,. (8.2.8)

HereYo andv~arethe Diracmatricesfor 4 Euclideandimensions,andmaybe representedin termsof

the Pauli matricesr,. as

= (° ~), ~, = (° ~), (8.2.9)

78 Mi Duffel at, Kaluza—Klein supergravily

and y~= ~YoY1Y2Y3. In termsof this decomposition,the condition ~ = 0, with Cab given by (8.2.4),reducesto

y5®1~=n, yoy,.®1~i1®r,,~. (8.2.10)

We nowadopttheconventionthat if A is a4X 4 matrix andB is a 2x 2 matrix, thenthe componentsofthe 8 x 8 matrix A® B areobtainedby replacingeachelementA,.

1 of A by the 2 X 2 matrix A,1B.Thuswith the standardconventionsfor the Pauli matrices,(8.2.10)implies that ~ is given by

(8.2.11)

wheref is an arbitrary function.Havingsolved the integrabilitycondition (4.2.7),onenowhasto return to the original Killing spinor

equation(4.2.5). Substituting the explicit expressions(8.1.17) for the connection one-forminto theKilling spinorequation,onefinds that (8.2.11)satisfies

D,.fl = 3(4V5)~Fa~i, (8.2.12)

provided f = constant.Thus from (8.2.2) we see that ~ satisfies the Killing spinor equationDa7l =

~ where rn is the positiveroot of (8.2.2). Normalizing ij so that ~ = 1, we obtain

—ï(8.2.13)

wherethe factor of i ensuresthat it is Majorana.Substituting(8.2.11)into Dali = _3(4V5YhTafl, onefinds thatthereis no solutionfor f, andhenceno spinorsatisfyingD,.li = —~mF,.ij,wherem = 3(4V5)1.

This is in agreementwith the skew-whiffing theoremof section4.2,which showedthat only the roundseven-sphereadmitssolutionsof both D,.li+ = ~mF,.li+andDali- = —~mF,.li simultaneously.

Thesolution (8.2.13)was obtainedas a purely local solution, andso in order to be certainthat it isvalid globally one hasto check that it can be extendedwithout singularitiesover the entire seven-dimensionalmanifold. Oneway in which this extensionmight fail is if themanifold did not admit a spinstructure,in which caseno spinor fields could be globally defined.The existenceof aspin structureis a

Mi Duff eta!., Ka!uza—K!einsupergravily 79

topological property of a manifold M, characterizedby the vanishingof the secondSteifel—Whitneyclass [190] w2(M). For asphereof any dimension(d � 2), w2(S”) = 0, andso this certainlyprovidesnoobstruction in the caseof the squashedseven-sphere.In fact all the Stiefel—Whitneyclassesof S

7vanish,which is equivalentto the statementthat S7 is parallelizable. (Seechapter10.) This meansthattherecan neverby any globalobstructions.

In orderto establishconventions,it is convenientat this stageto rescalethe squashedspheremetricso that whenA2 = ~it satisfiesthe standardlynormalizedEinstein equationRa~= 6m2g,.bwhich we areusing in the discussionof spontaneouscompactification.We do this by first defining a new siebenbein

= cea, (8.2.14)

whereea is given by (8.1.16)and

= (2V7mA)’(l + 81,2 — 2A4)1’2, (8.2.15)

which impliesthat the new metric d~2= ~a~a hasRicci scalarR = 42m2 for all valuesof A. Thus,whenA2 = ~it satisfiesthe requiredEinstein equation.With respectto this rescaledsiebenbeinthe spinor ~given by (8.2.13) satisfiesthe Killing spinorequation(4.2.5) andhencegives rise to a four-dimensionalvacuumwith N = 1 supersymmetry.Reversingthe orientationof the sphere,for example by sendingea ...~ ë” implies that now li satisfiesnot (4.2.5)but (4.2.8),andthereforethefour-dimensionalvacuumhasN = 0 supersymmetry.Of course,one can alwayschangeconventionsin the original supergravitytheory or the Freund—Rubinansatzin suchaway that the rolesof the two orientationsof the squashedsphereareinterchanged(for exampleby replacingF,,,,,,,,,.= 3rne,,,,,,,~by F,.,,,~,,,= —3ms,,,,,,,~),but sincetherearetwo physicallydistinct four-dimensionalvacuait is convenientto establisha set of conventionswithin which onecan unambiguouslyspecifywhich vacuumis which. Accordingly, we will alwaystakethe Freund—Rubinansatzto be (3.1.7), and designatethe orientationof the spherewhich gives theN = 1 vacuumas the left-squashedsphere,and the orientationgiving the N = 0 vacuumas the right-squashed sphere.

Becauseof triality, therearethreeinequivalentS0(5)x SU(2) subgroupsof SO(8).The N = 1 solutionpicks out [50(5) x SU(2)],, underwhich 8,, goes into (5, 1) + (1, 3) andboth 8, and8~go into (4,2), whileright squashingpicks out [50(5)x SU(2)], underwhich 8, goesinto (5, 1) + (1, 3) andboth 8,, and8,. gointo (4,2). (A third subgroup[SO(5)x SU(2)],. may also be obtained by squashingS7 but in aninhomogeneousmannerwhich doesnot yield an Einsteinmetric).

8.3. Spontaneoussymmetrybreakinginterpretation:the “space invaders” scenario

We describe in this section how squashing the seven-spherecorrespondsin four-dimensionallanguageto a conventional Brout—Englert—Higgs—Kibble mechanismwhereby scalar fields developnon-zero VEVs. Let us first consider the breaking N = 8/SO(8)-~ N = 1/[SO(5) x SU(2)]~.First weobserve that when S0(8) breaks to [S0(5)x SU(2)],,, the masslessN = 8 multiplet of section 7.1decomposesas follows: 1-4(1,1); 8, -+ (4,2); 28~(10, 1) + (5,3)+ (1, 3); 56,—* (16, 2)+ (4,4)+ (4, 2);35,.—~(10,3)+ (5, 1); 35,,—* (14, 1) + (5,3)+ (1,5) + (1, 1). The statesby themselvescan neverform N = 1supermultiplets.In particularwe notethat all the eight gravitinosmust acquirea mass.It follows thatthe singlemasslessgravitino of the N = 1 phasemustcomefrom the massivesectorof the N = 8 phase.

To proceed,we recall from table 5 that masslessgravitinosin d = 4 arein one-to-onecorrespondence

80 Mi Duffel al., Kaluza—Kleinsupergravity

with —7m/2 modesof the Dirac operator4~= FmDm. In section8.4, therefore,we shall calculatetheDiracoperatorin the squashedS7 metric with arbitrary A andlook for the eigenmodeVi which, in thecaseA2 = 1/5, is thesolution~ of theKilling spinorequationDrnli = 0. Theresult is that Vi = ~ for all A2andthat the correspondingeigenvalueis

—~x7U2m(1+ 2A2)(1 + ~1,2 — 2A4)~2.

Thus the eigenvalue—7m12mode of ~ on the A2 = 1/5 spherehascomefrom a —9m/2 mode on theA2 = 1 sphere.Now from section 7.2 the Dirac eigenvalueson the round S7 are +(n + 7/2)m and—(n + 7/2)m, n � 0, the modesbeing in the (n, 0, 1, 0) and (n, 0, 0, 1) representationsrespectively.Thesinglet masslessgravitinoon the A2 = 1/5 squashedS7 has,therefore,comefrom the next-to-lowestlevelof Dirac eigenmodeson the round 57, which is a 56,,. It is, in fact, the singlet in the decomposition56,, -~ ((10, 1) + (10, 3) + (5, 3) + (1, 1) of 50(8)breakingdown to the [50(5)x SU(2)],, isometrygroupofthe left-squashedsphere.We refer to this mechanism,wherebystateswhich areapparentlyverymassiveM2 —~g2/G in the N = 8 phasezoom down from the Planckian sky to becomemasslessin the N = 1phaseas the “spaceinvadersscenario”.(Seetable 10.)

The Higgs andsuper-Higgseffect works as follows. Consider

gmn(x, y) = km~(y)+ hmn(X, y), (8.3.1)

Table 10The spaceinvadersscenario

Left squashed~7 Round S7 Right squashedS7

.42=1/5 .42=1 .42=1/5N=1 N=8 N=0lSO(5)x SU(2)],, SO(8) lSO(5)x SU(2)],

Gravitinomass (10,3) (4,4)

(5,3) //

\ \, /1

1~~ (16,2)

M—M~ ____________

(10,1) ______________ ~ 56,, ‘~ . (1,3)I,,

(4,2)

(4 2) _____________________/ ,‘ (5, 1)//I __~_~‘

M=0 (1,1).................f “ ___________

spaceinvader

Mi Duffel a!., Ka!uza—Kleinsupergravily 81

where~,,.(y) is the metric on the round57~Fourierexpandinghmn(X, y) in harmonicson the roundS7

hmn(X,y) = ~ Sk(X)Ymkn(y) (8.3.2)k

oneobtainsthe 35,.scalarsof the N = 8 masslesssectorfrom the lowestmodetogetherwith aninfinitetower of massivescalarsSk(x) where k denotesthose representationsof SO(8) appearingin the 0~sectorof table8. In the symmetricphase

(gmn(x, y)) = ~‘m,.(y) -~ all (Sk(x)) = 0, (8.3.3)

but deformationsof S7 away from its maximally symmetricgeometrycorrespondto non-zerovacuumexpectationvaluesfor someof thesescalars(see,however,section12.4)

(gmn(x,y)) � krnn(y) —~ some(Sk(x)) � 0. (8.3.4)

In order to determinewhich areresponsiblefor the breakingof SO(8)down to S0(5)x SU(2),wefirstnotethat in the orthonormalbasisof (8.1.16)the metric deformationhmn whichtakesthe roundS7 intothe squashed57 is of the form

hab -~diag(0,0,0, 0, 1, 1, 1). (8.3.5)

A straightforwardcalculationin the S7 backgroundshowsthat this is a Killing tensor

V(lhmn) = 0, (8.3.6)

of the kind discussedin section4.3 for all valuesof the squashingparameter.On round spheresall such Killing tensorsare built from productsof Killing vectors [214]. As

discussedin [211this meansthat thereare 336 symmetricrank-two Killing tensorson the round 57,

transformingas a 1, a 35~anda300 underSO(8).The 35~areassociatedwith the masslessscalars(in then = 0 level of table9) while the 1 andthe300 describemassivescalars(in the n = 2 level of table9). Thesinglet Killing tensoris just a constantmultiple of themetric andhasconstanttrace;the traceof the 35~gives a 35~of eigenfunctionsof the scalarLaplacianon S7, while the 300 is trace-free.From (8.3.5)weseethat h,.,, hasaconstanttrace andso mustbe a linear combinationof just the 1 and the 300. Thesinglet correspondsto an overall scalingof the metric andso the squashing(8.1.16) correspondsto anon-zeroVEV for the 300. As a consistencycheckwe notethat underSO(8)—s’50(5)x SU(2)we have300-~ (1, 1) + (5, 3)+ (1,5)+ (10,3)+ (14, 1) + (35, 1) + (35, 3)+ (14,5) andwe find as expectedthe singlet(1, 1) togetherwith the Goldstonebosons(5, 3) whichgive amassto the (5, 3) spin 1 gaugebosonsin thedecomposition28—~(10, 1)+ (1, 3)+ (5,3).

Now we turn our attention to the breaking N = 8/SO(8) -4 N = 0/[S0(5) x SU(2)],. In this casethemassless supermultiplet decomposesas 1-4 (1, 1); 8, -+ (5, 1) + (1, 3); 28-4(10,1) + (5, 3) + (1, 3);56, —~ (10, 1) + (10,3)+ (5, 3)+ (1, 1); 35,.-÷(10, 3)+ (5, 1); 35,,-9 (10,3)+ (5,1). All supersymmetriesarebrokenbut not at a commonscale; the (5, 1) gravitinoswill havea different massfrom the (1,3). Seetable10.

The Higgs effectworks in the sameway as the N = 1 solutionwith the 1 and300 acquiringnon-zeroVEVs. The mass spectrumfor spins 2, 1 and 0’ (which haveonly a v subscriptor no subscript) is

82 Mi Duff etat, Ka!uza—Kleinsupergravity

insensitiveto the handednessof squashingand will be the samefor both the left- and right-squashedspheres,but the spectrumfor spins ~, ~ and 0 (which have s or c subscripts)will be different. Inparticular,therewill be a single masslesspseudo-scalaron the right-squashed57 sincethe relevantmassoperatorhasa single zeroeigenvaluemode.Seethe discussionin section8.4 below.This correspondstoa 0 “invasion” sinceit belongsto the massive35, (in the n = 2 level of table9) of theN = 8 phase.

The stability of the left-squashedvacuum is guaranteedby the unbrokenN = 1 supersymmetry.Curiouslyenough,as explainedin section11.1, theright-squashedN = 0 solution is alsostable.We nowexaminethe massspectrumin both cases.

8.4. The massspectrum

In contrastto the round 57, the massspectrafor the squashed57 solutionsareknownonly partially.In this sectionwe list what is known to dateof the relevantmassoperators.Of the operatorsappearingin table 5 completeresultshavebeenobtainedonly for LID, ~~1/2 and LI~while partial resultshavebeenderivedfor LIL~

We start the discussionby considering the scalar Laplacian LI0. The eigenfunctionsoccur inrepresentationsat S0(5)x SU(2)which correspondto the decompositions

En/2]

(n, 0,0,0)-9 ~ (n — 2r, r; n — 2r), (8.4.1)

where(n, 0,0,0)is the Dynkin label for the SO(8)representationsand(p, q; t) denotesarepresentationof SO(5)x SU(2) with S0(5) Dynkin label (p, q) and SU(2) Dynkin label (t). [n/2] meansthe integerpart of n/2. The eigenvaluesof LI0 are derivedin [651for arbitrary squashingparameterA. For A

2 =

andwith the metric normalizedsuch that Rmn= 6m2gmntheresultcan bewritten as

LI~= ~m2C0, (8.4.2)

whereCG is the second-orderCasimiroperatorfor theisometrygroupG = 50(5)X SU(2). It is given by

C0= Cso(5)+ 3Csu(2), (8.4.3)

wherethe Casimir operatorfor SO(5) and SU(2) havebeen normalizedsuch that Cso(5)= 6 for theadjoint representationand Csu(2)= 2 for its adjoint.

Similarly for the Dirac operator.01/2 we needthe decompositions

[n/21

(n,0,0,1)—*~(n+1—2r,r;n+1—2r)+ ~ (n+1—2r,r;n—1—2r)

[n/2J [n/21

+ ~(n— 1—2r,r+1;n+1—2r)+~(n—1—2r,r;n—1—2r), (8.4.4)

and

Mi Duffel at, Ka!uza—K!ein supergravity 83

[n/2]+1 [(n—1)/21

(n,0,1,0)~~ (n—2r,r+1;n—2r)+ ~ (n—2r,r;n—2r)r=0 r=O

[n/21 ln/21—1

+ ~ (n—2r,r;n+2—2r)+ ~ (n—2r,r;n—2—2r). (8.4.5)r=O r0

From[65], wefind that eigenvaluescan bewritten as

‘P1/2 = —~m±~m\/5[C0 + 81]1/2 (8.4.6)

or

= ~m±~mV5[Co + 49]1/2 (8.4.7)

Theseformulaeareto be usedas follows: (8.4.6) gives the eigenvaluescorrespondingto the first setofSO(5)x SU(2) representationsin (8.4.4)and(8.4.5),while (8.4.7)gives the eigenvaluesfor the remainingthreesetsof representationsin (8.4.4)and(8.4.5). Furthermorethe plus signin (8.4.6)and(8.4.7)shouldbe chosenfor the representationsof (8.4.5),and the minussign for (8.4.4).

For vectors the relevant SO(5)x SU(2) representationsare obtained by decompositionof the(n, 1, 0, 0) representationsof SO(8). For the detailswe refer the readerto [92], from which one findsthat the eigenvaluesaregiven by (seesection4.3)

LI1 = ~m2[Co+~±(1/V5)(Co+~)h/2]. (8.4.8)

Similarly, the eigenfunctionsof the Lichnerowicz operator4L occursin the SO(S)x SU(2) represen-tations obtainedby decomposingthe (n, 2, 0, 0) of SO(8). The eigenvalueswere derived using thetechniquedescribedin section4.3. The resultsare [198]

LIL= ~m2[CG+~], (8.4.9)

or

LIL = fm2[C0 + ~±(2/\/5)(C0+ .t)1/2] (8.4.10)

The choiceof (8.4.9)or (8.4.10), andof the±sign in (8.4.10)dependsupon the particular50(5)x SU(2)representation,and is discussedin section4.3.

We now turn our attentionto the masslesssectorsof the two squashedseven-spherevacua.First werecall what has already been established.The N = 1 solution has as its unbroken gauge group[SO(S)x SU(2)],, under which the 8,, of S0(8)goes into (5, 1) + (1,3). Here, there is a massless2—supermultipletin the (1, 1) representation,and massless1 — ~supermultipletsin the (10, 1) + (1,3). TheN = 0 solutionhas[SO(S)x SU(2)]. underwhich the 8, of SO(8)goesinto (5, 1) + (1, 3). Here,thereis amasslessspin 2 in the (1, 1) representation,andmasslessspin 1 in the (10, 1) + (1, 3). This N = 0 solutionalsoadmitsa single masslesspseudoscalar[66].

Interestinglyenough,the appearanceof this masslesspseudoscalaris aspecialcaseof a moregeneralphenomenonoccurring with all thosenon-supersymmetricvacuaobtainedby skew-shiffingthe super-

84 Mi Duffel at, Kaluza—K!ein supergravily

symmetricones.If the supersymmetricvacuum admits N Killing spinors ‘q’, then it follows by theconstruction[22]

~ = ~‘Fmnpfl’ (8.4.11)

that there are ~N(N + 1) zero modes of the 0 mass matrix of table 5 in the non-supersymmetricvacuum.

Returningnow to the squashedS7, it remainsto be seenwhetherthereare any furthermasslessspin ~or spin 0 states.Regardingspin ~,direct inspectionof the .01/2 spectrum(8.4.6)and(8.4.7) showsthat nofurther masslessmodescan arise in the ~1) or ~4) towers. The problemof analyzingthe towers ~(2)

and~3), i.e. the spectrumof 03/2, can be reducedto a bosonproblemby writing an arbitrary mode Vi

m

as [58,25]

Vim = Amli + BmJ”lJ, (8.4.12)

where li is the Killing spinor. Substituting this into the ~(1), (2) mass operator, and demandingmasslessness,enablesone to deducethat Am = 0, Bm

m = 0, BFmn] = 0, VmBmn = 0, and that ALBmn=

4m2Bm,. in the N = 1 casewhile LILBmn = 7m2Bmn in the N = 0 case.From(8.4.9) and (8.4.10),we seethat either of theseeigenvaluescan occur.

In the0~sector, the absenceof ziL = 4m2 modesimmediately showsthat therecan be no masslessstatesin the 0~~2)tower. We can solve the ~ and0~°~towers completely, using (8.4.1) and (8.4.2).From table 5, O”~statescan neverbe massless,while 0~1)will be if LI

0 = 16m2 or 40 = 40m2. From

(8.4.2) thereareno 16m2 modes,but therearea (35,3) and (30, 1) of 40m2 modes.These135 masslessfields occur for both orientationsof the sphere. In the N = 1 casethey are membersof massiveWess—Zuminomultiplets; this samephenomenonwas found for the round 57 vacuum,wherethereare294 masslessscalarsin a massiveN = 8 supermultiplet[194].Indeed,the (35, 3) is a subsetof the 294,while the (30, 1) correspondsto a 0~“spaceinvasion”from level 4 [25].It is interestingto notealsothatwhereas112 0~statessaturatethe Breitenlohner—Freedmanbound M2� —m2 on the round 57 (seetable9), all 0~stateson the squashed57 exceedthe bound.The lowestmodeof 4L is the Killing tensorsquashingmodeof (8.3.5),whichbarely scrapesby with M2 = —~m2[57,25].

Finally, we note that the ratio of the 50(5) and SU(2) coupling constantshavebeencalculatedin[215], using[157].This resultdiffers from that obtainedfrom a conventionalYang—Mills—Higgs breakingof S0(8) to SO(5)x SU(2), owing to the non-minimal coupling of the scalarsto the vectorsin theYang—Mills kineticenergyterm.This is alwaysa featureof Kaluza—Klein theories.

9. Solutionswith otherM7 topologies

9.1. Ricci flat M7: T7 and K3 x T3

A specialcaseof the Freund—Rubinansatz(3.1.7)ariseswhen ~‘,,,,,,,.,.,. vanishes,althoughin this casethereis no preferentialcompactificationto d = 4. Indeedd = 11 Minkowski spaceis a perfectly validsolution. Solutionswith a compactM

7 are possible,of course,provided M7 is Ricci flat which in turnimplies a Minkowski spacetimein d = 4 rather than AdS. For M7 the only known solutionsare the

Mi Duffel at, Ka!uza—K!einsupergravily 85

seven-torusT7 andK3 x T3 whereK3 is Kummer’squarticsurfaceinCPl.* A reviewof the K3 literaturemaybe foundin [216].

When .kmn = 0 masslessfermionscorrespondto zeromodesof .01/2 and.03/2 andmasslessbosonstozeromodesof Ao, 4~,42, 4L and 0. This means,in particularthat the Betti numbersb~(p = 0, 1, 2, 3)now play an importantrole. Moreover,zero modesof 4L are in one-to-onecorrespondencewith thenumber ofparametersappearingin the M

7 metric.This is because4L describesthe first variationof the

Einsteintensorandso its zeromodespreserveRmn = 0.

First we considerT7 which, with its flat metric, has trivial holonomygroup ~C= 1 and, hence,incommonwith the round S7 yields N = 8. The masslessand massivestatesmust then fall into N = 8supermultipletsof Poincarésymmetry.In this casethe multipletshorteningof the massivestatesis dueto centralcharges[217].As an exercise,let us derive the masslesssector displayedin table 11. Thesingle ~ comesfrom thesingle zeromodeof LID. The7B,~from the 7 Killing vectorsof [U(1)]7andthe28 scalarsS from the 7 x 8/2 parametersof T7. The 8 Vi~comefrom the 8 Killing spinorson T7. Ofthe 56 x~48 correspondto transversevector-spinorzero modes of 03/2 and 8 to 01,2. The anti-symmetric tensorcount of 1, 7, 21 and 35 is simply given by the Betti numbersof T7. In order toobtain the standardcount of (1, 8, 28, 56, 35, 35) for particlesof spins(2, ~, 1, ~,0~,0) which oneexpectsin a masslessN = 8 multiplet, onemustperformduality transformationsto convertthe 21 axialvectorA,,,, to vectorsB,,,, and the 7 A,,,. to scalarsS [37].(A masslesssecondrank antisymmetrictensorgaugefield has1 degreeof freedom).Since the 28 S correspondto parametersof the M

7 metric, theycorrespondto Goldstone bosons and are describedby the coset GL(7, R)/S0(7). After dualitytransformations,this may be enlargedto the 35-dimensionalcoset SL(8,R)/SO(8) and then, byincluding the 35 A, to the 70 dimensionalE7/SU(8).The inclusion of the massivestatesrendersthescalarsonly pseudo-Goldstonebosonsas discussedin section 1.4. The massivespectrumis simplyobtainedfrom table5 by setting m = 0, andretaining all the different towers. This wasnot strictly thecasefor the masslesssector since table 5 was derived under the assumptionof vanishing first Bettinumberwhich is always true when m � 0. (Bearing in mind that masslessand massiveA,,,,,.’s havedifferentdegreesof freedom.)

Now we considerK3 X T3 [23]. The Ricci flat metric on K3 is not known explicitly but thereis an

existenceproof. Moreover, it is known to have58 parameters,to havea self-dualRiemanntensor,andno continuoussymmetries.Topologically,K3 hasEulernumberx = 24,Hirzebruchsignaturer = 16 andBetti numbers:

b0=1, b1=0, b2=22, b3=0, b4=1. (9.1.1)

This informationwill be sufficientfor usto determinethebosonicmasslesssectorof table11. The singleg,,,. comes from the single zero mode of the scalar Laplacian, the threeB,,, from the threeKillingvectorson T

3 (K3 hasno Killing vectors),andthe 64 scalarsS from the zeromodesof 4k.: 58 from K3(the 58 parameters)and6 from T3 (the 3 x 4/2 parameters).We notethat these6 areKilling tensorsbutthat the 58 arenot, apartfrom the dilation mode.

As far as fermionsareconcerned,we first note that sinceK3 is half-flat the holonomy groupis SU(2)ratherthan the SU(2)X SU(2)of a genericfour manifold andhenceit admitstwo covariantly constantspinors(i.e. Killing spinors)which areleft or right handedaccordingas K3 is self-dual or anti-self dual[218].The four Vi~comefrom the four Killing spinorson K3 x T3 (2 on K3 x 2 on T3). To obtainthe 92

* See however,thesectionon recentdevelopments.

86 Mi Duffelat, Ka!uza—K!einsupergravisy

spin 1/2 fields x we note that thereare40 zero-modesof the Rarita—Schwingeroperatoron K3: 38 ofwhich areF-trace-freeand2 of whicharenot but arecovariantlyconstant,while on T3 thereare6 suchzero-modeswhich are covariantly constantbut not F-trace-free.We note that these6 are Killingvector-spinorsbut that the 38 arenot. With theseconditionsthe 92 modes,given by the 40 on K3 x 2Killing spinorson T3 plus 6 on T3 x 2 Killing spinors on K3, will be zero-modesof the spin ~ massmatrix.

The numbersof A,,,,,~,,A,,,,., A,,,, and A fields are given by the Betti numbersb0, b1, b2 and b3 of

K3 xT3. But for a productmanifold M = M’ xM”

bp”~bW,~_r, (9.1.2)

where b,,, b and b are the pth Betti numbersof M, M’ and M” respectively.This is the Kunnethformula [190].Hencefrom (9.1.1)andtheBetti numbersb~= (~)for T3, we obtainthe numbersgiven intable 11. A detaileddiscussionof bosonandfermionzero-modeson K3 (andtheir relationto axial andconformal anomalies)maybe found in [218,219].

Table 11Masslessmodeson T7 and K3 x 1’.

d = 11 d = 4 Spin T7 K3 x

gMN g,,. 2 1B,. 1 7 3

S 0 28 64*51 *1~ 3/2 8 4

x 1/2 56 92

AMNP A,.,,,. — 1A,,,, 0 7 3A,. 1 21 25A 0 35 67

To summarize,the spin content is given by 1 spin 2, 4 spin ~, 28 spin 1, 92 spin ~, 67 scalarsand67pseudoscalars.This correspondsto an N = 4 supergravitymultiplet (1,4, 6, 4, 1 + 1) coupledto 22 N = 4spin 1 matter multiplets (1, 4, 3+ 3). Note that the numberof masslessdegreesof freedom (per d = 4

spacetimepoint) of thisN = 4 theory obtainedfrom K3 x T3, namely192+ 192, exceedsthe 128+ 128ofthe N = 8 theory obtainedfrom i’~.Just aswith T7, one expectsthat the scalarsmaybe assignedto acosetspace.The massivespectrumcannotbe determinedby grouptheory sinceK3 hasno symmetries.All we can sayis that the massivestatesmust fall into N = 4 supermultiplets.

9.2. Other solutions

We now discussthe othersolutionslisted in table6, beginningwith the M(m, n) spaces.

(i) M(m, n) spacesTheseare the homogeneousspaceswith SU(3)x SU(2)x U(1) symmetry’groupfirst consideredby

Witten [26].As mentionedin chapter6, we will characterizethesespacesby the two parametersm andn [16,201] ratherthan by the threeparametersp, q and r in the notation ~ introducedin [26] and

Mi Duff eta!., Kaluza—K!einsupergravily 87

subsequentlyusedin [30]. The reasonfor this is that thesespacesareall U(1) bundlesover CP2X S2and so the topology is characterizednaturally by specifying the winding numbersm and n of thecorrespondingU(1) monopolefield over CP2and~2 respectively.As shownin [201],when m andn arerelatively prime, the manifold M(m, n) is simply-connected,while if instead m and n havegreatestcommondivisor 1, then the manifold M(m, n) is not simply-connected,but hasfundamentalgroupZ,;i.e. M(m, n) = M(m/l, n/l)/Z,[201]. No furthergeneralityis addedby introducingthe (redundant)thirdparameterof the M”T notation,and furthermorethe rules for determiningthe fundamentalgroupofthe non-simply-connectedM”~ spacesare rather awkward [200]. The parametersare relatedbym/n = 3p/2q [201].

We add as a parentheticalremark at this point that the whole question of obtainingnon-simply-connectedspacesby taking a simply-connectedspaceand factoring by a discrete subgroupis verycomplicated. In particular, there seem to be many possibilities which, in the case of the M(m, n)(~M”~)spacesfor example,do not readily lendthemselvesto a descriptioneitherin the cosetlanguageof [26,30] or in termsof the U(1) bundleover CP2X 52 For instance,the specialcaseM(0, 1) = M~°=

C p2 x 53 hasnon-simply-connectedfactoringsCP2x (Si/F)whereF= Zk (lensspacesL(k; 1)), thebinarydihedral,tetrahedral,octahedralor isocosohedralgroups.Of thesepossibilities,only the simple caseofL(k; 0) is containedwithin the M(m, n) or M”~classification.Non-simply-connectedfactoringsdo notyet seemto play an important role in Kaluza_Klein.* Obtaining an exhaustiveclassification of allpossiblefactoringsfor all possiblecompactifyingspacesis likely to bea difficult task,which we shallnotconsiderfurther.

Following [16, 201], we obtain metricson the M(m, n) spacesusingthe following generalmethod.Let M

6 be a six-dimensionalmanifold with metric d.~2admitting a non-trivial U(1) bundlewith field

strengthF, given locally by a potentialA. Thenwe can write a metric ds2 on M7, the U(1) bundleover

M6, as

ds2 = c2(dr— A)2+ di2, (9.2.1)

wherec is aconstant,andr is the (seventh)coordinateon the U(1) fiber. Without loss of generality,weassignr to haveperiod41T. Let e1 (i = 1, . . . , 6) bean orthonormalbasisfor dS~2ande” (a = 0, . . . , 6) bean orthonormalbasisfor ds2, definedby

e°=c(dr—A), e’=ë’. (9.2.2)

A straightforwardcalculationthen showsthat the connectionforms Wab andRiemanntensorRabcd forM

7 arerelatedto ~ and R,.,k,on M.,, by:

W.J = + ~cF,.~e°,wo,. = —~cF,.1e’, (9.2.3)

RIJk! = Rilk! — ~c2(F,kF,— Fu

1~k+ 2F,JFk,), (9.2.4)

R0101= ~cF,.kFk, RVko= ~cVkF,.J, (9.2.5)

whereF = ~F,.~e1., ë’ = dA, A = A•ë1,andVk is definedwith thesix-dimensionalconnection~. The Ricci

* See,however,thesectionon recentdevelopments.

88 Mi Duffel at, Ka!uza—Kleinsupergravi!y

tensor,R,.1,, of M7 is relatedto that of M6 by

R1 = — ~cF,.kFfk, (9.2.6)

RW= k2F,

1F,1, (9.2.7)

R0, = —~c11F,1. (9.2.8)

For the M(m, n) spaces,M6 is CP2X S~endowedwith the productof standardEinsteinmetricson

CP2and S2 satisfyingR.1 = A48,1 (i, J = 1,. . . , 4) and R., = A28,1 (i, j = 5, 6) on CP

2 andS2 respectively[273]:

CP2: ds2= (6/A4)[d,a

2 + ~sin2~(o~+ o~+ cos2~2~.)] (9.2.9)

S2: ds2= (1/A2)(d0

2 + sin2 0 dm2), (9.2.10)

where0 � � ir/2, do-,. = ~O~2 A 03 etc., and0 and4’ are the usualpolarcoordinateson ~2 The existenceof non-trivial U(1) bundles correspondsto the existenceof harmonictwo-forms, since thesecan betakento representthe secondcohomologyclass.On CP2X 52 the mostgeneralsuch two-form is givenby an arbitrary constantlinear combinationof the Kãhler form on CP2andthe volume form on S2, andhencethe metric on M

7 is

ds2= c2(dT+ m sin2 ,ao-

3+ n cos0d4’)2+ (6/A

4)[d,a2 + ~ sin2j.~(o~+ o~+ cos2/J~o-

3)]

+ (1/A2)(d02+sin2 0d4’2). (9.2.11)

As discussedin [16,201], the requirementof regularityof the metric implies that the constantsm andnmustbe integers.The Ricci tensoris

= c2(~m2A ~+ ~n2A~) (9.2.12)

R0 = A4(1 — ~c

2m2A4)8,,, i, j = 1,2,3, 4, (9.2.13)

R.1 = A2(1 — ~c2n2A

2)8q, j, j = 5, 6. (9.2.14)

Solving the Einstein equationRat,= A8a~is thusreducedto a purely algebraicproblem. In termsof aparameterx, definedby x = A4/A2, one finds

4x A A2= A, (9.2.15)1+2x 1+2x

4’4m2 2+9 2’~

c2 = ‘A (9.2.16)9(1+2x)

(rn\~2= 9(2x — 1) (9.2.17)\nJ 4x2(3—2x)

M.J. Duffel at, Kaluza—Kleinsupergravity 89

Thus,given m and n, (9.2.17)is a cubic equationfor x, which hasexactlyonereal root. It lies in therange � x � ~. The Einsteinmetricson thesespaceswere first obtainedin [30].

The supersymmetryof thesesolutionswas also examinedin [30], andconvertedinto the notationofthis section,the result is that for M(3, 2) (correspondingto x = 1) thereis N = 2 supersymmetry,whilefor all otherM(m, n) no supersymmetrysurvives.

For all M(m, n) Freund—Rubincompactificationsthe masslesssectorof the four-dimensionaltheorycomprisesa masslessgraviton,the gaugebosonsof the isometry group(which is SU(3)x SU(2)x U(1)exceptfor M(1,0) = X 52 which hasSU(4)x SU(2)andM(0, 1) = CP2x 53 which hasSU(3)x SU(2)xSU(2)),anda singlet pseudovectorcorrespondingto the fact that the secondBetti numberof M(m, n) isone [199] (seetable 5). In the supersymmetriccaseM(3, 2) there are in addition the two gravitini,togetherwith the N = 2 superpartnersof the SU(3)x SU(2)gaugebosonsand the pseudovector[30].(Note that the U(1) gaugefield becomespart of the gravitonmultiplet.) Therearealso masslessscalarsin massiveN = 2 supermultiplets[199].Sincethe spectraof the operators.03/2, 4L and0 havenot beencompletelyanalyzed,it is not knownif therearefurthermasslessspin 0 or spin ~states.

(ii) Q(n1, n2, n3) spaces

Thesespacescan bedescribedas U(1) bundlesover ~2 X ~2 X S2 with n

1, n2 and n3 characterizingthewinding numbersof the U(1) field over the threetwo-spheres[16].They werefirst discussedin [205],

where theywere describedas the cosetspacesSU(2)x SU(2)x SU(2)/U(1)x U(1) and shown to haveisometrygroupSU(2)x SU(2)x SU(2)x U(1). From(9.2.1) the metric may be written as [16,201]

ds2= c2(dr+ n

1 cos01 d4’1 + n2 cos02d4i2+ n3 cos03 d4’3)2 + (1/A

1)(d012+ sin2 01 d4’

12)

+ (1/A2)(d02

2+ sin202 d4’2

2) + (1/A3)(d03

2+ sin2 03d4’3

2). (9.2.18)

They aresimply-connectedif the integersn1, n2 and n3 arerelativelyprime. Otherwise,Q(n1,n2, n3) =

Q(n1/l, n2./l, n3/l)/Z,, where I = gcd(n1, n2, n3) [207].Again, onecan calculatethe Ricci tensorfor thesemetrics,andverify that thereis exactlyoneEinsteinmetric for eachchoiceof the integersn1, n2, andn3[207].The casen1 = n2 = n3 = 1 was shown in [205] to admit an Einstein metric and to give N = 2supersymmetry.

For all Q(n1,n2, n3) spacecompactifications,the masslesssectorof the four-dimensionaltheory willincludea gravitonandthe gaugebosonsof SU(2)x SU(2)x SU(2) x U(1). In addition,sinceb2 = 2, therewill be two masslesspseudovectors.In the supersymmetriccasethereare also two masslessgravitini,and the N = 2 superpartnersof the SU(2)x SU(2)x SU(2)gaugebosons.Surprisingly, thereare alsotwo oppositelyU(1)-chargedmasslessWess—Zuminohypermultiplets,in the representation(2,2, 2) ofSU(2)x SU(2)x SU(2) [35].Thesearealsofurthermasslessscalarsin massivesupermultiplets[35].It isnot knownwhetherthis constitutesthe wholemasslesssector,sincethespectraof the operators03,2, 0and

4L arenot known.As a final commenton the Q(n

1, n2, n3) spaces,we note that the caseQ(0, 1, 1) is of particularinterest.It is the productof S

2 with the cosetspaceS0(4)/S0(2)[205].Now as discussedin [206],SO(4)/SO(2)= T

1S3, i.e. the bundleof unit tangentvectorsover S3, andsince S3 is parallelizablethe

bundleis trivial, i.e. T1S

3= S2x53~However,the metric is not isometricto the standardoneon S2x S3,but is insteada non-product Einsteinmetric on S2x S3 [206].

(iii) N(k, 1) spacesThesespacesare constructedas the cosetspacesSU(3)/U(1), wherethe U(1) subgroupconsistsof

9(1 Mi Duff eta!., Ka!uza—K!einsupergravily

matricesof the form [220]

re~’° 1

e”° . (9.2.19)L e~’~”~°°J

All of thesespaceshavebeenshownto admit two inequivalentEinsteinmetrics,with the exceptionofk = 0 (or equivalently 1 = 0 or k= —1) for which only one is known. In the specialcasek = 1, the twoEinsteinmetricswere found in [211];the situationhereis very analogousto the two Einsteinmetricsonthe seven-sphere,since the spaceN(1,1) can be regardedas an S0(3) bundle over CP2[204]. Forgeneral k and 1, it was demonstratedin [203] that an Einstein metric exists for each N(k, 1).Subsequently,this result was obtainedindependentlyin a more explicit form in [202].Later, it wasdiscoveredin [204] that there are in fact two inequivalentEinstein metricsfor all the N(k, I) spaces(except k = 0). Interestinglyenough,for generalk and I thereappearsto be no descriptionof thesespacesasbundlesover lower dimensionalspacesanalogousto theSO(3)bundleoverCP2of the N(1,1)space.As in the caseof the M(m, n) spaces,we arenot usingthe ~ classificationof [202]becausethethird parameteris redundant.Theparametersarerelatedby k/i = (3p + q)/(3p— q).

The Einstein metrics obtainedin [202]were shown to yield N = 3 supersymmetryin the caseofN(1, 1) andN = 1 for all otherN(k, 1). All theseEinsteinmetricson N(k, 1) aredenotedN(k, 1), in table6. Using the resultsin [202],thenewEinsteinmetricsobtainedin [204],which we denoteN(k, ~ wereshown to yield N = 1 supersymmetryfor all k and 1, including k = 1 = 1 [204].The isometrygroupsforN(1, 1)~andN(1, 1)~~are SU(3)x SU(2),while thosefor all otherN(k, I)~andN(k, i)~~are SU(3)x U(1)[202].

The full spectrumof masslessand massivestateshasbeen analyzedin [197],using fermion—bosonmassrelationsgiven in [221].Thesemassrelationsareequivalentto the energyrelationswithin N = 1multiplets given in [183],convertedto massrelationsusing the resultsof [80] (seesection3.2). In allcasesthe expectedmasslesssupermultipletsimplied by gaugesymmetryandsupersymmetryare present(togetherwith amasslesspseudovectorandits superpartnersin thecaseof the N(1, 1) compactifications,sincethose haveb

2= 1). As seemsto be the rule in supersymmetriccompactifications,thereare alsofurthermasslessscalarsin massivesupermultiplets[197].

To concludethis discussionof the N(k, 1) spaceswe presentthe explicit family of metricson N(1,1)obtainedin [204],in which N(1, 1) is viewed as an S0(3)bundleover CP

2:

ds2= A2[(..~1 — cosj.ur1)

2 + (.X

2 — cos~to-2)2+ (.~ — ~(1+ cos2,a)o

3)2] + d/L2

+ ~sin2 ,a(o1

2 + o-2~+ cos

2,.au3

2). (9.2.20)

This is analogousto the family of metrics (8.1.31) on 57 regardedas an SU(2) bundleover 54 Thecoordinate~ and SU(2)one-formsu, arethe sameas in the CP2metric (9.2.9),andthe one-forms~satisfy d2

1 = ~ A Z~,etc. and are left-invariant forms on the manifold SO(3). Just as for theseven-sphere,thereare two values of the “squashingparameter”A

2 in (9.2.20) which yield Einsteinmetrics. In this casethey are A2 = ~,which gives the N = 3 supersymmetricN

1(1, 1) and 1,2 ~ whichgives N11(1, 1) with N = 1 supersymmetry[204].

(iv) Product spacesOf the productspaceslisted in table6, someof them havealreadybeencoveredin this chapter,since

M.i Duffelat, Kaluza—K!einsupergravily 91

~5x ~2 = M(1, 0), S2x S2xS3 = 0(0, 0, 1) S2x T1S

3= Q(0, 1, 1), and CP2X S3= M(0, 1). The remainingthreeare S4x S3 [29], the twisted S2 bundle over S2x S~[153] and (SU(3)/SO(3)max)X 52 [200].TheEinstein metric on 53 is

S~: ds2 = dO2 + sin2 0 d4’2 + (d4’ + cos0 d4’)2, (9.2.21)

while the Einstein metric on 54 was given in (8.1.29).The S2bundleover S2 metric is given in [153,190]andthe spaceSU(3)/SO(3)max is discussedin [200].

(v) SO(5)/SO(3)max and V52

These are the remaining Freund—Rubin solutions listed in table 6. Vs,2 is the Stiefel manifoldSO(5)/SO(3) whereSO(3)C SO(4)C 50(5).Thesespacesarediscussedin [2001andwe will not considerthem furtherhere.We merely notethat V5,2 is a parallelizablespace.

(vi) Skew-whiffedsolutionsThe masslessstatescorrespondingto the skew-whiffedvacuaarethe samefor spins2, 1 and0~but

not ~,~and0. In particulartherewill be no masslessspin ~with theexceptionof the round57 for whichtherewill be 8. Therewill also be N(N + 1)/2 massless0 as discussedin section 8.2. (As usualtheroundS

7 is an exception.)In general,thereis no reasonto expectany masslessspin ~.

10. Non-vanishing Fmnpq

10.1. Generalremarks

All the solutionswehavediscussedso far obey the Freund—Rubinansatzof section3.1 andweshallnow considermore general solutions.In particular in this chapterwe shall relax the assumptionthatFmnpq is set equalto zero.The first solutionof this kind wasdiscoveredby Englert [43].

Let us write

= 2me,,,.,,,,,, (10.1.1)

1’mnpq�O, (10.1.2)

so (3.1.3)becomes(dropping~ superscriptsfrom now on)

= ~ (10.1.3)

or, in termsof differential forms on M7

d*F= —4mF. (10.1.4)

Notethe factor2 in (10.1.1)comparedwith (3.1.7). Thischangeis purely for convenience.Furthermore,(3.1.2) splitsinto

92 Mi Duff eta!., Ka!uza—K!ein supergravi!y

R,,,... = ~[— 16m2— i~FmnpqFm~~Ig,,,,., (10.1.5)

Rmn= ~[FmpqrFn~ — ~gmnF,~,,rsF~Ts + 8m2gmn]. (10.1.6)

Since maximal spacetimesymmetry demandsthat R,,.,= Ag,,,,. we must require that Fmn,,QFm~~beconstant.Note, however,that sinceFmnpqFm”~� 0, we arestill restrictedto A <0, i.e. to AdS.

In contrastto ground statesof the Freund—Rubintype the symmetry of the vacuumis no longergiven by the isometrygroupof M

7 but ratherby the group which leavesinvariant both g,,,,, and Fmnpq.

For example,the Englert solution[43] for Fmnpq on the roundS7 discussedin section10.2 breaksSO(8)

down to SO(7) [76, 58—61, 30]. This symmetry breakingis somewhatreminiscentof the squashingofchapter8, wherewe found that therewas a four-dimensionalspontaneoussymmetry-breakinginter-pretation.In fact, non-zeroFmnpq also has such an interpretationin termsof non-zeroVEVs for thepseudoscalarfields [45]. The Higgs and super-Higgseffect works as follows. Considerthe Fourierexpansionof Amnp(X,y) in harmonicson S7

Amnp(X,y) = ~ Pk(X)Y~’mnp(Y). (10.1.7)

Oneobtainsthe 35,, pseudoscalarsof the N = 8 masslesssectorfrom the lowestmodetogetherwith aninfinite tower of massivepseudoscalarswherek denotesthoserepresentationsof S0(8)appearingin the0 sectorof table8. In the symmetricphase

(Amnp(X,y)) = 0 -~ all (Pk(x))= 0, (10.1.8)

but when

(Amnp(X,y)) � 0 —~ some (Pk(x)) = 0, (10.1.9)

i.e., non-zeroVEVs for someof thesepseudoscalarsand hencea spontaneousbreakdownof parityinvariance.A similar interpretation exists for the other solutions presentedin this chapterand inchapter12.

A curiousconsequenceof solutionsof thiskind is the appearanceof a calculableCP-violatingangle0. The term in the d = 11 supergravityLagrangian(2.2.1)

M

1M2 M1~~‘ I:’ A� 1 M1 . . M~

1 M5. M8’’M9. . M11

yields a term in the d = 4 Lagrangianof the form [45]

f (D\L’ IJ~’ KL p~r

JIJKLI,’ Ji i,,,. ~, Ewheref is a function of the pseudoscalarsandf,,,,,.’~is theYang—Mills field strength.When (P~� 0, onecan obtaina OF,,.F,,,,,e’

1””° term.Finally,we notethat therearem = 0 solutions(d = 5 AdS) x M

6, whereM6 is Einstein—KählerandF isthe productof the Kähler form with itself [270,271].

Mi Duffelat, Kaluza—K!einsupergravity 93

10.2. Solutions from spaces with Killing spinors

Oneway to obtain a solutionof d = 11 supergravitywith Fmnpq� 0 is to startfrom a Freund—RubinsolutionadmittingaKilling spinor i~.Considerthe construction[45,22]

Amap= c~Fmnpt~, (10.2.1)

where c is a constant,which automaticallyimplies the gaugeVtm Amnp = 0. From the Killing spinorequation(4.2.5) it is straightforwardto checkthat the gauge-invariantfield strength

Fmnpq= 4i91mAnpqi (10.2.2)

satisfies[22]

VmFm1’~= ~ (10.2.3)

andfrom the Fierz identity (B.11) that [48]

FmpqrFn”T = 384c2m2gmn, (10.2.4)

wherewe havenormalized~ = 1. Since the Killing spinor equationrequires~ = 6m2gmnwe find

that the aboveconstructionsolvesthe Einsteinequations(10.1.5)and(10.1.6) with= —lOm2g,,,,., (10.2.5)

Rmn= 6m2gmn, (10.2.6)

provided

= ~. (10.2.7)

In otherwordsthe construction(10.2.1)providesa solutionof the equationsof section10.1.wereit notfor the sign differencebetween(10.1.3) and (10.2.3). However, this may be easily remediedby theprocessof “skew-whiffing” discussedin section4.2. Equivalently,we could keepthe orientationfixedbut replacethe ansatz(10.1.1) by F,,,,.,,

0.= —2mr,,,,.,,,,. Eitherway, thesesolutionshaveoppositesign forthe PagechargeP of section3.1 from their supersymmetriccounterparts.Comparisonof (10.2.1)with(8.4.11) enablesone to interpret thesesolutions as vacuafor which someof the N(N + 1)/2 masslesspseudoscalarsof the skew-whiffed non-supersymmetricvacuahaveacquirednon-zeroVEVs.

Note, incidentally,that thefactor of 2 ratherthan3 in (10.1.1)was chosento maintainRmn= 6m2gmn,

but the spacetimecosmologicalconstant of (10.2.5) is now —10m2 rather than —12m2. Of course,keeping Rm,, fixed when comparing solutions with and without non-zero Fmnpq was purely forconvenienceand no physical significanceshould be attachedto it. This raisesthe questionof whetherthereis anyphysicalquantity which shouldbe kept fixed whem comparingdifferent vacua.The answerto this questionis not at all clear to us, but onecould envisagea situationwherea bubbleof the “true”

94 Mi Duff et at, Kaluza—K!ein supergravi!y

vacuumcould form inside a “false” vacuumat leastwhen the two vacuahavethe sametopology,e.g.,when both are 57 In this casethe field equationsdemandthat the Pagechargeof (3.1.19)be constantacrossthe boundaryof the bubble.In this respect,all quantitiessuch asmassesandcouplingconstantsshould really be expressedin terms of the fundamentalparameters,namely the d = 11 gravitationalconstantandthe Pagecharge,but this could prove rathercumbersome.

The criterion for unbroken supersymmetry,i.e. (~PM)= 0, becomesmore complicatedwhen~ � 0. However,in the specialcase

Fmnpq = ±m~Fmnpq~ (10.2.8)

describedin this section,one can prove that all supersymmetriesare necessarilybroken [50, 72]. Theunbrokengaugegroupis, as mentionedin section10.1, that which leavesinvariant both ~ andFmnpq.For solutionsof the kind given by (10.2.8) the correspondingsupersymmetricFreund—Rubinsolutionhassymmetty G = S0(N)xK for somegroup K, where N is the numberof Killing spinors[30] (seechapter6). Since the construction(10.2.8) singlesout one of the N Killing spinors,it follows that thesolutionso obtainedhasresidualsymmetry SO(N— 1) x K.

Consideredasa rank-threetensorunderthe tangentspacegroupof transformations,it is interestingtonotethat the ~ of (10.2.1) remainsinvariant underthe G2 subgroupof SO(7)since, as discussedinsection4.2 on holonomy,02 is the stability groupof aspinor in sevendimensions.This is relatedto thefact that ~~Fab,,7/is simply the multiplication tableof the octonions,i.e. in a suitablebasis

‘ijFabc?) = aabc, (10.2.9)

where

°a°b = ~ab + aabcoc, (10.2.10)

andwhere°aarethe imaginaryunit octonions,i.e.,

a0,.;=~ a,.1~= ijk , a~~= ~IJk (10.2.11)

in the notation of section8.1. As is well-known, the groupof outerautomorphismsof the octonionsisG2. For further referenceson octonionssee [51,55, 56, 59, 89]. Note, however,that althoughmostofthis literatureis confined to S

7, the octonionicinterpretationgiven aboveis quitegeneral.This rank-threetensoralsohasan interestinggeometricalinterpretationas a Ricci-flattening torsion

[50]. To seethis let usfirst establishsomeidentitiesusing(10.2.1)and thepropertiesof Killing spinors,in particular (B.11). First define

5,,,,,,, = — miil,,,,,,,,rj, (10.2.12)

where,as usual, ~j satisfies(4.2.5). Then

VqSmnp= iliqSmnpj, (10.2.13)

which is just theequationof a Yano Killing tensorgiven in section4.3. Also

MJ. Duff etal., Kaluza—Klein supergravity 95

r c —c c 2j — (10214v m~~npq— ~‘mnt-’pq m ~gmpg,~ gmqgnp,. k

On the otherhand,if we now constructa non-RiemannianconnectionF~+ S~from the RiemannianconnectionF~anda totally antisymmetricbut otherwisearbitrary torsion tensorS~,

Rmnpq(T + S)= Rmnpq(F)+VpSmnq — VqSmnp + SmqtSnp— SmptSnqt. (10.2.15)

WhenSm~,obeys(10.2.14), therefore,

Rmnpq(F + S)= Rmnpq(F)— m2(gmpg,~— gmqgnp). (10.2.16)

But sinceRmn= 6m2gmn,we obtain

Rmnpq(F+S)= Cmnpq(F), (10.2.17)

and hence

Rmp(F+ S)= 0. (10.2.18)

Note that this Ricci-flattening property is true only for the choice of sign given in (10.2.12). It is notclearwhat physicalsignificance,if any, shouldbe attachedto this propertybut solutionswith non-zeroFmnpq are sometimesreferredto as solutionswith “torsion”. For a reformulationof the d = 11 theoryusingAMNP as a torsionfield, see[166,222].

Solutionsof the kind discussedin this section may be obtainedfrom any solutionswith Killingspinors,althoughthe round S7 as usual requiresa separateanalysis,to which we now turn. This is theEnglert [43] solution.

On the roundS7 we havethe luxury of employingeither the ~±‘sobeying (4.2.5)or the ~_‘sobeying(4.2.8) in constructingAmnpsince both orientationshave8 supersymmetries.The construction(10.2.1)with ~ = i~÷obeys(10.2.3)andwith ~ = ~ obeys(10.1.3).The symmetryof thissolutionmustbeSO(7)since we startedwith SO(8) and we are singling out one of the eight spinors,but in this case it isnecessaryto specifywhich embedding.Choosing~ = ~ (or else ~ = ~± togetherwith a skew-whiffingwhich interchangess and c labels), the relevantS0(7) is fixed by the decomposition8~-+1 + 7. This[SO(7)]~symmetrywasfirst establishedin [76,58—61,30], andthe breakingof all 8 supersymmetriesin[46,48]. Alternatively,we may constructa solutionof (10.1.3)as [21]

A — IJKL — I r J — K r L 19mnpC fl+lmnVi+71±1p71+,

where CIJKL is constantand anti-self-dual.It must be chosen to correspondto the singlet in thedecomposition35~—* 1 + 7+27in order to satisfy (10.2.4). Replacing~+ by ~. would yield a solutionof(10.2.3).

When SO(8) breaksto [SO(7)]~the masslessN = 8 supermultipletdecomposesas 1 —*1; 8~—~8;28 —~7+ 21, 56.—*8 + 48; 35~—* 35; 35~—* 1 +7+ 27. All eight gravitinos must acquire a mass and all 8supersymmetriesare broken at a commonscale.The round S7, having 35 masslesspseudoscalars,isagainan exceptionto the N(N + 1)/2of the skew-whiffedsolutionsdiscussedpreviously.To understandthe spontaneoussymmetry-breakingwe recall that Amnp is a Yano Killing tensor.One can showthat

96 MJ. Duff etaL, Kaluza—Kleinsupergravity

there are 70 rank-three totally antisymmetric Yano Killing tensorson the round 57, transforming as a35~and a 35~under SO(8). The 35~areassociatedwith the masslesspseudoscalars(in the n = 0 level oftable8), while the 35~describemassivepseudoscalars(in the n = 2 level of table8). The 35~obey (10.1.3)andthe 35~obey (10.2.3).Thus the roundS7 with “torsion” correspondsto a non-zeroVEV for the 35~(in the n = 0 level of table 8). As a consistencycheckwe note that under SO(8)—* [SO(7)J~,35~—*

1 + 7+ 27 andwe find as expectedthe singlet togetherwith the7 Goldstonebosonswhich give a massto the 7 spin 1 gaugebosonsin the decomposition28—*7+21.

Thetorsion tensorSmnpof (10.2.12)alsoacquiresanew significancein thecaseof the round57~SinceCmn~= 0, we seefrom (10.2.17) that it is not merelyRicci-flattening but Riemann-flattening,i.e., it isoneof the two parallelizing torsions(the otheris obtainedby replacing~+ by ~ andchangingthe signin the (10.2.1)).The fact that 57 admitssuch an “absoluteparallelism”, i.e., a flat connectionwith totallyantisymmetrictorsion(total antisymmetrymeansthat theF + S yields thesamegeodesicsas F) was firstpointed out by Cartanand Schouten[223]in 1926. Indeed,apart from group manifolds for which thetwo torsionsaregiven by ~ wherefmnp arethestructureconstants,57 is theonly compactmanifoldtoadmit an absoluteparallelism.In fact Englertdid not employ Killing spinorsto find his solution[43]butinvoked the 1926 constructionof CartanandSchouten.

Another special case of interest is the squashed57~The solution with Fmnpq ~ flFmnpqli on thesquashed57 was first suggestedin [22] andderivedexplicitly in [50, 53, 58]. In addition to the non-zeroVEVs for the 1 and 300 of scalars(in the n = 2 level of table8) occurringon the squashed57 withouttorsion,the35. of pseudoscalars(also in then = 2 level) haveacquiredanon-zeroVEVs as may be seenfrom (10.2.3).As a consistencycheckwe notethat under SO(8)—* [SO(5)x SU(2)]~,35~-~(14, 1) + (5, 3)+

(1,5)+ (1, 1) and we find the singlet as expected.Since the non-zerotorsion is a singlet under thevacuumsymmetryof zero-torsionvacuumthe unbrokengaugegroupremains[50(5)x SU(2)]~but thereis, in addition, a spontaneousbreakdownof parity. The pseudoscalaracquiring the non-zeroVEVs isthe single pseudoscalarwhich was masslessin the zero-torsionsquashed57 vacuum.

The constructionof this sectionmay also be appliedto the othersolutionswith Killing spinors.Inparticular, we note that the SU(3)X SU(2)X U(1) of the N = 0 M(3, 2) vacuum,obtained by skew-whiffing the N = 2 M(3, 2) vacuum,would break to SU(3)x SU(2).

In conclusion,we notethat the Fmnpqof (10.2.8)vanisheswhen m is set to zeroandso T7 andK3 x T3do not admit solutionsof this kind.

10.3. Solutionsfrom U(1) bundlesoverKdhlersix-manifolds

Not all ground-statesolutionswith Fmn,~� 0 takethe ijFm~,,,~flform discussedin section 10.2. Twootherclassesof solutionhavebeenfound to date,in which M

7 is a U(1) bundleover a KählermanifoldM6 [88, 224]. To discussthesesolutionswe needcertainpropertiesof six-dimensionalKählerspaces,which we review below. We will usean orthonormalsechsbeinë

t for M6 (i = 1~. . . , 6), in terms of

which the Kähler form J = ~J~Je1A e~satisfies

JijJ,k = &k, VI,J~k= 0, (10.3.1)

whereV~is the covariantderivativeon M6. The Ricci two-form P = A ë’ can then be defined,as

P~= ~R~JkIJk,, (10.3.2~

MJ. Duff et aL, Kaluza—Kleinsupergravity 97

wheEeRIJkI is the Riemanntensorfor M6. Using the integrability condition [Vi,VJ]Jk, = 0, which followsfrom (10.3.1), onecan then show that

P11 = JIkRk, = R,kJkJ. (10.3.3)

A fundamental property of a Kãhler manifold is that it admits a gauge-covariantly constant spinor ~satisfying

= — ieA~= 0, (10.3.4)

wheredA = P, the Ricci form, and D, is the spinorcovariantderivativeon M6 [225].This can beseen

by consideringthe integrability conditionfor (10.3.4);

[fl,~ = —~RIJkIFkl~— ieP~~= 0, (10.3.5)

whereF, arethe d = 6 Dirac matrices.Actingon (10.3.5)with F~andwith J~implies that 4e2= 1, and

making the choice2e = +1, it follows from (10.3.5)that

JI)FJ~= if~, F0~= i~, (10.3.6)

where F0 = I’123456.

Using ~ we now constructthe following complexthree-form K111.

KIJk = sëMFijk~ (10.3.7)

whereeM is the Majoranaconjugateof ~ From(10.3.4), with 2e= 1 it follows that

V,Kjk, = iAIKIkI, dK = iA A K. (10.3.8)

From (10.3.6)it follows that Ku,. is holomorphic,in the sensethat

JI1KIJk = iKIJk, (10.3.9)

and sincefrom (10.3.4)~ = constant,then up to a constantfactor k it is given by

K= kZ’ A A Z3, (10.3.10)

where (Z1, Z2,Z3) is a complex local basis of the form Z’ = e1 + ie2, 22 = e3 + ie4, Z3= e5 + ie6.Choosing k appropriately, KEIk can thereforebe normalizedto satisfy

K1kIK)k,= 0, KIkIKk, = — iJ~. (10.3.11)

We now constructthespaceM7, a U(1) bundleoverM6 with metric given by (9.2.1),with A beingthepotentialfor the Ricci form appearingin (10.3.4), andusethe orthonormalsiebenbeinea (a = 0, . . . , 6)for ds

2 as given by (9.2.2). Specializingto Kähler spacesM~for which the Ricci scalarR constant,

98 MI. Duff eta!., Kaluza—K!ein supergravity

(9.2.6), (9.2.7)and (9.2.8) imply that the non-vanishingRicci tensorcomponentson M7 are

R~= R — ~c2Rjk1~k, (10.3.12)

R~= ~~c2R

11R~. (10.3.13)

The three-form(10.3.7)on M6 can be lifted to a globally definedthree-formL on M7, given by

L e~K. (10.3.14)

From(10.3.8)it follows that dL = —(i/c)e°A L, andhencefrom (10.3.14)

*dL -iL. (10.3.15)

This complex three-form will now be usedto constructan ansatzfor thereal field strengthFmnpqOfl M7,by defining

F= f3e°A(L+ E)= if3cd(L—111), (10.3.16)

where /3 is a real constant.From (10.3.15), it satisfies

d*F= -iF. (10.3.17)

Comparison with (10.1.4) implies that we require

c = (4m)1. (10.3.18)

Substituting(10.3.16)into (10.1.6), andusing (10.3.11),the Ricci tensoron M7 mustsatisfy

R1 = 2(fl2 + 4m2)5

11, (10.3.19)

R~= ~(p2+ m2). (10.3.20)

On the otherhand,the Ricci tensorof M7 is given by (10.3.12)and(10.3.13),andso we obtain (10.3.19)

and (10.3.20) provided that these equations are compatibleand(10.3.18)holds.Oneclassof solutionsto theseequationscan beobtainedby taking the Kählermetric on M6 to be

Einstein,

14 = ~ (10.3.21)

which thereforeimplies

A = 16m2, $2 = 8m2, (10.3.22)

MJ. Duff eta!., Kaluza—Kleinsupergravuy 99

togetherwith (10.3.18).We shall refer to solutionsof this kind as “type I”. Remarkably,thereis asecondclassof solution,which werefer to as “type II”, for which the Ricci tensoron M6 satisfies

= ~ diag(5,5, 5, 5, 9, 9), (10.3.23)

and

/32 = Wm2, (10.3.24)

and againc is given by (10.3.18). In [224],wherethesesolutionsareconstructed,it is shown that theyarethe only possiblesolutions.

A type I solution existsfor any Einstein—KählerspaceM6 with positive Ricci scalar. The known

examplesare CP3, CP2x ~2, s2 x 52 X 52 and SU(3)/T, where T is the maximal torus in SU(3). The

correspondingmanifoldsM7 havethe topologiesof S

7, M(3, 2), Q(1, 1, 1) andN(1, 1), respectively.In allcasesthe metric on M~is “stretched”by a factor of V2 alongthe U(1) fibre direction comparedwiththe lengthwhich would yield an Einsteinmetric.

A typeII solutioncanbe constructedfor any KählerspacewhoseRicci tensorsatisfies(10.3.23).Theonly knownexamplesare CP2X 52 and52 x ~2 x S2, with the standardEinsteinmetricson the subspacesscaledappropriately.The topologiesof the correspondingM

7’s areM(3, 2) andQ(1, 1, 1), respectively.The symmetryof a typeI or II solution is simply given by the isometrygroupG of thecorresponding

KählerspaceM6. This follows from the fact that the isometry group of the metric on M7 is at leastG X U(1). [It may be larger, as in the caseof S

7 whereit can be 50(8)andnot merely SU(4)x U(1)].The spinor~ is a singletunder0, but is chargedwith respectto the U(1), andso the field strengthgivenby (10.3.16)is invariant only underG. In particular,the typeI solutionwith M

7 = S7 is SU(4)invariant,

andcorrespondsto a spontaneousbreaking of the N = 8, SO(8) invarianceof the round S7 down toN = 0 and SU(4). This will bediscussedfurther in chapter12. For a symmetry-breakinginterpretationof the othersolutionsin this section,see [224].

11. Vacuumstability

11.1. Freund—Rubinsolutions

In section5.2 it wasshown that the necessaryandsufficient condition for perturbativestability of aFreund—Rubin vacuum was given by ~lL�3m2, where ~iL is the Lichnerowicz operatoracting onsymmetric transverse,trace-freetensorshmn in M

7. In this sectionwe classify which of all knownFreund—Rubinsolutionsarestableandwhich are not. To this end,it is convenientto divide them intothreedisjoint categories.

(1) Supersymmetric spaces and their skew- whiffed counterparts.From the formal propertiesof thesupersymmetryalgebragiven in section 3.2, supersymmetricvacua are automaticallystable.Thus welearn that all M7 with Killing spinorshave ‘iL � 3m

2. The known examplesappearin table 6. Moresurprisingly,perhaps,the non-supersymmetricvacuaobtainedby skew-whiffing are alsostable,sincethespectrumof LIL is insensitiveto the orientation[25].

(2) Productmetrics on M7. Theseare all unstable[25,226]. To seethis, let uswrite M7 = M(l) xM(2)

andlet the dimensionsof M(j) and M(2) be n1 and n2 with n1 + n2 = 7. We assumea productEinstein

100 MJ. Duff eta!., Ka!uza—K!ein supergravigy

metric ~ with block diagonalcomponents~ and~ We prove theinstability of such solutionsby explicitly constructingan eigenmodeof LIL with eigenvaluezero.This modehmn hasblock diagonalcomponentse1g~~1and r2g~~2,wherer~,and r2 are constantschosento satisfy n1~1+ n2~2= 0 inorder that hmn be trace-free.It is manifestlytransverseandimmediatelyyields JiLhmn = 0, violating thestability criterion (5.2.1). Physically, this mode correspondsto an instability in which oneof the spacesin the productexpandsat the expenseof theother.This applies to all the productspaceslisted in table6. Note, however, that M7 could be topologically a productbut with a non-productEinstein metric inwhichcasethe aboveproof doesnot apply.

(3) All remainingM7. Theserequiremoredetailedtreatment,andthereappearsto be no alternativeto carryingout a case-by-caseanalysis. However,therearefrom table6 only two knownfamilies: TheM(m, n) spaceswith genericm and n andthe Q(p,q, r) spaceswith genericp, q, and r. Their stabilitypropertiesareanalyzedin detail in [201,227,207]. Sincetheyarehomogeneousspaces,we may employ(4.3.33)to deducethat the solution will bestableif the largesteigenvalueof the Riemanntensor,Kmax,

definedby (4.3.32)satisfies

Kmax�~m

2. (11.1.1)

In [201,207], this eigenvalueis calculatedexplicitly for both the M(m, n) and Q(p, q, r) spacesand,furthermore, it is shown that the correspondingeigentensoris a Killing tensor satisfying (4.3.34),therebysaturatingthe inequality (4.3.33). Therefore, thesespacesare stableif and only if (11.1.1) issatisfied.For the M(m, n) spaces,the criterion for stability is

j7~6112< mIni <4~(66)h/2, (11.1.2)

or in the ~ notation,

~6h/2 < Ip/qi <n~(66) . (11.1.3)

The resultsaremoredifficult to presentfor the Q(p, q, r) spacessince the local geometrydependsuponthe two ratiosp/r and q/r. One can show that the solutionsarestableif andonly if p, q and r lie in acertainregion for which pir and q/r aresufficiently closeto unity [207].

11.2. OthersolutionsA generalstability analysisis much more difficult when Fmn,~� 0. It was arguedin [61] that the

Englert solution on the round S7 should be unstable, and this was confirmed in [73] by exhibitingspecific modeswhich violatethe Breitenlohner—Freedmanstability bound.We recall from section10.2that in spacetimethe Englertsolutionmaybeinterpretedas a spontaneousbreakingof theN = 8 SO(8)of the round S7 without “torsion” down to N = 0 andS0(7)~via a non-zeroVEV for the 35~masslesspseudoscalars.Under the decomposition35~—* 1+ 7+ 27, we find the singlet as expectedand the 7Goldstonebosonswhich give a massto 7 gaugebosons.The stability analysisof [73]showedthat the27acquiredmass-squaredsufficiently negativeto violatethe stability bound.

One might expect that other solutionsconstructedfrom Killing spinors of the type consideredinsection lp.2 shouldexhibit similar instabilitiesif theyhaveanaloguesof the 27, i.e., originally masslesspseudoscalarsover and abovethe singlet which acquiresan expectationvalue, andover andabovetheGoldstonebosonsassociatedwith any breakingof gaugesymmetry.This will occur if the numberN of

MJ. Duff eta!., Kaluza—K!einsupergravily 101

Killing spinorson M7 is greaterthan one.To seethis recall that thereare~N(N + 1) massless0 in thecorrespondingFreund—Rubinsolution and the isometry group is G = SO(N)x K. The solutionswith“torsion” haveinvariancegroupSO(N— 1) x K andhenceN — 1 gaugebosonswill becomemassivebyeatingN — 1 pseudoscalars.Taking into accountthe singlet acquiring the non-zeroVEV, we areleftwith ~N(N — 1) pseudoscalarswhich might be expectedto give unstablemodes.(Note that the N = 8 ofthe round 57 is a specialcasesince thereare only 35 0 ratherthan 36 becauseij’Fa,,.r~= 0). It canindeedbe shown[228]that these~N(N — 1) 0- do violatethe stability boundandhenceall solutionsofthis kind with N> 1 areunstable.

N = 1 requiresa separateanalysis. In [228]is was arguedthat the N(k, 1) spaceswith “torsion”shouldbe unstableif k/i is sufficiently closeto oneand this hasbeenconfirmedby explicit calculation[226].It hasalso beenshown that the squashed57 with torsion is unstable[84]. However, it is notknownwhethertherearestableexamples.

The stability of solutions of the typediscussedin section10.3 hasnot yet beenanalyzed.

12. Relation between d = 11 andd = 4 supergravity

12.1. Thede Wit—Nicolai theory

As discussedin section 1.2, de Wit and Nicolai [38, 39] constructedin 1981 a four-dimensionalsupergravitytheory by gaugingthe SO(8)subgroupof E7 in the global E7x local SU(8)supergravityofCremmer and Julia [37]. In common with the Cremmer—Julia theory, this theory describedtheself-interaction of a single massless N = 8 supermultiplet of spins (2, ~, 1, ~, 0~,0) but with localSO(8) x local SU(8) invariance. In addition to the gravitational constant, there was a new parameter: theSO(8) gaugecoupling constante, and it was necessary to modify the Cremmer—Julia Lagrangian andtransformationrules by other e-dependentterms in order to preservethe N = 8 supersymmetry.Inparticular,thesewere Yukawa-likeinteractionsbetweenthe fermion andspin 0 fields, anda non-trivialeffective potential for the scalars.As in the Cremmer—Juliatheory, the SU(8) gaugepotentialswerecompositefieldswith no kinetic term of their own andwere thereforeexpressiblein termsof the spin0fields andtheir derivatives.The transformationof the variousfields underS0(8)x SU(8)is summarizedin table12. We note,in particular,that the fennionsarechiral underSU(8). The numberof spin 0fieldsis reducedto 133 by assigningthe u’s and v’s to a “sechsundfunfsigbein”,which is an elementof E7.The numberof physical spin 0 statesis 70 = 133—63 since63 fields may be gaugedaway by an SU(8)rotation. Since E7 is no longer a symmetry, however, we cannot identify these 70 scalarsas theGoldstone bosons of E7 breaking to SU(8).

Table 12SO(8)x SU(8) assignments

Spin 2 3/2 1 1/2 0 0

Field e,~ ~ B~” ~Yk ~~uf.’ vi~

SO(8) 1 1 28 1 28 28

SU(8) 1 8 1 56 28 28

102 MJ. Duff eta!., Kaluza—K!ein supergravity

The existenceof the effectivepotentialnaturally prompted the question of its extrema. As discussedby de Wit and Nicolai [391,the obvioussymmetricextremumfor which both the 35 scalarsand 35pseudoscalars have zero VEVs yields an SO(8) vacuum state with N = 8 supersymmetry.[SU(8)is not,of course, a symmetry of the vacuum]. The physical states belong to the familiar massless N = 8supermultipletwith (1, 8~,28,56~,35w, 35~)representationsof SO(8).The effectivepotential thenyields anon-zero cosmological constant for this vacuum given by (1.2.2.), namely A = —l6irGe2, and theYukawacouplingsyield a “massterm” for the gravitino. The relevantalgebrais thereforethe °Sp(418)de Sittersupersymmetryratherthanthe N = 8 Poincarésupersymmetryof the Cremmer—Juliavacuum.Note, however,that the de Wit—Nicolai theory goessmoothlyover to the Cremmer—Juliatheory in thelimit e—* 0.

For full details of thede Wit—Nicolai theory,see[38,39]. By truncation,onemay alsoobtaingaugedS0(N)supergravitiesfor N <8, in particular N = 5 and 6. The gaugedN ~ 4 theorieswere alreadyknown to exist [142,229, 230]. The existence of gauged extended supergravity for N > 4 was alreadystrongly suggestedbefore the explicit constructionof de Wit and Nicolai, by the discoverythat suchtheories would necessarily have vanishing one-loop /3 functions[178].This was strongly reminiscentofthe vanishing one-loop /3 function in N > 2 super Yang—Mills theories, and indeed both weresubsequentlyexplainedby the device of spin—momentsum rules [40, 231]. We shall return to this inchapter14.

12.2. Conjectureson the seven-sphere

Shortly after thede Wit—Nicolai theory was constructed,Duff andPope[21,29,40, 45] observedthatd = 11 supergravityadmittedvacuumsolutionsof the form AdSx S7, and that since 57 has isometrygroupS0(8)andadmits8 Killing spinors,thisgives rise via the Kaluza—Klein mechanismto an effectivefour-dimensional theory with local S0(8)invarianceandN = 8 supersymmetry.As describedin section7.1, it followed automaticallythat this d = 4 theory describedthe interactionof one masslessN = 8supermultipletof spins[2, ~, 1, ~, 0~,0] in SO(8)reps(1, 8~,28, 56~,35w, 35~)with an infinite tower ofmassiveN = 8 supermultiplets.It alsofollowed that in this N = 8 vacuumA was proportional to — Ge2with a calculable,but at the time not yet calculated,coefficient.

All this led to the conjecture[21,29, 40, 45] that the dynamicsof the masslessN = 8 supermultipletof this Kaluza—Klein theory was nothing but the N 8 theory of de Wit and Nicolai. Let us call thisconjecture I. The arrival of the Englertsolution [43] led to the observation[45] that the round 57 andthe round 57 with torsion were merely differentphasesof the samefour-dimensionaltheory,and thattorsioncorrespondedto a non-zeroVEV for the 35~pseudoscalars.This led to conjectureII: Justas thesymmetricN = 8 extremumof the de Wit—Nicolai effectivepotentialin d = 4 correspondsto the roundS7 without torsion, so non-symmetricextremacorrespondto other solutionsof the d = 11 theory withdifferentgeometrybut the same57 topology.Moreover,non-zeroVEVs for scalarsandpseudoscalarsind = 4 correspondto a deformation of and/or torsion on S7 in d = 11 [21, 45]. (Of course, thedeformation or torsion in question would have to correspondto the 35~scalarsand/or the 35~pseudoscalarsin the n = 0 level of table8. This means,in particular,that the squashed57 of chapter8would not admit any such de Wit—Nicolai interpretation,sincehereit is the 300 in the n = 2 level whichacquiresthe non-zeroVEV. This latterfact was not appreciateduntil later [53], however).

While some held the truth of theseconjecturesto be self-evident,others raisedobjections*. Theirverification would require a completenon-linear analysisof the d = 4 equationsof motion obtained

* Indeed, one physicist is on record as stating that if proved true, he would eat his hat.

ML Duff etaL, Kaluza—Kleinsupergravity 103

from d = 11 and, to date, only partial results are known mainly becauseof the complicatednon-polynomial dependenceon the scalarsandpseudoscalars*.Let us thereforelist someof theseobjectionsandseehowfar theyhavebeenovercometo date.First of all, implicit in conjectureI is the assumptionthat it is, in fact, possibleto truncatethe full Kaluza—Klein to its masslesssectorin a consistentfashion.While this is trivially possibleat the linearized level, one may entertaindoubtsabout the interactingtheory sinceproductsof Fouriermodesappearingin themasslesssectormightcouple into thoseof themassivesector.As it turns out, suchfearsare in generalwell founded.It is a remarkablepropertyof 57that, with the exception of T7, it provides the only examples of M

7 compactificationsof d = 11supergravityto admit such consistenttruncations.This is the subjectof chapter13.

A secondobjectionto conjectureI concernedthe cosmologicalconstant.A necessarycondition foragreementis that the spacetimecosmologicalconstantA in theround 57 vacuummustbe relatedto theSO(8)gaugecouplingconstante by the sameformula 4irGA = —3e

2 as obtainsin the N = 8 phaseofthe de Wit—Nicolai theory.Weinberg [157] has shown how the coupling constantmay be calculatedstartingfrom pureEinstein gravity in higher dimensions.Applied to the round S7 of radius m1, theformulagives e2 = 64irGm2which, combinedwith A = —12m2from (3.1.9),yields 16i~GA= —3e2 whichdisagreesby afactor of 4with the deWit—Nicolai value:an apparentdisasterfor conjectureI. However,it turns out that the presenceof the AMNP field in d = 11 supergravityleads to a modification ofWeinberg’spuregravity calculation[64].As describedin sections12.3 and 13.2,whenthis is takenintoaccountit changesthe relation betweene2 and m2 to e2 = 16ITGm2, which yields precisely the deWit—Nicolai relation.

A third objectionto conjectureI, closely relatedto the othertwo, involved the masslessansatzandthe supersymmetrytransformationrules. One importantconsistencycheck, pointed out in [45], waswhether the d = 11 supersymmetrytransformationrules consistentlyyielded the correct d = 4 trans-formationruleswhenthe ansatzfor the masslesssupermultipletwas substitutedin. This masslessansatzwas known in full at the linearizedlevel [21,48] and is given in section 12.3, soone could checkthetransformationrules to the same order of approximation. In fact as we shall see in section 12.3,everything works out well [67,69] despite earlier claims that there was a mismatch in the y dependence.

A surprisingresult, not anticipatedat the timeconjectureI was made,is the appearanceof the294wmassless0~in the round S7 spectrumnot belonging to the masslessN = 8 supermultiplet[194].Theinclusion of these fields in the masslessansatzwould clearly violate supersymmetryand lead toinconsistencies.When discussingconjectureI it is alwaysimportant thereforeto distinguishthe ansatzfor the masslesssupermultipletfrom the ansatzfor the masslesssector.

At the time conjectureII was made,only one non-symmetricextremumof de Wit—Nicolai theorywas known [60]. This was Warner’s SO(3)x SO(3) extremumwith N = 0. Since then Warnerhasdiscoveredmanymoreandhasclassifiedall thosewhich haveat leastSU(3) symmetry(seesection12.4).

Until recently, these results posed the gravestthreatto conjectureII. Foralthoughall comparisonsofthe SO(7)~extremumin d = 4 with the Englert solution in d = 11 proved positive [61, 73], the othernon-symmetricextremadefied all attemptsto yield to a d = 11 origin. It was even proved that noFreund—Rubinsolutionwith orwithout “torsion” couldhaveG

2 symmetryunlessit was theroundS7 [68].

Theresolutionof thisproblemis thesubjectof section12.4andturnsoutto havefar-reachingconsequencesfor Kaluza—Klein.

12.3. Linearizedmasslessansatzandtransformationrules

The ansatz for the massless supermultiplet at the linearized level can be obtained by retaining only

* See,however, the section on recent developments.

104 MJ. Duff eta!., Kaluza—Kleinsupergravi!y

the lowest modes in the Fourier expansionof section 5.1, with the harmonicsspecializedto 57~

Historically, however,the ansatzwasfirst given in a differentgauge,which we shalluse for the purposesof thissection.Making the field redefinitions

I.’ —1. .i! L.qf&MN — r&MN 2gMNnq ,

= hIM — ~f’MF~frq, (12.3.2)

the ansatzis [21,48]

h~~(x,y) = h,~,,(x), (12.3.3)

~ y) = ~ (12.3.4)

h~~(x,y)=~ (12.3.5)

~ y) = ~ — ~h7), (12.3.6)

~ y) = (24m)1e~~~,V~Vqht, (12.3.7)

f,~1*q(x,y) = —(2m)

1~V”V[~h~]”, (12.3.8)

4~ I \ — 1 ~1 DIJKL I \ IJKL

J,jnpqt,X,Y)211,jJ. !~X)flnpq ,

fmnpq(X, y) = 2mPIJKL(x)~mnhI(L, (12.3.10)

i/i,~(x,y)=t/I~’(x)~’, (12.3.11)

t/4,,(x, y) XIJK(X)h7mI~I~. (12.3.12)

The y-dependenttensors,~aredefinedin termsof the Killing spinorsi~,satisfying (4.2.4), as:

II — - I r J IJKL = [IJ KL]Tlm — 77 mfl , Tlm,s 77m 7ln

(12.3.13)IJKL — [II KL] IJKL — [If KLI

7lmnp — 77[mn 77p) , flmnpq — 17[mn 1Jpqj

wherewe alsodefine 77mn~~= 77~1’mn71~The vector-spinor~ is definedby

= ~,7[IflJK1 (12.3.14)

The vectors77m’~are in fact the 28 Killing vectors,77mnpL~~are Yano Killing tensors,and flm~~1eare

Killing vector-spinors,as definedin section4.3.The expansionof the supersymmetryparameter,which we shall modify later, will for now be taken

to be

e(x,y) e’(x)q’. (12.3.15)

MJ. Duff etaL, Kaluza—K!einsupergravity 105

The ideanowis to substitutetheseansãtzeinto the d = 11 supersymmetrytransformationrules (2.2.23),(2.2.24) and(2.2.25),in orderto verify to first order in fields that four-dimensionalsupersymmetryrulesare obtainable,i.e., that the y-dependenceof the left- andright-handsidesof eachof theseequationsmatches.Following [67], we start from (2.2.23) by first converting it into a transformationlaw for thelinearizedmetric

3hMN = —2i~f~h’N), (12.3.16)

wheret$~)is obtainedfrom tA using the backgroundvielbein. The calculationshereare straightfor-ward, and indeed yield equationsin which the y-dependencematches.There are three cases toconsider,correspondingto MN = 1av, ~n and mn, andso dropping the y-dependencethesegive, aftersomealgebra,

= —2ië’y(,~c)’, = —21E”75t/i/1— 2iëKy~XIat, ~SIJKL =

(12.3.17)

wherein the final equation[IJKL]+ denotesthe self-dualprojectionof [IJKL]. This projectionfollowsfrom the duality propertyof 77mn” definedin (12.3.13).

We now convert(2.2.25) into a transformationlaw for FMNPO andso after linearizationwe obtain

= 6b[M(et0)NP~Q]). (12.3.18)

On substitutingthe ansätzeinto (12.3.18),oneagainfinds after somestraightforwardbut tediousalgebrathat the y-dependencematches.Therearefive casesto consider,correspondingto the variouspossiblecombinationsof spacetimeandinternalindices.All exceptthe casesof 3 or 4internalindicesreproducethe previouslyobtainedtransformationlaws (12.3.17). The remaining casesproducethe last bosonictransformationlaw

~pIJKL = —2ë”x~”~1-. (12.3.19)

We now turn to the fermion transformationlaw (2.2.24). This is the equationwhich at first sightappearsto give rise to a mismatchof the y-dependence,and hencean inconsistentresult, andsohereweshalldescribethe calculationin greaterdetail. Linearizing(2.2.24), we obtain

= DME — ~(2eANb[MVNIB — eA eB°eMDPVQD)FE

I~I 14 A

11j(0) NPQR ~ Q~ N j(0)POR\f— t,lF 1~P)~1 M ~ OUM 1 JJNPQRE

— j~j14411j(1) NPQR ~ ~ N j(1)PQR\~’(O)~ ~ M ~ °~‘M I j~ NPQRE, .

where VMA is the fluctuation around the backgroundvielbein eMA correspondingto the metricfluctuation hMN. It is convenientto imposeLorentz gaugecondition Vm” = 0. The matricesF~)MI~~I~OR

andF(~oRare the terms linear in VMA which result from converting from tangentspaceto worldindices on the Dirac matricesFABCDE andpCDE~

We now substitutethe ansãtzeinto (12.3.20)andconsiderfirst the caseM = m. On generalgroundswe expect the variation &4!m to involve terms containingB,~”,V1~B~1Ut,9~S,o~P,S and P. Detailed

106 Mi Duff eta!., Kaluza—K!einsupergravity

calculation shows that the term in B,. vanishes,while aftera Fierz transformationthe y-dependenceofthe term in V1,.B~1is seento matchwith the left-handside of (12.3.20). For the scalarandpseudoscalarterms, we requirethe Fierz transformation

144(tY 7im~M] + ~Fm3I[J7fKLMI) = Fm~T1’77npq~’~M+ 6F~77h7)mflpll~I~~M+ 18F~~TiI77mn~M

— 2Pm7)17! KLMgP(l + Fmr~~jI

77~p[JK77qLMI g’~~_2771~ IJK 77LM] g~*P

(12.3.21)

wherewe havedroppedthe “°“ superscripton the d = 7 F matrices.It is crucial to note that the firsttwo termson the right-handsideof (12.3.21)areanti-self-dualin JKLM, sincethey areassociatedwiththe 35~of three-formsof the pseudoscalaransatz,while the remainingterms are self-dual in JKLM,sincetheyareassociatedwith the 35~of Killing tensorsof the scalaransatz.Thetermsin &l/Im involvingt9,.S and t9,.P arepreciselythe self-dualand anti-self-dualpartsof (12.3.21),respectively,andhencethey-dependenceof thesetermsmatcheswith SI//rn; theyoccur in the combination 9~.(S+ iy5P).

Repeatingthe procedurefor the S andP termsin SI/im, one runsinto an apparentinconsistency.Therelevanttermsin the transformationrule are

+ ~FrnyIIJK) = (m/72)S~~~~MeI(_5Fm1*~1 77~~[fK1)qLM] gn~— 2Fmll’ 77~,(JJKLMgP(?

— 18F” 771 flmn~LM) + (im/72)P~LM Y5E’(8Fm”~ 711 77JKLM

— 12F~~1mf(~~M). (12.3.22)

Using (12.3.21),one can now showthat the y-dependenceon eachsideof (12.3.22)doesnot match.Toremedythis, we note that the ansatz(12.3.15) for r(x, y) implies that 15mg = 0, but t~atthe ansatzcanbe modified by the addition of 5- and P-dependentterms in such a mannerthat DmE now producespreciselythe requiredextratermsin (12.3.22)in order that the y-dependencecoincideswith that of theFierz equation(12.3.21).The modified ansatzis

e(x, y) = [1+ ~

5JKLM (_277rnnJKLM gmfl + 377mn~”’

77~LM1g”Fm)

+ (i/72)y5PhI~L

M( 37~TrnnpJKLMFm°1’)]E1 (x)77’. (12.3.23)

Note that this doesnot upsetany of the previouslyderivedlinearizedtransformationrules,sincethere~alwaysappearsin termsalreadylinear in fields. Summarizing,the transformationrule for St/fm gives,intermsof F,J1 2V

1,.B~11’,

SXIIK = ~F,.’y~”s’~~— 2y5ô,.(S’~”+ iy5PUu1~L)y~eL— 4m(SH~~~~+ iy5PhJM~)rL. (12.3.24)

Finally, substitutingthe ansãtzefor the masslessfields, andthe modified ansatz(12.3.23) for e(x,y),

into the transformationrule for SI/i,., one finds that the y-dependencematcheson each side of theequation,and

St/i,.’ = — 2mB,!’e3— [~(1— y

5)F~J— ~(1+ y5)F~’]ye~, (12.3.25)

whereF,.~,.= ~(F,.,. ±*F,~,,)and5F = (i/2)e,.~,,.~F’~.The derivative15,. is the anti-de Sitter covariant

Mi Duffeta!., Kaluza—K!einsupergravity 107

derivativeLI,. + my,.y5including termsup to first orderin fluctuations.The transformationrulesmust,of course,be S0(8)gaugecovariant,anddefining an S0(8)gauge-covariantderivative 0,. by

= D,.e’ — eB,.” e’, (12.3.26)

then (12.3.25)maybe written as

St/i,.’ = L5,.~’ — [~(1— y5)F,.~”— ~(1+ y5)F,.;’~]e~, (12.3.27)

providedthe SO(8) gaugecouplingconstante is chosento be

e = 2m. (12.3.28)

Herewe havedefinedO,. in thesamemanneras15,., i.e. O,. = D,. + my,.y5.(As we shall see in section13.2, the normalizationchosenfor the masslessansatzof this sectionis such that 4irG = 1, whereG isthe d = 4 Newton’sconstant.This implies e

2 = l6irGm2, i.e. 4rrGA = —3e2 as requiredfor consistencywith conjectureI).

It is important to check that the redefinitionof s(x,y) in (12.3.23)doesnot upsetany of our previousresults.In particular,the criterion for unbrokensupersymmetry,(~‘I’M) = 0, given in chapter4, remainsunaltered.This is because(S) (P) = 0 in the roundS7 background.

Thus we see that the truncationof the d = 11 theory to includejust the masslesssupermultipletisindeed consistent with the transformation rules at this order of approximation. Moreover, to the sameorderof approximation,the d = 4 transformationrulescoincidewith thoseof thede Wit—Nicolai theoryas maybe seenafter sometrivial rescalingsof the fields. In particular,note the SO(8)covariantizationof the derivative in (12.3.27) and the appearancethroughoutof the combinationof fields (S+ iy

5P).Whenoneconsidersthe verydifferentd = 11 origins of S andP1 thislatterfact is truly remarkableandissuggestive of some deeper connection between gMN and AMNP in d = 11 which may explain the SU(8)symmetry in d = 4.

One may wonder how the linearized ansatz can give information about coupling constants andcovariant derivatives. The answer is that, although we worked to only linear order in fields, thetransformationrules involve products of fields and supersymmetryparameters.Hence they yieldinformationaboutcubic termsin the Lagrangian.Thus consistencyof the abovetransformationrules isalreadyprobing the non-linearstructure.It is to the non-lineartheory that we now turn.

12.4. New S7 solutionsand Warner’sextrema

Warner[71] hasnow completedthe list of all the extremaof the de Wit—Nicolai potential in d = 4which include a breakingof SO(8) to somesmaller groupcontainingSU(3). It is not yet known howmany otherextremathereare, but the original exampleof SO(3)x SO(3) is known as mentionedinsection12.2. The knownextremaareshownin table13.

Let usnow reexamineconjectureII in the light of theseextrema.All comparisonswhich havebeenmade to date regarding gauge symmetries, supersymmetries,cosmologicalconstants,and partial massspectraare consistent with the identification of: E

0 with the Duff—Pope S7 solution of section 7.1 [21,29,

40, 45]; E1 with the EnglertS

7solutionof section10.2[43]; andE4 with the P~pe—WarnerS

7 solutionofsection 10.3 [88].In all cases,non-zeroVEVs for the pseudoscalarsP in d = 4 correspondto non-zero

108 Mi Duff eta!., Kaluza—K!einsupergravily

Table 13Known extrema of the scalar potential

Extremum Gauge symmetry Supersymmetry Non-zero VEVs

F0 S0(8) N=8 —

E1 SO(7)~ N=0 PE2 S0(7), N=0 SE3 G2 N=1 S,PE4 SU(4) N=0 PE5 SU(3) x U(1) N = 2 S,P

S0(3)x SO(3) N = 0 S,P

Frnn,~in d = 11. The fact that (5) = 0 does not necessarilyimply the round S7 geometry;e.g. the

seven-sphereis “stretched” in the Pope—Warnersolution.This reflectsthe fact that the pseudoscalarswill enterat the non-linearlevel in the ansatzfor gmn(x,y) [95].

Much more mysteriousfrom the point of view of conjectureII are those extremawith (S)� 0,namely E

2, E3, E5 and E6. Indeed,it can be proved that with the usual productansatzof chapter3,there is no Freund—Rubin type solution, with or without “torsion”, capableof reproducing theseextrema[68].This led to claims that conjectureII was wrong.It was realizedin [77]that the resolutionto theproblemwas the “warp factor” f(y) [274,68, 113, 232, 95] discussedin section3.1, which we haveignoreduntil now. In otherwords,oneshouldgeneralizetheground-stateansatzfor themetricgMN to be

(j~(x,y)) = ~ ]. (12.4.1)0

Onecan nowfind an S7 solution with S0(7)~symmetryand N = 0 which can be identified with the E

2extremum[77].In [95],it was shownthat this de Wit—Nicolai solutioncan bewritten in the simple form

f(y) = a (1 — ~ sin2 a)21’3, k

m~dytm dy” = bft”2 sin2ad11~+ cfda2 (12.4.2)

wheredQ~is the metric on the roundS6, anda is the seventhcoordinatewith 0 sa � ir, anda, b andcareconstants.The E

3 extremumcan be explainedin a similar way [95], as can presumablythe E5 andE6. Clearly Fmnpq� 0 no longer implies the breakingof all supersymmetrieswhen f(y) � 1; indeedunbrokensupersymmetrynow seemsto require it. It is also interesting to note that all the knownsolutionswith f(y) � 1 correspondto inhomogeneousdeformationsof S

7, i.e., theyarenot cosetspaces.This is becauseany extremumwith (5) � 0 correspondsto a deformationhmn of S7 inducedby the 35~Killing tensors.As explainedin chapter8, the tracehmm is non-constantfor the35~(in contrastto the 1and300) andhencethe deformedS7 is inhomogeneous.

As for stability, E0, E3 andE5 areautomaticallystable;E1 andE2 areknown to be unstable;the rest

havenot yet beenanalyzed.

Mi Duff eta!., Ka!uza—K!einsupergravity 109

13. Consistencyof the Kaluza—Klein ansatz

13.1. GenericKaluza—Kleintheories

In attemptingto establishthe truth of conjecturesI andII in chapter12, we areforced to re-examinesomeof the basic assumptionsof Kaluza—Klein theoriesin general. Implicit in much of the Kaluza—Klein literature is the assumptionthat it is in fact possible to truncate the full theory to its masslesssectorin a consistentfashion.This is the subjectof [78,96] andof this chapter.

Let us begin by recalling that in modern approachesto Kaluza—Klein theories, the extra (k)dimensionsaretreatedas physicalandarenot to be regardedmerely as a mathematicaldevice. In thisframework, therefore, it is essentialthat at every stage in the derivation of the effective four-dimensionalfield theory one maintainsconsistencywith the higher-dimensionalfield equations.Toderive this effective theory one selects the groundstateof (spacetime)x (compactmanifold Mk) andperformsa generalizedFourierexpansionof all the fields in terms of harmonicson Mk. Providedoneretainsall the modesin this expansion(i.e., all massivestates)then no such problemsof inconsistencycan arise.Moreover,it is well-known that if Mk hasisometrygroup0 thenthe theory includesmasslessYang—Mills bosonswith gaugegroup0 (assuming,aswe shall for the time being, that anyothermatterfields non-zeroin the groundstatearesingletsunderG).

In practice, however,one is often interestedin extracting an effective “low-energy” theory bydiscardingall but a finite numberof statesincludingthe masslessgraviton, themasslessgaugefields, andothermatterfieldswhich areusually(but not necessarily)massless.For historicalreasonsthisprocedureis often called the “Kaluza—Klein ansatz”.Despitethe name,this should not be an ad hoc procedurebut should correspondto retaining only the appropriateFourier modes, for example, the zeroeigenvaluemodesfor the massoperatorsin the caseof the masslessparticles. It is generallybelievedthat the correctansatzfor the metric tensor~ y) is

g,.~(x,y) = g,.~(x)+ A,.i(x)ApI(x)Kmi(y)KnJ(y)~~~(y)

g,.~(x,y) = A,.i(x)Kmi(y)~*~~(y), gmn(x, y) = ~mn(Y), (13.1.1)

where the coordinatesx~(p. = 1,. . . , 4) refer to spacetimeand ym (m = 1,.. . , k) to the extradimensionsand~ is the metric on Mk. The quantitiesKmi(y) arethe Killing vectorscorrespond-ing to the isometriesof this metric and i runsover the dimensionof the isometrygroup G. The claimthat this is the correct ansatzis basedon the observationthat substituting(13.1.1) into the higherdimensionalEinstein action and integratingover y, oneobtainsthe four-dimensionalEinstein—Yang—Mills actionwith metric g,.,,(x) andgaugepotentialA,.1(x).

But, as we havealreadyemphasized,the correct Kaluza—Klein ansatzmust be consistentwith thehigher-dimensionalfield equationsand,as weshall nowdemonstrate,(13.1.1)doesnot in generalsatisfythis criterion. As hasalready been stressedin chapter1 oneobvious sourceof inconsistencyis theneglect of scalarfields in (13.1.1). For example,setting ~ = 1 in the d = 5 pure gravity theory isinconsistentwith the 55 componentof the Einstein equationwhich would force F,.,.Fw*~to vanish.In this case,the remedyis simple: one includesthe single masslessscalarfield ~ via ~ = ~‘(x), thenaftera suitableWeyl rescalingthe 55 equationbecomes

Ll(log ço)—

110 Mi Duffet aL, Kaluza—K!einsupergravity

In this example,the processof restoring consistencystopshere;thereis no needto includeany highermassivemodesin theFourierexpansion.

The purposeof this section is to point out a new, and muchmore serious,sourceof inconsistencyarising from (13.1.1). To illustrate it, let usconsiderthe field equationsof pure gravity with a positivecosmologicalconstantA in d = 4+k dimensions

RMN—AgMN. (13.1.2)

This theory admitsa classicalground-statesolutionof (de Sitterspacetime)X (compactmanifold Mk).(The fact that spacetimeis de Sitter ratherthananti-deSitteror Minkowski spaceneednot concernusheresincewe areconcernedonly with the mathematicalconsistencyof theansatz).Substituting(13.1.1)into (13.1.2)usingappendixA we find the d = 4 Einsteinequation

— ~g,.,,R+ Ag,.~= ~(F,.~’FY” — ~ F~”)K~’Ku’, (13.1.3)

whereF,.~’is the Yang—Mills field strength.The inconsistencyis now apparent.The left-hand side of(13.1.3) is independentof y while the right-handside in generaldependson y via the Killing vectorcombinationKm~K”. For example,when Mk = S” with its SO(k+ 1) invariant metric,

K~1(y)Kmi(y)= 3” + Y’(y), (13.1.4)

where Y’~(y) is that harmonic of the scalarLaplacian with next to lowest non-vanishingeigenvalue2A (k + 1)/(k— 1) belongingto the k(k + 3)/2 dimensionalrepresentationof SO(k+ 1).

How can consistencybe restored?It is not difficult to seethat including scalarsin the ansatz(13.1.1)is not sufficient to cure the problemwith the gaugefield’s stresstensor.One might try including thenext-to-lightestmassivegraviton which involves the harmonic Y’(y) in an attempt to provide thecorrecty-dependenceon the left-handside of (13.1.3).However, it is well known that only with zero[233]or an infinite numberof massivegravitonsis thetheory consistent.In otherwords,onewouldendup havingto put backan infinite numberof massivestates.

If oneinsists on making a truncationto a finite numberof statesincluding gaugebosons,onemayalternativelyselect a subgroupG’ of 0 with Killing vectorsKm’, where i’ runs over the dimensionof0’, for which

Km” K”’~” = 5”. (13.1.5)

Oneobviousway to achievethis ariseswhenMk is itself the non-Abeliangroupmanifold 0’ which, withits bi-invariant metric, hasisometrygroup0 = 0’ x 0’. Then the left-invariant or right-invariant vectorfields are Killing vectorswhich separatelyobey (13.1.5),but takentogetherthe crosstermsgenerateay-dependentpiece as in (13.1.4). Contrary to many claims in the literature, therefore,the consistentKaluza—Klein ansatzfor groupmanifoldsyields the gaugebosonsonly of G’ and not 0’ x G’. The fullinvarianceof G’ x 0’ is restoredonly by including the massivemodes.For Abeliangroupmanifolds0’,the isometry group is only 0’ and all Killing vectors automaticallysatisfy (13.1.5). So for com-pactificationon the k-torusT”, the ansatzis alwaysconsistent.

Another way to achieve(13.1.5) arises when Mk is a principal bundlewith G’ as the fibre. Forexample, Mk = 54n±3 (n = positive integer) is an SU(2) bundleover HP”. In this casethereare two

Mi Duff etaL, Kaluza—K!einsupergravity 111

Einstein metrics: the round sphere with G = SO(4n+ 4) and the squashedsphere with 0 =

Sp(n+ 1) x SU(2) [211].In eithercase,the SU(2)Killing vectorson the fibre satisfy (13.1.5).Onceagain,the residualgaugegroupis muchsmallerthanthe isometrygroup.

Note that (13.1.5)implies the existenceof everywherenon-vanishingvectorfieldsand so (13.1.5) canneverbe satisfiedon spacesnot admitting such fields, i.e., on spaceswith non-zeroEuler numberx(odd-dimensionalspacesalwayshavex = 0). For example52n hasx = 2, andCP” hasx = n + 1. In thesecases,the Kaluza—Klein ansatzwould involve no gaugebosonsat all. Since(13.1.5)is a sum of k terms,it follows that for a generalspace,the numberof Killing vectorssatisfying(13.1.5) is lessthan or equalto k, the dimensionof thespace,with equality for groupmanifolds.

Of course,having curedthe inconsistencyof the ~z’ componentsof the Einsteinequation,onemustalso ensurethe consistencyof the remainingcomponents.For Rmn, this would involve in generaltheinclusionof scalarsin the ansatzfor ~mn~

13.2. d = 11 supergravityand theseven-sphere

The situationwe have describedso far changesradically when we turn our attention to d = 11supergravity.The reasonfor this differenceis the presenceof the three-indexgauge field AMNp inaddition to the metric ~ Recall that the bosonicfield equationsare

z~ 1’ iS_líii’ I~’PQR 1’ si’2\1’MN — 2gMNI’ — 3t,~’MPQRFN — 8gMNu ),

7 I’MPQR — — L M

1 .. M5PQR i~’— 576 ~ 1 Mi.. . M~

1 M

5. . . M8,

whereFMNPQ = 4a[MA N~J. It is this ~ field, ratherthanan explicit cosmologicalconstant,which isresponsiblefor the spontaneouscompactificationto d = 4. In particular,let us considerthe Freund—Rubin type compactificationsof chapter3, with groundstateAdSx M~obtainedfrom

= ~ ~ = ~mn(Y), ~ = ~ (13.2.3)

with all othercomponentsvanishing,where~g,.,.is themetric on AdS and~ the Einsteinmetric onM7 satisfying.kmn = 6m

2kmn.

The crucialobservationis that the standardKaluza—Klein ansatz(13.1.1) for the gaugebosonsmustnowbe augmentedby the additionalansatz[21,67, 78]

= —~m1e,.~,~F~”V[pKq’]. (13.2.4)

(In the caseof the round S7 this can be seen from (12.3.8).) Since we are concernedprimarily withpossible inconsistenciesin the gaugefield sector,we shall postponethe inclusion of the scalarfields.Substituting(13.1.1) and(13.2.4) into (13.2.1)we find the d = 4 Einstein equation[78]

— ~g,.,.R— 12m2g,.~= ~(F,.~’F,f’ — ~ F’~)(Km’ K” + ~m_2V~K~iVmKfh). (13.2.5)

Comparing(13.2.5)with (13.1.3)we seethat insteadof the consistencycondition (13.1.5)which permitsat most7 Killing vectorswe now have

112 Mi Duffel aL, Kaluza—K!einsupergravity

Km1’ K”’~” + ~m_2VrnKn~’VtmK”” = 5”-”, (13.2.6)

which permitsat most28 sinceit is the sum of 28 terms. Underwhat circumstancesis (13.2.6)satisfied?Let usbeginby examiningthe maximally symmetricroundS7 solutionof chapter7. SinceS7 admitsthemaximum of 8 unbrokensupersymmetries,i.e., 8 Killing spinors 1)~(I = 1,.. . , 8) satisfyingD~n’=(m/2)F~~’,we may form all 28 Killing vectorsof SO(8)via [21]

Krn” = ~‘Frnfl’ (13.2.7)

and, for convenience,we choosethe 17’s to be orthonormali~’r~’= 5”. Using the Fierz identity for

d = 7 commutingMajoranaspinors[21] of appendixB,= ~(x~ — FmX~Fmt/’— ~F~flXçf’FmflI/~+ ~ (13.2.8)

one may verify that condition (13.2.6)is indeedsatisfiedfor all 28 Killing vectorsof SO(8)andthat theansätze(13.1.1)and (13.2.4) are thereforeconsistent.Of courseone must also include the scalarsandpseudoscalarsin order to verify complete consistency,not only of the p.,’ componentsof (13.2.1)but also the other componentsof (13.2.1) and all componentsof (13.2.2). One must also check thefermion ansatz.On the round57 the total ansatzfor the masslessN = 8 supermultipletof spins(2, ~, 1,~, 0~,0-) is not yet known to all ordersowing to the complicatednon-polynomialdependenceon thescalarsandpseudoscalars*.However, to the order to which it is known, everythingis consistent.Thisseemsto be guaranteedby the fact that the ansatzfor all spinscan be expressedin termsof the Killingspinors i~’ [21].

Indeed,wheneverM7 admitsN � 2 Killing spinors one may always find Killing vectors satisfying

(13.2.6), namely, those constructedin (13.2.7).For such compactificationsthe total gaugegroup G isalways of the form 0 = SO(N)x K for some group K [30]. The construction (13.2.7) singles out0’ = SO(N) as the subgroupsurviving in the consistentansatz.These are the gaugebosonsof theN-extendedgravity supermultiplet.

In fact the round S7 seemsto be the only M

7 which exploits the extrafreedompermittedby (13.2.6)as comparedwith (13.1.5).To seethis, we first rewrite (13.2.6)as

(4m2)’(Ll+ 16m2)Krn” Ktm.~’= 5k’’. (13.2.9)

Thus wheneverM7 admits Killing vectorsKm” satisfying(13.1.5),then~Km~’will also satisfy (13.2.9). In

order to see whether there are any more Killing vectorssatisfying (13.2.9) but not (13.1.5), let usobservethat in the caseof a generalcosetspacethe appropriatelynormalizedKilling vectorssatisfy

Krn” K”’~” = !~i~i’+ pl’J’(y), (13.2.10)

where tj”(y) is tracefree.The extra freedom of (13.2.9) over (13.1.5) is that s~Y”(y)can now be

non-zero,providedthat it satisfies

= 16m2~”. (13.2.11)

See,however,thesectionon recentdevelopments.

Mi Duff eta!., Ka!uza—Kleinsupergravity 113

Moreover,acase-by-caseanalysisof all homogeneousM7’s revealsthat, with the exceptionof theroundnoneof thempermits eigenfunctionsof the Laplacianin the right representationandwith theright

eigenvaluesto satisfy (13.2.11). Interestingly enough, condition (13.2.11) implies the existenceofmasslessscalarsin theO~”tower of table5. In the caseof the round57, theseare just the 35~masslessscalarsof the N = 8 supermultiplet.None of the other M7’s has masslessscalarsof this kind. Thesurviving gaugegroups0’ compatiblewith the consistentansatzarelisted in table 14 for sometypicalM7 spaces.

We haveseenthat all gaugebosonsappearingin the consistentansatzfor pure gravity in d = 11 willalsoappearin the consistentansatzfor supergravitybut with a normalizationdiffering by afactor of 2.Weremarkincidentallythat thisis responsiblefor themodification of Weinberg’s[157]formularelatingthe gaugecouplingconstantto the radiusof the extradimensions.Onefinds [64]

e2(supergravity)= ~e2(puregravity). (13.2.12)

Note that by integrating (13.1.3) and (13.2.5) over y, one may verify that Weinberg’s formula and(13.2.12) remain valid for all the gaugebosonsof 0 and not merely thoseof 0’. Note also that theansatz(13.1.1) togetherwith the normalizationof the Killing vectorschosenin (13.2.6) establishestheconvention4irG = 1 in the Einstein equation(13.2.5). This resultwas usedin section12.3 in derivingtherelatione2 = l6irGm2 on the roundS7, a resultwhichcould alsobeobtainedfrom (13.2.12)withoutreferenceto the transformationrules.To obtain the coupling constantdirectly, one can calculatetheYang—Mills structureconstantsfrom the algebraof the Killing vectorshaving onceestablishedtheirnormalization.

Table 14Gaugegroupssurviving in theconsistentmasslessansatz

M7 G G’

RoundS7 SO(8) SO(8)

SquashedS7 SO(5)X SU(2) SU(2)M(m, n) SU(3)x SU(2)x U(1) U(1)N(k, 1) SU(3) x U(1) U(1)N(1, 1) SU(3)x SU(2) SU(2)Q(p,q, r) [SU(2)l3X U(1) U(1)T7 [U(1)l7 [U(1)]7

13.3. Significanceof consistency

We concludethis chapterwith somecommentson the significanceof Kaluza—Klein consistency.(1) The consistencycondition (13.2.6) was derivedunder the assumptionof Freund—Rubincom-

pactificationsgiven by (13.2.3), but we might also considermore generalAdS solutionswith (gm,.,,) =

f(y)g,.~(x)and/or~ ~ 0. We havelittle to sayaboutthe consistencyof theKaluza—Kleinansatzforgeneralsolutionsof thiskind but we can makesomeplausibleconjecturesin thecaseof thosewhich aretopologically 57~These conjecturesare relatedto conjecturesI and II of chapter12. It seemshighlyplausiblethatjust asthe roundS7 ansatzfor the masslessfieldsof the deWit—Nicolai theory wasalreadycompletelyconsistentwithout any reductionof the vacuumsymmetryG, so the ansätzefor theotherS7

114 Mi Duff eta!., Kaluza—Kleinsupergravity

vacua correspondingto table 13 will also be completelyconsistentprovided we retain the samedeWit—Nicolai fields, someof which will now be massive.Indeed,it seemsthat S7 andd = 11 supergravitygo hand-in-hand.

(2) For a genericKaluza—Klein theory, it would seemthat our consistencyconditionis equivalenttothe requirementthat the entire Kaluza—Klein ansatz(and not merely the ground-statemetric) is leftinvariant under a transitively acting subgroupK of the isometry group. This is a version of what issometimescalled “K-invariance” [234—242]of the ansatz.For example,thiswould yield only the gaugebosonsof 0’ in the caseof a groupmanifold 0’ andonly the gaugebosonsof SU(2) in the caseS4”~3.The reason for this equivalenceis that the consistencycondition amountsto requiring that theLagrangianobtainedby substitutingthe ansatzinto the higher-dimensionalLagrangianbeindependentof y. Although thisis guaranteedby “K-invariance”, it is clearly too stronga condition as the exampleof S7 andd = 11 supergravityillustrates.(The smallestsubgroupof SO(8)which actstransitivelyon S7 isSO(S)andso“K-invariance” wouldyield only the gaugebosonsof SU(2)).

(3) In summary,we haveseenthat the standardKaluza—Klein ansatzfor masslessgaugebosonsis ingeneralinconsistentwith the higher-dimensionalfield equations.Consistencycan be restoredeither byputting backthe infinite towerof massivestatesor elseby reducingthegaugegroupGto somesubgroup0’. Exceptionsto this rule are provided by certain 57 compactificationsof d = 11 supergravity. Aninterestingquestion,to which we do not know the answer,is whetherthere areother Kaluza—Kleintheorieswith consistenttruncationsinvolving the full 0 symmetry.We emphasizethat we are notdisputingthe Yang—Mills contentof the completeKaluza—Klein theory but only the ability to truncateconsistentlywhile retainingthe full gaugesymmetry.

So far we havebeen concernedwith mathematicalconsistency.Does inconsistencymatter from aphysicalpoint of view? Here it is importantto distinguishbetweenKaluza—Klein theorieswhich predicta Minkowski spacetimeground stateand thosewhich predict AdS with a Planck-sizedcosmologicalconstant.In theformer casethe answeris probably no if we areinterestedin energiessmall comparedwith the compactificationscale,sincethe inconsistencywill only be relevantfor operatorsof dimension>4*. (We aregrateful to E. Wittenand S.Weinbergfor pointing thisout.) In the caseof a cosmologicalconstantA e2/G (typicalof d = 11 supergravityandall thoseKaluza—Klein theoriesfor which A is notfine-tunedto zero), then thisis no longer the case.For example,dimension-sixoperatorslike GRF,.,~F”may be converted to dimension-fouroperatorslike e2F,,~F~”on using the field equations Re2/G + . (This observationwas also relevantwhen computing the /3(e) function in gaugedsuper-gravity andits relationto the renormalizationof GA [178].)In this case,of course,it is no longer clearthat the masslesssector is in anycasea good approximationto the full theory.

In our opinion, until such time as the cosmologicalconstantproblem in Kaluza—Klein theorieshasbeensolved(asopposedto beingfine-tunedaway),the physicalsignificanceof the Kaluza—Kleinansatzremainsobscure.

14. The quantum theory

14.1. Divergencesand anomalies:the Chern—Simonsterm

Sofar we haveconfinedourattentionmainly to classicalconsiderationsandin this chapterwe discusssomeaspectsof quantizationin Kaluza—Klein theories.The first to spring to mind is that of ultraviolet

* See,however,the sectionon recentdevelopments.

MJ. Duff etaL, Ka!uza—K!einsupergravity 115

divergences[40].In commonwith four-dimensionalquantum-gravitytheories,Kaluza—Klein theoriesarepower-countingnon-renormalizable.In d dimensions,the superficial degreeof divergenceD of anL-loop Feynmandiagramis given by

D=(d—2)L+2. (14.1.1)

Thus D increaseswith increasingloop order for all d > 2: the disasterof non-remormalizability.Notethat in this respect there is no compelling reason for confining one’s attention to d = 4, unlikenon-gravitationaltheorieslike Yang—Mills. Experiencewith supersymmetry,of course,suggestsanalternativepossibility, namely that on-shell S-matrix elementsmay be actually finite order by orderin perturbationtheory.This seemsjust as likely or unlikely for field theoriesin d >4 as in d = 4.

As far as d = 11 supergravityis concerned,onewouldhaveto go beyondthe one-looplevel to detecta divergencesince, in commonwith most other odd-dimensionaltheoriesof gravity andmatter,it istrivially finite at odd-looporder [29].Possibleinvariantswhich might serveas countertermsdo appearat two loops and beyond, however[29]. Since two-loop calculationsin supergravityare as yet tootechnically difficult to perform the actual presenceof thesecountertermsremainsopen. There isevidencethat d = 10 superstringsmaybe finite to all orders,but a rigorousproofis still lacking. Shouldthisprove to be the caseand should d = 11 supergravityprove infinite, thiswould be a clearsign thatsuperstringswerebetter,irrespectiveof any low-energyphenomenologicalconsiderations.

Odd-dimensionaltheories,like d = 11 supergravity,are also free of gravitational trace anomalies[244—246]at odd-looporder [29,243] and, of course,sincechiral fermionsdo not exist for odd d, arealso free of chiral anomaliesin thedivergenceof the gaugecurrentsand the energy-momentumtensor[247].The problemof chiral anomaliesin evendimensionsis a fascinatingonewhich hasattractedagood deal of attentionlately [29,243, 247—250]especiallytheir calculationin certaind = 11 superstringtheories[15, 247]. These considerationsare vital if one wishesto obtain elementarychiral fermionsdirectly from the extra dimensions [9]. There is, however,a curious discrete-symmetryanomaly in4k — 1 dimensionsandin particularin d = 11 supergravity,wherebya finite topologicalChern—Simonsterm is induced at one loop [247, 251]. Its gravitational part is a linear combination ofR A R n R A R A R n w andothertermsof thesamedimensionformedfrom thed = 11 spinconnectionWM and curvatureR~.~(w).Its supersymmetriccompletion is not yet known but there is nocompelling reasonto doubt its existence.Note that such a term violates the discrete symmetry of(2.2.29).No newcoupling constantsareintroducedby includingsucha termin thetree-levelLagrangianof (2.2.1) becausethe relative coefficients of each term are fixed by the topology and the overallcoefficientis quantized[251].

We leaveit asan exercisefor the readerto rewrite thisPhysicsReportwith the d = 11 Lagrangiansomodified! Lots of questionsspringto mind, of course.Would it affect the chirality problem?Would itaffect the cosmologicalconstantproblem?Would it affect the ultraviolet divergenceproblem?

14.2. Casimir energies

It was first pointedout by Unwin [252] andsubsequentlydiscussedby Duff andToms [29,243] andby Pollard[253]that the compactextradimensionsof a Kaluza—Klein theory would necessarilygiveriseto an enormousquantumvacuumenergyvia the Casimireffect.

Following the work of Appeiquist andChodos[159,254, 255] on the quantizationof the d = 5 puregravity theory of chapter1, Candelasand Weinberg[158] attemptedto exploit this vacuumenergy,

116 Mi Duff eta!., Ka!uza—K!einsupergravity

usingit as the driving forcefor a one-loopquantumcompactificationin d > 5 theorieswhich admittedno such compactificationat the tree level. They pointed out, moreover,that in such a self-consistentapproachto compactification,the size of the extradimensionswould now be a calculablepurenumbertimesthe Plancklength.Consequentlythe gaugecouplingconstantwould now becalculable!

A somewhatdifferentattitudeto Casimirenergieswas takenin [29,45, 243, 253]. Hereit was arguedthat Casimir energiesare bad and that the virtue of a Kaluza—Klein ground state with an unbrokensupersymmetrywould be that the bosonand fermioncontributionswould cancel.This can beproved inPoincarésupergravityquite easily [253] by generalizingZumino’svacuumenergyarguments[256,257],and so Ricci-flat compactificationswith N � 1 do yield vanishingCasimir energy. In AdS supersym-metry, the problemis moresubtle[45]eventhoughthe E = ~Q2 argumentsof chapter4would seemtoimply that supersymmetryalwaysyields zeroenergy.For someof the problemsinvolved see [258,266,267]. A lessstrong conjecturewas that the Casimirenergyon the N = 8 round S7 backgroundwouldvanish [45]. This was basedon the vanishing/3-functionresultsof the d = 4 gaugedsupergravity[178]using spin-momentsum rules [40, 231], and noting that the sum rules continuedto hold when themassiveN = 8 multiplets were included. This conjecturehas now indeed been verified using spin-momentsum rules [81,82].

Critics of the Kaluza—Klein programme(e.g. Coleman,private communication)havearguedthat thequantizationof Kaluza—Klein theoriesis inherently untrustworthy.As we understandit, one way toparaphrasetheseobjectionsis as follows. The low-energysectordependson the choiceof groundstate.But quantumgravitationaleffectsmay sodrasticallychangethe natureof this ground stateand hencethe low-energysector, that any finite order of perturbationtheory is hopelesslyunreliableeven forlow-energyphysics.This remindsusof the gaugehierarchyproblem in GUTSand, it seemsto us, hasthe sameresolution:supersymmetry.With an unbrokensupersymmetryof the vacuum, the quantumCasimir energieswill not be allowed to spoil the natureof the tree-levelvacuum. This is yet anotherreasonwhy we believeKaluza—Klein andsupersymmetrygo handin hand.

14.3. Fermioncondensatesand the cosmologicalconstant

As discussedin section 10.1, no purely bosoniccompactificationof d = 11 supergravitywith m � 0can avoidthe enormousPlanck-sizedcosmologicalconstantfor spacetime.Onepossiblesolutionto thisproblem,which alsoincidentallyinvolvesaparity breakdown,would be spin ~fermioncondensates,i.e.nonzeroVEVs for fermionbilinearssuchas

(,~(x)ysX(x))= constant� 0. (14.3.1)

Within the semi-classicalapproachdiscussedby Duff and Orzalesi[47], the requirementof vanishingcosmologicalconstantfor spacetimeimposesthe following conditionson the curvaturesand torsionsofd = 11 supergravity:

= 0, ~kmn((Z~)= 0, PMNPQ = 0, (14.3.2)

where

Wabc = Wabc + 5abc — Sacb + 5cab, Sa#,c(Y) = (çlia(x, y)t6ifr~(x,y)),

5MNP = 0, otherwise.(14.3.3)

Mi Duffci a!., Ka!uza—K!einsupergravily 117

In otherwords, theremust exist a Ricci-flat connectionin the extradimensionsbut wherethe torsionS,,,~(y) is built of fermionicbilinears.(Theso-called“torsion” of chapter10 built out of bosonicAmnpfieldswill not do the trick.) From(14.3.3)we seethat the requiredfermioncondensatesarepreciselyofthe spin ~parity-violating form (14.3.1) sincethe index of the gravitino field i/fM lies only in the extradimensionsandsinceTa = Y5Ia.

Onetrivial solution is to haveS~= 0 and.kmn(~) = 0, i.e., Ricci-flat solutionslike T~[37]or K3 x[23]. If, on the other hand, we requirekmn(~) � 0 as we must to get non-Abelianelementarygaugefields, thenfor mostgeometriesthe problemhasno solution.However,as discussedin section10.2, M7with at leastoneKilling spinor 17 gives riseto a Ricci-flat torsion

S~= ±m77f’abcl7. (14.3.4)

(Sincewe are not obtainingthe torsion from Ama,,, it doesnot matterif it satisfies(10.1.3)ratherthan

(10.2.3).)Thus if it were to turn out that in the true vacuum(l/farbt/Ic) = ~ (14.3.5)

thenthe true vacuumwould have

A 0. (14.3.6)

Anothersolutionwith S~= S[,,,,,~] would be to takethe parallelizingtorsion on the round 57, but thiswould be somethingof an overkill with .kmnpq(ó.~)= 0 whereasonly kmp(pi) 0 is requiredfor A = 0.See [259]for earlierattemptsto exploit the parallelizability of groupmanifolds.Of course,(Illatbi/Ic) ~5

not necessarilytotally antisymmetric,andso a non-absoluteparallelismwith S,,.,~� ~ might be morenatural [91].Seesection10.2. Note, however,that this is possibleonly for parallelizablemanifoldssuchas 57 or 55 X 52 [47].

Of course,we havesucceededonly in swappingthe problemof why A = 0 for why (çfr~tbl/I~)shouldbe the Ricci-flattening torsion. The points we are making are that (a) in d = 11 supergravitythecosmologicalconstantof spacetimeis calculable; a necessarycondition for explainingA = 0. TheorieswhereA maybe “tuned” can neverexplain this. (b) Thespecialpropertiesof 57 andotherspaceswithKilling spinorspermit the possibilitythat A = 0. In mostgeometriesthisis forbidden.(c) If (l/Jatbl/Jc) hasany geometricalsignificance,(14.3.5)seemsthe mostnaturalone.Note,incidentally,that m would nowbe a calculablemultiple of the Planck mass.Thus,in commonwith the modelsuggestedby CandelasandWeinberg[158],onewouldbe able to calculatethe Yang—Mills couplingconstantsaspurenumbers.

Finally, we remark that non-zero VEVs for fermionic bilinears is not the only way in which onemight resolvethe cosmologicalconstantproblem. Hawking [260],in his study of spacetimefoam, hasobservedthat a cosmologicalconstantof orderunity (andnegative)at the Planckscalecan neverthelessbe consistentwith an essentiallyzero valueat everydaylengthscales.Indeed,from this point of view,havingA —M~in the fundamentaltheory might evenbe regardedas desirable.

14.4. Towardsa realistic theory?

In section1.1, we entertainedthe possibility thatthe bosonsandfermionsof the standardmodelmaybe bound statesformed from amongthe infinity of Kaluza—Klein preons.Let us first considerthepossibleorigin of SU(3)X SU(2)X U(1) given the assumptionof elementarySO(8)x compositeSU(8) asthe maximum possiblesymmetry.Thereare literally hundredsof waysof embeddingSU(3)x SU(2)x

118 M.J. Duff ci aL, Ka!uza—Kleinsupergravity

U(1) into S0(8)X SU(8)but someembeddingsappearmorenaturalthan others.One possibility discussedin [66] beginsby noting that the 50(5)X SU(2) of the N = 1 squashedS7

vacuumcontainsSU(2)x U(1). Therewerealsosuggestionsthat theSU(3)might be foundinside the G2

subgroupof SU(8). Extra support for this picture was given by the observationthat one could givemassesto the unwantedhelicities in the SU(8) boundstatesupermultipletof EGMZ [98] in a wayconsistentwith an unbrokenN = 1 andG2 invariance.Thus,in thispicture the electroweakinteractionshadtheir origin in the d = 11 generalcoordinategroupandthestronginteractionsin the d = 11 Lorentzgroup; quite the oppositeof the GUT idea. Onevirtue of this approachwas that the ~ termsofsection10.1 appearonly in the electroweaksectorandnot the strong, thusprovidinga possibleway outof the strong CP problem. The major drawbackwas the chirality problem; chiral fermion represen-tationsseemedvery difficult to explain within this framework.

In fact chirality would suggestthat we turn things aroundandlook for the electroweakinteractionsinside the chiral SU(8). This could be donein two ways: as a GUT theory usingSU(8)D SU(5),or elseby putting SU(3)inside the SO(8) isometry[280]thusexploiting the absenceof chirality for the stronginteractionrepresentations.This latter possibility might receive supportfrom the observationthat theN = 8 supergravityLagrangian suffers from SU(8) anomalies[261] and that the only anomaly-freesubgroupsof SU(8) which are not vector-like correspondto particular embeddingsof SU(2)x U(1)[262].However, the schemewhich to date comesclosestto phenomenologyis the confinementschemeof [79],basedon anearlierideaof [39].(Much of therelevantgrouptheoryhadalreadyappearedin [263].)It wasarguedthat the breakingof the elementaryS0(8)(N = 8) symmetryof the squashed57 (N = 1 orN = 0) correspondsto a breakingof the compositeSU(8) down to either SU(4)x SU(2) (N = 1) orSU(S)x SU(3)x U(1) (N = 0). It was thensuggestedthat for N = 0 the SO(S)x SU(2) actsas a confiningforceyielding boundstatesof SU(5)X SU(3)X U(1). By demandingthattheeffectiveboundstatetheorybeboth anomaly free (in the SU(8) sense)and asymptoticallyfree, one finds with the standardSU(5)embeddinga fermionspectrumwith 4 generationsof (5* + 10) togethera realisticHiggs sector.

In summary,we hope that we havesucceededin showing in this report that extra dimensionsandsupersymmetryareinseparable.However,it hasto beconcededthatthereissofarno convincingcandidateforarealisticmodel,sincetheaboveboundstatespeculationsreston toomanyunprovenassumptions.Theproblemsof ultravioletdivergences,vanishingcosmologicalconstantandrealisticfermionrepresentationsseemto remainthe greatestchallenges.

Appendices

A. Curvaturecalculations

Indices:World Tangentspace Coordinates

d = 11 spacetime M, N, P,... A,B, C,... zM

d=4 spacetime p., t’,p,... a,/3, y,...

d = 7 internalspace m,n,p,... a, b, c,... ym

Notation:

M,1 = -~ (AMI . . . M

0 + antisymmetricpermutations),

Mi Duff eta!., Kaluza—K!einsupergravity 119

A(MI . . = -~ (AMI. . . M~ + symmetricpermutations),

[x,y]=xy—yx, {x,y}=xy+yx,

5~:::~ 5~’... 5~, F~ = F~2”. . .

whereTA areDiracmatrices.

Curvature:We use the following derivative covariantwith respectto general reparametrizationsand local

tangentspacerotations:

~7 A_~ A B A r P AV%4eN — OMeN —eN (LIMB ~ ep

where (L)MAB = — WMBA is the spin connectionand T’MN = TNM~the (torsion-free)affine connection.IfeM in (A.1) is the vielbein thenthe condition

=0 (A.2)

is the analogueof the metric postulate.(A.2) can be usedto relatet’~MA andFM~.J’while the conditionfor zero torsion

~ DFMeN]A =0, (A.3)

maybe usedto expressWMAB in termsof the vielbein:

t0MAB = ~ I1MAB + ‘1ABM — ‘~BMA), (A.4)

where

= ~~2a[MeN]A. (A.5)

In a generalrepresentationthe Lorenzcovariantderivative,denotedDM, is given by

DM = 3M + (UMAB2~ (A.6)

where ~AB are the generatorsof the tangentspacegroup in the appropriaterepresentation.Forinstancein the vectorandspinorrepresentationswe have,respectively,

— AB

~ )CD — ~5CD, (A.7)

lAB = _1TAB (A.8)

In termsof the derivativeDM oneobtainsthe curvaturetensorby meansof the Ricci identity:

120 MI. Duff et a!., Ka!uza—K!ein supergravity

rim ri 1_n ‘(‘A.B

[1-’M~ L~NI — “MNAB~

That is, using(A.6)

R,.,,,.,~= 3MWNAB— I9NWMAB + t0MA WNCB — WNA WMCB. (A. 10)

The Ricci tensorandcurvaturescalararethen:

Li — P0~“MN — g “MPNO,

R=g~’R~.,, (A.12)

Differentialforms:Let eA be an orthonormalbasisfor the metric ds2:

ds2=e~~®eBfl.~. (A.13)

The connectionone-forml’LI~ is uniquelydefinedby

de~4.=_wABA eB, W~—WBA (A.14)

(comparewith (A.3).)Thecurvaturetwo-form 6~ABrelatedto theRiemann(curvature)tnesorRABCD by

i’m _lri C D— 2.~ABCDe A e ~

is thengivenby [comparewith (A.10)]

= dWAB + WA A wEB. (A.16)

Yang—Millsexample:Consider

d.~2= ea® ~ + (ea — K”At) ® (e” — K~’A’)5a6, (A.17)

where ea = e”(x) is a vierbein in d = 4 spacetime,ea = ea(y) is a vielbein in the internal space,K” = K”(y) = Kim(y)emdl(y)are the orthonormalcomponentsof a set of Killing vectorsK

1m(8/ay~)which generatesa Lie group G, andA = Ai(x) = ea(x)Ai~(x)is a Yang—Mills potentialwith gaugegroupG.

The Killing vectorssatisfy

[K’, K’] = f~kK”, (A.18)

wheref’,. arethe structureconstantsof G. The Yang—Mills field strengthF’ = ~F’~,,e”A e’3 is

F’ = dA’ + jf’~”A’ A A’~. (A.19)

MI. Duffel aL, Ka!uza—K!ein supergravity 121

Define anorthonormalbasis~A for d.~2:

= e”, ~a= e”—K”’A’. (A.20)

Thenthe connectionone-form~ definedby (A.14), is

= W~ + ~F’03Ka ~

t’~ab= t~1ab+ (VaK’b)A’, t~1~b= 2Kb F’0~ê~, (A.21)

whereW~j3andWab are the connectionone-formscalculatedfrom e” and e” respectively,using(A.14).

Thenon-zerocomponentsof the Riemanntensor,calculatedfrom (A.1S) and(A.16), aregiven by= ~ — ~K”K’a (F’0~F’~8 — F’0~F’~,,+ 2F’0~F’~~),R0bCd= Ra&d,

= ~(DyF’a~)K’d, ~ = F’a~(Vc Kd) F’~.,,F~7K’1~K,~1, (A.22)

togetherwith thoseimpliedby the symmetriesof theRiemanntensor.~ andRaI,.cd arethe Riemanntensors calculatedfrom w0~and Wab respectively,using (A.15) and (A.16). D~is the Yang—Millsgauge-covariantderivative,

= V~F’0~+ filkAl ~ (A.23)

TheRicci tensoris given by

= R0~— ~KiaKIaFIay.F1,/, Rab = R01, + ~K’aK~F’~~~1ab = E,,~=

(A.24)

andfinally the curvaturescalar

= R~+ R(k) — ~K’aK” F’~~F-”~, (A.25)

whereR(4)and R(k)arethe curvaturescalarsin spacetimeandthe internalspace,respectively.

B. F-algebra

The d = 11 Dirac matricestA satisfy

{tA, PB}= —277AB, (B.1)

where77AB = diag(—, +,. .., +), andso

1+ whenA = a time indexrA—±TA, j . . (B.2)

i— whenA = a spaceindex

Thesematricesmay be expressedin termsof d = 4 Dirac matricesY~and d 7 Dirac matricesTa:

122 Mi Duff eta!., Ka!uza—K!ein supergravjgy

t,,=y0Ø1, t0Y5®Ta, (B.3)

where

= a13YS~~~~ (B.4)

The Levi—Civita symbolsaredefinedby

EAI...Ad~E[AIAd], e012—+1 (B.5)

for all casesof interest, i.e., d = 11 and 4 Minkowski and d = 7 Euclideaninternal space.Generally,therefore,

~ (B.6)

(s= +1 Euclidean;s = —1 Minkowski). The d = 7 matricesareanti-Hermitianandsatisfy

{Fa, Tb} = 25ab, (B.7)

andin therepresentationof section8.2

Ta1 a7 = Ea1 . . . a7~. (B.8)

Of frequentusearethe d = 4 andd = 7 Fierz-identities:

= —~[i/i~A+ y~çIiç~y”A— ~y~~i/i~y”A + y~y5i/J~y”ySA+ y~i/i~ysA], (B.9)

A~~ç1i= ~[i/iç~A— Fai/~F”A — ~F06çfrç~F””A+ ~F~~&A], (B.10)

whereA, ~ and i/i are arbitrary (anti-commuting)d = 4 spinorsin the caseof (B.9) and (commuting)d = 7 spinorsin the caseof (B.10). Also, note that in the former case~ = ~ andin the latter case

= ~t.

In section4.2we usedthe identity for Majoranaspinors

TaX~Ta= - kX, (B. )

which is derivedfrom (B.10) by subtracting(B.10) with A = = x from (B.10)with A = = TaX. Otheruseful Fierz identitiesderivablefrom (B.10) are:

T”~” 1~T0b77K] = —2T”771’ 1~3T

077K] (B.12)

Tl~7)[~?jIT~6hid1 = 5771kh~Ta?7”~ +FaT” 7711 1~tT677K1, (B.13)

511) 77K Tab 71L] = 2ij’T1a711’,~“T

6177~+ ~iiITabTC 77E~~‘< T~~ (B.14)

Mi Duff eta!., Ka!uza—K!ein supergravity 123

where ~‘ (I = 1,. . . , 8) arethe eight Killing spinorsof the roundS7.Finally we give a numberof relationsbetweenthe F-matricesF”’~~. ap:

F Tap a~= ~ (B.15)

‘ / (7 )t,

TaTb = Tab — nab, (B.16)

FaJ’c = Ta6c —

2TEaSblc, (B.17)

TabcFd= Tabcd — 3T[abSc]d, (B.18)

TaJ”~= Tab’~+ 451a”T6i’~ — 25~, (B.19)

Tat,cT~= Fabc*k + 6T[0b”t S~— 6T[a5~], (B.20)

FabcT~~= Tabc”1 — 9T106~&~]— 18T[~~”5~+ ~ (B.21)

TaTaFd = 5Ta, T’~1ai,Fa= 3Fab, T”Ta6cTa = Ta6c. (B.22)

C. Modified massoperators

In this appendixwepresentthe resultobtainedby repeatingthe analysisof chapter5 for the bosonicpart of the d = 11 supergravityLagrangianto which we haveaddeda cosmologicalconstant,i.e.,

L = ~e(R— 2A) — 1~eFMNpOFMNPQ+ (~)2eeMt .MIIF . . .M4FM5. . .M8AM9.. . M~1. (C.1)

We will write A in termsof a parameter/3 as

A=54m2f3 (C.2)

andconsiderthe rangefrom /3 = 0 (which is the casestudiedin chapter5) to /3 = 1 (which is the valuefor which spacetimeis fiat).

For arbitrary/3 the massoperatorsare

Spin Massoperators2

(2) zl~+ 12m2±6m i1 + 4m

2)1~’2

0+(1~3) ~ + 44m2— 20m2/3±12m[~o+ (3— f3)2m2]1”2

&—4m(l+8/3)0~1).(2) Q2+6mQ+8m2(1—f3). (C.3)

Thetowerscontainingthe graviton andthe Yang—Mills gaugebosonsareunchangedas expected.In

124 Mi Duff et aL, Ka!uza—K!ein supergrauity

fact, from (C.3) we seethat this is true for all spin 2 and spin 1 towers. The O~and0~towers, on theotherhand, all becomedependentupon the parameter/3.

This meansthat the stability analysisof section 5.2 mustbe redone,now using the Breitenholner—Freedmanconditionfor spin 0

M2�-m2(1-/3). (C.4)

Of course,for other spinswe still havestability criterion M2� 0.We first observethat the possibilitiesof obtainingunstablemodeshaveincreasedsincefor the 0.~2)

tower the criterion for stability now reads

LIL � 3m2(1+ 11/3). (C.5)

Of moreinterestperhapsis the fact that with j3 ~ 0 the 0~2)tower is no longer the only tower thatmaycontain unstablemodes.This is also the casefor theQ(~and0~1)towers. In the formercasewe findfrom (C.3) and (C.4)that stability in this sectoris guaranteedif

(Q+3m)2�9m2J3 (Q<0). (C.6)

In flat spacetime(J3 = 1) (C.6) is violatedfor three-formsfor which —6m <0 <0, e.g., on the roundS7the three-formsa of tower ~ satisfy

Qa=—(p+3)ma, (p�l), (C.7)

which implies that the modeswith p = 1 andp = 2 violate(C.6). In the caseof the tower~ the modesthat saturatedthe /3 = 0 stability bound now become unstable. Instability of the round 57 corn-pactificationfor p = 1 was first demonstratedin [268].

Acknowledgements

Partof this work was carriedout at the Aspen Institutefor Physicsin the summerof 1984 andat theHigher DimensionalWorkshop,Institute for TheoreticalPhysics,SantaBarbara,autumn1984.We aregrateful to the directors and organizersfor their hospitality. We would also like to thank our manycolleagues,specially thosewho havecollaboratedwith uson work describedin the text.

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Recentdevelopments(added in proof)

The developmentsin extradimensionsandsupergravityover the last few monthshavebeenso rapidthat it hasnot beenpossibleto includethem in the text. In this sectionwe shallattemptto remedysomeof theseomissions.The moststriking of theseare the spectaculardevelopmentsin superstringswhichfollowed the observationson anomaly cancellationsof Green and Schwarz [15], in particular thediscoveryof the E

8x E8 heteroticstring [286]andthecompactificationof string theorieson Calabi—Yaumanifolds [287] discussedin section 2 below. Although this progressappears,at first sight, to runcounterto the original spirit of Kaluza—Klein advocatedin this PhysicsReport,we show in section3below that the gauge bosonsof the heteroticstring in fact admit a traditional Kaluza—K]ein inter-

Mi Duff et a!., Ka!uza—Klein supergravily 131

pretation.As a preliminary, however,we first considerrecentprogressin the questionof consistenttruncationsand generalizethe discussionto include spaceswith no continuoussymmetriesappropriateto string compactificationfrom d = 10 to d = 4.

1. Consistenttruncations

Let us first reviewthe statusof the conjectureof chapter12 that the masslesssupermultipletis aconsistenttruncationof the S7 compactificationof d = 11 supergravity.The complexitiesof the scalarandpseudoscalarsectorshavestill preventedan explicit constructionof thenon-linearansatz.However,as apreliminaryexerciseonecan sidestepthe spin0 problemby showingthat the bosonicsectorof theN = 3 truncationof N = 8 supergravity(i.e. Einstein coupledto SU(2) Yang—Mills) canbe embeddedind = 11 supergravityas an exact solution [288].Moreover, the appearanceof the scalarsin the metricand fermion ansatznecessaryto reproducethe de Wit—Nicolai theory hasnow been written downexplicitly to all orders [94, 95] and shown to be consistent in that there is a matchingof they-dependencein the transformationrules [94].An existenceproof for thefull non-linearansatzhasnowbeenclaimedby de Wit andNicolai [289].Theserecentdevelopmentsfollowed the proof of [290]thatthe d = 11 theory exhibitsa hiddenSU(8) invariance.It has beenconjecturedthat the hiddensymmetryis as largeas E

8 x SO(16)[291].As discussedin section12.2 the reasonwhy a genericKaluza—Klein truncationis inconsistentat the

non-linear level is becausethe y-dependenceassociatedwith discardedmodescan be generatedbyproductsof thosewhich areretained.In the caseof the S

7 compactificationdiscussedabove,the reasonswhy this doesnot happenarevery subtle and not fully understood,sinceit is not ruled out by SO(8)group theory alone. However, in certain other contextswe can use simple group theory to ensureconsistency,for examplethe “K-invariance” of section13.3. Hereconsistencyis guaranteedby retainingall those fields and only thosefields invariant under a subgroupK of the isometry group G, sinceobviouslynon-singletscannotbe generatedby productsof singlets[96].(This maybegeneralizedby thedeviceof congruencyclasses:we discardrepresentationsnot containedin productsof thosewe keep.)Of coursethis doesnot guaranteethat the truncatedtheory will describeonly a finite numberof states.To get a finite numberof statesin the four-dimensionaltheory we must further require that K be atransitively acting subgroupof the isometry group [96]. See section 4.1. For example S7 admits thetransitiveactionsof SO(5),SU(4), SO(7)andS0(8)andtheseyield consistenttruncationsof the d = 11theory with N = 3, 2, 1 and0 supersymmetries,respectively.Thesearedistinct from, andnot containedin, the N = 8 truncationsincethey involve both masslessandmassivestates[96].

Of coursethe problemof consistencyat the non-linear level is equallyrelevantwhen the compactmanifold Mk hasno continuoussymmetries,as, for example, in the Calabi—Yau compactificationofsection2 below, or evenwhenMk is a productmanifold with one of its factorshavingno continuoussymmetries,as, for example,in the K3 x T3 compactificationof section9.1. (We note,incidentally,thatthe existenceof compactsix-dimensionalRicci-fiat spacesY providesus with yet more solutionsofd = 11 supergravitynot included in table 6, namely M

7 = Y x S’ [94].) The question,once again, iswhetherproductsof the y-dependentharmonicson Mk correspondingto stateswhich areretainedcanyield harmonicscorrespondingto the states which are discarded.For example, in the K3 xcompactificationof d = 11 supergravity,consistencycannotbe achievedfor the entire masslesssectordescribedin section9.1. Consistencyis achievedby retainingonly thosefields whoseharmonicson K3are constantor covariantly constant.Thus we must discard 19 of the 22 zero modesof ~2, the 57tracefreezero-modesof /iL and the38 tracefreezero-modesof 03/2. The resultingconsistenttruncation

132 Mi Duffel a!., Ka!uza—Kleinsupergrauity

is N = 4 supergravitycoupledto only 3 of the original 22 N = 4 vectormultiplets. This theory is in fact atruncationof theT7 masslesssector.

Can one make senseof an inconsistentmasslessansatz?As we see it, the problem is one ofnon-uniqueness.Supposeone takesthe point of view that oneobtainsthe effective four-dimensionaltheory by substituting the ansatz into the d-dimensionalaction and integrating over the extra kcoordinates.(For an inconsistentansatz,thisis theonly possiblepointof view sincesubstitutioninto thed-dimensionalequationsof motion will not work.*) One will certainly obtain somefour-dimensionalLagrangianin thisway, but the interaction termswill not be uniquesincemasslessnessalonedoesnotuniquely determinethe correct ansatzbeyondthe linearized level. Although one tends to think of amasslessansatzas retainingonly the lowest(i.e. massless)termsin aFourierexpansion

~P(x,y) = ~ 4’k(x)Yk(y), (Li)

we could modify this prescriptionby including higher harmonicswhose coefficients are non-linearfunctionsof masslessmodes.The functionsmustbe compatiblewith the symmetriesbut are otherwisearbitrary. Any such modification would yield an equally valid masslessansatzsince the criterion formasslessnessinvolves only the linearized ansatz, but would yield a totally different interactionLagrangian.Note that becausethismodification involveshigher harmonicsone cannotundo it by fieldredefinitions amongst the masslessfour-dimensionalmodes. In fact anotherway of restating theproblemis to keepthe infinite tower of statesandthen makefield redefinitionswhich mix the masslessandmassivestatesbefore settingthe massivestatesto zero.

Physically,of course, the way to obtain the correct low-energy effective theory is not to set themassivestatesto zero,but rather to integratethem out, therebyobtaininga four-dimensionaltheorywhich is uniqueup to field redefinitionsamongstthe masslessfields. The resulting low-energyeffectivetheory correspondsto making a very specific choiceof non-linearmodificationsto the masslessansatz,with (in principle)calculablecoefficients(again, of course,up to masslessfield redefinitions).

In order to calculatethe coefficientsof the non-lineartermsin the ansatz,onefirst needsto know themassesand interactionsof the massivestates.In practice, if one were working with, say, a K3 orCalabi—Yau compactification, it is almost inconceivable that one could calculate the spectrumofmassivestates;apartfrom any otherdifficulties, the eigenvaluesof the massoperatorsdependupon themetric, andno onehassucceededin constructingan explicit Ricci-flat metricon any Calabi—Yauspace.Thus for all practicalpurposes,the coefficientsof the non-lineartermsin the masslessansatzmust beconsideredto be incalculable.In turn, this meansthat most of the interaction terms in the effectivefour-dimensionalLagrangianwill haveuncalculablecoefficients.

In section 13.3 we gave the impressionthat theseproblemsof inconsistencywere presentonly fordimension>4 operatorsand henceirrelevant for energiesmuch less than the compactificationscale.Indeed,the gaugeboson inconsistenciesof chapter13 are of this kind. However, therewould be oneimportant exceptionif therewere a coupling of a masslessscalar L to a heavy scalarH via a LLHinteraction in the four-dimensionalLagrangian.This would give rise to a dimension4 operatorL4 afterintegrating out the massivefield H (Such couplingsare, of course,ruled out when the truncation isconsistent becauseby construction discardedmodes can never appear linearly in the Lagrangian.WhetherL2H terms, asopposedto L3H, etc., can actuallyoccurwhen the truncationis inconsistentisunclear.)

* Equivalently, in thecaseof a supersymmetriccompactification,substitutioninto thetransformationruleswill not work.

Mi. Duff eta!., Ka!uza—K!einsupergravily 133

By contrast,the four-dimensionaleffectivetheory obtainedby a consistenttruncationis unique.Torepresenta goodlow-energyapproximation,of course,a truncationshouldretain all themasslessstates.Theideal situation,it seems,ariseswhenall the masslessstatesarecontainedin aconsistenttruncation.

2. Supersirings

Thelast few monthshaveseenseveraldramaticdevelopmentsin the theory of superstrings.First, thediscoveryof GreenandSchwarz[15]that thegravitational,mixed and Yang—Mills anomaliesof N = 1supergravityin d = 10 coupledto N = 1 Yang—Mills all cancelprovided the gaugegroup is not onlyS0(32)(asdiscussedin section1.1)but alsoE8 x E8. Secondly,the discoveryof a new closedsuperstringtheory,the heteroticstring, which is consistentonly for thesetwo groups[286].Seealso[292].Thirdly,the compactificationof d = 10 superstringson Calabi—Yaumanifolds [287]yielding chiral anomaly-freetheoriesin d = 4.

As a consequence,N = 1 supergravityin d = 11 now seemsratherunfashionablebut, asweshall nowdiscuss,manyof the techniquesset out in this PhysicsReportcontinueto haveapplication to d = 10superstrings.In fact we shall confineour attentionto the low-energylimit of theE8 x E8 heteroticstring[286]which seemsthe mostphenomenologicallypromising.This correspondsto the field theory of theN = 1 supergravitymultiplet in d = 10 (eMA, c

1’M, AMN, A, 4) coupledto the N = 1 Yang—Mills [293,294]multiplet (AMt,x’) with gaugegroupG = E

8x E8. As remarkedin section1.1, this theory is chiral (lefthanded ~I’M and x’ right handedA) but is neverthelessfree of both Yang—Mills and gravitationalanomalies.Thisallows the exciting possibilityof a chiral anomalyfree theory in d = 4 providedwe canfind the right chirality preservingcompactificationof the extra6 dimensions.

The first attemptsto find non-trivial compactificationsof N = 1 d = 10 supergravityyielded nullresults[295] which were later elevatedto a no-go theorem and extendedto include the Yang—Millscoupling [17]. It was not until the realization that the Chapline—Manton[294] Lagrangian must begeneralizedto include gravitational Chern—Simon terms (to cancel the anomalies [15]) and termsquadraticin the Riemanntensorthat a chiral preservingsolution was found by Candelaset a]. [287].

These authorslooked for compactificationswhich (i) were of the product form M4 x M6 with M4maximally symmetric(seesection 1.3), (ii) preservedN = 1 in d = 4 andhencerequiredcW = SU(3) and(iii) yieldedchiral fermionsin d = 4. Theserequirementsweresatisfiedby taking M6 to be Ricci fiat andKahler (the Calabi—Yau spaces)and by embeddingthe SU(3) spin connectionin the E8 gaugegroup.(The idea of solving the coupledEinstein—Yang—Millsequationsin Euclideansignatureby settingthevector potentialequalto the spin connection,and hencerelating the Yang—Mills Dirac index to thegravitationalEulernumberx was discussedin a different context by CharapandDuff [296].SeealsoWilczek [297].)They found, moreover,that unbrokensupersymmetryforcesM4 to beMinkowski spacesoA = 0, that E8 x E’8 is brokento E6 x E’8 andthat thenumberof families in d = 4 was given by IxI/

2.The relationbetweenKilling spinors,holonomyandsupersymmetryfor N = 1 supergravityin d = 10

proceedsalong lines similar to those in d = 11 discussedin section4.2. The transformationrules nowyield [287] Dm7

1 0 where m = 1 - . . 6 and ~ is a real four-componentspinor on M6, i.e., the 4+ 4representationof S0(6) SO(4). One again obtainsthe integrability condition (4.2.7) but wheretheindices now run over 1 to 6. In this casewe always have the Ricci-fiat condition Rmn= 0. Thecorrespondingvaluesof N for sometypicalchoicesof ~‘ aregiven in table 15.

To illustrateall this, let us considerthe specific four-family model of Candelaset a!. [297]for whichthe Betti numbersare(b~= b6~)

134 Mi Duff eta!., Kaluza—K!einsupergravity

Table 15Holonomy groups and unbrokensupersymmetriesin four dimensionscoming from N= 1 supergravity in

d= 10

4~ N~,

S0(6) 4 0SU(3) 1+3 1

SU(2) 1+1+2 21+1+1+1+1 4

b0=1, b1=0, b2=1, b3=12,

andhencex = —8. As with K3, the metric is not known explicitly. The zero-modecountproceedsalongsimilar lines to thoseof K3 x T~of section9.1 (K3 is alsoa Calabi—Yaumanifold [298])and the resultsarequotedin table16 for thegravity sectorandtable 17 for the Yang—Mills sector.For comparisonwehave also included the correspondingtable for M6 = T

6. (One could also consider K3 x T2. Thecompactificationd = 10 supergravityon K3 and K3 x T~hasbeen discussedin refs. [23], [300] and[301].)The resultingmass!essgravitationalsectorconsistsof N = 1 supergravitycoupledto 6 (=2— x12)Wess—Zuminomultiplets andonelinear multiplet (i.e., a multiplet for which oneof themasslessspin 0is describedby A,~ratherthan q5).

The Yang—Mills sectorconsistsof one E’8 Yang—Mills multiplet (the “hidden” sector)and oneE6

Yang—Mills multiplet coupledto four families in the 27 representationof F6 (five families of 27 andoneantifamily of 27).

Actually, theE6 is brokenfurtherwhentheCalabi—Yaumanifoldis not simplyconnected(which is thecasefor this x = —8 example)and so someof theseotherwisemasslessmodeswill be absent.N = 1supersymmetryremainsunbroken.

It hasbeensuggestedthat thissurvivingN = 1 supersymmetrymight bebrokenat somelower energyscaleby gluino condensatesin the hiddensector[302,303]. A similarmechanismhadbeensuggestedforbreakingthe survivingN = 1 resultingfrom compactificationof d = 11 supergravityon the squashed57

[66].However, the non-vanishingof ~ (fermions))would thenspoil the vanishingof the cosmologicalconstantand it was further suggestedthat this might be cancelled by the non-vanishingof (T,~,.(antisymmetrictensor)) [302].(Again reviving a similar mechanismsuggestedin d = 11 [47,66], exceptthat in d = 10 we havethe threeindex field strengthHMNP ratherthan the four index FMNPQ.) It stillremainsunclearwhetherany of the known (or yet to be discovered)Calabi—Yaumanifoldswould thenyield satisfactoryphenomenology.Perhapssomethingnew is required.In view of the apparentloss ofpredictivepowerin an inconsistentmasslesstruncation,onemayentertaindoubtsabouteverbeingableto makedetailedcomparisonswith experiment.

3. Kaluza—Kleinapproach to the heteroticstring

The most appealingfeature of Kaluza—Klein theories is that what we perceive to be internalsymmetriesarereally space-timesymmetriesin a higher dimension.In particular,it is not necessarytopostulatethe separateexistenceof Yang—Mills fields, they are an automaticconsequenceof gravity.

It is ironic thereforethat therecentspectacularsuccessesof superstringsseemto ignore thisbeautiful

Mi Duff et a!., Kaluza—K!ein supergravity 135

Table 16Massless modeson T

6 and Y (gravitysector)

d=10 d=4 Spin T6 Y

gMAT g,,,~ 2 1 1B,~ 1 6 0S 0 21 11

~l’M ~ 3/2 4 1x 1/2 24 6

AMN A,,~ 0 1 1A,~ 1 6 0A 0 15 1

A A 1/2 4 10 1 1

concept.For althoughformulatedin 10 ratherthan4 dimensions,the Yang—Mills fields of E8 x E8 or

SO(32) are presentas primary fields already in the d = 10 Lagrangian.Moreover, the favouredcompactificationto d = 4 yields no Kaluza—Klein gaugefields since Calabi—Yau manifolds have nocontinuoussymmetries.

In this sectionwe show [321],in fact, that Kaluza—Klein can provide an explanationfor the gaugefields of the heteroticstring [286],as might havebeensuspectedfrom the Kaluza—Klein like relationbetweenthe Yang—Mills coupling constantg, the gravitationalconstantK and the slopeparametera’,

namely, a’g2 —~-K~.(This is to be contrastedwith the correspondingrelationg2 — ica’ of the openstring

[12—14]which appearsto admit no such K—K interpretation.)The Yang—Mills gaugegroup will, asusual, be a subgroupof the higher dimensional general coordinate group. Although the theorycompactifieson a group manifold G, however,our Kaluza—Klein ansatzwill involve only the gaugebosonsof GL and not those of the full isometry group GL X GR. As discussedin chapter13 this isessentialfor the consistencyof the Kaluza—Klein ansatz.Surprisingly,consistencyis achievedwithoutthe introductionof Kaluza—Klein scalars.

Our startingpoint is the off-shell effectiveaction F of FradkinandTseytlin [304,305] for theinfiniteset of local fields correspondingto the modesof a free bosonicstring spectrum.F is the generatingfunctionalof all possibleoff-shell scatteringamplitudeson arbitrarybackgrounds.In the caseof closedorientedstringsF is defined(in Euclideansignature)by

F[t~(x),gMN(x), BMN(x), . . .] = ~ e~~xf [dya6]x[dX

MI e~, (3.1)x=2. 0, —2,...

whereo- is a dimensionlessparameterandwhere

d2z

5= J {V~ yab ~a xM ~Ng,.~,(x) + a’\/y R(y)ç~(x)+ ~ab

8aXM o~N~MN(x)+ . . .} (3.2)

Here xM(z) defines the embeddingof the two-dimensionalstring worldsheetM2 in a spacetimeMd(Al, N = 1, . . ., d); Z~z(a = 1, 2) are the coordinateson M2 R(y) is the curvaturescalarof the metricYa6 x is the Eulernumberof the worldsheet.The graviton gMN(x), the antisymmetrictensorBMN(x)andthedilaton ~(x) correspondto themasslessmodesof thebosonicstring spectrum.The dotsrefer to

136 M.i Duffeta!., Kaluza—Klein .cupergravity

Table 17Masslessmodeson T

6andY (E8 x E’8 Yang—Mills sector)

d=10 d=4 Spin T6 Y

A~ A,, 1 248 78 —

~ 0 248x6 (5x27x27)x278

x x 1/2 248x4 5x27+27

A/4 A,~ 1 248 248

/‘ 0 248x6 0

x’ x’ 1/2 248x4 248

termsdescribingthe higher spin massivemodesand the scalar “tachyon”. In Writing the action (3.2),andin theremainderof this section,we revertto Minkowski signaturefor both the worldsheetandforspacetime.

For string consistency,the two-dimensionaltheory must be conformally invariant and hencethetwo-dimensionalworldsheetstresstensormust be traceless.The generalstructureof the traceis givenby

= 134*VYR(y)+ ~3~~~Vyy ÔaX

8bX + I3MNEaaXobX, (3.3)

wheref34~,f3~and/3B arelocal functionalsof ~(x), gMN(X) andBMN(X). As notedby Polyakov[306]thefirst termis just the two-dimensionalgravitationaltraceanomaly[244—246]while the secondandthirdtermsarepresentevenfor a fiat worldsheetand arisefrom the self-interaction.The /3 functionsin thetreeapproximation(y = 2) havebeencalculatedby Callanet al. [307].They find*

161T2/3~= d-26 + [4(o~)2 - 4Ei~- + ~I~] + O(a’), (3.4)3a

I3~1N= RMN — ~IHM HNPQ+ 2~M9~+ O(a’), (3.5)

I3MN = VPH MN — 214 MN ~ + O(a’), (3.6)

whereHMNP = 3~[M~NP] andsatisfies

aIOHMNPI = 0. (3.7)

The absenceof a traceanomaly,i.e. the vanishingof ~ $AT and19B is thereforejust equivalentto the

Einstein-matterfield equationsobtainedfrom the effectiveaction

F Jd”x V~e2~[d_26 j~— 4(~)2+ ~142+ O(a’)]. (3.8)

One obvious solution of the field equationscorrespondsto (~)= constant, (gMN) =

71MN and

* The peculiar4*-dependencein (3.4) to (3.6) wasobtainedin superspacein [319].

M.i Duff eta!., Ka!uza —Klein supergravity 137

(HMNP) = 0, but this is valid only for d = 26. For d > 26, the cosmologicalterm in (3.4) obligesus tolook for solutions in which some of the dimensionsare compactified and we can now follow thetraditional Kaluza—Klein interpretation. Accordingly we split the indices xM = (XAT y

m) where x’~(1u = 1, . . . , n) refersto spacetimeandy

m (m = 1, -.., k) refers to the extradimensions.One solutionwhich suggestsitself correspondsto the casewherethe extradimensionsarea groupmanifold G withk = dim G. In this case

(Emn(X, y)) = g~~(y)= ~ (3.9)

where fm,,,, are the structureconstantsof the group, converted to world indices, and CA is the

second-orderCasimirin the adjoint representation.This will indeedbe a solutionprovided

(t~(x,y)) = constant, (3.10)

and

(14mnp(X,y)) = Hmnp(y)= mfmnp, (3.11)

where m is a constantwith dimensionsof mass.The left-invariant Killing vector fields K~~(y)are

normalizedbygmnKi K’~ = 3i~, (3.12)

andsatisfy

[K’, K’] = mf K”, (3.13)

where K’ = K’tm a~.With these normalizationsHmnp correspondsto the parallelizing torsion. Thecurvatureis given by

(Rmnpq)=~m2fmn’f~j, (3.14)

ID _~ 2,.-’ ~

—

4m ~

(R~)=~m2CAk, (3.16)

whereRk is the curvaturescalarof the k-dimensionalgroupmanifold. Substituting(3.9)—(3.11)into thefield equations(3.4)—(3.7)we seethat (3.5)—(3.7)areidentically satisfiedwhile (3.4) yields

d—26~ka’m2CA/2+O(a’2). (3.17)

Our aim is to obtain the bosonic sectorof the heteroticstring in ten dimensions.Neglectingthehigher derivative terms, this correspondsto the bosonic sectorof the Chapline—Manton[293,294]N = 1 supergravitywith fields ~(x), g,,,,(x), H,,,,~,(x)coupled to Yang—Mills fields A’~(x)with G =

138 Mi Duff eta!., Kaluza—Klein supergravity

E8x E8 or S0(32). This meansthat d = 506 andk = 496. We shall now write down a Kaluza—Kleinansatzwhich achievesthis goal,but for the sakeof generalitywe allow G andthe dimensionsn andd tobe arbitrary.For the scalar,wewrite

~(x, y) = constant+ 4~(x). (3.18)

For the metric ansatz,weemploy the one-forms~A = ~AM dxM for which

~A~B~MN — eM eN

71AB,

andwrite

~(X, y) = e~(x)dx”, (3.20)

?(x, y) = e”~(y)dytm— m’K”(y)A’,~(x)dx” - (3.21)

The quantities4(x), e”.~(x)and A’,~(x)will be interpretedasthe scalar,vielbein andYang—Mills gaugepotential in n-dimensionalspacetime.We must also makean ansatzfor HMNP(X, y). Experiencewithantisymmetrictensorsin d = 11 supergravity,(13.1.1), suggeststhat this must also involve the Yang—Mills fields as well as the d = 10 three-form~ Accordingly, we write

~ y) = Ha~y(X), (3.22)

1Z~~~(x,y) = —m1F’~(x)K1~(y), (3.23)

Habc(X, y) 0, (3.24)

Habc(X, y) = mfabc, (3.25)

whereF’,,,,~(x)is the Yang—Mills field strength

F’,,.,, = a,,A’,, — + f’~A’,, A”,,. (3.26)

It is important to note that (a, a) are tangentspaceindices and (bc, m) are world indices and thatfabc f ijk~a K~’bKk~.

Severalcommentsare now in order. That eqs. (3.18) to (3.25) correspondto the correct Kaluza—Klein ansatzcan only be justified a posteriori. First they must be mathematicallyconsistentand, asemphasizedin chapter13, this is a highly non-trivial requirement.A “consistent”ansatzis one forwhich all solutionsof the n-dimensionaltheory aresolutionsof theoriginal d-dimensionaltheory.For agenericKaluza—Klein theory with homogeneousextradimensions,a consistenttruncationis achievedby an ansatzwhich retainsall those fields, and only thosefields, invariant undera transitively actingsubgroupof the isometrygroup. In particular,for groupmanifolds the gaugebosonsareonly thoseofGL andnot thoseof the full isometrygroupGL X GR. In certaincases,it may not benecessaryto retainall the GR singlets. Indeed,surprisingly,our ansatz(3.18)—(3.25)is consistentin spiteof omitting theusual Kaluza—Klein scalars.Secondly,eqs. (3.18) to (3.25) should be physically consistentin that we

Mi Duff eta!., Ka!uza—K!ein supergravily 139

recoverthe desired n-dimensionaltheory. In our casethat means,for n = 10, that we recoverthelow-energylimit of the heteroticstring [2861,i.e., the Chapline—MantonN = 1 supergravitycoupledtoYang—Mills with gaugegroupG including,of course,thecorrectYang—Mills Chern—Simonsterms[293,294].

Note that we havenot specifiedthe ground-statevaluesof the spacetimefields 4,(x), g,,.,,(x), H,,,,,,,(x)or A,,(x). Since,by definition, a “consistent”ansatzis onefor which all solutionsof the n-dimensionalfield equationsare solutions of the original d-dimensionalfield equations,it is not necessaryat thisstageto single out any preferredconfigurationsuch as g,,.,. = i~,,,,, 4, = H,,,,~= A’,, = 0. (For example,therecan beothersolutionssuch as Calabi—Yaumanifolds.)

We now substitutethe ansatz(3.18) to (3.25) into the field equations(3.4) to (3.7). Most of therelevantcalculationsmay be foundin appendixA. The /3~,,equationbecomes

R,,,,— ~H,,.”H,,,,,,, + 2V,,a~4,— m2Ft,,,,F’J~’= 0. (3.27)

The fl~,,equationbecomes

D,,F’,L~— ~m2H,,” F’,,,, — 2F’,,” a~4,= 0, (3.28)

where D~is the Yang—Mills covariantderivative.Note that since our ansatzis invariant under thetransitive action of GR, all the y-dependenceof the ansatzdisappearsfrom the n-dimensionalequations.Note also that half the Yang—Mills contributionto the energy—momentumtensorcomesfrom the standardmetric ansatzandan equalcontributionfrom the antisymmetrictensoransatz.The/9~,,,equation is satisfied identically. (This is the equation in which one might have expected toencounterinconsistenciesby omitting the Kaluza—Klein scalarsSu(x) since in a generictheory oneobtainsan equationof the form ES0 F’,,,. F”~.In our case,however,the inclusionof the Yang—Millsfields in the ansatz(3.23)for ~ exactlycancelsthis right-handsideand henceS°may consistentlybeset to zero.)

The f3~,,equationbecomes

— 2H~,,,, a,,4, = 0, (3.29)

the$~equationsimply reproduces(3.28);while the f3~,.equationis satisfiedidentically.

Next we considerthe /3~equation,which becomes

4a,,4,a~4,— 4LJq5 — R + j~rH,,,,,,I-1~”+ ~m2F’,,.,,F’~~= 0 (3.30)

after using(3.17).Finally, eq. (3.7) becomes

aE,,H,,p,,]+ ~,n2F’1,,,.F’,,,,1= 0 (3.31)

or, in termsof differential forms,

dH + m2F’ n F’ = 0. (3.32)

In the casen = 10, the readerwill no doubt recognizeequations(3.27)—(3.32) as the bosonicsectorof

140 Mi Duff et al., Kaluza—K!einsupergravity

the Chapline—MantonN = 1 supergravitywith coupling constant ic coupledto N = 1 Yang—Mills withcoupling constantg, wherewe makethe identification

g2 = ~2m2. (3.33)

It is perhapsremarkablethat in (3.32)we correctly reproducethe Yang—Mills Chern—Simonsterm inn dimensionseven though therewas no such term in d dimensions.(The appearanceof the F’ A F’term on reducingfrom d to n dimensionsis known in the mathematicalliteratureas a transgression.)

Equation (3.33) is of the standardKaluza—Klein form since m1 is just the radiusof the extradimensions.As in d = 11 supergravity,however,the numericalcoefficient will differ from that obtainedin pure gravity Kaluza—Klein theoriesowing to the Yang—Mills content of the antisymmetrictensoransatz.

If, on the other hand,we compareeqs. (3.27)—(3.32)with the low-energylimit (�2 derivatives)of theheteroticstring [307],we find

m2a’=l. (3.34)

Hencewe recoverthe well-known relation[286]betweenK, g anda’ of the heteroticstring

a’g2”#c2. (3.35)

One major difference from generic Kaluza—Klein compactifications is that we can obtain ann-dimensionaltheory with zero cosmologicalconstant.The particular4, dependenceof the action (3.8)ensuresthat the Einstein equation(3.5) containsno cosmologicalterm, while the cosmologicalterm inthe d-dimensional4, equationcancelsout in the n-dimensional4, equation.

By ignoring the O(a’) termsin the d-dimensionalequationswe haveobtainedthe Chapline—Manton10-dimensionalequationsincluding the Yang—Mills Chern—Simonsterm.

Elsewherewe shall show that including the aFRMNPQRMNPQ ~ termsin d dimensions,we canobtainthe modified field equationsof Candelaset al. [287]in 10 dimensionswith their ~ —

F’,,,,F’~~)terms. This Lagrangianwill contain not only the Chapline—MantonYang—Mills Chern—Simons term but also the Green—Schwarz[15] gravitational Chern—Simonsterm ensuring bothYang—Mills and gravitationalanomalycancellationsin the caseG = E

8x E8 or SO(32); Le. eq. (3.32)becomes

dH+ a’(F’ A F’ — R”~A R”) = 0. (3.36)

Oneis thenin a positionto seeka secondcompactificationfrom n = 10 to n = 4 of the kind discussedby Candelaset a]. [287].If so desired,of course,one could descenddirectly from d = 506 to n = 4 inwhich casethe embeddingof the Yang—Mills potential in the spin connection[296,297] employedbyCandelaset al. [287] could be reinterpretedas a purely gravitational phenomenon.Since, in ourcompactificationfrom d = 506 to n = 10 the Hmnp backgroundwas non-trivial, moreover,thereseemstobeno compelling reasonto insist that it bezero in going from n = 10 to n = 4~*An interestingquestionis whetherone can attachphysicalsignificanceto compactificationswhich go directly from d = 506 to

* Might one alsoneedthenon-unit warpfactor of section1.3?

Mi Duff eta!., Kaluza—Klein supei-gravity 141

n = 4 and which do not admit any intermediaten = 10 interpretation.The answerto this questionwould seemto requirea better understandingof howthe fermionsareincorporated.

It is interestingto notethat whereasin d dimensionsthe a’ expansioncorrespondsto an expansionin numbersof derivatives(i.e. zero-slopelimit = low-energylimit), in n dimensionsthiscorrespondencebreaksdown. For example, we obtain a’F’,,.,,F”” terms in n dimensionseventhough we ignoredaRMNPQR termsin d dimensions.In fact we conjecturethat the groupmanifoldwith parallelizingtorsionsolvesthefield equationsto all ordersin a’ andthattheKaluza—Klein interpretationappliestothefull string theory andnot merelyto its field theory limit.

All this, of course,is entirely consistentwith previouswork on stringsin curvedspaceandnon-linearo- models [308—315].The Lagrangian (3.2) with the fields set equal to their backgroundvalues, i.e.4, = constant;g,.,,, the metric on the groupmanifold; and Ha,, the parallelizing torsion is nothingbutthe non-linearo- modelwith Wess—Zuminoterm.Accordingto Witten [309],this modelhasvanishing/3functions provided a’m2 = 1, which is just eq. (3.34). Furthermore,the critical dimensionof thesetheoriesis given by [312,313]

d—26= kCA/(2+ CA) (3.37)

(e.g. with G = SO(32), k = 496, CA = 60 and hence d = 506 and n = 10). On the other hand, ourequation(3.17)yields

d—26=~kCA+•~~ (3.38)

on using (3.34). This suggeststhat solving the /3~equationto all ordersin a’ would yield the exactequation(3.37).

Note, however,that the ordinary bosonicstring formulatedon the group manifold G would yieldstatesbelongingto representationsof thefull isometrygroupGL X 0R in contrastto the heteroticstringwherethe statesbelongonly to representationsof GL. In our Kaluza—Klein formulation this is takencare of by the consistencyof the truncationin the Kaluza—Klein ansatzwhich demandsthat we retainonly G~singlets.

In this Kaluza—Klein picture,the E8 x F8or SO(32)symmetryof theheteroticstring is a subgroupof

the generalcoordinategroupin d = 506. This is to be contrastedwith the Frenkel—Kac—Goddard—Olive[316,317] approachwherethe gaugesymmetry emergesfrom compactificationof the d = 26 string onthe maximal torusT

16 [292,286, 318]. No doubt the two approachesareultimately equivalentandit willbe interestingto establishthe appropriatedictionary. In particular one can ask whetherthe n = 10fermionscan beincorporatedin any naturalway into the d = 506bosonictheory.For example,one hasthefeeling that the gravitational,Yang—Mills andmixed anomalycancellationsof GreenandSchwarzinn = 10 [15] mayhaveasimpler puregravity interpretationin d = 506.

Recently,Casheret a]. [320]haveargued,following an earlier conjectureof Freund[292],that thefermionsof the d 10 superstringemergeas solitons of the d = 26 bosonicstring, andhencethat thebosonicstring maybe regardedas the fundamentaltheory.Coupledwith the ideasof this section,onearrives at the picture of a three-in-oneworld that can be describedequivalently by 10, 26 or 506dimensions.Kaluza—Klein is dead;long live Kaluza—Klein!

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