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The R2-action in d = 10 conformal supergravityRoo, M. de

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NUCLEARNuclearPhysicsB 372 (1992)243—269 P H VS I C S BNorth-Holland ________________

The R2-actionin d = 10 conformalsupergravity

M. de RooInstitutefor TheoreticalPhysics,Nijenborgh 4, 9747AG Groningen, TheNetherlands

Received25 June1991(Revised24 October1991)

Acceptedfor publication14 November1991

We presentthe invariant action for conformal supergravityin ten dimensions.We compareour result to d = 6, N = 2 conformal supergravity,and show that in d = 6 a superconformalinvariantbasedon the Gauss—Bonnetcombinationmustexist. Thecontributionsof the antisym-metric tensorgaugefield in d = 10 cannotbecompletelyexpressedin termsof torsion.

1. Introduction

In this paperwe derivethe invariantactionof d = 10 conformalsupergravityupto termsquartic in fermions. Startingfrom a linearisedinvariant, the non-linear

contributionsrequiredfor full supersymmetryareobtainedby the Noethermethod.The resultingaction is unique.The gravitationaldegreesof freedomdo not appearin the form of the Gauss—Bonnetcombination.

Conformal supergravityhas played an importantrole in the constructionofmatter couplingsin supergravitytheories.This has beenthe casein the develop-ment of phenomenologicalsupergravitymodels,but also in the systematicstudyof

supergravitytheories in higher dimensions. The important ingredient in theseapplicationsis the fact that the large superconformalsymmetry breaksup therepresentationsof Poincarésupergravityin smallerparts,which canthen be moreeasily put togetherto constructinvariant actions. Gauge choicesrelate the twoformulationsof supergravity,andmake it possibleto go from the superconformalformulation to the physically moreconvenientoff-shell Poincaréversion.

Conformal supergravityis also interestingin itself. Conformal gravity theoriesmay be consideredas fundamentaltheoriesof gravity (see ref. [1] for a review).The use of conformal supergravityin this context requiresthe constructionof aninvariant action for the superconformalgaugemultiplet, the Weyl multiplet, itself.

Such actions havebeen known for a long time for the N = 1 [2] and N = 2 [3]

0550-3213/92/$05.00© 1992 — ElsevierSciencePublishersB.V. All rights reserved

244 M. deRoo/ ConformalsupergraLily

conformalsupergravitytheoriesin four dimensions.The linearisedform of such anaction for N= 4 conformal supergravitywas givenin ref. [3].

In dimensions > 4 a complete, non-linearconformal supergravityaction wasconstructedonly for the d = 6, N = 2 theory [4]. For d = 10 only an actioninvariant under the linearised transformationsis known [51.The problems inproceedingin d = 10 to a fully non-linearinvariantwere discussedin ref. [4]. In

this paperwe take up this problem again, and solve the technical difficultiesinvolved in the constructionof the d = 10 superconformalaction.

The interestof this work is, besidesthe aspectsmentionedabove,the possible

relation betweenthe conformal supergravityactions and the low-energylimit ofsuperstringtheory.

The low-energylimit of superstringtheory is givenby a supergravitytheory inten dimensions. In the zero-slopelimit this theory is d = 10, N = 1 Poincarésupergravity,coupledto the supersymmetricYang—Mills multiplet [61.The super-stringinducesmodificationsto this theory,in particulartheLorentz Chern—Simonsterm,which were first discoveredby Green and Schwarz[7]. The purely bosoniccontributionto thesemodifications, which are required to cancel anomalies[7],breaks the supersymmetry.However, supersymmetrycan be restoredby theaddition of termswhich dependon the fermionic fields of d = 10 supergravity.In

the past years,much effort hasbeendevotedto the constructionof a supersym-metricversion of a d = 10, N= 1 supergravitytheorywhich includesthe LorentzChern—Simonsterms(see the extensivelists of referencesin ref. [81).

The work on conformal supergravityin d = 6 containedan importanthint on

the form of this effective action. It was found in ref. [41that in the d = 6, N = 2conformalsupergravitytheorythe field strengthH’~’ of the anti-symmetrictensorgaugefield B~occurs in the action as torsion, i.e. as a modification to the spinconnectiono~”.This was a crucial simplifying aspectin the constructionof thesuperconformalaction. Essentially, the proper combination of w and H trans-forms undersupersymmetryas a Yang—Mills field, so that the constructionof the

effectiveaction, known resultson the Yang—Mills action itself canbe used[9].The d = 10, N = 1 Poincarésupergravitytheory also contains a two-index

anti-symmetrictensorgaugefield, and therethe use of a spin-connectionwithtorsionturns out to beequallyuseful.The detailedconstructionof the componentform of the supersymmetriclow-energyeffectivestring action in d = 10 [101reliedheavilyon this analogybetweensupergravityandYang—Mills theories.

Thereis a secondversionof d = 10, N = 1 Poincarésupergravity,which canbeobtainedby a duality transformation[111.In this formulation one has the sameanomaly cancellation mechanism[121. The duality transformationreplacesthetwo-indexgaugefield B1~,,by a six-indexanti-symmetrictensorgaugefield A,1The R

2-actionhasbeenconstructedalso in this secondformulation [13]. SincetheWeyl multiplet in d = 10 also containssuch a six-index gaugefield, this secondformulation of supergravityappearsto be closer to Poincarésupergravity.The

M. deRoo/ Conformalsupergravity 245

relationship betweenPoincaréand conformal supergravitieswas exploredin ref.[5] (see also ref. [14]). However, in d = 10 conformal supergravityone doesnotreadily have compensatingmultiplets available with which one can make thetransitionbetweenPoincaréandconformalsupergravity.The structureof off-shellsupergravityin d = 10 hasstill not beenresolvedbeyondthe linear level [15]. Therelationshipbetweenthe off-shell superconformaland the on-shell PoincaréR2-action in d = 10 will be discussedbriefly in sect. 5.

This paperis devotedto the constructionof the superconformalaction for thed = 10 Weyl multiplet. The Weyl multiplet is an off-shell multiplet containingmassivespin-2 degreesof freedom. It contains 128 + 128 bosonic and fermionicdegreesof freedom,and the superconformalsymmetries(dilatations, conformalboosts, S-supersymmetry,besidesthe usual Poincarésupersymmetiytransforma-tions)canbe implementedon thesefields. The field strengthof the six-index gaugefield of the Weyl multiplet can be representedby a three-indextensor H,1VA.However,this doesnot imply that a torsioninterpretationin conformal supergrav-ity is possible,andindeedthe simplifying methodsof refs. [4,9] arenot applicable.

In the absenceof a torsion interpretation, the construction of the action istechnicallycomplicated.

In sect. 2 of this paperwe briefly review d = 10 conformal supergravity.It isimportantto note that theconstructionof the invariant is donein a suitablegauge,which eliminates two of the fields of the Weyl multiplet (the scalar and thespin-1/2 field). In sect. 2 we also discuss the connectionbetween differentformulationsof the Weyl multiplet. Conformal supergravityin d = 10 is somewhatspecialin the sensethat it is not basedon a superconformalalgebra,nor does ithavea satisfactorysuperspaceformulation (for a recentproposalon this lastpoint,andfurther references,seeref. [16]). In this paperwe consideronly the componentformulation presentedin sect. 2. Definitions and propertiesof fields and curva-turesthat we requirein this paperaregatheredin appendixA.

In thelinearisedversionof this theoryonecan, in an obviousway, constructtwo

independentactions,quadratic in the fields, which are invariant under the lin-earisedtransformationrules.One of these is the purely bosonic Gauss—Bonnet

combination,the other is a sumof bosonic andfermionic terms. It turns out thatonly one combination of these actions allows invariance under the completenon-linear transformationrules. This meansthat the Gauss—Bonnetcombinationis not separatelyinvariant. In d = 6 this situationwasnot yet clear,sincein ref. [41no conclusionsaboutthe existenceof a non-linearGauss—Bonnetinvariant weredrawn.

The constructionof the invariant is given in sect.3. We give the resultwithoutthe termscontainingexplicit gravitinos in (3.11).The completeresult is presented

in appendixB, eq.(B.2). In sect.4, we discussthe relationof our result to the workin d = 6, N = 2 conformal supergravity.In particular, we show that our resultimplies the existenceof the superconformalGauss—Bonnetinvariant in d = 6.

246 M. deRoo/ ConformalsupergraL’ily

2. Conformal supergravityin d = 10

The Weyl multiplet is definedas the smallestoff-shell multiplet containingthespin-2 and spin-3/2representationsof supergravity.In d = 10 this multiplet has128 bosonic and128 fermionic degreesof freedom[5,17].Thesecanbe describedin terms of the following fields: the zehnbein e,1a (45), the antisymmetric tensorgaugefield A,11 ,~ (84), and the gravitino ~‘,1 (a Majorana—Weylspinor, 144).Sincethesefields containtoo many degreesof freedom,a bosonicanda fermionic

constraintwhich eliminate 1 and 16 degreesof freedom, respectively,must beimposed.

The transformationrulesunderQ-supersymmetryread *:

=

~ ~L(w)� — + 3F(3)F,1)cI-1~

3),

35A,1,16= 4 x 6! (2.1)

We havedefined

Habc = ~ (2.2)

where i~(A)(7)is the supercovariantcurvatureof A(6). The Bianchi identity forR(A) leadsto the following supercovariantrestrictionon H:

D~Habc=0. (2.3)

Closureof the algebrarequiresthe useof the constraint

= 0, (2.4)

where ~I’abis the supercovariantgravitino curvature.The variationof (2.4) impliesthe bosoniccondition

I~(w) — ~HabcHabc = 0. (2.5)

In this way the superfluousbosonic andfermionic degreesof freedomare elimi-nated. Of coursethe identity (2.3) is also crucial for the off-shell closure: thismultiplet definitely requires a six-index gauge field, which we prefer, wherepossible,to representin the form (2.2).

In the aboveformulation of the Weyl multiplet the fields are invariant underscaletransformations(dilatations) and S-supersymmetiytransformations.These

* For spinorsand y-matricesin this paperwe use the conventionsof ref. [5].

M. deRoo/ Conformalsupergravity 247

local symmetriescan be introducedby adding a scalar4 and a spin-1/2 field A,with transformationrules

~ ~A=~7~ (2.6)

where AD and ii are the parametersof local dilatationsand S-transformations,respectively(thesefields are at this stage inert under Q-transformations).Notethat eqs.(2.1) and (2.6) still describe128 + 128 degreesof freedom, andthat thealgebraclosesoff-shell. However, the commutatorof two Q-transformationsnowcontainsfield-dependentD- and S-transformations,which arerequiredfor closureon 4 and A. One may also introduce gaugefields b,1 and for the D- andS-transformations,respectively,as well as K-symmetry (conformal boosts, withparameterAK’~)andthe correspondinggaugefield f,1a. It is convenientto choose

= — ~ + a,1AD + e,1AKa, (2.7)

andto turn and f,1a into dependentfields by the conventionalconstraints

D,1A = 0, D,1(4~~Da4)= 0. (2.8)

The steps(2.6)—(2.8) introduceno new degreesof freedom.If we now define a new zehnbeinandgravitino by

e,1 = ~1e,1, ~, = ~1/2~ + Fae,1a/A4, (2.9)

then e,1a~hasWeyl weight — 1, andthe new gravitino ~ Weyl weight — ~. Thegravitino also has the usual S-supersymmetrytransformation = Such

field redefinitions, as well as a redefinition of the Q-transformationswith localLorentz and S-transformations,lead to the formulation of d = 10 conformalsupergravitythat was presentedin ref. [5]. From the formulation of ref. [5] onegoesbackto (2.1) by the gaugechoice 4 = 1, A = 0.

The formulation (2.1) of conformalsupergravity,with its simpletransformationrules, is well suitedfor the purposeof constructinginvariant actions, andfor thecoupling of conformalsupergravityto matterfields (the d = 10, N= 1 Yang—Millsmultiplet). Further details about covariant curvaturesand their transformationrulesaregatheredin appendixA.

Ourstartingpoint in the constructionof an invariant is the following quadraticaction:

e’ ~

+ bOtfJ~bF~21C(W)Ifrab

+ ~ (2.10)

248 M. deRoo/ Conformalsupergravily

Note that a possibleterm a2R(w)2canbeeliminatedby usingthe constraint(2.5),

at the expenseof a quarticterm H4. Otherquadraticcombinationsof 91H and Rcan, by partial integration, be put in the form (2.10), or can be rewritten asnon-leading(cubicandquartic) termsby usingeqs.(2.3) and(2.5). Similarly, othercontractionsof two gravitino curvaturesand one derivativecanbe eliminatedbythe use of eq.(2.4) andthe Bianchi identity for

The requirementof invariance under global, linearisedsupersymmetryonly

determinesb0 and c0. The reasonis that the Gauss—Bonnetcombination

e = ~ R5~~’(w)

= Rabcd(W)Rcdab(W) — 4Rab(W)Rba(W) + R(w)2, (2.11)

transformsinto a total derivative(at the linear level) for any variationof w. This isdueto the Bianchi identity for the Riemanntensor(A.15).

The result of the linearisedcalculationis

b0 = ~(a1 + 4a0), c0 = —3b0. (2.12)

At this stage we therefore have a two-parameterstarting point for the fullnon-linearcalculationbasedon the transformationrules (2.1). This will be the

subjectof sect.3.

3. The construction of the action

In this sectionwe will discussthe constructionof the invariant action. Themethod is simple: we first makean ansatz,containing all possible terms (exceptquartic fermions),with arbitrarycoefficients, and we then fix the coefficientsbydemandinginvarianceunderthe transformations(2.1).

Let us howeverelaboratesomewhaton the starting point and on some of theintermediatestagesof the calculation.The linearisedresultfrom sect. 2 is

~ +a1R,1~(w)R~~(w)

+~(a1 ~

— ~(a1 + ~ (3.1)

Beforeparametrisingthe Noetherterms, it is useful to set out the proceduretobe followed. Terms in the action can be orderedin the following way. We canassigna level to eachfield, such that w, ~‘, and A eachhave level 1, and thezehnbeinand derivativeshave level 0. Then (3.1) is the action at level 2. The

M. deRoo/ Conformalsupergrauity 249

TABLE 1Theschematicform of thesupersymmetrytransformationrules

No. Transformation Level change

1 ~i~i=9J� —12 ~/J=HE 03 = ~-~J~(2)’~1A= 04 t~w—E~IJ(2) 05 hco~fiH 16 1~/I(2)=�R 07 t1~1I(2)—E~?TH 08 ~1.i/J(2)=eHH 19 ~e=~i 1

Thesymbol ~i representsthegravitino, ~‘(2) thegravitinocurvature.The fields w, /i and A havelevel 1.Thecompleteform of thesetransformationrules is given in eqs. (2.1) and (A.11)—(A.14).

different termsin the supersymmetrytransformations(2.1) and (A.11)—(A.14) arepresentedschematicallyin table 1. Only one contribution (i~1,1~ to thesetransformationsdecreasesthe level.

Wewill determinethe action up to termsquarticin fermions, so that theactionconsistsof purely bosonic terms, and of terms with two fermions. We can thenignore all variations of the action which are trilinear in fermions, since suchvariationswould also havecontributionsfrom the unknownquarticfermionsector.In the variation of bosonic terms we may ignore all contributionsthat are morethan linear in fermions. This meanswe can freely use identities for e.g. theRiemanntensor,modulo termsbilinear in fermions,but only in the variation oftheaction. In the termsin the action whicharebilinear in fermionsonly the fermionshaveto be varied.

In theconstructionof the actionwe alwaysuse w(e, ~ji), i.e. the spin connectionsatisfies

= ~ (3.2)

The ~!,2 contributionto (3.2) may be safelyignored in the variation of the action.All derivativesin the action are Lorentz-covariantderivatives~,1(w(e, i/i)). Also,we useeverywherethe supercovarianttensorHabc with threeLorentz indices.TheBianchi identity for H may be usedwith a Lorentz-covariantderivative ~, as in(A.19), in the variation of the action. Similarly, derivatives on are always

written in termsof the supercovariantgravitino curvature~/~ab (see eq.(A.22). TheRiemanntensoris not usedin supercovariantform.

It is useful to determinebeforehandhow partial integrationsin the variationofthe actionwill be performed.We alwaysintegratederivativesaway from e, so thatthe remaining independentterms in the variation of the action never contain

c-terms.

250 M. deRoo/ ConformalsupergraLily

TABLE 2Thedifferent structuresin thevariationof the action

No. Variation Level Remainder

(A) 2 R -~ H2

(B) i921/J,

2,R 2 ~q~,(2), R —~H2

(C) �~IJ(2).~l)9~H 2 -+ RH

(D) E9~/J(2)9~H 2

(E) ~~RR 3 R—~H2

(F) ~pR9~H 3 R-*H2(G) e~H~JR 3 R-*H2(H) El/J(

2)RH 3 R—* H2, ~~‘(2) coy.

(I) E~I/J(2)HH 3

(J) ~C~H)2 3 —

(K) ~ 3 .~H -~ RH(L) E~IJ(

2)H~2TH 3 ~‘(2) coy.

(M) ~RHH 4 R-~H2

(N) g~pHH~H 4 -

(0) EIIJ(2)HHH 4 ~‘(2) coy.

(P) i/JHHHH 5 -

The remaindersindicate terms that may be left over aftercancellation,since by the useof identities(2.5) and(A.20)—(A.22) they can be shifted to higher-levelcalculations.~sindicatesthe gravitino, ~‘(2)

the gravitino curvature.

Our ansatzfor the completeaction is presentedin appendixB (eq. (B.1)). Thevariation of (B.1) gives rise to 16 different combinationsof fields and derivatives,whicharepresentedschematicallyin table 2. For eachof thesestructureswe haveto obtain a cancellation,but somecontributionsto the calculationcanbemovedtoa higher-levelvariationby usingconstraintsandidentitiessatisfiedby the fields inthis theory.Therearefour suchrelations,which are indicatedin the lastcolumnof

table 2:(1) R —‘ HH. The constraint(2.5).(2) ~~2)• The Bianchi identity of I/fob, with a Lorentz covariantderivative. The

four termsin this Bianchi identity eachhavea higher level than andarepresentedin (A.20).

(3) ~ In some calculationswe encounterthe commutatorof two derivativeson H. This canbewritten as a combinationof RH-terms(eq. (A.21)).

(4) ~/‘~covariantization(coy.). From partial integration the combinationmay arise. This is rewritten, using eq. (A.22), as i//,1v~with the appropriatecovariantizations.The covariantizationscontributeto a higher-levelcalcula-tion.

The 16 differentvariationswill be referredto as (A)—(P), asin table 2. Of coursethe Bianchi identities (A.15) for R and (A.17) for H are also used in thecalculation,but thesedo not producehigher-levelcontributions.

M. deRoo/ Conformalsupergravity 251

TABLE 3All contributionsto thelevel-2 variations(A)—(D). Thenumbersin the tablecorrespondto the

supersymmetrytransformationsin table 1

Contribution (A) (B) (C) (D)

RR 4 — — —

— 6 — 7

~H~H - - 1 1~fn/J(

2)R 1 1 - -

~IJ~1J(2)~iTH - - 1 1

We will go in somedetail throughsomeof the calculationsat level 3. It is at thislevel that we obtain a relationbetweenthe coefficientsa0 and a1.This implies thatthereis only onenon-linearsuperconformalR

2 invariant in d = 10.Let us first reconsiderthe cancellationof the level-2 variations(A)—(D). The

contributions to this calculationare shown in table 3. The entries denote thesupersymmettytransformation(see table 1), which gives rise to the particularvariation..Contributionscome of coursefrom all level-2 terms in the action, andfrom the level-3 termswith an explicit gravitino. Since this is the lowest level, we

haveno contributionsfrom previouscalculations.The cancellationrequires two relationswhich are contractionsof the Bianchi

identity (A.15) for the Riemanntensor.Theseare

~,1(w)(eR~t~~c(w)) = 2eeAt0~A(w) Rb]~~(w) + bilinear fermions, (3.3)

~4(w)(eR(w)) = 2ee ~5(cv)R(w) + bilinear fermions. (3.4)

In both (3.3) and(3.4) all bilinear fermionsare dueto (3.2). We may ignore themin thevariationof the action.The termsresultingfrom (3.4) are movedto the nextlevel by the use of (2.5).

In eq. (B.1) we see that there are six independentterms of the type I/FIIJ(2)R.

Their six coefficientsm, n arenot completelydeterminedby thiscalculation.Theresult is

m1=—~(a1+4a0), n1=0,

m2= ~(a1 + 4a0)+ ~m3, n2 = 2m3,

m3= free, n3 = —2m3. (3.5)

252 M deRoo/ Conformalsupergravity

Using eq. (3.5), the terms in the action with m and n can be rewritten in thefollowing way:

Q~’ —II~ +4 ~‘ rabpz.lcdD— 4~ I a

0,~,1 ~P abcd

— 4m3c11,1FF IIJfgRabcd+ m3i/101bI/J°”R. (3.6)

The secondterm plays no role at level 2, since integratingaway from c in thevariationof the gravitinogives Bianchi identitiesbothfor R and I/’~21.The last termis also of a higher level. We shall seethat the coefficientm3 will be determinedatthe next level. Note that in casea1 = — 4a0 (the Gauss—Bonnetchoice) the usualNoetherterm is absent.

The p-terms are treated in a similar fashion.The remainingterms after thecancellations(C)—(D) havethe form

+ ~(a1 + 4a0)V H0l~~~{~I/JdyOI/JbC— ~I/1eYabcI/1de + ~I/’aYbceI/1de — ~I/’dYobeI/’ce}

(3.7)

Note that in the caseof the Gauss—Bonnetcombinationall theseterms are absent.In the p-sectorin the action one can also write terms which, like the second

term in eq. (3.6), do not contribute to the level-2 calculation. There are fourindependenttermsof this type,whichcanbe written as

~eHabc{ P l0~~abcTefghI/fgh + Pi 1~

1TabTefghTc~ghP12~~FaFefgh FbcI/fgh

+p13I/JfFefghFabc~gh}. (3.8)

The variation of the gravitino gives after partial integrationa term with ~~2JHwhich canbe rewritten as RH, anda term with in the form of the Bianchiidentity for the gravitino curvature.In this casethe coefficientsof thesetermswillnot be determinedby higher-levelcalculations.The reasonis that in (3.8) a partialintegration(with ~1”) can be performed,which expresses(3.8) in terms of otherlevel-3 and -4 contributions. Since all such terms alreadyappearin the ansatz,(B.1) would beovercompletewith (3.8). Thereforethe four coefficientsin (3.8) maybe set equal to zero, andthey do not appearin (B.1).

The overcompletenessof the ansatzcan be recognizedby the fact that freecoefficientsremain at the end of the calculation.A more critical aspectof thecalculationis to decidewhichof the contributionsto the variation of theaction areindependent.If a dependencebetweentwo variations is not recognized,oneobtains too many equationsfor the input coefficients, and no solution. For thisreasonit is extremelyuseful to “integrateaway from c”, sincethen the possibility

M deRoo/ Conformalsupergravity 253

TABLE 4All contributionsto thelevel-3variations(E)—(L)

Contribution (E) (F) (G) (H) (I) (J) (K) (L)

RR 9 — 5 — — — — —

2)~~/~(2) — — — — 8 — — —

— — — — — 9 — 4

6 7 — 2 — — — —

— 6 — — — 7 — 2RHH — — — 3 — — — 4ARR 3 - - 4 - - - -

— — — 6 — — — 7HH~H — — — — — — — 3

~JJçIJRH - 1 1 1 - - - -

qJçlJ(2)HH - — - — 1 — — 1

- - - - - 1 1 1

(A) - - - - - - - R(B) B2 B3 - B, R - - -

(C) - - - - - - -

(D) - B2 - - - B3 - B,

The last four lines indicate contributions which arise in the cancellation of the level-2 variations(A)—(D) (see column 4 of table 2). B, indicate contributions from the Bianchi identity (A.20) of thegravitino curvature.

of a complicateddependencethrough partial integrationsdoesnot have to beconsidered.

This concludesthe discussionof the cancellationof level-2 variations.Varia-

tions of level 3 arepresentedin table 4. Now we have,besidesthe straightforwardsupersymmetrytransformationsof terms in the action, also contributionsdue toremaindersfrom the level-2 cancellations.Theseare indicatedin the lower linesoftable 4. The first stage of the level-3 calculations is to advancetoward thecancellationof ~4’(2)RH-terms,called(H). As we seein table 4, thereare manycontributionsto this variation. However, the cancellations(E)—(G) alreadydeter-mine some of the coefficients involved, and will therefore be consideredfirst.Particularlyusefulin this respectis the (E)-cancellation.This variation is relatively

simple, anddeterminesthe coefficient m3 left over from previouswork, aswell asthe coefficientof the Lorentz Chern—Simonsterm in the action (all relativeto a0and a,). The result for this part of the action thenbecomes(only terms indepen-dentof H)

e~2’= aOR,1~,~l~(w)Rm~uTh(w)+ a,R,1~(o)R~v(w)

+ 4111,1F ((ai + 4a0)F~I/jc~~+ ~ —

+ ia0%~e1~ ~ (3.9)

254 M. deRoo/ Conformalsupergravi!y

Note that the contributionof the Lorentz Chern—Simonsterm wR (the last term

in eq.(3.9) can bewritten as coRH by a partial integration)is independentof a1,andthereforeoccursboth in the Gauss—Bonnetcombinationand in the (R,1~ab)

2

action.The intermediateresult (3.9) was alsogiven in ref. [4].— The (F)—(G) cancellationsaresufficient to determinethe coefficients t of theI/JçIJRH terms in termsof a

0 and a,. The (G)-cancellationrelatesthe t-coefficientsin such a way that the ~I/JH1T1JRvariationscontain 9/IR in the form of a Bianchiidentity. The (F)-cancellationthen fixes the valueof thesecoefficients.

The (H)-cancellationthen determinesthe d and e coefficients,but, in particu-lar, it fixes a, relativeto a0,

a, = —2a0. (3.10)

It is at this stagethat the remainingfree parameterin the action (one being an

arbitrary scaleis fixed.The cancellationsat level 3 ((I)—(L)), andthoseat levels4 lead to uniquevalues

for all the remainingcoefficients.The cancellationof the singlelevel-Svariation isthen a powerful check of the result. Theseand other calculationsin this section

havebeendone with the help of a computerprogramfor algebraicmanipulations.The contributions to the higher levels are presentedin table S. The completeresult,with the valueobtainedfor all coefficients,canbe found in (B.2).

The contributionsto the actionwithout explicit gravitinosare (a0 = 1)

~ — ~ + I/JOb~Yl/Jab—

+ ~1~e’ aj.ao~~~Ap~R abR ab

1 C a1...a6 ,1l~ Ap

+ V~IHa {6I/fFI/i — ~I/’adTbI/’cd — i~I/1deTabcI/1de

3~adTbde~ce— ~I/JdeTabcdf~ef}

+ 9R~il~HacdHbcd+ ~

+ ~HabeHcdeHacfHbdf. (3.11)

Somecontributionsto the action(comparethe ansatz(B.1)), aresimpler than theymight havebeen.For instance,of thethreeindependentcombinationsof H4-terms,two havea vanishingcoefficient.

Since (3.11) contains many explicit H-contributions, we should considerthepossibility of rewriting the actionwith H-torsion. We may define

fl,1 ~ab +YI~I,1th~ (3.12)

M deRoo/ Conformalsupergravity 255

TABLE 5The calculation at levels 4 ((M)—(0)) and level 5 (F). Weindicate the contributions from supersymmetry

variations (1—9) (see table 1), and those from previous cancellations(see table 2)

Contribution (M) (N) (0) (P)

RR - - - -

— — — —

~H~H — 5 — —

çlJqJ(2)R 8 - - -

- 8 - -

RHH 9 5 — —

ARR 5 - - -

clF(2)clJ(2)H — — 8 —

HH9i~H — 9 4 5

2 - - -

~InIJ(2)HH 6 7 2 8i/Jç1JH~1H - 2 - -

HHHH — — 3 9

~HHH - 1 1 2

(A) - - - -

(B) B4 - - -

(C) - - - -

(D) - B4 - -

(E) R - - R(F) - R - -

(G) - R - -

(H) coy. — R —

(I) B2 B3 B, B4(K) - - -

(L) — coy. — —

(M) - - - R(0) — — — coy.

so that

= R,1~1th(w)+ 2y.~[,1H~]1Th— 2y2H[,1°’H~JC’~,

Rab(Ll) =Rab(W) +y2HaCdHbcd,

R(fl) ‘R(w) +y2HabcHabc. (3.13)

From thelastequationin (3.13)we seethat in the constraint(2.5) the H2-termhasthe wrong sign to be interpreted as torsion. Also the H-contribution in the

256 M. deRoo/ ConformalsupergraL’ity

transformationrule of the gravitino cannotbe absorbedinto w. Notice that thequadraticaction for the Riemanntensorwith torsion takeson the form

~ = ~ + ~ (3.14)

so that in the quadraticaction (3.1) H cannotbe interpretedas torsionwith a realvaluefor y (we find y2 = — ~).Nevertheless,onecanalwaysrewrite some or all ofthe spin connectionswith torsion. However, if this cannotbe done such that allexplicit H-contributionsare absorbed,it doesnot truly simplify the result.

A last remarkabouttorsionconcernsthe Gauss—Bonnetcombination(2.11). Ineq. (2.11) w canbe replacedby 12 for any y, without disturbingthe invarianceatthe linear level. The questionthereforeariseswhy in eq.(2.12) we did not find athree-parameterfamily of solutions, one parametercorrespondingto y. Thereasonis that the (~H)2-termsarising from the Gauss—Bonnetcombinationwithtorsion cancel, so that the difference between(2.11) with w and (2.11) with 12consistsof termsof a higher level.

4. Comparisonwith d = 6, N = 2 conformalsupergravity

In ref. [4] an R2-invariantwasobtainedfor d = 6, N = 2 conformalsupergravity.In this sectionwe will discussthe implications of our result for d = 6. The mainpoint we want to establish is, that the (~H)2term which appearsin our d = 10action, when reducedto d = 6, can be absorbedinto the (Riemann)2 term astorsion, leavinga Gauss—Bonnetcombination.

In d = 6, N = 2 conformal supergravity[18] thereare two formulationsof theWeyl multiplet. In one case the multiplet contains a two-index antisymmetrictensorgaugefield B~,,~, as in d = 10, N = 1 Poincarésupergravity,in the otherformulation thesedegreesof freedomare representedby a three-index,anti-self-dual tensor. Only the first version allows the constructionof a superconformalinvariant [4].

The Weyl multiplet in d = 6, N = 2 conformal supergravityhas 40 + 40 degrees

of freedom. When the multiplet is formulated in terms of a scale-invariantsechsbeinand an S-invariant gravitino, as we did in d = 10, thesedegreesoffreedomare representedas follows. The bosonicfields are the sechsbein(15), thegaugefield B~(10), andthe SU(2)-gaugefield V~,symmetricin theupperindices(15). The chiral N = 2 gravitino has40 degreesof freedom.The leading terms inthe d = 6 action takethe form

e’2’=R,1 ab(Q )R~~(Q) + 2~ab

7~~I/, — J~1~,ujJ/,1”~ (4.1)

* In this section ten-dimensionalspace-timeand Lorentz indicesare written as M, N A, B

six-dimensionalindiceswill be denotedby j~,v a, b The four-dimensionalinternal indicesare s, t

M. deRoo / Conformalsupergravity 257

TABLE 6Reduction of the fields of the d = 10 Weyl multiplet d = 6

d=10 d=6 d.o.f.

hMN h,~,, 15x1hE* 5x4

lx (9+ 1)AMM ~ 0

1x4A,*IE

4*( Sx(3+3)

AE~”” 10x4A~~~~””’ lOx 1

80x1

16x4In this tablethe indicesM, N, ... take on the values1 10, ~s,v, ... thevalues1 6 ands, I,

thevalues 1 4. The numberof degreesof freedom(d.o.f.) in the last column are presentedas aproduct,with theinternal degreesof freedom(S0(4)) asthe secondfactor.

In eq. (4.1) we have 12,1 ab = — H,1cth, where H is the field strengthof B,1~.V,1 is the field strengthof V~’.The interpretationof H as torsion canbe madethroughoutthe action. In ref. [4] a start was made with the constructionof asecondinvariant in d = 6, of which the leading terms form the Gauss—Bonnetcombination.We will show that our result in d = 10 implies that this secondinvariant exists. In principle it could be obtained from (B.2) by dimensionalreduction.

In table 6 we present the d = 6 content of the fields of the d = 10 Weylmultiplet. The table contains 129 + 144 degreesof freedom, so the constraints

must still be imposed. Obviously, a truncationmust be made to arrive at the40 + 40 degreesof freedom in d = 6. In the linearised,globally supersymmetric

theory we can representthe sechsbeinof the d = 6 Weyl multiplet by ~ asymmetrictensorgaugefield. In A,1 . ~ the anti-symmetricupperindicescanberestrictedto be (anti)-self-dual.The self-dualpart is truncated,the anti-self-dualpartrepresentsthe 15 degreesof freedomof V,1

11• The field A,1~mcorrespondstothe 10 degreesof freedomof B,1~,,the internaldegreesfreedomobviously giving asinglet. The gravitino is a 16-componentfermion, as is the Majorana—Weylfermion in d = 10. In d = 6 we can split this fermion in two 8-componentchirald = 6 fermions, of which onechirality is truncated.Whenwritten in a Weyl basiswith four components,the N = 2 structurebecomesexplicit. We will not usethis

N = 2 basis,so that also the relationbetweenA,1, ,14St andthe SU(2) gaugefieldwill not be neededexplicitly.

The d = 6 transformationrules that follow from the transformationrules(A.1)

in d = 10 show that indeed the truncationsoutlined abovecan be done consis-tently. The field B,1~ EStUVA,1 canbe written astorsion in the transformation

258 M. deRoo / Conformalsupergrauity

rule of the gravitino, i.e. we find ~i/f,1 =~(12±)c+ ..., with Q±~w±T. Thetorsion T is given explicitly by

T,1VA = 90%I~a[,1B~5]. (4.2)

This requiresthat the d = 6 gravitino, I/’,~,is definedas a suitablelinear combina-

tio~iof 1/1,1 and I/J~ of table 6. To have the usual transformationrule of thes/chsbeinwe needto makea similar redefinition of h,1~:

= ~ + 2LV~ (4.3)

Theten-dimensionalconstraintscanbe resolvedin termsof thetraceof h4t, andof

the y-traceof I/i5: y5ç(15. Thus the constraintsno longer restrictthe super-gravita-tionaldegreesof freedom.We will comeback to the bosonicconstraintat the end

of this section.The leadingbosonictermsin the d = 10 action(B.2) are

.~= (R~(~))~— 2(RMN(w))2— 3~MHABC~MHABC. (4.4)

Considerthe term

—3~MHABC~MH~4~= (8 x 7!)~MR(A)Nl...N7~MR(A)~~1N7. (4.5)

We can work out the coefficient of the contributionof B,1~in this expression.Expressingthis in termsof the torsion(4.2) we find that the left-hand side of eq.(4.5) becomes

4(9~aTbcd)2 (4.6)

In d = 6 the terms containingthe Riemanntensorwith torsion canbe written as

(R,1~(fl))2= (R,1~(w))2 + 2(~aTbCd)2 — 2(~aTbcd)(~bTacd)

= (R~~b(w))2+ ~(~aTbcd)2, (4.7)

where in thelaststep theBianchi identity for the anti-symmetrictensorgaugefieldis used.There is still a crucial factor two between(4.6) and (4.7). This factor iscorrect,sincewe canrewrite our result (4.4), in d = 6, in theform

+ ~{(R,1P0b(w))2- 4(Rab(W))2}. (4.8)

Thereforeonly half of (4.7) is needed,andthe remainderin (4.8) correspondstothe first two termsof the Gauss—Bonnetcombination.

M. deRoo/ ConformalsupergraL’ily 259

In (4.8) the contribution of the Riemann scalar R2 to the Gauss—Bonnetcombination is still missing. It appearswhen h’ is used instead of h, and thebosonic constraintis resolvedfor hss. Let usdo this analysisin some detail. Thelinearisedform of the Riemanntensorand its contractionsin d = 10 is given by

RMNRS(h) = hNs,MR — hMsNR — hNR SM+ hMRsN),

R ‘‘‘~ 1f~ 1 1 r—il.MNV1) — 21”,MN — “MR,RN — “NR,RM L~j “MN

R(h) = oh — hMN,MN. (4.9)

We have used h hMM, and the notation hMNR 3RhMN. The contributionsof(4.9) to an action quadraticin R are

= (D hMN)2 — 2(0hMN)(hMRRN) + (hMN MN)’

4(RMN(h))2= (oh)2 — 2(0 h)(hM~~~MN)+ 2(o hMN)2

\ -~[j.MN‘ JY’MR,RN) ~“ MN)

(R(h))2 = (oh)2— 2(0 h)(hMNMN) + (hM~1MN). (4.10)

When the threeterms in eq. (4.10) areaddedin the Gauss—Bonnetcombination(2.11) the result vanishes[19]. Let us now reducethe threeterms in (4.10) fromd = 10 to d = 6. As before we truncateh,15 and the tracelesspart of h5’ (so wereplacefr” by ~85thu~). Thenwe get the following contributiondueto h55 in d = 6:

— (R~~(h))2+ ~(oh~)2,

4(RMN(h))2 —~ 4(R,1~(h))2+ 2(0 hss)R(h)+ ~(o h~)2,

(R(h))2 —~ (R(h))2+ 2(0 h~)R(h) + (0 h~~)2. (4.11)

The linearisedform of the constraint (2.5) in d = 10 is R(h) = 0. In d = 6 thisbecomes

R(h) + oh55 = 0, (4.12)

as we canreadoff from the last line in (4.11).Finally we reduceour combinationof (Riemann)2and (Ricci)2 terms in (4.4) to d = 6. Using (4.11) and (4.12), weobtain

~R2 = (R,1~5~(h))

2— 2(R,1~(h))2+ ~(0 h~)2

_IID (~\\2~if~j,SS\2

2’~~,1*’Ap~ 1) 8~ ‘ )

+ +{(R,1~AP(h))2- 4(R~(h))2+ (R(h))2}, (4.13)

260 M. deRoo/ ConformalsupergraLity

wherewe havekept the Gauss—Bonnetcombination(eventhoughit vanisheswhenlinearised)to show how it appearsin the reductionof (4.4) to d = 6. Finally wemust introduce the redefined field (4.3) for h’. This gives anothercontribution

containinghss.We find

(R,1~5~(h’))2= (R,1~

4~(h))2+ (0 hss)R(h)+ ~(0 ~

= (R,1~AP(h))2+ ~(0 hss)2, (4.14)

where we againusethe constraint(4.12). Finally we substituteeq.(4.14) into eq.(4.13), andfind that thecontributionof h55 cancels.The linearisedaction in d = 6reads

~R2 = ~(R,1~AP(h’)) + ~{(R,1~A~(h’))2 — 4(R,1~(h’))2+ (R(h’))2}. (4.15)

The scalar hSS has disappearedfrom the action (and from the transformationrules),andwe obtain the expectedform of the R2-action.As we discussedbefore,(4.15) receives the required torsion contributions from the (~H)2-partof theaction.

Thuswe seethat our result,reducedto d = 6, gives the known actionof ref. [41

with a torsion interpretation,and the Gauss—Bonnetcombination. This implies

that in d = 6, N = 2 conformal supergravityboth invariantsexist.

5. Discussion

The result of sect. 3 shows that the superconformalinvariant in d = 10 exists.This implies that the invariant in d = 4, N = 4 conformal supergravityalso exists,

sincethe two theoriesare relatedby dimensionalreduction.The relationto d = 6,N = 2 waspresentedin sect.4.

The connectionbetween d = 10 conformal supergravityand off-shell d = 10Poincarésupergravityis not known beyond the linear level [15]. The connectionwith the on-shell supergravitytheory is known, andwasdiscussedin detail in refs.[5,14]. If suitable restrictions are imposed on the fields of the Weyl multiplet theremainingcomponentsrepresentthe degreesof freedomof Poincarésupergravity.A sufficient constraintto trigger thisprocedureis, in our notation,

~[a”bcd] = (5.1)

which is the equationof motion ~r°1R(A)0 . . . a7 = 0 of the six-index gaugefield. By

supersymmetrythis implies that also

~[aTI/’hJc = 0, ~[ORb]C= 0. (5.2)

M deRoo/ Conformalsupergravity 261

In ref. [5] it was found that theseconditionsindeedproducethe on-shellPoincarédegreesof freedom.

In ref. [13] the PoincaréR2-actionwith a six-index antisymmetricgaugefield

was obtained.Schematically,the actionhasthe form

2’=R+aSfp(R2). (5.3)

This action is invariantonly to first orderin a,and requires0(a) modificationsofthe transformationrules. These arise, when in the variation of (5.3) a termproportional to an 0(a0) equationof motion (i.e. an equationof motion arisingfrom the R-action)is obtained.Thereforethe action ..~?~(R2)is invariantby itselfif theseequationsof motion areconsideredasconstraints.In conformalsupergrav-ity we encountereda similar situation,but with the weakerconstraints(2.4) and(2.5).

It was shown in ref. [5] that the transformationrules of d = 10 conformalsupergravitycan be put in exactly the same form as those of d = 10, N = 1Poincarésupergravity.The redefinitionswhich are requiredto put, e.g.(2.1) in theform used in ref. [13], were outlined in sect. 2. Let us assumethat theseredefinitionshavebeenperformed,also in the action(B.2). Then(B.2) is invariantunder the sametransformationrules as ~‘~(R2), but the invariance requiresweakerconstraints.Therefore,if we imposethestrongerconstraintscorrespondingto (5.1) and(5.2) on (B.2), with the redefinitionsmentionedabove,the result of ref.[13]will be obtained.

Is it possibleto makecontactbetweenthe resultspresentedin this paperandoff-shell Poincarésupergravity?The problemis that the transitionto the off-shellPoincarétheoryusingthe establishedmethodsrequiresthe presenceof compen-satingfields to break the superconformalinvariance.In the absenceof suitablecompensatingmultiplets this presentsinteresting, but thus far unsurmountableproblems.

It is a pleasureto thank E. Bergshoefffor severaluseful discussions.

AppendixA

CONFORMALSUPERGRAVITYIN THE ~ =1, A =0 GAUGE

In this appendixwe gatherthe relevant formulae of d = 10 conformal super-gravity. The transformationrulesare

15e,1 = pa~,

~I/’,1 _~Y,1(W)E + ~I~(F,1F(7) — 3F(7~F,1)eR(A)(7),

36A,1,16= 4 x 6! (A.1)

262 M. deRoo/ ConformalsupergraL’ity

Hereand elsewherewe suppressexplicit indiceswith the notation

In thispaperwe usethe notationof ref. [5].In transformationrules and action we prefer to work with the dual of the

supercovariantcurvatureR(A),

Habc = ~ (A.2)

The commutatorof two supersymmetrytransformationsgives, besidesthe generalcoordinatetransformation,a field-dependentLorentz transformation,

[~(c~), 8(c2)] ~

+ ~L(fl l~f—FcfHabc+ ~ (A.3)

Whenthe commutatoris evaluatedon the gravitinothe result contains,besidesthetransformations(A.3), the following additional terms:

[8(c,), ~(E2)]I/f,1 = (A.3) + ,92E2F,1E,F I/lab + 7~E2F~J’a,s~ 1/lcd

— l~l6~2T,1abcd , FahcdFefl/lef. (A.4)

Thereforewe haveto imposea constrainton the supercovariantgravitino curva-ture I/tab’ which in turn implies, by supersymmetry,a constrainton the Riemannscalar,

= 0 (A.5)

J~(w)— ~H°~”~HabC = 0. (A.6)

We now presenta list of definitions of some of the dependentfields andcurvaturesand their transformationrules:

ww,1(e, 1/i)

= w(e) — + — ~ (A.7)

= a(1) — 0~ab Wac(~cb+ ~ (A.8)

M. deRoo / Conformalsupergravity 263

~(A),1...,17 a[,11A,12 ,17] — 8 x6! ~ (A.9)

~ ~Ll/’~ ~V~/’,1 — ~/~(F[,1F(3) + 3T~13)F[,1)I/1~JH(

3), (A.10)

15(0,1 = ~r,1Ipab +

+ 2~�FcI/l,1H + ~F~~I/J,1Hcde, (A.11)

— 8x6! ~~al...a5~a6a7]’ (A.12)

lSHabc = ~/~J~Fabcefl/lef, (A.13)

= — cd~?ah(W)

+ ~J(F~a1~’(3) + 3F(3)Tia)EDb]H(

3)

+ 12 x 48 ~ + 3F~~~~a)

x (Fb]F~

3~’+ 3F~3~”Fb])cH(3)H(3)’ (A.14)

Here 1~(w)is the supercovariantversionof R(w).The derivativeD is supercovari-ant.

The curvaturessatisfy the following Bianchi identities:

= 0, (A.15)

DtaiRa2 a8] = 0, (A.16)

D’~Hacd= 0, (A.17)

DEal//bc] = p(3) + 31(

3)F[a)l/lbc]H(

3). (A.18)

In the calculationof the superconformalinvariantwe useeverywherethe Lorentz-

covariantderivative~. The Bianchi identity for H, eq.(A.17), is thereforeusedinthe form

e~a~,1(w)Ha= bilinearfermions. (A.19)

264 M. deRoo / ConformalsupergraLily

The Bianchi identity for the gravitino curvaturetakeson the form

~[cI/’ab] = ~ + 31(3)F[a)l/fbc]H(

3)

1~efi p4 ‘l’[c ab]ef

— ~1~\/~(F[aF(3) + 3F(3)T’~a)I/lb~c]H(

3)

+ 12 x 48 (~aT~

3~+ ~ + 3F~3~’Fb)4/lc]H(3)H(3)~

+ trilinear fermions. (A.20)

The four contributionsto (A.20) are referredto as B,—B4, respectively,in tables4and5.

The constraint(A.6), the identity (A.20), the relation

[.~,1(w), ~1~(w)] H~1~= _3R,1c[ad(W)Hl~~~]d, (A.21)

andthe supercovariantizationsin

= + + 3F(3~F[~)l/i~,]H(

3), (A.22)

link the different steps in the determination of the invariant action. This isexplainedin detail in sect. 3.

Appendix B

THE COMPLETERESULT

In this appendixwe presentthe completeresult for the d = 10 superconformalaction. The action is first written in the form of an ansatz,in which arbitrarycoefficientsappear.Thesehavebeendeterminedwith the procedureexplainedinsect. 3. Finally we write the action again, this time substituting the calculatedvaluesfor the coefficients.

The ansatzfor the N = 1, d = 10 conformal supergravityaction, in the gaugeA = 0, c/i = 1, reads

e ~‘= a0R,1,f~uiR~~l)+ ~ + bO1/J”~’~~i/lab+ c0~1,1HQ~~~1H

+R~{m,c/laFb~cd + m2~eFabel/lcd + m31/JaFcdel/Jbe)

M. deRoo / ConformalsupergraLily 265

+Rab{fl,I/IcFaI/lbc + n21/FaFc1/’be + n3ç!JcFcadl/Jbd}

+Habc{pI~dFa1/ibc +P2~aFb1/ldc

+P3~eFdaeI/lbc+ P4I/’e’~abeI/’dc+ P5I/’e”dab~ce+ P6~eTabcI/~de

+P7l/faFbce~de+P8~dFabe~ce+P91/JeFabcef~df}

+ih ~ R abR ab~ /LV Ap

+R~{tlHacdI/lbFcI/Jd + t

2Hacd~cFb1/ld+ t3HacdIPeFbcdef~f

+ t4HcdeIPaTbcdef~f}

+Ra~{t5HSb,I/J,I7~qld + tbHabel/lc~’eI/fd

+t7HabeI/JfFcdefg~g+ t8Haef~bFcdefgI/Jg+ t9Hefg~aFcdefg~b}

+Habc{dlIpabFd~cd + d2~adFb1/Jcd+ d3lPdeFabc~de

+d4I/JadFbce~de+ d5~adFbdel/lce+ d6lIldeFabcdf~ef}

+elR”~~’HabeHcde+ e2R~HacdHbcd

+f1 HdaeHbcegjdjqabc

+ ~ + ~

~

~ + q31/’5

1~~1/’b~+ q4I/JcFcadI/Jbd}

+H~mH~m{q51/JcFdI/’ab + ~

+ ~ + q9I/’a

1’cdeI/’be

~ + ~

+qI2~eFabc~fl/Idf+ q131/Jerabcdfc/lef}

+ HabcHdef{q14~aTbcd~ef+ q15~

Tbde~cf+ ~I6~a1defI/’bc

266 M. deRoo / ConformalsupergraLily

+q,7I/JmFabcdmc/Jef + q18I/lmTabcde1/lfm + q19I/1aFbcdemI/lf~

+q20I/JaFbdef~I/J~~+ q2,I/J,nTabcdemnI/lfn}

+HbCd~f~lHbcdUlI/laT~l/lf

+ H’~92JbHCdaU2I4f~

Tj I/If

+H9~dH1We{U3I/JaFbI/Ie + U4I/JbT~l/lC}

+H~bHCde{U5I/JaFbI/Jc + Uol/1b1

51/JC+ U7I/IaF~I/Jb)

+HC~~~dHl~euSI/JfFabcfgI/pg

+H~fbHCdeu9I/lfFabcfgI/fg

~ + U11lI/clabdegl/lg}

+H~f~cH~f{~12~a’~bcdegl//g + U 13I/’c”abdegI/’g + Ul4lIJdFabcegl/Jg}

+ H ~dHef~{u,5~fFabcde~g + Ul6lIJdFabcefI//g + U17~aFbcdef(//g

+U18~aFbdefg~c+ U191/JdFabefgl//c+ U20~iFabcdefgij~j}

+ l/IaFbI/lc{W HabeHmnpHmnp+ w2HabmHcnpHmnp

+W3HaCmHbnPHmnP+ W4HamnHbnPHcpm}

+ /le’ a/see f(w5HabcHmnpHmnp+ w6HabmHcflpHmflp

+ W7HamnHbnpHepm}

+ 1pepabcdf,//{W8HabCHdmnHemn+ w9HabmHcenHdmn)

+ ~ ~ + w1,HabcHdfmHegm

+wl2HabfHcdmHegm + Wl3HafgHbcmHdem}

+ qi,F~b~~fEhil//1w14HabcHdemHfgm. (B .1)

In the termsbilinear in the gravitino (thosewith coefficients t, u and w) no termswith F~

3~or appear.The reason is, that theseare symmetric in the twogravitinos, and thereforegive upon partial integration ~/‘~ +~~I/l,1 in the varia-tion of the action. Sincethereareno othersourcesfor variations,thesetermsallhaveto vanish.

M de Roo / ConformalsupergraLily 267

In (B.1) the coefficientsu4, u14, u15 andu18 canbe setequal to zerobecausebypartial integration the correspondingterms can be rexpressedin terms of othercontributions to the ansatz.A similar mechanismfor the p-contributionswasdiscussedin sect.3.

We now rewrite the aboveansatz,with the valueswe obtainedfor the coeffi-cients.The arbitraryscale(a0) hasbeenset equalto one.This next formula is themain resultof this paper:

e’ ~=R,1R’~’~’ — 2R,1~R~V+ ~ —

+R~”{ ~I/JaFbl/Jcd + ~/‘e~’abeI/~cd+ ~I/JaFcdel/lbe}

+ I/1a

1’ctPbc — I/1cT~ad4’1bd)

+ ~dHabc{3~/5p~ — f~/~e”abc~/~de+ ~I/’aT’bce’/’de — ~I/1dI’abe I/lce}

+ iV~e I ~ . . •t~6/LPAPAa...aSR,1~~RApab

+ ~R~”{~HacdlPbFcI/Jd — ~Hacdl/Jc FbI/Id + ~HacdI/JeFbcdef I/if

— ~HcdeI/IaFbcdef1/Jf}

+ V R’~”{ — ~HabeI/JeFcl/Jd — ~HabeI/IcFeI/id

— ~HaefI/JbF.cdefgI/Jg — ~HefgI/IaFcdefgI/ib)

+ V~H01C{6I//abFdl/Icd— ~)/‘ad~’bI/’cd — 17)/’deTabcI/’de

3~adFbde1frce— /IdeFabcdf~ef)

+ 9R~1’HaCdHbcd

+ 27~HdaeHbce~dHabc

+ ~HabeHcdeHacfHbdf

+ HmnPHmnp~baFbI/,ab

+H~~mnHbmn{~I/IcF~/IbC— ~I/a1’cI/”bc + ~/‘c1’cadI/’bd}

+H~~mH~~m{— 7I/~cT’d’/iab — llI/JCFal/Jbd

4’l’e abe’l’cd 4 ‘Pe ace’Pbd 4 ‘Pa cde’Pbe

268 M. deRoo/ ConformalsupergraLily

+ l41/laFbcel/Jde+ ~/~ef’abc I/Ide

+ ~eFabCef h/Idf — eFabedfl/Ief}

+ SI/Ia[’bdei/Icf + //a1~defI//bC

13 —— ~I/’m1’abcdmI/’ef — ~I/’m[’abcdeI/’fm — ~I/laFbcdern I/~fm

+ a[’bdefm~cm— f~mTabcdemnI/’fn)

~H~bHcd”I/IaI}l/if

— 4H~2YdH”~eI//aFbI/Jc

— H~~~1bHCdeI//aTbI/c

— ~H~iYdHbCeI//fFabcfgIPg

+ ~Ha e~1bHCdeI/JfFabCfgI/’g

+Hf~fH{~I/JaFbcdeg~g + 2~cFabdegI//g)

_H5bffCHdCfIPaT~hcdegI//g

+HabydI~Ief~{+ ~1/’a[’bcdefI/’g

+ ~/‘aTbdefgI/’c — Tj~I/1iFabcdefgij1/ij}

+

1[~paFbI/,c{ — ~HabcHmnpHm,ip + ~HabmHcnpHmnp

+3HaemHbnpHmnp— 4~HamnHbnpHcpm}

+ /1eFa~~~/1f{~SHabcHmnp Hmnp — ~HabmHCnpHmnp}

+ l~I/1e~~”I/1f{ — ~HabcHdmnHemn — ~HabmHcenHdmn}

+ f~a~1~g{ ~Habc HdemHfgm — ~HabcHdfm Hegm

+ ~HabfHcdm Hegm — 32’~afgH/scmHdem}

— ‘ /J.Fa(Jef~iJI/J.H~Hdem Hfgm~ (B .2)

For convenienceof thereaderthe termsin (B.2) areorderedin the samewayas inthe ansatz(B.1). Undoubtedlythe result canbe simplifiedsomewhatby absorbing

M. deRoo/ ConformalsupergraLily 269

some of the terms with explicit gravitinos into supercovariantizations,or bycombingtermsby usingidentitiesfor the F-matrices.Sincewe haveno systematicway of proceedingwith suchefforts,we preferto presentthe result in the form it

was obtained.

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