SUPERGRAVITY P. van NIEUWENHUIZEN To Joel Scherk 0370 ...

In memoriam Joel ScherkJoel Scherk died on May 16, 1980. Although 33 years old, he had made many very importantdiscoveries in dual string theory and supergravity, as will be manifest in this review. It is striking tonote how logical and sustained the development of his thinking was.His first important contribution was the renormalization of the one-loop open dual string amplitude.Then he developed the idea of the zero-slope limit when all masses except the lightest tend to infinity,and found that this leads to quantum field theories. In particular the closed string sector yielded generalrelativity, and from then on he was fascinated by gravitation. The one-loop contributions of the openstring model in d = 26 dimensions were shown to correspond to a new ghost-free set of states,corresponding to closed Strings and interpreted as Pomeron states. One of these states was described byan antisymmetric tensor gauge field, which is nowadays of interest. Then he started wondering whathappens with the extra dimensions in our four-dimensional world. This led him to the idea ofspontaneous compactification of space-time, extending ideas of Kalusza and Klein.In supergravity he started by solving the matter coupling problem and worked out the super-Higgseffect. Then he constructed the N = 3 supergravity, and both the SO(4) and the SU(4) versions of N = 4extended supergravity. The latter model was suggested by taking the zero slope limit of the d 10dimensional closed string dual model with fermions. The d = 10 open string model yielded in the zeroslope limit N = 4 supersymmetric Yang—Mills theory. In the SU(4) model he found a noncompactSU(1, 1) global symmetry, and in the N = 2, 3, 4 models he gave a complete treatment of the dualityand chiral symmetries. He then formulated N = 1 d = 11 supergravity, which yields the N = 8 d = 4model upon dimensional reduction. He also developed the idea of spontaneous symmetry breaking bymeans of dimensional reduction, and worked out the particle spectrum. There he found a masslessvector boson and proposed the idea that it yields the (testable) force of antigravity.Despite this impressive record, Joel was a relaxed and very friendly person. He was among the bestand most pedagogical speakers and his articles were crystal clear. He was always open to discussionand had a very developed sense of humour. He also contributed actively to the popularization oftheoretical physics by radio talks, lectures for lay-men and articles in popular scientific magazines.This short review was written by his closest friends. We will miss him deeply.

192 P. van Nieuwenhuizen, Supergravity5.4. Construction of multiplets using the closed gauge al- B. Majorana spinors 362gebra 321 C. Symmetries of covariants built from Majorana spinors 3645.5. The Wess—Zumino approach to supergravity 324 D. Fierz rearrangements 3645.6. Chiral superspace 329 E. Two-component formalism 3655.7. Gauge supersymmetry 335 F. Supermatrix algebra 3696. Extended supergravities 343 G. Superintegration 3716.1. The N = 2, 3, 4 models 343 H. The super Jacobian is the superdeterminant 3746.2. The N = 8 model in II dimensions 349 I. Elementa;y general relativity 3766.3. The N = 8 model in 4 dimensions by dimensional J. Duffin—Kemmer—Petiau formalism for supergravity 377reduction 353 K. Glossary 3796.4. Spontaneous symmetry breaking in the N = 8 model by L. Conventions 380dimensional reduction 355 References 3816.5. Auxiliary fields for N = 2 supergravity 359 Addendum7. Appendices 361 6.6. Supergravity and phenomenology 395A. Gamma matrices 361PrefaceSupergravity was discovered four years ago [234,120, 235] and interest in this new theory has beengreat from the beginning. However, the rapid development of the field and various technical barriers(such as properties of Majorana spinors) have prevented many interested physicists from starting activeresearch. This report hopes to provide an introduction which will allow the reader to go from noknowledge at all about supergravity up to the point where present research begins. It is written as acourse for graduate students and thus the focus is more on explaining than on reviewing. Mostimportantly: it is self-contained. We have tried to discuss all major areas of supergravity, including theadvanced aspects, but we have placed the emphasis on supergravity as a gauge theory. In this way, thisreport might be useful as the basis for a graduate course in gauge fields and quantum field theory.Rather than giving some general discussions or refer to the literature for details, we have tried to beexplicit. Thus we have not shied away from working out details, and this should be helpful to thereader. Actually, large parts of this report were tried out on students at Stony Brook, and in this way wediscovered where more explicit work was required. We must confess that as a result this report is ratherlong, and we have had nightmares in which the publisher provided it together with a wheelbarrow.In ~rder to describe the work of our colleagues in detail, we have visited many of them for longdiscussions. Thus the preparation of this report took more than two years. It is a pleasure to thank themfor their cooperation. In addition, we have asked all of our colleagues to send us a list of theirpublications as a basis for the references in this report. This report is dedicated to our superfriends, whohave created light (and heat) by their enthusiasm. Let us hope that nature is equally enthusiastic and hasreserved an important role for this most elegant of all gauge theories.1. Simple supergravity in ordinary space-time1.1. Introduction and outlookThe gravitational force is the oldest force known to man and the least understood. Some reasons forthis curious fact are that gravitational experiments are difficult to perform, due to the smallness of thegravitational constant K, while at the theoretical level the dimensionful character of the gravitationalconstant has prevented the construction of a predictive, i.e. renormalizable, quantum theory of gravity,

194 P. van Nieuwenhuizen, Supergravityquarks (color, flavor, lepton number, etc.), but differing from these by having spin 3/2 rather than spin1/2.Whether massive or massless, the new fermionic gravitational fields do not modify the classicalpredictions of general relativity because the exchange of fermions leads to a short-range potential. (Thisfollows, for example, from dispersion theory and is due to the necessity to exchange two fermions toproduce a potential.) On the other hand, at short distances and thus at high energies, supergravitydiffers radically from ordinary general relativity and is a better quantum theory. Infinities in theS-matrix in the first and second order quantum corrections cancel due to the symmetry betweenfermions and bosons. Whether these cancellations persist in all higher order quantum corrections is anopen question at present. Thus supergravity is not finite because infinities are renormalized away, as inthe nongravitational theories, but because in the S-matrix infinities cancel. Due to this finiteness,supergravity has predictive power, just as ordinary renormalizable models.In simple (N = 1) supergravity, bosons and fermions occur always in pairs, and these pairs are theirreducible representations of the supersymmetry algebra. One such pair is the graviton and gravitino.Other pairs are spin 1/2 and spin 1 doublets (“photon—neutrino system”) and spin 0 spin 1/2 doublets.One can add as many of such matter doublets to the spin 2 spin 3/2 gauge doublet as one wishes.However, a special feature arises when one adds one or more spin 1 spin 3/2 doublets to the spin 2 spin3/2 doublet. The resulting theories are the so-called extended (N = 2,. . . , 8) supergravity theorieswhich, in addition to the space-time symmetries which any gravitational theory possesses, have as manyFermi—Bose symmetries as there are gravitinos, namely N. In addition, they have ~N(N — 1) spin 1 realvector fields, and lower spins. Most importantly, they have a global U(N) group of combined chiral-dualsymmetries which has nothing to do with supersymmetry. Under these symmetries, fermions rotate intothemselves or y~times themselves, and curls of gauge fields rotate into their dual curvatures. The 0(N)part of this U(N) group rotates the gravitinos into each other according to the orthogonal group 0(N),and simultaneously the photons into each other, etc.The first of the extended supergravities is the N = 2 model [511]. It realizes Einstein’s dream ofunifying electromagnetism and gravity, and does so by adding two real (=one complex) gravitino to thephoton and the graviton. It is in this model that the breakthrough in finiteness of quantum supergravityoccurred: an explicit calculation of photon—photon scattering which was known to be d~vergentin thecoupled Maxwell—Einstein system yielded a dramatic result [512]:the new diagrams involving gravitinoscancelled the divergencies found previously (see fig. 1).One should view the N-extended supergravities as extensions of pure general relativity (i.e., withoutmatter) in which the single graviton, N gravitinos, ~N(N — 1) vectors etc., all rotate into each otherunder either supersyinmetry- or U(N) symmetries. Since these extensions are irreducible, one calls thesetheories pure extended supergravities. Thus the graviton is replaced by a new superparticle whose“polarizations” are the graviton, gravitinos, quarks, photons, electrons, etc. This unification of allparticles into one superparticle leads also to a unification of all forces, because forces arise by exchangeof particles. One can also couple matter to these theories, but only N-extended matter to N-extendedsupergravity (N-extended matter having before coupling N global supersymmetries and a global U(N)invariance).The importance of the extended supergravities lies in the fact that the global 0(N) group can begauged by the ~N(N — 1) vector fields which are present in the pure N-extended supergravities[108,225].This leads then to theories with two coupling constants: the gravitational coupling constant Kand the internal 0(N) coupling constant g. The latter should describe the nongravitational interactions,but in order that this be possible spontaneous symmetry breaking must first occur such that g splits up

P. van Nieuwenhuizen, Supergravity 1957 1’ V 7SPIN = 3/2c,Jc,JMAXWELL-EINSTEINTHEORY0))IIIIcSPIN = 3/25’ 5’ 7 7(137 1>T5’~>T~~~Finite probability for loop diagrams in a quantum theory of gravity is obtained by including gravitinos in the interaction. The diagrams shownFig. 1.here are those for the interaction between two photons. The first diagram, labeled Maxwell—Einstein theory, consolidates all the one-loop diagramsthat involve only gravitons and photons; the contribution from these diagrams is equal to an infinite quantity multiplied by the coefficient 137/60.Five one-loop diagrams involving gravitinos can be constructed; each of them is proportional to the same infinite term multiplied by the coefficientsshown in brackets. Only the sum of the diagrams is observable, and adding the coefficients shows that the sum is zero. Hence the infinitecontributions of the gravitinos cancel those of the gravitons and the diagrams have a finite probability [2491.into the different couplings of the electroweak and strong interactions. This has not yet been achieved,and is of prime importance for the phenomenological implications of supergravity. It is tempting tospeculate that supergravity is the high energy limit of the nongravitational interactions. In that case, onewould restore the symmetry by going to high energies and find that bosons and fermions have equal(vanishing?) masses.

196 P. van Nieuwenhuizen, SupergravityIn extended supergravity theories with unbroken supersymmetry all the companions of the graviton,including of course the vector gauge fields, but also the “matter fields” (spin 1 and spin 1/2) are (asthe graviton) massless. We already mentioned that even in pure extended supergravity models there is amechanism of symmetry breaking that gives masses to the gravitinos and also to vector and matterfields, leaving the graviton massless [404,405]. Even in the situation when supersymmetry is broken, thebreaking is soft and mass relations exist that insure that quantities such as the one-loop corrections tothe cosmological constant, which are finite when supersymmetry is unbroken, still remain finite whenspontaneous symmetry breaking is introduced [406,603]. Whether in these massive models the one- andtwo-loop corrections to the S-matrix are finite is not known.The N-extended supergravities are only viable for 0 N 8 because for N> 8 spin 5/2 andhigher-spin fields enter onto the stage, while also several spin 2 fields are needed. The reason is that themassless irreducible multiplets of N-extended supersymmetry constrains: one helicity + 2 (the graviton),N helicities + 3/2 (the gravitinos) and so on, according to a binomial series. For acceptable fieldtheories, one adds the CPT conjugate irreducible representation, starting with one helicity —2, Nhelicities —3/2 and so on. Only for N = 8 is one representation enough, since it starts with A = +2 andends with A = —2. For N> 8 the tower of helicities which starts with A = +2 extends below A = —2.Recently, interest in higher-spin theory has been revived [258,259], and acceptable free-field actions forall higher spins have been found. However, the coupling of spin 5/2 to gravity and to other spins isinconsistent [153]and there exists no satisfactory coupling of fields with spins exceeding 2. Thus natureseems to allow only spins ranging from 0 to 2, and we are stuck with the extended supergravities with0N8.The N = 8 theory with spontaneously broken symmetry comes rather close to being a unified fieldtheory of Nature. In the special case when some of the still arbitrary mass parameters are set equal, itsinvariance group is SU 3 x U1 x U1 (see fig. 2). The 8 gravitinos are massive and describe a spin 3/2-quarkand a spin 3/2 lepton. The spin 1/2 fields are all massive and contain precisely the up, down andcharmed quarks as well as an electron and gluinos (spin 1/2 color [i.e., SU3] octets) and sexy quarks (aspin 1/2 color sextet). (A strange quark is eaten by a massless gravitino with charge —1/3 according tothe super-Higgs effect.) This N = 8 model seems to be the first model that predicts rather than assumesfractional charges (the correct +2/3 and —1/3) for quarks and integer charges (1,0) for leptons. Of the28 vector fields, only 10 are massless, namely the photon and the gluons, while for the last masslessvector boson corresponding to the second U(1) factor, several interpretations exist: either it describesthe weak neutral current (Z°),or it describes “antigravity”. (At short distances the exchange of thisvector, being proportional to the gravitational constant K, cancels the gravitational exchange betweentwo particles and doubles it between two antiparticles. Hence it might describe a gravitational neutralcurrent. It makes testable predictions [391].)Great excitement has followed the recent discovery in this N = 8 model of an extra global E7 andlocal SU(8) symmetry. It is speculated that the local SU(8) group could even produce spin 1 and spin 1/2bound states which are needed for grand unification. (In its simplest form, W~,W, the p~and T leptonand all neutrinos would appear in this way. Moreover, among the fields which gauge this local SU(8)would be W~and W, just as the ordinary spin connection viewed as a function of other fields (tetrads,gravitinos) is not itself an independent propagating field, but still gauges Lorentz rotations.) If this canbe shown to be the case, the N = 8 model comes indeed very near to being “the model of Nature”.It seems then that supergravity is our best hope of unifying the forces of nature and the threeprinciples mentioned before. It is a tight scheme with very high predictive power, which is exciting butmight be its downfall because one cannot adjust masses or couplings at will to fit the experimental data

P. van Nieuwenhuizen, Supergravity 197J2 1(0) graviton~ U 3f~)graviquark U 1(0) gravitino18(0)gluons1 ~0)y U 3(.Z) •3(i) • 3(1)11(0)1 ~ —1)d3(—• 6(2) U 1(—1)e •3(~)u 18(0) • 3~)cgiuinos0 18(0) 3(2) i(—1) 6(~) 6(-~-) 1(—1—H~ U’ I I ~ •2m 3m E E E E Mass1(0) C’) c”i E E mI I + + +Fig. 2. Spectrum of the spontaneously broken, N = 8 model with SU(3)® U(1)® U(1) invariance. The SU(3)representations are underlined (ex: 3);the bracket indicates the electric charge [3911.without destroying local supersymmetry. Once one accepts the principle of a local supersymmetrybetween bosons and fermions, the theory follows in a unique way, and we shall derive it in this way. It isof course not excluded that supersymmetry is only one attempt at quantization and unification in a longhistorical row, and that it will eventually fail, so that one must consider another symmetry. All one cando is work out its consequences and hope for the best.From a theoretical point of view, supergravity is the gauge theory of the space-time symmetries andsupersymmetry. The tetrad e~ and spin connection w ~‘ gauge the space-time symmetries while thegravitino (the massless real spin 3/2 field) gauges supersymmetry. All these symmetries form anirreducible algebra that is an extension of the concept of ordinary Lie algebras because it containsanticommutators as well as commutators. This algebra is called the super Poincaré algebra. Thesymmetry between bosons and fermions is usually formulated in terms of anticommuting numbers(Grassmann variables, we will give matrix representations), but this is only a matter of conveniencesince one can formulate the symmetry also without them. A very conservative way of consideringsupergravity is to interpret it as just another quantum field theory in ordinary space-time, on the samefooting as for example quantum electrodynamics, the only new feature being that it contains spin 3/2fields.A very interesting and much less conservative description of supergravity it to consider it as a theoryin superspace. Superspace has as coordinates not only the ordinary x — y — z — t, but in addition 4Nspinorial anticommuting coordinates O’~(a = 1, 4 and i = 1, N). There are various approaches to

198 P. van Nieuwenhuizen, Supergravitysuperspace, based on different geometrical ideas and using different base and tangent manifolds andgroups, but they all have in common that the notion of Grassmann variables (anticommuting c-numbers) as coordinates is essential. The approaches using ordinary x — y — z — t space, are entirelyequivalent to the approaches using (f, 9~x)superspace, but superspace methods give a better geometricalinsight. They are also less accessible due to algebraic complications and for that reason we devote a fairamount of space to them.1.2. Physical arguments leading to supergravity1.2.1. SupersymmetrySupergravity is the gauge theory of supersymmetry and supersymmetry is the symmetry betweenbosonic and fermionic fields of certain Lagrangian field theory models. In this section we show howsimple physical arguments lead one directly to supergravity.Let us begin with supersymmetry. In order to be definite, consider the simplest model of globalsupersymmetry, namely the Wess—Zumino model [581]with a scalar A, a pseudoscalar B and spin 1/2field A. All fields are real and massless, and we choose a real representation of the Dirac matrices (i.e.,~1 ~2 ~,3 real while ~4 = iy°is purely imaginary). We shall later on define Majorana spinors asgeneral representations of the Dirac matrices; however, the Dirac matrices y’,.. . , are alwayshermitian. Thus, for real A, we define A = A at this point. We work with real (Majorana) spinors andthese are more fundamental than complex ones (two real make up a complex spinor). One could,however, also describe supersymmetry in terms of complex spinors; in our example one would needthen four real (or two complex) scalar fields.The action is a sum of the Klein—Gordon actions and the Dirac action2—= —~(8~A)~ — ~(d~B)~AJA, A = A t~4 (1)This action is globally supersymmetric, since it is invariant under the following set of transformationrules6A = ~iA, ~B = —~ iy5A, 5A = -~,8’(A— iB’y~)= ~t74 Ys = ~1~2~3~4 = Hermitian and imaginary. (2)(In fact this model is even invariant if one drops B everywhere. However, adding mass terms or a selfcoupling, invariance is lost. Also the algebra without B is different (see below).) Note that thetransformation rules preserve the reality:(eA)* = A = (A)a(7 4)as( yl = — ~( 7 4y’~(Ay~ = iA (3)since ~4 is antisymmetric in a real representation. We have already used that and A anticommute,about which more below. The action is clearly hermitian, since A a and A~ anticommute. This is requiredby the spin:statistics theorem, but it is amusing to note that if Aa and A ~ were commuting, theLagrangian A/A would be a total derivative. The proof that the action I = fd4x ~ is invariant, is rathersimple. Varying the first terms in ~‘, one has —~“A)t9~(ëA)while varying ÔA = ~IAe and ÔA =(ÔA)Ty4 = ~eT(/A)Ty4 = —~ë(/A),one find from the last term in (1) —~A//A+/ië(/A) (IA). Similarly

P. van Nieuwenhuizen, Supergravily 199for B. Hence the Lagrangian varies into a total derivative,= ~ K~= —~iy”[/(A — iy 5B)]A (4)as a little calculation shows, using ëÀ = A ,and ëy5A = Ay~~(see appendix C).The transformation rules in (2) reveal a set of general properties which any globally supersymmetricmodel has (by global we mean constant ). The general law which rotates a boson into a fermion, is ofthe form (suppressing indices).ÔB=iF. (5)Requiring covariance of this equation, i.e., requiring that both sides have the same properties, onederives(1) Statistics: Since fermions must be anticommuting because of the spin-statistics theorem, also e” areanticommuting parameters. An explicit matrix representation can be given in terms of Wigner—Jordanabsorption matrices a for fermions (not also creation operators) as follows4)a. (6)= ~ ~q3a1 ~a = ~ ajf3aj, i~aj= (13h’The Dirac bar does not act on a1, the sum over j runs over an infinite set, and the f3a~are ordinarynumbers. A similar representation can be used for spinor fields A, and one defines the t operation onthe product by (ajak)t = aka1, not a ~a. Since such spinor fields anticommute, we treat A (x) here as aclassical field, and not in second quantization form. Formally this is the h -40 limit in the quantummechanical (anti)commutators. From this explicit representation one reads off the following(anti)commutation relations between fermion fields F, boson fields B and parameters{~a ~$}= {Ea j~}= ~ F} = ~ B] = {F, F} = 0. (7)Thus the e” and Aa span a Grassmann algebra and not a Clifford algebra. (A Clifford algebra has “one”on the right hand side, for example the Dirac matrices span a Clifford algebra. Perhaps one could useClifford algebras for supersymmetry.)(2) Spin: Since bosons have integer spin and fermions have half-integer spin, the parameters “ havehalf-integer spin. The simplest case, spin 1/2, is realized in supergravity. Models where e’~has spin 3/2have negative energy states [44]. Also the gauge fields for parameters with spin 3/2 are spin 5/2 fieldsand for these no consistent coupling exists [538].(3) Representations: Since e has spin 1/2, the irreducible representations must involve at least oneboson and one fermion. It was shown by Salam and Strathdee that for massless states, this is in factenough: the representations consist of a boson—fermion doublet with adjacent helicities (A, A + 1/2). Forexample, in our model there are two irreducible representations, namely (A +iB, (1 + y5)A) and(A — iB, (1 — 75)A). As the reader may verify using (2), A + iB and (1 + y5)A transform into each other;similarly for the second doublet. These doublets are in fact the irreducible representations of a new kindof algebra, namely superalgebra, see below.(4) Dimensions: Boson fields have dimension one and fermion fields have dimension 3/2 in orderthat the action be dimensionless (in units h = c = 1). The reason is, of course, that boson fields have two

200 P. van Nieuwenhuizen. Supergravityderivatives in their action while fermion fields have only one (see eq. (1)). Hence the dimension of is—~ according to eq. (5). However, this means that in the inverse transformation oF — Be there is a gap ofone unit of dimension. The only dimensionful object available in flat space with massless fields to fill thisgap is a derivative. Thus, purely on dimensional grounds (omitting indices)oF=(8B) . (8)The reader may check this relation for the special case of eq. (2).(5) Commutator: Consider now two consecutive infinitesimal global supersymmetry transformationsof a boson field B (the same arguments hold for F). The first rotation transforms B into F, but thesecond rotates F back into 3MB. Thus two internalsupersymmetry transformations have led to a space-timetranslation. In any globally supersymmetric model one always finds the same relation[8(e1),8(e 2)]B = ~(i2y~Lei)8,~B. (9)For F the same result holds, see subsection 8. We leave it as an exercise to verify this relation for eq. (2)using the appendix. Hence, global supersymmetry is the square root of the translation operator, and oneexpects that local supersymmetry is the square root of general relativity, just as the Dirac equation is thesquare root of the Klein—Gordon equation. This we will show in more detail in subsection 3.5.We derived this extremely interesting result from the simple fact that bosons and fermions havedifferentdimensions. For a discussion of the complete algebra of symmetries of globally supersymmetricmodels, see below. For further details of global supersymmetry, see the Physics Report by P. Fayet andS. Ferrara.1.2.2. SupergravityWe now turn to local supersymmetry and ask what would happen if we make “For example, in eq. (2) we definelocal, hence “ (x).OA = ~(/(A — iy5B)) (x) without ~terms in OAsince A is considered as a matter field. (Quite generally, only gauge fields have in their transformationrules derivatives of parameters (in this case (x))and only on the parameters to which they belong. Forexample J~ewill only occur in the supersymmetry gauge field (the gravitino) but not in, say, theYang—Mills field transformation law.) Since for constant the action in (1) is invariant under (2), itfollows that for local e the variation of the action must be proportional to 3,~e(x)4x (3~,,i(x))j~. (10)SI = J dHerej~is the Noether current, and can be obtained from (2) and (4) as follows-. o.~’ 1-A. 1Indeed, for constant one has O5~’= ~ + (8./8~)5~= 3L~Ke~so that on shell (where

P. van Nieuwenhuizen, Supergravity 201= 3,.~’/8~) the Noether current is conserved. For local one ho &~‘= ~ + (8~,ë)s~withsome s”. Hence for local ,=(~— a~ + ~ [~-o~] = 8LLK~+(8~i)s’~ (12)and equating the 8 terms it follows that sM = j~.Therefore, as in any gauge theory, one may start aniterative procedure (“Noether method”) of adding extra terms to action and transformation laws suchthat in the end the complete action is invariant. The first term is, as always, the coupling of the gaugefield to the Noether current.Since gauge fields are obtained by adding a world index p~to the parameter, the gauge field ofsupersymmetry is a vectorial spinor (or spinorial vector, if one prefers) i/i,.~ called the gravitino. Indeed,its spin content is~ (13)where the two spin 1/2 parts correspond to 8 i/i and y~~/‘.(The decomposition into 3/2 + 1/2 + 1/2 isnonlocal. As far as the Lorentz group is concerned, 1/~~ transforms as(~,~)® [(i,0) + (0, ~)]= (1, ~)+ (~,1)+ (0, ~)+ (~,0).) (14a)This corresponds to the local decomposition(1 ±7~)(t/i,,, — ~/y. i/i) and (1 ±y~)(yy~i/i). (14b)Thus one adds to the action in eq. (1) the following Noether couplingIN = Jd~x(—Kç~j~) (15)and requires that ~ -= 8,~, (x)+ more. (Note that ~ is off-diagonal in the fermionic fields t/’~.and A;this is due to the fact that ~. is a gauge field.) However, since fermions have dimension 1 3/2 and has~ + more wheredimension —“more” should 1/2, bea fixed dimensional at the next coupling iterativeK appears. stages. We will write therefore St/i1. = KThe dimensionful coupling is the first indication that local supersymmetry needs gravity. A secondargument in this direction follows from the result in eq. (9). For local (x),one expects something of theform[8(e~(x)),0fr2(x))]B —~~(e_2(x)y1.e1(x)) 81.B. (16)(The exact result is given in subsection 9, eq. (8).)In other words, one expects translations 81. over distances d 1. = ~i 2(x) y1. 1(x) which differ frompoint to point. This is the notion of a general coordinate transformation and leads one to expect thatgravity must be present. Thus local supersymmetry should lead to gravity. The reverse is also expected,since if one starts with a constant parameter and performs a local Lorentz transformation, then this

202 P. tan Nieuwenhuizen, Supergravitvparameter will in general become space-time dependent as a result of this Lorentz transformation.Hence local supersymmetry and gravity imply each other and this is the reason for the namesupergravity. However, we stress that this is a heuristic argument only, and that, for example, constanttranslations when written in terms of curvilinear coordinates look also like general coordinate transformations.Nevertheless, the results confirm this heuristic reasoning.A third argument is more solid but requires a little bit of work. If one considers the variation of IN in(15) then one finds new terms due to SA, SB, OA in 5jN~For the terms quadratic in A and B one finds1.~(B)) (17)3(I+It.~) f d~x~(~1.y~ )(T1.~(A)+ Twhere T,,~(A)= 8 is1.A 8~A— ~S1.~(8AA) 2 the energy momentum tensor of the field A. (The AB termswill be discussed in subsection 3, eq. (10).) Such variations can only be canceled by adding a secondNoether coupling, of a new field g1.,. to the Noether current of translations, ~T1.P,and requiring thatunder local supersymmetryK K—Sg1.~= — ~- t/i~y,, — ~- t/i~y1. . (18)Since 1915 we know how to do this kind of Noether method to all orders in iteration: by covariantizingwith respect to gravity (81. -4 D1., °1.v -~ g1.~and a V’~in front of the Lagrangian). Varying in—~V’~ 81.A 8~.Ag 1.~’the metric, one finds ~Og 1.~. T 1.~(A)and the field g 1.,. in eq. (18) can be identified withthe metric tensor.We have now learned two things: First of all, gravity must be present and since in supersymmetryone necessarily has fermions present, the gravitational field is described by tetrads ~ rather than themetric field g1.~.The relation between them is g1.1. = e m 1.e”~8mn~ For details of tetrad formalism, seeWeinberg’s book. Secondly, the transformation law of the tetrad must satisfy (18). This does notdetermine 8e~uniquely, but consideration of the AA terms in 01N yields as unique result8e~= ~K~yml/i,1. (19)(see under (32)). For the transformation law of the gravitino one expects1D= K 1.E, D1. = ~ + ~1.mnUmn (20)since we are in curved space. Here v1. m7’ is a spin connection (see appendix I).The gauge fields of supersymmetry should in particular form a representation of global supersymmetry.(More precisely: the physical states with helicities (+2, +3/2) and (—2, —3/2) described by e ‘ andt/’~.should form two doublets.) This puts another constraint on 8e~and Ot/I~P,which is in agreement with(19) and (20) as we now discuss.The helicity (2, 3/2) representation of global supersymmetry is given by= ~ (iy1.çb~+ ~yvt/ip,), ~*1. = (W1.mht)ijTmn (21)

P. van Nieuwenhuizen, Supergravity 203where L stands for linearized and are constant. Since boson fields have dimension 1, one defines thespin 2 field by h 1.~= (g1.~— 81.~)/K in which case the K in (21) disappear. (Thus h1.~is the quantumgravitational field.) Thus (21) agrees with (19). This result was derived in ref. [281]by using only thealgebra of global supersymmetry, the so-called super-Poincaré algebra, which we now derive anddiscuss.If one defines for the fields A, B, A in eq. (1) 80(e)A = [A, ~aQ”]mPm] and 8M (A mn = ~(Amn~Fmn)a~A$etc.,= [A,~Am”Mmn] and defines the onePoincarécan findgenerators the commutators by S~(~)A of Q,=P 81.A and=M[A, using ~ the Jacobi identities. For example[S~( ~),8 0( 2)]A = ~(j2ym~1) 8mA = ~~2ym 1[A, Pm]=[A, 20],ët0] — [A, iTO], 20] = [A, [ë20, eQ]] (22)where we used the Jacobi-identity the last equation. Removing ë2 and ë1 (using that ~ = t 0C$~andthat and 0 anticommute) one finds that the following equation holds when acting on A{Q~Q$} = ~(ym)a 8$pm (23)8(C_l)where Cap~ = Oa5’ and CF = —C (see appendix B). In this way one derives the followingalgebraic commutators and anticommutators when acting on A or B. (For A one only finds the sameresults if one adds auxiliary fields, see subsection 8.)[Mmn, Mrs] = SnrMms +3 more terms[Pm, Mrsl = SmrPs — SmsPr, [Pm, P~]= 0 (24b)F/la 7) 1_A F/la Al 1—( \a çL’I ~T~JU, [~‘ ,lVJmnJk(Jmfl) gt~./(Qa Q$}(ymC_l)a$pm. (26)This is the super-Poincaré algebra. It is a closed algebra since all Jacobi identifies are satisfied (or since anexplicit matrix representation will be given in section 3).The results in (24) are of course the ordinary Poincaré algebra, but (25) states that the charges ~aare constant in space and time, and carry spin 1/2. An explicit representation of 0” for the model in (1)is 0” = fd3xj°~~” with the Noether current given in (11). The really interesting relation is (26); it is thealgebraic analogue of eq. (9). One may check (24—26) by expressing all charges in terms of theircorresponding Noether currents, and using canonical (anti)commutation rules. (For these rules forMajorana spinors, see subsection 2.14.) Since 0” have spin 1/2, a physicist is not too astonished to see in(26) anticommutators, but the appearance of anticommutators leads one outside the domain of ordinaryLie algebras and into superalgebras.1 =The matrix C in (26) must satisfy CYC 7m.T in order that the following Jacobi identities aresatisfied{[Mmn, 0”], Q$} + {[Mmn, Q$], Q’~}= [Mmn, {Qa, Q~}]. (27)Physicists recognize this matrix as the charge conjugation matrix; for further details see appendix B.(24a)

204 P. van Nieuwenhuizen, SupergraviiyOne important conclusion is that if eq. (25) holds, one cannot have a term with M 1.~in the right handside of eq. (26) since it would violate eq. (27) with M1.~replaced by P1..Thus, as follows from eq. (21), the law 81/il. = (1/K)D1.E combines both the local gauge aspectSI/il. = (i/K) ~ + more, and transformation law of the global spin (2, 3/2) multiplet. (A similarsituation occurs of course in Yang—Mills theory with 8w1.a = (D1.A)~= 81.A a + fa& W1.bA C) For thetetrad e~,one expects with eq. (21) that Se tm 1. ...i~jymt/, or Oe m 1. = ~(ëy!~t/Jm)ebl.. The difference is aLorentz rotation, Se tm 1. = ~y mt/,” y”t/Jm)eb —1., but fixing 8t/i1. = (1/K)D1. , one has no freedom indropping Lorentz transformation from Se tm,.. alone. One argument in favor of Setm tme~1.is not). However, there is a better1,. argument,= ~ and is that that it islinear obtained fields by gauging (ym is the constant, algebrabut of y1. global = y supersymmetry in eqs. (24—26) [586]. This we now discuss.Associating with every generator a gauge field and a parameter as followsV1. = e tm 1.P~— lwmnM + Kt/J1.aQ” (28)mPm + ~Am”Mmn + jQ0 (29)A = ~one defines gauge transformations bySgV1. = D1.A = — [V1.,A]. (30)Applying these rules to the super Poincaré algebra, one findsm~= 0. (31)8e~=.~~ymt/, ~ = 8l. +2(5h~0,~n , 8wThe minus sign in (28) was chosen such that w in St/i,. has a plus sign. The result Ow,.tm” = 0 will beexplained when we discuss the 1.5 order formalism. Hence, group theory leads one straight toSetm,. = ~K ymt/J,.. (32)Also the Noether method leads to this result. If one would put the Spin 1/2 action in eq. (1) in curvedspace-time, variation of the tetrad yields81(A) = (Oetm1.)T~~(A). (33)On the other hand, there are also AA terms coming from varying A and B in 1N, and again one is led to(32). It is a good exercise to derive these results.Finally we come back to the question how to use non-real representations.Majorana spinors are spinors which are real in a real representation of Dirac matrices. As shown inappendix B, a Majorana spinor is in general complex, but satisfies the following relationA~“A 8C~~‘A~(y 4)~~. (34)For a real representation, C -~ y~,but in general this relation does not hold. For gravitinos, ~,. =~pTc From now on we will use general (hermitian) matrices Ym and Majorana spinors. The diligentreader may check that all results of this section remain true if one replaces real spinors by Majoranaspinors.

P. van Nieuwenhuizen, Supergravizy 2051.3. The gauge action of simple supergravityAs we have seen from gauging the super Poincaré algebra, there are three fields in the gauge action:the tetrad e~,the gravitino t/i~ and the spin connection w ~ Since e,.m and t/’~ already form aboson—fermion doublet of states with adjacent helicities ±2and ±3/2,the field w,.~”should not bephysical, since otherwise there would be too many bosonic states. This is a familiar situation in ordinaryrelativity where the Einstein—Cartan theory starts out with two independent fields e,.m and w,.””’, butwhere subsequently w ~“ is eliminated as an independent field by solving its nonpropagating fieldequation, as a result of which w,.~”becomes a function of e~.This solution w,.tm”(e) happens to satisfythe tetrad postulate (“the tetrad is covariantly constant”)0,.ern~+w,.mn(e)en~_F~,.(g)ern~ =0. (1)(When one defines 00(e) by (1), one speaks of the tetrad postulate. We, however, do not postulate (1)but solve w from its field equation.) Also for supergravity we will use the same Einstein—Cartanformalism. The method of finding the dependence of w,.~”on other fields by first treating it as anindependent field in the action and then solving its (always nonpropagating) field equation is calledPalatini formalism.For the bosonic part of the gauge action of supergravity it thus seems appropriate to take the usualHubert action R (and not, for example, R2, since in this case w is propagating and cannot be eliminatedalgebraically). Let us forget for a moment the gravitinos and concentrate on R alone. Usually onedefines the Hubert action in terms of connections 1. (such that under parallel transport OA,. =F~PA~ dx” and under round-transport 8A~= ~ ~x~’dx”) as followsR =— .~r’-’ — .~F~+ r~r~— r~r~PpIJ — p~ ~ ~7 VP ~ vp ~ ~‘p’ Au.At this point one can use second order formalism and define F~.to be the (symmetric) Christoflelsymbol F~~(g), or one can treat F~.as an independent field (not necessarily symmetric) and solve itsfield equation 8I/8F”,.~= 0. The result is then F~,.= F~~(g) if there is no matter present. If there ismatter present, one gets a different answer, and ~ is no longer symmetric.From a more fundamental point of view, the connection F~,has less meaning than the spinconnection w,.””. As we shall see, group theory leads to the following curvatureR,.V”’” (w) =00mn — 300ma + tmC00 fl — 00tmC00PZ (3)There is an interesting connection between (2) and (3). If one relates the (nonsymmetric) symbol ~ tothe spin connection w~”(both considered as independent fields) by the same “tetrad postulate” as ineq. (1) (thus formally: the completely covariant derivative of the tetrad field vanishes)tm”e~~— ~ = 0 (4)3,.e’ + w,.then a direct but tedious calculation yields the following interesting resultR,.~~~(w) = R”,,.~(F)~ (5)

206 P. van Nieuwenhuizen. SupergravityIn particular, one can thus write for the Hilbert action with e = det e~ *~(2) = - ~-~\/~R (g, fl = — ~ eR (e, w)R(e, w) = em”e”~R(w) (6)The factor ~is a matter of convention (the kinetic term occurs then with the canonical normalization),but the K2 is needed to give the action the correct dimension. It is this form, in terms of e and w, whichwe will take as bosonic part for the gauge action of supergravity. Let us remark that if one solves fromeq. (6) (thus without matter) the field equation for w~”,then one finds w,.”” = w,.”~.”(e).With matter, theaction with w ,.“~“(e) yields a different result from the action which one obtains if one starts with 0),.”” asan independent field and then eliminates it by solving SI/Ow,.””’ = 0. There is no strict rule which forbidsany one of these choices. However, as we shall see, it is the choice w,.””’ (thus Palatini formalism) whichleads to the simplest description of supergravity.We now turn to the fermionic part of the gauge action. We expect that its linearized, free-field partshould be quadratic in ~c and contain only one derivative. As weshall show in subsection 13, there is onlyone such action which haspositive energy and that is the same action as writtendown in 1941 by Rarita andSchwinger in their Physical Review article2’ = —21.i/i,.y5y~8~i/,~. (7)It is unique up to field redefinitions i/i,, = tt’u + Aycry 4’ with A —~.We also expect that the fermionicpart of the gauge action of supergravity should have the local linearized gauge invariance 8’/’~=Indeed, the Rarita—Schwinger action as in eq. (7) (thus with A =0) has this invariance. This leads one totake for the full nonlinear gravitino action in curved space the extension of eq. (7) to curved space,~(3/2)= Pu~757~,D~ D~t/i~ = (0,, + ~~~“Om~)t//u. (8)Note that D,,t/i~occurs as a curl, due to the m inc-symbol. front of the Since action. ~““’ The is amatrix density 7~= (hence ~1~2~3~4 always is equals constant, 0,±1),we do not. need the density e =but y~depends on the tetrad field. Let detus e,. now discuss the curl D~t/i~ — Dull/p. In principle one needs twoconnections in D~t/i~, a Lorentz connection for the flat indices and a connection for the curved indices.The role of these connections is to ensure that Dat/i,., transforms as a tensor as indicated by its indices.One has four choices: w or w(e) for Lorentz connection, and F or F(g) for the other connection. Thechoice appropriate for local supersymmetry is w and F(g). Any other choice would do as well, but onewould need extra complicated terms in the action. Since F~(g)is symmetric in p, a~it cancels from thecurl.It gives faith in this series of arguments, that also the Noether method which we used to couple theWess—Zumino model to supergravity, leads to these results. We recall that we had arrived at the matteractionJ + JN = J d4x — [(8,.A8~A+ 8,.B8,.B)g1.” + Aø(w(e))A — Kt/J,. (/(A + iy5B ))y~A] (9)* A simple way of writing this action is 2(D”E”,,) (D’E”,,)— (D~E”)(D,.E”) with E,.~=

P. van Nieuwenhuizen. Supergravity 207and that we had found that with 8t/’,. = (i/K) 8,. + more and Se,.tm = (K/2)ëytml/J,. all terms of order K0cancelled, while also at the order K level the AA and BB terms cancelled. If one now varies A in theNoether term and considers the AB terms, one finds8(1+ JN) = d4x ~(1i,,.y5v~&”~’)(ys )(8,,A 8,B). (10)According to the Noether method, one must cancel these terms by adding either terms to the action orto the transformation laws. The only way to cancel eq. (10) is to partially integrate 8,,A or 8GB, and tointerpret D 1.’as the gravitino field equation. In that case, adding to Si/i,. a term(iK/4)7 0(i/,,.y5y~) ’°”56(A8,.B), one cancels part of the 21/iu/IA8B terms in to eq. the (10). action (The and other using part, the Si/i,. the terms= (i/K) with 3,.e leading 8 , arevariation.) cancelled by Inadding this way a term one again of thefinds formthat K the gravitino action is given by eq. (8). Strictly speaking, atthis point in the Noether procedure, one would have found in eq. (8) the spin connection w(e), but notyet w(e, t/r). However, at the next levels in K one would find the extra terms which turn w(e) intow(e, t/’). This possibility to derive the gauge action from requiring gauge invariance of the matter actionis peculiar to supergravity (it is, for example, not the case in Yang—Mills theory). It is due to the factthat matter fields appear in the gauge field transformation laws if one does not use auxiliary fields. Infact, auxiliary fields can be viewed as Lagrange multipliers which eliminate such matter terms, seesubsection 9.Thus we have arrived at the following proposal for the gauge action of simple supergravity. Theaction is the sum of the Hilbert action for the tetrad field and the Rarita—Schwinger action for thegravitino field. In both, the spin connection is considered a dependent field. We now turn to a proofthat this action is in fact invariant under the local supersymmetry transformation 8e~’= (K/2)ëyml/i,.and Oil’,. = (1/K)D,. in which one takes again the same spin connection w. No independent transformationlaw for w ~“ need be given since w is a function of tetrad and gravitino obtained by solvingits own field equations.1.4. Palatini formalismIn this subsection we solve the field equation of the spin connection. As a result, we will find torsioninduced by gravitinos. The analysis is very simple if one uses a trick *: the rewriting of the Hubert actionas~2) = _L. ~ (1)where we used that 1.””’ mncdetm5R,,U~d(w) = Dp800u~— D~8w,,~~ cd — ~ cd + ce~ d de~p (1~~ —t1p (1~u 00p 00ue (Op (0,, e~1.e”p= 2e(e~~ed” — eCed). Varying the spin connection, one findsPartially integrating, one finds (using the remark under (4) on page 224)* This trick is really writing differential forms with the indices explicitly shown.

208 P. van Nieuwenhuizen. SupergrarirvS2’( 2) = ~1. ~‘ mncd (D,,e~)e ~ 800cdD ~ tm~1~ mn (3,.e,. — 0,,e,. w,, er,..Note that the derivative D,, always contains only the spin connection, but not a connection F~,..As aresult, D,.e’ is non-zero.Next we vary the spin connection in the Rarita—Schwinger action.82’312 = ‘ ~“~~(tl/yyut/i) (80)cd) (4)Since l/i,.y5y~u~dt//,, can be written as vector terms and axial vector terms by decomposing Y~O~cd(seeappendix A)l/i,.757,ArcdtIJ,, = ~tlJ,.yS(e~~yd — ed~yC)llI,,+ ~et~,, bcdml/i,.ymt/,,, (5)and since l/’,.ylydt/i,, is symmetric in p~and o- while ~,.y’~i/i,, is antisymmetric (see appendix C), we find82’3/2 = — 8 1. ~“ ncdm (~ym~, )en ~ cd (6)Comparison of eqs. (6) and (3) thus yieldstm,, — D~etm,.= ~- (çj~ytmç/i) (7)D,.efor the field equation of the spin connection. Due to our rewriting of 2’2 as in eq. (1), one can easilyread off this result, since in eq. (6) and in eq. (1) the two -symbols are in common.With the first tetrad postulate (which is really a definition of w(e)) that 0,.em~+w,.m~(e)—= 0, one can proceed to solve from eq. (7) the spin connection ~ itself. To this purpose itis useful to introduce the contorsion tensor K,.””’ byma — mtm( ~+ ma(0,. —00,. ~, K,.w,.rn~(e) = ~e~”(8,.e~~ — 8~e~,.) — ~ (8,.emv — Opem,.) — ~em”en~(8~e~,, — 8,,e,~,)e’,.. (8)With (7) and the first tetrad postulate, one finds0,.etm~+ wtm(e) (,~ v) 0K2- 9— Kr,,,,. = ~ tli,,y~tl/~. ( )This equation is solved in the same way as one solves in general relativity for F”,.~(g)in terms of ~that is to say, one considers the identity— K~~,.)+~ — iç,.m)+ ~ — Içptm) = 2K,.,,,~. (10)

P. van Nieuwenhuizen, Supergravitv 209Substituting (9), one arrives at~ ~/i)= ~ — ~,.ynl/im + ~rny,.ll’n). (ii)Using the second tetrad postulate, 8,.e~+ ,.a~ — F~,.e”~ = 0 one can obtain an expression for F”,.~from eq. (ii). In general one defines torsion as the antisymmetric part of F”,.~= ~(F~— F~,.). (12)Clearly, F”,.~— F~,.= —K,.”~+ K~a,.,so that the torsion is given byS”,.~= —~-(t/i,.y”çfi,,). (13)In a similar manner as in eq. (10), one can solve from D,.(w(e))e’ —~s~-* ii = 0 the dependence ofw~,b(e)on the tetrad field. (Another way is to solve directly the tetrad postulate 8,.etm,, + w,.m”(e)e~~ —= 0.) Putting this result together with eq. (ii), one finds~ i/i) = ~ — R,.mn + Rmn,.)R,.p,m = —8,.e~~+ 8~e,,,,.+ ~- çl’,.y~i/i~. (14)The symbol R,.,b.m is defined to be ~ similarly for the other two R-functions. Clearly, the spinconnection w~”’(e,t/i) is supercovariant, by which is meant that if one transforms it under localsupersymmetry transformations, then it contains no 8 terms. Indeed, the terms (—~8,.ë)y”i/i,.comingfrom 8(—8,.em~)cancel against the 8 terms from Oil’,.. The symbols R,.~.rnare supercovariant bythemselves. That the solution of SI/Ow = 0 is supercovariant is an accident. For example, in theextended supergravities this equality does not hold, see subsection 6.2.1.5. Flat supergravity with torsionIn Einstein—Cartan theory, the Hubert action has a well-known symmetry f d~xeR(e, w(e)+ r) =f d~xe[R(e, 00(e)) — T + 1. (rAA,. )2] under w,.”~’1.V~T’~~0)ab + r,.~thwith T,.”~’= — ~ba but further arbitrary.The proof is trivial if one writes the terms linear in r in R,.,~flb(w(e)+r) as D,.(w(e))rpab — p.~*-~ 71 sinceone may add a Christoffel connection to D,.r~~5(because it cancels anyhow in the curl) and having thusobtained the full covariant deriv,ative, one can partially integrate to obtain a total derivative.In supergravity a similar identity holds. It is a matter of a small computation to show that [207]2’~ 2~(e, w(e, ~/~)+r)+2’312(e, t/’, w(e, t/i)+ r)= 2’~2~(e, w(e, I/i)) + 2’312(e, i/i, w(e, t/J)) + ~-~--~ (r,,~~r”’~ — (TAA,.)2) + total derivative. (1)Since w(e, I/i) is an extremum of the action, terms linear in r cancel.

210 P. van Nieuwenhuizen. SupergravityOne can now take a particular choice for r, namely ~ab =0),.ab (e, i/i). In this case, the Hubert actionvanishes, and the gravitino2/2)i/i,.y5/f~actioninloses eq. its (14) spin of the connections. last sectionFrom one finds the result that one forcan w,.” rewrite in terms the of action thecurls 3,.e”~— 8~e”,.— (Kof supergravity as an action without curvature R but with torsion termsI = J d4x [—~e~””çti,.y 1. + ~R~AA]. (2)5y~8~i/i,, — ~R~va — R,.paR”The objects ~ = —0,.e~,.+ Ovea,. + (K2/2)c~,.yat/Ipcan clearly be interpreted as the supercovariantizedtorsion tensor —D,.e~,,+ Dpea,. + (K2/2)tII,.yat/iv with vanishing spin connection. In other words, one canreinterpret supergravity (just as general relativity) as a teleparallelism theory: since w,.°1’ can beconsidered as being zero, one can define parallel transport over finite distances and not only in aninfinitesimal neighborhood. However, this is simply a rewriting of the same theory, and is rather acuriosity than a result of fundamental significance.1.6. The 1.5 orderformalismThe first proof of gauge invariance of the action used second order formalism. By starting withwab(e) Freedman, van Nieuwenhuizen and Ferrara found that extra il/i/i terms in 8t/~,.and extra K2(t/il/i)2terms in the action are needed by the Noether method to obtain complete invariance [500].The actionand transformation laws were the same as if one had started with 2’ = 2’(2) + ~3/2) and 8t/i,. =K1D,.(00) ,and replaced00ab by the w,.ab(e i/i) which solves 01/Ow,.” =0. From the expression derivedbefore for ~ t/i), one finds for the variation of ~ i/i) according to the chain rule (thus insecond order formalism)500,.ab(secofld order)= ~ (ë~yi,tI’,.a— Yal/i,.b — 7,.t/iab) (1)tm,. = (K/2)iytmI/i,..where ~/‘ab e~~e~(D,.i/i~ —A reformulation using D4,.) first order and Se formalism simplifies the action. By taking w,.” to be an independetitfield from the start, Deser and Zumino [120]showed that one can find a law for S00,..ab suchthat ,~(2) + ,~(3I2)viewed as a function of ea,., t/,a and w,.” is invariant. In addition to the previousresults Se”,. = (K/2)iy”t/f,. and Si/i,. = K 1D,.(w)e, they found that one needs5w,.a~(fir5torder) = ~E757,.Il’ab + ~ 75(7 lIlAbCa,. — 7~kil~Aaeb,.),— cdt’cd~1— Cab ‘,Since, as we shall see, the gravitino field equation impliesi/i,.,~+ y51/i,.p = 0, y~~p = 0, (3)it follows that on-shell (1) and (2) are equivalent. Off-shell they differ, however.The simplest formulation of supergravity is undoubtedly a mixed case, combining the virtues ofsecond and first order formalism [517, 586]. The basic observation is so obvious that it often causes

P. van Nieuwenhuizen, Supergravity 211problems to understand it. If the spin connection (or any field for that matter!) satisfies O1/0w,.~~0,then in the variation of the action one has according to the chain ruleOI(e, t/i, w(e, ç!i))= 8e~I + 51~!L1 +~, (Sw(e~~) ö~+t5~~(e, t/’~Se). (4)Se ~ e.w(e,~P) e.çi~ Si/i SeSince, however, w,.a~(e,t/i) satisfies its own field equation, one can drop the last term due to 01/Sw 0identically after inserting w = w(e, i//). Thus, one can replace in the variation of the action thecomplicated expression which results when one applies the chain rule to (o(e, i/i) by zero! In otherwords, one only needs to vary the explicit tetrad fields and gravitino fields, but, although w(e, t/’) iscertainly not invariant, one may put Ow = 0 in the action. This is the same result as found in subsection2 by gauging the super Poincaré algebra, where we found that Sw =0. Hence, it is the 1.5 orderformalism that makes contact with group theory.It should be noted that in the extended supergravities, this observation has been crucial in obtainingthe actions. When one deals not with actions, but, for example, with gauge algebras, then one still needsthe non-zero law 8w(e, i/i) in eq. (1).One small technical detail 1.l/i,..These in our notation. two differ When sincewe D,.e~is write ~I’abwe non-zero. mean ea e~(D,.ç(i~— Dpi/i,.) and notDaili~— The result D~çfr~with in (2) il,~= will e~ be needed when we discuss constrained Hamiltonian systems, hence we brieflydiscuss its derivations. Varying the spin connections in the Einstein action, one finds (see subsection 4,eq. (3))~ mn~pp~,o- \ira r\I!sUW~ I~ mnrs jy....,.e ,,,~2e,,Varying St/i,. = (1/K)D,J in the Rarita—Schwinger action and partially integrating, one picks up a term~ .P1~u(D,.empXiysymDpI/i,,). (6)The sum of both terms must vanish, and one solves Ow,.,,,, from501pm,, — ~ = E757m1/ipg,S00pab = abCdS(0p (7)in the same way as one solves for the Christoffel symbol..1.7. Explicit proofof gauge invarianceWe now present an explicit proof of the gauge invariance of the gauge action. Using the 1.5 orderformalism, we must vary the fields indicated by arrows. From now on we will put K = 1,2’ = ~ — ~ ~“ii,.y5y,,Dp(w)ili,,. (1)1~1~1’ t1~ 1’As we shall see, the variation of the Hilbert action yields as always the Einstein tensor times 8e”,.. Thevariation of the gravitino fields yields the commutator [Dr,D0.] ,hence a curvature times . Since there

212 P. van Nieuwenhuizen. Supergravilyare not enough indices available, one is guaranteed to find again the Einstein tensor times an expressionof the form i/i . Finally, as we shall see, the contribution of 8(y~)cancels the extra terms due to partiallyintegrating Si/i,. = D,.i. We now give the details.The variation of the explicit tetrads in the Hilbert action yields with Se”,. =82(2) = ~ (iy”i/i”)G,,,, G 1~= R~a— ~ (2)where R,,a = R,.va~e”~.We have used that Se”” = —~ -y”t/i° which follows from 3(e~,.e’~”)The variation of i/i,, yields a factor[D,,,D,,] = ~ *))~~~ (3)This one proves easily, using that the Lorentz generators ~ab satisfy [gab c,.cd] = 8bc0.ad + three moreterms, obtained by antisymmetrizing in (a, b) and (c, d). Varying i/i,. and partially integrating D,.E~onefinds again a commutator [D,.,D~]t/i,,plus an extra term due to the fact that D,. does not commute with~YV~For the two curvature terms one finds023/2 = ~“~[~YsY,,~’d— 75y~o~~ai/i,.]R~, (4)after relabeling of the indices in the second term. Using that for Majorana spinors (see appendix C)YSYVUCdI/i,. = t/i,.y~o~~~yu (5)= 0.one finds with YpOcd + Ucd7~ = cdbaysye,and using the identity= (S~’+ 8~’+ S~j — c ~ d)e (6)where 5~’= e,, ebeC”, that varying the gravitino fields yields823/2 = e (~1.ya )G (7)1.y” (see appendix C), the terms with the Einstein tensor in eqs. (2) and (7) clearlySince cancel. iy”i/P = —i/iNote that since we use 1.5 order formalism, we did not vary the spin connection in 2(2)• All that isleft is the terms due to 8(y~)and the terms with (D,.y,) obtained by partially integrating D,.E SinceD,.ë = ~91,L — ~ (8)after partial integration D,. commutes with ‘y~(since [‘I’s,0a~] =0) but for the commutator of D,. with y,.one findsD,.(y~D,,i/i,,)— y,,D,.D~i/i,,= {0,.y,, + ~y~]}D~i/i,,= (D,.e”~)y0D,,./i,,. (9)

P. van Nieuwenhuizen. Supergravitv 213In fact, as illustrated in this example, one has generally [D,.,‘yr] = (D,.e”,)y~.Thus the remaining termsare82312(rest) = ~ ‘~“ (i/i,.75yaDpi//u)(~iy”i/i~)+ ~ ‘~“’ (iysyaDpiIiu)(D,.e”~). (10)For the second term we substitute the torsion equation of subsection 4 eq. (7) and find~ “~“’ ( y5y~D~i/i,,)(~ifr,.y”i/i~). (ii)For the first term in eq. (10) we use a Fierz rearrangement as discussed in appendix D. The generalformula yields—~/i,.O1t/i~)(~ëy” )O1(—~ ~””y5y~D~i/i,,) (12)but since only for O~= ‘y, and 2iu~~the factor i/i,.O1t/i~is antisymmetric in ~i and v while y”o~~y~0(see appendix A), one finds for the first term in eq. (10) using y”y,y~y~= 2y,y~Clearly, eq. (13) plus eq. (ii) vanish.This concludes the proof of gauge invariance. We have shown that the action in eq. (1) is invariantunderSe”,. = ~Eyal/i,., St/i,. = D,.(w(e, çlr))c. (14)In the proof we have used 1.5 order formalism twice: Once by not varying w(e, t/’) in eq. (1) and oncemore by using the torsion equation in eq. (10).One can rewrite supergravity in terms of differential forms, and thus make the structure moretransparent. As an example we reproduce the preceding proof.The action is written as(13)2= ~~flkI A ~~‘I’s)’ A Di/i (15)where77k1 = ~O A O”( klmn det e), i/i = t/i,,O”, ‘y = 7,0”,Di/J = di/i + ~wa~o””t/1, 00ab ~mab° and [1k1 = ~Rmn~0~A 0”.One may think of 0” as e”,. dx 1..Since SDt/i = DOt/I and Sf’l” = Dow ‘~one has upon partial integration using Si/i = De,02= _~S0mA Ilk! A ~k! — ~71k!m A DO”’ A500k1 —Ys7 A DDçIi — ~ëy5y,,Dt/in Do tm— ~t/iA ~‘5y,,Di/iA ~ — I’~)~ A 0k,lfr)&o (16)since ‘Ilkim = OIik!. Now DDçfr = 0abt/h A f1ab and evaluating, as before, i/’ A y~yA o~~i/i one finds a term

214 P. van Nieuwenhuizen, Supergravityml/Iwith the Einstein tensorand a term with the dual of the Riemann tensor‘qk,m A 11~’~A ëyA Dt/I A 77~b~1-Using 00~= ~ the terms with the_Einstein tensor cancel. The w field equationexpresses the torsion DOtm (=—D,.e”, — Due”,.) in terms of t/’yt/iDOtm mi/i. (17)—~-~,~k!m1/i A ysycrkllIi t/yThus one is left with02 7,)’ A Dt/i A flab11 — ~iYSYm A Di/1DOtm — ~I/i A ysy~DçfrA ~ym~fr (18)From here on, the proof proceeds in the same way as before.The Lagrangian density varies into the following total derivativeS2”d,.K~, ,c1.—ëy1.a~”D~tfr,,. (19)This shows clearly that supersymmetry is not an internal symmetry, but a spacetime symmetry, justlike general coordinate transformations, where 02 = 8,.(2~~). In both cases 82 = 0 on-shell.Finally, we note that the linearized action is, of course, invariant under global transformations whichare linearized in the fields. However, not every action quadratic in fields which is invariant under globaltransformations which are linear in fields, can always be extended to an action which is invariant underlocal transformations (which usually then becomes nonlinear in fields). A counter example [478] isN = 1 supergravity in d = 11 dimensions with as fields a tetrad, a gravitino and a photon ~1.8. Auxiliary fields in global supersymmetlyIn globally supersymmetric models such as the Wess—Zumino model, one needs auxiliary fields inorder that the algebra of global symmetries closes. It is only with auxiliary fields that one can obtain atensor calculus, see section 4, while also the quantum theory becomes much simpler with auxiliaryfields, since in this case the transformation laws are linear and one can easily derive Ward identities forone-particle irreducible Green’s functions. For supergravity the auxiliary fields are not only needed forthe same reasons, but in addition one needs them in order to apply the usual covariant quantizationmethods of Feyman, De Witt, Faddeev—Popov, ‘t Hooft and Veltman and others.Let us begin with global sypersymmetry. That one needs two scalar auxiliary fields for theWess—Zumino model might be expected by counting field components4 fermionic components A”2 bosonic components A, B.Thus one needs at least two extra bosonic fields in order to have equal numbers of bosonic andfermionic field components off mass shell. The action reads2= —~[(o,.A)2+ (a,.B)2 + £8’A — F2 — G2] (1)and it is rather easy to show that it is invariant under

P. van Nieuwenhuizen, Supergravily 215SA = ~iA, SB = — ~iiy 5A, OF = ~UA, SG == ~(/(A — iy5B)) + ~(F+ iy5G)c. (2)Since the dimensions of the nonpropagating fields F and G are 2, one must have a law SF -~(OA )e andsince F = G = 0 are field equations, and field equations rotate into field equations, one understands theappearance of IA in SF and 8G. *Let us now consider the commutator algebra. There are three global symmetries: translations P,Lorentz rotations L, and global supersymmetry 0. To give an easy example[O~( ), S71(~~)]A= ~ = 0 (3)reflecting that [0”, P,.] = 0. Next considerm”),SQ( )]A= ~A~”OmnA) (4)[8L(Areflecting that [0”, M,.~]= (o-,.~)”~Q~.More explicitly: S thMab]0( )A= [A, ‘~Q] and SL(A”)A =and with the Jacobi identity[A, ~A’[A,jQ],~AaIab]_[A,~Aa~.M~ab],E_Q][A,[~Q,~A”t’.M’a~]]one finds using the (anti) commutators of the super-Poincaré algebra that[SL(Amfl), S~( )]= SQ(_~Am~~,,fl ) (5)in agreement with eq. (4).In general one has for generators XA and XB and structure constants fABC defined by [XA, XB] =fABCXC a realization of the algebra on fields[8B(aB), SA(aA )]~~,b= Sc(aC = _fABCaBaA)# (6)where 4’ can be any field or combination of fields and a~are parameters.For the commutator of two supersymmetry variations one finds without F and G[S( ~),8(2)]A= ~(e2’y~ i)0,.A, idem B.For A one finds, however,[8( ~), 8( 2)]A= ~(i18,.AX’y” 2)— ~(i1y50,.A )(y”y5 2) — 1 ~-* 2. (7)Upon a Fierz rearrangement, this becomes[5( ~),5( 2)]A= — /4101E2)(y”010,.A — y 1.’y 501y58,.A) —(1 ~-* 2). (8)* One can replace A, F and G by one antisymmetric tensor gauge field :~,,which has three components. In that case no auxiliary fields are needed toclose the algebra, but mass terms and self-couplings are difficult to write down.

216 P. van Nieuwenhuizen, SupergravityClearly, only the tensors with O~= y, survive since i 10~ is~only antisymmetric for 0~= y~ or ~7sO,.~7s= oP.,.,,. One findsbut[S( ~),S( 2)]A=~.(E2y~ 1)O,.A—~(i2y’ 1)(-yJA). (9)Thus, only on-shell where IA =0, does one find the same result as for A and B.In principle one could define a new symmetry S’A = ~‘yJA, 0’A = S’B =0. This would not besufficient to close the algebra, since new commutators involving 8’ will lead to new symmetries 0”, andso on. The only way to obtain a finite-dimensional closed algebra is to introduce F and G. Indeed, theircontribution to (8) is~(iJA) 2—~(i1ysIA)ys 2— 1 ~-*2 (10)and after a Fierz transformation, the IA terms in eqs. (10) and (9) cancel.For any set of fields and set of global symmetry operations which closes on-shell, one can try to find aset of extra fields which lead to closure off-shell. 2 theories) In general, they these propagate. new fields Therewill is some be nonpropagating confusion the inthe literature action, asbut to sometimes whether auxiliary (for example fields in canR have derivatives in the action, and can be gauge fields. Asimple example due to Ogievetski and Sokatchev suffices to clear this point up. Consider [347]2’ = 1.””T,.~O,,A,,. (11)Clearly, the A,, field equation is 1.””8~T,.~ = 0 and can be solved T,.~= t9,.t,, — 3~t,..Similarly, the T,.,,field equation yields A,, = O,,A. Also T,.,. and A,, are gauge fields since the action is invariant under0T,.~= O,.A,, — O,,A,. or SA,, = O,,A. In a Hamiltonian formalism, the moments for A0 and T,.() lead toso-called primary constraints (ph,, — “’“ T,.0 = 0) and this shows that there 2 poles. are noHence dynamical theremodesis noassociated propagationwith between A,~and the sources. T,.0. Also Thus, in the auxiliary propagators fields can onebefindsgaugeno fields k and carry derivatives in theaction.In order that the reader may test his understanding, we close this section with a second model ofglobal supersymmetry: the photon—neutrino system. The action is2’ = —-~F,.~2 — ~AIA+ ~D2, F,.~= O,.B,, — O~B,. (12)and is invariant under SB,. = —~iy,.A, SA = ~o~”F,.,+ ~iy5D , SD = ~iëy~IA.In this case the countingshows that there are four fermionic components A” and only three photon components, hence the needfor the single auxiliary field D. The reason that there are only three components of B,. is that there is agauge invariance in the algebra as follows from the commutator[S~( ,), 8o( 2)]41 = ~~27~ ! O,.4’ + Sgaugc(11 =Sgauge(11)B,. = 3,.A, Ogauge(1t )A = Sgauge(A )D = 0.(13)In general there are as many field components absent as there are local gauge parameters. This is ofimportance for supergravity, to which we now turn.

P. van Nieuwenhuizen, Supergravity 2171.9. Auxiliary fields for the gauge algebraAuxiliary fields are needed in order that the transformation rules of gauge fields do not depend onmatter fi~Ids[see subsections 11 and 12]. If they did, one could not add, without further modifications,two matter actions each of which has been coupled to the gauge action in an invariant way, such that thesum is again invariant. The reason is that the gauge field transformation rules of system I would notwork for system II and vice-versa. However, if one adds auxiliary fields, then the gauge fieldtransformation rules are always the same, independent of matter fields and valid for any mattercoupling system. Let us start by counting how many auxiliary fields we need.In supergravity there are three local gauge invariances: general coordinate transformations G withparameter ~, local Lorentz rotations L with parameter A”” and local supersymmetry transformationso with parameters “. Thus the counting of field components in the gauge action yields the followingresultl6em,. —4 gen. coord. —6 local Lorentz = 6 bosonic fields (116i/i”,. —4 local supersymm. = 12 bosonic fields.Hence there is a mismatch of six bosonic components. This suggests that the gauge algebra will not beclosed, and secondly, that one needs 6+ n bosonic auxiliary field components and n fermionic ones.There exist several sets of auxiliary fields, the most prominent being the set with n = 0, consisting of anaxial vector A,,, a scalar S and a pseudoscalar P.We begin by giving the transformation rules of this minimal set of auxiliary fields. Then we will showthat they close the gauge algebra. Finally, we will discuss how one obtains this result;Oem,. =~- ymi/i,.= -~-(D,. + ~-A,.y5) —(2)RCOVSAm = ey5(R~ — ~7m7 RCOV)‘1/ = —(S — iy5P — i,.4y~).For once we have explicitly shown the K dependence; from now on we will again put K = 1. The gaugeaction is invariant under these rules and reads2(2’(e, w) + 2312(e, i/i, w) — ~ (S2 + P2 — A2,,). (3)2(gauge) =

218 P. van Nieuwenhuizen. SupergravityIn this action S, P, Am are clearly nonpropagating, and one sees that, as in the Wess—Zumino model,1.~OVtheir dimension is 2. Hence they must rotate into the gravitino field equation. The symbolis theRgravitino field equation R1. = 1.”~y5y~D~t/i0 (which we will discuss in the next subsection) but with thesupercovariant derivatives. Hence1.,co~~= “ysy~(D~~~,,,R ~ A,,y — 5t/i,, + ~‘Yu7l1/i~). (4)One obtains the supercovariantization D,.COV of the derivative O,.A of some field A transforming asOA = EB by adding the connection —4iB. Indeed, in that case S(D,.COVA) = 8(3,,A — i/i,.B) is free from3,. terms. For the gravitino this rule yields all m” terms inside D~for eq. (4), two except reasons. that First, we did thenot curltry [Dr,Do.] to addconnections is already supercovariant for the variations by itself 8t/i~= (and Da covariantizing and Swa Si/i,, = D,, by adding a connection, the whole curlwould vanish). Secondly, the spin connection w~mn(ei/i) is supercovariant by itself because it wasalready supercovariant without S, P, A,. while in ~,.mn (e, i/i) no 8i/i terms appear, so that the extra termsin Si/i,. cannot produce 0 terms in 5~mn One finds upon explicit computation [224]= ~ë(y~i/’,.~’~°” — 7ai/’,.b’~°”— y~/i~~CoV)+ ~ë(u~~fl+ 7P~7ab)i/’,. (5)where i/i~’ is the curl between parentheses in eq. (4).Let us now consider the gauge algebra. Without auxiliary fields, all commutators on em,., i/i”,. closeexcept the commutator of two local sypersymmetry transformations on the gravitino [235]. For thiscommutator on the tetrad one finds[S( ~),0(2)]e~,.= ~i27”D,. 1— 1 ~ 2. (6)Guided by the global anticommutator [e~0, i20] = ~i2y~ 1P,.,we expect on the right hand side generalcoordinate transformations with parameters ~ = 2 2Y~ I. Thus we rewrite (6) as2eAO,.( 27e1) + ~(3,.e”AXe2YA i)+ 4 2[7,,,.cd]~1~cd= ~{3(i2~yAEl)}emA+ ~(E27 E1XOAe ,.) + ~(D,.e~A— DAem,.)i2yA l+ ~ 2yEIwAen,.. (7)The first two terms are clearly a general coordinate transformation on the tetrad, with parameter~ The last term we recognize as a local Lorentz rotation with parameter mA —DAem,. ~i2yA =~l/i,.yml/’A iwAmn. The asremaining ~~~~(e terms can be rewritten, using the torsion equation D,.e m,. with parameter~_~(e 2yA l)t/,Aymq,,. and clearly constitute a local supersymmetry variation of e2yA l)t/iA. Hence, we can summarize that, as far as the tetrad is concerned, the following commutatoris valid[8~(e~),So( 2)]= S~(~~)+ SL(~~w,.~”)+ ~(8)r 2 27~ 1.This is the local version of the super Poincaré algebra. As one sees, the P in {Q, 01 = P is replaced by

P. van Nieuwenhuizen, Supergravily 219general coordinate transformations, and this we anticipated in the beginning of this report when wederived supergravity. But there is more. One also finds on the right hand side the other two gaugesymmetries. We also see that the structure constants defined by this result are field-dependent. Onemight speak of structure functions. This is a property of supergravity not present in Yang—Mills theoryor gravity. The moral of this result is that one cannot simply deduce the local algebra from the globalalgebra.We now repeat this calculation for the gravitino, still without auxiliary fields[5( ~),S( 2)]çl’,. = ~(o~flE2XO( i)w,.) — 1 ~-* 2. (9)With the result of subsection 6, eq. (1) for 5~,.mn without auxiliary fields, one finds after Fierzing so thatthe parameters 2 and ~are in the same spinor traceth(ii 0i 2X2~ya~0jy~i/i,.a + OabOjY,.l/Jba] — 1 2. (10)Using that Ybl/i,,a — 7a1/1,.b is equal to7,~JI’ab+ R’ -terms (see the next subsection) one arrives atwhere[S( ~),S( 2)]lfr,. =~(e2yA lXDA4~,. —D,.4’A)+~(ëly” 2)~ +~(ëlo”’ 2)T,.P,,AR” (11)~ = ~ + 2eç~PAy5y (12)eT,.,,,,A = g,.pg,,A +~g,..Ao,,,,—~eE,,,,,.Ay5. (13)The functions V and T will be called the nonclosure functions. The first terms in eq. (11) produce againthe result in eq. (8). To see this, we rewrite these terms as~( 2y lXOAt/s,.)+~{O,.(e2yEl)}t/iA ~~~t9,.(e2~yA lt/,A)One again finds the general result of eq. (8) back, since5 lXwA. cri/i,. —00,. 01/iA). (14)+~(ë2y= ~A0Al/~,,. +(D,.~)l/iA, S0i/i,. =~i’.t~i —.1 si 1 5— tmflI~ ~0 ~ ~mnThus, without the auxiliary fields S, P, A,., the gauge algebra does not close, since there are extra termsproportional to the fermionic field equation in the commutator of two local supersymmetric variationsof the fermion. These results are identical to what happened in the Wess—Zumino model.The other commutators in the gauge algebra close and are given by[öo(fl”), oG(r)J = SG(~”3a77~— ~“3~~) (16)[OL(Wmn), OG(fli] = SL(flaO~Wmn) (17)(15)[SL(wm~), 5L([lmn)1 = oL(—W ~flPII + flm,.~~) (18)

220 P. ran Nieuwenhuizen. Supergraritv[S~( ”), OG(n”)] = S~(~”o~ ) (19)[8~( ”), OL(01m72)] = So(~w””’o,,~ ). (20)tm,. = w~~~ëy~t/i,.As a little help for the reader, we derive the last relation.while —OLOQe”,. =SOSLeoil’,.. The commutator [8~, 8L]e~,.is equal to ~ . wy”i/i,., which is indeed a local supersymmetryvariation of the form Setm,. = klymi/,,. with ~“ = w. To find ‘ we use ‘ = _C_t(i~)T andfind ‘ = ~w o~ .The reader who feels insecure in handling Majorana spinors can always flip ~- . wyml/i,.around to i/i,.ytmw o and compare this with Setm,. = —4Ey~i/i,..All these structure constants are fieldindependent. With auxiliary fields, these commutators remain the same. If one uses first orderformalism with w,.””’ an independent field (transforming under G and L the same way but under 0 witha different law as we discussed before) then eqs. (16)—(20) are unmodified, but for eq. (8) one finds thatit closes on the tetrad only modulo the w fIeld equation, on the gravitino only modulo the w and i/iequations and on the spin connection only modulo all three field equations.With auxiliary fields the gauge algebra closes. In this case the {Q, Q} anticommutator can be mosteasily found if one evaluates it on the tetrad; however, the result is valid for all fields[8~( ~),Oo( 2)] = OG(fl+ S~(—~~)+SL[~,.mn + ~E2umn(S— iysP) i](ü,.ab =00~ab — ~ ,.abcA, ~ = ~~:y~ l. (21)Hence the structure constants now also depend on the auxiliary fields.The geometrical meaning of closure of the gauge algebra is the following. Consider all gaugetransformations acting in the space spanned by all field components, with arbitrary parameters.Considering gauge transformations as vectors in this function space, they define hypersurlaces. Closurenow means that making successive gauge transformations, one never leaves a given hypersurface.The minimal set of auxiliary fields 5, P and A,. was discovered by Ferrara and van Nieuwenhuizen[525],and by Stelle and West [445]. An earlier nonminimal set by Breitenlohner involved two auxiliaryspin 1/2 fields, two axial vector fields, a vector field, and a pseudoscalar and a scalar ~68,69, 160, 70].Let us discuss how one arrives at this result. The tetrad e,.tm and gravitino i/i,.” fit into a vectorsuperfield (see subsection 4.1) as first observed by Ogievetski and Sokatchev [349]H,.(x, O)’ C,. + OZ,. + P,LOO + S,.OiysO + e,.mOy~iy5O+ 900iy5i/i,. +(OOXOO)A,.. (22)The components of H,. should transform under linear gauge transformations since e ~ and i/i,.”transform under linear gauge transformations G, L and 0. As we shall explainOH,. = Dy,.iy5A, I~(1+ y5)D15(1 — 75)A =0 (23)where A” is a spinor superfield and D” = a/80~+ ~(IO)”.look, one finds after field redefinitions thatSC,. = (~‘,., SZ,. = 2,., SP,. = P,. with o,.P’~’= 0, idem OS,.,Working out how the components of OH,.Oem,. = O,t~m + ~ + A’mn, Si/i = (9,.ê (24)

P. van Nieuwenhuizen, Supergravity 221where the hatted symbols are functions of the components of A”. One recognizes the expected generalcoordinate and Lorentz transformation on the tetrad, and the local linear supersymmetry rotation onthe gravitino. It follows that one can use the gauge freedom to eliminate in H the components C,., Z,.,,and the transversal parts of F,., S,.. Hence, one expects as auxiliary fields, 0,.S~,0,LP~and A,..To find how these fields rotate into each other under linear global supersymmetry transformations,one first performs the transformation OH,. = ~GF~L. (see subsection 5.3). Since this will produce nonzeroC,., Z,., etc., one subsequently performs a gauge transformation, which restores the gauge C,. = 0 etc.The sum of both transformations is then the global supersymmetry transformation, which leads to thelinearized version of (2). —To explain why local (linear) gauge transformations are of the form OH,. = Diy,.y4x is invariant under Oh,.~= O,,4,, 5A + we 0,4,. recall since that the inlinearizedsource satisfiesgeneralo,.T”’relativity the coupling f h,.,, T”’ d= 0. In global supersymmetry T”’ is part of a superfield of sources JAA (intwo-component notation) containing also the axial current and the Noether current of global supersymmetry.These sources satisfyDAJ~ = DAS (25)where ~5 is a chiral superfield (DAS = 0). (For the definition of DA, see again subsection 5.3.) IfDAJ~ = 0 would hold, f (DAAA — DAAA)JAA = 0 and one would expect that HAA (the two-componentform of H,.) is invariant underSHAA = (DAAA — DAAA). (26)However, DAJ’~ = DAS, hence we must restrict AA such that AADAS vanishes. All these termsappear in the action and using well-known results of global supersymmetryJ d4x d4O AA DAS = —f d4x d4O (DAAA)S= Jd’~xd2O~D”DA[(D~”AA)S]one arrives at the quoted result for OH,..1.10. Field equations and consistency= fd4x d2~(1YDAD’~AA)S (27)The field equations for the gauge action are obtained very easily by using 1.5 order formalism.Indeed, in the action2’ = —~eR(e,w)—~ ~““l/i,.y 2 — A~,) (1)5y~D~t/f,, — e(52 + Pwe may again consider w,.m”(e, i/i) as an inert field. Thus one easily derives for the field equations

222 P. van Nieuwenhuizen, SupergravitvSPAmO (2)= eG”” — lIAyyai/A~ — ~ e”’(S2 + P2 — A~,,) (3)= ~“~(y5y~D,.,i/i,, + h’syaili,,(Dpe”,,)). (4)The latter term is obtained by partially integrating i/i,.y5’y~D~Oifr,, = (D~8~,,)y5y4,. and vanishes (seeappendix C). Thus the field equations reduce toSPAmO, R~as ~”’~ysyvDpçbcr0 (5)eG”” — ~4’AySya~/iAv = ~, ~ = ~ — ~ (6)We now discuss the question of consistency [120].Since R~= 0 has open indices, one can once moredifferentiate. To show what might go wrong, consider first the case of a complex spin 3/2 field coupledminimally to electromagnetism. The field equation ~““’y~y~(3~— ieA~)i/i,,= 0 yields upon differentiationwith 3,. — ieA,. that F~”y,.i/i~=0 which puts as extra constraint for consistency that either i/i~=0or that the photon be a gauge excitation. In supergravity, no such problems arise. FromM = ‘~“ ys[ yvD,,Dpi/1,, + (yaDpi/i,,XD,.e°~)] (7)D,.Rand the torsion equation D,.e”,. — Due”,. ~l/I,.yal/,~ one findsDLLR~= ~ ~““’y5[y,.o~dlII,,R,.P’4 + (y0D~~,,Xi~,.y”ip~)]. (8)If one expands the product y,~o~cdas explained in appendix A and uses the cyclic identity with torsion2Ge,,y’i/i~+ ~‘/Ydt/1,, ~~”R,.P~(757 UcdlIJ,,) ~”R,.p’~=r~,.pvd~.W) — i~(f’A7d aVJ~9,lE~ i \_ IT ~ \ a$A,, (9)all terms in DIR” cancel on-shell. (Use the Einstein field equations and Fierz both undifferentiatedgravitinos together.) Hence supergravity is consistent.It is usually believed that consistency is a consequence of gauge invariance. This is not always so; forexample [125],adding to the gauge action of simple supergravity a mass term ei/I,.o””i/I~leads again to aconsistent theory although it is not locally supersymmetric. (For this one also needs a cosmologicalterm, see subsection 6.1.)Finally we give a list of different forms of the gravitino field equationR” = ““°ysy~Dpçlçr, YAIPA,. = R,. — . R, yA~A,.= —275R,.‘Il . R = 2o”~t/i,.~, t/i,.~,= D,.i/i~ — D~t/i,.7a*~,y + 7~i/Aya + )~ylf1~= ,.,~,y5R” (10)I 7 —— I — P”I~ 275~P,w — ~ , ‘~‘~‘— c” ~R,. — ~y,,y• R = ~ysy”çb,.,.— ~7”I/i,.,,.

4cr”‘’. To derive these results, use for example that 7,, ”””7s)’~= —2o,.~y~ ””~ = Also use— ~gvp~.j. ~Pav .j.. .~°~“P— (P. van Nieuwenhuizen, Supergravity 223,,,,E U~,, Ua$y Ua$y ~where S~= S~O”~S.It follows that on-shell GAA =0, but one should not conclude from this result that the gravitinoenergy momentum tensor is traceless. There are extra gravitino terms in G,.~which should beconsidered as part of the gravitino stress tensor, and their trace does not vanish. Thus, in second-orderformalism the action for the gravitino (including the torsion terms from the Hubert and Rarita—Schwinger actions) is not locally scale invariant on-shell. (We recall that a matter action 2(e, 4’) for amatter field 4’ varies into TAA, under Se”,. = Ae”,. and 84’ = A”çb with some power a, provided one ison-shell where the 4’ field equation is satisfied, so that 82/54’ = 0. See Weinberg’s book.)Sometimes one considers the gravitino as a fermion in an external gravitational field. The curl— D,~i/i~contains then only the connection w,.mn(e), not w,.m”(e, i/i). In this case the action isinvariant under Si/c,. = D,.(w(e)) provided the external gravitational field satisfies the Einstein equations(we leave the proof as an exercise). Also this action is not locally scale invariant, now becausealthough the gravitino field equation again reads R” = 0 (with w(e)), we can no longer use 1.5 orderformalism and there is an extra contribution to the stress tensor coming from varying 00(e) with respectto the tetrad field. However, in first-order formalism, the action —~ ““’°‘t/i,.yOw,.tm” = 0. This is obvious 5y~(30 since if the is2i/i,.,locally derivative scale 3 hits invariant A, theunder rest (4~’,.y t5e”,. = Ae”~and Si/i,. = A”+~w,,. cr)t/i,,5’y4,,) ””””vanishes due to the Majorana character of the gravitino.Again, one cannot say that the matter action is invariant, since the terms containing w coming from theHilbert action are not invariant.1.11. Matter coupling: the supersymmetric Maxwell—Einstein systemAfter having discussed the gauge action of simple (N = 1) supergravity, we now turn to the secondstage: matter coupling. As in any gauge theory, we start with a globally supersymmetric action, andcouple it to the gauge action by introducing couplings to the gauge fields. For ordinary gauge theoriesthe prescription is rather simple: replace ordinary derivatives by minimally covariant derivatives(3,. — ieA,. in electromagnetism). In supersymmetry this is not so simple and the analogous solution isbest obtained by the tensor calculus which we will discuss later. Here we will start with the Noethermethod since it is conceptually simpler (though algebraically more tedious).We will begin with the coupling of the spin (1, 1/2) photon—neutrino system to the gauge action ofsupergravity. Historically this was the first matter coupling [204].Another coupling exists betweenphotons and gravitons, namely the coupling of the spin (3/2, 1) systems to the gauge action (so-calledN = 2 extended supergravity) [203].This latter coupling comes nearer to a unification of gravity andelectromagnetism, since one can connect gravitons and photons by a series ofsymmetry operations. In themodel we are going to discuss, this is not possible. Forconvenience weconsider an abelian vectorfield, butall results easily generalize to the Yang—Mills case [202,237].The photon—neutrino Lagrangian density in curved space readscpO__1~’r ,.pvcr_~Av1oA E’ —~D_.~D— 4ez ,.~ ,.,,,g g 2 ‘‘ — ~ U,J.J,.where D~A= o,.A + ~w,.ab(e)cr~~ contains not yet i/i-torsion. 2°varies under the global supersymmetry

224 P. van Nieuwenhuizen, Supergravitvtransformations in flat space8B,.—~iy,.A, SA~cr~F , oFo”~F,.~ (2)into a total derivative, S2°=3,.[~F”~ëy,Jt+-Lky”cr . F ]. To show this, use the Bianchi identity~VPO8,F =0. The Noether current is thus given by —~o~F-y”A. Hence, we begin by adding theNoether term= ~ ~ Fy”A. (3)This ensures that the order ic°terms in 8(10 +1N) cancel,2)iy”i/f,.even in 1°and in curved fromspace.SB,. and SA in JN~Considerfirst Tothe order terms K, one of the has terms form ,cF2 coming . The fromMaxwell Se”,. = (K/action yields (K/2Xiy”l/i~)T,.~(B) while the Noethercoupling yields (K/2)l/i,.cr Fy”o- . F . The sum is equal to~uI \Iz~ _..i L I~a$4 kV~’#’Y5Y~ ~a rn4g,.~alSlSince, however, the second factor is identically zero, these variations cancel. (This identity can beproved by using that ~ antisymmetrized in (va/pcr) vanishes.)To ordcr K there are also K l/iAOA terms in 8(1°+ 1N)~From the Dirac action one finds5J1/2 = —e ~ ~ i/i(AØ°A)+~ ( y”/ia)Oty”1-T),.°A) ~ (Ay~cr”A)(2ëybl/i,.,,— 7,.l/iab). (5)The first term is clearly cancelled if one adds to Sit a new term SA = —(K/2)Ey . i/iA since this newvariation produces in 211(2 precisely the opposite. Clearly, quite generally terms in the v~zriedactionproportional to a field equation can always be cancelled by adding an extra term to the transformation lawof the field whose field equation appears. Using this observation,m is proportional we can to cancel the the gravitino last terms field in equation. eq. (5),since A Aymcr~I~A= second general ~ ~““Ay5y,~A mechanism and which *hI~z&yai/1bc= one uses 2y5R over and over again in the Noether method is theobservation that terms in the varied action with 3 can be cancelled by adding an extra term to the actionobtained by replacing 3,. by —i/i,., provided all gravitinos appear symmetrically in the end result. Forexample, the variation of B,. in the Noether coupling in eq. (3) yields~ ycr”~A8~ (ëy,3A) — ~(i~”y”A~t9a (ëy,.A) — 0,. (ëyait)]. (6)Partially integrating the first two terms, they are proportional to R” since Iy~ — 3 . i/i = ~ R and/i/i,. — 3,.y• i/i = ~y~3,.R”so that they are cancelled by an extra Si/i,.. But the last term is equal to~- (ii”y~AXa,.e)yaA + ~ (~“y”A)(ëy~3,.A) (7)and one clearly sees that the 3 terms can be cancelled by adding to the action ~(K 2I4Xl/i,.y,,AXi/i”y”A)since both gravitinos yield a 3,, variation and appear symmetrically.

P. van Nieuwenhuizen, Supergravity 225In this way one adds systematically new terms to action and transformation laws until one arrives at acompletely invariant action. There are general arguments that this iterative procedure must always stopexcept when there are spin 0 fields A present [202]; in that case the action can be infinite series in(KA) tm.The final result obtained in this way is given by2 —~F,.,F””—~,øA+~ çfr,.o .Fy”A+~—(çIi,.u”~y”AXi/i~y,,A)+~ K2(AAXAA)Sit = ~ Fco~~ , SB,. = — ~ëy,.A (8)Setm,. = ~- iymi/i,. Si/i,. = -~-D,. + ~- (~y5y”A)(g~,.+ o-,,,. )y~ .The spin connection in ØÀ contains only i/i-torsion, but no A-torsion. The symbol F~’ is thesupercovariant curlF~=8,.B~+~i/i,.y,A-,a4-*v. (9)The first suggestion that there should be an axial vector auxiliary field was due to this result [202].Clearly, the result for Si/i,. depends on the matter fields A and suggests replacing them by a new field A~.We know already from the gauge algebra that= i(D,. + ~A,.y5) + ~y,.(S — iy5P — iA~y5) (10)and hence one adds to the action the gauge action 2(gauge, e, i/i, 5, P, A,.) as well as a coupling+~~~y5yh1IA)A~. (11)Solving for A,,, and substituting the solution A,, =4 and A2 terms.—(31/8),c(Aysy,,A) back into the action and Si/i,., onereproduces Thus, in all theAsupersymmetric Maxwell—Einstein system, only A,. plays a role, but S and P do not.This is due to the fact that this system is superconformal invariant. As we shall see, A,. is the gauge fieldof chiral rotations. (In the example of the spin (0, 1/2) coupling, S and P do play a role.) The action canbe written in a suggestive manner as2=~2)+2’3/2_~1(S2+P2_A~j+21_~A(ø+~-1.4ys)A+~(~,.cr”~y”AXF~s +F~’)(12)SB,. = —~-iy11,A, Sit “‘so .Due to the auxiliary fields S, F, A,. one has only added matter terms to the action without changing the

226 P. van Nieuwenhuizen, Supergravitvgauge field transformation laws. Since the A field equation in the absence of A,. can only rotate into theB,. field equation and this latter field equation requires two derivatives, the variation of the A-fieldequation cannot contain 8 terms. Thus the A-field equation must be supercovariant by itself, and sincethe terms t/JAF in the action contain only one field A, they require an extra factor of 2 in order that theA field equation becomes ~ØCOVA.Finally, we discuss the role of the matter auxiliary fields. The globally supersymmetric spin (1, ~)system has a closed algebra if one adds a pseudoscalar field D (see subsection 8)SB,. = —~ëy,.A, 5A = ~cr F + ~ y 5D , SD = ~ eyJA. (13)One finds uniformly for A and D the usual result [S( ~),5( 2)]D= ~(i2y” 1)3,.D, but for B,. there is aMaxwell gauge transformation added: —3,. (~i2B~1). The flat space action becomes2. (14)2= —F~—~/A +~DIn curved spacetime, the Noether current and hence the Noether coupling are unchanged. Thematter transformation laws are now given bySB,. = —~iy,.A, SA = . F~°”+ iysD),. coy KSD = 7s7 ~ A + ~-A,.y5A0~+ iysD)ili,.. (15)D~VA= D,.A — ~- (o . PSince D is a matter field, one expects SD to vary without 8 , and indeed one finds a supercovariantderivative in SD.Summarizing, the total action reads2= ~2)+23/2+21+2h/2(D,. +~A,.y5)_~(S2+P2_A~)+D2+~ i~,.(cr F+u Fco~o)yMAand is invariant under eq. (15). The derivatives D,. contain only i/i-torsion.(16)1.12. The scalar multiplet couplingThe next example of the coupling of a globally supersymmetric matter system to the gauge action ofsimple supergravity is the coupling of the spin (~0~0) Wess—Zumino model [202,509]. The action,already covariantized with respect to curved spacetime, is given by2°= —~eg”(3,.Ac9,,A + 8,,B 8MB) — ~eAy”D,.°A+~e(F2+ G2) (1)where D,,°is the gravitationally covariant derivative without torsion. In curved space it varies into

P. van Nieuwenhuizen. Supergravitv 227SA=~iA, SB=—~y 5A, SF=~Ø°A,OG = ~ iy510°A, Sit = ~[I(A — iy5B)] + ~(F+ iysG)e, (2)02 = — ~ (D,,ë)(JA +4’iy5B)y”A + 3,. [—~ ~y”(IA —Ii75B)A + ~ (F + iy5B)y”AFrom now on we will omit the auxiliary fields F and G, but reinstate them at the end of thissubsection.Since the action 1°varies for local (x)into the Noether current, (see subsection 2) we add theNoether coupling= ~ ~,.[I(A÷ iy5B)]y”A. (3)The variations of A, B, A yield(i) AA and BB terms. These cancel with the vierbein variation in 2°(both are proportional to theenergy-momentum tensor, see subsection 2). —(ii) AB terms. These are of the form K(i/,.y~ ) ”’~3,,A 3,,B and upon partial integration the 8terms are cancelled by adding a term ~while the remainder is proportional tothe gravitino field equation and can tm,. be absorbed in 2°andby from an extra SA andSi/i,. SB-~y5 in 2NA3,,B cancel (see (after subsection tedious3).algebra),if one (iii) adds the AA Sit terms-~K1/IAcoming and Si/i fromAA Se terms to the action. One needs Fierz rearrangements, the identityIv. i/i — 3. i/i = b~,.R”and the identityDp(Ay”yay,.A)Irr AIyay,iA + AYa71JA + 2A(Oay,. — )‘aO,.)lt. (4)The new Sit and Si/i variations as well as the old variations must now be used in the new, order ,c 2 termsin the action. One finds -(iv) that the new Si/i -.-.‘cA OB ~fl 2N are cancelled by adding a term K2AAA t9B to the action.(v) that the new Si/i AA in 2’N yields K2A3(OA) variations while also the old SA ëA in the newterm in(iv) yields such terms. The sum is cancelled by adding a K2(AA)2 term to the action, and aSit = K2AAA term.(vi) Finally, many sources contribute to K21/12A OA variations, but only one source yields a term with3 . Partially integrating this term, all variations cancel if one adds a Si/i —~K2AA1/1 term.The complete answer is given by2= 2(gauge, e, i/c)+2°+2°’+ 2(contact)2(contact) = ~ ~y5y,,A — ~ A~~B]+ eIc2(Ays,flA)[_j~i/f~ysy,4i” ~A3TB ~4Ays7~AJ (5)where 2’(gauge) contains i/i-torsion, but the Dirac action for A contains no torsion. The transformation

228 P. van Nieuwenhuizen, Supergravityrules areOA=~iA, SB——~Ey 5A, Sem,.=~ëyml/1,.Si/i,. = -~-D,. + ~ Kys AD~B + ~ o-,.,,ys (Aysy”A) (6)Sit = ~EØ~V(A — iy5B)] + ~- 75A[AiysA — iBiA].The derivative D,. contains i/i-torsion, and D~(V is the supercovariant derivative (D,.A 0,.A —(K/2)l~,.A).So here we have a second model of matter coupling. By now the reader probably starts askinghimself the following questions: how can one eliminate the matter fields from Si/i,.; in particular, couldone have •found the minimal set of auxiliary fields 5, F, Am in this way. (We recall that theMaxwell—Einstein system was too simple: it is a conformal system and hence S and P did not play arole. This system is not conformal.) Indeed, one can rediscover 5, F, A,, as well as their transformationrules in this way, but, to be honest, only if one knows already the answer; otherwise one probablywould not have thought of all necessary subtleties. The next question is: how unique are such resultsobtained by the Noether method? Fortunately, the most general coupling of the scalar multiplet tosupergravity is known from the tensor calculus, and hence we can answer this question completely.The most general coupling (eq. (67) 2 of+B2) ref. [541])with times (a,.A)2 canonical + (3,.B)2 gauge where action f is and arbitrary, canonical andspin also1/2aaction nonpolynomial has as scalar potential kinetic forterms A and f(A B. Requiring canonical scalar kinetic terms (f = —~)and apolynomial action (which means G = —~zz* — ln ~IgI2in eq. (67) and letting g = constant —*0) one findsthat eqs. (5, 6) are unique; going back to the point in ref. [541]where one still had 5, P, A,. present, andwhere em,., i/i,.~,5, P, A,. transform as in subsection 9, this action takes a nonpolynomial form. (In goingfrom this version to eqs. (5, 6) complicated field redefinitions eliminate this nonpolynomiality but alsodestroy the canonical transformation rules of the gauge fields. Also S, F, A,. have been eliminatedthrough their (complicated) nonpropagating field equations.)Thus it would have been difficult to deduce the auxiliary fields from (5, 6). The action with 5, F, A,.which is equivalent to (5, 6) and invariant under the transformation rules of the scalar multiplet X = (A,B, it, F, G) as given by the tensor calculus,SA=~A,SB=—~Ey~ASit = ~EØcoV(A — iysB)] + ~(F+ i’y5G)SF = ~e(.øc0v — 1 A’y~)A+ ~SG = ~ieys(Øc0~~ — ~ ..4~s)A— ~ysqx (7)

P. van Nieuwenhuizen, Supergravitv 229where i~= —~(S— iym,., and i/i,.,reads as follows 5P — i.A’y5), and under the standard transformation rules for 5, F, A,., e2’ = exp(—x)[2(gauge)+ 2’°(1— x) + 2°l(1— x)+ ~ + 2(A,.)+ 2(S, P, F, G)]. (8)Here x = —~(A2+ B2), 2(gauge) is the gauge action of subsection 3 (with i/i-torsion but without 5, F,A,.), and 2°is the Wess—Zumino action and 21~~the Noether coupling of eq. (3).Interestingly enough, one finds the improved Noether coupling= —~(A— iy5B)o.~v(Do,.i/i~) . (9)(where D°,.does not contain i/i-torsion). In flat space it satisfies on-shell~1 (.,.•.j. \_ “‘~~“+ \—Ii ~10V,.~JN JN,impJ — 7 VN 1N,impJ — .Hence we have here a clear signal that the minimal auxiliary fields of supergravity are closely linked toconformal aspects. We will see more about this in the section on conformal supergravity — in fact, wewill show there how one can derive the tensor calculus from conformal supergravity.The remaining terms are= ~ A 2,. —~ A”(AD~B)— ~ (A,Xy 5A)(1 — x)+ 4-fermion terms + (A3~B)[-~(~-y5y~A )(1 — ~x)— ~~ (11)Finally,2) + ~(F2+ G2X1 — x) + ~F(AS + BP) + ~G(BS— AF)2(S, F, F, G) = ~52+ terms + F with AA and (AA)2. (12)The addition of masses and a p4-coupling in the scalar multiplet coupled to simple supergravity wasdone in ref. [221].A super-Riggs effect in this model, as well as the gauging of the N = 4 supergravity(the 0(4) version) and the scalar sector of this model, and also the coupling of super-QED tosupergravity was discussed in ref. [222].1.13. Free spin 2 and spin 3/2 field theoryThe physical fields of supergravity are the tetrad em,. and the gravitino i/,~,.It is interesting that thefree field limit of the gauge action of supergravity reduces to the sum of the Fierz—Pauli andRarita—Schwinger actions, which are the unique actions for the spin 2 and 3/2 without ghosts as we shallshow. Indeed,CD — 1,2 ~1L2 IL. 1. ~Iz.2.Z.FP — ~ ~ 21t,., — 2I’i,.fl.,. T~

230 P. van Nieuwenhuizen, Supergravisy2RS = ~~E””l/i,.y5yv8~,i/icr (2)where h,. = OAhA,. and h = hAA, and .%p is the linearized part of .1—\/jR d 4x.We first show that these actions indeed describe two physical modes with helicities ±2and ±3/2.Then we show that the only actions for h,,~and i/i,. with positive energy are those in eqs. (1) and (2).For spin 2 the field equations read— ~ — xv,,. + S,.p~,,,,=0, ~ = h,.~— ~ (3)where x,. = 3AX,.A, and x,.r and h,.~are symmetric. It is invariant under gauge transformations= 8,,4. + 34. — S,L,4”,~.Thus 8(8,.x,~~) = Di,, and one can choose the de-Donder gauge in which= 0. In this gauge the field equatiDn reduces to ~ = 0. The solutions are plane waves andtaking these along the z-axis, X,.~(x3— x4), one has from the de-Donder gauge thatX3,. = X4,.. (4)As always in differential gauges, one can once more use the gauge freedom provided LII~= 0 in ordernot to undo the de-Donder gauge. Thus ~ =x~+ 8,.A~.+ 8,.it,. — S,.~,,,.with Eit,. =0 still satisfies=0. Choosing A1 such that ~3i = ,y31 + 33A~= 0 one finds from 3,,4”V = 0 that ~4i = 0. Similarly, aproper choice of it2 yields ~32 = = 0. Finally, it3 and it4 are fixed by requiring that ~AA and ~ vanish.As a consequence, from (4) also X33 = x~=0. Thus, only X12 = X21 and x’ = ~X22 are nonzero. Sincethe polarization tensors for spin 2 fields with helicities ±2are given in terms of photon polarizationvectors ~ by2A~= ~ ~, A;~= ; p, “ ‘ “ (5)one sees that the Fierz—Pauli action describes two physical modes with helicities ±2.This follows easilyif one writes x,.~= A~~a~(k)e”~’ +h.c. because this shows that only Xii ~X22 and Xi2 are nonzero.We now repeat this analysis for spins 3/2 (ref. [522]).The spin 3/2 field equation in flat space readsR” = ““Y~7v3pi/’cr 0, OR” =0 if Si/icr = Ou . (6)One chooses the gauge y~i/i =0, which can be solved since S(y. i/i) =1 .FromI*,.—8,.y~çfrR,. ~Ri/i —8 i/i = 2o””8,.i/i,, = . R (7)it follows that on-shell in this gauge also 3 i/i = Ii/i~= 0, so that Eli/i,. = 0. Decomposing i/i,. in k-spaceon-shell into the complete set ~,., k,. and k,.+ +~ — -+‘- ~— ,. Ua ~ u0 r..,.Ua r~,.Vawhere k,. = (k, —k4) is the time reversal of k,., it follows from k~i/c =0 that Va =0. Also, from li/i,. 0we see that Iu~=Iu°=1u = 0.

P. van Nieuwenhuizen, Supergravily 231As before, we choose a second gauge, maintaining the previous gauge y~i/i ~0(9)One still has k . = = =0 on-shell. Thus, in particular(10)where we used that %~%~ =0 and defined °k = ~Jky 1yJ/(2i).Thus u is the spin projection operator, andu~has helicity + 1/2. Since e’~has helicity + 1, we conclude that ~ describes two physical modes withhelicities ±3/2.We now prove that (1) and (2) are unique. We only treat the spin 3/2 case. For the spin 2 case see forexample, Nucl. Phys. B 60 (1973) 478. The most general action with one derivative is given by2’ = i/i,.[ay,.t9~ + f3y~3,.+ y,,Jy~+ ~IO,.~]i/’~. (11)We introduce spin projection operatorsP3/2 — ‘~ ~ — — — ~ 2’fl~— ~ 3Y,.Yv, )‘~~— )1~ 00k, W~— (i~,V Li‘P 1/2\ — 1’ ‘ f 1/2\ — 1 — —I. Ii J,,v — ~ ~ 12 ),.v — ~,r 7,.00,~, !/~p — U~~v 01,.01v,(P~2),.~ = w4c, (P~2),.~ = w,.w~. (12)They have three properties(i) orthonormality: ~ =(ii) decomposition of unity: P312 + P ~ + F ~ = I. Hence the sum of the diagonal projection operatorsequals the unit operator I;(iii) completeness: the operators P~span the space of all field equations of the form (11). This isobvious since the five P~span the four-dimensional space in (11) and they must satisfy one linearrelation to cancel the 8,~8~/El terms. We have chosen this set, but one may use any other set satisfying(i), (ii), and (iii) (for example, with 5,.,, — (k,.~,,+ k,,le,.)/(k . k)).We now consider first massive spin 3/2 fields and then the massless case.Massive fields: On-shell one has y~i/c = 3 . i/i =0, hence on-shell i/i,. = ~ [use(ii)1. However, onecannot define as field equation (P3121 — M)i/i = 0 since P3”2 contains E’ singularities. The choice(P3”202 — M2)./i = 0 is local (i.e., free from El1 singularities) but a higher derivative equation. However,if one writes for the field equation (01— M)i/i = 0 such that 01 is a linear combination of theoperators in (12), which is local, then iteration yields OlOlili = M2i/i = E1P312i/i. Hence 01 must be asquare root of EJP3”2. This method is called the root method and is due to Ogievetski and Sokatchev.The most general square root is0=P312+a(P2+P~+$P~+l3’P~2). (13)

232 P. van Nieuwenhuizen, Supergravit’,’(To derive this result, it is convenient to use the identity {‘~‘,.,,e’} = 0.) Ohs local if1) (14)2(1 — a) = —\/~a(f3— p~(Special cases f~= 0 or f3~= 0 are included.) Thus one has obtained a one-parameter set of fieldequations of the form (11) which leads on-shell to y~i/i = 3 . i/i = 0. However, one cannot use as action= i/i,.0,.ji/i,, since the two gravitino fields do not appear symmetrically in this action, so that varyingi/i,. would reproduce the field equations, but variation of i/i,, would not.There is a way out of this dilemma of finding an action, and that is to shift the field in the fieldequation. With parameter 0 inçfr,,ço,.+oy,.yço (15)one has again a local field equation, and requiring that the new field operator 0~.is such that the newfield equationF,. = 0~,Jço,,—M(ço,.+cry,.y. ‘p)=O (16)has the property that ~,.F,.is_symmetric under a Majorana flip (see appendix C) one finds as action2’ = i/i,.0,~,,1i/i,,— M(i/i,.i/i,. + cri/i~yy~i/i) =0. The most general solution is again a one-parameter class,one element of which is given by (use o~= (ft + J3~)(2V3— p — 3p1y1 and let /3 —*2’ = —~ ~(P312— 2P~2)Ii/r—~ ~(F3t2— 2F~2—~~(p~2 + P~2))i/i (17)while the general solution is obtained if one shifts the field in this action according to (15). The result ineq. (17) yields the massive Rarita—Schwinger action. Adding a coupling to an external source in order toobtain below propagators, one has2’ = ~ “°‘ii,,ysyvi9pi/ic, —~Mi/i,.cr”~i/c,, — ~ (18)It is clear that the mass terms in the action in eq. (18) cannot be brought in the form i/i,.i/i,. by theredefinition in eq. (15), but that one can redefine the fields in the field equation such that there one doesfind a diagonal mass term M.From eq. (17) one immediately finds the propagator corresponding to eq. (18), since one only needsto invert separate spin blocks. One inverts the field equation F,. = J,. = (P3”2 + P~ + P ~),L~JI,and finds= [P3~(/~,,~vI) — \/.~M(P ~2 + P~2) — P~2(I — M)]JV. (19)Hence, the propagator is given by[(S — ê,,8,,)’v12) (I—M) + 4(y,. — 8,.M~XI+M) (,, — 8,M~)](El — M2~. (20)Physically this means that El and I in P312 have been replaced by M2 and M, so that one finds exactly

P. van Nieuwenhuizen, Supergravity 233spin 3/2 on-shell. Similar results hold for spin 1, 2, 5/2, etc. This is a general result, valid for any massivespin. (Note that (/+ M)I = (I + M)M if El = M 2.)Massless fields: For massless fields there are local gauge invariances and hence some of the spinblock matrices are singular. One finds the propagator which is sandwiched between sources by invertingthe maximal nonsingular submatrices of the spin blocks. Each gauge invariance implies a sourceconstraint and these source constraints are needed to cancel ghosts in the propagator. To prove that thelinearized Rarita—Schwinger action for real massless gravitinos is the only action (up to fieldredefinitions as in eq. (15)) which is free from ghosts, one must consider all possible cases contained ineq. (11): the rank of the 2 X 2 spin 1/2 matrix 0, 1, 2 and the rank of the 1 x 1 spin 3/2 matrix being 0 or1. If one requires that at the pole k2 = 0 the residue is positive definite, then only the Rarita—Schwingeraction is found. We shall not do this straightforward but tedious computation. Instead we show that (2)is without ghosts.From eq. (17) with M = 0 one finds the field equation with external source J,.(F3”2—2P~2)1i/i,.=J,.. (21)Clearly, there is the gauge invariance Si/i,. = (P~ where x~is arbitrary, since replacing i/i,. by Si/i,.the left hand side vanishes. Thus there is the gauge invariance Si/i,. = 8,. . Acting with P~2on thisequation, the left hand side again vanishes, and one finds the source constraint PJT~J= 0 (for j = 1, 2),which is equivalent to= 0. (22)Inverting the spinblocks, the propagator becomes in k-spaceiJ~(—k)C(P3”2—4F~2),.,, ~J,,(k) = 1. ~ (23)For a Majorana source 1(x) = JTC, one has JT(_k)c= ~J(k))ty4which one may define to be 1(k). Toshow that the residue J,.(k)y,.k’y,.J,,(k) is positive definite, we use the spin 1/2 result that X =u~u~+ ui1, and the decomposition of the Kronecker delta function5,.,, = ~±(~±)*+ ~~(~)* + (k,.le,, +1~,,k,,)(k . 1~’) (24)2 = 0. We rewrite the residue as ,..o,..,.y,,k’y,.&,,J,,. The terms in eq. (24) with k,,or valid k,. on-shell do not contribute where k since Ku~= 0 and k J = 0. For the rest one finds, usingcu4=0, ~u=O (25)that the residue is given byJ12 + ü~% J~2 (26)Hence, there are no ghosts in linearized Rarita—Schwinger theory, and two physical modes.

234 P. van Nieuwenhuizen, Supergravily1.14. Higher spin theoryOnly for N> 8 do the extended supergravity theories have a SU 3 x SU2 x th internal symmetrygroup. This is obvious since 0(9) contains 0(6) x 0(3) and 0(6) is isomorphic to SU(4) which containsSU3 x U,. However, for N> 8 one has also spin 5/2 fields and more than one graviton. The questionarises whether one can repeat history and construct consistent field theories for spin 5/2 and higher.Free field actions for any higher spin do indeed exist (for massive fields, see Hagen and Singh; formassless fields see Fang and Fronsdal [258,259]).We consider as spin 5/2 field a symmetric tensor spinor i/,~,,,= i/ic,.. Coupling it to an external sourceJ”,.,,, one can ask whether an action exists which leads to a propagator without ghosts or tachyons. (Thisis equivalent to requiring positive energy of the classical theory, and easier to prove.) The result is thatthere is an action, given for massless Majorana spinors by2’= —~i).j,,,1i/i,.,,— i/i,.,,yJy~ifr~,. + 2i/J,.,,y,,3Al/JA,.+~i/I~.Il/J,.,. i/IAA3,.7,,i/i,L,, + LJl~ (1)It is invariant under the following local gauge transformationsSi/i,.,, = 8,. ,,+ 8,,e,. with y” ,.= 0, (2)as first found by J. Schwinger.In fact, this action is unique as an analysis based on spin projection operators shows. The fieldequation can be written in a Christoffel-like way as~I_~i I —~ I —~ I —7 ~V,.I/)a$ OaI,V~/3 U$I~,.a —Since the free field action is invariant under local spin 3/2 gauge transformations, one might repeatthe analysis of supergravity and see whether minimal coupling to gravity by means of 3,. —* D,. leads to avariation of the action proportional only to the Einstein tensor. In that case one can add the Einsteinaction and define the tetrad variation such that the total system is invariant to lowest order (afterwardsone should then follow the Noether procedure). However, a simple calculation [153]shows that under= D,. ,,+ D,, ,. with y =0 one finds4i,.YA1PAb +SI = G,.b[~i,,y”l/i”~— ?‘y”i/i~+4j,,YSYcrl/i,.p t~P~~~ëA7vlfrnAeIJ~]+ Ep7alIJAbC — ~ p757al/JAb ”~’°’Cpgab. (4)The last two terms are proportional to the Weyl tensor and seem to exclude a consistent couplingbetween spin 2 and spin 5/2.Approaching the problem from the matter end, no matter systems have been found which areinvariant under transformations with constant parameters ,. satisfying y = 0 [153] (for such systemsone could couple i/i,.,, to the Noether current and iterate), while also taking for spin 5/2 a vierbein spinor(nonsymmetric i/i~,,.)did not improve matters [144].The massive spin 5/2 system contains a mass term for i/i,.,, plus an extra spin 1/2 field x and is, again,unique. Quite generally, for any higher spins, these extra auxiliary fields x couple only in the mass term

P. van Nieuwenhuizen, Supergravity 235to the i/c field; for spin 5/2 on has2’(mass) = —M(i/i,.4,.,, — ~. ‘y’y~i/iS — — —4~x). (5)Hence here there is already a so-called van Dam—Veltman mass-discontinuity in the action.For spin 3, a massless and massive theory exist, too. In this theory the limit m —*0 cannot even betaken in the sandwiched propagator. (For spin 2 there is a finite discontinuity.)Our conclusion is that it seems, at this moment, that Nature stops at spin 2.2. Quantum supergravity2.1. IntroductionSupergravity can be covariantly quantized by the same methods as any other gauge theory, providedthat the gauge algebra closes. This closure is obtained by adding auxiliary fields to the classical actionand these fields remain auxiliary fields in the effective quantum action, but become propagating in thecounter terms. The covariant quantization methods can be justified by a Hamiltonian path-integralapproach as we shall see, and, once quantized, unitarity and gauge invariance can be proven. Thesimplest way to do so is to use the Becchi—Rouet—Stora—Tyutin symmetry (BRST-symmetry) of thequantum action which is the quantum extension of the classical gauge invariance. We will also considerthe BRST formalism for gauge theories whose gauge algebra does not close, for example simplesupergravity without S. F, Am.Gravitational theories have a dimensional coupling constant, Newton’s constant, and this fact alone issufficient to rule out ordinary renormalizability for the following reasons. Due to the dimensionalcharacter of the gravitational coupling constant, one-loop (and higher loop) divergences have a differentfunctional form than the quantum action so that one cannot absorb these divergences back into theoriginal quantum action by rescaling of the physical parameters. For example, in Einstein gravity theone-loop divergences are on-shell proportional to2 and the two-loop divergences areproportional to (R,.,,aøR~”Rp,.t~) etc., whereas 5di~’= the original erR ~,,,+ /3R action is proportional to ,(2(g112R).Clearly, one cannot reabsorb the two-loop divergences back into 2’ by rescaling of g,.,, and K alone.(Expanding5di~’ into powers of h,.,, where Kh,.,, = g,.,, — ~, and subsequently replacing each field h,.,, byan infinite set of tree graphs with physical momenta and physical polarization tensors at the end of thetrees, one finds the divergences of the n-point Green’s functions on-shell.) One also finds that anyn-point Green’s function is divergent from the two-loop level on. At the one-loop level, i~2=0 sinceR,.,, = 0, where R,.,, = 0 because the fields g,.,, satisfy R,.,,(g,.,,) = 0 since they represent infinite sets oftree graphs with physical end-momenta and polarization tensors. For more details, we refer to ref. [6061,where the relation between normal field theory and the background field method is given.Thus gravitational theories are never renormalizable and only the two extremal cases are possible:finiteness of the S-matrix or the infinities in the S-matrix cannot be removed. The latter case isusually called “non-renormalizability of the S-matrix” but we stress that it is not the counterpart ofrenormalizability but of finiteness. Finiteness of the S-matrix is of course a much stronger property thanrenormalizability and one can expect it to occur only when “miraculous” cancellations of divergencesoccur due to a symmetry of the theory. It is here that supergravity has had impressive successes. First

236 P. van Nieuwenhuizen, Supergravitvdiscovered by a direct calculation and later explained by various equivalent theoretical proofs, it wasfound that pure simple supergravity as well as the pure-extended supergravities (irreducible supergravitieswith one graviton, N gravitinos and specified numbers of lower spin fields) are one-loop finite.They are even two-loop finite. This is a better result than obtained so far for pure ordinary Einsteingravity which is known to be one-loop finite (but which might be two-loop finite; this constitutes one ofthe major unsolved problems in quantum gravity). Thus supergravity has taken the lead in the loop raceand the reason is that the extra Fermi—Bose symmetry allows one to prove extra cancellations. Even ifultimately pure Einstein gravity will turn out to be two-loop finite, supergravity has the advantage thatthe pure extended supergravities contain not only gravitons but also gravitinos, vectors, spinors andscalars. Thus one has the possibility of a finite quantum theory describing gravity as well as the otherinteractions which might be phenomenologically correct.About the need to quantize gravity, and now also supergravity, various opinions exist. Some peoplefeel that one should not quantize at all. Others believe that Einstein’s theory of gravity is aphenomenological theory such as thermodynamics and that one should try to quantize the underlyingmicroscopical more fundamental theory. In this connection there have been endeavours to consider thegraviton as a bound state of two photons or of two gravitinos [310,311]. Another point of view is thatgravity is something like van der Waals forces, present where matter is but not existing as free radiationmodes. The recent results from a binary pulsar, which seems to lose energy through gravitationalradiation, seem to refute this proposal. Yet other ideas are to replace the points in spacetime by twistorsand to try to quantize those. A point of ordinary spacetime is described by the intersection of twotwistors, and hopefully the quantum fluctuations give rise to “fluttering” lightcones rather than to fuzzypoints in spacetime. Even further goes the proposal to use path-integral quantization in the space of allgeometries. In this connection by far the most appealing has been Hawking’s spacetime foam: verysmall strongly curved regions of spacetime which average out to flat spacetime over larger distances.Interesting as these suggestions are in their own right, it would be preferable if the same ideas ofquantum field theory which have worked so well in quantum electrodynamics and, later, in unifiedtheories of the weak and electromagnetic interactions, would also turn out to describe gravity with thesame success. Of course there are two major differences between gravity and other interactions. First ofall, the metric determines which points are spacelike and which are timelike and it is not at all clear thatone ends up with the same light cones due to the quantized metric as those due to the classical metricone starts with. A second problem is that only in special classical spacetimes (those with a timelikeKilling vector field) can one define the absorption and creation operators of second quantization. Thesetwo, and other related problems are serious, but our strategy is to just go ahead and see how far onegets. As it turns out, and we will see, elegant results are obtained, and sitting at one’s desk and seeingthe cancellations occur, one cannot resist the feeling that something important is going on. Maybe whatfollows is not the ultimate way of interpreting quantum gravity but it may be the solution in disguise.2.2. General covariant quantization of gauge theoriesWe begin by reviewing the covariant quantization of gauge theories. Suppose one has given aclassical action 1(d) which depends on a set of gauge fields ~‘ and which is invariant under local gaugetransformations with parameters ~= R’,~(4)r. (1)

P. van Nieuwenhuizen. Supergravitv 237We use DeWitt’s summation convention in which i and a denote indices as well as spacetime points andrefer to refs. [514,303, 304]. The symbol R may contain derivatives and depend on gauge fields, forexample it could represent the Yang—Mills law 8W,~= (~ô~+gfa~~W,~)flc. The parameters ~ dependon x~’and represent the parameters of general coordinate, local Lorentz and local supersymmetry.One adds a gauge fixing term for each local gauge invariance. We will consider this I(fix) to bequadratic in the gauge choices Fa(~)I(fix) ~F~(~)y~F 0(4). (2)3is taken to beThis independent is not necessary, of the quantum but usefulgauge in order fields to be able to define propagators. The matrix y’~’~. It may depend on external fields, namely when oneconsiders a quantized gravitino in a background gravitational field, but the correct theory for the casethat y’~depends on çb~is not yet known. The gauge fixing term is needed to remove the degeneracy inthe path integral. For practical purposes, its use is that the kinetic terms of the sum I(cl)+I(fix) arenonsingular, so that one can invert them to obtain the propagators of the theory.By varying the gauge choices one obtains the ghost action I(ghost):I(ghost) = ~ (3)Thus one varies the gauge function, replaces the gauge parameter in eq. (1) by a ghost field andmultiplies from the left with another ghost field which is denoted by C*~.In a path-integral formalism,this result is obtained for bosonic symmetries by exponentiating the normalization determinant, seesubsection 4, in front of the gauge-fixing delta function det (Fa,jR’~3)~(F~— b~)as followsdet ~= J [I dC~dCI exp(~~Integration over anticommuting variables being defined by f dC’~=0, f dC’~C~= ô~,one then finds,upon expanding the exponential, that only the term with four C1 and four C2 fields is nonzero and yieldsindeed the determinant of (FajR’~). The normalization factor i/2 cancels in computations.If one has both bosonic and fermionic symmetries, one defines a superdeterminant (see appendix) andstill arrives at eq. (3). Since, as we shall see, ghost fields C~have3opposite in (3) tostatistics the far right. from the parametersThe oneeffective must specify quantum whereaction to put is the thusC~.We given by will always put C’I(qu) = 1(d) +~Fay~’3F~+ C*~~Fc~jR~C’3. (4)It should be observed that the sum over a and f3 is over the whole gauge group. Thus, for example, insimple supergravity Lorentz ghosts interact with supersymmetry antighosts, etc.The quantum equivalent of the classical gauge invariance given by eq. (1) is the global nonlinearBecchi—Rouet—Stora—Tyutin transformations.t This quantum symmetry uses a constant anticommutingt See Proc. 1975 Erice Conf., eds Velo and wightman, pages 331, 332.

238 P. van Nieuwenhuizen. Supergravitvparameter A and reads= ~~fa~~CMC’3 (Sb)ÔC~= AF~y’3’~.The invariance of (4) under (5) is easy to demonstrate. 1(d) is invariant under eq. (5a). The variationof 1(fix) and of C*~yield~ =0 (6)which vanishes since the variation of an action is a bosonic object. The remaining terms come fromvarying Fa,JR”3C’3. The variation of F~~ in 1 (4) vanishes sinceF~Jk= (_1ykF~,k), R~C3AR1’3C’3.=(_l)(k±Ixi±1)±k±1±J±1(k~-*j) (7)so that the symmetries of both terms in (j, k) are opposite. For the variation of R”3C’3 one findsR~~,kRkYCMC’3 — ~f’3P~R~~’3CUACP =0. (8)Closure of the gauge algebra implies that the commutator of the local gauge transformations is again alocal gauge transformation, thus again of the form (1)(5a)(Sc)[~(~7),~5(~j]4~= R~a.kRksn’3e—(9)= R~f~(çb)r(5~~.The structure constants may depend on 4i~and will be called structure functions. These equations are thelocal equivalent of the (anti) commutators which define superalgebras. It is now clear why (8) vanishes.Replacing in eq. (9) i~’3by C’3A and ~ and CaA1 one finds the combination(C’3A)(CUA1)_ (C’3A1)(CaA) = 2(C’3ACa)A1. (10)(Note that interchanging ~ and i~in eq. (9) does not mean interchanging also their indices f3 and a!)Thus the first term in eq. (8) can be written as a commutator, and applying eq. (9), one completes theproof that eq. (4) is invariant under eq. (5).It is instructive to verify that indeed relations such as eq. (10) follow from the formal symmetries ofthe ghost fields for the case of Yang—Mills theory.We leave it as an exercise to verify thatf°~(D~C)”ACC isindeed antisymmetric in both ghosts although one of them is differentiated.BRST transformations are nilpotent. Applying eq. (5a) and eq. (5b) to (5a) one finds zero due to eq.(8). Variation of 6CC yields— ~f~R 8 1C~AC’3+ ~ C~AC’3+ ~fa~~CYA(f’3,,~C°~A1 C°). (11)

P. van Nieuwenhuizen, Supergravily 239Since the structure functions f’3~~are antisymmetric in (po) except when both denote fermionicsymmetries, one finds that eq. (11) is equal toJj_;~~ 2~, I ‘3y.k” Dk j~ ;~ (A \r’~A r’yAc’1 3 o-l-J j3AJ y3)L~111L~ I1~ ——That this expression vanishes follows from the double commutator of gauge transformations and willbe derived in subsection 8.For antighosts, the BRST transformation is not nilpotent as given by eq. (5), but one can obtain alsohere nilpotency by introducing auxiliary fields once again. For example, in Yang—Mills theory onedefines~f’(qu)= ~cl) + ~g2+ g(a - W) + C”t9,~D~C (13)and one defines 6g = 0, ÔC* = Ag. The general case is obvious.Thus, for any theory with a closed gauge algebra one can define nilpotent BRST transformationswhich are the quantum equivalent of the classical gauge invariances. By means of them one can derivethe Ward identities which are needed for the proofs of unitarity and gauge invariance. The nilpotency ofBRST transformations of CC and qY is used to derive Ward identities for one-particle irreducible Greenfunctions. The anticommuting nature of the constant A has led some authors to speculate on a deeprelation between supergravity and BRST transformations, but the two have really nothing to do witheach other: any gauge theory is BRST invariant at the quantum level.2.3. Covariant quantization of simple supergravityIn this subsection we apply the general formalism of the previous subsection to simple supergravity.Since we assumed that the gauge algebra closes, we consider as gauge fields not only the graviton andgravitino but also the minimal set of auxiliary fields S, P, A,~.There are three local symmetries and hence three pairs of ghost-antighost fields [236](i) general coordinate ghosts CC, C~(ii) local Lorentz ghosts Ctm”, ~(iii) local supersymmetry ghost CC, C~.The complex vector CC and the complex tensor C”~are anticommuting with themselves but thecomplex four-component spin 1/2 ghost C’~is commuting with itself. This not only follows from generalprinciples but can also be checked by verification of a Ward identity, as we shall show (page 257).As gauge fixing term a convenient choice is£~(fix)= _~(a~V’jg~)2 + a(ea,.. — eb~6~ô~)2 + ~y,J(y5/ç). (1)The advantage of this choice is that the propagators for graviton and gravitino become very simple. Thekinetic terms of the Hubert action+~(fix) read in terms of h~.with g,~,.= 6,~+ Kh,~~ h~.p+~h~.p+a(ca~—c,~a)2 (2)where ~ = 6~,.+ Kca,~.From hp,. = c,~.,.+ ~ + C(c2) one therefore finds for the ca,.. propagator for

240 P. van Nieuwenhuizen. Supergravitva -*p~) = (c~~c~) = ~ (6~6~+ ~ — ö~)k2. (3)For comparison we note that the spin 0 propagator reads P°= —ik2. The gravitino propagator isobtained from the terms bilinear in çl’,.~~3/2 = — ~ ç~[y5~yPy~ — yC~yPyIL]~ + ~= (4)where e is the determinant of e~.The field equation is thus ~Y~IY~~/’c. and its inverse yields thepropagator [510]P~L~ = (I4~L~C) = ~ (5)For comparison we note that the spin 1/2 propagator reads P”2 = —Kk2.Instead of (1), one sometimes uses the part linear in h,~,.of V~g’~”. Also one often chooses instead of‘y~”i/i,.. the linearization y’26”ac(’,,. with constant y~.They yield the same propagators. For a —~, theantisymmetric part of the vierbein is frozen out. However, as we shall see, the Lorentz ghosts derivedfrom the antisymmetric vierbein should not be neglected, even in the limit that a tends to infinity.In n dimensions with the indices of h,.~running from 1 to n, the graviton propagator acquires a factor2/(n —2) in front of ~ The gravitino propagator in eq. (4) in n-dimensions yields the n-dependentresult [556]= [y,.k’y,..+ (4— n){6,.~,,Jc’— 2kMk~k2X}]k~2. (6)However we keep all indices of l/J,.~and h,~always four-dimensional and only regularize by letting x~’andmomenta p’4 become n-dimensional (see subsection 7).We now turn to the ghost action. From ~‘(fix)we find the gauge choicesF~= {_a~(\/~g~u~), eav. — ~ — y ~‘}. (7)The ghost action is obtained by varying F~with respect to all gauge symmetries and sandwiching theresulting matrix with an antighost row and a ghost column. Thus one finds for infinitesimal variations6(Vgg’4”) = ~+ (~9,~~)g’4’~] + a(~Vgg’4~’)— s/~(ey’4~fr”+ ëy”~Y’)+~ey- fr\/~g’4’~. (8)6(y ~)= ~t9a(y ~)+ (0 +~,4y~- 2~)e+ ~A cry ~. (9)t5(ea~—ebC3,.~öa)={(9~~ )ea~+r~9,,ea~+Aa1’eb~~ (10)

P. van Nieuwenhuizen, Supergravity 241For the kinetic terms of the ghosts one clearly obtains after replacing r by CC, e’~by CC and A ab byCab..~(ghost)~= C~L1C’— COC+ 2C~b(Cab+ 3bCiA5a). (11)Thus the general coordinate vector ghost is itself no longer a gauge field. The bar on C denotesDirac’s bar: C = C 74 in our conventions. Clearly, one must rediagonalize the Lorentz ghost fieldCab + ~8bCa ~.8aCb C~.One then finds new vertices originating from the last term in eq. (9) [556]ofthe form (C~/I)a’4CP.The ghost propagators are easily found(C~C~) = —i6~~k 2 (CaC~)=IC” C’* \_j~ ~ _~ ~~ t~adUbcThe vertices of quantum simple supergravity can be worked out from the foregoing.Thus we have obtained at this point the propagators and vertices of the theory, and this definesquantum supergravity in a perturbation expansion. We note that in I(fix) = ~F~,y’~’3F~we indeed didchoose ya’3 to be field-independentya$ = {—~o”,aôaa’6’4’4’, ~jaa} (13)We now consider the role of the auxiliary fields in somewhat more detail. They appear in I(qu) dueto (9) as follows— ~(dete)(S2 +P2 — A~,,)—~ Xy5C+~(S — iy5P — ii4~75)]C. (14)Clearly, 5, F, Am are still nonpropagating at the quantum level and can be eliminated using theirquantum field equationseS=—CC, eP=Ciy5C, eA=~Ci-y5yC. (15)Reinserting the quantum field equations of the auxiliary fields in the quantum action, new four-ghostcouplings emerge,£~‘(fourghost) = ~ (S~+ P 2 A~~) = [(~C)(t~C) (C — ~ — 75C)(~75C)+ ~ (~y5ymC)(~y5ymC)1.(16)~Let us now bring ghosts and antighosts together in the same spinor covariants. To this purpose wedefine the Majorana conjugates of C by_C = C~,wherç ~ is the charge conjugation matrjx. Similarly,the Majorana conjugate of the spinor C is defined by C = C~’.In this case, CC = —CC since theseghosts commute. After a Fierz rearrangement, not forgetting an extra — sign since the C commute, onefinds the very simple resultS..~‘(four-ghost)= — ~mC). (17)(C7mC)(Cy

242 P. van Nieuwenhuizen, SupergravitvOne can also obtain BRST invariance of I(qu) without S, P, A,~.In addition to eq. (17), one onlyneeds to substitute eq. (15) into the transformation rules. This we will discuss in subsection 2.8.2.4. Path-integral quantization of gauge theoriesThe approach to covariant quantization sketched up to this point used the notion of BRST symmetrywhich can be used to derive Ward identities from which one may prove unitarity and gauge invariance.An alternative approach is path-integral quantization. Since it exhibits some quantization aspects in aclear way, and explains why I(qu) is as it is, we will also consider this approach.One starts with the path-integralZ = J [dqY]exp(iI(cl))= (em~,I/ía S, P, A,,,) (1)and multiplies by unity1 = d~[~ ô(F,,(j’(~))— a~)sdet(Fa 1R’o) (2)where qS(~)=4,’ +R’ar and sdet denotes the superdeterminant (see appendix). Using gauge invarianceand replacing in eq. (1) [d4i’]by [d4,’(~)],one obtains, after interchanging the [d4,’(~)]andd~integrations and dropping a field-independent factor, writing [d4,’]instead of [d4,’(a)]Z = J [d4,’dCC dC*~]exp{i[I(cl)+ I(ghost)]}6(F0 — a0). (3)Again one may multiply by unity1 = J [dba]exp(~ba7”3b0) (sdet ~)I/2 (4)If one can show that eq. (3) is independent of aa and if the matrix y’~’3is independent of quantum fields,one may replace 6(Fa — aa) by 6(F,. — 12,,) and integrating over [dba] one arrives at the same result asbeforeZ = J [d4i’ dC” dCi] (sdet ~)II2exp(iI(qu)). (5)Thus the restriction that y”3 be independent of 4,’ clearly emerges from this formalism.We now prove that (3) is indeed independent of a,,. A necessary and sufficient 3 criterion = 6~theisproduct that under ofaJacobian gauge and transformation the variation64,’ of the = R’,,N”~’A0with Faddeev—Popov (super) arbitrary determinant A0 and (F,,.1R’0)N’3 equals unity. As for example shownin ref. [607],this is equivalent to6(R’,4”)/ôçb’ +f”,, 0 = 0. (6)

P. van Nicuwenhuizen, Supergravity 243A direct calculation shows that (6) is indeed true, provided one uses as basic fields in addition to e~,I/ia S, P the contravariant vector field A’4 and not the world-scalar Am [446,556]. Perhaps a moresignificant basis is4,’ = (em,.,, Il/rn”, 5, F, Am), Il/rn” = e~’4i/í~”. (7)Also on this basis (6) holds, since6(&fr,,,a )/ôç(, 2= 6(6~~” )/6IlJ~”+ 6(ôe,n’4)/ôem’4 (8)m~with a similar relation with i/i~,~ Am and 4,~-~ A’4 so that the vierbein-terms cancel. In equations such as(6), tracesover fermionicsymmetries or fermionic fields acquire an extraminus sign. Right-differentiations6/64,’ in (6) are equivalent to left-differentiations, see appendix H.On the basis (etm14, l/i,~”, S, F, Am) there is thus a measure in the path-integral given by (det e) 1 since6(ln e’) = (—6e~)e~”while öem~= —em~(6e~~)e”~. If one integrates out 5, F, Am one finds theintegration measure[dem~ di/i~”(detg)2]. (9)The precise form of the measure has no physical significance, but it is important that one can choose ameasure at all. For example, if one would not have a closed gauge algebra (by dropping 5, P, Amaltogether), (6) does not hold and four-ghost terms must be added to the action.Later on we will derive Ward identities from the BRST invariance of I(qu). To that purpose onemust show that also the Jacobian equals unity for a BRST transformation [514].It is not alwaysappreciated that the criterion which allows Slavnov—Taylor identities which was given in (6) is also thecriterion for unit BRST Jacobian. Indeed, the first term is the contribution to the Jacobian from 4,’ ifone replaces r by C”A while the second term is precisely the contribution of the ghost fields. Theanti-ghosts do not contribute, since their BRST-law does not contain antighost fields. In Yang—Millstheory each term in (6) vanishes, but in supergravity only the sum vanishes, but it vanishes for all threelocal symmetries separately.Finally we come to an interesting aspect of path-integral quantization. If one considers a quantizedgravitino in a background gravitational field, one integrates only over [dlfr~”].The classical action is then~‘(cl)=and it is invariant under &/i,~ = D, alone, provided the gravitational field satisfies Ru,, = 0 and the spinconnection c~,.,”’(e) is free from torsions. Under these circumstances one may choose as gauge fixingterm~2~(fix) = ~ ~ yø~y.Il’so that at all stages one has kept the space-time symmetries manifestly preserved. In this case thenormalization factor (det ,Ø)~112leads to a new ghost, first discovered by Nielsen and Kallosh[343—345, 304]. Since a commuting Majorana spinor has vanishing action, one circumvents problemswith spin and statistics by replacing (det Ø)_1~l2 by (det 0)’ (det Ø)1~l2~In this case one finds a

244 P. van Nieuwenhuizen. Supergravitvreal anticommuting and a complex commuting ghost. The net effect is that there is one Majorana ghost.This Majorana ghost is crucial, for example, in order to obtain the correct axial anomaly.2.5. Gauge independence of supergravityCoupling the gauge fields 4,’ and ghost and antighost fields to sources as followsI(total) = I(qu)+ i(çb’J, + K 0C*’3 + L0C’3) (1)one finds, changing the integration variables of the path-integral by an infinitesimal BRST variation(whose Jacobian equals unity as we saw),(64,’J, + K06C*’3 + L0ÔC’3) = 0. (2)Left-differentiating with respect to K0 and right-differentiating with respect to .4, and then putting K0and L0 equal to zero, one finds(jC*’3(RIaC~~A)(t3~~ + jj4,k)~ AF,,y”3i4,”) = 0. (3)In the absence of S, F, Am, &,b’ contains extra three-ghost terms which should be added to R’aC”A. Inthat case, (3) not only relates the longitudinal part of the gauge field propagator to the ghostpropagator, but also to extra four-ghost terms.For linear F,, = F,,J~4,’,this Ward identity can be rewritten as (assuming that F,,, is a commutingconstant)((6Z/6Fflk)iA — C*O64,IJ,4,~5= 0. (4)On-shell the second m~= —4i~ymCA)or term does not otherwise contribute, contraction since either with a64,’physical contains polarization no one-particle tensor yields pole zero (forexample (for example for ôe the term 6~/í~ = 8~CA).Thus the S-matrix is gauge invariant under changes in F,,, [304].For general Fa which are nonlinear in 4,’, the proof may be extended as follows [Townsend and theauthor, unpublished]. Consider changing F,, into F,, + eG,, with infinitesimal . Then we add to (1) aterm iG,,J”, and obtain after right-differentiation with respect to J’3(jC*’3(Go,Ri,,C~~~A +AFa7”3jG0) +iC*’3(Ri,,CCA)iJ,Gp) = 0.(4a)The first two terms are (ÔZ/&)iA while .the last term does not contribute to the S-matrix for the samereasons as before. Thus, on-shell the S-matrix is independent of the choice of gauge fixing term F,,.We must now decide whether the S-matrix is also independent of y”3 [304]. Consider the followingWard identityJ [dqS’dCC dC~~(sdet 7)1/1 ~~C*’3 exp iJ(qu) = 0. (5)For one anticommuting variable, (8/aC)f(C) is C-independent because f(C) is at most linear in C, and

(5) follows from fdC=0 (only f dCC= 1). ThusP. van Nieuwenhuizen. Supergravity 245(6~+ (_1)(’3±lX~l)C*’3iF~ 1R1,,Ca) = 0. (6)As usual, /3 = 1 for fermionic symmetries and /3 = 0 for bosonic symmetries.Let us, for later use, return to (3) and multiply it by F k and — ~i(y~)~a. Then on-shell again(~C*’3(i[ ‘)o~F kRk,,CaA + ~fl~~J7~4,k) = 0. (7)Rewriting (6) as((—1)~’~6~ + C*’3iFEkRI,,C~)= 0 (8)one finds for the first term in (7)(~C*0(y~)ObF kRk,,CaA) = (y~) ~( — ) +1A) (9)Hence, on-shell we find the following Ward identity if one would not add a measure sdet ~integralin the path(6Z/67& + ~(y’)~~( — )E) = 0. (10)The last term is, however, precisely the variation of (sdet y)h/2 because6(sdet ~a0)1/2 = ~(7’)eô (ôy&) (~)~(sdet~aI3)1/2 (11)Thus the S-matrix of supergravity is indeed gauge-invariant under changes in ~ if one chooses themeasure as sdet ~1/2 as we did before.Thus also gauge invariance requires that one take into account the factor (sdet ~ )1/2. This factor isof course the Nielsen—Kallosh ghost which we found before. It plays a crucial role in obtaining thecorrect axial anomaly when one uses the Adler—Rosenberg method.2.6. Unitarity of supergravityClosely related to gauge invariance is unitarity of the S-matrix [522].Unitarity means that unphysicalmodes in the propagators do not contribute to S-matrix2, elements. the renormalizable Let us beginpropagator with the easier is 6~,,(—ik2). exampleof Yang—Mills theory. In the gauge ~(fix) = _~.The unitary propagator, on the other hand, reads W)(—ik2Xe~(e~)~ + ;( ~,) = (_ik2)(6,~,. — ~‘4). (1)This equation holds on-shell only, where k2 = 0.. The problem is now to show that the terms with k,, andits time reversal 1~,..cancel with the ghost contributions. Consider a two-particle cut

246 P. van Nieuwenhuizen, Supergravity= ÷ 01110Fig. 1. Unitarity in Yang—Mills theory.The proof of unitarity is based on the following Ward identities, which one derives by making a BRSTtransformation of the integration variables in the path-integral6BRsT(C*c~(x)c9.W”(v))=0=(3~W°(x)a. Wb(y)) (2)SBRS.,-(C(x) W~(y))= 0 = (i9~W~(x)W~(y)—C*~(x)e~C’~’(y)). (3)Graphically they are represented in fig.momentum-shell.2. In (2) we have omitted a term fl~since we are on~ ~= (a:::::‘: )q,.(~:‘.::: )~~~voFig. 2.Ward identities for unitarity.The second identity describes a conservation law, but note that one must hit all non-physical (but withphysical momenta) lines by momenta. The first identity shows what happens if one only hits one gaugefield: the gauge field becomes a ghost field which emits a momentum at the other line.Let us now return to fig. 1. Using= 6,.,~— (k~1e~+1çk~)(k ~y’ (4)we omit the k,., terms in one of the two propagators, since we can use yet another Ward identity(3• W”(x) W~’(y 1)...W~’,,(y,,))= 0 (conservation) (5)where all W~,(y,)fields are on-shell with physical polarizations. Thus we consider fig. 1 with 6,.,,. upstairsbut 6,.,,. downstairs in the first diagram. Using the first identity in fig. 2 once to cycle clockwise on theleft, and_once more to cycle clockwise on the right, one makes a full circle and ends up with a factork)’ times the final ghost momentum k,. This is exactly the Faddeev—Popov ghost plus minussign with counter clockwise orientations. Similarly, the term with —,k’~k,.(k R)’ can be used to make afull circle counter clockwise and yields the other orientation of the Faddeev—Popov ghost. Thus, asexpected, although after fixing the gauge all four modes of W”,~propagate for a given a, the complexworld-scalar ghost subtracts two modes, and one ends up with two physical modes in the quantumtheory as well.

P. van Nieuwenhuizen, Supergrav0y 247Let us now turn to supergravity. There are 16 real fields ,l/a so one expects after gauge fixing thatthere are 8 gravitino modes propagating. Indeed, there are 8 modes in the propagator as an explicitdecomposition shows [522].However, the complex spin 1/2 supersymmetry ghost is expected to subtractonly 4 modes. (Fermions have always half as many modes as bosons.) Where do the remaining 2 modesgo?The solution of this puzzle can be found in the proof of unitarity of ref. [522].To see what happens,we begin by stating that the unitary gravitino propagator is given by~&a(yXy 0)~~~k2. (6)The Ward identities now become (considering a gravitino loop)6BRST(C*Y. çfj) = 0 = ( ~ y,ø)(-y. i/i)) + (C*JC), cf (2) (7)hence (yx)i~fy)~y) = 0 on-shell, since there are extra factors I in (C*/C), and= 0 = (y• i/i(x) i/i~(y))+(C*a(x) O~C”(y)), cf. (3). (8)Notice now that y’4(4i,.,4,,.) = 2k,.k2 so that we may use conservation to replace in one of the two cutpropagators the unitary propagator in (6) by the renormalizable propagator ~y,.X’y~k2. (We also usethat a term k~k,,in (6) vanishes on-shell.) In the other propagator one has leftk2(F~(unit)— F~L~(ren)) = —~(k,.XXy~+ y,,k’R’k,. — 2k,Yk~)(k kr’. (9)The second term can be written as k~times a factor (—1~,.)(k . k)’ times the gravitino propagator~ Cycling as before with k,.,, once on the left, and once on the right, both times clockwise, oneends up with k,, times —t (k . ~ hence with precisely one ghost loop. Notice that contracting agravitino with its own momentum on one side of a blob first of all ffips this fermion line into a ghostline, but, as is clear from (8), it also obliterates the numerator of the gravitino propagator at the otherend, replacing it by the numerator of the ghost propagator.Thus the first two terms in (9) are just equal to the ghost contributions. That leaves us with the lastterm. It represents indeed two modes in the propagator (since R’ can be written as ~ + wóY withspinors w~,w~orthogonal to the physical spinors u~and u); but they decouplefrom the S-matrix. Thisfollows easily if one cycles once with the factor k,. on the left and then uses (7).The above sketch of the unitarity proofhasbeen extendedto general N-particle cuts, taking into accountall ghosts and antighosts. The results are unchanged. One can choose different commutation propertiesbetween ghosts without violating unitarity [553].2.7. Supersymmetric regularizationAll preceding manipulations only make sense if one would use a regularization scheme to givemeaning to the divergent loops. Preferably, but not necessarily, a scheme should be used whichmaintains manifest supersymmetry at all stages. Such a scheme exists. It is an extension of dimensionalregularization [417,425], but there are also some differences. It is based on the idea of dimensionalreduction.

248 P. van Nieuwenhuizen. SupergravityConsider as an example how the scheme works, quantum electrodynamics. The action is given by= —~9,.,A,.— ~9,.A~) 2 — ~y’~(9,~— ieA~)A. (1)We now continue only coordinates and momenta to n dimensions where n 4. However, the indices j~of A,., remain 4-dimensional, while also A remains a four-component spinor. Thus one can write theaction as= —~(t9,A1 3~4.) ieA 2— ~y’(3~— 2+ iety”AA~. (2)1)A — ~(o,A~)In other words, we have here normal QED in n-dimensions, plus extra scalars coupled to the spinors bya Yukawa coupling. Since 0 i n but n

P. van Nieuwenhuizen, Supergravity 249If one manipulates the Dirac algebras as follows=yJt’y,=(2—n)J( (7)= —2Xone finds that (6) is satisfied at the one-loop level to order if one uses either dimensionalregularization or dimensional reduction, but to order °only the latter scheme preserves (6) [556].In ordinary renormalizable models, this scheme respects renormalizability only if these theories aresupersymmetric. For example, the two coupling constants e in (2) are rescaled by different Z-factors.Thus for supersymmetric theories this scheme makes sense, but for non-supersymmetric ones it doesnot. One can also use it for supergravity, where the issue of renormalizability does not exist. One canuse this scheme to calculate anomalies.There exist other regularization schemes which are claimed to preserve global (and sometimes local)supersymmetry. One such scheme is the higher derivative scheme of Slavnov. For example, inYang—Mills theory one adds to the usual Yang—Mills action a term A2(DAG’4,.” )2 where A ultimatelytends to infinity. Propagators now fall off like k4 and this regularizes all loops except single loops. Forthose one sabstracts infinities by adding to the action gauge-invariant Pauli—Villars terms~ (62~1(A,.,,A )/ôA~6A,.)B’,.,B~. — m~(B~..)2 (8)(satisfying, as usual, ~ a, = ~ a,mcorresponding to D~(A)B’4’1= 0= 0) as well as anticommuting ghosts c’ for these new gauge fields B,.,.2? = D~(A)E’ D,,(A) c~— ~ (9)For global supersymmetry one can use superfields and find globally supersymmetric Yang—Mills-gaugefixing terms. For supergravity it is not clear whether this scheme can be applied.There is also the BPHZ-scheme, as applied to supersymmetry by Clark, Piguet, Sybold, where oneexpands Green’s functions in terms of the external momenta. Each coefficient turns out to besupersymmetric, but to prove this requires detailed analysis.2.8. BRST invariance and open algebrasSupergravity without auxiliary fields has an open gauge algebra, by which we mean that it closes onlyon the classical mass shell. On the other hand, for quantum calculations it is much easier to workwithout auxiliary fields. Thus the problem arises what new features arise at the quantum level for gaugetheories with open algebras. In particular, unitarity and gauge invariance of the S-matrix can be derivedfrom Ward identities, and to obtain these Ward identities one needs BRST invariance of the quantumaction. In this subsection we see how to modify the transformation rules and action to obtain BRSTinvariance if the gauge algebra is open [304, 1621. For closed gauge algebras we refer back to subsection2.

250 P. ran Nieuwenhuizen, SupergravitvThe gauge commutator for open algebras has the formR, k (\a$ i k — i fy + ~ I)a.k $ \ 1 /3k a — yJ ‘40 ~ ,f~!a13where I~’= 61”/64r’ with JCI the classical action, and we always consider in this section right-derivatives.In the variation of the usual quantum action of subsection 2, it is only at the last step that not all termscancel, but one is left with6I(quantum) = C*aF,,.~(~I~/i,~,.CvA )C’3. (2)As we discussed in the section on matter coupling, such terms proportional to I’~can always be canceledby adding an extra term to 64,. After bringing I’/ to the left, one finds64,’ (extra) = _~C*aFa.j7~yCYAC’3 ()(J+1)i (3)This new variation of 4,’ leads to extra terms in 61 (quantum). Keeping the same gauge fixing term,we find for the ghost action by variation of the gauge fixing terms and sandwiching the result with ghostsand antighosts the following resultJ = JCI~~p 7~’3J3~+ C*aF,, 1R10C’3 ~ (4)The new four-ghost coupling is due to 84,’ (extra) and one has a factor —1/4 rather than —1/2 since bothantighosts appear symmetrically due to symmetry properties of ‘q’ which we now discuss.From the gauge commutator it follows that 64, = I’~]i~-q’3~” is again a local gauge invariance of theaction. HenceJciJd~i~~$~a = 0. (5)This means that the nonclosure function ~~ is a sum of terms either with the symmetry~q_()ij+l~ji (6)(where i = 1for fermionic fields and i = 0 for bosonic fields), or with &~‘a gauge transformation itself,which then of course vanishes on-shell,= R1A(X)~~IcJ) (7)where XAI is some matrix which follows from (5). In supergravity we calculated the nonclosure function~ in subsection 1.9 and found that it is nonzero for i, j, a, /3 all supersymmetry indices, since only thecommutator of two local supersymmetry transformations on a gravitino gave a field equation, namelythe field equation of the gravitino. Thus for the tetrad 64,’ = 0, from which we conclude that XAJ in (7)vanishes. Hence the nonclosure function has the symmetries in (6), and this justifies the factor —1/4 in(4). This symmetry follows also directly from eqs. (12, 13) of subsection 1.9 if one replaces R” by_C_tR~~,Tand uses that y,,C’ is symmetric but y5y,,C~antisymmetric. In general one has to add the

P. van Nieuwenhuizen, Supergravily 251local symmetries Ô4,’ to the total set of gauge invariances in theories with open gauge algebras when64,’ is nonzero.It should be stressed that these new four-ghost couplings in (4) can no longer be written as aFaddeev—Popov determinant and hence that the usual Slavnov—Taylor method for deducing Wardidentities is not applicable here. Instead one should use BRST invariance, as we shall do.We must now use the new 4, variation in the old two-ghost action, and the new and old 4, variationas well as the ghost variation in the new four-ghost action. From now one we specialize to supergravitywhere ~~ depends only on tetrads and the indices i, j, a, /3 refer to gravitinos and supersymmetryparameters. In this case the new 4, variation does not contribute in the new ghost action sinceand since F,,, and F 0~,.in (4) do not depend on the gravitino field. (We exclude gauge choices F,.quadratic in 4’ from consideration. This is not necessary, but covers all useful cases.) For the remainingthree variations one findsC*~F,,,1Ri0,,( ~ ~C”/1 ~Y’)C’3— ~C*~FajC*’3F’3,kfl ~/~,(RIACAA )C’ tC~’_~C*aF,,,jC*$F$,kn~(_ ~r~C~AC”)C”. (9)Since C” in the first term is commuting, we symmetrize the first term in both antighosts and find6I(quantum)= _~C*AFA.IC*TFT,,F~4$aC~AC’3C’ (10)where the expression for P ‘~~$alooks very much like a commutator of an ordinary gauge transformation64,1 R’,, with a transformation 64i’ ~ ~ sinceF” — fly$,k ii k a j Iy,k77$a kl + Iy,kTI$a ki — 77y~j U $a.11Since the transformation 64, -~ ~ looks like one of the terms one finds in the commutator of twogauge transformation, one is led to consider the Jacobi identities for three consecutive gauge transformations.Consider the Jacobi identity6(~)[6(~), ô(~)]— [6(q), 6(~)]8(C)+ cyclic in ~ =0. (12)Substituting (1), and substituting in the result once more the identity (1), one findsRiAAA,,$~Y?~~ + I’~B”+ cyclic in ~ = 0 (13)where A and B are defined byAA __;A ;8 itA flk~ a$y — I ~äJ $y ~J a$.k~ yB” = —n ~f8,,0~1’3r~”+ 77 ,,$.kRk~,~77$~a— R ‘y.k(”77 ~ n’3r(~)“+Rç,,,77 Ic! ‘i~rc”()’~’ ±y) (15)(8)

252 P. van Nieuwenhuizen, SupergravityIf the algebra closes, B =0. In the second and third terms of B” one can bring ~“change in sign. Moreover, since qIk()Yk =77~k(yi it follows thatto the right withoutB” = B”(—~~’. (16)We now write r = C’~A1,770 = C’3j12 and ~“verify a very useful identityCM3 in order to make contact with (10). One may~ + two cyclic terms = C’3A 3C”C”(—)”(3A ,A2). (17)In other words, using ghost fields the cyclicity is automatic. With i~u—)” = —~j~and= 7l~$,kR,,C”AC’3C” (18)one finds upon contracting B” with two antighosts and dropping the common factor 3A1A2, thefollowing result (after substantial relabeling of indices)C*AFA,IC*TFT.,Bhi = C*)FA jC*TEI[T1~öfö$a — 77y$,kRa R’y.kfl~a+R’7kn~’,,(—)”]C’~’AC’3C”. (19)This result looks indeed very much like (10) if one compares (11) with the expression between squarebrackets in (19). In fact, these expressions are equal. To see this note that i + I is even since B has thesymmetry in (16) but the two antighosts in (19) have symmetry (—) 1X~±1)~Since always either i or j inB” is fermionic, see (15), ii = 1. Thus the terms with B” in (19), coming from the triple commutator ofgauge transformations, are indeed the same terms as what is left of 61 (quantum).If one could argue that B” = 0, then one would have completed the proof of BRST invariance.However, B is nonzero and we proceed to solve it from (13). On-shell R ~AAA...=0 and defining R A suchthat R’A 0 on-shell (parts vanishing on-shell are proportional to I.,,, and can be incorporated into thenon-closure function q), it follows that AA = I(cl),A”. Hence (13) becomes I(cl),(RIAAA 1 + B” +cyci.) = 0 which tells us that the second factor is a gauge invariance of the classical action. ThusR&AAAE.. + B” + cycl. = R’AX” + I(cl),1M”with I(cl),,I(cl).,M” =‘O and this yields finallyB” = (RIAXIA + IC] MJII) + (— )“~(j ~ 1) (20)where M” has the same symmetry in (ji) and (ii), namely that of n”. Thus M” is completely symmetric.We will now show that in supergravity the function M vanishes while X is only nonzero when Arefers to local Lorentz rotations. From (15) it follows that B” does not contain a~por aae terms. Indeed,i~ and fl~o,kand R’71, do not contain any derivatives, while R”7 does not contain a derivative of fieldsand ~ contains only a derivative of a field (of the tetrad) if r refers to local Lorentz rotations and a, /3to local supersymmetry (in this case the commutator is proportional to üj~m12(e, 4’) as we saw). Thus Mi”vanishes. The Jacobi identity thus becomes[R’AA”a’3y + IiR’AX~’p7(— )‘~ 1]C”AC’3C°( — )~= 0 (21)

P. van Nieuwenhuizen, Supergravity 253where X~ 87follows from (13), (15), (17). Since the R’A are independent, this implies[...fA~fa0 +f”a$kR”y— IcX~071(~7n’3~a + cyclic terms) = 0. (22)This is the open gauge commutator for fields which do not transform as 4,’ but which transform in theadjoint representation. For closed gauge algebras X = 0 and we find the identity which is needed fornilpotency of BRST transformations of ghost fields (eq. (12) of subsection 2.2).Since in supergravity R ‘A does not vanish on-shell, AA,0,, must vanish when I~]= 0. Inspection of (14)shows that this is only possible when A refers to local Lorentz invariance and a, /3, y to localsupersymmetry. In that case fAa0 is proportional to the spin connection (as we already discussed), whichrotates under local supersymmetry into the gravitino field equation plus terms which cancel against theproduct of the two structure functions in (14). Thus also the index I in (13) refers to the gravitino.The final conclusion is that the remaining terms in the variations of (4) under the usual BRST lawspIus (3) is given by (see (10), (19) and (20))SI(quantum) = ~C*~~~FA,,C*TFr.I(2R‘AX ~,,A07)C~~A C’3C”. (23)Since i + 1 = even and A = local Lorentz, one may place R’A next to FA,, without sign change. Hence wecan cancel (23) by adding an extra term to the ghost transformation law of the Lorentz ghost field,6(extra)C” = —~f”07(extra)C”AC’3f”~7(extra) = ~C*~Fj~X~oCs.This new term in the variation of the Lorentz ghost cancels 81 (quantum) when substituted in thetwo-ghost action, but it does not yield new variations in the four-ghost action since that depends onsupersymmetry ghosts only.Thus supergravity without auxiliary fields, i.e. with open gauge algebra, is indeed BRST invariant.The total quantum action is given in (4), and the transformation rules are the usual BRST rules forclosed gauge algebras plus the extra terms in (3) for the gauge fields and an extra term in thetransformation law of the Lorentz ghost. Those results are the same as if one had started with auxiliaryfields and eliminated them from the quantum action. In particular, the extra terms in the BRST law forthe gravitino come from elimination of the auxiliary fields. Also the extra term in the Lorentz ghosttransformation law are understood from this point of view, namely only in the Lorentz rotation which isproduced by two local supersymmetry transformations are there auxiliary fields.For the extended supergravities one might reverse these arguments, and obtain information aboutthe auxiliary fields by first establishing BRST invariance. Since this is one of the central problems insupergravity, we have been rather detailed in this section.We close with referring the reader to the literature for the most general case of theories with opengauge algebras [162]. In these cases, the process of adding extra tenus to the ghost action and to thegauge and ghost fields does not stop after one cycle, but one can get 6, 8, etc. ghost-couplings. At eachlevel one considers a Jacobi identity of two gauge transformations and one new transformation of 4,’(the first new case being 64,’ — ~ These new transformations are no new invariances of the action,but lead to results similar to the A, B analysis given above. One gets each time new M terms in 64,’ andnew X terms in ÔC”, but 6C*a remains unchanged. These new BRST laws are again nilpotent upon use

254 P. van Nieuwenhuizen. Supergravitvof the quantum field equations (now also of the ghosts) as derived from the full quantum action. Forexample, two BRST transformations on the antighost C*a yield a result proportional to ~5(F 0y’3”) =F0,,R’7C” which is indeed proportional to 6.2~(qu)/6C*a.(Nilpotency is important in renormalizabletheories to analyze the higher order quantum correction.) It is not known whether such theories with 6and higher ghost-couplings exist, and, if they exist, whether one can eliminate these ghost couplings byintroducing auxiliary fields.2.9. Finiteness of quantum supergravity?Quantum supergravity is more finite as far as ultraviolet divergences are concerned than ordinarygeneral relativity. Since this subject has been discussed in detail at many conferences, we refer fordetails to these sources as well as the original literature. For a review, see ref. [286].The first breakthrough [512]occurred in the N = 2 model, where an explicit computation of theone-loop correction to photon—photon scattering revealed that all divergences in the process cancel. Atheoretical explanation was given [512], based on the observation that(i) graviton—graviton 2 and on-shellscattering Rh,. = 0. is finite because possible counter terms are of the form £2? =aR~,.+ (ii) Using /3R (global) supersymmetry of the S-matrix, one can “rotate” gravitons away and replace bygravitinos, etc. This then shows that all four-point processes are finite. (Higher point processes have alsobeen shown to be finite in this way.)Of course one must show that starting from graviton—graviton scattering one can indeed reach allother processes. This has been done.An alternative proof, valid for all n-point functions, was given [123]by showing that no £2? existson-shell which is invariant under the on-shell transformations of supergravity. In particular it was shownthat no £2? can start with gravitino terms but must start with R~,.2and R2. Not all details of this analysishave been published.At the two-loop level, again both proofs of finiteness have been used. Since the only bosonic £2?which does not vanish on-shell is given by [503]£2? = a~2R,,0R~,.”R~’3 (1)and describes helicity-ifipping of graviton—graviton scattering, whereas supersymmetry (global as well aslocal) conserves helicity, a =0. Hence, as before, graviton—graviton scattering is two-loop finite insupergravity, and rotating gravitons away and replacing them by other particles, one shows that also atthe two-loop level supergravity is finite [283].The counter term analysis has matched also this result byclaiming that one cannot extend £2? in (1) to a full on-shell supersymmetnc invariant [123].For N = 1 supergravity, the tensor calculus [189],or, equivalently, superspace methods [191]havesuperseded these proofs and prove one- and two-loop finiteness. In particular, the counter termanalysis was performed only to order 4,2, whereas the tensor calculus and superspace give completeresults with considerable less algebra. For N = 2 supergravity similar results have recently beenobtained.Possible problems (but no more than that!) arise at the three-loop level. Namely, it seems that in allN-extended pure supergravities on-shell a supersymmetric invariant three-loop counter term exists[1291.Whether it actually will have nonvanishing coefficient is the big question. This £2? has in itsbosonic sector the Bel—Robinson tensor.

The Bel—Robinson tensor is defined byP. van Nicuwenhuizen, Supergravity 255T”’3”4 = R~~PAUR$P’4O. + *RaPAU *R$’4 + R*aPk~~R*$~’4~ + *R*aPAu *R*$’4 (2)where the stars denote left and right duals. Since *R*~ + Ra 0p,~,is proportional to R~,.— ~ and** = —1, it follows that on-shell (i.e., Ru,. + Ag,,,. = 0), left and right dual of the Riemann tensorcoincide. Also *R*‘~$ is on-shell proportional to the Einstein tensor. The B.R. tensor is symmetric inaf3, in ~ under pair exchange (a$) ~-* A~and on-shell it is conserved, Da Ta$A~~=0. (See the thesis ofL. Bel, for example.) If there is no cosmological constant A, one has on-shell that T~~0AMis completelysymmetric, traceless and conserved.It is now possible, either by the Noether procedure or (to all orders5to in Ka and full locally orders supersymmetricin ~fi,.~)by thetensor invariant. calculus, Thus, to at show the three that loop one can levelextend there the is a invariant £2’ for N T~03,8T”’3”’ = 1 supergravity on-shell which is on-shellinvariant under local supersymmetry and which poses a danger for the finiteness of supergravity.The crucial question whether N =8 supergravity is finite beyond two loops has been analyzed bymeans of superspace methods by Howe and Lindström. Just as for N = 1, 2, 3, 4 supergravity, alltorsions and curvatures are on-shell functions of only one superfield. For N = 1 this is a 3-spinor WABC(see section 5), for N = 2 a 2-spinor WAB, for N = 3 a one-spinor WA, for N = 4 a scalar W, while forN = 8 one has a scalar W,1k, where i, ..., 1 are SU(8) indices. Moreover, W,jkl is totally antisymmetricand self-dual. The 0 =0 components of W,Jk, represent the scalars in the N = 8 model after fixing thelocal SU(8) gauge (see section 6). The constraint D~,Wjk~m= öbAk~m1aputs W on-shell. These results arelinearizations of results obtained by Brink and Howe.The Weyl tensor C appears in W asD~D~D~D~Wmnpq~ (3)(In two-component notations, C is totally symmetric, see the appendix.) It is now easy to write down asupersymmetric invariantI = K~~Jd’~x ~ (4)2088h8where Note that g~,.= from y,.,’, (3) + Kh~,..Performing it follows that Wthe is dimensionless, 9-integration one so that endsone up finds with ainnonvanishing I a term of the product form of K Weyltensors (together with its supersymmetric extensions). Thus it would seem that N = 8 has nonvanishinginvanants which could serve as infinities in the S-matrix from 7 loops onwards. Increased understandingof how to build invariant actions with measures d”8 where n

256 P. van Nieuwenhuizen, Supergravit’yseems unlikely. Secondly, other N = 8 formulations might exist, notably formulations involvingantisymmetric tensor fields. However, it is known that different field representations for a given spin dolead to the same S-matrices (although they lead to different anomalies [560]).It should, however, not beoverlooked, that using antisymmetric tensor fields, the trace anomalies cancel “miraculously” in theN = 8 model [609].Something is going on here.Extra symmetries usually help to eliminate possible invariants. For N = 8, the maximal on-shellsymmetry group is U(8) (see subsection 3.3), and W,Jk, is manifestly SU(8) covariant, while a U(1)invariance would violate the self duality property. Thus there seem to be no on-shell extra symmetriesleft. (The chiral-dual symmetries first discovered in “miraculous” cancellations were later shown to bepart of the U(N) invariance. In addition, the N = 8 model has a self-duality, see section 6.)In the presence of central charges, the integration measure is no longer d320 since in that case thehighest component of a multiplet varies into a total derivative plus something more. However, centralcharges usually vanish on-shell (section 6).Yet another approach is to pin one’s hope on the spontaneously broken version of the N = 8 model(section 6). However, mimicking in superspace the dimensional reduction from the massless d = 5,N =8 theory to the massive d =4, N =8, one would expect to end up with a dangerous invariant £2? ind = 4 dimensions. In fact, the reverse might happen: it might be that massive d = 4, N = 8 is not evenone-loop finite. This is presently unknown and under study.Of course, miracles do sometimes happen, but they would have to happen at every ioop order. Itseems better to accept a fundamental property of gravitons: that no nearby massless theory exist (thevan Dam—Veltman theorem) and to use nonperturbative methods. Perhaps this is the way quantumsupergravity should go in the future, but it is easier to say that one should do nonperturbativecalculations than to do them.2.10. Explicit calculations2.10.1. Green’s functions are not finiteAs a first application of the Feynman rules derived in the last section we calculate the gravitonenergy in simple supergravity at the one-loop level. This yields at once an important result of quantumsupergravity: Green’s functions are not finite in general [510].Later we shall see that S-matrix elementsare sometimes finite.The general amputated two point function receives contributions from a spin 2 loop, a generalcoordinate (spin 1) ghost, a spin 3/2 loop and a supersymmetry (spin ~) ghost, see fig. 1. The divergent partsare local and the general form is (using dimensional regularization)D~,.,,,0.= —~-—~ [Ap~p,.p~p,, + (2B6~~6,.,, + ~ + (2Dp~p,.6,.,~+ 4Ep~pp6,.c,.)p 2]where one should still symmetrize in (j~ii),in (po) and under (~~v) ~ (,po). As regularization scheme we~graviton spin 1 ghost gravitino spin ~ghostFig. 1. Graviton self energy graphs.

P. van Nieuwenhuizen. Supergravirv 257may use either dimensional regularization or dimensional reduction for the one-loop divergences. Thelatter scheme should be used in general (see subsection 7).For the results one findsspin 2 spin 1 spin 3/2 spin 1/2A 13/15 —1/6 —3/40 1/12B 71/240 —1/24 23/480 0C 27/80 —7/48 —7/80 0D —19/60 1/8 7/80 0E —11/40 1/4.8 —13/480 —1/48We have added an extra minus sign to the spin 1 (ghost) and spin 3/2 (physical) loops, but not to thespin ~(ghost).Two conclusions can be drawn from these results:(i) The supersymmetry ghostfield commutes with itself, since only then is the Ward identity~L7)1’ ~Lvpuj-’satisfied. It is of course satisfied separately in the purely gravitational and in the extra half-integer-spinsector.(ii) Green’s functions are not finite. For example, the B coefficients do not sum up to zero. InS-matrix elements the two point functions vanish since p2 =0 and polarization tensors are transverse. Insupergravity, the S-matrix elements of higher point vertices are finite if one adds self energies, triangles,boxes. However, the sum of self energy graphs is separately not zero. In fact one findsB(photon) = , B(photon ghost) =0B(real electron) =1B(real1scalar)=Clearly, all B are positive as one might expect from a unitarity argument and no magical combination offields of different spins can yield zero for the graviton self energy. We have restricted our attentionhere to B because if one couples the gravitons to a traceless conserved source (photons) only Bcontributes.As a natural counterpart one may consider the gravitino self energy. Again the four-gravitinocouplings do not contribute, nor do the four-ghost couplings due to elimination of 5, F, Am from thequantum action. One finds the results in fig. 2. In the gauge y. 4’ the ghost loop is zero separately andthe self energy is again transverse. In more general gauges, the self energy graphs of the gravitino aretransverse independently of whether the supersymmetry and coordinate ghost commute or anticommutewith each other. This freedom has been explained theoretically [553].2.10.2. Matter couplings are not finiteSince Green’s functions are divergent, the next question is whether S-matrix elements are finite. We

258 P. van Nieuwenhuizen, Supergravitvpgravitino—graviton loopFig. 2. Gravitino self energy corrections.spin i—spin 1/2 ghost loopfirst consider matter coupling theories (i.e., not the extended supergravities) and we will later come backto the pure (simple and extended) supergravities.Only an explicit calculation can demonstrate that matter couplings diverge, as a moment of thoughtwill convince most readers. As an example of such a calculation, consider photon—photon scattering atthe one-loop in supersymmetric Maxwell—Einstein theory. The spin content of this system is (2, 3/2) +(1, 1/2), and the action reads= 2(gauge) + ~2’(kinetic) + ~ eç~u. Fy~LA+ (4,4, 4i2A2 A 4)terms.The process photon—photon scattering is considered since it yields the simplest diagrams; these aredepicted in fig. 3. The diagram “ME” stands for all diagrams which are already present in the ordinaryEinstein—Maxwell system and which contain only gravitons and photons. These one-loop divergenceswere evaluated with the background field method and with normal field theory. As expected, the sameresult was obtained. This result was rather amusing: (137/60) times (n — 4)_i times the energymomentumtensorof thephoton squared. The other diagrams in fig. 3 are what supergravityadds, and threeconclusions are drawn [510]:(i) The supersymmetric Maxwell—Einstein system is invariant under combined ~uality-chiralitytransformationsup to order K2 at least6A —aiy5A.The duality invariance of the ordinary Maxwell—Einstein system and the chirality invariance of themassless Dirac system is well-known. However, the Noether coupling is invariant under 2, sincecombinedF,,,.4 andduality-chirality transformations. This explains why all divergences are of the form T,,,.60 120 402Fig. 3.

P. van Nieuwenhuizen. Supergravity 259are not duality invariant. The combined chirality-duality invariance first found in this system (asa result of “miraculous regularities” in an explicit calculation!) has been found also in the extendedsupergravities as a complete invariance to all orders in K (see section 6).(ii) Mattercouplings are not finite. The sum of the coefficients in fig. 3 is 25/12 and hence the S-matrixdiverges. Also all other matter couplings have been found to be divergent. These include(a) in the (2, ~)+ (~,0~,0—) system photon—photon and scalar—scalar scattering is not finite [516].(b) in N = 4 extended supergravity with spin content (2, 3/2~,16, 1/2~,02) coupled to m matter 0(4)multiplets with spin content (1, 1/2~,06) one finds that gauge—photon gauge—photon scattering isone-loop finite (although virtual matter contributes!) [223]. This can be explained by helicityarguments [512, 515]. Anyprocess whose externalfields are related to the graviton by symmetries is oneand two-loop finite, even though virtual matter fields contribute. However, matter-photon matterphotonscattering is not finite [223].(iii) It is incorrect to state that unitarity prevents cancellation of all divergences (as the finitenessof extended supergravities shows). In a background field calculation, the coefficient of R,L,.2, R2, F,,,.’5 etc.are indeed positive, but on-shell G,,,. = T,,,. and for example R,LIF~”’F~~ can become negative. (For adiscussion about the meaning of G,L,,,. = T,~,.for diagrams, see ref. [607].)Thus we conclude that it seems likely that no matter coupled system exists with a finite S-matrix atthe one-loop level.2.10.3. Calculations of finite processesFor pure N = 1 supergravity, the simplest and at the same time only rigorous proof on one- andtwo-loop finiteness is by means of the tensor calculus. For N> 1, no rigorous proofs exist, but it israther probable that also here one- and two-loop finiteness exists. Historically, though, it was onlythrough explicit calculations of certain processes that one-loop finiteness was discovered. Since thesecalculations have been done up to N = 8, it is very probable that all extended supergravities areone-loop finite. About two-loop finiteness no rigorous results beyond order 1(2 are known because hereno tensor calculus is yet available. Also two-loop computations in supergravity are hard. Hence here atensor calculus might bring relief.N =2. The first successful result in quantum supergravity was a calculation of photon—photonscattering in N =2 extended supergravity [512].The only terms in the action which contribute inaddition to the N =2 gauge action is the Noether coupling= ~ I/F,L(F’~”+ ~ (4,,,. and ç,. are the two gravitons). (1)The set of diagrams is given in fig. 4.137 (23\~ (-i)Fig. 4.

260 P. van Nieuwenhuizen, Supergravio’All other diagrams are zero, for example the two diagrams with gravitino ghost loops.Again all divergences are proportional to T,,,.2 and again an exact chirality-duality invariance is theexplanation. We have indicated in fig. 4 the coefficients of these graphs.Since there are two local supersymmetries associated with the two gravitinos 4’,. and ~, we choosetwo gauge conditions y~4’ =0 and y~ =0. Since(2)one finds that 6( ~,)’y4, = 0, so that there is no mixing between ghosts related to 4’ and ghosts relatedto ç. In fact, these ghosts do not contribute. For example, the vertices due to the spin connection inCØC yield (Cy~”C)3beaM.Only the vector terms in Cy~~ra~~C survive, but those yield eAA or ÔAeA,,.. andthis in turn yields zero since T,~,.(photon) is conserved and traceless. There is no mixing betweenbosonic and fermionic ghosts since in that case a gravitino should emerge from the ghost loop and leadto a second loop. All gravitational ghost contributions are already contained in “ME”. Also thephoton self energy is contained in “ME”.N = 3. The N = 3 pure extended supergravity theory contains both the N = 2 case and thesupersymmetric Maxwell—Einstein case as consistent truncations (meaning that one can set some fieldsequal to zero, such that they not appear in the transformation rules of the rest of the fields). Againphoton—photon scattering is considered [516] and again only the Noether coupling contributes inaddition to the kinetic terms= ~eK(JI,~’o. Fy”A — ~gjk 2V’~i/i,,,’(eF~~” + !y,~Fs~~J)4’kAgain one finds finiteness, both for photons with the same index i and for i +j —~ i +j scattering. Thischecks also the nonzero result for the supersymmetric Maxwell—Einstein system.N = 4, 5, 6, 7, 8. Photon—photon scattering with all possible internal indices (“i” and “j”) as well asscalar—scalar scattering is one-loop finite [223].These calculations serve as a check on all othercalculations in gravity and in supergravity. Some extra “miraculous cancellation” in the calculations forN = 6 and N = 8 seem to indicate new symmetries, but these have not yet been found.We conclude by stating that, as in other areas of supergravity, there are very many details which thereader cannot find in the literature, nor could these details be explained here. The only way to discoverthem is to do calculations oneself. The reader be warned — it is addictive.2.11. Higher-order invariants, multiplets and auxiliary fieldsThe action of simple and extended supergravity should describe states which constitute multiplets ofglobal supersymmetry. In general, one does not know the auxiliary fields of the extended supergravities,but by assuming that they exist, one can get some knowledge about what they are by considering thephysical states of the following action [194]2 + more) + 13(Rw,,2 — ~R2+ more). (1),2?(kin) = —~y(R+ more) + a(RHere, “more” denotes terms with gravitinos, auxiliary fields, vector fields, etc., which are needed tomake each term separately locally supersymmetric.

P. van Nieuwenhuizen. Supergravitv 261For N = 1 supergravity, one finds that the R2-type invariants as far as kinetic terms are concerned,are given by (either by the Noether method or by the tensor calculus)~2’2=R2+R .yJy.R—4(~S)2—4(a,,P)2+4(a•A)2= (R,~,,2—~R2)—~I~(L16,W — 9,,~9,.XR”...~“y R)—~(8,~A,. —(2)For the gauge action we have, of course,~ =R+ifr R+~(S2+F2—A~,).To find the physical states in this system, it is advantageous to use spin projection operators in order todescribe the terms bilinear in fields. In this way one finds with (see subsection 1.13)RM,. = ~(EhM,. — h,,,. — h,.,, + h.,,,.) = ~(F2+ \/~p0.ts + PO.S)h,,,,,eR = L1O,~,.h,,,,.— ~h,,,.(L~F2— 2LIJFO.s),0.h,,,,. (3)R 2 = 3h(LI2P°”)h, R,~,.2—~R2 = ~h(LIFF2)hthat the graviton propagator becomes P2(~yE+ ~$L12Y’+ FOs(_~yE+ 3aE2)~’,so that the gravitonstates are:(i) a massive spin 2 with M2 = —~yIJ3and ghost if /3 0(iii) a massive spin 0 with M2 = ~y/a and physical if a > 0.For the gravitino part one finds (see subsection 1.13)4’ R = ~(P312— 2P1”2~)J4,~ yJY• R = ~121i(LIP1”2”)J4’ (4)4i~,LI&”~(R,, —~y,;y. R)~(EP3”2),~’4’that the propagator becomes —P3”2[J(f3L1 + ~y)]’ + P~’2[J(aEJ+ ~y)]~. Thus the gravitino states are:(i) two massive spin 3/2 with M2 = —~y//3and both ghosts if /3 0(iii) two massive spin 1/2 with M2 = i1syla and both physical if a > 0.Note that at E = 0 both P2 and ~O.S(and F3”2 and Pl~2~s) define the usual tetrad and gravitino states.To check these results, note that ~2’3 contains only highest spin (as expected from a conformalinvariant) while ~2’,yields the usual spin 2 and 3/2 propagators if sandwiched between conservedcurrents. The relative factors between the graviton and gravitino projection operators in ~2’2and ~2’3arethe same.If one considers the spin 1 and spin 0 states of the auxiliary fields (which propagate in 2’2 and 2’3!)one finds for the A,,, propagator P’(~y+ ~/3[])~’+F°(~y — 4aLJ). For the fields S and P one finds twomore massive spin 0 states. All these fields clearly constitute(i) the usual (2,~)massless physical multiplet;

262 P. van Nieuwenhuizen. Supergraviti’(ii) a massive (2, ~,~,1) ghost multiplet with M 2 =(iii) two massive (~,0~,0~)multiplets, the first containing 5, F and one mode of y~R. the othercontaining t9 A, the scalar in R + R2 and the second mode in y R. Both have M2 = ~y/a.If one would not have known these auxiliary fields, one would have predicted the existence of auxiliaryfields which lead to the encircled states(2, ~),(2, ~,~,CD), ~,0~,®)‘ U’ (I (ö~)). (5)Of course, this does not uniquely determine which field representation should yield these states, but theresults are so simple that one can hardly miss AM, S and F. It also explains that a chiral gauge field isneeded in N = 1 conformal supergravity.One can turn these arguments around and deduce some knowledge about auxiliary fields from therequirement that all states should fill up multiplets [168]. In the N = 2 model, one of the linearizedinvariants contains R2 + R . yJ’~y•R + 2[8M(t,,,. — 2~h’2(3MA~— 8~AM))]2where the auxiliary field t,,~rotates into the gravitino field equation! [169](A similar thing happens in the N = 8 model in 11dimensions [559].)Adding the Poincaré action, the states constitute one massless (2, ~,~,1) multiplet andthe massive (1, (~)4, 05) multiplet. Adding the invariant which starts with R~,., a massive(2, (~)4, (1)6, (~)4, 0~)multiplet is produced. A curious feature is that an entire (1, (~)4, (0)~)multipletdevelops, generated by the auxiliary fields. All N = 2 auxiliary fields occur in these multiplets (seesection 6).2.12. On-shell counterterms and non-linear invariancesOne approach to renormalizability has been to investigate which supersymmetric counterterms arenon-zero on-shell, based on the assumption that counterterms must be supersymmetric. This assumptionmay not be correct as the following counter examples show. In Yang—Mills theory, the Lorentz invariantW,. (W,. n FM,.) is not a Yang—Mills invariant, nor does it vanish on-shell, but its variation vanisheson-shell. If it would be produced in one-loop corrections as a counterterm, it would clearly be incorrectto first write down all possible off-shell invariants and then to study which of them vanish on-shell.As a second example [165], consider the globally supersymmetric Wess—Zumino model~ 2?~’= MA)2 )~2A/A+~F 2+~G2 (1)= —F(A2 + B2)+ 2GAB + ~(A + iy5B)A.Both ~ and ~2’g are separately invariant under t5(A + iB) = i(1 + y5)A, 8(F + iG) = iJ(1 + ‘y5)A, and= /(A — iy5B) + (F + iy5G)e. The one-loop divergences £2? are proportional to only ~2?~”; this is apeculiarity of this model which has been explained in several ways, but which fact we will take forgranted. It is now clear that if one eliminates F and G from £9~’~ + ,2?g, from £2? and from thetransformation rules by substituting the equations of motion obtained from .2?~”+ ,~2’g,then ~ + ,2’g isstill invariant but £2’ is no longer invariant. In fact one finds£2? = 2-(aMA) 2 ~(8MB)2— — ~IA + ~g2(A2+ B2)2 (2)6 £2’ = ig(A — iysB)21A with l.A = IA — 2g(A + iy5B)A. (3)

P. van Nieuwenhuizen, Supergravity 263One can also add masses and consider non-renormalizable models such as the Lang—Wess model. Inall these cases one finds the same phenomenon. Thus there are three possibilities for counterterms:(i) £2’ is off-shell invariant;(ii) £2? =0 on-shell but not off-shell invariant;(iii) 8(&2’) = 0 but &.2’~0 on-shell, nor off-shell invariant.It is easy to construct many examples of case (iii). Take any invariant action, thus 61 = ôçb’I,61,g + (&/i”,g)I,j 1 =0.Differentiating so that this identity with respect to some parameter g in the action, one finds81’.g vanishes on-shell but I~gitself is not invariant in general.Thus to prove the finiteness of the S-matrix it is incorrect to demonstrate that all invariant countertermsvanish on-shell. It is correct to show that on-shell no £2? exists which is on-shell invariant. In that case,one must use the on-shell transformation rules and, for example, to show that no £2’ exists withoutpurely gravitational sector, requires very detailed analysis. Details of such analysis have not yet beenpublished.From a more general point of view [165],the generating functional in the background field methodZ(~)=fdx exp i[I(cb +x) — I(cb) — I(~).~x] (4)explains these examples. If the original action is invariant under 4’ -+ g(4’) then in general thebackground functional is invariant under background symmetry transformations 4’-~g(çb) and x -~g(4’ + x) — g(çb), provided g(4’) is linear in 4’. (In that case I.~xis again an invariant.) In our examples,however, g(4) was after elimination of auxiliary fields no longer linear in fields.In order to connect these considerations to supergravity, one must find arguments which show thatcase (iii) counterterms do not occur there. In most of the analysis of on-shell invariants, auxiliary fieldswere not considered since theywere unknown at the time. Now, however, one can discuss the influenceof nonclosure of the algebra on invariance of counterterms. Since the commutator of two symmetrytransformations is again a symmetry of the original action, in supergravity with open algebra (withoutauxiliary fields) there are infinitely many field-dependent transformations under which any trueinvariant of the theory should be invariant. In most cases this leaves the original action as the only trueinvariant, and dangerous counterterms should then be of type (iii) with respect to these symmetries.It should be noted that in supergravity the transformation laws with auxiliary fields are still non-linear.However, the one- and two-loop finiteness remains true, since it was proved on-shell using helicityarguments [512,283] or Noether methods [123].2.13. Canonical decomposition of the actionThe Hamiltonian approach to supergravity, using path-integrals, first demonstrated four-ghostcouplings in supergravity (Fradkin and Vasiliev [226]).As an introduction we discuss the “p, q”decomposition of the action. This action analysis is the counterpart of the analysis of the field equations,and, of course, we will again find that there are two physical modes. However, unlike in the fieldequations, one does not need to choose a gauge for the actions.We consider the free linear spin 3/2 action first, and make a spacetime split [126]11= ~E”fIIiys( 7j494 — 749j)lIJk + E”~4y5%84rk. (1)

264 P. van Nieuwenhuizen. SupergravityThe Lagrange multiplier leads to the constrainthikyyTh/i = —2y4o~8j4,k= 0. (2)Thus only 4,,” appears in the action, since the longitudinal part~ô 4, vanishes in the first termin (1) after partial integration and use of (2). If we had worked with field equations, we would at thispoint have needed to introduce a gauge choice. Next we note that since we may replace 4’ by1/IT4’k -~ 8k L]= 0, ‘/‘T = (6/ — a,59’V~ 2)4, 1 (3)again due to the constraint in (2). Thus ‘y = 0 since I 8y’ is elliptic (Latin indices run alwaysfrom (1, 3)). Hence, only 4. appears in the action and is transverse both in ordinary space (a . i/i = 0) andin spinor space ~ 4. = 0). Thusa — (p3/2\a b = (~ a\TF 4‘PM — k’ MV) b’p,. — ~‘PM Iwhere F 312 is the spin 3/2 spin projection operator of subsection 1.13. The fields 41”’ and 41T are gaugeinvariant (i.e. invariant under 64,,,. = ~M~) so that the action has been decomposed into gauge invariantdynamical modes.The action becomes equal to7”’ (5)I = J d~x— ~“(y~84 + y’8~)4,if one uses the expansion of 7~~J7kand the double transversability of 4,,”~.Thus there are two physicalmodes, given already explicitly in subsection 1.13.Using electromagnetic notation, one can also rewrite the action as= —~./iy5(yA E + 70B) (6)whereE, = 4,1~rr, B’ = (7)so that the field equations read E + 75B =0.2.14. Dirac quantization of the free gravitinoAlthough a Majorana field is its own conjugate momentum, there is no problem in determininguniquely the canonical commutation relations, if one uses Dirac brackets { }D rather than Poissonbrackets in order to go from the classical to the6}Dquantum is nonzero theory. and Poisson in agreement brackets are with defined as usual,{4,M2, ~.,.b}D Inbut, second as quantized we will see, formthethere Dirac is no bracket problem {4’M’~’4’ at all, since one finds in ~ absorption operators a,, (k) andcreation operators a ~(k) (and not b~(k)since 4’Ma is a Majorana field) so that {4’M”, 4’V} evaluated at the

quantum level is nonzero. For example, in the gauge y• 4, =0 one hasP. van Nieuwenhuizen, Supergravity 265= ~ ~ [a,,(k) u~(k)eu1~x+ a ~(k) u~(k)*e~~] (1)where u~= ~u,,and A = + or —.Consider a theory with fermionic variables. We define canonical momenta as left derivatives of theLagrangianp = a~L/a4. (2)Then the Hamiltonian is independent of 4 ifone defines it asH~=4p—L, 5H~=4~p_5q(3eL/~q). (3)The equations of motion read (r = right)a’~H~/ap=4, a’H~/aq=—atLI3q=—ji (4)where in the second equation we have used the Euler—Lagrange equationsatL/aq = ~j (a~L/t94)= j5. (5)Afunctional F(q, p) evolves then in time asdF/dt = (3rHJ3p~~eF/~q) — (a’F/ap)(a’H~/aq)= {F, H~}. (6)This defines the Poisson bracket for two variables f and g if at least one of them is bosonic.In the general case, the Poisson bracket is defined as{f, g} = —(a’f/apXa~gIaq)+ (— )fa(arg/ap)(aef/aq) (7)where f = 0(1) if f is bosonic (fermionic). The extra sign (~)ft is needed in order that for two fermionic fand g one has {f, g} = ~g, f}. The overall sign is fixed by requiring that {f, gh} = {f, g}h + (— )~{g,fh}. Wenow consider the free spin ~field, following P. Senjanovië [608].Thus, for example, for the gravitino field {4’M”,4,b} = {*Ma lTb} =0 where we denote the conjugatemomentum by *~a rather than ITMa in order to obtain manifestly Lorentz covariant results. In particular3(x—y). (8){4,Ma(X, t), lTb(y, t)} = —6g6~oThe Hamiltonian is (as usual for fermionic systems) independent of time derivatives= Jd 3x iEM~rc~4’yya4, (r= 1,2,3). (9)

266 P. van Nieuwenhuizen, SupergravitvAll sixteen momentum definitions in (2) are primary constraints, and can be written as—! 0 ~O 0. (10)Thus, as stated above, the field variables are equal to their conjugate momenta.Adding the primary constraints to the canonical Hamiltonian leads to the total HamiltonianHT=Hc+~McM. (11)In terms of HT one has the usual Hamilton equations valid for unconstrained systems= —a’L/a4, = —~, a’HT/3ff = 4,. (12)The Lagrange multipliers CMare anticommuting functions. Consistency, i.e., 4’M =0 for all time, thus{4’M, HT}0, then leads to new, so-called secondary, constraints. From (d/dt)4” =0 one finds~(743r4,s— y,8,4i4) + j~krs7 = 0. (13)This equation fixes c,. but does not lead to secondary constraints. The solution is unique and can befound by multiplication by lcmhIymCk = “iêk4,4’ k~nn(~y~ôm4’n) _~j )l y4(a~4,k— 8k4,~). (14)From (d/dt)~° =0 one finds with {4’~,4’~}= 0 four secondary constraints,~a {~oa,Hc}elmhhiai(~mysyn)a 0. (15)These constraints hold whether or not one has substituted the solution for Ck. Since ,~.and H~do notdepend on momenta, consistency of ~ =0 leads to {~,4’k}ck 0. This seems to lead to yet newconstraints, but one finds{,~a4’kc,(} = _jE~dlmy4y5yk8lCm (16)which is just 8,. times (13). Hence the total set of constraints consists of the sixteen ~ in (10) and thefour ,~ in (15).One now divides these constraints into constraints which commute with all other constraints (firstclass constraints) and the rest (second class constraints). One finds eight first class constraints3k9!IIYSY (17)= IT,,, Xa + 8k4’,, = 3,.ir,, +~ Itmand twelve second class constraints (namely the çb” plus any linear combination of (17)). To check this

P. van Nieuwenhuizen, Supergravity 267result note that~ ~ = _~EkIm(Cy 5ym)a0. (18)First class constraints generate gauge transformations. For each first class constraint one chooses agauge condition. For our case we choose(4,°’~, ök4’I’~) = gauge conditions. (19)Defining A~= (wa, A~O~)as the first class constraints, gauge choices and second class constraints, thematrix C,,, = {A1, A} must be nonsingular. In that case one defines the Dirac bracket by{A, B}D = {A, B} — {A, A1}C~.0{A1,B}. (20)Clearly {A1, B}D = 0 for arbitrary B. Thus, asfar as the Dirac bracket is concerned, it does not matter thatfor Majorana fields the conjugate momenta are proportional to the fields (see (10)).Since the pair (4i°”, ff~) commutes with all other constraints, we may drop them. To evaluate theDirac bracket, one first evaluates it with respect to the 4”~,and then with respect to the other variablesA in A. Denoting the first bracket by { },, one has{A, B}D = {A, B}1 — {A, A,},({A, A},)”~’{A1,B},. (21)In this way one finds for the Dirac brackets{4,7(x, t), ‘ff~(y,t)}D = —~(F”’ 2)~~6~63(x —y) (22)where F312 is the spin 3/2 projection operator. The anticommutator for two 4, fields or momenta iTfollows from (21) and {A1, B}D = 0; in particular, the result for {4’, ‘fr}D is the same as when evaluatedusing (1). (In the gauge (19), the constraints imply y 4’ = 0 [107].For comparison one must use in (1) thepolarization tensorssatisfying y~u = lu,,, = 0.) Thus the total result is that replacingthe Poisson bracket bythe Dirac bracket replaces &b by ~(F”” 2)”b. (The factor ~is needed in order that 4,7 = {4,7, HT}D.)For path-integral quantization, one starts withZ =f d4i,, dñ~’d 3x (14M.frM4!i0 d~°ô(coa) ô(~a)ô(Oa) SdCt{~a,~b} (sdet{Oa, Ob}) exp if— ~‘T) (23)dif the gauge conditions are in convolution, {~,,,~}= 0. The product of both superdeterminants is just(sdet{A,, A1})”2 as one easily verifies, using that the Poisson bracket of p with 0 and itself vanishes.2.15. AnomaliesIn global supersymmetry, the axial and trace and supersymmetry anomalies (a . ~.4, T,,~and y Js)are part of a multiplet [299J.We discuss below the first two anomalies for a gravitino in an externalgravitational field.

268 P. Lan Nieuwenhui:en. SupergraLitvAxial anomaly — The action of a gravitino in a background gravitational field= ~ (1)is gauge invariant under 64,,, = D,,(w(e)) , if the gravitational field satisfies the Einstein equationsGMV(e) = 0. Thus one must fix the gauge. Two obvious choices are~2?’(fix) = ~ yjy• 4’, 2?”(fix) = ~ y,ø(w(e))y. 4,. (2)In the second choice, the quantum action is manifestly general coordinate invariant, but one must addNielsen—Kallosh ghosts (see subsection 2.4). In this gauge one can apply the well-known Adler—Rosenberg method for obtaining the axial anomaly. Denoting the matrix element for the axial currentgoing to two gravitons (a, /3, p’) and (y, 6, p2) by MM~~, 5(p, P2)’ general coordinate invariance= 0 relates divergent to convergent form factors, and the axial anomaly A =(Pl.M +p2,M)MM,,,,.,,.,,s = 0 can be expressed in terms of finite integrals. Clearly, the contributions of thecomplex and real spin 1/2 ghosts (both commuting) are known from the spin 1/2 axial anomaly, and oneonly has to apply the Adler—Rosenberg method to the gravitino loop. The answer is —21 times theanomaly for a real anticommuting spin 1/2 field, and was first obtained by Duff and Christensen by atopological method [172]. Here we will follow the Adler—Rosenberg method. For comparison betweenthe various existing methods, see ref. [530].We must begin by writing down the axial currents. The classical axial current is the Noether currentfor the global symmetry 64’, == —! ~M~P~74jy4j (3)and transforms under gauge transformations 64,,, = D,(w(e)) into= _ia,,E M~‘~(4,,.y,,e) — iiy 5R M (4)so that the classical axial charge is gauge invariant on-shell. At the quantum level the gravitino equationacquires an extra term from the gauge fixing term [344]— 3J~U/~q, = M~”y5y,.D~i/i,, — ~ yMØy. 4,. (5)At the quantum level we also use BRST transformations rather than gauge transformations, and with.SC = A (— ~4, y,~)one finds that the axial current due to the Faddeev—Popov ghost action (whichcorresponds to F,, = —y~4’) transforms under BRST transformations as followsj~= ieC’y My 5C, 6j~p= — ~ eçti y,Ø7M75CA (6)and the extra term in (5) is produced by (6) so that again j7~+j~gives on-shell a BRST-invariant

P. van Nieuwenhuizen. Supergravity 269charge. Although the chiral charge of the ghost C~is fixed by requiring this on-shell invariance, itagrees with the chiral charge which keeps the effective quantum action invariant (note that there areterms (CôMy~çfj)CM in the Faddeev—Popov ghost action).One is left with the contribution from the gauge fixing term and the new Nielsen—Kallosh ghost. Thefirst yieldsj~= (i/4)4r. ~~ M75~ 4,. At this point we use the Ward identity derived in subsection 6 whichstates that constructing two gravitinos with gamma matrices is equivalent to minus twice the sameprocess with a spin 1/2 field. (Insert in (8) of subsection 6 a factor 8MYMYSE_l and integrate over x = y.To check sign and factors consider free propagators.) Thus, at the one-loop level the chiral charge isBRST invariant if the chiral charge of the Nielsen—Kallosh ghost is opposite to that of the Faddeev-Fopovghosts. (Note that both closed ghost loops do not require a minus sign.)The result is now easy. For the gravitino loop one straightforwardly finds —20A, for the Faddeev—Popov ghost —2A and for the Nielsen—Kallosh ghost +1A, where A is the axial anomaly for a realelectron. It would be interesting to analyze what happens at the two-loop level, but this requires first tosolve the problem how to define BRST invariance with y” in ~F,,y”F~being dependent on dynamicalfields. It would also be interesting to redo the calculation for different gauge choices, to check that theaxial anomaly is gauge-invariant.Trace anomaly [172]—This anomaly can, in principle, be calculated by using again the Adler—Rosenberg method. It has also been calculated by using ‘t Hooft’s lemma for one-loop divergences,usingTi,, = _2g”2gMP 6W/ôgTM’ = £2’ (7)where W is the total one-loop effective action (W but not Ti,, contains nonlocal terms). The £2’result in (7) is valid for massiess theories. If the classical action is not invariant under local scaletransformations, as for example supergravity, which is only globally scale invariant as far as the termsquadratic in quantum fields are concerned, then the Ti,, in (7) includes the non-anomalous part as welland is infinite (in fact its infinity is proportional to 1L1R). Using dimensional regularization, oneobtains the anomaly as the difference between first letting n tend to 4 and then taking the trace, and thereverse order of these two operations.Here, however, we will present a method which gives the trace anomaly for all spins at the sametime. The method yields Ta,, off-shell, but only on-shell (with respect to the background field which mayinclude the extended supergravity fields) is Ti,, proportional to a total derivative (since £Sff in (7) isquadratic in field equations up to total derivatives). To obtain the anomaly due to a physical massless ormassive spin J particle, one considers fields 4’(A, B) which transform irreducibly under Lorentztransformations as the (A, B) representation and takes linear combinations such as to end up with thecorrect number of helicities. For example, for spin 3/2 one considers 4,(1, ~)and subtracts twice thecontribution from 4)(~,0). Working with 4’(1, ~)means choosing the unweighted gauge y~4, = 0, while24’(~,0) is the usual Faddeev—Popov ghost. One obtains the desired anomaly by computing thecontributions from the fields 4’(A, B) as obtained from the asymptotic expansion of the heat kernel; fordetails see ref. [172,174].The results for N massless spin J physical fields areI .~4— J.., [.....~L... I ,.~4 * D * DM”P”~ x g A — ~ L3~iT2~ ~ x g 1’MVP” ~x[—~ (N2 — N312 + N, — N,,2 + ~N0)+ (58N2 — 17N,,,2 — 2N1 — N112 + No)] (8)

270 P. van Nieuwenhuizen. Supergravitvf d~x\/~DMj~=~~ J d~xV~*R RMV~~~][21N 112 — N,,2]. (9)The result in (8) holds for RMV = 0 but (9) is valid even off-shell. Readers who are not sympathetic toheat kernels could have derived these results by direct though tedious computations. For supersymmetrictheories, the first term in (8) cancels, and one finds that 0(4) supermatter (N~= n, N,,2 = 4n,N,) = 6n) and 0(3) supergravityhave vanishing trace anomaly (note that although for theories which arenot locally scale invariant Ti,, (x) is infinite, its infinity, proportional to ER disappears upon integration).We remark that for self-dual spaces the left-hand side of (8) should be an integer, so that theEuler number x between square brackets in (8) is an integer times 24; also the numbers ~ are thenumber of zero spin 2 modes in a self-dual compact gravitational background. Vanishing of the axialanomaly is harder to achieve (although one might 4= define 0 evenappropriate in supersymmetric chiral weights) theories. and one notes inparticular that f Ti,, = 0 does not imply f DMJM~2.16. Chirality, self-duality and counter termsThe gauge action of N = 1 supergravity has a global chiral invariance under &/IM = i’Y54’M. This can beused to find restrictions on possible counter terms [175]. Since the curls DM4’,, —D4,,, = 4’M’ satisfyon-shell c~MV+ b’s’l’MV = 0, the combinations ~ = 4,,,,,. ± y54’,,,~transform as follows4’~,.—*(exp ~ iq)4i~,, if ~I’M—* exp(ifl’yS)4’~. (1)If the leading fermionic terms are only products of 4’MV (which is the case in all known models), then thischirality symmetry only allows as counter terms (4’±)k(4’_)k. In superspace, the possible counter termsare constructed from the Weyl superfield WABC (symmetric in the three dotted or undotted indices),and introducing again W~and W as above, one finds for the generic counterterms1~(DWr + w~w~[(Dw÷)~ +~. •]. (2)£2’= W+Wk+(DW+)This result is due to the fact that DW±is invariant under (1). This follows from the fact that W+ scalesby a factor, as in (1), while the derivative D scales in the opposite direction (see [498]).Since thesuperfields W±contain only self-dual Weyl tensors (and W_ only anti-self-dual Weyl tensors), it isclear that if W+ = 0 then the only possible counter terms are of the form (DW+)” with any number ofderivatives D. These, it is claimed, are total derivatives, and hence the only possible counter termshave as leading bosonic parts the following product of (anti) self-dual Weyl tensorsC~C~+C~C~+,k2 (3)possibly with arbitrarily many derivatives (parity requires the sums in (3)).If the theory would also have a chirality invariance of the Weyl tensorC±—*C+, C—*-C (4)then in (3) one only has k and n even. However, there is no reason to suppose that such an invariance ispresent, the only duality invariance being the one for 4,,,,,..

P. van Nieuwenhuizen, Supergravizy 271We now discuss the reason that in (3) one has k > 1. This is because in two-components notationthe Weyl tensor has four dotted or undotted indices and is completely symmetric, so that one can onlycontract self-dual and anti-self-dual Weyl tensor among themselves. Thus, since the Weyl tensor istraceless, one needs at least two of them to obtain a nonvanishing invariant.From these superspace arguments it follows that there are in N = 1 supergravity possible dangerouscounter terms at 3, 4, 5, etc. loops. All these counter terms vanish when the superfield Wabc is self-dualor anti-self-dual (except the one-loop topological counter term which starts with ~ — 4R~,,. + R 2which is nonvanishing in topologically nontrivial spaces). The result seems to be connected with thephenomenon of helicity conservation in supergravity [178].These dangerous counterterms alsofollow fromthe tensor calculus [189].2.17. Super index theoremsIndex theorems relate the number of zero eigenvalue modes of certain elliptic differential operators,such as the Euclidean Dirac operator and the Laplace operator to the topological invariantsx = ~—~Jd4x \/~*R*RMVPU~(1)where x is the Euler number and F the Pontrjagin number (while the Hirzebruch signature is equal to= ~P in four-dimensional compact space without boundary).Let A, be the nonzero eigenvalue of the operator ~, then i(s)= ~ A ~‘ is finite at s —* 0~and countsthe number m of nonzero eigenvalues. On the other handTr e~’= ~ e~” ~ BktU~_4~/2 (2)k =()wheredenotes an asymptotic expansion for small t. Thus, the t-independent terms yield B4 = n + mwhere n is the number of zero eigenvalues. The Bk are integral invariants formed from the Riemanntensor and its covariant derivatives, involving k derivatives. Thus2 + *RMVI,,, RM’~. (3)B4 = Jd4x ~g[a*R~,.,,,,.*RMVIJ~~] +$R~,,.+ yRIn order to obtain relations between the number of zero eigenvalues of two differential operatorswith the same nonzero eigenvalues, one subtracts the two corresponding B4 coefficients so thatB4—B4’=n —n’ [174].One considers fields 4’(A, B) which are irreducible (A, B) representations of the Lorentz group, andfields 4’[A, B] which have 2A dotted and 2B undotted indices but are reducible. Next one introduces aset ofM, operators ~ 0) = ~i(A,B) —DMDM + acting ~R while on 4’(A,B). ~(0,~)differs For example, because the ~i(0,0) connection is the Klein—Gordon DM is different. operator Furthermore,~ ~)= —D,,D~g,,,,.+ RM,.. It is more convenient to consider zl[A, B] acting on Ø[A, B],—DMDbecause one can show that the number of nonzero eigenvalues in [A,B] depends-~mlyon S = A + B but

272 P. van Nieuwenhuizen, Supergravitynot T = A — B. In this way one finds index theorems by considering for example B 4[A, B] —B4[A — ~, B + ~]= n[A, B] — n[A — ~, B + ~]since one can also compute the a, /3, y, e in (3) as functionsof A and B. This is the method for ordinary index theorems.For super index theorems, one can show that when the Riemann tensor is (anti) self-dual, thenm[A, B] is proportional to 4~.Linear combination of B4 functions for different spins (hence different S)can still give under these conditions index theorems, now relating zero bosonic and fermionic modes.For exampleB4[S, T] — 4s_s’B4[S~, T’] = n[S, T] — 4S_S’n[SF T’]. (4)To give a few examples of ordinary index theoremsandn[~, 0]— n[0, ~J= —~P (Atiyah—Singer)n[1, 01 — n[~,~]= ~ + ~P Hirzebruch signaturen[~,~]— n[0, 1] = ~ + ~P Gausz—Bonnet theoremspin 3/2 axial anomaly.A few super index theorems aren[~, 0]— 2n[0, 0] = —~(~‘+ ‘iP)n[~,J~]—2n[0,~1=—~(k’+~P)Christensen—Duff.Notice that on the left hand side one finds the same field content as in supersymmetric theories.3. Group theory3.1. The superalgebrasThe algebraic basis of supergravity is formed by the super algebras. The simplest kind (the only oneswe will discuss) contain even and odd generators, and one has always commutators in the algebra,except between two odd elements in which case one has an anticommutator. Since this is entirely as aphysicist would expect, we will forego an axiomatic definition. We will only consider algebras whosestructure constants are ordinary c-numbers. More general cases (anticommuting structure constants orcommuting structure constants containing nilpotent terms) will not be considered (see a forthcomingbook by B. DeWitt, P.C.West and the author.)The simple super algebras consist of two main families, the orthosymplectic OSp(n/m) and thesuper-unitary SU(n/m). In addition there are several exceptional (and hyper exceptional) algebrascorresponding to some (but not all!) of the exceptional Lie algebras G2, F4, E, E7, E8. In addition tosimple (super) algebras, there are semisimple super algebras and non-semisimple super algebras. Anexample of the latter we have already encountered, it is the super Poincaré algebra which has an ideal(an invariant subalgebra) spanned by the generators Fm. Simple (semisimple) super-algebras have no ideal(Abelian ideal).

areP. van Nieuwenhuizen, Supergravity 273In general we consider algebras with even (E) and odd (0) generators. The commutations relations[E 1.E1] = IkE,, [E,, 0,,] = g,,,”0,~, {0,, O~}= h,,~’E,. (1)In order that this is a consistent system, one must satisfy the Jacobi identities. Those for three evenelements are the usual Jacobi identities for ordinary Lie algebras. In addition there are these Jacobiidentities[E,,[E,,0,,]] — [E,,[E,,0,,]] = [E,, E,], 0,,] (2)[E,,{0,,,0~}]={[E,, 0,,]. 00}+{[E,, On], 0,,} (3)[0,,,{O~,O~}]= [{O,,,O~},O~]+ [{0,,,0~},0~]. (4)The first of these three identities says that the 0,, form a representation of the ordinary Lie algebraspanned by E,. (Consider the 0,, as vectors on which the E act.) The second is equivalent to the first ifthe Killing form is nonsingular in which case g,,,~= h,,~,.* The last identity restricts the possiblerepresentations 0,, of the ordinary Lie algebra. This relation is the reason that not every ordinary Liealgebra can be extended to a superalgebra.At this point we wish to make a distinction between graded algebras and superalgebras (the two areoften confused). A graded algebra is an algebra in which a grading exists, such as for example in theLorentz group, where the rotation generators, denoted by {L0}, and the boosts {L1} define a Z2 grading:[L,,L1] C L1+J,,,0d2.A superalgebra is an algebra with a Z2 grading (“even” and “odd” elements) such that (i) the bracket oftwo generators is always antisymmetric except for two odd elements where it is symmetric and (ii) theJacobi identities (2, 3, 4) are satisfied. The bracket relation between two generators need not always berepresented by a commutator or anticommutator (see (31) for an example), but one can always find arepresentation with this property (namely the regular (adjoint) representation).In order to familiarize the reader a little with super algebras, we first give some explicit representationsof the super Poincaré algebra. The defining representation is a 5 x 5 matrix representation-~yrn(1-y5) 0Pr,, = 0 , =000 00000 0 0000 0(Qa)nl = 0, (Q~)~= [—~(1+ ys)C_u]ma(Qa) = ~(1— y~,,,. (5)The matrix C is the charge conjugation matrix, satisfying Cy,,,C’ = —yT,,.The Killing form has only even—even and odd—odd parts. If it is nonsingular, the superalgebra is nonsingular (the converse is not true).

274 P. van Nieuwenhuizen, SupergravityOne easily verifies that the matrices in (5) satisfy the usual Poincaré algebra, plus [0”, Fm] =0, [0”, Mm,,] = (Omn)”~Q’~and {Q0 Qi9} = 4(ymC~l)asp Note that this matrix representation usesonly ordinary c-numbers, but anticommutators as well as commutators enter in the algebra. One candefine the algebra only in terms of commutators by introducing anticommuting elements in the matrix0, but this we shall not do.A representation in terms of differential operators is given by (see subsection 5.3)Fm = 8/3xm, 0° = 8/80,, —RIO)”Mmn = xm8,, — x,,öm + 0~mn8/t90. (6)The variables 0°are anticommuting variables, and derivatives are left derivatives. Thus= (C’)”~$~’ ~= ~ (7)The multiplets of the tensor calculus are simply irreducible nonlinear representations of the superalgebraconsidered. Acting with (6) on (xM, 0”) yields an 8-dimensional nonlinear representation.Casimir operators for the super Poincaré algebra can be defined: the supermass operator F20, and thesuperspin operator (KMF,. — K,.PM)2 whereKM = “~“°M,.~P,, + QyMy 5Q.By adding a chiral generator A_whose only nontrivial bracket is [00, A] = iytmy5OF,,, (F”~Fk)‘. These three Casimir operators 5”8Q” completely one can specify defineanother Casimir operator: A —any representation [442]of the iQy N = 1 super Poincaré algebra without central charges.The orthosymplectic super algebra OSp(N/M) contains as bosonic part the ordinary Lie algebrasSp(M) and S0(N). It can be defined as those linear transformations which leave invariant the bilinearreal formF—x’y’s50+0”~’9C,,,,, (i,j=1,N;a,f3=’l,M) (8)where 0 and x are anticommuting objects. (If one takes also 0 and x as real ordinary numbers, then xand 0 must transform differently from y and x.) The symbol C denotes an antisymmetric real metric. Thefirst term is of course invariant under SO(N). The second term is invariant under Sp(M). For M 4 (thecase of most interest), it is invariant under 0’ = (a’O,)O where 0~are the 16 Dirac matrices, if(a’O,)TC + C(a’O,) = 0. (9)Clearly 01 = {y0,, 0mn}. Since the form F is real, we need real 0~.In a Majorana representation we thusfind that Sp(4) is generated by(o~~i,icr~4,~ y,.,~y4), (k, I = 1,3). (10)This set of generators generates also the algebra 0(3, 2). Indeed, denoting the above generators by

P. van Nieuwenhuizen, Supergravity 275MAB(A

276 P. van Nieuwenhuizen, Supergravity[M,M]~M, [M,11]~H, [H,TI]-~M (IS)[M,S1~S, [H,S].--S, {S,S}-~M+H(with H ~—MMS and M ~—MM,.) and scales aH P and f3S Q. (One does not scale M since one wantsto keep the M, M — M Lorentz algebra.) Two possibilities arise:(i) a = /32 This leads to {Q, Q} — P and [P,P] = [P, Q] = 0 for a —‘O. Thus one recovers the superPoincaré algebra.(ii) a = /3. Now (0, Q} = [0, F] = [P, P] = 0. In this case the 0 are “outside charges” which rotateas spinors under M, but for the rest have nothing to do with spacetime groups.Above we have discussed some representations of OSp(1/4). The defining representation was 5 X 5dimensional. The adjoint representation (the one in terms of structure constants) is of course 14 x 14dimensional since there are 14 generators. However, not every representation of SO(N) x Sp(M) can beextended to a representation of OSp(N/M). For example, the set of matrices (y,,,, ow,,) and the set(y 0,y5, y3) both are a representation of Sp(4). Probably the second set cannot be extended to arepresentation of ,OSp(l/4).Supercanonical transformations.t Canonical transformations between creation and absorption operatorsfor bosons define the group Sp(M). (We recall that M is always even for Sp(M).) For fermions onefinds the group SO(N, N). As one might expect, the most general canonical transformations mixingbosons and fermions lead to OSp(N, NIM). This we now prove.Consider the infinitesimal transformationQB.I = aBI + (p~kaBk+ ~ + ~kaFk+ ThkaF.k0F.k ~kaF.k ~kaB.k EkaB.,,aF~— aF,, ikwhere , i~, E, H are Grassmann variables. The bosonic sector is symplectic (4’ symmetric, ç~antiHermitean)(16)~ ‘~‘~“), AJ+JAT=O, j= ( 0 I’~~ (17)i/i (p/S —IOi(A star denotes complex conjugation and T denotes transposition.) Also well-known is that the purelyfermionic sector is orthogonal (P antisymmetric, F antiHermitean):D~(~ *)~ DS+SDT=O, s=(~~). (18)S can be diagonalized by the matrix IØ(o-~+o-~)and has N eigenvalues +1 and N eigenvalues —1.Thus one finds the group SO(N, N) for the canonical transformations between fermionic variables.The transformations which mix bosonic and fermionic operators are canonical if H = _1 =E = — ~.Introducing matrices as before, one must not forget that in á~.1one finds ( IJaFJ)71TandI thank Professor Berezin for showing me these results and B. Voronov for discussions.

P. van Nieuwenhuizen, Supergravity 277One finds77 \ ç,(E H\ nc~rrT_n ~19— \, * — *)‘ ‘~ — k,H” E*)~ — .Thus the canonical transformations are generated by OSp(N, N/M) and the generators can berepresented by (M + 2N) x (M + 2N) matrices satisfying(A B\(J 0\~(J 0\(A B’~ T 0 (20)\.C DI \0 SI \0 SI \C DI1, _Bt, D’ (see appendix).where The the superunitarity supertransposition algebras, replaces to whichA, weB, now C, D, turn, by are AT, the C other main family of super algebras. Theyare denoted by SU(N/M) and contain as bosonic part the algebras SU(N) x SU(M) x U(1). In an(N + M) x (N + M) representation, the SU(N) and SU(M) algebras lie in the first N x N and lastM x M submatrices, while the U(1) lies along the (N + M) x (N + M) diagonal. As for the orthosymplecticcase, one can define the superunitary algebras as leaving invariant the formF (x1)*x~6,j+(ga)* 0”g,,,,,, (g,,~= ±o,,v) (21)where i = 1, N and a = 1, M Since the 0’~are complex and F is real, one can always diagonalize the0-metric and thus M and N can be even or odd. Hence, for SU(N/M) we do not need an antisymmetricmetric C,,~but we can use two diagonal metrics 6~,and g,,~.(If one takes 0 to be commuting, one simplyfinds the algebra SU(N + M); thus one must take 0 as anticommuting.) It is clear that again thesetransformations form a closed algebra under ordinary commutation relations provided the x —*0transformations contain anticommuting entries. Again we go over to representations with ordinarynumbers only by extracting the anticommuting parameters.The most interesting case is the superconformal algebra SU(2, 2/N). A representation of SU(2, 2) interms of 4 X 4 matrices is well-known. It is the set of matrices which leaves 00 =o~0l+ o~o~— 0~o~—O~O~ invariant. (For this representation of y4 see appendix A.) This is the set of 15Dirac matrices satisfying 0~y4+ y4O1 = 0. Thus the generators of SU(2,2) arePm = —~ym(1— y~). K,,, = ~ym(1+ y5), D = —~ys, M0,,, = Omn. (22)A representation of the generators G’ of the other unitary group and of the U(1) is given by theantiHermitean generators r’ for SU(N) and all matrices have zero supertrace. FurtherA=—~diag(1,1,1,1,~,~ (23)for U(1).* Finally there are 2 N X 4 generators 0” and S”’ which are the usual square roots of Pm andK,,,. They have entries only in the (4+ i)th columns and rows, and are given by(floi\k — r iii \~-‘--I1ko ~~‘ai\k — F!a’l — \r~—11ka~ ) 4+j — L2~’ + )‘5)U J , t,’) ) 4+i — I.2~’ )‘5)’..~ J4~k= ~(l— y~)”~, (S”’)4~’,,= —~(1+ ys)°,,. (24)(Q”’)For N = 4A is a central charge which may be omitted (and is absent in the S-matrix, see subsection 3.3) without violating the Jacobi identities.

278 P. t’an Nieuwenhuizen, SupergravityThe superconformal algebra SU(2, 2/1) will be discussed in subsection 4.1. In order to obtain theextension to SU(2, 2/N), one endows ~a and 5°with an extra index i, as in (24). If one evaluates thecommutators of 0”’ (and S°’)with G’, one finds structure constants which are a product of y~times thesymmetric matrices r’ (the antisymmetric matrices are multiplied by the unit matrix). Hence it isadvantageous to introduce chiral charges. One finds when with Q~’= ~(1+ y 5)°,3Q13, etc.[Q’L G”— ~ 3R/ [QRI LJLI G’—— J jkfQRkIc”’ ~—( ~ ~‘k J’~ i— (r) ,,, . (L~R J \Thus, QL and SR transform in the vector representation of SU(N), while OR and S1. transform in thecomplex conjugate of this representation. As explained in appendix E, we therefore have written theindices i of the (N) representation upstairs but those of the N* representation as subscripts. Moreover,0 ~ anticommutes with S ~ (and QR,i with SL.I). Since there are no generators which transform astensors under G’, this is to be expected. However, the product of (N) and (N*) contains scalars andvectors, as shown by{QL, SLJ} = ~(1+ y6~C~D— umnC~M,,,,,— iC’A) + 2(rkYJC’G~]{QRI, Sk} = ~(1—y5)[8~(~C~’D _O.mnC_ItvI,,,,, + iC’A) — 2(Tk)’JC’G~].(26)Also this is clear, since the invariant tensors 6~and (Tk)’j are the Clebsch—Gordon coefficients for thetwo irreducible representations contained in (N) x (N*). The anticommutator with right-handed charges1’ are antiHermitean.)followsOf course from complex the superconjugation. Poincaré algebra (Usingis Q* contained = —S, y~= in the—y~and superconformal that the Talgebra; therefore we havechosen the representations for Pm, Mmn and 0”’ the same. However, also the de-Sitter algebra iscontained in SU(2, 2/N) [197].To see this, take a general linear combination of all charges0’ = aQ~+/3QRI + YSLI + SSk (27)and require (0’, ‘} = ~(ymC_l)Pm6i1 + more. One finds Pm = af3Fm — ySKm. Next require that [P, 0] isagain proportional to 0. One finds only one condition, namely ay = /38. However, if one now evaluatescompletely {Q, Q} one finds that only the antisymmetric generators T~ appear on the right-hand side{Oai, 0~’}= {~ymC’’P6” — (ay + /36)O.m0C_lMmnö~i+ 4ay(rk,A)”C~’G~J°~. (28)This suggests that the maximal internal symmetry group of the super de-Sitter algebra is an SO(N)and not an SU(N) group. It also explains why the de-Sitter models are of the 0(3, 2) type and not0(4, 1): because the commutator [Pm,F,,] = kMmn has k = 4af376 > 0. The spin 1 fields of N-extendedde-Sitter supergravities gauge these 0(N) subgroups, and one has in these models always a cosmologicalconstant and a masslike term for the gravitinos. - -If one now goes from de-Sitter space to Minkovski space (i.e., [Q”,Pm] = 0) by means of a groupcontraction, the 0(N) charges can become central charges which commutewith all other generators. Thus,asexplained in the sectionon how to gauge algebras, the gauge coupling ofthe vector fieldsdisappears andone finds ~N(N — 1) Abelian vector fields. The process of contraction leads one to the super Poincaréalgebra. To this purposeone multiplies the {0, 0} relation by ~2, defines new charges0 = eO andhas then

P. van Nieuwenhuizen, Supergravity 279two possibilities for e -+0 and J 3 =(( 0), (e0)} ‘—‘ (~2~)+ e2(M) + 2(G), [( 0), G”] ... (EQ)(( 0),(~O)} (~2J3)+ 2(M)+ (~2Q) [( 0),( 2G”)] C2( O)(29)In the first case the internal charges G become outside charges: they are not reproduced on theright-hand sides, but they rotate the 0 under internal symmetry transformations. In the second case onehas central charges: now the G commute with all other generators, but they are produced on theright-hand sides. For example, OSp(2/4) has an 0(2) charge which rotates the two 0”’ into each other.Contracting to the N = 2 super Poincaré algebra with central charge, this central charge still produces the-~i’4,~, ,jtransformations of the photon under supersymmetry, but all physical states become inertunder the central charge (only ÔA,. = 8MA remains after contraction).Other super algebras. There exist other super algebras. Since they are little used in supergravity wewill be brief. The matrices with vanishing supertrace form of course a closed system since strM 1M2 = str M2M, (considering entries with anticommuting variables one has only commutators in thealgebra). This is the super algebra SL(M/N) which can be viewed as acting on a carrier space with Mfermionic and N bosonic coordinates. If M = N, the unit element has zero super trace, and one canomit it to obtain SSL(N/N) = SL(N/N)/AI. These algebras are simple.An example of a non semisimple algebra is for example given by the fermion operators E, a, at, ata.This is GL(1/1) and the ta, unit andelement SSL(1/1) Ebyisomitting the ideal the (anunit invariant E. subalgebra). Again one can constructSL(1/1) Somebyexceptional omitting asimple super algebras are F4 with 3 + 21 + 2 x 8 generators (with bosonic partSU2 x spin (7) where spin (7) are the twentyone 8 X 8 matrices a~ constructed from the gammamatrices in 7 dimensions), G3 with 3+ 14+2 x 8 generators (its bosonic part is SU2 X G2 the ordinaryexceptionalLie algebra G2 is obtainedby fixing one componentof spin (7) and hasthus 14 generators). Alsothere is an algebra D(2/1; a) (=OSp(2/4) if a = —1), as well as W, 5, S, H.In addition there are hyperexceptional algebras P(n), Q(n). For example 0(3) contains as evengenerators the SU3 generators along the diagonal, and as odd generators again the SU3 matricesEven = (‘~ ,~,), Odd = (,~‘b). (30)The bracket relations are usual commutators, except that the bracket for two odd generators is given by2}—~tr(O’O2) (31)(Q1 02)= {0’, O(which yields the d1,. coefficients of SU3). We have given this example to show that the bracket relationsneed not be only commutators or anticommutators, when one has a representation in terms of matrices.For further literature, seeL. Corwin, Y. Ne’eman and S. Sternberg, Rev. Mod. Phys. 47 (1975) 573.P. Freund and I. Kaplansky, J. Math. Phys. 17 (1976) 228.V.G. Kac, Funkc. analiz 9 (1975) 91 and Comm. math. Phys. 53 (1977) 31.Lecture notes in Math. 676, Advances in Math. 26 (1977) 8.

280 P. van Nieuwenhuizen, SupergraviyW. Nahm, V. Rittenberg and M. Scheunert, Phys. Lett. 61 B (1976) 383 and papers in J. Math. Phys.A. Pais and V. Rittenberg, J. Math. Phys. 16 (1975) 2062.3.2. The gauging of (super)algebras(Conformal) supergravity is the gauge theory of the super(conformal) Poincaré algebra. In order tobe complete we will therefore first discuss how one gauges in general a superalgebra.Let the algebra be given by generators XA which satisfy the following commutators or anticommutators[XA, XB] =!ABCXC. (1)The structure constantsf,~ are ordinary numbers.Corresponding to any generator, one introduces a gauge field, as shown by the following superalgebra-valuedvector field for the super Poincaré algebra= hMAXA = eMF 0, + ~cL Mmn + ~I’M0Q° (m

P. van Nieuwenhuizen. Supergravily 281Under arbitrary variations, curvatures transform as8RM,.A = D,.8hMA — DM6h,.A. (8)Consider now a Yang—Mills type of action,1— IA4 0 A— ~ x “~L~’0 Rf~M~po- 91~’po~s..IABwhere 0 may depend on fields. Under local gauge transformations81 fd~x{RM,.’~?R (—fCB”Q~~”+ fBA”Q~’)+ RM,.AR,,,~8gaugeQ~~}. (10)Under arbitrary variations81 = Jd4x [4(D,.6hM~R~Q~T + RM,,ARP~B6QMAT]. (11)Partially integrating D,,6hMA and using that always fCD~~~h,.l)8hMC’ = _6h,,Ch,.DfDj~~one finds [308]forarbitrary variations= +R:Ro;A PO~T~48hMARPU8 3~Q~T]. (12)In the frequent case that Q~A~B~” is a constant times ~ the last three terms vanish, due to the Bianchiidentity D,.R,,~ ~°°’ = [D,.,[D0,D~]] ~°~ = 0.Finally we derive a theorem, which is useful when constraints on curvatures are not gauge invariantunder all local symmetries [523].Suppose one can solve a constraint RMA,.I~MA~= 0 for a given field hM 4°.Then hMA0 is a functional of other fields, and transforms no longer as (DM )A0, but according to the chainrule. However, after solving the constraint, its variation vanishes identically. Denoting by ô’ thedifference between the actual transformation law (using the chain rule) of hMA0 and the group lawo(gauge), one has the followingTheorem:(8’h,/~°) 8h8A, (RM,.”F~~) = [_.f~AR,,,B~C— R,,,,.” 8(gauge)]F~,”. (13)In particular, if the constraint is invariant under 8(,gauge), then hMA0, although no longer an independentgauge field, still transforms according to o(gauge) just as when it was an independent field.Using the formalism of this subsection, McDowell and Mansouri [308] showed that N = 1 supergravitywith cosmological constant follows from gauging OSp(1/4) and Townsend and the author [517]showed the same for N = 2 and OSp(2/4). In the latter case, however, a term 84’~— FM,. ’ did not follow

282 P. van Nieuwenhuizen. Supergravilyfrom group theory alone. It might be that elimination of auxiliary fields would lead to such a term; thisis at present an open question.An interesting aspect of the N = 2 model is that it contains a central charge in the limit of vanishingcosmological constant (see section 6). This is due to the generator T in{Q~,Qfl = ~6, 1(ymC~)0$P,,, + ,1(C’)”~T. (14)Gauging of this OSp(2/4) superalgebra leads to a theory where the photon gauges T, andwhere physicalstates (namely the gravitinos) rotate under T. Making a group contraction, T no longer acts on physicalstates (it only acts on the photon, and only as a Maxwell gauge transformation) and T becomes in theglobal algebra a central charge. In the gauge algebra the commutator of two local supersymmetrytransformations now also contains a Maxwell transformation with parameter (E~ ~e,1).The actions for N = 1 supergravity takes in this approach the form3(ysC),,~] (15)I = J d~x[RM,.(M)mnR,(Mr M~~~ e~ + RM,.(Q)°R~(Q)’and invariance is obtained if one imposes the constraint RM,.(P)~= 0, which is solved by w = &(e, 4’).Hence, here we encounter the same result as obtained from 1.5 order formalism, but now by a grouptheoretical approach. The action for N = 2 is obtained by summing the 0-terms over i = 1, 2 andadding a Maxwell action.It is interesting to note that the Yang—Mills type of action in (14) actually only contains terms linearin the curvatures. This comes about becauseRMV(M)m” = Riemann tensor + (e ~e~— e ‘e~,)+ 4,Mum”4’,.. (16)The term quadratic in the Riemann tensor yields in (15) a total derivative (the Gausz—Bonnet invariant),while the cross terms yield the Einstein action. The remaining terms contain the cosmological constant.Similarly, after partial integration, one finds that the higher derivative terms in the gravitino actioncancel, and one finds the mass-like terms. (Without gravitinos, one thus also can write ordinary generalrelativity in Yang—Mills form.) To go to theories without cosmological terms, one performs a Wigner—InOnü group contraction.Finally we note that also the action and transformation rules for N = 1 conformal supergravity can beobtained [523]by gauging SU(2, 2/1), see subsection 4.2.A slightly different version is due to Regge and Ne’eman. They need a symmetric decomposition ofthe de-Sitter (not the Poincaré) superalgebra: (OL + M)+ (OR + F) and start out with a group manifold(see next subsection). They contract the horizontal part (OR + F) and find thus as field equationsR(P) = RR(Q) = 0 (omitting indices). Subsequently they add a new set of generators and fields(Q~+ M’) + (QL + F’) and do the same. At this point all torsions are zero, while only curvatures withtwo horizontal legs (see next subsection) are nonzero. The constraints R(Q) eliminate two of the fourgravitino fields 4,L, 4’R, 4’L, ç14 and one finally ends up with the same action as MacDowell andMansouri. Thus this approach does their construction twice: once in a left and once in a right basis. Thedoubling of generators is very reminiscent of the chiral superspace approach and we refer to subsection5.6.

3.3. The Haag—Lopuszanski—Sohnius theoremP. van Nieuwenhuizen, Supergravity 283One can, to a large extent, determine the possible extra symmetries in supersymmetric theories byanalyzing the commutator algebra. We stress that in what follows we do not analyze the gauge algebra,but only field-independent super algebras. Thus, for example, central charges which vanish on shell arenot discovered in this way. Nevertheless, it is amazing how much can be deduced from (seemingly)rather weak assumptions.Consider a super algebra which is an extension of the super Poincaré algebra. It contains thePoincaré algebra itself, and N conserved spin ~charges (i = 1, N)[Q”j, Fm] = 0, [O”i, Mmn] = ~ (1)In addition, the following (anti) commutators are present, using two-component notation (see appendix)(QA. QB} = ~I1~Z, (2){QA. ~l} = o~ Om)F ~ (QAEAB)* (3)[QA. B,] — (b,)/0A 1. (4)The Z, are a subset of the internal charges B, and, according to (2) these Z, commute with P0,. Wechoose these Z, to be a basis for the Z,1 = [1,’Z,,hence there are in general fewer Z, than Z,1. In fact, itfollows quite generally from the Coleman—Mandula theorem that all B, are Lorentz scalars whichcommute with F,,,. This theorem states that the most general bosonic extension of the Poincaré algebrais a direct sum of the Poincaré algebra, a semisimple Lie algebra and an Abelian algebra. Since theColeman—Mandula theorem holds only for symmetries of the S-matrix, the only symmetries discussed inthis subsection are those of the S-matrix. Since the Z, carry no Lorentz indices, the invariant tensor ~ARmust appear in (2) (in four-component notation this means that central charges appear as U, + iy5V,)and it follows that Ii’,, is antisymmetric in (ii). We choose all B, anti-Hermitean in order to preserveunitarity.From now on we will use the Jacobi identities to analyze the algebra further. We will denote theseidentities for three generators A, B and C by (ABC). Let us consider first (0, 0, F) and note that in (3)one cannot add a term with Mmn on the right-hand side. (In a de-Sitter space, [0”, P] is non-vanishingand one does indeed find terms with Mmn. Here we consider non-Sitter spaces where [Fm, F,,] = 0 andhence [QaP}0) -From (OBB) it follows that the b, are a representation of B,. From (BOO) it follows that(b,),” = _((b,)ki)* ~ b, anti-Hermitean (5)assuming that F,,, and B, (and hence Z,) are anti-Hermitean.The notation (Q~~,)* = ~ will now be justified. If a tensor T, transforms under a given group asMITJ let us define a tensor T’ to transform as (M.~.T)i~T1.(Note that M_l.T is again a representation ofthe same algebra of which M is a representation.) One could define two more kinds of tensors,transforming under M* and under Mlt. (These four sets of tensors correspond in two-componentformalism to XA,f, XA and x’~.)Since according to (5) the group elements generated by b, are unitary,

284 P. van Nieuwenhuizen. SupergravitvM~~T= M* so that (O~)*transforms as a tensor with index i up. This is the reason one writes for(07)* the symbol Q~,.For the spinor index B, see appendix E.The Kro’necker delta function in (3) is already the most general possibility. If one would have writtena matrix N~instead, complex conjugation of (3) would imply that N is Hermitean. Diagonalizing N bymeans of a unitarity transformation of the 0~,and noting that on states {Q~,(0~)~} is positive definite,one can rescale 0 such that N~is replaced by 8~.The (000) Jacobi identity tells us that f1’b, vanishes[1’, 1Z,,Q~]= (1’,J(b,)k”O~’= 0 (6)for any values of i, j, k, k’. Thus Z,1 Q’qZ~commutes with O~.It also commutes with Q~as followsfrom (5). From (OOB) we find the interesting matrix equationb,fl + QmbT = Ck,mf1l~ (7)where the structure constants c of the Lie algebra generated by B, are defined by [Bk,B,] = ck,B.We now turn to the commutator[2”,1Z~,B,] = flki.ck,mB (8)Using (7), the right-hand side is rewritten as[Z1~B,] = (b,11 + fl tmbT), 1Z,,, (9)since Q~vanishes for B,,, not equal to any of the Z,,, according to (2). This proves that the Z,~,form aninvariant subalgebra of the internal symmetry algebra generated by the B.Consider now the (0, 0, Z,1) Jacobi identity where we recall the definition Z~j= [2~Z.,. It leads to[Z,1,.4,] = 0 due to (6). Clearly, the Z~,form an Abelian subalgebra of B,. Going back to theColeman—Mandula theorem, it follows that the Z,1 cannot be part of the semisimple Lie algebra so thatthey must be part of the Abelian Lie algebra in this theorem. Consequently, the 4 commute with allB,. Also Z,1 commute with O~’and O~according to (6), and with B, as shown above, and with thePoincaré algebra according to the Coleman—Mandula theorem. Consequently, Z,1, and hence Z, arecentral charges.Let us now return to (8). Since the index kmrefers in (8) to must Tk (since vanish. (2” Thus vanishes one obtains for Bk the notimportant equal to Zk, result see(2)), it follows that the structure constants Ck,b,f2 + flb’T = 0. (10)In other words, (2’°~, is an invariant tensor under the internal symmetry group.For the particular (but frequently arising) case that 12’~is real we make an orthogonal transformationsuch that= Qk~J.!Q~,m (11)is of the form of a degenerate symplectic metric

P. van Nieuwenhuizen, Supergravity 285n=(~ ~). (12)It now follows from (10) and (5) that the internal symmetry representation b isUSp(2n, 2n)Ø U(m). (13)Thus, for example, the N = 2 supergravity with U(2) invariance cannot (and indeed does not) have acentral charge on-shell.For super de-Sitter algebras we are not aware of similar complete results.As far as the super conformal algebras are concerned, if one considers only internal charges whichare Poincaré scalars again, then in the {Q, Q} and {S, S} relations one finds no B, for dimensionalreasons alone, but in the {Q, S} anticommutators a group U(N) (but SU(4) for N = 4) must be producedand central charges are absent. For N = 4, an outside charge U(1) (i.e., a charge which appears on theleft-hand side but not on the right-hand side) can be present, so that Q and S still rotate under a fullU(4). This special role of the N = 4 case may explain certain cancellations of infinities in the N = 4globally supersymmetric Yang—Mills theory.For literature, see:S. Coleman and J. Mandula, Phys. Rev. 159 (1967) 1251.R. Haag, J. Lopuszanski and M. Sohnius, Nucl. Phys. B 88 (1975) 257.3.4. Supergravity as a gauge theory on the group manifoldAn approach to the gauging of superalgebras which differs from the approach in subsection 2 is basedon the notion of the group manifold [334,1]. In this approach one considers as “base manifold” a spacewith as many (bosonic and fermionic) coordinates as there are generators in the full superalgebra G.The superalgebra G is arbitrary, and can be non-semi-simple. Fields depend initially on all coordinates,but as a consequence of the field equations they actually turn out to depend on fewer coordinates thanare present in the group manifold. One chooses a particular sub(super) algebra H of the full algebra Gand requires that the action be invariant under general coordinate transformations in the full basemanifold and under local gauge transformations generated by H. It then turns out that these symmetriesbecome symmetries of the final theory in a smaller base manifold [608].It is at present not known when the program works and when not. Also, one knows at present onlyhow to obtain the final theory on-shell, but no guiding principle is known how to obtain the auxiliaryfields. Nevertheless, the approach seems promising.General coordinate transformations can be written as gauge transformations plus curvature~terms*,, jp’,, A....j-l1,, ~ A -, p11, AOgcI~c)flit — ç unnn ~ c’nc flu= Dn(~Hh 1j4)+~~~RrjnA (1)(D11 )”= ~3~A + hnB ~fCB~~.‘This result seems due to F.W. Hehl. P. von der Heyde and GD. Kerlick, Rev. Mod. Phys. 48 (1976) 393.

286 P. van Nieuwenhuizen. SupergravilyIntroducing one-forms hA = dy~~hAAand flat superparameters ~ = ~one has8gc(~)hA= (D~~+ h~~BR 8c~~. (2)These transformations will become in the end the local supersymmetry and general coordinatetransformations in ordinary four-dimensional spacetime.As we shall see, it sometimes happens as a result of the field equations that curvatures with one orboth lower indices in the subalgebra H vanish (RH = 0, horizontality). Moreover, sometimes R0 can beexpressed in terms of R11~(rheonomic symmetry) where I are the generators in the final base manifoldand 0 the generators which are neither in H nor in I. In such cases the transformations o~acting onh1” become transformations which involve only these fields themselves and their I-derivatives and arethus ordinary symmetries defined in the final base manifold I. For example, in N = 1 supergravity, G isthe super Poincaré algebra and the coordinates of the group manifold are y’ 4 = (XM, 0”, 77ab) Thesubgroup H is the Lorentz group and I = P0,, 0 = 0”.6gc becomes indeed the sum of general In this coordinate case oneand has horizontality local supersymmetry and rheonomic transformations.andsymmetryThose final symmetries can be systematically traced, once G, H and the action are given. Onechooses as action an expression linear in curvatures (this is thus an essential difference with the methodof subsection 2),I = J RA A h” A hK~CA.K, K, (3)where A ranges over the whole (super) algebra G and K, lie in G/H. The C are field-independentconstants such as ~ijk~and YSYm. The fields hA’4 are, for example, equal to emM(x, 0, re). In order that theaction be H-invariant, the C must be invariant tensors under H.Varying hKi the h Ki A CA... ~, do not transform in the coadjoint representation of the full group G.(Thus, fora tensor TA one has ÔTA = I~CT~Bsothat TATA is invariant.) Note that in general D,ICA K,,is not zero since the C’s are not invariant tensors under G.The manifold M over which one integrates is (n + 2)-dimensional since the integrand is clearly an(n +2) form. One has n +2< dimK = dim G/H. One uses a generalized variationalprinciple to obtainthe field equations: one varies both fields and surface. (Thus one considers all possible surfaces M in thegroup manifold, and not one particular surface.)The heart of the matter are the constants C. These are determined by requiring that the vacuum be asolution of all field equations, since this fixes the C to be the coefficients of a cohomology class of theLie algebra G. The equation for C to be solved is thusD[(g~dg)” A (g’ dgf”CA,K, K,~]= 0. (4a)Indeed, for the vacuum h = g’ dg and RA vanishes, so that one need only to vary RA yieldingÔRA = D5hA, and if (4a) holds, then 81 = 0. (One may partially integrate to obtain (4a) if the object in(4a) is in the coadjoint representation.) The solution of (4a) is obtained if the C in the integrand in (3)form a cohomology class of the G superalgebra. In fact, cohomology class means thatD(h” A A h”CAK K,,)”O (4b)

P. van Nieuwenhuizen, Supergravily 287is proportional to curvatures and is not itself the exterior covariant derivative of some other forms. Thefirst condition guarantees that while varying the action, one finds field equations linear and homogeneousin curvatures, while the second condition guarantees that the action does not vanish as aconsequence of the Bianchi identities. The useful mathematical result is that cohomology theory givesall solutions to (4b) and that there are only few such solutions.The h = hAXA are super Lie algebra valued one-forms, but h are not yet the usual connections(“principal connections on a fiber bundle”) because in that case the H-part of h should have the formh’~’= (g~ dg)’~’+ dyKhK~~ (5)with the curved K in G/H, meaning simply that the H-part of h’~is a gauge transformation under Hitself (gauge transformations with flat parameters ‘~ are defined as usual, 8h”~= (D y~).One calls theh” therefore pseudoconnections. If (5) holds, h” defines a principal connection on the fiber bundleG(G/H, H) and the rest of h, namely ~h”(with flat K) is what is called the soldering form (“thesuper-vielbein”).One now proceeds as follows. Varying the action in (3) with respect to all h 4’4, one finds the completeset of field equations. From here on we discuss the particular case of supergravity; this should enable thereader to deduce also the general case. Assume that one has found the coefficients C such that theaction readsI = J (R” A E” A E’ qk!+ 4’ A ys7,~4’A E,). (6)The R’ are the Lorentz curvatures dw” + ~w”A wj~’,E” is the vielbein one-form, 4”’. the gravitinoone-form and ~ is covariant with respect to H, i.e., it contains only the Lorentz connection. Theintegration is over a four-dimensional manifold since for the super Poincaré algebra there are only threecohomology classes of the algebra (not to be confused with the cohomology classes of manifolds), one ofwhich does not contain gravity, a second gives a theory with only the vacuum solution while the third isthe one used in (6). This proves the uniqueness of supergravity from the point of view of differentialgeometry.The field equations become8118w” = CilkiR n E’ = 0 (7)8118E’ = ~k,R” A E’ + ~çfrA y5y,R = 0 (8)6II8~fIa= (y5y,R)” A E — 1(yy4’)a A R’ = 0. (9)The symbols R” and R” denote the curvaturesof ifr” and E’~(thelattercontains a ihifr term). In deriving (9)we used the identity which we also used in the proof of the gauge invariance of the action insection 17~4’A 4’y’4’ = 0. (10)t. Aftersome Thelengthy interesting algebra point onenow finds is that the following one can project results: (7,8,9) onto the various components of dy’

288 P. van Nieuwenhuizen, Supergravity(i) Horizontality: all curvatures RABC with at least one vertical leg vanish. In other words, if Aand/or B equals H, RAB” = 0. The indices A and B are flat, due to contracting Rt. From this result one derives factorization.11.,’’ with inverse groupvielbeins (ii) Factorization: hA’ the fields hA with A in G/H have no components along d 1’ while for A = H thecomponent of hA along d 7777” is a gauge transformation= (g’ dg)” + do” w,,” + dx’s WM” (11)1 + dx” E~ (12)E = 0 + do” E,,4’bo +dO”4’,,”+dx”4’,,”. (13)This result states that after one has removed any dependence on ~“ (or its integral g”) of allcomponents w,,’1... 4’a” by a suitable local Lorentz rotation (as one can do always), the d771’ coefficientsof E’ and 4” vanish, while those of w” are those of a left-invariant one-form on H (the Lorentz group).This basically means that at this point one has a tangent and base space, i.e. we are down to superspace,and that H is a true symmetry because it acts only on the indices of the fields.Factorization also implies that g” only depends on 77” but not on x”, 0”. The reverse is also true: ifgkl = gkl(771)) and the d77” components of E’ and 4” vanish, then horizontality holds (as is proved in, forexample, Kobayashi and Nomitsu’s book on differential geometry).(iii) Field equations in the flat 0 directions: These will enable us to come down from superspace toordinary spac’~.As a result of horizontality we only need consider curvatures with legs in the (flat) x and0 directions. The results are (in the normalization of the Torino group)R’~~=0, R’~,,,=0 (14)R”,,,, =0, R”,,,,, =constantx(B”,,m +~8’,,,B”,,,, 5’,nB”c,n) (15)R”,,,~=0, R”~0,=0 (16)1,, = ~ (As usual, spinor indices are raised by C’ and lowered by C where Cwhere is the B” charge conjugation matrix.) Looking back at (1) and inverting (14)—(16) we see that with flatparameters ~cA = (~i ~a ~u1)one has= ~ + 2Em~”R”mn+ constant x Em~°(B”,,,,,+...) (17)dE’ = (D~)’— Eke’ —‘ ~I4’ (18)d4”’ = (Di)” — 2Em~”R”0,~ + ~(o~4’)”~”. (19)(iv) Final base-manifold: ordinary spacetime. The transformation rules in (17)—(19) look very muchlike the transformation rules of supergravity in ordinary spacetime. In principle, however, a problemmight arise, namely it could be that the flat-index curvatures (in particular those in the tensor B)contain curvatures with curved indices also in the 0 directions (namely, RAmn = hm’thn 11R 11’t”) and onemight find in (17)—(19) curvatures R~,.with curved fermionic a. In that case one would not only findderivatives ~/Of in the symmetries of supergravity in ordinary spacetime, but also a/ao” derivativesand one would not have arrived at ordinary spacetime. Explicit evaluation shows that this does not

P. van Nieuwenhuizen, Supergravity 289happen, fortunately. In fact, this is not accidental, but follows if one chooses a coordinate system insuperspace such that in (12) E’,, = 0 at 0 = 0. (A similar “gauge” we will discuss in superspace later.)This concludes the discussion of the action in (6).The results are likely to hold in more general cases. Whenever G/H is larger than the manifold Mover which one integrates, it may happen (as, for example, in the above case) that the curvatures with atleast one flat index out of M but in G/H (“outer directions”) are algebraic functions (as a consequenceof the equation of motion) of those curvatures whose flat indices are in M (“inner directions”). Whenthis happens, one calls the theory rheonomically symmetric, since it has an extra symmetry on spacetime6hA = D(e”°”h~’,,,) + hBCk0~tR~O,,B (20)so that 6hA only depends on fields with indices in M. This was pointed out for the first time by Reggeand Ne’eman.At this moment it is not yet understood when this program works and when not; samples are knownwhen a given CAKI.. .K, does not lead to a consistent theory and the program crashes.Since one starts out with a closed algebra (general coordinate transformations and gauge transformationsin the gauge manifolds), one might say that one has the maximal rather than the minimal setof auxiliary fields (most of the h,,~are auxiliary). In descending to fewer fields by eliminating them bymeans of the field equations there comes a point where closure is lost. Where precisely closure is lost isnot known (although one might discover this by explicit calculation). Perhaps the greatest virtue of theprogram is that one need not introduce constraints by hand from the outside. Some aspects are underfurther study: cohomology theory for non-compact groups and the notion of a manifold with anticommutingvariables.Another approach using fiber bundles has been developed by Mansouri and Schaer [326—328] and byMcDowell [309].To start with, they take the base manifold to be superspace and the fiber the full groupG. Then they make a 1-to-i identification of the G/H part of the fiber with the base manifold. In otherwords, P and 0 in the fiber and in the base manifold appear as separate entities to begin with, but theyeliminate this “double representing” not by imposing constraints on curvatures but by “soldering”. Inthis way they obtain non-linear realizations of local gauge symmetries, in which all the transformationlaws of the fields (including general coordinate transformations) follow from gauge transformations(defined by covariant derivatives, as usual). The method is equally applicable to Poincaré, de Sitter, andconformal supergravity.The strength of this approach rests in its manifest covariance with respect to all local (super-)symmetries. To take full advantage of this, one writes down invariant actions directly in superspace.The variation of each action leads to a set of field equations. Then to make contact with models in4-dimensional spacetime, one looks for 0-independent solutions of these equations. In other words, oneregards the 4-dimensional models as particular classes of solutions of superspace field equations. Thus,the correspondence between superspace and spacetime is established not directly between actions butthrough these solutions which extremize both actions.3.5. Supergravity and spinning spaceIt is possible to find the local analogue of the commutator algebra of global supersymmetry by usingHamiltonian methods [457]. The action of supergravity reads in Hamiltonian form (using Diracs

290 P. van Nieuwenhuizen, SupergravilyHamiltonian)I =Jd~[4p—A mC(p,q)]q — ~e ~, ~, , W, ~, —— f m , a mnl ~ 0, — 5 m I1eo, ‘PO,a, W0The p are the conjugate momenta, while C(p, q) denote the first class constraints (after eliminating thesecond class constraints by solving the latter) which generate gauge transformationsCm(P, q) = {~m,0”, J,,}. (2)All generators have flat indices. The proof that the action can be written in this way is well-known forpure gravity (due to work by Dirac) but the extension to supergravity which we discuss below is due toTeitelboim and Pilati.m and 8q = {q, Cm} m (where ~m are fieldindependentthe action localisparameters), to be invariant one finds under after 8/Li partially = {p, Cm} integrating (discarding surfaceIfterms)s5(4p) = 48p — (8q)j.5 = tmC. (3)_~mdC0,/dt = éVariation of Cm yields the Poisson bracket of the constraints8C {Cm,Cn}C”. (4)Defining the field-dependent structure constant f,,,,,” by(C, C,,} = fmn”Cp (strongequality) (5)it follows that in order for I to be invariant one must transform the Lagrange multipliers as6A~ ? +fmn”C”Am. (6)From the known transformation rules of em0, 4’o”, ~ as given in section 1 one can thus read off whatf,,P are. This is a very simple and elegant way to find the algebra of constraints, but one can, of course,also obtain the result by directly determining the charges Cm and then compute the anticommutators.In order that 6A has indeed the form as given above, one must combine a general coordinatetransformation with parameter ~ with a local Lorentz transformation with parameter ~m~0,kl and alocal supersymmetry transformation with parameter ~ (the same combination as found byFreedman and the author in the commutator of two local supersymmetry variations). We will call thesetransformations covariant translations.The transformation laws of the fields of supergravity in first order formalism as discussed in section 1read— tmJI—m, ~m ~Lie — 2EV yi,~— ~ ,,e ~84’,,” = ~ me~(D~4’,,”—D~4’,,”)+D~ ”~ . (8)

P. van Nieuwenhuizen, Supergravity 291= ~ me m” — ~mev,~ ~ (9)0,~R coy. mn + ~ + D,,A= given in subsection 1.6, eq. (2). (10)Since the variations of the objects on the right hand sides are independent of é if s 4, it follows thatthe expressions themselves in the right hand sides must be independent of the Lagrange multipliers for= 1, 2, 3. This follows from the general rule of varying an arbitrary function8G — ~ + k “~ + I .~L — 11— ÔA k (~ f m~ 6qk C dpk 3pk C oqk ( )and observing that only the explicit ~“ contains a time-derivative of e”.The reader who is not familiar with Hamiltonian methods, may at first wonder how, for example, H,,,,in 8~= ~ me”,Hm,, manages to depend only 4’,” and em 1, but not on the Lagrange multipliers 4’~”ande m 4. We now show this in detail,Hab = e,,eb’I’i,J + (ea 4e,,~— e,,’eb)I-I 41. (12)One should now express 84c~in terms of p and q by evaluating {4’,, HT}. An equivalent and quicker wayis to use the field equation y”’(D,,,4’~— D4,,) = 0. In this way one finds with Hmn = Amn”Hij4e,,’— ebea) y4y’/g44 (13)An,,” = eaeb’ — (e~with curved ~4 and y~.Now, ea4(g”t)”2 is the normal na, satisfying naea, = 0 and nan” = 1. Hence, na depends only on e”,.ClearlyAab” = (ea’eb’ — ebea’) — (flaeb’ — nbe~’XeCnd)yy (14)with flat y”,7dDecomposing em’ into the complete orthonormal set fla, eal as follows,e~’= a’n~+ $ 1”e,, 1 (15)3g” (the inverse of g,,) and therefore “good”. The a’ areit “bad” follows butupon cancel multiplication from Aab” if by onebl uses that flU =7d7c = 8dc + 2~,.dc and replaces e’~by a’n~for the 8’~”term. Forthe term with ~ one can replace e~’by f3”e~1.Hence, An,,” only depends on e,,, but not on the Lagrangemultipliers e,,4. Hence,= ~me~,H0,,, (16)indeed only depends on (p, q) for ~ = j.We can now read off the algebra of constraints from the laws (7—10). One finds for the nontrivial(anti)commutators{Q”(x), ~b(Y)} = (7m)a~(X) 8~(x— y) (17)

292 P. van Nieuwenhuizen. Supergravitv{~C(x), ~‘n(y)} = [~2,n,f”Jpq+ ~(Dm4’n — D,,4’,,,)] 63(x — y) (18){O”(x), ~‘ 3(x — y) (19)m(y)}= (EYsY,nDa4’b) ””Jcä 6where 12 contains the supercovariantized Riemann curvature. Indeed, in the flat space limit one findsback the global algebra. We stress that these results could have been found by a straightforwardHamiltonian analysis, in which case one would have found that in all these (anti)commutators only p’sand q’s appear.The truly interesting thing is the off-shell closure of the gauge algebra. To establish its relation to thegauge algebra when evaluated on fields F, we note that the Jacobi identity yields{F, {C, C,,}} = (F fm,,P}ç, +f,,,,,~{F,C~}. (20)Whenever (F, fm,,”} is nonzero, one finds extra terms in the gauge algebra proportional to C,,. These arethe equation of motion terms known from the Lagrangian approach (which can only be gotten rid of byintroducing the auxiliary fields into the theory). Apparently, in supergravity the structure constantsdepend on the canonical momenta. (In gravity they depend only on the canonical coordinates in thecostumary parallel-transverse basis, but they depend also on momenta if one uses the same as oneused above.)In the anticommutator (0~Q} ~ no fields appear in the structure functions as a result of defininginstead of general coordinate transformations the covariant translations with parameters f”. There arequite a number of fine points. For example, the brackets are really Dirac brackets, but we refer to theliterature for further study.Let us now discuss why the {Q, Q} - ~‘ relation implies that supergravity is the theory of spinningspace. We are led to consider the Dirac equation as a square root of the Klein—Gordon equation2)4’ = 0. (21)(ytm3 + m)~/i= 0, (111 — mMultiplying by y~and defining y~ym= 0”, 7ç = g5, we can introduce, in addition to the usual canonicalvariables (x, p,,) five anticommuting variables (0, 0~)satisfying{xm, p,,} = 8~’, {o”, o~}= —26~”, {o~,o”’} = 0, {o~,o~}= 2. (22)One now finds a set of first class (i.e. commuting) constraintsS = Otmp,,, + m05, ,9~?= p2 + m2 (23){S,S}=2~r, {s,H}=o, {~r,~r}=0. (24)An action for this system leading to these (anti)commutators can also be given. Hence one can take asquare root of the constraint ~‘.Just as taking the square root of the Klein—Gordon field leads to a spinning particle, taking thesquare root of (the generators Hm of) ordinary space leads to (the generators Q~of) spinningspace.

P. van Nieuwenhuizen. Supergravity 2934. Conformal simple supergravity4.1. Conformal supersymmetryConformal supergravity is the supersymmetric extension of Weyl’s action R2,,,. — ~R2 which isinvariant under local scale transformations 3g,.,~(x)= A(x) g,,,,(x). The theory is thus a higher derivativetheory, and for the time being it does not seem to be of physical interest. However, that may be due toour present underdeveloped knowledge how to treat higher derivative field theories. At the moreformal level, conformal supergravity is of the utmost interest, since it explains the structure of ordinarysupergravity: where the auxiliary fields come from and how to obtain a tensor calculus. Since(super)conformal methods are little known, we will give here a rather complete treatment. We begin bydiscussing conformal rigid symmetry.At the basis lies the superconformal algebra [581, 519]. It contains as bosonic part the conformalalgebra with (P0,, M0,,,, K,,,, D). Since F,,, and the conformal boosts Km play a rather symmetrical role, itcomes perhaps as no surprise that there are two fermionic spinorial charges, namely the usual squareroot of P0, called 0”, and a square root of K,,,, called henceforth S”9Pm (1){Q”, Q~}”’ +~(y”C~)”’{s”, S~}= _~(ymC_I)asK0,. (2)(In the literature on conformal supergravity one often finds the special representation where (C’)”” =~ see appendix.) The rather unexpected feature is that if one adds 0” and S”, one needs at thesame time one more bosonic generator A for chiral rotations{Q”, S~}= +~C~1”’~D— ~ — (iy5C’)”’~A (m > n). (3)The conformal algebra is given by the ordinary Poincaré algebra, discussed in subsection 2.1, eqs. (24)together with[Pm,D] Fm, [Km,D]” ~ [Km,Pn] 2(6mnD+Mmn). (4)The nonvanishing commutators linear in fermionic charges are[~:~Mmn] = (omn)”s(~:)~ ~ A] =rQ”1 I “ .. IC” D 1 — I \a $I DI_HJI I.”’ mj~7mj$ 3. (5)1s”’ — ~ [0”, Km] = (Ym)”$S’Rather than check explicitly the Jacobi identities, we prove that this is a closed algebraic system bygiving an explicit matrix representation. We construct 5 x 5 matrices, of which the 4 x 4 part containsthe conformal group, in the fifth row and fifth column one finds Q and S, and along the diagonal thereisA,

294 P. van Nieuwenhuizen, SupergravityMm,, = (-Tm,,, Pm = —h~m(1 7~),Km ~A=diag-~(1,1,1,1,4), D=—~y 5(Qa)/~= —~[(1 + 75)C’]”’, (Qa)5 = +~(1—(S~5= +~[(1— 75)C’]”’, (S”) 5 1 = — ~(1+ ~ (6)We will now write down a representation of this superalgebra on fields. Consider the scalar multiplet= [A, B, x~F, G]. (We hope the reader will not confuse the scalar field A(x) with the chiral generatorA.) We already have a representation for the global super Poincaré algebra. Defining 80A(x) =[A, EQ] = ~Jx, one finds with i,, == ~x, 60B = —~iiy5,y, 80F = ~iJx, 80G = ~i~y!JX8oX = ~J(A — iy5B) + ~(F+ iy5G) .(7)Thus [8(e~),8(e2)]A = [A, [ë2Q, e~0]] so that if one definesmPm] = ~m’9mA (8)6~A(x)= [A, ~then the {Q, Q} — F relation is satisfied.In order to find the representation of Mmn on fields we first define how Mmn acts on fields at theorigin. The only nontrivial result is8~~~f(0) = [x”(O),Am”Mmn] = Amn(o~m,,)a~X$ (m

P. van Nieuwenhuizen, Supergravity 295i5j the boost generator can be nonzero. Specifically, with SDC = ACA 13, we shall see that[AKm]””2’~t3mC, [A,Km]~AymZ. (12)In order to avoid confusion with the dilation generator D, the field i5 carries a hat.Consider next S-supersymmetry. From S = /~[Km, (ymQ)~] one finds that A, B are S-inert. But for xone finds a nonzero result using the Jacobi identityOne finds[K,,,,8bA] = [Km, [‘4, Pb]] = [Km, A], Pb] — [Km, Pb], A]. (13)8~x”A(A+iy5B) s, o~(~)=(1_A)(~~~~). (14)8~, we determine 8A from (0, S} anticommutator. One findsSince we know now 5~andIA\ A 1—B\ 1—3 A\. /F\ 1—3 A\f—G\ÔA(~B)= 2 ~..A )AA~ 6~Y= ~T’)’~~~’ ÔAI~G)= ~ F )AA. (15)These results for D, Ka, A are again only for the fields at x 0. Just as for Mm,,, one can find thetransformation rules of the fields away from zero by using the commutation rules of Pm with thesegenerators. However, these results we will not need, because in order to construct covariant derivatives,one only needs the transformations of fields at the origin (“the spin parts”; the orbital parts are neededto prove, for example, the invariance of the Dirac action under global Lorentz transformations). Thuswe have derived a representation of the superconformal algebra on the scalar multiplet.A representation of the superconformal algebra on the coordinates of superspace (x”’, 0”) is given byMm,, = X0,D,, — Xnt9m + O(Tmnt9I8O, P,,, = 3,,,0” = ~(a/8O—J0)”, D = x ö + ~ëa/aëA = —~i~y53/8~ (16)Km = 2x0,x2t9m — OX’7m3/30 ~(O0)28m — O0(07m3130)a — X5” = ~(O03/3O+0750753/30)” + ~(0y5ymO)(75yin8/80)~ + (—~a/aë+~ejo + ~0J0)”.The derivatives are always left derivatives and .~= ytmx0,. One can always choose the scale of 0 and thesign of the charge conjugation matrix such that 0” is as above. (Taylor expanding as ç(x, 0) = q’(x, 0) +O”(3/30”)p(x,O)+... and using 3/30” = C~$3/80$.)For completeness we also determine the representation of the superconformal algebra on the vectormultiplet V = [C, Z, H, K, B0,, A, D]. The transformation rules follow from ö(supsym)V(x, 0) =(Eiy5G)V(x, 0) with G” = a/a0~— (JO)” (see subsection 5.3; we have rescaled 0 here by a factor 2).

296 P. van Nieuwenhuizen, SupergravityUsingV(x, 0’) = C+ ~Z+ ~0H + ~OiyS0K+ (~ym7sOB + ~‘00(A+ ~JZ)+ ~o)2(ñ + ~L]C) (17)one finds easily8 0C = ~Jiy5Z= ~(iy5H — K — 0 + JCiy5)60W = ~Eiy5JZ+ ~Eiy5A80K=—~EJZ—~EA (18)1— 1—60B0, = ‘~ 3mZ— ~EymA80A ‘~(o~F+iy515)e,F~”r3,,,B,—d,B,.8~!3= +~Eiy5JA.Indeed, the (0, 0) -“ P relation holds with 8~C= ~ m3 0,C.The Lorentz transformation rules of V areobvious. The conformal weight is defined by 6DC = [C, ADD] = AADC (where C is the first componentof V and D denotes the dilation generation). Thus the scale of Z is A +~,that of H, K, B~is A + 1, thatoffl isA+~andthatofD=A+2.Under conformal boosts the only possible nonzero relations are[AKm]a3m~, [A,K0,]=f370,Z. (19)For example, [A K0,] Bm or [B0,,K,,] -~80,,,C have wrong parity. We will fix a, f3 below but continuefirst.For S-supersymmetry one finds with the [K, 0] -‘- S commutator and using [K, F] -‘- D + M as before8~C= 0,8~Z= (iy5 ~)AC= (A — 1 + fl/2)(~siy5Z)= (A — 1 + f3/2)(e~Z)8~B= ((A +2— fl)(EsymZ)= [~ f3y5H + K — (~ + ~),g + (~. —(20)= —A(e~iy5A)+((J3 — a/2)(Esiy5JZ).

R,.,,(D) = 23kb,,. + 4e0,,,fm,, + ~ (3)P. van Nieuwenhuizen, Supergravity 297Under chiral rotations C, B 0, and D are inert, but from the (0, 5) anticommutator one finds= (~—~—~)(~), 8,~Z=~((A+fl)—~)iySZAA. (21)Also from (0, 5) one finds finally13=—A, a2A. (22)4.2. The action of conformal simple supergravity [523]The superconformal algebra consisting of the conformal generators Fm, Mm,,, Km, D, the spinorialcharges 0”, 5” and the axial generator A was discussed in the last subsection. We now apply the resultsof subsection 3.2 to it.We define gauge fields and parameters byh,. = e m m”Mm,,+ ç~aQ”+ f’”,,Km + b,D + ,,S” + A,,A (1)4~,Pm+ W~yC = ~mPm + Am”M0,,, + E~Q”+ ~ + A13D + Es~S”+ AAA (2)with m > n and construct the curvatures,R,,,, m”(M) = 23,~mn — 2W~m~~~n — 4(e e”,,fm~)— m,,,f”~— 2~,m,,R, tm (F) = 2.9~em 1~— 2w~ m”e,,,,+ ~ + 2em 1,b~R,,,.]” (K) = 23f’~.— 2W~ m”f,,,.— ~— 2f”,,,b~R,,..,,~(Q) = (2D,4,., + 2~y~+ b,,ç~.— ~iA,4,,y5)~R,,.,,~(5) (2D4. — 2c~,,ymf”,,— b,4,,, + ~R,,,(A) = 20,,A,, — 2i~y5~~.All these curvatures are still to be antisymmetnzed in (,av). As we will see, there are constraints onthese curvatures. These have been found by considering the (unique) action for conformal simplesupergravity, but the results are so simple and suggestive, that it seems clear that a deeper geometricalorigin exists, independent from any dynamical model.The most general action, bilinear in curvatures, parity conserving and without dimensional constantsand affine (i.e., without explicit gauge fields) is given by

298 P. van Nieuwenhuizen, SupergravityI = J d~x[aR,,~, m”(M)R,,j”(M) 0,,,~+ $R,,,,(Q)y5(—C’)R,,,,~(S)+ yR,,,~(A)R~(D) + ~R,,.,(P)R,,.,.(K)]&”’” (4)(note that R,~(S)starts with ~ + ~ For internal symmetries such as electromagnetism, theaction is unique and non-affine, namely the Maxwell action. Hence, since the internal symmetriesconsist here of only the U(1) chiral invariance, we add its actionI = J d~x[8R,,~(A)R,,~(A)g”~g”~’Vg] (5)Under local M, D, A gauge transformations (given by oh”,. (D,,,e)”’ with e equal to Amn, AD, AA) theaction is invariant if one uses the rule of transformation of curvatures. Under local K-gauges, the actionis invariant if the P curvature vanishes and if the 0-curvature is chiral-dualR,.,, m(P) = 0, R,.~(Q)+ ~ ,.~“=R,,,~(Q)y = 50. (6)Local S-invariance follows then iffl=2iy=6=—8a, ~=0. (7)As we will discuss shortly, local 0-invariance follows if one imposes one more constraintR,,,,(0)o” = 0. (8)Thus all constraints are derived from a dynamical model by requiring various gauge invariances. Inprinciple, the transformation rules of fields are modified when one solves these constraints. However, asone easily verifies, all constraints are M, K, D, A, S invariant, and thus for these symmetries all fieldskeep their transformation rules 3h/~= (DE)A and the action remains M, K, D, A, S invariant.One can solve the constraints above. The P-curvature yields the torsion equation,my4’~)+(e”btm — em,,b”). (9)(0,, = _w mn(e)_.i(4’ — * yn4’m + ~pThus there is the same torsion as in ordinary supergravity, induced by gravitinos, plus torsion inducedby the dilation field.The solution of the 0-constraints eliminates the conformal supersymmetry gauge field= (y”(S~v+(7~~~v), 5,.,, = ~ /J,

P. van Nieuwenhuizen, Supergravity 299transformation rules for the dependent fields arise after solving the constraints (see subsection 3.2)= ~Rmn(0)y,. 0 (11)= y” (y5R,,,,,(A) + ~1~,,,,(A)) 0. (12)These results and a similar result for O’f”,. will be derived in subsection 4.We now discuss the 0-invariance of the action. If all fields would transform as given by the structureconstants alone, one would have81 = 8a Jd4x [1~~(0) y5y0,e0R,,,,~(K)+2iR~~(S)(ysR,.,,(A)+~R,.,,(A))eo— ë,.yy . 4’)R,,,,(A)R~~(A) + 2~0y”4’0R,.,,(A)R’”’(A)]. (13)The extra terms O’w,,”’ and 8’4i,. give an extra 8’I, whose general form was derived on page 281.However, since Qi”~ is constant and proportional to “~ for A = M, 5, the Bianchi identity holdsso that the D,,R,,. terms vanishes, and also the 3,,Q terms vanish. One is left withS’I = 8a J d 4x[28~w,.m~1~~”~(K) ~ + 41~”(S)y,~,y~5’4,, + 4ü~,.(y 5R”~(A) + ~1~”’(A))o’q 5,,]. (14)The R(Q) R(K) terms in 81 cancel against the R(K) term in 8’I if and only if 8lw,.mn is as given above.For this to be the case, one needs both 0-constraints; with the dual-chiral constraint alone nocancellation would occur. Furthermore, substituting O’~,,.all other terms in 81+ 8’I cancel at once.We now show that the dilation field cancels from the action. The remaining independent fields areem,,, 4’~,,,A,,, b,. and ftm,. while ffll,. is nonpropagating and is expressed in terms of other fields bysolving its own field equation (just as ~,.mn in ordinary supergravity). Let us consider how these fieldstransform under K-gauges. Only b,. transforms, and since 8Kb,. = 2~K,,. one can gauge away b,.altogether. This one might have expected since there are as many K-gauge degrees of freedom asb-components. However, since the action was K-invariant to begin with, it must therefore beindependent of b,,.The action is thus unique and determines in turn the constraints [523].In tact, as we shall see, thesolution of the ftm,, -field equation is at the same time the solution of an extra constraint [535].At thelinearized level, these results were obtained using superspace methods [191].Let us close this subsection by showing that there are equal numbers of boson and fermion states.Since the graviton and gravitino have higher derivative actions, one has two spin 2 states (El2 counts astwo El) and three spin ~states (Jill counts as three )T~.Indeed, one finds one massless (2,~)multipletand one massive (2, ~,~,1) multiplet [191,526].4.3. Constraints and gauge algebraFor ordinary supergravity, one can formulate the theory in second order formalism either byimposing a constraint on curvatures (R14,,tm (F) = 0) or by solving a field equation belonging to the gauge

300 P. van Nieuwenhuizen, Supergravityaction (namely OII8w,.mn = 0). Both methods lead to the same result, but the former has the advantagethat it is independent of any particular action. In conformal supergravity, the spin connection ispropagating, and the torsion is here fixed by the constraint R,,,,m(F) = 0, and not by a field equation.However, for the conformal boost field ftm,. this duality is again present: solving the field equation81/of”,. = 0 for f”,. one finds a result which can also be obtained as a constraint on the curvatures. Thisconstraint isR,.i,mn(M)emAen,. —~RA,.(Q)y,,~I/ +1~ ,,,,~,,,R”°~(A)= 0 (1)and it is again K, M, D, S, A invariant. Thus under these symmetries, O’f = 0 but as we shall deriveshortly, 8’~fr0. (It should be noted that as far as invariance of the action is concerned, all O’f areallowed to be non-vanishing since f satisfies its own field equation. See subsection 1.6.) Eq. (1) is thecovariantization of the “Einstein equations” R,,,,(M) = 0. The term with R(Q) is the connection neededto covariantize the derivative 3w inside the M-curvature. If one introduces a connection~~mn = ~,.mn ~e,,kCk0,?,,~A~~ (AC = chiral gauge field) (2)then the constraint even simplifies toR~(M,th)=0. (3)Thus there are three m,,,,(P) constraints = 0, not in a field conformal equation. simple supergravity:-(i) (ii) Torsion: Duality-chirality R and tracelessness: R,.,,(0)-y” = 0 which is equivalent to R,.,,(Q) + ~R,.,,y5= 0plus R,,,,(Q)o”~= 0.(iii) Einstein equation: R ~M, ~) = 0.It is probably not a coincidence that the generators for which the curvatures are constrained (F, 0, M)form the super Poincaré algebra.All constraints are gauge invariant under all 24 local symmetries, except under P-gauge and 0-gaugetransformations. The extra terms 8~~w,.mnand ~ were mentioned before. In a similar way one findsc,’ zm — 1 covi \ mp lflm,CovI \OoJ ,. — ~ ,,,. ~ jO~ C~— 8r.,. ~ j 7~C~where the Q-covariantization is due to the 34 terms in R(S).One can now understand geometrically why these constraints are needed. The global commutator{0, 0) — P must turn into {Q, Q} — general coordinate transformation plus more if one evaluates thecommutator of the two ordinary supersymmetry transformations on fields. This is to be expected sinceour action was by construction invariant under general coordinate transformation, and invariance underalso P-gauges would be double-counting. How does the theory manage to do this? Note that a generalcoordinate transformation on agauge field hA,. is related to local gauge transformation by the followingidentity8gen.coord(~~)h’~,.(D,.(~. h)~+ ~‘R”,.A. (5)

P. van Nieuwenhuizen, Supergravity 301Consider now the {Q, Q} anticommutator acting on em~.Since only physical fields appear at allstages, and physical fields have never a nonzero 8’ transformation law, the only way the theory managesto make P-gauges an invariance, is by requiring that R(P) in (5) vanishes. ThusF~I \~I \lm _~~tA BC_1-~oQ~el),uQ~E2)Je ~L~°PV BC 1E2 —2 2Y ~1m(F) = 0, this is according to (5) a sum of a general coordinate transformation plus otherand localsince symmetries. R This explains also why in the superconformal algebra the parameters in the {Q, Q}anticommutator are all of the form ë2y~’eitimes the gauge field h~.The other constraints play a similar role. For example, in the {Q, 0) anticommutator on ~ one findsnow an unphysical field, namely ~ due to ~ = D~ .Thus now the theory has a choice: eitherchoosing an appropriate ô’w (which again leads to the constraint R(P) 0), or by putting R(Q) = 0 as aconstraint. The latter choice is not possible since no field can be solved from R(Q) = 0, while the formerchoice is the correct one:[~fri), ôo 2)]~= ië y’~’ R’~ (Q) + 8P(2ymE1)~~~ (7)m= ~Rmfl(Q)YILEQ. Thus, for ordinary supergravity, the constraint R~~m (F) = 0 is enough.due But to for8~,w,~’ conformal supergravity one has one extra independent field to consider, namely A~.Now with= 1EQY5&[ÔQ( j), 60( 2)]A~ 8(lEyme)A — (ii2y5 ( ~)&— 1 ~-~2) (8)and indeed ~ is such that the last term turns into a term with R,~~(A) such that the sum is again ageneral coordinate transformation.Thus the constraints on curvatures have the geometrical significance of turning P-gauge transformations,defined by the group in the tangent space, into general coordinate transformations in the basemanifold.The gauge algebra is thus given by[o(e~), 5( ~)]h~= o(fA ~~eC)+fA~6~(eA)hBeC_ 1~-~2. (9)Since only ö’~is nonzero, only the (0, 0), (0, K), (0, S), (0, D), (0, M) or (Q, A) relations might bemodified. However, K, S, D, M, A rotate physical fields into physical fields so that only the (0, Q)anticommutator is modified. One finds uniformly on e~,~, A,. and b~,and hence (because of theJacobi identities) also on any function of these fields (in particular on q5~,f~,w~”~)F~( \ ~ I \] (~A\j ~ I_LA mn\j ~t”~~~ii,‘-‘Ok~2)J — ‘~gen.coord~S I ULorentz~ S W~ j (‘chira!~ 5+ ôo(—~frA)+6~(—~~) (10)11Awith ~ = ~E2yk 1. If one puts b,., equal to zero, as well as ob,~,the gauge algebra still closes but now anextra S-supersymmetry transformation is present,extra 1 —ô~ (~(~e~)ë1y~ — 1 ~-~2). (11)

302 P. van Nieuwenhuizen, SupergravityThus the gauge algebra of N = 1 conformal supergravity closes without the need to introduce auxiliaryfields [5231.This is no longer true for the N> 1 case.We now show how to derive8’W~m~and ô’4~without much work, using group theory. SinceR,,J~(P)= 0 identically, its variation vanishes, too. This variation consists of two contributions: namely,if all fields would transform according to the group, one could use the homogeneous rotations ofcurvatures8(group)R~,m(F) = _~R,.~,,(Q)yme. (12)However, ~ is modified because w~m”is expressed in terms of other fields by solving R~~m(F) = 0.Thus there is an extra term ô’W,~m~e~~— ô’co~m~e~,~. Solving in the same way as one solves for theChristoffel symbol, one finds ô’w,~m”.For ô’& we proceed as follows [1691.From the Bianchi identity= 0 = — ~‘°(R~~m(M)+ e~mR~(D)) (13)one finds two relations, using that R,~r(Q)o~’y’ = ~R,~,,(Q)y5yA ~~ = 0R~~(M)—R~(M)= —2R~(D) (14)Ra~~r(M) + 2 terms cyclic in af3y = —2ô~~R~,,(D) + 2 terms. (15)From the latter result one deduces~ (M) — ~ (4~)= o~,7R~8(D) + (a 4*13, y ~ 8). (16)As before, we now use that the variation of a solved constraint vanishes (E == 0 = y~’[~y5 R~(A)— R,~(D)+ ~ Om~ R~mn(M)]+ [2ô’qS~ + y,,y ô’4]. (17)It is now easy to deduce the result for ô’qS,.. quoted in subsection 2.The result for 6 1f”,.. in eq. (4) is obtained in a similar way, by varying the third constraint (theEinstein equations) and using the results for 8’w~m~and 6’&, as well as the Bianchi identityD~R~(Q) = 0. (This identity is needed since varying the Lorentz curvature one produces terms ofthe form 8ja3W’~m”.)4.4. Conformal tensor calculusSince all fields in conformal supergravity are gauge fields which couple minimally, it is straightforwardto obtain a tensor calculus for conformal supergravity: all one has to do is to replace ordinaryderivatives by covanant derivatives. In order to obtain a tensor calculus for ordinary supergravity, onehas to do two things. First of all, the transformation rules of ordinary supergravity are obtained from

P. van Nieuwenhuizen. Supergravity 303“Q + S rule” of subsection 4.5. Since the results then still depend on the conformal weight A, whereas thenotion of a weight is an alien concept in ordinary (nonconformal) theories, one then redefines fields suchthat the final results are independent of A.To illustrate the first step, consider for example the spin (1, ~)scalar multiplet with canonical weightA = ~ [527]W,. = [Aa, i(Ay 5)~,(0m”Fmn + ~y5D)pa,~Ai)a, ~(~iy5)a]. (1)Since ÔV,. = —fy,.A, ÔA = (r~’F,.~ + i-y5D)E, t5D +iiy~XA,the local version W. is obtained by replacinga,. by the superconformally covariant derivative D,.’D,LCA = (DM — ~iA,.y5— ~b,.)A — ~F,.—(a~V~+~4,~jcA)—p.4*v.+ iy5D)~.(2)This D,.c follows from subsection 1. As a rule, it is better to use flat indices; for example, it is Fmn whichtransforms simply, not F,.~= e m,.e~.Fmn.As a second example we consider the local analogue of the “kinetic multiplet” T(~)[531]. Thismultiplet will be used in the next subsection. The details which follow have not been published before.It is easy to show that the following is a scalar multiplet of rigid conformal supersymmetryT(~)= [F,-G, Ix~LJA, -EIIB]. (3)To make it a local multiplet, we replace ordinary derivatives by covariant derivatives and D,.CX isalready obtained from subsection 1. Since under S-supersymmetry 6~F= (1 — A)E~while the firstcomponent should be S-inert, it follows that one should take A = 1, ie., T(~)has canonical weight 2.We will now derive the conformal dalembertian EICA. For completeness we will derive the result forarbitrary A, although in T(~)one only can admit A = 1 as we showed.From subsection 1 we know the global superformal transformation of I; in particularCA_ A 1~ LA AC~ 4where A,.c is the chiral gauge field (A,.c = — ~4,.aux). In order to take the covariant derivative of thiscovanant derivative, we begin with the following well-known simple expression for the dalembertian ingeneral relativity~ (g~’eD,.’~A). (5)This takes care of the Lorentz and general coordinate connections (the latter replace as always P-gauge).We now write the other connection terms for 3,. (g~~~eD,.cA). First we state an important property:Ordinary derivatives in local objects must never be written as commutators with Pa if one wants toconstruct connections. For example, if one wants to determine the K-connection for the derivative ofa,.A then one finds zero, and it would be wrong to argue that [Kr, a,.A] = [Ku, [A, F,.]] = —[Kr, F,.], A].

304 P. van Nieuwenhuizen, SupergravityFor the connections one now findsS: y~+~ifr~(A+iy 5B)4,.—~~A-~&y5~fr~B. (6)Here we have used that 5sX = A(A + iy 5B)e~while ô~b,.= — ~and {Q~,S~}= ~C~’~D +more,as follows from 8h,.” = fcBh,.eA: ~ ~ (7)D: (2— A)b~D,.’~A. (8)tm,. and that 6DD,.CA = ADD,.CA. Thus under local scaleHere we have used that &,e”,. =transformation, the scale of the derivative —Ane is zero. (This makes sense, 3,. is not a field and one onlytransforms fields under local scale transformations.)K: —2AAr,.em~= 2AA(~R — IJ,.(T’~”’4)~). (9)The connection comes from 8Kb,. —2~7~em,.. All other fields are K-inert. In particular 8,.A is inert underlocal K-gauges (under global K-transformations, 3,.A is not inert since [K,.,Pr] ~ 0!). The symbol Rdenotes the scalar curvature with w(e, i/i, b).The Q-connection is obtained by varying eg~”and D,.CA. For the latter one finds= 80’~(3,.A— i/i,.,~y+ BA,. — Ab~A)m”Umn~=~e(a,.+~W,.— -~- Ey~A,.+ B(—ëiy5~,.)— (Eço,. )A — (i~)b,.. (10)Now one can writeasThus— ~4’,.6o’~X = — ~~Øc(A — iy5B)+ F + iy5G)e (11)E(Øc(A — iy5B) + F + iy5G)(—~i,.)+ ~i(ØcA)./,,.. (12)= ~&,.CX + ~ë(ø’~A)i~i,.. (13)

To this one must add the variation of ~P. van Nieuwenhuizen, Supergravily 305which yields an extra termM~~))(D,.cA)=(8oc(eg(~ey i/ig~~’ — — ~ey~r)D,.cA. (14)Interestingly, the non-covariantizable terms EØCAçIJ,. cancel, and the complete 0-connection is easilyfound to be0: ~ — . ~/iD,.~A. (15)Thus the full superconformal dalembertian is given byPfA~3r(g”~’eD,.’A)+~c~ y~—~A’~8B— ~A”~5x+~j-b .ACB2A + (2— A )b~D,.5~’+ AR + ~- Aii~zu~~’çbr— ~- (A,.c) ~/i”D,.’~y— — ~ ç~y çLcD,.CA. (16)Two interesting remarks are in order. First of all, although the separate connections are not inert underthose symmetries to which they do not refer, the sum (thus LJCA) is really covanant under allsymmetries. Secondly, as we have stressed so often, flat indices simplify transformation properties, andindeed,ô0’~(Da’A)= ~(EDa”X). (17)One can now at once write down the conformal coupling of the scalar multiplet to the gauge fields ofconformal supergravity. It is given by the action for the multiplet I ® T(I). The multiplication of twolocal multiplets is the same as the multiplication of two global multiplets of conformal sypersymmetryI1®12 (A1A2—B1B2, A1B2+A2B1,(A1+iy5B1)X2+(A2+iysB2)X1,—glx2-f A1F2+A2F1—B1G2—B2G1,A1G2+A2G1 +B1F2+B2F1+iX1y5X2). (18)One may verify explicitly that the product of two superconformal scalar multiplets with weights A1 andA2 respectively, as given by (18), is again a superconformal scalar multiplet with weight A1 + A2. Insuperfields this is immediately clear. In the next subsection we show how to construct an action for asuperconformal scalar multiplet.4.5. The origins of the auxiliary fields S, F, Am [471,472]m,., the gravitino(ordinary The independent supersymmetry (physical) gauge fields field) of ,fr’~,.,the conformal chiralsupergravity gauge field A,. are and the the tetrad dilation e b,.. Just as inordinary supergravity, one can again count field components in the action, taking into account that eachdegree of gauge invariance removes one field component. The count is as follows:

306 P. van Nieuwenhuizen, Supergravityl6em,. — 10 spacetime — 1 dilational gauge = 5 Bose fields16q/’,. — (4+4) ordinary and conformal supersymmetries = 8 Fermi fields4A,. — 1 chiral gauge = 3 Bose fields4b,. —4 conformal boosts = 0.Thus there are equalnumbersof Bose and Fermi fields off-shell, andno auxiliary fields are needed from thispoint of view. Indeed, as discussed in subsection 4.3, the gauge algebra of conformal simple supergravitycloses by itself. This has an important consequence. Since all fields of conformal simple supergravity aregauge field, they couple minimally to matter.Thus, one can at once write down the coupling of the globally superconformal multiplet discussed insubsection 1 to conformal supergravity. All one has to do is replace ordinary derivatives by superconformalderivatives, where the gauge connections are given in the usual minimal way.To see the relevance of this fact for ordinary supergravity, let us recall that the Hilbert action ofgeneral relativity (R) can be written in a locally scale invariant form by adding a scalar kinetic term withunphysical signs2’(Brans—Dicke) = — ~ R4o2 + ~ 3”4~34~ (1)Fixing the new degree of gauge freedom by fixing p = K1, one regains the Hilbert action. Orequivalently, one can choose a gauge in which the regauged metric g,.~p-2 is locally scale invariant.In supergravity the analogue of a scalar field ~ is a scalar multiplet I = [A, B, X, F, G] which wetake to be a superconformal scalar multiplet with weight A = 1 in order to interpret A as the analogueof ~. The action for I is in flat space= —~[(3,.A)2 + (3,.B)2 + ~/A — F2 — G2] (2)and it is invariant under 6(A + iB) = ë(1 + ys)X,= ~J(A — iy5B)e + ~(F + iy5G) and 8(F + iG) = ~E/(1+ y5)X.To couple we first obtain the supercovariant derivatives D,.C. From subsection 1 we find easilyD,.CA = 3,.A - ~ c,.x - Ab,.A + A,.B1- AD,.CB = 3,.B + ~~I’,.Y5X — Ab,.B — ~ A,.A= ~ øc(A_iy5B)cb,. —~(F+iy5G)iIi,.(3)—(A +~)b~~—(~—~)A,.(i~s~)—A(A +iy5B)ço,..

P. van N,euwenhuizen, Supergravity 307To obtain the supersymmetric extension of (2), one replaces 0,. by D,.c and uses the same productrules for multiplets as in global supersymmetry (since covariant derivatives satisfy the Leibniz rule, justas ordinary derivatives). Thus one constructs the multiplet I® 11 and uses the following actionformula, valid for any superconformal scalar multiplet I (in particular I ® 11) with weight A = 3 (sothat the F component has weight 4) [527]I =fd~x[eF+ ~ey,y + ~ei/i,.o-’~”(A — iy 5B)i/i~], (A = 3). (4)In this way one obtains supersymmetric Brans—Dicke theory.An alternative way to derive this action is as follows. The action 22(1) of global (conformal)supersymmetry can be obtained by first writing I as a vector multiplet [4481V(1) = [A, ~75,1~’,F, G, 8mB, 0, 0] (5)5-component) which isthen givenmultiplying by V(I) times itself, and finally taking the last component (the .12C2ZA+H2+(~B2m(8mC)2~IZ (6)Replacing 8,. by D,.C, and using the following action formula for a vector multiplet with weight A = 2 (sothat D has weight 4) [447]I =fd~xe[I3 — ~K2C(—~R— ~ + more] (7)onefinds the action from V(1) x V(1) + V(par 1) x V(~parI), wherepar I is the parity reflected multiplet(B, —A, —i-y5~,G’, —F’). The result is2(supersymmetric Brans—Dicke) = (—4e)x2+ B2XR + ~ + ~D,.cA)2+ ~D,.cB)2 — ~F2— ~G2+ .. •]. (8)[—MATo reduce to the case of ordinary supergravity, on~fixes the local scale invariance by A = constant, thelocal chiral invariance by B = 0, the local boosts by b,. = 0, and a linear combination of Q andS supersymmetry such that x = 0 as well as 8~= 0. The only linear combination of 0 and S supersymmetrywhich satisfies 8,~= 0 is 6oc(2E) + o~((—F— iy5G — (iI2)A~y5A)e)and the action reducesto2= —~(R+ Rr)_~(F2+ G2)+~A~A2. (9)If one identifies the chiral gauge field A,. with —2/3 times the auxiliary field A~X, and further F =G = —~.P,A = 1 then one recovers exactly ordinary supergravity. For example, 80(2 )F= eøcX andsince ÔSCF = 0 for A = 1, one finds, using that y = —~y,.R~ + ~ the result for ÔS in

308 P. van Nieuwenhuizen. Supergravityordinary supergravity. If we had initially included the locally scale (Weyl) invariant coupling (eA 4 +more), then one would have found Poincaré supergravity plus cosmological term.In particular, the supersymmetry transformation of ordinary supergravity is a linear combination of thetwo supersymmeeries of conformal supergravity [525]= ô~’~(e)+Ss’~i~e + ~.JJ ), ~ = —~(S— iy5P — iA~”y5). (10)Moreover, the chiral gauge field turns into the axial auxiliary field Am and the auxiliary fields F, G of thescalar multiplet become the auxiliary fields S, F [471].These results are not only valid for matter fields but hold generally. For example, for the field e”,.,~ A,. one has= ~EQym~,., oo~~A,.c= —iE0y5~,.~ / ~ 1 ~= ~D,. +5b,. —-i-A,. Y5~o (11)v5s~em,.= 0, ~ = 7,. s, 6~’~A,.’~ = i sYsl,lI,.and using the “Q + S rule” one finds the usual transformation m~and rules the for explicit the tetrad b,. and in 6~’~/s,. gravitino inordinary supergravity. (In particular the dilation terms in w,. cancelagainst the X in 6~.)It is thus clear also, why ~2 and F2 have negative signs and A~,has a positive sign in the gauge actionof ordinary simple supergravity: because the Brans—Dicke scalars have unphysical kinetic sign.4.6. Tensor calculusfor ordinary supergravityIn the preceding subsection it was shown how to obtain a tensor calculus for conformal simplesupergravity. In this section we show how one derives from these results a tensor calculus for ordinary,i.e. Poincaré, supergravity.Consider, as an example, a scalar multiplet with conformal weight A. Under conformal supergravityone hasöCfr)A = ~Ex, 80’~(e)B= ~7~X(1)and= ~(ØC(A — iy5B) +F + iy5G)= 8sC( )B= 0,osc(E)F = (1 — A)EX,6~( )x= A(A + iBy5)e(2)6~’~(e)G= —i(1 — A)ëy5~.These results follow from covarianting the results of global supersymmetry, which is unique as we havestressed already several times.

P. van Nieuwenhuizen, Supergravity 309Using the “0 + S rule”, the transformation rules of ordinary supergravity are given by= ~ix, 6~(e)B=(3)= ~(ø”(A — iy 5B) + F + iy5G)e + ~A(A + iBy5)tj + ~AX(B + iy5A)eplus results for F and G which we will discuss in a moment. The derivatives D~Aand D,.~Barecovariant only with respect to the super-Poincaré algebra, as required in ordinary supergravity, and allaxial auxiliary field terms cancel since ~ = —~(S—iy5P—i,(”y5)and A(,gauge)= —~A(aux).However,the result for 8~,4’is A-dependent. From the structure of ~ it is clear that a redefinition of fields ispossible such that the A-dependence disappears.F’= F—~A(AS+BF), G’= G-~A(BS—AP). (4)The final task is now to show that the variation of these fields F’ and G’ is again A-independent. Onehas= EØ’x + (1 — A)(Er~+ EX)x — E(S — iy5P)~— A i(A — iy~B)y. RCOV. (5)SinceD,.CX = D,.’~— ~ç4B + iy5~4A)çfr,.+ (~-— ~)i~,.iy5x — A(A + iy5B)çb,.and (subsection 4.2, eq. (10))(6)one finds that F’ is indeed A-independent, provided one uses also F’,finds for F’ and G’G’ on the right hand side. One= — ~ ,~auxy )~ + Eq~, 8~(e)G’= jgy5(ØP — ! — iiy5~. (7)These are the results for a local scalar multiplet in ordinary supergravity [527]. (In the literature oneoften omits the primes on F and G, but we use them to distinguish F from F’, see (4).) If one wouldhave known a priori that elimination of A by redefinition of fields was possible at all, then one couldhave obtained these results straight away by merely putting A = 0. The action is (cf. p. 307)I = d 4x [eF’+ ~ei/i y~+ ~efr,.o~’(A— Iy 5B)çls~,+ e(SA + PB)]. (8)

310 P. van Nieuwenhuizen, SupergravitvOne can apply these considerations for example to the kinetic multiplet T(I). Since its firstcomponent is the same in Poincaré as in conformal supergravity, one expects that ECA does not containb,., since b,. plays no role in Poincaré supergravity. Indeed, b,. cancels in ~CAThe tensor calculus for supergravity was developed by Ferrara and the author [526,527, 531] and byStelle and West [583,584, 585, 445] as well as others [297,167, 534]. There are much more details of thetensor calculus, but since an exhaustive review is available [539]we refer the reader to that source.5. Superspace5.1. Introduction to superspaceOne way to describe supergravity is to consider it as just another gauge theory in ordinary spacetime.The local symmetry between bosons and fermions can even be formulated without anticommutingparameters (for example, 84 = A~and ÔA” = I(A — iy 5B)~)and invariance of the action can then stillbe investigated; it has, in this case, an open index 13. Such a theory can be quantized in the usual wayand one can compute S-matrix elements. This is the approach we have been following up to now, and itleads to all results in which one is interested.There is, however, another approach to supergravity theory, and that is superspace. Actually, thereare many approaches, and we will discuss them in this order: (i) the approach which builds a bridgebetween the ordinary space and superspace, (ii) the Wess—Zumino approach, which is geometrical buthas too many field components in superspace and needs therefore constraints on the torsion from theoutside imposed. These constraints also serve to eliminate fields with spins exceeding two, (iii) the chiralsuperspace approach of Ogievetski, Sokatchev, Siegel, Gates and others, which is economical in that itdeals with two small chirally related superspaces but whose geometry is less usual, and might needconstraints for N> 1, (iv) gauge supersymmetry of Arnowitt and Nath which was the first superspaceapproach, but which contains ghosts and higher spin fields.A major advantage of superspace approaches is their application to quantum gravity, in particular toexplicit calculations. In the ordinary space approach one has gauged away many of the fields present inthe superspace approach, and consequently one cannot use globally supersymmetric gauges. Also, in theordinary space approach one cannot use a supersymmetnc background field formalism, in which theeffective quantum action is invariant under locally supersymmetric background field transformations. Inthe superspace approaches, these goals can be met. Finally, superspace methods allow the use ofsupergraphs, which contain many ordinary Feynman graphs, and are easier to use than ordinaryFeynman diagrams. However, as with any new formalism, many new problems had to be solved beforeit could be used which explains that up to now the ordinary space approach has yielded all results first.Another advantage of superspace methods is that it explains many results of ordinary space in asimple way. To mention a typical example, if one wants to know how a scalar multiplet with externalLorentz indices transforms, one can obtain the result directly in ordinary space [224], but a methodbased on the use of covariant derivatives in superspace gives more insight. However, if one needsexplicit formulae in terms of component fields, for example to check whether there exist supersymmetricextensions of topological invariants [534], then, in the author’s opinion, ordinary space methodsare more appropriate.We now discuss general properties of superspace in the remainder of this subsection.

P. van Nieuwenhuizen, Supergravity 311Consider a base manifold consisting of four bosonic coordinates x~and 4N fermionic coordinates O~where a = 1, 4 and i = 1, N. The index N refers to N-extended supergravity. For simplicity (and sincefor N> 1 only partial results are known) we restrict ourselves to N = 1. As in general relativity, oneerects at each point (xv, O~~) a local tangent frame. In the base manifold one considers generalcoordinate transformations with parameters E’ t where A = (JL, a). As we shall see, E’~generate theordinary general coordinate transformations but ~ generate local supersymmetry transformations! Thedifference between the various superspace approaches is often that a different symmetry group is chosenin tangent space. For example, in the theory of Arnowitt and Nath the tangent group is Osp(3, 1/4N),while in most other approaches the tangent group is the purely bosonic Lorentz group times unitary (ororthogonal) transformations acting on the index i, hence 0(3, 1) x SU(N).There are two geometrical superfields in superspace, namely the supervielbein (viel = many inGerman) and the superconnection. (In the chiral approach, both are functions of an axial vectorsuperfield. In the Wess—Zumino approach, the superconnections can be expressed in terms of thesupervielbein after imposing constraints, see subsection 5. All these approaches use thus second orderformalism.) In addition, there may or may not be matter superfields. The supervielbein VA~’(x,0)depends on all 8 coordinates x~,0’~,and the curved (world) index A runs over A = (u, a), while the flatindex A runs over A = (m, a). As the indices indicate, V,.tm, and vaa are bosonic objects, and V,.” andV~mare fermionic. That is to say, they are even and odd elements of a Grassmann algebra, respectively.The superconnection hAm~(x,0) is bosonic for A = ~uand fermionic for A = a. The two flat indices mnrefer to the Lorentz group. For N> 1 one has also an 0(N) connection. It follows that hAm~isantisymmetric in (mn).The base manifold and tangent manifold symmetries act on these two geometrical fields in astraightforward generalization of general relativity. Under general coordinate transformations G withparameters .E” (x, 0) one has= (8nE~r)V,~A+ .E~7r8ITVAA= (8AE~)h,,.m~+ S~8,rh,~”(1)where 8.,~ = 8/0x’~ is a left derivative. Contractions are defined without extra signs if the upper index ison the left. Each contraction is thus a sum over four bosonic and four (or 4N) fermionic indices, and 8Astands for (8/OxTM, 8/OOa). A scalar density D transforms as SD = 8A (EJID)(_)A as one easily checks forsdet V.The local Lorentz rotations L in the tangent frames with parameters L’3(x, 0) act on the two basicsuperfields asSL(L) VAtm =Lm~V;= ~(Lm~mfly1bVfl?~ (2)~ Ir~i. mn__~ tmnJL mryn~L aiim~.‘LI.. Jftfl — ~UflJ_~ nA 1 “A ~ rTensors with indices A upstairs and indices A downstairs transform per definition such that TA TA andT’~T4 are scalars. Thus if STA = LABTB and I5TA = LABTB, then LAB = _LBA. Using our contractionrule we rewrite this as STA = — TBLBA. The matrices LAB are diagonal (have no Bose—Fermi parts) and

312 P. van Nieuwenhuizen. SupergravilyLAB = ~Lmn(X~flY’Bacton tensors as in (2). One can noweasily show that SAB is an invariant tensor. (Sincethe parameters Lrs(x, 0) are bosonic it does not matter whether one writes them in front or in the back. Inthe theory of gauge supersymmetry with tangent group Osp(3, 1/4N)thereare fermionic parameters, whichleads to complicated extra signs.)There are 8 x 8 x 16 ordinary fields in VAA (since expansion in 0 yields sixteen components) and8 x 6 x 16 ordinary fields in hAm~.The number of local symmetries is 8 X 16 (for ~) plus 6 x 16 (forA ~fl)~ These numbers are much larger than for the ordinary space approach to supergravity, where onehas 22+ 16 fields (or 22 + 16 + 4 x 6 fields if W,.m~is an independent field) and 4+4 + 6 local symmetries.The question now arises what the relation between the superspace approach and the ordinary spaceapproach is. As we shall see, if one chooses a certain gauge in superspace in which to order 00V,.m = e,.m, v,.’~= ~ h,.ma = W,.mi~(e,~‘) (3)and if one imposes in addition constraints on the supertorsions by hand (i.e., from the outside; they arenot field equations), then one recovers the ordinary space approach.Let us conclude this introduction by defining torsion and curvature. The inverse of the supervielbeinis defined by~z A11 B — ~ BVA VA —11 = SAH. However,As a consequence, also VAB VB~r H~j B —‘ \(B+I1XA±B)T, B11 TIVB VA I~) VA VBwhere in the exponent H = 0 for bosonic values and 11 = 1 for fermionic values.Covariant derivatives are defined byDA =8A + ~hAmu1Xmfl,DA = VAD.4 (6)where Xmn are the Lorentz generators. These derivatives are covanant with respect to local Lorentzrotations, but only with respect to general coordinate transformations if one considers curls. They arethe kind of derivatives first introduced by Cartan, and one can introduce p-forms in superspace whichavoid any extra minus signs. We prefer to exhibit indices explicitly for pedagogical reasons.It should be stressed that these Cartan covariant derivatives are not the same as the covariantderivatives which one sometimes introduces in gauging groups. For example, in gauging the superPoincaré algebra, we defined in section 3= (D,.E)A O,. A + h,.BeCfcB~~ (7)and one sums over all connections h,.B, not only over the Lorentz connections. Thus, in this way ofgauging groups there is double counting: translations appear both in the tangent group and in the basemanifold (as general coordinate transformations).Torsion and curvature are defined by[DA, DB} = RABm~X~fl) — ~ (8)

P. van Nieuwenhuizen, Supergravily 313(The factor 1/2 is due to our summing over m > n and m

(A V,.”)00 = ~ s*a = (D,. + ~ A,.ym) —314 P. van Nieuwenhuizen. Supergravitywith the ordinary space transformation= 8,.re~”+ ~r8,.m + ~fiytmf// + A ~ (3)In order to do so, one identifies the parameters as followsL””’(O=O)=Am”. (4)Compatibility for the terms with ~“ and A tmfl is manifest, but for the terms in “ one finds(8,. ~1)Vam(0= 0) + ~a (8aV,im ~ = ytm~f; (5)Thus, V~m(0 = 0) = 0 and V,.tm = e,.m + ~ëy”i~j,. + . One must thus choosea gauge such that these resultshold. Whether this can be done in all order 0 cases considered below, has not been proven but seemsplausible.Repeating the same procedure for the gravitino, we have (see page 217)(AV,.i)= (8,.Er)V,~~+(8~E”)Va°+Er8rV,.a +Ea8aV,.ui +~(L. ~7)abIIIb(6)In this case, 8(sup)*,. must be equal to the 0 = 0 terms in (8,.Ea)V,.~~+ E~8V,.~ and one findsva”(0 = 0) = ~a ~ = ~a + ~ — ~,. )o” +... (7)where m~= —~(S+ iy5F — 2i,.4y5) and ~uses ,~and ~i instead of ~ and w)is the improved spin connection (the algebra simplifies if onema — ma( ,\_ mnp~~ ‘8(O~ —Ui,. l~C~I~I~J3 ~L p~m~to order 0, and finds hatm “ (0 = 0)= 0. The gauges VaA(0 = 0) = h~~(oSimilarlyone can find h,. = 0)= 0can always be implemented by choosing the order 0 parts of EA and A ma approximately.Since Va”(O = 0) = 6,,” and V,,m(0 = 0) = h,,m~(0= 0) = 0, one can require also compatibility for thesecomponents. One finds then to order 0 =0= 0 = (0a~”)~II,.” + (8,,E~)6fl°+ r8,.&r,a + ~8~V,,”+~Atmh2(OmflY~b6,,b. (9)Clearly one needs to know first the parameters to order 0, and then one can determine Vaa to order 0from (9).The superparameters to order 0’~’follow from the knowledge of the superparameters to order0k byrequiring compatibility of the parameter composition rules in ordinary space and in superspace. Insuperspace a symmetry operation with (E2A L2 m~)followed by an operation with (,E 1A, L1 tm~)minus14*2 yields again a symmetry operation with composite parameters

P. van Nieuwenhuizen, Supergravity 315fl_(~A 8 ~~II+~Z~II\_(14*2—12 ~2 A1 12 / I.L12 m~= (~A8Lmn + L ma) — (1 2). (10)2rn(L17a + S1L2Since we work in a special gauge, the superparameters will in general depend on em,., cu,.”, etc., and onemust take the variation of these fields as dictated by the ordinary space approach. This is indicated bythe symbol Si. In ordinary space the parameter composition law is as discussed in section 1 (pages 219, 220)= (~2r8,41TM+ ~e2y~ ’)— (14*2)~12 = (~2”8r l”+~A2mut(O.maEir— ~ 2y 1çb,. ) —(14*2) (11)A12 = (e2~8~Ama + A 2 mA ~ + AE2YTM 1Ui,. + ~j 2o.m~~(5 — iy5P)e1) —(14*2).Thus we need in ordinary space a closed algebra, and since this has only been achieved in second orderformalism, we use u = co(e, i/i).As a result one findsm11, theand superparameters solving these equations. to orderUsing 0 bythese equating order~12~(00 parameters, = 0) = ~12~(0 one can = 0) obtain andsimilarly the remaining for E12~and order 0L12 components of the supervielbein. The results are given in the following table:TableEM = ~ + E” = “ + ~iy”O,~r,,”— kA . ~~orL’”” = A’”” +vA_fe M+2Oy~—‘ 4~(&Y’”)rEv~A~’”” + lu”(S — iy~P)]~‘+l(c~,.o.—,3y,,)9In order to obtain parameters to order 02, one evaluates [42, 4i} on, say, the field V,.tm, and in ~24i V,.tmthe variation 42 not only acts on V,.tm as dictated by superspace but also on the fields of ordinary spaceaccording to 82. It is essential to consider always the full groups in superspace and ordinary space. Forexample, [A(E”), A(Lm’”)] is again a Lorentz rotation in superspace, but in ordinary space [Sfr~),5(A ma)] is a supersymmetry transformation.The general procedure is to obtain first the parameters to order 0~’from those to order 0”. ThenV,.” and h,.tm~ to order0k+1 follow from the superparameters to order 0” and V,,A and h,, m~followfrom the superparameters to order g1c+1~It is remarkable that in solving for the superparameters onefinds differential equations whose equations must satisfy certain fermionic integrability conditions. Theyalways do. Also, the superparameter solutions are not unique but contain arbitrary integrationconstants. For example, solving28a,~I — 1 8,,,~ = 2~2YTM~1 (12)the general solution is.ETM(0) = — ~iy~0+ ëOH,. + iy5OK,. + ~15~0L (13)with arbitrary H, K and L. We will discuss below an example of conformal superspace supergravity

316 P. van Nieuwenhuizen. Supergravitvwhere these integration constants are nonzero and become fixed at the next 0-level. In most cases,however, they are zero.Suppose one has obtained the supervielbein to order 0, then one can construct the supertorsion toorder 0 = 0. Similarly, knowing the superconnection to order 0, one can find the supercurvatures toorder 0 =0. One can now find tensor relations between supertorsion components to all order in 0 bysimply first determining their relations to order 0 = 0 and then using that such relations must hold to allorder in 0 due to the following theorem.Theorem: a tensor which vanishes at 0 = 0 vanishes for all 0.The proof is simple: in the transformation rule of the tensor all terms vanish, except the fermionictransport term ( “8,,times the tensor). Thus, by induction, the tensor vanishes at all order in 0. Physicallythis important theorem is clear: the origin in one coordinate system is not the origin in another, and byrequiring that a tensor vanishes at the origin of any coordinate system, it vanishes everywhere.It turns out that (to order 0 =0 in any case) the supertorsion and supercurvature are independent ofthe integration constants. This is to be expected, but a proof is lacking. If one chooses w,. tm” =w,.mn(e, ~,) as in second order formalism, one finds, first to order 0 = 0 and thus to all order in 0, thefollowing tensor relationsTabC = Tar~’= Trs’ = Tab~+~(CyT)ah=0. (14)These are the constraints Wess and Zumino postulated for N = 1 superspace supergravity. If one usesinstead the improved spin connection ~mn one finds the constraints of Ogievetski and Sokatchev. Ifone were to be diligent, one could compute V,~Ato all orders in 0 (i.e., to order 0~),then obtain sdet V.and then obtain from the action f d~xd40 sdet V the ordinary space action (by taking the O~componentof sdet V). This action should then agree with the action of the ordinary space approach. (Incidentally,one can directly obtain sdet V to all orders in 0, since sdet V is also a tensor. Similarly one candetermine the supertorsions and supercurvatures to higher order in 0 once they are known to order0 = 0, without having to determine V~,since also these geometrical objects are tensors in superspace.)In this approach we have translated results of the ordinary space approach into results for thesuperspace approach (one could also go the other way). Crucial was the identification of E” (0 = 0) withthe field-independent parameters (~, e’~’).In other approaches [578] one sometimes chooses E~4=VAASA as field-independent. We have obtained the constraints of N = 1 superspace supergravity,assuming that the gauge algebra in ordinary space closes, i.e., with the presence of the auxiliary fields.For N = 2 the constraints were in a similar way found from the auxiliary fields for N = 2 supergravity.Some of them readT — T — TCk — TCk —1rs — 1a1.s 1 Ai,Ej — Ai,Bk —but at this moment it seems that constraints quadratic in T’s appear [555].For conformal supergravity in superspace, one can play the same game [549]. In superspace onehas now, in addition to the spacetime symmetries 4~(E)and ztL(L), two more symmetries, namely localchiral rotations 4A(AA) and local scale transformations 4D(AD). By requiring compatibility of the gaugealgebras in ordinary space and superspace and identifying AA(0 =0) = AA and AD(0 =0) = AD, one findsalso AA and AD to order 0. One must keep here integration constants proportional to &~.One finds thatthe S-supersymmetry parameter ~ of the ordinary space approach appears at the order 0 level in AA

P. van Nieuwenhuizen, Supergravity 317mand AD, butdoes not seem to appear in the superparameters. The~KAD such that for these terms AD ± (i/2)AA are chiral, i.e.,~ parameters appear in AA and(1 ~ ~ ~_(AD ±_AA) =0. (16)Requiring that this relation is generally true at the order 0-level (and not only for the ~ parameters)one finds compatibility for A V,.tm and Setm,. if one defines A(AA, AD) VAtm = AD V~mCompatibility forAV,.~and &/J,.L° as far as the ~çterms are concerned, can only be achieved if one defines chiral rotationsbyA(AA, AD)V.4” = l/~m7~ + Ys)2~(_AD ~ AA) + (~AAYs ~AD) VA”+ V4mymi(1_yS)~(_AD+!AA). (17)These transformations are just the transformations which Howe and Tucker [498]found in superspaceby requiring that they leave the Wess—Zumino constraints in (14) invariant (provided one replaces 8/00,,by Do). Thus the kinematics of N = 1 superspace supergravity is even conformal invariant, but thedynamics may or may not be, depending on whether one takes an action which is invariant under theseHowe—Tucker transformations or not. For the superconnection one finds analogous resultsm~= ~VA” V~118TIAD— m ~ n)+ VA ~CUmai(1 + Ys)-~(—AD — ~ AA) + c.c. (18)A(AD, A4hAFor the superdeterminant one finds with S sdet V = sdet V( VA’tSv~A)( — )A that 5(sdet V) =—2AD(sdet V).Since torsions and curvatures can be expressed in terms of V m~,one can also determinehow they transform under A(AA, AD). In particular, since, as we 4~”and shall see, h4 all torsions and curvaturesare functions of three arbitrary superfields R, G~ and WABC only, one can find how these threesuperfields transform. One finds*SR = (21* —41)R —~b(1— y~)DI*5Gm = _(I+I*)Gm +iDm(I* —I)SWABC = —3.EWABC, I = — AD — AA. (19)The superfields R, G~,WABC are the superspace equivalents of three similar multiplets in theordinary space approach, and are discussed further in subsection 5.A slightly different approach is due to Lindström and Roéek [365,367], who constructed ~ Stand (DADA + 4R) to all orders in 0 in terms of chiral coordinates by translating the kinetic multiplet ofordinary space discussed in subsection 4.4 into superspace.“Note that DAR = DA.X = DA(DMDA — 8R)U = 0 for any U.

318 P. van Nieuwenhuizen, Supergravity5.3. Flat superspace*Supertorsion and supercurvatures are defined by the (anti) commutator of two covariant derivativeswith flat indices. In global supersymmetry one can also define covariant derivatives, and from their(anti) commutators one finds that global supersymmetry has torsion but no curvature. This flat space limitof curved superspace was obtained in the last subsection by using gauge completion; in particular wefound that Va tm = ~ Here we will obtain the same result by deriving the covariant derivatives inglobal supersymmetry and by using the method of induced representations, following Salam andStrathdee.Consider a group element of the formgL(x, 0, i~)= exp(~Q+ xmP~+ ~maMmn) (1)Consider next a new group element, obtained by left multiplicationgL(x, 0’,~‘)= exp(~Q+ ~mPm ±~Amt2M~fl)g~(x, 0, ‘q). (2)Using the Baker—Hausdorfi formula, one finds an induced representation of the generators F, 0, M onthe coordinates x, 0, ~ of the group manifold, which is 14-dimensional and infinitely nonlinear.To find an induced representation on superspace with coordinates xm, 0” we consider the cosetspacesexp(~Q+ XtmPm) mod M. (3)By mod M we mean exp ~Atm”Mmn with any Am~Any group element of the form (1) belongs to a coset.To see this explicitly, note that, using= e_ RJ/2+ eAeB (4)where A = 00 + x P and B = . M, we can factor the M dependence out. This is possible, since inthe multiple commutators [A,B],B]... one always encounters only P and 0 while [M,M} -~M (Inother words, M ~ (P + 0) is a semidirect sum. The decomposition (M + QL)~ (P + OR) yields asymmetric algebra.) We now considergL(x,0)=exp( Q+~P+2A .M)exp(O. Q+x .P) modM (5)with antihermitian 0, P. M. Usingt2~ if A””’” =0 (6)eAeB =e~~~”eAeB =e~j~~eA jf~_~:_Ø (7)* I thank Dr. Ro~ekfor useful discussions concerning this subsection.t Eq. (6) is the usual form of the Baker—Hausdorif theorem. Eq. (7) follows if one expands eAeBe_A = exp(eAB e~4)in terms of B. Eq. (4)follows from (6) if one uses (6) twice.

P. van Nienwenhuizen, Supergravizy 319one finds the following induced representation of the generators P, 0, M on the coordinates x, 0~L( \~a — a ~L( \ m _3.— mUQk )U — uot,Ejx —4 )’S~(A)0”~S~)xm=~tm (8)5~(A)xm=A m~f.We have used the anticommutator {Qc~, Q~}= ~(ymC_I)aIsp Note that (8) is an 8-dimensionalnonlinear transitive representation of the supersymmetry algebra on (xm, 0”). The stability subgroup ofany fixed point (xt~,0~)of this representation space .is isomorphic to the Lorentz generators ~ (asone easily sees by considering the origin (0,0)). Consequently, the representation space x, 0 isisomorphic tothe coset space G/M P + 0. This coset approach forces one to the Lorentz group as tangent group insupergravity.Define now a scalar superfield 4(x, 0) byçf.i’(x’, 0!) = 4i(x, 0). (9)Let ~‘(x, 0) be related to ~(x,0) by a unitary transformation with operators Q~,pL ML which at thisstage have nothing to do with 0, P, M. Then, infinitesimally, the commutator= [ëQL + ~tmP~+ MmnM~~,cb(x, 0)] (10)is, according to (9), equal to= —(SO” 8/80~+ Sxm 8/Oxm)çb. (11)Substituting (8), one finds= [~QL ~j = _e(~+~7moa~)~ =~LA_F~mDL .L1_.up(p—[ç Em,W1 ç um8 8 (12)= ftA””’”M~,~b]= _(~Ama(o.~~o)a-+A m~x’”Before interpreting these results, we repeat the preceding analysis, but now using right multiplication.ThusgR(x, 0’) = (mod M) exp(ö~Q+x P) exp(—eQ — . P — ~A . M). (13)Now one finds= +~iym0=0, S~)xm= (14)= +~A. 8~(A)xm=

320 P. van Nieuwenhuizen. Supergravin~which again constitutes a representation of the supersymmetry algebra on (x””, 0”). Finally, with (9) andintroducing O’~,p~ M’~as in (10) one findsSR~(X0) = [~OR+ ~ + ~A””’”M~,P,.s~f(x,0)]= [~QR ~] = +~(-~._~m9~)~ = —( ~G)4~ ~(15)= ~ ~ — (~A. o,O~+A m~x’”As one may verify, the operators0R, pR MR satisfy the super-Poincaré algebra. In particular{G”, G~}= ~(,a’C’)”~= ~ (16)Also the operators in (12) form a representation of the supersymmetry algebra on fields. In particular{D”, D~}= ._J(ymc~-l)a~i3 = +~(ymC_I)a$p~,. (17)In fact, D” and G~anticommute, as one may easily verify directly. This is no accident but due to thefact that the induced representations on the coset space are associative[(exp ~O)g] exp ëRO = exp eLoEg(exp jRQ)], (18)Hence on fields(exp ERG) (exp — ELD) 4(exp ELD) (exp — ERG) = (same with D G) (19)from which it indeed follows that {D”, G’~}= 0.The interpretation of these results is the following. G generates supersymmetry transformations.(Note that on ordinary fields S~A= [A, P,.] = 8,,A.) For example, on asuperfield V(x, 0) = C + OZ + ~00Hetc. one has from 8( )V= EGV, upon equating the coefficients of definite powers of 0SC = EZ, SZ = (20)[S(e,), 5( 2)]C= ~E2y”” 13,,,C (21)and one has indeed a representation of the algebra of global supersymmetry on the components ofV(x, 0). Since D” commutes with G~,we identify D” with the spinor part of the covariant derivative.Indeed, covariant derivatives are per definition derivatives such that, for any tensor T, D”T transformsas T, and this means that D” TM. must (There commute are better with but G~.For morethe complicated vector part arguments.) of the covariant derivativeone easily finds that D,. = 8/ôx

P. van Nieuwenhuizen. Supergravity 321The connection with the covariant derivatives D..~of curved superspace is now easily establishedDA = (~4~ + ~(~/~a, ~D 4 =(~,~ (22)Note that we consider (x, 0) as “cartesian” coordinates for which no connection is needed in the basemanifold. (Even for rigid translations in curvilinear coordinates one needs connections.) The flatsupervielbein reads thusVAA(X,0)=(l(~m:) 30). (23)Since supervielbeins can be considered as determining the orientations of local superirames withrespect to the base manifold, this result states that a particular orientation (which is supercoordinatedependent) leads to derivatives DA which are very convenient (they commute with supersymmetry).This is indeed the flat limit of the supervielbein as obtained by gauge completion. This flat supervielbeinand the vanishing spin connection are a solution of the fields equations of supergravity (but not of thoseof gauge supersymmetry at the classical level). If one computes the supertorsion and supercurvatureaccording to the formulae in subsection 1, then one finds that there is a nonvanishing component ofsupertorsion (see page 316)tm = _~(Cym)ab. (24)TabThus, flat supersymmetry has torsion but no curvature.5.4. Construction of multiplets using the closed gauge algebraWe show here how one can use superspace methods to extend a tensor in ordinary space withexternal Lorentz or internal indices to a full multiplet of which this tensor is the first component. Forexample, in the supersymmetric Maxwell—Einstein system there is a scalar multiplet I,, which starts withA,,, but the other components do not rotate into each other in exactly the same way as the componentsof the canonical multiplet I = (A, B, x~F’, G’) of subsection 4.6. This one might already expect fromthe fact that the commutator of two local supersymmetry variations contains a local Lorentz rotationwhich does not act on A but does act on A,,. Hence local supersymmetry transformations for I~must bemodified.The method is based on the known anticommutation relations of the covariant derivatives DA. Sinceto order 0 = 0 in global supersymmetry D” is equal to the supersymmetry generator G” (see theprevious subsection) while in local supersymmetry this is still true for superfield tensors with flatindices, one defines, to begin with5~~~( A(x, ) 0 = 0) = (EC~D)A(x, ~ (1)We use complex notations where A stands for A — iB, and repeat that A may have external indices.Using a Fierz rearrangement (see appendix) one finds for the commutator of two spinorial covariant

322 P. van Nieuwenhuizen, Supergravilvderivatives, not forgetting while Fierzing that Da and Db do not anticommute[Da ”, Dbfl”l = — ~(jym~) (,‘,,,c’- ‘D) + (Eum~~~)(D(TmnC 1D)= C” (2Tab’Dc — Rabtmfl~Xmn)~+ internal symmetries. (2)We recall that Da starts with 8/80*2 and ~ = ~[ym y’”]. From the explicit expressions for thesupertorsions and super constraints of N = 1 superspace supergravity one finds upon equating thecoefficients of EO tm’” = [(_j/12)0y~A, mn3t + ~jc,m~(S— i-y1~(using that ha 5P)],. andm”= (Cu””’”)(S — ~YiP)ab — ~ (Cyr)ai,Asas follows easily from gauge completion) thatD’y~C’D= +Dm — ~ mapqG’”X” (3)Da~~~(1 +75)C’D = ~RX~ with X~= — ~ (4)RSiP+C(0), Gm”Am+C(O) (5)and a similar result with R replaced by R * and the + and — signs interchanged.From now on we consider chiral multiplets and chiral superfields. The superfield A is then restrictedby[(1+ y5)C~D]’~A(x, 0) =0. (6)From (4) one immediately deduces with (5) thatX~2A(x,0) = 0. (7)Since (1— y5)C~Dis not constrained, we define a new superfield X’” which has the same extra indicesasA1D]”A. (8)Xa = [~.(1— 75)CClearly x = ~(1— 75)X. From two superfields A(1) and A(2) one can also construct the productA(12)=A(1)A(2). Then (8) leads to X”(12)=A(1)X”(2)+A(2)X”(l). This is the beginning of theproduct formula of two multiplets.We now apply (3) to A. Since A is chiral, (3) yields with (7)D’ym (1 ‘Ys)X = +DmA — ~ (X~A)G”. (9)

Acting with the complex conjugate of (4) on A yieldsP. van Nieuwenhuizen, Supergravity 323Dt~mn~(1— Ys)x = — ~R*X(L)A (10)From (9) and (10) one knows how Da acts on y m~,YmYsX and ~define a new superfield F as followsThus, since x is chiral, we canF(x, O)= +D~(1—ys)~ (11)where the complex F(x, 0 = 0) corresponds to F — iG of the multiplet I of Poincaré supergravity. Forthe product rule we now find2). (12)F(12) = A(1) F(2) + A(2) F(1) — ~(1) (1 — y~) x(We can now determine how X” transforms by using (1). Combining (9), (10) and the definition in (11),we get— ‘ys)X = ~(DmA — ~ GnX~A)ym~(1+ y~)+ ~(F + R *x(L)ci.mn) ~(1— ~ (13)One sees here clearly that if A has extra Lorentz indices, then Ssup( )X, as given by the 0 =0 part of(13), is modified by terms proportional to LQrentz rotations on A.Next we determine how F transforms. As before we apply (3) and the complex conjugate of (4) to— Vs)X. We do this by acting on (13) with Dytm and D~mn~(1— 75) respectively and find(DytmC~D)~(1— ys)X = [Dytm(—y’”D~A+ . . . F)~(1— y5)(~C’)]T (14)(Do-”””~(1± y 1)]~. (15)5)C’D)~(1— Ys)X = [Do.m~(1± y~)(yD~A+ . . .) (—~c-Using on the left-hand side (3) and (4), one finds that DaF is fully determined (already by (14) alone,(15) yields a consistency test). The only unknown at this point is the double derivative on A, which,however, can also be obtained using thatDaDmA = DmDaA + [Da, D~]A. (16)From FDa, D~]= 2 Tam”Db + Ram~Xpq one sees that in the transformation law of x a term enters,proportional to the gravitino curl when A has external Lorentz indices.In this way we have a multiplet of superfields which transform into each other under the action ofD~.The correspondence with the multiplets in ordinary spacetime is simply that the order 0 = 0components of the superfields are the components while the action of D,, is a supersymmetry

324 P. van Nieuwenhuizen. Supergravitytransformation. Only if A has no external indices, one can see that F* + RA again satisfies the samechiral constraint as A does, namely eq. (6). Hence, it can be viewed as the lowest component of a newmultiplet, namely the kinetic multiplet.Similar ideas can be used for vector multiplets (non-chiral superfields A) or for linear multipletscorresponding to superfields and satisfyingDC 1DA = (Dy 1D)A 0. (17)5C~D)A= (D~~mnCFor applications of these ideas to supergravity see P. Breitenlohner [68,71].5.5. The Wess—Zumino approach to superspaceIn this approach, the tangent group is the ordinary Lorentz group. The geometry is thus: generalsupercoordinate transformation in the base manifold, local Lorentz rotations in the tangent manifold,and one has supervielbeins VAA and superconnection hA’”. These superfields are not arbitrary, butsatisfy the following constraints (as we deduced earlier from the ordinary spacetime approach but nowpostulate)pt_~r:_pc_pm~Yc’m\ —1rs — 1as — 1ab — 1ab ~4~’-~Y)ab —We now discuss how one might arrive at postulating them. First of all, notice [605]that for a tensor8A(sdet VwAVAA)(_)A = sdet V(DAw~ô+2wBTBAA)(_Y. (2)To prove this relation, note that 5(sdet V) = (sdet V) (5 VAA) VA’~(—)Atransformationso that for a general coordinate6(sdet V) = 8.,~(~A sdet V)(~)A (3)This expression can be written as11 VIJA)VA.4( )A(SV/~)VAA(_ )A =(Sfo11V1A + 8A5= 8A(SVfJ)VA’(~) +EH(8mV4A_(_)~~8nV,~(*)VA1t(_)1t. (4)LOne can replace 8A by DA in both terms, while the extra terms needed in replacing Oj~by D~cancel~nce (Xmn)AB is traceless in (A.B). Identifying wA with 511 Vil4 the result in (2) follows.Hence in order that one has just as in general relativityJ d4xd4Osdet VDAwA(_)~=0 (5)oj~ehas the requirement that TBAA( — )A = 0. For B = b this follows from the constraints, for B = m it istIue jf Tma” = 0. This is indeed the case and follows from the Bianchi identities [578]if one assumes theconstraints on the torsion. We will discuss this below.Another interpretation is that chiral superfields can live in superspace [224,421,452, 453]. Indeed, if

P. van Nieuwenhuizen, Supergravity 325(1 — ys)a,,J~b4,(x 0) = DA~(X, 0) = 0 (6)then also {DA, DB}4 = 0. From the definition of torsion and curvature this implies‘7’ C ‘1’1AB ——The first constraint” is part of T~= 0, the second follows from Ti,,,’ + ~(Cy’)~b= 0. Actually, ifDA~CD= 0 one also finds R.i,~m’”= 0 and also this constraint follows from the torsion constraints as aconsequence of the Bianchi identities. Thus the set of constraints imposed on the supertorsion seemsreasonable, but they are postulated, not derived from some general principle.The Bianchi identities follow from[DA, DB}, D~}(— )AC + 2 cyclic terms = 0 (8)by taking separately the terms proportional to DE or DE. Thus one findsin,, ,‘,‘r D ,nn —A BC TLi4~~ DC —DA TBCD + 2TABETEc~)+ ~RABCD = 0 (9)where one must add terms cyclic in ABC with the proper signs. The symbol RABc’~DD is RABm’”Dn ifDD is bosonic and RABm’”DD( kmn)°cif DD is fermionic. The minus signs are due to the fact that theindex D of DD is a lower index.As an example, consider the following identity[{DA, DB}, D~]+ [{DB, D~},DA] + [{D~, DA}, Dfi] = 0 (10)where A,B, C are 2-component spinor indices (see appendix). The terms proportional to DD yield with(C-2’ ),~— 0RABmn~[X~fl,D!~.]+ [(Cym)BCT~AI) + A 4*B]DD. (11)Since [Xmn, Da] =OmnY’a1)t, one has in 2-component notationMoreover[Xmn, DE] = ~(0m, ~ — m ‘-~ n)DF. (12)D mn — I m,EF n.FF~ .P~AB — 4O~RAB,EE.FE = EERABEF + EFRAB,EF. (13)We arrive at (symmetric parts are underlined)~ -= TBC,A.O + TAeBD. (14)Decomposing the torsion as

326 P. van Nieuwenhuizen, SupergravityTDGOA = T~j~+ TDO GA+ T~ DO+ oO GA(2iR) (15)and substituting this back and multiplying by ~AO~AG one finds TDO = 0. Similar manipulations lead toTGA = T~~j= 0. Finally, with D~in (10) replaced by Dm, a similar analysis yields~ =~(EEAEDO + DA EO)RREDGA=0. (16)This example shows how to use Bianchi identities; much work in superspace is based on deducingconsequences of Bianchi identities.After working out all Bianchi identities, one arrives at the following remarkable result [578]: allsupertorsions and supercurvatures can be expressed in terms of three superfieldsR, G~, WABC.Moreover G~.is Hermitean, WABC totally symmetric while1-’A1~— o — J—’A rs . ui ‘TBCD — g~ U, ~-‘ y-~Aç~ ~—‘AA . — o *DAW~C=DBEG~+B4*C(17)The conditions with DA state that R and WABC are chiral superfields; the last condition shows thatdifferentiating ~ (so that for example the curl of A,. enters without 0 factor) yields the same resultas differentiating G~.(Indeed, G,. (x, 0 = 0) = A,. and again its curl is produced.) The second relationhas a similar meaning; it corresponds in ordinary space-time to the fact that the trace of the Einsteintensor is proportional to the scalar curvature.The formulas expressing TA~and RABtm” in terms of R, G~and WABC are complicated and willnot be reproduced here.In Einstein theory the geometry is specified by stating that it is Riemannian. In superspacesupergravity we defined the geometry by the restriction that the tangent space parameters are theLorentz parameters (to appreciate how drastic this choice is, see subsection 7)n_rm b_1rmn~ ~a a— m_— ~ fl, ~a — 21~~ (9mn) b~ Em — Ca —and by the torsion constraints.Since the tangent group is degenerate the constraints in (1) on the supertorsion lead to two results(i) all spin connections can be expressed in terms of supervielbeins. Thus enough components ofare given to solve the torsion equation. Specifically, from Tr,’ = 0 one obtains hrs’, just as ingeneral relativity, while from T~.,j= 0 one finds h~,.,j(and from ~ = 0 one finds h,~,j).From thesix constraints ~ + TACB = 0 one finds the six ~ using that hBAC = h~ = 0.(ii) The other constraints are therefore constraints on the supervielbein. (In general relativity such aresult is not present since there the tangent group is non-degenerate.) In particular one can express V,,,’5in terms of Va’s. The V,,A can in turn be expressed into prepotentials [273].We now turn to dynamics. The action of N = 1 superspace supergravity is simply [574]1Id4~405~tV.(19)

P. van Nieuwenhuizen, Supergravity 327Let us see what its field equations are. From sdet V = expstr In V one has5I=Jd4xd4O(VA~~5V 4A)(_)*2sdetV. (20)but the variations SVA~must respect the torsion constraints. From the definition of TBCA given earlier itis straightforward to derive (using the index convention of page 313)STBCA = !DHA — 1( — )BCD HA + TBCDHDA H°TDCA + ( )BCHD TDBA + QBC — ( )13C1)A1BCwhere HAB = VAASVAB while ‘= VBA&,bACAi ‘~A(B±11)8AVfJ +U) VII ~PAB1jB~ A (22LJ4Vfl —11 Ti II .~VA — ~14 VA — (PAAThe signs follow by first making the purely bosonic sector agree, and then moving indices such that theyappear on the left and right hand side in the same order. As a consequence, the Leibniz rule8’5(vAuA) = (8’5vA)uA + (_)AAvA8u (23)(21)holds also for covariant derivatives. Of course the ~spin connections h~m~(compare with (18)).One can solve this equation for HAB and finds thencan be expressed in terms of the earlier definedSJd4xd4osdet V=fd4xd40sdet V[VtmTma” _RU_R*U*1 (24)where vtm and U are arbitrary superfields. Since Tm,,” is proportional to Gm, one has as field equationsGmRO. (25)These are the N = 1 superspace supergravity field equations. Since, as we shall show presently,R = S + iP +... and G~= A~X~ they agree with the ordinary space approach.To show how the Wess—Zumino approach can reproduce the ordinary space approach we establishthe bridge by requiring compatibility as before [577].For any superfield V (with any number of tangent space indices) a transformation S is defined bysv = ~DA V = ~A VAA 8A V+ ~~!hAmn(~X~,,V). (26)Hence, S is the sum of a general coordinate transformation with field-dependent parameter e” = ~A VAAand a local Lorentz transformation with field-dependent parameter ~AhAmn The advantage of thisdefinition is that the commutator is proportional to only supertorsion and super curvaturer.g.~1~z—~B~r1~B~A(0 mnly IT CL 1, 2J 525T1[ A, BfS2S1~ AB 2 mn AB C (27)

328 P. van Nicuwenhuizen, SupergravitvFor the vielbein itself one has an extra term due to the index A,SV/’ = DA~ + 2VA~TBCA. (28)As in subsection 2, we choose at 0 = 0 a gauge withV,.” = em,., V,.,° 14f a V,, 0 = 5a V,,m = 0. (29)m”(O = 0) agrees withUsing the connection the constraints ordinary on the supergravity. supertorsion, one finds that h~”(0= 0) = 0 while h,.Supersymmetry transformations are now defined by ~tm(0= 0) = 0 and ~0(0 = 0) ~a with °(x)theusual four-component spinorial parameter of ordinary supergravity. These transformations maintain thegauge V,tm = 0 if~ ~m~jC,~b~-’ Pfl_t1aS VaSlbChence with Tbam = — ~(Cym )b and Tb,, = 0 one finds~m(x,0)= ~m(x)+~iym + C(02) (31)precisely as before. On the other hand, the condition ~,,a= 5~is maintained provided the order 0 termin r(x, 0) vanishes. This also coincides with the Caltech approach, since ~a = ~A T~/~a=and e” = —~Ey~Owhile e’”(x, 0) = “ + ~,çb,.”(Ey~0).Let us now see how the tetrad and gravitino at order 0 transform. One finds for the tetrad:5t!~,a+ s,,”~”= ~ + 2e,.~~ l~Tb,,m. (32)m= _~(Cym)~and Tb,,tm = 0 this precisely reproduces the law Setm,. = ~Eym~/J,,.With Forthe theconstraints gravitino one T~ finds an interesting result= (D,. ~’+ 2em,. L~Tbm0. (33)The torsion components Tbm”fields R and Gmcan be expressed by means of the Bianchi identities in terms of the superTB,MM,AE13J~’4MAR‘7’ . . ~ . . ( )I B,MM.A ~‘ EMA BM — .)EBA~..JMM— .)EBM AM~Identifying at 0 = 0 the auxiliary fields by R -~ S + iP and Gm A~IXone finds the result of ordinarysupergravity back= (D~+ ~ A~~ys) + ky,. (S — iy5P — i,475) . (35)Thus the minimal set of auxiliary fields 5, P, Am follows from the superspace approach if one imposes

P. van Nieuwenhuizen, Supergravity 329the constraints on supertorsion. Moreover, the constraints themselves follow if one requires thatsupersymmetry transformations maintain the gauge at 0 = 0, and if one requires that superspacereproduces the transformation rules of ordinary supergravity.5.6. Chiral superspaceThe auxiliary fields of ordinary N = 1 supergravity in ordinary space fit into a real vector superfieldH”(x, 0)H”(x, 0) = C~+ £1Z~+ ~OOfr + ~0iy50e~~ + 60Olfr~+ O0O0A~. (1)50K~+ ~Oiym-yNamely P = a,.ii~,S = 8,.K~and C” = Z” = 0 in a special gauge. (To avoid confusion between thesuperfield H(x, 0) and the ordinary field H”(x), we have put a hat on the latter.) Thus an axial vectorsuperfield seems to have some significance for supergravity. The superspace approaches discussedbefore have the drawback of involving a large number of fields and a large symmetry group (generalcoordinate transformations in superspace in addition to ordinary Lorentz rotations) so that one mustchoose constraints and a particular gauge to establish compatibility with ordinary supergravity. (Forsome purposes, for example for the construction of invariant objects, the larger symmetry may beuseful.) An approach which uses from the start fewer fields and a smaller symmetry group is due toOgievetski and Sokatchev [346—360], and Siegel and Gates [407—428]. (See also Ro~ekand Lindström[368].)They consider two chiral complex superspaces which are related by complex conjugation. Suchspaces were also considered by Brink, Gell-Mann, Ramond and Schwarz [401].The coordinates in eachspace are complex and the symmetry group in each separate space is complex general coordinatetransformations(,. w’\ ~!‘~x L,t’ LI, uX LA—ss’ikXL,L’LJ, ~\ c”.’5’5a1L’t ~XL,UL.LThis approach is typically tailored for four dimensions, since it uses dotted and undotted indices. It isnot clear how to extend it to higher dimensions. The right-handed coordinates do not transformindependently but as the complex conjugate of these rules. One now identifies— Ill L \L1 L~ — 1111 — \tl’t* 11 e’ .i. ~ —(FL — ~ ~ ~5)U, ~R = ~2~1 )1~)v) ,X~L— X”R = 2iH~(x, 0)2~X L ~ X R) — X(3)where H~(x, 0) is a function of X”L + X”R, °L and °R~Thus, the imaginary part of the coordinates XL andXR is interpreted as an axial super field while the real part is identified with true spacetime. Of coursethese transformations form a group (two general coordinate transformations lead again to a generalcoordinate transformation). However, for Einstein supergravity one restricts the group to the supervolume preserving transformations in each of the chiral spaces, i.e., one requires that both superJacobians equal unity (see appendix),8X”L 0L)—~—A”(xL,OL)= 0. (4)Since super Jacobians satisfy the product rule, one can require that only the product or quotient of the

330 P. van Nieuwenhuizen Supergravitysuper Jacobians of each chiral space be constraint. This adds global chiral or scale transformations tothe symmetry group. If one has no constraint on AF’ and A” at all, one finds that this leads to conformalsupergravity.Thus the transformations in the chiral superspaces can be written as (see appendix F)= x” + ~A~(x”+ iH~(x,0), ~(1+ y~)0)+~AF’*(x~— iH”(x, 0), OR) (5)and similarly for 0””. For the axial super field one hasrJ’~’I ‘ a’\—_I ‘,. — ‘F’ \—_( F’ — F’ii ~X,v 1—2.~X L X L X R= H”(x, 0) + A~(x+ iH(x, 0), ~(1+ y~)o) + c.c.}. (6)The crucial point is now that by expressing x’ and O’in terms of x, 0 one gets an equation for SH”(x, 0)which determines how the components of H” (x, 0) transform. After choosing a gauge which puts thefirst two components of H” (x, 0) equal to zero, there remains still a class of transformations whichmaintains this gauge. As we shall see, these transformations are just general coordinate, local Lorentzand local supersymmetry transformations, while the remaining components of H” are the fields e m,.,~,a s~P, A,,, of simple pure supergravity with auxiliary fields. Thus the minimal set of auxiliary fieldsarises naturally from this chiral superspace approach. However, although this approach is perhaps theclosest possible to the ordinary approach in normal space-time, one still has to fix the gauge such thatH” takes the form mentioned above. On the other hand, no constraints are needed here, whereas inother superspace approaches they are needed (for one reason, to exclude spin 5/2, 3 etc.).The geometric picture is thus that one has in the combined 8+ 4 dimensional space (XL, XR, OL, OR) a4+4 dimensional hypersurface defined by,. — ,. ,.,. ,.1 ~ j~. ~ LIX L X R — “ ~L X R, ~L, (FRWhen one shifts points by a general coordinate transformation, the hypersurface itself is deformedaccording to (6) such that the new points lie on the new hypersurface.The idea to consider the imaginary part of the bosonic coordinates of the chiral superspaces as adynamical field, i.e. eq. (7), comes from global supersymmetry. There, a chiral superfield is defined byDa(1 — 7S)ab4,(x 0) = 0, D~= (8/80” + ~(ö%) 0) (8)whose genefal solution is easily seen to be47 = 47(x1., OL), X”L = X~— ~O7”7~0, OL = ~(1+ y~)0. (9)Usually one considers the X”L and X”R = x” + ~0y”y~Oas some convenient basis inside the 4+4dimensional space (x~,0”). However, in order to generalizeX”LX”R” 40770 (7a)

P. van Nieuwenhuizen, Supergravity 331to curved space, we consider (7a) as a flat hypersurface in the 4+8 dimensional space spanned by X”L,0L, O~.The global supersymmetry transformations on the coordinates (x”, 0”) in eq. (8) ofX”R,subsection 3 induce the transformationstm L ~m II— LI ~ Pfl _~ ?tlLI j— EL ~ Vt (YUL oX L — 4 ’)’ ~L ~ ~ It nX L= ER+ ~(A ~)*oR, SXinR = ~ + ~ + Am,,X”R. (10)By rewriting these results in two-component notation one finds that SXmR = (5xmL)* and 50R = (SOL)*.To go to the local case, one replaces these combined global transformations by one local function (justas in the case of ordinary gravity) as in (2). One requires also in the local case that right-handed andleft-handed transformations are related by complex conjugation. Finally, Oy~y~0being the flat spacelimit of H”(x, 0) in (1), it is natural to generalize (7a) to (7).Let us now describe how this chiral superspace approach indeed describes ordinary supergravity.First one performs a general super coordinate transformation such that the components of H” (x, 0)with no and one 0 factor are zero. The set of transformations which maintain this gauge haveparameterswhich are field-dependent. The same phenomenon one encounters in the previous approaches tosuperspace. The advantage of this gauge choice for H” (x, 0) is that then the transformation rule,SH”(x, 0)= ~(A”(x+ iH OL)— A*~(x—iH, OR))—~(A”(x+iH, OL)+A (x — iH, OR))8,J-I” (x, 0)— [A”(x +iH, OL)~—+A*~(x—iH, 0R)~~]H”(X~ 0) (11)which is a nonlinear realization of the left- and right-handed supergroup on H” (x, 0), becomespolynomial (though still nonlinear) in H”(x, 0). The parameters are expanded in terms of 0L (or 9~)asA” = —a” +ib” + OL~”L+ OLOL(5” +ip”), with °L°L= OLCOLA” = EL+ WL~O~L+OLOL?1L (12)where a”,. . ., vj’” still depend on X”L = X” + iH”. The bispinor ~L”~ is equal to ~(1+ y5)”PwL~(1+y)U; in other words, viewed as a 4 X 4 matrix it only has nonzero elements in the first two rows andcolumns in a representation in which y~is diagonal.Let us now consider the field-independent terms in 5H”SH” (x, 0) = ~- [2ib” + OL~”— OR~*l~+ OLOL(S” + ip”) — ëROR(s” — ip”)] +~.~, ë~,,.=(13)where all b”,. . . , p” now only depend on x. Clearly there is enough freedom in parameters to choose agauge such that C”(x) and Z”(x) in H”(x, 0) are zero. Let us assume this has been done. Then the full8ff” is polynomial, since third powers of H” vanish.The set of transformations which maintain this gauge have b” = 0 while for the vanishing of the order0 terms in SH”(x, 0) one finds the conditions

332 P. van Nieuwenhuizen, SupergravilyOLc,L0Rc,R = 2i[E”L$—+ “R~&—] [~ O0!i~plus KF’ and em” terms]. (14)Hence, the parameters c,’’ are expressed in terms of dynamical fields. These conditions are part of thetotal set of gauge conditions in this approach.The condition that volumes are preserved under coordinate transformations expresses someparameters in terms of others. One finds that8 8-~—~— s” (.rc) = -i---— p” (XL) = 0CJXL VXL0 (15)8X”L~~~ = 0.(If one requires that only the product (quotient) of both superdeterminants is constant, then one finds inthe trace of (VL$ in 2-component notation the extra constant term ib(b) which can be shown to generateglobal chiral (scale) transformations.) Thus s” and p” are transverse, and these transverse fields appearin SH” in the order 00 terms ast5H”(x, 0)=~O0p~ —~Oiy50s”~ (16)Hence, we can gauge away also the transversal parts of fi” and K”. This completes the set of gaugeconditions. We introduce new symbols for the longitudinal parts of H~and K”P=0,.I~I”, S=8,.K”. (17)0”. The remaining independent gaugeparameters At this point areall thus fields that remain are S. P, A”, e,,,” and cu“, a” and traceless part of WL~,. The latter is an element of SL(2, C), hence wecan expand it on a complete basis(Z)L$ _~!5au.)Y = (0.)aQmn (18)with real coefficients (1”” (see appendix). These f)mn generate local Lorentz rotations. Thus localLorentz rotations are not independent from but induced by the general coordinate transformations (justas in the Wess—Zumino approach local supersymmetry but not local Lorentz rotations are induced bygeneral coordinate transformations).tm” After are complicated just the parameters field redefinitions of local supersymmetry, one finds thengeneral following coordinate results. andThe local parameters Lorentz rotations&‘, a” and offl ordinary supergravity. The fields S, P, A,., e,.m, i/i,.” transform precisely as the minimal set of auxiliaryfields.The advantage of super space approaches is that the closure of the gauge algebra is evident. In thisapproach, the product of two volume preserving general coordinate transformations, each of whichkeeps the gauge in which the axial superfield H” (x, 0) has only terms of order 02 and higher and haslongitudinal components H” and K”, is again of this kind.

P. van Nieuwenhuizen. Supergrav:ty 333Spinor covariant derivatives are defined, and vector covanant derivatives follow then from theanticommutator of two spinor covariant derivatives. Invariant actions can be defined and generalcounter terms have been obtained [411,191].Summarizing one might say that the chiral approach uses fewer fields (64 in H”(x, 0) as compared to1024 in VA’5(x, 0)) and fewer parameters (40 volume preserving A” and A” as against 224 parametersE’t(x, 0) and Lm,,(x, 0)). Moreover, one only needs to fix 26 more gauges to find back the 38 fields etm,.,~ S, P, A,. while in the non-chiral approaches one simply does not have enough gauge parameters todo this (1024> 224). Therefore, one must impose in the non-chiral approach additional algebraicconstraints. These are satisfied automatically in the chiral approach.All supervielbeins and connections are expressed in terms of the axial H~(x, 0) above, and H” is anunconstrained field. As action one takes again the superdeterminant sdet V4”’. Varying with respect toH”, the field equations read GAA(X, 0) = 0, but not R = 0, as in the Wess—Zumino approach. R beingchiral, of course satisfies DAR = 0. Defining G~ to be determined by the supertorsion just as in theWess—Zumino approach1A.BB — AB(JBp •C ç’•Cone finds that (off-shell)DAGA~=DAR*. (20)Hence, on-shell, DAR = DAR = 0, so that R = constant. In order to understand this constant, one canconsider component fields. In the action the term eS2 = e(e_l8rnSm + (j/8)~frmO~tmfh/J,,)2appears. Varyingwith respect to Stm (not 5!) one finds 8m(e18nS” + (j/8)i/im~m”~frn) 0. Hence S = constant which isactually the same constant as found before since R starts with S. Inserting this result into the fieldequations for gravitino and graviton, one finds mass and cosmological terms. Note that the cosmologicalterms of the component approach e(S — (j/8)~IimOm”iIin)turns here into a total derivative if one expressesS in terms of 5=.At this point we anticipate our discussion of the work of Siegel and Gates, and discuss how in theirapproach a cosmological constant appears in the action. A chiral scalar density superfield V isintroduced such that gauge transformations now have an unconstrained superfield parameter, asopposedto the casewediscussed in subsection 1.9, eq. (23). (The parameternow generates superconformaltransformations.) The cosmological term in the action is now given by A f d4x d20 V+ h.c. =A f d4x d40 (E/R) + h .c. Varying with respect to V directly yields the equation of motion R = A (because inthe action supdet E the supervielbein is V-dependent).Both the chiral and the non-chiral approach can reproduce the results of the non-superspaceapproach to simple supergravity. Although the initial starting points look very different, each approachcan lead to the other.In fact, one can choose a special gauge (just as in general relativity) in which the components ofH”(x, 0) take on a particularly simple form at a given point. One finds that the first few componentsvanish, and the rest is a function of R, G~and WAPC alone. Using this special form of H”, it is easy toshow that the supertorsions and supercurvatures (which depend on VAA, which depend in turn only onderivatives of H”) satisfy identically the Wess—Zumino constraints.We discuss now the closely related superspace approach of Siegel and Gates. They consider complexcoordinate transformations. A scalar superfield 4i(x, 0) transforms as

334 P. van Nieuwenhuizen, Supergravityand47’(x, 0)= exp(A 118 11) 47(x, 0)= 47 +A 118 1147 ~ (21)11(x, 0) A *(x 0)11AChiral superfields are defined by1 8and depend thus on x and0A only (not yet on XLThus, with (21),(22)11 which maintain chirality.and OL). Consider those A8A(S47) = 8B(A’~8A+ AA8A + A”8,.)çb = 0. (23)It follows that AA and A” are chiral but AA is unrestricted since 8A4., = 0 from (22).Similarly, 47 ~ is antichiral, ~(1+ y~)(8/80)47 * = 8A47 * = 0, and requiring that also antichirality ismaintained, one finds 8BA *A = 8BA*F’ = 0. Again A *A is unrestricted. (The star operation is in a generalrepresentation not equal to complex conjugation C, but rather x” = C~y~{.In a Majorana representation* = C.)The chiral scalar superfields of Ogievetski and Sokatchev are obtained as follows47(XL, OL) = exp(iU) ~(x, gA)47*(xR OR) = exp(—iU) 47(x, OA)(24)where U = U’~(x, 0)8~with (U’~)* = UA. This field U transforms per definition as(exp iU)’ = (exp A *fI~~) (exp iU) (exp — A”811). (25)To lowest order in A and U one hasso that one can gauge away the spinor superfields UA and UA by fixing AA and A *A appropriately.Clearly, (25) preserves the reality (Majorana character) of U’5. In this gauge (24) yields= (exp iU)x”, X”R = (exp — iU)x” (27)as one may easily check. It follows that X”L — X”g = 2i U” +~... Hence, to lowest order, U” = H” isthe axial vector superfield discussed before. This is superconformal superspace supergravity.To obtain Poincaré superspace supergravity, consider a chiral scalar density VSV= A 118 11V+(8,.A” — 8AA’~)V. (28)(26)

The gauge choice V = constant leads then to further constraints on A 11P. van Nieuwenhuizen. Supergravity 3358,.A”~8AA”=0. (29)Clearly, 1/47 is again chiral, and f d~xd20 V47 is an invariant action.Siegel and Gates construct explicit constraint-free supervielbeins, and, after a conventional choice ofthe superconnection, they find that the Wess—Zumino constraints are satisfied identically.The motivation for writing the fields such that they appear in an exponential is to have a formalismwhich is analogous to the super Yang—Mills case. For the extended supergravities it seems unavoidablethat one needs constraints on H” itself. Thus here the distinction between the chiral and non-chiralapproach seems less clear even at the starting point.5.7. Gauge supersymmetryThe first approach to local supersymmetry is the theory of Arnowitt and Nath [17—43],called gaugesupersymmetry. Its tangent group is not, as in supergravity, 0(3, 1) x 0(N), but the larger supergroupOsp(3, 1/4N). The dynamical equations of motion are the Einstein equations with a cosmologicalconstant, generalized to a space with four boscnic and 4N fermionic coordinates. Due to thecosmological constant, the metric g,.~= 8,.,. is not a solution of the field equations, while globalsupersymmetry (see subsection 3) is only a solution at the tree level when N = 2. When quantumcorrections are taken into account, global supersymmetry could also be a solution for other values of N.The main problem of gauge supersymmetry is that it contains higher spin fields and ghosts. As such it isnot a good particle theory. Nevertheless, we feel it important to discuss also this theory here because itis different from supergravity in its geometrical structure.*We will begin by defining tensors. This is more complicated than in supergravity, since the tangentgroup has Bose—Fermi parts in gauge supersymmetry, but not in supergravity.The base manifold is z’~= x”, 0”’ where i = 1, N and N is arbitrary. The basic field is the metrictensor g~n(z)where the caret indicates that the two indices transform differently as we shall see.Coordinates transform as= — ~eA di’s = dz’~— ~ dz’ (1)where in this subsection we follow the literature and always use right derivatives. Requiring that the lineelement(ds)2 = dz’tgAn dz” with g~i~ = (~)“ ~“~‘5”gIIA (2)is invariant, one finds that the metric transforms under infinitesimal general supercoordinate transformationsas— Z \(Z+1)A 1 ISg~n—g~~.n+(—, ~ .AgIn+gAn,~ . (3)Note that left-up to right-below contractions are the usual contractions and are free from carets, butthat the (unusual) other contractions carry carets on the lower index. In fact, changing an index A to Aintroduces a factor (—)‘~as we shall discuss.* The following is an excellent exercise in tensor analysis. Dynamics starts at eq. (33).

336 P. van Nieuwenhuizen, SupergravityContravariant vectors with fiat and curved indices transform per definition as follows= 47B~A + 47A~A (4)5471% = ~~ 1%E47’ + 47A~I (5)where ERA are completely arbitrary functions of z. The law for 5471% follows from requiring that41%gn 11be a scalar. The internal symmetry group ~1I is at this point arbitrary.11~iCovanant vectors are defined by requiring that VJA47A, 47Acu,A cu/Açb1% and çb’~fr.jbe scalars. A scalar Stransforms as SS = ~81%S= (8AS)e. One finds, for example,— Bj ~j.(NPA~EA (PB~(PA.Ac— A Bf_\(A±B)A~ ~A(‘(PA ‘PBEA )Note that (— )A47A transforms as 47A. This allows one to ,switch to and from hatted indices, bymultiplication by (~)A.1?AE is introduced by requiring thatNext a tangent space metricA B47 ~i/i with ~ = (~ flBA (7)be a tangent and world scalar. This determines 5~j~and shows that ~moreover that the tangent metric be an invariant tensor, one findsis a tensor, but requiringe g~_ C ,‘ \B(B+C) Cj— U — EA 11CB — ?JAEEB vClearly, flAt is a constant tensor, and clearly, the tangent space parameters ~= EA~i~( — )B, must satisfy the following symmetrywhich are defined byEAB+ BAO. (9)This thus restricts the parameters for given flat metric.One can lower flat and curved indices of contravariant tensors in two ways, by putting gIJA and ~on the left or on the right of a given tensor. We define= 471g1,4, cbA = 47BnBA47A11A1147— 11 — Bçb1%—gnnçb ,Clearly, 47A = (— )‘~47~. These objects are good covariant tensors as defined above (6). For completenesswe give the transformation rules of curved covariant tensorsS4~A= 47~S~1,A+ 47A~SS~1— I Ii \jt±1 ( )8471% —çb1%g +ç6~~ A~)The last term can be rewritten as 47~I~1%in which case all indices match on both sides of the equation.

P. van Nieuwenhuizen, Supergravity 337In general one contracts indices as in the four contractions above (6). One first writes down anequation so that it is correct in the bosonic sector, and then adds signs so that it is correct for all sectors.As an exercise the reader may derive47An = (— ~ (12)We now come to the important point of specifying the group in tangent space, i.e., the determinationof ~. At this point a (weak) assumption is made: the tangent group contains ordinary constant Lorentzrotations. Under ordinary Lorentz rotations Sv = A m,,v” and Sv” = ~(A. ~y)~2~~b• In superspace 55A =SBEBA per definition. Hence for the bosonic part Am~= ,,m while for the fermionic part Eb” = ~(A .Notice that we always use gamma matrices with indices as in (ym )cd.O-)Taking in 8i~~in (8), A and B equal to m and n, one easily derives that urn,, is a multiple of theMinkowski metric. On the other hand, taking A = m and B = b one finds that the Bose—Fermi part ofi~ vanishes. For the Fermi—Fermi part of the tangent metric one finds with A = a, B = b from (8) withthese special Eb0b.— i,0~( ~ )“b = 0. (13)Hence, i~must commute with the Lorentz generator. The general solution is therefore1lcd = a[C(1 + ibys)]~d (14)where Cat, is the charge conjugation matrix.Hence, there is a 3-parameter class of solutions of ~ in (8) for the special case that is equal toconstant Lorentz rotations, namely [d~mn (Mink.), aC0~+ ib(Cy5)~~].We now pick one element of thisclass as the metric in tangent space. It is at this point that the difference with supergravity occurs,because in supergravity one keeps all 3 invariant metrics, as a result of which the tangent group ofsupergravity is the Lorentz group. Also the coset approach of page 319 leads to this result.Substituting this form for the tangent metric, one finds that the tangent group is defined by leavinginvariant d47 m n~,,(Mink.)~!i~ + 0(1 + iy 5b)~i.This shows that the local tangent group is Osp(3, 1/4N).(For b = 0 this is the standard definition for b~ 0, see below.)In gauge supersymmetry, the only field is the metric g,~. There are no independent matter fields; thecomponent matter fields appear in the 0 expansion of the metric. Therefore one can describe the theoryequally well in terms of the metric or of supervielbeins. However, in order to make contact with otherformalisms, we introduce a supervielbein V1%~’’by the definition— A \I7(I+B) Bgnn— Vn ua~() V,i . (15)The transformation properties of V1%” are as indicated by the indicesSVAA = ~(A+IXA+l)vA~I + ~ + VABER” (16)Knowing how V4”, g~and ,~ transform, one finds that the relation between g and V is a correcttensor relation. One can now scale the y~in the tangent metric away by rescaling the super-vielbeins.To see this, note that (a + iby5) = (a 2 + b2)1”2 exp(iay 2 + b2)112, and redefine 47 =5) with cos a = a(a

338 P. van Nieuwenhuizen, Supergravity(exp — (i/2)ays)47’. The rescaled super-vielbein is still arbitrary, except when a2 + b2 = 0 which case weexclude. We also rescale V,~tm such that the Bose—Bose metric becomes strictly equal to the Minkowskimetric. Thus the tangent metric takes on the simple form= [?lmn(MiTlk.), kCab], k >0. (17)(If k 0. (18)In other words, the tangent group is Osp(3, 1/4N) (not Osp(N/4)). By restricting the tangent group to thelocal Lorentz group, one can go over to supergravity.Up to this point we have exclusively considered the case N = 1. For N> 1 the flat indices areA = (m, ai) and a similar analysis shows that= [~mn(Mink.), kCabSq], (i, j = 1, N). (19)We now will first define what gauge supersymmetry is, then discuss its field equations and finallyconsider in what way global supersymmetry is part of gauge supersymmetry.A Riemannian space is defined by(i) the covariant derivative of the super-vielbein vanishes. The latter is defined by~j A — ~y A —‘ \X(A+4)r 4 ~i A L Ti B~ AVA .~ VA .~ %,J 1A ~V4 ~ VA J&B ~.Contrary to general usage, the index ~Ein the spin connection is written on the right. This ensures thatno extra signs appear in the term in (20) with 11. (In supergravity, one can equally well write £ on theleft of AB since A and B are there both bosonic or fermionic. Not so here.) Similarly, one defines forcovariant tensors TB and T,~the covariant derivative by requiring that T8TB and TBT’~are scalars forwhich the covariant derivative equals the ordinary derivative, and by requiring that the Leibniz ruleholds.(ii) the affinity IA1fl is symmetric (i.e., no torsion)FAIl = (_ )A +11+1(1% ~ (21)(We might note that defining FAIl1 to be the Christoffel connection would simplify the formulae.)As a consequence of (20), also the metric is covariantly constantgAn.~= g1%11,.~— (—) ~ —(— )I(fI+4)F~4,~g4fl= 0. (22)To prove this, use (15) and (20) and deduce that (22) implies that ~ = 0. This means that also thetangent space metric must be covariantly constant, and this is the case provided= ~ )~f2BA~ with t1AB-~ = (IA~17BO(23)

P. van Nieuwenhuizen, Supergravily 339From VAA;E = 0 and the transformation properties of tensors the transformation laws of the twoaffinities in (20) follow (see Nucl. Phys. B122 (1977) 301, eq. A(15)). Just as in general relativity withouttorsion one can solve the Christoffel symbol from (22) and the spin connection from (20). We leave it asan exercise to write down immediately the correct result for F 11’A in terms of g,-~ by adding signs to theChristoffel symbol of ordinary general relativity in the way discussed before.The inverse metric is defined by11= 5J’ where g’11 = (— )“1’g”. (24)g1%~g’(Incidentally, for the tensor gAll one has gAll = (— )A11g111%,shows that alsotoo.) As a consequence a little calculationHIg gSA—vA —o ,.~.Next one can solve DBAC — ( )AB+BC+CA11A from (20) but in order to find Ii itself one must lowerthe upper index by means of u~. Defining71B15ç~ ~D i,Ai \CAL~4BC — LA 1% V C ~. )and using (IABC= —( — )‘~QBAC one finds for the connectioni-s _IID ~i~I \BCD I \A(B+C)~— 27~-”ABC ‘ ~—) “ACB —I) — Ti Ai \C-I-ECrT,D I \D(A+1)+A+1+A1T,D 1*’VA !~) I~A ~ VI ,AJU) Yfi flDc.The inverse super vielbein is defined byTi A1, B — ~ BVA VAA remark about the signs: they follow easily by noting that one contracts as A A and A A~ Thereare more signs in the Riemannian geometry formulae than in the corresponding supergravity formulaesince the tangent space group is Osp(3, 1/4N) and not 0(3, 1) ® 0(N) (only the latter is truly bose).The curvature tensor is defined by the commutator of two covariant derivatives on, say, U1%and readsUA;fl~ (~‘UA;~Il= (~“ 4~4RA4n1U 4 (30)4~.~= —( — )“FA4~,~+ FA4n~+ (~)1~(4 ~F 11~F — 4~ (— )‘~ IrF 4~. (31)RA1% 11 1%i~F11(This definition differs by (—) from earlier definitions.) There are only two independent contractions, asin general relativityRAn =(—)4RA44fl and R=gAhI~fl~(_)ht~ (32)

340 P. van Nieuwenhuizen, SupergravityWe can now define the dynamics of gauge supersymmetry. It consists of the following equations ofmotionRAn = AgAn. (33)They follow from the following actionI=Jdmxcro\/_g[R+(n_m+2)A]. (34)11 = m — n.The symbol \‘—g is the superdeterminant (see appendix) and we used thatThere isgnAg”also the familiar Bianchi identity(_)A(RHA —~g’MR)~A=0. (35)Clearly (for A non-zero) flat space (gn11 Smn) is no solution.We turn now to the question whether gauge supersymmetry contains global supersymmetry. Thiscan be analyzed by looking at the vacuum expectation value of the field equations. Consider thatVA = (01 VA~4I0)is given byV,.m(O) = 5~, V,.”~= 0, VO,m(O) = _~(6Fm)0,, V0,”’~°~ = ~ (36)This is the most general form for V1%” which is invariant under constant transformations with~A(O)= (~i’F~0’, E”). (37)The functions (CF~)0~must be symmetric in (ai, /3j) in order that the metric in g(O)1%fl satisfy theEinstein equations (33). (A tree level statement.) Hence= y”F0(s) + iy”-y 5F 0(a). (38)Any global tensor QAH (i.e., a constant tensor invariant under ~~~0))is of the form= ki’q,.,,, Q,..a, = kl(0F,.)aA,= k2,11C~+ k1(OF~)~1(ëF,.)~1.Squaring Vm(O) according to g = V~V one finds for g~°~ the same result as in Q,,,. but with k1 = 1 andk2,0 = kSq where k is the k in i~. In the tree approximation g~hobeysprovidedRAn(g~)= Ag~)~ (40)—2k 2F,.F” = A, —k2 Tr(F~F,.)= An,.,.. (41)(39)

P. van Nieuwenhuizen, Supergravitv 341These equations follow from the Bose—Bose and Fermi—Fermi part of (40); the Bose—Fermi content isredundant.If one remains at the tree level, then this implies that N = 2 (trace the second equation). However, inthe presence of quantum loop correction, calculated with the globally supersymmetric gauge fixing term,the corrections to (0JgAn~0)are obtained by differentiating the effective potential with respect to g~°h.The result must again be of the form of Q,.,. but also R 1%~must be of this form (with calculablecoefficients). The only difference is that in (37) one must replace A by A — K’ and A — A’ respectively. In thiscase it may turn out that tracing of (37) 2 selects particular N values or perhaps none. In any case, (37)+ 1 unknowns (namely, the elements of F(s) and F(a), and A).represents Hence, if there N(N are + 1) solutions, + 1 equations therefor must N be extra symmetries.To investigate the residual symmetries of the vacuum, one solves the Killing equations(0) — — ~A+fl+AH42The general solution is (see Phys. Rev. 10 (1978) 2759):= a” +A”,.x~+EF”0= ~(Ao-r~o~’ + “‘ — M”’,~10~’ (43)where the matrix M must satisfy (CM) + (MTC) = 0, and [M, F”] = 0. Lorentz covariance impliesM”’,3J = (M0)~5~ + (M1)’1(i75)”,9 so that the two N x N matrices M0 and M1 are antisymmetric in i andj.The important point is now that corresponding to the global symmetries of the vacuum metric given byM, the full dynamical theory has the corresponding local symmetries. In particular, the component fieldsare found inside the metric tensors as followsg,.~(z)= g,.,.(x) + Otfr,.,.(x) += cui,.,ai(x)+ (~M’)a,47’,.+ (gym) e”(x)+.... (44)The fields qS~are Yang—Mills fields and I denotes the generators of the internal group M. Also, uponexplicit evaluation, it is found that the action in (34) has as unbroken symmetries only the spacetime andlocal supersymmetries, plus the Yang—Mills symmetries associated with M. All other symmetriescontained in ~ (z) are broken spontaneously, hence there are no fields with vanishing masses with spinsexceeding 2. One might call the symmetry group which is given by the Killing vectors in (39) the groupGsp(N/4)R.,.~since it contains the contracted symplectic group Sp(4) (hence the spacetime symmetries)plus the group 0 of Yang—Mills invariances plus its grading.Let us repeat the analysis performed above on the metric now on the super-vielbein. Requiring that= 0 with V/’~°~ and e” ~ given before one finds that= A~”(x), E~”’(Z)~o~ = 0Eajtm (z)~°~ = 0, Eaj”(Z)~°~= ~(A. O•)baSli + p,j.bi (45)In other words, of the whole original tangent symmetry group Osp(3, 1/N) only the diagonal part0(3, 1) x G remains unbroken.

342 P. van Nieuwenhuizen, SupergravityWe come now to the well known k —*0 limit. In this limit the group G(N) is expanded into the group0(N), and one hopes to find the N-extended supergravity models. To see how this limit is obtained, westate without proof that one may place F” by ‘y” in (36) and (39) provided one replaces in g~, 91thematrix S,~by K,1. In that case the condition becomes [M, y”] = 0 and [M, kK] = 0. Thus M”~1=&~(M0)’.,where (M0)~is antisymmetric, so that the internal symmetry group expands into 0(N).We conclude our discussion of gauge supersymmetry with a short discussion of four topics.Vanishing cosmological term. While the supercosmological constant A in (30) is nonzero, the ordinarycosmological constant (in the Einstein sector) vanishes as a consequence of the global supersymmetry ofthe vacuum metric below (35). (It may be stressed that other vacuum solutions might not be globallysupersymmetric, but this might induce an enormous cosmological constant if such breakdown occurredat the tree or the first loop level.) The reason is that if one substitutes in (30) g = g~+ h, due to (36)only terms at least linear in the quantum fields h survive. In more detailV I \ — DEinstein( \ L r”(°)r$(°) —~ ~x1 ,.~ a —and the second term cancels Ag~.Ultraviolet finiteness. One quantizes with the globally supersymmetric gauge fixing termgAll (0)CB(z) CA(z) (46)where CA(z) is the linearized harmonic gauge. The Faddeev—Popov ghost action is then obtained asusual. The propagator of the metric is obtained by solving the Ward identity(ShgA (z1) h1~(z2) + h11A (z1) 8hi~(z2)) = 0. (47)The result is/iIIA.I1’I(P, z1, z2) = exp(oip,.F”~)~ F~x1u(p,wiP~(~”) (48)where F,,, is a polynomial or order m in ~“ = ~(0’~’ + 0~)and a” = 0~’— 0~.Furthermorem(p, w”)—~~ ~ . . Wa’. (49)Fii 0Let us now consider the n-polygon in fig. 1. There are n vertices, and each vertex goes like k2.Q Fig. I.Its degree of divergence is given byd,, =4+2n—~(2+4N+n,). (50)

P. van Nieuwenhuizen, Supergravity 343Since ~ n, 4N(n — 1) because there are only n — 1 independent (012, W~,. . . , w,,~,it follows thatd_8 onehas particles with spins larger than two and more than one graviton (for N = 9 one has 10 gravitons, an“embarras de richesse”). Since it has been shown that the free field action for spin 5/2 is unique and

344 P. van Nieuwenhuizen, Supergravitycannot be coupled consistently to either gravity or to simple matter systems, it seems as if nature stopsat spin 2, and supergravity at N = 8 (see subsection 1.14).The simplest extended supergravity is the N = 2 model (Ferrara and van Nieuwenhuizen [203]). Itwas obtained by coupling the (2, 3/2) gauge action to the (3/2, 1) matter multiplet by means of theNoether method. It was only afterwards found that the model had a larger symmetry, namely (to beginwith) a manifest 0(2) invariance which rotates the two gravitinos into each other. This model unifieselectromagnetism with gravity. The action reads= — ~ R(e, co)— ~ ti,~F”D~(o4cui, — + 4V2 [e(F”‘~+ E””) + ~y 5(P~+ ft~a)]~b1~E11 (1)and is invariant under general coordinate, local Lorentz and Maxwell transformations (SA,. = 3,.A) aswell as under the following two local supersymmetriesse tm,, =!~L~iym~,i 5A,. (2)~Ii — I \ ‘~_..!L..4I~ A11 ~ A \ I(J(J1,. — ~~~(O~?E-‘ ‘. ,~AY ~ 2e~,.A7 751E.Indeed, each gravitino gauges one local supersymmetry as one sees from Si/i,,’ = a,.e’ + more. Thesymbol .F,,,. equals ~ and .F,,,. is the supercovariant photon curl= (a~A~— 2\/2 i/I~ifr~~Eu’)— (j ~-* v). (3)One may always take the spin connection co in the Hilbert action as that function of tetrads andgravitinos which solves SI/Sw = 0 (1.5 order formalism). If one would have started in the Hilbert actionwith (0+ T where r is arbitrary, then one finds only r 2 = terms upon expanding about co. These i/i4~p4terms are absorbed into the F,,,. tensors, and it is easy to argue why this must be possible (as of coursehas been checked explicitly). Since fermionic field equations rotate into bosonic field equations and theformer have only one derivative while the latter have two derivatives, no derivative can act on e. (Sinceotherwise there would no longer be two derivatives available for the bosonic field equations. Thisargument breaks down when there are nonpropagating (auxiliary) fields.) The gravitino field equation isobtained by varying all t,li fields, and since the terms with P have four i/i’s which appear symmetrically,one must take one half of the former in order that the field equation only contains the supercovariantF~,,,but not, for example, bare F,,,,’s. Note that this argument also tells one that the spin connection inthe gravitino field equation must be supercovariant. As it happens, w is itself supercovariant, henceboth in Hilbert and gravitino action one finds a. (In other cases, for example in d = 11 or d = 5dimensions, one finds in the gravitino action ~(w+ ó~)instead of w, but, based on our argument givenbefore, one always chooses co in the Hilbert action.)The same kind of argument shows that if auxiliary fields are absent the gauge algebra must close onbosonic fields, and this is also a helpful criterion in constructing theories. (It is difficult to credit oneparticular person, but certainly J. Scherk gave an important contribution.)This model was the model where finite quantum corrections were found for the first time (see ref.

P. van Nieuwenhuizen~Supergravity 345[2821). Some chiral—dual symmetries were first found in another model which was not finite butdisplayed miraculous cancellations [510]. For a complete treatment of dual—chiral symmetries, see ref.[3821.An explanation of this finiteness was given by using these symmetries in this model which wenow discuss.According to the analysis of Haag, Lopuszanski and Sohnius (subsection 3.3), the maximal symmetrygroup of the S-matrix of supersymmetry algebras containing the Poincaré algebra is U(2). In fact, thefield theory has this global symmetry only on-shell (i.e., only the field equations have an U(2) globalsymmetry). Off-shell, only an SU(2) remains. These symmetries read [382]547,.L = iw $,.‘- — ~ = ~(1+ y~)i/i,.= ~ ~ ~I’~— kg)’ ~,.R = ~(1— y 5)çb,.U(1): Si/i,. = —iy5~’,, and SE,.,, = ie ,.,.~F”. (4)Thus the SU(2) part rotates ~,L as (2) and i/,’~as (~),while the U(1) part is a combined chirality-dualitytransformation. To prove the U(2) invariance, note that in the gravitino action i/’L couples to çIITRC sothat these terms are SU(2) and of course chirally (U(1)) invariant. The torsion is separately U(2)invariant, and, finally, the remaining terms can be analyzed by eliminating the F~,.kinetic terms throughthe equation of motion for F,.,.. This cancels half of the Noether coupling with bareF,.,, and the result isthat the action is the sum of the tetrad and gravitino actions plus the following term4V2 çi’,.[eP”~ + ~y5F”~]çb,,’E”. (5)Clearly, also this last term is U(2) invariant. The SU(2) invariance holds off-shell, but for N >2 only the0(N) invariance holds off-shell.An interesting connection between the coupling of photons and the cosmological constant was foundby Das and Freedman [108]and Fradkin and Vasiliev [225]. When one couples the photons minimallyto the fermions one needs at the same time a cosmological constant and a mass like term in the action.Thus it would seem that electromagnetism is due to the curvature of a de-Sitter universe. Actually, thiscosmological constant is much too large, and can be eliminated altogether by spontaneous symmetrybreaking, as we shall discuss in subsection 5.For the N = 2 model one can find this extension of the action by adding a cosmological constant tothe action and finding the further modifications by the Noether method. The complete set of extra termsis given by.~‘(cosm.)= 6eg 2 + 2eifr,.’o-”~47,.’— ~ 1F””°(D~i/i,,.’+ ge”A,,i/i,~,”) (6)8./i,.’ = D,.(a(e,i/J))E’ + gy,. ’+ gE”A,.E”where we added the terms with D,. for comparison. Clearly, g is a dimensionless gauge coupling, but itis remarkable that also a mass-like term is needed (as well as a cosmological term) when one couples thephoton to the gravitinos in an electromagnetic manner. Note that the theory still has the same numberof local invariances. The mass term is actually needed in order that i/i,.’ still be massless in de-Sitter

346 P. van Nieuwenhuizen, Supergravityspace [1251,and for g 0 there is only an SO 2 symmetry on and off-shell, since the mass term breaks theSU2 invariance.In fact, even in simple (N = 1) supergravity one needs a mass term if one adds a cosmological termeven though here there is no photon [474].As the reader may easily verify, the action2 + 2egifr,,o”~./i,, (7)= — ~ R(e, w)— F”~D~i/i~, + 6egis invariant under Setm,. = ~ëy~./i,. and Si/i,. = (D,. + g-y,)E. This cosmological term plus mass termfollows easily from the tensor calculus; * it is simply the action belonging to the unit scalar multiplet= (1, 0, 0, 0, 0). This cosmological constant is positive because in the N = 1 gauge action one finds— ~52 + aS with a arbitrary. On the other hand, in the Wess—Zumino model one finds a term + ~F2(withF the usual auxiliary field) and by coupling this model to supergravity, one can cancel the cosmologicalterm [532,541].The next extended supergravity we discuss is the N = 3 model. It was obtained by Freedman [240]andFerrara, Scherk and Zumino [201].Here a new feature is that the Abelian photon gauge invariance turnsinto anon-Abelian Yang—Mills invariance in de-Sitter space. In other words, as in theN = 2 case, couplingthe triplet of photons to the gravitinos, oneneeds atthe same time a cosmologicalconstant and amass term.The action readsZ = 2’(kinetic terms for em,,, i/i,,’, A,.’, A with i = 1, 3)+ ~a~hhIC1 + ~ + (eq. (6) with “A,. —* &“A,.’) (8)where the spin connection depends on i/~,,1 and A and solves SI/Sw = 0. (We thus again use 1.5 orderformalism.) The bare (i.e. non-supercovanantized) Noether coupling is a sum of the Noether couplingsof the (~,1) and (1, ~)systems to the (2, ~)system.= — ~ çi,,1(eF”~”+ ~y5P”~”)./Ij’E” + ~(.~‘o”~y~A)(F~). (9)The supercovariantized Noether coupling follows from the local supersymmetry transformation rules,which read= ~jIym~I SA = ~(~“~E~) (E,.,,’)5A,.~= E”E’l/i,.” —5./,,.’ = D,.(Cd(e, i/I, A))& + E’(o~°y,.E”)(E,,~)+ ~ (y~E’)+ (.fr,.’y~y5A)(y~y” ”)]1. (10)+~(Ay5y”A)(y~y,.y5E’)+gE”A,.’E” +gy,.ENote that the expression for Si/i, is itself no longer supercovariant. The noncovariant terms involvethe spin ~ fields, and this is a general (but not understood) feature: without spin ~ fields, all* Use eq. (8) on page 307.

P. van Nieuwenhuizen, Supergrauity 347transformation rules are always supercovariant. This means that there must be extra terms in thecommutator of two local supersymmetry transformations. These terms are: an extra local Lorentztransformation with parameterI’)~/i\1 ilk f—i k ‘j .Le —i k I ‘11~ ~E 2Ei ma 2E275 1 ma)and an extra local supersymmetry transformation with parameter(2V2)- ~ (E’2e~A — E’275 i y5A) (12)and an 0(3) Yang—Mills gauge transformation with parameterA’ = (2V2)~ ”(E’~e~) —SA,. = 9,~A’+ ge’~”~A~,.A”, &m,. = 0 (13)5i/J~,.= gekikçfri,.A’c, SA = 0.Note that the parameter composition rules do not depend on g (or only through F’,,,,, as in (11))although the transformation rules depend on g explicitly.There is a global U(3) invariance for g = 0 on-shell. Off-shell only the 0(3) invariance remains, sinceall other symmetries involve not only chiral transformations but also duality transformations. Theprecise rules follow by truncation from the U(4) group of the S0(4) version of the N = 4 model (seebelow). For g 0, the 0(3) invariance remains valid off-shell, but since the Yang—Mills coupling involvesbare A,, fields and the mass term breaks chiral invariance, the U(3) invariance is again lost for g~0 onand off-shell.The gauge parameter in (13) contains a nonvanishing term in the global limit (when fields are setequal to zero). Hence, there is a central charge in the super Poincaré algebra (i.e. for g = 0), but thischarge does not act on physical states. Also in the super de-Sitter algebra (with g 0) there is an“electric” charge in the (0, Q} anticommutator, but now this charge is no longer a central charge, sinceit does not commute with supersymmetry. As a result S(gauge) i/i,. 0, so that for g 0 the electriccharge does act on physical states.From the N = 3 theory (or any other theory) one can obtain other theories by consistent truncation.A truncation means putting certain fields equal to zero, and 2, consistency = i/is,. = A = 0requires which yields that the alsoN their = 2variations then vanish. Two consistent truncations are: A~= model [511’],while A2,. A= A3,, = i/it = i/r~.= 0 yields the (2, ~)+ (1, ~) Maxwell—Einstein systems [507].We now discuss the N = 4 extended supergravity. New here is the occurrence of scalars, for exampleA, which appear in a nonpolynomial way. This is possible since KA is dimensionless. For photons, KA,.is also dimensionless, but Maxwell gauge invariance forbids an infinite series in photons [202]while onecan also show that fermion fields cannot appear nonpolynomially [202].There are actually two versions of the N = 4 model known, the S0(4) model [109,90, 91] withfields (e, i/i”, V,”, A”, A, B’) and the SU4 model [190]with fields (e, i/i’, A,”, B,.”, A’, 4, B). The sixfields V,” are all vector fields, but the three A,.” are vector fields while the B,” are axial vector fields.One can obtain the SU(4) model by reduction_of the N = 1 model in d = 10 dimensions. Spinors ind = 10 can satisfy both the Majorana condition A’~= AM and the Weyl condition A = ~(1± y11)A. Thus,

348 P. van Nieuwenhuizen, Supergravityin d = 10, i/i,, has 16 components and becomes in d = 4 equal to four gravitinos. However, one finds ind = 4 matter multiplets, and splitting in d = 4 these matter couplings from the pure N = 4 gauge actionis rather hard and has not been attempted. Rather, the SU(4) model was constructed directly. Theappearance of an off-shell SU(4) symmetry is obvious since the Lorentz group 0(9, 1) splits into theusual Lorentz group 0(3, 1) of d = 4 and an 0(6) which is equivalent to SU(4). Since the pseudoscalarfield B is obtained by a duality transformation on an antisymmetric tensor field (which can only appearpolynomially in d = 10 and d = 4), B necessarily has opposite parity and appears polynomially;however, i/i does appear nonpolynomially.At the classical level the two theories are equivalent because one can find a transformation law of thefield which turns the one action into the other. For the axial and vector fields the correspondence is thefollowing. Denoting by G,.,.” the tensor which gives the equation of motion of V~,hence D’G,,,,” = 0(since only the curls F~,.of V~appear in the action, G,,,.” can be defined) one has for thetransformation laws of the curls of A,.” and B,,”~ = (F~,.+ ~F~’,, ”,,)(14)[3~B~,. =(G~,,.—where a and /3 are certain numerical matrices. The first equations can be carried over directly onto thegauge fields themselves, but the second equation together with the Bianchi identity D”B,W~= 0 yields(D”G~,.)—~ ‘k,(D”G,,~) = 0. (15)Hence, one-half of the V’~field equations must be satisfied and one is partly on-shell. It is now alsoclear why in the SU(4) model there are three axial vector fields, namely due to the e,.J’~symbol in (14).The SU(4) model is the simpler model. In the action one only finds nonpolynomiality in the scalarfield 4,, and only by means of exponentials of 4,. In the SO(4) model, on the other hand, the scalarkinetic term reads [1 — .c2(A2 + B2)]1 [(8,.A)2+ (3,B)2], which, incidentally, limits the range of KA and KBbetween —1 and +1 [221,222].In the SU(4) model there is a group of global SU(4) invariances off-shell, and a further global SU(1, 1)on-shell. The off-shell SU(4) transformations transform the fields A~’,. and B”,, into each other andcontains also chiral rotations of i/i,.’, and of x~while 4,, and B and em,, are inert.The SU(1, 1) global symmetry group leaves the scalar kinetic terms invariant. These are given by(8,. Y8’Y) (Y + Y)2 where Y = exp(—24,) — 2iB and the transformations Y -* (aY + if3) (iyY + 5)_lwith aS + fly = 1 leave it invariant. To see that these transformations are really SU(1, 1), note that thetransformations8(z) = = (a i/3’1(zi\Z21 \1y Sj\z2can be written as exp(iw1r1 + a2r2 + w3’r3) and leave z * r1z invariant. Since this noncompact group isrealized nonlinearly it does not lead to ghosts. Again, this SU(1, 1) contains duality transformations of theform SA,.j’ = G,,,”(A) and idem for B,.”.

P. van Nieuwenhuizen, Supergravity 349We now turn to the SO(4) version of the N = 4 model. Here the global SU(4) consists of a manifestS0(4) symmetry plus transformations generated by symmetric antihermitian matrices iAj”. More indetail~,L,j_~4j1L.kLap, — 111 k’(’,.*SXLSI = —i(ASA=BtrA,1k — S’k tr A )x” (16)SB=-AtrASF~,.= A G ik —The tensor D’G,,,.”‘ is again the field equation of V,,”‘.All SU(4) symmetries except the 0(4) hold only on-shell. The same is true for the N 3 model, asone easily sees by reduction (i.e., A = B = 0, etc.). There does not seem to be a consistent truncation toN = 2 or N = 3 supergravity theories of the SU(4) version of the N = 4 theory.One can gauge both the SO(4) and the SU(4) models. The SO(4) model was gauged by Das, Fischlerand Roéek [111]and has a potential with an indifferent equilibrium. Of the SU(4) model, having only 6spin 1 fields, one can at best gauge an SU(2) X SU(2) subgroup and this was done by Freedman andSchwarz [244].However, in this case the potential has no stationary point.Let us now jump to the N = 8 theory. One might expect that this theory is prohibitivelycomplicated — but one can formulate it in a very simple way in d = 11 dimensions as N = 1 (simple)supergravity. If one then uses dimensional reduction to d = 4 dimensions, the full theory emergesautomatically.6.2. The N = 8 model in 11 dimensions [93]Dimensional reduction means that all fields are assumed to depend only on x1,. . . , x~instead ofx1,.. . , x~.A tetrad in d = 11 splits up then in d = 4 into one tetrad e,.m (m = 1, 4), 7 vectorsea,. (a = 5, 11) and 7 X 7 scalar fields e~(a = 5, 11) which describe ~8 x 7 scalar particles (because the local0(7) symmetry, the residue of the 0(10, 1) Lorentz symmetry, eliminates the antisymmetric parts of e~).Dimensional reduction has been extremely fruitful in supergravity (and in global supersymmetry).We now explain why the simplest form of N = 8 model is in d = 11 dimensions, and then constructthis theory in d = 11 in the remainder of this subsection. Dimensional reduction and spontaneoussymmetry breaking will be discussed in following subsections.The example of the tetrad just given shows that by going to higher dimensions, supergravity theoriescan be reformulated in terms of fewer fields. The N = 8 theory in 4 dimensions has 8 gravitinos, henceone expects that its simplest form appears in d = 10 or d = 11 dimensions. Indeed, in d = 10 or d = 11,spinors have 32 components. The simplest version is the d = 11 theory, with only an 11 x 11 “elfbein”e,.m, a32 x 11 gravitino i/i,.” anda “photon” A,.,.~(antisymmetric in ~tvp).That one needs A,,,,0 follows fromcountingm transversalof statesand traceless: ~9x 10— 1 = 44e,.,/,,.~transversal in gauge y i/i = 0: (9 x 32—32) x = 128A,.,,,, transversal: (~)= 84.

350 P. van Nieuwenhuizen. Supergravity(A choice A,. 1.. ~6 has also (~)= ~) = 84 states, but does not lead to a consistent theory [559].)Thus thereare indeed equal numbers of bosons and fermions.Having decided upon 11 dimensions we note that the signature of this space must be (10, 1) in orderthat one does not find ghosts in the dimensionally reduced action. With more than one time coordinate,some of the scalars e would have kinetic terms with the wrong sign.Since A,.,.,, is transversal it must be a gauge field with gauge invariance SA,.,.,, = 8,,A,.,, + cyclic terms,with A,,,, = —Ar,.. Hence, the action must start with= — ~ R(e, w)— ~ ~,.F”~’D~(w)çfr,1, — ~ (1)where F,,,,,,,. = 8,.A,,,,,,. plus 23 terms. (‘The symbol F”~°has strength one, F” = F~FpI~Uantisymmetrized.)The gravitino must be real, in order that the counting of states works, or rather it must satisfy aMajorana condition. (Since d = 11 is odd, we cannot invoke a Weyl condition to halve the number ofcomplex fields.)One may ~provein general or by giving1 =an explicit representation that in 10+ 1 dimensions there— ~ ~ follows that in d = 11 just as in d = 4exists a real representation and that C,SS,C(AF~~~ F~’,~’) = (~)~F” . . . F”A). (2)To construct the action, we begin by postulating= ~j~fl1i/J,., 8./i,, = D,,,(w) . (3)Using 1.5 order formalism, one finds for the variation of the Hubert action— ~ (4)For variation of i/i,,. in the gravitino action, we get— ~(.~,.F~PFmnE)R,,,,,~,,(), (Fm~= 2um”). (5)On the other hand, varying i/i,. and partially integrating, neglecting torsion termsD,.em,, — D,,em,.since these are of higher order in i/i, we find+ ~ ~ (6)With the symmetry in (2), the sum of these two variations equals

P. van Nieuwenhuizen, Supergravity 351m~&}~froR~jpmn•E{~~’,(7)fIn the anticommutator only the tenns with five and one gamma survive but the three gamma termscancel. The five gamma term leads to a complete antisymmetrization of R~pmn and hence we againneglect these terms, since they are of higher order in ifr (without torsion the cyclic identity says thatthese terms vanish). The one-gamma terms are~ —~ + mo”T~)— (m n) (8)and one finally finds that these terms again lead to an Einstein tensor. In fact, all terms cancel! Thus,this justifies our choice for öe~.This is a nontrivial result because it relies on the fact that no threegamma terms are present. If, for example, in d = 11 one would have Cy,~C1= + ~ then one wouldhave found a commutator [F~°, f~fl] and no invariance to order ~ti could have been achieved. (In 3+2dimensions, CFAC~= ~ and one finds a commutator. Hence, although one can define in 3 + 2dimensions a Majorana spinor as usual, no supergravity exists. In 4+ 1 dimensions one still hasCFAC1 = +TA.T but here one defines a Majorana spinor with internal indices and one finds now ananticommutator.)To find the higher order terms, we recall that if o is a solution of ôI(total)/& = 0, thenj(2)(~ + r) +I~312~(w+ r) = I(2)(,) +I~312~(w) + r2 terms. (9)Hence, without loss of generality, we can take the Hilbert action as eR(e, w) but we must then use 1.5order formalism (i.e., w = w(e, ~fr) is a solution of ~5I/&u = 0). The r2 terms are of order ç1/~and must befound later.Taking the most general laws for öA~ iëifr and S~, KFE and the most general terms for Kç!13F inthe action, onefinds a unique solution. Apeculiar term bilinear in the Maxwell curls and linear in a barephoton field is needed as well. It is gauge invariant! Thus onehas the theory up to terms trilinearin cli in thevariation laws andup to termsquartic in cl’ in the action. The remainingtriuinear terms are fixed by requiringthat 5~.be supercovariant. Indeed, since in [ô(e1),ö( 2)]~.the ö~terms cancel onemust make ô~/i,.itselfsupercovariant. (In the commutator one finds a general coordinate contribution (e~’).Ii~and asupersymmetry contribution 8~(—~)with r = ~ ~2~ 1. From the {Q, Q} anticommutator on vierbeinswe know this. The 9~’terms cancel.) This yields= Di,, (~) + (gamma matrices) t~~~~ (10)where ft is supercovanant, while óis the supercovariant extension of w(e), which is thus the sameexpression as in d = 4A~m~ = c’.’p~mn(e)+ ~(i/~ym~fr~ — + i~y~fi~). (11)Finally the quartic terms in the action are fixed by requiring that the gravitino field equation be itselfsupercovariant.The final result for the N = 8 action in d = ills very simple. It is polynomial sipce a power series in

352 P. van Nieuwenhuizen, SupergravityKA,~ is not allowed (it would violate Maxwell gauge invariance)~‘= -~R(e,~— e~[~F’~’clr~ + ~ EFai 3,,ô + ~$Y8] + CKEL~ôe m&fr~.K (12)K1D~(W)E+ D(F~”~. — 86~f’~”~) fta~y&= EiF[,~4~1. (C, D, E are constants.)Remarkably enough, the ~ terms are obtained by putting in the gravitino action ~(w+ ~)as connectionand also taking ~(F+ F) in the Noether-type coupling. I do not know why this “minimal” solution yieldsall cl’ terms. (In N = 2, d = 4 supergravity, and also in the d = 4 Maxwell—Einstein system one finds thesame features.) Roughly speaking, the covariantization terms in F have twice as many ~‘s as the termswith F, so one needs half of them to obtain a supercovariant field equation, but unlike the N = 2 model,the four cl”s in the F terms do not appear symmetrically, so that this argument is not complete.The spin connection w solves ôI/öw = 0 and is given by(Z)~mn= A~m~ — ~ (13)In d = 4, a. = c~,but here they differ. This illustrates once again that by going to more complicatedtheories one finds out which equalities are a coincidence and which are not.The gauge algebra is the usual one except that the Lorentz parameter istm’2 = ~W + 2(T~ — 24em’2e1F~)Pa,~ (14)Awhile two supersymmetry transformations on A~ lead to a gauge transformation 5A~,,.=3~A~+ cyclic terms, where= —~E 2r,~v — 1 ~ (15)Terms such as ~always come from converging P in {Q, Q} = P + more, into~ I.!.”\ ~ /.~ALA0gen.coord.~ ) — °gauge~~ ~A—The gauge algebra reads[5~( ~), ôQ(E2)] = ögen.coord.(~’) + 50(—r~)+ ÔL(Atm’2 in (14)) + 6M~welI(A,~in (15)) (17)and this suggests strongly that the N = 1, d = 11 model can be obtained by gauging Osp(1/32) sinceSp(32) is spanned by Jtm, F~’2,F”2’2’~.(These are the matrices satisfying the definition of symplecticmatrices CM + MTC = 0 with antisymmetric C. We recall that CPA ~ One would like to

P. van Nieuwenhuizen, Supergravity 353consider e~and w~”2as the gauge fields of Ftm, and Fm’2. However, A,~ cannot be considered as thegauge field of fm’2~st(it does not have enough indices and it is totally antisymmetric), and this is aproblem for future research [559].6.3. The N = 8 model in 4 dimensions by dimensional reductionIn this subsection we discuss the important and pioneering work of Cremmer and Julia [96,99]. Aswe saw, in ordinary dimensional reduction all fields depend only on x1, . . . , x~instead of on all elevencoordinates. One then proceeds to compute the various quantities appearing in the action: inversevielbein and connections. One writes the four-dimensional part of the elfbeins as ö~’em1.where ô is thedeterminant of the 7 X 7 part of the elfbein, and by choosing y appropriately, the Einstein action of thereduced theory has the usual form.In this section we discuss how the massless N = 8 model in d = 4 dimensions is obtained by ordinary(i.e. not accompanied by spontaneous symmetry breaking) dimensional reduction. This analysis will alsolead us to deduce that the d = 4 theory has as internal symmetry an E(7) global x SU(8) local. However,the SU(8) local has not 63 gauge fields, so that this SU(8) cannot serve as a grand unification ofSU(3)x SU(2) x U(l). Rather, the connections are nonlinear functions of other fields, in the same wayas the spin connection w(e) gauges SO(3, 1) but is not an independent gauge field.In d = 11 dimensions, one has general coordinate transformationsOVA =~“öHVA+(öA~”)Vll, (A,H=1,11). (1)Splitting A = (A, a) with A, ~u,v = 1, 4 and a, f3, y = 5, 11 one has under ordinary dimensional reduction1,... , x’~)restricted to ~ . , x~)a global symmetry group generated bywith fields ~(x= (~= 0, ~“ = M”~x~). (2)Indeed, OVA again depends only on x’,. . . , x4 as follows fromOV~0,OVa =(~a~)V13M~aV1~. (3)One must in fact require that oar = M~ = 0 in order that the reduced Hilbert action is still0Ainvariant (peR)under since the thisaction specialis class a scalar of general density. coordinate Usually one transformations. discards this total To see derivative this, note since thatcoordinates O(eR) =tend toinfinity, but now 7 coordinates have become little circles and one must require that 9,~’2= 0 = M’~a.Thus, in d = 4 one expects a global group SL(7, R).The Lorentz group 0(10, 1) in d = 11 acts on tetrads asOeM = ~M~N (4)Splitting the flat index M into (m, a) with m, n,~ = 1, 4 and a, b, c,= 5, 11, we haveoem = wmen + meaôea = wae’2 + (J)aeb (5)Thus we can fix the parameters w’~,.= _~~a by putting e’2~= 0. In this case the vielbein has the formtn aM.I~L ~Len ‘~0 e,~ (6)

354 P. van Nieuwenhuizen SupergravityThe remaining symmetry group is now 0(3, 1) x 0(7), and one expects in d = 4 a local group 0(7).Let us now do the reduction of the d 11, N = 1 fields down to d = 4. If one reduces from d = ii tod = 4, one finds for the bosonic fields (a, a, f3,. . . internal indices)eMn —~ 1 tetrad, 7 photons e~,~7 x 8 scalarse~~ —* ~ 21 photons A,~a 8,7 scalars A,~,, (~) = 35 scalars Aapy. (7)The field A,~, in d = 4 is pure gauge, while ~ is a scalar since ~ = ,,,,ifr° so that O~’F,L,~= 0implies ~ = O,4i. This is indeed the bosonic sector of the N = 8, d = 4 model, but all bosons have only amanifest 0(7) symmetry (from 0(10, 1) —* 0(3, 1) x 0(7)) while the 0(8) invariance is hidden.The fermionic sector is reduced as follows. One splits the d = 11 gravitino ~ into eight gravitinoswith A = = 1,4 and A = (1, 32)= ai with a = 1, 4 and i = 1, 8 and 56 spin ~ fields Aaa with a = 5, 11.One finds thus also the fermionic fields on N = 8, d = 4 theory in this way. The numbers (1, 8, 28, 56,70) are the dimensions of antisymmetric tensor representations of SU(8). (Under SO(8) the scalars A ijkland BulkI, with A self dual and B anti-self dual, transform into themselves, but the 4)4k1complex with ~ 4)~)~C(A +iB)~~c~ •.~8form and çbk/an irreducible AihI~t—iB”~’.)Since representation one expects of SU(8) that andthere satisfy are (4) no*)u~~=central charges presenton-shell, one expects that there is on-shell a global SU(8) symmetry.*The first nontrivial extension of the expected SL(7, R) global x 0(7) local occurred when it was foundthat there was a set of global scale invariances for the action in d 4 which extended SL(7, R) toGL(7, R). Secondly, it was found that the action had a local SU(8) invariance, of which the local 0(7)was a part. (Historically, Cremmer and Julia conjectured, when they tried to extend 0(7) local to anexpected 0(8) local, that in fact an SU(8) local would be possible. Upon reduction to the N = 4 model(the SU(4) version of the N = 4 model we have discussed) this gauge gets fixed and no SU(4) local is left.)General theory of nonlinear realization tells us that the scalars can be written as an exponentialwith in the exponential the generators of a global group. Since one can eliminate 63 scalars by fixing thelocal SU(8), and there are 70 scalar particles in the N 8 model, this global group must have63+ 70 = 133 generators. To make these arguments more familiar, we compare with the usual tetrade”,~.It describes 10 particles (if we forget the general coordinate invariance for a moment). The localgroup is SO(3, 1) with 6 generators, and hence the global group has 10 + 6= 16 generators. This group isof course GL(4, R). The local group must be a subgroup of the global noncompact group in order toeliminate ghosts.Cremmer and Julia found the Lie algebra with 133 generators in Cartan’s work: E(7)! This algebra isdefined by the real matrices A and ~= A 2~k) — A~kx”+ l iJaI - -= flky — Atykl + ~ijkIX(8)with A’,, _Aki, A, = 0. (The Ak’ are not the transposed of A’,,.) The ~ are totally antisymmetric. Thevectors x” = —x” and y” = —y” have each 28 components, and this56 dimensional representation is in factthe fundamental representation of E(7). The 63 A ‘s generate SL(8, R) (and the previous GL(7, R) is thus* Not a U(8) symmetry since the .symbolis only an SU(8) invariant tensor.

P. van Nieuwenhuizen, Supergravity 355contained in E 7) while the remaining (~)= 70.X’s generate the rest of E7. Also the expected SU(8)global canbe found by considering (x~+ iYjk) (not to be confused with the SU(8) local). This SU(8) is the maximalcompact subgroup of this form of E7.The global E(7) symmetry acts on the curls of the photons, not on the photon fields themselves. It isan on-shell symmetry only, of the same kind as the dual-chiral symmetries in the N = 2, 3, 4 models. Infact, truncating the N = 8, d = 4 model to the N = 4, d = 4 model, one finds back the SU(4)x SU(i, 1)global symmetry of the so-called SU(4) model.The reader may have wondered what happens with the general coordinate transformations whoseparameters are in d = 11 given by r(x’,. . . , x~)where a = 5, 11. The answer is that they becomeMaxwell gauge parameters for the photons A,,” which are defined by e,,” = A,~”e~”. As one easilychecks, OA!L” = 3~”,while Oe m 1. = Oe”~= 0.There are much more fascinating details. We regretfully refer to the literature for this masslessN = 8, d = 4 model, and now turn to a massive version of it.6.4. Spontaneous symmetry breaking in the N =8 model by dimensional reductionThe work which we now discuss is due to Cremmer, Scherk and Schwarz [404,405, 406]. It solves along-standing puzzle of how to obtain spontaneous symmetry breaking in the extended supergravities.For simple supergravity, spontaneous symmetry breaking can occur as soon as S or P are nonzero(which may or may not imply that the scalar fields A and B have nonzero vacuum expectation values[541]).However, for N> 1, many supergravity practitioners were afraid that no analogous results wouldbe obtained. As usual, matters turned out to be better than expected.A most interesting way to give masses to the particles of the N = 8 model in d = 4 dimensions is tofirst reduce the d = 11, N = 1 theory to the d = 5, N = 8 massless theory. Then, in d = 5 dimensions,one writes fields as5) = exp(M’x5)~c(x’,. . . , x’~) (1), xwhere M is in general a matrix; see below. In this way, for example, a massless scalar field in d = 5 canbecome massive in d = 4= ~,*(~_pJ2)~p (2)Note that this kind of symmetry breaking is not such that the theory itself determines the minimum of aHiggs potential, but, rather, it is we who (arbitrarily) prescribe that fields must have the factored formas in (1). Since coordinates are dimensional, the substitution in (1) introduces masses in a natural way.In general, the procedure for constructing a D-dimensional theory with spontaneously brokensymmetry is to first consider a theory with unbroken symmetry in D + E dimensions and to wrap theextra E dimensions into circles with such small radii that only the lowest Fourier coefficients have finitemass. If we write fields as in (1), then the Lagrangian density in D + E dimensions is independent of M(and hence remains gauge invariant in D + E and in D dimensions) as long as the transformation4) = exp(Mx5)i,1i corresponds to a symmetry. In practice, the useful symmetries are the global symmetries.How do we find global symmetries in D + E dimensions? A very interesting idea is to consider yet a

356 P. van Nieuwenhuizen, Supergravityhigher dimension, say D + E + F, and to use local symmetries there to obtain global symmetries inD + E dimensions. For our applications we start with the N = 1 model in D + E + F = 11 dimensions,and want to obtain a global symmetry of as high a rank as possible in D + E = 5 dimensions. The localsymmetry we use is general coordinate invariance in d = 11. (In order that the method works also forsystems without fermions, we do not study whether local Lorentz invariance can be used.) As wediscussed in the last section, the expected global symmetry group is SL(6, R). Now no extra scaletransformations are present which extend this SL(6,R) to GL(6, R). (Note that E 6 does not containGL(6, R) but that E(7) contains GL(7, R.) Basically the reason is that there are two types2(F~)2 of termsandintheE~ abcdefF,,vFpgArt.action which cannot Thus be the expected made scale-invariant global symmetry at isthe SL(6, same R). We time, refer namely to the previous 3” section.The Lorentz group S0(D + E + F — 1, 1) splits up into SO(D + E — 1, 1) x S0(F) if we fix part of thislocal symmetry by casting the vielbein in triangular form. This we discussed in the previous section.Then, naively we expect for the N = 8 model in d = 5 dimensions as internal symmetry group SO(6)local® SL(6, R) global.However, things are more interesting. The particle spectrum in d = 5 consists of irreduciblerepresentations of Sp(8) rather than of SO(6). This particle spectrum is: one tetrad e~(m,ji = 1, 5), eightgravitinos ~i~(a = 1, 8), 27 vector fields, 48 spin ~fields and 42 scalar fields. Indeed, all fields can berepresented by completely antisymmetric tensors which are traceless with respect to the symplecticmetric. For example, the photons are described by the 27-dimensional Sp(8) representation A,,~”9=—At” with ~ = 0 (a, j3 = 1, 8) which is not an irreducible S0(6) representation.This particle spectrum is obtained from the fields enM, t/.~,AAfII(M, A, [1, ~ = 1, 11 and B = 1, 32)by ordinary dimensional reduction as follows:-+ e~,6 vectors e~,~(6x 7) = 21 scalars e~—*8t/i~and 48 spin ~fields tfr~.(i = 1, 8; a = 5, 11) 3—~ sca ar ~ vec ors ,~,, vec ors ~20 scalars A~,3,,.The field A,,,~,,has as field equation DMF,LP,,., = 0 and since F,,~ = 8[,,A~,,,,1= ~ it follows that= 3,.~’and A,,,,,,, describe vectors (since O[p~AvpIa = F,,,~pa= ~ one has ~‘p~= O~”pa~. Insertingthis into F yields a Maxwell action). As a check one may verify that, as in d = 11, there are 128 bosonicand 128 fermionic states (in d = 5 the little group of a massless particle is SO(3) and hence a particle ofspin J has 2J+ 1 polarizations).Thus one expects that one can write the fields as completely antisymmetric and symplectic tracelesstensors e~,~, Afl”, x”, 4)abcd where 1lai,A~= 0 and I? = 14 X ir2 is the antisymmetric 8 x 8 metric ofSp(8).The reality conditions of the bosonic fields are (A,,~~!~)* = A,,ab and idem for 4)abcd where indices areraised and lowered by 11. Note that under Sp(8) a vector va transforms as ov” = A bV” and that= 0. Hence, OVa = — Vi,A”a where V,, = Vaflab. Only if (V~)*transforms as V0 can weimpose the reality conditions mentioned above. Now O( V°)* = (A ab)*( V~~)*. Hence one must requirethat (A ab)* = ~j b In other words, in addition to being symplectic, the matrices must be antihermitian. Thelocal symmetry group is Usp(8), not T[2 Sp(8). + tiM However,= 0. Multiplying they have bythe Ii same one has generators. also tiMT + Mfl = 0 sinceSymplectic = —1 and taking matrices the satisfy real andMimaginary parts of the sum and differences of these equations, one seesthat Usp(N, C) has as many generators as Sp(N, R). In fact, for N = 8, Usp(N, C) = Sp(P’~,R).

P. van Nieuwenhuizen, Supergravity 357In order to discuss the reality conditions for the spinors, we want to define Majorana spinors. Ind = 5 one has CIAC-1 = +FA.T with A = 1, 5. (In a Majorana representation C = ~4~5• In d = 5, C isantisymmetric.) Thus we define(~D\a — (~c*\t( 4\a — ~ a~~’flab~ ~since, as we discussed, (Aa)* transforms as A0 under Usp(8), and since we can only equate tensors withindices in the same position. We now define a Majorana spinor by )~1)= a~Mwith a arbitrary. We leaveas an exercise to show that taking the complex conjugate of this relation and reinserting in this resultagain that ~D = a~, one finds indeed a consistent result for c~= 1. Crucial for this result is that= —1. (In fact, the most general solution of such (1 is probably equivalent to the symplectic metricwe have assumed so far.) Thus the reality conditions for the fermions are(,~abc)* = x~bCC, idem cl’~. (5)We can now again use the theory of nonlinear representations to predict what the global symmetrygroup will be. Since Sp(8) has ~(8x 9) = 36 generators and there are 42 scalars, one needs a Lie algebrawith 78 generators: E6!The Lie algebra E6 is most easily defined by giving its fundamental 27-dimensional representation1~Z’~— A1kz” + .X’klZ (6)Or” = Awhere A’,, is antiHennitian and symplectic (~QA+ ATA2 = 0 meaning that A, = A11) and ~ is totallyantisymmetric, traceless with respect to D~and (.~iJkj)* = ~ The tetrad is of course a scalar underboth groups (Sp(8) local and E6 global), but gravitinos are in the 8 representation of Sp(8) (anddenoted by iJ’) while photons are in the 27 of E6 and denoted by A~(and thus Sp(8) scalars).Finally, the scalars are written as Va,9~’where the indices a, ~3are E6 indices and a, b are Sp(8)indices.One writes cy ~ as an exponential of E6 generatorso~p r I de,4 j ViIk!\1 ab— ~exp~c ~, u Cjj~ JJajlwhere c are coefficients and A, .~ indicate the various generators which are matrices. For example(A \ ab — ‘~ [a~bJ (A \ a — fl ~a‘~ cdJa$ — ~. cdj[a “$1~ 2, cd/a — [ca d].1O,LLVSince, in general, 7/~where (0, )cd0b = Q,,[0[cOb)dIlies in the algebra (it is not a group element), one has V~8,,’V= Q,. + j’,,is a linear combination of Sp(8) generators. Since~ a_~~a~,4a C~i b c ~ib j~c’b0,. is the Sp(8) gauge field, and one can define Sp(8) covariant derivatives. For example, 1I1D,~=(Just as for the spin connection in d = 4, N = 1 theory, one can find the exact form of 0 by solving itsfield equation OI/OQ,. = 0. This is again 1.5 order formalism.)We will not write down the complete action; for that we refer to the literature. However, let us now

358 P. van Nieuwenhuizen, Supergravityturn to the mass generation. One writes only the fields with curved indices in the form of (1). namely~ (x, x 5) = U” 0 (x 4)U~(x5)A,,”~’(x)U”~(x5)= exp(Mx5) (10)where M is a Sp(8) matrix, which can be taken as0 m1-m1 0M= ••..• . (11)0 m4-m4 0In principle any E6 matrix could have been chosen as mass matrix, but in order to exclude a cosmologicalconstant, one selects the maximal compact subgroup of E6. As it happens, this is also a Sp(8) group. (This isthusaglobal Sp(8) andhas nothing to do with the Sp(8) local which extendedthe 0(6) local.) Since Sp(8) hasrank 4, there are precisely four arbitrary mass parameters generated.Inserting (10) into the Maxwell action, one finds as mass term for the photons“2“2~(photons)= A~~A~(( — )“m10121 + (— )~m~12j). (12)This leads in d = 4 dimensions to 3 massless photons 2 where(A12, i, j =A34, 1, 4 A56 (eachand mass A78, is satisfying doubly degenerate). ~ =A0)and fourth 24 massless massive vector photonfields is obtained with masses by the(m1 reduction± rn,) of the vielbein from 5 to 4 dimensions. One can takern1 = m2 = m3 in which caseonefinds sixmore massless photons, yielding atotal of 10. These represent the 8gluons, the ordinary photon and the Z°or the antigraviton, see Scherk in ref. [391].For the fermions, masses are due to the Sp(8) connections Q~in the covariant derivatives, while thescalars obtain masses through P5. We recall that5) = Vsa~~(x)(U ‘(x5))~ 1(X5))8p (13)0(U—c/t~(x xso that one finds for ‘V’35’V the result9$’)°Va$ = e~M A e” = M . A + [M A, c.~]+... (14)— Vcd~(2MaO’in the special Sp(8) gauge where there are no A generators in the exponent of V. Clearly, 05 starts withM A and hence Q~yields masses to the fermions.The masses of the scalars come from P 1D”‘V = — tr P,,P”and because P 5 because the scalar kinetic matrix is tr D,.7/~5 is linear in masses and scalar fields: P5 = [M A, c.X1 +.... In fact, the E6 algebra hasa Z2 grading: [A, A] -~ A, [A, £] .~, [I, ~]-~ A. (This is thus a Cartan grading of an ordinary Liealgebra. Superalgebras have also a Z2 grading, but the bracket relation of two odd elements is forsuperalgebras symmetric whereas it is here still, as usual, antisymmetric.) Thus the Q,. are even in thenumber of scalar fields c, while the P,. are odd.This concludes our discussion of the way in which one can introduce masses into the N = 8 model in

P. van Nieuwenhuizen, Supergravity 359four dimensions. One obtains a massive action without cosmological constant (which previously wasalways disastrously large) and with a potential which is bounded from below.* It is a beautiful method,but it may not be the last word. Hence (in the author’s opinion at least) it is not impossible that one canalso construct different mass spectra. Perhaps not everything we need for physical applications descendsupon us from higher dimensions. Therefore we refer the interested reader to a result in the literaturewhich gives the most general spontaneous symmetry breaking which can take place in the N = 1 modelin 4 dimensions.As shown by Cremmer, Julia, Scherk, Ferrara, Girardello and the author [95], one can indeed breaksupersymmetry spontaneously (at least in N = 1 supergravity) such that the super-Higgs effect occurs,the potential is bounded from below, there is no cosmological constant (this fixes only part of thefreedom in the action) and a mass formula ~2J + 1) (~)~‘(mj)2 = 0 is found. All masses are furthertotally free, and the model still contains an arbitrary function of two variables. Clearly, localsupersymmetry is much less restrictive than was thought in the beginning of supergravity.Finally, we mention one of the outstanding problems for the N = 8 model: in how far are thequantum corrections still finite if one adds masses? It has been shown that all one-loop divergences inthe corrections to the cosmological constant vanish, leaving only a finite constant correction. However,whether also physical processes are finite is an open question. If they are, one might call the symmetrybreaking discussed in this section spontaneous, since for spontaneously broken theories the theorybehaves in the ultraviolet region as if there are no masses. Let us conclude with an optimistic note.Perhaps the N = 8 model with masses is the ultimate finite quantum theory of gravity, and it will beshown that all particles of nature appear as bound states of this model.6.5. Auxiliary fieldsfor N=2 supergravityRecently, sets of auxiliary fields for N = 2 supergravity have been found. A first set for Poincarésupergravity was found by Fradkin and Vasiliev [228,229], and de Wit and van Holten [164,169], bydirect means. Breitenlohner and Sohnius found a set by starting with the causal group (Poincaré pluslocal scale transformation) [72].All thiswork has beenclarified by superconformal methods. Interestingly,these superconformal methods are quite similar to the corresponding N = 1 case which we discussed insection 4, and it is hoped that this approach will also be useful for the N = 8 case.The original N = 2 Poincaré auxiliary fields constituted the following setBose fields: Am, Am”, Vm”, S, P°,tmn’1, Vm, M”, N”Fermi fields: x’~A. (1)The bars denote symmetric and traceless tensors, the hooks antisymmetric tensors. In the action theyappear as~p.~2jj Ij\2jJj,,ij \2~ A2 ~1I if\2~Ij~lJiI\2 1j’~j \2— ‘-7 22, / 82,&mn) m 42, m) 42, 2’ mJ 22, mJ— ~(M~)2— ~(N~)2 + 2A‘t~”+ ,ØAi) + trilinear terms. (2)In order to make the reduction to N = 1 supergravity, one starts with Setm,. = ësees how two multiplets develop:mt,b~,+ e2ymIfr~,and1y* The one-loop corrections to the cosmological term are finite 1406] due to the mass relations I( — )‘(2J + 1)mi” for n = 0. 1, 2, 3 [208].

360 P. van Nieuwenhuizen, Supergravity(em,., ~ x” A1, S, P11, V,,,, A~,,1,A~)(‘I’~,x2~A2, B,1,2, t,1,~,,,P52. M12, N’2, ~ (3)The first multiplet is, in fact, the nonminimal set of auxiliary fields of refs. [68, 69, 160] for N = 1supergravity. The second multiplet is a spin (~,1) multiplet, which, however, does not coincide with thespin (~,1) multiplet of ref. [347],one characteristic difference being that in the latter there are auxiliarygauge fields (see subsection 1.8, eq. (11)). A check on the set in (1) is that by covariantly quantizing thetheory with auxiliary fields according to the procedure of subsection 2.2, and then eliminating theauxiliary fields by means of their field equation, one obtains the same result as when one quantizes thetheory without auxiliary fields [473] (see subsection 2.8). This equivalence was the basis of theconstruction of refs. [228,229].As expected from the N = 2 model without auxiliary fields, the transformation rules as well as thePoincaré action (but not the de-Sitter action) are manifestly SU(2) invariant. From now on we use SU(2)notation.Based on the auxiliary fields one can define the following multiplets(i) scalar multiplets: (A”,, x”. F”,). These multiplets have nonvanishing central charges (a = 1, 2);(ii) chiral multiplets (in superspace: (1 + y5)D”14) = 0) with complex and chiral (A,, B,, C, Fmn =~,A,);(iii) vector multiplets (the sum of (0, ~)and (~,1) multiplets) containing (A”, B”, F”, G”, tfr’, W,.);(iv) linear multiplets: which are a sum of an ordinary (0, ~)multiplet and a (0, ~)muitiplet in which anantisymmetric tensor represents the spin 0 field (the simplest case is (T~,A, B”, di’, F, G));(v) nonlinear multiplets: which is the first known case where matter fields transform into squares ofmatter fields.Due to space limitations, we cannot discuss these multiplets in further detail, see refs. [170, 171].We now discuss how the notion of N = 2 superconformal gravity sheds light on the structure of theseresults. In N = 2 conformal supergravity one expects as fields which gauge the algebra SU(2, 2/2)(em,., wm0 b,,, f,., ip~,~, V,,’,, A,,) with V,.’, = _(V,,)*~,. (4)One imposes the same constraints on the (modified) R(P), R(M), R(0) curvatures as discussed insection 4. Elementary counting then reveals that one lacks 1 auxiliary spinor field (at least). The auxiliaryfields which one must add, turn out to be: T,d,” (6 fields) + D~(1 field) +x~(—8 fields). Thus one has atthis point a closed algebra for N = 2 conformal supergravity with (24 + 24) fields. (The Weyl multiplet.)All previously mentioned multiplets can be coupled to this gauge action in a locally N = 2 supercovariantway. If one couples the vector multiplet, one finds (32 + 32) fields and a field-dependentcentral charge in the {Q, Q} anticommui,ator which only acts on the photon (as a Maxwell transformation).One can then also couple a scalar multiplet (which needs a central charge). Fixing thenon-Poincaré symmetries by fixing most of the components of the scalar multiplet and some of the Weylmultiplet, one finds a different set of auxiliary fields from the one in (1), although it also contains(40 + 40) fields. As action one finds the sum of the vector multiplet action and the kinetic multipletaction for the scalar multiplet (see section 4). The de-Sitter N = 2 action results if one adds the massterm of the scalar multiplet.The original set in (1) results if one couples a vector and a nonlinear multiplet to superconformalN = 2 gravity.

P. van Nieuwenhuizen, Supergravity 361In both, inequivalent, (40 + 40) sets, Poincaré supersymmetry follows a 0 + S rule (except that for (1)it becomes a Q + S + K rule and for the set which yields the de-Sitter action, it becomes even aQ+5+K+U(1)rule).Aconformal tensor calculus has been developed forN = 2, from which onededuces the N = 2 Poincarétensor calculus. Similar results for the Poincaré tensor calculus based on the set (1) have been obtainedby Breitenhohner and Sohnius from superspace methods [72].It is gratifying to see how starting from a larger symmetry (N = 2 superconformal theory) with fewerfields, one can descend to smaller symmetries with more fields in a systematic fashion. Although rathertechnical, all these results reflect the N = 1 results, and the reader should have no conceptual problemsafter studying section 4.Subsection 6.6 is added as an addendum7. AppendicesA. Gamma matricesThe Dirac matrices we use satisfy {y,,, y~}= 2g,,~with p, i’ = 1, 4 and ys = Y1Y2Y3Y4 with y~= 1. Allfive matrices are Hermitean. A basis for 4 x 4 matrices is given by the 16 elementsO,=(1,y,.,2io,.,,iy 5y,.,y5) with p’(v (1)which satisfies tr(O,O,) = 4S~.We define r,.,, = ~[y,.,y~]so that they satisfy the Lorentz algebra[o,.~, cr,,~,]= g,,~ a~,.,,+ 3 other terms. (2)tm0Umn. Gamma matrices with Latin indices areFurthermore, constant, but with we Greek define indices 0 = D,.y”, one has and~,,. A .~ = A The matrix y~is always constant.Some useful formulae areC abcdU = 2Y50~cd (3)YaYbYc + YcYbYa = 2(c50acyb) (4)0bYC + ObcYa —YaYbYc — YcYbYa = 2 abcdY5Y’~, (~1234= +1). (5)The result for YaYbYC follows by adding the last two equations. ThusY,’Uab = ~(e0,.y6 — eb,.ya + ec,,eO&dy5y~i). (6)Other identities often used are= U”~YaUcd = 0 (7)= ~3, O~”tT~dO~b = (8)

362 P. van Nieuwenhuizen. Supergravi~’yAn often used matrix representation of the gamma matrices is/0 lcTk\0 )‘ ~11 0’—i)’\lUk/0 —10For two-component spinor formalism one uses another representation, see appendix E. A Majoranarepresentation is, for example,10 —10B. Majorana spinors10 iU2\ 11 072 0 )‘ “~ ‘\0 —1/0 1U3‘~‘~~iU3 0~A four-component spinor A”(a = 1,4) transforms under Lorentz transforms actions as(A’)” = = ~[y,~’ ‘y,,] (1)4 are purely imaginary (in our conventions s, i’ = 1, 4 and {y,,, y~}= 28,.,,). Thewhere Dirac conjugate w’~are real spinor and w’ A4y30 (A ~ (y 0 transforms thus as(~)~ = A,~(exp— (2)because o,.,. are antihermitian (y,. is always taken to be hermitian). The Majorana conjugate spinor A T isdefined byAT 0 = A 0C 00 (3)and is required to transform in the same way as the Dirac conjugate spinor. Note that A and AT are ingeneral complex. We will always write the index of A as a superscript and that of A as a subscript. FromoX0 = A0(C )“ (exp ~(4)it follows that the matrix C01,must satisfy (note the position of the indices)C00(o,,~~~(C’)”~ =(5)The most general solution of this equation isC= a(1+ y5)Co+/3(1— y5)Co(6)where C0 is a special solution, usually called “the charge conjugation matrix”, and discussed below, anda and /3 are arbitrary constants. In two-component formalism, C0 is diagonal and equal to ( ,.~,,, ~4B),and since dotted and undotted spinors transform independently, while cr,,,. is also diagonal in dotted—undotted space, the appearance of two arbitrary constants is clear.A Majorana spinor is per definition a spinor whose Dirac conjugate is proportional to its Majoranaconjugate (note again the position of indices)

P. van Nieuwenhuizen, Supergravity 363A,. = A~C 1,0b, b arbitrary. (7)Hence AT= AtC*yb* = ATCY4C*7(b*b) from which it follows that CY4C*Y must be a positivenumber times the unit matrix. This is indeed the case in four dimensions (see below), but, for example,in five dimensions this is not the case and there one needs an internal index in order to define Majoranaspinors. We will always scale C such thatTC. (8)A=AIn order that the relation A = ATC is maintained in time, one has for a spinor whose field equation isthe Dirac equation (y”ö,. + M)A = 0C C-1—— T 9X(C-’7~TCä,.+M)OJ )‘L — ‘/~~ ()For massless spinors, one finds the weaker conditionC~y,.Cl=a~y,.T, a2+1. (10)(One easily proves that a2 = +1 from (5).) In four dimensions one finds solutions with a = +1 and —1, but,for example, in five dimensions a = +1. However, although in five dimensions a matrix C exists, one needsmore than one spinor in order to define a Majorana spinor, as we have seen above.Since the dotted and undotted components of A are mixed in the Dirac equation, one now gets thestronger condition Cy,.C-1 = ±y,,Tfor C, and it is clear that if a C exists, it is unique up to a constant.This matrix C is the charge conjugation matrix. That a C exists follows from the fact that the matrices(—y4,,”) (as well as +y4,,T) satisfy the same group multiplication table as y,,. Hence, since according toSchur’s lemma there is only one inequivalent four-dimensional representation of the finite group with 32elements spanned by y-matrices, y,. and ~,.T are indeed equivalent, as are y,. and +y,.T.For spin 3/2 fields one can essentially repeat their steps, and the same results are obtained.In a general representation of {y,,, y,.} = 20,,,. with, however, hermitian y,,, the charge conjugationmatrix C is antisymmetricCr=_C. (11)However, only in special representations is it true that (C- 1)as = — C,.1, (as should already be anticipatedfrom the position of the indices). To see this, note that using Cy,.C-’ = ay,,T twice, one findsTCy,.C-lCr= y,,. (12)(Cl)Hence C-1C~commutes with y,,. Thus C-ICr= kI, or CT= kC so that (cl)T C= k2C and k = ±1.Thus (Cy,)T= ak(Cy,.), (Q,.)T k(Cu) (Cy5y,.)T —akCy5y,. and (Cy5)T= kCy5. Clearly a = +1and a = —1 are both allowed (C = y~y~and C = y~in a Majorana representation). Since the 16 Diracmatrices are linearly independent and there can be at most 6 antisymmetric matrices, k = —1 and C isantisymmetric.Taking the hermitian conjugate of Cy,.C-’ = ay,,T one finds that CrC= 11 and clearly 1>0. Thus.

364 P. van Nieuwenhuizen, Supergravitvone can scale C such that C is unitarity. It follows that C* = —C~.Only if a = —1 is Cy 4C*y~apositive multiple of the unit matrix. For representations in which y,. is symmetric 1 = —corHowever, antisymmetric, If Ya does onehas not C(Cy,,C~)C-’= have this symmetry y,, then so that thisone not cantrue choose in general. the scaleSummarizingof C such that C-C’~=-C, CC=CC=I, C*C= CC* = C’y,,C-1 = (13)but C-1—C only in special cases.In two-component formalism (see below), the Majorana condition A,. = A1,C0,. becomes(s .\* —— /2. EJ~4(14)which is sometimes written as AA = AA.C. Symmetries of covariants built from Majorana spinorsFrom Cy,,C-’ =1,C7T and ~,.= A 1,,. and idem for x’ one may derive the following symmetriesA~=~A, A7=x= XYmA,AY~7mX= XY5YmA, ~k75X = ~y5A.)tUmnX =(1)In generalAymjym,~Xym,yn,iA()”.(2)These relations are easily proven. For example,= A 1,C 0 (since A andy anticommute)1,0~”= —fC~A= ~“C,,0A1, (since Cis antisymmetric)= kA.Note that if Cy~C’= +y,0T. then no extra minus signs appear.D. Fierz rearrangementsConsider four Majorana spinors A, x~i/i, ~ (they may have external indices, for example t/i,,). Letthem be coupled as follows (again suppressing possible extra indices)I = (~,.M”sx 1,)(~~N~co8). (1)

P. van Nieuwenhuizen, Supergravity 365We will show thatI = (~MOjNço)(ç~O,~) (2)where the 160, were given before. The proof is based on the observation that the matrix M” 1,N~canbe viewed for fixed a and 0 as an ordinary 4 X 4 matrix which can be expanded into the complete set of0,hA” ~T7 — IIVI /31V sc1~2, J)~.The coefficients c follow from the orthonormality propertyThustr(O,0,) = 40,,. (4)I = (A,.x$*vco~~)(~M01N)0s(0j)Ts (5)and one finds the desired results if one takes into account that all spinors anticommute.As an application one may show by “Fierzing” twice thatand(~A)A”= ~y5A)(ysA)” = y5ymA)(y5ytmA)0(~y5A)A= —(~A)(y5A) (6)(AY5YmA)A =~A)y5ymA.An important identity which follows by Fierzing once is(c~,,y”tfr,.)(yal/’p)E~” = 0. (7)It is used to show that torsion terms in the gravitino field equation cancel.E. Two-component formalismSince SO(3, 1) — SL(2, C) and four-component spinors transform as the (~,0) + (0, ~)representation ofSL(2, C), it is advantageous to work with the irreducible 2-dimensional spinor representations. One canalways go from four-component to two component XA = ~(1+ y5~0A”,~A = ~(1— Y5)A0A 1,, but in orderto solve algebraical constraints like those in superspace, two-component formalism is indispensable.Consider the particular hermitian representation of the Clifford algebra {y,,, y,.} = 20,,,. (js, ii = 1, 4)in which y~is diagonal and in which the rotation generators 0-,~= ~(y,yj — y~y,) are proportional to the

366 P. van Nieuwenhuizen, Supergravitvusual SO 3 generators (cr1, cr2, cr3). This representation is unique up to the signs of y~and y5. We choose/0 1cr,,\ /0 1\ 7~~j /1 —1 0Ykl~j~,, 0 )‘ ~ o)’The Lorentz generators o,.,, are diagonal, and under Lorentz transformations (A P)0 = (exp ~w cr)”,,A b(x’Y = [exp(w+ iv)(io./2)]~~BxB, (~‘)A= [exp(4~,— iv)(iu/2)]A8~B. (1)We define (fl’)A and(A’~by ~x’Y(n’)A = X”fl~and (A’Y(C’)A = AA~A.Note that w’ = k11~wJkand v’ = iw4are real. Thus (xF)A = LABXB with L E SL(2, C) and ~A transforms as ~‘A = 1AB~Awith lAB the Hermitianinverse of L.it is straightforward to prove that ~, ,.~, ~, ~4j~are invariant tensors under SL(2, C). (Afortiori, they are invariant tensors of the subgroup SU(2), and ~A = ~B BA transforms as ~ ForSU(2) one therefore denotes tensors which transform as the complex conjugate of tA by tA, and forSU(N) a similar convention is adopted although for N >2 no EBA exists. Instead one has the sameresult that (~A)*transforms as ~A =11BA~For SL(2, C) the transformation~Bçj~~ properties if oneof considers the groups USp(N) with symplectic metric(~~~)* ~.A differ from those of ~A, but here anotherrelation exists, namely (A,j* transforms as ‘~~B.)We make the at first sight unusual normalization— A.B...... •. AE.....1 ~ — —~—f —--~ ~—- —i vOI’ — , —6AB and C is a Hermitian matrix. The tensors are what the metric ~ is in fourThus ( AB)tdimensions.=They raise=and lower indices as follows XA = XBEBA XA = ~~XB (idem with A, B). In orderthat CAB = cA DB, one finds that 12 = 12, and we always contract as indicated: from left-above toright-below (defining XA = XB BA would give the opposite sign).It is straightforward to prove that XA transforms as (~A)*, and ~A as (x~4~*.This is really thejustification of denoting complex conjugated spinors by dotting their indices. Thus we define generaltensors XA and ~A by(x’Y = (L~~~B)*xB _LAi~xBIr’\ _iT.B\*r _i Bs-~)A2,’A )~BtA~B.(3)Another invariant tensor is OAB and SAB. In fact ABEBC = —O’~’c,so that the 0 and tensors arereally the same abstract tensors. The charge conjugation matrix in this special representation is diagonal/ jU3 0\C=y4~y2=k0 ~AA). (4)Thus a Majorana spinor satisfies ~A = (C4*.The gamma matrices define 2 x 2 matrices which map an upper dotted index into lower undottedindex, and vice-versa,— ./ 0 ~(5)

P. van Nieuwenhuizen, Supergravity 367One thus finds= (o’, il)’ 48 and (U,,)AB = (u, —iI)~ 8. (6)Since (cr,. )AB are Hermitian (which means antiHermitian for s = 4 in our conventions), lowering itsindices to (O,,)c~one finds the transpose (or complex conjugate) of (0,L)AB, namelyNote that(u~)co (cr” )‘4~ ~ = U~Dc. (7)48(o~”)nc= 0”~c (no sum over it). (8)(cr,.)’Raising of the index ~ is done by 0”~= (+, +, +, +). with ,a, ii = 1,4 in our conventions.The matrices (o~J’~and (o~,.)’48are invariant tensors of SL(2, C). That is to say that, for example,(cr~’~48)’ = L’4cL~L~,.u~’~ = U~~AE (9)One can thus express tensors with Lorentz indices into tensors with spinor indices as followsT~..:-* T~: (U,. rM(O-”)RRT~..:. (10)Often one writes TRR ~ as TRSRS, but one should not confuse this with TSRRS.The connections (cr,.~8 satisfy the following completeness relations~CD D C(o,,)An(o ) —20 A0 g(cr,. )AB(U~)cD = 2 A~BD= (ci,,)BA(O~)Dc (see (7))(U,.)AB(U~~)BA= 20~= (u,.)AB(ci~)~~ = (0,.)~8(0.o)~4B(cr,,Y~ 8(cr~)’~ = 2 ~ B1~~ = (cr,,)~(o.1~f The reason we defined ~~B= —1 for A = 1, E = 2, is that with this definition no minus signs occur inthese relations. One can use them to go back and forth:r’ I AR CI)i,r,,,. — CT,. FABCD(12)FARCD = F,.,XJ~AE1Y = ~(11)where FBADC= F,.,.(o~”)8’4(o-”)DC.In fact, quite generally VAJ3 = VEA, as one easily proves usingUMAB = ciBA and o~~4B=0RA (13)An antisymmetric tensor GAB = — GRA is clearly proportional to EAR. Hence, for an antisymmetric

368 P. van Nieuwenhuizen, Supergravitvtensor F,,,. = —F,.,. one hasF,,,. -* FAA CD = EACPm5 + ~jTAC (14)where PADand ‘r~care symmetric. Repeated application yields for antisymmetric tensor ~“~“with12~~t = +1 the resultfM!1~RSS~~1*~(cr,,rM ... ~ = 4( MN RS MR N — ~ (15)To check this expression, note that it vanishes when two pairs of 2-component indices are equal, andthat it agrees for a particular choice of M, M, ... , S. Also one may verify that 0~goes over into 20~0~.We now give a few applications. If F,.,. is self dual~ then p~j=O. (16)The proof is simple: multiply by ~ )AB(O~)cD and use the completeness relations to prove thatj~RRSS = LP°’I \RRj \SS — RR’ SS’ RR’ S.~’~i —i 2,U,,,~ 2,O~) — I R’R’S’S’where FRRSS = F,,,.(cr”)RR(o-~)ss.Inserting the expression for ~~“’one finds indeed that4F,,,. = 0, then FMMNN = ~ (18)F,.,. +For anti selfdual tensors it is of course p,~.,which survives.If F,.,. is real, then it follows with the Hermiticity property [(o~~)AB] * = (.)s**4(ci~ )RA thatTAC(flAC). (19)The minus sign is due to (EAB)* =Consider now a tensor spinor R~,.with the spinor index cv running from 1 to 4. Substituting theexpressions for y~and y,, one finds with R”,,,. = Rfl’,. + R,,,.A that~(1+ y5)(y”)”1,R~,.= (_iU~)ACT,,MMU,.~~RA.MMNN(20)RAMMN!.J =M54IAJ~4J~1 + ~TAMN (idem with A).Thus the constraint ~(1+ 75)y”R,,,. = 0 is equivalent toAM.. ~4. —CANE PA,MN — ONTAAN —Hence, the constraint y~R,.,.= 0 says that R”,.,. is in two-component formalism a sum of a function withthree undotted indices, and a function with three dotted indices, both completely symmetric.Two weaker constraints one often encounters are now discussed. If cr”~R,,,.= 0 this implies that pand r are symmetric=0 then PAAR = TA,AB = 0. (22)

P. van Nieuwenhuizen, Supergravity 369The chirality-duality condition R,,,. + ~y 5R,,,.= 0 yields two relations if one projects with ~(1± y~)D A — I DPO’A — ~ .~,, A . —1%,,,. 2 ~,’po~ — ~ MN —1 DPP’—15—~., . —~ 2~,.po~2”A — V 7’ TA,MN —D. L(23)Hence these two conditions together yield the same result as y”R,,,. = 0 alone. In four-componentformalism this is easier to see: if y~R,,,.= 0 then from cr”1,o”~R,,,.= 0 it follows upon expanding cr”1,o-~~as2c,”1,o~~= {~“1,, o~”~} + [CT”1,, CT”] (24)that indeed R,,,. + ~y5 ,.,.,,.,R”= 0.F. Supermatrix algebraThe following subsections deal with algebra and analysis using anticommuting variables. For moredetails, see a forthcoming book by B.S. DeWitt, P.C. West and P. van Nieuwenhuizen.Supermatrices are matrices M = (~)whose Bose—Bose parts A and Fermi—Fermi parts D are evenelements of a Grassmann algebra and whose Bose—Fermi parts B and Fermi—Bose parts C are oddelements. They are multiplied as ordinary matrices; this multiplication rule is associative and theproduct is again supermatrix. There is the obvious unit element. In order to see whether they form agroup, we construct an inverse by first decomposing M asM=ST,5~CfA I)’ 0\ T-~— /1 A’B 0 D-CA’BM-UV, U-~- /I B\ /A-BD 1C 00 DI’ ~ D’C IM’ — T’S’ — /1 —A~B(D— CA’B)’\/ A~ 0— — ~0 (D—CA’B)~ )k,—CA’ IM~—V~U’—/ 1C(A—BD1C)~ (A—BD’C)’ 0\fI lAO —BD~ D’— ~—DNote that M1ff = (D — CA’B)’. Thus an inverse exists if Mb. and M’~ are non-singular, or if M~bband Mff are nonsingular. Let us define A(ord) as A minus its nilpotent part A(nil). Thus A =A(ord) + A(nil). Then the inverse of A exists if A(ord) has an inverse because one can expand A1 as[A(ord)]’ times a power series in the nilpotent elements (this power series is in fact a finite series):A1 = A(ord)’[l —A(ord)A(nil)+ [A(ord)A(nil)]2 ...] (2)Since in D — CA1B the terms CA1B are nilpotent, it follows that an inverse exists if and only ifA(ord) and D(ord) have an inverse. In other words, M has an inverse if and only if M(ord) has aninverse.

370 P. van Nieuwenhuizen, SupergravityThe superdeterminant is defined bysdet M = det Mbb det M~,,= det A det’(D —(3)= det 1 M1bb det~1M 1C) deFt D.1, = det(A — BDIt is sometimes called the Berezinian. In other words, the superdeterminant is the determinant of theBose—Bose part of M times the determinant of the Fermi—Fermi part of the inverse of M. Thisdefinition should of course be shown to be the one needed in the applications. We shall show below thatin path integrals the Faddeev—Popov determinant is indeed this superdeterminant. One usually replacesthis definition by generalizing the well-known formula det A = exp(tr ln A)sdetM = exp(str ln M)str N = ~(—)“N”~ (4)ln M = ln(1 + M — 1) = ~(M — 1)k ()k_I k’where a = 0 for A = bosonic and a = 1 for A = fermionic. (These minus signs for the trace overfermionic elements are reminiscent of the minus signs for closed loops of fermions in Feynmandiagrams.) With this definition of the supertrace one has strM1M2= strM2M1. If M1,2= expN,,2, onecan use the Baker—Hausdorff rule to obtain M,M2 = exp(N, + N2 + multiple commutators) and itfollows from the definitions of the supertrace thatsdet M1 sdet M2 = sdetM,M2. (5)Using the decomposition M = ST and computing str ln S = tr in A, str in T = —tr ln(D —finds the result for sdet M From the definition of sdetM in terms of str in M, it follows thatCA ‘B), one0(sdetM) = (sdet M) str(M’OM) (6)as long as M is a supermatrix. (ln M = in M(ord) + ln(1 + M~’(ord)M(nil)) exists if M ‘(ord) exists.)Another nice way of deriving these results is to start fromM=(’~ ,g)[(~ ~)+K], K=(D21c A~B) (7)so that sdet M = det A(det D)’ sdet(1 + K). Since K is nilpotent, ln(1 + K) is well-defined, and againwith sdet(1 + K) = exp str ln(1 + K) (easy to check directly in this case) one finds1C). (8)sdet(1 + K) = det(1 — A’BDThus one finds again the result for sdet MFor supermatrices onemay define a transition rule satisfying (M1M2)T = M2TM1T. This relation holds ifone defines

P. van Nieuwenhuizen, Supergravily 371MT=(BTA T ~,-~(fDT), cr=±1. (9)It follows that sdet MT = sdet M and (M_~)T= (MT)_l. This transposition rule follows naturally from1’ = M’,x’ and 1’ = x1(MT)1I. In our conventions, A is the Bose—Bose part of M and cr = +1. (If onewould have defined that D is the Bose—Bose part then one would have found that o~= —1.)G. SuperintegrationIn path-integrals, the integration over anticommuting physical fields must be translation invariant inorder that the expectation of the field equation vanishes. For example, for a Dirac electron i/iZ = J d~tidi~expI(’q, ~)/ 0 \ 1 —3I~—~_n)ZJ dçlsrdi/i—(expl)=0.with I(q, ~) —i~Iifr+ ~fr+(1)Thus, for anticommuting variables 0 one requires f dO(d/dO)f(O) = 0, and since quite generally f(O) =a + bO, one hasf dO=O, fdOO=1. (2)The number 1 in the second integral is an arbitrary normalization. Indeed, the integral is nowtranslationally invariantdof(O) = b =f dof(0 + ). (3)Thus, on physical grounds, one requires translation of invariance of the 0-integral.A closely related argument is that if translational invariance holds, then one can complete squares inthe exponent in the path integral and perform the integration. One is then left with ~(J)’~ in theexponent, and functionally differentiating with respect to ~, one recovers the usual Feynman rules.Anticommuting ghosts appear in path integrals when one exponentiates a Faddeev—Popov determinantas followsdet M = f [I dC~dC exp(iC~,M~1C’2). (4)If one defines C~to be anticommuting and defines the integrals as for 0, only the term (C1MC2r in theexpansion of the exponential contributes and yields det M (up to an irrelevant constant). (The minussigns in the expression for det M come about correctly because the C’s anticommute.) Thus, also forFaddeev—Popov anticommuting ghosts the same integration rules hold but now derived from a differentphysical requirement.

372 P. van Nieuwenhuizen. SupergravityIn what follows we will show that(I) fixing the gauge in the path integral one finds as a generalization of the Faddeev—Popovdeterminant a superdeterminant;(ii) exponentiating this superdeterminant by means of a super Gaussian integral one finds thegeneralized ghost actions.Consider a path-integral with bosonic and fermionic symmetries. We want to multiply by unitywriting as usual1 = Jd~”d~ 1,0(Fa foe) o(B 1, — b1,)J (5)where ~(~)are the bosonic (fermionic) gauge parameters, B(F) the gauge choices and b(f) ordinary(Grassmann) variables. We must first define the Dirac delta function for fermionic arguments. SincefdOO = 1, it follows that 0(O)= 0. Thus 0(F,. —f,.) is equal to (F,. —f,.) for fermionic F,. —f,,. Making achange of integration variables (~,~)-÷ (B, F), it follows that J is the Jacobian for the change ofintegration variables (B, F)—~~ (~,~j).Theorem: The Jacobian for x” —~ x’(x’, 0’),9”’ —* 0”(x’, 0’) is given by the superdeterminant1,/3x’)] (6)—det[~x”/3x’— (19f/o0”) det(t90”’/t991,) (30’/39)~”1,(89’where it does not matter whether one uses left or right derivatives except that lJx”/90” is a rightderivative. (In all other cases we use left derivatives.)We will begin by considering linear transformation(x’\(A B\fx 7~o’i ~C DROand derive that J = sdet M In this case it is clear that 3x”/30” is a right derivative. Consider first thecase of linear changes of anticommuting variables 0’ (i = 1, N). FromJ d9~ .dONOSJ 61=1=JdOhI...dOlNO7~’~=Jdo1...deNJo~N...o~1(8)with 0” = D’,O’ it follows that J = det D’, just the inverse oL what one has for bosonic variables. Forgeneral linear transformations involving commuting x’ and anticommuting 90 one has/x’\ fA B\ fx\ /1 BD1\ fA—BD’C 0\ / I O\ fx 9~o’) = ~C D) ~e) = ~o i I ~ 0 D) ~D1C i) ~o ()The first and last transformations are shifts of integration variables 0” and x’ respectively, and since theintegrals are translationally invariant these Jacobians equal unity. The second change of integration

P. van Nieuwenhuizen, Supergravity 373variables is diagonal in x and 0, and one finds indeed that for linear changes of integration variables thesuper Jacobian is equal to the superdeterminant.For nonlinear changes of integration variables, the super Jacobian is given by the result previouslygiven, or by the equivalent resultJ = det(3x”/ôx’) det’(3O”’/aO1, — 3O”’/9x~(Dx’/ax)1~3x”/301,) (10)and this result is derived in the next subsection.Finally we discuss how to exponentiate the superdeterminant. We claimsdetM= JdCexpiciJvieiC~, dC=fldC~dC~ (11)where C~denote commuting as well as anticommuting ghosts, namelyC~M,,C’2= C~A,,,.C2”+ C~B,,,.C2”+ C~CO,.C~ + C~’DO1,C~ (12)and C~’are anticommuting while C’ are commuting. The proof is based on translational invariance0= (—~(0M‘)“ fdC (exp(iC1MC2))= (—Y(0M 1)0 J dC[iM,1— M,kC~C~M,,]exp(iC1MC2)= fdC[istr(OM 1M)— C~OMIk C~]exp(1C 1MC2) (13)since interchanging [(8M 1)MC 2]’ and (C,My cancels the factor (—)‘ and yields an extra factor (—1)since ghosts have different statistics.WithI = fdC exp(iC1MC2) and str(0M’M) = —str(OMM~) (14)one finds thus thatOlnI=OlnsdetM (15)which is indeed what we intended to prove.One often rewrites the ghost action as1. (16)..~‘(ghost)= C*IM,1CIndeed, the Feynman rules are the same, but since (for example in Yang—Mills theory) M is not

374 P. van Nieuwenhuizen, SupergravityHermitian, it is not true that the extra terms C 1MC~+ C2MC2 vanish (their kinetic parts vanish). Notethat we did not have to assume any special properties for MSummarizing, we have the following results1 =Jfldz’~o(F~(z)—a~)sdet(8F~/3z), ~A = (x”, 0”)sdet(3FA/OzB) = fri dC~dC~exp(iC~(DFA/3z”)C~’).(17)H. The super Jacobian is the superdeterminantIn the covariant quantization of supergravity using path integrals, an essential ingredient is the superJacobian. We show here that for a general change of integration variables (x, 0) to 1(x, 0), O(x, 0) thesuper Jacobian is given by the superdeterminant:m(x, 0)/ar). (1)J = det(Ox’(i, ë)/91’) det(80Defining the supermatrices M and M’ byM — (Dx/81 8x/30\ M’ — (31/Ox 31/30 2— \a0/81 89/30)’ — ~09/8x 80/80one need specify only for Ox/0O and 31/00 whether one has left or right directives. One needs rightderivatives, in order that (Ox/31)(31/Ox) + (Ox/00)(00/Ox) be the unit matrix. Hence the super Jacobian isthe determinant of the Bose—Bose part of M (the usual result) times the determinant of theFermi—Fermi part of the inverse of M For infinitesimal changes one has J = ~ Ox’/81’ — Oem/Oem, and aproof that J equals the superdeterminant has been given by DeWitt, based on the principle thatexponentiation of this linearized result gives the superdeterminant. Here we give a direct proof valid forfinite transformations and due to Fung. -We consider the change of variables(x, 0) —* (1, 0) in two steps(i) we change 9~,... ~ to em, ... , 0~,keeping all xfixed;(ii) we change x’ x”~to 1’,. . . , I”, keeping all 0 fixed.The second change of variables has an ordinary Jacobiandet(Ox’ (1, ë)/D1’).Notice that it is essential that this change of variables is performed when the Fermi variables are alreadybarred — otherwise one would find the mixed determinant det(0x’(x, 0)101’). We now consider (i).Consider an integral of the formI=Jdxl...dxkdom...de1P(x,o)— — (3)= J dI’ . . . dl” dotm .. d~1JP(x(1, 9), 0(1, 9))

P. van Nieuwenhuizen. Supergravity 375where P is a polynomial in 0 and an arbitrary function of x, and J the super Jacobian to be determined.One can for any given i decompose P(x, 0) = A .+ O’B, where A and B, are independent of 0’. Thus,with i = 1whereI dxdO(A,+01B,)=fdxd0mf- - (4)dx dotm ••~dO2do1(A,+O1B,)~_fdx dOtm •~•d02(~9Ü’/~9ö’)B,0’ = ~‘(x’ X~z,~ . .. , 02, ~1) (5)is obtained by inverting 0’ = O’(x1, ... , x~,em, ... , 0’) for fixed x and em, ..., 0’. Thus, in going fromdo’ to di’, the Jacobian equals (Oö’/0O’)’.Next we change 02 to ë2 keeping x, 0~ o~and O~fixed. Again one can write (0O’/801)P asA2 + 02B2 where A2 and B2 do not depend on g2 and defining2(x’, ... , x”, ~m , 0~ë2 ~•‘) (6)02 = ~one finds a Jacobian (002/002)1. The total Jacobian for (i) is thus(aö’/a~)]1. (7)It is now a general theorem that for any change of variables y’ = y’(9) with i = 1, m one can writethe Jacobian as followsdet(0y’/89~)= (09m/09m)... (091/891) (8)where 9m(7) = ym(9) and 9m_1, . . , 9’ are obtained by successively inverting y = y(9) as followsyl=9l(9l,y2,.~~,ym)y2=92(7l92ySytm)yrn=9m(9l9m)(9)The proof is simple. Define F’(y1, ... , ym, 91, ... 5?fl) = — y’. Thendet(Oy’/89’) = (_)m det(OF’189’) deF’(OF’/Oy’)097091 0 ~ —1 891/0y2~ o9~/a9’ 892/a92 0... ÷ 0 —1 092/0y3.= (091/091). .. (09m/05m) (10)

376 P. van Nieuwenhuizen, SupergravityApplying this theorem to the transformations under (i), the proof of the super Jacobian is completed.(Incidentally — the general theorem is in fact a proof that the ordinary Jacobian equals the ordinarydeterminant.)I. Elementary general relativityThe -tensors ~“~and ,,,.,,,, are always +1 or —1 ( 12~ = 1234 = +1). They are a density and anantidensity, respectively. Thus ~ is obtained from by lowering the indices by means of the metric anddividing by (det g,,,.). The tensors ‘~“~“and abcd are also ±1and have density-weight zero. Thus E,.b~,,isobtained from ““ by lowering the indices with the Minkovski metric (0,,,. = + + + +) in our conventions).Consider a four-dimensional manifold with(i) connections which define parallel transportOv” = —F,,~ 0v”dxv,(ii) a metric g,,,. defining distances (ds)2 = g,,,. dx” dx”.Hence the covariant derivative is D~v”= O~v”+ f’1,~”v1,.Length is preserved under parallel transport ifand only ifDog,.,. = 0 = 8~g,.,.— f,,~”g,.,. — T,.~”g,,,,. (1)Parallel transporting an infinitesimal vector u” along an infinitesimal v1,, and v1, along u”, theparallelogram closes if and only if F,.,.” — F,.,.” = 0. Torsion is defined by the torsion tensor S,,”,. =— fe,,). One usually requires in general relativity that length is preserved and torsion is absent. Inthis case F”,,,. equals the Christoffel symbol {,~‘,.}but let us relax these restrictions.If length is preserved, but torsion is present, one can solve (1) to obtain=VA = Ap(ça ~ 0 +S~i~’ 5 ‘~ ~p5o,. ~,.5op ,.p,5~(2)The tensor KA,,,. is often called the contorsion tensor. Conversely, if p Ag,. is given by (2) with arbitrarySt,,, then length is preserved.If length is not preserved but changes proportional to dx” and to the length itself, one has0(ds)2 = (~,,dx”)(ds)2. (3)The covariant vector /,,. is the Weyl vector and is by us interpreted as the dilation gauge field. NowD0g,.,. = 4,.g,.,. (4)and

P. van Nieuwenhuizen, Supergravily 377pA,.,. = + KA,,,. + ~g~(çb,,g,.,. + ~ — çb~g,4. (5)Conversely, for such j’A~,. length is changed according to (3).So far, nuilness is preserved. The most general relation readsDog,,,. = 4~g,.,,+ 7,,.,. (6)and does not preserve (ds2) = 0. One can clearly require that T,.,,~= T,.~,.and g””T,.,.,. = 0 by redefiningthe Weyl field. 1~ is called the shear tensor; it stretches and shrinks length. The connection is given by(5) with ~,,g,.,. replaced by ~ + T,,,.,,. (Instead of 1,,t = 0 one can make T,.,,,,, completely symmetric.)Under parallel round transport a vector v” is changed intoOv” = —~R”,.~v”~x’~dx°R”,.~ ~ (7)When spinors are present, covariant derivatives are defined by adding connections for local Lorentztransformations. For a spin 1/2 field D,.~= 0iiX + ~W,,tm0UmnX see Weinberg’s book. All that is requiredis that co,,tm” transforms as a world tensor under general coordinate transformations= ~“0,.w,.”’0+ (0,.~v)W,.m0 (8a)and as a connection under local Lorentz rotations= —(8,.Am1’ + w,.mt~Apn+ W,7AmP). (8b)In this case, (D,.~)°transforms as a tensor T,,”. There are many ~‘s satisfying (8a) and (8b). Theparticular connection w,.’8(e, ~r) used in supergravity has the advantage that it is a tensor under localsupersymmetry transformations, i.e., its variation contains no 0 terms.In supergravity one need never specify f~,.;one only needs w,.mfl This is possible since one alwaysdeals with world curls such as D,,I/J~— Dpi/i,, for which no general coordinate connection is needed (onecould, however, add one. This would not lead to supergravity). However, one can define F”,,,, in termsw,,m”, just as one defines in ordinary Einstein—Cartan theory {,‘,,} in terms of ø,.m0(e). Namely, onepostulates that the tetrad is covariantly constantm — ,~ m ra m ma —— ~e ,. ~ ,,~e a w~ e0,, —If this postulate is assumed, then one can rewrite the Riemaun tensor as in subsection 1.3.The derivative D,. satisfies the Leibniz rule and D,.y5 = 0. In the presence of torsion one still hasD,. ,.&d = D,. ,.,,,,.,. = 0.J. Duffin—Kemmer—Petiau formalism for supergravityIn this formalism the action for bosons has one derivative only, so that bosons and fermions have the

378 P. van Nieuwenhuizen, Supergraviiysame dimension. One might expect that this is the formalism best suited for supersymmetry. Let us takea closer look.The action for a real spin 0 field is given by~E=—~A(p”o,,+M)A, A=AT~. (1)The matrices f3” are supposed to be Hermitian and to satisfyf3~/3~J3A+ 13Ap~p~= 3~~13A+ 0PA13~~. (2)Just as the equation (a,.u” — 1)(b — 1 is solved by the Dirac matrices a,,, = b~= y,,, in thesame way the f3” are a particular 0.u” solution + 1) =of* uTuT(a,~u”u°’ + c~u~— 1)(bAu” + 1) = u~uT— 1. (3)The Lorentz generators are Utm” = ~[13m 13”] and satisfy the Lorentz algebra as a consequence of (2).One can define A = At~by requiring that AA is Lorentz invariant. This leads to[~,Umn]”O and {~,cr~4}=0withm,n=1,3.A Majorana conjugate boson A = ATn is defined by requiring that A transform as A. This leads toflo~,,,vfl_l= ~ A Majorana boson is defined by A = A. In order that this relation is maintained intime under (1), one needs -113~,T =There exists a 5 X 5 representation of (2) for spin 0 fields and a 10 X 10 representation for spin 1fields. A Majorana representation of the former is given by4\ —(~ )5j — (J) )j5 — 01, I,p )54 — ~(‘P )4~— 1.— — ~k io~\ — 1nIn this representation ~ = ~= 2~3~/3” + 1 = diag(1, 1, 1, —1, —1) commutes with 134 and anticommuteswith13k• Also /3~L1 = 3~,T However, ~T = ~ whereas for fermions C’~= —cThe field equations of (1) read in this representation MA,. + OkA5 = 0 and MA4 — iO4A5 = 0 so that Aequals the five-vector (OkA5, —i04A5, —MA5). Inserting these equations into the action, one finds theKlein—Gordon action for ~ = iA5q~iA5. (5)Consider now as a model of supersymmetry a Majorana boson A and a Majorana fermion ~!i.Theaction reads= —~A(8”a,.+ M)A — ~y”0,, + M)~/i. (6)It is invariant under* I thank Dr. Dresden for pointing this out to me.

P. van Nieuwenhuizen, Supergravily 379OA = (iO,.i/i)(J3”u) + (ëy”O,.ifr)u= —(üO,,,A)(y” )— (ü/3”0,.A) . (7)The five-vector u = diag(O, 0,0,0, 1) is Lorentz invariant. To prove the invariance, one may useAB = BA, A13”B = —Bf3”A for Majorana bosons, and f3”f3”u = 0””u while ü$”f3” = ÜÔ”’.The algebra closes only on-shell, even if one adds a pseudoscalar B just as in the usual formulationof the Wess—Zumino model. Off-shell closure requires six auxiliary spin 0 fields and two auxil~aryMajorana spinors. Since this number is considerably higher than the two auxiliary fields F and G in theusual formulation, it is to be expected that Kemmer—Duffin—Petiau formalism does not lead to a simplerformulation of supergravity.For further literature seeR. Gastmans and W. Troost, Leuven preprint.P. Roman, Theory of elementary particles (North-Holland, Amsterdam, 1961).H. Umezawa, Quantum field theory (North-Holland, Amsterdam, 1956).A. I. Akhiezer and V. B. Berestetskii, Quantum Electrodynamics (Interscience, New York, 1965).N. G. Deshpande and P. C. MacNamee, Phys. Rev. D5 (1972) 1389.K. GlossarySupersymmetric transformation laws (SSTL) — A set of continuous transformations which transformclassical, commuting, Bose, real or complex, integer spin fields, into classical, anticommuting, Fermi,real (Majorana) or complex (Dirac), half integer spin fields and vice versa. In the usual technicalsense, a supersymmetric generator transforms a real Bose field into a real (Majorana) Fermi field.Supersymmetric model (SM)—A classical Lagrangian field theory in flat space-time, whose action isinvariant under one-global (x-independent) SSTL.Extended supersymmetric model (ESM) — The same as before, with “one” replaced by N (N 2).Supersymmetry — The domain of Mathematical Physics which studies SSTL’s, SM’s and ESM’s.Superspace — A set of coordinates z” which generalizes ordinary space-time through the introduction ofextra Fermionic anticommuting coordinates 0”’ (cv = 1,.. . ,4; i = 1,. . . , N) carrying spin ~. Thisgeneralizes the usual concept of a “point” to the concept of a “spinning point”, referring “spin” tospacetime rather than to the fields.Superfield — A function O(zA) with or without indices.Supermultiplet — An irreducible representation of simple (N = 1) or extended (N 2) supersymmetry interms of Poincaré covariant fields.Matter supermultiplet — A supermultiplet with maximal spin JMAx = 1 or ~. For JMAx = 1, they are calledvector multiplets. For JMAX = ~, they are called scalar multiplets.Supersymmetric Yang—Mills theories (SSYM) — The self-interacting field theory based on a vectorsupermultiplet, with dimensionless gauge coupling constant g, containing as a special case theYang—Mills theory.Goldstino — The massless, spin ~ fermion associated with the spontaneous breaking of supersymmetryinvariance.Gluinos — Supersymmetric partners of the gluons, having spin ~ (colour octets of zero electric charge).Majorana representation — A representation of the y matrices where they are all real or imaginary(depending upon the metric convention) such that the Dirac operator is real.

380 P. van Nieuwenhuizen. SupergravilyMajorana spinor — Aspinor whose Majorana and Dirac conjugate are equal. In a Majorana representationfor the y matrices, it is a real spinor.Weyl spinors — Left- or right-handed spinors, i.e. eigenfunctions of (1 ± ys) with ~ = +1Majorana—Weyl spinors — Have both of the previous properties. This is possible only for space-times ofdimension D = 2 modulo 8.Rigid supersymmetry — Supersymmetry with constant parameters. The more familiar term globalsupersymmetry has the drawback that it is not meant to refer to global topological properties.Global supersymmetiy — See above.Local supersymmetry — Supersymmetry with spacetime dependent parameters.Simple supergravity models — Supersymmetric models in curved space-time, invariant under one SSTL.Extended supergravity models — Same as before, with “one” replaced by N (N 2).Supergravity — The domain of Mathematical Physics which studies simple (N = 1) or extended (N ~ 2)supergravity models.Gravitinos — The gauge particles of spin ~, associated with local, simple or extended supersymmetry.Gauge supermultiplet — The supermultiplet containing the graviton, the gravitinos and eventually (forN 2), lower spin fields.Pure supergravity — Simple or extended supergravity models based on a gauge supermultiplet with a selfcoupling K, of dimension the inverse of a mass, uncoupled to any matter multiplet.Super Higgs effect — The supersymmetric analog of the Higgs effect whereby, when supersymmetry isspontaneously broken, a gravitino (or several of them) “eats up” a goldstino (or several) andbecomes massive.Antigravity — A vectorial force, due to the exchange of a vector particle which is a member of anextended (N 2) gauge supermultiplet and which between two particles exactly cancels the gravitationalforce in the static limit, and doubles it between a particle and an antiparticle.Tetrad — The gauge fields of spin 2 which must be used instead of the metric when fermions arepresent.Vierbein — Same as tetrad.Supervielbein — Vierbein in superspace (viel = many in German).Graded Lie algebras — A Lie algebra with a grading. For example, the Lorentz group with k and j has aZ 2 grading.Superalgebras — A Z2 graded Lie algebra whose elements are “even” or “odd” such that the bracketrelation for two odd elements is symmetric, while also the super-Jacobi identity is satisfied. (Hence asuperalgebra is a graded algebra, but not the reverse.)Fierzing — Performing a Fierz rearrangement on four spinors (see appendix D).Vector multiplet — A multiplet containing a vector field. In N = 1 supergravity, it contains a real vector, aMajorana spinor and an (auxiliary) scalar D.Scalar multiplet — A multiplet containing scalar fields. In N = 1 supergravity it contains a Majoranaspinor and two propagating and two auxiliary spin 0 fields.Multiplets — Irreducible representations of superalgebras. In global supersymmetry, these representationsare linear, in supergravity they are nonlinear in fields.Supercovariant — An object whose supersymmetry variation does not contain terms with 0,, .L. ConventionsIn order to facilitate comparison between the conventions used in this report and those used by otherauthors, we give here a table of translations

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P. van Nieuwenhuizen, Supergravily 395Addendum6.6. Supergravity and phenomenologyThe nongravitational forces around 102 GeV seem to be adequately described by SU 3 (color) x SU2 xU1. If one uses the renormalization group 2, they techniques seem to come determine together the dependence at iO’~GeV of(the “grand corresponding unifiedthree mass” running M). Although couplingthis constants is not on far Qfrom the Planck mass (h/G)1”2 c512 = 1019 GeV, it is commonlyassumed that the nongravitational forces first combine at M into the simple group SU5. Since SU5 (orother candidates like E6 and SO~~) have many free parameters, while also gravity is left out, theseresults cannot be the conclusive end point of unification.Supergravity opens the possibility to include also gravity; moreover, it has very few free parameters.The massless N = 8 model has only a global SU8 symmetry of the physical particles on-shell, while it isnot even clear whether the subgroup SO8 can be gauged off-shell. Even if it could, this local SO8 is toosmall to contain SU3 x SU2 x U1. However, there are now several recent proposals to circumvent this.One fascinating proposal by Ellis, Gaillard, Maiani and Zumino, though containing many loose endsand based on rather uncertain assumptions, nevertheless comes up with the result that we have seen allquarks (except, at this moment of writing the top quark). The argument goes as follows.The basic ansatz is that the local off-shell SU8 symmetry of the classical (preon) action in d = 4dimensions (comparable to the local Usp(8) symmetry in d = 5 dimensions, see section 6.3) can give riseat the nonperturbative quantum level to bound states which form an N = 8 supersymmetry multiplet atthe Planck mass, in which particles of a given spin form St]8 multiplets, such that the theory with boundstates plus the graviton (which is a preon) has a local SU8 symmetry. At the classical level there is only alocal SU8 symmetry if one adds to the 70 physical spin 0 fields 63 extra nonphysical fields (just as onecan add to the metric ~ extra antisymmetric fields such that one ends up with tetrads). This is thus afake symmetry. The gauge fields of this local SU8 are composites of scalar fields (compare with the spinconnection w (e)). However, it is known that 2’ = ~ + iA5.)~2 which has the same features (one cansolve A5. from its algebraic field equation, but with A~,the theory has a local U(1) symmetry), yields aspin 1 massless bound state (poles in the 2-point Green’s function in a 1/N expansion in thepath-integral) and that the fermions also form bound states. Actually, in this model the fermionic boundstates have the global SU(N) symmetry of the classical action, not only the local symmetry, which forthe N = 8 model would mean that the bound states form E7 multiplets. Since E7 is noncompact, thesemultiplets would be infinite dimensional; we will not discuss this possibility any further.Thus one assumes that the bound states form an N = 8 massless supersymmetry multiplet. Whichmultiplet? To answer this question, note that the N = 1 massive multiplet [2, 3/2, 3/2, 1] splits into themassless [2, 3/2] + [3/2, 1] multiplets, while for N = 2 the massive [2, (3/2)~,(1)6, (1/2)~,0] contains amassless [(3/2)2, (1)~,(1/2)2]. These cases suggest the following N = 8 supersymmetry multiplet in whichthe vectors are in the adjoint representation of SU(8) rather than U(8) (see subsection 6.3)helicity 3/2 1 1/2 0 —1/2 —1 —2 —5/2jiCtuç’A TJA r’A ~A r’Ai,j V ~ F BC ~) BCD 1’ BCDEcomponents 8 63 216 420 504(+1) (+8) (+28) (+56)Since one expects St]8 rather than U8 multiplets, one drops the traces; these are indicated in brackets.

396 P. van Nieuwenhuizen. SupergravizyTo this multiplet one adds the CIP conjugate multiplet which starts at helicity + 5/2, as well as thegraviton which is not considered as a bound state but rather as a preon. (Since it has no SU 8 quantumnumbers, it will presumably not be confined inside bound states.) One already sees that there are morethan the expected 63 Yang—Mills fields, but the second spin 1 multiplet will be argued away below.The idea is now to consider for which subgroup of SU8 there are spin 0 fields in SABCD (and SA)which transform as scalars under this subgroup. This is a necessary condition for SU8 to be able to bebroken at (or below) the Planck mass down to this subgroup by means of the acquisition of a nonzerovacuum expectation value of one or more spin 0 fields. We will show in detail below that the firstpossible grand unified group is SU5!Having found SU5, one then also decomposes the SU8 multiplets for spin 1/2 and spin 1 into SU5multiplets. One drops the spin 3/2, 2, 5/2 fields, as well as the vector fields outside the 24 of SU5 on thebasis of Veltman’s theorem. Actually, Veltman’s theorem only says that if one has a renormalizabletheory to begin with, and one considers a subset of light particles and looks at low (1015 GeV) energies,then only those particles can remain light which form a renormalizable theory. The rest disappears tohigh energies (the Planck mass). In our case, the theory with higher spins is not strictly renormalizable(instead, supergravity is finite) but Veitman’s theorem is nevertheless used. On the basis of this theoremone also drops those spin 1 fields which do not gauge SU5, as we already said.The theorem is used a third time for the spin 1/2 sector. One considers only those sets of SU5 spin 1/2multiplets in which the sum of all axial anomalies cancels. Since one can consider all multiplets asleft-handed by replacing right-handed multiplets in a representation R by their left-handed complexconjugatedR, one can easily calculate these axial anomalies. Axial anomolies spoil renormalizability,and those spin 1/2 which cause axial anomalies will disappear again to the Planck mass. There remainseveral possible anomaly-free sets of spin 1/2 SU5 multiplets, but one must require more. The QCDand EM interactions are purely vector-like (which means that fermions couple to the gauge fields of SU3(color) and U1 (em) only with y5. but not with y5.y~).Thus one must require that in the anomaly-freesets, when decomposed with respect to SU3 (color) and U1 (em), there exists for every complexrepresentation R an R representation. Under these conditions one finds a unique largest set of spin1/2 SU5 multiplets. This largest set contains many fermions which can (and therefore do so accordingto the party line) acquire masses (left-handed fermions can acquire a Majorana mass 5p~(R)T~5~,~(R) iffor every R there is an R, or if R is real). However, there are precisely 3 families (10 + 5) of leptonswhich cannot acquire masses (“chiral leptons”), and these are massless at 1015 GeV, and are the leptonswe see at 102 GeV! (In the first family 5 contains dL, CL and VL, while 10 contains UL, UL, dL and eL,where u and d are the up and down quark.)Let us now show in more detail how SU5 arrives from an analysis of SABCD, and how one calculatesaxial anomalies for the fields in FABC.The spin 0 St]8 multiplet SABCD can be decomposed under SU5 by splitting each index A = 1, 8 intoa = 1, 5 and j = 6, 8. Thus one easily finds for the SU5 x SU3 multiplets (SU3 is a bookkeeping device,but might be the generation group)420= SABCD = (24, ~)+ (5, 3)+ (5, ~)+(45, 3)+ (40, 1)+ (10, ~)+ (5, 1)+ (10, 1)+ (10, 8)+ (1, ~).Thus there are indeed SU5 scalars, namely the three Sf678 (note that S’,78 = —S 6 678 since we dropped thetrace parts in SABCD). If one decomposes SU8 under SU7 or SU6, one finds no scalars. One cannotconsider E6 or SO10, since they are not subgroups of SU8 (because E6 has more (78) generators than

P. van Nieuwenhuizen, SupergravOy 397SU 8, while if S0~owere a subgroup of SU8 then the 8 of SU8 would have to split up in ten l’s of SO~~which is impossible). So supergravity rules out SU6, SU7, SU8, E6 and SO10, but allows SU5 as the grandunified group.Let us finally discuss how to determine which subsets of SU5 spin 1/2 multiplets have vanishing axialanomaly. The axial anomaly of a triangle graph (this is enough to consider) due to fermions in an St]5representation R which are minimally coupled to the SU5 gauge fields, is given byAabc(R) = tr[{Ta(R), Tb(R)}TC(R)] = dabcA(R)where dabc = tr[{Aa, A6}A~]and Ta(R) and Aa are the SU5 generators in the R and adjoint representation,respectively. To compute A(R) for the various spin 1/2 multiplets in216 = (10,3) + (5, ~)+ (45, 1) + (1,3) + (5, 1) + (1,6) + (24,3) + (5,8)504= (45, 3)+ ~40,3)+ ~24,1)+ (15, 1)+ (10, 1)+ (10,8)+ (10,6)+ (10,3)+ (5,3)+ (5,3)one uses that A(R1 + R2) = A(R1)+A(R2), while A(R1 ®R2) = A(R1) x dim R2 + dim R1 x A(R2).Since all representations of compact Lie algebras can be taken unitary, A(R) = —A(R). With theserules one can ex ress all A(R) in terms of A(S) and A(10). For example, A(10®5)= A(10)+A(40),where 9 ®~=~+9~and= 40 = — trace (5 x 10— 10 = 40 components).The reader easily computes the other A(R), and relates them to elements of FABC or PA. As iswell known, for SU5 one has A(5) = A(10). To prove this by elementary means, note that 5 splits upunder SU3 into 3 + 1+ 1, while 10 (= 5 X 5 antisymmetrized) splits up under St]3 into 3 + 3 + 3 + 1.Consider now in Aai,c(R) the indices a, b, c to be of the subgroup SU3, and make a similaritytransformation such that in Ta(R) the St]3 matrices appear in the left upper corner. The dabc are, ofcourse, invariant under this change of basis, but dabc (SU5) = dabc (SU3) for these a, b, c. Consequently,A(5, SU5) = A(3, SU3) and A(10, SU5) = A(3, SU3).With these results one finds several subsets which are free from axial anomalies (one of which isgiven below). However, decomposing each set under St]3 (color) X U1 (em) and requiring vector-likecouplings to these groups, there is a unique subset which has more chiral fermions than all others(45 + 45)+ 4(24) + 9(10 + 10) + 3(5 + 5) + 3(5 + 10) + 9(1).All fermions in this set can acquire masses (between 1015 and 10’~GeV), except the three chiral (5+ 10)multiplets. This is the result mentioned above.There are other approaches to phenomenology. One of them (due to E. Witten) starts from thed = 11 dimensional model of section 6.2, and assumes that the 11 —4 = 7 dimensional internal space is acoset space G/H. If one takes G = SU3 x SU2 X U1, the minimal H which still allows a nontrivial actionof each of the factors SU3, St]2 and U1 of G on G/H is H= SU2 x U1 x U1. Remarkably, this space isexactly 7-dimensional. If one decomposes the elfmetric 2K’~a (y), g~in where the the wayrectangular we have discussed matrix K’~a(y) in section are 6.3, theand Killing replaces vectors theof7 the vector 7-dimensional fields A5.a (x, y-spacey) by A5. (a (x~ = 1, 12 and a = 1, 7), and if one substitutes this ansatz

398 P. van Nieuwenhuizen, Supergravityback into the Einstein action, one finds the required number of Yang—Mills actions needed to gaugeSt] 3 x SU2 x U1 in d = 4. The y-dependence drops from the action after integration over y,but thenontrivial dependence of K on y provides the extra spin 1 modes. In ordinary dimensional reduction, Kis a y-independent square matrix, where as in section 6.4 K was equal to the square matrix U(y)exp My. This is a different approach to phenomenology, which also circumvents the problem that SO8 istoo small to contain St]3 X SU2 X U1 (Proof: if it did, the 8 of SO8 would split up under St]3 into3+ 3 + 1+ 1 since it must be real, but then SU2 could not act on these St]3 multiplets), but in thisapproach one does not need to invoke bound states. Whether these ideas, which have been worked outfor the metric only, can be extended to supergravity is not known.This concludes our report on supergravity. It is certainly (at least in the author’s mind) three-loopfinite and possibly all-loop finite, which means something, although what exactly is not clear sincenonperturbative effects should be important. It may unify grand unified theories or the 102 GeVinteractions with gravity, leading to a model with almost no free parameters. It is the unique theory witha local gauge symmetry between fermions and bosons. It is the most beautiful gauge theory known, sobeautiful, in fact, that Nature should be aware of it!