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

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.