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

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