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

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