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

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