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

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)