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

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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)
P. van Nieuwenhuizen, Supergravily 339From VAA;E = 0 and the transformation properties of tensors the transformation laws of the twoaffinities in (20) follow (see Nucl. Phys. B122 (1977) 301, eq. A(15)). Just as in general relativity withouttorsion one can solve the Christoffel symbol from (22) and the spin connection from (20). We leave it asan exercise to write down immediately the correct result for F 11’A in terms of g,-~ by adding signs to theChristoffel symbol of ordinary general relativity in the way discussed before.The inverse metric is defined by11= 5J’ where g’11 = (— )“1’g”. (24)g1%~g’(Incidentally, for the tensor gAll one has gAll = (— )A11g111%,shows that alsotoo.) As a consequence a little calculationHIg gSA—vA —o ,.~.Next one can solve DBAC — ( )AB+BC+CA11A from (20) but in order to find Ii itself one must lowerthe upper index by means of u~. Defining71B15ç~ ~D i,Ai \CAL~4BC — LA 1% V C ~. )and using (IABC= —( — )‘~QBAC one finds for the connectioni-s _IID ~i~I \BCD I \A(B+C)~— 27~-”ABC ‘ ~—) “ACB —I) — Ti Ai \C-I-ECrT,D I \D(A+1)+A+1+A1T,D 1*’VA !~) I~A ~ VI ,AJU) Yfi flDc.The inverse super vielbein is defined byTi A1, B — ~ BVA VAA remark about the signs: they follow easily by noting that one contracts as A A and A A~ Thereare more signs in the Riemannian geometry formulae than in the corresponding supergravity formulaesince the tangent space group is Osp(3, 1/4N) and not 0(3, 1) ® 0(N) (only the latter is truly bose).The curvature tensor is defined by the commutator of two covariant derivatives on, say, U1%and readsUA;fl~ (~‘UA;~Il= (~“ 4~4RA4n1U 4 (30)4~.~= —( — )“FA4~,~+ FA4n~+ (~)1~(4 ~F 11~F — 4~ (— )‘~ IrF 4~. (31)RA1% 11 1%i~F11(This definition differs by (—) from earlier definitions.) There are only two independent contractions, asin general relativityRAn =(—)4RA44fl and R=gAhI~fl~(_)ht~ (32)
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