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

P. van Nieuwenhuizen, Supergravity 253where X~ 87follows from (13), (15), (17). Since the R’A are independent, this implies[...fA~fa0 +f”a$kR”y— IcX~071(~7n’3~a + cyclic terms) = 0. (22)This is the open gauge commutator for fields which do not transform as 4,’ but which transform in theadjoint representation. For closed gauge algebras X = 0 and we find the identity which is needed fornilpotency of BRST transformations of ghost fields (eq. (12) of subsection 2.2).Since in supergravity R ‘A does not vanish on-shell, AA,0,, must vanish when I~]= 0. Inspection of (14)shows that this is only possible when A refers to local Lorentz invariance and a, /3, y to localsupersymmetry. In that case fAa0 is proportional to the spin connection (as we already discussed), whichrotates under local supersymmetry into the gravitino field equation plus terms which cancel against theproduct of the two structure functions in (14). Thus also the index I in (13) refers to the gravitino.The final conclusion is that the remaining terms in the variations of (4) under the usual BRST lawspIus (3) is given by (see (10), (19) and (20))SI(quantum) = ~C*~~~FA,,C*TFr.I(2R‘AX ~,,A07)C~~A C’3C”. (23)Since i + 1 = even and A = local Lorentz, one may place R’A next to FA,, without sign change. Hence wecan cancel (23) by adding an extra term to the ghost transformation law of the Lorentz ghost field,6(extra)C” = —~f”07(extra)C”AC’3f”~7(extra) = ~C*~Fj~X~oCs.This new term in the variation of the Lorentz ghost cancels 81 (quantum) when substituted in thetwo-ghost action, but it does not yield new variations in the four-ghost action since that depends onsupersymmetry ghosts only.Thus supergravity without auxiliary fields, i.e. with open gauge algebra, is indeed BRST invariant.The total quantum action is given in (4), and the transformation rules are the usual BRST rules forclosed gauge algebras plus the extra terms in (3) for the gauge fields and an extra term in thetransformation law of the Lorentz ghost. Those results are the same as if one had started with auxiliaryfields and eliminated them from the quantum action. In particular, the extra terms in the BRST law forthe gravitino come from elimination of the auxiliary fields. Also the extra term in the Lorentz ghosttransformation law are understood from this point of view, namely only in the Lorentz rotation which isproduced by two local supersymmetry transformations are there auxiliary fields.For the extended supergravities one might reverse these arguments, and obtain information aboutthe auxiliary fields by first establishing BRST invariance. Since this is one of the central problems insupergravity, we have been rather detailed in this section.We close with referring the reader to the literature for the most general case of theories with opengauge algebras [162]. In these cases, the process of adding extra tenus to the ghost action and to thegauge and ghost fields does not stop after one cycle, but one can get 6, 8, etc. ghost-couplings. At eachlevel one considers a Jacobi identity of two gauge transformations and one new transformation of 4,’(the first new case being 64,’ — ~ These new transformations are no new invariances of the action,but lead to results similar to the A, B analysis given above. One gets each time new M terms in 64,’ andnew X terms in ÔC”, but 6C*a remains unchanged. These new BRST laws are again nilpotent upon use