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

P. van Nieuwenhuizen, Supergravily 251local symmetries Ô4,’ to the total set of gauge invariances in theories with open gauge algebras when64,’ is nonzero.It should be stressed that these new four-ghost couplings in (4) can no longer be written as aFaddeev—Popov determinant and hence that the usual Slavnov—Taylor method for deducing Wardidentities is not applicable here. Instead one should use BRST invariance, as we shall do.We must now use the new 4, variation in the old two-ghost action, and the new and old 4, variationas well as the ghost variation in the new four-ghost action. From now one we specialize to supergravitywhere ~~ depends only on tetrads and the indices i, j, a, /3 refer to gravitinos and supersymmetryparameters. In this case the new 4, variation does not contribute in the new ghost action sinceand since F,,, and F 0~,.in (4) do not depend on the gravitino field. (We exclude gauge choices F,.quadratic in 4’ from consideration. This is not necessary, but covers all useful cases.) For the remainingthree variations one findsC*~F,,,1Ri0,,( ~ ~C”/1 ~Y’)C’3— ~C*~FajC*’3F’3,kfl ~/~,(RIACAA )C’ tC~’_~C*aF,,,jC*$F$,kn~(_ ~r~C~AC”)C”. (9)Since C” in the first term is commuting, we symmetrize the first term in both antighosts and find6I(quantum)= _~C*AFA.IC*TFT,,F~4$aC~AC’3C’ (10)where the expression for P ‘~~$alooks very much like a commutator of an ordinary gauge transformation64,1 R’,, with a transformation 64i’ ~ ~ sinceF” — fly$,k ii k a j Iy,k77$a kl + Iy,kTI$a ki — 77y~j U $a.11Since the transformation 64, -~ ~ looks like one of the terms one finds in the commutator of twogauge transformation, one is led to consider the Jacobi identities for three consecutive gauge transformations.Consider the Jacobi identity6(~)[6(~), ô(~)]— [6(q), 6(~)]8(C)+ cyclic in ~ =0. (12)Substituting (1), and substituting in the result once more the identity (1), one findsRiAAA,,$~Y?~~ + I’~B”+ cyclic in ~ = 0 (13)where A and B are defined byAA __;A ;8 itA flk~ a$y — I ~äJ $y ~J a$.k~ yB” = —n ~f8,,0~1’3r~”+ 77 ,,$.kRk~,~77$~a— R ‘y.k(”77 ~ n’3r(~)“+Rç,,,77 Ic! ‘i~rc”()’~’ ±y) (15)(8)