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

(5) follows from fdC=0 (only f dCC= 1). ThusP. van Nieuwenhuizen. Supergravity 245(6~+ (_1)(’3±lX~l)C*’3iF~ 1R1,,Ca) = 0. (6)As usual, /3 = 1 for fermionic symmetries and /3 = 0 for bosonic symmetries.Let us, for later use, return to (3) and multiply it by F k and — ~i(y~)~a. Then on-shell again(~C*’3(i[ ‘)o~F kRk,,CaA + ~fl~~J7~4,k) = 0. (7)Rewriting (6) as((—1)~’~6~ + C*’3iFEkRI,,C~)= 0 (8)one finds for the first term in (7)(~C*0(y~)ObF kRk,,CaA) = (y~) ~( — ) +1A) (9)Hence, on-shell we find the following Ward identity if one would not add a measure sdet ~integralin the path(6Z/67& + ~(y’)~~( — )E) = 0. (10)The last term is, however, precisely the variation of (sdet y)h/2 because6(sdet ~a0)1/2 = ~(7’)eô (ôy&) (~)~(sdet~aI3)1/2 (11)Thus the S-matrix of supergravity is indeed gauge-invariant under changes in ~ if one chooses themeasure as sdet ~1/2 as we did before.Thus also gauge invariance requires that one take into account the factor (sdet ~ )1/2. This factor isof course the Nielsen—Kallosh ghost which we found before. It plays a crucial role in obtaining thecorrect axial anomaly when one uses the Adler—Rosenberg method.2.6. Unitarity of supergravityClosely related to gauge invariance is unitarity of the S-matrix [522].Unitarity means that unphysicalmodes in the propagators do not contribute to S-matrix2, elements. the renormalizable Let us beginpropagator with the easier is 6~,,(—ik2). exampleof Yang—Mills theory. In the gauge ~(fix) = _~.The unitary propagator, on the other hand, reads W)(—ik2Xe~(e~)~ + ;( ~,) = (_ik2)(6,~,. — ~‘4). (1)This equation holds on-shell only, where k2 = 0.. The problem is now to show that the terms with k,, andits time reversal 1~,..cancel with the ghost contributions. Consider a two-particle cut