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

P. van Nieuwenhuizen, Supergravity 241For the kinetic terms of the ghosts one clearly obtains after replacing r by CC, e’~by CC and A ab byCab..~(ghost)~= C~L1C’— COC+ 2C~b(Cab+ 3bCiA5a). (11)Thus the general coordinate vector ghost is itself no longer a gauge field. The bar on C denotesDirac’s bar: C = C 74 in our conventions. Clearly, one must rediagonalize the Lorentz ghost fieldCab + ~8bCa ~.8aCb C~.One then finds new vertices originating from the last term in eq. (9) [556]ofthe form (C~/I)a’4CP.The ghost propagators are easily found(C~C~) = —i6~~k 2 (CaC~)=IC” C’* \_j~ ~ _~ ~~ t~adUbcThe vertices of quantum simple supergravity can be worked out from the foregoing.Thus we have obtained at this point the propagators and vertices of the theory, and this definesquantum supergravity in a perturbation expansion. We note that in I(fix) = ~F~,y’~’3F~we indeed didchoose ya’3 to be field-independentya$ = {—~o”,aôaa’6’4’4’, ~jaa} (13)We now consider the role of the auxiliary fields in somewhat more detail. They appear in I(qu) dueto (9) as follows— ~(dete)(S2 +P2 — A~,,)—~ Xy5C+~(S — iy5P — ii4~75)]C. (14)Clearly, 5, F, Am are still nonpropagating at the quantum level and can be eliminated using theirquantum field equationseS=—CC, eP=Ciy5C, eA=~Ci-y5yC. (15)Reinserting the quantum field equations of the auxiliary fields in the quantum action, new four-ghostcouplings emerge,£~‘(fourghost) = ~ (S~+ P 2 A~~) = [(~C)(t~C) (C — ~ — 75C)(~75C)+ ~ (~y5ymC)(~y5ymC)1.(16)~Let us now bring ghosts and antighosts together in the same spinor covariants. To this purpose wedefine the Majorana conjugates of C by_C = C~,wherç ~ is the charge conjugation matrjx. Similarly,the Majorana conjugate of the spinor C is defined by C = C~’.In this case, CC = —CC since theseghosts commute. After a Fierz rearrangement, not forgetting an extra — sign since the C commute, onefinds the very simple resultS..~‘(four-ghost)= — ~mC). (17)(C7mC)(Cy