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

308 P. van Nieuwenhuizen. Supergravityordinary supergravity. If we had initially included the locally scale (Weyl) invariant coupling (eA 4 +more), then one would have found Poincaré supergravity plus cosmological term.In particular, the supersymmetry transformation of ordinary supergravity is a linear combination of thetwo supersymmeeries of conformal supergravity [525]= ô~’~(e)+Ss’~i~e + ~.JJ ), ~ = —~(S— iy5P — iA~”y5). (10)Moreover, the chiral gauge field turns into the axial auxiliary field Am and the auxiliary fields F, G of thescalar multiplet become the auxiliary fields S, F [471].These results are not only valid for matter fields but hold generally. For example, for the field e”,.,~ A,. one has= ~EQym~,., oo~~A,.c= —iE0y5~,.~ / ~ 1 ~= ~D,. +5b,. —-i-A,. Y5~o (11)v5s~em,.= 0, ~ = 7,. s, 6~’~A,.’~ = i sYsl,lI,.and using the “Q + S rule” one finds the usual transformation m~and rules the for explicit the tetrad b,. and in 6~’~/s,. gravitino inordinary supergravity. (In particular the dilation terms in w,. cancelagainst the X in 6~.)It is thus clear also, why ~2 and F2 have negative signs and A~,has a positive sign in the gauge actionof ordinary simple supergravity: because the Brans—Dicke scalars have unphysical kinetic sign.4.6. Tensor calculusfor ordinary supergravityIn the preceding subsection it was shown how to obtain a tensor calculus for conformal simplesupergravity. In this section we show how one derives from these results a tensor calculus for ordinary,i.e. Poincaré, supergravity.Consider, as an example, a scalar multiplet with conformal weight A. Under conformal supergravityone hasöCfr)A = ~Ex, 80’~(e)B= ~7~X(1)and= ~(ØC(A — iy5B) +F + iy5G)= 8sC( )B= 0,osc(E)F = (1 — A)EX,6~( )x= A(A + iBy5)e(2)6~’~(e)G= —i(1 — A)ëy5~.These results follow from covarianting the results of global supersymmetry, which is unique as we havestressed already several times.