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

R,.,,(D) = 23kb,,. + 4e0,,,fm,, + ~ (3)P. van Nieuwenhuizen, Supergravity 297Under chiral rotations C, B 0, and D are inert, but from the (0, 5) anticommutator one finds= (~—~—~)(~), 8,~Z=~((A+fl)—~)iySZAA. (21)Also from (0, 5) one finds finally13=—A, a2A. (22)4.2. The action of conformal simple supergravity [523]The superconformal algebra consisting of the conformal generators Fm, Mm,,, Km, D, the spinorialcharges 0”, 5” and the axial generator A was discussed in the last subsection. We now apply the resultsof subsection 3.2 to it.We define gauge fields and parameters byh,. = e m m”Mm,,+ ç~aQ”+ f’”,,Km + b,D + ,,S” + A,,A (1)4~,Pm+ W~yC = ~mPm + Am”M0,,, + E~Q”+ ~ + A13D + Es~S”+ AAA (2)with m > n and construct the curvatures,R,,,, m”(M) = 23,~mn — 2W~m~~~n — 4(e e”,,fm~)— m,,,f”~— 2~,m,,R, tm (F) = 2.9~em 1~— 2w~ m”e,,,,+ ~ + 2em 1,b~R,,,.]” (K) = 23f’~.— 2W~ m”f,,,.— ~— 2f”,,,b~R,,..,,~(Q) = (2D,4,., + 2~y~+ b,,ç~.— ~iA,4,,y5)~R,,.,,~(5) (2D4. — 2c~,,ymf”,,— b,4,,, + ~R,,,(A) = 20,,A,, — 2i~y5~~.All these curvatures are still to be antisymmetnzed in (,av). As we will see, there are constraints onthese curvatures. These have been found by considering the (unique) action for conformal simplesupergravity, but the results are so simple and suggestive, that it seems clear that a deeper geometricalorigin exists, independent from any dynamical model.The most general action, bilinear in curvatures, parity conserving and without dimensional constantsand affine (i.e., without explicit gauge fields) is given by