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

To this one must add the variation of ~P. van Nieuwenhuizen, Supergravily 305which yields an extra termM~~))(D,.cA)=(8oc(eg(~ey i/ig~~’ — — ~ey~r)D,.cA. (14)Interestingly, the non-covariantizable terms EØCAçIJ,. cancel, and the complete 0-connection is easilyfound to be0: ~ — . ~/iD,.~A. (15)Thus the full superconformal dalembertian is given byPfA~3r(g”~’eD,.’A)+~c~ y~—~A’~8B— ~A”~5x+~j-b .ACB2A + (2— A )b~D,.5~’+ AR + ~- Aii~zu~~’çbr— ~- (A,.c) ~/i”D,.’~y— — ~ ç~y çLcD,.CA. (16)Two interesting remarks are in order. First of all, although the separate connections are not inert underthose symmetries to which they do not refer, the sum (thus LJCA) is really covanant under allsymmetries. Secondly, as we have stressed so often, flat indices simplify transformation properties, andindeed,ô0’~(Da’A)= ~(EDa”X). (17)One can now at once write down the conformal coupling of the scalar multiplet to the gauge fields ofconformal supergravity. It is given by the action for the multiplet I ® T(I). The multiplication of twolocal multiplets is the same as the multiplication of two global multiplets of conformal sypersymmetryI1®12 (A1A2—B1B2, A1B2+A2B1,(A1+iy5B1)X2+(A2+iysB2)X1,—glx2-f A1F2+A2F1—B1G2—B2G1,A1G2+A2G1 +B1F2+B2F1+iX1y5X2). (18)One may verify explicitly that the product of two superconformal scalar multiplets with weights A1 andA2 respectively, as given by (18), is again a superconformal scalar multiplet with weight A1 + A2. Insuperfields this is immediately clear. In the next subsection we show how to construct an action for asuperconformal scalar multiplet.4.5. The origins of the auxiliary fields S, F, Am [471,472]m,., the gravitino(ordinary The independent supersymmetry (physical) gauge fields field) of ,fr’~,.,the conformal chiralsupergravity gauge field A,. are and the the tetrad dilation e b,.. Just as inordinary supergravity, one can again count field components in the action, taking into account that eachdegree of gauge invariance removes one field component. The count is as follows: