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

302 P. van Nieuwenhuizen, SupergravityThus the gauge algebra of N = 1 conformal supergravity closes without the need to introduce auxiliaryfields [5231.This is no longer true for the N> 1 case.We now show how to derive8’W~m~and ô’4~without much work, using group theory. SinceR,,J~(P)= 0 identically, its variation vanishes, too. This variation consists of two contributions: namely,if all fields would transform according to the group, one could use the homogeneous rotations ofcurvatures8(group)R~,m(F) = _~R,.~,,(Q)yme. (12)However, ~ is modified because w~m”is expressed in terms of other fields by solving R~~m(F) = 0.Thus there is an extra term ô’W,~m~e~~— ô’co~m~e~,~. Solving in the same way as one solves for theChristoffel symbol, one finds ô’w,~m”.For ô’& we proceed as follows [1691.From the Bianchi identity= 0 = — ~‘°(R~~m(M)+ e~mR~(D)) (13)one finds two relations, using that R,~r(Q)o~’y’ = ~R,~,,(Q)y5yA ~~ = 0R~~(M)—R~(M)= —2R~(D) (14)Ra~~r(M) + 2 terms cyclic in af3y = —2ô~~R~,,(D) + 2 terms. (15)From the latter result one deduces~ (M) — ~ (4~)= o~,7R~8(D) + (a 4*13, y ~ 8). (16)As before, we now use that the variation of a solved constraint vanishes (E == 0 = y~’[~y5 R~(A)— R,~(D)+ ~ Om~ R~mn(M)]+ [2ô’qS~ + y,,y ô’4]. (17)It is now easy to deduce the result for ô’qS,.. quoted in subsection 2.The result for 6 1f”,.. in eq. (4) is obtained in a similar way, by varying the third constraint (theEinstein equations) and using the results for 8’w~m~and 6’&, as well as the Bianchi identityD~R~(Q) = 0. (This identity is needed since varying the Lorentz curvature one produces terms ofthe form 8ja3W’~m”.)4.4. Conformal tensor calculusSince all fields in conformal supergravity are gauge fields which couple minimally, it is straightforwardto obtain a tensor calculus for conformal supergravity: all one has to do is to replace ordinaryderivatives by covanant derivatives. In order to obtain a tensor calculus for ordinary supergravity, onehas to do two things. First of all, the transformation rules of ordinary supergravity are obtained from