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

P. van Nieuwenhuizen, Supergravily 287is proportional to curvatures and is not itself the exterior covariant derivative of some other forms. Thefirst condition guarantees that while varying the action, one finds field equations linear and homogeneousin curvatures, while the second condition guarantees that the action does not vanish as aconsequence of the Bianchi identities. The useful mathematical result is that cohomology theory givesall solutions to (4b) and that there are only few such solutions.The h = hAXA are super Lie algebra valued one-forms, but h are not yet the usual connections(“principal connections on a fiber bundle”) because in that case the H-part of h should have the formh’~’= (g~ dg)’~’+ dyKhK~~ (5)with the curved K in G/H, meaning simply that the H-part of h’~is a gauge transformation under Hitself (gauge transformations with flat parameters ‘~ are defined as usual, 8h”~= (D y~).One calls theh” therefore pseudoconnections. If (5) holds, h” defines a principal connection on the fiber bundleG(G/H, H) and the rest of h, namely ~h”(with flat K) is what is called the soldering form (“thesuper-vielbein”).One now proceeds as follows. Varying the action in (3) with respect to all h 4’4, one finds the completeset of field equations. From here on we discuss the particular case of supergravity; this should enable thereader to deduce also the general case. Assume that one has found the coefficients C such that theaction readsI = J (R” A E” A E’ qk!+ 4’ A ys7,~4’A E,). (6)The R’ are the Lorentz curvatures dw” + ~w”A wj~’,E” is the vielbein one-form, 4”’. the gravitinoone-form and ~ is covariant with respect to H, i.e., it contains only the Lorentz connection. Theintegration is over a four-dimensional manifold since for the super Poincaré algebra there are only threecohomology classes of the algebra (not to be confused with the cohomology classes of manifolds), one ofwhich does not contain gravity, a second gives a theory with only the vacuum solution while the third isthe one used in (6). This proves the uniqueness of supergravity from the point of view of differentialgeometry.The field equations become8118w” = CilkiR n E’ = 0 (7)8118E’ = ~k,R” A E’ + ~çfrA y5y,R = 0 (8)6II8~fIa= (y5y,R)” A E — 1(yy4’)a A R’ = 0. (9)The symbols R” and R” denote the curvaturesof ifr” and E’~(thelattercontains a ihifr term). In deriving (9)we used the identity which we also used in the proof of the gauge invariance of the action insection 17~4’A 4’y’4’ = 0. (10)t. Aftersome Thelengthy interesting algebra point onenow finds is that the following one can project results: (7,8,9) onto the various components of dy’