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

344 P. van Nieuwenhuizen, Supergravitycannot be coupled consistently to either gravity or to simple matter systems, it seems as if nature stopsat spin 2, and supergravity at N = 8 (see subsection 1.14).The simplest extended supergravity is the N = 2 model (Ferrara and van Nieuwenhuizen [203]). Itwas obtained by coupling the (2, 3/2) gauge action to the (3/2, 1) matter multiplet by means of theNoether method. It was only afterwards found that the model had a larger symmetry, namely (to beginwith) a manifest 0(2) invariance which rotates the two gravitinos into each other. This model unifieselectromagnetism with gravity. The action reads= — ~ R(e, co)— ~ ti,~F”D~(o4cui, — + 4V2 [e(F”‘~+ E””) + ~y 5(P~+ ft~a)]~b1~E11 (1)and is invariant under general coordinate, local Lorentz and Maxwell transformations (SA,. = 3,.A) aswell as under the following two local supersymmetriesse tm,, =!~L~iym~,i 5A,. (2)~Ii — I \ ‘~_..!L..4I~ A11 ~ A \ I(J(J1,. — ~~~(O~?E-‘ ‘. ,~AY ~ 2e~,.A7 751E.Indeed, each gravitino gauges one local supersymmetry as one sees from Si/i,,’ = a,.e’ + more. Thesymbol .F,,,. equals ~ and .F,,,. is the supercovariant photon curl= (a~A~— 2\/2 i/I~ifr~~Eu’)— (j ~-* v). (3)One may always take the spin connection co in the Hilbert action as that function of tetrads andgravitinos which solves SI/Sw = 0 (1.5 order formalism). If one would have started in the Hilbert actionwith (0+ T where r is arbitrary, then one finds only r 2 = terms upon expanding about co. These i/i4~p4terms are absorbed into the F,,,. tensors, and it is easy to argue why this must be possible (as of coursehas been checked explicitly). Since fermionic field equations rotate into bosonic field equations and theformer have only one derivative while the latter have two derivatives, no derivative can act on e. (Sinceotherwise there would no longer be two derivatives available for the bosonic field equations. Thisargument breaks down when there are nonpropagating (auxiliary) fields.) The gravitino field equation isobtained by varying all t,li fields, and since the terms with P have four i/i’s which appear symmetrically,one must take one half of the former in order that the field equation only contains the supercovariantF~,,,but not, for example, bare F,,,,’s. Note that this argument also tells one that the spin connection inthe gravitino field equation must be supercovariant. As it happens, w is itself supercovariant, henceboth in Hilbert and gravitino action one finds a. (In other cases, for example in d = 11 or d = 5dimensions, one finds in the gravitino action ~(w+ ó~)instead of w, but, based on our argument givenbefore, one always chooses co in the Hilbert action.)The same kind of argument shows that if auxiliary fields are absent the gauge algebra must close onbosonic fields, and this is also a helpful criterion in constructing theories. (It is difficult to credit oneparticular person, but certainly J. Scherk gave an important contribution.)This model was the model where finite quantum corrections were found for the first time (see ref.