From table 5.1 it can be seen that in the case of extended supersymmetry of type N, a supermultiplet must contain states of helicity h such that |h| ≥N/4. This means that for N > 2 we have no supermultiplets with scalars and spin one-half fermions only; for N > 4 we must exceed helicity one, (thus no Yang–Mills supermultiplets for N > 4) and as of N ≥ 9 states of helicity larger than two make an appearance (the critical values N = 2, 4, 8 correspond to the CPT self-conjugate cases mentioned in chapter 5). There are strong reasons to believe (though really no proof as yet) that nontrivial interacting theories containing fields of spin five-halves and higher in the lagrangian are inconsistent (Aragone & Deser 1979, Curtright 1979 Berends, van Holten, de Wit & van Nieuwenhuizen 1980). Hence one is limited to supergravities with N ≤ 8.

The ‘natural’ N = 1 supergravity in four space–time dimensions has just been presented. Here we ask for the extended supergravities (N > 1). The corresponding lagrangians can be constructed. We shall not do so here, but will content ourselves with a few remarks on these lagrangians and on the phenomenology of the N = 8 theory. First of all, the elegant construction (MacDowell & Mansouri 1977) we presented for the N = 1 case, already fails at the N = 2 level, as in addition to the types of terms in the ansatz (22.5) it requires terms explicitly containing the vierbein or metric (Townsend & van Nieuwenhuizen 1977).