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

P. van Nieuwenhuizen, Supergravity 359four dimensions. One obtains a massive action without cosmological constant (which previously wasalways disastrously large) and with a potential which is bounded from below.* It is a beautiful method,but it may not be the last word. Hence (in the author’s opinion at least) it is not impossible that one canalso construct different mass spectra. Perhaps not everything we need for physical applications descendsupon us from higher dimensions. Therefore we refer the interested reader to a result in the literaturewhich gives the most general spontaneous symmetry breaking which can take place in the N = 1 modelin 4 dimensions.As shown by Cremmer, Julia, Scherk, Ferrara, Girardello and the author [95], one can indeed breaksupersymmetry spontaneously (at least in N = 1 supergravity) such that the super-Higgs effect occurs,the potential is bounded from below, there is no cosmological constant (this fixes only part of thefreedom in the action) and a mass formula ~2J + 1) (~)~‘(mj)2 = 0 is found. All masses are furthertotally free, and the model still contains an arbitrary function of two variables. Clearly, localsupersymmetry is much less restrictive than was thought in the beginning of supergravity.Finally, we mention one of the outstanding problems for the N = 8 model: in how far are thequantum corrections still finite if one adds masses? It has been shown that all one-loop divergences inthe corrections to the cosmological constant vanish, leaving only a finite constant correction. However,whether also physical processes are finite is an open question. If they are, one might call the symmetrybreaking discussed in this section spontaneous, since for spontaneously broken theories the theorybehaves in the ultraviolet region as if there are no masses. Let us conclude with an optimistic note.Perhaps the N = 8 model with masses is the ultimate finite quantum theory of gravity, and it will beshown that all particles of nature appear as bound states of this model.6.5. Auxiliary fieldsfor N=2 supergravityRecently, sets of auxiliary fields for N = 2 supergravity have been found. A first set for Poincarésupergravity was found by Fradkin and Vasiliev [228,229], and de Wit and van Holten [164,169], bydirect means. Breitenlohner and Sohnius found a set by starting with the causal group (Poincaré pluslocal scale transformation) [72].All thiswork has beenclarified by superconformal methods. Interestingly,these superconformal methods are quite similar to the corresponding N = 1 case which we discussed insection 4, and it is hoped that this approach will also be useful for the N = 8 case.The original N = 2 Poincaré auxiliary fields constituted the following setBose fields: Am, Am”, Vm”, S, P°,tmn’1, Vm, M”, N”Fermi fields: x’~A. (1)The bars denote symmetric and traceless tensors, the hooks antisymmetric tensors. In the action theyappear as~p.~2jj Ij\2jJj,,ij \2~ A2 ~1I if\2~Ij~lJiI\2 1j’~j \2— ‘-7 22, / 82,&mn) m 42, m) 42, 2’ mJ 22, mJ— ~(M~)2— ~(N~)2 + 2A‘t~”+ ,ØAi) + trilinear terms. (2)In order to make the reduction to N = 1 supergravity, one starts with Setm,. = ësees how two multiplets develop:mt,b~,+ e2ymIfr~,and1y* The one-loop corrections to the cosmological term are finite 1406] due to the mass relations I( — )‘(2J + 1)mi” for n = 0. 1, 2, 3 [208].