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

P. van Nieuwenhuizen, Supergravity 355contained in E 7) while the remaining (~)= 70.X’s generate the rest of E7. Also the expected SU(8)global canbe found by considering (x~+ iYjk) (not to be confused with the SU(8) local). This SU(8) is the maximalcompact subgroup of this form of E7.The global E(7) symmetry acts on the curls of the photons, not on the photon fields themselves. It isan on-shell symmetry only, of the same kind as the dual-chiral symmetries in the N = 2, 3, 4 models. Infact, truncating the N = 8, d = 4 model to the N = 4, d = 4 model, one finds back the SU(4)x SU(i, 1)global symmetry of the so-called SU(4) model.The reader may have wondered what happens with the general coordinate transformations whoseparameters are in d = 11 given by r(x’,. . . , x~)where a = 5, 11. The answer is that they becomeMaxwell gauge parameters for the photons A,,” which are defined by e,,” = A,~”e~”. As one easilychecks, OA!L” = 3~”,while Oe m 1. = Oe”~= 0.There are much more fascinating details. We regretfully refer to the literature for this masslessN = 8, d = 4 model, and now turn to a massive version of it.6.4. Spontaneous symmetry breaking in the N =8 model by dimensional reductionThe work which we now discuss is due to Cremmer, Scherk and Schwarz [404,405, 406]. It solves along-standing puzzle of how to obtain spontaneous symmetry breaking in the extended supergravities.For simple supergravity, spontaneous symmetry breaking can occur as soon as S or P are nonzero(which may or may not imply that the scalar fields A and B have nonzero vacuum expectation values[541]).However, for N> 1, many supergravity practitioners were afraid that no analogous results wouldbe obtained. As usual, matters turned out to be better than expected.A most interesting way to give masses to the particles of the N = 8 model in d = 4 dimensions is tofirst reduce the d = 11, N = 1 theory to the d = 5, N = 8 massless theory. Then, in d = 5 dimensions,one writes fields as5) = exp(M’x5)~c(x’,. . . , x’~) (1), xwhere M is in general a matrix; see below. In this way, for example, a massless scalar field in d = 5 canbecome massive in d = 4= ~,*(~_pJ2)~p (2)Note that this kind of symmetry breaking is not such that the theory itself determines the minimum of aHiggs potential, but, rather, it is we who (arbitrarily) prescribe that fields must have the factored formas in (1). Since coordinates are dimensional, the substitution in (1) introduces masses in a natural way.In general, the procedure for constructing a D-dimensional theory with spontaneously brokensymmetry is to first consider a theory with unbroken symmetry in D + E dimensions and to wrap theextra E dimensions into circles with such small radii that only the lowest Fourier coefficients have finitemass. If we write fields as in (1), then the Lagrangian density in D + E dimensions is independent of M(and hence remains gauge invariant in D + E and in D dimensions) as long as the transformation4) = exp(Mx5)i,1i corresponds to a symmetry. In practice, the useful symmetries are the global symmetries.How do we find global symmetries in D + E dimensions? A very interesting idea is to consider yet a