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

P. van Nieuwenhuizen, Supergravity 197J2 1(0) graviton~ U 3f~)graviquark U 1(0) gravitino18(0)gluons1 ~0)y U 3(.Z) •3(i) • 3(1)11(0)1 ~ —1)d3(—• 6(2) U 1(—1)e •3(~)u 18(0) • 3~)cgiuinos0 18(0) 3(2) i(—1) 6(~) 6(-~-) 1(—1—H~ U’ I I ~ •2m 3m E E E E Mass1(0) C’) c”i E E mI I + + +Fig. 2. Spectrum of the spontaneously broken, N = 8 model with SU(3)® U(1)® U(1) invariance. The SU(3)representations are underlined (ex: 3);the bracket indicates the electric charge [3911.without destroying local supersymmetry. Once one accepts the principle of a local supersymmetrybetween bosons and fermions, the theory follows in a unique way, and we shall derive it in this way. It isof course not excluded that supersymmetry is only one attempt at quantization and unification in a longhistorical row, and that it will eventually fail, so that one must consider another symmetry. All one cando is work out its consequences and hope for the best.From a theoretical point of view, supergravity is the gauge theory of the space-time symmetries andsupersymmetry. The tetrad e~ and spin connection w ~‘ gauge the space-time symmetries while thegravitino (the massless real spin 3/2 field) gauges supersymmetry. All these symmetries form anirreducible algebra that is an extension of the concept of ordinary Lie algebras because it containsanticommutators as well as commutators. This algebra is called the super Poincaré algebra. Thesymmetry between bosons and fermions is usually formulated in terms of anticommuting numbers(Grassmann variables, we will give matrix representations), but this is only a matter of conveniencesince one can formulate the symmetry also without them. A very conservative way of consideringsupergravity is to interpret it as just another quantum field theory in ordinary space-time, on the samefooting as for example quantum electrodynamics, the only new feature being that it contains spin 3/2fields.A very interesting and much less conservative description of supergravity it to consider it as a theoryin superspace. Superspace has as coordinates not only the ordinary x — y — z — t, but in addition 4Nspinorial anticommuting coordinates O’~(a = 1, 4 and i = 1, N). There are various approaches to