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

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194 P. van Nieuwenhuizen, Supergravityquarks (color, flavor, lepton number, etc.), but differing from these by having spin 3/2 rather than spin1/2.Whether massive or massless, the new fermionic gravitational fields do not modify the classicalpredictions of general relativity because the exchange of fermions leads to a short-range potential. (Thisfollows, for example, from dispersion theory and is due to the necessity to exchange two fermions toproduce a potential.) On the other hand, at short distances and thus at high energies, supergravitydiffers radically from ordinary general relativity and is a better quantum theory. Infinities in theS-matrix in the first and second order quantum corrections cancel due to the symmetry betweenfermions and bosons. Whether these cancellations persist in all higher order quantum corrections is anopen question at present. Thus supergravity is not finite because infinities are renormalized away, as inthe nongravitational theories, but because in the S-matrix infinities cancel. Due to this finiteness,supergravity has predictive power, just as ordinary renormalizable models.In simple (N = 1) supergravity, bosons and fermions occur always in pairs, and these pairs are theirreducible representations of the supersymmetry algebra. One such pair is the graviton and gravitino.Other pairs are spin 1/2 and spin 1 doublets (“photon—neutrino system”) and spin 0 spin 1/2 doublets.One can add as many of such matter doublets to the spin 2 spin 3/2 gauge doublet as one wishes.However, a special feature arises when one adds one or more spin 1 spin 3/2 doublets to the spin 2 spin3/2 doublet. The resulting theories are the so-called extended (N = 2,. . . , 8) supergravity theorieswhich, in addition to the space-time symmetries which any gravitational theory possesses, have as manyFermi—Bose symmetries as there are gravitinos, namely N. In addition, they have ~N(N — 1) spin 1 realvector fields, and lower spins. Most importantly, they have a global U(N) group of combined chiral-dualsymmetries which has nothing to do with supersymmetry. Under these symmetries, fermions rotate intothemselves or y~times themselves, and curls of gauge fields rotate into their dual curvatures. The 0(N)part of this U(N) group rotates the gravitinos into each other according to the orthogonal group 0(N),and simultaneously the photons into each other, etc.The first of the extended supergravities is the N = 2 model [511]. It realizes Einstein’s dream ofunifying electromagnetism and gravity, and does so by adding two real (=one complex) gravitino to thephoton and the graviton. It is in this model that the breakthrough in finiteness of quantum supergravityoccurred: an explicit calculation of photon—photon scattering which was known to be d~vergentin thecoupled Maxwell—Einstein system yielded a dramatic result [512]:the new diagrams involving gravitinoscancelled the divergencies found previously (see fig. 1).One should view the N-extended supergravities as extensions of pure general relativity (i.e., withoutmatter) in which the single graviton, N gravitinos, ~N(N — 1) vectors etc., all rotate into each otherunder either supersyinmetry- or U(N) symmetries. Since these extensions are irreducible, one calls thesetheories pure extended supergravities. Thus the graviton is replaced by a new superparticle whose“polarizations” are the graviton, gravitinos, quarks, photons, electrons, etc. This unification of allparticles into one superparticle leads also to a unification of all forces, because forces arise by exchangeof particles. One can also couple matter to these theories, but only N-extended matter to N-extendedsupergravity (N-extended matter having before coupling N global supersymmetries and a global U(N)invariance).The importance of the extended supergravities lies in the fact that the global 0(N) group can begauged by the ~N(N — 1) vector fields which are present in the pure N-extended supergravities[108,225].This leads then to theories with two coupling constants: the gravitational coupling constant Kand the internal 0(N) coupling constant g. The latter should describe the nongravitational interactions,but in order that this be possible spontaneous symmetry breaking must first occur such that g splits up
P. van Nieuwenhuizen, Supergravity 1957 1’ V 7SPIN = 3/2c,Jc,JMAXWELL-EINSTEINTHEORY0))IIIIcSPIN = 3/25’ 5’ 7 7(137 1>T5’~>T~~~Finite probability for loop diagrams in a quantum theory of gravity is obtained by including gravitinos in the interaction. The diagrams shownFig. 1.here are those for the interaction between two photons. The first diagram, labeled Maxwell—Einstein theory, consolidates all the one-loop diagramsthat involve only gravitons and photons; the contribution from these diagrams is equal to an infinite quantity multiplied by the coefficient 137/60.Five one-loop diagrams involving gravitinos can be constructed; each of them is proportional to the same infinite term multiplied by the coefficientsshown in brackets. Only the sum of the diagrams is observable, and adding the coefficients shows that the sum is zero. Hence the infinitecontributions of the gravitinos cancel those of the gravitons and the diagrams have a finite probability [2491.into the different couplings of the electroweak and strong interactions. This has not yet been achieved,and is of prime importance for the phenomenological implications of supergravity. It is tempting tospeculate that supergravity is the high energy limit of the nongravitational interactions. In that case, onewould restore the symmetry by going to high energies and find that bosons and fermions have equal(vanishing?) masses.
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