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

310 P. van Nieuwenhuizen, SupergravitvOne can apply these considerations for example to the kinetic multiplet T(I). Since its firstcomponent is the same in Poincaré as in conformal supergravity, one expects that ECA does not containb,., since b,. plays no role in Poincaré supergravity. Indeed, b,. cancels in ~CAThe tensor calculus for supergravity was developed by Ferrara and the author [526,527, 531] and byStelle and West [583,584, 585, 445] as well as others [297,167, 534]. There are much more details of thetensor calculus, but since an exhaustive review is available [539]we refer the reader to that source.5. Superspace5.1. Introduction to superspaceOne way to describe supergravity is to consider it as just another gauge theory in ordinary spacetime.The local symmetry between bosons and fermions can even be formulated without anticommutingparameters (for example, 84 = A~and ÔA” = I(A — iy 5B)~)and invariance of the action can then stillbe investigated; it has, in this case, an open index 13. Such a theory can be quantized in the usual wayand one can compute S-matrix elements. This is the approach we have been following up to now, and itleads to all results in which one is interested.There is, however, another approach to supergravity theory, and that is superspace. Actually, thereare many approaches, and we will discuss them in this order: (i) the approach which builds a bridgebetween the ordinary space and superspace, (ii) the Wess—Zumino approach, which is geometrical buthas too many field components in superspace and needs therefore constraints on the torsion from theoutside imposed. These constraints also serve to eliminate fields with spins exceeding two, (iii) the chiralsuperspace approach of Ogievetski, Sokatchev, Siegel, Gates and others, which is economical in that itdeals with two small chirally related superspaces but whose geometry is less usual, and might needconstraints for N> 1, (iv) gauge supersymmetry of Arnowitt and Nath which was the first superspaceapproach, but which contains ghosts and higher spin fields.A major advantage of superspace approaches is their application to quantum gravity, in particular toexplicit calculations. In the ordinary space approach one has gauged away many of the fields present inthe superspace approach, and consequently one cannot use globally supersymmetric gauges. Also, in theordinary space approach one cannot use a supersymmetnc background field formalism, in which theeffective quantum action is invariant under locally supersymmetric background field transformations. Inthe superspace approaches, these goals can be met. Finally, superspace methods allow the use ofsupergraphs, which contain many ordinary Feynman graphs, and are easier to use than ordinaryFeynman diagrams. However, as with any new formalism, many new problems had to be solved beforeit could be used which explains that up to now the ordinary space approach has yielded all results first.Another advantage of superspace methods is that it explains many results of ordinary space in asimple way. To mention a typical example, if one wants to know how a scalar multiplet with externalLorentz indices transforms, one can obtain the result directly in ordinary space [224], but a methodbased on the use of covariant derivatives in superspace gives more insight. However, if one needsexplicit formulae in terms of component fields, for example to check whether there exist supersymmetricextensions of topological invariants [534], then, in the author’s opinion, ordinary space methodsare more appropriate.We now discuss general properties of superspace in the remainder of this subsection.