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

P. van Nieuwenhuizen. Supergrav:ty 333Spinor covariant derivatives are defined, and vector covanant derivatives follow then from theanticommutator of two spinor covariant derivatives. Invariant actions can be defined and generalcounter terms have been obtained [411,191].Summarizing one might say that the chiral approach uses fewer fields (64 in H”(x, 0) as compared to1024 in VA’5(x, 0)) and fewer parameters (40 volume preserving A” and A” as against 224 parametersE’t(x, 0) and Lm,,(x, 0)). Moreover, one only needs to fix 26 more gauges to find back the 38 fields etm,.,~ S, P, A,. while in the non-chiral approaches one simply does not have enough gauge parameters todo this (1024> 224). Therefore, one must impose in the non-chiral approach additional algebraicconstraints. These are satisfied automatically in the chiral approach.All supervielbeins and connections are expressed in terms of the axial H~(x, 0) above, and H” is anunconstrained field. As action one takes again the superdeterminant sdet V4”’. Varying with respect toH”, the field equations read GAA(X, 0) = 0, but not R = 0, as in the Wess—Zumino approach. R beingchiral, of course satisfies DAR = 0. Defining G~ to be determined by the supertorsion just as in theWess—Zumino approach1A.BB — AB(JBp •C ç’•Cone finds that (off-shell)DAGA~=DAR*. (20)Hence, on-shell, DAR = DAR = 0, so that R = constant. In order to understand this constant, one canconsider component fields. In the action the term eS2 = e(e_l8rnSm + (j/8)~frmO~tmfh/J,,)2appears. Varyingwith respect to Stm (not 5!) one finds 8m(e18nS” + (j/8)i/im~m”~frn) 0. Hence S = constant which isactually the same constant as found before since R starts with S. Inserting this result into the fieldequations for gravitino and graviton, one finds mass and cosmological terms. Note that the cosmologicalterms of the component approach e(S — (j/8)~IimOm”iIin)turns here into a total derivative if one expressesS in terms of 5=.At this point we anticipate our discussion of the work of Siegel and Gates, and discuss how in theirapproach a cosmological constant appears in the action. A chiral scalar density superfield V isintroduced such that gauge transformations now have an unconstrained superfield parameter, asopposedto the casewediscussed in subsection 1.9, eq. (23). (The parameternow generates superconformaltransformations.) The cosmological term in the action is now given by A f d4x d20 V+ h.c. =A f d4x d40 (E/R) + h .c. Varying with respect to V directly yields the equation of motion R = A (because inthe action supdet E the supervielbein is V-dependent).Both the chiral and the non-chiral approach can reproduce the results of the non-superspaceapproach to simple supergravity. Although the initial starting points look very different, each approachcan lead to the other.In fact, one can choose a special gauge (just as in general relativity) in which the components ofH”(x, 0) take on a particularly simple form at a given point. One finds that the first few componentsvanish, and the rest is a function of R, G~and WAPC alone. Using this special form of H”, it is easy toshow that the supertorsions and supercurvatures (which depend on VAA, which depend in turn only onderivatives of H”) satisfy identically the Wess—Zumino constraints.We discuss now the closely related superspace approach of Siegel and Gates. They consider complexcoordinate transformations. A scalar superfield 4i(x, 0) transforms as