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

316 P. van Nieuwenhuizen. Supergravitvwhere these integration constants are nonzero and become fixed at the next 0-level. In most cases,however, they are zero.Suppose one has obtained the supervielbein to order 0, then one can construct the supertorsion toorder 0 = 0. Similarly, knowing the superconnection to order 0, one can find the supercurvatures toorder 0 =0. One can now find tensor relations between supertorsion components to all order in 0 bysimply first determining their relations to order 0 = 0 and then using that such relations must hold to allorder in 0 due to the following theorem.Theorem: a tensor which vanishes at 0 = 0 vanishes for all 0.The proof is simple: in the transformation rule of the tensor all terms vanish, except the fermionictransport term ( “8,,times the tensor). Thus, by induction, the tensor vanishes at all order in 0. Physicallythis important theorem is clear: the origin in one coordinate system is not the origin in another, and byrequiring that a tensor vanishes at the origin of any coordinate system, it vanishes everywhere.It turns out that (to order 0 =0 in any case) the supertorsion and supercurvature are independent ofthe integration constants. This is to be expected, but a proof is lacking. If one chooses w,. tm” =w,.mn(e, ~,) as in second order formalism, one finds, first to order 0 = 0 and thus to all order in 0, thefollowing tensor relationsTabC = Tar~’= Trs’ = Tab~+~(CyT)ah=0. (14)These are the constraints Wess and Zumino postulated for N = 1 superspace supergravity. If one usesinstead the improved spin connection ~mn one finds the constraints of Ogievetski and Sokatchev. Ifone were to be diligent, one could compute V,~Ato all orders in 0 (i.e., to order 0~),then obtain sdet V.and then obtain from the action f d~xd40 sdet V the ordinary space action (by taking the O~componentof sdet V). This action should then agree with the action of the ordinary space approach. (Incidentally,one can directly obtain sdet V to all orders in 0, since sdet V is also a tensor. Similarly one candetermine the supertorsions and supercurvatures to higher order in 0 once they are known to order0 = 0, without having to determine V~,since also these geometrical objects are tensors in superspace.)In this approach we have translated results of the ordinary space approach into results for thesuperspace approach (one could also go the other way). Crucial was the identification of E” (0 = 0) withthe field-independent parameters (~, e’~’).In other approaches [578] one sometimes chooses E~4=VAASA as field-independent. We have obtained the constraints of N = 1 superspace supergravity,assuming that the gauge algebra in ordinary space closes, i.e., with the presence of the auxiliary fields.For N = 2 the constraints were in a similar way found from the auxiliary fields for N = 2 supergravity.Some of them readT — T — TCk — TCk —1rs — 1a1.s 1 Ai,Ej — Ai,Bk —but at this moment it seems that constraints quadratic in T’s appear [555].For conformal supergravity in superspace, one can play the same game [549]. In superspace onehas now, in addition to the spacetime symmetries 4~(E)and ztL(L), two more symmetries, namely localchiral rotations 4A(AA) and local scale transformations 4D(AD). By requiring compatibility of the gaugealgebras in ordinary space and superspace and identifying AA(0 =0) = AA and AD(0 =0) = AD, one findsalso AA and AD to order 0. One must keep here integration constants proportional to &~.One finds thatthe S-supersymmetry parameter ~ of the ordinary space approach appears at the order 0 level in AA