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

332 P. van Nieuwenhuizen, SupergravilyOLc,L0Rc,R = 2i[E”L$—+ “R~&—] [~ O0!i~plus KF’ and em” terms]. (14)Hence, the parameters c,’’ are expressed in terms of dynamical fields. These conditions are part of thetotal set of gauge conditions in this approach.The condition that volumes are preserved under coordinate transformations expresses someparameters in terms of others. One finds that8 8-~—~— s” (.rc) = -i---— p” (XL) = 0CJXL VXL0 (15)8X”L~~~ = 0.(If one requires that only the product (quotient) of both superdeterminants is constant, then one finds inthe trace of (VL$ in 2-component notation the extra constant term ib(b) which can be shown to generateglobal chiral (scale) transformations.) Thus s” and p” are transverse, and these transverse fields appearin SH” in the order 00 terms ast5H”(x, 0)=~O0p~ —~Oiy50s”~ (16)Hence, we can gauge away also the transversal parts of fi” and K”. This completes the set of gaugeconditions. We introduce new symbols for the longitudinal parts of H~and K”P=0,.I~I”, S=8,.K”. (17)0”. The remaining independent gaugeparameters At this point areall thus fields that remain are S. P, A”, e,,,” and cu“, a” and traceless part of WL~,. The latter is an element of SL(2, C), hence wecan expand it on a complete basis(Z)L$ _~!5au.)Y = (0.)aQmn (18)with real coefficients (1”” (see appendix). These f)mn generate local Lorentz rotations. Thus localLorentz rotations are not independent from but induced by the general coordinate transformations (justas in the Wess—Zumino approach local supersymmetry but not local Lorentz rotations are induced bygeneral coordinate transformations).tm” After are complicated just the parameters field redefinitions of local supersymmetry, one finds thengeneral following coordinate results. andThe local parameters Lorentz rotations&‘, a” and offl ordinary supergravity. The fields S, P, A,., e,.m, i/i,.” transform precisely as the minimal set of auxiliaryfields.The advantage of super space approaches is that the closure of the gauge algebra is evident. In thisapproach, the product of two volume preserving general coordinate transformations, each of whichkeeps the gauge in which the axial superfield H” (x, 0) has only terms of order 02 and higher and haslongitudinal components H” and K”, is again of this kind.