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

376 P. van Nieuwenhuizen, SupergravityApplying this theorem to the transformations under (i), the proof of the super Jacobian is completed.(Incidentally — the general theorem is in fact a proof that the ordinary Jacobian equals the ordinarydeterminant.)I. Elementary general relativityThe -tensors ~“~and ,,,.,,,, are always +1 or —1 ( 12~ = 1234 = +1). They are a density and anantidensity, respectively. Thus ~ is obtained from by lowering the indices by means of the metric anddividing by (det g,,,.). The tensors ‘~“~“and abcd are also ±1and have density-weight zero. Thus E,.b~,,isobtained from ““ by lowering the indices with the Minkovski metric (0,,,. = + + + +) in our conventions).Consider a four-dimensional manifold with(i) connections which define parallel transportOv” = —F,,~ 0v”dxv,(ii) a metric g,,,. defining distances (ds)2 = g,,,. dx” dx”.Hence the covariant derivative is D~v”= O~v”+ f’1,~”v1,.Length is preserved under parallel transport ifand only ifDog,.,. = 0 = 8~g,.,.— f,,~”g,.,. — T,.~”g,,,,. (1)Parallel transporting an infinitesimal vector u” along an infinitesimal v1,, and v1, along u”, theparallelogram closes if and only if F,.,.” — F,.,.” = 0. Torsion is defined by the torsion tensor S,,”,. =— fe,,). One usually requires in general relativity that length is preserved and torsion is absent. Inthis case F”,,,. equals the Christoffel symbol {,~‘,.}but let us relax these restrictions.If length is preserved, but torsion is present, one can solve (1) to obtain=VA = Ap(ça ~ 0 +S~i~’ 5 ‘~ ~p5o,. ~,.5op ,.p,5~(2)The tensor KA,,,. is often called the contorsion tensor. Conversely, if p Ag,. is given by (2) with arbitrarySt,,, then length is preserved.If length is not preserved but changes proportional to dx” and to the length itself, one has0(ds)2 = (~,,dx”)(ds)2. (3)The covariant vector /,,. is the Weyl vector and is by us interpreted as the dilation gauge field. NowD0g,.,. = 4,.g,.,. (4)and