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

P. van Nieuwenhuizen, Supergravity 329the constraints on supertorsion. Moreover, the constraints themselves follow if one requires thatsupersymmetry transformations maintain the gauge at 0 = 0, and if one requires that superspacereproduces the transformation rules of ordinary supergravity.5.6. Chiral superspaceThe auxiliary fields of ordinary N = 1 supergravity in ordinary space fit into a real vector superfieldH”(x, 0)H”(x, 0) = C~+ £1Z~+ ~OOfr + ~0iy50e~~ + 60Olfr~+ O0O0A~. (1)50K~+ ~Oiym-yNamely P = a,.ii~,S = 8,.K~and C” = Z” = 0 in a special gauge. (To avoid confusion between thesuperfield H(x, 0) and the ordinary field H”(x), we have put a hat on the latter.) Thus an axial vectorsuperfield seems to have some significance for supergravity. The superspace approaches discussedbefore have the drawback of involving a large number of fields and a large symmetry group (generalcoordinate transformations in superspace in addition to ordinary Lorentz rotations) so that one mustchoose constraints and a particular gauge to establish compatibility with ordinary supergravity. (Forsome purposes, for example for the construction of invariant objects, the larger symmetry may beuseful.) An approach which uses from the start fewer fields and a smaller symmetry group is due toOgievetski and Sokatchev [346—360], and Siegel and Gates [407—428]. (See also Ro~ekand Lindström[368].)They consider two chiral complex superspaces which are related by complex conjugation. Suchspaces were also considered by Brink, Gell-Mann, Ramond and Schwarz [401].The coordinates in eachspace are complex and the symmetry group in each separate space is complex general coordinatetransformations(,. w’\ ~!‘~x L,t’ LI, uX LA—ss’ikXL,L’LJ, ~\ c”.’5’5a1L’t ~XL,UL.LThis approach is typically tailored for four dimensions, since it uses dotted and undotted indices. It isnot clear how to extend it to higher dimensions. The right-handed coordinates do not transformindependently but as the complex conjugate of these rules. One now identifies— Ill L \L1 L~ — 1111 — \tl’t* 11 e’ .i. ~ —(FL — ~ ~ ~5)U, ~R = ~2~1 )1~)v) ,X~L— X”R = 2iH~(x, 0)2~X L ~ X R) — X(3)where H~(x, 0) is a function of X”L + X”R, °L and °R~Thus, the imaginary part of the coordinates XL andXR is interpreted as an axial super field while the real part is identified with true spacetime. Of coursethese transformations form a group (two general coordinate transformations lead again to a generalcoordinate transformation). However, for Einstein supergravity one restricts the group to the supervolume preserving transformations in each of the chiral spaces, i.e., one requires that both superJacobians equal unity (see appendix),8X”L 0L)—~—A”(xL,OL)= 0. (4)Since super Jacobians satisfy the product rule, one can require that only the product or quotient of the