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

P. van Nieuwenhuizen, Supergravity 337In general one contracts indices as in the four contractions above (6). One first writes down anequation so that it is correct in the bosonic sector, and then adds signs so that it is correct for all sectors.As an exercise the reader may derive47An = (— ~ (12)We now come to the important point of specifying the group in tangent space, i.e., the determinationof ~. At this point a (weak) assumption is made: the tangent group contains ordinary constant Lorentzrotations. Under ordinary Lorentz rotations Sv = A m,,v” and Sv” = ~(A. ~y)~2~~b• In superspace 55A =SBEBA per definition. Hence for the bosonic part Am~= ,,m while for the fermionic part Eb” = ~(A .Notice that we always use gamma matrices with indices as in (ym )cd.O-)Taking in 8i~~in (8), A and B equal to m and n, one easily derives that urn,, is a multiple of theMinkowski metric. On the other hand, taking A = m and B = b one finds that the Bose—Fermi part ofi~ vanishes. For the Fermi—Fermi part of the tangent metric one finds with A = a, B = b from (8) withthese special Eb0b.— i,0~( ~ )“b = 0. (13)Hence, i~must commute with the Lorentz generator. The general solution is therefore1lcd = a[C(1 + ibys)]~d (14)where Cat, is the charge conjugation matrix.Hence, there is a 3-parameter class of solutions of ~ in (8) for the special case that is equal toconstant Lorentz rotations, namely [d~mn (Mink.), aC0~+ ib(Cy5)~~].We now pick one element of thisclass as the metric in tangent space. It is at this point that the difference with supergravity occurs,because in supergravity one keeps all 3 invariant metrics, as a result of which the tangent group ofsupergravity is the Lorentz group. Also the coset approach of page 319 leads to this result.Substituting this form for the tangent metric, one finds that the tangent group is defined by leavinginvariant d47 m n~,,(Mink.)~!i~ + 0(1 + iy 5b)~i.This shows that the local tangent group is Osp(3, 1/4N).(For b = 0 this is the standard definition for b~ 0, see below.)In gauge supersymmetry, the only field is the metric g,~. There are no independent matter fields; thecomponent matter fields appear in the 0 expansion of the metric. Therefore one can describe the theoryequally well in terms of the metric or of supervielbeins. However, in order to make contact with otherformalisms, we introduce a supervielbein V1%~’’by the definition— A \I7(I+B) Bgnn— Vn ua~() V,i . (15)The transformation properties of V1%” are as indicated by the indicesSVAA = ~(A+IXA+l)vA~I + ~ + VABER” (16)Knowing how V4”, g~and ,~ transform, one finds that the relation between g and V is a correcttensor relation. One can now scale the y~in the tangent metric away by rescaling the super-vielbeins.To see this, note that (a + iby5) = (a 2 + b2)1”2 exp(iay 2 + b2)112, and redefine 47 =5) with cos a = a(a