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

212 P. van Nieuwenhuizen. Supergravilyare not enough indices available, one is guaranteed to find again the Einstein tensor times an expressionof the form i/i . Finally, as we shall see, the contribution of 8(y~)cancels the extra terms due to partiallyintegrating Si/i,. = D,.i. We now give the details.The variation of the explicit tetrads in the Hilbert action yields with Se”,. =82(2) = ~ (iy”i/i”)G,,,, G 1~= R~a— ~ (2)where R,,a = R,.va~e”~.We have used that Se”” = —~ -y”t/i° which follows from 3(e~,.e’~”)The variation of i/i,, yields a factor[D,,,D,,] = ~ *))~~~ (3)This one proves easily, using that the Lorentz generators ~ab satisfy [gab c,.cd] = 8bc0.ad + three moreterms, obtained by antisymmetrizing in (a, b) and (c, d). Varying i/i,. and partially integrating D,.E~onefinds again a commutator [D,.,D~]t/i,,plus an extra term due to the fact that D,. does not commute with~YV~For the two curvature terms one finds023/2 = ~“~[~YsY,,~’d— 75y~o~~ai/i,.]R~, (4)after relabeling of the indices in the second term. Using that for Majorana spinors (see appendix C)YSYVUCdI/i,. = t/i,.y~o~~~yu (5)= 0.one finds with YpOcd + Ucd7~ = cdbaysye,and using the identity= (S~’+ 8~’+ S~j — c ~ d)e (6)where 5~’= e,, ebeC”, that varying the gravitino fields yields823/2 = e (~1.ya )G (7)1.y” (see appendix C), the terms with the Einstein tensor in eqs. (2) and (7) clearlySince cancel. iy”i/P = —i/iNote that since we use 1.5 order formalism, we did not vary the spin connection in 2(2)• All that isleft is the terms due to 8(y~)and the terms with (D,.y,) obtained by partially integrating D,.E SinceD,.ë = ~91,L — ~ (8)after partial integration D,. commutes with ‘y~(since [‘I’s,0a~] =0) but for the commutator of D,. with y,.one findsD,.(y~D,,i/i,,)— y,,D,.D~i/i,,= {0,.y,, + ~y~]}D~i/i,,= (D,.e”~)y0D,,./i,,. (9)