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

P. van Nieuwenhuizen. Supergravitv 213In fact, as illustrated in this example, one has generally [D,.,‘yr] = (D,.e”,)y~.Thus the remaining termsare82312(rest) = ~ ‘~“ (i/i,.75yaDpi//u)(~iy”i/i~)+ ~ ‘~“’ (iysyaDpiIiu)(D,.e”~). (10)For the second term we substitute the torsion equation of subsection 4 eq. (7) and find~ “~“’ ( y5y~D~i/i,,)(~ifr,.y”i/i~). (ii)For the first term in eq. (10) we use a Fierz rearrangement as discussed in appendix D. The generalformula yields—~/i,.O1t/i~)(~ëy” )O1(—~ ~””y5y~D~i/i,,) (12)but since only for O~= ‘y, and 2iu~~the factor i/i,.O1t/i~is antisymmetric in ~i and v while y”o~~y~0(see appendix A), one finds for the first term in eq. (10) using y”y,y~y~= 2y,y~Clearly, eq. (13) plus eq. (ii) vanish.This concludes the proof of gauge invariance. We have shown that the action in eq. (1) is invariantunderSe”,. = ~Eyal/i,., St/i,. = D,.(w(e, çlr))c. (14)In the proof we have used 1.5 order formalism twice: Once by not varying w(e, t/’) in eq. (1) and oncemore by using the torsion equation in eq. (10).One can rewrite supergravity in terms of differential forms, and thus make the structure moretransparent. As an example we reproduce the preceding proof.The action is written as(13)2= ~~flkI A ~~‘I’s)’ A Di/i (15)where77k1 = ~O A O”( klmn det e), i/i = t/i,,O”, ‘y = 7,0”,Di/J = di/i + ~wa~o””t/1, 00ab ~mab° and [1k1 = ~Rmn~0~A 0”.One may think of 0” as e”,. dx 1..Since SDt/i = DOt/I and Sf’l” = Dow ‘~one has upon partial integration using Si/i = De,02= _~S0mA Ilk! A ~k! — ~71k!m A DO”’ A500k1 —Ys7 A DDçIi — ~ëy5y,,Dt/in Do tm— ~t/iA ~‘5y,,Di/iA ~ — I’~)~ A 0k,lfr)&o (16)since ‘Ilkim = OIik!. Now DDçfr = 0abt/h A f1ab and evaluating, as before, i/’ A y~yA o~~i/i one finds a term