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

P. van Nieuwenhuizen, Supergravity 369The chirality-duality condition R,,,. + ~y 5R,,,.= 0 yields two relations if one projects with ~(1± y~)D A — I DPO’A — ~ .~,, A . —1%,,,. 2 ~,’po~ — ~ MN —1 DPP’—15—~., . —~ 2~,.po~2”A — V 7’ TA,MN —D. L(23)Hence these two conditions together yield the same result as y”R,,,. = 0 alone. In four-componentformalism this is easier to see: if y~R,,,.= 0 then from cr”1,o”~R,,,.= 0 it follows upon expanding cr”1,o-~~as2c,”1,o~~= {~“1,, o~”~} + [CT”1,, CT”] (24)that indeed R,,,. + ~y5 ,.,.,,.,R”= 0.F. Supermatrix algebraThe following subsections deal with algebra and analysis using anticommuting variables. For moredetails, see a forthcoming book by B.S. DeWitt, P.C. West and P. van Nieuwenhuizen.Supermatrices are matrices M = (~)whose Bose—Bose parts A and Fermi—Fermi parts D are evenelements of a Grassmann algebra and whose Bose—Fermi parts B and Fermi—Bose parts C are oddelements. They are multiplied as ordinary matrices; this multiplication rule is associative and theproduct is again supermatrix. There is the obvious unit element. In order to see whether they form agroup, we construct an inverse by first decomposing M asM=ST,5~CfA I)’ 0\ T-~— /1 A’B 0 D-CA’BM-UV, U-~- /I B\ /A-BD 1C 00 DI’ ~ D’C IM’ — T’S’ — /1 —A~B(D— CA’B)’\/ A~ 0— — ~0 (D—CA’B)~ )k,—CA’ IM~—V~U’—/ 1C(A—BD1C)~ (A—BD’C)’ 0\fI lAO —BD~ D’— ~—DNote that M1ff = (D — CA’B)’. Thus an inverse exists if Mb. and M’~ are non-singular, or if M~bband Mff are nonsingular. Let us define A(ord) as A minus its nilpotent part A(nil). Thus A =A(ord) + A(nil). Then the inverse of A exists if A(ord) has an inverse because one can expand A1 as[A(ord)]’ times a power series in the nilpotent elements (this power series is in fact a finite series):A1 = A(ord)’[l —A(ord)A(nil)+ [A(ord)A(nil)]2 ...] (2)Since in D — CA1B the terms CA1B are nilpotent, it follows that an inverse exists if and only ifA(ord) and D(ord) have an inverse. In other words, M has an inverse if and only if M(ord) has aninverse.