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

P. van Nieuwenhuizen, Supergravity 277One finds77 \ ç,(E H\ nc~rrT_n ~19— \, * — *)‘ ‘~ — k,H” E*)~ — .Thus the canonical transformations are generated by OSp(N, N/M) and the generators can berepresented by (M + 2N) x (M + 2N) matrices satisfying(A B\(J 0\~(J 0\(A B’~ T 0 (20)\.C DI \0 SI \0 SI \C DI1, _Bt, D’ (see appendix).where The the superunitarity supertransposition algebras, replaces to whichA, weB, now C, D, turn, by are AT, the C other main family of super algebras. Theyare denoted by SU(N/M) and contain as bosonic part the algebras SU(N) x SU(M) x U(1). In an(N + M) x (N + M) representation, the SU(N) and SU(M) algebras lie in the first N x N and lastM x M submatrices, while the U(1) lies along the (N + M) x (N + M) diagonal. As for the orthosymplecticcase, one can define the superunitary algebras as leaving invariant the formF (x1)*x~6,j+(ga)* 0”g,,,,,, (g,,~= ±o,,v) (21)where i = 1, N and a = 1, M Since the 0’~are complex and F is real, one can always diagonalize the0-metric and thus M and N can be even or odd. Hence, for SU(N/M) we do not need an antisymmetricmetric C,,~but we can use two diagonal metrics 6~,and g,,~.(If one takes 0 to be commuting, one simplyfinds the algebra SU(N + M); thus one must take 0 as anticommuting.) It is clear that again thesetransformations form a closed algebra under ordinary commutation relations provided the x —*0transformations contain anticommuting entries. Again we go over to representations with ordinarynumbers only by extracting the anticommuting parameters.The most interesting case is the superconformal algebra SU(2, 2/N). A representation of SU(2, 2) interms of 4 X 4 matrices is well-known. It is the set of matrices which leaves 00 =o~0l+ o~o~— 0~o~—O~O~ invariant. (For this representation of y4 see appendix A.) This is the set of 15Dirac matrices satisfying 0~y4+ y4O1 = 0. Thus the generators of SU(2,2) arePm = —~ym(1— y~). K,,, = ~ym(1+ y5), D = —~ys, M0,,, = Omn. (22)A representation of the generators G’ of the other unitary group and of the U(1) is given by theantiHermitean generators r’ for SU(N) and all matrices have zero supertrace. FurtherA=—~diag(1,1,1,1,~,~ (23)for U(1).* Finally there are 2 N X 4 generators 0” and S”’ which are the usual square roots of Pm andK,,,. They have entries only in the (4+ i)th columns and rows, and are given by(floi\k — r iii \~-‘--I1ko ~~‘ai\k — F!a’l — \r~—11ka~ ) 4+j — L2~’ + )‘5)U J , t,’) ) 4+i — I.2~’ )‘5)’..~ J4~k= ~(l— y~)”~, (S”’)4~’,,= —~(1+ ys)°,,. (24)(Q”’)For N = 4A is a central charge which may be omitted (and is absent in the S-matrix, see subsection 3.3) without violating the Jacobi identities.