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

276 P. van Nieuwenhuizen, Supergravity[M,M]~M, [M,11]~H, [H,TI]-~M (IS)[M,S1~S, [H,S].--S, {S,S}-~M+H(with H ~—MMS and M ~—MM,.) and scales aH P and f3S Q. (One does not scale M since one wantsto keep the M, M — M Lorentz algebra.) Two possibilities arise:(i) a = /32 This leads to {Q, Q} — P and [P,P] = [P, Q] = 0 for a —‘O. Thus one recovers the superPoincaré algebra.(ii) a = /3. Now (0, Q} = [0, F] = [P, P] = 0. In this case the 0 are “outside charges” which rotateas spinors under M, but for the rest have nothing to do with spacetime groups.Above we have discussed some representations of OSp(1/4). The defining representation was 5 X 5dimensional. The adjoint representation (the one in terms of structure constants) is of course 14 x 14dimensional since there are 14 generators. However, not every representation of SO(N) x Sp(M) can beextended to a representation of OSp(N/M). For example, the set of matrices (y,,,, ow,,) and the set(y 0,y5, y3) both are a representation of Sp(4). Probably the second set cannot be extended to arepresentation of ,OSp(l/4).Supercanonical transformations.t Canonical transformations between creation and absorption operatorsfor bosons define the group Sp(M). (We recall that M is always even for Sp(M).) For fermions onefinds the group SO(N, N). As one might expect, the most general canonical transformations mixingbosons and fermions lead to OSp(N, NIM). This we now prove.Consider the infinitesimal transformationQB.I = aBI + (p~kaBk+ ~ + ~kaFk+ ThkaF.k0F.k ~kaF.k ~kaB.k EkaB.,,aF~— aF,, ikwhere , i~, E, H are Grassmann variables. The bosonic sector is symplectic (4’ symmetric, ç~antiHermitean)(16)~ ‘~‘~“), AJ+JAT=O, j= ( 0 I’~~ (17)i/i (p/S —IOi(A star denotes complex conjugation and T denotes transposition.) Also well-known is that the purelyfermionic sector is orthogonal (P antisymmetric, F antiHermitean):D~(~ *)~ DS+SDT=O, s=(~~). (18)S can be diagonalized by the matrix IØ(o-~+o-~)and has N eigenvalues +1 and N eigenvalues —1.Thus one finds the group SO(N, N) for the canonical transformations between fermionic variables.The transformations which mix bosonic and fermionic operators are canonical if H = _1 =E = — ~.Introducing matrices as before, one must not forget that in á~.1one finds ( IJaFJ)71TandI thank Professor Berezin for showing me these results and B. Voronov for discussions.