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

274 P. van Nieuwenhuizen, SupergravityOne easily verifies that the matrices in (5) satisfy the usual Poincaré algebra, plus [0”, Fm] =0, [0”, Mm,,] = (Omn)”~Q’~and {Q0 Qi9} = 4(ymC~l)asp Note that this matrix representation usesonly ordinary c-numbers, but anticommutators as well as commutators enter in the algebra. One candefine the algebra only in terms of commutators by introducing anticommuting elements in the matrix0, but this we shall not do.A representation in terms of differential operators is given by (see subsection 5.3)Fm = 8/3xm, 0° = 8/80,, —RIO)”Mmn = xm8,, — x,,öm + 0~mn8/t90. (6)The variables 0°are anticommuting variables, and derivatives are left derivatives. Thus= (C’)”~$~’ ~= ~ (7)The multiplets of the tensor calculus are simply irreducible nonlinear representations of the superalgebraconsidered. Acting with (6) on (xM, 0”) yields an 8-dimensional nonlinear representation.Casimir operators for the super Poincaré algebra can be defined: the supermass operator F20, and thesuperspin operator (KMF,. — K,.PM)2 whereKM = “~“°M,.~P,, + QyMy 5Q.By adding a chiral generator A_whose only nontrivial bracket is [00, A] = iytmy5OF,,, (F”~Fk)‘. These three Casimir operators 5”8Q” completely one can specify defineanother Casimir operator: A —any representation [442]of the iQy N = 1 super Poincaré algebra without central charges.The orthosymplectic super algebra OSp(N/M) contains as bosonic part the ordinary Lie algebrasSp(M) and S0(N). It can be defined as those linear transformations which leave invariant the bilinearreal formF—x’y’s50+0”~’9C,,,,, (i,j=1,N;a,f3=’l,M) (8)where 0 and x are anticommuting objects. (If one takes also 0 and x as real ordinary numbers, then xand 0 must transform differently from y and x.) The symbol C denotes an antisymmetric real metric. Thefirst term is of course invariant under SO(N). The second term is invariant under Sp(M). For M 4 (thecase of most interest), it is invariant under 0’ = (a’O,)O where 0~are the 16 Dirac matrices, if(a’O,)TC + C(a’O,) = 0. (9)Clearly 01 = {y0,, 0mn}. Since the form F is real, we need real 0~.In a Majorana representation we thusfind that Sp(4) is generated by(o~~i,icr~4,~ y,.,~y4), (k, I = 1,3). (10)This set of generators generates also the algebra 0(3, 2). Indeed, denoting the above generators by