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

P. van Nieuwenhuizen, Supergravity 301Consider now the {Q, Q} anticommutator acting on em~.Since only physical fields appear at allstages, and physical fields have never a nonzero 8’ transformation law, the only way the theory managesto make P-gauges an invariance, is by requiring that R(P) in (5) vanishes. ThusF~I \~I \lm _~~tA BC_1-~oQ~el),uQ~E2)Je ~L~°PV BC 1E2 —2 2Y ~1m(F) = 0, this is according to (5) a sum of a general coordinate transformation plus otherand localsince symmetries. R This explains also why in the superconformal algebra the parameters in the {Q, Q}anticommutator are all of the form ë2y~’eitimes the gauge field h~.The other constraints play a similar role. For example, in the {Q, 0) anticommutator on ~ one findsnow an unphysical field, namely ~ due to ~ = D~ .Thus now the theory has a choice: eitherchoosing an appropriate ô’w (which again leads to the constraint R(P) 0), or by putting R(Q) = 0 as aconstraint. The latter choice is not possible since no field can be solved from R(Q) = 0, while the formerchoice is the correct one:[~fri), ôo 2)]~= ië y’~’ R’~ (Q) + 8P(2ymE1)~~~ (7)m= ~Rmfl(Q)YILEQ. Thus, for ordinary supergravity, the constraint R~~m (F) = 0 is enough.due But to for8~,w,~’ conformal supergravity one has one extra independent field to consider, namely A~.Now with= 1EQY5&[ÔQ( j), 60( 2)]A~ 8(lEyme)A — (ii2y5 ( ~)&— 1 ~-~2) (8)and indeed ~ is such that the last term turns into a term with R,~~(A) such that the sum is again ageneral coordinate transformation.Thus the constraints on curvatures have the geometrical significance of turning P-gauge transformations,defined by the group in the tangent space, into general coordinate transformations in the basemanifold.The gauge algebra is thus given by[o(e~), 5( ~)]h~= o(fA ~~eC)+fA~6~(eA)hBeC_ 1~-~2. (9)Since only ö’~is nonzero, only the (0, 0), (0, K), (0, S), (0, D), (0, M) or (Q, A) relations might bemodified. However, K, S, D, M, A rotate physical fields into physical fields so that only the (0, Q)anticommutator is modified. One finds uniformly on e~,~, A,. and b~,and hence (because of theJacobi identities) also on any function of these fields (in particular on q5~,f~,w~”~)F~( \ ~ I \] (~A\j ~ I_LA mn\j ~t”~~~ii,‘-‘Ok~2)J — ‘~gen.coord~S I ULorentz~ S W~ j (‘chira!~ 5+ ôo(—~frA)+6~(—~~) (10)11Awith ~ = ~E2yk 1. If one puts b,., equal to zero, as well as ob,~,the gauge algebra still closes but now anextra S-supersymmetry transformation is present,extra 1 —ô~ (~(~e~)ë1y~ — 1 ~-~2). (11)