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

244 P. van Nieuwenhuizen. Supergravitvreal anticommuting and a complex commuting ghost. The net effect is that there is one Majorana ghost.This Majorana ghost is crucial, for example, in order to obtain the correct axial anomaly.2.5. Gauge independence of supergravityCoupling the gauge fields 4,’ and ghost and antighost fields to sources as followsI(total) = I(qu)+ i(çb’J, + K 0C*’3 + L0C’3) (1)one finds, changing the integration variables of the path-integral by an infinitesimal BRST variation(whose Jacobian equals unity as we saw),(64,’J, + K06C*’3 + L0ÔC’3) = 0. (2)Left-differentiating with respect to K0 and right-differentiating with respect to .4, and then putting K0and L0 equal to zero, one finds(jC*’3(RIaC~~A)(t3~~ + jj4,k)~ AF,,y”3i4,”) = 0. (3)In the absence of S, F, Am, &,b’ contains extra three-ghost terms which should be added to R’aC”A. Inthat case, (3) not only relates the longitudinal part of the gauge field propagator to the ghostpropagator, but also to extra four-ghost terms.For linear F,, = F,,J~4,’,this Ward identity can be rewritten as (assuming that F,,, is a commutingconstant)((6Z/6Fflk)iA — C*O64,IJ,4,~5= 0. (4)On-shell the second m~= —4i~ymCA)or term does not otherwise contribute, contraction since either with a64,’physical contains polarization no one-particle tensor yields pole zero (forexample (for example for ôe the term 6~/í~ = 8~CA).Thus the S-matrix is gauge invariant under changes in F,,, [304].For general Fa which are nonlinear in 4,’, the proof may be extended as follows [Townsend and theauthor, unpublished]. Consider changing F,, into F,, + eG,, with infinitesimal . Then we add to (1) aterm iG,,J”, and obtain after right-differentiation with respect to J’3(jC*’3(Go,Ri,,C~~~A +AFa7”3jG0) +iC*’3(Ri,,CCA)iJ,Gp) = 0.(4a)The first two terms are (ÔZ/&)iA while .the last term does not contribute to the S-matrix for the samereasons as before. Thus, on-shell the S-matrix is independent of the choice of gauge fixing term F,,.We must now decide whether the S-matrix is also independent of y”3 [304]. Consider the followingWard identityJ [dqS’dCC dC~~(sdet 7)1/1 ~~C*’3 exp iJ(qu) = 0. (5)For one anticommuting variable, (8/aC)f(C) is C-independent because f(C) is at most linear in C, and