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

P. van Nieuwenhuizen, Supergravily 239Since the structure functions f’3~~are antisymmetric in (po) except when both denote fermionicsymmetries, one finds that eq. (11) is equal toJj_;~~ 2~, I ‘3y.k” Dk j~ ;~ (A \r’~A r’yAc’1 3 o-l-J j3AJ y3)L~111L~ I1~ ——That this expression vanishes follows from the double commutator of gauge transformations and willbe derived in subsection 8.For antighosts, the BRST transformation is not nilpotent as given by eq. (5), but one can obtain alsohere nilpotency by introducing auxiliary fields once again. For example, in Yang—Mills theory onedefines~f’(qu)= ~cl) + ~g2+ g(a - W) + C”t9,~D~C (13)and one defines 6g = 0, ÔC* = Ag. The general case is obvious.Thus, for any theory with a closed gauge algebra one can define nilpotent BRST transformationswhich are the quantum equivalent of the classical gauge invariances. By means of them one can derivethe Ward identities which are needed for the proofs of unitarity and gauge invariance. The nilpotency ofBRST transformations of CC and qY is used to derive Ward identities for one-particle irreducible Greenfunctions. The anticommuting nature of the constant A has led some authors to speculate on a deeprelation between supergravity and BRST transformations, but the two have really nothing to do witheach other: any gauge theory is BRST invariant at the quantum level.2.3. Covariant quantization of simple supergravityIn this subsection we apply the general formalism of the previous subsection to simple supergravity.Since we assumed that the gauge algebra closes, we consider as gauge fields not only the graviton andgravitino but also the minimal set of auxiliary fields S, P, A,~.There are three local symmetries and hence three pairs of ghost-antighost fields [236](i) general coordinate ghosts CC, C~(ii) local Lorentz ghosts Ctm”, ~(iii) local supersymmetry ghost CC, C~.The complex vector CC and the complex tensor C”~are anticommuting with themselves but thecomplex four-component spin 1/2 ghost C’~is commuting with itself. This not only follows from generalprinciples but can also be checked by verification of a Ward identity, as we shall show (page 257).As gauge fixing term a convenient choice is£~(fix)= _~(a~V’jg~)2 + a(ea,.. — eb~6~ô~)2 + ~y,J(y5/ç). (1)The advantage of this choice is that the propagators for graviton and gravitino become very simple. Thekinetic terms of the Hubert action+~(fix) read in terms of h~.with g,~,.= 6,~+ Kh,~~ h~.p+~h~.p+a(ca~—c,~a)2 (2)where ~ = 6~,.+ Kca,~.From hp,. = c,~.,.+ ~ + C(c2) one therefore finds for the ca,.. propagator for