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

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238 P. van Nieuwenhuizen. Supergravitvparameter A and reads= ~~fa~~CMC’3 (Sb)ÔC~= AF~y’3’~.The invariance of (4) under (5) is easy to demonstrate. 1(d) is invariant under eq. (5a). The variationof 1(fix) and of C*~yield~ =0 (6)which vanishes since the variation of an action is a bosonic object. The remaining terms come fromvarying Fa,JR”3C’3. The variation of F~~ in 1 (4) vanishes sinceF~Jk= (_1ykF~,k), R~C3AR1’3C’3.=(_l)(k±Ixi±1)±k±1±J±1(k~-*j) (7)so that the symmetries of both terms in (j, k) are opposite. For the variation of R”3C’3 one findsR~~,kRkYCMC’3 — ~f’3P~R~~’3CUACP =0. (8)Closure of the gauge algebra implies that the commutator of the local gauge transformations is again alocal gauge transformation, thus again of the form (1)(5a)(Sc)[~(~7),~5(~j]4~= R~a.kRksn’3e—(9)= R~f~(çb)r(5~~.The structure constants may depend on 4i~and will be called structure functions. These equations are thelocal equivalent of the (anti) commutators which define superalgebras. It is now clear why (8) vanishes.Replacing in eq. (9) i~’3by C’3A and ~ and CaA1 one finds the combination(C’3A)(CUA1)_ (C’3A1)(CaA) = 2(C’3ACa)A1. (10)(Note that interchanging ~ and i~in eq. (9) does not mean interchanging also their indices f3 and a!)Thus the first term in eq. (8) can be written as a commutator, and applying eq. (9), one completes theproof that eq. (4) is invariant under eq. (5).It is instructive to verify that indeed relations such as eq. (10) follow from the formal symmetries ofthe ghost fields for the case of Yang—Mills theory.We leave it as an exercise to verify thatf°~(D~C)”ACC isindeed antisymmetric in both ghosts although one of them is differentiated.BRST transformations are nilpotent. Applying eq. (5a) and eq. (5b) to (5a) one finds zero due to eq.(8). Variation of 6CC yields— ~f~R 8 1C~AC’3+ ~ C~AC’3+ ~fa~~CYA(f’3,,~C°~A1 C°). (11)
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
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