Quantum Gravity

72 COVARIANT APPROACHES TO QUANTUM GRAVITYcf. (1.20), and set Λ = 0. The action (2.142) is not only invariant under generalcoordinate transformations and local Poincaré transformations, but also underlocal SUSY transformations which for vierbein and gravitino field read√δe m µ = 1 2 8πG¯ɛ α γαβψ m µ β ,δψµ α = √ 1 D µ ɛ α , (2.143)8πGwhere ɛ α is an anticommuting parameter function and ¯ɛ α its complex conjugate.Note that the factors √ G are needed already for dimensional reasons.What can now be said about the divergence properties of a quantum SUGRAperturbation theory? The situation is improved, but basically the same featuresas in Section 2.2 hold: the theory is non-renormalizable (the occurrence of thedimensionful coupling G due to the equivalence principle), and there is in generalno cancellation of divergences (Deser 2000). To give a short summary of thesituation, in n = 4 there are no one-loop or two-loop counterterms (due to SUSYWard identities), but divergences can occur in principle from three loops on. Thecalculation of counterterms was, however, only possible after powerful methodsfrom string theory have been used, establishing a relation between gravity andYang–Mills theory, see Bern (2002) and references therein. It turns out that inn =4,N < 8-theories are three-loop infinite, while N = 8-theories are fiveloopinfinite. The same seems to be true for dimensions 4