Kiefer C. Quantum gravity

72 COVARIANT APPROACHES TO QUANTUM GRAVITY
cf. (1.20), and set Λ = 0. The action (2.142) is not only invariant under general
coordinate transformations and local Poincaré transformations, but also under
local 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)
8πG
where 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 SUGRA
perturbation theory? The situation is improved, but basically the same features
as in Section 2.2 hold: the theory is non-renormalizable (the occurrence of the
dimensionful coupling G due to the equivalence principle), and there is in general
no cancellation of divergences (Deser 2000). To give a short summary of the
situation, in n = 4 there are no one-loop or two-loop counterterms (due to SUSY
Ward identities), but divergences can occur in principle from three loops on. The
calculation of counterterms was, however, only possible after powerful methods
from string theory have been used, establishing a relation between gravity and
Yang–Mills theory, see Bern (2002) and references therein. It turns out that in
n =4,N < 8-theories are three-loop infinite, while N = 8-theories are fiveloop
infinite. The same seems to be true for dimensions 4