ar X iv : h ep - t h / 99 05 15 4 v 1 2 1 M ay 1 99 9 Two loop and all loop finite 4 - metrics

In pure Einstein theory, Ricci flat Lorentzian 4-metrics of Petrov types III or N have vanishing counter terms up to and including two loops. Moreover for pp-waves and type-N spacetimes of Kundt's class which admit a non-twisting, non expanding, null congruence all possible invariants formed from the Weyl tensor and its covariant derivatives vanish. Thus these Lorentzian metrics suffer no quantum corrections to all loop orders. By contrast for complete non-singular Riemannian metrics the two loop counter term vanishes only if the metric is flat. Solutions of classical field equations for which the counter terms required to regularize quantum fluctuations vanish are of particular importance because they offer insights into the behaviour of the full quantum theory. Conversely, classical solutions for which quantum corrections do not vanish afford little or no insight into the strongly quantum regime since quantum corrections are expected to be large. Thus in gravity theories, in plane wave spacetimes, which are of Petrov type N, all counter terms vanish on shell and so they suffer no quantum corrections [1, 2], This corresponds to the fact that the graviton remains massless in the quantum theory and presumably remains a physically valid concept no matter how large quantum effects are. Another important class of examples occur in supersymmetric theories when one has a supersymmetric, so-called BPS, solution of the classical equations. In four-dimensional supergravity theories, examples of BPS solutions are provided by positive definite metrics whose curvature is self-dual or anti-self dual, or equivalently SU (2) ≡ Sp(1) holonomy. An interesting question is whether there are non-trivial positive definite metrics which are not supersymmetric and for which quantum corrections nevertheless vanish in pure gravity. One purpose of the present note is to show that, subject to regularity, there are none.