In perturbative QFT in flat spacetime the perturbation expansion typically does not converge, and estimates of the large order behaviour of perturbative amplitudes reveals ambiguity of the perturbative expansion of the order exp(-1/g^2) where g is the expansion parameter. This ambiguity in turn is related to the existence of asymptotically Euclidean classical solutions (instantons) which contribute to these correlation functions and whose contribution resolves the ambiguity in the perturbative expansion and allows for a non-perturbative completion of the theory. All this well-known stuff is a prelude to my question about gravity. Naively all the statements about the perturbative expansion still hold, at least if one can resolve the problems arising from non-renormalizability of the theory (in other words define the individual terms in the series). Optimistically, perhaps for N=8 SUGRA that should be possible.
This brings to mind the question of the existence of instantons, namely: Do non-trivial asymptotically Euclidean solutions exists in theories of gravity? Now, there are well-known objects that are called gravitational instantons, but those are not asymptotically Euclidean. Rather they are asymptotically locally Euclidean - they asymptote to a quotient of Euclidean flat space. The difference means that these objects do not actually contribute to correlation functions (or more to the point S-matrix elements) around flat spacetime. My question is whether objects that do contribute exist in some (perhaps unconventional) theories of gravity.