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.

  • $\begingroup$ What is "the perturbative expansion" of gravity? There is no single obvious quantum field theory of gravity. Ordinary perturbative QFT builds on a background Minkowski spacetime, so this is not going to work here. $\endgroup$
    – Turion
    Sep 18, 2011 at 19:58
  • 1
    $\begingroup$ My question is about properties of the perturbative expansion of gravity theories around Minkowski spacetime, this is a well-defined procedure and I ask a well-defined (perhaps no so interesting) question about it. $\endgroup$
    – user566
    Sep 18, 2011 at 20:03
  • $\begingroup$ I agree that this can only be interesting if there are field theories where the perturbative expansion can be made sense of. There are some indications that N=8 SUGRA in four dimensions might be perturbatively finite which is what brought this question to mind. $\endgroup$
    – user566
    Sep 18, 2011 at 20:13
  • $\begingroup$ Is there a reason why you're looking specifically for instantons? My understanding was that in field theory there are various renormalon divergences that aren't necessarily related to semiclassical instantons. Also, I don't see why semiclassical solutions localized in space but not time can't contribute to this sort of thing. It seems like black holes should be relevant for the S-matrix in a gravitational theory, right? $\endgroup$
    – Matt Reece
    Sep 18, 2011 at 21:30
  • $\begingroup$ @Matt, there could be other ways to obtain non-perturbative amplitudes but instantons are the easiest one. As for solitons, in my mind their Euclidean action is infinite due to time translation invariance so they do not contribute to the partition function. This should include static solutions with or without horizons. $\endgroup$
    – user566
    Sep 18, 2011 at 21:35

1 Answer 1


The answer is yes in dimensions where there exists an exotic sphere. So, the answer is yes in dimensions 7,8,9,10,11,13,14,15... (In 4 dimensions the existence of such an exotic sphere hinges upon the resolution of the smooth 4-dimensional Poincare conjecture.) The logic as to why this is the case is as follows...

For any Euclidian Yang-Mills instanton I there always exists an "anti-instanton" -I such that the instanton I when "widely" separated from the anti-instanton -I yields a guage field I - I that is homotopic to the trivial gauge field A=0.

As I - I is homotopic to the trivial gauge field A=0, one must include I - I in path integrals. In such path integrals I may be centered at x and -I may be centered at y. If x and y are very distant, then this produces, by cluster decomposition, the same result as an isolated instanton I at x. This is why instantons play a role in path integrals.

Applying this logic to gravity one wishes to find an instanton J and an anti-instanton -J such that J - J is diffeomorphic to the original manifold. If there exists such a pair, then J should be interpreted as an instanton and -J as an anti-instanton.

The set of exotic spheres form a group under connected sum. Hence, for any exotic sphere E there exists an inverse exotic sphere -E such that the connected sum of E and -E is the standard sphere.

Consider now a manifold M of dimension n=7,8,9,10,11,13,14,15... As M is of this dimension, there exists an exotic sphere E of dimension n and an inverse exotic sphere -E such that the connected sum of E and -E is the standard sphere. As the connected sum of the standard sphere and M is diffeomorphic to M, these exotic spheres can be interpreted as instantons in n dimensions vis-a-vis our above argument.

This logic was first presented in section III of Witten's article Global gravitational anomalies.

  • $\begingroup$ Thanks! I am collecting quite the reading list here already... $\endgroup$
    – user566
    Sep 18, 2011 at 21:37

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy