Physical instabilities and the phase of the Euclidean path integral

A bstract We compute the phase of the Euclidean gravity partition function on manifolds of the form S p × M q . We find that the total phase is equal to the phase in pure gravity on S p times an extra phase that arises from negative mass squared fields that we obtain when we perform a Kaluza-Klein reduction to S p . The latter can be matched to the phase expected for physical negative modes seen by a static path observer in dS p . In the case of S p × S q the answer can be interpreted in terms of a computation in the static patch of dS p or dS q . We also provide the phase when we have a product of many spheres. We clarify the procedure for determining the precise phase factor. We discuss some aspects of the interpretation of this phase.