Physical instabilities and the phase of the Euclidean path integral

TL;DR

The paper analyzes the phase of the one-loop Euclidean gravity partition function on manifolds of the form $S^p\times M_q$, showing that the phase equals the pure $S^p$ gravity phase $i^{p+2}$ times an extra factor $(-i)^N$ arising from physical negative modes produced by KK reductions. It provides a concrete counting framework for these negative modes, including volume, KK scalar, and transverse-traceless sectors, and demonstrates how contour rotation procedures yield a gauge-invariant result (favoring Procedure I). The authors derive explicit phase formulas for products of spheres, including a clean cancellation for $S^p\times S^q$, and extend the analysis to multi-sphere products with careful treatment of $S^2$ factors and zero modes. A detailed discussion connects the partition function phase to growing quasinormal modes in the static patch, two-dimensional dilaton gravity reductions in special cases, and the norm of the Hartle-Hawking wavefunction, highlighting the physical interpretation of these phases and their potential implications for cosmology and holography.

Abstract

We compute the phase of the Euclidean gravity partition function on manifolds of the form $S^p \times M_q$. 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 $S^p \times S^q$ 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.