Physical instabilities and the phase of the Euclidean path integral

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.

💡 Research Summary

The authors investigate the complex phase of the Euclidean gravity partition function when the background geometry is a product manifold of the form (S^{p}\times M_{q}). They define the phase by analytically continuing Newton’s constant (or (\hbar)) as (G_{N}\to G_{N}(1-i\epsilon)) with (\epsilon>0). In this prescription the partition function acquires a universal factor (i^{p+2}) multiplied by an additional factor ((-i)^{N}), where (N) counts the number of “physical” negative modes present in the background. These negative modes are precisely those that correspond to exponentially growing quasinormal modes in the Lorentzian static patch of de Sitter space, i.e. modes behaving as (e^{\kappa t}).

Section 2 serves as a warm‑up, showing how a single inverted harmonic oscillator contributes a factor ((-i)^{2n+1}) to the Euclidean path integral, with the integer (n) determined by the product (\kappa\beta). The authors then treat a free scalar field with negative mass‑squared on de Sitter space in two equivalent ways: (i) by counting growing quasinormal modes in the static patch, and (ii) by expanding the field in spherical harmonics on the Euclidean sphere (S^{D}). Both approaches give the same total phase ((-i)^{N}), where (N) is the sum over all modes that become tachyonic in the Euclidean problem.

In Section 3 the main analysis is performed using Kaluza‑Klein (KK) reduction of pure gravity on a background (dS_{p}\times M_{q}). The D‑dimensional graviton decomposes into spin‑2, spin‑1 and spin‑0 fields on the de Sitter factor. The spin‑2 and spin‑1 sectors only produce massless gauge modes or massive modes that respect the Higuchi bound, so they do not generate negative‑mass‑squared fields. All potentially unstable modes are scalars on (dS_{p}) and arise from three distinct sources:

1. Volume modulus (\phi_{0}) – a uniform fluctuation of the internal space volume. Its mass‑squared is (m_{0}^{2}=-2(p-1)/r_{p}^{2}), which is sufficiently negative that both the (\ell=0) and (\ell=1) Euclidean spherical harmonics become tachyonic. This contributes (p+2) negative modes (one from (\ell=0) and (p+1) from (\ell=1) polarizations).

2. Higher scalar Laplacian modes (F_{n}) – eigenfunctions of the internal Laplacian (-\nabla_{q}^{2}) with eigenvalues (\lambda_{n}). Their effective de Sitter masses are (m_{n}^{2}= (p-1)-2 + b\lambda_{n}/(q-1)), where (b) is a known constant (the lower bound of the Laplacian spectrum). Only those with (\lambda_{n}<2(q-1)) lead to negative (m_{n}^{2}), and each such eigenvalue contributes a single Euclidean negative mode (the (\ell=0) harmonic). The total number of such modes is denoted (N_{L,q}).

3. Transverse‑traceless tensor modes (\phi_{I}) – internal metric fluctuations that are TT on (M_{q}). They satisfy (