Convergence criterion for conformally invariant scalar path integrals on complex metrics

Determine the precise condition on the complex metric g_{μν} that guarantees convergence of the path integral over a real, conformally invariant scalar field with higher-derivative action coupled to Weyl-squared gravity, that is, establish a metric-allowability criterion ensuring convergence of ∫ Dφ exp(i S_φ[φ, g_{μν}]).

Background

The authors discuss the Kontsevich–Segal criterion, which constrains complex metrics by demanding convergence of generic matter path integrals. A simple contour around the powerball branch cut appears to violate this bound, potentially excluding such metrics if one demands convergence for arbitrary matter fields.

However, near the conformal core the relevant gravitational theory is effectively Weyl-squared, prompting the authors to focus on conformally invariant matter. They present a specific conformally coupled scalar action but state that the condition on g_{μν} ensuring convergence of the associated path integral is unknown. Deriving this condition would determine whether powerball metrics are allowable in a gravitational path integral with conformally invariant matter.

References

The condition on $g_{\mu\nu}$ for the convergence of $\int\mathcal{D}\phi\,\exp(iS_\phi[\phi,g_{\mu\nu}])$ is not known (and most likely highly nontrivial);

— Conformal Cores of Quantum Black Holes in Quadratic Gravity  (2411.19311 - Liu et al., 2024) in Section 6.2 (Allowance of the metric)