Euclidean geometric representation of the non-local statistical gravitational path integral

Ascertain whether the non-local statistical path integral derived for the fluctuating hydrodynamics of nested causal diamonds admits a representation as an integral over fields on a smooth Euclidean manifold or as a double-cone–type wormhole geometry, and determine the nature of such a representation if it exists.

Background

Integrating over entropy-current fluctuations yields a non-local contribution to the effective action that is localized along the light-cone directions but non-local across transverse dimensions. The resulting statistical path integral governs the hydrodynamic evolution of maximal-area surfaces along a geodesic in a background spacetime.

While certain fixed-area horizons (e.g., eternal black holes) admit Lorentzian double-cone wormhole descriptions of connected spectral form factors, the authors emphasize that for the general non-local statistical path integral they constructed, it remains unclear whether there exists a corresponding smooth Euclidean field theory or wormhole geometry representation akin to the double cone or double trumpet examples in lower-dimensional gravity contexts.