Gravitational treatment of non-subregion subsystems

Ascertain whether subsystems of a gravitational theory that are not associated with spatial subregions—such as subsets of field species or mode collections—admit a consistent semiclassical gravitational description, and determine how their entanglement entropy should be captured via replica path integrals in the absence of codimension-two quantum extremal surfaces.

Background

The standard holographic entanglement prescriptions (RT/HRT/QES) are formulated for spatial subregions, relying on extremal or quantum extremal surfaces. For subsystems defined by algebraic or non-geometric criteria—e.g., particular fields or modes—no established geometric extremization principle exists.

From the path-integral perspective in QFT, replica constructions for non-subregion subsystems can be implemented by gluing only the degrees of freedom belonging to the chosen subsystem. Extending this to gravity is unclear: saddles may not correspond to codimension-two extremal surfaces, and a universal geometric prescription is lacking. The authors explicitly state that understanding the gravitational and semiclassical treatment of such subsystems remains unresolved.