Superselection Sectors of Gravitational Subregions

TL;DR

This work addresses how to define entanglement of the graviton by partitioning GR's phase space across subregions with an imaginary boundary, ensuring the separating surface carries no physical degrees of freedom. It shows that such subregions must be bounded by extremal-area surfaces and identifies the centre variables as the conformal class of the boundary metric constrained by the traceless extrinsic curvature, effectively discarding nearby deformations of extremal surfaces. Through a gravity-specific symplectic analysis and a discussion of Jacobi deformations, it argues that the graviton entanglement structure naturally decomposes into superselection sectors tied to boundary data, with potential implications for holographic entropy prescriptions and quantum corrections to black hole entropy. The results illuminate the interplay between gauge freedom, edge modes, and the geometry of entangling surfaces in gravity, while highlighting open questions about extending beyond extremal surfaces in dynamical settings.

Abstract

Motivated by the problem of defining the entanglement entropy of the graviton, we study the division of the phase space of general relativity across subregions. Our key requirement is demanding that the separation into subregions is imaginary---i.e., that entangling surfaces are not physical. This translates into a certain condition on the symplectic form. We find that gravitational subregions that satisfy this condition are bounded by surfaces of extremal area. We characterise the 'centre variables' of the phase space of the graviton in such subsystems, which can be taken to be the conformal class of the induced metric in the boundary, subject to a constraint involving the traceless part of the extrinsic curvature. We argue that this condition works to discard local deformations of the boundary surface to infinitesimally nearby extremal surfaces, that are otherwise available for generic codimension-2 extremal surfaces of dimension $\geq$ 2.