We consider the algebra of gravitational perturbations of the subregion of spacetime corresponding to the exterior of a black hole or the static patch of de Sitter. The algebra of gravitational perturbations are subject to constraints arising from the group of large gauge transformations of the subregion. Physically, these constraints arise due to the backreaction of gravitational perturbations on the spacetime geometry. Previous work has shown that the quantization of a single mode (the horizon area) arising from the time translation symmetry yields a Type II algebra whose entropy is equivalent to the generalized entropy (up to a state independent constant). 

In this lecture, we consider the gravitational constraints arising from the remaining set of constraints. In addition to fluctuations of the angular momentum and total area of the black hole, we show that gravitational back reaction will, in general, perturb the area of the horizon in an angle dependent way. These ​“edge modes” are, in turn, related to an infinite dimensional “boost super translation symmetry” of the horizon. In this talk I will construct the algebra of gravitational perturbations satisfying this infinite family of constraints. We prove that this algebra is Type II and the trace takes a universal form. The entropy is the generalized entropy with an additional “edge modes” contribution. In de Sitter, the static patch is defined relative to the worldline of a localized “observer”. We show that a consistent quantization of the static-patch algebra requires a more realistic model of the observer, in which higher multipole moments perturb the “shape” of the cosmological horizon. We argue that a proper account of the observer’s rotational kinetic energy and (non-gravitational) binding energy implies that the algebra is of Type II_1 and thereby admits a maximum entropy state.