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Generalized Entropy is von Neumann Entropy II: The complete symmetry group and edge modes

We consider the algebra of observables of perturbative quantum gravity in the exterior of a stationary black hole or the static patch of de Sitter spacetime. It was previously argued that the backreaction of gravitons on the spacetime perturbs the area of the horizon at second-order which gives rise to a non-trivial constraint on the algebra of physical observables in the subregion. The corresponding "dressed" algebra including fluctuations of the total horizon area admits a well-defined trace and is Type II. In this paper we show that, at the same perturbative order at which the horizon area (and angular momentum) fluctuates, gravitational backreaction also perturbs the horizon area in an angle-dependent way. These fluctuations are encoded in horizon charges -- i.e., "edge modes" -- which are related to an infinite dimensional "boost supertranslation" symmetry of the horizon. Together, these charges impose an infinite family of nontrivial constraints on the gravitational algebra. We construct the full algebra of observables which satisfies these constraints. We argue that the resulting algebra is Type II and its trace is shown to take a universal form. The entropy of any "semiclassical state" is the generalized entropy with an additional "edge mode" contribution as well as a state-independent constant. For any black hole spacetime, the algebra has no maximum entropy state and is Type II$_{\infty}$. 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.