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

Abstract: 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.

• The paper introduces a method to incorporate gravitational backreaction by constructing a full horizon symmetry group, leading to a Type II dressed algebra.
• It rigorously implements gravitational constraints using algebraic crossed products to ensure well-defined traces and generalized entropy computations.
• The work distinguishes entropy types in black hole and de Sitter contexts, highlighting universal and context-specific edge mode contributions.

Generalized Entropy, Symmetry Groups, and Edge Modes in Gravitational Subregions

Overview

This work systematically analyzes the algebraic and entropic structure of perturbative quantum gravity in spacetime subregions outside stationary black holes and within static patches of de Sitter space. It extends the algebraic approach to quantum field theory in curved spacetimes by rigorously imposing gravitational constraints induced by horizon backreaction on quantum observables. The key technical advances are the explicit construction of the full symmetry group acting at the horizon—including the infinite-dimensional group of angle-dependent ('boost supertranslation') symmetries—and the demonstration that the associated "dressed" gravitational algebras remain of Type II, thus enabling well-defined traces and entropies even when standard QFT algebras are of the more pathological Type III.

Gravitational Constraints and Symmetry Structure

Quantum fields in a fixed curved spacetime background yield local von Neumann algebras of observables ($\mathfrak{A}(\mathscr{R})$) that for standard subregions are of Type III$_1$, lacking pure states, traces, or well-defined density matrices. The inclusion of gravity, however, fundamentally changes this picture: gravitational backreaction and diffeomorphism invariance enforce global ('edge') constraints on what observables may be assigned to a region. At second order in perturbation theory, fluctuations of the horizon induce not just global charges (e.g., total area and angular momentum), but an infinite family of angle-dependent charges conjugate to a group of "boost supertranslations." This symmetry group is shown to be

$G = H_{\mathrm{isom}} \ltimes \mathcal{S}$

where $H_{\mathrm{isom}}$ is the horizon's isometry group ($\mathrm{SO}(3)$, $\mathrm{U}(1)$, etc.) and $\mathcal{S}$ is an infinite-dimensional abelian group of smooth functions parameterizing angle-dependent horizon translations.

These charges impose nontrivial operator constraints on first-order gravitational phase space observables, physically corresponding to the necessity of gauge-invariant, diffeomorphism-invariant observables. Concretely, the algebra of physical ('dressed') observables must commute with all the symmetry generators acting on the complementary region.

Dressed von Neumann Algebras: Algebraic Crossed Products

The central mathematical construction is the "dressed" algebra $\mathfrak{A}_{\mathrm{dress.}}(\mathscr{R};G)$, defined as a crossed product of the naive QFT algebra with the full symmetry group: $$\mathfrak{A}_{\mathrm{dress.}}(\mathscr{R};G) \sim \mathfrak{A}(\mathscr{R}) \rtimes G$$ This algebra implements the constraints associated with all horizon symmetries, including the infinite-dimensional component, and represents observables "dressed" to be invariant under horizon diffeomorphisms. The formal apparatus of constructing operator-valued weights and group von Neumann algebras is utilized to handle the infinite-dimensional structure and the absence of a Haar measure, relying—where possible—on Gaussian quasi-invariant measures over the group.

An important technical result is that, even when G is infinite-dimensional and/or noncompact, the essential structure persists: the dressed algebra is Type II (semi-finite) or, in some instances, Type II$_1$ (finite trace), contrasting with the undressed Type III situation.

Edge Modes and Entropy Structure

The inclusion of the full symmetry group, and correspondingly the full set of horizon edge modes, modifies the entropy formula for quantum states. For any algebraic state over $\mathfrak{A}_{\mathrm{dress.}}(\mathscr{R};G)$, the existence of a trace allows one to define a renormalized von Neumann entropy: $S_{\mathrm{vN}}(\rho) = - \tau(\rho \log \rho)$ For a broad class of semiclassical states, the entropy is shown to take a universal form: $$S_{\mathrm{vN}}(\rho_{\Psi}) \simeq S_{\mathrm{gen}} + S_{\mathrm{edge}} + C$$ where:

• $S_{\mathrm{gen}}$ is the standard generalized entropy — the sum of the horizon area (in Planck units) and the QFT entropy of fields outside the horizon.
• $S_{\mathrm{edge}}$ is a new, calculable contribution from the state-dependent fluctuations of the horizon charges, i.e., the edge mode entropy associated to the symmetry group G. For matter with compact gauge group, $S_{\mathrm{edge}}$ exactly matches known edge mode contributions in gauge theory (cf. [Donnelly, Wall, (Donnelly et al., 2014)]).
• $C$ is a state-independent constant encoding scheme-dependent additive ambiguities.

The explicit form of the edge mode entropy depends on the 'charge sector' of the state and, for noncompact/infinite-dimensional groups, reflects the nonfactorizable structure of the Hilbert space—no artificial UV cutoff or Hilbert space factorization is required.

Type II$_{\infty}$ vs. Type II$_1$: Black Holes and de Sitter

A major result is the precise classification of the dressed algebra's type:

• Generic Black Holes: The algebra is shown to be Type II$_{\infty}$, i.e., it admits a semifinite trace but no maximum entropy state. The spectrum of area/charge fluctuations is unbounded above due to the gravitational degrees of freedom.
• de Sitter Static Patch: For an observer-defined static patch in de Sitter, accounting for the full set of constraints and realistic observer modeling—including rotational kinetic energy and binding energy—makes the total charge spectrum bounded below, yielding a Type II$_1$ algebra. Here, the existence of a maximum entropy state (formally the 'tracial state') is restored, matching expectations from 'finite entropy' arguments in semiclassical gravity.

Edge Mode Generalization and Matter

The analysis extends to include matter degrees of freedom with gauge symmetries, demonstrating that similar large gauge constraints lead to infinite-dimensional edge mode contributions. The framework is robust provided the (potentially non-abelian, noncompact) large-gauge transformation group satisfies technical conditions on unitary implementability of group actions and the existence of faithful, semifinite, normal "neutral weights". This includes, for example, electromagnetism and Yang-Mills theory in regions with boundaries.

Implications and Future Directions

By fully incorporating the infinite-dimensional symmetry structure at horizons, this work ensures that generalized entropy is defined in a mathematically precise way for gravitational subregions, even when the underlying quantum observables would be pathological (Type III). The explicit construction clarifies how gravitational edge modes and large diffeomorphism constraints regulate UV divergences, providing a universal formula for entropy that seamlessly merges horizon area, bulk QFT, and edge contributions.

The demonstration that Type II$_{\infty}$ structure is universal for black holes but specialized modeling of observers in de Sitter gives Type II$_1$ reflects deep distinctions in the phase space of gravitational degrees of freedom within different cosmological contexts.

Practically, this framework opens several research avenues:

• Algebraic AdS/CFT: The symmetry and edge mode structure outlined here may be related to the gluing and modular aspects of AdS/CFT, particularly regarding the role of bulk symmetries at the boundary and in hydrodynamic effective descriptions [cf. Knysh et al., (Chatziparaschis et al., 2024); Liu et al., upcoming].
• Generic Extremal Surfaces: The ideas generalize to subregions bounded by arbitrary extremal (not just Killing) surfaces, intersecting with the 'island' program and quantum extremal surface prescription in holography.
• Entropy in Quantum Gravity: The operator-algebraic setting and explicit edge mode entropy pave the way for nonperturbative entropy calculations in quantum gravity (see links to recent algebraic QFT work, e.g., Witten 2023 (Du et al., 2023)).

Conclusion

This work establishes, with mathematical precision, how the full group of horizon symmetries induces a Type II algebraic structure in gravitational subregions, permitting a universal definition and computation of generalized entropy, including all edge mode contributions. The techniques developed here provide both a unifying perspective and a practical computational framework for entropy and information in the semiclassical regimes of quantum gravity, with immediate applications to black hole thermodynamics, cosmological horizons, and the structure of entanglement in gravity.

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Key Numerical and Structural Results:

Context	Symmetry Group $G$	Algebra Type	Max Entropy State	Entropy Structure
Black hole exterior	$H_{\mathrm{isom}} \ltimes \mathcal{S}$ (infinite-dim)	Type II$_{\infty}$	No	$S_{\mathrm{gen}} + S_{\mathrm{edge}} + C$
de Sitter static patch	" " plus observer constraints	Type II$_1$	Yes	" "
Field theory (no gravity)	— (local QFT region)	Type III$_1$	No	Undefined/generalized entropy diverges
Gauge theory matter	Large gauge group ($\mathcal{G}$)	Type II/II$_1$	Context-dependent	Includes edge modes; matches known formulas

References:

• "Generalized Entropy is von Neumann Entropy II: The complete symmetry group and edge modes" (2601.07910).
• Donnelly, Wall, "Entanglement entropy of electromagnetic edge modes" (Donnelly et al., 2014).