Gravitational Edge Modes (GrEMs)

Updated 9 November 2025

Gravitational Edge Modes (GrEMs) are boundary-specific degrees of freedom emerging from unfixed gauge and diffeomorphism invariance in finite gravitational systems.
They play a crucial role in explaining black hole entropy, holography, and the entanglement structure by encoding physical data at interfaces such as horizons and corners.
GrEMs are analyzed through diverse formulations (metric, tetrad, Chern-Simons, BF) and techniques including quantization, coadjoint orbits, and matrix regularizations to bridge classical and quantum gravity insights.

Gravitational Edge Modes (GrEMs) are boundary-localized degrees of freedom that emerge in the phase space of gravitational theories when considering finite regions or subregions bounded by interfaces, corners, or horizons. Unlike gauge-invariant bulk data, these modes are the remnant of diffeomorphism or gauge freedom that cannot be fixed locally at the boundary without introducing new, physical variables. GrEMs are central to formulating the quantum and classical theory of gravitational subsystems, explaining black hole entropy, topological entanglement, and the correct representation of boundary symmetries and charges. Their precise algebraic and physical content depends on the spacetime dimension, the structure of the theory (metric, tetrad, Chern-Simons, or BF formulations), and the nature of the boundary (spatial, null, or timelike).

1. Definitions and Phase Space Origin

In diffeomorphism-invariant theories, the physical phase space associated with a region $\mathcal{M}$ with boundary splits as

$\Omega = \Omega_\Sigma + \Omega_S,$

where $\Sigma$ is a spatial Cauchy slice and $S = \partial\Sigma$ is a codimension-2 surface (“corner”) or the entangling/horizon interface. The bulk $\Omega_\Sigma$ involves metric (or tetrad/connection) data, while $\Omega_S$ encodes additional canonical pairs—edge fields such as coordinate embeddings $X^\mu(\sigma)$ , Lorentz frames $\Lambda^\alpha_\mu$ , or group elements $g(\sigma)$ —restoring full gauge/diffeomorphism invariance at $S$ (Freidel et al., 2020, Donnelly et al., 2022).

In the first-order connection–vierbein formalism, for instance, the presymplectic structure with boundary is

$\Omega = \int_M \delta A^i \wedge \delta \Sigma_i + \frac{1}{2\kappa\gamma} \int_S \delta e^i \wedge \delta e_i,$

with $A^i$ the SU(2) connection and $e^i$ the co-frame one-forms. The presence of $S$ necessitates the introduction of boundary data (edge modes) so functional differentiability and gauge invariance are maintained (Freidel et al., 2019).

2. Corner Symmetry Algebras and Classification

The central algebraic structure is the “corner group” $G = \mathrm{Diff}(S) \ltimes SL(2,\mathbb{R})^S$ (or appropriate internal group), acting on the edge mode fields through

$\left\{P[\xi], P[\zeta]\right\} = P\left([\xi,\zeta]\right),\,\, \left\{N[\alpha], N[\beta]\right\} = N([\alpha, \beta]),$

and

$\left\{P[\xi], N[\alpha]\right\} = N(L_\xi\alpha),$

where $P[\xi]$ and $N[\alpha]$ are diffeomorphism and internal symmetry charges, respectively (Donnelly et al., 2020).

On $S^2$ , the coadjoint orbits of this group are classified by:

the total area $A = \int_S \sqrt{q}$ (the analogue of mass, a Casimir),
an infinite family of “spin” Casimirs built from the curvature of the normal bundle (“dressed vorticity” $\bar w$ ), analogous to Wigner’s classification for Poincaré orbits (Donnelly et al., 2020, Donnelly et al., 2022).

The corner symmetry algebra can be realized as $\mathfrak{sdiff}(S^2)\ltimes \mathfrak{sl}(2,\mathbb{R})^{S^2}$ in 3+1d, or as Kac-Moody and Virasoro extensions for lower dimensions (Freidel et al., 2019, Donnelly et al., 2022).

3. Realizations in Various Theories and Dimensions

3D Gravity (Chern-Simons and Metric)

Chern-Simons formulation with $SL(2,\mathbb{R})_L\times SL(2,\mathbb{R})_R$ admits no local degrees of freedom in the bulk, so all physically relevant data localize at the boundary as edge modes. The minimal factorization map in Chern-Simons theory reveals a quantum group structure for the edge algebra, with bulk states factorizing over irreducible representations of $U_q(sl(2,\mathbb{R}))$ after imposing a singlet condition under the quantum group “surface symmetry” (Mertens et al., 1 May 2025, Joung et al., 11 Sep 2025). The edge states realize anyonic statistics governed by the universal $R$ -matrix, admitting a direct link to topological entanglement and (in the black hole sector) the Bekenstein-Hawking entropy: $S_{EE} \approx \log\dim_q j_* \approx \frac{\text{Area}}{4G_N}.$

2D JT Gravity

In Jackiw-Teitelboim (JT) gravity, the only true dynamical variables are boundary-reparametrization modes (“wiggle” field $\theta(t)$ ) which survive as edge modes and are governed by the Schwarzian action. In supersymmetric extensions ( $\mathcal{N}=1$ JT), the super-Schwarzian action controls the dynamics of superspace edge modes (Lee et al., 25 Mar 2024, Liu et al., 3 Nov 2025). For geometries with corners or defects, intricate coupling between gravitational and gauge edge modes occurs, with canonical pairs realizing $\mathfrak{sl}(2)$ algebras and discrete Casimirs (e.g., parallelogram area-squared) (Liu et al., 3 Nov 2025).

4D Gravity

In higher dimensions, GrEMs at codimension-2 surfaces (entangling surfaces, corners) carry the action of infinite-dimensional groups. The classical phase space splits into bulk gauge data and edge variables (Stueckelberg fields for boundary diffeomorphisms and Lorentz rotations), and the symplectic structure reflects this decomposition (Donnelly et al., 2020, Donnelly et al., 2022). The quantization problem involves, for instance, matrix regularizations ( $SU(N)$ , $SU(N,N)$ ) for the continuum symmetry algebra $\mathfrak{sdiff}(S^2)\ltimes \mathfrak{sl}(2,\mathbb{R})^{S^2}$ , with quantum Casimirs matching classical invariants in the large- $N$ limit (Donnelly et al., 2022).

Null boundaries (light cones) also yield SL $(2,\mathbb{R})$ edge-mode algebras, with bulk radiative modes coupled to boundary degrees of freedom through gluing constraints, and quantization proceeds via effective reductions or skeletonizations (Wieland, 2021).

4. Entanglement, Area Law, and Black Hole Entropy

Gauss law constraints in gravity and gauge theory obstruct naive Hilbert space factorization across entangling surfaces. The correct factorization introduces edge-mode degrees of freedom labeled by normal components of the field strength (Maxwell) or Riemann tensor (gravity), leading to a direct sum structure—superselection sectors—on the reduced density matrix (David et al., 2022). The edge entropy is identified as the Shannon entropy over these labels and exactly matches the “contact term” or “edge character” contribution to entanglement entropy, as computed from partition functions (e.g., via the conical or replica trick) (Blommaert et al., 20 May 2024, Nair, 2022). For black holes, GrEM counting yields the Bekenstein-Hawking entropy, as the area is canonically conjugate to the "boost angle" at the corner (Takayanagi et al., 2019).

Table: Entanglement and GrEMs

Setting	Edge Mode Label	Entropy Contribution
Maxwell	$E_r\|_{S^{d-2}}$ (normal electric flux)	$-\frac{1}{3}\ln(R/\epsilon)$ (in $d=4$ )
Linearized gravity	$R_{0r\,ij}$ , $R_{0r\,0i}$ (normal Riemann)	$-\frac{16}{3}\ln(R/\epsilon)$ (4d)
3D Gravity/CS	Quantum group index $j$	$\log\dim_q j_*\sim$ Area

5. Quantization and Representation Theory

Quantization of the edge-mode sector (e.g., via coadjoint orbits or matrix regularizations) yields Hilbert spaces carrying representations of the corner symmetry algebra:

For LQG and similar approaches, spin network punctures are lifted to charge networks, carrying not only SU(2) flux but also higher Kac-Moody/Virasoro or Poincaré representations, encoding missing gravitational observables (Freidel et al., 2019, Freidel et al., 2018).
In the quantum group context, edge modes transform in representations of $U_q(sl(2,\mathbb{R}))$ for Chern-Simons gravity (Mertens et al., 1 May 2025).
Matrix quantization on the sphere replaces the classical infinite-dimensional algebra with a finite-dimensional one ( $SU(N), SU(N,N)$ ), with area spectra and Casimirs determined by group representations and recovering continuum results as $N\to\infty$ (Donnelly et al., 2022).

6. Edge Modes as Dynamical Reference Frames

Recent developments interpret GrEMs as dynamical reference frames, providing a relational encoding of subregion location and gauge-invariant boundary condition imposition (Carrozza et al., 2022, Giesel et al., 22 Oct 2024). In this “frame-dressed” approach, diffeomorphisms acting on the subregion are still gauge, while edge-mode reorientations carry nontrivial physical charges, giving a sharp separation between gauge and physical boundary symmetries. Hamiltonian charge algebras for boundary-preserving symmetries close as in $\mathrm{diff}(S)$ , with possible central extensions when additional structures (e.g., translations) are included.

7. Applications, Physical Implications, and Outlook

Black hole entropy: The microcanonical entropy counting for horizons arises from GrEMs, with the area law matching via edge-mode quantization (Takayanagi et al., 2019, Mertens et al., 1 May 2025).
Holography: In AdS/CFT and AdS/BCFT, GrEMs capture non-additivity of the gravitational action and encode boundary degrees of freedom required for dual CFT partition function factorization (Takayanagi et al., 2019).
Entanglement structure: Edge modes account for nonlocal operator algebras, log-coefficient universalities in entropy, and restore the correct factorization of Hilbert spaces under subregion decompositions (Blommaert et al., 20 May 2024, David et al., 2022).
Quantum geometry: Enhancing LQG with GrEMs yields an enriched quantum geometry (tube networks, Poincaré and Kac-Moody symmetries) and bridges classical, loop, and Chern-Simons/BF descriptions (Freidel et al., 2019, Freidel et al., 2018).
Fluid/gravity correspondence: The corner algebra reductions exhibit deep isomorphisms between gravitational and hydrodynamical symmetries, with Casimirs (area, enstrophies) matching in both contexts (Donnelly et al., 2020).
Dynamical locality and subsystem theory: Subregion symplectic forms, uniquely defined under post-selection, clarify the algebra of charges, conservation, and the physical status of observables and reference frames (Carrozza et al., 2022, Giesel et al., 22 Oct 2024).

In summary, gravitational edge modes provide the essential physical and mathematical infrastructure for understanding gravitational subsystems, entropy, and boundary symmetries in both classical and quantum regimes, underpinning developments in quantum gravity, holography, and black hole physics.