Gauge Edge Modes in Quantum Gravity

• Gauge Edge Modes (GaEMs) are boundary-localized degrees of freedom that ensure gauge invariance by compensating for broken symmetry at boundaries and corners.
• Canonical analysis reveals GaEMs add independent symplectic pairs with first-class constraints, linking bulk topology to edge dynamics in models like JT gravity.
• Algebraic structures of GaEMs, including SL(2,ℝ) representations in Chern-Simons theory, underpin key insights into black hole entropy and topological entanglement.

Gauge Edge Modes (GaEMs) are boundary-localized degrees of freedom in gauge-invariant theories—such as diffeomorphism-invariant gravities or locally gauge-invariant BF/Chern-Simons models—that become physical on lower-dimensional loci (boundaries, corners) where naive gauge-invariance is obstructed by the presence of nontrivial gluing conditions or boundaries. In gravity, and especially in topological models and models with boundaries or defects, GaEMs often interplay nontrivially with “gravitational edge modes” (GrEMs), yielding a rich geometric and algebraic structure at the boundary or corner, crucial for the quantum description of black holes, wormholes, or AdS boundaries.

1. Structural Distinction: Gravitational vs. Gauge Edge Modes in Gravity

In first-order BF-type formulations of gravity, both gravitational and gauge edge modes arise at codimension-1 boundaries and codimension-2 corners. Gravitational edge modes (GrEMs) typically encode diffeomorphism redundancies translated to the boundary—e.g., wiggling or reparametrization degrees of freedom (such as Virasoro or Schwarzian modes in JT/AdS2)—that become dynamical when boundary conditions break part of the gauge group. Gauge edge modes (GaEMs), by contrast, arise in the presence of an internal gauge group (e.g., SO(2,1) in JT gravity or SL(2,ℝ)×SL(2,ℝ) in 3D Chern-Simons), and manifest as residual gauge parameters or fields required to restore gauge invariance when patching together different regions at corners.

For instance, in Jackiw-Teitelboim gravity with defects (Liu et al., 3 Nov 2025), the boundary “wiggling” GrEM parameterizes the location/shape of the boundary curve and has Schwarzian-type dynamics, while at a corner formed by the meeting of two boundary segments, extra SO(2,1) GaEMs must be introduced to maintain gauge invariance under gluing. These GaEMs are a priori independent variables with their own symplectic structure and constraint algebra.

2. Canonical Analysis and First-Class Constraint Structure

In the extended Hamiltonian analysis of gravity+gauge theories with corners, canonical pairs describing GaEMs typically supplement the bulk phase space. For example, in 2D JT gravity with conical or wormhole defects (Liu et al., 3 Nov 2025), at each corner one introduces canonical pairs $(e_{μa},\,\bar B^{μa})$ and $(n^a,\,N_a)$ satisfying nontrivial Dirac brackets:

• $$\{e_{μa},\,\bar B^{νb}\}_D = -\delta^\nu_μ\delta^b_a$$
• $\{n_a, N^b\}_D = \delta_a^b - n_a n^b$, $\{N_a,N^b\}_D = n^b N_a - n_a N^b$

A set of three first-class constraints—enforcing norm and orthogonality of corner normals and connecting to the dynamical GrEM via a constraint on the quadratic Casimir—are imposed:

1. $N^a n_a \approx 0$ (angle conjugacy)
2. $n_a n^a - 1 \approx 0$ (norm constraint)
3. $N^a N_a - V(e,B,n) \approx 0$ (linking to boundary reparametrization via $V$).

The single physical GaEM degree of freedom per corner in JT gravity thus encodes the necessary would-be gauge data to achieve full gauge-invariant gluing.

3. Algebraic and Representation-Theoretic Aspects

The corner GaEM variables in low-dimensional gravity typically form nontrivial representation algebras. In 2D JT (Liu et al., 3 Nov 2025), combinations of corner variables $(e,B)$ furnish three families of SL(2,ℝ) generators,

$$j_0 = \tfrac{1}{2}g_{μν}e^{μa}B^\nu{}_a, \quad j_1 = \tfrac{1}{2}g_{μν} e^{μa}e^\nu{}_a, \quad j_X = -\tfrac{1}{2}g_{μν} B^{μa} B^\nu{}_a,$$

with algebra $\{j_0, j_1\}_D = +j_1$, $\{j_0, j_X\}_D = -j_X$, $\{j_1, j_X\}_D = 2j_0$ and quadratic Casimir $\mathcal{C} = j_0^2 - j_1j_X$ (interpreted as the area squared of a parallelogram spanned by $e,B$). In unitary representations one finds a discrete spectrum for $\mathcal{C}$, mirroring the quantization of area in higher-dimensional loop gravity.

Similar structures occur in the Chern-Simons formulation of 3D gravity, where anyonic edge modes—realized as quantum-group representation labels and Sklyanin-algebra generators—provide a minimal factorization of the bulk Hilbert space and encode all gauge-charged boundary data (Mertens et al., 1 May 2025).

4. Maurer–Cartan Packaging and Gluing

Restoring gauge invariance at the intersection (corner) of adjacent boundary segments requires a precise gluing condition relating GaEMs across the interface. This is elegantly formulated via the Maurer–Cartan form $\lambda^{ab} = \theta^{-1} \delta \theta$, which manifests the extrinsic normal-tangent information in a Lorentz-covariant way. The extended symplectic potential splits variations into physical and pure-gauge Lorentz parts, supporting two independent SL(2,ℝ) algebras at the corner and defining a gluing law (e.g. (Liu et al., 3 Nov 2025), Eq. 4.49): $$\alpha_{IJ} B^{IJ} + f_\alpha V(q) q - (\theta^{-1} \alpha \theta)_{ab} B^{ab} = 0,$$ ensuring consistency of the full extended system and integrability of the charges.

5. Linking GrEMs and GaEMs: Constraint-Induced Correlations

While in principle GaEMs and GrEMs are independent, the physical constraints at the corner typically link the two—specifically, the constraint on the norm of the corner GaEM (quadratic Casimir) is fixed by the dynamical boundary GrEM reparametrization: $N^a N_a = V(e,B,n),$ where $V$ is determined by boundary data (e.g., by the local value of $\theta'(t)$ or the Hawking temperature and horizon shift for defects in JT (Liu et al., 3 Nov 2025)). Thus, the boundary dynamics “imprints” itself onto the corner GaEM algebra, embedding bulk topology and boundary shape within the corner symplectic structure.

6. Extensions and Physical Consequences

The structure and representation theory of GaEMs generalizes to quantum-group anyonic edge algebra in 3D Chern-Simons gravity (Mertens et al., 1 May 2025), higher-spin models, matrix-regularized models of edge symmetry (Donnelly et al., 2022), and supergravity generalizations (super-BF, OSp(2|1) (Lee et al., 25 Mar 2024)). In all cases, GaEMs provide the algebraic backbone for entanglement entropy counting, the universal area law ($e^{A/4G}$), and the precise accounting of black-hole microstates and topological entanglement.

In summary, Gauge Edge Modes are boundary-corner-localized dynamical degrees of freedom necessary to maintain global gauge invariance in the presence of subregion decomposition, providing a representation of the corner symmetry group and interlocking with gravitational edge modes via first-class constraints and gluing conditions. Their quantization controls the microscopic structure of horizon entropy and underpins the operator algebra of quantum gravity in bounded regions.