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Gravitational Edge Modes, Coadjoint Orbits, and Hydrodynamics

William Donnelly, Laurent Freidel, Seyed Faroogh Moosavian, Antony J. Speranza

TL;DR

The paper develops a Kirillov orbit method analysis of gravitational edge modes, identifying the corner symmetry group G_SL(2,R)(S) for a spherical boundary and classifying its positive-area coadjoint orbits. It shows the total area is a Casimir and constructs an infinite family of Casimirs from the outer curvature via a dressed vorticity, with orbit invariants encoded by measured Reeb graphs; the hydrodynamical group emerges as a reduction, unifying the orbit structure across groups. A moment map links these corner orbits to the GR phase space, connecting edge-mode charges to the gravitational data and laying a foundation for geometric quantization and a deeper understanding of entanglement entropy in gravitational subsystems. The results illuminate gravity–hydrodynamics parallels, clarify the geometric origin of invariants, and provide a structured framework toward nonperturbative quantization of gravitational edge modes and their role in quantum gravity.

Abstract

The phase space of general relativity in a finite subregion is characterized by edge modes localized at the codimension-2 boundary, transforming under an infinite-dimensional group of symmetries. The quantization of this symmetry algebra is conjectured to be an important aspect of quantum gravity. As a step towards quantization, we derive a complete classification of the positive-area coadjoint orbits of this group for boundaries that are topologically a 2-sphere. This classification parallels Wigner's famous classification of representations of the Poincaré group since both groups have the structure of a semidirect product. We find that the total area is a Casimir of the algebra, analogous to mass in the Poincaré group. A further infinite family of Casimirs can be constructed from the curvature of the normal bundle of the boundary surface. These arise as invariants of the little group, which is the group of area-preserving diffeomorphisms, and are the analogues of spin. Additionally, we show that the symmetry group of hydrodynamics appears as a reduction of the corner symmetries of general relativity. Coadjoint orbits of both groups are classified by the same set of invariants, and, in the case of the hydrodynamical group, the invariants are interpreted as the generalized enstrophies of the fluid.

Gravitational Edge Modes, Coadjoint Orbits, and Hydrodynamics

TL;DR

The paper develops a Kirillov orbit method analysis of gravitational edge modes, identifying the corner symmetry group G_SL(2,R)(S) for a spherical boundary and classifying its positive-area coadjoint orbits. It shows the total area is a Casimir and constructs an infinite family of Casimirs from the outer curvature via a dressed vorticity, with orbit invariants encoded by measured Reeb graphs; the hydrodynamical group emerges as a reduction, unifying the orbit structure across groups. A moment map links these corner orbits to the GR phase space, connecting edge-mode charges to the gravitational data and laying a foundation for geometric quantization and a deeper understanding of entanglement entropy in gravitational subsystems. The results illuminate gravity–hydrodynamics parallels, clarify the geometric origin of invariants, and provide a structured framework toward nonperturbative quantization of gravitational edge modes and their role in quantum gravity.

Abstract

The phase space of general relativity in a finite subregion is characterized by edge modes localized at the codimension-2 boundary, transforming under an infinite-dimensional group of symmetries. The quantization of this symmetry algebra is conjectured to be an important aspect of quantum gravity. As a step towards quantization, we derive a complete classification of the positive-area coadjoint orbits of this group for boundaries that are topologically a 2-sphere. This classification parallels Wigner's famous classification of representations of the Poincaré group since both groups have the structure of a semidirect product. We find that the total area is a Casimir of the algebra, analogous to mass in the Poincaré group. A further infinite family of Casimirs can be constructed from the curvature of the normal bundle of the boundary surface. These arise as invariants of the little group, which is the group of area-preserving diffeomorphisms, and are the analogues of spin. Additionally, we show that the symmetry group of hydrodynamics appears as a reduction of the corner symmetries of general relativity. Coadjoint orbits of both groups are classified by the same set of invariants, and, in the case of the hydrodynamical group, the invariants are interpreted as the generalized enstrophies of the fluid.

Paper Structure

This paper contains 38 sections, 204 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction and summary of results
  2. Conventions
  3. Generalities of coadjoint orbits
  4. Coadjoint actions and coadjoint orbits
  5. The canonical symplectic form on coadjoint orbits
  6. Coadjoint orbits of the Poincaré group
  7. Coadjoint orbits of the hydrodynamical group
  8. Hydrodynamical group
  9. Area-preserving diffeomorphisms
  10. An example of a Reeb graph
  11. Construction of invariants
  12. Coadjoint orbits of corner symmetry group
  13. Orbit reduction
  14. Lifting invariants from the little group
  15. Corner symmetries in general relativity
  16. ...and 23 more sections

Figures (1)

  • Figure 1: An example function $f$\ref{['example-function']}, its Reeb graph, and the measure on the Reeb graph. Figure (a) shows the contours of $f$ in the Lambert azimuthal equal-area coordinates. When plotted as a height function (b) we clearly see the saddle point and two maxima. The corresponding Reeb graph (c) shows the splitting of the orbits as the height is varied, going from a single orbit below the critical orbit to a pair of orbits above. The measure on the Reeb graph, whose density is displayed in (d), displays clearly the logarithmic singularities at $f = 0$ associated with the saddle point.