Gravitational Edge Modes, Coadjoint Orbits, and Hydrodynamics

TL;DR

This paper classifies the coadjoint orbits of boundary symmetry groups in general relativity, revealing connections to hydrodynamics and providing insights into quantum gravity through the structure of edge modes and their invariants.

Contribution

It provides a complete classification of positive-area coadjoint orbits for boundary symmetries in GR, linking them to hydrodynamics and invariants like area and curvature.

Findings

Total boundary area acts as a Casimir, similar to mass in Poincaré group.

Invariants from curvature relate to spin-like quantities.

Hydrodynamics symmetry group emerges as a reduction of gravitational corner symmetries.

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 Poincar\'e 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 Poincar\'e 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.